University of Southern California Dissertations and Theses

Investigation of the electrode-tissue interface of retinal prostheses

(USC Thesis Other)

Investigation of the electrode-tissue interface of retinal prostheses

PDF

Download a page range

Download transcript

Copy asset link

Request this asset

Transcript (if available)

Content INVESTIGATION OF THE ELECTRODE-TISSUE INTERFACE OF RETINAL PROSTHESES Copyright 2016 by Boshuo Wang A Dissertation Presented to the FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (BIOMEDICAL ENGINEERING) May2016 Boshuo Wang Epigraph One Generation The dark night gave me dark eyes; yet with them, I seek the lights. -J8J1Jm GU Cheng (1966-1993) English translation by Boshuo Wang To the bright future of humanity, generations to come. 11 Acknowledgments It has been a long and rewarding journey to complete my Ph.D., and I am grateful to all the guidance, help, and support that I have received over the past five and a half years to make it happen. First and foremost, I would like to thank my advisor Dr. James D. Weiland for his great and kind mentoring. Jim has given me freedom and independence in research, while keeping me focused and motivated. He has always been available to me and the other students, and provided me with rigorous supervision and inspiration. Over the years, he has shared many of his experiences in academia, whether big or small, and encouraged me to continue my pursuit of a career in scientific research. I have entered the field of biomedical engineering under his advising, and I owe my deepest gratitude to him as I further my career. Besides my professional development, Jim also cared for my personal well-being, and provided many amiable moments of Zen regarding life- both graduate and beyond. I am also thankful to Dr. Mark S. Humayun, whose pioneering work on retinal prostheses has provided such a great opportunity for me to be part of this large collaborative group. His exuberance has greatly infected me. I would like to thank the rest of my dissertation committee, Dr. Robert H. Chow and Dr. Dong Song, for their valuable input and feedback on my research during busy semesters, and for critically reviewing this dissertation. I would like to thank the collaborators within the NIH Multilevel Modeling project, including, here at USC, Dr. Theodore W. Berger, Dr. Song, and Dr. Jean-Marie C. Bouteiller, and at the University of Utah, Dr. Gianluca Lazzi, Kyle M. Loizos, and Mavin Martin. I would also like to thank many of the professors of the Biomedical 111 Engineering Department of USC, including, Dr. Michael C. K. Khoo, who signed my admission to USC and also served on my qualifying exam committee, Dr. Gerald E. Loeb and Dr. Terrance D. Sanger for mentoring my lab rotation projects, and Dr. Noberto M. Grzywacz as well as all the aforementioned USC professors for their wonderful lectures. Next, I would like to thank the colleagues of the lab, past and present, for their helpful discussions and assistance with my research, and their friendship that kept my balance between work and life. Specifically, I am thankful to Dr. John Jack Whalen, III, Dr. Artin Petrossians, Dr. Andrew C. Weitz, Dr. Yi Zhang, and Dr. Lan Yue; Dr. Alice K. Cho, Dr. Navya S. Davuluri, Dr. Samantha I. Scott (nee Cunningham), Dr. Kiran Nirnrnagadda, and Mort Arditti; my fellow "Fab Five" classmates Karthik Murali, Steven T. Walston, Arninat Adebiyi, and Nii Q. Mante, who all joined the lab at the same time; and Alejandra G. Calle, Yao-Chuan Chang, Christopher B. C. Girard, and Sahar Elyahoodayan. I would like to thank for all the technical support I received during my research, from Xiaopeng Wang of the Ophthalmology Department, the Center for Electron Microscopy and Microanalysis (CEMMA), the Viterbi/Dornsife Machine Shop, the Department of Animal Resources, the Cell and Tissue Imaging Core of USC Norris Comprehensive Cancer Center, and the Keck Photonic Laboratory. I also appreciate all the administrative support for my graduate studies at USC, especially from the Biomedical Engineering Department, the Ophthalmology Department, and the Biomirnetic MicroElectronic Systems Engineering Research Center (BMES ERC), including Mischa! C. Diasanta, Doris Lee, Ellis F. Troy, Diana Sabogal, Juan Sepulveda, and Erick Bonilla. IV Besides my doctoral research, a significant part of my graduate study was with the newly established Health Technology and Engineering (HTE@USC) program. I would like to thank Dr. Sanger and Dr. George Tolomiczenko for bringing me into this program to be among the first class of students, and also the numerous M.D. 's who have shown me the world of medicine. I would like to thank Jack, who has been an advisor and friend to me and our HTE project group. I also would like to thank my group mates Thomas Cummins, Mike Kwon, and Brian Quinn, who have or soon will receive their respective Ph.D. or M.D. degrees. It has been a valuable and glad experience to work as a team. I would like to thank my parents, Dr. Xiao-Ying Lii and Dr. Zhi-Gong Wang. Thank you for all the love, support, and education throughout my life, and for being my role model and aspiration since I was little. I am grateful to my brother Boduo, my grandparents, and my aunt for their encouragement all these year. Last and most importantly, I would like to thank Yiwen Zhang, my wonderful and beloved wife. I am very fortunate that we have met here at USC, and my Ph.D. journey would be incomplete without our love, care, and support for each other. v Table of Contents Epigraph ........................................................................................................................ ii Acknowledgments ........................................................................................................ iii List ofTables ................................................................................................................. xi List of Figures .............................................................................................................. xii List of Abbreviations ................................................................................................. xvii Abstract ....................................................................................................................... xx Chapter 1 Introdnction ................................................................................................ 1 1.1 History of the Visual Prostheses-From Cortex to Retina ............................................ 2 1.2 Review of Relevant Anatomy, Physiology, and Pathology ............................................ 6 1.2.1 Overview of the Visual Systern ......................................................................... 6 1.2.2 Anatomy of the Retina ...................................................................................... 7 1.2.3 Electrophysiology of the Retina ...................................................................... 12 1.2.4 Retinal Degeneration ...................................................................................... 13 1.2.4.1 Age-relatedMacular Degeneration ......................................................... 14 1.2.4.2 Retinitis Pigrnentosa ............................................................................... 16 1.2.4.3 Inner Retinal Survival and Remodeling in Retinal Degeneration ............. 17 1.2.4.4 Animal Models of Retinal Degeneration ................................................. 17 1.3 Retinal Prostheses ...................................................................................................... 18 1.3.1 Types of Retinal Prostheses ............................................................................ 19 1.3.1.1 Epiretinal Prostheses .............................................................................. 19 1.3.1.2 Subretinal Prostheses .............................................................................. 21 1.3.1.3 Suprachoroidal Retinal Prostheses .......................................................... 23 1.3.2 Common Components of Retinal Prostheses ................................................... 24 1.3.2.1 Multielectrode Array (MEA) .................................................................. 24 1.3.2.2 Electrical Stimulator, Integrated Circuits, and Packaging ........................ 25 1.3.2.3 Power and Signal Telemetry ................................................................... 27 1.3.2.4 Camera and Image Processing ................................................................ 29 1.3.3 Safety Concerns ............................................................................................. 32 1.3.3.1 Biological Safety .................................................................................... 32 1.3.3.2 Device Safety ......................................................................................... 33 1.3.4 Clinical Trials and Commercialization ............................................................ 37 1.3.4.1 Epiretinal Prostheses .............................................................................. 37 VI 1.3.4.2 Subretinal Prostheses .............................................................................. 43 1.3.4.3 Suprachoroidal Retinal Prosthesis ........................................................... 46 1.3.5 Research Areas and Challenges in (Epi-) Retinal Prostheses ........................... 47 1.3.5.1 Electrode-Tissue Interface and Electric Field Distributions ..................... 47 1.3.5.2 Visual Acuity ......................................................................................... 49 1.3.5.3 Visual Field ............................................................................................ 51 1.3.5.4 Spatial and Temporal Activation Patterns of Retinal Ganglion Cells ........ 52 1.3.5.5 Color Vision ........................................................................................... 58 1.3.5.6 Effects of Retinal Degeneration .............................................................. 59 1.4 Conclusion ................................................................................................................ 60 1. 5 Th es is Overview ........................................................................................................ 61 1. 5 .1 Basic Approach and Sununary of Results ........................................................ 61 1.5.2 Structure of the Thesis .................................................................................... 64 Chapter 2 Reduction of Edge Effect on Disk Electrodes .......................................... 66 2.1 Edge Effect on Disk Electrodes .................................................................................. 66 2.1.1 Efforts to Reduce the Edge Effect ................................................................... 67 2.1.2 Primary and Secondary Distribution ............................................................... 69 2.1.3 Utilizing the Secondary Distribution ............................................................... 70 2.2 Finite Element Model ................................................................................................ 70 2.2.1 Model Geometry and Basic Parameters ........................................................... 70 2.2.2 Input Current Waveforms ............................................................................... 73 2.2.3 Symbols and Conventions .............................................................................. 74 2.2.4 System of Equations and Boundary Conditions ............................................... 75 2.2.5 System Solving and Data Analysis .................................................................. 76 2.2.6 Performance Quantification of Different Waveforms ....................................... 77 2.2.7 Results ........................................................................................................... 78 2.2. 7.1 Current Density and Nonuniformity ....................................................... 78 2.2. 7.2 Charge and Potential Profiles ................................................................. 80 2.2. 7.3 Quantification of Edge Effect and Comparison of Different Waveforms .. 82 2.3 Mathematical Analysis ............................................................................................... 84 2.3.1 Current Step Response of Disk Electrodes ...................................................... 84 2.3.2 Constructing Arbitrary Response from the Step Response ............................... 88 2.3.3 Computational Optimization of the Waveform ................................................ 89 2.3.4 Results ........................................................................................................... 90 2.3.4.1 Steady State Response and Parallel Conductance .................................... 90 2.3.4.2 Transient Response ................................................................................ 93 2.3.4.3 Optimal Waveform and Approximation .................................................. 97 2.4 Pulsing Experiment ................................................................................................... 98 2.4.1 Experiment Setup ........................................................................................... 98 2.4.1.1 Circuit for Waveform Generation ........................................................... 98 vu 2.4.1.2 Electrodes .............................................................................................. 98 2.4.1.3 Pulsing Protocol ................................................................................... 100 2.4.1.4 Surface Analysis ................................................................................... 101 2.4.2 Results ......................................................................................................... 102 2. 5 Discussion and Conclusion ...................................................................................... 105 2. 5 .1 Comparison on Different Approaches ........................................................... 105 2.5.2 Trade-Off Considerations ............................................................................. 106 2.5.3 Implementation Considerations .................................................................... 108 2.6 Summary ................................................................................................................. 109 Chapter 3 Analysis of the Peak Resistance Frequency Method ............................. 110 3.1 Introduction ............................................................................................................. 110 3.2 Methods .................................................................................................................. 113 3.2.1 The Electrode-Tissue Interface Model .......................................................... 113 3.2.1.1 Tissue Resistance ................................................................................. 114 3.2.1.2 Charge Transfer Resistance .................................................................. 115 3.2.1.3 Electrode Double Layer ........................................................................ 116 3.2.1.4 Parasitic Capacitance ............................................................................ 116 3.2.2 Analysis of the PRF Method ......................................................................... 117 3.2.3 Electrode Parameters for Simulation ............................................................. 117 3.2.4 Simulation of PRF Method ........................................................................... 120 3.2.5 Analysis on Sampling Density and Noise ...................................................... 121 3.2.6 Comparison with Least Squares Method ....................................................... 122 3.3 Results .................................................................................................................... 123 3.3.1 Analytical Calculation .................................................................................. 123 3.3.1.1 Model with CoL and RcT ................................................................... 123 3.3.1.2 Models with CPE ................................................................................. 124 3.3.2 Simulation .................................................................................................... 126 3.3.2.1 Model with CoL and RcT ................................................................... 126 3.3.2.2 Models with CPE ................................................................................. 127 3.3.3 Frequency Sampling and Noise Analysis ...................................................... 130 3.3.4 Comparison with CNLS ............................................................................... 132 3 .4 Discussion ............................................................................................................... 133 3.4.1 Analytical Solution ....................................................................................... 133 3.4.2 Idealness of Tissue-Electrode Interface ......................................................... 135 3.4.3 Dependence of CPE on Tissue Resistance ..................................................... 136 3.4.4 Considerations for Implementation ............................................................... 137 3.5 Conclusions ............................................................................................................. 138 Vlll Chapter 4 Electrical Mapping of the Retina ........................................................... 139 4.1 Introduction ............................................................................................................. 139 4.2 Experiment Setup .................................................................................................... 140 4.2.1 Recording Chamber and Experimental Set-up ............................................... 140 4.2.2 Retina Preparation and Histology .................................................................. 142 4.2.3 Electrodes and Model of Electrode Interface ................................................. 143 4.2.4 Impedance Measurement using the PRF Method ........................................... 144 4.2.5 Data Processing and Analysis ....................................................................... 144 4.3 Results .................................................................................................................... 146 4.3.1 ImpedanceMeasurernent .............................................................................. 146 4.3.2 Histology ..................................................................................................... 146 4.3.3 Comparison .................................................................................................. 148 4.4 Discussion ............................................................................................................... 149 4. 5 Conclusion .............................................................................................................. 150 Chapter 5 Summary ................................................................................................. 151 5.1 Key Findings and Significance ................................................................................ 151 5.2 Implications for Epiretinal Prostheses ...................................................................... 152 5.2.1 Current Density Distribution ......................................................................... 152 5.2.2 Electrode Impedance Monitoring .................................................................. 153 5.2.3 Dealing with Retinal Degeneration in Computational Modeling .................... 154 5.3 Future Studies ......................................................................................................... 155 5.3.1 Inclusion ofFaradaic Reaction in FEM Studies ............................................. 155 5.3.2 Testing the Tissue Dependence of CPE ......................................................... 156 5.3.3 Intraretinal Potential Mapping and Validation of Computational Models ....... 157 5.3.4 Experimental Validation of Multilevel Modeling of Neuronal Activation ...... 158 Chapter 6 Supplementary Materials ....................................................................... 159 6.1 Introduction ............................................................................................................. 159 6.2 Model and System of Equations ............................................................................... 162 6.3 General Form of Solution ........................................................................................ 165 6.3.1 Rotational Elliptic Coordinates ..................................................................... 165 6.3.2 General Form of Solution ............................................................................. 167 6.3.3 Symbol Naming ........................................................................................... 170 6.4 Primary Distribution ................................................................................................ 171 6.5 FrequencyDispersion .............................................................................................. 174 6.6 Transient Response to Current Step Input ................................................................ 177 6.6.1 Decomposition of the Solution ...................................................................... 177 6.6.2 Steady State Response .................................................................................. 180 6.6.3 Eigensolutions of the Transient Response ..................................................... 184 6.6.4 Transient Response ....................................................................................... 188 IX 6. 7 Transient Response to Voltage Step Input. ................................................................ 190 6. 7.1 Steady State Response .................................................................................. 191 6.7.2 Eigensolutions of the Transient Response ..................................................... 191 6. 7.3 Transient Response ....................................................................................... 194 6.8 Accuracy ofSolution ............................................................................................... 196 6.9 Appendix A-Legendre Functions on the Imaginary Axis ........................................ 199 6.9.1 Legendre Functions ...................................................................................... 199 6.9.2 Evaluating Legendre Functions on the Imaginary Axis .................................. 200 6.10 Appendix B-Tables of Numerical Calculations ...................................................... 206 References .................................................................................................................. 211 x List of Tables 1.1 Charge injection limits for some electrode materials ......................................................... 34 2.1 Maximum overshoot percentage of current density ........................................................... 84 2.2 Normalized COI for different waveforms and transition times .......................................... 86 2.3 First ten eigenvalues for transient solution of current step response of disk electrodes ...... 95 2.4 Parameters and COI of optimized and fitted ExpCDF waveforms ..................................... 97 2. 5 Time constants of platinum electrodes placed on retina or in vitreous ............................. 108 3.1 FHC bipolar electrode parameters from EIS ................................................................... 120 6.1 List of symbols, subscripts and their meaning ................................................................ 170 6.2 List of superscripts and their meaning ............................................................................ 171 6.3 Coefficients for constructing Legendre functions on the imaginary axis .......................... 206 6.4 Matrices AH, A 0 , and A ............................................................................................... 207 6.5 Matrices MH and M .................................................................................................... 207 6.6 Coefficients sss n for the steady state response .............................................................. 208 6.7 Eigenvalues ACO and coefficients B~i) for the eigensolution of current step response .. 209 6.8 Coefficients cCO for the transient response of current step response .............................. 209 6.9 Eigenvalues ACO and coefficients B~i) for the eigensolution of voltage step response .. 210 6.10 Coefficients cCO for the transient response of voltage step response ............................. 210 XI List of Figures 1.1 Galvani's experiment on bioelectricity ............................................................................... 3 1.2 Cortical implant by Brindley and Lewin ............................................................................. 4 1.3 Estimated bandwidth of sensory modalities ........................................................................ 8 1.4 Cross section of the eye ..................................................................................................... 8 1. 5 Cross section of human retina ............................................................................................ 9 1.6 Distribution of rod and cone photoreceptors in human retina ............................................ 10 1. 7 Diversity of morphology among retinal cell types ............................................................ 11 1. 8 Illustration of phototransduction cascade .......................................................................... 13 1.9 Simulated vision of AMD and RP .................................................................................... 15 1.10 Histology sections of WT, rdl, and rdJO mouse retina ...................................................... 19 1.11 Electrode array location for different types ofretinal prostheses ....................................... 20 1.12 Argus epiretinal prosthesis ............................................................................................... 22 1.13 Heat molded and annealed retinal electrode array retaining spherical curvature ................ 25 1.14 IC packaging methods ...................................................................................................... 26 1.15 Methods of telemetry ....................................................................................................... 28 1.16 Intraocular camera implanted in the lens .......................................................................... 30 1.17 Simulated prosthetic vision demonstrating image processing ............................................ 31 1.18 Charge density and charge per phase as cofactors in determining tissue damage ............... 34 1.19 Biphasic current pulse with interphase gap ....................................................................... 36 XU 1.20 Fundus photographs of 6 types ofretinal prostheses ......................................................... 38 1.21 External and internal components of the Argus II device .................................................. 39 1.22 Alpha IMS prosthesis ....................................................................................................... 45 1.23 Simulated vision of text, office scene, and face ................................................................ 50 1.24 Wide field electrode array ................................................................................................ 53 1.25 Different visual percepts depending on activated retinal area ............................................ 54 1.26 Phosphenes drawn by Argus II subject ............................................................................. 55 2.1 Primary current density distribution of disk electrodes ..................................................... 67 2.2 Corrosion and tissue damage due to edge effect on electrodes .......................................... 68 2.3 Previous method to reduce edge effect by altering electrode geometry .............................. 69 2.4 Geometry of FEM model ................................................................................................. 71 2. 5 Meshing of the FEM model ............................................................................................. 72 2.6 Three candidate waveforms ............................................................................................. 75 2. 7 Current density profiles comparison for control and Ramp waveforms ............................. 79 2. 8 Current nonuniformity versus time for control and Ramp waveforms ............................... 80 2.9 Double layer charge profile for control and Ramp waveforms .......................................... 81 2.10 Potential profiles for control and Ramp waveforms .......................................................... 82 2.11 Normalized current density maxima for all waveforms and transition time ....................... 83 2.12 Current overload versus time for all waveforms and transition time .................................. 85 2.13 Illustration of rotational elliptic coordinates ..................................................................... 86 2.14 Flow chart for the algorithm deriving the optimal waveform input ................................... 91 Xlll 2.15 1-dimensional example illustrating the algorithm ............................................................. 92 2.16 Steady state distribution of step response ......................................................................... 93 2.17 Normalized current density for first ten eigensolutions ..................................................... 95 2.18 Optimized current waveforms .......................................................................................... 97 2.19 Electrical circuit for waveform conversion ....................................................................... 99 2.20 Breadboard prototype of conversion circuit.. .................................................................... 99 2.21 Adtech 64-Electrode array for pulsing experiment.. ........................................................ 100 2.22 EIS of Adtech electrodes ................................................................................................ 100 2.23 Typical corrosion profile after pulsing ............................................................................ 102 2.24 Atomic count percentage of oxygen on pulsed disk electrodes ........................................ 103 2.25 Example of current waveform modification at both pulse onset and end ......................... 107 3.1 Example impedance spectrum demonstrating the PRF method ....................................... 112 3.2 Tissue-electrode interface model .................................................................................... 114 3.3 Configuration ofFHC bipolar microelectrodes ............................................................... 118 3.4 EIS data ofFHC bipolar microelectrodes ....................................................................... 119 3.5 Photo ofFHC bipolar microelectrode in syringe ............................................................. 120 3.6 Simulated impedance spectra for double layer capacitance model .................................. 127 3. 7 Simulated impedance spectra for dependent CPE model... .............................................. 128 3.8 Simulated impedance spectra for independent CPE model... ........................................... 129 3.9 Log-log plots of PRF versus resistivity for the CPE models ............................................ 131 3.10 Relative errors ofresistivity extracted using the PRF method ......................................... 132 XIV 3.11 Relative errors ofresistivity extracted using the least squares method ............................. 133 4.1 Eye-cup/retina recording chamber and illustration of experiment set-up ......................... 141 4.2 Experimental set-up for resistivity measurement ............................................................ 142 4.3 Close-up look of the electrode and eye cup in the experiment set-up .............................. 143 4.4 Test results of the PRF method in agar ........................................................................... 145 4.5 Resistivity profiles of WT, rdl, rdJOretina ..................................................................... 147 4.6 Typical histology staining of WT, rdl, rdl Oretina .......................................................... 148 4. 7 Thickness comparison of the resistivity profiles and histology data ................................ 149 6.1 Coordinate system for analyzing the disk electrode ........................................................ 162 6.2 Current density components on the surface of the electrode ............................................ 164 6.3 Disk electrode and electrolyte space in rotational elliptic coordinates ............................. 166 6.4 Potential field of some eigenfunctions of the disk electrode ............................................ 169 6.5 Surface distribution of potential and current density for the eigenfunctions ..................... 170 6.6 Primary current density distribution ............................................................................... 173 6. 7 Impedance spectrum of disk electrode ............................................................................ 178 6.8 Equivalent series resistance and double layer capacitance .............................................. 178 6.9 Surface distribution of potential and current density for different frequencies ................. 179 6.10 Steady state surface distribution of potential and current density of current step input. .... 182 6.11 Steady state voltage on the electrode for current step input.. ........................................... 183 6.12 Charge redistribution of the double layer via different pathways during the transient ...... 186 6.13 Surface distribution of potential and current density of the eigensolutions of the xv transient response to current step input.. ......................................................................... 187 6.14 Initial surface distribution of potential and current density of the transient response to current step input ........................................................................................................... 189 6.15 Scaling factor of the steady state response of the voltage step input.. .............................. 192 6.16 Steady state surface distribution of potential and current density of voltage step input .... 192 6.17 Surface distribution of potential and current density of the eigensolutions of the transient response to voltage step input .......................................................................... 194 6.18 Initial surface distribution of potential and current density of the transient response to voltage step input. .......................................................................................................... 195 6.19 Eigenvalues of the transient response of the steady step input.. ....................................... 197 6.20 Unprocessed initial surface distribution of current density of the transient response to current step input ........................................................................................................... 198 6.21 Legendre functions of the first and second kind .............................................................. 201 6.22 Partial solution M 2 n(0 to the disk electrode ................................................................ 204 XVI List of Abbreviations a-SiC AC AMD ANO VA ASR CE cGMP CNLS co COI CPE CT DI DL EDS EIS ExpCDF Expinc FDA amorphous silicon carbide ................................................................................... 37 alternate current .................................................................................................. 14 age-related macular degeneration ........................................................................ 27 analysis of variance ........................................................................................... 103 Artificial Silicon Retina ..................................................................................... 431 Conformite Europeenne (European Conformity) .................................................... 1 cyclic guanosine monophosphate ......................................................................... 13 complex non-linear least squares ....................................................................... 122 current overload .................................................................................................. 78 current overload index ......................................................................................... 78 constant phase element ...................................................................................... 116 charge transfer ................................................................................................... 115 de-ionized ......................................................................................................... 101 double layer ........................................................................................................ 73 energy-dispersive X-ray spectroscopy ................................................................ 101 electrochemical impedance spectroscopy ............................................................. 87 cumulative distribution function of exponential distribution ................................. 73 exponential increase ............................................................................................ 73 Food and Drug Administration .............................................................................. 1 xvn FEM GCL H&E HSD IACUC IC ILM IMI INL IPG IPL IR logMAR LS MEA MPDA NIH ONL OPL OSL OTCM finite element modeling/method .......................................................................... 70 ganglion cell layer ................................................................................................. 9 hernatoxylin and eosin ....................................................................................... 143 honestly significant difference ........................................................................... 104 Institutional Animal Care and Use Committee ................................................... 142 integrated circuit ................................................................................................. 24 inner limiting membrane ....................................................................................... 9 Intelligent Medical Implants, GrnbH .................................................................... 37 inner nuclear layer ................................................................................................. 9 interphase gap ..................................................................................................... 36 inner plexiform layer ............................................................................................. 9 infrared ............................................................................................................... 28 logarithm of the minimum angle ofresolution ..................................................... 40 least squares method .......................................................................................... 122 multi electrode array ............................................................................................. 24 microphotodiode array ........................................................................................ 22 National Institutes of Health .............................................................................. 158 outer nuclear layer ................................................................................................. 9 outer plexiform layer ............................................................................................. 9 overshoot limit .................................................................................................... 90 one-time-constant model ..................................................................................... 87 XVlll PBS PDE PEDOT PRF RCS RD RF RGC RI ROC RP RPE SEM SR SS SSMP STS TZ VPU WT phosphate buffered saline .................................................................................... 99 phosphodiesterase ............................................................................................... 13 poly (3,4-ethylenedioxythiophene) ...................................................................... 32 peak resistance frequency .................................................................................... xx Royal College of Surgeons .................................................................................. 18 retinal degeneration ............................................................................................. 14 radio frequency ................................................................................................... 27 retinal ganglion cell ............................................................................................. 10 Retina Implant AG .............................................................................................. 43 region of convergence ....................................................................................... 202 retinitis pigrnentosa ............................................................................................. 14 retinal pigment epithelium ................................................................................... 14 scanning electron microscope ............................................................................ 101 step response ....................................................................................................... 87 steady state (response) ......................................................................................... 87 Second Sight Medical Products, Inc ..................................................................... 37 suprachoroidal transretinal stimulation ................................................................ 23 transient (response) ............................................................................................. 87 video processing unit. .......................................................................................... 38 wild-type ........................................................................................................... 142 XIX Abstract The aim of this thesis is to attain better understanding of the electrode-tissue interface of the epiretinal prosthesis and neural prosthetic devices in general. Three studies explore this interface from different aspects. The overall goal is to improve this interface to produce safe and efficient electrical stimulation and also provide the basis for computational models to better understand electrical stimulation targeted at the retina. A novel method is described and analyzed to design current waveform input to reduce the edge effect-the primary reason for non-uniform current density distribution on electrodes. Finite element modeling and mathematical analysis showed that waveform design can reduce the edge effect on disk electrode without the need to alter the electrode's geometry. Current waveforms with a slower rise to steady-state level compared to the abrupt rectangular step can reduce the current density non-uniformity by allowing current density to redistribute over time. Numeric method optimized the design for the waveform, which can be approximated via a RC circuit. The approximation of the optimized waveforms was tested in a pulsing experiment. The results showed reduced corrosion on the edge of platinum disk electrodes, therefore demonstrating the effectiveness of the waveform shaping method. The peak resistance frequency (PRF) method-a simple method to extract tissue resistance from impedance spectroscopy measurement of the electrode-tissue interface-was explored for its mechanisms and inherent properties. The PRF method uses a variable frequency point at which the impedance phase is most resistive to estimate the tissue resistance. The previous study that first xx proposed the method showed that it works very accurately, compared to the large deviation of estimations from fixed frequency points. In this study, theoretical analysis and computational simulation reveal that the PRF method is only a good approximation for the tissue resistance. The PRF method has an inherent deviation that varies depending on the idealness of the electrode-tissue interface but is nevertheless correctable. Further simulations tested the realistic limitation of measurement noise and frequency sampling, and showed that the PRF method works reasonably well and reliable under realistic conditions. This work provides a solid theoretical foundation for the PRF method and means to correct the results when the parameters of the interface are not ideal. As a first step of electrical mapping within the retina tissue, the resistivity profiles of healthy and degenerate retina were measured, utilizing the aforementioned PRF method. The study provided data on both healthy and degenerate mice retinas for the first time, which are relevant models for neurophysiology study of retinal prostheses. The experimental results show that the peak resistivity decreases with degeneration. Also, the resistivity profiles were thinner in degenerate retina and the thinning agreed with histology data. Therefore, the changes in tissue properties need to be taken into account for modeling study of retinal prosthesis. XX! Chapter 1 Introduction The Argus II Epiretinal Prosthesis has become the first commercially available visual prosthesis, first receiving the CE mark in March 2011, and then FDA approval in February 2013 (Zrenner, 2013a). This is a significant milestone in the history ofbioelectrical visual prostheses. In this long-lasting effort to restore vision to the blind (Dagnelie, 2012; Margalit et al., 2002; Weiland et al., 2005, 2011; Weiland and Humayun, 2014a; Weitz and Weiland, 2014; Zrenner, 2013a), humankind has substituted electrical current for light as the source of stimuli to activate neurons in the visual system in order to create a perception of vision. Prosthesis users are now able to perceive light and perform simple tasks. With the current development in technology, optimistic projections suggest that facial recognition and fairly fluent reading will become feasible in next generation retinal prostheses with a resolution of a thousand electrodes. "We're going to need a word for 'formerly blind."' is a statement of such confidence (USC Keck Hospital advertisement). However, continuous research and development is needed to overcome the challenges to reach this goal. The work presented in this dissertation is but a small building block among the collective effort of many. In this chapter, an introduction is given on retinal prostheses and related topics. A brief history of visual prostheses is given to show where retinal prosthesis fits in the framework of artificial vision (section 1.1). Before going into the details of the retinal prostheses, the relevant anatomy, physiology, and pathology of the eye and retina are given (section 1.2). The primary focus of this chapter is retinal prostheses, especially the epiretinal prosthesis (section 1.3). The challenges of the current retinal prostheses are discussed (section 1.3.5), along with possible strategies to address 1 them. Conclusions (section 1.4) and a brief overview of the dissertation (section 1.5) are given in the end. 1.1 History of the Visual Prostheses-From Cortex to Retina Medical devices and treatment using electrical recording and stimulation trace their origin to the 18th century when bioelectricity was first discovered and explored. For example, Luigi A. Galvani observed during a dissection in 1780 that the leg muscles of dead frogs twitched when struck by a spark (Figure 1.1) (Galvani, 1791). Although Galvani made some wrong interpretation about the nature of this phenomenon, it was among the first investigation into the field of bioelectricity, which studies the electrical recording and stimulation of biological tissue. Since then, scientists and engineers have been developing a myriad of ideas to interact with the human body using electricity. The neural effects of electrical stimulation have been known from very early on in recorded history, and treatments with electricity have been explored as early as mid- l 700s (Veratti, 1748) for various purposes. Among these efforts, Charles LeRoy, a French chemist and physician, most likely documented the first account of visual phosphene-light perception without actual light stimuli to the retina-perceived by a blind subject, when a Layden jar was discharged to a coil wrapped around the subject's head and the subject reported "seeing" a flame rapidly descend "before his eyes" (LeRoy, 1755). Attempts for artificial vision through electrical stimulation succeeded only in the modern era of the 20th century. In 1929, Otfrid Foerster, a German neurosurgeon, placed a stimulating electrode over the visual cortex (posterior pole of the occipital cortex) of a healthy subject under local 2 anesthesia When electrical stimulation was applied, the subject reported a light spot \\hose location was dependent on the electrode's placement over the cortex (Foerster, 1929). Similar results were soon reported in a blind patient \\ho lost vision for eight years (Krause and Schum, 1931 ). Figure 1.1: One of Galvani's experiments on bioelectricity. The leg of a dead frog twitched \Wen struck by a spark. (Image from Internet Archive: https://archive. org/.) These early attempts demonstrated that cortical electrical stimulation can successfully evoke light perception, and such perceptions have a spatially-specific mapping in the cortex. Also vital to the concept of visual prostheses is that blind patients retain the ability to perceive light perception from electrical stimulation, as long as certain neuronal circuits remain intact. Now it is known that this capability persists for decades (Dobelle et al., 1974; Karny, 1975; Schmidt et al., 1996). 3 The first chronic prosthetic visual implant was developed in the late 1960s. This device consisted of an eleclrode array of 80 platinum electrodes, and was placed over the surface of the visual cortex of a blind woman (Figure 1. 2) (Brindley and Lewin, 1968). Half the electrodes elicited phcsphenes, and the device was still working to a certain extent after 6 years (Kamy, 1975). Figure 1.2: The cortical implant created by Brindley and Lewin (Brindley and Lewin, 1968). Left: The eleclrode array is shown on the left, while the larger receiver array is to the right. Right: X-ray image of the device post implantation. The receiver array was placed beneath the pericranium and secured to the skull. The intracranial electrode array lies between the medial surface of the occipital pole of the right hemisphere and the falx cerebri. In 1972, a second implantation in a blind patient with retinitis pigmentosa was performed bilaterally. The patient could read Braille letters with little training at a rate of 8.5 characters per minute (Kamy, 1975). In the 1970s, William Dobelle designed a visual coctex prosthesis that allowed patients to recognize simple patterns (Dobelle et al., 1974). Electrical signals were converted from images recorded by a television camera and delivered to the medial occipital cortex by electrodes. These attempts yielded impressive results; however they were far from becoming feasible medical devices. The crosstalk due to the large electrode sizes (around 1 mm 2 ) and 4 consequently high current levels ( 0.8 to 4 mA thresholds) limited the patients from seemg distinct phosphenes when adjacent electrodes were simultaneously activated, therefore rendering the devices impractical (Dobelle et al., 1974). Visual prostheses had a major development milestone when researchers discovered that intracortical stimulation has much lower stimulation thresholds compared to surface stimulation, and the spatial resolution for individual phosphenes was possible for two electrodes placed as close as 700 µm (Bak et al., 1990). High density arrays with small electrode sizes also became feasible as fabrication technology advanced (Maynard, 2001; Weiland and Humayun, 2014a). And in the meantime, the success of cochlear implants showed promise for commercial neural prosthetics. All these factors encouraged the research of modern-day visual prostheses. Although cortical stimulation has been investigated early on and showed promise, the visual cortex is not the only part of the visual system that can be targeted for electrical activation. Actually, its complex functional specialization such as color, orientation, and shape etc. (Margalit et al., 2002), spatial organization, and location inside the skull create numerous challenges. Other locations along the visual pathway, namely the optic nerve (Chai et al., 2008; Fang et al., 2006; Veraart et al., 1998) and retina (Tassicker, 1956), have been proposed and investigated, and the retina stood out as a suitable target due to several reasons. The natural retinotopic projection of the visual field means that patterned stimulation could be performed easily. And stimulating the earliest part of the visual pathway also allows the implant to utilize the early visual processing in the retina, which remains somewhat viable even in degenerate retinas. Finally, the relative ease of accessibility to the retina compared to the latter part in the visual processing pathway results in less complicated surgical 5 procedures. Therefore the retinal prostheses have received the most attention recently and are the furthest along the development pathway. In the late 1980s, two research groups in the United States started to investigate in retinal prosthesis simultaneously, one at the North Carolina State University and Duke University, and the other at the Massachusetts Eye and Ear Infirrnaryffiarvard Medical School and the Massachusetts Institute of Technology (Rizzo et al., 2011). Experiments in early 1990s demonstrated the feasibility of electrically stimulating the retina. In 1991, an acute experiment was performed on a patient with retinitis pigmentosa in an operating room at Duke University Eye Center. While the patient was awake and under local anesthesia, a small wire electrode was inserted into the intraocular space to test whether stimulating the blind retina could yield phosphenes feasible for constructing useful images (Humayun and de Juan, 1991; Dagnelie, 2012). The results were positive and this study stands at the basis of retinal prostheses. Since then, several US and German research groups have pushed retinal prosthesis from a scientific concept through laboratories prototypes into clinical testing. Two devices, Argus II and Alpha IMS, have entered the market now (Zrenner, 2013a). Meanwhile, global interest in retinal prostheses has grown and many more groups have been joining the field (see 1.3.4). 1.2 Review of Relevant Anatomy, Physiology, and Pathology 1.2.1 Overview of the Visual System Vision is arguably the most important and most complex of human sensory systems. It provides the predominant amount of information input (Figure 1.3), and it is estimated that 50% of 6 the neural processing power of the brain is dedicated to vision and vision related computation (Walker, 2009). The eye is the sensory component of the visual system receiving the light input from the outside world. Figure 1.4 shows the cross section of human eyes, which is an approximately spherical structure consisting of several tissue layers on the outside and vitreous humor in the inside. The external layers of the eye include the cornea in the very front and sclera covering the rest of the eyeball. The transparent cornea allows light to pass through and enter the eyeball through the pupil, which is an aperture formed by the center opening in the iris. The light is focused by the cornea and the lens, projected through the vitreous humor, and forms an image on the retina. The retina lines the back and inner surface of the eye. The eyeballs are rotated by extraocular muscles to allow the image of interest be aligned on the fovea centralis. The fovea is the central part of the retina with the highest visual acuity, while the peripheral retina attends to edges of the visual field, albeit with lower acuity. The retina converts the incoming light into electrical signals through a process called phototransduction (see section 1.2.3). The neural signal is carried by the optic nerve through the back of the eye to the visual cortex. 1.2.2 Anatomy of the Retina The retina is a complex laminated structure about 300-µrn thick (Bonanomi et al., 2006), interfacing with the retinal pigment epithelium which provides functional and metabolic support. There are five major layers in the retina concerning neural circuitry-three contain cell bodies of retinal neurons and two contain synaptic connections (Figure 1.5). 7 1 250 M81s 125 Meis ...... USBkey Hearing Smell Figme 1.3: Estimated "bandwidth" of how much information each sensory modality processes (infographic presenting by Tor N01Tetranders). Choroid Cornea ~ Ciliary body Figme 1.4: Cross section of the eye. The cornea and lens focus the light passing through the pupil, and create an image on the retina, which lines the inner back of the eyeball. The visual information created by the retina is then transmitted to the cortex by the optic nerve. (Image from Webvision, http J/webvision.med.utah.edu/.) 8 photo- receptors cones OLM ONL scret~I~~ OPL horizontal florizontal cells bipolar cells /NL bipolar amacrine cells cells a~:fi~ne IPL ganglion cfi.lls 1 axons Figure 1.5: Left Light microscope of cross section of the central human retina. The three layers of neuronal soma are the outer nuclear layer (ONL), the inner nuclear layer (INL), and the ganglion cell layer (GCL). The two layers containing synaptic com1ections are the outer plexiform layer (OPL) and the illi1er plexiform layer (IPL). The inner limiting membrane (ILM) is formed by glia and defines the im1er border between the retina and the vitreous humor. Right: Simple diagram of retinal organization. Light enters the retina from the ganglion cell side, and travels through the retina before it is absorbed and converted into electrical signals by the photoreceptors. (Image from Webvision, http://webvision. med. utah. edu/.) Specialized sensory cells, the photoreceptors, are located in the outermost layer. They include two types of cells, the rods and the cones. Rods are very sensitive, capable of detecting single photons. They operate in dimmer light, and therefore are almost entirely responsible for night (scotopic) vision. There are about 100 million rod cells in the human retina on average (Curcio et al., 1990), and they are mostly located in tl1e peripheral retina. The cones operate under ambient daylight levels (photopic). Three types of cones (red, green and blue), which are sensitive to photons of different wavelengths, provides color vision. On average, tl1e human retina has about 4.6 million cones (Curcio et al., 1990), highly concentrated in tl1e fovea in the central region of the macula where we have our highest visual acuity (Figure 1.6). The foveal region has a very specific neural circuitry that typically connects cone to tl1e im1er retina through one-to-one pathways, while in the periphery the output layer receives converging input from many photoreceptors. 9 - - ~ 1 00 "' ~ 80 a ... ll eo ~ ~ ~o 20 70 M S(I ~n J9 'lO f...,,1>"'31 IMma ' . • • • • . ' ' I . ' • • ' ' . ' 'ai .... ;g, (U ' ~· .g. w 0 w ~ ~ ~ ~ " ~ 10 ~ E QQelltnolly l'I i.lual !l•M• Figure 1.1>: Distribution ii Md and t:<ine oPhQtPt eceptors in human retina. C¢nes are c¢trce®.ated in the:f~yeal region, resp. onsi ble for high acuity vis i ol) and color Rods-are i<hsent fri; the.f~yea; and lrave their perut densit¥ in ,the P.arafoveal regioo about 20. degrees frmn , the • fovea. (Ini.age from Wehmsion, httpi// web.vis10n1 medutah.e.du/,) The photorec eptort; form synap. tic c :onnections• in tjie· outer pf exil'onn fayer with the frip;) lar cells. 'Tb:e bi pillar ce)ls span the inner nuclear faye;r t <i interface with reti-n'll ganglion cells (EGC) at the inner ple$jfqnn layer. The inner ;plexif . onn layei· h<i.s sub-layers wher:e the dendii't. e• ol' tliffererit subtype· s.- gf bipolar ~clls and ganglioo c. ell stratif y and form sy11apses, thus· separati!lg different otisual ,Pathway.s T)le. hori?:ontal o;elts and am 'acrine cell~. wh i-oh are also: located In tlte inner nucfear layer, fprm loc@ fee.d·f orward: and feedback neural networks at the . outer an\lli'tmer pl exif orrq hwer~. re· sp.ectivel!f, 'L'4e. I{GCs, which are the. la~t layer in the· retinal cirm itcy, p .i;.o: jects axond ntb the axon fiber layer in the1nrier most part 9f the retipa, w,hi'oh ec0nver1e attl~ ~pti~ dis k anq leair e the eye fQ1ming. the ¢ptitt nerve. Each ret!Aal cell class has a.gi·:eat. diversity in mcrph~bgy and fonctii)tt (Figure L7) i:TuI$ lau.d, 10 2001). These distinct neuron subtypes create specific pathways and circuits that allows for parallel visual processing in the retina For example, subtypes of ganglion cells process specific visual information, forming different visual information channels for contrast, color, etc. (Masland, 2001). By the time action potentials leave the retina through the optic nerve, the early processing has already compressed the visual signals into relevant information flow that is of interest for the cortex. Photoreceptors Horizontal cells Bipolar cells Amacrin e cells , . .J~ ~. ~ !I! LII1llI1II -t- _i:T.1 :r~ ·~ ,,.,..J...~ ~~~~ ~4 . ~t /~ f.~t ----±_ ff.~ :,;n, ;-·-, " tr r 1--i /~:::_-_,:1.::-_~::·:.._ •, I, •,I I I } -~~~--·"·-- =:::-=---- ---'--=---- =--~---..:=-=·. -=-=-=-~--- -~-L _: ---=""~ Ganglion cells =~j:::: '- - ·-~,~ -~-· --=-:.- - - --=-=-- -"'c- ----- Figure 1. 7: Example of the diversity of morphology is shown for each retinal cell type. The structural classification is correlated with functional specifications. (Masland, 2001) 11 1.2.3 Electrophysiology of the Retina When light hits the retina, the photons pass through the inner retinal layers, and initiate a process known as phototransduction when they are absorbed in the photoreceptors (Figure 1.8). Opsins, a specific type of G-protein coupled receptors, signal a second-messenger cascade in response to the photon absorption. This amplifies the incoming light signal through several stages of biochemical reactions, and in the final step converts the information into electrical signals by closing ion channels on the cell membrane of the photoreceptors. Photoreceptors hyperpolarize as the dark current, which is maintained when no light stimulus is present, is reduced. This is quite unique, as most other cell types depolarize in response to external stimuli. The retina is the most metabolically active tissue in the body, due to the high turnover of the phototransduction machinery. The hyperpolarization of the photoreceptors reduces the glutamate release at their synaptic terminals. Fluctuations of neurotransmitter release are detected by the bipolar cells, and with contributions of the local neural networks of horizontal cells and amacrine cells, are transmitted as a graded electrical signal down converging neural pathways towards the RGCs. Excitatory input from bipolar cells leads to RGCs firing action potentials, which are transmitted by their axons to the early visual processing nuclei, e.g. the lateral geniculate nucleus in the thalamus. The visual signal is then further transmitted to the primary visual cortex, and then higher visual centers in the cerebral cortex. The processing and sensory association of these cortical centers creates the sense of vision. 12 T * a. + cytoplasm - ::······ Na»Ca 2 • ......... CNG closed IPM Figure 1.8: Illustration of the phototransduction cascade in rods: 1. An incident photon is absorbed by the ll-cis retinal bound to the rhodopsin (R) on the rod's disk membrane, and the retinal undergoes isomerization from l l-cis to all-trans configuration. 2. The rhodopsin in response then activates transducin proteins, whose alpha subunit splits off. 3. The alpha subunit actives phosphodiesterase (PDE). 4. PDE breaks down cG MP to 5' -G MP, lowering the concentration of cytoplasmic cGMP. 5. The cGMP-gated sodium channels closes in response of the lower concentration, completing the transduction from light signal to electrical signal. The intracellular potential hyperpolarizes due to the reduced transmembrane sodium ion current. 6. Hyperpolerization closes the voltage-gated calcium channels, and reduces intracellular calcium. 7. The release of neurotransmitter in rods, which is regulated by calcium concentration, changes and neural signals are created in the postsynaptic neurons in the retinal network. (Image from Webvision, http://webvision.med.utah.edu/.) 1.2.4 Retinal Degeneration Visual impairment can take form in a wide range of symptoms, including blurry vision, reduced visual acuity and/or visual field, with complete blindness being most severe. Due to importance of vision, visual impairment is a very debilitating condition, and blindness has been 13 considered worse than death (Rose, 2006). Visual impairment can arise from injury, degeneration and malfunction of different parts in the visual system, including the optical pathways (cornea, lens, and vitreous humor), retina, optic nerve, and central visual pathways and cortex. A significant part of the visual impairment is related to degenerative diseases of the retina such as age-related macular degeneration (Gehrs et al., 2006), which could be remedied by electrical prostheses. Retinal degeneration (RD) is the deterioration of the retina caused by progressive photorecptor death in the retina. RD impairs vision of the patient in the form of night blindness, reduced visual acuity, tunnel vision etc. They most commonly arise from inherited genetic mutations, but could also be results of artery or vein occlusion, and diabetic retinopathy. The most prevalent types of retinal degeneration are age-related macular degeneration (MID) and retinitis pigmentosa (RP). Together, they account for millions cases of blindness worldwide. 1.2.4.1 Age-related Macular Degeneration MID is a progressive degenerative ocular disease which usually affects older adults (mostly over the age of 55). Vision loss due to secondary dysfunction or loss of photoreceptors in the central retina (macular region) is caused by senescence and dysfunction of the retinal pigment epithelium (RPE), accumulation of subretinal drusen deposits, and in some patients, patchy loss of RPE or subretinal choroidal neovascularization (Medeiros and Curcio, 2001 ). The patients lose their visual field of highest acuity, but the peripheral vision is usually spared (Figure 1.9, center). This makes it difficult and often impossible for patients to perform visual activities requiring fine resolution, such as reading, driving, and face recognition; and they also suffer from poor fixation stability, 14 orientation discrimination, and shape discrimination (Bedell et al., 2009; Neelam et al., 2009; Wang et al., 2002). Figure 1.9: Simulated vision of AMD (middle) and RP (right) compared to normal (left). (Image from National Eye Institute, http://www.nei.nih.gov/) Approximately 40 million people are affected by AMD worldwide. The World Health Organization estimates that 14 million people are legally blind or severely impaired, accounting for 8.7% of worldwide blindness (Gehrs et al., 2006). AMD is mostly seen in Europe and in countries with European-based populations like the USA, Canada, and Australia. In the USA, for example, it is estimated that about 2 million Americans above the age of 55 have AMD (Chader et al., 2009). Approximately 700,000 new cases occur each year in the United States alone, with 10% of the patients becoming legally blind (Curcio et al., 1996; Margalit and Sadda, 2003). The exact causes of AMD is unknown, but risk factors include age, family history, genetic mutations, hypertension, obesity, smoking etc. (Wang et al., 2007). AMD occurs in two forms: neovascular (wet, exudative) and non-neovascular (dry, nonexudative). The wet form is more severe, and involves abnormal blood vessel growth from the choroid behind the retina, causing subretinal scarring and retinal detachment. In the dry form of AMD, the RPE atrophies and remodels, and extracellular deposits of debris known as drusen accumulated between the retina and 15 the choroid. Treatment for either form of AMD is limited. For wet AMD, antiangiogenic drugs are administered through repeated injections into the eye to prevent the abnormal proliferation of blood vessels (Menon and Walters, 2009). Nutritional supplements with antioxidant, anti-inflammatory, and cell protective effects have been used in dry AMD; however they show limited effectiveness only (Tan et al., 2008). Stern cell therapies are experimental options under investigation. 1.2.4.2 Retinitis Pigrnentosa RP is the second most prevalent retinal degeneration that affects about 1.5 million people worldwide (Berson, 1993; Hartong et al., 2006). The incident rate is 1:400 live births overall, however could vary depending on ethnicity (Berson, 1993; Hartong et al., 2006; Margalit and Sadda, 2003). RP is a group of inherited eye diseases that mostly affects rod photoreceptors, causing night blindness and tunnel vision (Figure 1.9, right). The RP phenotype manifests through mutations in rhodopsin genes, with more than 100 genetic mutations identified (Phelan and Bok, 2000; Heckenlively, 1988; Hartong et al., 2006). More than half of all RP cases are autosornal recessive traits, a third are autosornal dominant, and the remainder are X-linked (Hartong et al., 2006; Heckenlively et al., 1988). Nutritional supplement of vitamin Ahas shown mitigation and postpone the disease progression (Berson, 1993 ). Other than the approved Argus II Epiretinal Prosthesis (and other potential prosthetic devices), treatment for RP is very limited. Ongoing research efforts involve retinal transplantation (Sagdullaev et al., 2003), gene therapy (Stein et al., 2011), and stern cells therapy (MacLaren et al., 2006). 16 1.2.4.3 Inner Retinal Survival and Remodeling in Retinal Degeneration In either AMD or RP, photoreceptors are lost as the disease progresses. However, much of the inner retina survives. In two studies, the inner retina of AMD patients is well preserved, with 70o/a- 93% of RGC survival and no significant difference in the inner nuclear layer cells (Kirn et al., 2002a, 2002b ). However, when comparing the two types of AMD, the exudative form has shown 50% ganglion cell loss, while the nonexudative form has no significant changes (Medeiros and Curcio, 2001). In RP, there is a varying degree of cell preservation, with about 78-88% of bipolar cells and 30-75 % of ganglion cells surviving (Santos et al., 1997; Hurnayun et al., 1999b ). This allows the prosthesis to stimulate remaining functional retinal neurons to provide some artificial vision, which was demonstrated by experiments for the first time in 1991 (Hurnayun and de Juan, 1991). Due to the degeneration of the photoreceptors in RD, the retina tissue undergoes significant remodeling. Abnormalities such as glial hypertrophy, neurite sprouting, neuron migration, and rewiring of the remaining neural network occur in response to the loss of sensory input (Fariss et al., 2000; Marc and Jones, 2003; Marc et al., 2003; Jones et al., 2012; Jones and Normann, 1997). The remodeling will depend on the disease progression and occurs in phases (Jones et al., 2012); and variability between individuals is always a possibility. 1.2.4.4 Animal Models of Retinal Degeneration The retinal remodeling due to RD poses a challenge to study the retinal prosthesis in the laboratory envirornnent. Animal models that mimic the degeneration in human retina are required 17 for electrophysiology studies, because the use of healthy animal/tissue could not reflect the realistic situations that the prostheses face in patient users. Models of retinal disease exist in dog, cat, chicken, mouse, and rat (Chader, 2002). For example, the Abyssinian cat is an animal model of retinal degeneration that closely mimics recessive human RP (Narfstr6m, 1999), and the Irish Setter exhibits a genetic defect also found in rd mouse and recessive RP in human (Suber et al., 1993). Many animal models have been developed in rodents to study the pathology, etiology, and electrophysiology of RD. For example, the Royal College of Surgeons (RCS) rat is the first known animal with inherited retinal degeneration and is widely used for research in hereditary retinal dystrophies (Bourne et al., 1938; Pu et al., 2006). The S334ter rat and P23H rat are transgenic models to express autosomal dominant mutations of the rhodopsin protein similar to that found in human RP patients (Lee et al., 2003). A few dozens of mouse models are available for RP and MID research, which have different genetic mutations, severity and progression of degeneration (Chang et al., 2002). The rdl and rdl 0 are widely studied models which reflect autosomal recessive traits of RP The rdl mice have a much faster onset of retinal degeneration and normal development never occurs; while in rdl 0 mice, the degeneration starts slower, and early retinal development is normal (Figure 1.10) (Chang et al., 2002, 2007). 1. 3 Retinal Prostheses The retinal prosthesis is a biomedical device that restores some sight in visually impaired patients by activating the retina via electrical stimulation. Depending on the location of the 18 electrode array (Figure l.U), retinal prosthesis could be divided into epiretinal, subretinal and suprachoroidal retinal prostheses (\Veiland and Humayun, 2014a; Zrenner, 2013a),. First, a brief overview of these ~ee types of pr.osJheses is given The conu:non COI)Jponents and S:afety concerns of retinal prosthes. es are. then presented. Clinical trial and conunercialization eff01ts of several prostheti.c devices are introduced. And finally challenges tlrat the current te1,1hnology fac· e are· dis~ussed. rd1/rd1 at 21 days of ago rd10/ft110at 24 Clays of age Figure 1.10: Histology sections of WT {left), rdl (tniddle),,and rd JD (right) mouse retina are shown side by side. The. rd.J retina has signifrcant loss of the, photoi'e.ceptor including -0uter nuclear layer. The rd.JO retina suffers from slower degeneration. '\Vbile the ·outer and inner segments · of the photorecepfors are complete! y degenerate cl, the loss of the outer nuclear layer is not s'o sever-e in rd.JO as compared to rdJ. (Chang et al., 2002) 1.3.1 Types efR!etinal Prostheses 1.3 .1.1 Epiretina{ Pr. osthes es· In the epiretinal prostl1esis, th!!' electrode.a:rr-ay is locate~d 9n th!!' inner surface of th~ reJjna (tlie inner limitin,g membr;me), with the ele9tr. Qdes being closest t0, the g'anglion cell layer. This configuration provides dir\;1;t ac!!ess t9 th~ RGCs-the output layer of th~ retina, while stimulation 0,fthe furth~ bjp. -;>hlt cel1 is stil{ p9Ssii,}e (Weitz, 2013). an OJ.le itaQ<I, this is advantageous because 19 this results tn the lowest stimulation threshold (Fried and Jensen, 2011): and ganglion cell are spiking neurons, thus the short latency response. ta direct stimulation of ganglion cells could be rell'ably reproduced at high frequency without adaptation (Freeman et al., 201lb). On the other hand, the close proximity betwe. en the stimulating electrodes · and ganglion cell axon. scan cause in axonal activation, which create irregular phosphene shapes and degrades the spatial resolution (Behrend et al, 2011: Nanduri et al, 2012). VJsu111 cot tu)( .;..--- Mb1 r11lllfif'l/\C\_..,"' Figure 1.11: A schematic showing the location of electrode array in relation tO: the retina for the three types ofretinal prostheses. (Zrenner,, 2013a) 20 For the epiretinal prostheses, a tight placement of the electrode array on the retinal surface is a challenge for both the engineering of the array and the clinical procedure. In some devices such as the Argus II (Figure 1.12), the electrode array is tacked to the retina and necessarily creates limited retinal damage at the tack site (Basinger et al., 2009). If not positioned correctly, the retinal activation threshold on some electrodes could become too high for safe and effective stimulation, rendering them non-functional and degrading the overall resolution (de Balthasar et al., 2008; Humayun et al., 2012). Some experimental devices are exploring protruding electrodes to improve contact with retina (Rossler et al., 2009), or using 3-D electrodes with one design even aiming to penetrate into the bipolar cell layer from the epiretinal side (Zrerrner, 2013b). The electronics of the epiretinal implants are placed in the orbit of the eye and/or the vitreous humor. The relatively large space is beneficial for avoiding heat and mechanical damage from the electronics (Lakhanpal et al., 2003). Also, a majority of the electronics of the device can be incorporated into the associated external components, allowing for a smaller implant and some upgrades capabilities without additional modification to the implanted components (Weiland et al., 2005). 1.3.1.2 Subretinal Prostheses In the subretinal prosthesis, the electrode array is inserted in the subretinal space between the RPE/choroid layer and sensory retina, in the place of the absent photoreceptors (Zrenner et al., 1999). The implantation is achieved typically through an incision an incision of the sclera, choroid and RPE (ab externo) but could also be done using an incision on the retina (ab interno) (Weiland 21 et aI., 2005). The ekctrodes are closest to the bitwlar cells, and could stimulate' any remaining photoreceptor. This could Work iii favor for th~ devic'e' by · acc'essing th~ . remaining ·peural petworks, but ·could also be a ·challenge due to the remodeling in degenerated retina. Some subretinal ptostheses utiliz~ 11 mict, ophotodfod'e airay (MPDA) f9 <;'Onvert the incoming light info electrical stimulation, therefore mimicking the function ti'f photoreceptor· s. ~ OfWc 1'11efVE! r tnbraln .....--- Implant tadr.l!d tof'ftln.d Figure 1..12: A schematic of the Argµs epiretinal prosthesis. A camera mounted on the glass te~r9fd visual .infonnation, ""1llch is sent to the implant tlu·ough wireless telemetty after being pfQcessed by an image processor. The implant ·electronics generate stirrrulation current pulses, which are delivered fo the electrodes on the epii:etinal array (Weiland. et al., 2005) Subretinal stimulation targets bipolar cells. This utilizes some of the neural proces.sing of the inner retina, and avoids activafron of ganglion cells axon bun< lles ""1llch could create. irregular 22 phosphene shape. The position of the array in the subretinal space eliminates the necessity to secure the array by mechanical methods (such as a tack); however, the design has to considered thermal damage to the retina by the electronics (Weiland et al., 2005), and the size limitation on the implant. Possible complications include the risk of retinal detachment due to the surgical procedure to insert the array in place and the long-term presence of the array under the retina. 1.3.1.3 Suprachoroidal Retinal Prostheses The suprachoroidal retinal prosthesis has an electrode array located in the suprachoroidal space between the choroid and sclera (Sakaguchi et al., 2004). The mode of its operation is sometimes known as suprachoroidal transretinal stimulation (STS) as the electrical current is passing through the entire retina to a return electrode inserted into the vitreous humor (Fujikado et al., 2011). The electrode array's location can be easily accessed by surgery, therefore simplifying the implantation procedure (Sakaguchi et al., 2004). The choroid and sclera are more robust and less prone to mechanical disturbance compared to the retina. The only intraocular component (if any) is the return electrode, therefore reducing the risk related with inserting complex components into the eye. Although the choroid layer thins in RP patients (Ayton et al., 2013), the distance between the electrode array and its stimulation targets, whether bipolar cells or RGCs, is larger compared to epiretinal or subretinal implants. This increases the threshold of stimulation and therefore power consumption; Focal stimulation, which is required for high resolution artificial vision, is a challenge due to the wider spread of electrical field and current at those distances (Weiland et al., 2005). 23 1.3.2 Common Components of Retinal Prostheses Due to the variable configuration, different types of prostheses don't necessarily share all the same components. The most common components are listed in considering their proximity to the neuronal targets in the retina, and are discussed in the context of the applicable type of devices. 1.3.2.1 Multielectrode Array (MEA) The electrode array is the direct interface between the device and the retina that delivers the electrical stimulation. It may be placed on the inner side of the retina ( epiretinal), in the subretinal space, or outside the choroid (suprachoroidal). Considerations for each type of placement have been mentioned in the previous section on the different types of retinal prostheses, and will be further discussed here. Compared to older implants such as cochlear implants and deep brain stirnulators, retinal prostheses demonstrated a significant advancement in electrode technology. Instead of hand-made electrodes, electrode arrays in retinal prostheses utilize microfabrication and photolithography techniques from the integrated circuit (IC) industry to achieve small electrode feature size and high contact density (Figure 1.13) (Weiland and Hurnayun, 2014a). Protruding electrodes have been designed or utilized (Fujikado et al., 2011; Palanker et al., 2005), and complex shape and geometry may become possible for future designs. For epiretinal and suprachoroidal prostheses, which have the electrode array separate from the implant electronics, the array are fabricated of flexible materials-polymers such as silicone, polyimide, or parylene (Zhou and Greenberg, 2009). For subretinal prostheses that utilize 24 photodiodes as the signal source, the electrodes are fabricated on the surface of the chip that contains the photodiodes and relevant electronics (Chow et al., 2004). .. .. • • • ii •• .. . . . . 2.5mm Figure 1.13: Heat molded and annealed retinal electrode array with retained spherical curvature (Rodger et al., 2008). 1.3.2.2 Electrical Stimulator, Integrated Circuits, and Packaging The electrical stimulator refers to the electronic components that generates the current/voltage pulses delivered to the electrodes. While often referred to as a "chip'', which implies a single integrated circuit, even modem retinal prosthesis require multiple electronic components to perform all the required functions while maintaining a small volume. The stimulator must produce high voltage to inject suprathreshold current across the electrode-tissue interface to the neuronal targets, and should provide certain programmability in the pulse parameters (pulse rate, pulse width, interphase gap, etc.). The design also needs to trade-off between stimulation efficiency, power consumption, and size (Weiland et al., 2005; Weiland and Humayun, 2014a). For retinal prostheses without implanted light detectors, the stimulator needs to receive 25 stimulation data, control signal, and power from other components. This has been achieved through radio frequency and infrared wireless links (see 1.3.2.3). The stimulator component is implanted, and therefore must be hermetically sealed to avoid malfunctioning and destruction due to the leakage of conductive bodily fluid into the system (Jiang and Zhou, 2009). The leads pass through the packing via feedthroughs to connectto the electrodes (Figure 1.14, top). Achieving high density feedthrough remains a limiting factor for size of the implant (Weiland and Humayun, 2014a) and is an active research area (Gill et al., 2013). For intraocular implantation, the vitreous humor also serves as a heat sink for the electronics (Piyathaisere et al., 2003 ), while devices located in the orbit of the eye also have efficient heat dissipation due to the relatively large space. Hermetic; Case Silicon Microchip Seal - SYicon Mioroch1p Feedthrough Platform • ----- Conformal Coating Fee<:!throoghs Figure 1.14: Examples of IC packaging methods involving hermetic encasing or coating and feedthroughs (Weiland and Humayun, 2014a). In subretinal prostheses utilizing photodiodes, the stimulators are integrated with the photodiodes and co-localized with their electrodes. This provides a major advantage because the stimulation units are locally grouped with their respective signal source (the photodiodes); hence 26 there is no need for external signal transmission and wiring of leads from electrodes to a central implanted signal processing unit, allowing a high density of stimulation sites. However, hermetic packaging is a challenge (Figure 1.14, bottom), and polymer coating have not been able to protect the electronics for long period of use (Sting! et al., 2013). Also, completely passive photodiode can't produce enough photocurrent for stimulation as shown in previous studies (Palanker et al., 2005), and active amplification is needed. Therefore power and control signal transmission from outside is still needed, while stimulation amplitude for each channel is controlled by incoming light. 1.3.2.3 Power and Signal Telemetry Commercial retinal prostheses avoid direct electrical connection across the sclera or skin, since this is a path for infection Therefore wireless transmission is necessary to deliver the visual data (if the camera is external), control signals, and power from external unit t to the implanted electrical stimulator. This mechanism also provides additional safety to the device, as the implanted component does not contain elements of high energy density, and the implanted device can be de- powered rapidly by removing the external source. Inductive coupling using radio frequency (RF) between two coils is one solution (Figure 1.15, left). One coil is with the electrical stimulator in vivo and the other one is on the outside of the body in close proximity with the first one. Magnetic field is created as AC flow through the external (primary) coil, and this induces a voltage on the internal (secondary) coil. The rate and efficiency of transmission should be sufficient for delivering the data and power. This could become challenging as the number of individual channels increase with large electrode array and more 27 e1ectmae&in 11\e f\ittu·e. D~ fi:'eqtleney bands "'7.c¢'1:1. esigrted iosc;bieve optittBI ttansmissiop fur tio1hpewer'.8!ld chta(C!ien eta!., 20 I Q ) andint!Udingba:ckwardtelemetty(Tran a at, 20 Iii) allows fee!ll:rai::k o fotheintertal information of the imptant's operation T rdtd· oo1 Sl;J i... • il"f:!Ji<4 ,.,.., ! ~ 11!'i~i\rt!. 0--trlin.g Figurel.15: Different methods. of. telemetry RF (left) (Rii:zo et~,.2.007) an4 II\ (tigl'tt) (Palanl,;er et.al, 200$). infrared (IR ) light is aneUrer prethod of transmission of data, and it utilizt:s· !he palural on.lie system 0 f!he eye(Figtlre l.15., right~. Alight souroeontheoutside, for =r!¢e.onapair 0 f glasses, projects J.WdtJ!a!ed infrared hgl)t.through flte pupil onto the retina where the s. tinmlatoris locate;;\, For ei ther ey1- or subretiral. irrgi.lants, phot.Qcliodes convert 11\e illfhlre4 lightinto elec!Jical s1gtals . . One advantage is that the. ttansmission is Interrupted by eyelid' closing. mmtcking llie norrrral situation where l)O light p&ception is )lt~ent tl\lling eye clostu·e. Since pasAiv.e ill)piant carillot gerretate ei1ough stinwtati:on ll!U1Cllt through pliotodiocl~, a. ctive atnplificatiort is requited .and in:c!uctive· coufiling is used for po.wet te!emetty Still, !he power ofinfrar~d ttarisrriission )las to. ·~e tarefullyl imited, so that the. heal effect doesn't . cause any datrage to. the ey~. :28 1.3.2.4 Camera and Image Processing An external camera is used to capture the visual input for the device. The video camera is incorporated on the frame of a pair of glasses for the patient to wear. Miniature cameras have become a fairly mature technology and sufficiently small for the prostheses purpose. The resolution of the camera is not a limitation as the number of electrodes of the prostheses is now far less than the pixel count (million pixels for smartphones ), and will stay so for the foreseeable future .. An external camera is not required only when an implanted microphotodiode array is used to directly convert incoming natural light into stimulation currents. Mounting the camera on the glasses requires the patient to move the head to scan the envirornnent or an object of interest. Although it helps with integrating information from the limited channels, this is an unnatural behavior that requires learning and adaptation by the patient, and it slows the performance of patient using the device (Stiles et al., 2011). For next generation of prostheses, miniature intraocular cameras are under development (Chai et al., 2008; Stiles et al., 2011). The device is small enough to be implanted in the lens of the eyeball, and therefore will allow natural target tracking using coordinated eye and head movement (Figure 1.16). A wide field camera together with eye tracking to select regions of interest from the scenes is another possible solution. The image processing unit receives the visual input from the camera, and processes it before sending the stimulation pattern to the stimulator. The image processing unit is usually part of the external components, which also serve as the general control unit for the entire device; however it could become part of the internal electronics as future devices become fully implanted (Humayun 29 et al., 2001) The imaging processing at feast involves down sampling and compressing the information in the irnage to match the limited ele:::trode numbers. Gray-scaling, contrast Enhancement, edge and motion detection are amoog I.he many fearures that can he implemented. Computer vision can provide optimization (Chen et al, 2009b; vanRheede et al., zorn) and manyaddttional features for patient users (Figure L 17). Inforrnation analysis can reduce redUtldant info1mat1on I n thepixelized image (Hallum et al.,, 2008). Salii:ncy-based cues have shown to improve mobility and search task in simulated prosthetic vision (Parikh et al., 2013). 1t&fi1aa«tct1!~ 5Mlllllff•i119 Figure 1.16: An intra ocular camera is implanted in the .lens. This will allow prosthesis users to achieve natural' gaze :control with f!j€ movement, which improves the v. isual capability compared to using head motion in case of an externally-mounted camera (Stiles et al. , 2 011) As many prostheses stimulate the ganglion cells and bipolar cells, image processing that mimic retinal processing could improve stimulation. Models have been developed to predict the pattern of action potentials that RGCs fire in response to any light stimuli (Eckmiller et al., 2005; Jepson et al., 2014b; Nirenberg and Pandarinath, 2012). Applying these models, sophisticated image processing could be implemented to create "natural" stimulation pattern delivered to ganglion cells. Theoretically, this method will provide the best visual percepts for patient users if it is implemented with high resolution electrode arrays and good fidelity. Figure 1.17: Example of simulated prosthetic vision showing original image in top left, processed image in the middle column, and corresponding phosphenized output on the right (Chen et al., 2009b ). From top to bottom, the image was processed with filtering, histogram equalization, and edge detection, respectively. 31 1.3.3 Safety Concerns Many safety issues have to be considered for an implanted device with active electrical stimulation. Specific design controls have to be implemented, so that the benefits for the patient users outweigh the risk. For a retinal prosthesis, a safe implantation surgery without complications is only the first step for the use of many years to come. Biological safety is a major consideration for the device to work properly without harming the patients. On the other hand, the device itself needs to stay functional for a long time; and for this purpose, the electrochemical safety and packaging is a major consideration for the implant. 1.3.3.1 Biological Safety The implanted device has to be biocompatible with human tissue. The electrodes, the substrate of the array, the leads, and packaging of the electronics are all made of inert materials whose biocompatibility is well-known. The electrodes generally made ofbiocompatible materials that are metals (platinum, gold, titanium, or tungsten) (Merrill et al., 2005), metal compounds (iridium oxide, titanium nitride) (McCreery et al., 2006; Schaldach et al., 1990) or conductive polymer such as poly(3,4-ethylenedioxythiophene) (PEDOT) (Cui and Zhou, 2007). The substrate is made of biocompatible compounds such as silicone, polyimide, or parylene (Zhou and Greenberg, 2009), which are also used for protection coating of some devices (Hornig et al., 2007). Any novel materials, such as alloys or coating for electrodes, are required to go through rigorous bench testing and animal studies to establish their biocompatibility to prevent inflammation and adverse reactions. Mechanical damage to the retina and eye is also possible, if the electrode exerts mechanical 32 pressure on the retina (Colodetti et al., 2007; Ray et al., 2009), or if the substrate materials are too rigid. The need to detach the retina for subretinal implants is a consideration that limits the size of the array, as large area of detachment have severe consequences for the health of the retinal tissue. Epiretinal implants are constrained by size of the incisions that can be safely performed on the eyewall during the surgical procedure, and the placement of the retinal tack for securing the array should be carefully performed for minimal tissue damage. Thermal damage is another safety concern, as heat dissipated by the electronics could potentially injure neural tissue. For retinal prostheses, locating the electronics further away from the retina allows a larger electronics case and a greater power budget, since heat can be dissipated over a larger surface (case surface) and heat effects on the retina are minimal because of the distance. (Piyathaisere et al., 2001) Besides the passive materials, the active electrical stimulation must operate within safe levels for stimulation. Applying too much current or charge to a certain region could potentially damage the tissue. Safety levels have been extensively studied (McCreery et al., 1990; Shannon, 1992; Merrill et al., 2005) for charge and charge density (charge per electrode surface area), and stimulation parameters are designed to stay within such boundaries (Figure 1.18). 1.3.3.2 Device Safety Electrochemical safety is a concern due to the mechanism of how electrical currents generated by the device pass through the electrode-tissue interface to reach the neuronal targets. The current injection mechanism involves capacitive charging of the interface and also electrochemical redox reactions (Faradaic) (Also see section 1.3.5.1). 33 ......... L a. 2.-.0 ...... .., ' ru e u I 00 k~?.O ' u ~ - >.. kc: l.5 ........ ...._ ·- en c Q) 0 Qt1 :o k-1 _0 OJ L 0 5 £ u 0.5 ] ro Chof"'-ge pe ..... phase (µC/phl Figure 1.18: Early studies showed charge density and charge per phase as cofactors in determining tissue damage in neural stimulation (Shannon, 1992). However, stimulation safety is a complex issue depending on many other factors too. Table 1.1: Charge injection limits for some electrode materials. Material PEDOT (Jan et al, 2009; Yamato et al., 1995) Carbon nanotube (Wang et al., 2006) Platinum (Pt) (Rose and Robblee, 1990) Platinum gray (Zhou et al., 2013) Iridium oxide (IrOx) (Cogan et al., 2009) Titanium nitride (TiN) (Weiland et al., 2002) Tantalum pentoxide (Ta20s) (Rose et al., 1985) Titanium Oxide (Ti02) (Rose et al., 1985) Charge injection capacity in mC/cm 2 11.3 1- 1.6 0.05 - 0.15 1 1-9 0.9 0.3 0.6 34 Developing electrode materials with higher charge injection limits gives larger safety margin, especially as electrodes decrease in size. Materials science research has been providing a significant increase in charge injection limits (Table I. I), and the decrease in impedance of electrode material can reduce the power consumption. Platinum/Iridium (Pt/Ir) coating on metal electrodes has increased the charge injection capacitance of the base metal by orders of magnitude (Petrossians et al., 20 I la), and improves neural stimulation efficacy significantly (Petrossians et al., 2014). Iridium oxide films can achieve a large anodic charge delivery capacity from 40 mC / cm 2 up to 157 mC/cm 2 (thickness depending) after electrochemical activation (Wessling et al., 2008), however such high values were obtained from cyclic voltammetry and not tested with electrical pulsing. Charge balanced biphasic pulse have been used in neural stimulation in order to stay within the safety limit for repeated stimulation (Lilly, 1961). The first phase, which is typically cathodic current, injects charge to evoke responses in the tissue; while the second phase recovers the injected charged and reverses the electrochemical process, therefore allowing consequent pulses to be applied. Using current pulses allows simpler control of the charge for each phase, while more considering is giving for balancing the charge of voltage pulses. For example, a blocking capacitor is typically used to ensure charge balance for voltage pulses. And specific electronic elements or circuits have been designed to reduce charge imbalance (Sit and Sarpeshkar, 2007), release accumulated charge and control the residual voltage by calibrating the current level (Krishnan and Kelly, 2012; Teixeira et al., 2015). Even with the use of charge balancing, the charge threshold for neural stimulation could exceed the electrochemical safety limit during the first phase of each pulse 35 electt'oi.1es' c-oi11d :n0, lpnger be ~afu)y Used fa· st'i.tnul' atian in :these· s1t~1atians: an. d fat l'a-h~l b"Otll b1olQ gieal and e!ectrochernical · safety; an/ii a1so ioft.fease battery life in prosth~ie devices Fer t,F" 1g41re 1 19~ fWeitt er aL, 2011) Wavefmns designs a:r. e act.iv e re;eart:h iill'eas that r.eJu,e 07 2010:1 a.'1.d eniil<lp'OYsiq1y a:ohieve ~herstimulatidn.features(Gr!!l andJy1ottiiner, 1 997) OµA pulse width I amplltude . f nterphsse gap Figure 1.1 9'. Biphasic pulses hav.e lreai wi.dei.y used irt rreµral engineering, The charge balancing prevents · cha:rge. ace umulation foc· repeated pulsing. The twei phases• d n not ne-ces· sarily have the same pulse wi<lth and · arnyliti,1cle. rAn irterphase gap bEtween the two phaf'es reduces neo.t'lll acti~ation threshol<l QWeitz ~al , 2011). Cherru~Land mec!iani:cal sta1Ji1ity· areal?,o 1-rey $011S!deratiPns fortlw l~ngevityoftl\:e del!i ce. Ordonez et al., 2012), and other novel materials such as multilayer multi-material films (Weiland et al., 2013), ultra-nanocrystalline diamond (Hadjinicolaou et al., 2012; Xiao et al., 2006), and amorphous silicon carbide (a-SiC) (Cogan et al., 2003; Sharma et al., 2012) have been developed and explored to achieve long-term protection. 1.3.4 Clinical Trials and Commercialization Among the many efforts for visual prostheses 1 over the past decades (Zrenner, 2013b), a few retinal prosthetic devices have entered/finished clinical trials (Weiland and Humayun, 2014a ), and two have received regulatory approval (Zrenner, 2013a ). This section introduces the clinical trials and commercialization of some of these devices (Figure 1.20). With the current research and development momentum, it can be expected that more products will enter clinical testing and become available on the market in the coming decade. 1.3.4.1 Epiretinal Prostheses Epiretinal prostheses that have been investigated in clinical trials include devices from Second Sight Medical Products, Inc. (SSMP, Sylmar, CA, USA), Intelligent Medical Implants GmbH (IMI Intelligent Medical Implants GmbH, Bonn, Germany) and EpiRet (EpiRet GmbH, Giefien, Germany). A schematic of the SSMP epiretinal prosthesis is shown in Figure 1.21. A camera mounted on a pair of glasses records the visual information from the environment in front of the patient user, 1 There are two online sources that maintain a listing of visual prostheses research efforts: http ://www. eye-tue bingen. de/zrenner/retim plantlist/ http ://www. io.m ei. ti tech. ac. jp/research/retina/index.htm l#Links 37 w}Uc. h i-> s. ent to the implanted el@ctroni.es th.tough wile less telemetzy ~r beirg process. ed by a. video processillgunit( VPU). 'I'h? s· tinulator th?11'gereiates cumld pulses, which are deliv~d to the electro:ies on the epiretimlmay. Ficure 120': Fun!us ph:>tograpJ. of6 miral -th?ses (Weiland et al., 2011). Image from u~ followug prostlwis graips : A Artificial Silicon Retu .. (white.,,..,,.,) (Optcl>icmics, In: . . ). B. Argus I (Secon! Sight Medjcal Pwhlcts, Ire.). C. Active subteli1ul de-nee (Retu" 1""4ri!, Gnl>H). D. Epi-Ret 25 electrode device (EpiRet, Gnl>H). E. Foey.riine electrode epiretimldevice (ll1lellig.nt hledical lniplms ). F . Argus Il (Second Sight MedicalP10ducts, Inc.). Tie· fxrst gerera.tion ~vice from SS MP, tl'e ~s I in¥ilard (also knO#na.s Argus 16 fm- th? sim of the electrode a.rra.y) was tes~d.in 6 sub~·cts with RPbetwee112002 ard'.2004 in th? Uni~d Slaws (de Balthasaret al., JJOB; Weiland .,W Hun..}Ul\ J)l4a). Ile extraoculuelectronics w.,. a.I., JJ. 03), A c.able lun; along the ten1ple irilo tl"te 0>:bit, termirut~ a.t tl-e il1l~oolar IvlEA w hi: h 1w a4x4grid ofl6pla.tirumeled10des-of 260 µm or 520 µm diam?ters. 38· A Coil/ B Camera ~ VPU I Glasses Suture tabs Electrode cable .------Array r Sciera I band Electronics case / ~Implant coil Figure 1.21: The external (A) and implanted (B) components of the Argus II device (Humayun et al., 2012). 39 The device enabled the first demonstration of the safety of long-term retinal stimulation in humans. The stimulation thresholds of most electrodes were within safety limit of platinum (de Balthasar et al., 2008). Subjects were able to perceive light, detect and discriminate objects from a small set, and detect motion of a moving bar (Yanai et al., 2007). The phosphenes produced from single electrodes were in general small and circular shaped (de Balthasar et al., 2008; Horsager et al., 2009), although irregular shape of elongated phosphenes have also been reported (Nanduri et al., 2008). Increases in stimulus amplitude tended to increase the brightness and size (de Balthasar et al., 2008; Greenwald et al., 2009; Humayun et al., 2003). The best possible visual acuity given the electrode spacing of Argus I is 2.21 logMAR (20/3240), which was measured in one subject (Capsi et al., 2009). Threshold, electrode impedance, and electrode-retina distance were measured, and threshold was correlated only to proximity of the electrode and the retina, and not with the other factors (de Balthasar et al., 2008). Therefore, maintaining a tight position of the array is critical. A long term follow-up study of up to ten years in one patient demonstrated the stability and durability of the device (Yue et al., in press), which is also promising evidence of longevity for the next generation implant. The second generation device built by SSMP is Argus II, which has an array of 60 electrodes (200 µm diameter and 525 µm pitch), covering a visual field of approximately 20 degrees diagonally (Zhou et al., 2013). The electronics were implanted in the orbit in a hermetic package, with the epiretinal electrode array connect by a transscleral cable. The multi-center international clinical trial on the Argus II device is the largest one so far on any retinal prostheses. The trial showed encouraging results, reporting a total of 45.6 cumulative subject years when the interim 40 results were published in 2012 (Humayun et al., 2012). Thirty subjects were implanted between 2007 and 2009. Comprehensive studies were tested on 28 subjects, with one device removal due to recurrent conjunctiva! erosion and one subject unavailable for testing (Humayun et al., 2012; Dorn et al., 2013; Weiland and Humayun, 2014a). All subjects perceived light during electrical stimulation, and the best visual acuity achieved was 20/1260. Object localization, motion detection, and letter reading were tested: With the device on, 27 out of28 subjects performed better to located and touch a white square on a computer screen (Ahuja et al., 2011 ), 16 out of 28 subjects performed better at identifying the direction of a moving bar on a computer screen. 22 subjects could correctly identify a set of eight high-contrast letters with 72.6% success rate, versus 16.8% when the device was off. Subjects were allowed to take time to identify the letters; reading speed was slow but improved somewhat with practice (da Cruz et al., 2013). Patients continued to use the device in home. The Argus II device demonstrated safety and the true feasibility of high-resolution prosthetic vision. It received CE mark in March 2011 and FD A approval in February 2013, becoming the first commercially available retinal prosthesis in the world. As a high-tech medical device, Argus II is relative expensive (about $100,000 for device and surgery, as of2014), however, simulation studies have shown that it is a cost-effective treatment for RP in the long term (Vaidya et al., 2014). The IMI device is similar to Argus II, such as in terms of external components and their placement, like the camera and telemetry components mounted on a pair eyeglasses, processing unit carried on a waist belt, and the epiretinal location of an array of 49 iridium oxide electrodes (Hornig et al., 2007). It uses inductive coupling for power transmission, but infrared light for visual 41 information. The IMI device uses an image processing algorithm, Retina Encoder, to predict firing pattern of ganglion cells, and produces stimulation to imitate this pattern. After acute clinical testing in 20 RP patients (Keserii et al., 2012), chronic implant study started in 2005 with 7 subjects (Weiland et al., 2011; Weiland and Humayun, 2014a). No external camera was used and the device was only activated in clinical settings. Patients were able to identify phosphenes and patterns created by different electrodes. The thresholds were within safety limit, and the device was stable and well tolerated. The IMI company has been acquired by Pixium Vision, and the IMI device has become IRIS® (Zrerrner, 2013b). According to the company website, the current version of 49 electrodes is under clinical testing, and a product with a 150-electrode array is planned for commercialization in Europe 2 . The EpiRet latest device, EpiRet3, is an epiretinal prosthesis with a hexagonal array of 25 electrodes (Klauke et al., 2011). The implanted electronics components are miniaturized to fit entirely inside the eye; while the camera and the image processor would be outside the body mounted on a pair of glasses. Inductive wireless telemetry is used for power transmission. The electrodes were fabricated from gold and covered with iridium oxide. The electrodes had a 3D design of 100 µm in diameter and 25 µm height, which improved contact with the retina and lowered perceptual thresholds. The intraocular implantation eliminated the use of a transscleral cable for the connection to electronics The device has been implanted in 6 patients in 2006, for a semi-chronic ( 4 weeks) implantation study (Rossler et al., 2009). Experimental signal transfer was 2 Retrieved Oct. 2015: http:/ /www. pixiurn-vision. corn/en/technology-1 /iris-vision-restoration-systern 42 used instead of a camera for the study. Thresholds were low and within safety limits of electrical stimulation, and subjects could discriminate between unique pairs of electrodes and identify simple shapes. However the company has ceased operation (Zrenner, 2013b ). 1.3.4.2 Subretinal Prostheses Two subretinal prostheses have been tested in clinical trials by Optobionics, Inc. (Glen Ellyn, IL, USA) and Retina Implant AG (RI, Reutlingen, Germany). Both of the implants were microphotodiode arrays (MPDA), however the first was passive and the second is an active device. A brief introduction is also given to two subretinal prostheses in preclinical stage. Optobionics' device, the Artificial Silicon Retina (ASR), is a chip of 3 mm diameter. Its microphotodiode array contains about 3500 units, with no external power supply (Chow et al., 2004). The device was implanted in 30 RP subjects, and safety was demonstrated (Weiland et al., 2011). The passive photodiodes produced electrical currents on the level ofnanoamperes, which are thousands times below thresholds sufficient for neural stimulation (Palanker et al., 2005). Improvement in visual perception was reported by some patients; however, animal studies show that this was a neurotrophic effect due to the presence of the device (DeMarco et al., 2007; Pardue et al., 2005). The focus of device development has shifted from eliciting phosphenes to neurotrophic rescuing retinal and visual functions (Chow, 2013; Chow et al., 2010). The original company was closed as the ASR was not able to demonstrated efficacy and its future remains uncertain. Retina Implant developed an active MPDA with 1500 units (Figure 1.22). Each unit has a 43 photodiode, an amplifier and a 70 µm titanium-nitride electrode. External power is transmitted to amplify the photocurrent produced by the photodiodes. This was achieved via a percutaneous cable for the experimental device in the pilot study, and by inductive telemetry for the commercial device (Alpha IMS). The array was about 3 mm in diameter and covers about 15 degree of the visual field (Besch et al., 2008; Sting! et al., 2013; Zrerrner et al., 2011). An algorithm that calculates optimal implant position has also been developed, improving the device-tissue interface (Kusnyerik et al., 2012). The experimental device of RI has been implanted in 11 patients (Zrenner, 2013a). It has several direct stimulation electrodes besides the photodiode array for testing, and these electrodes demonstrated feasibility of visual perception in 8 of the 11 subjects. Problems with hardware reliability and hermetic packaging led to failure in the early implants, and the MPDA was tested in detail only in the last 3 of the 12 subjects who received improved device with longer lifetime (Zrenner et al., 2011). Subjects could perform several visual tasks such as identifying and locating objects, and one subject could recognize letters. The wireless Alpha IMS implant has been tested in 9 subjects (Sting! et al., 2013). Eight out of nine (8/9) patients could perceive light, detect and identify objects. Light localization (7/9), motion detection (5/9), and grating acuity measurement (6/9) were measured. Three subjects could read letters spontaneously and one after some training. Best Snellen visual acuity was measured up to 20/546 for one patient. Mechanical stress on the intraocular cable resulted in functional failure in three subjects after three to nine months of implantation, which was solved to prevent further cable breaks in the remaining patients. Hermetic sealing was another issue that led to device failure 44 in 3 different subjects after about 250 days post-implantation (Sting! et al., 2013). This proves t. o be a challerrge as the MPDArequires a thin transparent package that maximizes the light intensity transmitted. The device has receivt d CE mark in rnid-2013, and a multicenter trial with 19 RP patients is ongoing (Zrenner, 2013a). 1 . Subretinal chip 2.Foitwlth wtres / I Through 4.Rerum electrode L """'""----· - 7, Subdermal wires ( 3 . Subdefmal recei ver nd e.lectronics antenna a 1' f l Subrecinal 1 j orblc J Subcutaneous to retro-auricular area - Eprscrecaf S . Transn1itter antenna 6. Power box and control unlr for brightness and contrast Figure 1.22: The Alpha Il'JIS prosthesis from Retina Implant (Zrenner, 2013a). The implanted part of the device is shown in the top panel, with description of the individual components . and their locations in vi\10. Ad et. ailed view of the MPDAis shown in the middle panel. The 1 ower left shows an X-ray illustration of where the individual components are lo cat eel, and the lower right shows the external components of the device. Asubretinal prosthesis is developed by the Boston Retina Implant Project (Rizzo et al., 2011), which uses a MEA instead of MPDA This device has only been tested in animals; however a human-grade device with 256 channels of protruding iridium oxide electrode is under development for clinical testing (Kelly et al., 2013). Another subretinal implant with MPDA approach started in Stanford (now as Prima© under Pixium Vision, according to Zrenner, 2013b) which also explored the use of penetrating electrode. This novel approach provides good proximity to bipolar and/or ganglion cells, and therefore should result in lower stimulation thresholds and more focused spatial activation (Palanker et al., 2007). However, the risk of tissue damage and the clinical feasibility must be addressed for such method to become practical. 1.3.4.3 Suprachoroidal Retinal Prosthesis A suprachoroidal retinal prosthesis has been tested in humans for a limited time ( 4 weeks) (Fujikado et al., 2011, 2012). The implant consists of a 49 channel platinum electrode array with 9 active electrodes, each 0.5 mm in diameter and 0.5 mm in height protruding from the silicon base. The array was inserted into a scleral pocket, and the return electrode was inserted in the vitreous cavity. Two RP patients were implanted, and showed better than chance performs in object detection, object discrimination. Object grasping and touch panel tasks were tested on the second subjects, showing success/ accuracy rate. It was also shown that 20 Hz stimulation shows brighter phosphenes than 10 or 50 Hz for the same current level. The Bionic Vision Australia group has been developing suprachoroidal retinal prostheses (Villalobos et al., 2012), and preclinical testing showed the feasibility of long-term implantation in 46 animals (Villalobos et al., 2013; Nayagam et al., 2014). A pilot clinical trial has been completed for initial safety and efficacy evaluation of the prototype in human ( clintrial. gov NCTO 1603 576). The preliminary results from three subjects with end-stage RP showed long-term stability over 18 months; all subjects can perceive phosphenes, and performed better with device on versus off for laboratory based visual function tests (Ayton et al., 2014). 1.3.5 Research Areas and Challenges in (Epi-) Retinal Prostheses This section will address some of the challenges faced by retinal prostheses, with the main focus on epiretinal implants. These challenges include findings from the aforementioned clinical trials and also implications from preclinical device testing and animal studies. 1.3.5.1 Electrode-Tissue Interface and Electric Field Distributions The electrical activation of retinal neurons requires an electrical field to be established in the tissue. The electrode-tissue interface is the first step in this process, and influences the performance of prosthetic devices and the properties of the stimulation. To form useful visual perception, multiple electrodes need to be activated on the electrode array, either simultaneously or within short time intervals. With simultaneous stimulation, the electric field generated by each electrode will overlap spatially and temporally to contribute to the activation of retinal neurons. Electrode crosstalk therefore increases the spatial distribution of activation and limits resolution (Wilke et al., 20lla, 20llb). Temporal interactions between neighboring electrodes can also impede the resolution of multielectrode stimulation (Horsager et al., 2010, 2011). Besides cross-talk, electrochemistry and heating also limit phosphene density as a function of distance from the 47 electrode surface (Palanker et al., 2005). Maintaining good contact and small distance between the electrode array and the retina is also important for the proper functioning of epiretinal electrodes. The Argus device sometimes suffers from reduced resolution in patients because not all electrodes may reliably and safely produce a phosphene due to increased distance to the retina after the implantation (Humayun et al., 2012). Additional crosstalk via the highly conductive vitreous (or the saline that replacs it) could be significant (Tran et al., 2012). The distribution of current and potential from the electrical stimulation by the electrodes of the prosthesis is a critical factor determining the response of retinal neurons. So far, no direct study has addressed this distribution, and simulations were performed based on the physical models of the eye tissue (Kasi et al., 20lla, 20llb). The resistivity of retinal layers has been estimated (Brindley, 1956; Heynen and van Norren, 1985; Ogden and Ito, 1971) and are used in these models. However, the previous studies have several shortcomings: ( 1) The studies were, with one exception, carried out in amphibians or birds, whose retinal structure differs from mammals; (2) The current flow was not well controlled; (3) Only one ocular location was measured. The sclera thickness varies from 0.3 to 0.8 mm and the density ofretinal cells changes dependent on eccentricity; And ( 4) the retinal prosthesis will interface with a diseased retina-in RP the retina may be half the thickness of a normal retina (Berson, 1993), and tissue impedance may be significantly different. Our lab has recently published a study of impedance measurements in the eye (Ray et al., 2011; Shah et al., 2007). The data show a clear increase in impedance when the electrode is in close proximity to the retina versus in the vitreous cavity, suggesting a change in current flow for electrodes close to the retina. The analyses of these data lumped retina, sclera, RPE, and choroid 48 into a single impedance, while they most likely have different properties. 1.3.5.2 Visual Acuity High visual acuity is necessary for restoring central vision in tasks like reading and facial recognition. Each electrode can be considered as a pixel of the artificial vision. Although the relationship between each electrode and thephosphene itmight create is variable (size, shape, color, etc., see 1.3.5.4 and 1.3.5.5), the number and size of the electrodes on the array is an important factor for the spatial resolution and visual acuity (Freeman et al., 201 lb). For example, Argus I, the first epiretinal implant in clinical trial had only a 4x4 array of 16 electrodes with 2 60 µm or 520 µm diameter and the second generation has a higher resolution with an array of JOx6 electrodes with 200 µm diameter (Weiland and Humayun, 2014a); Future generations aim at orders of magnitude more electrodes (Chader et al., 2009; Stronks and Dagnelie, 2014). As technological development makes it feasible for larger arrays with smaller electrodes and electronics with more channels (Weiland and Humayun, 2014a), the number of electrodes and electrode density required for useful vision is a critical question for retinal prostheses. Investigation with animal models can provide prima facie evidence of the spatial resolution of electrical stimulation of the retina. Calcium fluorescence imaging studies with in vitro salamander retina found that the smallest activation area by a single electrode was 150 µm in diameter (corresponding to 20/660), regardless of electrode size as small as 10 µm (Behrend et al., 2011). This suggests that electrode size was not the limiting factor for achieving high visual acuity. In other studies with monkey retina, however, it was found that single RGCs could be 49 stimulated with 10 µm electrodes (Jepson et al., 2013; Sekirnjak et al., 2008), inducing single spike with sub-millisecond timing precision; also spatially patterned stimulation could further enhance the spatial resolution (Jepson et al., 2014a). Simulations of prosthetic vision can also provide prediction of the visual acuity achievable (Figure 1.23). Pixelized/phosphenized vision have shown that an electrode density of 625 electrodes per square centimeter in the foveal region could achieve a visual acuity of approximately 20/30, and is sufficient for reading and guided mobility (Cha et al., 1992a, 1992b, 1992c). And a 20/80 visual acuity has been suggested as the target of visual prosthesis which corresponds to 20 µm pixel/phosphene sizes (Palanker et al., 2005). A B c Figure 1.23: Simulated vision of text (A), office scene (B), and face (C), with doubling of spatial detail from left to right, and the original image on the right (Chen et al., 2009a). However, subjects in clinical trials have used head movement to scan the camera of the prosthesis across their environment, and use the integrated visual information to compensate for 50 the limitation of the electrode numbers (Hayes et al., 2003; Thompson et al., 2003; Dagnelie et al., 2007); this could be the underlying reason for the hyperacuity beyond the theoretical limit in the Argus II trial (Stronks and Dagnelie, 2014). Simulations have also shown that reliable face recognition with crude pixelized grids is possible with training (Thompson et al., 2003). Therefore functional criteria are more useful to patients for determining the effectiveness of the device, which should be clinically evaluated for different visual tasks. From both animal studies and simulations, spatial resolution and visual acuity can be expected to improve for future prosthesis users. However, issues such as electrode interaction ( 1.3.5.1), axonal stimulation (1.3.5.4), and retinal degeneration (1.3.5.6) still need to be addressed. Another limiting factor is the foveal region, which has the highest neuronal density. Approximately half of the primary visual cortex is dedicated to processing visual information from the fovea, (Rodieck, 1998) and it is responsible for our best visual acuity. Whereas location-dependent image processing (Palanker et al., 2005) and biomimetic inhomogeneous electrode distribution (Rodger et al., 2008; Waschkowski et al., 2014) may improve the performance of the prostheses by utilizing the spatial properties of the fovea, selective stimulation is extremely difficult due to the stacking of RGCs in this area. Therefore, it is possible that the acuity of prosthetic vision for patients can only be restored to a certain limit. 1.3.5.3 Visual Field Visual field is another equally important consideration for the spatial distribution of prosthetic vision. While current prostheses mostly focus on the central visual field, a larger visual field can 51 cover the peripheral retina for better mobility and interaction with the envirornnent, which also reduces the need of head scanning. Subretinal implants span less than 10 degrees of visual angle, due to the aforementioned risk of detaching the retina; current epiretinal devices cover up to 20 degrees of visual field, but size is limited by the incisions on the eye wall; suprachoroidal prosthesis have less surgical limitations, but still need to conform with the curvature of the eyeball. To increase the visual field, wide field arrays (Figure 1.24) for epiretinal prosthesis have been developed and tested in dummy and animal models (Arneri et al., 2009; Rodger et al., 2008; Waschkowski et al., 2014). Foldable arrays made of flexible polymer substrates can be inserted through the eye wall and then expand when in the eye. A visual field of approximately 35-40 degrees could be covered. Considering the different roles of center versus peripheral vision, the electrode sizes and densities in these two areas could be designed differently to accommodate engineering limitation of the device. 1.3.5.4 Spatial and Temporal Activation Patterns of Retinal Ganglion Cells Electrical activation of the retina could recruit various cell types and result in different patterns of visual perception (Figure 1.25). RGCs could be directly stimulated via activation of their soma or axon, or indirectly activated by the synaptic connection of the inner retina (bipolar cells and amacrine cells). Direct activation of the RGC usually results in one action potential per stimulus pulse, which is time-locked with short (milliseconds) latency (Freeman et al., 20 lib; Fried et al., 2006; Jepson et al., 2013; Sekirnjak et al., 2008). Indirect stimulation via presynaptic retinal neurons generally results in a burst of spikes with longer (tens of milliseconds) and 'jittery" 52 latencies (Freeman, 2010; Freeman and Fried, 2011; Fried et al., 2006; Jensen et al., 2005a; Lee et al., 2013), resembling the response to light stimulus. Both types of activation come with their advantages and disadvantages. Figure 1.24: Afundus image of a wide field array with biomimetic electrode spacing matching the local RGC density. (Rodger et al., 2008). RGCs can follow direct stimulation pulses at fairly high rates (200 Hz up to 500 Hz) with one action potential per pulse (Ahuja et al., 2008; Fried et al., 2006; Jensen and Rizzo, 2007; Sekintjak et al., 2006), and therefore "natural response" may be evoked by encoding stimulus 53 pulses to have the same terq>oral pattern as natural firing (Eclaniller et al .• 1005; Jepson et al, 2014b; Nirenberg and Pandwinath, 2012) . However, direct stirrulatioo often results in <Luna! activation for epiretinal stirrulation, as lhe <L"Cons are located between the ganglion cell body layer w1d the epiretinal electrode array (Figure 1.25). The activated axons convey infom111tion to the higher visual processing centers, 'Which perceive them as if they originated from the ganglion cell somas further " upstream " (farther from the optic disk) of the stimulation site and results in irregular phosphenes shape (Greenberg et al., 1999; Schiefer and Grill, 2006f). ---- \). Electrode II (Point of c lil Optic disk stimulation) _,....... ~ Activated area ~ :z:.. (a) Ganglion cells { n Figure 1.25: Illustration of the <ifferent visnal percepts (a) and activated retinal. area (b) when RGCs are stim.tlated via the axorn (I), somas (II), or de. ndrites and presynaptic neurons (III) (Greenberg et al., 1999). J\.'{onal stimulation (I) results in streak-like phosphenes. 54 A model of ganglion cell axon layout on the fundus (Nanduri, 2011) and threshold measurements of ganglion cell somas and axons (Fried et al., 2009; Weitz, 2013) support this conclusion. And indeed, clinical studies with the Argus epiretinal prosthesis reveal that the phosphenes sometimes have irregular elongated shape instead of the ideal round ones depending on stimulation amplitude and electrode location (Figure 1.26) (Nanduri, 2011). This poses a challenge to provide the patients with a consistent good quality image of high resolution, especially for later generation devices with higher number of electrodes. Figure 1.26: Phosphenes (black shades) drawn by one Argus II subject, superimposed over the corresponding stimulation electrodes (red circles) (Nanduri, 2011). :Many phosphenes are long streaks, while only a few have a relative round shapes. The blue square marks the approximate location of the fovea. 55 Indirect activation recruits some of the remaining neural processing mechanism in the inner retina, and therefore generates spiking pattern in RGCs that might better represent visual information. It reduces the burden to reproduce the natural responses directly in ganglion cells, and may produce better spatial resolution as the bipolar cell recruit only RGCs with direct connection and avoids the diffuse activation due to axonal stimulation. However, indirect activation favor longer pulse widths and only follows lower frequency stimulation (less than 10 Hz) as a result of different dynamic properties of cell membrane, synaptic transmission and ion channels in bipolar cells (Freeman et al., 20lla; Jensen et al., 2005b; Weitz, 2013). Therefore, indirect activation usually has a larger charge threshold which longer pulses typically require (Geddes, 2004; Jensen et al., 2005b; Weitz, 2013). Also, indirect activation desensitize with repetitive stimulation, for example starting when paired pulses are less than 400 ms apart and becoming significant when rate of pulse trains increase beyond 10 pulse/seconds (Ahuja et al., 2008; Freeman and Fried, 2011; Jensen and Rizzo, 2007; Ryu et al., 2009; Sekirnjak et al., 2006). This phenomenon is attributed to inhibition from amacrine cells and other unknown mechanisms. The desensitization limits the temporal resolution for indirect activation (Freeman et al., 2011 b ), and is thought to be the underlying reason for phosphene fading in human subjects such as those demonstrated by Argus II subjects (Perez Pornos et al., 2012). Animal experiments (Freeman and Fried, 2011) revealed two components to the fading-a fast component lasting hundreds of milliseconds, and a slow component lasting several seconds-and the results are in agreement with those observed in human subjects. In subretinal implants the natural micro-saccades of the eye prevent fading to some extent (Zrenner et al., 2011). For epiretinal implants, fading of 56 images can be counteracted by head movements, and the development of intraocular camera for epiretinal prosthesis will allow saccadic eye movement to mitigate percept fading. Another approach to overcoming desensitization is to vary the stimulus from pulse to pulse, for example by varying the pulsed width. Therefore the retina is exposed to a new type of stimulus for each pulse, and different neuron populations are recruited (Davuluri and Weiland, 2014 ). Direct and indirect response often co-exist, and epiretinal stimulation most likely activates RGCs and the inner retina simultaneously. Separating the two types of response can increase the selectivity of the stimulation, and however tradeoff between spatial and temporal properties of activation. Low frequency sinusoidal stimulus have been shown to preferentially activate the inner retina (Freeman et al., 2010), and different pulse widths were explored as a factor for avoiding axonal stimulation. Threshold maps ofRGCs obtained by calcium fluorescence imaging of in vitro rat retinal preparation have shown that the threshold relationship of ganglion cell soma activation versus axonal activation differs with pulse widths (Weitz, 2013; Weitz et al., 2013). Longer stimulation pulse widths (over 16 milliseconds) activate the irrner retina only, and are suggested as a potential solution to axonal stimulation. The temporal resolution of a healthy human eye can range from about 15 Hz (rod-mediated scotopic vision) up to 50 Hz (cone-mediated photopic vision), depending on factors such as illumination, amplitude, adaption, and wavelength, etc. (Hart, 1987). The temporal resolution in prosthetic implants is usually much lower (e.g. 5 to 20 Hz) compared to these numbers due to two factors. The electrical stimulation mediates membrane potentials of retinal neurons in a different manner compared to neurotransmitter release, and high amplitude stimulation may render 57 retinal neurons unresponsive for a certain period of time. Also, repetitive electrical stimulation is required in order to form a meaningful "visual stream" in prosthesis users and to compensate for higher thresholds due to retinal degeneration as well. 1.3.5.5 Color Vision Given the relative large size of electrodes in contrast with the very small dimensions of the three types of cones and the associated color pathways, selective activation of specific color components is not possible with the current devices. In subretinal and suprachoroidal stimulation, each electrode will stimulate a large number of bipolar cells and rods (if there are any remaining), and patients typically report shades of gray (Zrenner, 2013a ). In epiretinal prosthesis, the electrode's proximity to the ganglion cells will result in stimulation of color-specific ganglion cell subtype. Due to the processing of the inner retina to maximize information transmission to cortex, color presentation at the ganglion cell layer consists of opponent hues (red versus green and blue versus yellow). Therefore direct activation of ganglion cells could result in color sensation. Most of the phosphene color reported by Argus II subjects were yellow or white in color, likely due to activation of many ganglion cells a single electrode, however some Argus II subjects were able to perceive colored phosphenes including red, orange, yellow, green, blue, pink, gray, and white. The perceived color depended on the electrode activated, and stimulation parameters used, but results were repeatable (Stanga et al., 2011). Different colors could be perceived simultaneously (Stanga et al., 2012), and phosphene could appear darker than background (Horsager et al., 2009; Perez Pornos et al., 2012).In the clinical study of the IMI 58 prosthesis, phosphenes appeared in a wide range of colors (Hornig et al., 2007). White, yellow, and blue were the predominant ones, and red and green were rare. This could be a result of specific neuronal microcircuitry of blue-yellow ganglion cells (Calkins et al., 1998). While subjects sometimes could perceive color due to random factors in use, explicit color perception will remain a challenge for prosthetic vision. 1.3.5.6 Effects of Retinal Degeneration Retinal degeneration causes significant changes in retinal structure, cell density, morphology and physiology. Therefore, it is expected that the performance of the prosthesis is affected by the specific condition of each patient, such as type of RD and disease progression. In agreement with studies in human subjects (Humayun et al., 1999a; Jensen et al., 2003), most animal studies report that RD results in higher stimulation thresholds. For example, rat models of RD have revealed that as the density of surviving retinal neurons decreases, stimulation threshold increases and becomes higher than normal (Chan et al., 2011). Results in mice show consistent increase in threshold (Jensen and Rizzo, 2008, 2009). A recent study, however, found no significant difference between thresholds of RGCs in RD and normal rat retinas (Sekirnjak et al., 2009). The use of small electrodes and explicit direct stimulation ofRGC in this study suggests direct activation threshold ofRGCs is not altered during RD, while threshold increase observed in earlier studies was due to threshold changes of the indirect stimulation via bipolar cells. Threshold mappings via calcium imaging have arrived at similar conclusion (Weitz, 2013). Therefore, good coupling between the electrode and the retina and selective direct activation of RGCs could lower the stimulation 59 threshold in prosthetic devices. 1.4 Conclusion The past two decades have witnessed how retinal prostheses progressed from early concepts and translated from laboratory research through clinical trials to commercial approval. This great success of restoring partial vision through artificial means has demonstrated a feasible treatment for the blind affected by outer retinal degeneration. Hundreds to thousands of patients are expected to benefit with improved mobility and object localization in the near future. Despite these great achievements, a great many of challenges remains to be solved. The research and development of retinal prostheses has been and will continue to be a highly interdisciplinary effort. At this time, a major goal is to improve the acuity and resolution of prosthetic vision, which is very limited and far from comparable with normal vision. Both acuity and resolution are expected to improve with relentless engineering improvement, which will allow a larger number of smaller and more tightly spaced stimulation electrodes. Also, smarter stimulation strategies based on research in neural electrophysiology will provide more focal and selective activation of the retina without invoking temporal desensitization. Although there might be an ultimate limit of how much vision could be restored due to the retinal and cortical remodeling of the degenerative diseases, further improvements building on the encouraging achievements of current generation prosthetic devices should allow future patients to read readily and recognize faces and details of objects. With such optimistic and realistic projections, retinal prostheses have the potential to restore sufficient vision to blind patients 60 allowing them to regain independence in their everyday lives. 1.5 Thesis Overview Given the challenges that current prosthetic devices still face and the significant room for improvement (see 1.3.5), this thesis aims to attain a better understanding of the electrode-tissue interface of the epiretinal prosthesis and neural prosthetic devices in general. The goal is to improve this interface to produce safe and efficient electrical stimulation and also provide the basis for computational models to improve electrical stimulation of the retina. Three topics are addressed: I. Reducing edge effect on electrode surface (Chapter 2). This project describes and analyzes a novel method to design current waveform input to reduce the edge effect-the primary reason for non-uniform current density distribution on electrodes. 2. Analysis on the Peak Resistance Frequency method (Chapter 3). This study explores the mathematical basis for a simple method to extract tissue resistance from impedance spectroscopy measurement of the electrode-tissue interface. 3. Electrical mapping of the retina tissue (Chapter 4). This experimental work measures the resistivity profiles of healthy and RD retina, providing input for computational modeling of retinal stimulation. 1.5.1 Basic Approach and Summary of Results My research has explored and combined a significantly wide range of methods to cover the different theoretical and experimental work. For the study on disk electrodes, the objectives were to investigate the method of shaping 61 current waveform to reduce periphery current density on disk electrodes. Three approaches were used: finite element modeling, mathematical analysis and numeric optimization, and pulsing experiment. Our lab has developed a finite element model for studying voltage response of disk electrodes (Behrend et al., 2008). I have improved this model for my study to explore the dynamic response of different current inputs (Wang et al., 2014; Wang and Weiland, 2012). The results demonstrated the changes in current density and charge distribution as the waveform was modified from the ideal step, and shows that the ExpCDF waveform has the best performance among the several candidate waveforms. It was also revealed that longer transition time between the pulse start to reaching the steady-state current level reduces the edge effect more. For the mathematical analysis, I have taken advantage of complex analytical solutions developed by the electrochemical society for the disk electrodes (Newman, 1966b; Nisancioglu and Newman, 1973a), which haven't been much taken into the context of biomedical electrodes. Building on those solutions, the response to any arbitrary current input waveform could be given. By analyzing the contribution of the different components of the response, I have laid the theoretical foundation for understanding the principle behind the current waveform shaping method and provided insights for the optimization. Converting the analytical solution into a numeric method, I obtained an optimized current waveform that best reduces the edge effect. Implementing an approximation of such waveforms via a first-order circuit, I pulsed disk electrodes with both the control rectangular waveform and the optimized waveform, and showed reduced corrosion on the edge of the electrodes, therefore demonstrating the effectiveness of the waveform shaping method. In my second study (Wang and Weiland, in press), I explored the mechanisms and inherent 62 properties of the peak resistance frequency (PRF) method (Mercanzini et al., 2009), which is a simple way to extract tissue resistance from impedance spectroscopy. My analysis involved theoretical analysis and computational simulation. Differential equations were solved to explain the behavior of the equivalent circuit model for electrode-tissue interface, and my results demonstrated for the first time that the PRF method is only a good approximation for the tissue resistance, and has inherent deviation that vary depending on the idealness of the electrode-tissue interface. The results also showed that the deviation is correctable and doesn't render the PRF method useless. A simulation study to investigate impedance spectra using parameters from a bipolar electrode reveals how accurate the analysis was and confirmed the condition for how accurate the PRF method works. Further Monte Carlo simulations were used to test the realistic limitation of noise and frequency sampling, and also to compare the PRF method with least squares methods. The results show that the PRF method works reasonably well and reliable under realistic conditions. I conducted a third study (Wang and Weiland, 2015) to measure the resistive profiles of retina, both healthy and degenerate. Our lab has pioneered in correlating electrode-retina distance with electrode impedance at single frequency (Ray et al., 2011). By expanding the frequency to a wider spectrum, I used the PRF method in my study to extract the apparent resistance of the tissue, and recorded retinal resistance profiles in mice retina. Previously this type of profile has only been obtained in healthy rats, chick embryos and macaque monkeys. My study provided important data in both healthy and degenerate mice retinas, which are the relevant models for our neurophysiology study of retinal prostheses. The experimental results show that in degenerate retina, the profiles 63 were thinner, and the thinning agreed with histology data. The peak resistivity also decreased with degeneration, however, was mostly affected by the changes in dimension. The two RD models have similar results, confirming that the onset and time course of the retinal degeneration results in little structural differentiation once the degeneration is "complete" in its end phase (Marc, 2010; Pennesi et al., 2012). For prosthesis research, this indicates that the end-stage of degeneration should be focus of study when using RD animal models. Also, the experiment set-up serves as a platform for further studies of intraretinal mapping of electrical stimulation. 1. 5. 2 Structure of the Thesis Chapter 2 presents the method of reducing the edge effect on disk electrodes by shaping the waveform of the current input to the electrodes. It first introduces previous literature on efforts of reducing the edge effect, and then gives details on our approach, which includes: a finite element model and mathematical analysis on the electrode-electrolyte system, a numeric optimization of the waveforms, and pulsing experiments showing results that validate method. Chapter 3 shows the analysis of the peak resistance frequency method, which is a simple and effective way to extract the resistance of tissue from the tissue electrode interface via impedance spectroscopy. This analysis lays the theoretical foundation for utilizing the PRF method to measure retinal resistivity profiles. Chapter 4 describes the study to map electrical properties of the retina, particularly the resistivity profiles as a function of depth in the retina. The resistivity profiles were compared for healthy and RD retina, and correlated with thickness measured from histology. 64 Chapter 5 summanzes the key findings of my studies, discusses the significance of my findings, and provides suggestions for further experimentation in the future. Chapter 6 provides supplementary materials for the disk electrode's analysis in Chapter 2. It is a comprehensive technical review on the paper series on the disk electrode written by Newman and his colleagues (Marathe and Newman, 1969; Newman, 1966a, 1966b, 1970; Nisancioglu and Newman, 1973a, 1973b ). 65 Chapter 2 Reduction of Edge Effect on Disk Electrodes 2.1 Edge Effect on Disk Electrodes Disk electrodes are commonly used in biomedical and electrochemical applications. Current density and potential distributions on the electrode-electrolyte interface have been carefully examined (Newman, 1966b, 1970; Nisancioglu and Newman, 1973a, 1973b; Wiley and Webster, 1982). In an ideal infinite space of electrolyte with homogenous resistivity, Newman's analytical solution concludes that, at the very beginning of a current or voltage step input (at t = o+), the electric field and thus current density are both sharply enhanced at the periphery of the electrode (Newman, 1966b). This distribution is termed the primary distribution. Newman showed that the primary distribution of current density exhibits a singularity at the very edge, whereas being only half the average value at the center of the disk ((2.1), and see Figure 2.1). Given in cylindrical coordinates (r,rp,z) with the origin at the center of the disk and z axis pointing perpendicular from the electrode surface into the electrolyte, the primary current density distribution along the radial axis is lo fz(r,rp,z)lz=O =Jo(r,rp) = V 2 2 2 a - r (2.1) where ](r, rp,z) = lr(r, rp,z) · f + ] 2 (r, rp,z) · z is the current density in the electrolyte, ] 0 (r, cf>) is the current density on electrode surface and lo is the average value, and a is electrode radius. Due to the rotational symmetry, all physical quantities are independent of the azimuth rp, thus this coordinate is omitted in the rest of the chapter for simplicity. 66 1.6 1.4 1.2 ....... 1.0 0.8 0.6 0.4 0.1? i/iavg for uniform polonfial 0 0 02 0.4 0.6 0.8 1.0 r / r'? - >. Position c .~ e U) ..... c :::J Q) UCI Position Figure 2.1: Left: primary current density distribution on disk electrodes as a function of radial location (Newman, 1966b). The current density at the center is only half the geometric average while having a singularity at the edge due to the abrupt change in border condition. Right: simulation of current density in a 3-D model showing the entire spatial profile over the disk (Ahuja, 2007). The current density doesn't reach infinity on the edge due to the limitation of the computational solution. While methods like charge balanced biphasic stimulation etc. have become the norm to reduce charge accumulation, it doesn't prevent the locally enhanced current density to drive electrochemical reactions at the disk's edge and thus corrode the periphery of the electrode (Figure 2.2, left). Tissue damage due to localized stimulation is of concern in biomedical applications (Figure 2.2, right) (Kim et al., 1986; Overmyer et al., 1979; Pagan-Carlo et al., 1997) 2.1.1 Efforts to Reduce the Edge Effect Several different approaches to reduce this undesired behavior have been proposed, mainly by altering the geometric design of the electrode (Figure 2.3) (Kim et al., 1986; Ksienski, 1992; Papazov et al., 2002; Rubinstein et al., 1987; Suesserman et al., 1991; Wiley and Webster, 1982). The segmented circular electrode consists of a set of annular rings around a central circular 67 electrode. By placing resistors in series with the conducting line to each segment, current can be distributed more evenly (Newman, 1966b; Wiley and Webster, 1982). Gradient of resistivity on surface achieves the same goal (Kim et al., 1986; Meyer et al., 2005). Designs of electrode recessed into the substrate have been discussed by several groups, with the focus on reducing the non- uniformity of current density on the surface of the electrode but not necessarily at the aperture of the recess (Ksienski, 1992; Papazov et al., 2002; Rubinstein et al., 1987; Suesserman et al., 1991; Tungjitkusolmun et al, 2000). Application of a resistive layer over the electrode surface has shown a more uniform current density distribution in certain applications (Mcintyre and Grill, 2001; Papazov et al., 2002; Sui et al., 2012); however this might increase the overall impedance to an unacceptable level. Figure 2.2: Left: the edge effect leads to corrosion on the edge of pulsed gold electrode, showing as a ring of dark coloration due to oxidation (Ahuja, 2007). Right: Skin burn in the shape of a rectangular form, following the shape of the electrode used for trans thoracic cardioversion (Pagan Carlo et al., 1997). The periphery has a deeper coloration and more severe lesion due to the edge effect. 68 h Ip "' saJine .... , Figure: 2.3! Various methods to. re.duce the 'edge effect. Top l'Cft: segmente.d. ·circular · electrode. achie,:ves more uniform current distributi<in over the elements by le:veraging the voltage on each segment (Wilc:y an.d W~bster, 1982). Bott9m left: a hemispherical elecJr9de tlrat a\lhiev~s unif orm current .density by utilizing. the symmetry of the geometry (K.sienski;. 1992). Right: recessed el~cl:r<Yd!ls tlrat r~duce the cuttept density n9n-uniformity on the surface of the ele'ctr9d~, but not ne. c essary at the aperture (Suess erman 'Ct al.,. 1991 ). 2,1. Z Primary and Sec1mclary Distribution In these m\ltho~s, th~ main fo~'US- is 'alt~ th~ primary \;uttent djsttil;>.ution, whik tl1e sec~mdary dis . tributi<in due'to"the· double layer c~pacitanc .e and. Faratlaic· rea, <;tions ar. e m~'t takenint.o considerations. 'I'he.se· two components of the electro· de-'Clectrolyte interface· cause· the. primaiy 69 current distribution to redistribute to a more uniform secondary profile (My land and Oldham, 2005; Nisancioglu and Newman, 1973a; Oldham, 2004), but these components have only recently been considered in modeling studies (Behrend et al., 2008; Cantrell et al., 2008; Cantrell and Troy, 2009). 2.1.3 Utilizing the Secondary Distribution Behrend et al. have proposed an approach to achieve more uniform current density profiles by re-shaping, i.e. slowing, the rising edge of the input current pulse (Behrend et al., 2008), while a similar idea was explored for voltage pulses (Cantrell and Troy, 2009). In this study, I have quantitatively explored this approach utilizing fmite element modeling (FEM) technique and shown that a simple ramp (in place of a step) can reduce the edge effect. Given positive results, an optimized current distribution method was then achieved: First: the FEM study compared the effect of different waveforms and transition lengths. Mathematical analysis further demonstrated the spatial and temporal factors contributing to the non-uniformity of current distribution, leading to an optimal current waveform via numerical solution. Finally, an approximation of the optimized waveform was implemented and applied to platinum disk electrodes, which showed a significant reduction of corrosion on the periphery (versus electrodes pulsed with rectangular pulses). 2.2 Finite Element Model 2.2.1 Model Geometry and Basic Parameters A finite element model was built (Wang et al., 2014; Wang and Weiland, 2012) with improvements compared to the one used by Behrend et al. (2008). Utilizing the rotational symmetry, the disk electrode model is built in cylindrical coordinates in COMSOL Multiphysics v4.2a 70 (CO,MSDL, Burlil'lgJ\)n, MA), with a disk electrode of radius a qnJ,, e.dtled in 11nd e.oplan;ir with a tlat insulating su}lstrate (Figure 2.4). The semi-infinite space· is limikd to 'a hemisphere with a tadius equal to ZOa for simulation The me,Sh · element size. · of the model is ' constrnined' to of solution space (Figure 2.5}. This configµratiou achieves spatial tes. olution as well as minimum degree1 uf freedom tn the.system (Behrend et al., 200_8). I (t·) - Rren1 Electrode Interface 1z <..J._;> ell< IS o r , symmetry r -----.. 2oa • Figiuie 2.4: Illus. tration of the· geoinetty for the, model. The, simulation volume was liinit~d :to ;i.O tim~s tlw ~~ctrode radius, with the rest 9( the el\;\;tro)y\e rep.laced by a series reslstamre cwm~pte-d to the equipotential. boundary. The ground was set at die electrode for convenience of modeling, and was moved to the e. quivalent infinity foi' data anal~is. Ideally~ the <Series resistance> R., repre'5enting the electrolyte/tissue abt>ve the electrode· is fheoreti-cal(y given 'b¥' (Newman,. 1966b) 71 1 Rs=-- , 41w where K is the conductivity of the semi-infinite medium assuming homogeneity. a (2.2) 20a Figure2.5: Meshing of the electrolyte space, showing the densest and smallest elements near the edge of the electrode where the highest resolution is needed. for an accurate solution for the boundary. However, the semi-infinite space is limited to a hemisphere with a radius eqqal to 20a for simulation, and the resistance of the electrolyte is therefore s. eparated into two portions: distributed 72 and lumped. The distributed region covers the simulation space from the electrode surface up to an approximately equipotential hemispherical boundary. The approximation introduces an error of ±0.002% in electric field magnitude on this surface (Myland and Oldham, 2005). The medium from here up to infinity is assumed to be equipotential at any radius, and the resistance is integrated into one lumped series resistance (rem for "remaining") Rrem = 2~K (r:in - r:aJ = 40~Ka (2.3) with Tmin = 20a and Tmax = +oo. The electrode-electrolyte interface consisted of a double layer (DL) capacitor C 0 L, modeled as distributed impedance with a capacitive component y as the double layer capacitance per unit area. Therefore CoL = yna2 (2.4) This gives the electrode's characteristic time constant T defined by Rs and CoL (2.5) 2.2.2 Input Current Waveforms The input current waveforms applied to the disk electrode were given by functions specified within the model. Input waveforms include the standard step input as control and waveforms with different rising edges, respectively. Three different waveforms were tested in combination with three different transition times T, which were set tor /2, T or 2r. The test waveforms include linear ramp (Ramp), exponential increase (Expinc), and the cumulative distribution function of exponential distribution (ExpCDF), which are described by (2.6). All current pulses reached the 73 same steady state amplitude I 0 . Figure 2.6 shows the three different waveforms with transition time equal to T for instance. The Ramp has abrupt transitions at both beginning and end of the ramp. In contrast, the Expinc and ExpCDF waveforms were chosen to test the effect of having smooth versus abrupt transitions at the beginning or end of ramping phase. Their time constants Tx were both set to 1/6 of the corresponding transition length. This choice of the time constant makes the Expinc waveform start at less than 0.25% of I 0 , and the ExpCDF waveform reach 99.75% of I 0 , to smooth the beginning and end of transition, respectively. The waveforms themselves would assure abrupt transition on the other side. In the modeling and mathematical analysis, waveform shaping was studied on only the leading edge of the step input, and results for both monophasic and biphasic rectangular waveforms used in neural stimulation can be easily obtained utilizing the linearity of the system. t - Ramp T I(t) Io c-T) exp -- Tx Exp Inc tE[O,T] (2.6) 1- exp(~:) ExpCDF 2.2.3 Symbols and Conventions Voltage on the electrode and infinity are given the symbol V, while potentials in the electrolyte space are <p. Variables on the electrode surface (r :S a and z = 0) are denoted with a subscript of 0, e.g. <p 0 (r, t) = <p (r, z, t )lz=o.rsa and V 0 ; whereas specifically for current density, the subscript also indicates the component perpendicular to the surface, i.e. in the z direction, e.g. fo(r, t) = ] 2 (r,z, t)lz=O.rsa· 74 '-' --, -;:;- "" " 0 ~ " c "' 0 N ~ a ~ 0 ~ 0,8 0,6 0-4 0,2 Candidate ci:rrcnt V.'avcforms .. - - - - - - -.. : .. :=-----,-:.--------------! ~-~ ~· / / / .. ·' .. .. .. .. .. Control ---Hamp •••••••••••• Explnc ---·-· ExpCDF ····•··•··•• o'-"'--~~···~··""-~~~~~~~~~~~~_J 0 0.2 0.4 0.6 0.8 1.2 1.4 1.6 1.8 ::\orn1alizcd tin1c t/r Figure 2.6: Three different current waveforms are shown along with the control. Test waveforms are given for one transition length as an example. Time constants of the Ex pine and ExpCDF waveforms are one sixth ofthe corresponding transition length. For the convenience of setting up the model in software, the voltage at the electrode is held at ground, while the hemisphere surface and hence the return at infinity are floating potentials to be solved. However, for comparison with the theoretical calculation and other results, the reference point of the potentials for further data analysis and presentation is shifted to the infinity point. Therefore, the symbols used for the potentials are given a tilde for those used in the COMSOL model and solved by the simulation, to distinguish them from the potentials further on without the tilde. The current density values remain unchanged throughout this process, and simple symbols without tildes are used throughout the chapter. 2.2.4 System of Equations and Boundary Conditions Laplace equation holds for the hemispheric simulation space and zero current density is 75 enforced at the insulator base. As mentioned above, the voltage at the electrode is held at ground for the simulation v 0 (t) = o , (2.7) while the hemisphere surface is a floating potential to be solved. The voltage at infinity is then Vw (t) = qi(r, Z, t) l,ir'+z'=ZOa - I (t) · Rrem (2.8) This allows the charging of the double layer at the electrode surface to be easily implemented with distributed impedance and could be described with ] 0 (r,t)=y:t(q;(r,t)-V 0 (t)) 2.2. 5 System Solving and Data Analysis (2.9) To accurately solve the equations at the disk edge, the time steps are chosen based on the spatial resolution. Increments in time are 1/500 of the time constant T from pulse start to T /2 after. This is about one eighth of the initial local time constant at r = a - a/30000 (Oldham, 2004). For the rest of the simulation the time steps are increased to T /100 to reduce computation. Data is output at time steps of T /100, processed with MATLAB (The Math Works, Natick, MA). As mentioned above, the reference point of the potentials in the presented data is shifted to the infinity point: V 00 (t) = 0. Thus in all following data <p(r,z, t) = qi(r, z, t) - V 00 (t) Vo(t) =Vo - Vw(t) (2.10) (2.11) For generality, the data are presented in dimensionless parameters: Space and time variables were normalized to the disk's radius a, and the electrode's characteristic time constant r; Current 76 density and potentials are normalized to corresponding parameters of the average geometric current density and the theoretical initial voltage - Io fo= na2 Io v: -- 0 - 4Ka (2.12) (2.13) respectively. The theoretical initial voltage utilizes the same relationship from a voltage step problem, as initial conditions at the input onset are identical. 2.2.6 Performance Quantification of Different Waveforms To show the overall time-dependent non-uniformity of the surface current density, the nonuniformity is evaluated as an area-weighted standard deviation from the geometric average value J~(t): 1 fa( - )2 - 2 fo(r, t) - ] 0 (t) 2nrdr na 0 (2.14) To avoid the singularity at the edge of the disk electrode, the integral was performed up to r = 0.9995a for numeric calculations. Two more quantities were calculated to compare the current density overshoot above the average current density at steady state (Wang et al., 2014). Again, a small annular region on the edge was excluded due to the singularity. The maximal current density at any given time was analyzed using jlf1 3 x(t) = max(J 0 (r,t)) r (2.15) Besides the maxima, the area-weighted average of surplus current density compared to steady 77 state average value was calculated and normalized to present the current amplitude that exceeded the desired uniform steady state distribution. This dimensionless value is defined as current overload (CO) 1 L - CO(t) = - (J 0 (r, t) - ] 0 )2nrdr) lo (rlfo(r.t)>Jol (2.16) Current overload index (COi) is the accumulated CO tw COi =Jo CO(t)dt (2.17) which gives an indication of how much damage would be caused by the edge enhancement if average current density was to be set to the safety limits of the electrode. A more comprehensive analysis could also include corrosion or other damage as a function of the current overload into the weight for the calculation of CO, which is not discussed here. 2.2.7 Results To illustrate the principle of the waveform modification, the linear ramp with transition time of T = T is used as an example and preliminary exploration. The profiles of current density distribution, voltage distribution, and charge density distribution, and current nonuniformity are shown in comparison to the ideal step input for a preliminary discussion. Then, the results for the different combination of waveforms and transition times are shown to further explore which factors of the waveform design have better performance in terms of current density reduction on the edge. 2.2.7.1 Current Density and Nonuniformity The normal current density at the disk surface ] 0 (r, t) shows how the test input reduces the current edge enhancement of the electrode (Figure 2. 7, right). The overshoot above the average 78 current density close to the edge is significantly smaller for the modified input, although at the pulse onset the current density would still theoretically reach infinity due to the singularity at the boundary of the disk. I~ Control step 2.5 ~-----~-~~ 2 t/r (J.00 O.OfJ O.lfJ o~---~--~-~ 0 0.5 I~adial position r/a 1.2 0.6 0.4 Ramp T T t/r 1.00 O.!JO I -------:,,.-~92:::-1.20 t- :1.00 O.fiO 0.:10 O.lfJ 0.2~--~ (J.01 o'--~~~~~=<-__J 0 0.5 I~adial position r/a Figure 2. 7: Current density profiles along radial axis at different time points. Edge enhancement beyond steady average is significantly reduced. The nonuniformity diminishes as the current input pulse stays on (Figure 2.8). The nonuniformity for the Ramp is not always smaller compared the step input, but this is mostly due to the fact that the modified input is still in transition while the step input has been redistribution charge along the radial direction since the pulse start. When the Ramp input current reaches / 0 , it has a smaller nonuniformity compared to the step input at the pulse beginning, therefore proving that its current enhancement above the steady state average is smaller. 79 0.8 i 1~ 0.7 < ' ~ ~ ~0.61 t. I > .. ·~· 0.5 I d ~ ·2 0.4 ~ c ~ 0.3 .... [:; 0.2 c,; 0.1 ::\ormalized current nonuniformity versus time .... 0.5 0.8 0.6 0 1 2 1.5 ::\"ormalizcd time t/T 2 - - - Control step --Ramp T= T 2.5 3 Figure 2.8: Current nonuniformity versus time. The decay of nonuniformity of the edge retarded input starts at smaller value when the input reaches / 0 , and decays faster if properly aligned as seen in the ins et. Meanwhile the results of the step input provide verification with the previous studies for primary current profile, primary potential distribution, and Rs (Newman, 1966b, 1970), as well as initial local time constant value (Oldham, 2004) r(r, t = 0) = 2rJ1 - r 2 /a 2 (2.18) 2.2.7.2 Charge and Potential Profiles Charge density profiles could be calculated by the local overpotential on the double layer a(r, t) = rVnL(r, t) = y(V 0 (t) - q:i 0 (r, t)) . (2.19) Figure 2.9 reveals that though the current periphery enhancement is reduced by waveform 80 modification, the charge accumulation still exhibits the same enhancement due to the integration of nonuniformity over time as well as charge relaxation along the radial direction. Charge density accumulation is slower for the Ramp input as expected. Control step 1.3 ~-----~--~ <}. c c ~ 1.1F----- " c "' c ~ ~ c c " 0 0.5 I~adial position r/a t/r (J.00 (J.01 O.OfJ O.lfJ 0.:10 :1.00 Ramp T T 1.2,_ __ _ 0.8 0.6 0.4 0.2 r-------- 0.5 I~adial position r/a t/r :1.00 1.00 O.!JO O.fiO 0.:10 O.lfJ Figure 2.9: Double layer charge profile along radial axis at different time points. For side to side comparison, normalization on both sides utilize the average total charge density at 3T for the ideal step input. Figure 2.10 shows potential profiles in the two situations exhibit identical asymptotic behavior reaching a steady state, which is expected from the analytical studies of(Nisancioglu and Newman, 1973a), although they follow different time courses towards this asymptote starting from different initial conditions. 81 Control step Ramp T T 1.2 t/r 1.2 t/r 3.00 3.50 0 J --------- --------- ·-. "" 3.00 " 2.40 c --------- 0.8 --------- 0.8 ;..., .-<;::: 2.40 " 1.80 " c --------- "' c 0.6 --------- 0.6 oc 1.80 ~ " 1.20 -" c --------- ~ 0.4 c --------- 0.4 ~' 1.20 .s 0.60 -3 --------- c 0.2 --------- 0.30 0.2 " 0.60 --------- 0.05 --------- - : 0.30 0 0 0 0.5 0 0.5 I~adial position r/a I~adial position r/a Figure 2.10: Potential profiles along radial axis "' different time points, showing the same asymptotic distribution. 2.2.7.3 Quantification of Edge Effect and Comparison of Different Waveforms The normalized maximal current density (}giax(t)/kJ is plotted versus time in Figure 2.11. The peak maxima are compared to the average steady state current density k, and expressed as an excess percentage m<>X(/!,"""(t))- J; P max = t X 100o/o lo (2.20) The control input resulted in peak overshoot of almost 4 time; that of the steady state (364.9%) before decaying to the average value. All the different test current waveforms have ITilJch smaller peak maximal current densitiei, with detailed values of Pmax for each waveform and transition length combination given in Table 2.1. The Exp Inc has the least improvement of all three candidate waveforms at each transition lengths, with peak overshoot at least half the steady state; the Ramp 82 and ExpCDF are markedly better than theExpinc. The Ramp is best for short transition length and has slightly less improvement compared to the ExpCDF for intermediate and long transition lengths. TheExpCDF has an extremely low peak overshoot for 2-r transition length. 0 0.5 1 T T/'2 1.5 T T 2 2.5 3 ~ ~~.------~.~-----_--_--~.---------~,------~.~------~.--------~ .5 1 Control ------ Ramp ~ 4 .. , a " "C 1~ ,, c.; , • I\ .~ ~ 2 1-1 \ "Ce~ I' ~·3·t.... ........... . 8 ~ I ,,•""'"'"--~-:-:;-::-::::s:-/~==-... --------------------------i ::3 it= ,; - ,, .. , ..... ·· y; - 0 i~ ·····•'** l . ................ Explnc -·-·-·-· ExpCDF . I • • 0 0.5 1 1.5 2 2.5 3 T '2 x T • • . ' • • 0 0.5 1 1.5 2 2.5 3 :\ormalizcd time t/T Figure 2.11: Normalized current density maxima within r $ 0.9995a versus time for three different transition I engths and different waveforms. All converge to unity as secondary distribution prevails through different time course. 83 Table 2.1: Maximum overshoot percentage of current density maxima Pmax Ramp Exp Inc ExpCDF T = r/2 48.39% 143.95% 62.18% T=T 25.23% 102.18% 23.97% T = 2r 9.50% 50.05% 2.49% The CO as function of time is given in Figure 2.12. The COi for the current step is equal to 0.0585, and the reduction for each combination of waveform and transition length is given in Table 2.2. A similar trend is seen for COi compared to the peak overshoot. However, the difference between the Explnc and the other two waveforms are not as large, and the Ramp has better performance compared to the ExpCDF for intermediate transition length. Again, the ExpCDF performed extremely well for 2r transition length. 2.3 Mathematical Analysis 2.3.1 Current Step Response of Disk Electrodes Nisancioglu and Newman analyzed the step response of a disk electrode subject to current input utilizing the transition from cylindrical coordinates to rotational elliptic coordinates (Figure 2.13) and gave analytical solutions. Materials relevant to the waveform modification method are given here. For details of the process of solving the system, please refer to the supplementary materials in section 6.6 of Chapter 6. Both double layer capacitance and Faradaic reactions were included. The Faradaic reaction in the analysis was represented as a conductance in parallel which we label BF. It relates to the kinetics of the Faradaic reaction with linear approximation to the Butler-Volmer equation for small surface overpotential (Newman, 1970; Nisancioglu and Newman, 1973a) 84 (2.21) where aa and ac are the anodic and cathodic charge transfer coefficients, respectively, j 0 is exchange current density, n is the number of electrons involved in the reaction, F is the Faraday constant, R is the uni versa! gas constant, and T is the absolute temperature. c v 0.2 0.1 0 "C 0.2 ce ..£ t ;. c 0.1 8 '"' '"' := I \ I ' \ 0 0 • ' I\ 0.2 .. ,, 0.1 I \ I \ ... \ \ I ' I 0 • 0 .. 0.5 1 - - - - Control 0.5 1 • . .. .. .............. - - " 0.5 1 T r /'2 1.5 T 'T 1.5 T 2 x r . . 1.5 2 2.5 ---Ramp ················ Explnc -·-·-·-· ExpCDF 2 2.5 • • ~ .................. . 2 2.5 \"ornrnlizcd time t/r 3 3 . . 3 Figure 2.12: Cwrent Overload (CO) versus time for three different transition lengths and different waveforms. All decay to zero as secondary distribution prevails, however accumulation differs. The CO of ExpCDF waveform with 2-r transition length is very small and barely visible in the bottom graph. 85 Table2.2: Normalized CO! for different wavefonns and transition cm,.., x 100% Ramp Exp Inc Exp GDF CO!ctr1 T = r/2 67.43% 81.35% 75.72% T =r 39.49% 72.34% 45.27% T = 2r 21.92% 55.16% 5.50% .... .... 77~0.2 .......... ' ..... / ......... / \ ~~o ......... ,,. ' / F ' ...... / '- .., / / , \ ' ... / I J I ' ~.;--, / 1 / ' ' / I \ ' ·tt ' ,,. I I \ ' / I I ? -+ ',. / I \ ..::?> \ 0' / I . cS> '' I I ~ ' , I 0 \ , ·,p \ , Figure 2.13: Illustration of the rotational elliptic coordinates used by Newman (Newman, l 966b). Same coloration as Figure 2.4. Switching from cy!indri cal coordinates to this coordination system allows the decomposition of the Laplace equation into two ordinary second-order differential equations. Thus analytical solutions could be obtained for the system. 86 Given the average conductance of the electrolyte determined from the electrode surface Gs 1 4K na 2 na 2 Rs TI a the normalized charge transfer conductance is defined as G =BF Bs (2.22) (2.23) This conductance corresponds to the "dimensionless exchange current density" ] as defined by Newman and Nisancioglu (Newman, 1970; Nisancioglu and Newman, 1973a). However using the average electrolyte conductance for normalization, G is scaled by n/4 compared to"]". In the one-time-constant model (OTCM) for electrodes, a version of the Randles electrode interface model, the parallel conductance represents the Faradaic reactions. In platinum electrodes, the surface binding of hydrogen is Faradaic in nature; however, as platinum electrodes are considered pseudocapacitive, we include this charge injection mechanism into the capacitance instead of the conductance. To avoid confusion, we use "parallel conductance" and the symbol G to refer to charge transfer conductance, especially irreversible ones. And "double layer capacitance" includes pseudocapacitive charge storage if present, which could be determined experimentally from electrochemical impedance spectroscopy (EIS) data. Solving for the step response (SR) involves decomposing it into steady-state (SS) and transient (TZ) components (Nisancioglu and Newman, 1973a), and here with the current density given as example (All the results are shown in the cylindrical coordinates for ease of presentation and interpretation): (2.24) The first term represents the stable response to a constant current of amplitude 1 0 , 1.e. 87 secondary distribution. The transient response doesn't contribute to the overall current input to the electrode. However, it is responsible for the actual distribution of current density on the electrode surface and its shift from the initial condition immediately after the input onset (t = o+) to steady state distribution, i.e. primary to secondary. As the boundary condition of the electrode-electrolyte interface is described with a first order differential equation with respect to time, the transient component is further decomposed into eigensolutions. This will be further discussed in the Results section (2.3.3). 2.3.2 Constructing Arbitrary Response from the Step Response For any input x(t), the output y(t) could be given by the convolution of the input and the impulse response h(t) as y(t) = x(t) * h(t) , (2.25) or alternatively by the step response g(t) = J h(t)dt as y(t) = x(O) · g(t) + x(t) * g(t) (2.26) The desired current waveform will reduce the current density non-uniformity; therefore the input l(t) = 1 0 ·x(t) shouldsatisfythefollowingconditions: { x(t) = 0 , t :S 0 x(t) iE C(!Rl.) , t 2 0 x(t)-->l , withinshortt (2.27) The second criterion states that x(t) is continuous and increases monotonically. The last criterion indicates a reasonably short transition time period for the input current to converge to its final amplitude, so that the modification to the rectangular pulses commonly used in neural stimulation would not distort the overall pulse shape too much. Otherwise it may alter the neural 88 response to stimulation. In these applications the initial value of the input is conveniently set to zero, further simplifying (2.26) to y(t) = x(t) * g(t) (2.28) Applying (2.28) to the results of (2.24) above, the current density response to the arbitrary input I (t) could be given as J8"'(r, t) = 1gs. x(t) - f{,Z(r, t) * x(t) (2.29) which provides the mathematical foundation of the waveform shaping method. The first term shows that the time domain contribution of the steady-state component in the output is just scaled by the input waveform, and quickly converges to the same secondary distribution as the step response. The contribution of the transient component, which the non-uniformity of the overall current density profile is due to, is in the second term. Hence the high peripheral current density could be reduced if the input's time derivative x(t) is small in amplitude, i.e. smooth transition. The transient component diminishes as condition (2.27) indicates that x(t) --. o (2.30) 2.3.3 Computational Optimization of the Waveform Due to the complexity of the analytical solution, a numeric solution has been developed to estimate the optimum waveform to reduce the local current density overshoot above the steady state situation. The numeric method is based on a discrete-time version of (2.26). The output is built by stacking scaled unit step responses carefully so that the total response does not exceed a certain 89 level (e.g. k). The input is therefore determined by coefficients used to the scale the unit step response. To implement this, the normalized step response (simulated or analytical solution) is sampled with time intervals M = O.Olr over a long enough time (e.g. t 2 3r): (2.31) The calculation was again limited to r :S 0.9995a. An overshoot limit (OSL) is introduced to allow some tradeoff between overshoot above the steady state current density k during the transition and the length of the transition time. The overshoot limit is a number equal to or larger than 1, and restricts the current density within the analyzed region to J8"'(r, t) :S OSL · J-;; , Vr, t . (2.32) The flowchart for the algorithm is shown in Figure 2.14, and an example is given Figure 2.15 in to show the concept of composing the input and output. Optimal current waveforms, as well as the best-fit ExpCDF waveforms were calculated for several different OSLs. 2.3.4 Results 2.3.4.1 Steady State Response and Parallel Conductance Analysis of the steady state solution showed that the smaller the parallel conductance, the smaller the current density enhancement at the disk periphery, with ideally uniform steady state current density for G = 0 (Figure 2.16). To have a moderate acceptable current peripheral enhancement for an input current pulse that reaches a steady state (first term of (2.29)), the normalized conductance G should be less than unity G < 0(10°) = 1 (2.33) 90 I Stm I Iuitiaiize luptitx(lm) = 0, V'm > 0 lnitia.lizeoutput]Out(r,tm) = O,v1n £ 0 .. I i = 1 I " No x(t,) < 1? !*+ ' Yes .I. . . OS Xj[J~ (J r,ti ( ( L - ]'"'( )) ) ilx(t1) =mm m;n jgi'(r,to) ,l-x(t,_J J. x (t 1 ) = x(t,_,) + Llx(t 1 ) .. J8ut(r, t'i:+rn) = ;g11t(r, ti+m) + .6x(t1) 'J~R(1", tm), vm > 0 I x(tm) = 1, \Im > i "' End Figure 2.14: Flow chart for deriving the numerical optimal waveform. 91 '? 0:25 ~ - ... ~ C;, .9 '<:' b 0.2 0.15 ~ 0 . ;.:::i • I <:' a 5 0.05 ?: __ J _ _ _ - - -L · -Q- · - -- - . . · -- - I ~ 1·:J.·· u f 'ta) --~------ --------- • I A.c~ · 'l!(t f';!) ---L-------------------- :+~.Tl. u.(l - tl) O '--~-&-~--'~~_._~_.,~~""'-~--'-~~"'---' -· 1 0 1 2 3 5 6 liSL'-< SS 1\ I ' · l' .., I " ' -t ' \ ..... .. ~ _..........._.--.._..._..~ .... ~-·~ .... ,- ............ .._ ...... "" ........ _..._~ ... - -·~ ..... -...-.---t SS ' " ..... . . . ... '- ' I ..._ I + ~ J'.: -.I " .. - ' • !>~.!.'~ .J y(I - 1;J-... - - - - - ' 1 ~\r/(I - t':l) ..... "" ""-r _ ---. __ - _ - .... ... --- --- ----- .... -- . ... -- ll) - -- - -- ------- 1~.t1·g(( !J(I = o ~ ·~---.(lto-~_..~~_._~~...._~ ........ ~~_._~__,.__~ -1 0 2 3 4 5 6 Nnn1v1Jiz.;d t uJH' (.,,/ dt Figure2.15: 1-elimensional example plptexplcining,theaigoihhm in F1 g111e 2. 14 The bla ck.dashed lipe re.pterents the 0plimized input and 9utpt1t with the sampliri, g points (circles) The grey dotted lines are the· underlying step inputs and step re. sp:onses that the s· ardp led input.and 1111 1;).l.ut are b.uilt of At each time. step,. the. size of.the. additional step of input is first determined by the 6ut.piJt, and then a scaled step and step re>'P0nse ar·e added t& the total lnpu!and 01.!tput. The· lines SS and SS X QSL ~n the bottoin figureindi.cated tlte ni>rmalized• stea.cly staterespori~e attd th:e aU~wed .011ersh~Pt: which is . set. to 20% \1ere for exmiple; (tlSL = l:J) Typical mtcro-electrodes in neural engineering applicattoo Work with puls'es in the micro- .to mi!1isen 0 nd range; with Faradaic rea~ti0ns having much srp· a!ler conducb;m:c:e compared ta the e\ectr '.Qfy. te copductancec In the OTCM pf eiectrode, the p arallel r esistance is· in the rn-egaohm to 92 g'i'[!aobln range (i..e. a lll>v conouctance),,c.o~par:ed to kiloo'hi:n values· of the series resistance of the· eh:cb·o1yti: (~.e. higher conductance"). Also, most: eltcttodes use.tl in ntural i1np.la})tS: ari: highly capacitive (or pseudo'C;!paciti:ve) as Fai:;adai'c 1-ea, ctjoii$ ru.-c:· .n.ut <ksi.,red dui: to tl;ieir iµ-e,ver,sibili,ty nnd t>otential .harm to tissue. The- abavc crittri'lin i's therefore easily met,. and further analysis 9.llfl G < O( l0-i3) I~ 0 (Z.'34) whi1;l1 · gJves uniform sfeaJy state current depsitr <;listribution. 2.5 2 I~ --.... ~ -1 .5 '" ..., 0 -= l c: = 1(1 Q'--~~~----'~~~~-'-~~~~_._~~~~-'-~~~----' 0 0.2 0.4 0.6 0.8 1 n.11rl41J p ns1tf(ID · ('I (I Fig_lll'e· 1.16: Steady state. dis:tr· ibutidn to s b:p.rtsponsc for thtee no11naliz· e. d parallel · conductance, generated acc.011ding. l:e Nisancioglu arul Ni:wman; I97Ja. Smail flactuati'ons towards tm. · edg. e are. due 'to the• accura. <;y 1. imitation of'the· nurnerkmethod. '.f.;.msient Response 'Ih~ t:ransiqlt resp011se, qf is given by a ful'thcr de4. Vfllposlti'!>n info expor!4lntial de1;ayir1 _g. eigensolutions with spatial profiles ]~i) (r) and time constants rCO (Nisancioglu and Newman, 1973a): (2.35) Each eigensolution corresponds to an eigenvalue ll (i), which is related to its time constant by (2.36) where A (i) is the normalized eigenvalue (2.37) Due to the normalization with r, the normalized eigenvalues are scaled by n/4 compared to the original values given by Nisancioglu and Newman (Nisancioglu and Newman, 1973a). Normalized Eigenvalues are calculated in increasing order (assigned according to increasing value), and their corresponding eigensolutions are solved from the system of equations. The first ten normalized eigenvalues are given in Table 2.3, and the corresponding eigensolutions are given in Figure 2.17 (generated according to Nisancioglu and Newman, 1973a). The normalized eigenvalues and the profiles of these eigensolutions are independent of the F aradaic conductance, but as (2.36) shows the time constants of the decays are not. Under the condition G "' 0, the relationship between the time constants and the eigenvalues are simplified to T 1 t..CO (2.38) It can be observed that the current density at the edge is not the highest for each eigensolution. The current densities towards the center of the disk have larger amplitude but alter in sign and virtually cancel out each other when they are scaled by the coefficients cCO and added up to form 94 the transient solution, At the edge however, tfre current density monotonically increases in amplitude with tbe order afthe cig-ensolution, i.e. with increasing eigenvalues or corresponcfog decreasing time constants. This leads to the high peripheral current density of the primary dlstribution. Therefore the desired input current waveformshouldreduce the contribution ofhigher order eigensoluti ons to the transientresponse·'as much as possible. 400 300 ·~ 200 ,__ .. ~· 100 ~ o~~~A,~~~~~ <! Cl =-- -100 - :_: -200 -300 -400 0 02 0.4 0.6 08 R i\rLi\l fl";:.lt 1011 • 11 Fig(lre 2.17: Normalized current densityv~su,~"Tadius for tile fir. st tm normalized eigensotutions, gener.ated according to NisanciogJu andN ewman, 1973a. Table 2.3: First ten normalized eigenvalues ,. generated according fo Nisancioglu aod Newman. 19J3a. i 1 2 3 4 :. J \ iJ) 3.237 5.766 8.260 10.74 13.22 i 6 7 8 9 10 AltJ 13.69 18.17 20.64 23.11 25.58 Equatio n (2 .38) incicates that tlie time con11taht Qf eigenso1utions decreases with increasing 95 order. Therefore higher order eigensolutions have broader spectrum and contain more high frequency components. To suppress the higher order eigensolutions, the derivative of the input x(t) should be a low pass filter. One simple implementation is to use an ExpCDF as explored in the modeling section, whose derivative is x(t) = xe-x' · u(t) (2.39) with X = Tx 1 . Smaller X reduces the amplitude of x(t), and hence the overall transient component in the response as discussed in (2.29). Furthermore, it can be demonstrated that for any i, the smaller X is compared to ll CO, the less this eigensolution contributes to the overall response (derivation not shown). Therefore for effective reduction of higher order eigensolutions, the exponential decay of the input should have X<J!CO (2.40) with i 0 being a small integer, e.g. i 0 = 1 or 2. Therefore the corresponding time constant 1 Tx > J!(O (2.41) In the modeling section, ExpCDF waveforms have times constants of 1/6 of the transition time lengths. Hence for the three transition time lengths used, the i 0 's are 5, 3, and 1 respectively: 1 {r/12 w<Tx=T/6 ll r/3 i ?. 5 for T = T /2 i?. 3 for T = T i ?. 1 for T = 2r (2.42) The results showed that the ExpCDF waveforms with transition time length of 2r indeed reduced the current density maxima and current overload index to an extremely low level. This is further verified in the computational optimization section. 96 2.3.4.3 Optimal Waveform and Approximation Based on the algorithm, the optimal waveforms for a few OSL value; are given as solid lines in Figure 2.18, with the dashed lines being their beit-fitExpCDFwavefonns. The correspondingX's and the COi of optimal and beit-fit waveforms are shown in Table 2.4 2.0 1.7 1.4 0.8 ~ ~ ' 0.6 '-' ·-. "" ~ 0.4 0 0.5 1.2 1.1 ::\orn1alizrd tin1r t/r OSL 1.0 -Optimized •••••••Fitted ExpCDF 1.5 2 Figure 2.18: Optimized current waveforms obtained from mathematical analysis for reduction of current density on the periphery of disk electrodes, and their best-fit ExpCDF waveforms. Both axes are normalized. Table 2.4: Parameters and COi of optimized waveforms and their fitted ExpCDF waveforms for several OSL. OSL 1.0 1.1 1.2 1.4 1.7 2.0 COi opt /Co1ctrl O.Oo/o 17.8% 32.3% 52.0% 68.2% 77.1% X ·T 2.61 3.43 4.21 5.80 8.24 10.80 Tx/T 0.383 0.292 0.237 0.173 0.121 0.093 COIX /COlctrl 1.38% 10.10% 21.47% 40.51% 58.59% 69.28% 97 Comparing X with the eigenvalues, there will be almost no overshoot within r :S 0.9995a, for X < J!Cll. For X < J!C 2 l, the overshoot is less than 40% (OSL = 1.4) of the average steady state current density. The fact that the optimum waveforms obtained by the algorithm resemble ExpCDF waveforms allows a relative simple circuit to generate such stimulation waveforms (next section). 2.4 Pulsing Experiment 2.4.1 Experiment Setup 2.4.1.1 Circuit for Waveform Generation The best-fit ExpCDF of the optimum current waveform is implemented by converting a square voltage pulse from a Model 2100 Isolated Pulse Stimulator (A-M Systems, Sequim, WA) into the desired waveform with a RC circuit, which was then converted into a current pulse of the same shape by operational amplifier and delivered to the electrode. A simplified circuit diagram is shown in Figure 2.19, which was implemented on breadboard Figure 2.20. The RC circuit has a variable resistor adjusting its time constant, hence the transition time of the waveform. A buffer stage was added in between for stability and impedance matching. For the control group, the RC circuit was omitted, and the voltage pulses were directly applied to the buffer stage. 2.4.1.2 Electrodes An Epilepsy/Long-Term Monitoring Subdural Electrode Array (Ad-Tech Medical Instrument Corporation, Racine, WI) was used for its consistent fabrication quality and large electrode size (Figure 2.21). The 8x8 array contained platinum electrodes of 4 mm diameter, with the center of 98 2.3 mm diameter being exposed. Electrochemical Impedance Spectroscopy (EIS) (Figure 2.22) using Gamry Reference600™ (Gam:cy Instruments, Warminster, PA) showed that the average access resistance of the electrode is 2 20 fl in phosphate-buffered saline (PBS), and the average double layer capacitance is 3.4 µF. The characteristic time constant is therefore approximately 0.75 ms. The parallel resistance is on the order of 100 Mfl. Electrode Vaut Return I -/ Figure 2.19: A diagram of the circuit to convert square voltage pulses to the desired current waveform. • • e • a I • • • e • I I • Figure 2.20: A breadboard prototype implementation of the conversion circuit. 99 FG64D-SPIOX- OOO & F G6-tD-SSI O X-OOO 64-contact, 8-tails LTM grid Figure 2.21: The Adtech electrode array used for the pulsing experiment. Each electrode site has a diameter of 2. 3 mm. A<lted164 Electrode Array 102...._~~~..._~~~...._~~...-... ......... ~~~ .......... ~~~ ........... ~~~ .......... 10-I ,-.. -20 fN ·· .• J -40 C) rr. ce ..d -60 0... 10 1 10 2 10 3 Freq n(')lcy f (IIz) Figure 2.22: EIS of the Adtech 64-electrode array, mean and standard deviation of 16 electrodes. 2.4.1.3 Pulsing Protocol The electrodes were pulsed in PBS with a thick platinum wire as return electrode. The pulsing 100 protocol consisted of cathodic-first biphasic pulses at a frequency of 50 Hz for 20 minutes (Rose and Robblee, 1990). The pulse width of each phase was 2 ms in duration. The control group was pulsed with rectangular pulses. The test group was pulsed with ExpCDF current pulse with a time constant of 0.4 ms, about half of the electrode's time constant. In both groups, the current amplitude was set so that the potential excursion of the electrode-electrolyte interface reached to ±1.5 V at the end of each phase. As a result, potential on the electrodes' surface would exceed the water window ( -0.6 V to 0.9 V) to cause corrosion (Rose and Robblee, 1990). For the test waveform, the potential excursion across the electrode-electrolyte interface was verified by subtracting the potential caused by ohmic drop due to electrolyte resistance from the total measured potential. The corresponding current levels for both situations were on average 5 mA, equivalent to 240 µC/cm 2 per phase. The sizes of the test and control groups were 5 and 4 electrodes respectively, while 2 more unused electrodes were used for baseline in the surface analysis. 2.4.1.4 Surface Analysis After current pulsing, the electrodes were detached from the substrate and cleaned with deionized (DI) water. Scanning electron microscope (SEM) was used to investigate the surface morphologies of the electrodes (JEOL JSM-6610, JEOL USA, Inc., Peabody, MA). Chemical compositional analyses of the pulsed electrodes were performed using energy-dispersive X-ray spectroscopy (EDS). For each electrode, three evenly distributed diameters across the exposed surface were randomly chosen and scanned. For each diameter, EDS data were collected from 20 evenly spaced spots with 15 kV acceleration voltage and 25 seconds collection time at each spot. IOI TI1e data is equivalently from 6 radii, with 10 data points per radius. Atomic count percentage of platinum and oxygen on the electrode surface were analyzed by EDAX Genesis program (EDAX Inc., Mahwah, NJ), and transferred to MA.1LAB for further analysis. 2.4.2 Results Cwrent pulses were applied to platinum electrodes to validate the model, as described in methods. Rectangular pulses were applied to 4 electrodes and ExpCDF pulses applied to 5 electrodes. After pulsing, the surface of the disk electrodes showed yellowish coloration on the periphery in both groups, indicating corrosion due to oxidation of platinum (Figure 2.23, arrows). This coloration was evident in both rectangular and ExpCDF pulses. Figure 2.23: Typical corrosion profile after pulsing. 102 EDS was performed on the surface, giving the atomic count percentage of the two major identified elements, platinum and oxygen. Figure 2.24 shows the atomic count percentage values averaged across the radii within each group of electrodes versus the radial location with the error bars showing standard deviation. At each location, the mean oxidation levels for the test group (black solid line), the control group (black dashed line), and the baseline (gray solid line) were compared using one-way analysis of variance (ANO VA), with a= 0.05. The null-hypothesis that the three groups have equal mean oxidation levels was rejected at all locations (p '.'O: 0.002). 30 --0-1"est (ExpCD1'') - o - Control (l\ert1tngnl1tr) 25 B1tseline (Not l'nlsed) ' ' ' ' ' ' ' ' ' ' ' ' ' ' < ' _, ' ---~ - -- ~ -- ' ' ' ' ' ' ' ' ~ 5 ~ "- 0 0 0.2 0.4 ' ' ' ' ' , ' ' , ' ' ~ ' '/• - -, - - - ...-ti - ' ' ' ' ' ' ' ' 1 0.6 0.8 :-\or111alizrd radial positio11 r/a ' * * ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' .... ~ .., , ' ' , ' ' ' ' ' ' ' ' , ,. ' Figure 2.24: Atomic count percentage of oxygen along disk radius. The data points were slightly offset at each location for clarity. Error bars show standard deviation. Asterisks indicate significant difference between test and control groups. Filled and empty markers indicated center and periphery subgroups within each group. The grey marker of test group is transition between center and periphery subgroups. 103 Further multiple comparison tests were performed using the Tukey's honestly significant difference (HSD) criterion to compare group means and determine the source of the significant difference(s ). The oxidation levels of the test group were significantly lower on the periphery (asterisks) compared to the control group, suggesting that the periphery current density was indeed reduced with the test waveform. As expected, the oxidation level for the baseline group was significantly lower, compared to either test group or control group at any location. ANOVA within groups shows change of oxidation level across the radius for the test and control groups. Based on the significant difference from ANOVAanalysis, the radial locations were divided into two subgroups, the center and periphery. For the control group, the two outer points (asterisks) and the 8 center points showed significant difference in oxidation level, which almost doubled from about 9.89% on average at the center to 18.07% on average on the edge. For the electrodes tested with the modified pulse, the 4 outer points differ from the 5 center points significantly, with one transition point in between. The transition point was statistically different from only to 2 of the 4 peripheral points. The oxidation level stayed relatively uniform around 9.46% in the center, only slightly increasing by one third to 12.12% on the edge. Hence the increase of oxidation level for the control group is much stronger within a narrower region compared to the test group. The baseline has a relatively uniform oxidation level on average of 5.53% with no significant fluctuation. The above results indicate that the pulsing induced oxidation in both control and test pulse groups, however greater oxidation was noted at the edge of the control pulse electrode. 104 2.5 Discussion and Conclusion This work provides comprehensive examination of the current waveform shaping method to reduce periphery current density on disk electrodes. Analytical and numerical calculations yielded a prediction for an optimal waveform and transition length. Pulsing experiment verified the effectiveness of the method on platinum electrodes. 2.5.1 Comparison on Different Approaches Finite element modeling and analytical studies provided insight and quantitative description to explain the mechanisms behind the current waveform shaping method. The double layer capacitance is responsible for the current density redistribution from the highly non-uniform primary distribution to the more uniform secondary distribution. The waveform shaping exploits this and increases its effect, allowing the current density to redistribute over longer time. From the mathematical standpoint, current waveform shaping suppresses contribution of the higher order eigensolutions to the transient response, which the edge effect mostly relies on. The modeling and analysis assume ideal conditions, and therefore have certain limitations (Mcintyre and Grill, 2001 ). The electrodes were assumed to have perfectly smooth surfaces and an ideal seal at electrode-insulator interface. A rough surface would increase the current density at locations of small curvature, therefore increasing the overall current non-uniformity. This is however a small factor compared to the ideally infinite edge effect unless the insulation of the electrode was very defective due to manufacturing failure. A non-perfect electrode-insulator interface would cause leakage of current originating from the electrode-electrolyte interface under 105 the insulation. Non-ideal electrical properties at the edge decrease the edge effect. However, an edge effect would exist and the current shaping method would still help to reduce the non- uniformity. The electrode-electrolyte interface has been simplified as well. The double layer capacitance was assumed to be an ideal capacitor, although it is often better described as a constant phase element (see 3.2.1.3), whose parameter might vary depending on many factors like frequency and bias potential. The Faradaic components were excluded from the analysis and the FEM study. This leads to further deviation from reality, especially for large overpotentials, such as those used in the pulsing experiment. The electrode was symmetric in geometry and referenced to an infinite ground, and the conducting media was isotropic. In reality, the electrolyte space is finite and of arbitrary geometry. In neural stimulation applications, the complex environment of surrounding tissue may change the electrical field and current distribution. Nevertheless, the experimental results demonstrated the effectiveness of the current shaping method despite the above mentioned simplifications and other factors such as the slight recession of the electrodes used. 2.5.2 Trade-Off Considerations Utilization of the proposed method must consider several trade-offs. To achieve the reduction in current density at the periphery will require a rising leading edge, not a step. To minimize edge effects throughout the entire pulse, a transition zone should be added wherever a discontinuity exists in the waveform, which for a rectangular pulse is at the beginning and end of the pulse as 106 edge effect is noted at pulse offset as well (Figure 2.25). The total charge injection woold remain the same for the modified pulse, only increasing the overall pulse width. This has the effect of making the pulse longer. A longer pulse is generally less efficient at stiITilJlating nerve;, so more charge may be required fcr the modified pulse to have the same effect as the square pulse. While the peak current density may be minimized, the stimulation could be less efficient. '-' ·-. Cl? 0 8 "' . " c ~ " 0.6 c "' c N ~ 0.4 a ~ c ~ 0.2 0 0 0.5 1.5 2 2.5 3 ::\orn1alizrd tin1r t/r Figure 2.25: Illustration of current waveform modification for pulse onsd and pulse end. A secondary consideration when increasing pulse width is the limitation this may place on pulse rate (the number of pulse; per second). For example, a pulse width of 2 ms (1 ms/phase) can berepeated at up to 500 pps. Lengthening the pulse would reduce the rate at ....+tichpulses can be applied. However, rarely are pulse; applied so rapidly, thus this issue is unlikely to be a significant limitation. For example, a micro-rough platinum electrode array with 200 µm diameter has a time constant of 0.43 ms tested in vitreous cr 0.98 ms when placed on the rttina 107 in vitro (Table 2.5) (Shah et al., 2007). The same type of electrodes in the Argus II Retinal Prosthesis use pulse widths of 0.5 ms at rates of 20 pulses per second. The pulse intervals are therefore 50 ms on average, and there is enough room to accommodate a modification to the pulses. Table 2.5: Time constants of micro-rough platinum electrode with different electrode diameters when placed in vitreous or on retina. Calculated from impedance data from Shah et al. (2007). Electrode diameter 200 µm 100 µm 50 µm Time constant in vitreous 0.43 ms 0.13 ms 0.06ms 2.5.3 Implementation Considerations Time constant on retina 0.98ms 1.03 ms 0.58ms Reducing the edge effect may improve device safety since neural damage is related to current and charge density. The biological safety limit of neural stimulation is variable, and depends on both charge per phase and charge density (Shannon, 1992). An analysis of experimental data has shown that stimulation is not damaging to neurons with higher charge density when using smaller charge per phase. On the other hand the electrochemical safety limit is a constant charge density. Therefore given a small charge per phase, smaller electrode could reach electrochemical safety limit before biological safety limit, necessitating protection through methods like waveform shaping. Larger electrodes may be restricted by biological safety (versus electrochemical safety) limits, so waveform shaping may be less important in these application. The high current density is transient in time and limited to near the surface of the electrode. Current nonuniformity decays rapidly away from the electrode surface, even in the primary 108 distribution. Therefore it is reasonable to expect that waveform shaping will have little effect on the spatial extent of neural response, especially in the situations where the electrodes could not be placed close enough to the stimulation target. Retinal prostheses use circular electrodes with rectangular pulses, but the subjects do not describe annular perceptions (Nanduri et al., 2008). Calcium fluorescence imaging studies of in vitro retina do not show preferential neural activation at the edges of 0.2 mm diameter electrodes (Behrend et al., 2011; Weitz et al., 2013). In contrast, annular skin burns have been noted after defibrillation, but these electrodes are large (several centimeters) and the skin is adjacent to the surface of the electrode (Pagan-Carlo et al., 1997; Papazov et al., 2002). With microelectrodes (less than 1 mm in diameter), the edge effect is limited in time and space. 2.6 Summary The objectives of this study were to investigate the method of shaping current waveform to reduce periphery current density on disk electrodes. Three approaches were used: fmite element modeling, mathematical analysis and numeric optimization, and pulsing experiment. The first two gave consistent conclusion on the optimal waveform and transition length, and the experiment verified the effectiveness of the method on platinum electrodes. Further development of this method includes exploring its efficiency on electrodes of different shapes and sizes (for example, cylindrical electrodes as used in deep brain stimulation or cardiac pacing), engineering simple and feasible implementation of the current source, and studying the possible benefits in potential biomedical applications. 109 Chapter 3 Analysis of the Peak Resistance Frequency Method 3.1 Introduction Neuroscience and neural engineering applications rely heavily on electrodes and electrode arrays. Cuff-electrodes, microelectrodes, and multi-electrode arrays have been utilized to study neural pathways, neural activity patterns, and neural response to outside stimuli (Buzsaki et al., 2012). Several types of medical devices, such as deep brain stimulators, cochlear implants, and retinal prostheses etc., have been developed to treat neurological and neural degenerative diseases using electrodes to stimulate specific targets within the nervous system (Panescu, 2008; Weiland et al., 2011; Wilson and Dorman, 2008). The tissue-electrode interface is critical to the proper functioning of electrodes in many of these aforementioned situations, and the tissue resistance plays an important role in this interface. As this resistance could change over time due to tissue encapsulation and inflammation (Grill and Mortimer, 1994; Mercanzini et al., 2009; Polikov et al., 2005) or repositioning of the electrode (Mahadevappa et al., 2005) etc., monitoring the tissue impedance is prudent in many applications. An increase in tissue resistance due to encapsulation could attenuate the recorded signal amplitude, and/or increase noise levels (Vetter et al., 2004); in stimulation, it affects the shape and magnitude of the electric field, and therefore neuronal activation thresholds (Grill and Mortimer, 1994). Decreases in tissue impedance could indicate electrode movement away from the target, which increases stimulation thresholds (Mahadevappa et al., 2005; Ray et al., 2011), and could reduce 110 recording sensitivity (Henze et al., 2000). Several methods have been used to determine the tissue resistance through impedance spectroscopy. Many impedance measuring devices are only set at 1 kHz (Mahadevappa et al., 2005; Williams et al., 2007), and electrode impedances are consequently often reported at this frequency (Blum et al., 1991; Mahadevappa et al., 2005). This is because the spectrum of action potentials and many stimulation pulses are centered roughly at 1 kHz. For most electrodes, however, the impedance at this frequency reflects the capacitive component of the electrode, whereas the tissue/electrolyte resistance dominates the impedance at higher frequency (Geddes, 1997; Ray et al., 2011). For example, impedance at 100 kHz was correlated with proximity to the retina. (Ray et al., 2011) and showed better sensitivity compared to the measurement at lower frequency (1 kHz) (Mahadevappa et al., 2005). Ideally, impedance will represent tissue resistance when measured at high frequency, where the double layer capacitor acts like a short circuit. However, parasitic capacitance in the system introduces a non-negligible factor that could change the ideal behavior (Mercanzini et al., 2009; Shah et al., 2007), therefore rendering the fixed frequency approach unreliable. Nonlinear least squares fitting is a viable approach to extract parameters from impedance spectrum that exhibits arbitrarily complex behavior by adding more model elements. However, it relies on post-processing of complete measurements covering a sufficient frequency range to be accurate. Mercanzini et al. described a method that utilizes a variable frequency point to estimate tissue resistance (Mercanzini et al., 2009). This method has been named the peak resistance frequency (PRF) method, as it chooses the frequency at which the complex impedance is the most resistive, 111 i.e. where the phase angle is closest to zero, to evaluate the impedance magnitude (Figure 3.1). They used electrode parameters from lhcit microelectrode array to simulate the jrnpedance spectrum when interfacing with tissue of v~g resistance. The results demonstrated high accuracy of the PRF methocl whereas fixed frequencies resulted in large deviation ' from tbe "true " value. l / (Jwr 1 . ) ~ IZPHrl - ';}' it . .... ~ 10~ - - - - - - - - - - - - - - - - - - - - - _ -_::::"'::::=-""" _- ...... __ ...._ __ _:.:::.__ 10~ lit 10" 20 · BO - - -~--=-,,.,...------ 1~ l~ 1~ FtiC'< tllt<U " v /' ( 1U I Flgnre 3.1: An example impedance spectrum of an electrod,e-tissue interface demonstrating the PRF mdhod, adapted from (Mt1"canzini et al., 2009). ln the phase panel; the frequency with the phase anB)e clos;est: to zero i'S determined as PRF (dashed vertical line). On the nlag,nitude panel; thePRF is localed between the two freqqency ranges dominated by the doUble layer and parasitic capacitance (dashed lines with negative slope). The magnitude at PRF represents the tis£ue resistance (horizontal dashed line) fairly accurately. Seven! interesting issues remain unresolved thou,gh. First, the principles af the PRF method 112 have not been explained and there are no analytical validations in the literature so far. Also, Mercanzini et al. discussed the double layer capacitance in terms of a constant phase element, however, used the interfacial capacitance to approximate the "admittance" of the CPE (Mercanzini et al., 2009). Whereas this approximation is acceptable when the exponent of the CPE is very close to 1, corrections and conversion between the double layer capacitance and the CPE are necessary when this condition is not met (Hsu and Mansfeld, 2001 ). Last, the CPE is not an independent component of the electrode-tissue interface model, but is related to other components (Brug et al., 1984). This complicates the discussion of how the PRF method should be interpreted when the impedance spectroscopy shows a double layer with non-ideal behavior. In this paper, we present mathematical analysis and simulations on the PRF method, and also explored both representations (capacitance or CPE) of modeling the double layer. The results demonstrate the validity of the PRF method, while also pointing out its limitation in accuracy when the tissue-electrode interface behaves non-ideally and results in noticeable deviation. The results show that the PRF method is a simple and effective method to measure tissue resistance and achieves reasonable precision given sufficient frequency sampling and low noise. 3.2 Methods 3.2.1 The Electrode-Tissue Interface Model There has been extensive modeling of the electrode-tissue interface that well characterizes the equivalent components of this interface (Geddes, 1997; McAdams et al., 1995; Merrill et al., 2005; Pollak, 1974). We include the following elements (Figure 3.2), some of which were discussed in 113 the previous chapter: the tissue resistance, the electrode double layer, the charge transfer resistance, and the parasitic capacitance. Adhering to the original method, the Warburg impedance was excluded and the tissue was modeled as a simple resistance (Mercanzini et al., 2009). The Warburg impedance is negligible in the frequency range of interest (100 Hz to 1 MHz) (McAdams et al., 1995). Tissue can be described using the Cole-Cole model or Fricke model (Lempka et al., 2009; Wilson et al., 2014), which takes the cell membrane capacitance and extra- versus intracellular electrolyte into account. The capacitance of the tissue is shown to slightly affect neural stimulation, (Butson and Mcintyre, 2005). These models are not adopted, however, because of the dependence on the specific tissue types and unnecessary complication of the interface model. Electrode Tissue/Electrolyte Rs CPE orCoL Cp Figure 3.2: The tissue-electrode interface model with parasitic capacitance from the leads and other components in the system 3.2.1.1 Tissue Resistance The tissue resistance Rs is in series with the electrode interface. Given the small dimension 114 of microelectrodes used, this resistance could be related to the resistivity p of the surrounding tissue/ electrolyte by Rs= k · p (3.1) where k is the cell constant of the electrode (Kasi et al., 20llc). This linear relationship assumes that the resistivity changes gradually in space, and the resistance is dominated by the environment in close vicinity of the electrode site (Pollak, 1974), and also frequency-independent in the frequency range of interest (Gabriel et al., 1996). The cell constant can be used to calculate local tissue resistivity after tissue resistance has been extracted with the PRF method. 3.2.1.2 Charge Transfer Resistance The charge transfer (CT) resistance RcT arises from the electron-ion charge transfer due to the Faradaic redox reactions at the electrode-electrolyte interface (Merrill et al., 2005). The charge transfer is a non-linear phenomenon described by the Butler-Volmer equation, and both small signal and large signal approximations for this resistance are available (e.g. (2.21)). For electrode characterization and tissue impedance measurement using electrochemical impedance spectroscopy (EIS), small voltages are usually applied (e.g. 10-25 mV) and the small signal approximation should be used (McAdams et al., 1995). For biomedical electrodes which in general avoid Faradaic reactions for biocompatibility considerations, RcT is much larger compared to Rs for voltages and frequencies practical in neuroengineering application (Wang et al., 2014). A dimensionless ratio of the two resistances of the interface is given as R - s 10- 3 10- 6 XR - -- "' - RcT (3.2) 115 3.2.1.3 Electrode Double Layer Charge separation occurs when a metal comes into contact with electrolyte (Merrill et al., 2005) and is modeled as an interfacial capacitance CoL which is in parallel with the charge transfer resistance. However, the double layer often deviates from purely capacitive behavior and is better described as a constant phase element (CPE), whose frequency dependence is (3.3) where the exponent n describes the constant phase angle BcPE = -nn/2 , and Y is the "admittance" of the CPE. An important difference between Y and CoL is that they have different units: the former is sn · f:C 1 , whereas the latter is F = s · n- 1 (Hsu and Mansfeld, 2001). Therefore correction is needed when relating these two quantities. Brug et al. described the following relationship (Brug et al., 1984, pp. 3) in electrochemical applications c _ ~(R-1 + R-1)n-1y DL - S CT / (3.4) which indicates that the CPE is coupled to both Rs and RcT· However, under the condition (3.2), where RcT » Rs, the relationship becomes { n~ CoL "' ~Rs Y Y _en Rn-1 - DL S (3.5) 3.2.1.4 Parasitic Capacitance The parasitic capacitance Cp is an undesired capacitance of the system in parallel with the tissue-electrode model, due to the dielectric linkage between the leads and the environment especially when long wires or cable are used. A second dimensionless quantity Xe is introduced to quantify the ratio between Cp and CoL: 116 c - p 10- 1 10- 4 Xe - - "' - CoL (3.6) 3.2.2 Analysis of the PRF Method The mathematical analysis to find the PRF starts with the complex impedance of the tissue- electrode interface model, which is given as Z(w) = (Z 0 L(w) II RcT +Rs) II Zp(w) (3.7) in which the symbol II denotes parallel connection of two impedance and can be given as (3.8) The phase angle is the principle argument of the impedance <p(w) = arg(Z(w)) , (3.9) and is in the range of [-n/2, OJ for the tissue-electrode interface model. The PRF is the maxima of the phase in the high frequency range, and is the root (or the larger of the roots) of the condition d<p(w) -,...---= 0 dw The magnitude of the impedance is then determined at this frequency UJPRF which is a good approximation of the tissue impedance. (3.10) (3.11) Two models are explored. The first uses a regular capacitance for the double layer and includes RcT· The second uses a CPE for the double layer and excludes RcT as the first model shows that it is negligible in the frequency range covering PRF. 3.2.3 Electrode Parameters for Sinmlation For simulation of the PRF method, the parameters of concentric bipolar rnicroelectrodes with 117 pencil-like hps (FHC Inc Bowdoin, ME, USA) were used This type of electrodes was used for impedance measurements of retinal l!ssue and consists of an inner pole of 80-20 Pt!Ir alloy cone entric with and insulated from the stain! es s steel outer pole The c onfigural!on and d.imensi ons of the electrodes are shown in Figure 3 3 I 125µm ' Figure 3.3: The configural!on of the microelectrodes The inn er pole consists of Pt/Ir alloy, and the outer pole is stainless steel A total of six electrodes were analyzed in PBS (Sigma-Aldrich, St Louis, MO, USA) by EIS (1 MHz to 100 Hz) using a potenl!ostat (Reference600 and Interface 1000, Gamry Instruments, Warminster, PA, USA) with a Ag/AgCl reference electrode A thick plal!num wire was used as return electrode for monopolar modes and the outer pole served as the return electrode for bipolar mode The frequency sampling was 10 points per decade The impedance spectra were averaged (Figure 3 4), and the electrode parameters were extracted or calculated from 1 east squares fithng of the interface model with a CPE Table 3 1 shows 118 the mean and standard error of the parameters, in which the parameters in bold were directly fitted. The equivalent CoL was calculated using (3.5). The conductivity of the PBS was measured as 15.6 ± 0.1 mS · cm- 1 by a conductivity meter (VWR, Radnor, PA, USA), and used to calculate k of the electrode using (3.1). The exponent n is not as close to 1 as some platinum electrodes (n ~ 0.9) (Franks et al., 2005), however its value around 0.8 is consistent with other biomedical electrode systems (IvlcAdams et al., 1995). The Rcr was in the megaohm to gigaohm range, but could not be reliably extracted from the EIS data even when frequency reached as low as 0.1 Hz. Its variance between electrodes was also very large, and therefore Rcr is not included in the table. A vernge of electrodes 106 ,....=-.---...--~......-TTT~-.----,..--.-.-....-......-~--.-......... -.-TT ....... ...-~...---.-........ ~ ....... ~ ] 104 :: r:: bC ce ~ ,...-; - InnerPole --.Outer Pole ~Bipolar 102 ,_____,.._.._ .................. ---. ........ __.. ......................... ~ ........................................... ~ ............................ ...... 102 104 Freqnency f (Hz) Figure 3.4: EIS data ofFHC bipolar electrodes. 119 Figure 3.5: Photo of FHC bipolar electrode mounted in a syringe. Table 3.1: Electrode parameters extracted from EIS in PBS (Mean and standard error of the Me.an). Rs (k!l) k (cm- 1 ) Y (n.Q · s-n) n CDL (pF) Cp (pF) Bipolar Mode 5.14 (0.15) 78.8 (2.2) 6.9 (0.3) 0.77 (0.02) 323.4 (31.0) 26.6 (2.1) 3.2.4 Simulation of PRF Method Monopolar Mode - Inner Pole 5.37 (0.12) 82.4 (1.8) 6.3 (0.3) 0.76 (0.02) 313.3 (52.3) 16.6 (0.3) Monopolar Mode - Outer Pole 0.96 (0.05) 14.7 (0.7) 21.7 (3.2) 0.84 (0.01) 2765.4 (491.8) 177.7 (9.8) Impedance spectra were simulated using the electrode-tissue interface model as described in Figure 3.2 and the bipolar parameters of the microelectrode from Table 3.1. The simulation and data analysis were performed in Matlab (Mathworks, Natick, lv1A, USA). The tissue/electrolyte resistance was varied and calculated with (3.1) using resistivity that covered 0.5 to 32 n · m (see Figure 3.6), spanning the range of neural tissue from low resistivity (cerebral spinal fluid) to high resistivity ( dura mater or epidural fat) (Wesselink et al, 1998). Although it is unlikely that the resistivity of tissue of an electrode interface changes such dramatically in a single specific applications, this wide range is chosen to demonstrate the robustness of the PRF method and also 120 covers the majority of other tissue types (Geddes and Baker, 1967; Faes et al., 1999). The frequency range of the simulation is 1 MHz to 100 Hz. The PRF and IZPRF I were extracted from the simulated spectra and compared to the theoretical values from the calculation. For simulation, all frequencies are given in units of Hertz. Three models were simulated. The first uses CoL for the double layer. In this case, RcT was included in the model to show its influence in the low frequency region ( 50 Mfl, from EIS data). The other two simulations use a CPE and RcT was excluded as the results of the first model showed that RcT does not affect PRF. One model used a dependent CPE assuming that Y is a parametric fit and not a real physical quantity. Hence for different Rs, the interfacial capacitance should be used to calculate the corresponding Y using (3.5) .. The other was an independent CPE model which assumed that Y is a real parameter that does not change as Rs changes. These two situations gave different simulation results, which should be verified against experimental data. 3.2.5 Analysis on Sampling Density and Noise To further explore the PRF method under noisy experimental conditions, the grid density of frequency sampling and the influence of noise were studied with simulations. A total of 100 simulation trials were run as follows in Matlab. First, a tissue resistivity was randomly generated for each trial in the range of 0.5 to 32 fl· m. The corresponding tissue resistance was calculated using (3.1 ), and with the other bipolar parameters of the microelectrode from Table 3.1, impedance spectra were generated in the range of 1 MHz to 100 Hz with frequency sampling densities of 5, 10, and 20 points per decade. Then, a complex Gaussian noise 121 was added to the spectra at each sampling point, with standard deviations of 1 %, 5%, and 10% of the impedance magnitude at each point. This approach overestimates the noise as previous studies have shown that noise from electrode-tissue interface is proportional to only the real part of the impedance (Liu et al., 2008). For each three by four combination of sampling density and noise level (including zero noise), the PRF method was applied to the spectrum and IZPRF I was extracted. The corresponding resistivity PPRF was extracted using (3.1), and the percentage error compared to the 'true' tissue resistivity was then calculated. This error was averaged for each condition over all the different trials, and the standard deviation for the error is also calculated. 3.2.6 Comparison with Least Squares Method Another commonly used method to extract the tissue resistance parameter of the tissue- electrode interface is using least squares (LS) fitting. Particularly the complex nonlinear least squares (CNLS) method applied to relative errors is used in consideration of the nonlinear nature of the interface model and the orders-of-magnitude difference of the impedance across the spectrum. To explore how the PRF method compares to LS methods, the relative error version of the CNLS method was applied to the same simulation data in the previous section: ~ _ . L [IZ(ll,w;) -ZMeasCw;)ll 2 8 - argmm I ( )I o . Z 8,w; ' (3.12) where 8 is the vector of parameters of the model (e.g. 8 = [R 5 , C 0 L, RcT, Rp]T for the first model), and i is the index of frequency points of the sampled spectrum. The same statistics were calculated for comparison. 122 3.3 Results 3.3.1 Analytical Calculation 3.3.1.1 Model with CoL and RcT The complex impedance of the first model is (3.13) Solving the corresponding condition of (3.10) gives a quartic equation and yields two positive roots: x( 1 +1- (xR + x~) ± jcxc: 1 +1+xR)(x( 1 +1- 7xR - Bx~) vlzRsCoL (3.14) Using Taylor expansion and neglecting second and higher order terms of xR, the larger root can be given as (3.15) Considering (3.2) and (3.6), (3.15) simplifies and yields j1 + x( 1 + 20(xR) UJPRF = R c . S DL (3.16) The relative error from the approximation steps in (3.15)-(3.16) is jo(xR)/(1 + x( 1 ) , which would be at most 1 % according to (3.2) and (3.6). Further calculation using (3.16) then shows that under this condition 1 Rs IZPRF I = -- "' Rs 1 +xc <i'PRF = - arctan ( 2 (1 + xc) / j 1 + x( 1 ) (3.17) 123 Thus far, the mathematical foundation of the PRF method is clearly established. Equations (3.16)-(3.17) explicitly explain that: (1) The PRF is inversely proportional to Rs, and their relationship on a log-log plot has a slope of -1; (2) IZPRF I is linearly proportional to Rs, with a ratio slightly smaller than I. A good approximation is obtained only when Xe « 1. Otherwise it is necessary to scale the impedance magnitude by (1 + Xe) to find the real tissue resistance. This could be achieved by either calculating Xe with parameters extracted from EIS, or calibrating the measurement system using electrolyte of known resistivity; (3) <i'PRF is constant and independent of Rs, which could be used for experimental validation of this model. 3.3.1.2 Models with CPE The complex impedance of the model using CPE is ( ) 1 + (iwrYRs z UJ = ~~--~~------ (iw)ny + (iw)n+ 1 YRsCp + iwCp (3.18) Using Euler's formula to separate the real and imaginary components of (iwr, the corresponding condition of (3.10) yields cos(~") (1 - n) + RsY wn [ 1 - 2n + 2cos 2 (~") + R~Y 2 w 2 n(3 - n) cos(~")] +R£Y 3 w 3 n = R~Y 2 w 2 nnsin (~") /(wRsCp) (3.19) This pseudo-cubic equation contains nth, 2nth, 3nth and (2n - l)th order terms of w, which is impossible to solve analytically. However, as CoL and CPE describe the double layer's behavior very similarly, the PRF of one model approximates that of the other. Therefore using (3.16) from the first model to approximate w in the denominator on the right side of (3.19), a cubic equation with an approximate solution can be obtained: 124 X 3 + [(3 - n) cos CTI) - nsi~T)l X 2 + [ 1 - 2n + 2 cos 2 Crr)] X 2 Xe 2 +cos C 2 TI) (1 - n) = O (3.20) in which X = Rs Y wn is dimensionless variable. Given the condition that n is close to 1 and Xe « 1, an analysis of the roots shows that this equation has one real root which 1s approximately equal to the only coefficient term that is significantly different than zero: (3.21) And the PRF is therefore given as _.!. 1 _ __!__ UJ = NR ny-n:x 2n PRF S C (3.22) ' where N = (n sin(nn))ii:. Due to the approximation steps, this equation contains both representations of the double layer (Y, and CoL in the form of Xe). Therefore (3.5) is used to substitute one for another and two expressions of the PRF were obtained: 1 I l-2n 1-3n Nc-zn y zn2 R zn2 UJ - p s PRF - 1 1-2n NC -zn C 2il R- 1 P DL S (3.23) Using the CPE or capacitance for calculating UJPRF raises a discrepancy on the dependence of UJPRF versus Rs, which will show slightly different slopes on a log-log plot. The issue arises from whether to assume that the CPE is an element independent from Rs. Both cases are simulated and further discussion is given in the next section. Using similar techniques in approximation, IZPRF I can be obtained as 125 ( nn) z cot - 1 Nz 1 N 3 1 I PRF I l + 2 2 2 -n: T2n + O( ) ___ ,,, X --X --X Xe Rs n e 2 e n e (3.24) with all terms of Xe with orders equal or higher than 1 are neglected. This gives an upper bound of error at 10% according to (3.6); however because the neglected terms do not all have the same signs, the real error would be smaller as these terms partially cancel out. Again, Rs is approximated accurately when Xe « 1 and n "' 1. When this condition is not quite met, whether IZPRF I over- or under-estimates Rs is not apparent from simple observation as is the case with C 0 L, and further calculation is needed. Calibration using an electrolyte of known resistivity would be the simplest way to correct the results if needed. 3.3.2 Simulation 3.3.2.1 Model with CoL and ReT The impedance spectra simulated with different tissue resistance are shown in Figure 3.6. As predicted by (3 .17), <i'PRF stays constant. For the PRF extracted from the phase plot, the simulation and calculation agree with each other within an accuracy of 0.05%. The IZPRF I from the magnitude plot shows an approximately - 7.6% difference from the "true" resistance used for simulation, confirming the relationship in (3.17). The log-log plots of PRF versus resistance (not shown, but see Figure 3.9 for similar results of models with CPE) has slopes of -1 (R 2 > 1 - 10- 7 ), further validating the calculation. The - 7 .6% difference is relatively large, and reveals that the PRF method has an inherent deviation when electrode parameters are non-ideal, i.e. when Xe « 1 or n being close to 1 is not satisfied. However, due to the constant relative error regardless of actual tissue resistance value, 126 calibra.Tion c1iuld be ea siiy done' either using. (3 .17) or measurin-g salin~ o f known tesistivl.ty. -111 - 20 -10 ~ -40 C,; ~ -50 ~ ~o -11:1 Simula.tiou wi\.h \.11L 1 0• F'rNjO !:!lJey f (I.hi) Figure 3.6,: Simulated impedance: spectra fur the. · do1i'ble. layer c . apacitance: meclel with varying tissR e resisti. vity (p = 0.5, 1, 2, 4 B, 16, and 32 fl • m; arrow indicates increasing resistivity) . . I ZPS.FI and PPRF 'are' markl!~Jwith asterisk~- ¢.PRF is indi:penclent of the tissue· resisti'lity. 3.3.2 .. 2 :Models with CP.E Two situations• were simulate.cl, both with Rc-r igr1 ore.cl. TI1e first assumes that the CPE ailmittance: is not a real element, and· the.interfacial capacitance should be used with the vruying 127 Res, to cah;ulate the equivalqit QPE. The S~Co!ld asswt1es. the' ovp. osite: tlrat tl~e CPE is an !ndepenilent element and do. esn''t clrarrge with the tissue'fele.ctrolyte' envirO'runent. TI1e· sinntlatcd impedancepl-uts under the two situati'011S al"e.sh!.l'Wn in Figure 3.7 and Figure 3.8,.respectivefy; Sfmulat.ioo wit.h dependent. CPE l!I : S i.tnnlaLtnH 'i. : Oalrnllitii.m ~ f I I 1ai'-~~~~~..._...._~~---~_.._...__..~~~~~~"'"-'~~~~~- m-L j(I c..-~~~~~~~~~~~~~~~~~~~~~~~~~~~---. - I (l * : SimulaLimJ -80 -00L_~~~~~~~~.___._. __ _._~.____.~~-'-~--====::::;::;:~ 10;;: 10~ 1 a5 F'rC'<J • l l'\J (.',Y l t U. z.) Figure:3.7: ' $1inulated impedance spectt'a forthe dep. entlent double layer OPE model .Mth varying tissue re$istivity (same as Figure 3.6). Ute PIW an4 I ZPRF. I are' · given for l;i. l)th simulation and cal'culation (upp· er panel, squar· es and circles with center dots, respectively), and only the simulated tRPRF is shown (lower panel, asterisks). tRPRF is indepmd!!nt of the· : tissue resistance; .the 128 :SUimlA.1 .[cm wit c. imlepE>nd011f, CP 1. ,~ ,..;::-~~~~..-,...,.-~~--...~-,-,.----~~~~~.-....... ~--~~~~n l!l : &imula.ticm ~ Gakn latfi)1;1. 11r-~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 10 * ; fi u1 1ttlatinrt pr -f;O -sol-~~~~~~~~~~~..__~~~~~~~--======~ H J2. '°' F\<'411e 11r,v f /llz) Figure3.8: Simulate!l.impedance .spe~trafot•theindep·endent double.layer CPEmodel with varying tissue l'esistivity (same as Figure 3.9). Markers are the same as Figur~ 3.7. lflP-RF i11crea:ses with tissue resistance; impe. daiwe magnitude at fow frec{ll· ency·is independent of tissue resistivity. dep endMt CPE. shows a foa. d (fl f'RF •similar to. the model v..ith CD!t• whereas the independent CPE has a lflt>RF tllP. ving closer to zero a:s tissqe resistance 'increas~s. F.or the dep. endent ' CPE, the impe. danc-e magnitil:de. diverges at low frequencies. , · dt,ie to dependency of the· admittance on ·the tis.sue resis,tanc· e. On the uther band,. an independent CPE .shoms. .spectra v..ithout. this b· ehavi'or ; 119 These characteristic might be used to experimentally identify which model is more accurate. The validity of the CPE models is shown by the relative errors comparing calculation to simulation. For PRF, the analytical approximation has a stable relative error of 12.4% in the dependent CPE model, not varying with tissue resistance; The independent CPE model exhibits a calculation error in a similar range of 12.1% to 17.4%, increasing with the resistance. For IZPRFI, the dependent CPE model has a constant relative error of 8.0% for simulation and 0.3% for calculation versus "true" resistance; the independent CPE model has an estimation error in the rangeof 8.1% to 6.1% for simulation, and -0.1% to 2.5% for calculation. Thesrnallererrors for the calculation are due to the approximation steps (3.20), (3.21), and (3.24). The log-log plots of PRF versus tissue resistance/resistivity are shown in Figure 3.9. According to (3.23), the dependent CPE model has a slope of -1, whereas the independent model has a slope of (1 - 3n)/(2n 2 ) which is steeper when n E (0.5,1). The results again show very good agreement. The simulation results show that the PRF method has a positive deviation for the non-ideal double layer, opposite from that of the ideal capacitive double layer. The error is constant (dependent CPE) or relatively stable (independent CPE) for different resistance values, therefore calibration can be similarly implemented. 3.3.3 Frequency Sampling and Noise Analysis The calculation and simulation so far have shown that the PRF method has an inherent deviation which needs correction when parasitic capacitance exists. Limited frequency sampling and measurement noise might increase the error further. Further simulations with such conditions 130 were perfonned, and the relative errors from 100 simulated impedance spectra are shown in Figure 3.10. The mean and standard deviation of the relative errors are plotted against noise level and grouped according to models and grid density of frequencies. ·~~~.~~--~~~~~~~~ ....... 10~ - Slop<•s: Si mu Int.ion Depe11denl, CPE: I .00000 l 11cl1 114 11ol11 11 ! ~ 'P F. . I t n I" i ~ Ilr·1w111l• llt CPE: Simulaf io11 -·)(·-I )r p1·11dn1t ('PE: C'alt·11l;1tiou - + - hu kp• 111 lr111 C'PE: Simulation · · +· hidq)(•ml<•ut <'PE: C'ak11h~tio11 10 1 ··True" t'C'sist.ivit~· I' (U · m) C.:ak ulat ion 1.ooo:w I ll tl70 Figure 3.9: The log-log plots of PRF versus resistivity for the CPE models. R 2 values for log-log fitting are above 0.99999. The mean relative errors stay relatively stable for each of the three models regardless of measurement condition, and are also in agreement with the errors from the previous simulation under ideal conditions (-7.6%, 8.0%, and 8.1% to 6.1% respectively for the three models). Therefore statistically speaking, realistic limitations do not add additional error to the PRF method on average. However, the standard deviation of errors increases with higher noise level and decreases with higher grid density of the measurement frequencies. This indicates that any 131 indi~<fual meast1tement is. prone 10' hlghQ' QTor"with noise' ahdsparse sllil:lpling. Ten po.ints-p' cr' decade is a reasonable· minimum, ' and higher SaDllli"l\g of the specJrutnis desired. :1: he measwem~ the inherent dtvl:itioo 11s shown hy the, 11r1aly.sis. ,f potiUt~ !)\IT' I J1 1('1"\1 H) 1nli!ili l •l~T' 1h•"1Hh & .25 zo ;J'I • Ir. I lfi 251 20· lo ltl v '- 5 5 - a -- - - - -·-· . - 0 --------· . ~ " ,(, -5 r£i .fl} IO I ·IS 15 ~ () t i1 -2:0 0 OPE .. Dt~ci • ••Ii• .~ ... ~ ·a .<!,:;.----~----· a s Ill D s Ill 5 IG N11iw l1'Yl.'1 I', I Figure~ .10: TI1erelatlve cttors (1ll~an and·stand;u-d deviati9n) 9fthe:resisthlit}" txb'_ac'fe'tl using th!} PRF method from 100 s'imufated impeilanc.e spe , 'Ctta with · differi:n'.t freqReney-. samplitlg il~nsjt .y and:t1.sise levels: The c . urves where slightly shifted in the horiwolal direction to :av.sid . nver1 11p of the, errot: hars. 3.3.4 Comparison with CNLS Ihe CNLS rnet11od 'W;ls ap,p,lied to ea !)h conditi911 of th\: same 100 .sirtllll:ation trials. and the results are shi'lwn in Figiu·e 3.11. The mean rel:afive errnc is· close to ze.ro for all models 1111.d .resisti: vity ahd. d,oes not ret1uit:e 'calib.rati, 1.1, n like the PRF meth.x d. The' stamlat:d 'deriati<in ' of ' errors Show ~.milar 1Fttul a£ the trnisi: and ·samplihg claang, es. bUt have· small er vfilu· es 'COmpat" ed fo the 133 PRE mctnod. The> CNLS me(hO'<i is mor~· tolei;ant aga:iilst noise~ at 'the cost of increased. mcasUr(:tllent time and computatiorL 1J) J•lli lll i ·•·r d' •'if·r 11 l11 '. .lJ ) puT nk P• ""'T lh11~u.d1 • ID IQ to---· II ~ n h ~ \J • -· ~ . ~ . ,, - ~ "° :3' a - ~ r - ,_. • - . , _ -~ e ~ 2 ~ ' J a ~~ - ~ (:J ~ ... ~ ... r -1 . .... -4 -.\ (! .., !! • . ~ () ,- ,"l~ ! 0 tiPlt. iQ~·•''"'uLq ,. ~ 0 • ·~ ·R. J dJih ci u•h ~I -111 -tn -ilA 0 & i{l II S" 111' ti 5 1 0 1'41 , , 1 , 1 1111 1 l' ~I FJgure3.ll: The rclative · errors (me~ and standarfi deViation) oftheresisti:vityel(tract~d,1usingfh~ !eastsquares.m~tlwd 3.4 Discussion This sbfdyp.rovides t<?mpi:ehe11sive• analysis'.of ttlC'PRFmcthod fot: llX.t,ractiqg. tissue t;esisfalice frQffiimpeJlamie; specttr oscopy . Mathematie,al S!Ylliti Qt!S validateil 'fhe FRF method an.ii re:veal'Ci! 'that the, PRF method. ' l,ihs· deviati-011s when electt: o'des hav:e, nun.ideal behaviors. Simulati, ons .sh 'ow that frequt:nty sarripling and ;r1oise do cJ1ot limit the pracfa:al ttse of ther PRF cnethotl 3.4,1 Analytical Solution estabfrshetl the· mathematical pri'nciple• behind the PRF method. However; the e : quatiott could only be solved explicitly for the ideal double layer capacitance. Nevertheless, this model provided important insight and results that became useful later in the non-ideal model. For example, the ideal model includes RcT, which complicated the analysis. Due to its non- linearity, RcT varies depending on the applied stimulation. When larger voltages are used during stimulation, a large signal approximation should be used, giving different RcT values. However, this doesn't change its relationship with the tissue resistance, which is smaller by orders of magnitudes. Therefore RcT only influences the EIS in low frequency ranges, and plays an insignificant role in determining the PRF. The mathematical calculation validates this conclusion, justifying the exclusion of RcT in the other models. The model in which the double layer is represented by a CPE is very complex, and becomes impossible to solve with the irregularities in the exponents. The PRF derived for the ideal double layer model provides an approximation for the solution, which can be used to eliminate the irregularity and yield a cubic equation. However, the complexity of an exact solution to the cubic equation would not reveal useful information, and a simple solution capturing the principal interaction between tissue resistance and the double layer was derived via further approximations. The simulation data then showed that these approximation steps resulted in a fairly accurate description of the CPE model, despite that the electrode parameters used for simulations have Xe "' 0.082 and n = 0.77, which can be argued as not quite being "xc « 1 and n close to 1": (1) The log-log slope of PRF versus tissue resistance/resistivity shows high accuracy (within 1 % ) for calculation versus simulation (Figure 3.9); (2) The calculated PRF shows a non-negligible difference (12% to 18%) when compared to the simulation results; (3) IZPRF I shows an opposite 134 deviation (around -8% to-3%) between calculation and simulation, with calculation giving results closer to the "true" value. 3.4.2 Idealness of Tissue-Electrode Interface The electrode parameters used in this work revealed the limitation of the PRF method for non- ideal electrodes. Although the calculation and simulation of IZPRF I agree, they both show a significant deviation from the "true" resistance for all three models. This type of behavior was not shown by Mercanzini et al., mainly because their electrode's parameters (xe "' 0.009 and n = 0.9) were too close to ideal. Using their parameters (detailed results not shown), the relative error for IZPRF I is only on the order of 0.1 % for the double layer capacitance model, and around 1 % for either case of the CPE model; also, the PRF method performs closer to the CNLS method for their parameters, confirming that the idealness of the electrode is also an important consideration for accuracy. Our results provide means to correct the impedance extracted with the PRF method using either (3.17) or (3.24). Using electrolyte of known resistivity is also a quick way to obtain the correction factor for a given electrode. Nevertheless, the consistency in the deviation compared to "true" resistance indicates that the PRF method is especially applicable when the relative changes in resistance are important but their absolute values are not. Overall, it is generally desirable to have electrodes with ideal behavior (xe « 1 and n close to 1) in order to use the PRF method without the concerns discussed above. The first condition relates to the parasitic capacitance, which is usually very small and can go unidentified or be attributed to other reasons (Franks et al., 2005, Fig. 2). Electronics design and device fabrication 135 should anu to limit this shuut capacitauce or its iuflueuce usmg methods such as ou-chip compeusatiou, bufferiug, or sigual processiug. However, as srualler (less thau 30 µm iu diameter) micro-electrodes have beeu fabricated (Behreud et al., 2011 ), the decrease of the surface area of the iuterface could result iu double layer capacitauce of similar maguitude compared to the parasitic capacitauce. The secoud couditiou relates to the double layer iuterface, which depeuds ou the surface iuhomogeueity of electrode ruaterials (Brug et al., 1984). To achieve higher charge iujectiou capacity iu ueural stimulatiou, mauy electrode ruaterials (Petrossiaus et al., 20llb) with rough surfaces have beeu developed to iucrease the real microscopic area of the electrode iuterface without iucreasiug the geometric area of the electrode. These rough electrodes typically have uou- ideal behavior of CPEs. 3.4.3 Depeudeuce of CPE ou Tissue Resistauce For the two CPE models, it reruaius uukuowu which oue describes the iuteractiou ofCPE aud tissue impedauce more accurately. The characteristic behaviors of these two models could be easily observed from simulatiou results, as the simulatiou covered a very wide rauge of tissue resistivity for demoustratiou purpose. However it might be difficult to test this experimeutally. Iu real devices, the chauge iu resistauce electrodes face is typically much smaller. For example, ueural tissue has resistivity of 2 to 10 fl· m (Geddes aud Baker, 1967; Goucalve et al., 2003; Kasi et al., 2011c; Li et al., 1968), with some variatiou due to the auisotropy of uerve fibers. Body fluids such as cerebral spiual fluid or vitreous humor have resistivity iu the rauge of 0.5 to 1 fl · ru, (Baumauu et al., 1997; Geddes aud Baker, 1967). This chauge of ouly oue order of maguitude could result iu 136 impedance spectra too close to distinguish the aforementioned characteristics, considering that the measurements are also prone to noise in recording, especially when performed in vivo. 3.4.4 Considerations for Implementation For application of the PRF method, the grid density and frequency distribution of the spectrum affects the accuracy of the measurement. For most devices, a reasonably dense frequency sampling (more than 10 points per decade) could be specified. If the approximate location of the PRF is unclear initially, the overall shape of the spectrum could be obtained from a first scan with lower sampling. After identifying the region containing the PRF, subsequent routine measurements just focusing on this region could achieve sufficient accuracy by reducing the frequency range and increasing the sampling. Overall, this becomes less of an issue if electrodes with ideal behavior are used, as they have wider and flatter impedance magnitude and phase curves at frequencies near the PRF, which reduces the sensitivity requirement in obtaining a very accurate PRF, allowing the measurement with less frequency points and faster measurement speed. The PRF method allows rapid measurement, as data acquisition can be terminated as soon as the PRF is identified. From the previous simulations, the PRF is mostly located within the range between 5 kHz to 500 kHz, therefore a frequency sweep only needs to cover one decade or two. For a particular application, this range might be even smaller as resistance changes over time or due to electrode location are limited. On the other hand, the CNLS method needs data from a full spectrum for post-processing, resulting in longer time for measurement. For high accuracy, a wide frequency range is needed. Also, the CNLS fitting process itself is time-consuming and 137 computational expensive, whereas the calibration for the PRF method is a simple multiplication. The CNLS method requires some a prior knowledge of the models to be used, whereas our analyses show that the PRF method works intuitively regardless of which of the three models is used. Therefore, the PRF method has advantages in terms of simplicity, measurement speed, and easy data reading, with trade-off in accuracy for non-ideal electrode interfaces in terms of standard deviation of error. 3.5 Conclusions The validity of the PRF method to extract tissue resistance through impedance spectroscopy has been established via mathematical analysis, and the deviation resulting from non-ideal electrode interfaces has been demonstrated. The analytical results provide means to correct the tissue resistance extracted by the PRF method, if the non-ideal tissue-electrode interface results in noticeable deviation. Comparison between models ofregular capacitance and CPE for the double layer shows that specific consideration should be taken if the impedance spectroscopy shows a double layer with CPE behavior. Simulations also show that the PRF method works reasonably well when considering realistic frequency sampling and noise. Further work will include experimentation to identify whether the independent or dependent model of the CPE is a more accurate description of tissue-electrode interfaces when tissue impedance varies over time or with electrode location. 138 Chapter 4 Electrical Mapping of the Retina 4.1 Introduction The past two decades have witnessed the translation of retinal prosthesis from laboratory to clinic (Weiland and Humayun, 2014b), restoring partial vision to the blind affected by retinal degeneration. Despite the progress, ongoing research aims to improve the quality of visual percepts created by these prosthetic devices. An important aspect is the modeling and simulation of electrical activation of retinal neurons (Kasi et al., 20 llb) to understand spatial patterns of stimulation and therefore selective stimulate certain regions and neuron populations. The distribution of current and potential within the retina is a critical factor determining the response of retinal neurons. Anisotropic and heterogeneous resistivity has been shown to affect distribution in other neural structures (Astr6m et al., 2011). So far, no direct experimental studies have addressed this distribution, and simulations were performed based on physical models of eye tissues (Kasi et al., 20llb). The resistivity of retinal layers used in these modeling studies was drawn from previous studies (Heynen and van Norren, 1985; Brindley, 1956; Ogden and Ito, 1971 ). There are several shortcomings: 1) in some studies the current flow was not well-controlled; 2) the retinal prosthesis will interface with diseased retinas-in which degeneration and remodeling significantly changes the tissue and the impedance may be very different; and 3) some studies were carried out in amphibians or birds, whose retina differs from mammals. While there are studies showing resistivity data of rat retina (Kasi et al., 20llc), no results are available in the mouse, 139 which is an important model for retinal prosthesis research. In this study, I measured resistivity profiles of healthy retina and two RD models using bipolar microelectrodes, to compare the terminal effect of different onset and time course of retinal degeneration. The peak resistance frequency (PRF) method (Mercanzini et al., 2009) is utilized for extracting tissue resistance from the impedance spectrum. Histology was performed to compare the changes in thickness of the different retina models. 4.2 Experiment Setup 4.2.1 Recording Chamber and Experimental Set-up An in vitro retina/ eyecup recording chamber has been fabricated (Viterbi/Dornsife Machine Shop, University of Southern California, CA, USA) after (Newman and Bartosch, 1999) (Figure 4.1, left). The chamber consists of an upper part to hold the physiological saline immersing the tissue, and a lower part holding the eyecup/retina. A polycarbonate sheet with a center hole is placed between the upper and lower components, which are made of acrylic. The sheet secures the tissue in place, while allowing access from above. The lower chamber contains a channel in the center to hold the return/ground electrode for measurement. Before assembling the chamber, agar (Sigma- Aldrich, St. Louis, MO, USA) is pipetted in this channel and solidifies in the form of a gel. The agar (1) provides a dome-shaped surface on which to place the retina, (2) gives a safe space of operation for the probing electrode once the tip penetrates the retina and reaches below it, while (3) forming a conductive media for the current to pass. Vacuum grease is applied between the upper and lower pieces to form a tight seal preventing leakage. 140 Saline Agar I ___ _ ReturnL~--- Figure 4.1: Left: Custom-made eyecup/retina recording chamber. Right: a cartoon illustration of the experement set-up: a probing electrode is controlled by a linear platform; reference electrode is placed in the saline, and the return electrode is located within the agar in the bottom. A 3-D linear platform (Newport Corporation, Irvine, CA, USA) has been utilizied to hold different types of electrodes or arrays, which probe the retina from above. A digital micrometer (Newport Corporation) on the vertical axis was used to record the location of the electrode. A bipolar electrode (see section 3.2.3 and 4.2.3 for details) was moved down through the saline, approaching the retina. Impedance spectra were recorded using Gamry potentiostat (Reference600 and Interface 1000, Ganlfy Instruments, Warminster, PA, USA) at regular intervals depth and more frequently when impedance began to increase as the tip neared and entered the retina. The procedure was terminated when several measurements showed that the impedance had returned to baseline as the electrode reached the agar beneath the retina. A microscope (United Scope LLC, Tustin, CA, USA) connected to a computer was mounted obliquely next to the linear platform and used to monitor the movement of the electrode tip. The entrance point of the electrode through the retina was chosen in the central area excluding a small region of the optic disk. The right panel of Figure 4.1 is an illustration of the set-up with a probing electrode, reference electrode, and return 141 electrode. Figure 4.2 and Figure 4. 3 show the experimental set-up and a close up look through the microscope. Figure 4.2: Experimental set-up with linear platform for controlling the movement of the electrode, and a microscope for monitoring the position of the electrode. 4.2.2 Retina Preparation and Histology Fresh retinal tissues were isolated from mouse eyes of female wild-type (WT), rdl, and rdlO mice (strain #000664, #004766, and #004297, The Jackson Laboratory, Bar Harbor, :ME, USA) at ca. 12-14 weeks postnatal following euthanasia under an approved IACUC protocol. After enucleation of the eyeball, the retina was dissected in oxygenated Ames solution (Sigma-Aldrich) and placed on top of the agar in the custom -made recording chamber with the RGC side up. After 142 ass. erilbling tlie upJ"r part of the recordil)g: cha~•~ •. U . ie whole mountretina was then transfe,red to tl\e· expeiimen! platform for recordi~ . The otlll't •J"b. all was also enucleatecl,, and standard procedures offixalipn MdH&E.staining ""re ils€d to perforn vhistology,examinatiors. TM sli&s were digitally·~ca1U'leda11d thef\ exarajned,in Ap. erio1mageScopj (Leica Biosyst~ms Inc., Buftalu 'Gtov.e, IL). f igure 4.J': Close-up look of the electro& tip.md roo. un!e:d e.Y"cup lhi;oUgh the 'microscope. T-lie platinum wire used as ·ref ere.nee e1ectro& is vislllla ta the right in the backgroUJjjl. 4 2 . 3 E lectrodes, am!Moilel of Electrode Interlace· ConM.ntric bij:<>lo~microele~tto"des tFHC fnc. Bowtlam, ME, USA}¥-itll tencil-like ti];; were U<!ld. T.he electrode consist. s pf an inner p:ile of 80-20 PV!r alloy of co~cal sh'IP'· Till' stain]a,,. stee.Iouter p:>le is concen1l'ic• v1itfrand ins.ulate. d'from the inner j:<>le. The configuration, diritensio!lS, and f."l•lloeters of th~ e!actro&s.are shown in t~ prwiou$ se~t!on (3.23, Figure 3.~. Fi~re 3S, \43 and Table 3.1). 4.2.4 Impedance Measurement using the PRF Method The Peak Resistance Frequency (PRF) method (Mercanzini et al., 2009) was used to extract the resistance of the tissue and/ or electrolyte. The method finds the frequency at which the impedance of the interface is most resistive (i.e. phase angle closest to zero degree), and records (or interpolates) the impedance magnitude at this frequency. See the previous section (Chapter 3) for a detailed analysis on the PRF method. The PRF method is an efficient method to observe impedance changes when the electrode is advancing through the tissue, not requiring post-processing of the impedance spectra. However, as our modeling studies revealed that the PRF method requires calibration under certain conditions (see 3.3), the conductivity of Ames solution was also measured to serve as the baseline for data normalization. Its value was determined to be slightly lower than PBS at 15.2 mS · cm- 1 . Preliminary testing was conducted on phantom material that mimic biological tissue. It consisted of different layers of agar, each made with different concentration of PBS (regular or diluted). The results showed that the PRF method could capture the change in the resistivity as the electrode was pushed through the agar phantom (Figure 4.4). 4.2.5 Data Processing and Analysis Impedance data were imported into MATLAB (Mathworks, Natick, MA, USA) for processing. The saline/tissue resistance was extracted using the PRF method, and correlated with the electrode location as documented from the micrometer readings. The resistances were then converted to 144 resistivity according to the cell constant ofthe electrode, and the resistivity profiles of each tissue type were aligned according to their peak location and averaged. 4.5 - 4 - 3.5 - ~ 3 - >. 2.5 - :: .::: .... T. 2 - "[ 1.5 - 1 - 0.5 - --e-- T1·,;1 -e--....... , :? • - - -PBS . . . . . . . . o...._~~~~·~~~~.._·~~~~·~~~~...._·~~~~·~~~ ....... ·...._~~__. 0.5 1 1.5 2 2.5 3 3.5 4 Dt'pt Ii .: l u11n) Figure 4.4: Test results of the PRF method using the FHC bipolar electrode and different layers of agar made from regular and diluted PBS. One test (green curve) was performed with a layers of agar with diluted PBS on top, hence having higher resistivity than the layer below with regular concentration PBS. The other (blue curve) had the diluted PBS agar on bottom, however, the concentration was not as diluted as the first test, and the resistivity increase was not as large as compared to the initial resistivity of the :first test. Because the electrode lip could not be visualized after penetrating the tissue, and due to the observation that the resistance at PRF started to increase even before the electrode lip touches the retina (Ray et al., 2011), the boundaries of the retinal influence on the resistivity profiles were delineated by :finding the location where the resistance increased by more than 10% above the 145 baseline from either direction. The apparent thickness of the retina's resistivity profile was determined by the distance between these two boundaries. 4.3 Results 4.3.1 Impedance Measurement The resistivity profiles of WT, rdl, and rdl 0 retina are shown in Figure 4.5. The RGC side is on the left of the graph and is aligned for all three groups, while the photoreceptor side is towards the right side, with each boundary shown as a dashed line. The WT retina profile shows an increase in resistivity starting from the RGC side inward, from 0.6 ± 0.1 fl·m to a peak resistivity of 3.7 ± 0.5 fl·m, and then decreasing down to baseline values on the photoreceptor side. The peak is located at about 70% thickness, which is consistent with literature data on rat and embryonic chick retina (Kasi et al., 20llc). The WT retina profile has an apparent thickness of 350 µm. Compared to the WT retina, the rd] and rd] 0 resistivity profiles are much thinner as expected due to degeneration (210 µm and 215 µm, respectively). The peak resistivity is lower (2.0 ± 0.2 fl·m and 1.5 ± 0.2 fl·m, respectively), but still in the nominal range of neural tissue, while the location is at about 60% thickness. The apparent retinal thickness extracted from the average profiles for each animal model is shown on the left side of Figure 4.7. 4.3.2 Histology Histology results are in agreement with established records of the retinal degeneration models (Figure 1.10, Chang et al., 2002). Typical results for each animal model are shown in Figure 4.6. Compared to healthy retina, retinal degeneration causes complete loss in the photoreceptor layer 146 and thinning of the· QUtetplexiform layer, and also shows lower cejl <1ens1t;res Ill bipolar c:efl layer 0 -----------------~L-= 50 100 II ~ I {50 200 Dt<pt ll ( 1i1.i1 ) 250 300 3 50 Figure 4.5: The resistivity profiles of WT, rdl, and rdl O retina. Shaded ar·ea indicates standard deviati0n. The dashed horizontal line .shows the baseline resistivity of saline. The vertical dashed line on the left shows the aligned boundaries 0n ,the RGC side, while the ones 0n .the right and in the middle ar:e the b <:lu1u lari es Pn the photoret ept.Pt sid e (showing the s· am e color c:ottesponding t<i· the· resistivity profile) The retinal thielmess measured from histology is shown on the right side ofFiglire 4.7,,in agreement with (Fennesi et al i; ~\1 '12) , The healthy retina from WT mice has an a;verage thickrres~ :Qf 202.1 ± 9.0 ~qn . The retina . thi: ckness o(the rdl and rdlO models ar.e 97.6± 3,6 ~im and 94.i ± 4.9.µm, respeetively, only abQut halt 1>f th: e IH>rmal value. Both models are sigrtificantly thi.rtner than WT, with no significant difference betwe: ep tlu~m, 147 4.3.3 Comparison Comparison of the thickness between retinal resistivity profiles and histology data reveals that the relative changes in the width of the resistivity profiles are in general agreement with the loss of thickness in the histology measurement. The resistivity profiles of rdl and rdlO retina are about 60% thick compared to the healthy retina, close to the 50% change in histology. One factor for the discrepancies between the absolute values of the two data sets is due to the loss of tissue volume during the process of tissue fixation, resulting in smaller thickness values in histology measurements. WT rd1 rd10 Figure 4.6: Typical histological staining of WT, rdl, and rdlO retina with the ganglion cell side on top and sclera on bottom. The rdl and rdlO retinas show complete loss of photorecq>tors and thinning of the outer plexiform layer. The two RD models have similar results, confirming that the onset and time course of the retinal degeneration results in little structural differentiation once the degeneration is "complete" in its end phase (Marc, 2010; Pennesi et al., 2012). For prosthesis research, this indicates that the end·stage of degeneration should be focus of study when using RD animal models. 148 ,-... 300 s ~ -- -..__. 100 0'--- Resitivity Profile -~TT - rdl rdlO • Histology Figure 4.7: 'Thickness comparison of the resistivity profiles and histol ogy data. For thickness extracted either fr om resistivity profiles or histology, \VT retina is around twice as thick as the two RD models, which are similar in thickness. The discrepancy in the absolute valu e arises due to the limits of the resistivity profile measurement, and tissue shrinkage du e to tissue fixa tion. 4.4 Discussion Tue resistivity profil es of the three animal models reveal that the apparent resistivity experienced by the electrode increases with depth from the tissue surface. The trend matches for the three profiles until the degenerate retinas reach their peak resistivity close to the center of the tissue. Aligning the profiles according to their photoreceptor side (not shown) demonstrates that they match well on this side as well, until the degenerate retinas reach their peaks. These observations suggest that the resistance measurement is mostly influenced by the geometry and dimensions of the tissue and the electrode. The resistance extracted with the PRF method is a spatial average of the tissu e/electrolyte 149 surrounding the electrode's tip. While its value is dominated by a small volume enclosing the tip, it reflects the changes of tissue resistance on a mesoscopic scale and is also affected by the saline surrounding the tissue. Hence, the thickness extracted from the resistivity profiles included additional thickness when accounting for the influence of retina when the electrode tip was outside of the retina or had partially penetrated the tissue; while the resistivity was lower due to the influence of the saline. Therefore, the PRF method is spatially limited by the size of the electrode used, and microscopic changes due to cellular structure and retinal remodeling due to degeneration could not be detected. Smaller electrodes or a different method with finer spatial resolution is required to measure the effect of retinal remodeling on tissue resistance. 4.5 Conclusion The resistivity profiles of WT, rd], and rd] 0 retina have been measured. In degenerate retina, the profiles were thinner, and the thinning agreed with histology data. The peak resistivity also decreased with degeneration, however, was mostly affected by the changes in dimension (thickness). The apparent resistivity determined with the PRF method is on a mesoscopic scale comparable to the electrode size. Therefore it was mostly influenced by geometric factors, while tissue structure at the cellular level, e.g. retinal remodeling and glial hypertrophy, could not be detected. Measurements with higher spatial resolution will be needed to assess the impact of these phenomena. 150 Chapter 5 Summary 5.1 Key Findings and Significance I have investigated several aspects of the electrode-tissue interface via mathematical and computational modeling, and performed experiments to validate the models and measure electrical properties of the retina. The use of sophisticated mathematical tools enabled deeper understanding of the interface, and developed the theories for interfaces of neural prosthetic devices. The key findings of my research projects are: I. The edge effect of current density on disk electrodes and those of other shapes could be controlled and reduced by designing optimal current input waveforms, which have a slower rising or falling edge compared to the rectangular waveforms. Such waveform design could reduce preferential corrosion on the edge of electrodes when pulsed with high current levels. 2. The PRF method for extracting the tissue resistance from impedance spectroscopy is a valid and simple method, although it is subject to inherent deviation and also influence from noise and sampling density. The PRF method performs well when electrodes with ideal behavior are used, and could be easily calibrated when electrode conditions are non-ideal. 3. Tissue resistance of retina reduces as the retina undergoes degeneration and loses volume and thickness from death of photoreceptor. This could affect the distribution of current density in the retina, when comparing healthy and degenerate animal models, and therefore end-stage RD models that closely mimic the disease stage of implant patients should be used for 151 modeling studies and in vivo or in vitro experiments. Overall, my work showed that electrode-tissue interface is of great importance for the functioning of retinal prostheses, and provides the physical properties for multi-level modeling to investigate neuronal activation. 5.2 Implications for Epiretinal Prostheses 5.2.1 Current Density Distribution Retinal prostheses use circular electrodes with rectangular pulses, but the subjects do not describe annular perceptions (Nanduri et al., 2008); also calcium fluorescence imaging studies of in vitro invertebrate retina (Behrend et al., 2011; Weitz et al., 2013) do not show preferential neural activation at the edges of 0.2 mm diameter electrodes. For prosthetic devices, a consistent tight placement of the electrode array on the retina is still an issue, resulting in some distance between the electrode site and the stimulation target. Therefore, the edge effect is more of a concern for electrochemical safety, rather than for undesired neural stimulation effect. Also, a preliminary testing with the calcium imaging set-up showed that the current waveform modification didn't change the spatial distribution of the activated RGC, therefore indicating that the waveform did not have significant effect on the threshold in those specific stimulation set-ups. These results could be explained by the distance between the electrode and RGC targets together with limited spatial extend of non-uniform current distribution. For future high resolution prosthesis, the distance between the electrode and the RGC should be minimized to allow focal and selective stimulation. Protruding or penetrating electrode could be utilized that have their electrode 152 site within the retina layers, such as those used by EpiRet3 (Rossler et al., 2009) and proposed by NanoRetina (Zrenner, 2013b). In those situation, careful design of electrode geometry and current waveform might be necessary to avoid high current density in the vicinity of neurons to prevent physiological damage. 5.2.2 Electrode Impedance Monitoring The analysis of the PRF method showed that the method could determine the series resistance of the electrolyte/tissue with fairly high accuracy and simplicity. Therefore it could potentially be used in the retinal prosthesis to monitor the electrode tissue interface. However, due to the limited resources and computation power of the implanted electronics, such measurement could be difficult to achieve with the current device or new ones in the near future. For example, the Argus I Epiretinal Prosthesis has functionality to measure the electrode impedance and transmit the measurement to external systems via back telemetry (Mahadevappa et al., 2005). This diagnostic function of the implant is used when the subject is in clinic for visits during the clinical trial and for long-term follow-up (Yue et al., in press). The implant uses a single frequency sine wave of 1 kHz to measure the impedance, and some correlation between the thresholds or electrode distances versus the impedance has been obtained. This frequency is fixed due to the software and hardware designed for such measurement, and the system does not report phase information. Therefore the PRF method could not be implemented even if just a small frequency range would need to be scanned for identifying the PRF. Later in vivo studies have shown that higher frequency measurement (100 kHz) has better 153 sensitivity for correlating impedance with electrode-retina distance (Ray et al., 2011). Therefore future devices should better provide an impedance measuring option at high frequency. This additional information could be very helpful in more accurately determining the performance of the electrode and the electrode-retina distance. To fully implement the PRF method, a frequency scanning mechanism needs to be included, and phase information also needs to be reported by the implanted electronics. Given the results from the simulation in 3.3.2 and 3.3.3, the frequency range to be covered doesn't need to be large, and could be determined from in vitro experiments with cow or pig eyes. Even just 5 points covering the decade including the PRF regions should reliably provide an accurate enough measurement of the series resistance in the device. Also, the inherent deviation of the PRF method is not much of an issue as the relative change of resistance is the concern for the long-term monitoring of the electrodes. 5.2.3 Dealing with Retinal Degeneration in Computational Modeling To understand how prosthetic stimulation works and how it could be improved, many studies have been performed on RD animal models that mimic the disease in human patients. Changes of neuronal responses in these models, such as threshold and activation pattern, have been known (Margolis and Detwiler, 2011) and show the challenges that prosthesis stimulation needs to address as compared to electrophysiological study of health neural tissue. However, there has been limited research on the changes of physical and electrical properties on mesoscopic to macroscopic scales in RD retina. Reduced resistance and thickness of the RD retina could affect the field distribution 154 under stimulation when compared to normal healthy retina, and account for some of the threshold changes observed. Computational modeling to simulate prosthetic stimulation can provide much insight on how retinal prostheses perform and how they could be improved. However, such studies have been explored mostly using models built based on healthy retina (Kasi et al., 20lla, 20llb). Modeling studies need to take the physical changes of the RD tissue into account in order to better describe the behavior of neural stimulation relevant to the prosthetic device and clinical applications. 5.3 Future Studies 5.3.1 Inclusion ofFaradaic Reaction in FEM Studies In Chapter 2, the role of the Faradaic components of the interface were limited. It was simplified to a small signal approximation in the mathematical analysis and excluded from the FEM study for the disk electrode study. Although this provided simplicity for the optimization process, it led to deviation from reality, especially for large overpotentials, such as those used in the pulsing experiment. It has since been shown that modeling the non-linear behavior of the Butler-Volmer equation is feasible in COMSOL software (Sue et al., 2013). Sue et al. simulated FEM models of herni- cylindrical cochlear implant electrodes of two different sizes and include a non-linear Faradaic reaction (oxygen reduction and platinum oxidation) as well as constant phase element instead of double layer capacitance on the interface. They explored currents of either polarities as input (25 µs duration and 0.5 mA amplitude for both electrode sizes). The results showed that for anodic 155 current, the Faradaic currents play little role in the overall current density non-uniformity, which drops as the primary distribution shifts to the secondary distribution. The Faradaic component in the anodic current contributes to less than 0.02% and 0.002% of the total current by the end of 25 µs for the two electrode sizes. This is consistent with my mathematical analyses based on Newman's work. However for the cathodic current, the current non-uniformity might not drop monotonically. For the small electrode they modeled, it dropped first and increased as the pulse get longer. Towards the end of the pulse, the Faradaic component increased rapidly and reached as much as 38% of the total current in the end. As this is a relatively short pulse width, the effects can be expected to be much larger for longer pulses. On the other hand for the larger electrode, the conclusion were similar to those with anodic current, only having less than 0.5% Faradaic current in the end. The underlying reason for the asymmetry of cathodic and anodic stimulation pulses is due to the different anodic and cathodic charge transfer coefficients for a pair of reduction-oxidation reactions. Therefore, the inclusion of the F aradaic component could yield in more accurate results and should be considered for studying the electrode-tissue interface in the time domain, especially when large electrical currents are simulated for electrical stimulation. 5.3.2 Testing the Tissue Dependence ofCPE In Chapter 3, two different behaviors were explored for the CPE representation of the electrode double layer: one that has a "conductance" Y independent of change in tissue resistance, and another that is dependent. It remains unknown which one describes the interaction of CPE and 156 tissue impedance more accurately. Besides the characteristics of the dependent or independent CPE mentioned in the analysis, it should be further tested whether the exponent n of the CPE is an independent variable for the double layer, which was assumed so in the PRF analysis. It could be the case that the CPE description of the double layer is highly variable as the resistance of the interfacing tissue changes, and therefore even the exponent is not a constant for a given electrode geometry and material. Although this does not change the conclusion of the analysis on the PRF method, further analyses could provide a more comprehensive description of the CPE component of the tissue-electrode interface, and shed light on the underlying reason for such non-ideal behavior. The retinal impedance data already collected might be sufficient for such analyses, but it is possible that more experimental data is needed. 5.3.3 Intraretinal Potential Mapping and Validation of Computational Models Chapter 4 provided measurement data of the electrical properties of different types of retinal tissue. The results are vital for multilevel modeling, as the first step of such modeling is to simulate the electrical field and current density in the tissue and especially in the region of interest for stimulation. As an effort to improve electrical neural simulation, computational modeling that provide predictive cases for the design of stimulation parameters should be as accurate as possible. This not only requires good modeling efforts, but also needs experimental validation. For the simulation of physics (electrical fields, voltage, and currents) in such models, there should be experimental work to map the intraretinal voltage and currents under electrical stimulation. This would determine whether the modeling inputs, such as the resistivity measurements, were accurate 157 and a good representation of retinal tissue. From the data obtained so far, it is likely that the resistivity obtained needs improvement, as the values determined with the PRF method reflect the properties on a mesoscopic scale comparable to the electrode size. Smaller electrodes or a different method with finer spatial resolution might be required to obtain more accurate results and to investigate the structural changes of retinal remodeling on the cellular level. Computational modeling and its experimental validation would provide useful information on the quality of the resistivity measurement. 5.3.4 Experimental Validation of Multilevel Modeling ofNeuronal Activation As part of the collaboration in the NIH Multilevel Modeling project, experimental work will validate not only the simulation of physical quantities as mentioned above, but also the predicted stimulation schemes that result from the modeling. Ongoing experimental methods in our lab, such as patch-clamping, extracellular multi-array recording, and calcium imaging, should be used in these model validation studies. Both WT and RD mouse models should be used, specifically the rdl 0 mouse model. Patch clamping can be used to study in detail the response of single cell (Cho, 2014). Extracellular array recording can provide information on temporal characteristics of cells in the neighborhood of the stimulating electrode (Ahuja et al., 2008). And the calcium imaging technique using transparent indium tin oxide electrodes allows the recording cell activity directly over the stimulating electrodes (Behrend et al., 2009, 2011; Weitz et al., 2013). Combined, these methods can provide a comprehensive picture of retinal activation and will be used to validate the model predictions. 158 Chapter 6 Supplementary Materials-Technical Review on Newman's Series of Analytical Calculation on Disk Electrodes 6.1 Introduction Dr. John Newman, an expert in electrochemistry, studied potential and current distribution of the conunonly used disk electrode extensively in the 1960s and 1970s. He and his colleagues published the results in a series of papers in the Journal of the Electrochemical Society (Marathe and Newman, 1969; Newman, 1966a, 1966b, 1970; Nisancioglu and Newman, 1973a, 1973b). This seminal series on this topic was consistent and well-written, and has been cited by many. In electrochemistry and closely related fields, the results are well-known. However, the papers, especially the later ones in the series, enjoined less familiarity in other fields, including biomedical engineering in which electrodes became widely used in neural stimulation. The purpose of this section is to review Newman's work on disk electrodes together. More mathematical details are supplemented to the original calculation to help the readers follow the derivation more easily, and a mathematical appendix is given. Three major adjustments are made to Newman's original analyses. First, the equation sets are summarized into matrix form, which is helpful in showing the similarity and differences between different inputs discussed in the papers regarding sinusoidal voltage, current step, and voltage step input (numbered [4]-[6] in the list below), and also provides easier implementation with the help of nowadays computer programs, such as MATLAB. Also, the normalization factors to give dimensionless variables have been 159 slightly scaled by n/ 4 compared to the original papers, which endows them the representation of physical quantities, instead of just mere normalization purpose. Lastly a consistent symbol naming system is used to refer to the results from different papers. The review will provide a comprehensive understanding of the original papers, which are listed below with comments given. [l] Resistance for Flow of Current to a Disk (vol. 113, pp. 501, 1966). This is the l" paper on this topic and gives the steady state solution to an ideal disk electrode without considering overpotentials on the electrode surface related to the double layer capacitance and any Faradaic reaction. The solution is the primary current density distribution of the following situations of current/voltage step input. The voltage-current relationship gives the resistance of the electrolyte using the interrupter technique, and also provides normalization factors for the following problems. The rotational elliptic coordinates are introduced to solve the Laplace equation, however due to the simplicity of the solution, few details are given on how to solve the partial differential equation. This paper's high citation volume (617 as of Oct. 2015) was discussed by Newman's colleagues Orazem and Tribollet (2009). [2] Current Distribution on a Rotating Disk below the Limiting Current (vol. 113, pp. 1235, 1966). This paper addresses how the diffusion layer may shape the current density profile, when the Faradaic reaction but not the double layer potential is considered. Two limiting situation will give current density at the disk when the potential just outside the diffusion layer at the disk surface is uniform (primary current distribution) and vice versa. If the current density is small compared to the limiting current, the concentration overpotential could be neglected and secondary current distribution comes into effect. Assuming the current density doesn't become 160 limited by the diffusion layer, its effect is not studied later in the series. This paper gives more detail on how to solve the Laplace equation in rotational elliptic coordinates, which is useful to understand calculations in later papers. [3] Current Distribution on a Rotating Disk (vol. 116, pp. 1704, 1969). Experiment verification of the solution in [2]. [ 4] Frequency Dispersion in Capacity Measurements at a Disk Electrode (vol. 117, pp.198, 1970). This papers studies sinusoidal voltage input to the disk electrode and it includes effect of double layer capacitance and a linearized Faradaic reaction current. Results give effective resistance and capacitance as a function of frequency. More details on numeric solution of the Laplace equation are given, which, compared with the following two situations, will show consistency in the solutions when using the matrix format. [5] The Transient Response of a Disk Electrode (vol. 120, pp. 1339, 1973). This paper describes the response to a current step input, and includes effect of double layer capacitance and a linearized Faradaic reaction current. The method decomposes response into steady state response and transient response, and discusses the different time constants associated with the transient response. [6] The Transient Response of a Disk Electrode with Controlled Potential (vol. 120, pp. 1356, 1973). This paper describes the response to a voltage step input, and also includes effect of double layer capacitance and a linearized Faradaic reaction current. The study discusses the different time constant associated with the transient response, especially the zeroth time constant that is unique to the voltage step input. 161 6.2 Model and System of Equations A disk electrode of radius a is embedded in an infinite large insulator substrate under a semi- infinite large space of electrolyte of conductivity K. Voltage or current is applied to the metal part of the electrode, and ground is set at infinity. Cylindrical coordinate system is established with the origin at the disk center and the z axis pointing perpendicular into the electrolyte space (Figure 6.1, modified from Wiley and Webster, 1982). disk electrode 0 I T a z electrolyte of conductivity K Figure 6.1: Coordinate system for the analyzing the disk electrode. Laplace equation holds for the electric potential cp(r, ¢, z, t) in the space of the electrolyte r,z ~ 0 (6.1) The second term is zero as the system is axisyrnmetric, and the azimuth ¢ is omitted in all further analysis. The time variable is only specifically shown when the system is time-dependent and its inclusion is necessary for disambiguation. General boundary conditions that apply to all the situations studied include \ l'IJ(r,z)I < +oo <p(r, z) = 0 fz(r,z) = Jo(r) fz(r,z) = 0 r ~ O,z ~ 0 .J r2 + z2 --7 +co z=O+,r~a z = 0,r >a (6.2) 162 where ] 0 (r) and ] 2 (r,o+) denote the current density trough the electrode surface and inunediately above the surface in the electrolyte. These conditions state that the potential within the electrolyte is finite, the ground is at infinity, the current density on the electrode surface is continuous with that in the electrolyte, and the insulator has no current flow across its boundary. The above boundary conditions do not directly relate the potential <p (r, z) in the electrolyte with the voltage or current input applied to the electrode. To complete the description of the disk electrode, the voltage V is related to the potential right above the electrode surface 'Po (r) = <p(r, o+ ). The difference between the two is defined as the surface overpotential V 0 p(r) = V - <p 0 (r) , r :Sa (6.3) The surface overpotential is related to the double layer charging and Faradaic reactions on the electrode surface, which both occur within a very thin layer (ideally of infinitesimal thickness) (Figure 6.2). Assuming the current densities are small and not limited by the diffusion layer, the concentration overpotential is neglected from the overpotential (Newman, 1966a). The current density due to the charging or discharging of the double layer is a(V 0 p(r)) foL(r) = Y at , T :Sa · (6.4) with y being the double layer capacitance per unit area. And the current density of the Faradaic reaction is fF(r) = gFVop(r) , r :Sa . (6.5) where 163 (6.6) relates to the kinetics of the Faradaic reaction with linear approximation of the Butler-Volmer equation for small surface overpotential V 0 p. The parameters aa and ac are the anodic and cathodic charge transfer coefficients, respectively, j 0 is the exchange current density, n is the number of electrons involved in the reaction, F is the Faraday constant, R is the universal gas constant, and T is the absolute temperature. 0 I r z 0 z =O . L aye r of double layer capitance and Faradaic Reactions J. z =o+ Figure 6.2: The current density components on the the surface of the electrode. Together, the current densities of the double layer and Faradaic components are part of the boundary condition of current continuity iJ<p(r,z) Jo(T) = JoL(r) + }F(r) = Jz(T,Z) = -K i)z , Z = 0+, T:::; a , (6.7) Combining (6.3)-(6.5) into (6.7), the electrode voltage V and the electrolyte potential <p 0 (r) are therefore related by J ( ) - a(v - ({Jo(r)) (v ( )) iJ<p(r,z) 0 T - y ~ +BF - (/Jo T = -K , z = o+, T :::; a ut i)z (6.8) The total current passing through the electrode, whether directly applied to the electrode or as 164 a response to an applied voltage input, is given by I(t) = Laj 0 (r, t)2nrdr (6.9) Thus, the disk electrode system becomes solvable given (6.1), (6.2), (6.8) and (6.9). 6.3 General Form of Solution 6.3.1 Rotational Elliptic Coordinates To decrease the complicity of the equations and boundary system, the rotational elliptic coordinates are introduced { r =a. -./(1 + (2)(1-172) z =a·(·17 with 1J E [O, 1] and ( E [O, +oo ). Boundaries and the axis of symmetry are related by l z = o+,r :Sa i( = 0+,17 E [0,1] z = 0,r >a 17 = 0,( E (O,+oo) )r2 + z2 __. +oo ~ (--> +oo, 17 E [0,1] · r = O,z 2 0 1J = l,( E [O,+oo) (6.10) (6.11) These line and other equal ( lines and equal 17 lines are shown in the following Figure 6.3 (modified from Newman, 1966a). Laplace equation in rotational elliptic coordinates with axial symmetry is (6.12) Current density right above the electrode would be given by lo(r) = -K aq;(r,z)I = - K aq;(17,()I az z=o+ a17 a( <=o+ r :S a or 17 E [0,1] , (6.13) with 1Jlz=o+ = )l-(r/a) 2 , r E [O,a]. 165 On the other hand on the insulator base iJrp(r,z)I K iJrp(rJ,01 fz (T,0) =-JC = - = 0 iJz z=O a~ iJri ii=o with ~lz=O = ,/ (r /a) 2 - 1, TE [a, +oo). -- - --- I I I I I I I I I , ¢> '•o 1 ~ I . o• I I \ \ \ \ \ \ ~ \ . ' 0 \ ".s;> \ T > aor~ ~ 0 , ---- --- ... 17'-~.;-- '-- ..... '... ..... .... "" .... '... J-40 " ........ . " ... ~ ... o'' ·6 ' Figure 6.3: The disk electrode and electrolyte space in rotational elliptic coordinates. Integration on the disk surface in rotational elliptic coordinates ( ~ = 0) is given by (6. 14) (6. 15) which would be used to calculate current density or charge density on the electrode·eleclrolyte interface. 166 6.3.2 General Form of Solution Using separation of variables, the potential in the solution is set to <pery,n = Nery)Men . (6.16) The Laplace equation then becomes two ordinary second order differential equations d [ dNery)l dry el - ry 2 )~ +},,Nery)= 0 (6.17) d [ 2 dMff)] _ d( ei+( )~ -},,Men-o . (6.18) The solutions are Legendre functions for ( 6.17) and Legendre functions with imaginary argument for (6.18). Let A= zez + 1), then { N1 ery) = crP P1 ery) + c~QQ1 ery) M1en = c['1P P1eUO + c~QQ1eUO (6.19) with P 1 ery) and Q 1 e11) being the Ith order Legendre functions of the first and second kind, respectively. The boundary conditions (6.2) in the new coordination system give the following conditions lqiery,()I < + 00 - K aqiery, ()I = o a( ary ry~O lim qiery, () = 0 (---';+oo => IN1ery)I, IM1e()I < +oo => dP1ery) I = 0 dry ry~O => l = 2n, n E ru 0 (6.20) => lim M2ne() = 0 {---cl+oo which apply to all situations. For M 1 en, the Legendre functions are evaluated on the imaginary axis. The complex coefficients ck1,;' and c!::,Q are chosen so that M 2 n en is a real function on ( E [O, +oo) with the first condition in (6.20) satisfied and normalized so that M 2 n(O) = 1. See Appendix A(section 6.9) for details on the Legendre functions and derivation for M 2 n(0. 167 Combining and renaming the coefficients, the general solution can be written in a form of summation +oo +oo <p(17,() = L N2n(17)M2n(() = L BnP2n(17)M2n(() (6.21) n=O n=O In this summation, P 2 n(~)M 2 n(0 are dimensionless and normalized eigenfunctions of the disk electrode system. The value of coefficients Bn, typically in units of Volts, will be determined for each specific input voltage/current applied to the disk electrode by the equivalent form of boundary condition (6.8) in rotational elliptic coordinates. The "potential" field in the electrolyte are given for a few eigenfunctions in Figure 6.4. Due to the distortion of linearity in rotational elliptic coordinates, the coordinates are converted back to the original cylindrical coordinates in most figures of the following chapters. The disk electrode is shown in black in the figures with the flanking insulator substrate in white. The influence in the aqueous medium is determined by M 2 n (~) of each eigenfunction. Almost all non-trivial behaviors of the solutions will be limited within very close proximity-less than one times the radius-of the electrode surface, except for the zeroth solution which decays slower and extends its influence to about one order of magnitude further into the solution. Besides penetrating its field into the medium, the zeroth eigenfunction is critical in delivering the current. Notice that P 0 (TJ) = 1, and (6.22) Therefore the total current through the electrode is only dependent on the zeroth term 168 (6.23) Eigenfunction n = 0 Eigenfunction n = 1 3 6 0.5 <::! 2 --... 0.5 .. ~ "Si ·s 0 :r:: 1 0 0 -0.5 -0.5 -2 -1 0 1 2 6 Radial position r/a - 0 -6 -4 -2 0 2 4 Radial position r/a Eigenfunction n = 4 Eigenfunction n = 9 1.5 <::! --... .. 1 0.5 0.5 fo ~ 0.5 0 0 0 -0.5 -0.5 -1 -0.5 0 0.5 -1 -0.5 0 0.5 Radial position r/a Radial position r/a Figure 6.4: The dimensionless and normalized field distribution of some eigenfunctions (n = 0, 1, 4, and 9) in the electrolyte space. The black bar on the bottom of each graph represents the disk electrode. The field of zeroth eigenfunction penetrates deep into the electrolyte space, while higher order eigenfunction are limited to the proximity of the electrode surface. The surface "potential" and "current density" distributions of the first 11 eigenfunctions are shown in Figure 6.5. The zeroth term indicates a steady state of uniform "potential" distribution on the electrode surface. All the other higher order terms will determine how much the actual distribution will deviate from this uniform distribution without altering the mean value. Interestingly, the current distribution of higher order solutions have higher magnitude at both the center and the periphery of the disk, while the lower order ones mainly have high current density at the periphery. 169 "Potential" and "current density" of eigenfunctions on electrode surface (dimensionless and normalized) ~ ~ c c " 0.5 0 ' -0.5 n 10 0 0.2 n II 0.4 0.6 0.8 Badia! position r/a ~ ·-. 15 n 111 f " " 10 ;;..;' ~ 5 " " s 0 ri..' -~' -5 c : -10 ~ ~ -15 0 0.2 0.4 0.6 0.8 Badia! position r/a Figure 6.5: The ''potential" and "current density" distribution on the the surface of the electrode of the first 11 eigenfunctions. The "current density" has a singularity at the edge for each eigenfunction. 6.3.3 Symbol Naming In the coming sections, the follo.....ring convention has been adopted for variable naming, superscripts and subscripts, which also applies to the analysis above. Table 6.1: List of symbols, subscripts, and their meaning <p(r,z) 'Po(r) f(r,z) fo(r) v Potential in the electrolyte space Potential in the electrolyte immediately above electrode: <fJo(T) = cp(r, o+) Current density in the electrolyte space: f (r, z) = i · f z(T, z) + f · fr(r, z) Current density through the double layer of the electrode: fo(T) = f z(T, o+) Voltage on the electrode's metal surface (z = o-) Amplitude of voltage step input, or primary voltage response to current step input Total current flo.....ring across electrode-electrolyte interface Amplitude of current step input, or primary current response to voltage step input 170 Table 6.2: List of superscripts and their meaning None General situation, e.g. <p(r, z) P Value corresponding to primary distribution, e.g. <pp (r, z) H Amplitude of harmonic oscillations, e.g. fz (r, z, t) = JJ! (r, z)eiwt SS Steady state solution, e.g. ]gs (r) TZ Transient solution, e.g. VTZ (t) (i) ith eigensolution of the transient response, e.g. B~i) 6.4 Primary Distribution This solution was given in (Newman, 1966b) to disk electrode without considering the surface overpotential related to the double layer and reaction currents (infinite large double layer capacitance or reaction conductance depending on type of input). The potential and current distribution is completely determined by tbe ohmic drop in the electrolyte. It is the primary distribution at t = o+ for a disk electrode applied with step input of current or voltage. The potential on tbe electrode is constant at V 0 , and the boundary condition of the electrolyte potential immediately next to the electrode is <{Jb (17) = Vo , 1J E [0,1] (6.24) indicating that (6.25) which is also obvious from the analysis for (6.23). The solution is the zeroth eigenfunction (6.26) 171 With the boundary condition, the coefficient B/j = V 0 . The solution could be written in several equivalent forms: <pp (ry, 0 2 1 2 2 1 = -arctan(<') = 1 - -arctan(O = -arcsin( ) Vo TI ~ TI TI .J 1 + (2 The current density on the disk and total current are therefore I 0 = 4KaB/j = 4KaV 0 The effective series resistance of the aqueous medium is V 0 1 R =-=- s I 0 4Ka The surface current density can also be given in the cylindrical coordinates as PC ) 2 KVo lo r = Tiya2-r2 2..jl - (r/a) 2 with the average current density given as - I 0 4KV 0 fo=-=-- na2 na (6.27) (6.28) (6.29) (6.30) (6.31) (6.32) Two characteristic quantities are introduced for normalization in the calculations of frequency dispersion and current and voltage step input. The average access conductance as seen from the electrode surface ( ( = o+) to infinity ( ( --> +oo) is _ Gs 1 4 K g-------- - Tia2 - Tia 2 R - TI a s (6.33) 172 I~ - - !> a.- "' >. - - 1 ·,5 .,, ~ ':"' ·3 -a 1 E ~ ~ 0.5 0 0 0.2 O,<f 0,6 0,8 Figilrt 6.6~ Primaly · cut•mt dist1ij)ution• . wya = - - ·- · "'* K. (6.'3'4) with the·t9tal doub.'le lay :er· capa. citance seen at s~ead ,y .stat~ , given -as (6,.35) 'Ihe cl1arac.t~ristit time constant wouldhavetypieal num, bers in the taqge of mict'oseeonds· to milliseconds dep.endin ,g .on 1he siz. e and material of the electr9de,. as well as-· the eleetr.olyte . .As will be sem i'n tlwnext set\tilit1,. strictly speal<lng the time ·constant is . frequency dtp. enden,t. as b.t>th the resistance. and ciipacitance are. Also local charging and discharging :tates diffef' ftmn the global process,. givirt'g rise· ~o lot al time constan~s. The validity of the overall time • ct>'n"Stan.t is discussed 173 by Oldham (2004), and T represents the average local time constant weighted by both area and current density. 6.5 Frequency Dispersion With AC input on electrode, the current can pass from the electrode to the electrolyte by either double layer capacitance or Faradaic reaction. The overall equivalent circuit can be modeled by a capacitive interface (sometimes characterized as a constant phase element in electrochernistry) and resistor in series, which are both dependent of the frequency. In this model, there is no separate component for the F aradaic conductance, and its effect is grouped into the resistive part of the circuit. The electrode potential is given as (6.36) and the potential in the electrolyte is (6.37) For simplification, two dimensionless quantities are introduced. The dimensionless frequency nya fl= TUJ = --UJ 4 K and the dimensionless Faradaic conductance G =BF g (6.38) (6.39) The dimensionless Faradaic conductance will also be used in the analysis of transient response 174 to current and potential step inputs. In practice, typical Faradaic reactions have much smaller conductance compared to the electrolyte conductance. In the Randles model for electrodes, the parallel resistance of the Faradaic reactions is in the megaohm to gigaohrn range, compared to kiloohrn values of the series resistance of the electrolyte. This gives several orders of magnitudes difference between the two quantities and hence G is typically very small (G < 0(10- 3 )). However calculations for larger G are still performed to identify the general trend. With these quantities, the boundary condition (6.8) on the electrode therefore becomes auH(ry,()I __ 4(in + c) ( _ He )) ai: - TI "// 1 Uo "// ~ <=D+ (6.40) Obviously UH (ry, (} also satisfy the Laplace equation and the same boundary conditions ( 6.2) as <p(ry, ().Therefore +w uH(ry,() = L Bi!P2nC"f/)M2nCD (6.41) n=O and ( 6 .40) is rewritten as H , _ 4(ifl + G) H +w ( +w ) ~ Bn M2n (O)P2n (ry) - - TI "// 1 -~ Bn P2n (ry) (6.42) The coefficients Bi( are functions of the input frequency ( w or fl). To determine their value, especially the zeroth one B~, the equation is multiplied by P 2 m ("//) and integrated with respect to "// over 0 to I. Utilizing the orthogonality of Legendre polynomials, this results in an infinite set of equations for Bi( H , Dmn 4(ifl + G) H +w 1 ( +w ) ~ Bn M2n(O) 4 m + l = - TI l "// 1-~ Bn P2n("//) P2m("//)dry (6.43) which can be written in matrix format 175 where BH = [B~,Br,-·-,Bf,-··f,and MH is a diagonal matrix H . (TIM~m(O)) M = d1ag 4(4m + 1) , m E N° and AH and A~ are matrixes defined as with AH - [A ] - m,n +oox+oo AH - [A ] 0 - m,O +ooxl m, n E N° m E N° Am.n = An.m = fo"r1P2m (17)P2n (17)d17 P2n (0) A - A - ------- o.n - n.o - 2(2n - l)(n + 1) m, n E N° n E N° See Table 6.4 and Table 6.5 in section 6.10 for the numeric values of these matrices. Solving the coefficient of the eigenfunctions gives H ( H MH )-1 H B = A - (G + ifl) A 0 (6.44) (6.45) (6.46) (6.47) (6.48) which is practical only with the matrix indices truncated to a fmite number nmax of rows and/or colunrns (see section 6.8 for discussion on the accuracy of the solution). With B~ calculated, the AC current through the electrode is given according to (6.23) as (6.49) The complex impedance of the electrode-electrolyte system is therefore (6.50) with 176 Rr me(s~) Rs IB~lz (6.51) from comparing (6.50) with (6.30) and (6.35). Hence the frequency dispersion could be obtained. Figure 6. 7 shows the normalized impedance spectrum of the disk electrode, for a few selected values of G. The frequency dispersion of resistive and capacitive impedance are also shown in Figure 6.8. The capacitive impedance peaks at fl= G With all Bi( (n 2 0), the potential and current density distribution are given in Figure 6.9. For fl--> +oo, the distributions converge to the primary distributions regardless of G, a trend which is already seen for fl= 10, and would be evident for larger fl. For fl= 0, the distributions are the same as the steady state response for voltage step input (shown in section 6.7). For G--> +oo, the distributions converge to the primary distributions regardless of fl. 6.6 Transient Response to Current Step Input The transient response to current step input l(t) = 1 0 • u(t) (6.52) applied to the electrode has the same boundary condition (6.8) as in the sinusoidal situation. The dimensionless Faradaic conduction G (6.39) is again utilized to simplify the equation later. 6.6.1 Decomposition of the Solution The solution can be given as a decomposition of a steady-state response and transient contribution, 177 :i; :r. -30 -[ Impedance spectrnscopy of disk electrnde 100 :'llonunli:i:t-'cl frt-'C)llt-'lH:y ~ ~ Figure 6.7: Impedance spectrum of the disk electrode for different Faradaic conductance values. ~· . ~._ 10 1Uu!.====i~=tr===~~5.;i;,;;;==-.~J ;::;· ~o'~-----~-----~ ~ .::> '0 1 '-' ~ ..s -~ '00 :; ~ 0. "' ..::: 10·1 ~ :.::I a i=! ?. 10·:.i ...._ _____ _._ _____ _, 10·<! 10u :Xo:·1ualiil'cl :·r<'<P<'t11:y 0 1 a:t 10·:t , Ou ='l<•ru111~~<·cl fn·q11:·uc:.v !? Figure 6.8: Equivalent resistive and capacitive impedance of the disk electrode for different Faradaic conductance values. 178 G-111 '$. 0.8 --- "1:}'. 0.61,::--:--------~ G-1 0.2 G-ll.l G-ll o~========================.......l 0 0.2 0.4 0.6 0.8 n 0.1 2.5 ~--~--~--~--~--~ G - ll &. G - ll.l o~--~--~---'---~--~ 0 0.2 0.4 0.6 0.8 Radial position r/a Ra<lial posit.ion r/a G- lll '$. 0.8 --- G- l ~ t:::===-~----- "1:}'.0.6 G- ll "2 ~ 0.4 c ,::., 0.2 o._ __ .._ __ _._ __ _._ __ _._ __ __, ~- 0.8 ~ "1:}'. 0.6 "2 ~ 0.4 c ,::., 0.2 0 0.2 0.4 0.6 0.8 Ra<lial posit.ion r/a G-ll&.G-1 G- lll o._ __ .._ __ _._ __ _._ __ _._ __ _. 0 0.2 0.4 0.6 0.8 Ra,Jial posit.ion r/a n 1 2.5 ~--~--~--~--~--~ 112..= <_ 2 ~ ~= -~' 1.5 ~ ~ .Q .. ;;;,,I.~ .................................. .. d G l ~ ~'----~------- 2 0.5 G- lll c; o~--~--~---'---~--~ 0 0.2 0.4 0.6 0.8 Ra<lial posit.ion r / a n 10 2.5 ~--~--~--~--~--~ I~~ 2 ~ ~= -~' 1.5 ~ CJ '"C ~ ~ ~----~ a o.5 c; G- lll o~--~--~---'---~--~ 0 0.2 0.4 0.6 0.8 Ra<lial position r/ a Figure 6.9: The potential and current density distribution on the the surface of the electrode for different frequencies and Faradaic reaction conductance. <p(r,z, t) = <pss(r,z) · u(t) - <pTZ(r,z, t) (6.53) in which the former includes the contribution of the entire input current which stays constant 179 (I (t) = I 0 = Iss), and the later represents the shift from the initial condition to the steady-state solution and doesn't contribute to the current input on the electrode. The initial condition immediately after the input onset ( t = 0 +) is easily shown to be identical to the primary distribution, therefore the current and potential values I 0 and V 0 for the primary distribution are used as normalization factors for the solution afterwards. This gives the convenience to set the coefficient of the zeroth term in the steady-state solution SS +oo <fJ (17, () = '\' 8 ss p ( )M (') Vo L n 2n 17 2n .., n=O (6.54) to unity ( BijS = 1 ), as the total current input is always the same during the shift. The voltage on the electrode as well as other variables could also be decomposed in the same manner V(t) = vss · u(t) - vTz(t) (6.55) 6.6.2 Steady State Response Analyzing the general boundary condition ( 6. 8) for the steady state gives a<pss (17, n I = - 4G (vss - SS ( )) 8' TI 1J </Jo 1J ~ s=D+ (6.56) Utilizing the same method for the frequency dispersion problem, the equation is multiplied by P 2 m(1'/) and integrated with respect to 1J over 0 to 1 after substituting (6.54) into (6.56): ( +w ) vss 1 nM' (O)Bss - - -- A BSS - 2m m V: - A L m.n n 4G ( 4m + 1) ' O m,O n=O (6.57) For m = 0, with BiJS = 1 and Ao.a = 1/2, the condition (6.57) becomes ( +w ) +w vss 1 nM' 0 Bss 1 -=- '\'sssA - oC) o =1+-+2'\' A sss V: A L n o.n 4G G L o.n n o 0,0 n=O n=l (6.58) 180 For m E ru+, (6.58) is substituted to the left side of(6.57), and gives +w '\:""' [ TIM~m (O)Omn] ss Am.a L Am.n -2Am.o ·Ao.n - 4 G( 4 m + l) Bn = G · n=l (6.59) Let Bss = [B,Ss, Bis,···, B~s, ··· r the group of equations of (6.59) can be written in matrix form (c(A- 2AoAl) -M)BSS =Ao where A, A 0 and M are matrixes defined as A= AT= [A ] m,n +oox+oo A - [A ] 0 - m,O +ooxl . ( nM~m (0) ) M = d1ag 4(4m + 1) m,n E ru+ m E ru+ m E ru+ (6.60) (6.61) (6.62) which are submatricies of AH, A~, and MH, respectively (See Table 6.4 and Table 6. 5 in 6.10 for the numeric values of these matrices.). Solving the coefficient of the eigenfunctions of the Laplace equation with the matrix indices truncated to a fmite number nmax of rows and/ or columns gives (6.63) and therefore the steady state solution of the electric field is obtained via (6.54). (See Table 6.6 in 6.10 for the numeric values of sss.) The potential and current density distributions are shown in Figure 6.10. For G --> +oo, the steady-state distributions converge to the primary distributions, while for G = 0, the current density is uniform. 181 " cr.= " c c JS " " c "' c ~ ~ c c 1.5 ~-----~----~ G- ll G G lll ·································································~ " "" 0.5 c ] 6 c " o~~~~~~~~~~~ 0 0.5 I~adial position r/a I~ 2.s ~-----~-----, ·-. " " c ;..., ~ ·g c "' " c ~ " 1.5 0.5 G- ll l G-1 ~ G- lll c 0 ~-----~-----~ 0 0.5 I~adial position r/a Figure 6.10: The steady state potential and current density distribution on the the surface of the electrode for different Faradaic reaction conductance. The steady state voltage is given according to (6.58) as (6.64) The steady state voltage is shown in Figure 6.11 as a function of the Faradaic conductance, and the solution indicates that a steady state voltage cou Id only be reached on the electrode if G > 0. In the special case with an ideally polarizable electrode, where G = 0, the solution 1s unbounded. Intuitively, the current could only pass through the system by constantly charging the double layer capacitance. While the current density and potential distributions in the electrolyte will reach an asymptote, the voltage on the electrode will continue to grow linearly. The current on the interface will shift from the primary distribution to a uniform one over time, and the boundary condition (6.8) becomes 182 = :Xorruulizc•J 1i'arudfilc· cnudur: 111JJce U fi'gure 6.11: The· stead~ .state llol~g e-of the· e l ec,trO'd.e for cli ffer~t Far· adai c reaotlO'Q'. C :Oll c!uctanee. The coeffi: c1 ents· .oou1CI be: oaletilare el as or (d.Ji7) which ~0\1111 aJ so b:e. dit-eo tiy o btain:ell from: (6 . .6 3) "Vtth (] -t .P:. Tlre:1rotentiill on the e 1e: ctro de will conb nUHo ,groW' 1itrear 1 y as 1011g as tiie. step 11\pl1t 1s mairitained, and its ,. steady-state" cmilc t be 1 0 ·t - vss (t) = __ + qigs CoL where the average potential in the solution above the electrode is <i'~s = ( 2na 2 fo 1 ~ B~ 5 P 2n (17) 17d17) /(na 2 ) = 1 + 2 ~ B~ 5 An.o +w 4 _ _ T _ 1 _ '\' (4n + l)P 2 n(O) _ ~ - 1 2AoM Ao - L (2n - 1)2(n + 1)2 3n2 . n=O Therefore, the "steady-state" voltage on an ideally polarizable electrode is given as v 55 (t) t 32 t 32 --=--+-=-+- Vo Rs CoL 3n 2 T 3n 2 6.6.3 Eigensolutions of the Transient Response (6.68) (6.69) (6.70) The transient response doesn't contribute to the current input, and only redistributes the potential and current density throughout the entire space from their initial state to the steady state. Because the boundary condition (6.8) is a first order differential equation in terms of time, the solution could be assumed to be exponential decays of certain spatial eigensolutions using the method of variable separation TZ r+w 1 qi (r, z, t) . _ ' . V: = LcC'le ~uC'l(r,z) ·u(t) 0 i=l (6. 71) where rCO is the time constant of the decay, cCO is the coefficient for the decay, and uCil(r,z) is the spatial distribution of the field associated with the ith decay. Switching to rotational elliptical coordination and introducing the dimensionless time and dimensionless eigenvalue t 4K 8=-=-t r nay (6.72) 184 (6.73) the solution to the potential in the electrolyte is (6.74) with uCO (ry, () given as +w urn(ry,() = L B~i) P2nC"f/)M2nCD (6.75) n=l The transient part of the voltage on the electrode can be given as (6.76) with equal coefficients cCO for i > 0. This is possible as the spatial distributions uCO (ry, () could be scaled by their coefficients B~i). These terms correspond to the charge redistribution of the double layer capacitors through the electrolyte (Figure 6.12). The zeroth term is unique to the electrode voltage, and represents the local charge redistribution via Faradaic reaction with time constant rC 0 l = r/G (equivalent eigenvalue t.C 0 l = 0, with eigensolution for potential and current density in electrolyte equal zero). If there is no Faradaic reaction (G = 0), then cC 0 l = 0. For each transient decay, the general boundary condition (6.8) becomes (i) I au (ry,() = 4ry Aco (i - ucoc )) ai: TI O "// ' ~ <=D+ (6.77) with B~i) = 0 from the analysis. The specific boundary condition (6.77) then becomes (6.78) 185 ., ., z = O ? .? ? 1 6 ° ) 1 6 °) 1 6 0) z = o+ 1 6'\r) Figure 6.12: Charge redistribution of the double layer via different pathways during the transient. The zeroth eigensolution distribute charge through local Faradaic reaction, while higher order eigensolution distribute charge through currents in the electrolyte. Again, the equation is multiplied by P 2 m(1J) and integrated with respect to 1J over 0to1, giving and +oo '\' (i) - - 1 L Ao,nBn - Ao,o - 2 n=l , m=O , I +oo [ rr.M' (O)o ] . A + Zm mn 3Ci) = A m E ru+ . m,n 4AC 0(4m+1) n m,o , n = l Let nCO = [sCO 3CO ·· · B(i) .. · ]T and (6 80) can be written in matrix format 1 ' 2 ' ' n ' ' · ( A + M ) nCO = A 0 AC0 , (6.79) (6.80) (6.81) with A, A 0 and M already defined in (6.61)-(6.62). And the coefficients could be expressed as (6.82) Substituting (6.82) into (6.79) gives (6.83) 186 Due to the inverse operation of the matrix, (6.83) is a polynomial equation of A(i) with infinite order. For numeric calculation, all malrix:es are lruncated to nmax of rows and colunms, and (6.83) becomes a polynomial equation of order nmax . Hence the first ~ax eigenvalues could be obtained with ascending value (descending value fer rCi)), and the ccrresponding coefficients B(i) can then be chtained from (6.82). (See Table 6.7 in 6.10 for the numeric values As can be observed from ( 6. 77)-( 6. 83), the eigenvalues and spatial distribution of the transient response's eigensolutions are independent of the presence or magnitude of the Faradaic re.action. The "potential" and "current density distributions" of the eigensolutions are shown in Figure 6.13. 30 c..: c..: ..s 20 ..... .... er. c..: 10 "C c ..... ..... c..: c..: "'(; r:: c ~ -10 ~ c..: ~ -20 -30 ........_ _____ _._ _____ __. 0 0.5 1 Radial position r/a Y -4UU 0 0.5 1 Radial position r/a Figure 6.13: The "pctential" and "current density" distribution (normalized and dimensionless) on the the surface of the electrode of the eigensolution of the transient response to current step input. The current density on the electrode is propcrtional to vrz (8) - <p~z (11, 8) or 1 - U~i)(11) as seen from (6.77). The current density of one eigensolution is orthogonal to the potential of 187 another: 1 +oo I f uCO(n) (1- uUl(n))nctn = Iii}'\' TIMznCO) (sC0)2 o ., 0 ., ., ., t.Crl L 4(4n + 1) n O n=l = liiJ (BW)T MB(il t.Crl ' (6.84) which is utilized to obtain the coefficient cCO for the exponential decay. 6.6.4 Transient Response The initial condition after input onset (t = o+) is the primary distribution. For the potential, (6.53) could be evaluated at t = o+ on the electrode surface (z = ( = 0) giving <p (r, 0, 0+) = l = [<{JSS (r, 0) _ <pTZ(r, 0, o+)] · u(O+) Vo Vo Vo +w L ccou~o(ri) , i=l Multiplying by 1 - U~) (ry), j E ru+ and utilizing the relationship (6.84) yields . fo 1 I~':\ B~s P2n (ry) (1 - u~l(11)) rydry cm = -----~---'---~---'--- f01 u~°Cri) ( 1 - u~l Cri)) rictri 1 '<'+oo nM~n (0) B(J) BSS t:Jli Ln=l 4( 4n + 1) n n L+w 8ij L+w nM~n (0) (sC0)2 i=l A (i) n=l 4(4n + 1) n (B 55 )T MBCil (BCil)T MBCi) . (6.85) (6.86) Although eigensolution of the transient components are independent of the Faradaic reaction, the corresponding coefficients and the decay time constants are not, as B 55 is dependent on G. With all the coefficients cCO given for i > 0 (Table 6.8), the spatial distribution of the transient solution is solved. Its potential and current density distributions on the electrode surface at t = o+ 188 are shown in Figure 6.14. Due to accuracy issue in the solution, the constructing the transient response from eigensolutions shows ripples similar to the Gibbs effect (See section 6.8). Therefore, the current distribution was spatially filtered. 0.3 G 0 0. 2 !,:;---;------· G 1 G 111 -0.2 -0.::1 ~-----~-----~ 0 0.5 I~adi.-'ll position r/a G II 0.5 ~'~=::::::=:::::::::::,,,._ G 1 --.._ ~I~ '-' --....:_ c~ c+ ,.s = " 00 .... ] ,; -0.5 ;§ ~= -1 ~-----~----~ 0 0.5 I~adial position r/a Figure 6.14: The potential and current density distribution of the transient response to current step input on the the surface of the electrode at t = o+ for different Faradaic reaction conductance. For the electrode voltage, however, cC 0 l remains to be solved, and could be obtained by evaluating (6.55) and (6.76) at e = 0 { ,, ++ cCol = vv: - 1 - I cCJl 0 j=l 0 (6.87) G=O However an easier way is through analysis of the current density components on the electrode- electrolyte interface. The steady state solution only contains Faradaic current as the double layer is charged to its asymptotic value. On the other hand, the transient solution contains both Faradaic current and capacitive current component that are not equal in magnitude at any location. The initial 189 condition is the primary distribution in the electrolyte at t = o+, and as the overpotential is zero, there is only capacitive current. Thus the initial capacitive current of the transient solution equals the primary current distribution on the electrode a (vTZ(t)-<pTZ(r,z,t)) Y at = -J/;(r, 0) (6.88) which is (6.89) Multiplying both side by "// and integrating over [0,1 ], the summation equals zero for i * 0 utilizing (6.84), therefore yielding 1 ~ C(o) = _ G (6.90) The transient solution is dominated by the eigensolutions of lower order components of A CO, i 2 1 (See Table 6.8 in 6.10 for the numeric values of cCO). As Faradaic reactions rate increases, the coefficients of cW, i 2 1 decreases. This indicates that more transient current runs through local loops versus through the medium. 6.7 Transient Response to Voltage Step Input The transient response to a voltage step input V(t) = V 0 • u(t) (6.91) applied to the electrode is similarly decomposed to a steady state solution and a transient solution. 190 <p(r,z, t) = <p 55 (r,z) · u(t) - <pTz(r,z, t) (6.92) 6. 7.1 Steady State Response The steady state solution for the voltage step input has a different current compared to the initial primary response ! 55 * 1 0 , and the transient response contributes a net current JTZ (t) that results in the difference between the primary current value and the steady state value. Nevertheless the steady state solution of this system could be easily given by scaling the results of the previous calculation for the current step input. Since the voltage is forced to stay at its initial value V (t) = V 0 = V 55 , all the other steady state values ( <p 55 (r, z), /8 5 (r) etc.) will be scaled by a factor (6.93) which is the inverse of the right side of (6.64). The scaling factor is plotted in Figure 6.15 as a function of the Faradaic reaction G. If G = 0, then this scaling factor will become zero, and all the steady state values except for V 55 are also zero as well. This is intuitive as the steady state without Faradaic reaction to discharge the double layer capacitance will result in fully charged capacitance with no current following in the electrolyte. The steady state potential and current density distributions on the electrode surface is shown in Figure 6.16. 6.7.2 Eigensolutions of the Transient Response In the voltage step input situation, the transient response will follow a different course to connect the primary distribution to the steady state, and is the focus of the analysis. Using the same notation in (6.72)-(6.73), the transience is given as 191 ; o -2 "-----~----....._ ____ ..__ ___ __. 10· 2 10· 1 10° 10 1 10~ N11t·mA!Ji11'1 Fl'l.tntlfl.!I' rnml t~"1•rntrn G Figure 6 . . 15: Scalin& factor the sfeady state. resp-Onse '0f the v0ltaae..step input c0rnpared to tliat of the turrenntep input as a foncti ~n 9f F aradai c reaction conductance. - J., 0.8 ·~ ~ 0.2 E \. If) r; _ 1 '1 J ':! Q.. (. = l) O'-~~--'-~~~~~~~---' 0 O .& c :ilt1dial lJt<'iilJtm I ,:" n .... i.il p.,,;jlj.;IJ f"/.; Figure 6.16: The· steady state potential an<l ciJrrent aens1t.,v distrtbi.iti-im on the .. tlre s1irt 'ace• of:tlte: ele~tmde ·for different Faradaic rea~tton con<li.ictance. 192 (6.94) with the summation starting from 0, indicating the net current. Obviously VTZ = 0, and the boundary condition (6.8) is aucocn ')I 4n . . ,,~ +-·' t..COuC'l( ) = o --a-,-- TI O "// ~ s=D+ (6.95) Again, setting +w uW(ry,() = L B~i) P2nC"f/)M2nCD (6.96) n=O yields +w '\:""' [ TIM~m (O)limn l (i) L Am.n + 4A(i)(4m + 1) Bn = 0 , m E N° n=O (6.97) To solve the set of equations, the first coefficient of uCO (TJ, ~) is set to B~i) = 1 for normalization, and the eigensolutions are scaled by the coefficients cCO. Then applying the same technique as in ( 6. 79)-( 6. 82) yields the equations for all the eigenvalues A CO i E l\1° (6.98) and for the corresponding coefficient B~i) (6.99) All eigenvalues, except for A (o), have a counterpart similar in value for the current step and voltage step input problem. The zeroth eigenvalue that is distinctively different from the other eigenvalues becomes non-trial due to the difference on the right side of the equations ( 6. 98) and (6.83). (See Table 6.9 in 6.10 for the numeric values of A CO and BCil). 193 The potential and current density distributions of the eigensolutions are sho\Vll in Figure 6.17. ~ 1 0 '"""'"' 1\\-1\ c ~ ~ c c 7 0 ..... . " c ~ -10 " c c '" -20 ' c c JS " " c "' ' 1 1 c ~ ~ c c " " c -400 ~-----~----~ 0 0.5 0 0.5 Badia! position r/a Badia! position r/a Flgure 6.17: The "potential" and "current density'' distribution on the the surface of the electrode of the eigensolution of the transient response to current step input. The current density on the electrode is proportional to cp~0(17,8) or u~0(17) as seen from ( 6 .9 5). The eigensolutions therefore satisfy an orthogonality described as (6.100) 6.7.3 Transient Response Similar to (6.86), the coefficients cCi) could be determined by the initial condition of the potential. Now that the steady state solution is scaled (B5s = F), the coefficients are given by F - z(Bssf MBCO -1 cCi) = --~~~--~- 1 - 2(BC0)T M BCO (6.101) Or utilizing the same analysis of the current density components, the same conclusion holds for the pctential step input, i.e. the initial capacitive current of the transient solution equals the 194 primary current distribution on the electrode. 8(0-<t>TZ(r,z,t)) Y at = -]/; (r,O) (6.102) which is (6.103) Using (6.100) yields (6.104) (NO+ G)(2(BCO)T MBCO -1) . The spatial distribution of the transient solution thus is solved (See Table 6.10 in 6.10 for the numeric values of cCO .) Its potential and current density distributions on the electrode surface at t = o+ are sho\VIlinFigure 6.18. 0 ] -0.2 " c c~ ::::.-~- -0.4 t ±:::' ,.s - ~ .... -0 6 er. • • -1 G G G G 0 111 ~ 1 ~ () .1 II l 0.5 Badia! position r/a 0.5 G 111 ;..., 0 .-<::::: 1 " , " G () .1 c "' -0.5 ~ G II " c ~ -1 ~I~ '-' --....:_ c ~ c " -1.5 JS " " ~ ct ;.. -2 :oN ·a~= >-< • -2.5 -3~~~~~~~~~~~~ 0 0.5 Badia! position r/a Flgure 6.18: The potential and current density distribution of the transient response to voltage step input on the the surface of the electrode at t = o+ for different Famdaic reaction conductance. Curves were processed after the construction from the eigensolution. 195 With B~i) = 1, the total current is given as (6.105) 6.8 Accuracy of Solution Numeric solution for the frequency dispersion, current step response and voltage step response all involve an infinite set of equations, summarized in the form of matrices. The matrices need to be truncated for the solution to be possible, and therefore may result in inaccurate solutions. To solve the coefficients BH for the frequency dispersion, (6.48) involves inverse operation of a matrix. However, as the terms of A~ on the right hand side of the equation decrease with increasing n, the accuracy of the calculated terms B~ could be guaranteed by setting nmax so that the nth term of A~, i.e. Ao.n is small enough. For the calculation, nmax was set to 200, which has an accuracy of 10- 10 (relative difference when increasing nmax by 1). Also, the coefficients Bi( (n > 0) decay fast in amplitude, allowing accurate calculation of the potential and current density distribution with only a few eigenfunctions. For the current step response and voltage step response, however, the truncation of (6.83) and (6.98) reduces the order of the polynomial equation for the eigenvalues. A truncation of nmax gives nmax eigenvalues, and therefore the accuracy of the solution is reduces. This is further complicated by the complexities and difficulties in solving symbolic inversion of large matrix. To investigate which of the eigenvalues are accurate, the eigenvalues of current step response are plotted from solutions with a few nmax (10, 20, 30, 40, 50, 80) (Figure 6.19). The accuracy could 196 be checked by comparing the same order eigenvalues calculated from different nmax, with a relative difference less than 0.001 % is considered accurate. Apparently, as nmax increases, more eigenvalues become accurate enough to be included for later calculations. The accurate eigenvalues seem to increase linearly, and about 60% of each nmax cou Id be considered accurate, which is an empirical rule. The ratio of accurate eigenvalues could be related to the golden ratio 0.618, however this is only speculation and needs further exploration. Similar results could be obtained for the voltage step response. 300..--~~~~~~--....!~~~~l..-~~~ l"""T""~~~-,-~~~-.-~~-...,. f..-~~---. I • I I I 250 ;: 200 .< c.; ..., ~ 150 6 c.; bC t£i 100 50 I I I ; + i I i i J i ! t f f i ; i i f I + ~ f */ i i ; f ** - i ! + -i- / ----- t + f J / - ... R 2 = O.!l!l!l!l!l!lOl i i .i Jt-f AU>= 0.859 + 2..170 xi j + T ;lf'T .j.. • -;(-+ + Accurate j I + InaccumLc * o lq:'.l.~~--'~~~--L~~~--'-~~~-'-~~~_._~~~-'-~~~"'--~~---' 0 10 20 30 40 Order i 50 60 70 80 Figure 6.19: The eigenvalues of the transient response of the current step input, solved with different size of matrix truncation. Horizontal axis is the order. Markers in blue show eigenvalues considered accurate after comparing the values solved from different size of matrix truncation. Markers in red show eigenvalues considered inaccurate when comparing with those solved by matrix truncated to larger size. The dotted line connect the eigenvalues solved from the same matrix truncation size. The dashed line shows linear regression to the accurate eigenvalues, showing highly linear behaviour. Another aspect of accuracy arises when assembling the potential and current density 197 distribution on the elec1r 0 cle surface fn:irp. the eigens· olutions. 'Tl' ci· s is especially prominent fot' the aunent deiuit;y i!ue to the cli~conti'1luity. 'F(gllfelt20 $ hows ripple effect. toward~ the edge of the .electro.de, muah;Jikethe Gith•s effe. ct in.tlrerecpnsthiction of Fourier series. it toillcl possibly be ret\ucecl by increasing n"'"" frotn smaller wues that. are otlierM'rise slreaily sufficient for the· ~ohntial distribution The cµrrent density distri~uti:ol) v;rere spatially filtere. d to get nd o[ th. e oscillation \vith t! eqiienties« higherthan 10 cytlesp:erunitlength, while. ke. eping in mind thatitleally fhere is a'Singulpi1ry at the"Very edge. 1 . 0 = 0 -2 .._ _ _.._ _ __.__ _ __._ _ _____.. _ _..... I 0 0 .. 2 0.4 06 0.6 H ad1iil pm;1t lnu r / 11 .Rigure6.10: The current d.E:lisify tlistribu.tion ofthe.1ransi.ent response. to iro 1tage-stw iQPtit on. the the stll'fat:e of: the ele. c Jrocle at t = O' for different Faradai:c reaction concluctanc.e: The unprocessed data for the· tmrent density si\Qw ripp1ei in the. etu11e ; indicating inacciuacy dt!e . to a pl\enomenasimilar to the Di.bbs effect 6. 9 Appendix A-Legendre Functions on the Imaginary Axis To solve (6.18), the Legendre functions are extended to the imaginary axis. The Legendre function are introduced, with background knowledge only related to the supplementary materials given. The detailed solution to (6.18) is further discussed, to complete the analysis in 6.3.2. 6.9.1 Legendre Functions Legendre functions are solution to the Legendre equation, a second order differential equation derived from Laplace equation in spherical coordinates (r, cf>, 8). Using separation of variables, the equation for e = arccos x under axial symmetric condition is the Legendre equation d~ [ (1- x 2 ) d:~)l + l(l + l)f(x) = 0 , lxl :S 1 . (6.106) The solution to the Legendre equation is (6.107) with P 1 (x) and Q 1 (x) being the Legendre function of the first and second kind. Generally speaking l could be complex, and usually P 1 (x) and Q 1 (x) are complex and not bounded at the points x = ±1. If l E l\1°, then P 1 (x) becomes a polynomial, and with normalization P 1 (1) = 1: l1/2J ( ) L (-l)m(zl - 2m)! 1 _ 2 Pix= x m 21 · m! (I - m)! (l - 2m)! · m=O (6.108) Q 1 (x) is obtained from P 1 (x) by the method of reduction of order: (6.109) with 199 J x dx u1(x) = (1- x2)?i2(x) (6.110) Q 1 (x) is unbounded at x = ± 1 and could also be written as (6.111) where W 1 ( x) denotes a polynomial of order l - 1. Legendre functions have several characteristics, with the following two important for the disk electrode analysis: • Orthogonality: Legendre functions are orthogonal polynomials over the interval [-1, 1] f 1 2 Pn(x)Pm(x) dx = - 1 --Bnm -1 2 + 1 (6.112) • Recurrent generation: Higher order Legendre functions could be generated by lower order ones (also applies to Q 1 (x) and W 1 (x)): (l + l)P 1 + 1 (x) = (21 + l)xP 1 (x)-lP 1 _ 1 (x) . Starting functions for recurrent generation are given as follows: 1 (1 + x) Q 0 (x) =-In -- 2 1- x (6.113) 1 P 0 (x) = 1 P 1 (x) = x 3x 2 - 1 ' x (l + x) Q 1(x)=2ln l-x -1 1 W 0 (x) = 0 W 1 (x) = 1 ' 3x · P 2 (x)= 2 ( ) (3x 2 - 1) (1 + x) 3x Q 2 x = In -- -- 4 1-x 2 W2 (x) =z The lower order functions are given in Figure 6.21 for even and odd number of l. 6.9.2 Evaluating Legendre Functions on the Imaginary Axis To solve (6.18), the variable ( could be substituted by a purely imaginary one 200 0.5 ,,...... "" ~ 0 ct -0.5 -1 1.5 0.5 + 0 0 -0.5 -1 -1.5 0 -0.5 0 Legendre functions of the first kind 0.5 ,,...... "" ~ t 0 ct -0.5 0.5 -1 -0.5 0 0.5 :r Legendre functions of the second kind 1.5 0.5 + 0 & -0.5 -1 -1.5 -2--~~...._~~-'-~~-'-~~--' -2'--~~-'-~~-'-~~-'-~~--' -1 -0.5 0 0.5 -1 -0.5 0 0.5 :r Figure 6.21: Legendre functions of the first and second kind, plotted separately for even and odd orders. ~ = t(() = i( , (6.114) and then the function M(O satisfies 201 and l M(() = M (t(()) = M(() dM~() dM(~)d~ = idM(() · d( d( d( d( Thereforefor each ll = l(l + 1), substituting (6.115) into (6.18) gives Equation ( 6.116) indicates that M ( () is a Legendre function, hence (6.115) (6.116) (6.117) (6.118) Before the Legendre functions are evaluated on the imaginary axis, the variable is first extended to the complex plane z = x + i · y E IC, and P 1 (z) and Q 1 (z) become complex. With l E l\1°, P 1 (z) is a polynomial, and Q 1 (z) could also be obtained by (6.109)-(6.111). The corresponding regions of convergence (ROC) are z E IC\{z = oo} for P 1 (z), and z E IC\{z = ±1, oo} for Q 1 (z), respectively. The latter could be inferred from the recurrent generation of Q 1 (z) and the expression of the first term Q 0 (z) 1 [(1 + x) 2 + y 2 ] i ( 2y ) Qo(z) = 4 ln (1 - x)2 + yz + 2arctan 1 - x2 - yz (6.119) Modifying the equations ( 6.109)-( 6.111) gives (6.120) and 202 (6.121) The functions P 1 ((fi) and Q 1 ((/i) are bounded on ( E [O, +oo), and unbounded at +oo. l = 0 is the only exception when both functions converge for +oo. The first few functions are given as { Po(Ui) = 1 {Q 0 ((/i) = -iarctan(O P, ((ii)= (ii Q 1 ((/i) = -(( arctan(O + 1) 3 2 +1' i P 2 ((/i)= ( 2 Q 2 ((/i)=z[(3('+1)arctan(0+3fl { Wo(Ui) = o w,((11) = 1 , 3i( W,((fi) = -- 2 With l = 2n, P 2 n ((Ii) is real and Q 2 n ((Ii) and W 2 n ((Ii) are imaginary. For M 2 n (0 to be real, crnQ is imaginary. To satisfy the conditions ofrelevant to M 2 n (~) (6.122) the following relationship must be true: (6.123) The coefficient ck1/ (Table 6.3) could be easily calculated as the reciprocal of P 2 n (0): MP_ 1 = (-l)n(zn · n!) 2 (-1)n(2n)!! c,n - P 2 n(O) (2n)! (2n -1)!! (6.124) The limit for the coefficient crnQ could be calculated from (6.120) as (6.125) Therefore MQ 2i 2i ( -l)n(zn · n!) 2 c,n - --cMP - - - TI 2 n - TI (2n)! (6.126) 203 The functions are shown in logarithmic scale in Figure 6.22. It should be noticed that the decay to zero is very fast even for small n. For example when n ~ 1 the function decays to less than 10- 2 within x ~ 3. Hence the zeroth order of the solution will dominate the middle to far field in the conductive medium, and the electrode would be perceived as a point source from far away. Higher order functions are decaying extremely fast and any contribution would be very limited to the origin of the { axis, i.e. very close to the electrode-electrolyte interface. Latitude component of the general solution 10·6 L~~~-"'--~~~"---~ 10· 3 10' 10' 10' ( Figure 6.22: The solution M 2 n({) on log-log scale, obtained from evaluating the Legendre function on the imaginary axis. The high order function decay very quickly, indicating that they don't influence the field distribution in the electrolyte very much. Only the zeros order solution extends far into the electrolyte space. The derivative of M 2 n ({) at the origin, which is used several times such as for calculating the current density on the electrode surface (6.23) and the matrix MH (6.45), is also calculated utilizing the relationship (6.120) and represents the first term in the series expansion of Q 2 n({ /i) 204 2 czn. n!)4 TI [(2n)!]2 (6.127) For the matrices MH and M, the diagonal elements have an asymptote that could be given by using the Stirling's approximation for large n (6.128) as lim = lim - TIM~n (0) TI ( 2 (2n · n!) 4 ) n~+w 4(4n + 1) n~+w 4(4n + 1) TI [(2n)!]2 . -z•n. 4TI2n2 (~)'n . (6.129) = hm -------'-"-'-~ n~+w (2n)4n 2(4n+1)·2TI·2n e = -TI/8 . (6.129) 205 6.10 Appendix B-Tables of Numerical Calculations Table 6.3: Coefficient for constructing tbe Legendre functions on tbe imaginary axis. The first 50 of c~t are given, where as MQ and M~n(O) could be easily calculated from CMP by using Czn 2n (6.126) and (6.127). n CMP 2n n CMP 2n n CMP 2n 0 1.00000 17 -7.36194 34 10.37316 -2.00000 18 7.57228 35 -10.52349 2 2.66667 19 -7.77694 36 10.67171 3 -3.20000 20 7.97635 37 -10.81790 4 3.65714 21 -8.17089 38 10.96214 5 -4 06349 22 8.36091 39 -11.10450 6 4.43290 23 -8.54671 40 11.24507 7 -4.77389 24 8.72855 41 -11.38390 8 5.09215 25 -8.90669 42 11.52105 9 -5.39169 26 9 08133 43 -11.65659 10 5.67546 27 -9.25268 44 11.79058 11 -5.94572 28 9.42091 45 -11.92305 12 6.20423 29 -9.58618 46 12 05408 13 -6.45240 30 9.74866 47 -12.18369 14 6.69138 31 -9.90848 48 12.31194 15 -6.92212 32 10.06575 49 -12.43887 16 7.14541 33 -10.22061 206 Table 6.4: AH and Ao,o, Ao and A in the form of AH =[Ao.a , Ao A/;] A. 0 2 3 4 5 6 7 8 9 0 0.5000 ! 0.1250 -0.0208 0.0078 -0.0039 0.0023 -0.0015 0.0010 -0.0007 0.0005 ---------!---------~-----------------~-----------------------------------~-----------------~------· 0.1250 0.1250 0.0339 -0.0063 0.0025 -0.0013 0.0008 -0.0005 0.0004 -0.0003 2 -0.0208 0.0339 0.0703 0.0202 -0.0037 0.0015 -0.0008 0.0005 -0.0003 0.0002 3 0.0078 -0.0063 0.0202 0.0488 0.0145 -0.0027 0.0011 -0.0006 0.0004 -0.0002 4 -0.0039 0.0025 -0.0037 0.0145 0.0374 0.0113 -0.0021 0.0009 -0.0005 0.0003 5 0.0023 -0.0013 0.0015 -0.0027 0.0113 0.0303 0.0093 -0.0018 0.0007 -0.0004 6 -0.0015 0.0008 -0.0008 0.0011 -0.0021 0.0093 0.0254 0.0079 -0.0015 0.0006 7 0.0010 -0.0005 0.0005 -0.0006 0.0009 -0.0018 0.0079 0.0219 0.0069 -0.0013 8 -0.0007 0.0004 -0.0003 0.0004 -0.0005 0.0007 -0.0015 0.0069 0.0193 0.0061 9 0.0005 -0.0003 0.0002 -0.0002 0.0003 -0.0004 0.0006 -0.0013 0.0061 0.0172 Table 6.5: MH and M are both diagonal rnatrixes, with MH = [-v 2 ~] - The diagonal elements are given. n M;{,n n M~n n M;{,n 0 -0.50000 11 -0.39280 22 -0.39272 1 -0.40000 12 -0.39278 23 -0.39272 2 -0.39506 13 -0.39277 24 -0.39272 3 -0.39385 14 -0.39276 25 -0.39272 4 -0.39337 15 -0.39275 26 -0.39272 5 -0.39314 16 -0.39275 27 -0.39272 6 -0.39301 17 -0.39274 28 -0.39271 7 -0.39293 18 -0.39274 29 -0.39271 8 -0.39288 19 -0.39273 30 -0.39271 9 -0.39284 20 -0.39273 10 -0.39282 21 -0.39273 i-oo -rr/8 207 Table 6.6: Coefficients B~ 5 for constructing the steady state solution of the current step input. G=O G = 0.1 G=l G = 10 Bis 1.0 1.0 1.0 1.0 B~s 0.31250 0.30592 0.25912 0.11754 B§s -0.05273 -0.05491 -0.06795 -0.07024 B~s 0.01984 0.02055 0.02610 0.04093 B~s -0.00993 -0.01026 -0.01291 -0.02454 B~s 0.00580 0.00598 0.00746 0.01542 B~s -0.00373 -0.00384 -0.00476 -0.01018 B~s 0.00256 0.00264 0.00325 0.00704 B§s -0.00185 -0.00190 -0.00234 -0.00507 Bss 10 0.00139 0.00143 0.00175 0.00378 Bss 11 -0.00107 -0.00110 -0.00135 -0.00290 Bss 12 0.00085 0.00087 0.00107 0.00228 Bss 13 -0.00069 -0.00071 -0.00086 -0.00183 Bss 14 0.00056 0.00058 0.00071 0.00149 Bss 15 -0.00047 -0.00048 -0.00059 -0.00124 Bss 16 0.00040 0.00041 0.00050 0.00104 ---------- ----------------------------------------------~---------------- vss /Vo 11.07922 2.06818 1.13327 208 Table 6.7: Eigenvalues ACO and coefficients sCil n for constructing the eigensolutions of the transient response of current step input. 2 3 6 7 9 10 A_Ci) 3.23686 5.76645 8.26009 10.74212 13.21888 15.69280 18.16501 20.63610 23.10643 25.57620 -----------------~---------------------------------------------------- Beil ' 4.56973 3.77405 3.44403 3.25860 3.13835 3.05343 2.98996 2.94056 2.90091 2.86832 Beil ' 3.58511 -3.70788 -4.65165 -4.79056 -4.75592 -4.67961 -4.59717 -4.51906 -4.44807 -4.38446 BCil 0.51738 -7.51661 -0.26793 2.76529 4.12700 4.79007 5.12852 5.30285 5.38887 5.42520 7 Beil ' 0.10883 -2.89555 9.61985 5.38646 1.50528 -1.06637 -2.72102 -3.80111 -4.52140 -5.01134 BCil 7 -0.03142 -0.67827 6.80910 -8.19370 -8.96686 -6.36996 -3.57332 -1.25152 0.55192 1.92822 Beil ' 0.02274 -0.02899 2.44679 -10.7731 3.13093 8.87179 9.40950 7.93245 5.90871 3.91150 Beil 7 -0.01587 -0.03991 0.44950 -5.76314 12.71019 3.84651 -4.60764 -8.76264 -9.88246 -9.36918 BCil 0.01161 0.02427 0.11317 -1.72889 10.11847 -10.9723 -9.83824 -2.30309 4.13705 8.05674 ' Beil ' -0.00879 -0.01882 -0.02225 -0.42934 4.33792 -14.0743 5.29853 12.04577 8.89869 2.86824 Beil rn 0.00684 0.01470 0.02444 -0.02855 1.34838 -8.32077 15.72582 2.78725 -9.11201 -12.08159 BCil -0.00544 -0.01174 -0.01900 -0.03497 0.25080 -3.31803 12.97479 -13.5525 -10.4004 1.88896 n Beil " 0.00441 0.00955 0.01553 0.02140 0.07855 -0.89858 6.63451 -16.8432 7.28305 14.42925 Beil B -0.00364 -0.00789 -0.01285 -0.01847 -0.01763 -0.23866 2.34443 -11.13101 18.13177 1.66850 BCil 0.00304 0.00660 0.01078 0.01545 0.02133 -0.01123 0.69463 -4.95840 15.96042 -15.44917 " Beil " -0.00257 -0.00559 -0.00914 -0.01313 -0.01740 -0.02767 0.12123 -1.73843 8.88252 -19.64680 Table 6.8: Coefficients cCiJ of the eigensolutions of the transient response of current step input. n G=O G = 0.1 G=l G = 10 cCoJ 100 1.0 0.10 ------ -------------------------------~--------------------------------- cClJ 0.03692 0.03582 0.02821 0.00903 cC2l 0.01356 0.01333 0.01156 0.00496 cC3) 0.00710 0.00702 0.00634 0.00321 cC•l 0.00438 0.00434 0.00401 0.00227 cCsJ 0.00298 0.00296 0.00277 0.00170 cC6) 0.00216 0.00215 0.00203 0.00132 cC7J 0.00164 0.00163 0.00155 0.00106 cCsJ 0.00129 0.00128 0.00123 0.00087 cC9) 0.00104 0.00103 0.00099 0.00072 cC10J 0.00085 0.00085 0.00082 0.00061 209 Table 6.9: Eigenvalues A (i) and coefficients B~i) for constructing the eigensolutions of the transient response of voltage step input. 0 2 3 6 7 9 Ji.Cil o.90931 3.39041 5.85921 8.32702 10.79460 13.26209 15.72954 18.19698 20.66440 23.13182 B~;J i.o i.o i.o i.o i.o i.o i.o i.o i.o i.o B~;J o.39451 -3.30704 -3.20144 -3.08673 -3.00260 -2.94030 -2.89258 -2.85489 -2.82433 -2.79903 B;il -0.01974 -3.09446 2.69232 3.87544 4.20749 4.29990 4.30959 4.28807 4.25489 4.21818 B;n 0.01259 -o.s2so2 6.45944 o.65745 -2.15584 -3.53763 -4.26428 -4.66685 -4.89691 -5.02954 Bdil -0.00657 -0.10223 2.64610 -8.32546 -5.09803 -1.69133 0.10001 2.29885 3.37463 4.11142 B;n 0.00393 0.02410 o.63787 -6.16120 7.06425 s.21140 6.07128 3.56954 1.41585 -0.29728 B~;J -0.00256 -0.01843 o.03554 -2.21oso 9.75696 -2.49614 -8.00976 -8.76740 -7.55547 -5.74426 B;il 0.00118 0.01289 o.03502 -0.43176 5.33232 -11.52214 -3.84585 3.99848 8.05051 9.27630 B~;J -0.00129 -0.00946 -0.02056 -0.10618 1.62963 -9.36729 9.90614 9.29270 2.43707 -3.64253 B~;J 0.00097 0.00118 o.01605 o.01863 0.40730 -4.07287 13.04355 -4.63473 -11.23179 -8.54562 B~2 -0.00015 -0.00559 -0.01255 -0.02158 o.02998 -1.27762 7.80961 -14.57467 -2.89008 8.37870 B~2 0.00060 o.00446 0.01003 o.01672 0.03202 -0.24198 3.14134 -12.18478 12.51967 9.96900 B~;l -0.00043 -0.00362 -0.00817 -0.01368 -0.01915 -0.07429 o.85897 -6.28195 15.82526 -6.60770 B~~l 0.00040 0.00298 o.00675 0.01133 o.01661 o.01564 o.22817 -2.23599 10.54522 -17.02880 B~2 -0.00033 -0.00250 -0.00565 -0.00951 -0.01390 -0.01951 0.01206 -0.66515 4.72715 -15.12894 B~2 0.00028 0.00211 0.00479 0.00807 0.01181 o.01587 o.02580 -0.11780 1.66486 -8.47037 Table 6.10: Coefficients cCO of the eigensolutions of the transient response of voltage step input. n c(l) cC2J cC3) c(4) cCS) cC6) cC7) [CB) cC9) cC10J G=O -0.88889 -0 05700 -0 01865 -0 00914 -0 00541 -0 00357 -0 00253 -0 00189 -0 00146 -0 00116 -0 00095 G = 0.1 -0.80082 -0 05537 -0 01834 -0 00903 -0 00536 -0 00354 -0 00251 -0 00188 -0 00145 -0 00116 -0 00095 G=l G = 10 -0.42334 -0.07409 -0 04402 -0 01443 -0 01593 -0 00689 -0 00816 -0 00415 -0 00495 -0 00281 -0 00332 -0 00203 -0.00238 -0 00155 -0 00179 -0 00122 -0 00139 -0 00098 -0 00112 -0 00081 -0 00091 -0 00068 210 References Ahuja, A K. An in vitro model of a retinal prosthesis. Ph.D. Thesis. University of Southern California, Los Angeles, CA, USA, 2007. Ahuja, A K., Behrend, M. R., Kuroda, M., Hurnayun, M. S., and Weiland, J. D. An In Vitro Model of a Retinal Prosthesis. IEEE Transactions on Biomedical Engineering, 55(6), pp. 1744-1753, 2008. Ahuja, A K., Dorn, J. D., Caspi, A, McMahon, M. J., Dagnelie, G., DaCruz, L., Stanga, P. E., Hurnayun, M. S., Greenberg, R. J., and Argus II Study Group. Blind subjects implanted with the Argus II retinal prosthesis are able to improve performance in a spatial-motor task. British Journal of Ophthalmology, 95(4), pp. 539-543, 2011. Arneri, H., Ratanapakorn, T., Ufer, S., Eckhardt, H., Hurnayun, M. S., and Weiland, J. D. Toward a wide-field retinal prosthesis. Journal of Neural Engineering, 6(3), pp. 035002, 2009. Astr6rn, M., Lemaire, J.-J., and Wardell, K. Influence of heterogeneous and anisotropic tissue conductivity on electric field distribution in deep brain stimulation. Medical & Biological Engineering & Computing, 50(1), pp. 23-32, 2011. Ayton, L. N., Blarney, P. J., Sinclair, N. C., Shivdasani, M. N., Petoe, M. A., McCarthy, C., Barnes, N., Allen, P. J., Luu, C. D., and Guyrner, R.H. Preliminary Results of the Bionic Vision Australia Suprachoroidal Visual Prosthesis Pilot Trial. Investigative Ophthalmology & Visual Science, 55(13), pp. 1813-1813, 2014. Ayton, L. N., Guyrner, R. H., and Luu, C. D. Choroidal thickness profiles in retinitis pigrnentosa. Clinical & Experimental Ophthalmology, 41(4), pp. 396-403, 2013. Bak, M., Girvin, J. P., Hambrecht, F. T., Kufta, C. V, Loeb, G. E., and Schmidt, E. M. Visual sensations produced by intracortical rnicrostirnulation of the human occipital cortex. Medical and Biological Engineering and Computing, 28(3), pp. 257-259, 1990. Basinger, B. C., Rowley, A P., Chen, K., Hurnayun, M. S., and Weiland, J. D. Finite element modeling of retinal prosthesis mechanics. Journal a/Neural Engineering, 6(5), pp. 055006, 2009. Baumann, S. B., Wozny, D.R., Kelly, S. K., and Meno, F. M The electrical conductivity of human cerebrospinal fluid at body temperature. IEEE Transactions on Biomedical Engineering, 44(3), pp. 220-223, 1997. Bedell, H. E., Tong, J., Woo, S. Y, House, J. R., and Nguyen, T. Orientation Discrimination with Macular Changes Associated with Early AMD: Optometry and Vision Science, 86(5), pp. 485- 491, 2009. Behrend, M. R., Ahuja, AK., Hurnayun, M. S., Chow, R.H., and Weiland, J. D. Resolution of the 211 Epiretinal Prosthesis is not Limited by Electrode Size. IEEE Transactions on Neural Systems and Rehabilitation Engineering, 19(4), pp. 436-442, 2011. Behrend, M. R., Ahuja, A K., Humayun, M. S., Weiland, J. D., and Chow, R. H. Selective labeling of retinal ganglion cells with calcium indicators by retrograde loading in vitro. Journal of Neuroscience Methods, 179(2), pp. 166-172, 2009. Behrend, M. R., Ahuja, A. K., and Weiland, J. D. Dynamic Current Density of the Disk Electrode Double-Layer. IEEE Transactions on Biomedical Engineering, 55(3), pp. 1056-1062, 2008. Berson, E. L. Retinitis pigrnentosa. The Friedenwald Lecture. Investigative Ophthalmology & Visual Science, 34(5), pp. 1659-1676, 1993. Besch, D., Sachs, H., Szurman, P., Giilicher, D., Wilke, R., Reinert, S., Zrenner, E., Bartz-Schmidt, K. U., and Gekeler, F. Extraocular surgery for implantation of an active subretinal visual prosthesis with external connections: feasibility and outcome in seven patients. British Journal of Ophthalmology, 92(10), pp. 1361-1368, 2008. Blum, N. A, Carkhuff, B. G., Charles, Jr., H. K., Edwards, R. L., and Meyer, R. A. Multisite microprobes for neural recordings. IEEE Transactions on Biomedical Engineering, 38(1), pp. 68-74, 1991. Bonanomi, M. T. B. C., Nicoletti, A G. B., Carricondo, P. C., Buzalaf, F., Kara-Jose Jr, N., Gomes, A M. V, and Nakashima, Y Retinal thickness assessed by optical coherence tomography (OCT) in pseudophakic macular edema. Arquivos Brasileiros de Oftalmologia, 69(4), pp. 539-544, 2006. Bourne, M. C., Campbell, D. A, and Tansley, K. Hereditary degeneration of the rat retina. The British Journal of Ophthalmology, 22(10), pp. 613-623, 1938. Brindley, G. S. The passive electrical properties of the frog's retina, choroid and sclera for radial fields and currents. The Journal of Physiology, 134(2), pp. 339-352, 1956. Brindley, G. S. and Lewin, W. S. The sensations produced by electrical stimulation of the visual cortex. The Journal of Physiology, 196(2), pp. 479-493, 1968. Brug, G. J., van den Eeden, A L. G., Sluyters-Rehbach, M., and Sluyters, J. H. The analysis of electrode impedances complicated by the presence of a constant phase element. Journal of Electroanalytical Chemistry and lnterfacial Electrochemistry, 176(1-2), pp. 275-295, 1984. Butson, C. R. and Mcintyre, C. C. Tissue and electrode capacitance reduce neural activation volumes during deep brain stimulation. Clinical Neurophysiology, 116(10), pp. 2490-2500, 2005. Buzsaki, G., Anastassiou, C. A., and Koch, C. The origin of extracellular fields and currents - EEG, ECoG, LFP and spikes. Nature Reviews Neuroscience, 13(6), pp. 407-420, 2012. Calkins, D. J., Tsukamoto, Y, and Sterling, P. Microcircuitry and Mosaic of a Blue-Yellow Ganglion Cell in the Primate Retina. The Journal of Neuroscience, 18(9), pp. 3373-3385, 1998. 212 Cantrell, D.R., Inayat, S., Taflove, A., Ruoff, R. S., and Troy, J.B. Incorporation of the electrode electrolyte interface into finite-element models of metal microelectrodes. Journal of Neural Engineering, 5(1), pp. 54-67, 2008. Cantrell, D. R. and Troy, J. B. Extracellular stimulation of mouse retinal ganglion cells with non rectangular voltage-controlled waveforms. In Engineering in Medicine and Biology Society, 2009. EMBC 2009. Annual International Conference of the IEEE, pp. 642-645. IEEE, Minneapolis, MN, USA, 2009. Capsi, A., Dorn, J. D., McClure, K. H., Humayun, M. S., Greenberg, R. J., and McMahon, M. J. Feasibility study of a retinal prosthesis: Spatial vision with a 16-electrode implant. Archives of Ophthalmology, 127(4), pp. 398-401, 2009. Chader, G. J. Animal models in research on retinal degenerations: past progress and future hope. Vision Research, 42(4), pp. 393-399, 2002. Chader, G. J., Weiland, J. D., Humayun, M. S., Verhaagen, J., Ho!, E. M, Huitenga, I., Wijnholds, J., Bergen, A. B., Gerald J. Boer, and Swaab, D. F. Artificial vision: needs, functioning, and testing of a retinal electronic prosthesis. In Progress in Brain Research, vol. 175, pp. 317-332. Elsevier, 2009. Chai, X., Li, L., Wu, K., Zhou, C., Cao, P, and Ren, Q. C-Sight Visual Prostheses for the Blind. IEEE Engineering in Medicine and Biology Magazine, 27(5), pp. 20-28, 2008. Cha, K., Horch, K. W., and Normann, R. A. Mobility performance with a pixelized vision system. Vision Research, 32(7), pp. 1367-1372, 1992a. Cha, K., Horch, K. W., and Normann, R. A. Simulation of a phosphene-based visual field: Visual acuity in a pixelized vision system. Annals of Biomedical Engineering, 20( 4), pp. 439-449, 1992b. Cha, K., Horch, K. W., Normann, R. A., and Boman, D. K. Reading speed with a pixelized vision system. Journal of the Optical Society of America A, 9(5), pp. 673--677, 1992c. Chang, B., Hawes, N. L., Hurd, R. E., Davisson, M. T., Nusinowitz, S., and Heckenlively, J. R. Retinal degeneration mutants in the mouse. Vision Research, 42( 4), pp. 517-525, 2002. Chang, B., Hawes, N. L., Pardue, M. T., German, A. M., Hurd, R. E., Davisson, M. T., Nusinowitz, S., Rengarajan, K., Boyd, A. P, Sidney, S. S., Phillips, M. J., Stewart, R. E., Chaudhury, R., Nickerson, J. M., Heckenlively, J. R., and Boatright, J. H. Two mouse retinal degenerations caused by missense mutations in the ~-subunit of rod cGMP phosphodiesterase gene. Vision Research, 47(5), pp. 624-633, 2007. Chan, L. L. H., Lee, E.-J., Humayun, M. S., and Weiland, J. D. Both electrical stimulation thresholds and SMI-32-immunoreactive retinal ganglion cell density correlate with age in S334ter line 3 rat retina. Journal of Neurophysiology, 105(6), pp. 2687-2697, 2011. 213 Chen, K., Yang, Z., Hoang, L., Weiland, J., Humayun, M., and Liu, W. An Integrated 256-Channel Epiretinal Prosthesis. IEEE Journal of Solid-State Circuits, 45(9), pp. 1946-1956, 2010. Chen, S. C., Suaning, G. J., Morley, J. W., and Lovell, N. H. Simulating prosthetic vision: II. Measuring functional capacity. Vision Research, 49(19), pp. 2329-2343, 2009a. Chen, S. C., Suaning, G. J., Morley, J. W., and Lovell, N. H. Simulating prosthetic vision: I. Visual models ofphosphenes. Vision Research, 49(12), pp. 1493-1506, 2009b. Cho, A K. Understanding the degenerate retina :S response to electrical stimulation: An in vitro approach. Ph.D. Thesis. University of Southern California, Los Angeles, CA, USA, 2014. Chow, A Y Retinal Prostheses Development in Retinitis Pigmentosa Patients-Progress and Comparison: Asia-Pacific Journal of Ophthalmology, 2( 4), pp. 253-268, 2013. Chow, A Y, Bittner, A K., and Pardue, M. T. The Artificial Silicon Retina in Retinitis Pigrnentosa Patients (An American Ophthalrnological Association Thesis). Transactions of the American Ophthalmological Society, 108, pp. 120, 2010. Chow, A. Y, Chow, V. Y, Packo, K. H., Pollack, J. S., Peyman, G. A, and Schuchard, R. The artificial silicon retina microchip for the treatment of vision loss from retinitis pigrnentosa. Archives of Ophthalmology, 122(4), pp. 460-469, 2004. Cogan, S. F., Edell, D. J., Guzelian, A. A., Ping Liu, Y, and Edell, R. Plasma-enhanced chemical vapor deposited silicon carbide as an implantable dielectric coating. Journal of Biomedical Materials Research Part A, 67A(3), pp. 856-867, 2003. Cogan, S. F., Ehrlich, J., Plante, T. D., Srnirnov, A, Shire, D. B., Gingerich, M., and Rizzo, J. F. Sputtered iridium oxide films for neural stimulation electrodes. Journal of Biomedical Materials Research Part B: Applied Biomaterials, 89B(2), pp. 353-361, 2009. Colodetti, L., Weiland, J. D., Colodetti, S., Ray, A, Seiler, M. J., Hinton, D. R., and Humayun, M. S. Pathology of damaging electrical stimulation in the retina. Experimental Eye Research, 85(1), pp. 23-33, 2007. Cui, X. T. and Zhou, D. D. Poly (3,4-Ethylenedioxythiophene) for Chronic Neural Stimulation. IEEE Transactions on Neural Systems and Rehabilitation Engineering, 15( 4), pp. 502-508, 2007. Curcio, C. A, Medeiros, N. E., and Millican, C. L. Photoreceptor loss in age-related macular degeneration. Investigative Ophthalmology & Visual Science, 37(7), pp. 1236-1249, 1996. Curcio, C. A., Sloan, K. R., Kalina, R. E., and Hendrickson, A. E. Human photoreceptor topography. The Journal of Comparative Neurology, 292( 4), pp. 497-523, 1990. da Cruz, L., Coley, B. F., Dorn, J., Merlini, F., Filley, E., Christopher, P., Chen, F. K., Wuyyuru, V., Sahel, J., Stanga, P., Humayun, M., Greenberg, R. J., and Dagnelie, G. The Argus II epiretinal prosthesis system allows letter and word reading and long-term function in patients with profound vision loss. British Journal of Ophthalmology, pp. 301525, 2013. 214 Dagnelie, G. Retinal implants: emergence of a multidisciplinary field. Current Opinion in Neurology, 25(1), pp. 67-75, 2012. Dagnelie, G., Keane, P., Narla, V., Yang, L., Weiland, J., and Hurnayun, M. S. Real and virtual mobility performance in simulated prosthetic vision. Journal of Neural Engineering, 4(1), pp. S92-SIOI, 2007. Davuluri, N. S. and Weiland, J. D. Time-varying pulse trains limit retinal desensitization caused by continuous electrical stimulation. In 2014 36th Annual International Conference of the IEEE Engineering in Medicine and Biology Society (EMBC), pp. 414-417, 2014. de Balthasar, C., Patel, S., Roy, A., Freda, R., Greenwald, S., Horsager, A., Mahadevappa, M, Yanai, D., McMahon, M. J., Humayun, M. S., Greenberg, R. J., Weiland, J. D., and Fine, I. Factors Affecting Perceptual Thresholds in Epiretinal Prostheses. Investigative Ophthalmology & Visual Science, 49(6), pp. 2303-2314, 2008. DeMarco, P. J., Yarbrough, G. L., Yee, C. W., McLean, G. Y, Sagdullaev, B. T., Ball, S. L., and McCall, M A Stimulation via a Subretinally Placed Prosthetic Elicits Central Activity and Induces a Trophic Effect on Visual Responses. Investigative Ophthalmology & Visual Science, 48(2), pp. 916-926, 2007. Dobelle, W. H., Mladejovsky, M. G., and Girvin, J. P. Artificial Vision for the Blind: Electrical Stimulation of Visual Cortex Offers Hope for a Functional Prosthesis. Science, 183( 4123), pp. 440-444, 1974. Dorn, J. D., Ahuja, A. K., Caspi, A, da Cruz, L., Dagnelie, G., Sahel, J. A, Greenberg, R. J., McMahon, M. J., and Argus II Study Group. The Detection of Motion by Blind Subjects With the Epiretinal 60-Electrode (Argus II) Retinal Prosthesis. JAMA Ophthalmology, 131(2), pp. 183-189, 2013. Eckrniller, R., Neumann, D., and Baruth, 0. Tunable retina encoders for retina implants: why and how. Journal a/Neural Engineering, 2(1), pp. S91-Sl04, 2005. Faes, T. J. C., van der Meij, H. A., de Muncie, J. C., and Heethaar, R. M. The electric resistivity of human tissues (100 Hz-10 MHz): a meta-analysis ofreview studies. Physiological Measurement, 20(4), pp. RI-RIO, 1999. Fang, X., Sakaguchi, H., Fujikado, T., Osanai, M., Ilcuno, Y, Kamei, M., Ohji, M., Yagi, T., and Tano, Y Electrophysiological and histological studies of chronically implanted intrapapillary microelectrodes in rabbit eyes. Graefe :S Archive for Clinical and Experimental Ophthalmology, 244(3), pp. 364-375, 2006. Fariss, R. N., Li, Z.-Y, and Milam, A H. Abnormalities in rod photoreceptors, arnacrine cells, and horizontal cells in human retinas with retinitis pigmentosa. American Journal of Ophthalmology, 129(2), pp. 215-223, 2000. Foerster, 0. Beitriige zur pathophysiologie der sehbahn und der spehsphare. Journal of Psychology 215 and Neurology, pp. 435-363, 1929. Foutz, T. J. and Mcintyre, C. C. Evaluation of novel stimulus waveforms for deep brain stimulation. Journal of Neural Engineering, 7(6), pp. 066008, 2010. Franks, W., Schenker, I., Schmutz, P., and Hierlemann, A. Impedance characterization and modeling of electrodes for biomedical applications. IEEE Transactions on Biomedical Engineering, 52(7), pp. 1295-1302, 2005. Freeman, D. Electric stimulation with sinusoids and white noise for neural prostheses. Frontiers in Neuroscience, 2010. Freeman, D. K., Eddington, D. K., Rizzo, III, J. F., and Fried, S. I. Selective Activation of Neuronal Targets With Sinusoidal Electric Stimulation. Journal of Neurophysiology, 104(5), pp. 2778- 2791, 2010. Freeman, D. K. and Fried, S. I. Multiple components of ganglion cell desensitization in response to prosthetic stimulation. Journal of Neural Engineering, 8(1), pp. 016008, 2011. Freeman, D. K., Jeng, J. S., Kelly, S. K., Hartveit, E., and Fried, S. I. Calcium channel dynamics limit synaptic release in response to prosthetic stimulation with sinusoidal waveforms. Journal of Neural Engineering, 8(4), pp. 046005, 20lla. Freeman, D. K., Rizzo, III, J. F., and Fried, S. I. Encoding visual information in retinal ganglion cells with prosthetic stimulation. Journal of Neural Engineering, 8(3), pp. 035005, 20llb. Fried, S. I., Hsueh, H. A., and Werblin, F. S. A Method for Generating Precise Temporal Patterns of Retinal Spiking Using Prosthetic Stimulation. Journal of Neurophysiology, 95(2), pp. 970- 978, 2006. Fried, S. I. and Jens en, R. J. The Response of Retinal Neurons to Electrical Stimulation: AS urnrnary ofln Vitro and In Vivo Animal Studies. In Visual Prosthetics, pp. 229-258. Springer US, 2011. Fried, S. I., Lasker, A. C. W., Desai, N. J., Eddington, D. K., and Rizzo, III, J. F. Axonal Sodium Channel Bands Shape the Response to Electric Stimulation in Retinal Ganglion Cells. Journal of Neurophysiology, 101(4), pp. 1972-1987, 2009. Fujikado, T., Kamei, M., Sakaguchi, H., Kanda, H., Morimoto, T., Ikuno, Y, Nishida, K., Kishima, H., Maruo, T., Konoma, K., Ozawa, M., and Nishida, K. Testing of Semichronically Implanted Retinal Prosthesis by Suprachoroidal-Transretinal Stimulation in Patients with Retinitis Pigrnentosa. Investigative Ophthalmology & Visual Science, 52(7), pp. 4726-4733, 2011. Fujikado, T., Kamei, M., Sakaguchi, H., Kanda, H., Morimoto, T., Ikuno, Y, Nishida, K., Kishima, H., Maruo, T., Sawai, H., Miyoshi, T., Osawa, K., and Ozawa, M Clinical Trial of Chronic Implantation of Suprachoroidal-Transretinal Stimulation System for Retinal Prosthesis. Sensors and Materials, 24(4), pp. 181-187, 2012. Gabriel, S., Lau, R. W., and Gabriel, C. The dielectric properties of biological tissues: II. 216 Measurements in the frequency range 10 Hz to 20 GHz. Physics in Medicine and Biology, 41(11), pp. 2251, 1996. Galvani, L. De viribw; electricitatis in motu mw;culari commentariw;. Ex Typographia Instituti Scientiarum, Bononire (Bologna), 1791. Geddes, L. A Historical evolution of circuit models for the electrode-electrolyte interface. Annals of Biomedical Engineering, 25(1), pp. 1-14, 1997. Geddes, L. A. Accuracy limitations of chronaxie values. IEEE Transactions on Biomedical Engineering, 51(1), pp. 176-181, 2004. Geddes, L. A and Baker, L. E. The specific resistance of biological material-A compendium of data for the biomedical engineer and physiologist. Medical and Biological Engineering, 5(3), pp. 271-293, 1967. Gehrs, K. M., Anderson, D. H., Johnson, L. V., and Hageman, G. S. Age-related rnacular degeneration-emerging pathogenetic and therapeutic concepts. Annals of Medicine, 38(7), pp. 450-471, 2006. Gill, E. C., Antalek, J., Kimock, F. M, Nasiatka, P. J., Mcintosh, B. P., Tanguay, A R. J., and Weiland, J. D. High-Density Feedthrough Technology for Hermetic Biomedical Micropackaging. In Symposium SS -Bioelectronics-Materials, Interfaces and Applications, vol. 1572, 2013. Goncalve, S., de Muncie, J. C., Verbunt, J. P. A, Heethaar, R. M., and da Silva, F. H. L. In vivo measurement of the brain and skull resistivities using an EIT-based method and the combined analysis of SEF/SEP data. IEEE Transactions on Biomedical Engineering, 50(9), pp. 1124-1128, 2003. Greenberg, R. J., Velte, T. J., Hurnayun, M. S., Scarlatis, G. N., and de Juan, Jr., E. A computational model of electrical stimulation of the retinal ganglion cell. IEEE Transactions on Biomedical Engineering, 46( 5), pp. 505-514, 1999. Greenwald, S. H., Horsager, A., Hurnayun, M. S., Greenberg, R. J., McMahon, M. J., and Fine, I. Brightness as a Function of Current Amplitude in Human Retinal Electrical Stimulation. Investigative Ophthalmology & Visual Science, 50(11), pp. 5017-5025, 2009. Grill, W. M. and Mortimer, D. J. T. Electrical properties of implant encapsulation tissue. Annals of Biomedical Engineering, 22(1), pp. 23-33, 1994. Grill, W. M. and Mortimer, J. T. Inversion of the current-distance relationship by transient depolarization. IEEE Transactions on Biomedical Engineering, 44(1), pp. 1-9, 1997. Hadjinicolaou, A. E., Leung, R. T., Garrett, D. J., Ganesan, K., Fox, K., Nayagam, D. A X., Shivdasani, M. N., Meffin, H., Ibbotson, M. R., Prawer, S., and O'Brien, B. J. Electrical stimulation of retinal ganglion cells with diamond and the development of an all diamond retinal prosthesis. Biomaterials, 33(24), pp. 5812-5820, 2012. 217 Hallum, L. E., Cloherty, S. L., and Lovell, N. H. Image Analysis for Microelectronic Retinal Prosthesis. IEEE Transactions on Biomedical Engineering, 55(1), pp. 344-346, 2008. Hartong, D. T., Berson, E. L., and Dryja, T. P. Retinitis pigmentosa. The Lancet, 368(9549), pp. 1795-1809, 2006. Hart, W. M. The temporal responsiveness of vision. In Adler :S Physiology of the Eye. The C. V. Mosby Company, St. Louis, 1987. Hayes, J. S., Yin, V. T., Piyathaisere, D., Weiland, J. D., Hurnayun, M. S., and Dagnelie, G. Visually Guided Performance of Simple Tasks Using Simulated Prosthetic Vision. Artificial Organs, 27(11), pp. 1016-1028, 2003. Heckenlively, J. R. Retinitis pigmentosa. J. B. Lippincott Company, Philadelphia, 1988. Heckenlively, J. R., Boughman, J. A, and Friedman, L. H. Pedigree Analysis. In Retinitis pigmentosa, pp. 14-24. J.B. Lippincott Company, Philadelphia, 1988. Henze, D. A, Borhegyi, Z., Csicsvari, J., Mamiya, A, Harris, K. D., and Buzsaki, G. Intracellular Features Predicted by Extracellular Recordings in the Hippocampus In Vivo. Journal of Neurophysiology, 84(1), pp. 390-400, 2000. Heynen, H. and van Norren, D. Origin of the electroretinogram in the intact macaque eye-II: Current source-density analysis. Vision Research, 25(5), pp. 709-715, 1985. Hornig, R., Zehnder, T., Velikay-Parel, M., Laube, T., Feucht, M, and Richard, G. The IMI Retinal Implant System. In Artificial Sight: Basic Research, Biomedical Engineering, and Clinical Advances, pp. 111-128. Springer, New York, NY, 2007. Horsager, A, Boynton, G. M., Greenberg, R. J., and Fine, I. Temporal Interactions during Paired Electrode Stimulation in Two Retinal Prosthesis Subjects. Investigative Ophthalmology & Visual Science, 52(1), pp. 549-557, 2011. Horsager, A, Greenberg, R. J., and Fine, I. Spatiotemporal Interactions in Retinal Prosthesis Subjects. Investigative Ophthalmology & Visual Science, 51(2), pp. 1223-1233, 2010. Horsager, A., Greenwald, S. H., Weiland, J. D., Humayun, M. S., Greenberg, R. J., McMahon, M. J., Boynton, G. M., and Fine, I. Predicting Visual Sensitivity in Retinal Prosthesis Patients. Investigative Ophthalmology & Visual Science, 50(4), pp. 1483-1491, 2009. Hsu, C. H. and Mansfeld, F. Technical Note: Concerning the Conversion of the Constant Phase Element Parameter YO into a Capacitance. Corrosion, 57(9), pp. 747-748, 2001. Humayun, M S. and de Juan, Jr., E. The Development of a Functionallntraocular Visual Prosthesis for the Severely Visually Handicapped. Duke University Eye Center, 1991. Humayun, M. S., de Juan, Jr., E., Weiland, J. D., Dagnelie, G., Katona, S., Greenberg, R., and Suzuki, S. Pattern electrical stimulation of the human retina. Vision Research, 39(15), pp. 2569- 218 2576, 1999a. Hurnayun, M. S., Dorn, J. D., da Cruz., L., Dagnelie, G., Sahel, J.-A., Stanga, P. E., Cideciyan, A. V, Duncan, J. L., Eliott, D., Filley, E., Ho, A. C., Santos, A., Safran, A. B., Arditi, A., Del Priore, L. V, and Greenberg, R. J. Interim Results from the International Trial of Second Sight's Visual Prosthesis. Ophthalmology, 119(4), pp. 779-788, 2012. Hurnayun, M. S., Prince, M., de Juan, Jr., E., Barron, Y, Moskowitz, M., Klock, I. B., and Milam, A. H. Morphornetric analysis of the extrarnacular retina from postmortem eyes with retinitis pigrnentosa. Investigative Ophthalmology & Visual Science, 40(1), pp. 143-148, 1999b. Hurnayun, M. S., Weiland, J. D., Fujii, G. Y, Greenberg, R., Williamson, R., Little, J., Mech, B., Cirnrnarusti, V, Van Boerne!, G., Dagnelie, G., and de Juan, Jr., E. Visual perception in a blind subject with a chronic microelectronic retinal prosthesis. Vision Research, 43(24), pp. 2573- 2581, 2003. Hurnayun, M S., Weiland, J. D., Justus, B., Merrit, C., John J. Whalen, Piyathaisere, D. V, Chen, S. J., Margalit, E., Fujii, G. Y, Greenberg, R. J., de Juan, Jr., E., Scribner, D., and Liu, W. Towards a completely implantable, light-sensitive intraocular retinal prosthesis. In Engineering in Medicine and Biology Society, 2001. Proceedings of the 23rd Annual International Conference of the IEEE, vol. 4, pp. 3422-3425 vol.4. IEEE, Istanbul, Turkey, 2001. Jan, E., Hendricks, J. L., Husaini, V, Richardson-Burns, S. M., Sereno, A., Martin, D. C., and Kotov, N. A. Layered Carbon Nanotube-Polyelectrolyte Electrodes Outperform Traditional Neural Interface Materials. Nano Letters, 9(12), pp. 4012-4018, 2009. Jensen, R. J., Rizzo, III, J. F., Ziv, 0. R., Grumet, A., and Wyatt, J. Thresholds for Activation of Rabbit Retinal Ganglion Cells with an Ultrafrne, Extracellular Microelectrode. Investigative Ophthalmology & Visual Science, 44(8), pp. 3533-3543, 2003. Jensen, R. J. and Rizzo, J. F. Responses of ganglion cells to repetitive electrical stimulation of the retina. Journal of Neural Engineering, 4(1), pp. Sl-S6, 2007. Jensen, R. J. and Rizzo, J. F. Activation of retinal ganglion cells in wild-type and rdl mice through electrical stimulation of the retinal neural network. Vision Research, 48(14), pp. 1562-1568, 2008. Jensen, R. J. and Rizzo, J. F. Activation of ganglion cells in wild-type and rdl mouse retinas with rnonophasic and biphasic current pulses. Journal of Neural Engineering, 6(3), pp. 035004, 2009. Jensen, R. J., Ziv, 0. R., and Rizzo, III, J. F. Responses of rabbit retinal ganglion cells to electrical stimulation with an epiretinal electrode. Journal of Neural Engineering, 2(1), pp. S 16, 2005a. Jensen, R. J., Ziv, 0. R., and Rizzo, J. F. Thresholds for Activation of Rabbit Retinal Ganglion Cells with Relatively Large, Extracellular Microelectrodes. Investigative Ophthalmology & Visual Science, 46(4), pp. 1486-1496, 2005b. 219 Jepson, L. H., Hottowy, P., Mathieson, K., Gunning, D. E., Dqbrowski, W., Litke, A M., and Chichilnisky, E. J. Focal Electrical Stimulation of Major Ganglion Cell Types in the Primate Retina for the Design of Visual Prostheses. The Journal a/Neuroscience, 33(17), pp. 7194-7205, 2013. Jepson, L. H., Hottowy, P., Mathieson, K., Gunning, D. E., Dqbrowski, W., Litke, A M., and Chichilnisky, E. J. Spatially Patterned Electrical Stimulation to Enhance Resolution of Retinal Prostheses. The Journal a/Neuroscience, 34(14), pp. 4871-4881, 2014a. Jepson, L. H., Hottowy, P., Weiner, G. A, Dabrowski, W., Litke, A M., and Chichilnisky, E. J. High-Fidelity Reproduction of Spatiotemporal Visual Signals for Retinal Prosthesis. Neuron, 83( 1 ), pp. 87-92, 20 l 4b. Jiang, G., Mishler, D., Davis, R., Mobley, J. P., and Schulman, J. H. Zirconia to Ti-6Al-4V braze joint for implantable biomedical device. Journal of Biomedical Materials Research Part B: Applied Biomaterials, 72B(2), pp. 316-321, 2005. Jiang, G. and Zhou, D. D. Technology Advances and Challenges in Hermetic Packaging for Implantable Medical Devices. In Implantable Neural Prostheses 2, pp. 27-61. Springer New York, 2009. Jones, B. W., Kondo, M., Terasaki, H., Lin, Y, McCall, M., and Marc, R. E. Retinal remodeling. Japanese Journal of Ophthalmology, 56(4), pp. 289-306, 2012. Jones, K. E. and Normann, R. A. An Advanced Demultiplexing System for Physiological Stimulation. IEEE Transactions on Biomedical Engineering, 44(12), pp. 1210-1220, 1997. Karny, H. Clinical and physiological aspects of the cortical visual prosthesis. Survey of Ophthalmology, 20(1), pp. 47-58, 1975. Kasi, H., Bertsch, A, Guyomard, J.-L., Kolomiets, B., Picaud, S., Pelizzone, M., and Renaud, P. Simulations to study spatial extent of stimulation and effect of electrode-tissue gap in subretinal implants. Medical Engineering & Physics, 33(6), pp. 755-763, 20lla. Kasi, H., Hasenkamp, W., Cosendai, G., Bertsch, A., and Renaud, P. Simulation of epiretinal prostheses - Evaluation of geometrical factors affecting stimulation thresholds. Journal of NeuroEngineering and Rehabilitation, 8(1), pp. 44, 20llb. Kasi, H., Meissner, R., Babalian, A, van Lintel, H., Bertsch, A., and Renaud, P. Direct localised measurement of electrical resistivity profile in rat and embryonic chick retinas usrng a microprobe. Journal of Electrical Bioimpedance, 1(1), pp. 84-92, 20llc. Kelly, S. K., Shire, D. B., Chen, J., Gingerich, M. D., Cogan, S. F., Drohan, W. A, Ellersick, W., Krishnan, A, Behan, S., Wyatt, J. L., and Rizzo, III, J. F. Developments on the Boston 256- channel retinal implant. In Multimedia and Expo Workshops (ICMEW), 2013 IEEE International Conference on, pp. 1--0. IEEE, San Jose, CA, USA, 2013. 220 Keserii, M., Feucht, M., Bomfeld, N., Laube, T., Walter, P., Rossler, G., Velikay-Parel, M., Hornig, R., and Richard, G. Acute electrical stimulation of the human retina with an epiretinal electrode array. Acta Ophthalmologica, 90(1), pp. el-e8, 2012. Kirn, S. Y, Sadda, S., Hurnayun, M. S., de Juan, E., Melia, B. M., and Green, W.R. Morphometric analysis of the rnacula in eyes with geographic atrophy due to age-related rnacular degeneration. Retina (Philadelphia, Pa.), 22(4), pp. 464-470, 2002a. Kirn, S. Y, Sadda, S., Pearlman, J., Humayun, M. S., de Juan, Jr., E., Melia, B. M., and Green, W. R. Morphometric analysis of the macula in eyes with disciform age-related macular degeneration. Retina August 2002, 22(4), pp. 471-477, 2002b. Kirn, Y, Fahy, J. B., and Tupper, B. J. Optimal Electrode Designs for Electrosurgery, Defibrillation, and External Cardiac Pacing. IEEE Transactions on Biomedical Engineering, BME-33(9), pp. 845-853, 1986. Klauke, S., Goertz, M., Rein, S., Hoehl, D., Thomas, U., Eckhorn, R., Bremmer, F., and Wachtler, T. Stimulation with a Wireless Intraocular Epiretinal Implant Elicits Visual Percepts in Blind Humans. Investigative Ophthalmology & Visual Science, 52(1), pp. 449-455, 2011. Krause, F. and Schum, H. Die epileptischen Erkrankungen. Neue Deutsche Chirurgie, 49, pp. 482- 486, 1931. Krishnan, A and Kelly, S. K. On the cause and control of residual voltage generated by electrical stimulation of neural tissue. In Engineering in Medicine and Biology Society (EMBC), 2012 Annuallnternational Conference of the IEEE, pp. 3899-3902. IEEE, San Diego, CA, USA, 2012. Ksienski, D. A A Minimum Profile Uniform Current Density Electrode. IEEE Transactions on Biomedical Engineering, 39(7), pp. 682--092, 1992. Kusnyerik, A, Greppmaier, U., Wilke, R., Gekeler, F., Wilhelm, B., Sachs, H. G., Bartz-Schmidt, K. U., Klose, U., Sting!, K., Resch, M. D., Hekrnat, A., Brockmann, A., Karacs, K., Nemeth, J., Suveges, I., and Zrerrner, E. Positioning of Electronic Subretinal Implants in Blind Retinitis Pigrnentosa Patients Through Multimodal Assessment of Retinal Structures. Investigative Ophthalmology & Visual Science, 53(7), pp. 3748-3755, 2012. Lakhanpal, R. R., Yanai, D., Weiland, J. D., Fujii, G. Y, Caffey, S., Greenberg, R. J., de Juan, E., and Humayun, M. S. Advances in the development of visual prostheses. Current Opinion in Ophthalmology, 14(3), pp. 122-127, 2003. Lee, D., Geller, S., Walsh, N., Valter, K., Yasumura, D., Matthes, M., La Vail, M., and Stone, J. Photoreceptor Degeneration in Pro23His and S334ter Transgenic Rats. In Retinal Degenerations, pp. 297-302. Springer US, 2003. Lee, S. W., Eddington, D. K., and Fried, S. I. Responses to pulsatile subretinal electric stimulation: effects of amplitude and duration. Journal of Neurophysiology, 109(7), pp. 1954-1968, 2013. 221 Lempka, S. F., Miocinovic, S., Johnson, M. D., Vitek, J. L., and Mcintyre, C. C. In vivo impedance spectroscopy of deep brain stimulation electrodes. Journal of Neural Engineering, 6( 4), pp. 046001, 2009. LeRoy, C. O'u !'on rend compte de quelques tentatives que !'on a faites pour gu' erir plusieurs maladies par l''electricit'e. Hist. Acad. Roy Sciences (Paris), M'emoires Math. Phys., 60, pp. 87-95, 1755. Li, C., Bak, A F., and Parker, L. 0. Specific resistivity of the cerebral cortex and white matter. Experimental Neurology, 20(4), pp. 544-557, 1968. Lilly, J. C. Injury and excitation by electric currents. In Electrical stimulation of the brain, pp. 60- 64. University of Texas Press Austin, 1961. Liu, X., Demosthenous, A, and Donaldson, N. Platinum electrode noise in the ENG spectrum. Medical & Biological Engineering & Computing, 46(10), pp. 997-1003, 2008. MacLaren, R. E., Pearson, R. A, MacNeil, A, Douglas, R. H., Salt, T. E., Akimoto, M., Swaroop, A, Sowden, J. C., and Ali, R. R. Retinal repair by transplantation of photoreceptor precursors. Nature, 444(7116), pp. 203-207, 2006. Mahadevappa, M., Weiland, J. D., Yanai, D., Fine, I., Greenberg, R. J., and Humayun, M. S. Perceptual thresholds and electrode impedance in three retinal prosthesis subjects. IEEE Transactions on Neural Systems and Rehabilitation Engineering, 13(2), pp. 201-206, 2005. Marathe, V and Newman, J. Current Distribution on a Rotating Disk Electrode. Journal of The Electrochemical Society, 116(12), pp. 1704-1707, 1969. Marc, R. E. Injury and repair: retinal remodeling. Encyclopedia of the Eye, 2, pp. 414-420, 2010. Marc, R. E. and Jones, B. W. Retinal remodeling in inherited photoreceptor degenerations. Molecular Neurobiology, 28(2), pp. 139-147, 2003. Marc, R. E., Jones, B. W., Watt, C. B., and Strettoi, E. Neural remodeling in retinal degeneration. Progress in Retinal and Eye Research, 22(5), pp. 607-655, 2003. Margalit, E., Maia, M., Weiland, J. D., Greenberg, R. J., Fujii, G. Y, Torres, G., Piyathaisere, D. V, O'Hearn, T. M., Liu, W., Lazzi, G., Dagnelie, G., Scribner, D. A., de Juan, Jr., E., and Humayun, M. S. Retinal Prosthesis for the Blind. Survey of Ophthalmology, 47(4), pp. 335-356, 2002. Margalit, E. and Sadda, S. R. Retinal and Optic Nerve Diseases. Artificial Organs, 27(11), pp. 963- 974, 2003. Margolis, D. J. and Detwiler, P B. Cellular Origin of Spontaneous Ganglion Cell Spike Activity in Animal Models of Retinitis Pigmentosa. Journal of Ophthalmology, 2011, pp. 1--0, 2011. Masland, R. H. Neuronal diversity in the retina. Current Opinion in Neurobiology, 11( 4), pp. 431- 436, 2001. 222 Maynard, E. M. Visual Prostheses. Annual Review of Biomedical Engineering, 3(1), pp. 145-168, 2001. McAdams, E. T., Lackermeier, A, McLaughlin, J. A., Macken, D., and Jossinet, J. The linear and non-linear electrical properties of the electrode-electrolyte interface. Biosensors and Bioelectronics, 10(1-2), pp. 67-74, 1995. McCreery, D. B., Agnew, W. F., Yuen, T. G. H., and Bullara, L. Charge density and charge per phase as cofactors in neural injury induced by electrical stimulation. IEEE Transactions on Biomedical Engineering, 37(10), pp. 996-1001, 1990. McCreery, D., Lossinsky, A., Pikov, V, and Liu, X. Microelectrode array for chronic deep-brain microstimulation and recording. IEEE Transactions on Biomedical Engineering, 53( 4), pp. 726- 737, 2006. Mcintyre, C. C. and Grill, W. M Finite Element Analysis of the Current-Density and Electric Field Generated by Metal Microelectrodes. Annals of Biomedical Engineering, 29(3), pp. 227-235, 2001. Medeiros, N. E. and Curcio, C. A Preservation of Ganglion Cell Layer Neurons in Age-Related Macular Degeneration. Investigative Ophthalmology & Visual Science, 42(3), pp. 795-803, 200 I. Menon, G. and Walters, G. New paradigms in the treatment of wet AMD: the impact of anti-VEGF therapy. Eye, 23, pp. S l-S7, 2009. Mercanzini, A, Colin, P., Bensadoun, J.-C., Bertsch, A, and Renaud, P. In Vivo Electrical Impedance Spectroscopy of Tissue Reaction to Microelectrode Arrays. IEEE Transactions on Biomedical Engineering, 56(7), pp. 1909-1918, 2009. Merrill, D. R., Bikson, M., and Jefferys, J. G. R. Electrical stimulation of excitable tissue: design of efficacious and safe protocols. Journal a/Neuroscience Methods, 141(2), pp. 171-198, 2005. Meyer, D. P. F., Gadsby, P. D., Sickle, D. V, Schoenlein, W. E., Foster, K. S., and Graber, G. P. Impedance-gradient electrode reduces skin irritation induced by transthoracic defibrillation. Medical and Biological Engineering and Computing, 43(2), pp. 225-229, 2005. Myland, J. C. and Oldham, K. B. How does the double layer at a disk electrode charge? Journal of Electroanalytical Chemistry, 575(1), pp. 81-93, 2005. Nanduri, D. Prosthetic vision in blind human patients: Predicting the percepts of epiretinal stimulation. Ph.D. Thesis. University of Southern California, Los Angeles, CA, USA, 2011. Nanduri, D., Fine, I., Horsager, A., Boynton, G. M., Hurnayun, M. S., Greenberg, R. J., and Weiland, J. D. Frequency and Amplitude Modulation Have Different Effects on the Percepts Elicited by Retinal Stimulation. Investigative Ophthalmology & Visual Science, 53(1), pp. 205-214, 2012. Nanduri, D., Humayun, M S., Greenberg, R. J., McMahon, M. J., and Weiland, J. D. Retinal prosthesis phosphene shape analysis. In Engineering in Medicine and Biology Society, 2008. 223 EMBS 2008. 30th Annual International Conference of the IEEE, pp. 1785-1788. IEEE, Vancouver, Canada, 2008. Narfstr6m, K. Hereditary and Congenital Ocular Disease in the Cat. Journal of Feline Medicine and Surgery, 1(3), pp. 135-141, 1999. Nayagam, D. A. X., Williams, R. A., Allen, P J., Shivdasani, M. N., Luu, C. D., Salinas-LaRosa, C. M, Finch, S., Ayton, L. N., Saunders, A. L., McPhedran, M., McGowan, C., Villalobos, J., Fallon, J. B., Wise, A. K., Yeoh, J., Xu, J., Feng, H., Millard, R., McWade, M., Thieu, P C., Williams, C. E., and Shepherd, R. K. Chronic Electrical Stimulation with a Suprachoroidal Retinal Prosthesis: APreclinical Safety and Efficacy Study. PLoS ONE, 9(5), pp. e97182, 2014. Neelam, K., Nolan, J., Chakravarthy, U., and Beatty, S. Psychophysical Function in Age-related Maculopathy. Survey of Ophthalmology, 54(2), pp. 167-210, 2009. Newman, E. A. and Bartosch, R. An eyecup preparation for the rat and mouse. Journal of Neuroscience Methods, 93(2), pp. 169-175, 1999. Newman, J. Current Distribution on a Rotating Disk below the Limiting Current. Journal of The Electrochemical Society, 113(12), pp. 1235-1241, 1966a. Newman, J. Resistance for Flow of Current to a Disk Journal of The Electrochemical Society, 113(5), pp. 501-502, 1966b. Newman, J. Frequency Dispersion in Capacity Measurements at a Disk Electrode. Journal of The Electrochemical Society, 117(2), pp. 198-203, 1970. Nirenberg, S. and Pandarinath, C. Retinal prosthetic strategy with the capacity to restore normal vision. Proceedings of the National Academy of Sciences, 109(37), pp. 15012-15017, 2012. Nisancioglu, K. and Newman, J. The Transient Response of a Disk Electrode. Journal of The Electrochemical Society, 120(10), pp. 1339-1346, 1973a. Nisancioglu, K. and Newman, J. The Transient Response of a Disk Electrode with Controlled Potential. Journal ofThe Electrochemical Society, 120(10), pp. 1356-1358, 1973b. Ogden, T. E. and Ito, H. Avian retina. II. An evaluation of retinal electrical anisotropy. Journal of Neurophysiology, 34(3), pp. 367-373, 1971. Oldham, K. B. The RC time "constant" at a disk electrode. Electrochemistry Communications, 6(2), pp. 210-214, 2004. Ordonez, J., Dante!, P, Schnettler, M., and Stieglitz, T. Hermetic glass soldered micro-packages for a vision prosthesis. In Engineering in Medicine and Biology Society (EMBC), 2012 Annual International Conference of the IEEE, pp. 2784-2787. IEEE, San Diego, CA, USA, 2012. Orazern, M. E. and Tribollet, B. Perspectives on Newman's Work on Resistance for Flow of Current to a Disk. The Electrochemical Society Interface, 18(1), pp. 56-58, 2009. 224 Overmyer, K. M., Pearce, J. A, and DeWitt, D. P. Measurements of Temperature Distributions at Electro-Surgical Dispersive Electrode Sites. Journal ofBiomechanical Engineering, 101(1), pp. 66-72, 1979. Pagan-Carlo, L.A., Stone, M. S., and Kerber, R. E. Nature and Determinants of Skin "Burns" After Transthoracic Cardioversion. The American Journal of Cardiology, 79(5), pp. 689--091, 1997. Palanker, D., Vankov, A, Huie, P., and Baccus, S. Design of a high-resolution optoelectronic retinal prosthesis. Journal of Neural Engineering, 2(1), pp. S 105-Sl20, 2005. Palanker, D., Vankov, A., Huie, P., Butterwick, A, Chan, I., Marmor, M. F., and Blumenkranz, M. S. High-Resolution Opto-Electronic Retinal Prosthesis: Physical Limitations and Design. In Artificial Sight, pp. 255-277. Springer New York, 2007. Panescu, D. Emerging technologies - Implantable neurostimulation devices. IEEE Engineering in Medicine and Biology Magazine, 27(5), pp. 100-105, 113, 2008. Papazov, S., Kostov, Z., and Daskalov, I. Electrical current distribution under transthoracic defibrillation and pacing electrodes. Journal of Medical Engineering & Technology, 26(1), pp. 22-27, 2002. Pardue, M. T., Phillips, M. J., Yin, H., Fernandes, A, Cheng, Y, Chow, A. Y, and Ball, S. L. Possible sources of neuroprotection following subretinal silicon chip implantation in RCS rats. Journal of Neural Engineering, 2(1), pp. S39-S47, 2005. Parikh, N., Itti, L., Hurnayun, M. S., and Weiland, J. D. Performance of visually guided tasks using simulated prosthetic vision and saliency-based cues. Journal of Neural Engineering, 10(2), pp. 026017, 2013. Pennesi, M. E., Michaels, K. V, Magee, S.S., Maricle, A, Davin, S. P., Garg, AK., Gale, M. J., Tu, D. C., Wen, Y, Erker, L. R., and Francis, P. J. Long-Term Characterization of Retinal Degeneration in rd! and rdlO Mice Using Spectral Domain Optical Coherence Tomography. Investigative Ophthalmology & Visual Science, 53(8), pp. 4644-4656, 2012. Perez Pornos, A., Sornrnerhalder, J., Cruz, L. da, Sahel, J. A, Mohand-Said, S., Hafezi, F., and Pelizzone, M. Temporal Properties of Visual Perception on Electrical Stimulation of the Retina. Investigative Ophthalmology & Visual Science, 53(6), pp. 2720-2731, 2012. Petrossians, A, Davuluri, N., Whalen, J. J., Mansfeld, F., and Weiland, J. D. Improved Biphasic Pulsing Power Efficiency with Pt-Ir Coated Microelectrodes. MRS Proceedings, 1621, pp. 249- 257, 2014. Petrossians, A, Whalen, J. J., Weiland, J. D., and Mansfeld, F. Electrodeposition and Characterization of Thin-Film Platinum-Iridium Alloys for Biological Interfaces. Journal ofThe Electrochemical Society, 158(5), pp. D269-D276, 20lla. Petrossians, A, Whalen, J. J., Weiland, J. D., and Mansfeld, F. Surface modification of neural 225 stirnulatin!irecording electrodes with high surface area platinum-iridium alloy coatings. In Engineering in Medicine and Biology Society, EMBC, 2011 Annual International Conference of the IEEE, pp. 3001-3004. IEEE, Boston, MA, 20llb. Phelan, J. K. and Bok, D. A brief review of retinitis pigrnentosa and the identified retinitis pigrnentosa genes. Molecular Vision, 6, pp. 116-124, 2000. Piyathaisere, D. V, Margalit, E., Chen, S. J., Shyu, J.-S., D' Anna, S. A, Hurnayun, M. S., Weiland, J. D., Grebe, R.R., Grebe, L., and Kirn, S. Y Effects of short-term exposure to heat on the retina. Investigative Ophthalmology & Visual Science, 42(2), pp. S814, 2001. Piyathaisere, D. V, Margalit, E., Chen, S.-J., Shyu, J.-S., D' Anna, S. A, Weiland, J. D., Grebe, R. R., Grebe, L., Fujii, G., Kirn, S. Y, Greenberg, R. J., De Juan, E., and Hurnayun, M. S. Heat effects on the retina. Ophthalmic Surgery, Lasers and Imaging, 34(2), pp. 114-20, 2003. Polikov, V S., Tresco, P. A, and Reichert, W. M Response of brain tissue to chronically implanted neural electrodes. Journal ofNeuroscienceMethods, 148(1), pp. 1-18, 2005. Pollak, V An equivalent diagram for the interface impedance of metal needle electrodes. Medical and Biological Engineering, 12(4), pp. 454-459, 1974. Pu, M., Xu, L., and Zhang, H. Visual Response Properties of Retinal Ganglion Cells in the Royal College of Surgeons Dystrophic Rat. Investigative Ophthalmology & Visual Science, 47(8), pp. 3579-3585, 2006. Ray, A., Chan, L. L. H., Gonzalez, A., Hurnayun, M. S., and Weiland, J. D. Impedance as a Method to Sense Proximity at the Electrode-Retina Interface. IEEE Transactions on Neural Systems and Rehabilitation Engineering, 19(6), pp. 696--099, 2011. Ray, A, Colodetti, L., Weiland, J. D., Hinton, D. R., Hurnayun, M. S., and Lee, E.-J. Irnrnunocytochernical analysis of retinal neurons under electrical stimulation. Brain Research, 1255, pp. 89-97, 2009. Rizzo, J. F., Shire, D. B., Kelly, S. K., Troyk, P., Gingerich, M., McKee, B., Priplata, A, Chen, J., Drohan, W., Doyle, P., Mendoza, 0., Theogarajan, L., Cogan, S. F., and Wyatt, J. L. Development of the Boston Retinal Prosthesis. In Engineering in Medicine and Biology Society, EMBC, 2011 Annuallnternational Conference of the IEEE, pp. 7492-7495. IEEE, Boston, MA, 2011. Rizzo, J. F., Snebold, L., and Kenney, M. Development of a Visual Prosthesis - A Review of the Field and an Overview of the Boston Retinal Implant Project. In Visual Prosthesis and Ophthalmic Devices: New Hope in Sight, pp. 71-93. Springer, New York, NY, 2007. Rodger, D. C., Fong, A J., Li, W., Arneri, H., Ahuja, A. K., Gutierrez, C., Lavrov, I., Zhong, H., Menon, P. R., Meng, E., Burdick, J. W., Roy, R. R., Edgerton, V R., Weiland, J. D., Hurnayun, M. S., and Tai, Y-C. Flexible parylene-based rnultielectrode array technology for high-density neural stimulation and recording. Sensors and Actuators B: Chemical, 132(2), pp. 449-460, 2008. 226 Rodieck, R. W. The first steps in seeing, xi. Sinauer Associates, Sunderland, MA, US, 1998. Rose, M. L. History of disability: ancient west. In Encyclopedia of Disability, vol. 2, pp. 852-854. Sage Publication, Thousand Oaks, CA, 2006. Rose, T. L., Kelliher, E. M., and Robblee, L. S. Assessment of capacitor electrodes for intracortical neural stimulation. Journal a/Neuroscience Methods, 12(3), pp. 181-193, 1985. Rose, T. L. and Robblee, L. S. Electrical stimulation with Pt electrodes. VIII. Electrochemically safe charge injection limits with 0.2 ms pulses (neuronal application). IEEE Transactions on Biomedical Engineering, 37(11), pp. 1118-1120, 1990. Rossler, G., Laube, T., Brockmann, C., Kirschkamp, T., Mazinani, B., Goertz, M, Koch, C., Krisch, I., Sellhaus, B., Trieu, H. K., Weis, J., Bornfeld, N., Rothgen, H., Messner, A., Mokwa, W., and Walter, P. Implantation and Explantation of a Wireless Epiretinal Retina Implant Device: Observations during the EPIRET3 Prospective Clinical Trial. Investigative Ophthalmology & Visual Science, 50(6), pp. 3003-3008, 2009. Rubinstein, J. T., Spelman, F. A., Soma, M, and Suesserman, M. F. Current Density Profiles of Surface Mounted and Recessed Electrodes for Neural Prostheses. IEEE Transactions on Biomedical Engineering, BME-34(11), pp. 864-875, 1987. Ryu, S. B., Ye, J. H., Lee, J. S., Goo, Y S., Kirn, C.H., and Kirn, K. H. Electrically-evoked Neural Activities of rd! Mice Retinal Ganglion Cells by Repetitive Pulse Stimulation. The Korean Journal of Physiology and Pharmacology, 13(6), pp. 443, 2009. Sagdullaev, B. T., Aramant, R. B., Seiler, M. J., Woch, G., and McCall, M. A. Retinal Transplantation-Induced Recovery of Retinotectal Visual Function in a Rodent Model of Retinitis Pigrnentosa. Investigative Ophthalmology & Visual Science, 44( 4), pp. 1686-1695, 2003. Sakaguchi, H., Fujikado, T., Fang, X., Kanda, H., Osanai, M., Nakauchi, K., Ikuno, Y, Kamei, M., Yagi, T., Nishimura, S., Ohji, M., Yagi, T., and Tano, Y Transretinal Electrical Stimulation with a Suprachoroidal Multichannel Electrode in Rabbit Eyes. Japanese Journal of Ophthalmology, 48(3), pp. 256-261, 2004. Santos, A., Humayun, M. S., de Juan, Jr., E., Greenberg, R. J., Marsh, M. J., Klock, I. B., and Milam, A.H. Preservation of the inner retina in retinitis pigrnentosa: Amorphometric analysis. Archives of Ophthalmology, 115(4), pp. 511-515, 1997. Schaldach, M., Ruhmann, M., Weik!, A., and Hardt, R. Sputter-Deposited TiN Electrode Coatings for Superior Sensing and Pacing Performance. Pacing and Clinical Electrophysiology, 13(12), pp. 1891-1895, 1990. Schiefer, M. A. and Grill, W. M. Sites of neuronal excitation by epiretinal electrical stimulation. IEEE Transactions on Neural Systems and Rehabilitation Engineering, 14(1), pp. 5-13, 2006. 227 Schmidt, E. M, Bak, M. J., Hambrecht, F. T., Kufta, C. V, O'Rourke, D. K., and Vallabhanath, P. Feasibility of a visual prosthesis for the blind based on intracortical micro stimulation of the visual cortex. Brain, 119(2), pp. 507-522, 1996. Sekirnjak, C., Hottowy, P., Sher, A, Dabrowski, W., Litke, A M., and Chichilnisky, E. J. Electrical Stimulation of Mammalian Retinal Ganglion Cells With Multielectrode Arrays. Journal of Neurophysiology, 95(6), pp. 3311-3327, 2006. Sekirnjak, C., Hottowy, P., Sher, A., Dabrowski, W., Litke, A M., and Chichilnisky, E. J. High Resolution Electrical Stimulation of Primate Retina for Epiretinal Implant Design. The Journal of Neuroscience, 28(17), pp. 4446-4456, 2008. Sekirnjak, C., Hulse, C., Jepson, L. H., Hottowy, P., Sher, A, Dabrowski, W., Litke, A M., and Chichilnisky, E. J. Loss of Responses to Visual But Not Electrical Stimulation in Ganglion Cells of Rats With Severe Photoreceptor Degeneration.Journal ofN europhysiology, 102( 6), pp. 3260- 3269, 2009. Shah, S., Hines, A, Zhou, D., Greenberg, R. J., Humayun, M. S., and Weiland, J. D. Electrical properties of retinal-electrode interface. Journal ofN eural Engineering, 4( I), pp. S24-S29, 2007. Shannon, R. V Amodei of safe levels for electrical stimulation. IEEE Transactions on Biomedical Engineering, 39( 4), pp. 424-426, 1992. Shanna, A, Rieth, L., Tathireddy, P., Harrison, R., Oppermann, H., Klein, M., Topper, M., Jung, E., Normann, R., Clark, G., and Solzbacher, F. Evaluation of the packaging and encapsulation reliability in fully integrated, fully wireless 100 channel Utah Slant Electrode Array (USEA): Implications for long term functionality. Sensors and Actuators A: Physical, 188, pp. 167-172, 2012. Sit, J.-J. and Sarpeshkar, R. A Low-Power Blocking-Capacitor-Free Charge-Balanced Electrode Stimulator Chip With Less Than 6 nA DC Error for 1-mA Full-Scale Stimulation. IEEE Transactions on Biomedical Circuits and Systems, 1(3), pp. 172-183, 2007. Stanga, P. E., Hafezi, F., Sahel, J.-A., da Cruz, L., Merlini, F., Coley, B., and Greenberg, R. J. Patients blinded by outer retinal dystrophies are able to perceive color using the Argus II Retinal Prosthesis. InARVO 2011, 2011. Stanga, P. E., Jose A. Sahel, J., daCruz, L., Hafezi, F., Merlini, F., Coley, B., Greenberg, R. J., and Group, A I. S. Patients blinded by outer retinal dystrophies are able to perceive simultaneous colors using the Argus® II Retinal Prosthesis System. Investigative Ophthalmology & Visual Science, 53(14), pp. 6952--0952, 2012. Stein, L., Roy, K., Lei, L., and Kaushal, S. Clinical gene therapy for the treatment of RPE65- associated Leber congenital amaurosis. Expert Opinion on Biological Therapy, 11(3), pp. 429- 439, 2011. Stiles, N. R. B., Mcintosh, B. P., Nasiatka, P. J., Hauer, M. C., Weiland, J. D., Humayun, M. S., and 228 Tanguay, Jr., A R. An intraocular camera for retinal prostheses: Restoring sight to the blind. In Optical Processes In Microparticles And Nanostructures: A Festschrift Dedicated to Richard Kounai Chang on His Retirement from Yale University, vol. 6, pp. 385-429. World Scientific Publishing Co. Pte. Ltd., Singapore, 2011. Sting!, K., Bartz-Schmidt, K. U., Besch, D., Braun, A, Bruckrnann, A., Gekeler, F., Greppmaier, U., Hipp, S., H6rtd6rfer, G., Kemstock, C., Koitschev, A, Kusnyerik, A, Sachs, H., Schatz, A, Sting!, K. T., Peters, T., Wilhelm, B., and Zrenner, E. Artificial vision with wirelessly powered subretinal electronic implant alpha-IMS. Proceedings of the Royal Society B: Biological Sciences, 280(1757), pp. 20130077, 2013. Stronks, H. C. and Dagnelie, G. The functional performance of the Argus II retinal prosthesis. Expert Review of Medical Devices, 11(1), pp. 23-30, 2014. Suber, M L., Pittler, S. J., Qin, N., Wright, G. C., Holcombe, V, Lee, R. H., Craft, C. M., Lolley, R. N., Baehr, W., and Hurwitz, R. L. Irish setter dogs affected with rod/cone dysplasia contain a nonsense mutation in the rod cGMP phosphodiesterase beta-subunit gene. Proceedings of the National Academy of Sciences, 90(9), pp. 3968-3972, 1993. Sue, A, Tran, P., Wong, P., Li, Q., and Carter, P. Time-domain fmite element models of electrochemistry in intracochlear electrodes. In 2013 35th Annual International Conference of the IEEE Engineering in Medicine and Biology Society (EMBC), pp. 1554-1557, 2013. Suesserman, M. F., Spelman, F. A., and Rubinstein, J. T. In vitro measurement and characterization of current density profiles produced by nonrecessed, simple recessed, and radially varying recessed stimulating electrodes. IEEE Transactions on Biomedical Engineering, 38(5), pp. 401- 408, 1991. Sui, X., Huang, Y, Chai, X., and Wang, G. 3D fmite element analysis of the electric field generated by epi-retinal MEMS electrodes. In 2012 IEEE Biomedical Circuits and Systems Conference (BioCAS), pp. 37-40, 2012. Tan, J. S. L., Wang, J. J., Flood, V, Rochtchina, E., Smith, W., and Mitchell, P. Dietary Antioxidants and the Long-term Incidence of Age-Related Macular Degeneration: The Blue Mountains Eye Study. Ophthalmology, 115(2), pp. 334-341, 2008. Tassicker, G. E. Retinal stimulator. Washington, DC, 1956. Teixeira, L., Rodrigues, C., and Prior, C. A charge-redistribution based controller for keeping charge balance in neural stimulation. In 2015 IEEE 58th International Midwest Symposium on Circuits and Systems (MWSCAS), pp. 1-4, 2015. Thompson, R. W., Barnett, G. D., Hurnayun, M. S., and Dagnelie, G. Facial Recognition Using Simulated Prosthetic Pixelized Vision. Investigative Ophthalmology & Visual Science, 44(11), pp. 5035-5042, 2003. Tran, N., Bai, S., Yang, J., Chun, H., Kavehei, 0., Yang, Y, Muktarnath, V, Ng, D., Meffin, H., 229 Halpern, M., and Skafidas, E. A Complete 256-Electrode Retinal Prosthesis Chip. IEEE Journal of Solid-State Circuits, 49(3), pp. 751-765, 2014. Tran, N., Halpern, M., Bai, S., and Skafidas, E. Crosstalk current measurements using multi electrode arrays in saline. InEngineering in Medicine and Biology Society (EMBC), 2012Annual International Conference of the IEEE, pp. 3021-3024. IEEE, San Diego, CA, USA, 2012. Tungjitkusolrnun, S., Woo, E. J., Cao, H., Tsai, J.-Z., Vorperian, V R., and Webster, J. G. Finite element analyses of uniform current density electrodes for radio-frequency cardiac ablation. IEEE Transactions on Biomedical Engineering, 47(1), pp. 32-40, 2000. Vaidya, A, Borgonovi, E., Taylor, R. S., Sahel, J.-A, Rizzo, S., Stanga, P E., Kukreja, A, and Walter, P The cost-effectiveness of the Argus II retinal prosthesis in Retinitis Pigrnentosa patients. BMC Ophthalmology, 14, pp. 49, 2014. van Rheede, J. J., Kerman!, C., and Hicks, S. L. Simulating prosthetic vision: Optimizing the information content ofa limited visual display. Journal of Vision, 10(14), pp. 32, 1-15, 2010. Veraart, C., Raftopoulos, C., Mortimer, J. T., Delbeke, J., Pins, D., Michaux, G., Vanlierde, A, Parrini, S., and Wanet-Defalque, M.-C. Visual sensations produced by optic nerve stimulation using an implanted self-sizing spiral cuff electrode. Brain Research, 813( 1 ), pp. 181-186, 1998. Veratti, G. Osservazioni fisico-mediche intorno alla elettricita. Lelio dalla Volpe, Bologna, 1748. Vetter, R. J., Williams, J. C., Hetke, J. F., Nunamaker, E. A, and Kipke, D. R. Chronic neural recording using silicon-substrate microelectrode arrays implanted in cerebral cortex. IEEE Transactions on Biomedical Engineering, 51(6), pp. 896-904, 2004. Villalobos, J., Allen, P J., McCombe, M. F., Ulaganathan, M., Zamir, E., Ng, D. C., Shepherd, R. K., and Williams, C. E. Development of a surgical approach for a wide-view suprachoroidal retinal prosthesis: evaluation of implantation trauma. Grae/e's Archive for Clinical and Experimental Ophthalmology, 250(3), pp. 399-407, 2012. Villalobos, J., Nayagam, D. A X., Allen, P J., McKelvie, P, Luu, C. D., Ayton, L. N., Freemantle, A L., McPhedran, M., Basa, M., McGowan, C. C., Shepherd, R. K., and Williams, C. E. A Wide Field Suprachoroidal Retinal Prosthesis Is Stable and Well Tolerated Following Chronic Implantation. Investigative Ophthalmology & Visual Science, 54(5), pp. 3751-3762, 2013. Walker, J. S. Neuropsychological Assessment. In Wiley Encyclopedia of Forensic Science. John Wiley & Sons, Ltd, 2009. Wang, B., Petrossians, A, and Weiland, J. D. Reduction of Edge Effect on Disk Electrodes by Optimized Current Waveform. IEEE Transactions on Biomedical Engineering, 61(8), pp. 2254- 2263, 2014. Wang, B., and Weiland, J. D. Analysis of the Peak Resistance Frequency Method for Extracting Tissue Resistance from Impedance Spectroscopy. IEEE Transactions on Biomedical 230 Engineering, in press. Wang, B. and Weiland, J. D. Reduction of current density at disk electrode periphery by shaping current pulse edges. In 2012 Annual International Conference of the IEEE Engineering in Medicine and Biology Society (EMBC), pp. 5138-5141. IEEE, San Diego, CA, USA, 2012. Wang, B., and Weiland, J. D. Resistivity Profiles of Wild-type, rdl, and rdl 0 Mouse Retina. In 2015 Annuallnternational Conference of the IEEE Engineering in Medicine and Biology Society (EMBC), pp. 5138-5141. IEEE, Milan, Italy, 2015. Wang, J. J., Mitchell, P., and Klein, R. Epidemiology of Age-Related Macular Degeneration Early in the 21st Century. In Retinal Degenerations, pp. 23-59. Humana Press, 2007. Wang, K., Fishman, H. A, Dai, H., and Harris, J. S. Neural Stimulation with a Carbon Nanotube Microelectrode Array. Nano Letters, 6(9), pp. 2043-2048, 2006. Wang, Y-Z., Wilson, E., Locke, K. G., and Edwards, A. 0. Shape Discrimination in Age-Related Macular Degeneration. Investigative Ophthalmology & Visual Science, 43(6), pp. 2055-2062, 2002. Waschkowski, F., Hesse, S., Rieck, A. C., Lohmann, T., Brockmann, C., Laube, T., Bomfeld, N., Thumann, G., Walter, P., Mokwa, W., Johnen, S., and Roessler, G. Development of very large electrode arrays for epiretinal stimulation (VLARS). Biomedical Engineering Online, 13(1), pp. 11, 2014. Weiland, J. D., Anderson, D. J., and Humayun, M. S. In vitro electrical properties for iridium oxide versus titanium nitride stimulating electrodes. IEEE Transactions on Biomedical Engineering, 49(12), pp. 1574-1579, 2002. Weiland, J. D., Cho, AK., and Humayun, M. S. Retinal Prostheses: Current Clinical Results and Future Needs. Ophthalmology, 118(11), pp. 2227-2237, 2011. Weiland, J. D. and Humayun, M. S. Retinal Prosthesis. IEEE Transactions on Biomedical Engineering, 61(5), pp. 1412-1424, 2014a. Weiland, J. D. and Humayun, M. S. Retinal Prosthesis. IEEE Transactions on Biomedical Engineering, 61(5), pp. 1412-1424, 2014b. Weiland, J. D., Kimock, F. M., Yehoda, J.E., Gill, E., Mcintosh, B. P., Nasiatka, P. J., and Tanguay, A R. Chip-scale packaging for bioelectronic implants. In 2013 6th International IEEE/EMBS Conference on Neural Engineering (NER), pp. 931-936, 2013. Weiland, J. D., Liu, W., and Humayun, M. S. Retinal Prosthesis. Annual Review of Biomedical Engineering, 7(1), pp. 361-401, 2005. Weitz, A C. Improving stimulation strategies for epiretinal prostheses. Ph.D. Thesis. University of Southern California, Los Angeles, CA, USA, 2013. 231 Weitz, A. C., Behrend, M. R., Humayun, M. S., Chow, R. H., and Weiland, J. D. Interphase gap decreases electrical stimulation threshold of retinal ganglion cells. In Engineering in Medicine and Biology Society, EMBC, 2011 Annual International Conference of the IEEE, pp. 6725-6728. IEEE, Boston, MA, 2011. Weitz, A. C., Behrend, M. R., Lee, N. S., Klein, R. L., Chiodo, V A., Hauswirth, W. W., Humayun, M. S., Weiland, J. D., and Chow, R.H. Imaging the response of the retina to electrical stimulation with genetically encoded calcium indicators. Journal of Neurophysiology, 109(7), pp. 1979- 1988, 2013. Weitz, A. C. and Weiland, J. D. Visual Prostheses. In Neural Computation, Neural Devices, and Neural Prosthesis, pp. 157-188. Springer New York, 2014. Wesselink, W. A., Holsheimer, J., and Boom, H. B. K. Analysis of current density and related parameters in spinal cord stimulation. IEEE Transactions on Rehabilitation Engineering, 6(2), pp. 200-207, 1998. Wessling, B., Mokwa, W., and Schnakenberg, U. Sputtered Ir Films Evaluated for Electrochemical Performance I. Experimental Results. Journal of The Electrochemical Society, 155(5), pp. F61- F65, 2008. Wiley, J. D. and Webster, J. G. Analysis and Control of the Current Distribution under Circular Dispersive Electrodes. IEEE Transactions on Biomedical Engineering, BME-29(5), pp. 381-385, 1982. Wilke, R., Gabel, V-P., Sachs, H., Schmidt, K.-U. B., Gekeler, F., Besch, D., Szurman, P., Stett, A., Wilhelm, B., Peters, T., Harscher, A., Greppmaier, U., Kibbe!, S., Benav, H., Bruckrnann, A., Sting!, K., Kusnyerik, A., and Zrerrner, E. Spatial Resolution and Perception of Patterns Mediated by a Subretinal 16-Electrode Array in Patients Blinded by Hereditary Retinal Dystrophies. Investigative Ophthalmology & Visual Science, 52(8), pp. 5995--0003, 20lla. Wilke, R. G. H., Moghadam, G. K., Lovell, N. H., Suaning, G. J., and Dokos, S. Electric crosstalk impairs spatial resolution of multi-electrode arrays in retinal implants. Journal of Neural Engineering, 8(4), pp. 046016, 20llb. Williams, J. C., Hippensteel, J. A., Dilgen, J., Shain, W., and Kipke, D. R. Complex impedance spectroscopy for monitoring tissue responses to inserted neural implants. Journal of Neural Engineering, 4(4), pp. 410-423, 2007. Wilson, B. S. and Dorman, M. F. Cochlear implants: current designs and future possibilities. Journal of Rehabilitation Research and Development, 45(5), pp. 695-730, 2008. Wilson, M T., Elbohouty, M., Voss, L. J., and Steyn-Ross, D. A. Electrical impedance of mouse brain cortex in vitro from 4.7 kHz to 2.0 MHz. Physiological Measurement, 35(2), pp. 267-281, 2014. Wongsarnpigoon, A., Woock, J. P., and Grill, W. M. Efficiency Analysis of Waveform Shape for 232 Electrical Excitation of Nerve Fibers. IEEE Transactions on Neural Systems and Rehabilitation Engineering, 18(3), pp. 319-328, 2010. Xiao, X., Wang, J., Liu, C., Carlisle, J. A, Mech, B., Greenberg, R., Guven, D., Freda, R., Humayun, M. S., Weiland, J. D., and Auciello, 0. In vitro and in vivo evaluation of ultrananocrystalline diamond for coating of implantable retinal microchips. Journal of Biomedical Materials Research Part B: Applied Biomaterials, 77B(2), pp. 273-281, 2006. Yamato, H., Ohwa, M., and Wernet, W. Stability of polypyrrole and poly(3,4- ethylenedioxythiophene) for biosensor application. Journal of Electroanalytical Chemistry, 397(1-2), pp. 163-170, 1995. Yanai, D., Weiland, J. D., Mahadevappa, M., Greenberg, R. J., Fine, I., and Humayun, M. S. Visual Performance Using a Retinal Prosthesis in Three Subjects With Retinitis Pigrnentosa. American Journal of Ophthalmology, 143(5), pp. 820-827.e2, 2007. Yue, L., Falabella, P, Christopher, P, Wuyyuru, V, Dorn, J., Schor, P, Greenberg, R. J., Weiland, J. D., and Humayun, M. S. Ten-Year Follow-up of a Blind Patient Chronically Implanted with Epiretinal Prosthesis Argus I. Ophthalmology, in press. Zhou, D. D., Dorn, J. D., and Greenberg, R. J. The Argus® II retinal prosthesis system: An overview. In Multimedia and Expo Workshops (ICMEW), 2013 IEEE International Conference on, pp. 1- 6. IEEE, San Jose, CA, USA, 2013. Zhou, D. D. and Greenberg, R. J. Microelectronic Visual Prostheses. In Implantable Neural Prostheses 1, pp. 1-42. Springer US, 2009. Zrenner, E. Fighting Blindness with Microelectronics. Science Translational Medicine, 5(210), pp. 210psl6-210psl6, 2013a. Zrenner, E. List of Retinal Implants: Developments in Electronic Visual Prosthetics (as of 05-12- 2013). Institute for Ophthalmic Research Tuebingen (http://www.eye- tuebingen.de/zrenner/retimplantlist/), 2013b. Zrenner, E., Bartz-Schmidt, K. U., Benav, H., Besch, D., Brockmann, A, Gabel, V-P, Gekeler, F., Greppmaier, U., Harscher, A., Kibbe!, S., Koch, J., Kusnyerik, A, Peters, T., Sting!, K., Sachs, H., Stett, A, Szurman, P, Wilhelm, B., and Wilke, R. Subretinal electronic chips allow blind patients to read letters and combine them to words. Proceedings of the Royal Society B: Biological Sciences, 278(1711), pp. 1489-1497, 2011. Zrenner, E., Stett, A, Weiss, S., Aramant, R. B., Guenther, E., Kohler, K., Miliczek, K.-D., Seiler, M. J., and Haernrnerle, H. Can subretinal microphotodiodes successfully replace degenerated photoreceptors? Vision Research, 39(15), pp. 2555-2567, 1999. 233

Abstract (if available)

Abstract The aim of this thesis is to attain better understanding of the electrode-tissue interface of the epiretinal prosthesis and neural prosthetic devices in general. Three studies explore this interface from different aspects. The overall goal is to improve this interface to produce safe and efficient electrical stimulation and also provide the basis for computational models to better understand electrical stimulation targeted at the retina. ❧ A novel method is described and analyzed to design current waveform input to reduce the edge effect—the primary reason for non-uniform current density distribution on electrodes. Finite element modeling and mathematical analysis showed that waveform design can reduce the edge effect on disk electrode without the need to alter the electrode’s geometry. Current waveforms with a slower rise to steady-state level compared to the abrupt rectangular step can reduce the current density non-uniformity by allowing current density to redistribute over time. Numeric method optimized the design for the waveform, which can be approximated via a RC circuit. The approximation of the optimized waveforms was tested in a pulsing experiment. The results showed reduced corrosion on the edge of platinum disk electrodes, therefore demonstrating the effectiveness of the waveform shaping method. ❧ The peak resistance frequency (PRF) method—a simple method to extract tissue resistance from impedance spectroscopy measurement of the electrode-tissue interface—was explored for its mechanisms and inherent properties. The PRF method uses a variable frequency point at which the impedance phase is most resistive to estimate the tissue resistance. The previous study that first proposed the method showed that it works very accurately, compared to the large deviation of estimations from fixed frequency points. In this study, theoretical analysis and computational simulation reveal that the PRF method is only a good approximation for the tissue resistance. The PRF method has an inherent deviation that varies depending on the idealness of the electrode-tissue interface but is nevertheless correctable. Further simulations tested the realistic limitation of measurement noise and frequency sampling, and showed that the PRF method works reasonably well and reliable under realistic conditions. This work provides a solid theoretical foundation for the PRF method and means to correct the results when the parameters of the interface are not ideal. ❧ As a first step of electrical mapping within the retina tissue, the resistivity profiles of healthy and degenerate retina were measured, utilizing the aforementioned PRF method. The study provided data on both healthy and degenerate mice retinas for the first time, which are relevant models for neurophysiology study of retinal prostheses. The experimental results show that the peak resistivity decreases with degeneration. Also, the resistivity profiles were thinner in degenerate retina and the thinning agreed with histology data. Therefore, the changes in tissue properties need to be taken into account for modeling study of retinal prosthesis.

Linked assets

University of Southern California Dissertations and Theses

Conceptually similar

PDF

Towards a high resolution retinal implant

PDF

Manipulation of RGCs response using different stimulation strategies for retinal prosthesis

PDF

Prosthetic vision in blind human patients: Predicting the percepts of epiretinal stimulation

PDF

Saliency based image processing to aid retinal prosthesis recipients

PDF

Stimulation strategies to improve efficiency and temporal resolution of epiretinal prostheses

PDF

Effect of continuous electrical stimulation on retinal structure and function

PDF

Cortical and subcortical responses to electrical stimulation of rat retina

PDF

Improving stimulation strategies for epiretinal prostheses

PDF

Electrical stimulation of degenerate retina

PDF

An in vitro model of a retinal prosthesis

PDF

Next generation neural interfaces for neural modulation and prosthesis

PDF

Strategies for improving mechanical and biochemical interfaces between medical implants and tissue

PDF

Oxygen therapy for the treatment of retinal ischemia

PDF

Viscoelastic characterization of soft biological materials

PDF

Viewing the picture we paint

PDF

Vibrational sum frequency generation studies of electrode interfaces

PDF

Dependence of rabbit retinal synchrony on visual stimulation parameters

PDF

Understanding the degenerate retina's response to electrical stimulation: an in vitro approach

PDF

Electrical signals to assess cardiovascular phenomenon

PDF

Multi-region recordings from the hippocampus with conformal multi-electrode arrays

Asset Metadata

Creator Wang, Boshuo (author)

Core Title Investigation of the electrode-tissue interface of retinal prostheses

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Biomedical Engineering

Publication Date 02/16/2016

Defense Date 11/19/2015

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag biomedical electrodes,electrode-tissue interface,impedance spectroscopy,neural engineering,OAI-PMH Harvest,retina degeneration,retinal prostheses,tissue resistance

Format application/pdf (imt)

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Weiland, James D. (committee chair), Chow, Robert H. (committee member), Humayun, Mark S. (committee member), Song, Dong (committee member)

Creator Email boshuowa@usc.edu,boshuowang@gmail.com

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c40-208403

Unique identifier UC11277147

Identifier etd-WangBoshuo-4101.pdf (filename),usctheses-c40-208403 (legacy record id)

Legacy Identifier etd-WangBoshuo-4101.pdf

Dmrecord 208403

Document Type Dissertation

Format application/pdf (imt)

Rights Wang, Boshuo

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the a...

Repository Name University of Southern California Digital Library

Repository Location USC Digital Library, University of Southern California, University Park Campus MC 2810, 3434 South Grand Avenue, 2nd Floor, Los Angeles, California 90089-2810, USA