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Computational approaches to establish safety and efficacy assessments of electrical stimulation to peripheral nerve

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Computational approaches to establish safety and efficacy assessments of electrical stimulation to peripheral nerve

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Content Computational Approaches to Establish Safety and
Efficacy Assessments of Electrical Stimulation to
Peripheral Nerve
by
Jinze Du
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
(ELECTRICAL ENGINEERING)
December 2024
Copyright 2024 Jinze Du

With profound gratitude, I dedicate this thesis to those who have been
instrumental in this journey. To Dr. Lazzi, for his insightful guidance and
unwavering support; to my parents, whose encouragement has been a constant
source of strength; to my wife, Shanshan, for her enduring love and
understanding; and to my daughter, Celia, whose joy and innocence inspire
me every day. . .
ii

Table of Contents
Dedication ............................................................................................. ii
List of Tables .......................................................................................... vi
List of Figures ......................................................................................... vii
Abstract................................................................................................ xv
Chapter 1: Introduction............................................................................... 1
1.1 Electrical Impulses in the Human Body ................................................... 1
1.2 Nerve Electrical Stimulation ................................................................ 2
1.3 Peripheral Nerve Stimulation ............................................................... 4
1.4 Safety Matters ............................................................................... 6
1.4.1 Shannon Criteria .................................................................... 7
1.4.2 Limitation of Shannon Criteria..................................................... 8
1.5 Ideal properties of an updated PNS safety criteria ....................................... 10
Chapter 2: Computational Platform for Peripheral Nerve Stimulation ........................... 13
2.1 Introduction.................................................................................. 14
2.2 Design and Implementation................................................................. 19
2.2.1 Overview of the PAM pipeline ..................................................... 19
2.2.2 3D Model Construction ............................................................. 20
2.2.2.1 Nerve Model Construction ............................................... 21
2.2.2.2 Cuff Electrode Model Construction ..................................... 21
2.2.3 Model Characterization and Adaptive Meshing ................................... 22
2.2.4 Electric Network Generation ....................................................... 24
2.2.5 Python Admittance Solver.......................................................... 26
2.2.6 PAM to NEURON .................................................................. 27
2.3 Results........................................................................................ 30
iii

2.3.1 Accuracy Validation against point-source approximation and communication
with NEURON ...................................................................... 31
2.3.2 Validation against COMSOL ....................................................... 32
2.3.3 PAM’s realistic peripheral nerve modeling vs other computational models..... 34
2.3.3.1 Model construction ....................................................... 34
2.3.3.2 Results comparison ....................................................... 36
2.3.4 PAM’s communication with NEURON ............................................ 37
2.4 Data Availability............................................................................. 38
2.5 Discussion .................................................................................... 38
Chapter 3: Computational modeling for the distance between cuff electrodes’ effect on current
density distribution....................................................................... 43
3.1 INTRODUCTION ........................................................................... 44
3.2 METHODS .................................................................................. 47
3.2.1 Building Nerve Model using CNN Segmentation of Peripheral Nerve Crosssectional Images ..................................................................... 47
3.2.2 Model Building and Admittance Method.......................................... 47
3.2.3 Current Density: Interpolation of Voltage Values ................................ 50
3.3 RESULTS .................................................................................... 51
3.3.1 Current Distribution inside the Nerve ............................................. 51
3.3.2 Current Penetration Depth and Electrode Separation............................ 52
3.4 DISCUSSION ................................................................................ 52
Chapter 4: Electrical Stimulation Induced Current Distribution in Peripheral Nerves Varies
Significantly with the Extent of Nerve Damage: A Computational Study Utilizing
Convolutional Neural Network and Realistic Nerve Models........................... 55
4.1 Introduction.................................................................................. 56
4.2 Methods ...................................................................................... 60
4.2.1 CNN Segmentation of Peripheral Nerve Cross-sections .......................... 61
4.2.2 Cell-wise Damage Evaluation Metrics ............................................. 62
4.2.3 Nerve Image Selection .............................................................. 63
4.2.4 Model Building and Admittance Method.......................................... 64
4.3 Results........................................................................................ 68
4.3.1 Current Distribution inside Two Nerve Models ................................... 68
4.3.2 Current Density in Different Tissue Types ........................................ 68
4.3.3 Current Density Values to Charge Density Per Phase............................ 69
iv

4.3.4 Nerve Damage Analysis............................................................. 71
4.3.5 Charge Density Versus Damage Extent............................................ 72
4.4 Discussion .................................................................................... 72
Chapter 5: Toward Safety Protocols for Peripheral Nerve Stimulation (PNS): a Computational
and Experimental Approach ............................................................. 77
5.1 Introduction.................................................................................. 78
5.1.1 Existing safety protocol for PNS ................................................... 79
5.1.2 Limitations of the existing protocol................................................ 81
5.1.3 Ideal properties of an updated safety criteria ..................................... 83
5.2 Methods ...................................................................................... 84
5.2.1 Experiments and microscopy ....................................................... 86
5.2.2 Image segmentation and Damage Analysis........................................ 87
5.2.3 Charge density from AM modeling and MLP prediction......................... 88
5.2.3.1 Nerve Sample selection(Build training dataset) ........................ 89
5.2.3.2 AM Modeling on the selected samples .................................. 90
5.2.3.3 Neural network training and charge density prediction ................ 92
5.3 Results........................................................................................ 95
5.3.1 Cell-wise charge density prediction results ........................................ 96
5.3.2 Correlation curves between cell damage and charge density ..................... 97
5.3.3 Comparison with Shannon Criteria and future safety criteria ................... 100
5.4 Discussion .................................................................................... 101
Chapter 6: Summary and Future Steps.............................................................. 104
v

List of Tables
2.1 TISSUE PROPERTIES ..................................................................... 36
3.1 TISSUE PROPERTIES ..................................................................... 48
5.1 TISSUE PROPERTIES ..................................................................... 91
vi

List of Figures
1.1 Electrical signaling in human neurons during an action potential. A stimulus causes
sodium ions (Na) to flow into the neuron, generating an electrical impulse by depolarizing the membrane potential. This triggers potassium ions (K) to exit the cell,
repolarizing the membrane and restoring the electrical balance. Ion pumps then reset
the ion distribution, preparing the neuron for the next electrical signal. Figure from
https://www.uwa.edu.au/; .................................................................. 1
1.2 Nerve electrical stimulation to help relieve pain. Figure from https://www.njpaindoc.com; 5
1.3 An illustration of Shannon Criteria Usage, K is the Shannon value and generally a
k>2 is considered unsafe. ................................................................... 7
1.4 McCreery et al. manually assessed the fiber damage and summarized the experimental data into charts. Figure from McCreery et al. (1990); ................................ 9
1.5 A study from Warren Grill et al in 2009 designed six different electrodes with same
surface area but different perimeter, by Shannon’s assumption these six electrode
would have the same charge density in his equation. However, the results suggested
otherwise. Figure from Wei and Grill (2009); ............................................. 10
1.6 Another study showed similar conclusion with microelectrodes with different tip radius, although these microelectrodes have the same surface area, their induced current
densities varied a lot. Figure from McIntyre and Grill (2001);........................... 11
1.7 What are the ideal properties are for an updated PNS safety criteria? ................. 12
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2.1 Overview of PAM’s process for creating and simulating biophysical nerve models with
customized electrodes and seamlessly integrating corresponding NEURON models.
(A) The pipeline begins with arbitrary nerve microscopy images. Utilizing a previously developed Convolutional Neural Network (CNN) for segmentation, PAM constructs the nerve model from these images, incorporating customizable electrodes to
match the user’s desired configuration. Following this, PAM performs material characterization, meshing, and electrical property analysis. (B) Starting from the same
segmented images, PAM identifies precise fiber locations through image processing,
enabling the creation of neurons at these exact points using NEURON modeling.
This precision allows the NEURON model to be accurately stimulated by extracellular voltages calculated by the PAM solver, facilitating bi-directional communication.
(C) Further insights into PAM’s capability to begin with nerve microscopy images,
constructing realistic nerve models with varied electrode shapes and placements, as
well as matching the biophysical nerve model to the NEURON model in a realistic
setup, based on actual nerve microscopy images. ........................................ 17
2.2 Example 2D slice of a meshed model of a peripheral nerve, including (A) an unmeshed
slice of the model derived from nerve images, using uniform cubic voxels, and (B)
illustration of the same model after applying the multiresolution meshing algorithm
with a maximum size of 64 voxels. Boundary voxels remain a relatively low resolution
(C): Illustration of the mesh simplification process on a 3D mesh model of a rat sciatic
nerve cross-section: In locations proximal to the contact electrode voxel size is kept
small and the voxel size is increased further away from the nerve periphery where
fine resolution is unnecessary. In this example, the mesh size is reduced by over 90
percent........................................................................................ 19
2.3 AM platform structure overview: after the Admittance Method model network is
constructed, the PAM solver constructs the sparse admittance matrix, applies multithreading, and performs data post-processing and interpolation. Results can be used
for a subsequent time step for time-dependent problems in AM or passed as input to
other computational strategies, such as NEURON in the case of external stimulation. 24
2.4 Other computational platforms resorting to AM communicate with NEURON through
an intermediate step, as shown in (A). This intermediate step significantly affects the
overall computation efficiency. With our proposed Python solver AM platform, neuron simulation tools can directly communicate with the Admittance Method through
Python’s shared memory, where all the nodal values and model information are
stored, as shown in (B). .................................................................... 25
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2.5 An example used to validate the proposed PAM platform, (A): models of a simplified
peripheral nerve are constructed and meshed for both COMSOL and PAM. The nerve
model is constructed by positioning smaller cylinders inside a larger cylinder cover.
Three cuff electrodes surround the nerve model, with the top electrode serving as
the source electrode and the bottom electrode serving as the current return. A 2mA
current source is applied. In order to compare the two simulation results, voltage
distribution on the XY plane at the source electrode level from COMSOL (B) and
from PAM (C) are shown. Similarly, the voltage distribution along YZ plane at the
nerve center from COMSOL (D) and from PAM (E) are shown. ....................... 27
2.6 To assess the PAM platform’s precision, we utilized a basic sphere model with a core
source node surrounded by a spherical external ground, mirroring the point source
approximation (PSA) used as a ground truth. (A) A 2D slice of the sphere model
illustrates the comparison between PAM and the benchmark through a heatmap,
with voltage values marked by dots in varying color shades for visual clarity. (B)
To further clarify local field potential accuracy, a line plot extracted along the 2D
slice’s center is shown to the right. This plot’s analysis indicates a normalized root
mean square error (NRMSE) of 0.065 between PAM and the benchmark, affirming
the model’s accuracy. ....................................................................... 30
2.7 Leveraging the same spherical model, our objective was to further validate the efficacy of PAM’s integration with NEURON software. Within the delineated yellow box,
each dot signifies a source node selected for interaction with the single-compartment
NEURON model, as depicted in the diagram to the left. (A) This phase entails the
activation of the single-compartment NEURON model by each source node within
the yellow box, employing both PAM and point source approximation (PSA) techniques for stimulation. (B) The NEURON model’s reactions to these stimulations are
graphically displayed on the extreme right, revealing minimal discrepancy between
the outcomes of PAM and PSA approaches. This juxtaposition effectively affirms the
robustness of PAM’s communication protocol with NEURON software, underscoring
its accuracy and dependability in facilitating neural simulations. ....................... 33
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2.8 In order to construct a realistic peripheral nerve model, a Cross-sectional image of
rat sciatic nerve (A) is taken and segmented (B) into image of nerve cross-section
containing multiple fascicles populated by axons of various radii and myelin(grey
represents myelin and white represents axon). Both realistic nerve model(C) and
traditionally used simplified models(D) are constructed and simulation results are
plotted(E)(F). To further illustrate the difference between PAM results and traditional results, (G)(H) further enlarged a section of current density distribution details inside both models at the same location. It is shown that using PAM a great
amount of current density details are captured whereas inside the traditional nerve
modeling such details are missing. The solving times for the two different models are
illustrated respectively. ...................................................................... 35
2.9 Illustration of PAM’s Bi-Directional Communication Capabilities. (A) Showcases
the model involving a basic sphere and a single-compartment neuron, chosen for
its simplicity to facilitate understanding and quick validation of bi-directional communication. (B) Displays NEURON’s membrane current and electric potential responses when stimulated by points within the yellow box, representing source node
interactions as previously seen in Fig. 2.7. This figure highlights PAM’s backward
communication by directly utilizing NEURON’s computed membrane currents from
shared memory, enhancing computational efficiency and simplifying the input process to PAM’s solver. The use of shared memory notably reduces process interaction
time by more than 80%, with expectations of further reductions as model complexity
increases. ..................................................................................... 39
3.1 CNN segmentation of cross-sectional image of peripheral nerve. (a) a cross-sectional
image of rat sciatic nerve, (b) segmented image of nerve cross-section containing
multiple fascicles populated by axons of various radii and myelin(grey represents
myelin and white represents axon). ........................................................ 46
3.2 Simulation models consisting of the nerve and three cuff electrodes (shown in blue
color). Source electrodes are at the top, return electrodes are at the bottom and
floating electrodes are in the middle. Distances between source and return electrodes
are: (a) 4 mm, (b) 2 mm, (c) 1 mm. ....................................................... 49
3.3 Interpolation process to get current density value from Admittance Method output,
∆s represents unit voxel size................................................................ 50
3.4 Current density distribution in 4 mm model configuration, the xy cross-sectional
layers are plotted on different slices: (a) on the source electrode level, (b) on the
floating electrode level, (c) on the return electrode level. Unit: mA/m2
. ............... 50
x

3.5 Line plot of radial average values extracted from slices with floating electrode from
all three models.............................................................................. 53
4.1 CNN segmentation of cross-sectional image of peripheral nerve. (A) a cross-sectional
image of rat sciatic nerve, (B) segmented image of nerve cross-section containing multiple fascicles populated by axons and myelin of various morphologies (grey represents
myelin and white represents axon), (C) CNN U-Net architecture used to generate the
segmented image. ............................................................................ 57
4.2 Steps for histological measurements (A-F). (A) Confocal image of rat sciatic nerve
at 80× magnification. (B) Microscopy images were segmented into axon and myelin
labels using AxonDeepSeg. (C) Individual fibers were labeled using the watershed
algorithm. (D) Fascicles area(grey) were manually labeled from the confocal images.
(E) Fiber packing was then measured by the ratio of the area of cells within the
measurement window to the area of the fascicle(green) within that same window. (F)
Fiber density was measured by counting the number of cells(represented by different
colors) within the measurement window divided by the area of the fascicle within
that same window. When the window cuts through a fiber, the fraction of the fiber
area within the window is counted. ........................................................ 62
4.3 Sample selection for healthy and damaged nerve models. To ensure the samples used
for modeling are good general representations of their class, we use fiber density and
fiber packing to select one healthy nerve sample and one damaged nerve sample from
a total of six samples. All of the similar types of cells fall within the range of twice
the standard deviation....................................................................... 65
4.4 (A)(F) Representative nerve microscopic images are used to generate one healthy
and one damaged nerve model. (B)(G) Convolutional neural network was applied
to the images to segment the myelin and axons. The damaged nerve image has a
lower cell density. (C)(H) The segmented nerve images used to reconstructed the
nerve structures under stimulation. (E) 3D model with source and ground electrodes
is constructed. (D)(I) Current density distributions near the source electrode are
plotted for both models. It shows that damaged nerve models have a higher current
density at the nerve periphery than the healthy nerve model due to the decrease of
the cell population. .......................................................................... 67
4.5 Current density values at different tissue types of (A) healthy and (B) damaged nerve
models. The current density values on axoplasm in the damaged model are much
higher than in the healthy nerve model. ................................................... 69
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4.6 (A)(E)One healthy and one damaged segmented microscopic nerve images are selected to visualize different damaging metrics: (B)(F) Fiber packing uses the same
window but divides the total cell area inside by the window area. (D)(H) G-ratio calculates the ratio between the inner and the outer diameter of each cell. (C)(G) Fiber
density is the number of cells inside (cells cut by the window are counted fractionally)
divided by a sliding window area. .......................................................... 70
4.7 (A) Cells from two nerve samples are grouped according to their local charge density. Using two nerve samples and more than 30,000 individual cells, these cells are
grouped into two groups based on the cell-specific charge density values. (B)Empirical
distribution function plot of these two classes of cells against one of their respective
damaging indicators, which is fiber density in this case. ................................. 73
5.1 Summary of our methodology involving (1) image segmentation and algorithm-based
nerve damage quantification for comprehensive assessment of structural and morphological axonal damage, and (2) computational modeling and neural network training
for detailed exploration of cell-specific charge density distribution. The correlation
between axon damage and local stimulation charge density leads to a safety criterion
for PNS. ...................................................................................... 80
5.2 Block diagram of the image analysis approach. (a) Experimentally derived nerve
samples from Sprague Dawley rats, comprising healthy and damaged specimens, are
used for high-resolution microscopy. (b) These images are segmented using a Convolutional Neural Network (CNN) and a detailed image damage analysis for computing
cell-specific damage indicators. Panel (c) illustrates the detailed process using a small
detail of the nerve cross-section. Specifically, (c)(A) shows the microscopy image of
the cells, (c)(B) shows the corresponding cell segmentation mask, while (c)(C) and
(c)(E) demonstrate the fiber density (both manual and automatic), calculated as the
number of cells (fractionally accounted for those cut by the window) divided by the
window area. Further, (c)(D) indicates the fascicle area, representing the area occupied by each cell; (c)(F) and (c)(G) depict axon and myelin packing respectively,
calculated as the total axon/myelin area within the window divided by the window
area; and (c)(H) fiber Nearest-Neighbor (NN) area, a crucial metric that assesses the
area of each myelinated fiber’s cell within the fascicle partition. Panel (d) shows the
obtained damage metrics visualized on the segmented nerve images for clarity......... 82
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5.3 (a)(b) Selection and segmentation of two representative nerve samples, one healthy
and one damaged, based on fiber packing and fiber density metrics. (c) Construction and computational modeling of the two selected nerve samples for Admittance
Method (AM) simulation. (d) Training of a multi-layer perceptron (MLP) model
using the dataset derived from the AM simulation of the two representative samples
to predict cell-wise charge density values. The MLP model, consisting of a fourlayer fully connected neural network with hidden layers of 512, 256, and 128 neurons
respectively, utilizes a softmax activation function for charge density classification. ... 85
5.4 Depiction of the neural network model training and performance assessment. (a) Illustrates the training and validation process for our multi-layer perceptron network,
performed over 1,000 epochs. This exhaustive training led to a peak model accuracy of approximately 90%, demonstrating its ability to discern underlying patterns
and avoid overfitting. (b) Showcases the classification performance of our model
on a diverse dataset of over 100,000 cells, represented through a confusion matrix.
The matrix elucidates the model’s misclassifications primarily between neighboring
classes, with notably fewer misclassifications in the highest charge density range - a
critical determinant for gauging potential cellular damage. .............................. 93
5.5 Analysis of cell groupings based on charge density against various damage metrics.
(a) Axon packing, (b) Axon size, (c) Fiber packing, and (d) Fiber Nearest-Neighbor
(NN) ratio. The empirical distribution function plots visually represent seven distinct cellular groups, including stimulated cells, control cells, sham cells, and five
subcategories within the stimulated cell group defined by their charge density. The
curves indicate that control and sham cell groups, typically resembling healthy cells,
tend to reside on the right, whereas cell groups with higher charge densities skew
left, suggestive of morphological changes and potential damage. Most notably, cells
with a local charge density per phase exceeding 3.5 nC/cm2
consistently deviate from
other categories across all damage metrics, underscoring the potential safety threshold. 94
5.6 Comparative analysis of proposed safety criteria and traditional Shannon Criteria
for nerve stimulation. The figure illustrates the proportion of cells for each nerve
sample, identified by a unique Shannon k value, exceeding a local charge density per
phase of 3.5 nC/cm2
. The plot reveals nerve samples with Shannon k=2.11, k=0.55,
and k=0.11 and the proportion of cells within each that surpass the proposed safety
threshold. Our criteria unveil potential risks not identified by the Shannon Criteria,
providing a more nuanced understanding of the damage risk under stimulation. The
visualization underscores the advantages of our proposed method, offering a more
granular perspective that can assist experimentalists in making more informed decisions on experimental parameters. ......................................................... 98
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5.7 Empirical model demonstrating the relationship between stimulation intensity and
the risk of excessive charge density to nerve cells. This curve-fitting process visualizes
the correlation between the controllable stimulation intensity from the source cuff
electrode (x-axis) and the percentage of nerve cells exceeding our proposed safety
limit (y-axis), providing a tool for practitioners to estimate potential cell damage
during peripheral nerve stimulation. ....................................................... 99
xiv

Abstract
Peripheral Nerve Stimulation (PNS) is a widely established neurostimulation technique used
for several medical conditions, including motor function recovery, chronic pain relief, spinal cord
injury pain management, and treatment of complex regional pain syndrome. Despite the advent of
numerous neural stimulation devices, defining safe electrical stimulation limits on peripheral nerves
is still under debate. Establishing clear safety guidelines is crucial for optimizing patient outcomes
and advancing neuroprosthetic system research and development.
In this dissertation, we utilized a novel machine learning and computational bio-electromagnetics
modeling platform to establish a safety criterion that captures the effects of fields and currents induced on axons. Our approach is comprised of three steps: experimentation, model creation, and
predictive simulation. We collected high-resolution images of control and stimulated rat sciatic nerve
samples at varying stimulation intensities and performed high-resolution image segmentation. These
segmented images were used to train machine learning tools for automatic classification of morphological properties of control and stimulated PNS nerves. Concurrently, we utilized our quasi-static
Admittance Method-NEURON (AM-NEURON) computational platform to create realistic nervemodels and calculate induced currents and charges, both critical elements of nerve safety criteria.
These steps culminate in a cellular-level correlation between morphological changes and electrical
stimulation parameters. This correlation informs the determination of thresholds of electrical parameters that are found to be associated with damage, such as maximum cell charge density. The
proposed methodology and resulting criteria combines experimental findings with computational
modeling to generate a safety threshold curve which captures the relationship between stimulation
current and the potential for axonal damage. Although focused on a specific exposure condition,
xv

the approach presented here marks a step towards developing context-specific safety criteria in PNS
neurostimulation, encouraging similar analyses across varied neurostimulation scenarios.
While this research marks a significant step toward developing comprehensive safety guidelines
for peripheral nerve stimulation, it remains an early phase. Future work will include data from
additional species, such as the pig vagus nerve, to ensure that the safety standards we develop are
applicable across a range of nerve types and more representative of human physiology. Ultimately,
these efforts aim to refine and expand our models, contributing to the establishment of robust,
evidence-based safety guidelines for PNS, advancing the field of neurostimulation, and enabling the
safer design of neuroprosthetic systems for human application.
xvi

Chapter 1: Introduction
1.1 Electrical Impulses in the Human Body
The human body operates through a complex network of electrical impulses, much like an
intricate electrical circuit. Neurons and muscle cells, known as "excitable cells," play a crucial
role in generating and transmitting these signals. Similar to electrical circuits, these cells maintain
a polarized state, with the inside of the cell carrying a negative charge compared to the outside
due to the unequal distribution of ions. The movement of these ions across the cell membrane
creates an electrical signal, known as an action potential, which is essential for communication
within the nervous system. These biological signals mirror the principles of electricity, as they
propagate through the body, allowing us to perform critical functions such as movement, sensation,
and cognition.
The rapid transmission of these electrical impulses, which can reach speeds over 100 meters
per second, is essential for survival—allowing us to react to stimuli, regulate vital processes like
heartbeat and respiration, and coordinate muscle movements. Any interference with this natural
Figure 1.1: Electrical signaling in human neurons during an action potential. A stimulus causes
sodium ions (Na) to flow into the neuron, generating an electrical impulse by depolarizing the
membrane potential. This triggers potassium ions (K) to exit the cell, repolarizing the membrane
and restoring the electrical balance. Ion pumps then reset the ion distribution, preparing the neuron
for the next electrical signal. Figure from https://www.uwa.edu.au/;
1

electrical system, whether from injury, disease, or toxins, can disrupt normal bodily functions, much
like a fault in an electrical circuit can disable a device. Yet, the similarities between biological and
artificial electrical systems also open doors to therapeutic interventions. For instance, neurotoxins like Botulinum toxin can modulate electrical signaling in the body for medical and cosmetic
purposes. Furthermore, groundbreaking studies, such as those on the giant axon of squid, have
provided valuable insights into the electrical properties of nerve cells, advancing our understanding
of how the body’s electrical system functions.
This close analogy between the body’s electrical signaling and external electrical systems underpins modern medical technologies like electrical nerve stimulation. By mimicking the body’s natural
electrical impulses, these technologies offer potential solutions for managing chronic pain, restoring
muscle function, and treating neurological disorders, highlighting the profound connection between
biological and artificial electricity.
1.2 Nerve Electrical Stimulation
Nerve Electrical Stimulation (NES) has a fascinating history that spans millennia, intertwining
ancient folklore with modern scientific discoveries. The roots of NES can be traced back to ancient
Egypt around 2500 BCE, where stone carvings suggest that electric fish were used therapeutically,
possibly to treat pain. This early use of bioelectricity is also noted in texts from other ancient
cultures, including those of Greece and Rome, where species like the electric ray (torpedo fish) were
applied to various parts of the body to relieve ailments ranging from headaches to gout.
The scientific journey of NES began to take shape in the 18th century with the advent of the
Leyden jar, an early form of a capacitor, which allowed for the storage and discharge of electric
charge. Pioneers like Luigi Galvani and later, his nephew Giovanni Aldini, used this technology
in experiments that demonstrated the electrical basis of nerve impulses. Galvani’s famous 1780
experiment, in which he used electrical currents to cause a frog’s leg to twitch, laid the groundwork
for electrophysiology. This not only sparked further research into bioelectricity but also popularized
the idea that electricity could have therapeutic applications.
However, it was not until the 20th century that NES gained widespread scientific credence,
particularly with the development of the gate control theory of pain in 1965 by Ronald Melzack and
Patrick Wall. This theory proposed that certain patterns of nerve stimulation could actually block
the transmission of pain signals to the brain, offering a scientific rationale for the analgesic effects
of NES. This led to the invention of transcutaneous electrical nerve stimulation (TENS) devices in
the 1970s, which are widely used today.
2

From these ancient beginnings to modern clinical applications, NES has evolved dramatically.
Its development reflects a deepening understanding of the body’s electrical properties and their
therapeutic potentials, illustrating a unique blend of historical practices and cutting-edge science.
Today NES is a rapidly evolving field that has gained significant attention in recent years as a
promising therapeutic tool for a wide range of physiological and pathological conditions. The fundamental principle behind NES involves the initiation and propagation of action potentials, which
are critical for neural communication. By modulating ion channels, NES influences these electrical impulses, enhancing or suppressing neural activity based on the stimulation parameters such
as frequency and amplitude. This modulation facilitates synaptic plasticity, potentially improving synaptic connections through mechanisms like long-term potentiation and promoting axonal
growth, which are vital for neural repair and functional recovery. Based on such, neural stimulation
devices are useful biomedical devices for neuroscience research and have shown considerable potential for therapeutic applications for various conditions caused by illness or injury(Altinok et al.,
2009; Kathawate and Acharya, 2008; Kumar et al., 2006; Roy, 2018). For instance, deep brain
stimulators have demonstrated a therapeutic potential in memory restoration, Parkinson’s disease,
essential tremor and dystonia(Buhlmann et al., 2011; Mcintyre and Foutz, 2013; Schmidt and van
Rienen, 2012). Peripheral nerve stimulators (PNS) have been shown to be effective in the treatment
of chronic pain, motor function loss, and epilepsy (Goodall et al., 1995; Kosta et al., 2020b; Lubba
et al., 2019; Reilly, 1989). Retinal stimulation could partially restore blind patients’ vision (Loizos
et al., 2018; Paknahad et al., 2022a, 2021a, 2020a,b, 2021b, 2022b; Stang et al., 2019a). In addition, studies have demonstrated that electrical stimulation can direct optic nerve growth and
recovery (Gokoffski et al., 2021).
Beyond pain management, NES has also shown great promise in promoting wound healing and
enhancing blood flow, particularly in treatments of non-healing ulcers and in diabetic care. By
stimulating angiogenesis—the growth of new blood vessels—NES supports essential physiological
processes that accelerate healing, underscoring its utility in complex clinical scenarios and its role
in advancing regenerative medicine.
Additionally, NES plays a pivotal role in motor rehabilitation for patients recovering from neurological disorders or nerve injuries, such as strokes or spinal cord injuries. By aiding the retraining of
muscles and neural pathways, NES facilitates the restoration of movement and coordination. This
application not only highlights the adaptability of NES to various therapeutic needs but also its
profound impact on improving motor functions and overall patient independence. Through ongoing
research and clinical applications, NES continues to enhance therapeutic outcomes, exemplifying
its enduring value in healthcare and rehabilitation.
3

1.3 Peripheral Nerve Stimulation
Peripheral nerve stimulation (PNS) is an established medical intervention method that has
been extensively utilized for over six decades. PNS is acknowledged for its substantial therapeutic
potential and is used to treat various medical conditions including chronic pain, impaired motor
functions, and epilepsy Campbell and Long (1976); Mobbs et al. (2007); Nashold Jr et al. (1982).
The fundamental principle behind PNS involves the modulation of neural activity through electrical stimulation, influencing action potentials and synaptic plasticity. This modulation has shown
promise in various therapeutic applications, from pain management to promoting nerve regeneration. Despite the advent of numerous neural stimulation devices, defining safe electrical stimulation
limits on peripheral nerves remains a significant challenge in the field. Most importantly, the lack of
comprehensive safety guidelines poses a critical issue in optimizing patient outcomes and advancing
neuroprosthetic system research and development. Current safety considerations for PNS encompass a range of factors, including stimulation threshold, intensity, duration, electrode placement,
and patient-specific variables. Yet, there is a notable absence of standardized, evidence-based safety
protocols that account for the complex interplay between these factors and their long-term effects
on nerve tissue.
This gap in knowledge not only limits the full therapeutic potential of PNS but also raises
concerns about potential adverse effects from prolonged or improperly administered stimulation.
The variability in patient responses and the diversity of clinical applications further complicate the
establishment of universal safety parameters.
In this dissertation, we address this critical need by developing computational models that depict
the impact of electrical stimulation on peripheral nerves at a microscopic level. By integrating
high-performance computing, machine learning algorithms, and detailed 3D nerve models, we aim
to provide a comprehensive framework for evaluating stimulation safety and examining potential
nerve damage. Our approach combines the multiscale Admittance Method with experimental data
to predict electric current flow through nerves and accurately assess stimulation-induced effects.
This multifaceted methodology allows for a nuanced understanding of how factors such as current
amplitude and electrode position influence tissue response and potential damage.
Through this research, we aim to contribute to the development of robust, evidence-based safety
guidelines for PNS. Our findings have the potential to enhance the efficacy and safety of PNS
therapies, ultimately leading to improved patient outcomes and accelerating the advancement of
neuroprosthetic technologies.
4

Figure 1.2: Nerve electrical stimulation to help relieve pain. Figure from
https://www.njpaindoc.com;
As we look to the future, Peripheral Nerve Stimulation (PNS) stands out as a pivotal therapeutic
tool, offering a both invasive and non-invasive and effective solution with relatively minimal risks
for the treatment of chronic pain and other conditions. With ongoing technological advancements,
PNS has become safer and more efficient, increasingly positioning itself as a critical option for
patients who have not found relief through traditional treatments. The promise of PNS extends
into the realm of continuous improvement and innovation. Further research is essential to deepen
our understanding of PNS mechanisms and to develop new, more effective techniques. This pursuit
not only aims to enhance the therapy itself but also to expand its applicability and efficacy, ensuring
that PNS remains at the forefront of therapeutic options, ready to improve the quality of life for
patients around the globe.
5

1.4 Safety Matters
As the therapeutic applications of PNS expand, it becomes crucial to understand the factors
influencing its safe implementation. Ensuring patient safety and maximizing treatment efficacy are
paramount in any medical treatment, and PNS is no exception. Despite its long history of use, the
safe parameters for induced electric fields and currents through PNS remain poorly defined Günter
et al. (2019); Helm et al. (2021); Silberstein et al. (2012).
Recent advancements in biocompatible materials and precision microfabrication technologies
have enabled the development of electrodes that not only minimize potential adverse effects but
also increase the precision of electrical stimulation. This ensures targeted nerve engagement while
reducing the likelihood of stimulating surrounding tissues unnecessarily, which can lead to discomfort
or other negative side effects.
Furthermore, the safety of PNS has been significantly bolstered by detailed attention to the
electrical parameters of stimulation, such as the level, intensity, and waveform of the electrical
current, as well as the size and placement of electrodes. Precise control over these parameters is
crucial as it helps prevent nerve damage and other complications associated with excessive electrical
stimulation. The development of smaller, more efficient electrodes has been particularly impactful,
as they can be placed more accurately and cause less tissue disruption during implantation, which
is crucial for reducing the risk of infection and other postoperative complications.
Additionally, computational models have become invaluable in the field of PNS by providing predictive insights into the outcomes of nerve stimulation. These models use patient-specific anatomical
and physiological data to simulate how electrical currents will distribute through various tissues,
thus allowing for the customization of stimulation parameters tailored to each patient’s unique
needs. Such modeling helps in optimizing the safety and effectiveness of PNS by ensuring that the
stimulation is both adequate to achieve therapeutic effects and limited enough to avoid unwanted
outcomes .
The continual refinement of electrode design and stimulation waveform, along with advances in
computational modeling and precision manufacturing, underscores a broader trend towards more
personalized and safer approaches in the application of PNS technologies.
Presently, both researchers and clinicians primarily adhere to the "Shannon Criteria" Shannon
(1992), established from extensive empirical research and theoretical models. This criterion has
become a cornerstone in determining safe stimulation parameters for PNS. While these advancements have significantly improved the safety and efficacy of PNS, there remains a need for more
comprehensive and nuanced safety guidelines.
6

Figure 1.3: An illustration of Shannon Criteria Usage, K is the Shannon value and generally a
k>2 is considered unsafe.
In the following section, we will delve into a detailed examination of the Shannon Criteria,
exploring its origins, applications, and limitations in the context of PNS safety. This analysis will
provide a foundation for understanding current safety standards and highlight areas where further
research is needed to enhance the safe implementation of PNS technologies.
1.4.1 Shannon Criteria
The aforementioned safety criteria are a crucial component in determining the appropriate parameters for PNS, but their application in real-world situations can be challenging for practitioners.
As previously mentioned, this criterion is based on experimental data collected by McCreery and his
colleagues on cat brains, which involved the manual assessment of fiber damage and summarization
7

of the data into charts. Based on this data, Shannon concluded the relationship between the charge
density per phase (D) and the charge per phase (Q) with the equation log(D) = k – log(Q). Where
k is the adjustable parameter indicating damaging risk, and it is generally considered safe if k < 1.5.
The charge density per phase is calculated by dividing the charge per phase (Q) by the electrode
surface area.
The Shannon Criterion serves as a pivotal benchmark in the realm of electrical stimulation,
particularly due to its foundational role in establishing quantifiable safety limits. This criterion
allows clinicians and researchers to gauge the appropriate levels of electrical charge and density that
can be safely applied without risking neural damage. By quantifying the thresholds where electrical
stimulation may become harmful, the Shannon Criterion provides a critical tool for designing safer
and more effective peripheral nerve stimulation (PNS) protocols. It essentially bridges experimental
findings with clinical practice, offering a scientific basis for parameter setting that enhances both
the efficacy and safety of neurostimulation therapies. This has made the criterion a cornerstone
in the development and refinement of PNS technologies, enabling practitioners to apply complex
therapies with a greater degree of confidence in their safety profiles.
However, as we delve deeper into the complexities of PNS, it becomes apparent that while the
Shannon Criteria has been invaluable, it may not fully address the unique challenges posed by
peripheral nerve stimulation.
1.4.2 Limitation of Shannon Criteria
Despite its wide acceptance in the field of electrical stimulation, the Shannon criterion is not
devoid of limitations, which become apparent when applying this standard outside of its original
context. Originally derived from studies on cat brain neurons, the Shannon criterion may not seamlessly translate to peripheral nerve stimulation (PNS) due to significant physiological distinctions.
Peripheral nerves differ from brain neurons in structural properties and myelination patterns, critical factors that influence how electrical stimulation affects the tissue. This discrepancy underscores
the necessity for developing safety guidelines based on direct data from peripheral nerves to enhance
both the precision and relevance of safety standards for PNS applications.
Another key limitation arises from the assumptions underlying the Shannon criterion regarding
charge density distribution. The criterion simplifies the distribution across the electrode surface
as uniform, represented by D = Q/A, where D is the charge density, Q is the charge per phase,
and A is the electrode surface area. However, this assumption has been challenged by various
studies demonstrating non-uniform charge distribution across the electrode surfaces McIntyre and
Grill (2001); Wei and Grill (2009). For instance, research by W. Grill et al. used six electrodes
8

Figure 1.4: McCreery et al. manually assessed the fiber damage and summarized the experimental
data into charts. Figure from McCreery et al. (1990);
of identical surface area but differing perimeters, finding significant variations in charge density
Wei and Grill (2009). These findings indicate that Shannon’s assumption may not hold true in
practical scenarios, which could potentially lead to unsafe levels of stimulation in areas of higher
charge concentration.
Additionally, the methodology employed in deriving the Shannon criterion involves manual damage analysis, which is inherently subjective and susceptible to inaccuracies. This subjective element
can introduce variability in the interpretation and application of the criterion, particularly when
generalized across different types of neural tissue.
Given these constraints, it becomes evident that while the Shannon criterion has provided a
valuable framework for assessing the safety of electrical stimulation parameters, it also exhibits significant shortcomings that limit its utility in advancing PNS. The development of a more robust and
precise safety assessment protocol, specifically tailored for peripheral nerve applications, is essential.
Such advancements are crucial not only for optimizing patient safety and therapeutic outcomes but
also for pushing the frontiers of PNS technology and clinical practice. This initiative will require
collaborative research efforts, leveraging both experimental and computational approaches to establish a comprehensive set of guidelines that better address the unique challenges and needs of
PNS.
9

Figure 1.5: A study from Warren Grill et al in 2009 designed six different electrodes with same
surface area but different perimeter, by Shannon’s assumption these six electrode would have the
same charge density in his equation. However, the results suggested otherwise. Figure from Wei
and Grill (2009);
1.5 Ideal properties of an updated PNS safety criteria
As we recognize the limitations of the Shannon Criteria, it becomes imperative to define the
ideal properties of an updated safety framework for Peripheral Nerve Stimulation (PNS). These
properties should not only address the shortcomings of current standards but also accommodate
the unique characteristics of peripheral nerve stimulation. Our work has identified several key
attributes that an updated safety criteria should possess and has aimed to incorporate these into a
more comprehensive and relevant safety criteria for PNS.
The first ideal property of an updated PNS safety criteria is that it should be based on peripheral
nerve data. The anatomy and physiology of peripheral nerves can vary greatly from one individual
to another, and it is important to have safety criteria that take these differences into account. A
safety criteria that is based on peripheral nerve data would allow practitioners to make informed
decisions about the appropriate stimulation parameters for each patient, taking into account the
specific characteristics of each patient’s peripheral nerves. This is particularly important because the
electrical stimulation may have different effects on different individuals, depending on the anatomy
and physiology of their peripheral nerves.
The second ideal property of an updated PNS safety criteria is that it should be algorithm based
tissue damage analysis. This means that the safety criteria should be based on a mathematical
10

Figure 1.6: Another study showed similar conclusion with microelectrodes with different tip
radius, although these microelectrodes have the same surface area, their induced current densities
varied a lot. Figure from McIntyre and Grill (2001);
algorithm that can accurately predict the potential for tissue damage based on the stimulation
parameters. The use of algorithms in safety criteria will provide a systematic approach to assessing
the risk of tissue damage and will help to ensure that practitioners have a clear understanding of
the risk associated with each stimulation parameter. Additionally, the use of algorithms will allow
for rapid and accurate assessment of tissue damage, reducing the time and resources required for
manual assessments.
The third ideal property of an updated PNS safety criteria is that it should have accurate
current density simulation. Accurate current density simulation is essential in order to predict the
distribution of electrical current in the tissue, and to assess the potential for tissue damage based on
the stimulation parameters. The use of accurate current density simulation will provide practitioners
with a more complete understanding of the potential effects of electrical stimulation on the tissue,
allowing them to make informed decisions about the appropriate stimulation parameters to use for
each patient. Accurate current density simulation can also be used to optimize the stimulation
parameters in real-time, ensuring that the stimulation is both safe and effective for each individual
patient.
In conclusion, an improved PNS safety criteria would be based on peripheral nerve data using
tissue damage assessment and analysis in conjunction with accurate local current density estimation.
By having a clear and accurate safety criteria, practitioners will be better equipped to provide safe
and effective PNS therapy to their patients. The implementation of these ideal properties will ensure
11

Figure 1.7: What are the ideal properties are for an updated PNS safety criteria?
that PNS remains a safe and effective treatment option for various medical conditions, particularly
chronic pain.
This dissertation aims to address the limitations highlighted above by leveraging computational
modeling and machine learning approaches. The goal is to propose a new, more robust safety criterion for peripheral nerve stimulation (PNS). By integrating empirical nerve data with advanced
computational techniques, the research will develop algorithms that accurately predict tissue damage and simulate current density in real-time. Machine learning will enhance these models’ precision,
enabling them to adapt and improve as more clinical data becomes available. Ultimately, my dissertation will establish a dynamic, updated safety criterion that enhances therapeutic outcomes and
ensures optimal safety tailored to individual patient profiles. This project involves cross-university
and interdisciplinary collaborations, beginning with electrical stimulation experiments on rat sciatic
nerves conducted by our collaborator at the University of Utah. Post-experiment, the nerve samples
were sent to a collaborator in Spain for high-resolution microscopy, and the resulting images were
analyzed by our team to conduct multi-scale computational studies.
12

Chapter 2: Computational Platform for
Peripheral Nerve Stimulation
© 2024 Scientific Reports paper(In Submission).
13

Abstract
Electrical stimulation of the nervous system has been a medical therapy option for more than
60 years and is gaining importance in treating various conditions, including chronic pain relief,
motor function restoration, epilepsy treatment, and more. However, stimulation settings must be
changed depending on the structure of the regions to be treated to generate the desired stimulation
effects. For decades, computational evaluation of the efficacy and safety of such neural stimulation
devices was challenging due to the lack of capable platform. To close this gap, we introduce a highperformance computational modeling platform: Python-based Admittance Method (PAM), which
is capable of accurately constructing realistic nerve models with customized electrode locations and
calculating their electric fields distribution resulting from external stimulation. This platform takes
advantage of the advanced features of newest CPU architectures with multi-threading capabilities,
thereby enabling simulation of realistic high-resolution models and allowing communication with
other Python-based neuronal simulation software, such as NEURON, through a shared memory
paradigm with minimal impact on performance. This paper presents the core concepts of PAM,
describes the steps involved in multi-scale modeling using PAM, and provides examples for the
simulation flow of peripheral nerve models and stimulation setups. Further, we validate PAM with
ground truth and quantitatively compare the results obtained. In addition, we present simulation
results on example nerve models using PAM and demonstrate its straight-forward mechanism when
there is need of communicating with NEURON software.
2.1 Introduction
Neural stimulation devices are useful biomedical devices for neuroscience research and have
shown considerable potential for therapeutic applications for various conditions caused by illness or
14

injury (Altinok et al., 2009; Kathawate and Acharya, 2008; Kumar et al., 2006; Roy, 2018). For
instance, deep brain stimulators have demonstrated a therapeutic potential in memory restoration,
Parkinson’s disease, essential tremor and dystonia (Buhlmann et al., 2011; Mcintyre and Foutz,
2013; Schmidt and van Rienen, 2012). Peripheral nerve stimulators (PNS) have been shown to be
effective in the treatment of chronic pain, motor function loss, and epilepsy (Goodall et al., 1995;
Kosta et al., 2020b; Lubba et al., 2019; Reilly, 1989). Retinal stimulation could partially restore
blind patients’ vision (Loizos et al., 2018; Paknahad et al., 2022a, 2021a, 2020a,b, 2021b, 2022b;
Stang et al., 2019a). In addition, studies have demonstrated that electrical stimulation can direct
optic nerve growth and recovery (Gokoffski et al., 2021).
Regardless of the application, achieving optimal stimulation efficacy and safety in different neural
architectures often requires multiple stimulator designs, distinct nerve-interfacing electrodes and/or
stimulation settings. Moreover, differences in the anatomy of the stimulated neurons across individuals may cause variability in the response to the stimulation. The nerve morphology, the types
of fibers, the electrode location and the electrode geometry all have a significant impact on the
response of the nerve fibers to a particular set of stimulation settings, and it remains challenging to
elicit the same responses across species and individuals. The ranges within which the stimulation
parameters may be varied and the complex interplay between their respective values on the outcome of the stimulation makes this process a daunting experience. Ultimately, outcomes of clinical
trials may outline a limited therapeutic effect - thereby bringing the developers back to the drawing
board. To help remedy this, decrease the attrition rate and speed up the development of efficacious
therapeutic strategies, computational approaches can provide early on in the development process
significant insights on the consequences of design decisions and stimulation parameters. Facilitated
by recent advances in CPU development and high-performance multi-node clusters, simulation of
large multi-scale computational models can help predict the electric and magnetic fields resulting
15

from the stimulation in complex and customized systems, and the neural response elicited by these
stimuli. The knowledge of voltage and current distributions in the tissue represents a critical asset
for the design of electrical stimulation devices and the assessment of their safety. The capacity to
tailor the model constructed with different nerve types and electrode locations is vital to bridge the
gap between research and clinical trials. In addition, for very complex and highly segmented volumes, model heterogeneity plays a critical role in electrical properties distribution, and the resulting
activation pattern (Iwamoto et al., 2016; Pelot et al., 2017; Zhang and Grill, 2010), several computational approaches have been proposed to adress these specific bioelectromagnetic problems (Lindén
et al., 2014; Musselman et al., 2021; Shifman and Lewis, 2019), however, none of these computational
platforms are able to capture the realistic inner structure of the biological bodies. Consequently,
these studies were limited in their ability to incorporate the realistic high-resolution anatomy of the
internal nerve structure and preserve sufficient level of detail (Goodall et al., 1995). Most of these
studies use either a simplified nerve model with homogeneous fascicles or heterogeneous fascicles
populated by axons with artificial radii and locations (Kosta et al., 2020b, 2019; RamRakhyani
et al., 2015; Raspopovic et al., 2011). Other investigations have already demonstrated that a higher
resolution mesh almost always yields more accurate results and greater predictive power in highly
heterogeneous models(Ambrosi et al., 2011; Du et al., 2022b; Lorenzo et al., 2016; Mourdoukoutas
et al., 2018; Southern et al., 2008). These results clearly demonstrate the importance of developing
a computational platform capable of handling the complexity and heterogeneity nature of the stimulated biological tissue, and especially with the need of exploring different nerve type and electrode
setup.
For many years, the Admittance Method (AM) solver has been utilized as an accurate bioelectromagnetic simulation method in which the problem space is discretely partitioned: In this
method, the computational model is represented as a network of admittances generated from the
16

Figure 2.1: Overview of PAM’s process for creating and simulating biophysical nerve models with
customized electrodes and seamlessly integrating corresponding NEURON models. (A) The pipeline
begins with arbitrary nerve microscopy images. Utilizing a previously developed Convolutional
Neural Network (CNN) for segmentation, PAM constructs the nerve model from these images,
incorporating customizable electrodes to match the user’s desired configuration. Following this,
PAM performs material characterization, meshing, and electrical property analysis. (B) Starting
from the same segmented images, PAM identifies precise fiber locations through image processing,
enabling the creation of neurons at these exact points using NEURON modeling. This precision
allows the NEURON model to be accurately stimulated by extracellular voltages calculated by the
PAM solver, facilitating bi-directional communication. (C) Further insights into PAM’s capability
to begin with nerve microscopy images, constructing realistic nerve models with varied electrode
shapes and placements, as well as matching the biophysical nerve model to the NEURON model in
a realistic setup, based on actual nerve microscopy images.
17

dielectric properties of different materials. It has been widely used for computational modeling of
bioelectromagnetic problems including safety assessment of wearable or implantable devices and
neural response to electric and magnetic field (Bingham et al., 2018, 2020; Cela, 2010; Kosta et al.,
2021, 2020a; Paknahad et al., 2021a; Stang et al., 2019b).Despite its widespread application, the
AM approach encounters several operational challenges: firstly, the necessity for users to complete
model construction and mesh generation independently prior to utilizing AM, which adds complexity; secondly, the original implementation of AM has not leveraged the capabilities of modern CPUs
for multiprocessing and parallel computing, thus limiting its computational speed; and thirdly, the
process of post-simulation analysis and the integration of results with other simulation platforms is
cumbersome and time-consuming.
To address these challenges and build upon the robust capabilities of the Admittance Method,
we developed a Python-based AM simulation platform (PAM). This platform is designed to streamline large-scale, high-resolution computational modeling of electrical stimulation in nervous tissue,
incorporating customizable electrode setups, adjustable nerve model resolution and adaptive meshing, enhanced solving efficiency through advanced computing techniques, and seamless integration
with other software tools such as NEURON(Carnevale and Hines, 2006; Hendrickson et al., 2016).
PAM advances the field of nerve modeling by enabling high-resolution, realistic simulations that
were previously unattainable. With its user-defined electrode configurations, it not only facilitates
intricate nerve modeling but also achieves seamless communication with NEURON through its
architecture. This integration enables PAM to guide real-world experiments with its simulation
ability, effectively narrowing the gap between theoretical development and practical implementation.
Herein, we detail the platform’s design, its key features, and validate its accuracy, showcasing its
using flow through illustrative examples.
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Figure 2.2: Example 2D slice of a meshed model of a peripheral nerve, including (A) an unmeshed
slice of the model derived from nerve images, using uniform cubic voxels, and (B) illustration of the
same model after applying the multiresolution meshing algorithm with a maximum size of 64 voxels.
Boundary voxels remain a relatively low resolution (C): Illustration of the mesh simplification
process on a 3D mesh model of a rat sciatic nerve cross-section: In locations proximal to the
contact electrode voxel size is kept small and the voxel size is increased further away from the nerve
periphery where fine resolution is unnecessary. In this example, the mesh size is reduced by over
90 percent.
2.2 Design and Implementation
2.2.1 Overview of the PAM pipeline
As a biophysical simulation platform, PAM excels in high-resolution nerve simulation, creating
three-dimensional nerve models from cross-sectional microscopy images. It allows for the customization of electrode configurations and their precise placement on the models. Users can specify electrode shapes and positions to tailor stimulation scenarios, streamline mesh generation, and calculate
electrical properties for specific stimulation intensities. Additionally, PAM facilitates the creation
of NEURON models at exact locations based on nerve imagery, enabling accurate simulations and
19

communication via Python’s shared memory. PAM currently uses Python version 3.8 and NEURON
version 7.8.
As shown in Fig. 2.1, the PAM process initiates with input from nerve microscopy images.
Through our previously developed image segmentation using a Convolutional Neural Network
(CNN)(Morales et al., 2024), it constructs ultra high-resolution realistic nerve models. This step
is followed by adding desired electrode models and then material characterization, meshing, and
electrical analysis. PAM then processes the segmented images to identify specific fiber locations,
create nerve fiber models at the corresponding locations, enabling the precise placement of neurons
in NEURON models and more accurate simulations. This meticulous alignment allows for the NEURON models to interact accurately with the extracellular voltages determined by PAM, supporting
bi-directional communication and enhancing simulation accuracy.
The user-driven workflow is managed via a JSON configuration file, where user can speficy the
parameters they want such as electrode size, location, stimulation intensity, model resolution, etc.
2.2.2 3D Model Construction
The Python Admittance Method modeling(PAM) for nerve analysis is distinctly designed to
accommodate individual samples through the segmentation of cross-sectional imagery. PAM diverges from traditional methods that employ generically modeled nerve structures, which often fail
to capture the detailed and variable anatomy of real nerve tissues(Kosta et al., 2020b, 2019; RamRakhyani et al., 2015; Raspopovic et al., 2011). This approach focuses on the detailed examination
of crucial structures, including myelin, axons, and other essential elements, while also providing the
flexibility to design and position electrodes at various locations within the nerve model, enhancing
the specificity and applicability of the analysis.
20

The multi-scale computational model to be employed for nerve stimulation simulations consists
of two main components: the segmented nerve model and the model of cuff electrode.
2.2.2.1 Nerve Model Construction
Upon segmentation of the obtained nerve microscopy images using CNN, detailed 3D models
of nerves are accurately constructed using the Python-based Admittance Method Simulation Platform(PAM), including the intricate details of myelin thickness and axon diameter, among others.
This detailed representation forms the basis for generating the realistic 3D nerve models, where each
segmented component is discretized to model the electrical properties of the nerve tissue accurately.
The model is then extruded along the z-axis, maintaining the fidelity of the cross-sectional anatomy
over the length of the nerve.
This standardized approach is implemented in Python, it offers users a range of customizable
options, from model dimensions and resolutions to specific voxel sizes, enabling the adaptation of
the model based on the quality of nerve imagery and the computational resources at hand. This
flexibility allows researchers to tailor the nerve model precisely to their requirements, ensuring that
both the fidelity of the representation and the efficiency of the computational process are optimized.
2.2.2.2 Cuff Electrode Model Construction
The design of cuff electrodes is conceptualized based on a common commercial cuff electrode,
in this modeling approach, only the metallic components of the electrodes are taken into account,
disregarding non-conductive parts not in direct contact with the tissue, such as the insulation
layer surrounding the electrodes. This simplification is justified by the negligible effect these nonconductive elements have on the distribution of current through the tissue, as shown in Fig. 2.1
21

(C). Most common commercial cuff electrodes have three metal contact wires, one of these wire
functions as the source cuff electrode, while the other serves as the ground cuff electrode. With
the customization ability, PAM allows for adjustments in their placement on the nerve model,
thickness, and other parameters, to meet specific experimental or therapeutic requirements. This
level of customization facilitates precise control over the electrode’s interaction with the neural
tissue, enhancing the effectiveness and specificity of neural stimulation or recording.
2.2.3 Model Characterization and Adaptive Meshing
In a separate input (.in) file, the electrical properties of various materials and the locations of
stimulation current sources are defined. The simulation introduces an electric current at a specific
source electrode. At the return electrode, a zero-voltage boundary condition is set, ensuring that
the voltage at this node is zero. PAM also supports different types of stimulation configurations,
such as monopolar, bipolar, and multipolar. An example of bipolar stimulation involves applying
+1 mA at one electrode contact and -1 mA at another contact.
To enhance the efficiency of the computational model, especially during the Admittance Method
step, PAM determines the electrical properties of all materials involved, including air, electrodes,
and various nerve tissues like the surrounding medium, endoneurium, epineurium, and perineurium.
After establishing these properties, PAM employs an adaptive meshing algorithm. This algorithm
adjusts the computational mesh based on the distribution of different materials in the model. The
goal is to reduce computational complexity without significantly sacrificing accuracy.
To be more specific, in the AM (Armitage et al., 1983) (or the analogous form, the impedance
method (Gandhi et al., 1984)), the system of interest is reconstructed and discretized as unitsized cubic voxels. Based on the material property of each cubic voxel, the model is represented
22

as an admittance network. Biological tissue can be comprised of numerous structures of differing
electrical properties. Accurate modeling of these systems requires a high spatial resolution at the
interfaces/boundaries where these structures meet. However, biological tissue can also have large
volumes in which the electrical properties are homogeneous, for which a low spatial resolution is
suitable. The multi-resolution mesh allows for the reduction of computational time by lowering
the local meshing resolution to match these requirements in biological tissue, thereby retaining
high resolution where strictly necessary (e.g., near the tissue-electrode interfaces or to properly
represent the finest biological details, such as the myelin of axons). In its most general version, the
clustering algorithm automatically maintains minimal resolution at tissue boundaries and increases
the voxel size further away from where the finest resolution is unnecessary(Loizos et al., 2014a). Twodimensional examples of uniform and multiresolution meshing grids are shown in Fig. 2.2(A)(B)
(Eberdt et al., 2003), along with the memory savings of the automated multiresolution clustering
strategy. Occasionally, it is essential to maintain a user-defined resolution not only based on model
feature size, but also to maintain the desired resolution in specific regions of the model, such as
regions of interest to the user. For example, in the case of peripheral neurostimulators such as cuff
electrodes, accurate modeling of the nerve periphery is generally of greater interest compared to
the center of the nerve, as higher current densities are likely confined to the region proximal to the
electrode; as shown in Fig. 2.2(C) nerve model, in locations proximal to the contact electrode the
model resolution is high, while it is increased further away from the nerve periphery where high
resolution tends to be unnecessary. In this way, the number of nodes is decreased significantly, and
the computational complexity is greatly reduced. Accuracy was assessed on single-fascicle nerve
bundles, demonstrating that the complexity of solving is reduced by over 60% with only a 10%
increase in error. This trade-off is particularly beneficial for large models previously unsolvable.
By employing efficient multiresolution meshing techniques, it is feasible to retain detail near areas
23

Figure 2.3: AM platform structure overview: after the Admittance Method model network is
constructed, the PAM solver constructs the sparse admittance matrix, applies multi-threading, and
performs data post-processing and interpolation. Results can be used for a subsequent time step
for time-dependent problems in AM or passed as input to other computational strategies, such as
NEURON in the case of external stimulation.
of interest and lower resolution in other areas, thus creating the most accurate model achievable
within our computational limits.(Cela et al., 2011).
2.2.4 Electric Network Generation
Similar to [14]–[16], an equivalent admittance network is derived by placing network nodes on
the vertices of each voxel. Lumped circuit components are implemented in different directions
using the corresponding admittance: their values are determined depending on the vertex, edge,
voxel size, and material attributes. To develop an equivalent circuit network model, for example,
first generate an electrical description of a single voxel in the model and compute its node and
24

Figure 2.4: Other computational platforms resorting to AM communicate with NEURON through
an intermediate step, as shown in (A). This intermediate step significantly affects the overall computation efficiency. With our proposed Python solver AM platform, neuron simulation tools can
directly communicate with the Admittance Method through Python’s shared memory, where all
the nodal values and model information are stored, as shown in (B).
edge values. As illustrated in Fig. refinterp, the vertices of each voxel are designated as network
nodes, and admittance values are computed for all the edges linking distinct vertices/nodes. Edges
shared by two or more voxels are also evaluated concurrently. AM creates a matrix representing
the general admittance (G), or conductance, across the model once all the nodes and edges have
been computed and accounted for. The G matrix is a sparse matrix because nonzero values exist
only when nodes are linked, and each node can only be connected to a maximum of six additional
nodes. The admittance values between two separate nodes are determined using conductivity and
distance in the x, y, and z directions, as indicated in Equation 1:
g
i,j,k
x = σ
i,j,k
x
∆y∆z
∆x
(2.1)
The generated admittance network information is stored in a text file for later use. The first several lines of the network file include the general information of the model, such as overall dimension,
electrode locations, and other model characteristics.
25

2.2.5 Python Admittance Solver
With the computational model network constructed, PAM has the following information ready
to use: The sparse matrix representation of internodal admittance denoted as matrix G, a current
vector (I) defined with current values applied to the nodes that contain a source. PAM performs the
following steps to import the above constructed network: (i) reconstruct the network by reading the
G matrix file and (ii) define the desired pre-conditioning method and specific numerical iteration
solver(conjugate gradient, bi-conjugate gradient, etc) to solve the nodal voltage values at their
desired accuracy/speed. PAM’s goal is to compute the voltage vector (V), which can be solved
using G and I in Equation 2, calculating nodal voltages according to Ohm’s law. With more
realistic nerve models the complexity of solving this linear equation is often overwhelming. Given
the matrix G’s size can exceed one billion by one billion elements, PAM employs multiprocessing
and Message Passing Interface (MPI) to expedite the solution process, offering users a choice of
iterative solvers.
GV = I (2.2)
Moreover, Python’s widespread use in neural simulation environments, including NEURON, has
motivated PAM’s development in Python. This choice facilitates a customizable interface with
NEURON and other computational methods. An efficient multi-threaded approach for solving
the Admittance Matrix (G) equation was implemented to handle large bioelectromagnetic models
with significant computational cell resolution variations. The Python Admittance Solver reads
the Admittance network file to construct G using a compressed sparse matrix format for efficient
storage. After building G, the equation GV = I is solved, where I represents the current source
values, and V contains the resulting nodal voltage values. A multi-threaded solver, such as the biconjugate gradient or conjugate gradient method, efficiently resolves this equation, with the solved
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Figure 2.5: An example used to validate the proposed PAM platform, (A): models of a simplified
peripheral nerve are constructed and meshed for both COMSOL and PAM. The nerve model is
constructed by positioning smaller cylinders inside a larger cylinder cover. Three cuff electrodes
surround the nerve model, with the top electrode serving as the source electrode and the bottom
electrode serving as the current return. A 2mA current source is applied. In order to compare
the two simulation results, voltage distribution on the XY plane at the source electrode level from
COMSOL (B) and from PAM (C) are shown. Similarly, the voltage distribution along YZ plane at
the nerve center from COMSOL (D) and from PAM (E) are shown.
V vector’s values corresponding to node coordinates. This streamlined approach highlights PAM’s
capacity to handle complex bioelectromagnetic models and integrate with neural simulation tools
like NEURON, enhancing its utility in bioelectromagnetic research. The overview of the PAM solver
code is shown in Fig. 5.1.
2.2.6 PAM to NEURON
In the field of computational neuroscience, the PAM platform excels by enabling the creation of
not only accurate biophysical models, surpassing previous efforts, but also NEURON models that
precisely replicate the anatomical locations of neural fibers. Similar to its approach in constructing
nerve biophysical models, the PAM platform also creates NEURON models from high-resolution,
27

segmented nerve microscopy images. Utilizing image processing algorithms, PAM defines fiber locations based on actual nerve cross-sectional images, enhancing the accuracy of the NEURON models’
representation of real fibers. Following image segmentation with a Convolutional Neural Network
(CNN), it produces clear mask images. These images reveal each fiber’s (x, y)-location within the
nerve cross-section, detailing both myelinated and unmyelinated fibers, including their positions
and myelin thickness. NEURON fiber models are then constructed at these precise locations and
shaped according to the 3D nerve model’s length. Thus, PAM not only develops realistic nerve
biophysical but also constructs NEURON models in a more accurate way.
In addtion to the accuracy of the created Admittance Method(AM) and NEURON models,
PAM also employs Python’s shared memory for seamless communication between the AM solver
and NEURON, facilitating effective bidirectional communication.
The communication mechanism operates as follows: After precisely constructing the AM and
NEURON models to simulate neural responses to electrical stimulation with specific electrodes
and settings, and completing the AM solution, the solved voltage values are stored in Python’s
shared memory as a list containing the voltage of each model voxel. During NEURON simulation,
these voxel center voltage values are assigned to the nearest NEURON compartments, applying
a voltage source, and then solving the NEURON simulation. Once the NEURON models are
solved, the compartmental current values are stored using Python’s shared memory, enabling direct
access by the AM solving process. These current values are assigned to the nearest AM voxel
or node and then utilized as an independent new current source to solve the AM process. This
algorithm could operate cyclically in the simulation, starting with solving the Admittance Method
model, applying the estimated field to the NEURON model, and then solving a NEURON model
timestep. The resulting compartmental currents are then fed back to the Admittance mesh, setting
28

up for the next simulation timestep. Spatial alignment is achieved using one of two methods:
interpolating/splitting to/from a source or shifting a compartment or source to the nearest AM
mesh node, with these approaches detailed graphically in (Bingham et al., 2018). However unlike
the previous implementation, where the bi-directional communication involves multiple time steps,
now with PAM the communication is conducted seamlessly within a single process.
This bidirectional communication is crucial for accurately predicting neuronal responses to stimulation, which, in turn, can alter the activity of adjacent neurons based on the conductivity of the
surrounding tissue. High conductivity can enable the weak electric fields generated by individual neurons to affect their neighbors’ spiking activity, an interaction that’s critical to model for a
thorough understanding of neural behavior in densely packed neuron environments. There are a
few tools that allow the communication between simulation and NEURON to be modeled like ASCENT and LFPy. They could integrate the coupling effect into its simulation environment(Lindén
et al., 2014)(Shifman and Lewis, 2019). However, these tools, along with earlier AM-NEURON
implementations, fall short in enabling direct communication within a single process. Each of these
approaches necessitates convoluted external process calls and the sluggish exchange of data, as
illustrated in Fig. 2.4(A). In contrast, our PAM platform uniquely captures the complex nature
of biological bodies in external electrical simulations and establishes seamless communication with
NEURON via shared memory, as depicted in Fig. 2.4(B). This feature significantly enhances the
efficiency of our simulation tool.
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Figure 2.6: To assess the PAM platform’s precision, we utilized a basic sphere model with a core
source node surrounded by a spherical external ground, mirroring the point source approximation
(PSA) used as a ground truth. (A) A 2D slice of the sphere model illustrates the comparison
between PAM and the benchmark through a heatmap, with voltage values marked by dots in
varying color shades for visual clarity. (B) To further clarify local field potential accuracy, a line
plot extracted along the 2D slice’s center is shown to the right. This plot’s analysis indicates a
normalized root mean square error (NRMSE) of 0.065 between PAM and the benchmark, affirming
the model’s accuracy.
2.3 Results
First, we designed test simulations to assess the biophysical modeling capabilities of PAM. These
tests involved a simple sphere model to emulate and compare with the point-source numerical solution. The comparison revealed a strong agreement in local field potential value distributions between
the PAM-generated results and the numerical solution. These local field potential results were then
integrated with a simplistic illustrative neuron single compartment model, using solved voltage values stored in a list within Python’s shared memory. These values were directly accessed from a
NEURON Python script, and the resulting NEURON responses closely matched, demonstrating
the accuracy of PAM’s solutions and the efficacy of the communication mechanism via Python’s
shared memory.
Moreover, to highlight PAM’s versatility in customizing nerve models and electrode setups
throughout the entire workflow, we constructed a simple pseudo-realistic nerve model using both
PAM and COMSOL. We compared current density value distributions between these two methods.
The models constructed were identical, and the results closely matched, indicating that PAM not
30

only matches the accuracy of COMSOL simulations but also offers greater flexibility in electrode
customization for certain biophysical simulations.
We then demonstrated a capability that neither COMSOL nor other nerve simulation platforms
could achieve: large-scale, high-resolution nerve modeling using realistic rat sciatic nerve microscopy
images. By simulating rat sciatic nerve models stimulated by cuff electrodes, we compared the
outcomes of realistic nerve modeling with conventional pseudo-nerve modeling. The PAM results
showcased detailed variations in current density within the nerve, surpassing what other models
could achieve. Furthermore, PAM’s multiprocessing ability also showcased improved solving speed.
Lastly, we demonstrated PAM’s bi-directional communication capability through an illustrative
graph.
2.3.1 Accuracy Validation against point-source approximation and communication with NEURON
To simulate point-source electrical stimulation within nerve tissue, we developed a sphere model
composed of two key elements: an outer layer of superconductive metal acting as the ground and
an inner sphere imbued with dielectric properties. By positioning a point source at the sphere’s
core and grounding the outer layer, our aim was to replicate the electrical dynamics characteristic
of point-source stimulation. This approach enables a straightforward comparison with numerical
benchmarks, as illustrated in Fig. 2.6(A). Here, we analyzed voltage values across a 2D crosssectional slice of the sphere, employing a heatmap to delineate the simulation results of PAM against
those of the benchmark. Distinct shades represent varying voltage levels for clearer visualization.
The resulting Normalized Root Mean Square Error (NRMSE) of 0.065 between PAM and the
benchmark underscores the model’s precision.
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Further examination of local field potential accuracy was conducted through a line plot along
the 2D slice’s central axis, as depicted in Fig. 2.6(B). The comparative analysis of current density
distributions along this axis between PAM and the point source ground truth indicated a normalized
root mean square (NRMS) difference under 0.05, affirming a close alignment with the numerical
standard. Although slight variances in PAM’s outcomes were noted, these were largely due to meshing and interpolation errors—necessary steps that can introduce minor inaccuracies. Nevertheless,
the overall consistency in current density values attests to PAM’s capability for precise electrical
field simulation.
Building on this foundation, we aimed to further assess PAM’s integration with the NEURON
software. In the context delineated by a yellow box in Fig. 2.7(A), individual dots represent
source nodes engaging with a single-compartment NEURON model. This setup facilitated the
comparison of PAM and point source approximation (PSA) techniques in activating the NEURON
model. The responses, illustrated in Fig. 2.7(B), exhibit minimal differences between PAM and
PSA methodologies, thereby validating the effectiveness of PAM’s communication with NEURON
software. This comparison not only highlights PAM’s precision and reliability but also its potential
to advance neural simulation practices.
2.3.2 Validation against COMSOL
With PAM’s solving accuracy validated against a point-source numerical solution, we utilized
both the PAM computation platform and COMSOL to simulate a pseudo nerve model. This involved
positioning smaller cylinders inside a larger cylinder to represent the nerve structure. The outer
cylinder symbolizes the nerve membrane, while the inner cylinders, embodying fascicles, contain
numerous axons, each surrounded by myelination. In COMSOL, the nerve model was crafted using
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Figure 2.7: Leveraging the same spherical model, our objective was to further validate the efficacy
of PAM’s integration with NEURON software. Within the delineated yellow box, each dot signifies
a source node selected for interaction with the single-compartment NEURON model, as depicted
in the diagram to the left. (A) This phase entails the activation of the single-compartment NEURON model by each source node within the yellow box, employing both PAM and point source
approximation (PSA) techniques for stimulation. (B) The NEURON model’s reactions to these
stimulations are graphically displayed on the extreme right, revealing minimal discrepancy between
the outcomes of PAM and PSA approaches. This juxtaposition effectively affirms the robustness
of PAM’s communication protocol with NEURON software, underscoring its accuracy and dependability in facilitating neural simulations.
its 3D geometry generation tool, whereas PAM generated the 3D nerve model through a code that
translates 2D slice images of the cylinders. The nerve created was surrounded by three customized
cuff electrodes, as shown in Fig. 2.5(A). The top cuff electrode served as a source electrode, while
the bottom cuff electrode served as a current return. A current source of 2 mA was applied to
stimulate both the COMSOL model and the PAM model. As shown in Fig. 2.5(B)(C), the voltage
distribution along XY slices was identical for both COMSOL and PAM. Also, the voltage values
along the YZ slice plane are plotted in Fig. 2.5(D)(E). The voltage distribution pattern shows
an almost identical distribution pattern. We further calculated the cumulative error of voltage
distribution values between the two simulations. PAM exhibited a mean difference of less than 3
percent when compared to COMSOL, primarily due to different meshing strategies. These results
highlight PAM’s the capability to customize simulations like in COMSOL, as well as further validate
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its accuracy in realistic settings.
2.3.3 PAM’s realistic peripheral nerve modeling vs other computational models
None of the current modeling studies have been able to reproduce the complex inner structures
of peripheral nerves fully. Convolutional neural network(CNN) is a powerful tool to segment and
recognize image components (Cai et al., 2018; Chen et al., 2018; Zaimi et al., 2018), we had previously
utilized it to perform segmentation of cross-sectional nerve images and generated a corresponding
AM model to study the effect of cuff electrode positions on peripheral nerve stimulation (Cela,
2010). Peripheral nerve tissue was represented by an electrically-heterogeneous, high-resolution
nerve model based on segmented cross-sectional images of rat sciatic nerve and included fine details
such as axons and myelin. However, the resulting realistic nerve model’s extreme computational
intensity is hard to handle with existing computation platforms. With the help of PAM, we have been
able to compute the electric field values and current distributions inside the nerve; we investigated
the effect of the distance between the source and return cuff electrodes on the internal current
distribution within the nerve. This study provided a basis for application-specific electrode design.
2.3.3.1 Model construction
Previous nerve modeling studies were unable to reproduce the anatomy of the internal peripheral
nerve structure due to its complexity and a lack of methodology that could extract and preserve these
details (Goodall et al., 1995). Most studies use either a simplified nerve model with homogeneous
fascicles or heterogeneous fascicles populated by axons with artificial radii and locations (Kosta et al.,
2020b, 2019; RamRakhyani et al., 2015; Raspopovic et al., 2011). Thus, we constructed a peripheral
nerve model using a high-resolution cross-sectional image of a rat’s sciatic nerve. To accomplish this,
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Figure 2.8: In order to construct a realistic peripheral nerve model, a Cross-sectional image of rat
sciatic nerve (A) is taken and segmented (B) into image of nerve cross-section containing multiple
fascicles populated by axons of various radii and myelin(grey represents myelin and white represents
axon). Both realistic nerve model(C) and traditionally used simplified models(D) are constructed
and simulation results are plotted(E)(F). To further illustrate the difference between PAM results
and traditional results, (G)(H) further enlarged a section of current density distribution details
inside both models at the same location. It is shown that using PAM a great amount of current
density details are captured whereas inside the traditional nerve modeling such details are missing.
The solving times for the two different models are illustrated respectively.
35

Table 2.1: TISSUE PROPERTIES
Tissue Type Conductivity (σx, σy, σz)S/m
Perineurium (0.01, 0.01, 0.01)
Myelination (2 × 10−4
, 5 × 10−9
, 5 × 10−9
)
Intracellular space (0.91, 0.91, 0.91)
Extracellular space (0.33, 0.33, 0.33)
we utilized a confocal image of the rat sciatic nerve and performed image segmentation to highlight
axons and myelin. Image segmentation was performed using AxonDeepSeg, an open-source easily
trained convolutional neural network (CNN) described in (Zaimi et al., 2018). Next, the segmented
cross-sectional image was discretized and extruded to build the pseudo-3D length of the nerve for
electrical stimulation modeling. The original nerve confocal image Fig. 2.8(A) and the resulting
segmented image are shown in Fig. 2.8(B) respectively. The model size was 1300x1300x2000 voxels
in x, y, z dimensions, and the resolution was 2 µm in all three dimensions. The model was discretized
in cubic voxels with a unique material index representing each voxel. The material properties of
the nerve model are taken from (Butson et al., 2011; McNeal, 1976) and listed in Table 1.
2.3.3.2 Results comparison
The model constructed from the CNN segmented image contains great detail, as shown in
Fig. 2.8(C). In contrast, the conventional model of the peripheral nerve, shown for example in
Fig. 2.8(D) typically assumed homogeneous inner fascicle tissues (Kosta et al., 2020b, 2019; RamRakhyani et al., 2015; Raspopovic et al., 2011). From the voltage and current distribution of the
two models shown in Fig. 2.8(E)(F), we can see that the high-resolution anatomical features of
the peripheral nerve significantly affect the current distribution inside the nerve. When comparing
the two simulation results, simulations with the CNN-built heterogeneous model show significant
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currents flowing through the edge of the inner fascicle, confirming experimental findings that peripheral nerve electrical damage tends to occur first near the nerve periphery (Du et al., 2021; Lago
et al., 2005). Modeling employing the traditional homogeneous three fascicles simplified model will
not capture such phenomena. These results suggest that a very high resolution and heterogeneous
model of the peripheral nerve is necessary to accurately capture the response of the peripheral
nerve to electrical stimulation, and the new PAM method enables such high-resolution modeling
to be performed. Besides predictions on the recruitment of target fibers, the significant variations
due to the model heterogeneity also allow us to predict the thresholds of stimulation that do not
induce axonal damage. Thus, the proposed method and models can help design and optimize target
effects and identify safety limits for stimulation protocols and electrode design. Such realistic nerve
simulation could not be done without PAM.
2.3.4 PAM’s communication with NEURON
The results previously demonstrated PAM’s capability for forward communication. To highlight
its backward communication abilities, we used the same basic sphere and a single-compartment
neuron model as shown in Fig. 2.9(A). This model was chosen for its simplicity, making it easier to
understand and quicker to validate bi-directional communication.
In particular, Fig. 2.9(B) shows the communication mechanism of membrane current responses
and electric potential responses from NEURON when stimulated by points within the highlighted
yellow box. Each point represents a source node interacting with NEURON, as illustrated earlier
in Fig. 2.7. These results are vital as they showcase PAM’s backward communication capability by
utilizing NEURON’s computed membrane currents directly from shared memory. This direct use
facilitates a seamless and intermediary-free input to the PAM solver, enhancing both computational
37

speed and efficiency. This simple illustrative example showcases the use of shared memory to
communicate the stimulation between PAM and NEURON, this way the wall time between two
processes is reduced more than 80%.
The use of shared memory allows PAM to instantly access membrane current values calculated
by NEURON and reintegrate them into its solver, bypassing the delays associated with traditional
file-based data exchange. This immediate data retrieval not only speeds up the computation but
also simplifies the simulation workflow, leading to a more streamlined process that is less prone to
errors.
2.4 Data Availability
The PAM pipeline is publicly available for download to any Linux, Mac or Windows machine: the
release associated with this paper (v1.0.0), as well as other archived releases, are available through
GitHub (https://github.com/djz52w/PAM). The code is documented in a git wiki associated with
the code repository (https://github.com/djz52w/PAM). The project is licensed under the MIT
License for non-commercial use (see the LICENSE file in the root of the repository for details).
2.5 Discussion
We have presented a novel simulation platform, PAM, for predicting electric fields resulting from
external electrical stimulation in large-scale models of various biological tissue.
We demonstrated that this platform had two critical advantages: 1) It enables the simulation
of large-scale high resolution biological models. Both the scale and resolution of the models that
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Figure 2.9: Illustration of PAM’s Bi-Directional Communication Capabilities. (A) Showcases
the model involving a basic sphere and a single-compartment neuron, chosen for its simplicity
to facilitate understanding and quick validation of bi-directional communication. (B) Displays
NEURON’s membrane current and electric potential responses when stimulated by points within
the yellow box, representing source node interactions as previously seen in Fig. 2.7. This figure
highlights PAM’s backward communication by directly utilizing NEURON’s computed membrane
currents from shared memory, enhancing computational efficiency and simplifying the input process
to PAM’s solver. The use of shared memory notably reduces process interaction time by more than
80%, with expectations of further reductions as model complexity increases.
may be simulated in PAM were previously unattainable. Indeed, the previous solutions fell short
for modeling large and complex model networks. Similarly, they could not incorporate many of the
details, which could result in inaccuracies in the prediction of electric field distribution in the tissue.
Uniform grid models that are typically used in traditional AM methods, or conformal grids, typical
of methods such as FEM, struggle to perform simulations of such complex models due to memory
and execution time. Even the most recent state-of-art nerve computational modeling platforms
do not capture the detailed nerve structure (Musselman et al., 2021). Here, the PAM platform
enables the simulation of larger spaces while incorporating a level of details that may be critical
for accurately predicting the distribution of the electric field in the tissue. 2) PAM has an open
architecture and can be integrated seamlessly with other external computational platforms, such
as NEURON in the cases of neurostimulation applications. The proposed implementation provides
39

seamless communication and integration with other platforms thanks to Python shared memory
mapping.
Neural tissue and biological structures in general are known for their detailed heterogeneous
structures. Leveraging parallel computing and multi-threading, PAM constitutes an excellent solution to simulate highly detailed complex models such as the rat sciatic nerve model presented
above. Like many other peripheral nerves, the rat sciatic nerve consists of a large number of packed
fibers. Simulating nerve models with such degree of realism has been extremely challenging until
now - forcing modelers to use highly simplified substitutes - resulting in crude approximations of
the electric field. The peripheral nerve model presented here consists of tens of thousands of fibers,
with each fiber having a different shape, size, and myelination, resulting in a model that represents
the biological system with an unparalleled degree of realism. Myelination is an essential feature
that must be captured as the resistivity of myelin and axon are very different, thereby affecting
current distribution inside the nerve, thus potentially resulting in different activation patterns. In
this work, we have demonstrated the necessity of capturing heterogeneous tissue properties and the
PAM’s ability to make accurate predictions using these highly realistic datasets. In parallel, the
example of the rat sciatic nerve also demonstrates PAM’s potential to simultaneously consider fields
and neural activation in the same platform via Python’s shared memory.
Notably, Python was chosen as the programming language because it is increasingly becoming
a universal interface to many neural simulation software such as NEURON, which has excellent
Python bindings and APIs (application programming interfaces). Consequently, a Python implementation results in simpler and more efficient interfacing and makes simulations that leverage
the functionalities of other software a lot more convenient. An example of co-simulation in which
40

bi-directional communication is an essential factor to consider is when modeling the effects of ephaptic coupling. We demonstrated that bi-directional communication between the Python NEURON
model and PAM using the shared memory paradigm was fast and resulted in little to no overhead,
highlighting the overall efficiency of PAM’s co-simulations paradigm.
While PAM is faster than earlier implementations, the detailed and complex nature of computational neuroscience still makes running multi-time step simulations with bi-directional capabilities
on realistic nerve models a lengthy process. Hence, we used a more straightforward single compartment model for demonstrating its communication features.
Future developments of the platform will address limitations, including reducing excessive computer memory usage on large high-resolution models, minimizing interpolation errors between the
border of different materials and streamlining the AM model generation process. In addition, future work will also include the bi-directional communication between PAM and NEURON for more
complicated realistic nerve models.
The development of novel therapies based on the electrical stimulation of biological tissue and
the nervous system in particular is an active field that is still gaining momentum. Regardless
of the application, achieving optimal stimulation efficacy and safety in different tissues and neural architectures often requires distinct designs, with different nerve-interfacing electrodes and/or
stimulation settings. The ranges within which these parameters may be varied, and the complex
interplay between their values and the physical nature of the stimulating electrodes and the tissue
with which they interact results in a complex and high-dimensional parameter space. Computational
approaches can provide significant insights on the consequences of design decisions and stimulation
parameters on the desired outcome. Knowing voltage and current distributions in the tissue represents a critical asset to guide the design of electrical stimulation devices and assess their safety.
41

The novel PAM platform enables such predictions on tissue models that comprise very complex and
highly segmented volumes with an unprecedented level of realism, resulting in highly accurate predictions - thereby constituting a powerful tool to support the rapid identification and development
of efficacious therapeutic strategies.
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Chapter 3: Computational modeling for the
distance between cuff electrodes’ effect on
current density distribution
© 2021 EMBC. Du J, Morales A, Paknahad J, Kosta P, Paknahad J, Bouteiller JM, Lazzi
G. Electrode Spacing and Current Distribution in Electrical Stimulation of Peripheral Nerve: A
Computational Modeling Study using Realistic Nerve Models. Annual International Conference of
the IEEE Engineering in Medicine and Biology Society.IEEE Engineering in Medicine and Biology
Society.Annual International Conference (2021).
43

Abstract
Electrical stimulation of peripheral nerves has long been used and proven effective in restoring
function caused by disease or injury. Accurate placement of electrodes is often critical to properly
excite the nerve and yield the desired outcome. Computational modeling is becoming an important
tool that can guide the rapid development and optimization of such implantable neural stimulation
devices. Here, we developed a heterogeneous very high-resolution computational model of a realistic
peripheral nerve stimulated by a current source through cuff electrodes. We then calculated the
current distribution inside the nerve and investigated the effect of electrodes spacing on current
penetration. In the present study, we first describe model implementation and calibration; we then
detail the methodology we use to calculate current distribution and apply it to characterize the
effect of electrodes distance on current penetration. Our computational results indicate that when
the source and return cuff electrodes are placed close to each other, the penetration depth in the
nerve is shallower than the cases in which the electrode distance is larger. This study outlines the
utility of the proposed computational methods and anatomically correct high-resolution models in
guiding and optimizing experimental nerve stimulation protocols.
3.1 INTRODUCTION
Electrical stimulation of peripheral nerve is used for various applications including pain treatment (Weiner, 2000), restoration of motor functions following spinal cord injury or stroke (Grill and
Kirsch, 2000; Stein et al., 1992), and treatment of epilepsy by vagus nerve stimulation (Salinsky,
1995). Different electrode designs have been developed for efficient nerve stimulation, such as spiral
electrodes (Sweeney et al., 1990), cuff electrodes (Larsen et al., 1998), intrafascicular interfaces with
44

thin wires (Bowman et al., 1979; McNaughton and Horch, 1996), and silicon probes (Barth et al.,
1985; Veltink et al., 1989). Cuff electrodes are amongst the most widely used electrodes for peripheral nerves since they have several advantages compared to others: 1) they allow for the reduction
of input current intensity, and thus, minimize the possibility of nerve damage, 2) they allow for
correct positioning of electrode leads to minimize mechanical distortion and damage, and 3) provide
selective stimulation of nerve fascicles.
There have been numerous neurophysiological studies designed to evaluate the effect of electrode
position on neural excitation. Early studies by Simmons and Glattke (Simmons and Glattke, 1972)
and Walloch and Cowden (Walloch and Cowden, 1974) demonstrated that electrical stimulation is
more efficient in exciting nerve fibres than Scala tympani stimulation when electrodes were placed
directly into the nerve. Shepherd et al. (Shepherd et al., 1993) found that the optimal electrode
position for auditory nerve excitation results in significant threshold reduction. However, few studies
focused on parametric assessment of the effectiveness of peripheral nerve stimulation using very-large
scale, anatomically correct, peripheral nerve models.
In this paper, we have utilized convolutional neural network segmentations of nerve cross sectional images along with a multi-scale, computational model of field distribution to study the effect
of cuff electrode positions on peripheral nerve stimulation. Our multi-scale computational modeling platform is based on the Admittance Method (AM) (Cela, 2010) to predict the electric fields
generated inside peripheral nerve tissue. Peripheral nerve tissue is represented by a heterogeneous
very high-resolution nerve model that is based on segmented cross-sectional images of rat sciatic
nerve and includes fine details such as axons and myelin. Using AM computed electric field values
and current distribution inside the nerve, we investigated the effect of the distance that separates
the source and return cuff electrodes on the internal current distribution, which provide a basis for
45

Figure 3.1: CNN segmentation of cross-sectional image of peripheral nerve. (a) a cross-sectional
image of rat sciatic nerve, (b) segmented image of nerve cross-section containing multiple fascicles
populated by axons of various radii and myelin(grey represents myelin and white represents axon).
application-specific electrode design.
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3.2 METHODS
3.2.1 Building Nerve Model using CNN Segmentation of Peripheral Nerve
Cross-sectional Images
Due to the complexity and the densely populated inner nerve structures, to the best of our
knowledge, none of the currently used computational models of peripheral nerve stimulation are
based on realistic very high resolution model replicated from true peripheral nerves (Goodall et al.,
1995). In fact, most of the studies use either simplified nerve model with homogeneous fascicles or
heterogeneous fascicles populated by axons with artificial radii and locations (Kosta et al., 2020b,
2019; RamRakhyani et al., 2015; Raspopovic et al., 2011). Thus, we constructed a peripheral nerve
model using high-resolution cross-sectional image of a real peripheral nerve. To accomplish this, we
utilized a confocal image of rat sciatic nerve and performed image segmentation to highlight axons
and myelin. Image segmentation is performed using AxonDeepSeg, an open source easily trained
convolutional neural network (CNN) described in (Zaimi et al., 2018). The confocal image and
the resulting segmented image are shown in Fig. 1. Next, the segmented cross-sectional image is
discretized and extruded to build the pseudo-3D length of nerve for electrical stimulation modeling.
3.2.2 Model Building and Admittance Method
The multi-scale model considered for peripheral nerve stimulation consists of two main components: the segmented nerve model and the model of cuff electrode. The cuff electrodes are modeled
based on the cuff electrode design from typical commercial cuff electrode, with 3 metal contact
wires in each cuff electrode. Only the metal contact part of the electrode is modeled while the other
non-conductive elements that are not in touch with the tissue, such as the surrounding insulation
47

Table 3.1: TISSUE PROPERTIES
Tissue Type Conductivity (σx, σy, σz)S/m
Perineurium (0.01, 0.01, 0.01)
Myelination (2 × 10−4
, 5 × 10−9
, 5 × 10−9
)
Intracellular space (0.91, 0.91, 0.91)
Extracellular space (0.33, 0.33, 0.33)
layer, are discarded since they do not impact the current distribution in the tissue. Three simulation
models with different electrode positions are considered in this study as visualized in Fig. 3.2. The
size of the model is 300x300x550 voxels in x, y, z dimensions and the resolution is 8 µm in all three
dimensions. The model is discretized in cubic voxels and each voxel is represented by a unique
material index. The material properties of the nerve model are taken from (Butson et al., 2011;
McNeal, 1976) and described in Table 1.
The multi-resolution admittance method (AM) (Cela, 2010) is used to compute electric field
values at each node of the computation grid (Paknahad et al., 2020a,b, 2021b; Stang et al., 2019a).
AM defines a matrix describing the admittance (G), or resistance, throughout the model. The
resistance of each node is defined by the diagonal components of the matrix, while the surrounding
values define the resistances between nodes, producing a sparse, diagonal matrix. The admittance
values are computed using the conductivity and the distance between nodes in the x, y, and z
directions, as described in Equation 1.
g
i,j,k
x = σ
i,j,k
x
∆y∆z
∆x
(3.1)
A current vector (I) is defined with current values applied to whichever nodes contain a source.
A voltage vector (V) can then be solved for using G and I in Equation 2. A multi-threaded Python
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Figure 3.2: Simulation models consisting of the nerve and three cuff electrodes (shown in blue
color). Source electrodes are at the top, return electrodes are at the bottom and floating electrodes
are in the middle. Distances between source and return electrodes are: (a) 4 mm, (b) 2 mm, (c) 1
mm.
program using a biconjugate gradient algorithm is developed to construct the matrices and solve
the matrix equation with accelerated speed.
GV = I (3.2)
A 3D multi-resolution meshing algorithm is executed prior to the field simulations in order to
reduce the complexity of the problem without impacting the accuracy of the solution. In this, a high
level of details and fine resolution is maintained near the nerve periphery, in locations proximal to
the contact electrode, and the voxel size is increased further away from nerve periphery where fine
resolution is unnecessary. Therefore, along the center of the nerve bundle, the resolution would be
coarser, whereas along the periphery of the nerve (i.e. closer to the electrodes and fascicle edges),
the resolution would remain fine. In this way, the number of nodes and edges are decreased and the
computational complexity of the system is reduced.
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3.2.3 Current Density: Interpolation of Voltage Values
AM simulations provide voltage values at every node of the model. In our multi-resolution
scheme, network nodes are located at the vertices of voxels. Because conductivity value is considered
constant inside each voxel, trilinear interpolation is used to calculate the voltage at arbitrary points
inside a voxel from the values at its vertices. Once voltage values have been interpolated back to
Figure 3.3: Interpolation process to get current density value from Admittance Method output,
∆s represents unit voxel size.
Figure 3.4: Current density distribution in 4 mm model configuration, the xy cross-sectional
layers are plotted on different slices: (a) on the source electrode level, (b) on the floating electrode
level, (c) on the return electrode level. Unit: mA/m2
.
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unit voxel, electric field, charge density and current density could be calculated at any point in the
model. The entire interpolation process is depicted in Fig. 3.3.
3.3 RESULTS
3.3.1 Current Distribution inside the Nerve
A Python program is developed to interpolate AM computed electric field and to compute
current density distributions inside all three simulation models. As an example, the current density
distribution inside the nerve for simulation model with 4 mm electrode spacing is shown in Fig. 3.4.
A 3D perspective view of the simulation model is shown on the top panel of the figure. In the bottom
panels, the current density distribution inside the nerve is plotted at three different XY-slices, each
located at three electrodes positions. Current distribution on XY-slices corresponding to source
and return electrode locations (Fig. 3.4(a) and (c)) demonstrate that the current density values are
higher near the nerve periphery and near the edges of fascicles which are close to electrodes. Further,
we can see the hot spots at the edge of fiber fascicles in the slices with electrodes. Conversely, for the
XY-slice located at the non-functional floating electrode (Fig. 3.4(b)), far from source and return
electrodes, inside the fiber fascicles there is higher current flowing inside the axon intracellular
space due to its low resistivity. The axon intracellular space works as a conductive channel that can
deliver current from source electrode to return electrode throughout entire nerve. In comparison to
extracellular space, a lower current flows through the extracellular space.
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3.3.2 Current Penetration Depth and Electrode Separation
All three simulation models (with 4 mm, 2 mm and 1 mm electrodes spacing) consider a stimulation level of 100 µA current amplitude. Every other detail of these models is ensured to be
the same except the electrodes locations. For all three simulation models, we computed the radial
average values of the current density on the floating electrode slices. Fig. 3.5 presents the plots
of these radial average current density values with respect to the radial distance, R, ranging from
the nerve center to the nerve edge. Results presented in Fig. 3.5 suggest that when the source
and return electrodes are placed at 1 mm separation distance, the current flowing in the center of
the nerve is significantly lower than that of 2 mm and 4 mm separation distances. Therefore, these
results suggest that when the source and return electrodes are placed too close to each other, current
penetration into the nerve center is less compared to the cases when source and return electrodes
are separated farther. In addition, the current distribution peak for the 1 mm electrode distance
setup is near the nerve bundle edge, which implies that the current is not flowing deep enough into
the nerve center to recuit a large number of fibers. Rather current is flowing along the nerve edge
area directly from the source electrode to the return electrode without entering deep into the center.
Finally, the curves in Fig. 3.5 suggest that the heterogeneity of the nerve model significantly affects
the current distribution inside the nerve. Therefore, realistic heterogeneous peripheral nerve models
are essential for the analysis and design of peripheral neurostimulation electrodes.
3.4 DISCUSSION
A computational study using the Admittance Method multi-scale computational platform was
conducted to further our understanding of the effect of electrical stimulation site in the peripheral
nerve electrical stimulation. We first performed the image segmentation of nerve cross-sectional
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Figure 3.5: Line plot of radial average values extracted from slices with floating electrode from
all three models
images using convolutional neural network and then built realistic peripheral nerve models from the
segmentation results. We also developed realistic cuff electrode models with different separation
distances between electrodes. We found that the current distribution values around the nerve
center are lower when the source electrode and return electrode are placed in a closer vicinity.
Our results suggest that the accurate, high-resolution anatomical features of the peripheral nerve
significantly affect the current distribution inside the nerve. This suggests that a very high resolution
and accurate model of the peripheral nerve plays a critical role in the design and optimization of
53

neurostimulating electrodes. Besides implications on recruitment of target fibers, the significant
variations due to the model heterogeneity have implications on the level of stimulation that are
considered safe and do not induce axonal damage. Thus, with the proposed method and models,
criteria for safe and effective peripheral neurostimulations can be established.
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Chapter 4: Electrical Stimulation Induced
Current Distribution in Peripheral Nerves Varies
Significantly with the Extent of Nerve Damage:
A Computational Study Utilizing Convolutional
Neural Network and Realistic Nerve Models
© Du, J., Morales, A., Kosta, P., Bouteiller, J-M.C., Martinez-Navarrete, G., Warren, D.J.,
Fernandez, E., Lazzi, G.International Journal of Neural Systems 33, 2350022 (2023) .
55

Abstract
Electrical stimulation of the peripheral nervous system is a promising therapeutic option for several
conditions; however, its effects on tissue and the safety of the stimulation remain poorly understood. In order to devise stimulation protocols that enhance therapeutic efficacy without the risk
of causing tissue damage, we constructed computational models of peripheral nerve and stimulation cuffs based on extremely high-resolution cross-sectional images of the nerves using the most
recent advances in computing power and machine learning techniques. We developed nerve models
using non-stimulated (healthy) and over-stimulated (damaged) rat sciatic nerves to explore how
nerve damage affects the induced current density distribution. Using our in-house computational,
quasi-static, platform, the Admittance Method (AM), we estimated the induced current distribution
within the nerves and compared it for healthy and damaged nerves. We also estimated the extent of
localized cell damage in both healthy and damaged nerve samples. When the nerve is damaged, as
demonstrated principally by the decreased nerve fiber packing, the current penetrates deeper into
the over-stimulated nerve than in the healthy sample. As safety limits for electrical stimulation of
peripheral nerves still refer to the Shannon criterion to distinguish between safe and unsafe stimulation, the capability this work demonstrated is an important step toward the development of safety
criteria that are specific to peripheral nerve and make use of the latest advances in computational
bioelectromagnetics and machine learning, such as Python-based Admittance Method and CNN
based nerve image segmentation
4.1 Introduction
Electrical neural stimulation has been shown to be effective in the treatment of various medical
conditions, such as retinal stimulation for vision restoration in Retinitis Pigmentosa patients and
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Figure 4.1: CNN segmentation of cross-sectional image of peripheral nerve. (A) a cross-sectional
image of rat sciatic nerve, (B) segmented image of nerve cross-section containing multiple fascicles
populated by axons and myelin of various morphologies (grey represents myelin and white represents
axon), (C) CNN U-Net architecture used to generate the segmented image.
deep brain stimulation for tremors, epilepsy, and Parkinson’s disease (Benabid et al., 2009; Nelson
et al., 2011; Yue et al., 2016). Electrical stimulation of peripheral nerves has been used to treat
chronic pain and aid the treatment of many diseases(Doucet et al., 2012; Grill and Kirsch, 2000;
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Kimiskidis et al., 2013; Pineau et al., 2009; Rajdev et al., 2011; Shi et al., 2021; Stein et al., 1992;
Weiner, 2000; Yaqub et al., 2022; Zhang et al., 2016). There have been numerous investigations into
the tissue damage induced by electrical nerve stimulation (Agnew and McCreery, 1990). However,
existing safety criteria only account for the stimulus waveform and the tissue-electrode contact
area, ignoring the actual distribution of current across distinct tissue regions (McCreery et al.,
1990; Shannon, 1992). Further, current safety criteria for electrical stimulation of the nervous
system rely on data collected in the Central Nervous System, and it is uncertain whether these
are applicable to the Peripheral Nervous System. Based on the material properties of the various
tissue types, electrical stimulation causes different current densities within the tissue. For instance,
healthy nerve fibers are frequently surrounded by highly resistant myelin sheath, which makes it
more difficult for the stimulation current to enter the axon region. When the myelin layer of a fiber is
damaged and exposed, the internal current distribution will be significantly changed. Consequently,
the heterogeneity of the nerve plays a crucial role in predicting induced currents and, in turn,
the likelihood of tissue injury. An accurate model of the nerve and the neural stimulation system
could shed light on how stimulation parameters influence the safety and efficacy of stimulation of
specific tissues and nerve fibers. There are numerous computational models of peripheral nerves;
however, the majority of these models are oversimplified and represent nerve fascicles using effective
(average dielectric) media properties(Kosta et al., 2020b; RamRakhyani et al., 2015; Raspopovic
et al., 2011). Therefore, for more accurate predictions of induced fields in the peripheral nerve
due to neurostimulators, improved computational models that contain heterogeneous fascicles with
accurate anatomical features such as axons of various size and myelination are required.
Convolutional neural network (CNN) is a powerful image segmentation and recognition method.
With the use of CNN for the segmentation of cross-sectional nerve images (Acharya et al., 2018a,b;
Ansari et al., 2019; Cai et al., 2018; Chen et al., 2018; Feng et al., 2021; Hassanpour et al., 2019;
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Manzanera et al., 2019; Martinez-Murcia et al., 2018; Miraglia et al., 2022; Nogay and Adeli, 2020;
Peng et al., 2021; Thomas et al., 2020; Wang et al., 2022; Xu et al., 2022; Xue et al., 2021; Zaimi
et al., 2018), in our prior work we constructed a multi-scale computational model of a healthy
rat sciatic nerve(Du et al., 2021). Using this model, we simulated the application of electrical
stimulation to the nerve using cuff electrodes and highlighted how the distance between the cuff
electrodes had a substantial effect on the current penetration depth within the nerve. In this work,
using cross-sectional images of non-stimulated and stimulated nerve cross-sections provided by our
collaborators from University of Utah, where rat experiments are conducted, we construct two
distinct computational models to represent healthy and damaged nerves. Animal procedures were
approved by the University of Utah Institutional Animal Care and Use Committee. One male, adult,
approximately 300 g Sprague Dawley rat underwent 3 phases in the experiment: implantation,
stimulation, and euthanasia. Under isoflurane anesthesia, the left sciatic nerve was implanted
with a multi-electrode cuff array (MECA, Microprobes for Life, Gaithersburg MD, USA, 2.0-mm
ID, two 100-um Pt contacts separated by 3 mm). The MECA’s connector was attached to the
skull, the wire bundle was transcutaneously routed to the nerve, and the cuff was placed on and
sutured to the nerve. One week later, the rat underwent an electrical stimulation under isoflurane
anesthesia. The contacts were attached to a stimulator (STG-4002-16 mA, Multichannel Systems,
Baden-Württemberg, DE) that applied charge balanced, constant current, biphasic pulses at 1.1
mA for 100 us per phase, and 400 us interpulse period. The nerve was stimulated at 50 Hz for 4
hours. One week later, the rat was perfused under deep isoflurane anesthesia by cardiac puncture
with phosphate-buffered saline (PBS), followed by 4% formaldehyde and 2% glutaraldehyde in PBS
(PFA+Glut+PBS). The region of the nerve with the implant was excised, and a similar region from
the contralateral (unimplanted) side was imposed. Both nerve samples were placed in a vial filled
with PFA+Glut+PBS and stored in a 4C refrigerator for one week. They were transferred into a
59

vial filled with 0.02% sodium azide in PBS and shipped for histological analysis. With the nerve
cross sectional images from the actual experiments obtained, first we utilize CNN segmentation to
identify the different tissue types in the nerve; then we create anatomically correct, heterogeneous,
high-resolution nerve models comprising fine anatomical features, such as axons and myelin. The
morphological features of the nerves, such as fiber density and fiber packing, are then quantified
to characterize the morphological and structural changes in the damaged nerve with respect to the
healthy nerve. Finally, we employ a multi- scale computational platform based on the Admittance
Method (AM) (Cela, 2010) to calculate the current distribution within healthy and damaged nerves
in response to cuff electrode stimulation. This approach allows us to evaluate how the distribution
of induced current density, and the likelihood of subsequent tissue damage, are affected by early
electrically-induced morphological and structural changes of the nerve.
4.2 Methods
Computational platforms serve as valuable tools that can predict the neural response as well as
the possibility of tissue damage due to electrical stimulation (Gaur et al., 2019; Geminiani et al.,
2018; Mollet et al., 2013; Sutherland, 1990; Villar et al., 2016). Simulation studies based on accurate
computational models are crucial to direct in vivo experiments and clinical trials and to reduce the
number of iterations required for the design of neural stimulation systems. Due to recent advances
in computing cluster development and with the help of machine learning approaches, multi-scale
computational models have now demonstrated the ability to accurately predict the fields induced
in complex biological media with unprecedented resolution. This section discusses the modeling
approaches and techniques that we used to develop accurate, ultra high-resolution computational
models of peripheral nerves and stimulation systems.
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4.2.1 CNN Segmentation of Peripheral Nerve Cross-sections
Most of the currently available computational models of peripheral nerves are based on heterogeneous fascicles populated by artificially generated axons or simplified homogeneous fascicles
(Kosta et al., 2020b, 2019; RamRakhyani et al., 2015; Raspopovic et al., 2011). To build more
realistic nerve models, we used high-resolution confocal imaging to acquire cross-sectional images of
stimulated and non-stimulated rat sciatic nerves. To replicate the details of axon location and morphology from these tissue samples, the histological images were segmented using a convolutional
neural network (CNN) known as AxonDeepSeg (Zaimi et al., 2018). As shown in Fig. 4.1, the
architecture consists of 7 "blocks", each containing 3 convolutional layers. The blocks on the top
path, feeding to the right, are followed by a down-convolutional layer to further expand the field of
view of the subsequent block. Symmetrically, the blocks on the bottom path, feeding back to the
left, are preceded by up-convolutional layers that also use the output of the corresponding block in
the top path. This U-Net architecture allows the network to make structural analysis over large
fields of view while localizing the tissue segmentations to pixel-wise resolution. Once the network
was trained on a relatively small sample of manually segmented images, we obtained fast pixel-wise
labeling of myelin and axon. The network was trained using multiple patches sized 256×256px, 459
patches for training and 116 patches for validation (70/30 split). After training the network was
validated with an accuracy 95% (F1/Dice Score: Axon:0.847, Myelin: 0.831). Our ground-truth
labels for training and validation datasets were manual segmentations performed and checked by
multiple expert histologists. The segmented cross-sectional slice of the nerve was then ready to be
discretized and extruded to build a pseudo-3D nerve model for electrical stimulation simulation.
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Figure 4.2: Steps for histological measurements (A-F). (A) Confocal image of rat sciatic nerve
at 80× magnification. (B) Microscopy images were segmented into axon and myelin labels using
AxonDeepSeg. (C) Individual fibers were labeled using the watershed algorithm. (D) Fascicles
area(grey) were manually labeled from the confocal images. (E) Fiber packing was then measured by
the ratio of the area of cells within the measurement window to the area of the fascicle(green) within
that same window. (F) Fiber density was measured by counting the number of cells(represented
by different colors) within the measurement window divided by the area of the fascicle within that
same window. When the window cuts through a fiber, the fraction of the fiber area within the
window is counted.
4.2.2 Cell-wise Damage Evaluation Metrics
The steps to characterize the health of the nerve, based on cross-sectional histological nerve
images, are summarized in Fig. 4.2. Fig. 4.2(A) presents a sample of a confocal image of a rat
sciatic nerve at 80× magnification. Using the semantic segmentations of axon and myelin tissue and
the watershed algorithm, we obtained CNN segmentations of individual fibers (axon and associated
myelin, as depicted in Fig. 4.2(B)). The fascicular area was manually masked for each nerve sample.
The labeled fibers (shown in Fig. 4.2(C)) and fascicle masks (shown in Fig. 4.2(D)) were used to
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calculate structural measurements of nerve health using parameters such as fiber packing and fiber
density. A measurement of the packing and density of the tissue surrounding each labeled fiber was
made by centering a window on each fiber, as illustrated in Fig. 4.2(E) and 4.2(F), respectively.
Fiber packing was measured by calculating the ratio of the area of cells within the measurement
window to the area of the fascicle within that same window. In order to avoid double counting the
cells truncated between two adjacent windows, for each cell not fully contained within a block, it
contributes only the fraction of it’s area contained to the cell count of that block. A cell with a
third of it’s area in the block is only counted as a third of a cell, with the remaining two-thirds
counted towards the neighboring block, and thus avoids the very valid concern of double counting.As
a result, fiber packing is a unitless quantity, with values ranging from 0 to 1. Fiber density was
instead computed by counting the number of cells within the measurement window divided by the
area of the fascicle within that same window. When the window cuts through a fiber, the fraction
of the fiber area within the window is counted. The value for window area was adjusted to account
for when windows included regions outside the fascicles or occupied by blood vessels where neurons
would not be expected. For what it is worth the best is to
4.2.3 Nerve Image Selection
In order to compare the current distribution inside healthy nerves and damaged nerves, two
different nerve models are constructed from two nerve microscopy images from actual rat sciatic
nerve stimulation experiments; one presumed healthy (that is of unremarkable morphological and
structural changes compared to that of a non-stimulated rat) and one with damage (that is, with
measurable morphological and structural changes compared to that of a non-stimulated rat). To
further rule out the individual differences between different samples, we considered fiber packing
and fiber density as metrics to help ensure that the selected images are a valid representation of
63

their own individual classes. As shown in Fig. 4.2, using a 200µm × 200µm (1600px × 1600px)
window, fiber density is the number of cells in a window (cells cut by the window are counted
fractionally) divided by the window area. Fiber packing uses the same window but divides the total
cell area by the window area. As shown in Fig. 4.3, we obtained fiber density ranges from 5k to
15k fibers/mm2
for damaged samples and from 20k to 28k fibers/mm2
for healthy samples, fiber
packing ranges from 0.3 to 0.4 in damaged nerve samples and 0.6 to 0.7 for healthy nerve samples.
These two metrics are essential in evaluating structural changes of the nerve as well as changes in
the distribution of the electrical properties of the nerve(Christensen and Tresco, 2015; Comin et al.,
2014; Sandell and Peters, 2001). Making use of these metrics, we selected one healthy and one
damaged nerve sample from a total of six samples within twice of standard deviation, as presented
in Fig. 4.3. The selected healthy nerve image is taken from control samples with implanted but
not stimulated cuff electrodes. whereas the selected damaged nerve image is taken from a sample
stimulated using a 1.2mA current source.
4.2.4 Model Building and Admittance Method
The multi-scale computational model to be employed for peripheral nerve stimulation simulations
consists of two main components: the segmented nerve model and the model of cuff electrode. The
cuff electrodes are modeled based on the design of a typical commercial cuff electrode with an inner
diameter of about 2mm and , two metal contact wires (as shown in Fig. 4.4(E)). One of the metal
wires works as a source cuff electrode, and the other one works as a ground cuff electrode. Only
the metal parts of the electrode are modeled and the other non-conductive elements that are not in
contact with the tissue (such as the surrounding insulation layer) are discarded since they do not
have a significant impact on the current distribution in the tissue. During the in vivo experiments,
the nerve is sutured to the electrode; therefore, the nerve model is placed in close contact with one
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Figure 4.3: Sample selection for healthy and damaged nerve models. To ensure the samples used
for modeling are good general representations of their class, we use fiber density and fiber packing
to select one healthy nerve sample and one damaged nerve sample from a total of six samples. All
of the similar types of cells fall within the range of twice the standard deviation.
TISSUE PROPERTIES
Tissue Type σx σy σz Unit
Perineurium 0.01 0.01 0.01 S/m
Myelination 2 × 10−4 5 × 10−9 5 × 10−9 S/m
Intracellular space 0.33 0.33 0.33 S/m
Axoplasm 0.91 0.91 0.91 S/m
Epineurium 0.1 0.1 0.1 S/m
Nerve membrane 0.02 0.02 0.02 S/m
Saline solution 1.45 1.45 1.45 S/m
Extracellular space 0.33 0.33 0.33 S/m
side of the nerve with saline solution filled in, as shown in Fig. 4.4(C)(H). For both healthy and
damaged nerve models, the cuff electrodes and their position are the same. The model is discretized
in cubic voxels, and a unique material index represents each voxel. The model size is 800×800×1000
voxels in x, y, and z dimensions, respectively, and the resolution is 3.1 µm in all three dimensions.
The material properties of the nerve model are taken from (Butson et al., 2011; McNeal, 1976) and
described in Table 1.
The multi-resolution Admittance Method (AM) (Bingham et al., 2020; Cela, 2010; Eberdt et al.,
65

2003; Loizos et al., 2014b) is used to compute electric field values at each node of the computation
grid (Paknahad et al., 2020a,b, 2021b; Stang et al., 2019a). AM defines a matrix describing the
admittances (G) throughout the model. The model networks consists of two major components:
The admittance of each node which is defined by the diagonal components of the matrix, and the
admittance between nodes which is defined by the off diagonal components. The nature of the
network model makes the admittance matrix (G) a sparse, diagonal matrix. The admittance values
are computed using the conductivity and the distance between nodes in the x, y, and z directions,
as described in Equation (1)-(3),where σx, σy, σz are the conductivities along x, y, z directions
respectively and ∆x, ∆y, ∆z are the distance along different directions between two nodes.
g
i,j,k
x = σ
i,j,k
x
∆y∆z
∆x
(4.1)
g
i,j,k
y = σ
i,j,k
y
∆x∆z
∆y
(4.2)
g
i,j,k
z = σ
i,j,k
z
∆y∆x
∆z
(4.3)
The vector, I, contains current values with non-zero elements for the source nodes. After defining
the admittance matrix (G) and current vector (I), the matrix equation (4) is solved to compute
the induced voltage vector, V, at all nodes of the model. To efficiently generate matrices and
solve matrix equations, a multithreaded Python computational platform leveraging a biconjugate
gradient method is developed.
GV = I (4.4)
Prior to the electric field simulations, a 3D multi-resolution meshing approach is used to reduce
the computational time without impacting the ac- curacy of the results. Considering that currents,
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Figure 4.4: (A)(F) Representative nerve microscopic images are used to generate one healthy
and one damaged nerve model. (B)(G) Convolutional neural network was applied to the images
to segment the myelin and axons. The damaged nerve image has a lower cell density. (C)(H)
The segmented nerve images used to reconstructed the nerve structures under stimulation. (E)
3D model with source and ground electrodes is constructed. (D)(I) Current density distributions
near the source electrode are plotted for both models. It shows that damaged nerve models have a
higher current density at the nerve periphery than the healthy nerve model due to the decrease of
the cell population.
for this configuration of electrical stimulators, are unlikely to penetrate deep into the nerve, a
fine resolution is not required in regions that are distal from the nerve periphery. Therefore, the
computational resolution is coarser around the center of the nerve, while it is finer near the nerve’s
periphery (i.e., closer to the electrodes and fascicle borders). This helped reducing the computational
time of the simulations by decreasing the total number of nodes in the model, thus reducing the
matrix size considerably. Voltage values are computed at each node of the model, and these nodes
are defined at the vertices of voxels. Trilinear interpolation is used to compute the voltage at
various points within a voxel based on the values at its vertices since the conductivity value within
each voxel is considered to be constant. Using the interpolated voltage values at unit voxels, other
parameters - such as electric field, charge density, and current density - can be calculated at any
point in the model.
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4.3 Results
We leveraged the AM for large-scale simulation to predict the current density distribution within
realistic peripheral nerve models. We constructed two separate models (healthy and damaged) of
the nerve using CNN segmentation to identify the difference between the distribution patterns of
induced currents in healthy and damaged nerves. We validated our simulation results by correlating
the charge densities to their respective cell-specific damage measures; the correlation shows that
there are more pronounced markers of damage in the cell group with higher induced charge densities.
4.3.1 Current Distribution inside Two Nerve Models
The initial CNN segmentation is performed in order to identify the axons and myelin in the
nerve, beginning with the two selected images shown in Fig. 4.4(A) and (F), which are representative
examples of their own category. The damaged nerve model has a lower cell density closer to its
nerve boundary compared to the healthy nerve model. The distribution of current density inside
the two nerve models is computed and plotted in Fig. 4.4(D)(I). The 2D slices are chosen in close
proximity to the source electrode. It is shown that the current density within both nerves is greatest
towards the nerve periphery, which is closest to the electrode, and decreases towards the center.
Due to the decrease in the cell density, the damaged nerve model exhibits a higher current density
at the nerve periphery than the healthy nerve model.
4.3.2 Current Density in Different Tissue Types
Using the constructed ultra high-resolution model with realistic nerve morphology, in addition
to the general current distribution pattern illustrated above, we are able to extract current density
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Figure 4.5: Current density values at different tissue types of (A) healthy and (B) damaged nerve
models. The current density values on axoplasm in the damaged model are much higher than in
the healthy nerve model.
val- ues in different tissue types. Fig. 4.5(A) illustrates the current density values for the healthy
nerve model, and Fig. 4.5(B) shows the current density values for the damaged nerve model. Four
tissue types are considered: myelin, intracellular space, axoplasm, and epineurium. The comparison
shows that the cur- rent density values in the axoplasm of the damaged model are significantly
higher than in the healthy nerve model due to the decrease of the highly resistive myelinated fiber
population. Further, due to the lower density of myelinated axons, the current penetrates deeper
into the nerve’s center and, therefore, enters the more conductive axoplasm in the damaged model.
Thus, the current density in the axoplasm is considerably higher in the damaged nerve model than in
the healthy one. Furthermore, the current density values generally decrease with the distance from
the electrode. Very little current is observed in the myelin, despite the large population number,
due to the high resistivity of myelin tissue, making it difficult for the current to flow through.
4.3.3 Current Density Values to Charge Density Per Phase
The “Shannon criteria", a safety criterion based on data collected by McCreery and his colleagues’
experiments using brain stimulation in feline, is still the most widely employed safety criterion by
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Figure 4.6: (A)(E)One healthy and one damaged segmented microscopic nerve images are selected
to visualize different damaging metrics: (B)(F) Fiber packing uses the same window but divides
the total cell area inside by the window area. (D)(H) G-ratio calculates the ratio between the inner
and the outer diameter of each cell. (C)(G) Fiber density is the number of cells inside (cells cut by
the window are counted fractionally) divided by a sliding window area.
practitioners. (Shannon, 1992). McCreery et al. manually evaluated the extent of fiber damage and
graphed the experimental results to summarize the gathered data (McCreery et al., 1990).Then,
applying his well-known equation, Shannon identified the two most important factors to estimate
the safety of electrical stimulated tissue: the charge density per phase and the charge per phase.
Traditionally, the charge density per phase and charge per phase employed in the Shannon Criteria were simply an average estimation on the electrode surface. However, such estimation of charge
density is extremely constrained when it comes to safety analysis. Specifically, multiple studies have
demonstrated that electrode surface charge density varies significantly with the electrode geometry
(McIntyre and Grill, 2001; Wei and Grill, 2009). With our realistic nerve model and computational
platform, we have demonstrated our ability to calculate localized current density values within the
nerve. In order to translate the previously determined current density into the safety-related parameter charge density Q, it is necessary to multiply the phase length t and axon area A by the
70

current density J. In this instance, we have used a pulse duration of t=400 uS. Using this parameter, we were able to obtain the charge density per phase and charge per phase at each local cell
of stimulated peripheral nerve samples. The accuracy of such local calculation is an improvement
compared to the average estimation used in the original Shannon’s equation.
Q = J × t × A (4.5)
4.3.4 Nerve Damage Analysis
Once the charge density per phase values for each cell of two nerve samples using the AM
modeling described above is obtained, the next step is to explore the potential relationship between
cell damage and cell charge density values. With this relationship, we are able to validate the
assumption that higher charge density values could induce morphological changes and, therefore,
potentially induce more damage due to the ability of the current to penetrate deeper in the nerve.
First, we calculated the damage indicators as described in the methods section for each of the cells
in both nerve samples, including fiber packing, fiber density, and G-ratio. As shown in Fig. 4.6, the
fiber packing, fiber density, and G-ratio of each cell from both samples are illustrated with varying
hues reflecting the degree to which its morphology is altered. For the healthy nerve sample we can
observe, for example, a significant number of cells in the yellow color range whereas, for the damaged
nerve sample, there are a greater number of cells in the red color range, indicating more significant
morphological and structural changes and, therefore, a possibility for additional current-induced
damage. Fig. 4.6 shows that a damaged nerve model may cause additional currents to flow into the
nerve center, hence causing additional damage.
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4.3.5 Charge Density Versus Damage Extent
To further investigate the potential association between cell damage and charge density, we
decided to group each nerve cell according to its local charge density. Using two nerve samples and
more than 30,000 individual cells, we grouped these cells based on cell-specific charge density values
obtained from AM simulations. Each class of cells can be derived from different nerve samples
as long as their local charge density values fall within the same class. We then have two sets of
cells with varying local charge densities per phase, as shown in Fig. 4.7(A). Using the empirical
distribution function plot, we plotted each of these two sets of cells against their respective damage
metrics, which is fiber density. The group with a charge density above 3nC/cm2
exhibited significant
morphological differences compared to those with a lower charge density. Moreover, we can see from
Fig. 4.7(B) that for the cell group with a different charge density value, the curve is migrating to
the left, further indicating potential damage happening within the over-stimulated cells. Thus, the
conclusion is that more currents entering the nerve fiber area in the damaged model appear to result
in additional damage.
4.4 Discussion
Current density has been considered one of the crucial factors in order to assess nerve damage
caused by electrical stimulation. However, knowledge of the accurate current density distribution
inside the nerve is challenging to estimate. We conducted a computational study using the Admittance Method (AM) multi- scale computational platform to help gain insights into the limits for
tissue safety during electrical stimulation. We segmented cross-sectional images of healthy and damaged sciatic nerves of rats using a convolutional neural network and built realistic peripheral nerve
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Figure 4.7: (A) Cells from two nerve samples are grouped according to their local charge density.
Using two nerve samples and more than 30,000 individual cells, these cells are grouped into two
groups based on the cell-specific charge density values. (B)Empirical distribution function plot of
these two classes of cells against one of their respective damaging indicators, which is fiber density
in this case.
models from the segmentation results. Further, we modeled electrical stimulation using realistic
cuff electrodes considering different levels of tissue damage.
We evaluated the extent of damage due to electrical stimulation in three healthy nerve samples
and three damaged nerve samples using a variety of morphological and structural measures. All
samples are within two times of the standard deviations of their respective classes. In addition,
we chose samples that are representative of the middle of their respective class in order to better
represent the class itself.
Simulations show that the current distribution values in the axoplasm are significantly higher
in the damaged model compared to the healthy model. Our findings suggest that the exact, highresolution anatomical features of the peripheral nerve have a significant impact on the distribution
of current within the nerve. In assessing the efficacy and safety of neurostimulation devices, an
extremely precise and high-resolution model of the peripheral nerve may be necessary.
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Besides implications on the recruitment of target fibers, the significant variations due to the
damage extent of models have implications on the level of stimulation that are considered safe and
do not result in axonal damage. Our results demonstrated the nerve morphology analysis of a
nerve sample stimulated by a 1.2mA current source with a Shannon value of k=0.11, which falls
within the safe range according to Shannon’s standard, as k=2 is the upper limit. The results of
the study revealed a significant difference between healthy nerve samples in the control group in
terms of fiber density and fiber packing. Our results are consistent with those of other studies,
which have shown that Shannon’s value is not an effective indicator of damage in the context of
peripheral nerve stimulation.Our results also demonstrated the results from the powerful automatic
algorithm-based cell damage analysis tool. With the damage analysis tool we performed cell-wise
damage evaluation using different metrics. These cell-wise damage evaluations, combined with our
charge density distribution from AM simulation results, allow us to establish criteria for safety
and effective peripheral neurostimulation, especially for long-term chronic stimulation where nerve
morphology might be altered due to structural damage.
The objective of our study is to achieve an accurate simulation of nerve stimulation conditions.
In order to achieve this objective, we have employed a multi-faceted approach in our modeling.
Specifically, we have based our cuff electrode model on the design of commonly used commercial
cuff electrodes. Additionally, we have incorporated a representation of the experimental setup of
the nerve during modeling, which includes factors such as the suturing of the electrode to the nerve.
This approach has resulted in the placement of the nerve model in close proximity to the nerve,
with a saline solution filling the space between them. This methodology is aimed at providing an
accurate representation of the experimental setup, ultimately leading to the accurate simulation of
nerve stimulation conditions. One limitation is neglecting the srrounding tissues around the nerve,
although our modeling involved saline solution all around the nerve model, which is pretty much
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equivalent to what the fat or muscle tissue does, the geometrical relationship between the nerve
and the surrounding tissues can also play a role. For example, if the nerve is surrounded by a large
amount of tissue, this can lead to an increased distance between the electrode and the nerve, which
can in turn increase the stimulation threshold. Also, The 3D model used in our study is extruded
from 2D model, as creating a pseudo-3D model from 2D models can be computationally demanding
especially considering the tortuosity of the nerve, and it can dramatically increase the complexity
and computational resources needed to run the simulation. One limitation of pseudo-3D models
is that they are based on a single 2D cross-section or slice, and therefore do not fully capture the
variation that may exist in the system across different cross-sections or slices. This can lead to
inaccuracies in the simulation, especially if the cross-section or slice used to create the model does
not represent the average or typical condition of the system. Another limitation is that these models
do not account for the interactions between the different layers or slices, which can be important
in certain systems such as nerve stimulation. It’s worth noting that we are working on addressing
these limitations by creating real 3D nerve models with realistic tortuosity as a continuation of this
work.
Other computational platforms with similar functionalities (Musselman et al., 2021; Raspopovic
et al., 2011) mostly use either a simplified nerve model with homogeneous fascicles or heterogeneous
fascicles populated by axons with artificial radii and locations. However, for the design and development of neural stimulation devices, our computational platform’s ability to capture accurate and
detailed bioelectrical behaviors, such as the electrode-tissue interface between implanted stimulation devices and neural tissue, is highly desired. In addition, the Python-based architecture of our
platform makes it easier to integrate with other computational approaches, such as NEURON, for
multiphysics bioelectromagnetic simulations. Limitations include, with the tremendous amout of
details inside realisitc nerve models, such computational modeling could be rather time consuming.
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Due to the complexity of rat experiments, the stringent procedures required to obtain highresolution microscopy images of stimulated samples, and the computational intensity of such a
large-scale modeling study, obtaining results on multiple samples is challenging. The method we
used to ensure the samples used for modeling are a good general representations of their class rely on
the use of two metrics of damage to select representative healthy nerve and damaged nerve samples.
In a future study, with additional experimental data, we aim to conduct a similar analysis on a
larger dataset and further establish a correlation between local charge density values and cell-wise
morphological damage. Our long term goal would be to make use of our computational results
to help with clinical practices make treatment plan. As our database gets larger, for one specific
patient we can conduct modeling based on comparable nerve models from our database and thus
provide insights to the possible current distributions and safety limits of stimulation to the current
patient.
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Chapter 5: Toward Safety Protocols for
Peripheral Nerve Stimulation (PNS): a
Computational and Experimental Approach
© Du, J., Morales, A., Kosta, P., Bouteiller, J-M.C., Martinez-Navarrete, G., Warren, D.J.,
Fernandez, E., Lazzi, G. BioElectromagnetic Special Issue on Neurostimulation(Under Submission
2024)
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Abstract
As the clinical applicability of Peripheral Nerve Stimulation (PNS) expands, the need for PNSspecific safety criteria becomes pressing. This study addresses this need, utilizing a novel machine
learning and computational bio-electromagnetics modeling platform to establish a safety criterion
that captures the effects of fields and currents induced on axons. Our approach comprises of three
steps: experimentation, model creation, and predictive simulation. We collected high-resolution
images of control and stimulated rat sciatic nerve samples at varying stimulation intensities and
performed high-resolution image segmentation. These segmented images were used to train machine
learning tools for automatic classification of morphological properties of control and stimulated PNS
nerves. Concurrently, we utilized our quasi-static Admittance Method-NEURON (AM-NEURON)
computational platform to create realistic nerve models and calculate induced currents and charges,
both critical elements of nerve safety criteria. These steps culminate in a cellular-level correlation
between morphological changes and electrical stimulation parameters. This correlation informs the
determination of thresholds of electrical parameters that are found to be determinant of damage,
such as maximum cell charge density. The proposed methodology and resulting criteria combines
experimental findings with computational modeling to generate a safety threshold curve which
captures the relationship between stimulation current and the potential for axonal damage. This
curve offers a practical mean to ensure safer, while still effective, PNS stimulation, marking an
important step towards specific safety criteria in PNS neurostimulation.
5.1 Introduction
Electrical stimulation of nerves represents a compelling frontier in contemporary medical treatment, offering unique therapeutic avenues for an array of neurological and physiological disorders
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(Benabid et al., 2009; Nelson et al., 2011; Yue et al., 2016). This technique leverages our growing
understanding of the nervous system, using precisely controlled electrical currents to modulate nerve
activity, thereby influencing the functions they control (Doucet et al., 2012; Grill and Kirsch, 2000;
Stein et al., 1992; Weiner, 2000). From the brain’s intricate network of neurons to the fine nerve
endings spread throughout the body, electrical stimulation has demonstrated potential in various
contexts, including pain management, motor function restoration, and even cognitive enhancement
(Kimiskidis et al., 2013; Pineau et al., 2009; Rajdev et al., 2011; Shi et al., 2021; Yaqub et al., 2022;
Zhang et al., 2016).
While the use of electrical stimulation spans the entirety of the nervous system, one particular
area of focus is the peripheral nerve stimulation (PNS). The PNS, comprising nerves outside the
brain and spinal cord, plays crucial roles in motor control, sensory information processing, and
autonomic function regulation (Chakravarthy et al., 2016; Eldabe et al., 2016; Nayak and Banik,
2018; Slavin, 2008; ?). Electrical stimulation to the peripheral nervous system—holds significant
promise due to its potential for direct, localized intervention. As we delve deeper into understanding
the capabilities and safety parameters of this promising technique, peripheral nerve stimulation
emerges as an exciting and pivotal area of exploration in modern neurotherapeutics.
5.1.1 Existing safety protocol for PNS
Peripheral nerve stimulation (PNS) is a method of medical intervention that has witnessed over
six decades of widespread clinical usage. Recognized for its significant therapeutic potential, it is
employed as a mode of treatment for a number of medical conditions including chronic pain, lost
motor functionality, and epilepsy (Campbell and Long, 1976; Mobbs et al., 2007; Nashold Jr et al.,
1982).
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Figure 5.1: Summary of our methodology involving (1) image segmentation and algorithm-based
nerve damage quantification for comprehensive assessment of structural and morphological axonal
damage, and (2) computational modeling and neural network training for detailed exploration of
cell-specific charge density distribution. The correlation between axon damage and local stimulation
charge density leads to a safety criterion for PNS.
With the growing recognition of the therapeutic effects of PNS and the consequential expansion
in its application, it becomes increasingly vital to understand the parameters that affect its safe
utilization. Ensuring patient safety and optimizing treatment outcomes constitute a critical objective of any therapeutic intervention, and PNS is no exception. However, despite its long-standing
usage, safe limits of induced electric fields and currents due to PNS remain unclear (Günter et al.,
2019; Helm et al., 2021; Silberstein et al., 2012).
Currently, both researchers and practitioners largely utilize the "Shannon Criteria"(Shannon,
1992), which is derived from extensive empirical observations and theoretical modeling. The Shannon criteria were developed based on experimental data derived from cat brains, collected by McCreery and colleagues(McCreery et al., 1990). It provided a safety guideline summarized by the
equation log(D) = k - log(Q), which was derived by visually assessing axonal fiber damage. In this
equation, D represents the charge density per phase, calculated by dividing the charge per phase (Q)
by the electrode surface area. The variable ’k’, indicating the risk of damage, is typically considered
to represent a threshold for safety when it is lower than 1.5. This criterion provides an essential
linkage between stimulus charge per phase and charge density per phase, which then relates to the
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likelihood of inducing nerve damage.
5.1.2 Limitations of the existing protocol
Despite its wide acceptance, the Shannon criterion is not without limitations.
The first limitation of the Shannon criterion is that it was originally derived from data obtained
from cat brain neurons - not peripheral nerves. Given the substantial physiological differences
between these nerve types, including their structural properties and myelination patterns, the applicability of the Shannon criterion to PNS is questionable. This observation highlights the need for
safety guidelines based directly on peripheral nerve data to ensure more precise and relevant safety
standards for PNS.
Furthermore, the Shannon criterion assumes uniform charge density distribution across the
electrode surface, represented by D equals Q divided by the electrode surface area (A). However,
numerous studies have raised concerns about this assumption, owing to the demonstration of nonuniform charge distribution across the electrode surface (McIntyre and Grill, 2001; Wei and Grill,
2009). For instance, a study by W. Grill et al. employed six electrodes with the same surface area
but different perimeters (Wei and Grill, 2009). Contrary to Shannon’s assumption, the study found
varying charge densities across the electrodes.
Finally, the damage analysis of the Shannon criterion is performed manually, which may introduce subjectivity and potential inaccuracies. This, coupled with the criterion’s origination from
non-PNS data, further underscores the need for the development of more refined, PNS-specific safety
guidelines.
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Figure 5.2: Block diagram of the image analysis approach. (a) Experimentally derived nerve
samples from Sprague Dawley rats, comprising healthy and damaged specimens, are used for highresolution microscopy. (b) These images are segmented using a Convolutional Neural Network
(CNN) and a detailed image damage analysis for computing cell-specific damage indicators. Panel
(c) illustrates the detailed process using a small detail of the nerve cross-section. Specifically, (c)(A)
shows the microscopy image of the cells, (c)(B) shows the corresponding cell segmentation mask,
while (c)(C) and (c)(E) demonstrate the fiber density (both manual and automatic), calculated as
the number of cells (fractionally accounted for those cut by the window) divided by the window
area. Further, (c)(D) indicates the fascicle area, representing the area occupied by each cell; (c)(F)
and (c)(G) depict axon and myelin packing respectively, calculated as the total axon/myelin area
within the window divided by the window area; and (c)(H) fiber Nearest-Neighbor (NN) area, a
crucial metric that assesses the area of each myelinated fiber’s cell within the fascicle partition.
Panel (d) shows the obtained damage metrics visualized on the segmented nerve images for clarity.
While the Shannon criterion has served as a beneficial starting point for safety guidelines, it is
clear that its limitations necessitate the formulation of a more robust protocol for safety assessment.
The development of an improved criterion is essential for optimizing patient safety, therapeutic
outcomes, and advancing the field of PNS.
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5.1.3 Ideal properties of an updated safety criteria
The objective of this study is to develop an updated safety criteria specific for PNS that can
address the above limitations. Ideal safety guidelines for peripheral nerve stimulation should be
anchored in data derived directly from peripheral nerves, utilize algorithm-based tissue damage
evaluation to ensure consistency and objectivity, and incorporate accurate current density calculations that account for non-uniform distribution across the electrode surface.
Firstly, the safety guidelines should be directly informed by data derived from the actual peripheral nerves, rather than extrapolated or migrated from other parts of the nervous system. This
approach ensures the specific nuances and unique characteristics of the peripheral nerve system are
taken into consideration, thereby enhancing the accuracy and relevance of the safety thresholds.
Secondly, the evaluation of tissue damage should be algorithm-based, leveraging advances in
computational neuroscience and data analysis. By replacing or supplementing manual damage
analysis with algorithmic methods, we can minimize human error, ensure consistency, and potentially uncover nuanced patterns or thresholds that may be overlooked in manual assessments. This
shift towards a more objective and reproducible analysis is likely to improve the reliability of safety
evaluations.
As a final note, it is crucial that ideal safety guidelines for peripheral nerve stimulation consider
accurate current and charge density calculations in lieu of broad approximations. Many existing
computational models of peripheral nerves use simplified homogeneous fascicles, which lack accuracy(Kosta et al., 2020b, 2019; RamRakhyani et al., 2015; Raspopovic et al., 2011). As such, there
is a clear need for a more realistic nerve computational model to achieve the necessary precision.
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Our study aims to develop a new safety criterion through a comprehensive study integrating
experimentation, automated model generation, and simulation. We focus on incorporating the key
attributes mentioned earlier to create a comprehensive safety guideline through the integration of
direct peripheral nerve data, algorithm-based tissue damage analysis, and current density computations using realistic heterogeneous nerve morphologies. These guiding principles are devised to
direct the evolution of safety guidelines, aligning them with the progress made in the field of peripheral nerve stimulation, with the goal of emphasizing patient safety and treatment effectiveness.
This study intends to provide the PNS community with improved safety guidelines, embodying
a comprehensive, multi-institutional, multi-disciplinary approach that harmonizes experimentation
with modeling and simulation methods.
5.2 Methods
In this study, we began with microscopic optical images of stimulated rat sciatic nerves. From
this initial point, our methodology followed two parallel paths, specifically designed to highlight
peripheral nerve stimulation (PNS) safety from the point of view of structural damage. The block
diagram of our study is shown in Fig. 5.1.
The first step focuses on image segmentation and analysis, employing a novel algorithm for
automatic nerve segmentation to identify the structural and morphological characteristics of myelin
and axons. We developed an algorithm-based nerve damage quantification, which enabled us to
identify and measure cellular-level damage within the stimulated nerve samples when compared to
healthy and control nerve samples. Complementing this evaluation, we integrated a wide range of
nerve damage metrics, such as fiber packing and fiber density, to provide a comprehensive picture
of the localized extent of damage across different nerve samples.
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Figure 5.3: (a)(b) Selection and segmentation of two representative nerve samples, one healthy
and one damaged, based on fiber packing and fiber density metrics. (c) Construction and computational modeling of the two selected nerve samples for Admittance Method (AM) simulation. (d)
Training of a multi-layer perceptron (MLP) model using the dataset derived from the AM simulation of the two representative samples to predict cell-wise charge density values. The MLP model,
consisting of a four-layer fully connected neural network with hidden layers of 512, 256, and 128
neurons respectively, utilizes a softmax activation function for charge density classification.
Simultaneously, we utilized computational modeling, via the Admittance Method (Bingham
et al., 2020; Cela, 2010; Eberdt et al., 2003; Loizos et al., 2014b; Paknahad et al., 2020a,b, 2021b;
Stang et al., 2019a). In this process, we used segmented images to develop realistic computational
models of the segmented nerves. These computational models, employed to replicate the experiments
(electrodes, nerves, and stimulation levels), unveiled detailed information about current and charge
distribution inside the nerve during stimulation. We further leveraged these simulated samples to
train a neural network capable of accurately grouping each cell by the simulated charge intensity.
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This led us to obtain local charge density values for all cells across all nerve samples during stimulation, offering a granular understanding of cell-specific current or charge density distribution within
the nerves, using morphologies and structural characteristics of the imaged stimulated nerves.
By combining results from these two approaches, we compiled a detailed dataset for every cell in
our nerve sample. This dataset shows both the degree of damage and the local stimulation charge
density for each cell. We used this information to explore the relationship between cell damage
and charge density. Studying this connection is key to better understanding PNS safety and move
towards an updated safety criterion.
5.2.1 Experiments and microscopy
To correlate experimentally stimulated samples and corresponding computational models, a
series of rigorously controlled animal experiments were designed and executed. Utilizing adult male
Sprague Dawley rats, a multi-stage process including implantation and stimulation at different
levels was developed. These procedures adhered to ethical guidelines set by the University of Utah
Institutional Animal Care and Use Committee.
During the implantation phase, the left sciatic nerve of each isoflurane-anesthetized rat was
fitted with a multi-electrode cuff array (MECA), manufactured by Microprobes for Life Science.
The MECA connector was securely affixed to the rat’s skull, with the wire bundle routed transcutaneously to the nerve, and the cuff sutured in place.
One week post-implantation, stimulation was conducted under isoflurane anesthesia. The contacts of the implanted MECA were connected to a stimulator delivering biphasic pulses at different
current levels for 100 µs per phase, with a 400 µs interpulse period. The nerve was stimulated at a
frequency of 50 Hz for a duration of four hours.
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A week later, during the final euthanasia phase, the rat was perfused under deep isoflurane anesthesia, first with phosphate-buffered saline (PBS), followed by a fixative solution of 4% formaldehyde
and 2% glutaraldehyde in PBS. The region of the nerve with the implant, along with a similar region
from the contralateral, unimplanted side, were carefully excised. The samples were then preserved
at 4°C in a solution of PFA+Glut+PBS for a week, subsequently transferred into a vial filled with
0.02% sodium azide in PBS, and sent for histological analysis.
High-resolution microscopy was used to examine the nerve samples. From the images the work
included nerve image analysis as well as the use of a quasi-static computational method called the
Admittance Method (AM) for modeling.
5.2.2 Image segmentation and Damage Analysis
Initially, this study involves an in-depth analysis of nerve cross-sections; the overview of this
component of the work is illustrated in Fig. 5.2. Leveraging a Convolutional Neural Network
(CNN) coupled with image analysis algorithms, we achieved precise segmentation of individual
fibers of the nerve images samples obtained from experiments (Fig. 5.2(a)(b)). The segmentation
of axon and myelin tissues was conducted semantically using CNN, supplemented by the watershed
algorithm, resulting in individual fiber segmentation of axon and myelin. Detailed procedures of
this segmentation are provided in our previous study(Du et al., 2023).
Upon segmentation of the nerve cross-sectional images, we proceeded to employing these to
calculate specific damage indicators at the cellular level. This process is summarized graphically
in Fig. 5.2(c), where the microscopy image of cells, segmented masks, and various fiber damaging
metrics are computed (Fig. 5.2(c)(A-G)). To be more specific, the fiber damaging metrics calculated
include fiber density, fascicle area, axon packing, myelin packing, and fiber Nearest-Neighbor (NN)
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area. Fiber packing was quantified by computing the ratio of cell area to fascicle area within the
same window. Double-counting risks involving cells partially intersecting two adjacent windows were
mitigated by proportionally attributing cell area to each block. As an example, a cell having onethird of its area in a block was counted as one-third of a cell. Thus, fiber packing is a dimensionless
measure, ranging from 0 to 1. Conversely, fiber density was deduced by dividing the cell count
within the measurement window by the corresponding fascicle area within that window. Only the
fraction of fiber area within the window was counted when a window intersected a fiber. Window
area values were adjusted to account for overlaps with regions outside fascicles or those occupied
by blood vessels, areas typically devoid of neurons. NN area is another important damage metrics
that we employed, which calculates the area of the voronoi cell, of the partition of fascicle area, for
each myelinated fiber. We finalized this stage by mapping the resulting damage metrics onto the
segmented nerve images as shown in Fig. 5.2(d), thereby offering a precise visualization of nerve
damage.
Achieving comprehensive segmentations and computing damage metrics for each cell across all
nerve samples provides the necessary data for the computation of cell-wise charge density values,
establish a correlation between induced electrical parameters and damage, and ultimately present
a criterion for the safety analysis of PNS system.
5.2.3 Charge density from AM modeling and MLP prediction
Our objective is to apply computational modeling to enable the development of a new safety
criterion for PNS based on experimental data. In order to achieve this, both a cell-by-cell damage
analysis and cell-by- cell current density values from computational modeling are needed. However,
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given the excessive computational cost of the Admittance Method (AM) simulation for extremely
high-resolution nerve images, it’s impractical to run AM simulations for each nerve sample.
Thus, an effective solution is to apply a neural network-based predictive model. As detailed
in our previous study (referenced as (Du et al., 2023)), we found that three key factors primarily
influence the charge density induced in the computational cells of the tissue: stimulation intensity,
the distance between the considered location in the tissue and the electrode, and the dielectric
properties of the considered cell. Using this knowledge, we performed AM modeling on two selected
nerve sample and proposed a multi-layer perceptron model which utilizes the information from the
simulated datasets to predict charge density values in unknown datasets based on these factors.
5.2.3.1 Nerve Sample selection(Build training dataset)
The first step towards constructing such a model is assembling a suitable training dataset. We
have so far gathered 38 nerve samples, each exhibiting varying levels of damage. To ensure that our
training dataset represents the whole spectrum of possible variations while avoiding bias, we need
to select representative samples for AM modeling.
To address possible discrepancies between individual samples, we focused on fiber packing and
fiber density, metrics that have been validated as useful indicators of nerve damage in previous studies (Christensen and Tresco, 2015; Comin et al., 2014; Sandell and Peters, 2001). These indicators
were key in ensuring that the chosen images correctly represent their respective classes.
Using these metrics, we have chosen two illustrative samples from our collection: one healthy
(with no significant morphological and structural changes relative to a non-stimulated rat), and
one damaged (displaying noticeable morphological and structural changes when compared to a nonstimulated rat). The selected healthy nerve image is sourced from control samples with implanted,
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yet non-stimulated cuff electrodes, while the selected damaged nerve image is taken from a sample
stimulated using a 1.2mA current source, as illustrated in Fig. 5.3(a)
5.2.3.2 AM Modeling on the selected samples
Upon selecting our two samples, we employed the admittance method to conduct further simulations.
The peripheral nerve stimulation simulations utilized a multi-scale computational model that
comprises two principal components: a segmented nerve model and a cuff electrode model. The
cuff electrodes are designed to represent typical commercial cuff electrodes, with an inner diameter
of approximately 2mm and two metal contact wires, one serving as a source cuff electrode and
the other as a ground cuff electrode. The computational model includes only the metallic parts of
the electrode, ignoring non-conductive elements not in direct contact with the tissue, such as the
surrounding insulation layer. These components have negligible effects on the current distribution
in the tissue. In the in vivo experiments, the nerve is sutured to the electrode, placing the nerve
model in close contact with nerve. The final constructed model is depicted in Fig. 5.3(c).
The simulation setup remains consistent for both healthy and damaged nerve models, maintaining identical cuff electrodes and position. The models are partitioned into cubic voxels, with
each voxel corresponding to a distinct material index. The dimensions of the final model are
800×800×1000 voxels in the x, y, and z directions respectively, with a resolution of 3.1 µm across
all three dimensions. The nerve model’s material attributes are extracted from the studies by
(Butson et al., 2011; McNeal, 1976), as outlined in Table 1.
The AM with its multi-resolution feature detailed in (Bingham et al., 2020; Cela, 2010; Eberdt
et al., 2003; Loizos et al., 2014b), is employed to calculate the electric field values at each computation
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Table 5.1: TISSUE PROPERTIES
Tissue Type σx σy σz Unit
Perineurium 0.01 0.01 0.01 S/m
Myelination 2 × 10−4 5 × 10−9 5 × 10−9 S/m
Intracellular space 0.33 0.33 0.33 S/m
Axoplasm 0.91 0.91 0.91 S/m
Epineurium 0.1 0.1 0.1 S/m
Nerve membrane 0.02 0.02 0.02 S/m
Saline solution 1.45 1.45 1.45 S/m
Extracellular space 0.33 0.33 0.33 S/m
grid node (Paknahad et al., 2020a,b, 2021b; Stang et al., 2019a). Briefly, this method relies on the
construction of a matrix, G, representing the admittances throughout the model, which encompasses
two primary aspects: the admittances connected to each node (depicted by diagonal components of
the matrix) and the internodal admittance (expressed by the off-diagonal elements). Owing to the
inherent structure of the network model, the admittance matrix, G, manifests as a sparse, symmetric
matrix. Admittance values are determined by considering the conductivity and the nodal distance
in the x, y, and z axes, as elaborated in Equations (1)-(3). In these equations, the conductivities
along the x, y, z directions are represented by σx, σy, σz respectively, and ∆x, ∆y, ∆z stand for the
inter-nodal distances along corresponding directions.
g
i,j,k
x = σ
i,j,k
x
∆y∆z
∆x
(5.1)
g
i,j,k
y = σ
i,j,k
y
∆x∆z
∆y
(5.2)
g
i,j,k
z = σ
i,j,k
z
∆y∆x
∆z
(5.3)
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For effective matrix generation and equation resolution, we have crafted a multithreaded computational platform in Python, employing a biconjugate gradient method. Upon defining the admittance matrix (G) and the current vector (I), we solve the equation GV = I to determine the
vector of induced voltage, V, at every node of the model.
Prior to running electric field simulations, we utilize a 3D multi-resolution meshing method to
diminish computational time, while retaining accuracy in the region with relatively higher currents
. Given, for the considered set-up, the low currents in the inner parts of the the nerve, the need
for fine resolution diminishes in distal regions from the nerve periphery. Hence, the computational
mesh resolution is coarser at the nerve’s core and finer near the nerve’s edge (i.e., proximal to
the electrodes and fascicle boundaries). This method significantly curtails computational time by
reducing the total node count in the model.
Voltage values are computed at every node of the model, situated at the voxel vertices. A
trilinear interpolation is applied to estimate the voltage at various points within a voxel, basing
calculations on the values at its vertices, as the conductivity value within each voxel is considered
constant. Employing these interpolated voltage values, additional parameters such as electric field,
charge density, and current density, at any location within the model can be calculated.
5.2.3.3 Neural network training and charge density prediction
The AM simulation results are used as the training dataset for the predictive model. In the
imaging dataset, cell-by-cell current density values are unknown. However, the distance of each
cell from the electrode, its material type, and stimulation level are known. With this information,
it is possible to train a neural network to predict the current density values for each unknown
computational cell.
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Figure 5.4: Depiction of the neural network model training and performance assessment. (a)
Illustrates the training and validation process for our multi-layer perceptron network, performed
over 1,000 epochs. This exhaustive training led to a peak model accuracy of approximately 90%,
demonstrating its ability to discern underlying patterns and avoid overfitting. (b) Showcases the
classification performance of our model on a diverse dataset of over 100,000 cells, represented
through a confusion matrix. The matrix elucidates the model’s misclassifications primarily between
neighboring classes, with notably fewer misclassifications in the highest charge density range - a
critical determinant for gauging potential cellular damage.
Our training dataset, based on two sample types, provides us with over one billion simulated
voxels, offering a vast volume of training data. From this, our analysis identifies several primary
factors contributing significantly to the cellular-level distribution of current density: the cell proximity to the electrode, the surrounding material type, and the stimulation level. These insights
are supported by our prior studies(Du et al., 2023)(Du et al., 2021)(Du et al., 2022a). Despite
the undetermined exact cell-level current density values within our imaging dataset, we leverage
these cellular attributes to construct a multi-layer perceptron (MLP) (Ruck et al., 1990). Our MLP
consists of a four-layer fully connected neural network that accepts three attributes as input. It
comprises hidden layers with 512, 256, and 128 neurons respectively and uses a softmax activation function in the output layer for charge density value classification, as illustrated in Fig. 3(d).
The training data input was pre-sorted into five classes according to the charge density ranges.
Through this method, we successfully approximated cell-wise charge density values across various
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Figure 5.5: Analysis of cell groupings based on charge density against various damage metrics.
(a) Axon packing, (b) Axon size, (c) Fiber packing, and (d) Fiber Nearest-Neighbor (NN) ratio.
The empirical distribution function plots visually represent seven distinct cellular groups, including
stimulated cells, control cells, sham cells, and five subcategories within the stimulated cell group
defined by their charge density. The curves indicate that control and sham cell groups, typically
resembling healthy cells, tend to reside on the right, whereas cell groups with higher charge densities
skew left, suggestive of morphological changes and potential damage. Most notably, cells with a
local charge density per phase exceeding 3.5 nC/cm2
consistently deviate from other categories
across all damage metrics, underscoring the potential safety threshold.
nerve samples, a task that would otherwise be impractical.
Leveraging a dataset of approximately 30,000 individual cells derived from two nerve samples,
we divided the data using a 70/30 split for training and validation purposes. The performance
of the resulting model was evaluated against an independent nerve model with full-resolution AM
simulation.
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5.3 Results
In summary, our methodology centers around a systematic analysis of cell damage in response to
varying charge densities. We initiated our process with image segmentation, subsequently carrying
out a damage assessment for each image, thereby enabling us to calculate the extent of damage on
a cell-by-cell basis across the entire image set. Simultaneously, we utilized the Admittance Method
(AM) modeling and neural network prediction on the segmented images, associating each cell with a
distinct charge density value. This facilitated the formulation of a relationship between the cell-wise
damage indicators and charge density values. Having established this correlation, we were then able
to examine the extent of damage across a spectrum, ranging from unstimulated (healthy) cells to
cells exposed under varying levels of stimulation, based on their specific cell-wise charge densities.
This analysis empowered us to propose safety criteria based on the correlation between a cell’s
charge density value and its extent of damage.
The results we present include the categorization of nerve cells according to their corresponding
charge densities, along with the accuracy of these classifications. By deploying empirical distribution
curves, we demonstrate a range of damage indicators specific to each cell group, distinguished by
their unique charge density levels. Examining these distribution curves facilitated the pinpointing of
the charge density group displaying the most substantial morphological alterations, thereby enabling
us to propose a potential safety threshold. This proposed boundary is rooted in the observation that
cells at this specific charge density exhibit an increased likelihood of damage, thereby warranting
the establishment of safety criteria centered around this particular level.
In contrast to the traditional Shannon Criteria, our proposed safety limit offers a more comprehensive risk assessment. By providing quantitative data corresponding to different stimulation
intensities, we’re able to pave the way for safer practices and strategies. This vital information
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could also contribute to the design of future nerve stimulation therapies and devices, advancing
both efficacy and safety within the field.
5.3.1 Cell-wise charge density prediction results
In our study, we selected two representative nerve samples for comprehensive modelling. Through
this process, we procured over 10,000 cells, forming a rich dataset for our analysis. We strategically
partitioned this dataset into three subsets: 70% allocated for training the model, 15% for validation
to fine-tune the model parameters, and the remaining 15% reserved for final testing to assess the
model’s performance on unseen data.
Our model, a multi-layer perceptron network, was trained over 1,000 epochs. This prolonged
training process allowed the model to learn intricate patterns and relationships within the data.
We observed that, following this training regimen, the accuracy for both training and validation
datasets plateaued, reaching a commendable level of approximately 90%, as shown in Fig. 5.4(a).
This level of performance indicated that the model was able to generalize well and was not overfitting
the data, i.e., the model was not merely memorizing the training examples but effectively learning
underlying patterns, enabling accurate predictions on unseen data.
Our neural network allows us to organize cells from our large set of nerve samples systematically
by calculating current density values. The dataset in question is extensive, encompassing more than
100,000 individual cells across 38 distinct nerve samples. The neural network aids in allocating each
cell to one of five categories, each representing a specific range of charge densities: 0-0.5, 0.5-1.5, 1.5-
2.5, 2.5-3.5, and 3.5nC/cm2
and above. The five categories are determined based on our simulations
as well as the results from our previous study (Du et al., 2023).
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We further tested the model on the test dataset, revealing a similar accuracy level to the training
and validation sets. This consistency underscored the model’s robustness and ability to maintain
performance across various data subsets. As we were dealing with a multi-class classification problem involving five distinct classes, we generated a confusion matrix to analyze the classification
performance in detail, as shown in Fig. 5.4(b).
The confusion matrix is a powerful tool for visualizing the performance of a classification model,
revealing the precise locations of misclassifications. Our matrix showed that the primary source of
misclassifications was between neighboring classes, which is a common occurrence in classification
problems due to similarities between adjacent classes.
Remarkably, we found that the class representing the highest charge density range exhibited very
few misclassifications. This finding is particularly crucial for our study, as accurate classification
of cells within this range is vital for understanding potential cellular damage due to high charge
densities.
5.3.2 Correlation curves between cell damage and charge density
Following the neural network classification process, we next turned our focus towards the visual
representation and analysis of these cell groupings, employing empirical distribution function plots
mapped against various damage metrics.
We again categorized the cells according to their charge density values, underscoring the significance of local charge density rather than the original nerve sample for the groupings. Notably,
the categorization of cells is independent of their originating nerve sample, as cells from disparate
samples can affiliate with the same group if their local charge density values align closely. During
the initial segmentation of our dataset, three principal groups surfaced: stimulated cells, control
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Figure 5.6: Comparative analysis of proposed safety criteria and traditional Shannon Criteria for
nerve stimulation. The figure illustrates the proportion of cells for each nerve sample, identified by a
unique Shannon k value, exceeding a local charge density per phase of 3.5 nC/cm2
. The plot reveals
nerve samples with Shannon k=2.11, k=0.55, and k=0.11 and the proportion of cells within each
that surpass the proposed safety threshold. Our criteria unveil potential risks not identified by the
Shannon Criteria, providing a more nuanced understanding of the damage risk under stimulation.
The visualization underscores the advantages of our proposed method, offering a more granular
perspective that can assist experimentalists in making more informed decisions on experimental
parameters.
cells, and sham cells. Further exploration within the stimulated cell group unveiled five distinct
subcategories, each characterized by their individual charge densities as previously enumerated.
Subsequently, we deployed empirical distribution function plots to visually represent these seven
cellular groups against various damage metrics. The interpretation of these plots uncovers intriguing
patterns: the distribution curves for control and sham cells predominantly reside on the right,
thus representing the archetype of a healthy cell distribution. Conversely, cell groups displaying
elevated charge density values display a left-leaning curve, indicative of morphological alterations
and potential damage, as shown in Fig. 5.5.
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These curve distributions across different charge density levels can fluctuate based on the selected
damage metrics. However, one consistent feature prevails: the curve corresponding to cells with
a local charge density per phase of 3.5 nC/cm2
or greater is consistently positioned far to the
left, indicating significant differences from other curves. These cells markedly diverge from other
categories across all damage metrics, suggesting an increased risk of damage. Given this result,
it appears advisable to introduce a safety threshold for the local charge density per phase of 3.5
nC/cm2
since this could induce substantial morphological changes in the cell.
Figure 5.7: Empirical model demonstrating the relationship between stimulation intensity and
the risk of excessive charge density to nerve cells. This curve-fitting process visualizes the correlation between the controllable stimulation intensity from the source cuff electrode (x-axis) and the
percentage of nerve cells exceeding our proposed safety limit (y-axis), providing a tool for practitioners to estimate potential cell damage during peripheral nerve stimulation.
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5.3.3 Comparison with Shannon Criteria and future safety criteria
Our proposed safety limit offers a compelling comparative perspective to the traditional Shannon
Criteria. To elucidate this, for each nerve sample identified by a unique Shannon k value ascertained
at the time of the experiment, we plot the proportion of cells for each sample that exhibit a local
charge density per phase exceeding 3.5 nC/cm2
.
For the nerve sample characterized by Shannon k=2.11, a remarkable 98.88% of cells display a
local charge density value beyond 3.5 nC/cm2
. This high proportion signifies a substantial risk of
damage for virtually all cells in this nerve sample, aligning with Shannon Criteria that deems a K
value of 2 as surpassing the generally accepted safety limit of 1.5.
In contrast, the nerve sample with Shannon k=0.55, generally considered as relatively safe by
Shannon Criteria, exhibits an unexpected pattern under our criteria. Over 59% of the cells within
this sample exceed the safety limit. A similar trend is observed for the nerve sample with Shannon
k=0.11, where 12.42% of cells surpass a local charge density value of 3.5 nC/cm2
, implying a
potential risk of damage for roughly 12% of cells, a level far from being considered very safe, as
shown in Fig. 5.6.
These findings underscore the capability of our proposed criteria to shed light on the potential
damaging risk of a nerve under stimulation, presenting a more nuanced understanding than what
the traditional Shannon Criteria can offer. Compared to the Shannon Criteria, our proposed stimulation safety limit of 3.5 nC/cm2
provides an enhanced level of detail regarding the safety of nerve
stimulation at given intensities. While the Shannon Criteria presents a single value and somewhat
ambiguous limit, our proposed method delivers a quantitative measure of damage risk. This granular data enables experimentalists to make more informed decisions regarding their experimental
parameters.
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To make our safety criteria easier to use, we fitted our stimulated samples to a curve that could
be used as a model for practitioners in the case of stimulating nerve samples using cuff electrodes
and subsequently quantifying the percentage of cells with a charge density exceeding the proposed
safety limit, as shown in Fig. 5.7.
In this figure, the y-axis represents the percentage of nerve cells exceeding our safety limit,
while the x-axis corresponds to the controllable stimulation intensity from the source cuff electrode.
Utilizing these parameters and data points, we performed curve-fitting procedures to ascertain
the relationship between these two variables. The resulting curve presents an empirical model,
encapsulating the observed correlation between stimulation intensity and the risk of excessive charge
density to nerve cells.
Although still in its early stages, this safety curve provides valuable insight into a potential tool
we aim to develop for practitioners, particularly those working with peripheral nerve stimulation
via cuff electrodes and more. The usability and significance of this approach are evident even at
this preliminary stage. Looking ahead, the proposed approach, with a more expansive dataset
and increased variety in experimental procedures, nerve and electrode types, etc., can provide an
updated and more robust neural network, paving the way for broader application in the field.
5.4 Discussion
The aim of this research is to advance the development of a more accurate and up-to-date safety
standard for peripheral nerve stimulation. Despite its long-standing use, the Shannon criteria, a
conventional safety guideline often used for such stimulation, is burdened with shortcomings. These
include a lack of data specific to the peripheral nervous system, an over-reliance on manual damage
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analysis, which can be prone to inconsistency, and a flawed assumption of uniform charge density.
To create an improved safety standard for peripheral nerve stimulation, several elements should be
considered: utilization of data that is specific to peripheral nerves, the application of algorithmbased assessments of tissue damage to reduce human (i.e., subjective) errors, and the inclusion of
accurate current density computations, rather than assuming a uniform distribution. In our efforts
to define this next-generation safety standard, we have incorporated these important considerations.
Evaluating nerve damage induced by electrical stimulation often hinges on the crucial factor of
current density. However, obtaining accurate estimations of current density distribution within a
nerve poses a significant challenge. To overcome this, we utilize a high-resolution computational
approach, the Admittance Method (AM), which constitutes a significant part of our multi-scale
computational platform. This approach is capable of producing exceptionally detailed results for
current density calculations. Despite its efficacy, applying the AM to the current high-resolution
nerve models necessitated the development of a multi-resolution adaptive meshing algorithm to
reduce the computing complexity along with an averaging filter to improve the accuracy.
The overwhelming computational complexity of conducting Admittance Method (AM) modeling
for each nerve sample necessitates an alternative approach. The neural network prediction approach
described in this study constitutes a viable alternative; it allowed us to identify several important
factors - such as distance from the electrode, material type, and stimulation level - that influence
cell-by-cell current density distribution. These characteristics can assist us in predicting current
density on cells without conducting full AM simulations. Despite the promising results, our current
predictive model exhibits limitations rooted in the homogeneity of the nerve samples under study.
Our dataset primarily consists of identical nerve types subjected to similar experimental procedures,
conditions under which the neural network thrived. However, if the experimental environment or
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nerve sample characteristics vary significantly, the performance of our current model might waver.
To overcome this limitation and enhance our methodology, we intend to leverage larger, more diverse
datasets in future studies. Incorporating variables such as differing experimental procedures, various
nerve and electrode types will enrich our data collection, introducing a new dimension of complexity
to our model. Consequently, a robust and versatile neural network will emerge, capable of delivering
more reliable predictions with a wider range of applicability. The potential of our approach lies in
its inherent adaptability to new datasets and conditions, signifying its capability to evolve with
expanding data resources. As we advance and refine our models, we are optimistic about unveiling
further insights into the intricate relationships between charge density induced by Peripheral Nerve
Stimulation (PNS) and nerve cell health.
Our research marks a significant but still initial step toward developing comprehensive safety
guidelines for peripheral nerve stimulation. Although our safety criteria are in their early stages,
we have a solid framework in place to build upon. The present study primarily utilizes data from
the sciatic nerves of rats. While this has provided essential insights, it may limit the applicability
of our results to other nerve types or species. Therefore, we plan to incorporate data from the pig
vagus nerve in future studies to include a wider range of species and nerve types. This expansion
is crucial to our overarching goal: enhancing the practical utility of our research, and ultimately
developing robust safety criteria applicable to PNS stimulation in human subjects.
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Chapter 6: Summary and Future Steps
This research aims to establish a more accurate and comprehensive safety standard for peripheral
nerve stimulation, addressing the limitations of the widely used Shannon criteria. The Shannon
criteria, while a longstanding benchmark, fall short due to a lack of peripheral nerve-specific data,
the reliance on manual and potentially inconsistent damage assessments, and the assumption of
uniform charge density across nerve tissues. In response, we have sought to create an improved
safety framework that integrates three critical elements: peripheral nerve-specific data, algorithmbased tissue damage assessments to minimize human error, and precise current density calculations,
moving beyond the assumption of uniformity.
A key aspect of this work has been the accurate estimation of current density within nerve tissue,
a crucial factor in evaluating nerve damage induced by electrical stimulation. We employed the
Admittance Method (AM), a high-resolution computational approach, to achieve detailed current
density distribution predictions. However, due to the computational demands of applying AM to
high-resolution nerve models, we developed a multi-resolution adaptive meshing algorithm, coupled
with an averaging filter, to reduce 50 percent of complexity while keeping the accuracy within 10
percent of loss.
In addition to the AM approach, we explored neural network-based predictions as a viable
alternative for estimating current density across cells. This approach identified several influential
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factors, such as electrode distance, material composition, and stimulation level, which shape the
current density distribution. Although our model demonstrated promise, its current limitations
stem from the homogeneity of the nerve samples studied. The dataset, largely composed of similar
nerve types subjected to consistent experimental conditions, allowed the neural network to perform
well. However, its predictive accuracy may diminish when applied to diverse nerve samples or
experimental environments.
Moreover, our findings suggest that factors like fiber packing and myelination levels significantly
influence micro-dosimetric exposure, but further studies are needed to establish causality. The
reliance on high-resolution models, while informative, may overlook variabilities in tissue properties,
the formation of scar tissue, and capacitive effects at the nerve-electrode interface. To address these
limitations, future studies will expand the dataset to include more diverse nerve types, experimental
procedures, and electrode configurations. This expansion will enhance the robustness and versatility
of the neural network, allowing for more reliable predictions across a broader range of conditions.
As our dataset grows, so too will the adaptability of our approach, opening the door to further
insights into the complex relationships between charge density and nerve cell health in the context
of Peripheral Nerve Stimulation (PNS).
In this study, the rat sciatic nerve served as our initial model due to its well-characterized
anatomy and suitability for experimental manipulation, making it a standard in peripheral nerve
research. The sciatic nerve’s mix of myelinated and unmyelinated fibers provided an ideal foundation for developing and validating our computational models. However, further analyses involving
nerves with different proportions of myelinated versus unmyelinated fibers will be necessary to determine the broader applicability of our predictive algorithms, particularly in relation to differences
in electrical conductivity and tissue response.
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While this research marks a significant step toward developing comprehensive safety guidelines
for peripheral nerve stimulation, it remains an early phase. Future work will include data from
additional species, such as the pig vagus nerve, as well as data obtained from different experimental
setups, to ensure that the safety standards we develop are applicable across a range of nerve types
and more representative of human physiology. Ultimately, these efforts aim to refine and expand
our models, contributing to the establishment of robust, evidence-based safety guidelines for PNS,
advancing the field of neurostimulation, and enabling the design of safer neuroprosthetic systems
for human application.
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Abstract (if available)

Abstract Peripheral Nerve Stimulation (PNS) is a widely established neurostimulation technique used for several medical conditions, including motor function recovery, chronic pain relief, spinal cord injury pain management, and treatment of complex regional pain syndrome. Despite the advent of numerous neural stimulation devices, defining safe electrical stimulation limits on peripheral nerves is still under debate. Establishing clear safety guidelines is crucial for optimizing patient outcomes and advancing neuroprosthetic system research and development.

In this dissertation, we utilized a novel machine learning and computational bio-electromagnetics modeling platform to establish a safety criterion that captures the effects of fields and currents induced on axons. Our approach is comprised of three steps: experimentation, model creation, and predictive simulation. We collected high-resolution images of control and stimulated rat sciatic nerve samples at varying stimulation intensities and performed high-resolution image segmentation. These segmented images were used to train machine learning tools for automatic classification of morphological properties of control and stimulated PNS nerves. Concurrently, we utilized our quasi-static Admittance Method-NEURON (AM-NEURON) computational platform to create realistic nervemodels and calculate induced currents and charges, both critical elements of nerve safety criteria. These steps culminate in a cellular-level correlation between morphological changes and electrical stimulation parameters. This correlation informs the determination of thresholds of electrical parameters that are found to be associated with damage, such as maximum cell charge density. The proposed methodology and resulting criteria combines experimental findings with computational modeling to generate a safety threshold curve which captures the relationship between stimulation current and the potential for axonal damage. Although focused on a specific exposure condition, the approach presented here marks a step towards developing context-specific safety criteria in PNS neurostimulation, encouraging similar analyses across varied neurostimulation scenarios.

While this research marks a significant step toward developing comprehensive safety guidelines for peripheral nerve stimulation, it remains an early phase. Future work will include data from additional species, such as the pig vagus nerve, to ensure that the safety standards we develop are applicable across a range of nerve types and more representative of human physiology. Ultimately, these efforts aim to refine and expand our models, contributing to the establishment of robust, evidence-based safety guidelines for PNS, advancing the field of neurostimulation, and enabling the safer design of neuroprosthetic systems for human application.

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Asset Metadata

Creator Du, Jinze (author)

Core Title Computational approaches to establish safety and efficacy assessments of electrical stimulation to peripheral nerve

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Electrical Engineering

Degree Conferral Date 2024-12

Publication Date 12/10/2024

Defense Date 10/08/2024

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

Tag computational modeling,electrical stimulation,OAI-PMH Harvest,safety and efficacy

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Language English

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Advisor Lazzi, Glianluca (committee chair), Bouteiller, Jean Marie Charles (committee member), D'Argenio, David (committee member), Moghaddam, Mahta (committee member)

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