University of Southern California Dissertations and Theses

Feature and model based biomedical system characterization of cancer

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Feature and model based biomedical system characterization of cancer

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Content FEATURE AND MODEL BASED BIOMEDICAL SYSTEM CHARACTERIZATION OF CANCER by Zaki Hasnain A dissertation presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulllment of the Requirements for the Degree of DOCTOR OF PHILOSOPHY (Mechanical Engineering) December, 2018 Dedicated to my grandparents. Forever in their shadow. Forever in their light. Acknowledgments Every new doctoral student is terried by the prospect of having a bad advisor. After many years of giving me intellectual freedom and timely direction in exchange for the occasional piece of chewing gum, I am proud to say my advisor Dr. Paul K. Newton is one of the best. I also thank Dr. Peter Kuhn and Dr. Jorge Nieva for trusting me to perform in a domain where I had no previous knowledge and showing me how important academic research is for the lives of cancer patients. After nding the perfect advisor, a doctoral student proceeds to worrying about lab-mates. I have been fortunate in this regard as well. I thank Dr. Angie Lee for setting the benchmark for composure during the latter years of doctoral studies; and thank her for keeping the thermostat xed at 55 as it made me more productive than I ever thought was possible. I thank Dr. Jeremy Mason for always putting people rst, especially in the quantitative four-cornered world where many of us take refuge from that very thing. I thank Dr. Jerey West for his casually serious approach to life and research, and for being by my side in the trenches. I thank my oce-mate Jonathan Hoy for being a loyal friend and being exponentially less distracting during the nal year of my studies. I thank my oce-mate Xiangyu Gao for his fortitude and wisdom. I thank my teammates in the USC Taekwondo club for never asking me about my research but always leaving me ready to face my ercest opponent for one more round. Finally, I thank my family for reminding me to be grateful for studying in sunny southern California by drawing me away to visit them during cold winter holidays in Virginia. I also thank my family in Pakistan for reminding me that life is often nonlinear, as the temperature-productivity relationship suggested in the previous statement, among many other things, does not hold true for scorching south-Asian summers, where this story began. ii CONTENTS Contents Dedication i Acknowledgements ii List of Figures vi List of Tables viii Abstract ix 1 Introduction 1 1.1 Biomedical system characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Feature based study of biomedical data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2.1 Feature types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2.2 Distance functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2.3 Dynamic time warping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2.4 The multivariate case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2.5 Dimensionality reduction and feature selection . . . . . . . . . . . . . . . . . . . . . . 8 1.3 Model based study of biomedical data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.3.1 Modeling techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.3.2 Point distribution model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.3.3 Hidden Markov model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.3.4 Machine learning models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.4 Chapter summaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2 Quantied kinematics from in clinic movements dierentiates patient risk for chemother- apy complications 18 2.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.3 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.3.1 Trial Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.3.2 Participants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.3.3 Clinical exercises and motion capture . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.3.4 Physical Activity measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.3.5 Statistical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.4.1 Patient cohort . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.4.2 Unexpected hospitalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.4.3 Physical activity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.6 Acknowledgement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.7 Appendix A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.7.1 Calculating the Emetogenicity of Multiple Agent Chemotherapy/Biotherapy Regimens 28 2.8 Appendix B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.8.1 Sagittal angle calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.8.2 Angular velocity calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.9 Appendix C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.10 Appendix D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 iii CONTENTS 3 Low-dimensional characterization of human performance of cancer patients using motion data 39 3.1 abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.3 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3.3.1 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.3.2 Data preprocessing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.3.3 Feature extraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 3.3.4 Time series similarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.3.5 Dimensionality reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.3.6 Unsupervised clustering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.3.7 Supervised classication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.4 Results & Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.4.1 Current clinical parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.4.2 Validation of preprocessing and choice of DTW . . . . . . . . . . . . . . . . . . . . . 46 3.4.3 Low dimension representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.4.4 Detecting performance clusters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.4.5 Learning physician ECOG performance status . . . . . . . . . . . . . . . . . . . . . . 52 3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 4 Machine learning models for long-term outcome prediction in bladder cancer 55 4.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 4.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 4.3 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 4.3.1 Data summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 4.3.2 Statistics and information theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 4.3.3 Machine learning approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 4.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 4.4.1 Survival statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 4.4.2 Correlations among predictors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 4.4.3 Correlations with long-term outcomes . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 4.4.4 Predicting post-cystectomy recurrence . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 4.4.5 Predicting post-cystectomy survival . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 4.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 4.6 Supporting information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 5 Feature and model based characterization of spatial and temporal organoid dynamics 81 5.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 5.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 5.3 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 5.3.1 Preprocessing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 5.3.2 Feature extraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 5.3.3 Low dimensional representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 5.3.4 Time series clustering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 5.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 5.4.1 Feature-based description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 5.4.2 Clustering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 5.4.3 Dynamical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 5.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 5.6 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 5.6.1 S1 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 5.6.2 S2 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 5.6.3 S3 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 iv 6 Temporal dierentiation of phenotypes using a Hidden Markov model of organoid shape dynamics 104 6.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 6.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 6.3 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 6.3.1 Dataset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 6.3.2 Model of organoid shape . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 6.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 6.4.1 Perimeter-area space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 6.4.2 Model selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 6.4.3 Basal and FGF2 hidden Markov models . . . . . . . . . . . . . . . . . . . . . . . . . . 109 6.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 7 Future work 116 8 Appendix I: The prisoner's dilemma as a cancer model 117 8.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 8.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 8.3 The prisoner's dilemma evolutionary game . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 8.4 A tumor growth model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 8.4.1 Mutations and heritability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 8.4.2 The tness landscape . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 8.4.3 Heterogeneity drives growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 8.5 Simulated drug dosing strategies and therapeutic response . . . . . . . . . . . . . . . . . . . . 128 8.6 Mathematical modeling and tumor analytics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 9 Appendix II: An evolutionary model of tumor cell kinetics and the emergence of molec- ular heterogeneity driving Gompertzian growth 135 9.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 9.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 9.3 Description of the model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 9.3.1 The Moran birth-death process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 9.3.2 The prisoner's dilemma payo matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 9.3.3 The tness landscape . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 9.3.4 Passenger and driver mutations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 9.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 9.4.1 Gompertzian tumor growth and three growth regimes . . . . . . . . . . . . . . . . . . 145 9.4.2 Heterogeneity and growth via statistical mechanics . . . . . . . . . . . . . . . . . . . . 147 9.4.3 Quantitative measures of tumor heterogeneity and growth . . . . . . . . . . . . . . . . 149 9.4.4 Dynamic phylogenetic trees and evolution of tness . . . . . . . . . . . . . . . . . . . 151 9.4.5 A comparison of early vs. late therapy . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 9.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 9.6 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 10 Publications 157 10.1 First author publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 10.2 Second author publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 11 References 158 v List of Figures 1 Biomedical dataset analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2 Dynamic time warping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3 Point distribution model example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 4 Hidden Markov model architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 5 Markov chain types for hidden Markov models . . . . . . . . . . . . . . . . . . . . . . . . . . 14 6 Illustration of chair-to-table exam maneuver . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 7 Schematic of anatomical sites detected in motion capture and two-joint sections . . . . . . . . 22 8 Top kinematic features which dierentiate patients by unexpected hospitalizations . . . . . . 25 9 Top kinematic features which dierentiate patients by physical activity . . . . . . . . . . . . . 26 10 Chemotherapy Emetogenicity Table. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 11 T-test scores for kinematic features which dierentiate patients by hospitalizations . . . . . . 33 12 Box plots of kinematic features which dierentiate by hospitalizations: features 1-20 . . . . . 35 13 Box plots of kinematic features which dierentiate by hospitalizations: features 21-40 . . . . . 36 14 Box plots of kinematic features which dierentiate by hospitalizations: features 41-55 . . . . . 36 15 T-test scores for kinematic features which dierentiate patients by physical activity . . . . . . 37 16 Box plots of kinematic features which dierentiate by physical activity: features 1-20 . . . . . 37 17 Box plots of kinematic features which dierentiate by physical activity: features 21-28 . . . . 38 18 Schematic for analysis of human motion data . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 19 Motion capture skeleton and exercise examples . . . . . . . . . . . . . . . . . . . . . . . . . . 42 20 Change in patient weight and tness scores . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 21 Sample motion capture time series and DTW distances . . . . . . . . . . . . . . . . . . . . . 48 22 Clustering of exercise kinematic features . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 23 Patient clustering based on kinematic motion capture data . . . . . . . . . . . . . . . . . . . 51 24 Correlation between physicain assigned tness and kinematic features . . . . . . . . . . . . . 53 25 Survival analysis by disease status . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 26 Survival analysis by age group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 27 Survival analysis by clinical T stage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 28 Survival analysis by patholgic stage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 29 Mutual information matrix between bladder cancer features and outcomes . . . . . . . . . . . 63 30 Association between predictors and the two long term outcomes . . . . . . . . . . . . . . . . . 65 31 History of discrepancy between clinical and pathologic staging . . . . . . . . . . . . . . . . . . 71 32 Survival analysis by stage subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 33 Survival analysis by lymphovascular invasion . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 34 Survival analysis by soft tissue surgical margin . . . . . . . . . . . . . . . . . . . . . . . . . . 73 35 Survival analysis by adjuvant chemotherapy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 36 Predictors ranked by Cramer's V. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 37 Schematic of organoid feature based clustering and model based analysis . . . . . . . . . . . . 83 38 Organoid segmentation and contour time series examples . . . . . . . . . . . . . . . . . . . . 93 39 Organoid feature time series and curve ts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 40 Area and perimeter relationship between all organoids . . . . . . . . . . . . . . . . . . . . . . 95 41 Organoid shape dynamics in principal component space . . . . . . . . . . . . . . . . . . . . . 96 42 Hierarchical clustering of organoids. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 43 Point distribution models of organoid shape variation. . . . . . . . . . . . . . . . . . . . . . . 98 44 Schematic of hidden Markov model for organoid shape . . . . . . . . . . . . . . . . . . . . . . 106 45 Organoid area and perimeter versus time and treatment type . . . . . . . . . . . . . . . . . . 108 46 Perimeter and area relationship of 39 organoids . . . . . . . . . . . . . . . . . . . . . . . . . . 109 47 Model selection for organoid hidden Markov models . . . . . . . . . . . . . . . . . . . . . . . 110 48 Basal hidden Markov Model results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 49 FGF2 Hidden Markov Model results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 50 Schematic of the Moran Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 51 Emergence of Gompertzian growth via selection . . . . . . . . . . . . . . . . . . . . . . . . . . 123 52 Markov Point Mutation Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 vi LIST OF FIGURES 53 Tumor tness drives tumor growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 54 Moran Process t to Gompertzian Growth Data . . . . . . . . . . . . . . . . . . . . . . . . . 127 55 Sample Dendritic Phylogenetic Tree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 56 Eects of varied dose density for early-stage, mid-stage, and late-stage therapies . . . . . . . 133 57 Growth-dependent tumor regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 58 Stochastic Moran birth-death process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 59 Fitness as a function of the selection parameter ww H w C . . . . . . . . . . . . . . . . . 143 60 Markov Point Mutation Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 61 Gompertzian equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 62 Moran birth-death process with selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 63 Tumor initiation prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 64 Comparison of stochastic Moran birth-death process, Gompertzian, and Shannon entropy growth curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 65 Emergence of genetic heterogeneity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 66 Simulated therapy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 67 A ow chart of the Moran process with selection and mutation algorithm . . . . . . . . . . . 156 vii List of Tables 1 Examples of descriptive features for biomedical systems . . . . . . . . . . . . . . . . . . . . . 3 2 Commonly used distance functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 3 Examples of models for biomedical systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 4 Three relevant paradigms of machine learning . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 5 Baseline characteristics of participants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 6 Top ten kinematic features correlated with hospitalizations . . . . . . . . . . . . . . . . . . . 24 7 Top ten kinematic features correlated with physical activity . . . . . . . . . . . . . . . . . . . 24 8 Full list of kinematic features which dierentiate between patients by unexpected hospitalizations 34 9 Full list of kinematic features which dierentiate between patients by physical activity . . . . 35 10 Patient tness score summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 11 Patient clustering comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 12 The ranked list of predictors by importance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 13 Performance of machine learning models for predicting recurrence and survival. . . . . . . . . 67 14 Bladder cancer dataset description. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 15 Chi-squared ranking of bladder cancer feature importance: feature 1-36 . . . . . . . . . . . . 75 16 Chi-squared ranking of bladder cancer feature importance: feature 37-72 . . . . . . . . . . . . 76 17 Bladder cancer features ranked by Cramer's V: feature 1:36 . . . . . . . . . . . . . . . . . . . 78 18 Bladder cancer features ranked by Cramer's V: feature 37-72 . . . . . . . . . . . . . . . . . . 79 19 P-value for clinical T stage Kaplan-Meier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 20 P-value for pathologic T stage Kaplan-Meier . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 21 Features extracted from organoid contours . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 22 Prisoner's dilemma model mutation paramter combinations . . . . . . . . . . . . . . . . . . . 148 viii LIST OF TABLES Abstract Progression of cancer occurs across multiple length and time scales and encompasses biodynamical processes with intrinsically unique implications for the host patient. The compartmentalized study of these processes has not only revealed scale-specic biomedical questions but also led to an equally diverse set of quantitative approaches derived from various elds in the applied spirit of engineering. Due to advances in imaging, experimental, and patient engagement technologies, a growing subset of cancer progression studies are em- pirical in nature and yield data which is often spatio-temporal. The focus of this thesis is to explore a general design approach for developing quantitative analysis tools which leverage data from these studies to provide quantied answers to biomedical questions across time and length scales, and to apply the methodology to three case studies representing three dierent scales. Biodynamical processes or systems observed via experiments, clinical studies, and longitudinal medical assessments span the physical spectrum from nanometer scale molecular phenomena, human body locomotion at the meter scale, to population level disease trends. These processes may be characterized by features from statistics, mechanics, signal processing, or any domain-specic life science. The direct use of these features, optionally in combination with dimensionality reduction and distance metrics, forms the rst arm of biodynamical analytical study design called the feature based approach. This is the more direct approach and leads to an intuitive understanding of a process' role in cancer progression, for instance, by comparing, correlating, and ranking samples or features themselves. The second arm of biodynamical analytical study design, the model based approach, uses stochastic, physics, or statistical modeling techniques to create models of the biodynamical processes or system features. This is the more complex design arm and requires a greater amount of data, but is distinct from the feature based approach due the more advanced questions the models are able to answer. Both approaches' ultimate goal is to provide a quantied output in the form of one or a mixture of the following: i) clustering or dierentiating samples, ii) classifying samples as a type, iii) forecasting or predicting a numerical sample characteristic, and iv) characterizing the behavior of a given sample. The rst case study of morphogenesis of 40 organoids focuses on extracting geometric and signal pro- cessing based features from time series of images at the micrometer scale. The feature set is used to cluster organoids into groups of similar phenotypes in order to separate invasive and normal behaving organoids. Secondly, two model based approaches are used to understand the dynamics of growth: i) a statistical model of shape is created to corroborate the feature based clustering, and ii) a hidden Markov model of organoid shape is used to discover unmeasured genomic states and their corresponding shape dynamics. A standardized code is provided in order to apply the feature based clustering to image data from organoid ix experiments. The second case study is a human performance assessment of 37 cancer patients where a non-invasive motion capture camera is used to record patients performing clinically supervised exercises. The goal of the study is to identify patients with higher risk of deteriorating health while undergoing a chemotherapy regimen. A kinematic feature based approach is used to cluster patients into groups of low and high risk of unexpected hospitalization and limited physical activity. Secondly, a machine learning model based on kinematic and demographic features is used to classify individual patient tness levels based on a single exercise instance. This work serves as a proof-of-concept of a non-invasive human performance based risk calculator of clinical utility. The nal case study uses a feature based approach to understand the importance of clinical measures in a longitudinal bladder cancer dataset of 3500 patients, and machine learning models to stratify patient survival and disease recurrence post-surgery. The combination of feature and model based approaches developed here can be applied to most clinical datasets which contain survival data, without additional domain specic knowledge of the cancer type. The accuracy and temporal resolution of predictions made with the developed models can dramatically change post-cystectomy patient care. x 1 INTRODUCTION 1 Introduction 1.1 Biomedical system characterization The analytical toolsets developed in this thesis are motivated by scientic and clinical questions which arise through biomedical experiments, clinical studies, and medical assessments. Fig.1 shows the relevant spectrum of biological data sources of various length scales and the subsequent process of designing tools to produce quantied outputs. The projects presented in this thesis include data from i) organoid laboratory experiments at the m scale, ii) clinical human motion capture studies at the m scale, and iii) preoperative imaging and operative data from surgery at a population scale. Indeed the range of biomedical data is vast (Fig.1), spanning nanometer scale molecular data to population scale demographics in space, and fractions of a second to years in time. Consequently the choice of study design is dependent, in part on whether a dataset is multi-scale or not. For instance, model based approaches including stochastic, physics, and statistical models are dicult to adapt to multi-scale data, where as feature based approaches easily combine data from various scales. Furthermore biomedical data collection methods are often fraught with high noise, which lead to lower resolution and less informative models. Nevertheless, feature based studies fail to describe or simulate biomedical behavior. Generally, feature based studies are more suitable for clinical applications where multi-modal data is ubiquitous and ultimate decision making must involve the corresponding complex higher level interactions which are dicult to model. In contrast, model based studies provide an opportunity to discover and describe biomechanisms and behavior which remain relevant to future studies. The two approaches are not entirely independent, and the ultimate goal of biomedical characterization studies is to provide a quantied summary output from a single or series of empirical observations. The next sections provide brief introductions to feature extraction (Section 1.2), dimensionality reduction (Section 1.2), model based approaches (Section 1.3), and clustering and classication (Section 1.3) topics relevant to studying biomedical datasets as shown in Fig.1. 1.2 Feature based study of biomedical data 1.2.1 Feature types A complete or sucient set of descriptive features of a biomedical system or process may be derived from a combination of dierent disciplines including biology, mechanics, geometry, information theory, time series analysis, statistics, and signal processing in addition to domain-specic biomedical parameters. Table 1 lists some of the commonly used features extracted from biomedical time series or spatio-temporal data to describe a system. Although the source disciplines of a set of features may be diverse, generally speaking the 1 1 INTRODUCTION Time Length nm μm mm cm m Spatio-Temporal/time series data Feature based Model based Natural phenomena Empirical observation Study design Quantified output Cluster/ differentiate Classify Numerical forecast Behavior characterization Dimensionality reduction  Statistics  Mechanics  Signal processing  Domain based Distance metrics  Probabilistic  Physics  Statistical Figure 1: Schematic of analytical tool design options for studying biomedical processes of cancer progression. Steps include empirical observation of processes across various scales, feature or model based study design, and quantied output options. features themselves can be characterized as being in either the time or frequency domain, and being scalar or time series. Furthermore, scalar features may be taken over a moving window to produce time series, e.g. minimum of every ve time points. The number of feature types required to describe a system may be known a priori, or be discovered as part of the analysis. Nevertheless, due to the complexity of biomedical systems, often a multitude of perspectives are helpful to form a complete understanding or to discover dierences among a set of samples. Once a set of features is selected, it may be used directly to draw conclusions, or may be processed with dimensionality reduction, distance metrics, or be used to train a model (Figure 1). 2 1 INTRODUCTION System feature types Source discipline Feature Statistics and Probability min/max/median/mean/mode standard deviation pearson correlation moment: skewness/kurtosis/etc Kullback{Leibler divergence Mechanics (linear/angular) velocity (linear/angular) acceleration (linear/angular) momentum kinematic/potential energy kinetics Geometry perimeter/area/volume circularity curvature Information theory mutual information entropy Time series analysis ARIMA orders autoregressive parameters moving average parameters autocorrelation function Signal processing DFT/FFT frequencies DFT/FFT amplitude cross-correlation energy Table 1: Examples of descriptive features for biomedical system time series and spatio-temporal data 1.2.2 Distance functions In mathematics, a metric is a function g :XX! [0;1) which denes the distance between two points x;y2X subject to the following four conditions: I: g(x;y) 0 (1) II: g(x;y) =g(y;x) (2) III: g(x;y) = 0()x =y (3) IV: g(x;z)g(x;y) +g(y;z) (4) known as the non-negativity, symmetry, identity of indiscernibles, and the triangle inequality conditions respectively. The rst and third conditions together are required for positive deniteness. Although a metric must satisfy all four conditions, the notion of distance is maintained even if a subset of conditions is met, in which case the distance measure may be classied as a quasimetric and pseudometric for example. A quasimetric is a distance function which satises the triangle inequality but is not symmetric, and a 3 1 INTRODUCTION pseudometric relaxes the identity condition by allowing g(x;y) = 0;x6=y. A list of commonly used distance functions are cataloged in Table 2. Time series distance functions Name formula metric conditions ` 0 , discrete P i 1 X(i)=Y(i) IIV ` 1 , absolute, taxi cab, city block, Manhattan P i jX(i)Y (i)j IIV MAE: mean absolute error (1=n) P i jX(i)Y (i)j IIV ` 2 , Euclidean P i (X(i)Y (i)) 2 1=2 IIV SSE: sum squared error P i (X(i)Y (i)) 2 IIII MSE: mean square error (1=n) P i (X(i)Y (i)) 2 IIII ` p ( P i (X(i)Y (i)) p ) 1=p IIV Minkowski distance ( P i jX(i)Y (i)j p ) 1=p IIV Cosine similarity XY=kXkkYk I;II;IV (absolute) Pearson's distance 1j X;Y j I;II Mahalanobis distance (XY ) T 1 (XY ) 1=2 IIV Table 2: Commonly used distance functions for comparison of real-valued time series In practice, the choice of the optimal distance metric is not always apparent, however, empirical tests [72] suggest various metrics outperform the often standard choice of Euclidean distance. Furthermore, Keough et al showed that these metrics may be enhanced by employing correction factors which account for the discrepancy in complexity of the two time series being compared [14]. By considering time series as stochastic signals, several statistical divergence measures [13] may also be employed as distance measures [115].These functions can either provide distances directly by acting on raw time series data, the corresponding derivatives [101], and any extracted features described in section 1.2, or be used in a nested scheme such as the DTW algorithm. 1.2.3 Dynamic time warping Misaligned and temporally deformed time series cannot be compared directly using a distance function due to mismatch between local features, and classication or clustering pipelines built on such erroneous comparisons will fail to detect or group similar sources. When two time series are out of phase, a global alignment may reduce error in the direct application of a distance function, however, subsequence osets pose a more complex problem which cannot be solved via a global alignment. Consequently, the non- 4 1 INTRODUCTION linear mapping learned from the DTW algorithm [18] leads to a substantial reduction in sensitivity to such misalignments. In the classic DTW algorithm the similarity operators in 1.2.2 act on a pair of points (x i ;y j ) from the two time series X and Y being compared, and a warped path of matching points is found by minimizing the total distance using dynamic programming, Fig.2A. Figure 2: A) Two time seriesX andY of lengthsn andm respectively may be aligned using the dynamic time warping (DTW) algorithm. B) The cost matrix C based on the l 1 distance function. C) The accumulated cost matrix D created using eqs.14-16 The resulting DTW distance, DTW (X;Y ) = X (i;j)2p g(x i ;y j ) (5) is the sum of the distance function g over all pairs of matched points (x i ;y j ) in the optimal warping path p of length L [18]. The warping path must satisfy the i) boundary conditions, p(1) =(x 1 ;y 1 ) (6) p(L) =(x n ;y m ) (7) the ii) monotonicity condition, p(l) = (x i ;y j ) (8) i 1 i 2 :::i L (9) j 1 j 2 :::j L (10) (11) 5 1 INTRODUCTION and iii) the step-size condition, p =p(l + 1)p(l) (12) p2f(1; 0); (0; 1); (1; 1)jl2 [1;L 1]g (13) while assigning an element of X to one from Y . The boundary conditions prevent only a subsequence from one of the time series being used. The monotonicity conditions ensure that a feature from one time series is not repeatedly matched to the second time series, and the step-size conditions ensure that some features of one series are not entirely omitted in the warped path. Two optional, slope and window constraints, prevent matching between large and small sections and large warpings. Furthermore the search for the warping path may be altered by using local or global constraints which apply weights to certain path selections in the accumulated cost matrix. The dynamic programming algorithm used to nd the DTW distance and the warping path involves creating a cost matrix C of the pairwise distances (Fig.2B) and an accumulated cost matrix D (Fig.2C) [128], whose elements are calculated as D(a; 1) = a X i=1 g(x i ;y 1 ) a2 [1;n] (14) D(1;b) = b X j=1 g(x 1 ;y j ) b2 [1;m] (15) D(a;b) = minfD(a 1;b 1); D(a 1;b); D(a;b 1)g +g(x a ;y b ) a2 (1;n];b2 (1;m] (16) where the nal entry D(n;m) =DTW (X;Y ) is the DTW distance between the time series and the warping path p is recovered by tracing back from D(n;m) to D(1; 1), or, equivalently, keeping track of the choice made in the min function in eq.16. The DTW distance for two perfectly similar time series is zero, however, there is no upper bound or intrinsic scale to this measure and more dissimilar time series have larger DTW distances. DTW satises the non-negativity, symmetry, and identity of indiscernibles metric conditions (eqs. 1-3) but not the triangle inequality condition (eq. 4), therefore it is not a metric. Nevertheless, it is a vastly popular similarity measure and several modied versions have been adapted to distinct applications [68]. Typically the distance metric used to nd the warping path is also used to report the nal distance, however, a secondary metric may be applied to the matched points such as the Malhabonis distance which in turn may be applied to neighboring subsequences of the matched points [217]. The time complexity of the dynamic programming algorithm is O(nm) when assessing similarity between two time series of lengths n 6 1 INTRODUCTION andm, however empirical results suggest O(n) complexity can be achieved using lower bounding techniques [161]. In addition to the broader tasks of clustering and classication, DTW may also be used for i) nding the average of set of time series [150], ii) sub-sequence alignment [169] which is the search of an alignment of a short time series to a much longer time series, and iii) query systems [218]. 1.2.4 The multivariate case Although unviariate time series may be compared to one another directly using the classic DTW algorithm, multivariate, or multidimensional time series require modication of the similarity measures and distance matrix, due not only to the higher dimensionality but also the correlation between those dimensions. GA ten Holt et al [89] introduced an intuitive generalization of the univariate DTW algorithm for K-dimensional multivariate time series involving the sum of the distances between all dimensions of a time series while constructing the accumulated cost matrix D2R mn , D(a; 1) = a X i=1 K X k=1 g(x i;k ;y 1;k ) a2 [1;n] (17) D(1;b) = b X j=1 K X k=1 g(x 1;k ;y j;k ) b2 [1;m] (18) D(a;b) = minfD(a 1;b 1); D(a 1;b); D(a;b 1)g + K X k=1 g(x a;k ;y b;k ) a2 (1;n];b2 (1;m] (19) and then searching for the warped path as usual, wherex a;k andy b;k are thek th dimensions of time series X and Y at time points a and b respectively. Here, X and Y may be of dierent lengths m and n, however allK dimensions of a time series must be of the same length. TheK dimensions must be normalized to zero mean and unit variance due to the summation of distances between each dimension of the two time series. This is not the only generalization of DTW for multidimensional time series. Shokoohi-Yekta et al [174] simulate and test two paradigms of DTW similarity for multivariate time series: dependent DTW and independent DTW. Dependent DTW, which is the technique proposed by GA ten Holt et al [89] and described in eqs.17-19, matches two multivariate signals by nding one warped path from a single accumulated cost matrix D for all the dimensions thereby generating a single distance, whereas independent DTW matches each pair of dimensions separately using K accumulated cost matrices, DTW (X;Y ) = K X k=1 D k (n;m); D k 2R mn (20) and aggregates the correspondingK pairwise warped path distances D k (n;m) into a single resultant distance. 7 1 INTRODUCTION The underlying dierence is that dependent DTW algorithms assume that all dimensions of the time series are warped by the same function, and independent DTW algorithms assume that each dimensions may be warped in a dierent way. Clearly, dependent DTW is appropriate to use when learning one nonlinear mapping between all the dimensions of two time series, and independent DTW ought to be used when a new mapping is responsible for misalignment between each dimension of two time series. 1.2.5 Dimensionality reduction and feature selection The number of descriptive features of a biomedical system may be too large in certain cases and prove to be a hindrance towards achieving the ultimate goal of creating a simple quantied output. This high dimensionality of a system requires many samples for each combination of features to fully link the feature space to sample behavior and collecting the sucient amount of data is nearly always impractical due to high cost of biomedical observations and experiments. Furthermore, a high dimensional scenario where the number of features in a dataset far exceeds the number of samples, leads to erroneous overtting of prediction and classication models, and lower condence predictions, thereby reducing clinically applicability of any analyses. These issues necessitate dimensionality reduction and feature selection techniques which reduce the total number of features required to describe a system. Although there is no optimal way of reducing the feature set, most methods either use a transformation of the entire original feature set and embed into a lower dimensional space, or build a smaller feature set using either exclusion or inclusion decision rules. The most widely used transformation technique is principal component analysis (PCA) which maximizes the variance of the data in the resulting dimensions by using a linear mapping. PCA can be performed using singular value decomposition (SVD) of a data matrix X2R np with n samples and p features, X = USV T (21) to get a reduced data matrix Y r , Y r = XV2R nr (22) = USV T V (23) = US (24) where r n is the number of principal components. Decision rule based feature selection methods use correlation metrics between features and target outcome variables to either select the most relevant features or remove the least informative features iteratively. The choice of correlation metrics depends on the nature 8 1 INTRODUCTION of the feature, for instance, categorical features may be selected using a the chi-squared test of independence and continuous variables may be ranked using Pearson's correlation. 1.3 Model based study of biomedical data 1.3.1 Modeling techniques Mathematical modeling using biomedical data can be achieved using many dierent approaches as shown in Table 3. Some of these modeling techniques are tailored to very specic applications such as compartmen- tal models for disease propagation, and rigid body kinematics for understanding biomechanical behavior, however, other more generic techniques such as time series analysis (Table 3) may be adopted to various datasets. Furthermore, statistical techniques for regression and classication such as support vector ma- chines and random forest, require preprocessing of time series data. Whereas graphical models are more readily applicable to sequential data. Unlike feature based characterization of biomedical systems, where multiple disciplines are often combined in an analysis, model based approaches typically rely completely on a single type of a model for a given biomedical dataset. The choice of a modeling technique may be obvious if a conned system is being observed and an associated physical model exists, however, in general, model characterizations may assist this decision. Specically, models may be characterized as i) linear or non-linear, ii) static or dynamic, iii) discrete or continuous, iv) deterministic or stochastic. Once a modeling technique is decided, the best set of model parameters can be identied using model selection techniques which optimize a selection criteria such as likelihood, error between generated and observed instances, and prediction or classication accuracy. In the following sections, statistical and graphical models which are employed in subsequent chapters of this thesis are introduced and discussed in more detail. System model types Discipline Examples Statistical regression, support vector ma- chines, active shape model, ran- dom forest Graphical hidden Markov model, neural network, (switching) linear dy- namical systems Mechanics rigid body kinematics, spring- mass models, hemodynamics Time series analysis ARIMA, ARCH, GARCH, EWMA Ordinary/Parital dierential equations epidemiology compartmental models, mass action model Table 3: Examples of models for biomedical system time series and spatio-temporal data 9 1 INTRODUCTION 1.3.2 Point distribution model Morphology is a direct indicator of phenotype in biology and can be measured using imaging technologies resulting in spatio-temporal data. In such situations, a point distribution model (PDM) [36, 37] is especially applicable as it requires only the discrete point sets which dene the morphology and outputs statistical modes of geometric variation. Additionally, these models may even assist segmentation tasks when used as subroutines in active appearance models and active shape models. PDM can be used to study two- dimensional (d = 2) or three-dimensional (d = 3) shapes of length k, Q = 8 > > < > > : (~ x;~ y) T ; d = 2; Q2R 2k (~ x;~ y;~ z) T ; d = 3; Q2R 3k (25) to create a simulated model Q sim = Q + P ~ b (26) where Q is the mean shape over a sequence of samples, P2R dkp are the p principal modes of variation, and ~ b2 R p are the coecients of the principal modes. The mean shape is found by averaging M sample shapes, Q = 1 M M X k Q k ; (27) and is used to create a covariance matrix, S = 1 M 1 M X k (Q k Q)(Q Q) T : (28) whose eigendecomposition denes the principal modes of geometric variation. Specically the eigenvectors ~ u i are the principal modes of variation, and eigenvalues i are the variance explained by the principal modes (i = [1;dk]). Using alli =dk of the principal to generate new shapes Q sim is not only numerically expensive but also not necessary as in practice a handful of modes create shapes which are suciently representative of the samples in the observation set Q i ;i = [1;M]. Bounds for the principal mode coecients can be determined from the set of observed shapes to generate shapes within an acceptable range. The benet of using the eigendcomposition is that the principal modes of variation are orthogonal and Q sim can represent novel combinations of these morphology dimensions resulting in shapes which may not necessarily be present 10 1 INTRODUCTION in the observation set. Figure 3 shows an example of a point distribution model created an observation set of M = 5 concentric circles (Figure 3A) whose covariance matrix (Figure 3B) has one signicant principal mode of variation (Figure 3C) with an eigenvalue 1 = 1573 and a second insignicant mode of variation with an eigenvalue 2 = 0. Using this single principal mode and a coecient ~ b = (b 1 ; 0) = (7; 0), a new simulated shape Q sim is created in Figure 3D. Figure 3: Example point distribution model. (A) A training set of ve concentric circles. (B) The covariance matrix of the shapes. (C) Eigenvectors of the covariance matrix. (D) A simulated shape using the rst eigenvector. 1.3.3 Hidden Markov model The basic hidden Markov model [157, 156] architecture shown in Fig.4A is described by a discrete hidden random variable X t which is an element from a set of Q hidden states, and an output observation vector Y t 11 1 INTRODUCTION which may be discrete or continuous. Consequently, the hidden Markov model is a doubly stochastic process where the hidden generating sources evolve via a stochastic process and the observed sequence is generated via a second stochastic process conditioned on the hidden states[157]. The model parameters consist of a Figure 4: Hidden Markov model architecture. Hidden states X t are shown in white circles, and observed variablesY t are shown in solid gray circles. A) Standard hidden Markov model with parameters: transition matrix A, emission distribution B, and initial probability . B) Hidden Markov model with mixture of Gaussians as emissions. In addition to Hidden Markov model parameters, the emission Gaussian mean values , and covariance matrices are required. stochastic matrix A2R QQ which dictates transition probabilities between Q hidden states, A(i;j) =P (X t =jjX t1 =i) (29) an emission or output probability distribution B which decides the output vector given a hidden state, B =P (Y t jX t ) (30) and initial probability 2R Q which is a distribution over hidden states, =P (X 1 =j) (31) for a sequence of length T . The hidden state at time t depends only on the previous state X t1 , and the observation at time t depends only on the hidden state at time t. Multinomial distributions are a common choice for B when the outputs are discrete, whereas Gaussian and exponential distributions are popular choices for B in the case of continuous outputs. More complex distributions, and consequently more robust 12 1 INTRODUCTION outputs can be modeled by using a mixture of Gaussians with dierent mean and covariance matrices as in Gaussian mixture models. A hidden Markov model with mixture of Gaussians (GMM-HMM) as emissions is shown in Fig.4B, where the basic hidden Markov model (Fig.4A) parameters and A are augmented by a stochastic emission matrix B2R QM B(i;j) =P (M t =ijX t =j) (32) which contains probabilities ofQ hidden states emittingM Gaussians, and theMQ Gaussians' means and covariance matrices. Therefore in a GMM-HMM, each hidden state has a unique mixture of M Gaussians from which it emits sequence values. Regardless of the choice of B, each hidden state has a unique emission distribution corresponding to a portion of the dynamical space spanned by the observation sequences if the observation sequences are time series. The transitions between the dynamical subspaces then is directly governed by the hidden layer Markov process, which can be as varied as the structure of the Markov chain. Fig.5 shows three examples of Markov chains used in hidden Markov models, where ergodic or non-absorbing models allow each hidden state to transition to any other state. Less exible, yet more ordered models like the upper triangle (5), and more general left-to-right models are more suitable for time ordered sequences, as these models require higher numbered states to be preceeded by lower numbered states [157]. There are three functions or operations of a hidden Markov models and there exist three algorithms which are used to perform these functions[94]. The learning, training, or equivalently parameter estimation function is performed using the Baum-Welch algorithm [15] based on the Forward-Backward algorithm. Given any number of observation sequences, the hidden Markov model parameters can be learned using the Baum-Welch algorithm. The second function of a hidden Markov model is the evaluation or likelihood computation function which computes the likelihood of a given observation sequence, and this is performed using the Forward algorithm. Therefore, given a model the Forward algorithm can be used to nd the likelihood of an observation sequence. Furthermore, classication can be done by nding the model which yields the highest likelihood for a test sequence. Finally, the hidden state assignment or decoding function is the task of assigning a sequence of hidden states to an observation sequence, and this is performed using the Viterbi algorithm. Note that the structure of the hidden layer Markov chain and choice of emission parameters such as the number of Gaussians in GMM-HMM must be predetermined before training, and further model selection techniques are required to nd the optimal parameter set. Pohle et al [154] outline best practices of model selection for hidden Markov models, and conclude that despite the direct applicability of selection criteria 13 1 INTRODUCTION Figure 5: Examples of Markov chain structures for Hidden Markov model. such as Akaike information criterion (AIC) and Bayesian information criterion (BIC) care must be taken to avoid certain pitfalls. Therefore, to nd the best parameters for hidden Markov models with mixture of Gaussians emissions, we use a combination of heuristics and BIC, BIC =2 logL(Y ;) +n p logn o (33) logL(Y ;) : Log-likelihood of observations given model (34) where the parameter set = (; A; B;; ),n p is the number of model parameters, andn o is the number of observations used to train the model. The total number of parameters in the model depends on the choice of Markov chain structure, n p =Q +QM +QMd +QMd 2 + 8 > > > > > > < > > > > > > : 2Q 1 left-to-right (1=2)Q(Q 1) upper-triangle 2Q 1 bifurcated (35) where d is the dimensionality of the multivariate observation time series. 14 1 INTRODUCTION 1.3.4 Machine learning models The feature based approach of extracting as many independent descriptions of a system feeds directly into machine learning models which use a set of features for i) prediction via regression or classication, and ii) detection via clustering or anomaly identication (Table 4). Therefore machine learning models can be trained to predict or detect by example if training samples with labeled outcomes are available, this is called supervised learning. However, in the absence of labeled target or outcome variables, unsupervised learning techniques can be used to cluster similar samples. There are other paradigms of machine learning such as re- inforcement learning, but these do not have as direct applicability to biomedical problems as supervised and unsupervised techniques. For instance, supervised learning techniques can be used to predict probability of survival at the patient scale using demographic and human performance data, and classify pathological states at cellular and tissue scales using image data. In scenarios where a subset of the samples have labeled output data, e.g. if only a few patients' survival data is available in a longitudinal dataset, then semi-supervised techniques (Table 4) may be used to learn from partial knowledge of outcomes. Regardless of paradigm, machine learning models often employ a pipeline of steps including some or all of the following: i) prepro- cessing the data to replace missing data and to remove irrelevant features and samples, ii) a dimensionality reduction step to avoid curse of dimensionality ([132]), iii) calculating distance between features, particularly time series, iv) training, validation, and testing using one or a combination of algorithms. For instance, a set of time series features may be compared to one another or to a reference time series via dynamic time warping rst and fed into a K-means or K-medoids based algorithm for clustering. Machine learning paradigms Supervised Semi-supervised Unsupervised Paradigm Ground truth is known Mixed bag Unknown ground truth Data (input, output) (input, output) (input) (input) Tasks Regression Regression Clustering Classication Classication Dimensionality reduction Clustering Algorithms Nearest neighbor Generative models K-means/medoids Decision tree Self-training Mixture models Naive Bayes Transductive SVM Hierarchical clustering Support vector machine (SVM) Graph-based methods Principal component analysis Random forest Neural network/auto-encoders Articial neural network Graph model Table 4: Three relevant paradigms of machine learning and examples of algorithms. In biomedical applications of machine learning [16, 107, 216], model selection is performed by maximizing performance metrics such as area under the curve (AUC) of a receiver operating characteristic (ROC) curve 15 1 INTRODUCTION for classiers, and minimizing root mean square error (RMSE) for regression models. Clustering models can be selected by comparing concordance between a given clustering and some benchmark expert clustering. Sensitivity and specicity, precision and recall, and F1 scores are also prevalent performance metrics in biomedical studies where it is most important to detect pathological cases while avoiding false positives. As such, developments in machine learning have lead to data pipelines which are highly successful at prediction and detection tasks in the biomedical eld. 1.4 Chapter summaries In Chapter 2 human performance motion capture data of from patients performing supervised exercises in a clinical setting are studied using a feature based pipeline of tools to classify patients by unexpected hospitalizations and low physical activity at home using statistical features extracted from the raw motion time series and subsequent kinematic feature time series. Kinematic feature extraction is explained, and a derivation of angular velocity of two-joint sections is shown. A short list of kinematic features which classify low and high risk hospitalization and physical activity patient groups are provided. In Chapter 3 two model based pipelines are employed using the same human performance data from Chapter 2. The rst model based pipeline uses kinematic features from the captured motion, however, this pipeline uses the dynamic time warping algorithm to compare time series in order to predict physician assigned tness scores. The third pipeline also uses the dynamic time warping distances between kinematic time series features, but uses dimensionality reduction to learn a patient clustering and compares it with qualitative physician assigned clusterings of patients. In Chapter 4 a clinical dataset containing preoperative and operative features is used to create classiers of post-cystectomy survival and disease recurrence. Furthermore information theory based correlation tech- niques are used to understand dependencies among the feature set. Feature extraction is not performed here as the dataset is relatively high dimensional. The classiers developed in this chapter make predictions for individual patients using the population level dataset. In Chapter 5 a dataset of organoid morphogenesis images is studied using a feature based pipeline as well as a statistical model based pipeline. Contour time series are extracted from sequences of two-dimensional images of basal and growth factor treated organoids. Geometrical and signal processing based time series features are extracted from the contour time series and used to cluster organoids into groups of similar spatio- temporal behavior. Secondly, a point distribution statistical model of shape is learned from the organoid contours which is used to understand the behavior of the clusters detected using the feature based pipeline. In Chapter 6 the same dataset as Chapter 5 is used to create a stochastic model of organoid area and 16 1 INTRODUCTION perimeter in order to discover various organoid subtypes and their dynamics. To do this, a hidden Markov model is used which is trained with the area and perimeter time series for basal and growth factor treated organoids. The goal of this strictly model based approach is to reveal the hidden genomic states responsible for the observed breadth of area and perimeter phenotypes. Pre-thesis work is shown in Chapter 8 and Chapter 9, where a game theory model of cancer growth and tumor cell kinetics based on the prisoner's dilemma game is developed. 17 2 QUANTIFIED KINEMATICS FROM IN CLINIC MOVEMENTS DIFFERENTIATES PATIENT RISK FOR CHEMOTHERAPY COMPLICATIONS 2 Quantied kinematics from in clinic movements dierentiates patient risk for chemotherapy complications 2.1 Abstract Cancer patients, particularly those treated with dicult chemotherapy regimens are at high risk for hospital- ization during and between treatment cycles. Existing metrics based on physician observation of patients in the oce are qualitative, subjective, and lack agreement among observers. Imaging tools have the potential to provide an objective and veriable measurement of physician observations of patients in the oce. To de- termine whether an in-oce movement tracker, in cancer patients receiving highly emetogenic chemotherapy, can identify patients at risk for future unexpected hospital visits and low activity levels at home. Multicenter, single arm, observational clinical trial. A non-invasive motion capture system measured 1) patient movement from chair-to-table and 2) get-up-and-walk test. Activity levels were recorded using a wearable sensor over a two-month period with an additional 90-day follow-up. Kinematic signatures from motion capture data and subsequent statistics are tested for correlation with hospitalization and physical activity groups. A 15-hour (during study period) threshold was used to group patients into low and high activity groups. Secondly, patients were grouped if they had one or more unexpected healthcare encounters during the observation and follow-up period. Kinematic features from chair-to-table form the best classiers for unexpected visits (AUC = 0.816) and physical activity (AUC = 0.735). Chair-to-table acceleration of non-pivoting knee (P = 0.001) and hip (P =0.002) are most correlated with unexpected hospitalizations. Angular velocity of the hip (P = 0.022) and non-pivoting leg (P = 0.024) are most correlated with physical activity. Physician observations of patient movement in the examination room can be quantied in a routine objective metric . Motion capture systems and wearable sensors can evaluate kinematic features that are correlated with unexpected hospital visits and physical activity. 2.2 Introduction Each patient has specic and individual needs for optimal supportive care during cancer treatment. Pre- dicting these needs and providing specic solutions has the opportunity to both improve outcomes and the experience during treatment. Poor patient outcomes, patient satisfaction, quality of life, and economic cost are associated with unexpected hospitalizations with patients actively receiving chemotherapy [164]. A recent survey of US oncology nurses found that 61% of nurses cared for patients who had to go to the emergency room or were hospitalized due to chemotherapy induced nausea and vomiting (CINV) [34]. These CINV hospitalization costs were estimated to be over $15,000 per occurrence [164, 34]. Readily available tools and 18 2 QUANTIFIED KINEMATICS FROM IN CLINIC MOVEMENTS DIFFERENTIATES PATIENT RISK FOR CHEMOTHERAPY COMPLICATIONS metrics such as ECOG performance status, Body Mass Index (BMI), Mini Mental State Exam (MMSE), and Charlson Comorbidity Index (CCI), are part of a comprehensive geriatric assessment, however few physicians perform the complete assessment, as they are time consuming. There is emerging data that a comprehensive geriatric assessment can predict complications and side eects from treatment [64, 57, 92, 58, 159, 79, 38]. Currently, the most routine assessment is the ECOG performance status. It is well known that in metastatic cancer such as lung origin, ECOG strongly predicts survival independent of treatment and usually guides if treatment should even be given if poor performance status [97, 76, 78, 114, 187, 148]. Clinical assessment of performance status and risk of toxicity from cancer therapy includes observation of patient movement as part of the physician examination within a clinic room environment. This has been routine practice for many years, and while it has been recognized for a long time, oncologists and patients substantially dier in their assessment of performance status with most oncologists being overly optimistic on the patient's performance status [6, 191]. Technological advances in low cost spatial cameras, such as Microsoft Kinect, have the potential to objectively dene and categorize patients with varying levels of mobility at home or in the clinic [136]. Similarly, low cost activity trackers containing accelerometers, such as Microsoft Band, can capture daily movement in the clinic and at home, assessing dynamic changes related to exertion or to physical challenges such as the chemotherapy cycle. These consumer technologies have the capacity to bring objectivity to the assessment of mobility and performance status of patients on chemotherapy. The utility of activity trackers has been evaluated in areas outside of cancer medicine and demonstrated correlation with clinical outcomes in a wide variety of other disease settings [35, 28]. For example, in COPD, increasing additional steps correlates with reduced COPD hospitalizations [51] and formal exercise capacity evaluation such as the six-minute walk distance predicted COPD-related hospitalization [53]. After cardiac surgery, it was observed using an accelerometer that inpatient step count appears to predict repeat hospitalization [188]. In elderly hospitalizations it was found that mobility after hospital discharge could predict 30-day hospital readmissions [60]. To improve our understanding unexpected hospital visits in cancer patients receiving chemotherapy we conducted an observational study to evaluate the eect of physical activity as measured by a motion-capture system and wearable movement sensor and their relationship to unexpected healthcare encounters. 2.3 Methods 2.3.1 Trial Design This study was a multicenter, single arm, observational trial conducted in the United States between July 2016 to July 2017. It was designed to compare kinematic signatures obtained from motion-capture systems 19 2 QUANTIFIED KINEMATICS FROM IN CLINIC MOVEMENTS DIFFERENTIATES PATIENT RISK FOR CHEMOTHERAPY COMPLICATIONS (e.g. Microsoft Kinect) and wearable motion sensors (e.g. Microsoft Band) and see if there was a correlation with unexpected hospital visits and physical activity at home and is described elsewhere [81, 120]. The study protocol was approved by the institutional review boards at all participating sites. Written informed consent was obtained from all participants. 2.3.2 Participants Brie y, patients were eligible for the study if they were > 18 years of age, had a diagnosis of a solid tumor, and undergoing 2 planned cycles of highly emetogenic chemotherapy (see Appendix A), could ambulate without an assistive device, and had 2 separate kinematic evaluations successfully completed. Patient demographics Number of patients 36 Age Median 48 Range 24-72 Gender Male 18 Female 18 Ethnicity Hispanic 22 Non-Hispanic 14 Goal of treatment Curative 30 Palliative 6 Table 5: Baseline characteristics of participants 2.3.3 Clinical exercises and motion capture Patients underwent two clinically supervised tasks including chair-to-table (CTT) and get-up-and-walk (GUP) [129, 203]. CTT task begins with patients standing up from a chair while rotating the hip and left leg and pivoting on the right leg. Therefore, the CTT task design requires larger range of motion from the left lower extremities . The GUP task requires patients to stand up and walk to a marker 8 feet away, turn, and walk back to the starting position. We analyze the entire CTT task and the walking portion of GUP using the motion capture system. The two tasks are performed by the cohort of cancer patients once pre-treatment (visit-1) and once post-treatment (visit-2). The Microsoft Kinect, a depth-sensing motion capture camera is used record the exercises, and three-dimensional positions of 25 anatomical sites (Figure 7) are extracted [136], from which six types of kinematic features are calculated: 1) velocity, 2) acceleration, 3) specic kinetic energy, 4) specic potential energy, 5) sagittal angle, 6) angular velocity (Appendix B). We exclude wrist, hand, ankle, and 20 2 QUANTIFIED KINEMATICS FROM IN CLINIC MOVEMENTS DIFFERENTIATES PATIENT RISK FOR CHEMOTHERAPY COMPLICATIONS Figure 6: Illustration of chair-to-table exam maneuver demonstrating (A) sitting at rest (B) standing, rotating hip, pivoting on right leg, and using left (non-pivoting) leg to move onto exam table (C) sitting on exam table foot joints (Figure 7) from statistical analysis as the motion capture signal for these joints is less reliable. The combination of selected joints and kinematic features capture the underlying biomechanics of patient movement and are therefore selected for inter-patient comparison. Each patient has a pre- and post-treatment pair of samples of each feature, and four statistics (minimum, maximum, mean, median) from each visit's time series kinematic feature are averaged (mean) over the two samples. For the remainder of the paper, we refer to the mean-(minimum, maximum, mean, median) over the two visits simply as the minimum, maximum, mean, and median. 2.3.4 Physical Activity measure Patient outcomes were grouped by activity level and unexpected hospital visits. During the study period that spanned for 60 days while receiving chemotherapy and a 90-day follow-up period, patients wore a wrist motion sensor to track their overall daily physical activity. We recorded the number of hours spent above low physical activity (LPA) for each patient over this period. Patients were considered high activity, rather than low activity, if they met greater than a 15-hour during the study period physical activity threshold. Patients with more than 15 hours of activity above LPA (HALPA = 0) and patients with 15 hours or less active time than LPA form the two HALPA groups. Likewise, patients were grouped if they had one or more unexpected hospital visits compared to those that did not have any. Four types of unexpected hospital visits were tracked including: 1) Unplanned triage/infusion center visits, 2) urgent oce visits, 3) urgent hospitalizations, and 4) ER visits. Patients with zero unexpected hospitalizations (UHV = 0) and patients with one or more unexpected hospital visits 21 2 QUANTIFIED KINEMATICS FROM IN CLINIC MOVEMENTS DIFFERENTIATES PATIENT RISK FOR CHEMOTHERAPY COMPLICATIONS are (UHV = 1) form the UHV groups. 2.3.5 Statistical Analysis We dierentiate patients by the average of visit-1 and visit-2 statistics for the set of kinematic features and correlate to two binarized clinical outcome UHV and HALPA. The Welch's t-test is used to test whether the mean value of the four averaged statistics is dierent for the UHV or HALPA groups, thereby revealing kinematic features which distinguish between UHV = 0 and UHV = 1 patients, and similarly HALPA = 0 and HALPA = 1 patients. The Welch's t-test also known as the unequal variance t-test allows the central tendency of two groups of unequal sizes and unequal variance to be tested for equivalence [167]. Secondly, we calculate the receiver operating characteristic (ROC) curve and use the corresponding area under the curve (AUC) as a metric of a feature's ability to classify patients into risk groups. Figure 7: Schematic of anatomical sites detected in motion capture, along with six two-joint sections (red) whose angular velocities are extracted from raw kinematic data. 22 2 QUANTIFIED KINEMATICS FROM IN CLINIC MOVEMENTS DIFFERENTIATES PATIENT RISK FOR CHEMOTHERAPY COMPLICATIONS 2.4 Results 2.4.1 Patient cohort Of the 60 persons screened and agreed to participate in the study, 36 persons completed the study without drop out and had associated unexpected hospital visits and physical activity results. Overall the mean age of participates were 47.8 years old, and 50% were men. Breast, testicular, and head and neck cancer, comprised most of study participants . Chemotherapy was primarily of curative intent for most patients . Presumed reasons for higher than expected study drop out were likely due to a large proportion of persons being recruited from the Los Angeles County Hospital uninsured patient population combined with a large proportion being young males receiving chemotherapy for testicular cancer. These factors may explain why there was not a higher percentage of patients could complete the ve-month study period. There are 16 UHV = 0 patients and 20 UHV = 1 patients for a total of N = 36 patients for whom hospitalization data is collected. Similarly, there are 17 HALPA = 0 patients and 18 HALPA = 1 patients for a total of N = 35 patients for whom physical activity data is collected. 2.4.2 Unexpected hospitalizations We report the kinematic features that correlate most with unexpected hospital visits according to i) t-test and ii) ROC analysis in Table 6. CTT features dominate the list of UHV dierentiating kinematic features and GUP features were less associated with the two outcomes. The full list of 55 features with signicant t-test scores (p-value < 0.05) are listed in Appendix C. Hip and left side joints are the top UHV features due to the pivot on the right side, and resulting large left side motion of CTT (see Figure 6). Figure 8A shows the ROC curves for the features with the highest AUC values for UHV where the maximum left leg angular velocity about the y-axis during CTT forms the best classier of UHV (AUC = 0.816). The top three UHV dierentiating features according to the t-test are plotted in Figure 8B, which shows the left knee, left hip, and the spine base mean accelerations during CTT are all generally higher for patients with no unexpected hospitalizations compared to patients with one or more unexpected hospitalizations. 2.4.3 Physical activity Kinematic features that correlate most with physical activity according to i) t-test and ii) ROC analysis in Table 7. Unlike UHV, both CTT and GUP features appear in the list of HALPA dierentiating kinematic features. The full list of 15 features with signicant t-test scores (p-value < 0.05) are listed in Appendix D. Angular velocities, particularly those of the hip, dierentiate HALPA groups the most. Nevertheless, 23 2 QUANTIFIED KINEMATICS FROM IN CLINIC MOVEMENTS DIFFERENTIATES PATIENT RISK FOR CHEMOTHERAPY COMPLICATIONS Welch's t-test ROC analysis Feature t-test p-value Feature AUC 1 Left knee: mean CTT acc 3.735 0.001 1 Left leg: max CTT av-y 0.816 2 Left hip: mean CTT acc 3.398 0.002 2 Left knee: mean CTT acc 0.806 3 Spine base: mean CTT acc 3.258 0.003 3 Left elbow: max CTT pe 0.781 4 Left knee: mean CTT vel 3.177 0.003 4 Left hip: max CTT acc 0.781 5 Left knee: mean CTT ke 3.14 0.004 5 Spine base: mean CTT acc 0.775 6 Left elbow: max CTT pe 2.988 0.005 6 Left hip: mean CTT acc 0.775 7 Right hip: mean CTT acc 2.928 0.006 7 Left knee: mean CTT ke 0.775 8 Left hip: max CTT acc 2.925 0.006 8 Right leg: min CTT av-x 0.759 9 Left hip: mean CTT ke 2.921 0.006 9 Hip: min CTT av-z 0.756 10 Right arm: mean GUP av-y 2.91 0.006 10 Left hip: mean CTT ke 0.753 Table 6: Top ten kinematic features from Welch's t-test (ranked by absolute value of two-sample t-test scores) and top ten kinematic features with highest AUC for dierentiating between patients with no unexpected hospitalizations (UHV = 0) and patients with one or more unexpected hospitalizations (UHV = 1). (vel: velocity; acc: acceleration; pe: potential energy; ke: kinetic energy; sa: sagittal angle; av-x, av-y, av-z: angular velocity about x,y, or z axes). kinematic features from the clinical exercises are less correlated with HALPA groups than UHV groups as both t-test scores and AUC values are generally lower in Table 7 compared to Table 6. Figure 9A shows the ROC curves for the features with the highest AUC values for HALPA where the mean hip angular velocity about the vertical axis during CTT forms the best classier of HALPA (AUC = 0.735). Mean hip and minimum left leg angular velocities during GUP are both larger (absolute value) for higher activity patients as seen in Figure 9B. Welch's t-test ROC analysis Feature t-test p-value Feature AUC 1 Hip: mean GUP av-x -2.414 0.022 1 Hip: mean CTT av-z 0.735 2 Left leg: min GUP av-x -2.379 0.024 2 Hip: mean CTT av-y 0.729 3 Back: mean CTT sa -2.331 0.026 3 Left arm: mean GUP av-y 0.725 4 Left arm: min GUP av-y -2.328 0.032 4 Left knee: median GUP ke 0.722 5 Right leg: mean GUP av-z 2.224 0.033 5 Left leg: min GUP av-x 0.722 6 Left hip: mean CTT acc 2.221 0.033 6 Spine mid: mean CTT acc 0.719 7 Back: median CTT sa -2.219 0.034 7 Right leg: median CTT av-y 0.719 8 Hip: mean CTT av-x -2.193 0.035 8 Back: mean CTT sa 0.716 9 Left knee: median GUP ke 2.185 0.039 9 Shoulder: median CTT av-x 0.712 10 Right leg: median CTT av-y -2.184 0.037 10 Hip: mean CTT av-x 0.706 Table 7: Top ten kinematic features from Welch's t-test (ranked by absolute value of two-sample t-test scores) and top ten kinematic features with highest AUC for dierentiating between patients with more than 15 hours of activity above LPA (HALPA = 0) and patients with 15 hours or less activity above LPA (HALPA = 1). (vel: velocity; acc: acceleration; pe: potential energy; ke: kinetic energy; sa: sagittal angle; av-x, av-y, av-z: angular velocity about x,y, or z axes). 24 2 QUANTIFIED KINEMATICS FROM IN CLINIC MOVEMENTS DIFFERENTIATES PATIENT RISK FOR CHEMOTHERAPY COMPLICATIONS Figure 8: Kinematic features which dierentiate patients with zero unexpected hospitalizations from patients with one or more hospitalizations. (A) ROC curves for features with the highest AUC. (B) Boxplots for features with the highest t-test scores (UHV = 0: gray, UHV = 1: red). (vel: velocity; acc: acceleration; pe: potential energy; ke: kinetic energy; sa: sagittal angle; av-x, av-y, av-z: angular velocity about x,y, or z axes). 2.5 Discussion This study corroborates other studies by demonstrating that using a motion capture system and wearable motion sensor is feasible and can give kinematic data that can correlate and possibly be predictive of important clinical outcomes such as unexpected healthcare encounters [24, 77]. However, it seems to be clear that not all kinematic features are relevant while performing chair-to-table and get-up-and-go exams. Clinical assessment is of value, and what this study attempted to accomplish was to put a quantitative metric on the clinical assessment. As mentioned earlier, the kinematic features were based o of 25 anatomical sites that include head, arms, spine, hips, knees, and feet. The kinematic features of the get-up-and-go exam did not correlate with unexpected hospital visits however ve kinematic features of the chair-to-table exam did. Interestingly, the anatomic sites that were statistically signicant were left (non-pivoting) knee and hip, as 25 2 QUANTIFIED KINEMATICS FROM IN CLINIC MOVEMENTS DIFFERENTIATES PATIENT RISK FOR CHEMOTHERAPY COMPLICATIONS Figure 9: Top three kinematic features which dierentiate patients with 15 hours or more of activity above LPA from patients with 15 hours or less of activity above LPA. A) ROC curves for features with the highest AUC. B) Boxplots for features with the highest t-test scores (HALPA = 0: gray, HALPA = 1: red). (vel: velocity; acc: acceleration; pe: potential energy; ke: kinetic energy; sa: sagittal angle; av-x, av-y, av-z: angular velocity about x,y, or z axes). well as the spine base. The spine base velocity re ects the movement of a majority of the patient's mass that is not subject to high variability such as the distal hands or feet. Logically it would be safe to assume that patients can have a wide range of how they move their hands or feet during an exam maneuver which would not correlate with their overall physical tness and hence not correlate with hospitalization risk . The association between high physical activity level and kinematic features also seem to revolve around leg, knee, hip, and back movement. Similarly to above, these areas of the body intuitively carry the majority of a patient's mass and lower extremities generally are a more predictive measurement of a persons overall physical activity. This is supported by the calculated kinematic features (Table 6) noted above. The mean hip and minimum left leg angular velocities about the x-axis during get-up-and-go are the two best dierentiators of HALPA groups (Figure 9), and both these angular velocities are greater for patients 26 2 QUANTIFIED KINEMATICS FROM IN CLINIC MOVEMENTS DIFFERENTIATES PATIENT RISK FOR CHEMOTHERAPY COMPLICATIONS with higher physical activity compared to patients grouped in the low activity group. Mean sagittal angle during CTT is generally lower for patients with higher physical activity, which may be due to the increased ability of more active patients to crouch lower in the seated position before standing up and after reaching the medical table. Ultimately the main goal of many recent studies including this one is to identify high-risk patients in a quantitative manner thus increasing reproducibility and reliability. Other modalities such as an Actigraph, a triaxial accelerometer, or using patient reported outcomes, have also recently been studied to better understand this issue [153, 184]. A range of dierent metrics have been evaluated that usually comprise of age, ECOG performance status, cognitive function (mini-mental state exam), nutritional status, comedications, comorbidities, autonomy, creatinine clearance, albumin, or other lab abnormalities [64, 57, 92, 58, 159, 79, 38]. Identifying high-risk patients is postulated as only one component of reducing costly preventable hospitalizations in cancer patients. Other proven strategies include enhancing access and care coordination, standardize clinical pathways for symptom management, availability of urgent cancer care, and early use of palliative care [184, 204, 26, 80]. Interestingly, although performance status likely still has value in predicting side eects and hospitalization risk that in the era of targeted therapy and immunotherapy performance status is less important in aecting response to novel therapies [33]. This study has limitations due to the small sample size and observational design compared to a traditional large randomized controlled trial. Conrmation of the nding would best be accomplished by integrating the tool into randomized clinical trials. The eventual goal would be to identify high-risk patients that have low activity levels or high risk for hospitalizations that appropriate interventions could be performed to reduce poor clinical outcomes. These interventions could include more frequent outpatient visits, IV hydration, or closer nursing/clinical navigator support. In conclusion, patient performance and physical activity can be reliably quantied using camera based kinematic analysis. Modern sensor technology makes such as assessment rapid and low cost. The current platform requires a separate sensor and computer, and incorporation of these into a single hand-held device would be necessary to garner adoption of these algorithms for predicting adverse outcomes. Having such a tool available that quanties what the physician sees during the in-clinic examination has the potential to harmonize ndings among dierent physicians, specialists, researchers and families who all rely on a uniform assessment of patient tness for receiving dicult cancer treatments. 27 2 QUANTIFIED KINEMATICS FROM IN CLINIC MOVEMENTS DIFFERENTIATES PATIENT RISK FOR CHEMOTHERAPY COMPLICATIONS 2.6 Acknowledgement The project described was supported in part by award number P30CA014089 from the National Cancer Institute, and by Noetic Contract FFP 012016 (USA 051) Collaborative Initiative on Human Performance Optimization. The content is solely the responsibility of the authors and does not necessarily represent the ocial views of the National Cancer Institute or the National Institutes of Health. 2.7 Appendix A 2.7.1 Calculating the Emetogenicity of Multiple Agent Chemotherapy/Biotherapy Regimens Steps and Guidelines: 1. List each agent contained within the multiple agent regimen. 2. Identify the agent with the highest emetogenic level 3. Determine the contribution of the remaining agents using the following guidelines: a. Level 1 agents do not contributor to emetogenicity in combination regimens. Examples: level 1 + 1 = 0 2 + 1 = 2 3 + 1 = 3 4 + 1 = 4 b. Adding one or more level 2 agents increases the highest level by 1 in combination regimens. Examples: level 2 + 2 = 3 3 + 2 = 4 2 + 2 + 2 = 3 3 + 2 + 2 = 4 28 2 QUANTIFIED KINEMATICS FROM IN CLINIC MOVEMENTS DIFFERENTIATES PATIENT RISK FOR CHEMOTHERAPY COMPLICATIONS c. Adding level 3 or 4 agents increase the highest level by 1 per each agent in combination regimens. Examples: level 3 + 3 = 4 3 + 3 + 3 = 5 4 + 3 = 5 2.8 Appendix B Details of kinematic feature extraction from the raw three-dimensional position motion capture data are described here. Anatomical site position vectors~ r i =h~ x i ;~ y i ;~ z i i are three-dimensional time series constructed from position at each time point, r i (t) = (x i (t);y i (t);z i (t)) fori = 25 anatomical sites. The position vectors are used to calculate velocity magnitude, ~ v i = ( _ r x T _ r x + _ r y T _ r y + _ r z T _ r z ) 1=2 (36) and acceleration magnitude ~ a i = ( r x T r x + r y T r y + r z T r z ) 1=2 (37) of each anatomical site using the mean-value theorem. Due to the lack of distribution of mass information, specic kinetic energy ~ T i =~ v T i ~ v i =2 (38) and specic potential energy ~ U i =g~ z i =g(~ z i ~ z i (t =t 1 )) (39) We dene sagittal angle as the angle formed between v 1;m the vector originating at the spine base and pointing in the direction of motion, and v 1;3 the vector connecting anatomical site 1 (spine base) and 3 (neck) at each time point. 29 2 QUANTIFIED KINEMATICS FROM IN CLINIC MOVEMENTS DIFFERENTIATES PATIENT RISK FOR CHEMOTHERAPY COMPLICATIONS 2.8.1 Sagittal angle calculation We dene sagittal angle as the angle formed between v 1;m the vector originating at the spine base and pointing in the direction of motion, and v 1;3 the vector connecting anatomical site 1 (spine base) and 3 (neck) at each time point. The sagittal angle is calculated using the inverse tangent of the ratio of the cross product and dot product of v 1;m ) and v 1;3 , s = tan 1 kv 1;m v 1;3 k v 1;m v 1;3 (40) 2.8.2 Angular velocity calculation The angular velocity of the sections dened in Figure 7 are calculated using three-dimensional rigid body kinematic equations for relative motion. A section (Figure 7) is treated as a rigid bar and is dened by two anatomical points (e.g. left and right hips dene the hip section) and we refer generically to these two ends as point A and point B. We calculate the velocities of these two points from the position vectors using the mean-value theorem as mentioned previously. Therefore, using these two velocities, the angular velocity of the section ! AB can be isolated in the relative velocity vector equation ~ v B ~ v B =~ ! AB ~ r AB = (v x ; v y ; v z ) (41) where~ r AB is the vector from point A to point B, ~ r AB =~ r B ~ r A = (r AB;x ;r AB;y ;r AB;z ) (42) This vector equation has three components corresponding to the three directions and require an additional equation to solve for the three components of the angular velocity. Consequently, we use a kinematic restriction equation, ~ ! AB ~ r AB = 0 (43) 30 2 QUANTIFIED KINEMATICS FROM IN CLINIC MOVEMENTS DIFFERENTIATES PATIENT RISK FOR CHEMOTHERAPY COMPLICATIONS because the angular motion of the section along the axis of the section does not aect its action. This allows for a solution to the three components of the angular velocity vector ! AB = (! x ;! y ;! z ): ! x = v z r AB;y v y r AB;z r 2 AB;x +r 2 AB;y +r 2 AB;z (44) ! y = 1 r AB;x (r AB;y ! x v z ) (45) ! z = 1 r AB;y (r AB;z ! y v x ) (46) These equations are solved at each time point to get the time series of angular velocities for each section in Figure 7. 2.9 Appendix C Two-sample t-tests are done to determine if mean values of kinematic features are dierent for patients with zero unexpected hospitalizations (UHV = 0) and patients with one or more hospitalizations (UHV = 1), and the distribution of the resulting t-test scores and signicance values for the entire set of 526 features is shown in Appendix Figure 11. Full list of 55 signicant (p-value < 0.05) t-test scores is shown in Appendix Table 8, and boxplots of these signicantly dierentiating kinematic features is shown in Appendix Figures 12, 13, and 14. 31 2 QUANTIFIED KINEMATICS FROM IN CLINIC MOVEMENTS DIFFERENTIATES PATIENT RISK FOR CHEMOTHERAPY COMPLICATIONS Figure 10: Chemotherapy Emetogenicity Table. 32 2 QUANTIFIED KINEMATICS FROM IN CLINIC MOVEMENTS DIFFERENTIATES PATIENT RISK FOR CHEMOTHERAPY COMPLICATIONS Figure 11: Distribution of t-test scores and signicance values from two-sample t-tests for dierences in mean values of kinematic features between patients with no unexpected hospitalizations (UHV = 0) and patients with one or more unexpected hospitalizations (UHV = 1). 33 2 QUANTIFIED KINEMATICS FROM IN CLINIC MOVEMENTS DIFFERENTIATES PATIENT RISK FOR CHEMOTHERAPY COMPLICATIONS Feature t-test p-value Feature t-test p-value 1 Left knee: mean CTT acc 3.735 0.001 29 Spine base: max CTT vel 2.322 0.027 2 Left hip: mean CTT acc 3.398 0.002 30 Neck: max CTT pe 2.315 0.027 3 Spine base: mean CTT acc 3.258 0.003 31 Shoulder: median GUP av-z 2.29 0.028 4 Left knee: mean CTT vel 3.177 0.003 32 Left elbow: mean CTT pe 2.26 0.031 5 Left knee: mean CTT ke 3.14 0.004 33 Left hip: mean CTT pe 2.257 0.031 6 Left elbow: max CTT pe 2.988 0.005 34 Spine base: median CTT acc 2.233 0.033 7 Right hip: mean CTT acc 2.928 0.006 35 Right elbow: median CTT acc 2.232 0.032 8 Left hip: max CTT acc 2.925 0.006 36 Spine base: mean CTT pe 2.229 0.033 9 Left hip: mean CTT ke 2.921 0.006 37 Left shoulder: mean CTT pe 2.228 0.033 10 Right arm: mean GUP av-y 2.91 0.006 38 Left leg: median GUP av-x -2.227 0.033 11 Left knee: median CTT acc 2.844 0.008 39 Left knee: max CTT acc 2.195 0.037 12 Spine base: mean CTT ke 2.764 0.01 40 Left hip: median CTT acc 2.19 0.036 13 Left leg: min CTT av-x -2.759 0.011 41 Right hip: mean CTT pe 2.186 0.036 14 Spine base: max CTT pe 2.745 0.01 42 Right elbow: mean CTT vel 2.186 0.036 15 Right hip: max CTT pe 2.725 0.01 43 Right leg: max CTT av-x 2.181 0.037 16 Left hip: mean CTT vel 2.671 0.012 44 Right knee: mean CTT vel 2.161 0.038 17 Spine base: max CTT acc 2.658 0.012 45 Right shoulder: max CTT pe 2.151 0.04 18 Left shoulder: max CTT pe 2.654 0.013 46 Spine mid: mean CTT acc 2.15 0.039 19 Left hip: max CTT pe 2.65 0.012 47 Left elbow: mean CTT vel 2.149 0.039 20 Spine base: mean CTT vel 2.591 0.014 48 Left shoulder: median CTT pe 2.143 0.04 21 Right leg: min CTT av-x -2.566 0.017 49 Left elbow: median CTT acc 2.137 0.041 22 Right arm: max GUP av-y 2.542 0.02 50 Right hip: max CTT acc 2.13 0.04 23 Right hip: mean CTT ke 2.486 0.019 51 Left hip: max CTT vel 2.103 0.043 24 Spine mid: max CTT pe 2.456 0.02 52 Head: max CTT pe 2.095 0.044 25 Right hip: mean CTT vel 2.442 0.02 53 Left elbow: median CTT vel 2.078 0.046 26 Hip: median CTT av-m 2.396 0.023 54 Spine mid: mean CTT pe 2.071 0.046 27 Shoulder: median CTT av-m 2.363 0.024 55 Right hip: median CTT acc 2.062 0.047 28 Spine shoulder: max CTT pe 2.356 0.025 Table 8: Full list of kinematic features which signicantly (p-value < 0.05) dierentiate between patients with no unexpected hospitalizations (UHV = 0) and patients with one or more unexpected hospitalizations (UHV = 1). Ranked by absolute value of two-sample t-test scores. (vel: velocity; acc: acceleration; pe: potential energy; ke: kinetic energy; sa: sagittal angle; av-x, av-y, av-z: angular velocity about x,y, or z axes). 2.10 Appendix D Two-sample t-tests are done to determine if mean values of kinematic features are dierent for patients with 15 hours or more of activity above LPA (HALPA = 0) from patients with 15 hours or less of activity above LPA (HALPA = 1), and the distribution of the resulting t-test scores and signicance values for the entire set of 526 features is shown in Appendix Figure 15. Full list of 28 signicant (p-value <0.05) t-test scores is shown in Appendix Table 9, and boxplots of these signicantly dierentiating kinematic features is shown in Appendix Figures 16 and 17. 34 2 QUANTIFIED KINEMATICS FROM IN CLINIC MOVEMENTS DIFFERENTIATES PATIENT RISK FOR CHEMOTHERAPY COMPLICATIONS Figure 12: Box plots of kinematic features which signicantly dierentiate between patients with no unex- pected hospitalizations (UHV = 0, gray) and patients with one or more unexpected hospitalizations (UHV = 1, red). Kinematic features 1-20. Feature t-test p-value Feature t-test p-value 1 Hip: mean GUP av-x -2.414 0.022 15 Spine base: mean CTT acc 2.039 0.05 2 Left leg: min GUP av-x -2.379 0.024 16 Right hip: mean CTT acc 1.987 0.055 3 Back: mean CTT sa -2.331 0.026 17 Hip: mean CTT av-y 1.96 0.065 4 Left arm: min GUP av-y -2.328 0.032 18 Right leg: median GUP av-x -1.96 0.06 5 Right leg: mean GUP av-z 2.224 0.033 19 Head: mean CTT acc 1.879 0.071 6 Left hip: mean CTT acc 2.221 0.033 20 Left arm: max GUP av-x 1.838 0.076 7 Back: median CTT sa -2.219 0.034 21 Hip: max CTT av-y 1.837 0.084 8 Hip: mean CTT av-x -2.193 0.035 22 Shoulder: mean CTT av-x -1.805 0.083 9 Left knee: median GUP ke 2.185 0.039 23 Right arm: median CTT av-x -1.775 0.086 10 Right leg: median CTT av-y -2.184 0.037 24 Left leg: median GUP av-m -1.775 0.086 11 Spine mid: mean CTT acc 2.181 0.037 25 Left knee: median GUP vel 1.763 0.091 12 Spine shoulder: mean CTT acc 2.136 0.042 26 Right leg: mean GUP av-x -1.742 0.091 13 Neck: mean CTT acc 2.125 0.043 27 Spine mid: max CTT acc 1.727 0.094 14 Shoulder: median CTT av-x -2.115 0.042 28 Left hip: mean CTT ke 1.702 0.098 Table 9: Full list of kinematic features which (feature 1-15: p-value < 0.05, feature 16-28: 0.05 < p-value < 0.10) dierentiate between patients with no unexpected hospitalizations (UHV = 0) and patients with one or more unexpected hospitalizations (UHV = 1). Ranked by absolute value of two-sample t-test scores. (vel: velocity; acc: acceleration; pe: potential energy; ke: kinetic energy; sa: sagittal angle; av-x, av-y, av-z: angular velocity about x,y, or z axes). 35 2 QUANTIFIED KINEMATICS FROM IN CLINIC MOVEMENTS DIFFERENTIATES PATIENT RISK FOR CHEMOTHERAPY COMPLICATIONS Figure 13: Box plots of kinematic features which signicantly dierentiate between patients with no unex- pected hospitalizations (UHV = 0, gray) and patients with one or more unexpected hospitalizations (UHV = 1, red). Kinematic features 1-20. Figure 14: Box plots of kinematic features which signicantly dierentiate between patients with no unex- pected hospitalizations (UHV = 0, gray) and patients with one or more unexpected hospitalizations (UHV = 1, red). Kinematic features 1-20. 36 2 QUANTIFIED KINEMATICS FROM IN CLINIC MOVEMENTS DIFFERENTIATES PATIENT RISK FOR CHEMOTHERAPY COMPLICATIONS Figure 15: Distribution of t-test scores and signicance values from two-sample t-tests for dierences in mean values of kinematic features between patients with 15 hours or more of activity above LPA (HALPA = 0) from patients with 15 hours or less of activity above LPA (HALPA = 1). Figure 16: Box plots of kinematic features which signicantly dierentiate between patients with 15 hours or more of activity above LPA (HALPA = 0, gray) from patients with 15 hours or less of activity above LPA (HALPA = 1, red). Kinematic features 1-20. 37 2 QUANTIFIED KINEMATICS FROM IN CLINIC MOVEMENTS DIFFERENTIATES PATIENT RISK FOR CHEMOTHERAPY COMPLICATIONS Figure 17: Box plots of kinematic features which signicantly dierentiate between patients with 15 hours or more of activity above LPA (HALPA = 0, gray) from patients with 15 hours or less of activity above LPA (HALPA = 1, red). Kinematic features 21-28 38 3 LOW-DIMENSIONAL CHARACTERIZATION OF HUMAN PERFORMANCE OF CANCER PATIENTS USING MOTION DATA 3 Low-dimensional characterization of human performance of can- cer patients using motion data 3.1 abstract Biomechanical characterization of human performance with respect to fatigue and tness is relevant in many settings, however is usually limited to either fully qualitative assessments or invasive methods which require a signicant experimental setup consisting of numerous sensors, force plates, and motion detectors. Qualitative assessments are dicult to standardize due to their intrinsic subjective nature, on the other hand, invasive methods provide reliable metrics but are not feasible for large scale applications. Presented here is a dynamical toolset for detecting performance groups using a non-invasive system based on the Microsoft Kinect motion capture sensor, and a case study of 37 cancer patients performing two clinically monitored tasks before and after therapy regimens. Dynamical features are extracted from the motion time series data and evaluated based on their ability to i) cluster patients into coherent tness groups using unsupervised learning algorithms and to ii) predict Eastern Cooperative Oncology Group performance status via supervised learning. The unsupervised patient clustering is comparable to clustering based on physician assigned Eastern Cooperative Oncology Group status in that they both have similar concordance with change in weight before and after therapy as well as unexpected hospitalizations throughout the study. The extracted dynamical features can predict physician, coordinator, and patient Eastern Cooperative Oncology Group status with an accuracy of approximately 80%. The non-invasive Microsoft Kinect sensor and the proposed dynamical toolset comprised of data preprocessing, feature extraction, dimensionality reduction, and machine learning oers a low-cost and general method for performance segregation and can complement existing qualitative clinical assessments. 3.2 Introduction In oncologic practice, clinical assessments of performance stratify patients into subgroups and inform deci- sions about the intensity and timing of therapy as well as cohort selection for clinical trials. The Karnofsky performance status (KPS) [96] and the ECOG/World Health Organization (WHO) performance status [147] are two equally prevalent measures of the impact of disease on a patient's physical ability to function. The Karnofsky score is an 11-tier measure ranging from 0 (dead) to 100 (healthy) whereas the ECOG score is a simplied 6-tier score summarizing physical ability, activity, and self-care: 0 (fully active), 1 (ambulatory), 2 (no work activities), 3 (partially conned to bed), 4 (totally conned to bed), 5 (deceased) [147] . Although these metrics have been employed for many decades due the practicality, standardization of 39 3 LOW-DIMENSIONAL CHARACTERIZATION OF HUMAN PERFORMANCE OF CANCER PATIENTS USING MOTION DATA patient stratication, and speed of assessment, prospective studies have revealed inter- and intra-observer variability [151], gender discrepancies [20], sources of subjectivity in physician assigned performance assess- ments [151], and a lack of standard conversion between the two dierent scales [25]. Nevertheless performance status provides clinical utility because it is able to dierentiate patient survival [158, 98]. Consequently, the existing protocol of assigning a performance status based on an inherently subjective assessment must be rened to achieve a more objective classication of a patient's physical function. In contrast to the qualitative and relatively practical nature of physician assessments in the clinic, labo- ratory based invasive methods have been developed to biomechanically quantify elements of human perfor- mance. Many of these eorts have conducted gait analysis using accelerometer, gyroscope and other types of wearable sensors and motion capture systems [190] to detect and dierentiate conditions in patients with os- teoarthritis [198], neuromuscular disorders [67], and cerebral palsy [46]. The shortcomings of more extensive assessments such as gait analysis include high cost, time required to perform tests, and general diculty in interpreting results [176]. The need for new technologies has been emphasized, particularly in the oncology setting [99], to bridge the gap between subjective prognostication using KPS or ECOG performance status and objective, yet cumbersome assessments of performance. To this end, we propose a non-invasive motion-capture based performance assessment system which can (i) characterize performance groups using solely kinematic data and (ii) be trained to predict ECOG scores by learning from various physicians in order to reduce bias and intra-observer variability. The Microsoft Kinect is used as the motion-capture device due to its low cost, and ability to extract kinematic information without the need of invasive sensors. We describe and test a data processing and analysis pipeline using a cohort of 40 cancer patients who perform two clinically supervised tasks before and after therapy at USC Norris Comprehensive Cancer Center, Los Angeles County+USC Medical Center, and MD Anderson Cancer Center. 3.3 Methods A set of dynamical analysis and machine learning tools is developed to gather kinematic information from recordings of patients performing tasks (Fig.18) with the goal of validating the experiment design by per- forming unsupervised classication of performance categories (Fig.18, step 4a), as well as supervised learning of physician assigned ECOG performance status (Fig.18, step 4b). Although we illustrate the use of the toolset by exploring its application to an oncology cohort, the following methods are general and may be used to characterize patient performance in other settings. 40 3 LOW-DIMENSIONAL CHARACTERIZATION OF HUMAN PERFORMANCE OF CANCER PATIENTS USING MOTION DATA • Translational kinematics • Rotational kinematics • Kinetic and potential energy • Sagittal angle 3. Dynamical Feature Extraction 3D displacement time series for 25 anatomical sites: task 1 and task 2 exercises 1. Microsoft Kinect recordings § Segmentation § Coordinate transformation 2. Preprocessing Cluster patients into performance groups in extracted feature space 4a. Unsupervised learning Predict clinical parameters (ECOG) using extracted features 4b. Supervised learning Figure 18: Schematic of dynamical and machine learning analysis pipeline. Raw skeletal displacement data (Step 1) from two clinically monitored tasks are preprocessed (Step 2) before feature extraction (Step 3) and two mutually exclusive machine learning analyses are performed. Unsupervised clustering (Step 4a) of patients in a low dimensional space reveals the degree to which performance groups can by stratied using solely motion data. Supervised classication (Step 4b) tests the ability of motion data to evaluate patients similar to physician ECOG performance status. 3.3.1 Experimental setup The Kinect depth sensor employs an infrared laser projector to detect a representative skeleton composed of 25 anatomical points (Fig.19A) and recordings are post-processed using Microsoft Kinect SDK (v2.0) to extract 3-dimensional displacement time series data for the 25 points. The Microsoft Kinect sensor is used in the clinical setting to record patients performing two tasks: (i) task-1 requires patients, who start from a sitting a position, to stand up and sit down on an adjacent elevated medical table (Fig.19B), (ii) task-2 requires patients to walk 8 ft towards the Kinect sensor, turn, and return to the original position (Fig.19C). Both tasks are performed by each patient before and after a therapy cycle, providing two samples for each task for a total of four time series per patient. In both tasks the Kinect camera is secured to a tripod on a table, and oriented so as to capture the entire gure. Details about the data collection, skeletal data extraction, and experimental setup are described by Nguyen and Hasnain in [135]. 3.3.2 Data preprocessing Due to irregularities in the positioning of the Kinect camera across dierent experiments, time series for task-2 is distorted such that a level plane (e.g. clinic oor) appears sloped in the recordings. To resolve this, an automated element rotation about the x-axis is performed. The angle of distortion ranges between 5-20 in the time series studied. The second preprocessing step involves manually segmenting the series to trim irrelevant data in the beginning and end of each task while the patient is stationary. 41 3 LOW-DIMENSIONAL CHARACTERIZATION OF HUMAN PERFORMANCE OF CANCER PATIENTS USING MOTION DATA t = 1 t = 2 t = 3 t = 4 A B t = 1 t = 2 t = 3 t = 4 t = 5 t = 6 Elbow (R) Wrist (R) Hand (R) Hand tip (R) Thumb (R) Hand tip (L) Thumb (L) Knee (L) Hand (L) Spine mid Hip (L) 23 21 7 4 9 5 2 1 6 22 8 12 24 25 17 13 14 15 16 20 19 18 3 Spine base Head Neck Spine Shoulder Shoulder (L) Elbow (L) Wrist (L) Shoulder (R) Hip (R) Knee (R) Ankle (L) Foot (L) Foot (R) Ankle (R) 10 11 C Figure 19: (A) Kinect recordings are post-processed using Microsoft Kinect SDK (v2.0) to extract displace- ment time series data for a set of 25 anatomical joints and sites. (B) Task-1 requires a patient to stand up from a chair and to sit at a medical table, a sample time series is shown. (C) Task-2 sample time series, patient starts from a standing position (t = 1) and walks to a mark 8 ft away (t = 6) and returns to original position (not shown). 3.3.3 Feature extraction The position vector, ~ r i (t) =hx i (t);y i (t);z i (t)i for an anatomical joint i is used to calculate its velocity magnitude, v i (t) = ~ _ r i (t) (47) and acceleration magnitude, a i (t) = ~ r i (t) (48) 42 3 LOW-DIMENSIONAL CHARACTERIZATION OF HUMAN PERFORMANCE OF CANCER PATIENTS USING MOTION DATA using the mean-value theorem. In the absence of distribution of mass information, specic kinetic energy, ke i (t) = 1 2 v 2 i (t) (49) and specic potential energy, pe i (t) =gz i =g(z i (t)z i (t = 1)) (50) quantities are used to describe the energy signature of each anatomical joint. The sagittal angle, s (t), is dened as the angle formed between the vector originating at the spine base and pointing in the direction of motion, and the vector connecting points 1 and 3 (Fig.19A) at each time point t. Time series corresponding to the hand (7, 8, 11, 12, 22, 23, 24, 25 in Fig.19A) and feet (15, 16, 19, 20 in Fig.19A) joints are relatively noisy therefore these time series are precluded from analysis, yielding 13 joints of interest in Fig.19. In summary, the list of extracted features for each task performed by a patient during a single visit includes: v i ;a i ;ke i ;pe i ; s for i = 1;:::; 13 anatomical joints resulting in 53 time series features per task, and K = 106 time series features per visit. 3.3.4 Time series similarity For a given patient and task the before- and after-therapy time series of each feature are compared using a Euclidean metric dynamic time warping (DTW), which assigns a distance of zero for completely identical series and larger distances for more dissimilar series. Although DTW provides a distance which does not satisfy the triangle inequality and therefore is not a metric, it has been used extensively for time series clustering and classication [160]. In the present work, DTW is used to describe changes in the extracted features where it is necessary to detect similar series despite noise and distortions which are intrinsic to the Kinect sensor and subsequent skeleton extraction. Consequently, the pair of before and after-therapy time series are assigned a DTW distance, d DTW (p;k), for each patient p and feature k: D(p;k) =d DTW (p;k) = DTW X p 1;k ;X p 2;k (51) where X p 1;k and X p 2;k are the time series of patient p's feature k for visits 1 and 2 respectively. Calculating the DTW distance between before- and after-therapy visits forP patients andK features results in a matrix D2R PK of DTW distances. This matrix captures the changes in the dynamical feature set before and after therapy. Feature distance vectors ~ d k 02R P fork 0 = 1;:::;K, whose entries ared DTW (p;k 0 ), are columns of D . 43 3 LOW-DIMENSIONAL CHARACTERIZATION OF HUMAN PERFORMANCE OF CANCER PATIENTS USING MOTION DATA 3.3.5 Dimensionality reduction In practical applications, and the clinical case study presented here the number of patients who completed both visits, P = 37 << K, therefore further dimensionality reduction is required before implementing learning algorithms based on the matrix D in order to avoid overtting and the curse of dimensionality [50]. Here, we use principal component analysis (PCA) to recast D into a lower dimension space while still maintaining most of the variance in the data. The scale of DTW distances are feature dependent, therefore column-wise standardization of D is performed prior to PCA. This process results in a reduced distance matrix D r 2R PN , comprised of N principal components, where NP . 3.3.6 Unsupervised clustering Performance groups are detected in reduced principal component space by employing the K-medoids al- gorithm, where number of clusters, k, corresponds to the number of performance groups detected. The K-medoids algorithm is chosen as the unsupervised algorithm for its insensitivity to outliers and fast imple- mentation for the small dataset studied. The overall quality of the resulting clusters is assessed by varying (i) the number of clusters in the K- medoids algorithm, and (ii) the number of principal components N in the low dimensional distance matrix D r and measuring the silhouette s of the resulting clusterings as well as the concordance between a given learned clustering and three clinical clusterings based on changes in weight before and after therapy, weight = weight After therapy weight Before therapy (52) and change in ECOG performance status before and after therapy, ECOG = ECOG After therapy ECOG Before therapy (53) and the number of unexpected hospital visits (UHV) where patients are grouped into 0,1,> 1 UHV over the course of the entire study. Changes in physician and coordinator assigned ECOG scores are used in this comparison. The similarity between the ECOG, weight, UHV and Kinect based unsupervised clusterings is mea- sured using the Rand index (RI): RI(C;C 0 ) = a +b n 2 (54) 44 3 LOW-DIMENSIONAL CHARACTERIZATION OF HUMAN PERFORMANCE OF CANCER PATIENTS USING MOTION DATA whereC andC 0 are two clusterings ofn objects,a is the number of objects in the same clusters inC andC 0 , andb is the number of objects in separate clusters inC andC 0 . RI = 0 when there is complete disagreement between two clusterings, and RI = 1 for identical clusterings. 3.3.7 Supervised classication Instead of comparing a given patient's before- and after-therapy time series samples directly to each other as described in section section 3.3.4, a reference time series from a prototypical sample can be used to compare to the before- and after-therapy series separately. This approach allows for the construction of a distance matrix from a single patient visit, and enables subsequent machine learning models of the corresponding physician, coordinator, and patient assigned ECOG performance status. Three healthy subjects perform tasks 1 and 2 to generate the prototypical samples which serve as the reference points for patient performance, and DTW distances between a patient's time series data and the prototypical samples oers a standardized measure of performance. The DTW distances to each of the L = 3 prototypical healthy samples are averaged for patient p's extracted feature k from visit v, X p v;k , d v DTW (p;k) = 1 L L X l=1 DTW X p v;k ;X l k (55) where X l k is the l'th prototypical sample's feature k, and visit v = 1 is the before-therapy sample and visit v = 2 is the after-therapy sample. Subsequently a standardized DTW distance matrix, D s 2 R 2PK , is formulated in which each patient contributes a total of two rows for the two visits. D s represents the task- 1 and task-2 average DTW distance between a patient's performance and the three prototypical samples. Along with a R 2P vector of a ground-truth target variable, D s can be used to develop supervised learning models. Here, we use the physician, coordinator, and patient assigned ECOG performance status as the target variables in three separate models. 3.4 Results & Discussion 3.4.1 Current clinical parameters 37 patients completed the before and after therapy visits, and the corresponding physician and coordinator ECOG scores were recorded for a total of 74 visits, while only 31 patients reported ECOG scores (Table 1). For the subset of 57 cases of patient reported ECOG scores, from either one or both visits, the mutual information (MI) association between physician ECOG and patient ECOG scores is MI = 0:0653, while the association between coordinator ECOG and patient ECOG is MI = 0:1661. Consequently, there is a 45 3 LOW-DIMENSIONAL CHARACTERIZATION OF HUMAN PERFORMANCE OF CANCER PATIENTS USING MOTION DATA ECOG score distribution 0 1 2 3 Total Physician 35 37 2 74 Coordinator 37 35 2 74 Patient 19 27 9 2 57 Table 10: ECOG scores assigned to patients by physicians, coordinators, and patients themselves for the before and after therapy visits. larger agreement between the coordinator and patient scores in the current experiment, however, more data needs to be collected to verify this trend. Fig.20A shows the relationship between physician ECOG (Eq.53) and change in weight over therapy, where change in weight follows a normal distribution (Fig.20B). However, due to the large spread of change in weight for ECOG = 0 group in Fig.20C, there is no clear relation between ECOG and weight, suggesting either the patients are unhindered even when undergoing large weight change (e.g. patients 5, 9, 11 Fig.20A) or that physicians consider other physical and expression cues more heavily while assigning ECOG scores. Binning patients by the percent change in weight into groups of those who lose weight after therapy (weight < 2%), maintain weight (2% weight 2%), and gain weight (weight > 2%) results in a weight based clustering of the patients, which has aRI = 0:509 (n = 37 patients) with the physician ECOG clustering. The UHV clusters are comprised of 16, 9, and 11 patients in the 0, 1, and > 1 UHV groups, and has a RI = 0:498 (n = 36) with the physician ECOG. Although the time points of the physician ECOG scores correspond to the before and after therapy visits, the UHV events are summed over the entire course of the study. The level of concordance between physician ECOG and existing clinical parameters serves as a benchmark for the unsupervised clustering in step 4a (Fig.18). 3.4.2 Validation of preprocessing and choice of DTW A hierarchical Euclidean distance based clustering of the the task-1 and task-2 feature distances ~ d k 0 (Fig.22A, feature names in Fig.22B) reveals that changes detected by DTW in most features are related mainly to other features of the same task, with the exception of a few features which correlate across tasks. These cross-task associated features include potential energies of the knee joints, and task 2 sagittal angle and left elbow potential energy to a lesser extent (purple, Fig.22A). The smaller clusters within the larger task 1 (blue, Fig.22A) and task 2 (red, Fig.22A) clusters validate the preprocessing and DTW calculations of across-therapy time series feature because anatomically related sites appear in coherent subclusters. For instance the potential energy of the left and right hips (joints 13 and 17, Fig.19A) and spine base (joint 1, Fig.19A) appear in the same subclusters for both tasks respectively. Furthermore, the velocities for the 46 3 LOW-DIMENSIONAL CHARACTERIZATION OF HUMAN PERFORMANCE OF CANCER PATIENTS USING MOTION DATA ● ● ● ● ●● ● ● ● ●● ● ●● ● ●● ●● ●●●●●●●● ● ● ● ● ● ● ● ● ● ● −0.15 −0.10 −0.05 0.00 0.05 0.10 0.15 % Change in Weight −8.3 −9.3 −3.9 −3.1 −3.3 −3.8 −2.6 −3.2 −2.4 −2.7 −2 −1.9 −2.7 −1.2 −0.8 −0.3 −0.2 −0.1 −0.1 0 0 0.1 0.1 0.2 0.3 0.4 0.4 0.7 1 1 1.1 1.3 2.1 2.2 5 5.7 7.3 5 9 1 37 24 55 38 48 54 19 53 10 14 31 18 3 43 15 42 7 40 50 45 12 23 27 41 17 26 34 8 39 49 35 6 11 36 ● ● ● ∆ECOG = −1 ∆ECOG = 0 ∆ECOG = 1 Patients 0 5 15 −0.15 0.00 0.15 Frequency % change in weight A B ● ● ● −0.10 −0.05 0.00 0.05 −1 0 1 ∆ECOG % change in weight C Figure 20: (A) Relation between change in physician assigned ECOG (eq.53) and percent change in weight before and after therapy. The absolute change in weight in kg is annotated above the circles, and patient ID is annotated below. (B) Histogram and normal distribution t to percent change in weight. (C) Boxplot of change in weight by ECOG groups. knee joints 14 and 18 are more closely related in the task-2 subcluster than the task-1 subcluster, which makes sense intuitively because the knees synchronously oscillate while walking in task-2, but perform unique functions in the task-1 twisting motion of turning towards and climbing the medical table. Therefore, the choice of using DTW, despite its dependence on the underlying scale of the time series being compared, is suitable for the subsequent unsupervised and supervised learning analyses. 3.4.3 Low dimension representation The distance matrix D consisting of 106 feature distance vectors ~ d k 0 (56 per task) shown in Fig.22A con- stitutes a high dimensional representation of changes in the biomechanical performance of tasks 1 and 2 before and after therapy because the number patient samples is much less than the number of features: P = 37<<K = 106. A low dimension representation is achieved by performing PCA on distance matrices D consisting of task- 1, task-2, and both task features to generate the reduced matrices D r for comparison (Fig.22C). In each 47 3 LOW-DIMENSIONAL CHARACTERIZATION OF HUMAN PERFORMANCE OF CANCER PATIENTS USING MOTION DATA 0 1 2 3 4 0 0.5 1 Velocity, (m/s) Spine base Task 1 d dtw = 9.5 0 1 2 3 4 0 2 4 6 Acceleration, (m/s 2 ) Spine base d dtw = 52 0 1 2 3 4 0 0.2 0.4 0.6 Kinetic Energy, J/kg Spine base d dtw = 3.24 0 1 2 3 4 0 2 4 6 Potential Energy, J/kg Spine base d dtw = 23.1 0 1 2 3 4 Time, s 40 60 80 Sagittal angle d dtw = 471 0 1 2 3 4 5 6 0 0.5 1 Task 2 d dtw = 15.8 0 1 2 3 4 5 6 0 2 4 6 d dtw = 78.2 0 1 2 3 4 5 6 0 0.1 0.2 0.3 0.4 d dtw = 6.14 0 1 2 3 4 5 6 -1 -0.5 0 0.5 d dtw = 18.9 0 1 2 3 4 5 6 Time, s 80 90 100 d dtw = 310 A F B G C H D I E J Figure 21: Example of extracted dynamical time series features (patient ID = 36). Before therapy (blue) and after therapy (red) feature time series are compared using DTW and the distances are annotated on the corresponding plots. (A-E) Task-1 features. (F-J) Task-2 features. case, a small number of reduced dimensions can explain a signicant portion of the variance in the high- dimensional space. Specically, reduced matrices for task-1 and task-2 each require 8 principal components to describe nearly 90% of variance in the corresponding high dimensional distance matrices. 13 principal components are required to capture a similar amount of variance when distance features from both tasks are included in D due to the relative lack of cross-task association between distance features shown in Fig.22A. Nevertheless, the subsequent results are based on D which contains features from both tasks so as to prevent loss of information, and the additional task features do not adversely aect the learning algorithms. 48 3 LOW-DIMENSIONAL CHARACTERIZATION OF HUMAN PERFORMANCE OF CANCER PATIENTS USING MOTION DATA Feature labels: (variable)_(point)_(task) Variable Point Task acc: acceleration 25 anatomical points T1: task 1, chair to table vel: velocity ke: kinetic enregy T2: task 2, get up and go pe: potential energy thetaSag: sagittal angle Cumulative variance explained T1 features T2 features T1 & T2 features Principal Component Fraction of Variance B C A 0 2 4 6 8 pe4_T1 pe3_T1 pe21_T1 pe2_T1 pe5_T1 pe9_T1 pe17_T1 pe1_T1 pe13_T1 pe6_T1 pe10_T1 acc3_T1 acc21_T1 ke4_T1 ke3_T1 ke21_T1 vel2_T1 vel4_T1 vel3_T1 vel21_T1 ke6_T1 vel6_T1 acc6_T1 ke10_T1 vel10_T1 acc10_T1 ke5_T1 ke9_T1 acc5_T1 acc9_T1 vel5_T1 vel9_T1 acc4_T1 ke14_T1 vel14_T1 acc14_T1 ke18_T1 vel18_T1 acc18_T1 acc13_T1 acc1_T1 acc17_T1 acc2_T1 vel13_T1 vel1_T1 vel17_T1 ke2_T1 ke13_T1 ke1_T1 ke17_T1 acc4_T2 ke4_T2 ke3_T2 ke21_T2 acc14_T2 vel14_T2 ke14_T2 vel18_T2 acc18_T2 ke18_T2 vel13_T2 vel1_T2 vel17_T2 ke13_T2 ke1_T2 ke17_T2 acc1_T2 acc13_T2 acc17_T2 acc5_T2 ke5_T2 vel4_T2 vel5_T2 vel9_T2 acc9_T2 ke9_T2 acc3_T2 acc21_T2 vel6_T2 acc6_T2 ke6_T2 vel10_T2 acc10_T2 ke10_T2 vel2_T2 vel3_T2 vel21_T2 acc2_T2 ke2_T2 pe14_T1 pe18_T1 pe14_T2 pe18_T2 pe6_T2 pe10_T2 pe2_T2 pe17_T2 pe1_T2 pe13_T2 pe9_T2 pe3_T2 pe21_T2 pe4_T2 pe5_T2 thetaSag_T1 thetaSag_T2 Features Task 1 Task 2 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0 5 10 15 20 25 30 35 0.0 0.2 0.4 0.6 0.8 1.0 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0 5 10 15 20 25 30 35 0.0 0.2 0.4 0.6 0.8 1.0 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0 5 10 15 20 25 30 35 0.0 0.2 0.4 0.6 0.8 1.0 ● ● ● Task 1 data Task 2 data Task 1 and Task 2 data Figure 22: (A) Hierarchical clustering dendrogram of task-1 and task-2 feature distance vectors ~ d k 0. All but a few features (highlighted in purple) cluster primarily by task. (B) Feature label nomenclature for the dendrogram in A. (C) Fraction of variance explained by principal components of (i) distance matrix D comprised of task-1 and task-2 (K = 106) features (black), (ii) distance matrix D comprised of task-1 (K = 53) features (blue), (iii) distance matrix D comprised of task-2 (K = 53) features (red). The feature clustering and dimensionality reduction analyses in Fig.22 illustrate the fact that both clinical tasks provide unique information and to use one test in the absence of the other would incur a loss of biomechanical information. 3.4.4 Detecting performance clusters Changes in the before- and after-therapy performance of tasks 1 and 2 are captured in D r , and the number of performance clusters detected in D r is a latent variable derived by selecting the number of clusters, k, in the K-medoids clustering algorithm which minimizes the distance to a representative cluster patient, or medoid, in the reduced low-dimensional space spanned by the N principal components. Therefore the choice of the number of performance clusters, from strictly the machine learning perspective, is dependent on the balance between N and the corresponding quality of the clustering which is measured by the silhouette s2 [1; 1]. Higher values of s indicate higher intra-cluster cohesion and lower inter-cluster cohesion for a given patient. 49 3 LOW-DIMENSIONAL CHARACTERIZATION OF HUMAN PERFORMANCE OF CANCER PATIENTS USING MOTION DATA This balance is shown in Fig.23, where greater number of dimensions in D r generally correspond to lower average s. For instance, no matter the choice of k, a N = 2 distance matrix D r cannot be clustered with a higher silhouette than a N = 1 distance matrix. Here we seek the number of performance clusters to be much less than the number of patients in order to validate the unsupervised clustering with the three cluster weight, ECOG, and UHV clusterings, however, in general, larger values of k result in a higher resolution performance clustering of the patient group which in turn can be compared to higher resolution clinical categorizations. From Fig.23A, we select theN =k = 3 clustering and visualize it on the rst two principal components in Fig.23B, and compare all of thek = 3 clusterings to the physician ECOG, Weight, and UHV clusterings in Fig.23C. The number of principal components used in D r to segregate patients in to k = 3 clusters is varied and the corresponding concordance, as measured by RI is shown on the left axis and the average silhouette on the right axis of Fig.23C. Although the average silhouette of the K-medoid clusterings decreases withN,RI(K-medoid, ECOG), RI(K-medoid, Weight), and RI(K-medoid, UHV) are maximized by N = 4, 17, and 19 principal components respectively. As the number of principal components in the low-dimension space is varied, the concordance between the K-medoids clusters and the physician ECOG (black), Weight (yellow), and the UHV (green) clusterings also changes (Fig.23C). The K-medoids clustering has a higher RI with the Weight clustering for most choices of N compared to the benchmark RI = 0:509 between physician ECOG and Weight (Fig.23C). As illustrated in Fig.23C, the ECOG, Weight, UHV, and the unsupervised clusterings all have a similar concordance, therefore, the distance matrix D of before- and after-therapy performance of task-1 and task-2 oers an objective platform to stratify patients, and, as shown by the RI metric, may achieve concordance with existing clinical measurements. The RI between the unsupervised K-medoid clusterings and the clinical clusterings including the physi- cian and coordinator assigned ECOG clusterings is shown in Table 2. The K-medoid clustering has the highest RI with the UHV clusters, and the second highest association with the Weight. The coordinator ECOG clusters are more associated with the Weight and UHV clusters than the physician ECOG clusters. Although these trends are particularly interesting,larger datasets are required to validate these RI values and to fully detect statistically signicant disparities. Nevertheless, Table 2 shows that the K-medoid unsupervised clusterings based on kinematic changes in task-1 and task-2 across therapy oers a unique but useful patient clustering. The added utility of the motion analysis based unsupervised clustering method shown here is the ability to achieve higher resolution clusterings of patients compared to physician or coordinator ECOG by increasing k in the K-medoids algorithm. Fig.23D shows a comparison between N = 3 K-medoids and Weight 50 3 LOW-DIMENSIONAL CHARACTERIZATION OF HUMAN PERFORMANCE OF CANCER PATIENTS USING MOTION DATA A B C D Average silhouette, s # of K-medoid clusters (fitness types) Average silhouette, s # of K-medoid clusters, bins in ΔWeight Rand Index # PCs used to cluster PC-2 PC-1 Rand Index ● ● ● ● ● ● ● ● ● 2 4 6 8 10 0.0 0.2 0.4 0.6 0.8 1.0 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● # PCAs = 1 # PCAs = 2 # PCAs = 3 # PCAs = 5 # PCAs = 10 # PCAs = 15 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● −10 −5 0 5 10 15 −10 −5 0 5 10 15 1 3 5 6 7 8 9 10 11 12 14 15 17 18 19 23 24 26 27 31 34 35 36 37 38 39 40 41 42 43 45 48 49 50 53 54 55 ● ● ● Cluster 1 Cluster 2 Cluster 3 5 10 15 20 0.3 0.4 0.5 0.6 0.7 0.8 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.0 0.2 0.4 0.6 0.8 1.0 ● K−medoids avg. silhouette width Rand Index( ∆Weight , ∆ECOG ) Rand Index( ∆ECOG , UHV) Rand Index( ∆Weight , UHV) Rand Index( K−medoids, ∆ECOG ) Rand Index( K−medoids, ∆Weight ) Rand Index( K−medoids, UHV) ● ● ● ● ● ● ● ● ● 2 4 6 8 10 0.3 0.4 0.5 0.6 0.7 0.8 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● Rand Index( ∆Weight , ∆ECOG ) Rand Index( K−medoids, ∆ECOG ) Rand Index( K−medoids, ∆Weight ) Figure 23: (A) Quality of K-medoids patient clusterings measured by average silhouette s compared to the number of clusters k for dierent numbers of principal components N in the reduced distance matrix D r . (B) N = 3 and k = 3 K-medoids unsupervised clustering of patients shown in the plane formed by the rst two principal components of D r . (C) The RI concordance between the k = 3 K-medoids clusterings and physician ECOG (black), Weight (yellow), and UHV (green) compared to benchmark RI associations among the ECOG, Weight, and UHV based clusterings (solid lines). Quality of the K-medoids clusterings (gray) is shown on the right axis. (D) Concordance between clusterings where the number of clusters in K-medoids and bins in Weight are increased. 51 3 LOW-DIMENSIONAL CHARACTERIZATION OF HUMAN PERFORMANCE OF CANCER PATIENTS USING MOTION DATA clusterings where k and the number of bins in the Weight clustering are increased which leads to an increasing RI between the two clusterings and a maximum RI = 0:737 is reached at k = 9 clusters. This further demonstrates the potential clinical utility of the pipeline of analytical tools developed. Rand Index between clusterings Weight 0.541 UHV 0.575 0.537 ECOG P 0.571 0.509 0.498 ECOG C 0.550 0.571 0.530 0.497 K-medoids Weight UHV ECOG P Table 11: Association between patient clusterings based on: 1) unsupervised K-medoids clustering (k = 3 clusters) of D r 2) Weight: change in weight before and after therapy 3) UHV: three clusters based on 0, 1, and >1 unexpected hospital visits over the course of the study 4) ECOG P : change in physician ECOG scores before and after therapy 5) ECOG C : change in coordinator ECOG scores before and after therapy 3.4.5 Learning physician ECOG performance status A natural application of the Kinect motion capture system is to use the extracted kinematic signature of a patient's task-1 and task-2 performance to learn the associated physician, coordinator, and patient assigned ECOG performance status, particularly if patients are examined by dierent physicians in order to reduce bias of the resulting model. To learn the ECOG scores in the cancer patient cohort, we use the standardized distance matrix D s (section 3.3.7) and perform dimensionality reduction via PCA with scaling and centering. As in the unsu- pervised model, dimensionality reduction is required due to the relatively small number of patients compared to the number of features. In the 74 physician and coordinator assessments of the P = 37 patients there are 2 ECOG = 2 cases. Since the majority of cases were ECOG = 0 or 1 scores, the two ECOG = 2 samples are excluded, and a binary classier is trained to predict a 0 or 1 ECOG status using the two visits from P = 37 patients (excluding two ECOG = 2 cases from two separate patients), which results in a distance matrix for the physician and coordinator ECOG classiers. For the patient classier a D s 2 R 57106 matrix is used and the ECOG = 1,2, and 3 categories are combined due to limited data. The association of the original features of D s and its principal components with the ECOG status is measured by the point biserial correlation coecient r pb , where positive r pb values indicate larger values of the feature are associated with ECOG = 1, and vice versa ( Fig.24). For the physician ECOG, age has a r pb = 0:186, as older patients were more likely to receive higher 52 3 LOW-DIMENSIONAL CHARACTERIZATION OF HUMAN PERFORMANCE OF CANCER PATIENTS USING MOTION DATA ●● ●● ●●● ● ● ●● ● ●● ●●●●●● ● ● ●●●●●●● ●●●●●●●●● ●●● ● ●●● ● ●● ● ●●●● ●●● ● ● ● ● ●●●● ●● ●● ● ● ●● ● ● ● ● ●●● ●● ● ● ●● ● ● ● ● ● ● ● ●●● ●● ●●●●● ● ● ●●● ● 0 20 40 60 80 100 120 −0.3 −0.2 −0.1 0.0 0.1 0.2 0.3 ● Kinematic features Principal components Predictors r pb r pb (ECOG, age) r pb (ECOG, bmi) Figure 24: Point biserial correlation between physician assigned ECOG and kinematic features of D s (black) and reduced space principal components of D s (blue). Larger r pb values indicate association with ECOG = 1, and vice versa. The correlations of age and BMI with ECOG (gray dashed lines) serve as a comparison for the r pb of the kinematic features and principal components. ECOG scores and in comparison, 10 principal components in the reduced D s have a higher absolute r pb . Principal components with the largestkr pb k are used to create and cross-validate classiers by leaving one patient's two samples out as the test set. 10 principal components are used in a mixture-of-experts model comprised of a SVM, logistic regression, and a KNN model to predict physician ECOG with an average cross-validated test set accuracy of 84.7% accuracy. The same accuracy was achieved using the same number of principal components to predict coordinator ECOG with a SVM model. Top 5 principal components were used to train a logistic regression model to predict patient ECOG = 0 or > 0 which performed at an accuracy of 80.7%. 3.5 Conclusions A non-invasive motion capture system is proposed to measure the kinematic signature of clinically supervised patient assessments of performance. A toolset to pre-process and extract dynamical features from skeletal displacement data is combined with complimentary unsupervised and supervised learning schemes. The unsupervised clusters reveal a new and valuable grouping of patients in a cancer cohort undergoing therapy. Additionally, the supervised learning model is able to predict physician, coordinator, and patient assigned 53 3 LOW-DIMENSIONAL CHARACTERIZATION OF HUMAN PERFORMANCE OF CANCER PATIENTS USING MOTION DATA ECOG scores using the kinematic signature with a high level of accuracy. In comparison to the low-resolution ECOG scale, the present toolset provides a pipeline to develop a high resolution performance grading. In general, the dynamical characterization toolset may be used for prognostication in various applications where biomechanical signatures are reasonably correlated with existing clinical measures. The present work is a proof of concept of a low-cost non-invasive method for objectively assessing human performance in the clinic. 54 4 MACHINE LEARNING MODELS FOR LONG-TERM OUTCOME PREDICTION IN BLADDER CANCER 4 Machine learning models for long-term outcome prediction in bladder cancer 4.1 Abstract Currently in patients with bladder cancer, various clinical evaluations (imaging, operative ndings at transurethral resection and radical cystectomy, pathology) are collectively used to determine disease status and prognosis, and recommend neoadjuvant, denitive and adjuvant treatments.We analyze the predictive power of these measurements in forecasting two key long-term outcomes following radical cystectomy, i.e., cancer recurrence and survival. Information theory and machine learning algorithms are employed to create predictive mod- els using a large prospective, continuously collected, temporally resolved, primary bladder cancer dataset comprised of 3503 patients (1971-2016). Patient recurrence and survival one, three, and ve years after cystectomy can be predicted with greater than 70% sensitivity and specicity. Such predictions may in- form patient monitoring schedules and post-cystectomy treatments. The machine learning models provide a benchmark for predicting oncologic outcomes in patients undergoing radical cystectomy and highlight opportunities for improving care using optimal preoperative and operative data collection. 4.2 Introduction Bladder cancer (BCa) is the 6th most common cancer in the U.S, with an estimated 79,030 new cases and 16,870 deaths in 2017 [175] and has a 5-year relative survival rate of 79% [30]. BCa staging is based on the TNM system (tumor, nodes, metastasis). In BCa, the \T" stage is dictated by how deep the tumor invades into the various layers of the bladder wall. Ta represents a noninvasive papillary tumor, while T1, T2, T3 and T4 stages represent more aggressive cancers invading the sub-epithelial tissue, muscle, peri- vesical fat and adjacent organs, respectively. Radical surgery is the primary treatment method for invasive cancer and may be augmented with other forms of therapy such as chemotherapy to treat more advanced and aggressive cancers [47]. Radical cystectomy, the recommended method for treating invasive BCa [183], is surgical removal of the bladder, regional lymph nodes and adjacent organs (prostate, uterus, etc.) which may contain cancer. Technical precision of this surgical operation can dictate long-term oncologic outcomes, for instance, post-cystectomy survival is higher when negative surgical margins are obtained and more than ten pelvic lymph nodes removed during radical cystectomy [87]. Conversely, cancer recurrence rates are higher with positive margins and removal of less than ten nodes.5 Furthermore, patients with organ-conned disease are less likely to relapse beyond 5 years, and unlikely beyond 10 years after cystectomy, even without adjuvant treatment [11]. 55 4 MACHINE LEARNING MODELS FOR LONG-TERM OUTCOME PREDICTION IN BLADDER CANCER These trends are derived from focused studies with disparate cohorts. The large size of the current dataset oers a chance to conrm and rene these relations. Beyond knowledge discovery, larger and electronically managed medical databases lend to predictive tool development. Consequently, machine learning techniques have been applied extensively on clinical, epidemiological, and molecular data to predict prognosis and outcome in various cancers. Cruz and Wishart [43], and more recently, Kourou et al. [108] oer a review of some of these studies which predict of susceptibility, recurrence and survival, where the merit of techniques and the quality of the data are quantied by prediction accuracies. In BCa, the most relevant existing study used a multi-institution dataset of 9000 patients, including 980 data points from the present dataset, to construct a nomogram for predicting 5-year recurrence which achieved a concordance index of 0.75 [21]. The present work focuses on: (i) using preoperative and operative BCa data to uncover patterns of long term outcomes and (ii) assessing the predictive power of BCa-specic factors in elucidating overall survival (OS) and recurrence. We employ the information theory concept of mutual information (MI) to uncover correlated parameters. We then stratify the set of predictors by correlation with recurrence and OS to quantify their relative importance. The prognostic power of these variables is assessed by developing a machine-learning classication pipeline to predict recurrence and survival after radical cystectomy, urinary diversion and extended lymphadenectomy, the standard-of-care for high-risk, muscle-invasive BCa. The models presented deliver a quantitative method for stratifying patients into higher resolution risk groups than is possible with current methods. 4.3 Methods 4.3.1 Data summary The original dataset (details in Table 14) comprised of 3503 patients is pruned to 3499 (mean age 67.8 years) patients by removing 4 cases with missing survival data. All patients underwent radical cystectomy at the USC Institute of Urology from 1971 to 2016. Statistical results based on this dataset up to 1997 were published by Stein et al. in 2001 [182] on a subset of 1054 urothelial carcinoma patients. Presently, this is one of the largest known single-institute datasets of BCa cystectomy patients in terms of sample size and the 45-year timespan over which the data were prospectively and continuously collected with institutional review board (IRB) approval. Consequently, the evolution of preoperative and operative assessments is also explored. In addition to information pertinent to BCa, comorbidity data were also collected to study the eect of preexisting diseases on progression of BCa. Remainder of the data is comprised of demographics, clinical diagnostic information prior to cystectomy, tumor markers prior to cystectomy, and pathologic and surgical data at time of cystectomy including adjuvant therapy treatment information. In the context of machine 56 4 MACHINE LEARNING MODELS FOR LONG-TERM OUTCOME PREDICTION IN BLADDER CANCER learning, these preoperative and operative measurements are called predictors, and the target variables are binary indicator variables for recurrence and survival after a given number of years post-cystectomy. 4.3.2 Statistics and information theory We perform survival analysis using the Kaplan-Meier estimator to dierentiate OS by various predictors. However, to develop an understanding of system-wide patterns between all the predictors, recurrence, and OS, a network approach is more suitable. Relevance, or correlation networks [29, 127, 90, 189] can be created using a similarity measure. Therefore, we create a mutual information (MI) network and subsequently a Euclidean distance based complete-linkage agglomerative hierarchical clustering of the most closely associated variables. Here, we use normalized MI which ranges from 0 to 1 for entirely unrelated to maximally related pairs of variables. Larger values of MI correspond to higher dependence between two variables. We normalize MI by the maximum entropy of the two variables being compared [121], and normalized MI will be abbreviated as MI. The set of all pairwise MI relations make up the adjacency matrix of the MI network, which is visualized as a clustered heat-map. The predictors are ranked by their association with the two long term outcomes, recurrence and OS, using the chi-squared test of independence which measures the association between two categorical variables. OS, age, and other continuous variables are discretized to perform the chi-squared test. The composite assessment identies higher delity variables and encapsulates the clinical relevance of the measurements. A composite predictor ranking, rank i = q ( 2 Rec;i ) 2 + ( 2 OS;i ) 2 (56) based on the chi-squared values for recurrence ( 2 Rec ), 2 Rec;i = 2 Rec;i 2 ;Rec (57) and the OS chi-squared values ( 2 OS ), 2 OS;i = 2 OS;i 2 ;OS (58) is used to identify predictor importance. The chi-squared values for both outcomes are normalized by their respective standard deviations (Eq. 1-3) to weigh the eect of both outcomes equally. 57 4 MACHINE LEARNING MODELS FOR LONG-TERM OUTCOME PREDICTION IN BLADDER CANCER 4.3.3 Machine learning approach The performance of multivariate predictive models is compared to univariate logistic regression models. To create the multivariate models, a series of base predictive models are employed, subsequently mixture- of-experts and stacking based ensemble models are trained using these base models. The base models consist of: support vector machines (SVM), bagged SVM, K-nearest neighbor (KNN), adaptive boosted trees (AdaBoost), random forest (RF), and gradient boosted trees (GBT). The mixture-of-experts models are based on hard-voting among the base models, whereas the stacking ensemble models perform dimensionality reduction of the base model predictions before employing a second logistic regression or support vector machine (SVM) model. For each prediction task, a dierent triplet of models forms the nal meta-classier, which is constructed by combining one each of the best base, mixture-of-experts, and stacking classiers using hard-voting. This method of combining various models is achieves the highest performance metrics. Patients who leave the study before the target year of the survival models are removed from the dataset, resulting in n = 3201, 3066, and 2780 patients for the 1-, 3-, and 5-year survival datasets respectively. However, only patients who have no recurrences and leave the study before the target year of the recurrence models are removed from the corresponding models' datasets; resulting in n = 3071, 2955, and 2695 patients in the 1-, 3-, and 5-year recurrence datasets respectively. To avoid class imbalance while training, the subset of patients who recur are randomly oversampled to yield an equal count of recurring and non-recurring patients in the training sets. Similarly, for the survival classiers, the fraction of surviving and non-surviving patients is balanced by random oversampling. The procedure for feature selection consists of two steps: removal of irrelevant predictors and removal of redundant predictors. To remove redundant predictors, the hierarchical clustering of the 73 predictors is used to dene 60 predictor clusters, and the predictor with the highest MI with the target variable is selected from each cluster. Subsequently, to remove irrelevant predictors from the dataset, predictors with low MI with the target variables are excluded from the dataset (MI<0.006 for predicting recurrence, MI<0.003 for predicting survival). These two feature selection steps yield a set of 52, 54, and 51 predictors for the 1-, 3-, and 5- year recurrence models respectively, and 42, 45, and 45 predictors for predicting 1-, 3-, and 5-year survival respectively. Final performance scores are found using nested cross-validation with ten outer folds and ve inner folds in which the SVM, RF, GBT, and AdaBoost hyper-parameters are tuned. The Scikit-Learn platform [149] is used to implement the models. 58 4 MACHINE LEARNING MODELS FOR LONG-TERM OUTCOME PREDICTION IN BLADDER CANCER 4.4 Results 4.4.1 Survival statistics 3503 patients' OS is outlined in Fig.25 and patients with unknown recurrence status are excluded from analysis pertaining to recurrence in the rest of the study. There is an exponential decay in survival by age groups in the ve-year period post-cystectomy, which suggests the burden of BCa diminishes signicantly within ve years for patients undergoing radical cystectomy (Fig.26). Consequently, our prediction tasks focus on 1-, 3-, and 5-year survival and recurrence. Figure 25: Relationship between disease status and survival. (A) Kaplan-Meier survival for patients who never recurred (green, n = 2112, 60.3%) have the highest mean OS 7.93 years (SD = 7.69), patients who were initially disease free but then had recurrence (red, n = 969, 27.7%) have mean OS 3.00 years (SD = 4.14), patients who were never free (orange, n = 259, 7.4%) of the disease have the lowest mean OS 0.81 years (SD = 1.03), and patients whose post-cystectomy progression is unknown (teal, n = 163, 4.7%). (B) Boxplot of OS by disease progression of patients. (C) Histogram of OS in cohort show a dramatic reduction in survival rates up to ve years post-cystectomy. 59 4 MACHINE LEARNING MODELS FOR LONG-TERM OUTCOME PREDICTION IN BLADDER CANCER Figure 26: Relationship between age and survival. patients are stratied into ve age groups, according to the Surveillance, Epidemiology, and End Results (SEER) program age standards for survival [4]. (A) Boxplot of OS for patients by age group. Age < 55 (n = 449 patients, 12.8%), 55 age < 65 (n = 829 patients, 23.7%), 65 age < 75 (n = 1308 patients, 37.3%), 75 age < 85 (n = 771 patients, 22.0%), 85 age (n = 146 patients, 4.2%). (B) Kaplan-Meier survival by disease progression shows disease burden greatly diminishing ve years after surgery. Comparing survival for clinical staging prior to surgery (Fig.27) and pathologic staging (pT staging: TNM 5th edition staging) at time of cystectomy (Fig.28) reveals the higher delity of pathologic staging. Clinical staging fails to separate staging as clearly as pathologic staging, for example clinical staging does not separate T2b and T3a patients as clearly as pathologic staging P2b and P3a patients. During the study (1971-2016) there is a 24% (811/3417) agreement between the two staging measures, with clinical staging over-estimating pathologic stage by 25% (865/3417 patients). Since 2010, there is a surge in stage over-estimation, with a corresponding decrease in under-estimation; however, overall concordance between the two staging measures has remained relatively constant over the decades studied (Fig.31). A graver consequence of the inferior resolution of clinical staging is that it underestimated pathologic stage in 51% of patients (1741/3417). Patients with organ-conned (OC), extra-vesical (EV) and node-positive (N+) BCa had 5-year survival rates of 0.750 (95% CI [0.729, 0.772]), 0.413 (95% CI [0.377, 0.452]), and 0.213 (95% CI [0.186, 0.243]), 60 4 MACHINE LEARNING MODELS FOR LONG-TERM OUTCOME PREDICTION IN BLADDER CANCER Figure 27: Relationship between survival and clinical T stage assigned prior to cystectomy. Based on a combination of imaging and transurethral resection. (A) Kaplan-Meier survival shows some degradation with tumor staging, but many stages overlap signicantly. T3a and T2b patients in the (B) boxplot of OS by T stage are not dierentiated. P-values in Table 19. respectively (p<0.001) (Fig.32). Patients with and without lympho-vascular invasion had 5-year survival rate of 0.299 (95% CI [0.271, 0.330]) and 0.637 (95% CI [0.617, 0.658]), respectively (p<0.001) (Fig.33). Patients with a negative soft tissue surgical margin had 5-year survival of 0.572 (95% CI [0.553, 0.591]) compared to 0.358 (95% CI [0.285, 0.448]) for patients with positive ureteral/urethral margins and 0.063 (95% CI [0.034, 0.117]) for patients with soft-tissue margins (p < 0.001) (Fig.34). Patients with carcinoma in situ had no discernable dierence in OS as well as 5-year probability of survival compared to other patients. 61 4 MACHINE LEARNING MODELS FOR LONG-TERM OUTCOME PREDICTION IN BLADDER CANCER Figure 28: Relationship between survival and pT stage (TNM 5th edition) assigned at time of cystectomy. (A) Kaplan-Meier survival shows a steadier degradation with tumor staging than clinical staging. (B) Boxplot of OS by pT stage at time of cystectomy clearly dierentiates P3a and P3b patients from stage two patients. P-values in Table 20. 4.4.2 Correlations among predictors Fig.29 shows the MI network adjacency matrix as a heat-map and the hierarchical clustering of variables in the dataset, where a four-cluster division is highlighted. Mean MI within the four clusters (purple, green, blue, red) is 0.629, 0.233, 0.0237, and 0.653, respectively, therefore the blue cluster is comprised of signicantly less correlated variables than the other clusters. The purple cluster consists of pathologic staging variables which form the most correlated set of BCa-specic predictors (1-6 in Fig.29). BCa variables (purple, green and blue clusters) have an average MI of 0.0337 with each other, and an average MI of 0.00575 with the set of comorbidity variables (red cluster), marking the rst division in the clustering. This suggests that patients' preexisting comorbidities are not strongly related to BCa variables. In contrast to comorbidity factors which are strongly associated with each other, the mean MI between 62 4 MACHINE LEARNING MODELS FOR LONG-TERM OUTCOME PREDICTION IN BLADDER CANCER Figure 29: MI between the set of predictors as well as recurrence and OS. Predictors are clustered into four groups using a hierarchical clustering algorithm to discover associated predictor groups. BCa predictors and long-term outcomes are contained in the purple (anatomic staging), green (histologic staging), and blue (treatment and OS and recurrence) clusters, whereas the comorbidity factors comprise a solitary (red) cluster. Correlations within the purple and red clusters are high, but correlations between the comorbidity cluster and other clusters is low. predictors and the binary recurrence target variable (56 in Fig.29) is 0.01028, and 0.01229 between predictors and discretized OS (57 in Fig.29), highlighting the diculty in predicting long term BCa outcomes by using only preoperative and operative data. The association between discretized OS and recurrence, MI = 0.0482, is much higher than the mean MI between the predictors and either long term outcome. Both long-term 63 4 MACHINE LEARNING MODELS FOR LONG-TERM OUTCOME PREDICTION IN BLADDER CANCER outcomes are in the blue cluster (Fig.29) where there is a lack of strong associations among the variables aside from three sets of variables related to neoadjuvant chemotherapy (20-22 in Fig.29), adjuvant chemotherapy (26-28 in Fig.29), and radiation (23-25 in Fig.29) which are highly related because they are clinical re- classications or sub-groupings of each other within the original variable's domain. Aside from TNM 7th edition staging (7 in Fig.29) which is nearly identical to pT staging (TNM 5th edition, 8 in Fig.29), the highest associations with pT staging are pathologic stage subgroup (2 in Fig.29, MI = 0.255) and presence of pathologic carcinoma in situ (12 in Fig.29, MI = 0.1825). In contrast, clinical T stage (44 in Fig.29) does not have equally high MI with any of the other predictors except for the regrouped clinical staging variable (41 in Fig.29). The MI based heat-map and clustering in Fig.29 provides a system-wide view of the entire medical database for BCa patients and correlations between the predictors can be used to assess the quality of clinical measurement techniques. 4.4.3 Correlations with long-term outcomes The chi-squared test of independence for recurrence (vertical axis) and OS (horizontal axis) in Fig.30 and Table 22 shows the relative importance of each predictor. The variance in chi-squared values is computed by singular value decomposition [5] and is shown by an ellipse whose axes are the standard deviations (SD) along the rst (SD = 195.4) and second (SD = 76.8) principal components (green lines) in Fig.30A. Some predictors intrinsically contain more information about survival than recurrence, and vice versa, because the variances along the principal components are of comparable size. For instance, urinary diversion (rank 10) and age (rank 12) are strongly correlated with OS but not recurrence, nevertheless, they rank high due to large eect on survival. Pathologic stage subgroup (rank 3), which indicates whether patients have OC, EV, or N+ disease at time of cystectomy, has the largest 2 Rec , and its 2 OS is the fourth highest. Pathologic stage (rank 1) is even more strongly associated with OS and comparing it to clinical staging (rank 17) reinforces the superiority of pathologic staging in dierentiating patients by outcome as observed in the Kaplan-Meier curves (Fig.27A and Fig.28A). The number of positive lymph nodes removed at time of cystectomy (rank 7) is signicantly more correlated with recurrence than the total number of lymph nodes removed (rank 16), however, further data may be required for clarication due to the high p-value corresponding to 2 Rec for total number of removed lymph nodes. The predictors ranked 4-8 in Fig.30 have similar correlations with either outcome, and these predictors are part of the highly associated (purple) cluster in Fig.29. The Cramer's V eect size of pathologic positive lymph nodes (rank 8) and lymphovascular invasion (rank 14) are particularly high for both outcomes (Fig.36). 2757 patients who did not receive adjuvant chemotherapy had 5-year survival rate of 0.563 (95% CI [0.543, 64 4 MACHINE LEARNING MODELS FOR LONG-TERM OUTCOME PREDICTION IN BLADDER CANCER Figure 30: Association between predictors and the two long term outcomes, recurrence and OS as measured by the chi-squared test of independence. The predictors which have a statistical signicance of p < 0.05 for both outcomes (43 predictors) are colored in red. Predictors of signicance (p < 0.05) with recurrence (5 predictors) or OS (24 predictors) are shown in blue and yellow, respectively. The solid (green) lines are the two principal axes computed using singular value decomposition and the singular values are used to dene the semi-major and semi-minor axes of the solid (green) ellipse. The horizontal and vertical dashed (gray) lines indicate the mean chi-squared for recurrence and OS respectively. 0.583]), and 633 patients who did receive adjuvant chemotherapy had a 5-year survival rate of 0.450 (95% CI [0.411, 0.493]) (Fig.35). Since the prescription of adjuvant chemotherapy is limited to a homogeneous set of patients who are node positive at time of cystectomy, the association of adjuvant chemotherapy with recurrence and OS may be articially high in this dataset. Although the predictors with below average chi-squared values are less important than the others, they may still dierentiate patients who are similar in the higher ranked variables. The lowest correlates of OS and recurrence are gender ( 2 OS = 9.53) and intravesical treatment ( 2 Rec = 0.1046) respectively. Overall the predictors have a weaker association with recurrence than OS. Since the MI between BCa specic predictors is small, even the predictors with small chi-squared values add new information about a patient. However, this new information may not necessarily inform OS and recurrence. 65 4 MACHINE LEARNING MODELS FOR LONG-TERM OUTCOME PREDICTION IN BLADDER CANCER Recurrence Overall survival Rank Predictor 2 p-value 2 p-value 1 pT stage TNM 7th Edition 344.3 0.000 713.0 0.000 2 pT stage TNM 5th Edition 340.3 0.000 717.0 0.000 3 pathologic stage subgroup (OC, EV, N+) 423.1 0.000 558.0 0.000 4 pM stage TNM 7th Edition 344.2 0.000 450.0 0.000 5 pN stage TNM 7th Edition 350.0 0.000 425.0 0.000 6 pN stage TNM 5th Edition 347.9 0.000 414.0 0.000 7 # of positive lymph nodes 350.6 0.000 392.0 0.000 8 pathologic positive lymph nodes 322.0 0.000 380.0 0.000 9 type of urinary diversion constructed 4.2 0.041 598.0 0.000 10 current urinary diversion performed 3.4 0.064 530.0 0.000 11 pathologic # of tumors 27.8 0.000 490.0 0.000 12 age at time of cystectomy (discretized) 11.7 0.230 467.0 0.000 13 type of surgical procedure 35.2 0.000 432.0 0.000 14 pathologic lymphovascular invasion 211.9 0.000 246.0 0.000 15 urinary diversion subgoups 17.9 0.000 380.0 0.000 16 # lymph nodes removed 7.2 0.407 374.0 0.000 17 clinical T stage (preoperative) 102.2 0.000 301.0 0.000 18 adjuvant chemotherapy subgroups 114.4 0.000 284.0 0.000 19 type of adjuvant chemotherapy 99.6 0.000 270.0 0.000 20 neoadjuvant chemotherapy subgroups 23.7 0.000 290.0 0.000 21 positive soft tissue surgical margin 26.8 0.000 251.0 0.000 22 pM stage TNM 5th Edition 6.8 0.034 234.0 0.000 23 intracorporeal diversion performed 15.6 0.000 225.0 0.000 24 # of adjuvant chemo cycles 70.8 0.000 187.0 0.000 25 # of neoadjuvant chemo cycles 21.7 0.003 213.0 0.000 26 clincial staging subgroup (preoperative) 57.3 0.000 191.0 0.000 27 robotic surgical procedure 14.9 0.000 207.0 0.000 28 pathologic predominant cell type 97.8 0.000 91.2 0.001 29 clinical N Stage (preoperative) 50.7 0.000 154.0 0.000 30 type of neoadjuvant chemotherapy 17.3 0.000 171.0 0.000 Table 12: The ranked list of predictors by importance. Association between predictors and the two long term outcomes, recurrence and OS as measured by the chi-squared test of independence. Predictors are ranked by Eq.56. 4.4.4 Predicting post-cystectomy recurrence We evaluate the performance of machine learning models to predict post-cystectomy disease recurrence using preoperative and operative data as well as the type and number of adjuvant therapy cycles administered (Table 13). Both univariate (logistic regression) and more complex multivariate models (meta-classiers in Table 13) are used to predict 1-, 3-, and 5-year recurrence. Pathologic stage subgroup (rank 3 in Fig.30) and pT stage (rank 2 in Fig.30) are used to create the univariate models and these have lower precision and F1 scores than the meta-classiers. Furthermore, the single predictor models tend to suer from imbalance between sensitivity and specicity. In contrast, all recurrence meta-classiers have sensitivities and speci- cities over 70%. F1 scores improve with year perhaps due to a more even number of positive and negative 66 4 MACHINE LEARNING MODELS FOR LONG-TERM OUTCOME PREDICTION IN BLADDER CANCER recurrence cases in the corresponding datasets. Post-cystectomy Model Year Test set performance metrics outcome Sensitivity Specicity Precision F1 Recurrence Meta-classier 1 0.739 0.714 0.388 0.508 pT stage TNM 5th Edition 1 0.761 0.653 0.349 0.478 pathologic stage subgroup 1 0.826 0.593 0.332 0.473 Meta-classier 3 0.720 0.708 0.535 0.613 pathologic stage subgroup 3 0.774 0.631 0.493 0.602 pT stage TNM 5th Edition 3 0.670 0.694 0.503 0.574 Meta-classier 5 0.700 0.702 0.588 0.636 pathologic stage subgroup 5 0.744 0.611 0.537 0.623 pT stage TNM 5th Edition 5 0.619 0.698 0.553 0.583 Survival Meta-classier 1 0.741 0.770 0.473 0.577 pT stage TNM 5th Edition 1 0.739 0.672 0.387 0.506 pathologic stage subgroup 1 0.805 0.602 0.362 0.499 Meta-classier 3 0.722 0.788 0.700 0.711 pathologic stage subgroup 3 0.762 0.691 0.628 0.688 pT stage TNM 5th Edition 3 0.696 0.739 0.646 0.670 Meta-classier 5 0.741 0.768 0.780 0.760 pathologic stage subgroup 5 0.730 0.717 0.742 0.735 pT stage TNM 5th Edition 5 0.664 0.766 0.759 0.708 Table 13: Performance of machine learning models for predicting recurrence and survival. Single predictor (pT stage and pathologic stage subgroup classiers) and multiple predictor (Meta-classier) models for predicting 1-, 3-, 5-year recurrence and survival after cystectomy. The performance of all models for a given year is ranked per F1 scores (2*precision*recall/(precision+recall)) as well as mean sensitivity, specicity, and precision on test sets from a 10-fold cross validation. 4.4.5 Predicting post-cystectomy survival Like the recurrence predictions, the meta-classiers outperform the univariate models in predicting survival (Table 13), however the disparity is greater as the meta-classiers have considerably higher performance metrics for all year predictions. Additionally, and unlike the recurrence models, the survival meta-classiers have comparable precision and probability of detection, except for the 1-year survival models. The combi- nation of high precision and sensitivity leads to signicantly higher F1 scores for the 3- and 5-year survival meta-classiers. 4.5 Discussion Although recurrence and OS are highly associated, preoperative and operative measurements generally do not relate equally to recurrence and OS, and the two outcomes should be assessed separately. The pri- mary predictors of long-term outcomes are pathologic stage and its subgrouping into localized or metastatic conditions. However, the machine learning pipeline developed here can leverage less powerful predictors to improve accuracy of long term predictions. The benet of having low MI between variables means that each 67 4 MACHINE LEARNING MODELS FOR LONG-TERM OUTCOME PREDICTION IN BLADDER CANCER variable oers unique information however the drawback is that each patient needs to be described by many variables and thus the prediction task becomes a higher dimensional problem, for which lack of data can greatly limit predictions of long-term outcomes. Clinical T stage oers a lower resolution signal than the true pathologic T stage, and this loss of information can be particularly impactful in cases where there is an underestimation of disease severity prior to surgery [185]. The sensitivity and specicity of all the survival meta-classiers, and the 1-year recurrence meta-classiers are considerably higher than 70%. Recurrence meta-classiers are less accurate, perhaps, because of unde- tected metastatic disease at the time of cystectomy. 1-year meta-classier predictions for both outcomes oer a better combination of sensitivity and specicity than the 3- and 5-year meta-classiers. However, the later year models may also be used in the clinical setting to dierentiate lower- and higher-risk patients due to higher precision scores. In current clinical practice, post-radical cystectomy prognostication in the individual patient is informed by the best-evidence found in the literature [182, 155] which re ect probable outcomes in cohorts not the individual, or the prognostic nomogram which only calculates a 5-year outcome [21]. To improve upon this, we employ machine-learning algorithms to construct novel, patient prognostication models for survival and recurrence. Presently the international bladder cancer nomogram has proven to be a validation of multivariate approaches in predicting long term outcomes in the clinical setting[21, 200], and the models developed here oer higher resolution predictions which can assist post-cystectomy treatment and screening decisions. Despite several machine learning research eorts in predicting outcomes of cancer patients there is a low penetration of such models in clinical practice [108]. There are two specic hurdles before current models can be deployed in a clinical setting, rst, because the performance reported here re ects the quality of data collected at one center and to ensure the generalizability of the models, data from other institutions should also be studied. Combining additional datasets, such as the international BCa dataset [21], may also improve the per- formance of the algorithms due to general sparsity and low frequency of certain combinations of predictors in the present data. Secondly, the recurrence and survival models use a total of 42-54 predictors, therefore the standardized collection of these parameters must be ensured before the machine learning models can be deployed successfully in a clinical setting. The accuracy of predicting cancer recurrence, which may depend on several evolutionary steps beyond cystectomy, can undoubtedly be improved by combining genomic and molecular data, and this would be fruitful direction to pursue. The quality of the dataset, coupled with machine learning models in the present work, oers a benchmark of the value of current preoperative and operative patient assessment standards 68 4 MACHINE LEARNING MODELS FOR LONG-TERM OUTCOME PREDICTION IN BLADDER CANCER with respect to forecasting long term outcomes during the most vulnerable 5-year timespan in BCa treat- ment post-cystectomy. Furthermore, due to the absence of widely recognized biomarkers for BCa [32], clinicopathological-based predictions of clinical outcomes as shown here set the standard for long-term per- sonalized predictions in BCa. If deployed correctly, machine learning models can transform preoperative and operative data into accurate predictions and mitigate post-cystectomy burden of BCa. 4.6 Supporting information All P0 Pa PIS P1 P2a P2b P3a P3b P4a P4b (3503) (334) (94) (404) (481) (300) (389) (357) (536) (451) (141) Age (yr) Median 68 66 67 67 66 67 67 70 69 70 68 Range 23-95 25-92 34-87 27-94 33-95 31-89 30-92 38-91 31-92 38-94 23-95 Gender (n) Female 757 66 16 62 91 52 99 97 171 70 30 Male 2746 268 78 342 390 248 290 260 365 381 111 Histology (n) Adeno 40 3 1 0 4 1 6 7 7 7 4 Neuro 42 7 0 0 0 3 5 5 12 10 0 Sarcoma 19 2 0 0 1 5 3 3 1 3 1 Squamous 90 3 0 2 5 5 16 7 27 13 11 Urothelial 3301 315 92 402 470 286 358 335 486 417 125 Other 11 4 1 0 1 0 1 0 3 1 0 pN stage (n) Nx 151 14 5 11 16 8 8 5 16 30 32 N0 2477 309 88 381 435 248 293 206 284 181 49 N1 220 3 0 5 14 18 29 41 59 37 11 N2 639 7 1 7 16 25 59 101 176 197 46 N3 16 1 0 0 0 1 0 4 1 6 3 Pathologic subgroup (n) OC 1812 323 93 391 450 254 299 0 0 0 0 EV 760 0 0 0 0 0 0 208 293 201 58 N+ 931 11 1 13 31 46 90 149 243 250 83 Radiation (n) Neoadjuvant 325 35 7 34 50 26 33 20 35 54 28 Adjuvant 54 0 0 0 2 2 1 5 14 8 20 Chemotherapy (n) Neoadjuvant 519 93 7 59 32 32 39 52 62 89 52 Adjuvant 633 8 3 12 26 37 80 106 178 142 36 Smoker (n) Current 674 52 19 61 88 71 86 61 127 74 35 Never 807 84 18 82 117 57 92 94 132 94 34 Previous 1891 188 55 249 255 157 195 189 264 267 63 Metastasis (n) Total Patients 1061 36 16 67 96 68 100 154 242 231 48 Bone 332 6 1 14 27 18 26 54 91 80 14 Pelvis 308 8 1 15 13 14 33 48 83 70 23 Lung 288 11 4 19 29 24 29 43 70 49 10 Liver 255 12 2 20 24 8 19 35 68 56 10 Retro 208 6 0 21 19 12 14 42 44 43 7 LN (reg) 134 3 2 9 16 7 9 23 27 30 7 LN (dist) 95 4 0 5 5 12 9 19 14 22 5 Peritoneum 93 7 0 6 2 5 8 10 18 28 9 Urethra 88 5 7 15 20 11 5 2 8 14 1 Brain 65 1 0 4 6 2 10 11 14 14 3 UT 61 3 5 11 12 11 2 2 8 7 0 Adrenal 34 0 0 3 2 2 2 6 10 6 3 Other 174 7 2 11 15 14 13 29 37 36 10 Table 14: Dataset description. Patient and tumor characteristics by pathologic T stage. S1 Appendix. Machine learning process for nding best model. The feature selection process for generating datasets for machine learning is described here. MI is calculated between p = 73 predictors and used to cluster variables into K = 60 clusters. MI is also calculated between p = 73 predictors and RecX (x = 1, 3, 5 years) and SurvX (x = 1, 3, 5 years). Correlated predictors are removed by selecting a single predictor from K = 60 clusters whose MI with the outcome is the highest. Irrelevant predictors are removed by pruning all predictors whose MI with the outcome is below a threshold (MI = 0.006 for RecX, MI = 0.003 for SurvX). These thresholds are selected heuristically, with some trial and error. The choice of number of 69 4 MACHINE LEARNING MODELS FOR LONG-TERM OUTCOME PREDICTION IN BLADDER CANCER clusters is another decision which can be explored with further trial and error computations, however this task was found to be too demanding. Therefore, it may be possible to achieve similar performance with more strict removal of correlated predictors by picking a smaller number of correlated clusters, K to prune from. All p = 73 predictors are categorical except for age, therefore discretized (10 bins with equal frequency) and continuous versions of age were tried separately. Class-imbalance was overcome by random oversampling to achieve a ratio of minority to majority class k = 0.8, 0.9, or 1.0. Number of patients with recurrences (1) and no recurrences (0) in the 1-, 3-, and 5-year recurrence datasets are: (0: 2469, 1: 602) for 1-year, (0: 2020, 1: 935) for 3-year, (0: 1683, 1: 1012) for 5-year. Number of surviving (0) and not surviving (1) patients in the 1-, 3-, and 5-year survival datasets are: (0: 2503, 1: 698) for 1-year, (0: 1821, 1: 1245) for 3-year, (0: 1318, 1: 1462) for 5-year. Thirteen base models include: SVM (polynomial, sigmoid, radial basis function kernels), KNN, AdaBoost, gradient boosted trees, random forest, pathologic stage based logistic regression, pathologic stage subgroup based logistic regression, and bagged versions of SVM and KNN. The main challenge in this study was to overcome the skewed performance of base models towards high sensitivity or high specicity. Consequently, a series of ensemble learning techniques are employed to take advantage of the base models. Nine dierent majority voting based mixture-of-expert (MOE) ensemble models, and 13 dierent logistic regression and SVM based stacking ensemble models are created using the results of the base models. Finally, the meta- classiers are created by using a triplet of: a base model, a MOE ensemble model, and a stacking ensemble model. There are 1287 such triplets, and we nd the performance of these models using nested k-fold cross validation, and the best models from this list are reported. These ensemble methods prove successful in overcoming the skewed base model performance. Due to the medial nature of the dataset, the predictors are categorical and the addition of numerical predictors directly from imaging and biopsy procedures may improve performance. S2 Appendix. Clinical vs Pathologic staging trend since 1970s. Clinical staging (staging based on a combination of imaging and transurethral resection) and pathological staging (staging done at time of cystectomy) do not always classify patients in the same stages, and clinical staging may underestimate (shown in blue), agree (shown in green) or overestimate (shown in red). Clinical T stages are: T0, Ta, TIS, T1, T2a, T2b, T3a, T3b, T4a, T4b. Pathologic stages are: P0, Pa, PIS, P1, P2a, P2b, P3a, P3b, P4a, P4b. TX and PX indicate patients whose stage was declared unknown, and these patients are removed for this analysis resulting in a dataset of N = 3417 patients for plementary Figure 1 and all concordance analyses between clinical and pathologic staging. 70 4 MACHINE LEARNING MODELS FOR LONG-TERM OUTCOME PREDICTION IN BLADDER CANCER Figure 31: Discrepancy in staging over time. (A) Histogram of number of patients undergoing cystectomy for each decade of data collection. (B) The agreement between clinical and pathological staging over the period of data collection (1971-2016). Figure 32: Bladder stage survival analysis. (A) Kaplan-Meier survival computed by bladder stage at time of cystectomy subgrouped into OC (T0, Ta, TIS, T1, T2a, T2b), EV (T3a, T3b, T4a, T4b), N+ irrespective of stage (metastasized) (p-value < 0.001). (B) Boxplot of OS by bladder stage at time of cystectomy. 71 4 MACHINE LEARNING MODELS FOR LONG-TERM OUTCOME PREDICTION IN BLADDER CANCER Figure 33: Lymphovascular invasion survival analysis. (A) Kaplan-Meier survival computed by lymphovas- cular invasion. (B) Boxplot of OS by lymphovascular invasion. S3 Appendix. Chi-squared test of independence: predictors vs. recurrence and overall sur- vival. Complete results of the Chi-squared test of independence between predictors and recurrence and OS are shown here in Table 15 and Table 16. The continuous variables age and OS are discretized into 10 bins of equal frequency. Predictors are ranked by (Eqs.56, 57, 58). The tests for association with OS are conducted on a dataset of 3499 patients, excluding the 4 patients who are missing survival data. The tests for recurrence exclude 163 patients whose recurrence status is unknown, resulting in a subset of the original data consisting of 3340 patients. Included here is a ranking of the predictors based on Cramer's V shown in Fig.36 and listed in Table 17 and Table 18. Cramer's V normalizes the chi-squared values by the number of categories in the predictors and oers an alternate ranking of the predictor importance with recurrence and overall survival. A composite predictor ranking, rank i = q ( V 2 Rec;i ) 2 + ( V 2 OS;i ) 2 (59) 72 4 MACHINE LEARNING MODELS FOR LONG-TERM OUTCOME PREDICTION IN BLADDER CANCER Figure 34: Surgical margin survival analysis. (A) Kaplan-Meier survival computed by soft tissue surgical margin. (B) Boxplot of OS by soft tissue surgical margin. based on Cramer's V values for recurrence (V Rec ), V Rec;i = V Rec;i VRec (60) and the Cramer's V values for overall survival (V OS ), V OS;i = V OS;i VOS (61) is used to identify predictor importance. The Cramer's V values for both outcomes are normalized by their respective standard deviations (Eqs.59, 60, 61) to weigh the eect of both outcomes equally. 73 4 MACHINE LEARNING MODELS FOR LONG-TERM OUTCOME PREDICTION IN BLADDER CANCER Figure 35: Adjuvant chemotherapy survival analysis. (A) Kaplan-Meier survival computed by receipt of adjuvant chemotherapy. (B) Boxplot of OS by receipt of adjuvant chemotherapy. 74 4 MACHINE LEARNING MODELS FOR LONG-TERM OUTCOME PREDICTION IN BLADDER CANCER Recurrence Overall survival Rank Predictor 2 p-value V 2 p-value V 1 pT stage TNM 7th Edition 344.3 0.000 0.321 713.0 0.000 0.150 2 pT stage TNM 5th Edition 340.3 0.000 0.319 717.0 0.000 0.151 3 pathologic stage subgroup (OC, EV, N+) 423.1 0.000 0.356 558.0 0.000 0.282 4 pM stage TNM 7th Edition 344.2 0.000 0.321 450.0 0.000 0.160 5 pN stage TNM 7th Edition 350.0 0.000 0.324 425.0 0.000 0.174 6 pN stage TNM 5th Edition 347.9 0.000 0.323 414.0 0.000 0.172 7 # of positive lymph nodes 350.6 0.000 0.324 392.0 0.000 0.137 8 pathologic positive lymph nodes 322.0 0.000 0.310 380.0 0.000 0.330 9 type of urinary diversion constructed 4.2 0.041 0.035 598.0 0.000 0.413 10 current urinary diversion performed 3.4 0.064 0.032 530.0 0.000 0.389 11 pathologic # of tumors 27.8 0.000 0.091 490.0 0.000 0.187 12 age at time of cystectomy (discretized) 11.7 0.230 0.059 467.0 0.000 0.122 13 type of surgical procedure 35.2 0.000 0.103 432.0 0.000 0.117 14 pathologic lymphovascular invasion 211.9 0.000 0.252 246.0 0.000 0.265 15 urinary diversion subgoups 17.9 0.000 0.073 380.0 0.000 0.190 16 # lymph nodes removed 7.2 0.407 0.046 374.0 0.000 0.124 17 clinical T stage (preoperative) 102.2 0.000 0.175 301.0 0.000 0.098 18 adjuvant chemotherapy subgroups 114.4 0.000 0.185 284.0 0.000 0.116 19 type of adjuvant chemotherapy 99.6 0.000 0.173 270.0 0.000 0.196 20 neoadjuvant chemotherapy subgroups 23.7 0.000 0.084 290.0 0.000 0.129 21 positive soft tissue surgical margin 26.8 0.000 0.090 251.0 0.000 0.190 22 pM stage TNM 5th Edition 6.8 0.034 0.045 234.0 0.000 0.183 23 intracorporeal diversion performed 15.6 0.000 0.068 225.0 0.000 0.180 24 # of adjuvant chemo cycles 70.8 0.000 0.146 187.0 0.000 0.094 25 # of neoadjuvant chemo cycles 21.7 0.003 0.081 213.0 0.000 0.093 26 clincial staging subgroup (preoperative) 57.3 0.000 0.131 191.0 0.000 0.165 27 robotic surgical procedure 14.9 0.000 0.067 207.0 0.000 0.243 28 pathologic predominant cell type 97.8 0.000 0.171 91.2 0.001 0.066 29 clinical N Stage (preoperative) 50.7 0.000 0.123 154.0 0.000 0.121 30 type of neoadjuvant chemotherapy 17.3 0.000 0.072 171.0 0.000 0.221 31 pathologic size of bladder tumor, dim-3 (cm) 19.9 0.006 0.077 154.0 0.000 0.079 32 pathologic size of bladder tumor, dim-2 (cm) 26.0 0.000 0.088 147.0 0.000 0.078 33 pathologic carcinoma in situ 88.6 0.000 0.163 46.1 0.000 0.081 34 # of neoadjuvant radiation to bladder 14.1 0.015 0.065 147.0 0.000 0.092 35 pathologic size of bladder tumor, dim-1 (cm) 6.2 0.290 0.043 144.0 0.000 0.091 36 clinical M stage (preoperative) 20.3 0.000 0.078 122.0 0.000 0.132 Table 15: Chi-squared test, ranked by Eq.1-3. Predictors 1-36. 75 4 MACHINE LEARNING MODELS FOR LONG-TERM OUTCOME PREDICTION IN BLADDER CANCER Recurrence Overall survival Rank Predictor 2 p-value V 2 p-value V 37 pathologic multifocal tumors 69.6 0.000 0.144 41.1 0.001 0.077 38 Charlson: renal disease, moderate or severe 1.9 0.397 0.024 110.0 0.000 0.125 39 pathologic other variant histology 19.4 0.007 0.076 97.9 0.003 0.063 40 Charlson: connective tissue disorder 2.2 0.337 0.026 102.0 0.000 0.121 41 pathologic tumor grade 57.8 0.000 0.132 36.8 0.006 0.072 42 Charlson: chronic obstructive pulmonary disease 12.1 0.002 0.060 96.2 0.000 0.117 43 pathologic squamous dierentiation 47.4 0.000 0.119 55.8 0.000 0.073 44 Charlson: diabetes 7.4 0.025 0.047 93.5 0.000 0.116 45 pathologic glandular dierentiation 51.0 0.000 0.124 34.4 0.156 0.057 46 predominant clinical cell type (preoperative) 20.5 0.005 0.078 81.6 0.057 0.058 47 indication for surgery 17.0 0.005 0.071 83.6 0.000 0.069 48 Charlson: any other malignancy 2.2 0.333 0.026 86.8 0.000 0.111 49 pathologic neuroendocrine dierentiation 41.5 0.000 0.111 27.5 0.070 0.063 50 Charlson: congestive heart failure 6.6 0.037 0.044 70.6 0.000 0.100 51 clinical CIS 1.5 0.216 0.021 69.9 0.000 0.141 52 Charlson: myocardial infarction 4.7 0.095 0.038 64.7 0.000 0.096 53 Charlson: metastatic disease for any other primary 5.2 0.073 0.040 57.4 0.000 0.091 54 Charlson: peripheral vascular disease 2.3 0.314 0.026 57.2 0.000 0.090 55 Charlson: cerebrovascular disease 1.9 0.390 0.024 57.0 0.000 0.090 56 Charlson: diabetes with end-organ damage 1.9 0.393 0.024 54.4 0.000 0.088 57 Charlson: leukemia 1.6 0.453 0.022 53.4 0.000 0.087 58 Charlson: dementia 1.5 0.483 0.021 53.2 0.000 0.087 59 Charlson: hemiplegia 3.0 0.226 0.030 52.5 0.000 0.087 60 Charlson: liver disease, moderate or severe 5.9 0.052 0.042 51.4 0.000 0.086 61 Charlson: liver disease mild 4.0 0.134 0.035 51.5 0.000 0.086 62 Charlson: lymphoma 1.4 0.494 0.021 50.9 0.000 0.085 63 Charlson: ulcer disease 3.1 0.213 0.030 50.3 0.000 0.085 64 any prior pelvic radiation 7.7 0.006 0.048 40.9 0.000 0.108 65 history of smoking 12.3 0.006 0.061 37.4 0.088 0.060 66 intravesicle BCG treatment 0.2 0.655 0.008 39.1 0.000 0.106 67 neoadjuvant bladder radiation 9.3 0.002 0.053 33.2 0.000 0.097 68 clinical hydronephrosis 0.4 0.802 0.011 31.4 0.026 0.067 69 adjuvant radiation post cystectomy 2.5 0.115 0.027 29.0 0.000 0.091 70 pathologic micropapillary 4.0 0.046 0.034 21.9 0.009 0.079 71 any type of intravesicle treatment 0.1 0.746 0.006 20.6 0.015 0.077 72 gender 9.9 0.002 0.054 9.5 0.390 0.052 Table 16: Chi-squared test, ranked by Eq.1-3. Predictors 37-72. 76 4 MACHINE LEARNING MODELS FOR LONG-TERM OUTCOME PREDICTION IN BLADDER CANCER Figure 36: Predictors ranked by Cramer's V. 77 4 MACHINE LEARNING MODELS FOR LONG-TERM OUTCOME PREDICTION IN BLADDER CANCER Recurrence Overall survival Rank Predictor 2 p-value V 2 p-value V 1 type of urinary diversion constructed 4.2 0.041 0.035 598.0 0.000 0.413 2 pathologic positive lymph nodes 322.0 0.000 0.310 380.0 0.000 0.330 3 pathologic stage subgroup (OC, EV, N+) 423.1 0.000 0.356 558.0 0.000 0.282 4 current urinary diversion performed 3.4 0.064 0.032 530.0 0.000 0.389 5 pathologic lymphovascular invasion 211.9 0.000 0.252 246.0 0.000 0.265 6 pN stage TNM 7th Edition 350.0 0.000 0.324 425.0 0.000 0.174 7 pN stage TNM 5th Edition 347.9 0.000 0.323 414.0 0.000 0.172 8 pM stage TNM 7th Edition 344.2 0.000 0.321 450.0 0.000 0.160 9 pT stage TNM 7th Edition 344.3 0.000 0.321 713.0 0.000 0.150 10 pT stage TNM 5th Edition 340.3 0.000 0.319 717.0 0.000 0.151 11 # of positive lymph nodes 350.6 0.000 0.324 392.0 0.000 0.137 12 robotic surgical procedure 14.9 0.000 0.067 207.0 0.000 0.243 13 type of adjuvant chemotherapy 99.6 0.000 0.173 270.0 0.000 0.196 14 type of neoadjuvant chemotherapy 17.3 0.000 0.072 171.0 0.000 0.221 15 positive soft tissue surgical margin 26.8 0.000 0.090 251.0 0.000 0.190 16 pathologic # of tumors 27.8 0.000 0.091 490.0 0.000 0.187 17 urinary diversion subgoups 17.9 0.000 0.073 380.0 0.000 0.190 18 clincial staging subgroup (preoperative) 57.3 0.000 0.131 191.0 0.000 0.165 19 intracorporeal diversion performed 15.6 0.000 0.068 225.0 0.000 0.180 20 pM stage TNM 5th Edition 6.8 0.034 0.045 234.0 0.000 0.183 21 adjuvant chemotherapy subgroups 114.4 0.000 0.185 284.0 0.000 0.116 22 clinical T stage (preoperative) 102.2 0.000 0.175 301.0 0.000 0.098 23 clinical N Stage (preoperative) 50.7 0.000 0.123 154.0 0.000 0.121 24 pathologic carcinoma in situ 88.6 0.000 0.163 46.1 0.000 0.081 25 pathologic predominant cell type 97.8 0.000 0.171 91.2 0.001 0.066 26 # of adjuvant chemo cycles 70.8 0.000 0.146 187.0 0.000 0.094 27 clinical M stage (preoperative) 20.3 0.000 0.078 122.0 0.000 0.132 28 neoadjuvant chemotherapy subgroups 23.7 0.000 0.084 290.0 0.000 0.129 29 clinical CIS 1.5 0.216 0.021 69.9 0.000 0.141 30 type of surgical procedure 35.2 0.000 0.103 432.0 0.000 0.117 31 pathologic multifocal tumors 69.6 0.000 0.144 41.1 0.001 0.077 32 age at time of cystectomy (discretized) 11.7 0.230 0.059 467.0 0.000 0.122 33 # lymph nodes removed 7.2 0.407 0.046 374.0 0.000 0.124 34 Charlson: renal disease, moderate or severe 1.9 0.397 0.024 110.0 0.000 0.125 35 Charlson: chronic obstructive pulmonary disease 12.1 0.002 0.060 96.2 0.000 0.117 36 pathologic tumor grade 57.8 0.000 0.132 36.8 0.006 0.072 Table 17: Chi-squared test, ranked by Eq.4-6. Predictors 1-36. 78 4 MACHINE LEARNING MODELS FOR LONG-TERM OUTCOME PREDICTION IN BLADDER CANCER Recurrence Overall survival Rank Predictor 2 p-value V 2 p-value V 37 Charlson: connective tissue disorder 2.2 0.337 0.026 102.0 0.000 0.121 38 Charlson: diabetes 7.4 0.025 0.047 93.5 0.000 0.116 39 pathologic squamous dierentiation 47.4 0.000 0.119 55.8 0.000 0.073 40 any prior pelvic radiation 7.7 0.006 0.048 40.9 0.000 0.108 41 Charlson: any other malignancy 2.2 0.333 0.026 86.8 0.000 0.111 42 # of neoadjuvant chemo cycles 21.7 0.003 0.081 213.0 0.000 0.093 43 pathologic glandular dierentiation 51.0 0.000 0.124 34.4 0.156 0.057 44 pathologic neuroendocrine dierentiation 41.5 0.000 0.111 27.5 0.070 0.063 45 Charlson: congestive heart failure 6.6 0.037 0.044 70.6 0.000 0.100 46 intravesicle BCG treatment 0.2 0.655 0.008 39.1 0.000 0.106 47 neoadjuvant bladder radiation 9.3 0.002 0.053 33.2 0.000 0.097 48 # of neoadjuvant radiation to bladder 14.1 0.015 0.065 147.0 0.000 0.092 49 pathologic size of bladder tumor, dim-2 (cm) 26.0 0.000 0.088 147.0 0.000 0.078 50 Charlson: myocardial infarction 4.7 0.095 0.038 64.7 0.000 0.096 51 pathologic size of bladder tumor, dim-3 (cm) 19.9 0.006 0.077 154.0 0.000 0.079 52 pathologic size of bladder tumor, dim-1 (cm) 6.2 0.290 0.043 144.0 0.000 0.091 53 Charlson: metastatic disease for any other primary 5.2 0.073 0.040 57.4 0.000 0.091 54 adjuvant radiation post cystectomy 2.5 0.115 0.027 29.0 0.000 0.091 55 Charlson: peripheral vascular disease 2.3 0.314 0.026 57.2 0.000 0.090 56 Charlson: cerebrovascular disease 1.9 0.390 0.024 57.0 0.000 0.090 57 Charlson: liver disease, moderate or severe 5.9 0.052 0.042 51.4 0.000 0.086 58 Charlson: diabetes with end-organ damage 1.9 0.393 0.024 54.4 0.000 0.088 59 Charlson: liver disease mild 4.0 0.134 0.035 51.5 0.000 0.086 60 Charlson: hemiplegia 3.0 0.226 0.030 52.5 0.000 0.087 61 Charlson: leukemia 1.6 0.453 0.022 53.4 0.000 0.087 62 Charlson: dementia 1.5 0.483 0.021 53.2 0.000 0.087 63 indication for surgery 17.0 0.005 0.071 83.6 0.000 0.069 64 Charlson: ulcer disease 3.1 0.213 0.030 50.3 0.000 0.085 65 Charlson: lymphoma 1.4 0.494 0.021 50.9 0.000 0.085 66 pathologic other variant histology 19.4 0.007 0.076 97.9 0.003 0.063 67 predominant clinical cell type (preoperative) 20.5 0.005 0.078 81.6 0.057 0.058 68 pathologic micropapillary 4.0 0.046 0.034 21.9 0.009 0.079 69 any type of intravesicle treatment 0.1 0.746 0.006 20.6 0.015 0.077 70 history of smoking 12.3 0.006 0.061 37.4 0.088 0.060 71 clinical hydronephrosis 0.4 0.802 0.011 31.4 0.026 0.067 72 gender 9.9 0.002 0.054 9.5 0.390 0.052 Table 18: Chi-squared test, ranked by Eq.4-6. Predictors 37-72. Ta TIS T1 T2a T2b T3a T3b T4a T4b T0 0.656 0.896 0.442 0.049 4.00E-05 0.086 3.48E-05 3.85E-05 1.06E-06 Ta ||| 0.505 0.732 0.030 1.12E-07 0.104 1.03E-07 3.10E-07 9.80E-08 TIS ||| ||| 0.152 6.29E-04 4.27E-12 0.029 2.48E-13 1.58E-11 6.78E-10 T1 ||| ||| ||| 2.03E-04 1.44E-15 0.078 0 9.44E-15 4.98E-10 T2a ||| ||| ||| ||| 8.71E-09 0.536 3.97E-09 5.56E-08 3.98E-06 T2b ||| ||| ||| ||| ||| 0.232 0.519 0.699 0.024 T3a ||| ||| ||| ||| ||| ||| 0.199 0.207 0.018 T3b ||| ||| ||| ||| ||| ||| ||| 0.928 0.055 T4a ||| ||| ||| ||| ||| ||| ||| ||| 0.061 Table 19: p-value of paired log rank tests for clinical T stage subgroups. 79 4 MACHINE LEARNING MODELS FOR LONG-TERM OUTCOME PREDICTION IN BLADDER CANCER Pa PIS P1 P2a P2b P3a P3b P4a P4b P0 0638 0666 0047 0027 247E-07 0 0 0 0 Pa ||| 0791 0357 0243 0005 345E-09 329E-10 0 0 PIS ||| ||| 0083 0062 107E-07 0 0 0 0 P1 ||| ||| ||| 0724 148E-04 0 0 0 0 P2a ||| ||| ||| ||| 0004 113E-14 0 0 0 P2b ||| ||| ||| ||| ||| 659E-09 518E-12 0 0 P3a ||| ||| ||| ||| ||| ||| 0239 313E-07 0 P3b ||| ||| ||| ||| ||| ||| ||| 812E-06 0 P4a ||| ||| ||| ||| ||| ||| ||| ||| 289E-13 Table 20: p-value of paired log rank tests for pathologic T stage subgroups. 80 5 FEATURE AND MODEL BASED CHARACTERIZATION OF SPATIAL AND TEMPORAL ORGANOID DYNAMICS 5 Feature and model based characterization of spatial and tem- poral organoid dynamics 5.1 Abstract Organoid model systems recapitulate key features of mammalian tissues and enable high throughout experi- ments. However, the impact of these experiments is limited by manual, qualitative phenotypic analysis. We developed an automated image analysis pipeline to quantify shape dynamics in mammary organoids. Our pipeline consists of i) segmentation of image series ii) preprocessing, iii) geometrical and signal processing fea- ture extraction, iv) dimensionality reduction to dierentiate dynamical paths, v) time series clustering, and vi) dynamical modeling using point distribution models to explain shape variation. The pipeline is applied to time series from 10 untreated and 30 FGF2-treated organoids, revealing three FGF2-treated subgroups characterized by i) high-area high-circularity, ii) low-area high-circularity, and iii) high-area low-circularity. Our pipeline can characterize, cluster, and model dierences among unique dynamical paths that dene diverse nal shapes, thus enabling quantitative analysis of the molecular basis of tissue development and disease. 5.2 Introduction Our modern understanding of cell, developmental, and cancer biology relies heavily upon the analysis of isolated cells cultured at low density on at, rigid substrates. These experiments are typically termed two- dimensional (2D) or conventional cell culture. These model systems have dened our understanding of subcellular compartments, single cell behavior, and the molecular regulation of cell structure and dynamics. However, 2D cell culture is inherently limited in its ability to model complex 3D tissue architecture. This problem is aggravated in mammalian systems due to the large size of the organs, the diversity of their cell types, their relatively slow development, and their relative inaccessibility deep inside the body. To overcome these challenges, a broad range of 3D culture environments and organoid techniques have been developed to provide more physiologically accurate models of tissues and organs ([172, 52]). The term organoid, dened variously by dierent groups, here connotes the use of primary tissue collected directly from a mammalian organ and cultured acutely in a manner that preserves important features of the in vivo cell and tissue state. The large size of mammalian organs becomes an advantage in these assays, as a single normal organ or tumor can generate thousands to hundreds of thousands of organoids, each representing a sample of the heterogeneity of cell states present in vivo. However, conventional approaches to quantify the phenotypes of these diverse cell samples have often been limited to qualitative scoring systems, 81 5 FEATURE AND MODEL BASED CHARACTERIZATION OF SPATIAL AND TEMPORAL ORGANOID DYNAMICS such as binned percentages or categorical scales. This has limited the utility of organoid assays relying on manual inspection, subjective judgment, and qualitative criteria. Conversely, microscopic automation enables sequential imaging of dozens to hundreds or thousands of organoids across multiwell or micro uidic plates containing diverse experimental conditions. With abundant tissue and automated imaging, analysis quickly becomes the rate limiting step for determining the molecular regulation of these biological systems. We sought to overcome these limitations by developing an automated imaging pipeline that can extract quantitative shape descriptor features from time series image data of organoids, analyze the dynamics of organoid shape descriptors over time, compare average dierences in features between experimental condi- tions, and classify the phenotypic heterogeneity across experimental conditions. At the core of the pipeline are geometric and signal processing based descriptive features of organoid shape, as these are often the focus of shape assessment studies ([168, 110, 19, 152, 125, 219, 124]). We also introduce local curvature shape descriptors, and leverage all these features concurrently in order to increase robustness of downstream pheno- typic analyses. As a proof of principle example, we focused on a 3D culture model of normal mouse mammary development ([56, 137]). Brie y, mouse mammary glands are mechanically and enzymatically processed to generate thousands of primary tissue organoids, each consisting of 100-300 epithelial cells ([137]). In basal culture medium, the organoids undergo growth arrest with minimal cell division. Addition of nanomolar concentrations of a growth factor, such as broblast growth factor 2 (FGF2), induces branching morphogen- esis in a majority of organoids. We selected this model as we had qualitatively observed large dierences in growth and shape between the basal and FGF2-treated organoid groups, but we were previously limited in our ability to robustly quantify and describe the diversity of the phenotypes we observed. In this study, we apply our novel imaging pipeline to characterize the developmental trajectory of organoids in both basal and FGF2-treated groups. We quantied the anticipated growth induction by FGF2 and quantitatively charac- terized the heterogeneity in growth and branching responses to FGF2. We anticipate that our pipeline will enable quantitative analysis of the molecular and mechanical regulators of tissue growth and invasion in a broad range of 3D culture and organoid assays. 5.3 Methods Mammary organoids were prepared and embedded in a 3D matrix composed of equal parts Collagen-I and Matrigel according to published protocols ([56, 137]). Organoids were cultured either in basal media (DMEM- F12 with 1% insulin-transferrin-selenium and 1% penicillin-streptomycin) or in media supplemented with 2.5nM FGF2. The organoids were maintained at 37oC and 5% CO2 for 130 hours and imaged every 30 minutes using Dierential Interference Contrast (DIC) microcopy for a total of 261 time points, creating a 82 5 FEATURE AND MODEL BASED CHARACTERIZATION OF SPATIAL AND TEMPORAL ORGANOID DYNAMICS time series of contours of each organoid's boundary. The following sections describe each step of the analysis pipeline shown in Fig.37. Figure 37: Schematic highlighting the main steps in the analytical tool pipeline. The three major steps in the pipeline are: (i) data preprocessing, (ii) feature extraction, and (iii) time series analyses. 5.3.1 Preprocessing Each organoid image is converted to a two dimensional contour using the Automated Contour Macro (ACM) to get contours (~ x;~ y)2R N2 consisting of N points. The automated contour detection method produces contours composed of horizontal and vertical line segments. In the third preprocessing step, these contours are smoothed by replacing the original set of vertices by midpoints of each original line segment (S1 Appendix). In addition to smoothing the boundary, this step also avoids in ating the area and perimeter calculations. Finally, each smoothed contour of perimeterP is redrawn using a constant point density =N=P = 6px=m by increasing the number of points in the contour N while maintaining an arc length of 1= between points. Fig. 38 shows examples of preprocessed untreated and FGF2-treated organoid contours. 5.3.2 Feature extraction Ten features are extracted from each contour (Table 21), where area A and perimeter P are the two basic descriptors of organoid size (S2 Appendix). Form factor a f and solidity a h are two commonly used and useful area ratios. The form factor a f is dened as the ratio of the organoid area and the area of a circle whose perimeter is equivalent to the 83 5 FEATURE AND MODEL BASED CHARACTERIZATION OF SPATIAL AND TEMPORAL ORGANOID DYNAMICS organoid's perimeter, a f = A P 2 =4 = 4A P 2 : (62) The solidity a h is dened as the ratio of the area of the organoid and the area of the convex hull for the contour, a h = A A Hull (63) Note that a f reaches a maximum of 1 for circles, and both a f and a h measure the area-packing eciency, or complexity, of a curve (for which a f < 1 and a h < 1). Another measure of contour irregularity is the polar moment of area J zz (S2 Appendix). While area and perimeter directly describe the size of an organoid, the area ratios capture the increasing complexity of contours due to localized protrusions and can identify organoids that do not maintain shape over time. The polar moment of area J zz furthers this concept by calculating explicitly the squared distances of each innitesimal unit of area from the normal axis at the contour's centroid and describes the distribution of an organoid's area. A greater granularity in contour description is achieved by studying the local curvature along the contour, with the fraction of convex f vex and collinear f col points in a contour serving as summary statistic features (S2 Appendix). The fraction of concave pointsf cav is omitted to maintain linear independence in the feature set. A nal set of features is derived from the discrete Fourier transform (DFT) of a contour (S2 Appendix). We nd the set of modes with the highest amplitude that comprise 90% of the total energy (sum of all mode amplitudes) of a signal,E T , to calculate i) the number of modesN 90 , ii) the mean mode amplitude A 90 , and iii) the standard deviation 90 of mode amplitudes in the 90% E T signal. The 90% E T signal attenuation serves to omit higher frequency modes that are more likely due to contour segmentation artifacts in the rst preprocessing step. The p features (Tab.21) are extracted from an organoid's time series of contours, and together they represent a multivariate time series X o 2 R Tp of length T = 261 and dimensionality p for an organoid o. These multivariate time series are compared directly for the clustering task (Fig.373b). The time series are concatenated for the low dimensional representation (Fig.373a), where the entire set of p features for n O = 40 organoids are stacked in a data matrix X2R n O Tp . Each column represents a single feature for all n O T organoid contours in the dataset, and each row is a single time point or contour of an organoid. 84 5 FEATURE AND MODEL BASED CHARACTERIZATION OF SPATIAL AND TEMPORAL ORGANOID DYNAMICS Extracted features 1 A area 2 P perimeter 3 a f form factor 4 a h solidity 5 J zz polar moment of area 6 f vex fraction of convex points 7 f col fraction of collinear points 8 N 90 number of modes in 90%E T 9 A 90 mean mode amplitude in 90%E T 10 90 standard deviation of mode amplitude in 90%E T Table 21: List of quantitative features extracted from organoid contours. 5.3.3 Low dimensional representation Even though the feature space of the data matrix X is small compared to the number of sample points, prin- cipal component analysis (PCA) is performed in order to chart the composite evolution of all the geometrical and DFT properties of the entire set of organoids. A reduced data matrix Y r = XV2R n O Tpr (64) = USV T V (65) = US (66) of dimension p r is calculated using singular value decomposition of the original matrix X = USV T , where V is the matrix of principal components ([103]). To obtain a reduced data matrix with p r <p, the number of columns of V are reduced to p r . The features A, P , J zz , and A 90 are log-transformed before the PCA in order to linearize relationships involving these features. 5.3.4 Time series clustering The total distance between two organoids i and j, D(Y i ;Y j ) = 1 p r pr X k=1 g Y i k ;Y j k (67) is the mean distance over all p r principal components as measured by the distance metric g. The distance matrix is symmetric, D2 R n O n O , where n O = 40 is the total number of organoids in the dataset. The principal components are not standardized prior to applying the distance functiong in order to maintain the variance they describe for the clustering algorithm. This distance matrix is used to cluster organoids, and the choice of p r is based on the variance explained in the original data matrix as well as the quality of the 85 5 FEATURE AND MODEL BASED CHARACTERIZATION OF SPATIAL AND TEMPORAL ORGANOID DYNAMICS resulting clusters. Here, organoids are grouped using a hierarchical agglomerative average-linkage clustering algorithm based on the cosine similarity measure (S3 Appendix). Although the cosine similarity measure is not a true metric as it does not satisfy the triangle inequality, it is among the commonly used notions of distance and similarity like the L p Minkowski and entropy based measures ([31]). Point distribution model Statistical models of variation of an individual organoid's spatio-temporal changes are created using the point distribution model (PDM) [36, 37] whose outputs are principal modes of variation about the mean organoid shape from a sequence of contours. Contour points are stacked in a column vector for M contours of length N, (x i ;y i ); i2 [1;N] (68) ~ q k = 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 x 1 . . . x N y 1 . . . y N 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 2R 2N ; k2 [1;M]: (69) The mean shape organoid contour over all M samples, ~ q = 1 M M X k ~ q k ; (70) is then used to calculate the covariance matrix, S = 1 M 1 M X k (~ q k ~ q)(~ q k ~ q) T : (71) The eigendecomposition of S, ( i ;~ u i ); i2 [1; 2N] (72) 86 5 FEATURE AND MODEL BASED CHARACTERIZATION OF SPATIAL AND TEMPORAL ORGANOID DYNAMICS reveals the principal modes of variation and the fraction of temporal variance described by a single mode, Fraction of variance ~ u i = i P i i : (73) More complex evolution dynamics require a larger number of modes to capture the temporal shape variance therefore we use the fraction of variance explained by rst eigenmode, and the area under the variance versus number of eigenmodes curve as metrics of dynamical complexity. A statistically viable shape, ~ q sim =~ q + K X i=1 b i ~ u i ; K << 2N (74) is simulated using the K eigenmodes which modulate the mean shape ~ q sim by scalar factors b. The K principal modes of variationu i are ranked in decreasing order of eigenvalues i . Simulated shapes generated by combinations ofK eigenmodes~ u i and scalar factorsb i describe the dynamical evolution of an organoid's shape. The choice of b i is guided by the range observed in the dataset. It is typically bounded within a range of three standard deviations [36], 3 p i b i 3 p i (75) because the eigenvalues i describe the variance of b i in the dataset. Alternatively, the range of mode coecients can be calculated directly from the dataset, ~ b = U T (~ q k ~ q) (76) for k = 1;::;M contours, where the matrix U has K columns, each being one of the K eigenmodes. 5.4 Results 5.4.1 Feature-based description The time series of extracted features (Table 21) for the untreated (green) and FGF2-treated (purple) organoids are depicted in Fig. 39. Fitted exponential curves A untreated = 1843e 0:00305t and A FGF2 = 1507e 0:0106t in Fig. 39A, and P untreated = 181e 0:00109t and P FGF2 = 153e 0:00636t in Fig. 39B show a substantially greater increase in area and perimeter for FGF2 organoids compared to basal types. The area ratio time series are noisy and do not show a signicant separation between the two phenotypes. The tted 87 5 FEATURE AND MODEL BASED CHARACTERIZATION OF SPATIAL AND TEMPORAL ORGANOID DYNAMICS curves have similar parameters in Fig. 39C, a f;untreated = 0:709e 0:000869t (77) a f;FGF2 = 0:803e 0:00212t ; (78) and in Fig. 39D a h;untreated = 0:903e 0:000460t (79) a h;FGF2 = 0:936e 0:000569t : (80) Nevertheless, the decreases in a f and a h re ect the transformation from generally elliptical contours to irregular shapes at later times in the organoid evolution. In contrast to the area ratios, second moment of area in Fig. 39E shows the greatest temporal separation between untreated and treated organoids, with corresponding curve ts J zz;untreated = 584500e 0:00598t and J zz;FGF2 = 373000e 0:0223t . The observed separation occurs because the area of the FGF2-treated organoids is distributed further away from the perpendicular centroid axis over time, while the untreated organoids' areas do not change with time as they maintain their original elliptical contours. The local curvature features in Figs. 39F and 39G also reveal an interesting divergence in the shape changes of the two treatment groups. The curve ts for convexity in Fig. 39F,f vex;untreated = 0:257e 0:000742t and f vex;FGF2 = 0:265e 0:000500t show a slight increase in convexity from FGF2 treatment and a greater de- crease in untreated organoids over time. The opposite trend is seen in the collinearity curve tsf col;untreated = 0:430e 0:000431t and f col;FGF2 = 0:427e 0:000528t in Fig. 39G. The ts for concavity are f cav;untreated = 0:311e 0:0000518t and f cav;FGF2 = 0:305e 0:000308t . Therefore, for untreated organoids, increase in local collinearity over time is matched by a slight decrease in concavity and a larger decrease in convexity. In contrast, FGF2-treated organoids become less collinear and more convex and concave over time as they grow complex features. The DFT features are shown in Fig. 39H-J. The mean N 90 is 41.1 and 46.2 for untreated and treated organoids respectively, and the trend for N 90 remains linear in time. Mean amplitude (Fig. 39I) shows an exponential increase for FGF2 types, where curve ts are A 90;untreated = 463e 0:00310t and A 90;FGF2 = 217e 0:01486t . The standard deviation of the mode amplitudes (Fig.39J) has a similar trend, 90;untreated = 699e 0:00347t and 90;FGF2 = 369e 0:01464t . The exponential increase in A 90 for FGF2-treated organoids is consistent with their size increases seen in Figs. 39A and 39B. The changing relationship between area and perimeter over time (Fig. 40) reveals the departure from 88 5 FEATURE AND MODEL BASED CHARACTERIZATION OF SPATIAL AND TEMPORAL ORGANOID DYNAMICS circularity that some organoids undergo. The untreated organoids remain in a low-area and low-perimeter region in Fig. 40, except for one organoid that behaves like the FGF2-treated organoids and is seen as an outlier in Fig. 39 as well. In contrast, the FGF2 organoids deviate further from the circular limit of A =P 2 =4 (solid black line, Fig. 40). Similarly, the temporal changes of the organoid shape projected into the composite principal component space (Fig. 41) of the whole feature set shows a clear distinction in the shape changes of untreated and treated organoids. As seen in Fig. 41A, the rst three principal components capture 92.4% of the variance of the entire feature set; the inset plot shows the feature weights of the rst two principal components. The feature 90 was removed from the dataset for the PCA due to strong linear correlation with A 90 . While other dependencies are present in the feature set, this choice of features results in a desirable clustering which separates FGF2 and untreated organoids and identies FGF2 subtypes in the next section. Plots of the rst two principal components and convex hulls of the entire data sets, Fig. 41B, show that the FGF2 types dier from the range of basal contours. The untreated and FGF2-treated organoids projected onto principal components are shown in Figs. 41C and D, colored to specify early (t 1 ), mid (t 2 ), and late (t 3 ) time periods. The 30 treated organoids occupy a larger region than the 9 basal types in the projected space because the mid and late period FGF2-treated organoids move away from the initial region, while the untreated organoids remain close to their initial region. Although the rst two principal components explain 80.9% of the variance in the organoid shapes, Figs. 41B-D illustrate that the untreated organoids do not change projected location substantially over time. In contrast, the FGF2 treatment may lead to distinct patterns of shape changes because thet 3 data points do not coalesce to a single region of the principal component space. We turn to clustering to explore general shape characteristics that distinguish the separate groups. 5.4.2 Clustering Hierarchical clustering of the organoids based on p r = 5 principal components is shown in Fig. 42. The rst bifurcation in the clustering (Fig. 42A) is between untreated and FGF2-treated organoids: cluster #1 (green) includes all but one of the untreated organoids, and clusters #2-4 (yellow, blue, and red respectively) are entirely FGF2-treated, with the exception of organoid #8 which is an outlier in the set of untreated. The overlapped rst and last contour drawings in Fig. 42A show that untreated organoids' areas decreases with time, except for #7 whose area remains nearly constant. Although the FGF2-treated organoids start in the same region of the principal component space as the untreated organoids (Fig. 41B-D), they evolve into three clusters which occupy distinct regions of the principal component space (Fig. 42B) and are characterized by unique feature combinations shown by the distributions of features during t 3 in Fig. 42C-F. All three 89 5 FEATURE AND MODEL BASED CHARACTERIZATION OF SPATIAL AND TEMPORAL ORGANOID DYNAMICS FGF2 clusters have higher perimeter and area than the untreated cluster. However, there is a gradation in this eect of treatment: clusters #3 and #4 have the highest area perimeter (Fig. 42C,D). The primary distinction between the large organoid clusters #3 and #4 is that cluster #4 organoids are relatively more circular as seen by the lowa f numbers in Fig. 42F. Cluster #2 includes the smallest and most circular of the FGF2-treated organoids. Cluster #3 organoids are large and most irregularly shaped organoids. Cluster #4 organoids are large yet remain circular. Dierences in local curvature also stratify organoid clusters. The untreated organoids tend to have the lowest f vex , Fig. 42E. Consequently, the dierences in clusters #2-4 reveal the distinct phenotypic outcome of the FGF2 treatment. 5.4.3 Dynamical model A rst-mode simulation of a single organoid from each of the four clusters in Fig. 42 is shown over four time intervals of the evolution in Fig. 43. The rst mode's coecient b 1 is taken as the median of the value from the organoid's contours during the corresponding time period. In Fig. 43A, organoid 7 from cluster #1 undergoes small uctuations but a negligible change in overall size. On the other hand, organoid 34 in Fig. 43B maintains its general shape and expands in time. A larger increase in size is observed for both organoids 29 and 40 in Fig. 43C and D respectively. However, organoid 29 elongates primarily along one axis, whereas organoid 40 expands at each point of convexity leading to a slightly greater circularity than organoid 29. Next, we generated and compared PDMs from each organoid's time series of contours, then examined the temporal variance described by the PDM modes. Generally, contours that undergo a more complex evolution require a greater number of modes K to fully describe the correspondingly large temporal variance. The fraction of variance explained by the rst mode in each organoid's PDM is compared in Fig. 42G for the four detected organoid clusters. The rst modes of cluster #3 organoids, which have the least regular boundaries, describe the least variance in their respective organoids' evolution. Aside from a few outliers, Fig. 42G provides evidence that cluster #2 organoids have simpler dynamics because this cluster has the highest fraction of the total variance explained by the rst mode. This observation is consistent with the previous visual inspection of cluster boundaries. The cumulative fraction of variance explained by PDM modes ~ u i reaches saturation near ve modes for all organoids. The area under the cumulative variance curve (AUC) also describes the complexity of the contour dynamics because a higher variance in a series of contours will have a smaller AUC as more modes are needed to capture all sources of temporal variance. The AUC for K = 5 modes is shown in Fig. 42H for the four detected clusters. Clusters #2 and #4 have higher AUCs as these clusters' organoids remain consistently circular throughout the evolution. Cluster #1 has a smaller AUC due to smaller uctuations in 90 5 FEATURE AND MODEL BASED CHARACTERIZATION OF SPATIAL AND TEMPORAL ORGANOID DYNAMICS shape; however, cluster #3 has the lowest AUC as these organoids transform signicantly over time. 5.5 Discussion We developed a computational pipeline that provides an automated platform for studying the spatial and temporal dynamics of shape in complex 3D organoids. For simplicity in this rst implementation, we re- stricted our analyses to 2D projections of 3D volumes and found that this was sucient to detect average dierences in organoid shape between conditions and to characterize the heterogeneity in response of in- dividual organoids within the same experimental condition. Since our organoid assays generate 1000s of genotypically equivalent replicates, this analytical framework was also useful for identifying precisely when and how the experimental organoids diverged in their shape trajectories from control organoids. Furthermore, the set of geometric and discrete Fourier transform features together provide phenotypic signatures of organoid morphological changes. The low-dimension representation of the extracted feature set summarizes the divergent paths taken by FGF2-treated organoids in comparison to untreated organoids. Particularly, the treated organoids span a larger region of principal component space due to their complex shaped contours at later times. Point distribution based on sequences of contours succinctly describe the shape dynamics with three or fewer modes for most organoids and are useful for ranking clusters of organoids by their dynamical complexity. Hierarchical clustering of organoids in the reduced feature set not only separates untreated and FGF2-treated organoids but reveals three phenotypic subgroups in the FGF2-treated set: i) low-area high-circularity, ii) high-area high-circularity, and iii) high-area low-circularity. Biologically, these sub-clusters suggest that some organoids do not respond morphologically to FGF2, others grow in response to FGF2 but do not branch, and others both grow and branch. Since a cell must express an appropriate FGF receptor in order to recognize and respond to FGF2, an possible explanation is that FGF receptor expression is low or absent in the non-responding organoids. Similarly, although the genetic state is identical across organoids, dierences in cell state dened by transcription or by epigenetics may be responsible for dierences in FGF2-response. Thus, the ability to characterize the growth and branching response of organoids to FGF2 enables future experiments to characterize the molecular basis of their dierential response. The underlying basic biology is relevant to branching morphogenesis in normal development and potentially to aberrant cell behavior in cancer. Though a simple proof-of-principle example, the discovery of patterns of organoid behavior highlights the power of automated image analysis to identify biologically cohesive phenotypes in large populations in an unbiased fashion, or at least in a fashion in which the phenotypic boundaries are clearly dened. The distinct phenotypic groups can then be recovered for molecular analysis. We anticipate that this approach 91 5 FEATURE AND MODEL BASED CHARACTERIZATION OF SPATIAL AND TEMPORAL ORGANOID DYNAMICS will nd broad utility in determining the molecular regulators of cell structure and function within epithelial tissues and tumors. 92 5 FEATURE AND MODEL BASED CHARACTERIZATION OF SPATIAL AND TEMPORAL ORGANOID DYNAMICS Figure 38: Examples of organoid images, segmentation, and observed spatio-temporal changes. A-G Original organoid image (top) and segmented boundary (bottom, green). A-C are untreated basal organoids, and D-G are FGF2 treated organoids at initial time points. A-D untreated and E-H FGF2- treated organoids' series of observed contours color coded by time point and overlapped to show the spatio- temporal changes from t = 0 to t f = 130 hours. 93 5 FEATURE AND MODEL BASED CHARACTERIZATION OF SPATIAL AND TEMPORAL ORGANOID DYNAMICS Figure 39: Extracted feature time series for all organoids. Basal: green; FGF2: purple. A, area; B, perimeter; C, form factor; D, solidity; E, log of second moment of area; F, fraction of convex points; G, fraction of collinear points; H, number of modes in the 90% E T ; I, log of mean mode amplitude in the 90% E T ; J, log of mode amplitude standard deviation in the 90% E T . 94 5 FEATURE AND MODEL BASED CHARACTERIZATION OF SPATIAL AND TEMPORAL ORGANOID DYNAMICS Figure 40: Relationship between area and perimeter over time. Temporal data is shown by a three tier encoding: 0 t 1 < 43:5 hr t 2 < 87:5 hr t 3 130:5 hr. The area-perimeter relationship for a circle, A circle = P 2 =4 is shown by the black curve. Quadratic curve ts for untreated organoids, A untreated 0:0360P 2 , and FGF2-treated organoids, A FGF2 0:0410P 2 are shown in green and purple respectively. 95 5 FEATURE AND MODEL BASED CHARACTERIZATION OF SPATIAL AND TEMPORAL ORGANOID DYNAMICS Figure 41: Organoid shape dynamics in reduced principal component space. A, fraction of variance explained by principal components in the original dataset and the feature weights for the rst two principal components. B, all FGF2-treated and untreated organoid data points plotted in subspace of rst two principal components, the solid purple and green lines are convex hulls of all the FGF2-treated and untreated organoids respectively. B, C, D, organoid data points plotted in subspace of rst two principal components for both FGF2-treated and untreated, only untreated, and only treated respectively. Temporal data in C and D is shown by a three tier encoding: 0t 1 < 43:5 hrt 2 < 87:5 hrt 3 130:5 hr. 96 5 FEATURE AND MODEL BASED CHARACTERIZATION OF SPATIAL AND TEMPORAL ORGANOID DYNAMICS Figure 42: Hierarchical clustering of organoids. A Organoids grouped into four clusters (1: green, 2: yellow, 3: blue, 4: red) with each organoid's rst (light shade) and last (dark shade) contour. Untreated organoids are shown in green and FGF2-treated organoids are shown in purple. B All data points are colored by clusters in the space of the rst two principal components. C,D,E,F Boxplots of perimeter, area, fraction of convex points, and form factor for the four clusters during the late regime t 3 > 87:5 hr. G Fraction of temporal variance in time series of contours explained by rst mode in the point distribution models. H AUC of the fraction of temporal variance curve for the rst ve modes. 97 5 FEATURE AND MODEL BASED CHARACTERIZATION OF SPATIAL AND TEMPORAL ORGANOID DYNAMICS Figure 43: Point distribution models of shape variation. Far left subplots show rst mode coecient b 1 over time (blue), and median values (black) over four time quarters (0 t 1 < 32:6 hr t 2 < 65.3 hr t 3 < 97:9 hrt 4 130.5 hr). A,B,C, and D show contour simulations ~ q sim for each time quarter using only the rst mode in the PDM for organoids 7, 34, 29, 40 respectively. Mean contours are shown by a black dashed line. 98 5 FEATURE AND MODEL BASED CHARACTERIZATION OF SPATIAL AND TEMPORAL ORGANOID DYNAMICS 5.6 Appendix 5.6.1 S1 Appendix Preprocessing algorithms. Algorithm 1: Contour scaling and mid-point smoothing Data: Contour (~ x;~ y)2 R N2 Result: Smoothed and rescaled contour (~ x smooth ;~ y smooth ) scaling factor c; ~ x =c~ x; ~ y =c~ y; for i = 1 :N 1 do ~ x smooth (i) = (~ x(i) +~ x(i + 1))=2; ~ y smooth (i) = (~ y(i) +~ y(i + 1))=2; end ~ x smooth (N) = (~ x(N) +~ x(1))=2; ~ y smooth (N) = (~ y(N) +~ y(1))=2; 5.6.2 S2 Appendix Derivation and description of extracted organoid contour features are given here. Area is calculated by applying Green's theorem A = ZZ dxdy = I C xdy = I C ydx (81) ) I C xdy = X i Z Ci xdy = X i A Ci (82) A Ci = 8 > > > > > > > < > > > > > > > : Z xi+1 xi mxdx = m 2 x 2 i+1 x 2 i if x i 6=x i+1 ;y i 6=y i+1 Z yi+1 yi x c dy =x c (y i+1 y i ) if x i =x i+1 =x c Z yi+1 yi xdy = 0 if y i =y i+1 =y c (83) 99 5 FEATURE AND MODEL BASED CHARACTERIZATION OF SPATIAL AND TEMPORAL ORGANOID DYNAMICS Algorithm 2: Resample contour with constant density Data: Contour (~ x;~ y)2 R N2 Result: Contour (~ x 0 ;~ y 0 )2 R N 0 2 with point density Residual length l r = 0; New contour length n = 1; for i = 1 :N 1 do Direction from point i to i + 1; u x =~ x(i + 1)~ x(i); u y =~ x(i + 1)~ x(i); u x =u x = q u 2 x +u 2 y ; u y =u y = q u 2 x +u 2 y ; Eective length; l e = q U 2 x +u 2 y +l r ; Number of points to add; n add =bl e c; Add rst point (may be closer than 1=; ~ x 0 (n + 1) =~ x(i) + ((1=)l r )u x ; ~ y 0 (n + 1) =~ y(i) + ((1=)l r )u y ; n add =n add 1; n =n + 1; Add remaining points; for k = 1 :n add do ~ x 0 (n + 1) =~ x 0 (n ) + (1=)u x ; ~ y 0 (n + 1) =~ y 0 (n ) + (1=)u y ; n =n + 1; end Left over distance on current segment; l r =l e modulo 1=; end 100 5 FEATURE AND MODEL BASED CHARACTERIZATION OF SPATIAL AND TEMPORAL ORGANOID DYNAMICS where each summand A Ci of the line integral is calculated for a single line segment component y =mx +b. The centroid of the contour is calculated similarly, x = RR xdxdy RR dxdy = 1 2A I C x 2 dy (84) ) 1 A I C x 2 2 dy = 1 A X i Z Ci x 2 2 dy = 1 A X i x i (85) x i = 8 > > > > > > > < > > > > > > > : Z xi+1 xi 1 2 mx 2 dx = m 6 x 3 i+1 x 3 i if x i 6=x i+1 ;y i 6=y i+1 Z yi+1 yi 1 2 x 2 c dy = x 2 c 2 (y i+1 y i ) if x i =x i+1 =x c Z yi+1 yi 1 2 x 2 dy = 0 if y i =y i+1 =y c (86) y = RR ydxdy RR dxdy = 1 2A I C y 2 dx (87) ) 1 A I C y 2 2 dx = 1 A X i Z Ci y 2 2 dx = 1 A X i y i (88) y i = 8 > > > > > > > > < > > > > > > > > : Z yi+1 yi y 2 2m dy = 1 6m y 3 i+1 y 3 i if x i 6=x i+1 ;y i 6=y i+1 Z xi+1 xi y 2 2 dx = 0 if x i =x i+1 =x c Z xi+1 xi y 2 c 2 dx = y 2 c 2 (x i+1 x i ) if y i =y i+1 =y c (89) Perimeter is calculated by summing along the line segments: P = N X i = (x i+1 x i ) 2 + (y i+1 y i ) 2 1=2 : (90) Second moment of area is calculated about both the axes, I xx = ZZ y 2 dA; (91) I yy = ZZ x 2 dA; (92) and only the polar moment of area, J zz =I xx +I yy (93) is maintained in the extracted feature set. 101 5 FEATURE AND MODEL BASED CHARACTERIZATION OF SPATIAL AND TEMPORAL ORGANOID DYNAMICS Locally convex, collinear, and concave points are dened using the orientation matrix, O = 2 6 6 6 6 4 1 x i1 y i1 1 x i y i 1 x i+1 y i+1 3 7 7 7 7 5 (94) and its determinant det(O) = (x i x i1 )(y i+1 y i1 ) (x i+1 x i1 )(y i y i1 ) (95) for pairs of line segments connected by point i. For a clockwise oriented contour, a point i is convex if det(O) < 0, collinear if det(O) = 0, or concave if det(O) > 0. The conditions for convexity and concavity are reversed for a counter-clockwise oriented contour. The set ofN points of a contour satisfy the relationship f vex +f col +f cav = 1 (96) from which the fraction of convex f vex and collinear f col points in a contour are used as summary statistic features, and fraction of concave points f cav is ignored to maintain linear independence in the feature set. The discrete Fourier transform (DFT) of a contour (~ x;~ y) is calculated by rst casting it into the complex plane, z n =x n +iy n (97) n = 1;:::;N (98) and then applying the transform, Z k = N1 X n=0 z n e i2kn=N (99) k = 0; 1;::;N 1; (100) which has a sampling period T p = 1=, frequency f s =, and DFT frequency domain resolution f s =N. The 102 5 FEATURE AND MODEL BASED CHARACTERIZATION OF SPATIAL AND TEMPORAL ORGANOID DYNAMICS total energy of a signal, E T , is the sum of each mode's amplitude, E T = N1 X k=1 kZ k k; (101) kZ k k = p Re(Z k ) 2 + Im(Z k ) 2 : (102) We sort the modes in order of descending amplitude and nd the set of modes with the highest amplitude that comprise 90%E T to calculate i) the number of modes N 90 , ii) the mean mode amplitude A 90 , and iii) the standard deviation 90 of mode amplitudes in the 90%E T signal. The 90%E T signal attenuation serves to omit higher frequency modes, which are more likely due to contour segmentation in the rst preprocessing step. 5.6.3 S3 Appendix The cosine similarity measure, cos() = A B kAkkBk = n P i=1 A i B i s n P i=1 A 2 i s n P i=1 B 2 i (103) g cosine (A;B) = 1cos() (104) acts on two time series A and B of length n and used in the time series clustering analysis as the distance metric. 103 6 TEMPORAL DIFFERENTIATION OF PHENOTYPES USING A HIDDEN MARKOV MODEL OF ORGANOID SHAPE DYNAMICS 6 Temporal dierentiation of phenotypes using a Hidden Markov model of organoid shape dynamics 6.1 Abstract Phenotype dierentiation is a central focus of analytical tools designed to study outcomes in organoid ex- periments where the goal is to understand and reveal biomechanisms by associating observable features with various organoid treatments. Feature based clustering techniques oer static dierentiation and may be adapted to study temporal dierences, however such tools are not suitable for describing phenotype dynam- ics. Presented here, is a hidden Markov model of organoid shape geometry, which not only dierentiates phenotypes but also links them to distinct dynamics by considering hidden states as unique states of genetic expression which produce similar phenotypes. Furthermore, the task of model selection for Hidden Markov model parameters provides a natural process of determining the number of distinct genetic states in a dataset. Models are trained using perimeter and area time series from two dimensional contours segmented from a sequence of microscopy images of basal and broblast growth factor (FGF2) treated organoids. Particularly, dynamical models of area and perimeter are created which nd i) the number of distinct phenotypes ii) the evolution of hidden states responsible for observed contour dynamics, and iii) the characteristic dynamics associated with each hidden state. The hidden Markov model of shape geometry is an informative dynamical model capable of detecting phenotype groups and corresponding dynamical behavior regimes. 6.2 Introduction Organoids are clusters of organ-specic cells, derived from either animal or human tissue [23], which mimic functionality of the target organ [113] and are grown in controlled environments to study the eects of various treatments and disruptions in bioprocesses [119]. The resulting behavior of the organoid is captured via imaging techniques which allow micro-scale assessment of organoid anatomy [162]. This combination of highly tailored experiments and ne resolution spatio-temporal observations provide an opportunity for quantitative analyses and dynamical modeling of organoid morphology. Such models can help bridge the gap towards understanding and controlling multiscale morphogensis of organoids, which is a current limitation in the eld of tissue engineering where biomimicry is of high precedence [119]. Particularly, questions regarding phenotype count, dierentiation, and behavior are appropriate given time series data from dierentially treated organoid experiments. We propose hidden Markov models to leverage spatio-temporal data from organoid experiments in order to shed light on organoid behavior and dynamics. Hidden Markov models are diversely applicable and have been used successfully in the elds of speech 104 6 TEMPORAL DIFFERENTIATION OF PHENOTYPES USING A HIDDEN MARKOV MODEL OF ORGANOID SHAPE DYNAMICS recognition,[199], genomics [181], motion recognition [214], and nance [82]. Although the application areas are varied, hidden Markov models are generally used in one or a combination of three ways: i) classication ii) forecasting iii) general inference [154]. Classication is made possible when separate models can be trained for each target category in a supervised learning fashion and subsequently be used to calculate the likelihood of an observed sequence. However, when labeled data is not available, or it is not possible to know the generating source of a sequence, hidden Markov models can be used to understand how the sequence is formed, analogous to the unsupervised learning paradigm. Forecasting applications emphasize accuracy and are less concerned with understanding the underlying generating process. Here, we use hidden Markov models for general inference to discover and understand temporal dierences in geometery of basal and FGF2 treated organoids. We hypothesize that distinct genomic expressions are responsible for the observed features of an organoid, and these unique states can be modeled as the hidden states in a hidden Markov model, thereby yielding a quantitative model which explains spatio-temporal heterogeneity of organoid phenotypes. Basal and FGF2 hidden Markov models are created using two steps, preprocessing and model selection (Fig.44), using a dataset of 39 organoids. Subsequently, the models are used to i) associate observed time series to sequences of hidden states and ii) generate synthetic time series samples which further elucidate the role and behavior of the hidden states learned during model selection. 6.3 Methods 6.3.1 Dataset The dataset is comprised of 9 basal organoids and 30 FGF2 treated organoids which are photographed at 30 min intervals for a timespan of 130 hours for a total of 261 time points. During preprocessing (Fig.44), the organoid contours are segmented at each time point into a set of two dimensional coordinates, which are then used to extract the area and perimeter of the organoid contour. Consequently, each of the O = 39 organoids is described by two time series, area and perimeter, of length L = 261. We express this as a multivariate time series X o 2R L2 for o = [1;O]. 6.3.2 Model of organoid shape To create dynamical models of organoid shape, we train basal and FGF2 hidden Markov models following the schematic in Fig.44 using the multivariate time series X o 2R L2 . Individual hidden states represent a single genomic expression state and observations generated from the same hidden state represent phenotypes of the same kind. Therefore, transitions between hidden states represent dynamical evolution from one genomic expression to another, and the hidden state Markov chain is the corresponding governing mechanism. 105 6 TEMPORAL DIFFERENTIATION OF PHENOTYPES USING A HIDDEN MARKOV MODEL OF ORGANOID SHAPE DYNAMICS Figure 44: Analysis pipeline. Modeling procedure consists of preprocesing (step 1) of organoid images and model selection (step 2), which yield two distinct models for Basal and FGF2 types. Viterbi and forward algorithms are used to perform the analyses in steps 3 and 4. Organoids whose geometery varies less throughout the examined time period remain in the same hidden states of genomic expression, whereas organoids which dierentiate substantially from the original state are hypothesized to transition through a series of hidden states in the hidden Markov model. The Markov chain order selection process yields an optimal number of hidden states, and in our application to organoid shape modeling, this provides the optimal number of genomic states responsible for the observed range of phenotypes. However, we seek the fewest number of hidden states which can capture the full range of geometric variation in a set of organoid shape time series, as well as a Markov chain whose absorption states model the observed dynamical endpoints in the data. We characterize organoid shape by perimeter and area, as these two descriptors together can be used to dene organoid circularity, a f = 4A P 2 (105) which dierentiates more regular shaped organoids from more complex shaped organoids that are more invasive. Therefore, circularity, and in turn the combination of perimeter and area forms a computationally 106 6 TEMPORAL DIFFERENTIATION OF PHENOTYPES USING A HIDDEN MARKOV MODEL OF ORGANOID SHAPE DYNAMICS simple and intuitive rst order of approximation of contour complexity. Nevertheless, the adaptation of hidden Markov models to model organoid dynamics described here is generalizable to any combination of organoid descriptors. For the purpose of model selection, the basal model n o = 9L and the FGF2 n o = 30L in the BIC expression (Eq.34), andd = 2 is the dimensionality of the multivariate observation time series in the number of parameters n p (Eq.35) 6.4 Results 6.4.1 Perimeter-area space Basal and FGF2 organoids' perimeter and area time series are shown in Fig.45. Median perimeter decreases for basal organoids Fig.45A and doubles for FGF2 organoids Fig.45C. Median area decreases by 30% for basal organoids Fig.45B, and increases 380% for FGF2 organoids Fig.45D. While the the basal types stagnate or decrease in size, the FGF2 treated organoids grow exponentially. A more expressive view of the discrepancy in morphogenesis is shown in Fig.46 in the perimeter-area space, where the circular limit of area per perimeter forms an upper bound for all observations. Although basal organoids' area and perimeter decreases, they remain conned to a smaller area of the perimeter-area subspace, and the best t quadratic has a coecient of 0.0529. In contrast, FGF2 organoids span a much larger portion of the perimeter-area space and the best t quadratic has a lower coecient of 0.0410 because more FGF2 organoids diverge from circularity. Furthermore, unlike the basal organoids which have less varied dynamics, the FGF2 organoids evolve towards a few distinct regions of the perimeter-area space. This heterogeneity in behaviour is captured by the optimal hidden Markov model in the following sections. 6.4.2 Model selection Selection of parameters for GMM-HMM via BIC, shown in Fig.47, is performed to nd the best combination of Markov chain structure A, number of hidden states Q, and number of Gaussians in the mixture of Gaussians M, for both the basal (Fig.47A-C) and FGF2 models (Fig.47D-F). In addition to left-to-right and upper-triangle, bifurcated Markov chain structures which are two-branch tree graphs, are also trained in Fig.47. Each combination of A, Q, and M is simulated 10 times and mean BIC values along with standard deviation bars are shown in Fig.47. BIC decreases with higher values ofQ except for basal bifurcated models with M = 3; 4; 5 Gaussians. Despite this decrease in BIC, the left-to-right, Q = 2, and M = 2 model is selected for the basal organoids because of its simplicity and the small basal dataset size of 9 organoids. Given a xed value ofQ, all the FGF2 model BIC values are approximately within one standard deviation of 107 6 TEMPORAL DIFFERENTIATION OF PHENOTYPES USING A HIDDEN MARKOV MODEL OF ORGANOID SHAPE DYNAMICS 0 50 100 100 150 200 250 0 50 100 500 1000 1500 2000 2500 3000 0 50 100 0 200 400 600 800 0 50 100 0 5000 10000 15000 A B C D Figure 45: Time series of organoid geometry. Basal organoid (A) perimeter, and (B) area. FGF2 treated organoids' (C) perimeter , and (D) area . Initial perimeter range for basal and FGF2 organoids are 140- 220 m (median = 170 m) and 102-236 m (median = 181 m) respectively. Initial area range for basal and FGF2 organoids are 1170-2680 m 2 (median = 1590 m 2 ) and 718-3150 m 2 (median = 1840 m 2 ) respectively. Final perimeter range for basal and FGF2 organoids are 121-183 m (median = 148 m) and 189-757 m (median = 379 m) respectively. Final area range for basal and FGF2 organoids are 868-1480 m (median = 1110 m) and 2210-14300 m (median = 6920 m) respectively. each other. Although their BIC values are comparable to models with other Markov structures, the left-to- right FGF2 models are not suitable because FGF2 organoids do not converge towards a single region of the perimeter-area space as seen in Fig.46. The bifurcated model holds an advantage over the left-to-right model in that it has two absorption states thus it has the potential to model two distinct dynamical endpoints. Furthermore, compared to upper-triangle structures, bifurcated Markov chain transition matrices A have fewer non-zero terms which results in a simpler model. Therefore, the bifurcated, Q = 5, and M = 2 model is selected for the FGF2 organoids as the other Q = 5 models' BIC values are similar, and the bifurcated 108 6 TEMPORAL DIFFERENTIATION OF PHENOTYPES USING A HIDDEN MARKOV MODEL OF ORGANOID SHAPE DYNAMICS Figure 46: Perimeter and area over three time periods. Three time periods of organoid dynamics shown by individually plotting each contour in a time series: 0 t 1 < 43:5 hr t 2 < 87:5 hr t 3 130:5 hr. The area-perimeter relationship for a circle, A circle = 1 4 P 2 = 0:08P 2 is shown by the black curve. Quadratic curve t for 9 basal organoids A Basal 0:0529P 2 , and 30 FGF2 organoids, A FGF2 0:0410P 2 are shown in green and purple respectively. Q = 7 BIC is not considerably lower. 6.4.3 Basal and FGF2 hidden Markov models Fig.48 and Fig.49 summarize the results of i) training, ii) nding the most likely sequence of hidden states associated with each observed organoid time series, and iii) time series sampling of the basal and FGF2 hidden Markov models respectively. Basal model prior distribution of hidden states, = 1 0 T (106) transition matrix, A = 0 B @ 0:995 0:005 0 1 1 C A (107) 109 6 TEMPORAL DIFFERENTIATION OF PHENOTYPES USING A HIDDEN MARKOV MODEL OF ORGANOID SHAPE DYNAMICS Figure 47: Model selection for basal and FGF2 Hidden Markov models. (A), (B), (C) show BIC values for varying number of hidden states Q, and number of Gaussians M in the emission mixture for left-to-right, upper-triangle and bifurcated hidden state Markov chain structures. (D), (E), (F) show the corresponding BIC values for FGF2 Hidden Markov models. and emission matrix, B = 0 B @ 0:598 0:402 0 1 1 C A (108) are learned from the 9 basal organoid observed time series. Similarly the FGF2 model is trained using the 110 6 TEMPORAL DIFFERENTIATION OF PHENOTYPES USING A HIDDEN MARKOV MODEL OF ORGANOID SHAPE DYNAMICS 30 organoid observed time series, resulting in the following prior, = 1 0 0 0 0 T (109) transition matrix, A = 0 B B B B B B B B B B @ 0:988 0:005 0 0:007 0 0 0:99 0:01 0 0 0 0 1 0 0 0 0 0 0:992 0:008 0 0 0 0 1 1 C C C C C C C C C C A (110) and emission matrix B = 0 B B B B B B B B B B @ 0:378 0:622 0:363 0:637 0:434 0:566 0:556 0:444 0:795 0:205 1 C C C C C C C C C C A (111) 500 points are sampled from the trained models to illustrate the region of the perimeter-area space occupied by each hidden state in Fig.48B and Fig.49B. Basal state X 1 generates larger area and perimeter organoids compared to state X 2 (Fig.48B), however, neither states' samples diverge away from the circular upper limit more than the other. The FGF2 initial state X 1 represents the smallest organoids in terms of perimeter and area, and the subsequent states' organoids are progressively larger (Fig.49B). Unlike the basal states, the FGF2 branch formed by statesX 2 andX 3 diverges further away from the circular limit line than theX 4 andX 5 branch (Fig.49B). Furthermore, the variance of the mixture of Gaussians in the FGF2 model increases for the later states, unlike the basal model. The Viterbi algorithm uses the learned models to label the observed perimeter and area time series with most likely hidden states in Fig.48C-D and Fig.49C-D for basal and FGF2 models respectively. All but two basal organoids transition to X 2 , as these organoids' nal area and perimeter is relatively smaller (Fig.48C- D). All FGF2 organoids transition out of X 1 : seven of the smaller organoids end up in intermediate states X 2 and X 4 and 23 larger organoids eventually reach states X 3 and X 5 (Fig.49C-D). Although both FGF2 absorbing states' (X 3 andX 4 ) organoids are larger than the preceding states' organoids, the organoids which have lower area and high perimeter transition to state X 3 and the organoids which are closer to the circular 111 6 TEMPORAL DIFFERENTIATION OF PHENOTYPES USING A HIDDEN MARKOV MODEL OF ORGANOID SHAPE DYNAMICS limit transition to X 5 . In this manner, organoids can be grouped into subcategories of similar phenotypes by labeling their times series with sequences of most likely hidden states using the Viterbi algorithm and the trained models. To further understand the behavior of the learned basal and FGF2 models, 1000 time series are sampled and averaged from the respective models using the forward algorithm. The generated time series are grouped and colored according to the nal hidden state achieved in Fig.48E-I and Fig.49E-I for basal and FGF2 models respectively. The resulting time-averaged trajectories in the perimeter-area space are plotted in Fig.48E and Fig.49E, where the basal stateX 1 remains conned to the initial point while X 2 moves along the circularity limit towards lower area and lower perimeter. On average, the basal X 1 states' shapes stagnate (Fig.48F-G) while organoids in state X 2 decrease in size. In contrast, the FGF2 branch states move towards larger perimeter and larger area, and X 5 diers from X 3 in that it produces time series with larger areas than X 3 and stays closer to the circular limit (Fig.49E). Therefore FGF2 state X 3 represents more complex and irregular shapes than the equally large X 5 organoids. This behavior is made clear in Fig.49F-G where state X 1 stagnates, intermediate states X 2 and X 4 grow to a fraction of the size achieved by organoids which transition to statesX 3 andX 5 which themselves have the similar mean perimeter but state X 5 does so with more regular larger area organoids. The average timing of transitions between the states is calculated by computing the mode of hidden states for each hidden state group of generated time series (Fig.48H and Fig.49H). The basal state samples which transition toX 2 do so mostly at the 40 hour mark (Fig.48H). Similarly, in the FGF2 model, on average, the organoids which reach absorption statesX 3 andX 5 transition earliest, before the 40 hour mark (Fig.48H) to the intermediate states, and then transition again at approximately 60 hours to reachX 3 andX 5 . Therefore, organoids must transition early to achieve large area and large perimeter. Mean circularity of the groups of generated time series summarizes the corresponding hidden state char- acteristics in Fig.48I and Fig.49I for basal and FGF2 models respectively. Although the two basal states dier in size, their generated time series have the same mean circularity and never become irregular (Fig.48I). Mean circularity of FGF2 generated time series groups vary considerably (Fig.49I): X 1 and X 4 time series groups maintain high circularity (a f 0:8),X 2 andX 5 time series groups' circularity converges to a f 0:6 and a f 0:7 respectively, with X 2 organoids becoming irregularly earlier than X 5 organoids. Finally, the most irregular shaped X 3 state time series' mean circularity decreases most rapidly towards a f < 0:5. 112 6 TEMPORAL DIFFERENTIATION OF PHENOTYPES USING A HIDDEN MARKOV MODEL OF ORGANOID SHAPE DYNAMICS 0 100 200 300 400 0 2000 4000 6000 0 20 40 60 80 100 120 140 160 180 0 20 40 60 80 100 120 1000 1500 2000 0 20 40 60 80 100 120 1 1.5 2 100 150 200 250 0 2000 4000 0 20 40 60 80 100 120 0.4 0.6 0.8 1 Circle 0 20 40 60 80 100 120 100 150 200 250 0 20 40 60 80 100 120 0 1000 2000 3000 A B C D E F G H I Figure 48: Basal Hidden Markov Model results. (A) left-to-right hidden layer Markov chain with Q = 2 hidden states (hidden state color key for the entire gure). (B) 500 individual perimeter and area paired samples fromQ = 2 hidden states in perimeter-area space. 9 observed organoid (C) perimeter and (D) area time series color coded by most likely hidden state as per Viterbi algorithm. 1000 time series are generated from the learned model and grouped by last hidden state: (E) mean trajectory in perimeter-area space, (F) mean perimeter time series, (G) mean area time series, (H) mode hidden state of all generated series, and (I) mean circularity time series. In (E-I) time series are grouped and colored by the nal hidden state at t = 130 hours. 6.5 Conclusion The basal and FGF2 hidden Markov models discover the number of unique genomic expression states re- sponsible for the set of observed contour geometry time series. The basal hidden Markov model is a relatively 113 6 TEMPORAL DIFFERENTIATION OF PHENOTYPES USING A HIDDEN MARKOV MODEL OF ORGANOID SHAPE DYNAMICS 0 200 400 600 800 0 5000 10000 15000 0 20 40 60 80 100 120 100 200 300 400 0 20 40 60 80 100 120 0 5000 10000 0 20 40 60 80 100 120 0 2 4 6 100 200 300 400 500 0 5000 10000 0 20 40 60 80 100 120 0.4 0.6 0.8 1 Circle 0 20 40 60 80 100 120 0 200 400 600 800 0 20 40 60 80 100 120 0 5000 10000 15000 A B C D E F G H I Figure 49: FGF2 Hidden Markov Model results. (A) bifurcated hidden layer Markov chain with Q = 5 hidden states (hidden state color key for the entire gure). (B) 500 individual perimeter and area paired samples fromQ = 5 hidden states in perimeter-area space. 30 observed organoid (C) perimeter and (D) area time series color coded by most likely hidden state as per Viterbi algorithm. 1000 time series are generated from the learned model and grouped by last hidden state: (E) mean trajectory in perimeter-area space, (F) mean perimeter time series, (G) mean area time series, (H) mode hidden state of all generated series, and (I) mean circularity time series. In (E-I) time series are grouped and colored by the nal hidden state at t = 130 hours. simple two state model where one state produces organoids whose perimeter and area stay constant and the organoids in the second state shrink to a smaller size than the initial perimeter and area. Both basal hidden states produce organoids of the same circularity, therefore the geometric variation among basal organoids is 114 6 TEMPORAL DIFFERENTIATION OF PHENOTYPES USING A HIDDEN MARKOV MODEL OF ORGANOID SHAPE DYNAMICS low. The FGF2 hidden Markov model is a more complex bifurcated model with one initial state which connects to two branches for a total of ve hidden states. The bifurcated Markov chain model is suitable for FGF2 treated organoids as there are at least two distinct dynamical end points in the perimeter-area space which a majority of the organoids progress towards. Sampling time series from this model reveals the major dierences between the two FGF2 phenotype branches. Particularly, large FGF2 organoids can be more circular with matching perimeter and area or irregularly shaped with high perimeter per area which is indicative of more invasive behavior. Furthermore the bifurcated model structure also captures behavior of the FGF2 organoids which do not complete morphogenesis towards the two dynamical extremes. Therefore, the dataset of FGF2 treated organoids contains organoids with ve distinct genetic expression states, and the Viterbi algorithm is used to nd that all 30 of the FGF2 organoids are in one of four genetic states at the end of the experiment. We have demonstrated here a novel modeling approach for understanding morphogenesis of organoids. Hidden Markov models for organoid shape time series can learn the number of distinct underlying genetic states in a set of organoids, and then temporally dierentiate the organoids into this set of states. Further- more, the models can be used to generate time series of organoid geometry based on the learned stochastic dynamical progression between the hidden genetic states. These three features provide an advantage over other methods which i) are not directly suited for spatio-temporal dierentiation, ii) do not have rigor- ous methods for detecting number of generating states, and iii) cannot be used to generate samples. The proof-of-concept model shown here with a dataset of 39 organoids can easily be adapted to larger, more het- erogeneous datasets which may require additional geometric features to detect more hidden genetic states. The modeling technique developed here can readily be used to study datasets generated from organoid growth experiments, and provide a common quantitative language to compare complex behavior across studies and organoid types. 115 7 FUTURE WORK 7 Future work Future work of each project presented in the previous chapters is discussed below with emphasis on extracting more relevant and insightful results from each data source and experiment type. The feature and model based techniques developed here to analyze organoid growth need to be employed on larger more heterogeneous datasets to further understand the utility of the feature set, and to identify need for additional features. The binary treatment dataset comprised of growth factor and untreated organoids studied here only shows the potential of these tools, and experiments where organoids are treated with a combination of agents can especially take advantage of the present work. Specically, the strength of the geometric and signal processing feature based clustering is to identify similar organoids, and the hidden Markov models of organoid shape reveal groups of organoids in similar genomic states and their respective dynamics. Standardizing the application of this analytically pipeline across organoid studies can provide a common quantitative basis with which organoid morpghogenesis can be studied. The ability to reliably record the motion of patients in the clinic is a crucial step in achieving predictive feature and model based tools which are clinically relevant. At present, there is no clear successor technology to the discontinued Microsoft Kinect camera used in this thesis, and this limits the opportunities for deriving motion data non-invasively. Nevertheless, motion capture data from patient exercises performed in the clinic can be used for various ailments besides cancer. Secondly, such data can be correlated with and consequently used to learn other risk factors besides subjective physician assigned tness scores. Therefore applying the toolset developed here to study patients suering non-cancer conditions, and learning the corresponding risk factors is a fruitful direction. However, to better understand the utility of a non-invasive motion capture based tness predictor, longer longitudinal studies with greater patient enrollment is required. Deploying in-home or mobile versions of this combination of motion imaging technology and performance predicting tools is perhaps the most forward-looking application, and needs to be a focus of future study. Long-term outcome predictors of bladder cancer cystectomy patients developed here need to be tested on patients at the facilities which provided the data used to train the models as well as external centers in order to truly test the accuracy of predicting survival and recurrence of disease. In order to do this, the models created here need to be deployed as clinician-friendly applications. Concurrently, similar datasets from external centers ought to be used to create similar models to verify the accuracy achieved with the USC dataset, and to fully verify the models' transformational potential of changing identifying at-risk patients. 116 8 APPENDIX I: THE PRISONER'S DILEMMA AS A CANCER MODEL 8 Appendix I: The prisoner's dilemma as a cancer model 8.1 Abstract Tumor development is an evolutionary process in which a heterogeneous population of cells with dierent growth capabilities compete for resources in order to gain a proliferative advantage. What are the minimal ingredients needed to recreate some of the emergent features of such a developing complex ecosystem? What is a tumor doing before we can detect it? We outline a mathematical model, driven by a stochastic Moran process, in which cancer cells and healthy cells compete for dominance in the population. Each are assigned payos according to a prisoner's dilemma evolutionary game where the healthy cells are the cooperators and the cancer cells are the defectors. With point mutational dynamics, heredity, and a tness landscape controlling birth and death rates, natural selection acts on the cell population and simulated `cancer-like' features emerge, such as Gompertzian tumor growth driven by heterogeneity, the log-kill law which (linearly) relates therapeutic dose density to the (log) probability of cancer cell survival, and the Norton{Simon hypothesis which (linearly) relates tumor regression rates to tumor growth rates. We highlight the utility, clarity, and power that such models provide, despite (and because of ) their simplicity and built-in assumptions. 8.2 Introduction Cancer is an evolutionary process taking place within a genetically and functionally heterogeneous population of cells that trac from one anatomical site to another via hematogenous and lymphatic routes [8, 41, 61, 186, 207]. The population of cells associated with the primary and metastatic tumors evolve, adapt, proliferate, and disseminate in an environment in which a tness landscape controls survival and replication [126]. Tumorigenesis occurs as the result of inherited and acquired genetic, epigenetic and other abnormalities accumulated over a long period of time in otherwise normal cells [122, 215]. Before we can typically detect the presence of a tumor, the cells are already competing for resources in a Darwinian struggle for existence in tissues that progressively age and evolve. It is well established that the regenerative capacity of individual cells within a tumor, and their ability to trac multi-directionally from the primary tumor to metastatic tumors all represent signicant challenges associated with the ecacy of dierent cancer treatments and our resulting ability to control systemic spread of many soft-tissue cancers [59, 208]. Details of the metastatic and evolutionary process are poorly understood, particularly in the subclinical stages when tumors are actively developing but not yet clinically visible [180]. It could be argued that in order to truly understand cancer progression at the level in which quantitative predictions become feasible, it is necessary to understand 117 8 APPENDIX I: THE PRISONER'S DILEMMA AS A CANCER MODEL how genetically and epigenetically heterogeneous populations of cells compete and evolve within the tumor environment well before the tumor is clinically detectable. Additionally, a better understanding of how these populations develop resistance to specic therapies [73, 93] might help in developing optimal strategies to attack the tumor, slow disease progression, or maintain it at a stable level. Evolutionary game theory is perhaps the best quantitative framework for modeling evolution and natural selection. It is a dynamic version of classical game theory in which a game between two (or more) competitors is played repeatedly, giving each participant the ability to adjust their strategy based on the outcome of the previous string of games. While this may seem like a minor variant of classical (static) game theory, as developed by the mathematicians von Neumann and Morgenstern in the 1940's [202], it is not. Developed mostly by the mathematical biologists John Maynard Smith and George Price in the 1970s [178, 179] and Martin Nowak and Karl Sigmund [141, 145] more recently, this dynamic generalization of classical game theory has proven to be one of the main quantitative tools available to evolutionary biologists (if coupled with a tness landscape) whose goal is to understand natural selection in evolving populations. In this biological context, a strategy is not necessarily a deliberate course of action, but an inheritable trait [171]. Instead of identifying Nash equilibria, as in the static setting [131, 130], one looks for evolutionary stable strategies (ESS) and xation probabilities [88, 141] of a subpopulation. This subpopulation might be traced to a specic cell with enhanced replicative capacity (high tness), for example, that has undergone a sequence of mutations and is in the process of clonally expanding [146]. A relevant question in that case is what is the probability of xation of that subpopulation? More explicitly, how does one subpopulation invade another in a developing colony of cells? One game in particular, the Prisoner's Dilemma game, has played a central role in cancer modeling (as well as other contexts such as political science and economics) [9, 10, 12, 48, 49, 69, 70, 83, 84, 88, 91, 22, 95, 142, 144, 145, 193, 197, 201, 206]. It was originally developed by Flood, Dresher and Tucker in the 1950s as an example of a game which shows how rational players might not cooperate, even if it seems to be in their best interest to do so. The evolutionary version of the Prisoner's Dilemma game has thus become a paradigm for the evolution of cooperation among a group of selsh individuals and thus plays a key role in understanding and modeling the evolution of altruistic behavior [9, 10]. Perhaps the best introductory discussion of these ideas is found in Dawkins' celebrated book, The Selsh Gene [44]. The framework of evolutionary game theory allows the modeler to track the relative frequencies of competing subpopulations with dierent traits within a bigger population by dening mutual payos among pairs within the group. From this, one can then dene a tness landscape over which the subpopulations evolve. The tness of dierent phenotypes is frequency dependent and is associated with reproductive prowess, while the `players' in the evolutionary game compete selshly for the largest share of descendants [88, 206]. Our goal in this article is provide a 118 8 APPENDIX I: THE PRISONER'S DILEMMA AS A CANCER MODEL brief introduction to how the Prisoner's Dilemma game can be used to model the interaction of competing subpopulations of cells, say healthy, and cancerous, in a developing tumor and beyond. 8.3 The prisoner's dilemma evolutionary game An evolutionary game between two players is dened by a 2 x 2 payo matrix which assigns a reward to each player (monetary reward, vacation time, reduced time in jail, etc.) on a given interaction. Let us call the two players A and B. In the case of a prisoner's dilemma game between cell types in an evolving population of cells, let there be two subpopulations of cell types which we will call `healthy', and `cancerous'. We can think of the healthy cells as the subpopulation that is cooperating, and the cancer cells as formerly cooperating cells that have defected via a sequence of somatic driver mutations. Imagine a sequence of `games' played between two cells (A and B) selected at random from the population, but chosen in proportion to their prevalence in the population pool. Think of a cancer-free organ or tissue as one in which a population of healthy cells are all cooperating, and the normal organ functions are able to proceed, with birth and death rates that statistically balance, so an equilibrium healthy population is maintained (on average). Now imagine a mutated cell introduced into the population with enhanced proliferative capability as encoded by its genome as represented as a binary sequence of 0's and 1's carrying forward its genetic information (which is passed on to daughter cells). A schematic diagram associated with this process is shown in Figure 1. We can think of this cancer cell as a formerly cooperating cell that has defected and begins to compete against the surrounding population of healthy cells for resources and reproductive prowess. From that point forward, one can imagine tumor development to be a competition between two distinct competing subpopulations of cells, healthy (cooperators) and cancerous (defectors). We are interested in the growth rates of a `tumor' made up of a collection of cancer cells within the entire population, or equivalently, we are interested in tracking the proportion of cancer cells, i(t), vs. the proportion of healthy cells, Ni(t), in a population of N cells comprising the simulated tissue region. To quantify how the interactions proceed, and how birth/death rates are ultimately assigned, we introduce the 2 x 2 prisoner's dilemma payo matrix: A = 0 B @ a b c d 1 C A = 0 B @ 3 0 5 1 1 C A: (112) What denes a prisoner's dilemma matrix are the inequalities c > a > d > b. The chosen values in (112) are relatively standard, but not unique 1 . The essence of the prisoner's dilemma game is the two players 1 A general investigation of how the values in the PD payo matrix aects evolutionary dynamics of the subpopulations is addressed in [205]. 119 8 APPENDIX I: THE PRISONER'S DILEMMA AS A CANCER MODEL compete against each other, and each has to decide what best strategy to adopt in order to maximize their payo. This 2 x 2 matrix assigns the payo (e.g. reward) to each player on each interaction. My options, as a strategy or, equivalently, as a cell type, are listed along the rows, with row 1 associated with my possible choice to cooperate, or equivalently my cell type being healthy, and row 2 associated with my possible choice to defect, or equivalently my cell type being cancerous. Your options are listed down the columns, with column 1 associated with your choice to cooperate (or you being a healthy cell), and column 2 associated with your choice to defect (or you being a cancer cell). The analysis of a rational player in a prisoner's dilemma game runs as follows. I do not know what strategy you will choose, but suppose you choose to cooperate (column 1). In that case, I am better o defecting (row 2) since I receive a payo of 5 instead of 3 (if I also cooperate). Suppose instead you choose to defect (column 2). In that case, I am also better o defecting (row 2) since I receive a payo of 1 instead of 0 (if I were to have cooperated). Therefore, no matter what you choose, I am better o (from a pure payo point of view) if I defect. What makes this game such a useful paradigm for strategic interactions ranging from economics, political science, biology, and even psychology [9, 178, 206] is the following additional observation. You will analyze the game in exactly the same way I did (just switch the roles of me and you in the previous rational analysis), so you will also decide to defect no matter what I do. The upshot if we both defect is that we will each receive a payo of 1, instead of each receiving a payo of 3 if we had both chosen to cooperate. The defect-defect combination is a Nash equilibrium [131, 130], and yet it is sub-optimal for both players and for the system as a whole. Rational thought rules out the cooperate-cooperate combination which would be better for each player (3 points each) and for both players combined (6 points). In fact, the Nash equilibrium strategy of defect-defect is the worst possible system wide choice, yielding a total payo of 2 points, compared to the cooperate-defect or defect-cooperate combination, which yields a total payo of 5 points, or the best system-wide strategy of cooperate-cooperate yielding a total payo of 6 points. The game becomes even more interesting if it is played repeatedly [206], stochastically [197], and with spatial structure [116] with each player allowed to decide what strategy to use on each interaction so as to accumulate a higher payo than the competition over a sequence of N games. In order to analyze this kind of an evolving set-up, a tness function must be introduced based on the payo matrix A. Let us now switch our terminology so that the relevance to tumor cell kinetics becomes clear. When modeling cell competition, one has to be careful about the meaning of the term `choosing a strategy'. Cells do not choose a strategy, but they do behave in dierent ways depending on whether they are normal healthy cells cooperating as a cohesive group, with birth and death rates that statistically balance, or whether they are cancer cells with an overactive cell division mechanism (as triggered by the presence of oncogenes) and an underactive `break' mechanism (as triggered by the absence of tumor suppressor genes) [207]. In our context, it is not 120 8 APPENDIX I: THE PRISONER'S DILEMMA AS A CANCER MODEL the strategies that evolve, as cells cannot change type based on strategy (only based on mutations), but the prevalence of each cell type in the population is evolving, with the winner identied as the sub-type that rst saturates in the population. 8.4 A tumor growth model Consider a population of N cells driven by a stochastic birth-death process as depicted in Figure 50, with red cells depicting cancer cells (higher tness) and blue cells depicting healthy cells (lower tness, but cooperative). We model the cell population as a stochastic Moran process [210] of N cells, `i' of which are cancerous, `Ni' of which are healthy. If each cell had equal tness, the birth-death rates would all be equal and a statistical balance would ensue. At each step, a cell is chosen (randomly but based on the prevalence in the population pool) and eliminated (death), while another is chosen to divide (birth). If all cells had equal tness, the birth/death rates of the cancer cells would be i=N, while those of the healthy cells would be (Ni)=N. With no mechanism for introducing a cancer cells in the population, the birth/death rates of the healthy cells would be equal, and no tumor would form. Now, introduce one cancer cell into the population of healthy cells, as shown in Figure 50A. At each step, there would be a certain probability of this cell dividing (P i;i+1 ), being eliminated (P i;i1 ), or simply not being chosen for either division or death (P i;i ). Based on this random process, it might be possible for the cancer cells to saturate the population, as shown by one simulation in Figure 51 depicting N = 1000 cells, with initially i = 1 cancer cell, and Ni = 999 healthy cells. However, the growth curve would not show any distinct shape (Figure 51 (black)), and might well become extinct after any number of cell divisions, as opposed to reaching saturation. But we emphasize that without mutational dynamics, heritability, and natural selection operating on the cell population, the shape of the growth curve would look random, and we know this is not how tumors tend to grow [111, 112]. By contrast, Figure 51 (red) shows a Gompertzian growth curve starting with exponential growth of the cancer cell subpopulation, followed by linear growth, ending with saturation. The growth rate is not constant throughout the full history of tumor development, but after an initial period of exponential growth, the rate decelerates until the region saturates with cancer cells. The basic ingredients necessary to sustain Gompertzian growth seem to be: an underlying stochastic engine of developing cells, mutational dynamics, heritability, and a tness landscape that governs birth and death rates giving rise to some sort of natural selection. 121 8 APPENDIX I: THE PRISONER'S DILEMMA AS A CANCER MODEL Figure 50: Schematic of the Moran Process | (a) The number of cancer cells,i, is dened on the state spacei = 0; 1;:::;N whereN is the total number of cells. The cancer population can change at most by one each time step, so a transition exists only between statei andi1,i, andi+1. (b) During each time step, a single cell is chosen for reproduction, where an exact replica is produced. With probability m (0m 1), a mutation may occur. 8.4.1 Mutations and heritability Each of theN cells in our simulated population carries with it a discrete packet of information that represents some form of molecular dierences among the cells. In our model, we code this information in the form of a 4-digit binary string from 0000 up to 1111, giving rise to a population made up of 16 distinct cell types. At each discrete step in the birth-death process, one of the digits in the binary string is able to undergo a point mutation [62, 122], where a digit spontaneously ips from 0 to 1, or 1 to 0, with probability p m . The mutation process is shown in Figure 50, while a mutation diagram is shown in Figure 52 in the form of a directed graph. This gure shows the possible mutational transitions that can occur in each cell, from step to step in a simulation. A typical simulation begins with a population of N healthy cells, all with identical binary strings 0000. The edges on the directed graph represent possible mutations that could occur on a given step. The rst 11 binary string values (0-10) represent healthy cells in our model that are at dierent stages in their evolutionary progression towards becoming a cancer cell (the exact details of this genotype 122 8 APPENDIX I: THE PRISONER'S DILEMMA AS A CANCER MODEL Figure 51: Emergence of Gompertzian growth via selection | Random drift (black) plotted for a single simulation of 10 3 cells for 4 10 4 generations shows no particular shape. A single simulation of the Moran process (red) with selection (w = 0:5) and mutations (m = 0:1) gives rise to the characteristic S-shaped curve associated with Gompertzian growth. to phenotype map do not matter much). Mutations strictly within this subpopulation are called passenger mutations as the cells all have the same tness characteristics. The rst driver mutation occurs when a binary string reaches value 11-15. The rst cell that transitions from the healthy state to the cancerous state is the renegade cell in the population that then has the potential to clonally expand and take over the population. How does this process occur? 8.4.2 The tness landscape At the heart of how the Prisoner's Dilemma evolutionary game dictates birth and death rates which in turn control tumor growth, is the denition of cell tness. Let us start by laying out the various probabilities of pairs of cells interacting and clearly dening payos when there are i cancer cells, and Ni healthy cells in the population. The probability that a healthy cell interacts with another healthy cell is given by (Ni 1)=(N 1), whereas the probability that a healthy cell interacts with a cancer cell is i=(N 1). The probability that a cancer cell interacts with a healthy cell is (Ni)=(N 1), whereas the probability that a cancer cell interacts with another cancer cell is (i 1)=(N 1). In a xed population of N cells, with i cancer cells, the number of healthy cells is given by Ni. The average payo of a single cell ( H ; C ), is dependent on the payo matrix value weighted by the relative frequency of types in the current population: 123 8 APPENDIX I: THE PRISONER'S DILEMMA AS A CANCER MODEL Figure 52: Markov Point Mutation Diagram | Diagram shows 16 genetic cell types based on 4-digit binary string and the eect of a point mutation on each cell type. Blue indicates healthy cell type (0000 | 1010), red indicates cancerous cell type (1011 | 1111). Black arrows indicate passenger mutations (healthy to healthy or cancer to cancer), red arrows indicate driver mutations (healthy to cancer). H i = a(Ni 1) +bi N 1 (113) C i = c(Ni) +d(i 1) N 1 (114) Here, a = 3, b = 0, c = 5, d = 1 are the entries in the Prisoner's Dilemma payo matrix (112). For the Prisoner's dilemma game, the average payo of a single cancer cell is always greater than the average payo for a healthy cell (Figure 53C). With the invasion of the rst cancer cell, the higher payo gives a higher probability of survival when in competition with a single healthy cell. Selection acts on the entire population of cells as it depends not on the payo, but on the eective tness of the subtype population. The eective tness of each cell type (f H , f C ) is given by the relative contribution of the payo of that cell type, weighted by the selection pressure: f H i = 1w +w H i (115) and the tness of the cancer cells as: 124 8 APPENDIX I: THE PRISONER'S DILEMMA AS A CANCER MODEL f C i = 1w +w C i (116) The probability of birthing a new cancer cell depends on the relative frequency (random drift) weighted by the eective tness, and the death rate is proportional to the relative frequency. The transition probabilities can be written: P i;i+1 = if C i if C i + (Ni)f H i Ni N (117) P i;i1 = (Ni)f H i if C i + (Ni)f H i i N (118) P i;i = 1P i;i+1 P i;i1 ; P 0;0 = 1; P N;N = 1: (119) In the event of the rst driver mutation, the rst cancer cell is birthed. At the beginning of the simulation, the eective tness of the healthy population is much greater than the tness of the cancer population (Figure 53B). This is because although the single cancer has a higher payo than any of the healthy cells, the number of healthy cells far outnumber the single cancer cells. That single cancer cell initiates a regime of explosive high growth and the tness of the cancer population steadily increases. Cancer cells are continually competing with healthy cells and receiving a higher payo in this regime (compare the payo entries of a cancer cell receiving c = 5 vs a healthy cell receiving b = 0). At later times, growth slows because cancer cells are competing in a population consisting mostly of other cancer cells. The payo for a cancer cell is dramatically lower when interacting with a cancer cell (observe the payo entry of both cancer cells receiving d = 1 when interacting). As the cancer population grows, the payo attainable decreases and growth slows. In addition, the average tness of the total population steadily declines because each interaction derives less total payo, fromc+b = 5 tod+d = 1. It is precisely the payo structure of the Prisoner's Dilemma matrix that produces this declining average population tness as the cancer cells saturate the population. Although they receive higher payos than healthy cells on pairwise interactions, these cancer-healthy interactions mostly take place early on in the evolution of the tumor. As the cancer cells take over the population, most of the interactions take place between pairs of cancer cells (i.e. they eventually start competing with each other) causing the population tness to decline. This complex process of competition among cell types and survival of subpopulations, where defection is selected over cooperation, produces a Gompertzian growth curve shown in Figure 54, and compared with a compilation based on a wide range of data rst shown in [111, 112]. It is now well established that tumor cell 125 8 APPENDIX I: THE PRISONER'S DILEMMA AS A CANCER MODEL Figure 53: Tumor tness drives tumor growth | (A) The average of 25 stochastic simulations (N = 1000 cells,w = 0:5,m = 0:1) is plotted for 20,000 cell divisions to show the cancer cell population (defectors) saturating. The pink lines delineate the regions of tumor growth (dened by the maximum and minimum points of the second-derivative of i(t)). (B) Fitness of the healthy population, cancer population, and total population plotted for the range cancer cell proportion. (C) Average payo of a single healthy cell, cancer cell, and all cells plotted for the range cancer cell proportion. populations (and other competing populations, such as bacteria and viral populations) generally follow this growth pattern, although the literature is complicated by the fact that dierent parts of the growth curve have vastly dierent growth rates [111, 112], and it is nearly impossible to follow the growth of a population of cancer cells in vivo from the rst cancer cell through to an entire tumor made up of O(10 9 10 12 ) cells. Growth rates are typically measured for a short clinical time period [111, 112], and then extrapolated back to the rst renegade cell, and forward to the fully developed tumor population. 126 8 APPENDIX I: THE PRISONER'S DILEMMA AS A CANCER MODEL Figure 54: Moran Process t to Gompertzian Growth Data | The mean and deviation of 25 stochastic simulations (N = 10 3 cells, w = 0:7, m = 0:3) is overlaid on data from a \normalized" Gompertzian [111, 112]. Values for m and w were chosen by implementing a least-squares t to the data over a range of m (0 m 1), and w (0 w 1). Pink lines delineate regions of growth (dened by the maximum and minimum points of the second-derivative of i(t)). 8.4.3 Heterogeneity drives growth Insights into the process by which growth rates vary and conspire to produce a Gompertzian shape can be achieved by positing that growth is related to molecular and cellular heterogeneity of the developing population [27, 100, 186]. Indeed, an outcome of the model is that molecular heterogeneity (i.e. the dynamical distribution of the 4-digit binary string 0000|1111 making up the population of cells) drives growth. Consider entropy [40, 134] of the cell population as a measure of heterogeneity: E(t) = N X i=1 p i log 2 p i (120) (here, log is dened as base 2). The probabilityp i measures the proportion of cells of typei, withi = 1;:::; 16 representing the distribution of binary strings ranging from 0000 to 1111. We typically course-grain this distribution further so that cells having strings ranging from 0000 up to 1010 are called `healthy', while those ranging from 1011 to 1111 are `cancerous'. Then growth is determined by: 127 8 APPENDIX I: THE PRISONER'S DILEMMA AS A CANCER MODEL dn E dt =E(t) (121) It follows from (133) that the cancer cell proportion n E (t) can be written in terms of entropy as: n E (t) = Z t 0 E(t)dt (122) This relationship between growth of the cancer cell population and entropy is pinned down and detailed in [210]. We consider it to be one of the key emergent features of our simple model. A typical example of the emergence of genetic heterogeneity in our model system is shown in the form of a phylogenetic tree in Figure 55. This particular tree is obtained via a simulation of only 30 healthy phenotypic cells (0000), which during the course of a simulation expand out (radially in time) to form a much more heterogeneous population of cells at the end of the simulation. In our model, the genetic time-history of each cell is tracked and the population can be statistically analyzed after the simulation nishes. 8.5 Simulated drug dosing strategies and therapeutic response Figure 56 shows the clear advantage of early stage therapy in our model system. We compare the eect of therapy given at an early stage, mid-stage, and late stages of the Gompertzian growth of the tumor. The dashed black Gompertzian curve is the freely growing cancer cell population. In each of the gures, we depict the eect of a range of drug dose densities, D, where D =ct: (123) The dose density is the product of the drug concentration, c, and the time over which the therapy is administered, t, (123). Here, the drug concentration value is a weighting (0c 1) which determines the intensity of the drug treatment. A higher value of c will alter the selection pressure in favor of healthy cells (and to the disadvantage of cancer cells) more dramatically. Figure 56 varies the drug dose density by varying the drug concentrations (c = 0:2; 0:4; 0:6; 0:8; 1:0) administered for a constant time (t = 5000 cell divisions, black arrow). The colored curves show the subsequent decline of the cancer cell population under therapeutic pressure. Clearly, to obtain the desired eect of driving the cancer cell population down to manageable levels, one needs to (i) use a sucient dose density, and (ii) initiate therapy early enough in the growth history. These gures are meant to paint a broad brush with respect to the simulated advantages of early therapy and to show the capability of the model with respect to addressing questions of this type in a quantitative way. A detailed investigation is left 128 8 APPENDIX I: THE PRISONER'S DILEMMA AS A CANCER MODEL Figure 55: Sample Dendritic Phylogenetic Tree | Sample dendritic phylogenetic tree tracking point mutations as time extends radially, depicting the emergence of molecular heterogeneity. The tree shows a simulation of 30 cells (all with genetic string 0000 at the beginning of the simulation) with strong selection (w = 1, m = 0:2). Pathways are color coded to indicate genetic cell type. for a separate publication. An established empirical law which relates drug dose density to its eectiveness in killing o cancer cells is known as the `log-kill' law [177]. The log kill law states that a given dose of chemotherapy kills the same fraction of tumor cells (as opposed to the same number of tumor cells), regardless of the size of the tumor at the time the therapy is administered [177], a consequence of exponential growth with a constant growth rate. This eect is best illustrated on a dose-response curve, plotting the dose density, D, with respect to the probability of tumor cell survival, P S . Thus, the log-kill law states the following: log(P S ) =D: (124) As an example, if there are 1000 cancer cells in a tumor population, and the rst therapy dose kills o 90% of them (i.e. = 0:9), then after the rst round of therapy there will be 100 cancer cells remaining. If 129 8 APPENDIX I: THE PRISONER'S DILEMMA AS A CANCER MODEL a second round of therapy is administered, exactly as the rst round, starting soon enough so that no new cancer cells have formed, then this next round will also kill o 90% of the cells, leaving 10 cells, and so on for each future round of therapy. In a sense, since the rst round killed 900 cells, while the second identical round killed only 90 cells, the population gets increasingly more dicult to kill o using the same treatment on each cycle. The log-kill law, a fundamentally static law (it does not say anything about the relationship of the fraction of cells killed vs. the growth rate of the tumor), is veried in our model system, as shown in the dose response curve in Figure 57D. On the x-axis, we increase the dose density D, and we plot the number of surviving cancer cells. The slope of this straight line (verifying the log-kill law) is the rate of regression of the tumor, . Alternatively, can be estimated using an exponential t of i(t) during therapy (i.e. i(t) =i 0 exp((tt 0 )), where i 0 is the initial tumor size and t 0 is the time therapy is initiated). So how is the rate of regression, , related to the growth rate of the tumor, ? This is relevant, since we know from the shape of the Gompertzian curve, the growth rate is highest (exponential) at the beginning stage of tumor development and lowest at the late saturation stage. Figure 57A shows therapy is more eective (i.e. a higher rate of regression, ) for earlier stage therapy. These early stage therapies correspond to a higher growth rate, shown in Figure57B. The Norton-Simon hypothesis [139, 140, 140] states that the rate of regression is proportional to the instantaneous growth rate for an untreated tumor of that size at the time therapy is rst administered. A faster growing tumor (early stage) should show a higher rate of regression than a more slowly growing tumor (late stage). This hypothesis is also veried in our model system, and shown clearly in Figure 57C. The reality of this growth-dependent tumor regression rate eect (where early stage faster growing tumors are more vulnerable to therapy than later stage, slowly growing tumors) dramatically reinforces the need to administer drug treatment early in tumor progression when growth rates are high and there are fewer cancer cells to kill o. 8.6 Mathematical modeling and tumor analytics It is important to keep in mind that no mathematical model captures all aspects of reality, so choices must be made which involve prioritizing the features that are most essential in capturing the essence of a complex process and which are not. Most experts now agree that the evolutionary processes in a tumor played out among subpopulations of competing cells are key to understanding aspects of growth and resistance to chemotherapy, which will ultimately lead the way toward a quantitative understanding of tumor growth and cancer progression [126, 208, 207]. The paradigm of the cancer cell subpopulation and the healthy cell subpopulation competing as the defectors and cooperators in a Prisoner's Dilemma evolutionary game has been useful in obtaining a quantitative handle on many of these processes and frames the problem in an 130 8 APPENDIX I: THE PRISONER'S DILEMMA AS A CANCER MODEL intuitive yet predictive way. Nonetheless, the mathematical `taste' of the modeler plays a role in what techniques are selected and ultimately where the spotlight shines. This fact makes clinicians uncomfortable and can lead to deep suspicion of the mathematical modeling enterprise as a whole. Aren't the outcomes and predictions of mathematical models a straightforward consequence of the model assumptions? Once those choices are made, isn't the cake already baked? So why should we be surprised if you tell us it tastes good? Why not simply use tried and true statistical tools like regression methods to curve-t the data directly, with no built in assumptions, and be satised with uncovering correlations and trends? Clinicians (and experimentalists, in general) feel that they are dealing directly with reality, so why mess around with `toy' systems based on possibly `ad hoc' or incorrect assumptions that create articial realities that may or may not be relevant? To a theoretician, calling their assumptions ad hoc, as opposed to natural, is as insulting as calling a clinician sloppy and uncaring (try this for yourself at the next conference you go to! But please use the term `somewhat ad hoc' to lessen the blow.) And if you want to deliver an even harsher insult, you could comment that the model seems like an exercise in curve tting. But the usefulness of mathematical models built on simplied assumptions is well established in the history of the physical sciences, as detailed beautifully in Peter Dear's book, The Intelligibility of Nature: How Science Makes Sense of the World [45]. Bohr's simple model of the structure of the atom was crucial in moving the community forward towards a deeper understanding of cause and eect, and ultimately pushing others to develop more realistic atomic models. The same could be said for many other important, but ultimately discarded models of reality (e.g. the notion of aether used as a vehicle to understand the mysterious notion of action-at-a-distance [45]) now relegated to footnotes in the history of the physical sciences. Lessons from this history highlight the importance of using the principle of Occam's razor (law of parsi- mony) as a heuristic guide in developing models: (1) keep things simple, but not too simple; (2) see what can be explained by using a given set of assumptions, and try to identify what is either wrong or cannot be explained; (3) add complexity to the model, but do this carefully. Since ultimately, the model will always be wrong (with respect to some well chosen and specic new question being posed about a system), it is important that it be useful as a vehicle of intelligibility [45] associated with the set of questions surrounding the phenomena it was built to explain. Answers to some new questions will be found using the model as a temporary crutch, and new questions will emerge in the process that had not yet been asked, as their relevance had never previously been realized. A new quantitative language will emerge in which aspects of the model will be associated with the underlying reality it is attempting to describe, predictions will be easier to frame and test, and shortcomings will be exposed. In his famous article [212], Eugene Wigner writes 131 8 APPENDIX I: THE PRISONER'S DILEMMA AS A CANCER MODEL compellingly that `the miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. We should be grateful for it and hope that it will remain valid in future research and that it will extend, for better or for worse, to our pleasure, even though perhaps also to our baement, to wide branches of learning.' In general, the more complex the model (as measured, for example, by the number of independent parameters associated with it), the less useful it will be, and the less likely it is to be adopted by the community at large. After all, if the model is as complex as the phenomena it was built to understand, why not stick with reality? Eective models can be thought of as low-dimensional approximations of reality, surrogates that help us bootstrap our way forward. They arise as the outcome of a complex balancing act between simplicity of the ingredients, and complexity of the reality the model is meant to describe. They generally do not arise in a vacuum, but are built in the context of informed and sustained discussions among people with dierent expertise. In the context of medical oncology, this means physical scientists developing ongoing interactions with clinical oncologists, radiologists, pathologists, molecular and cell biologists and other relevant medical specialists. Appropriate data is a necessary ingredient in developing and testing any successful model, and treasure troves of medical data sit unexamined in patient les and government databases across the country waiting to be put to good use. There is no doubt that they are telling an interesting and important story that we have yet to fully understand. It is not currently possible for the computer to simulate all of the complex, relevant, and systemic ingredients at play to faithfully recreate all aspects of cancer progression and treatment response in patients. It is hard to imagine that a deep and actionable understanding can ever be obtained without the combined use of data, models, and computer simulations to help guide us and highlight some of the underlying causal mechanisms of this complex and deadly disease. 132 8 APPENDIX I: THE PRISONER'S DILEMMA AS A CANCER MODEL Figure 56: Eects of varied dose density for early-stage, mid-stage, and late-stage therapies | An average of 25 stochastic simulations of unperturbed tumor growth (N = 10 3 cells, w = 0:5, m = 0:1, no therapy) is plotted (black dashed line). The eect of varied drug dose density (eqn. 123), is shown by administering a range of drug concentration values (c = 0:2; 0:4; 0:6; 0:8; 1:0) for constant length of time (t = 5000 cell divisions, black solid arrows). This process is repeated for (A) high growth, early-stage, (B) linear growth, mid-stage, and (C) slowed growth, late-stage. The kill eect is highest for high drug concentration values (i.e. high dose density) and early therapy. 133 8 APPENDIX I: THE PRISONER'S DILEMMA AS A CANCER MODEL Figure 57: Growth-dependent tumor regression | (A) An average of 25 stochastic simulations of unperturbed tumor growth (N = 10 3 cells, w = 0:5, m = 0:1, no therapy) is plotted (black dashed line) with (B) corresponding instantaneous growth rate, (t), of the unperturbed tumor (red). Tumor regression, , (estimated using an exponential t of i(t) during therapy, shown in legend) during therapy (constant dose density: c = 1:0, t = 2000) is calculated for a high growth, early-stage therapy (purple), linear-growth, mid-stage therapy (green), and late-stage, slowed growth (light blue); (C) This process is repeated for a full range of growth rates (between vertical blue dashed lines). Average values of are plotted with standard deviations. Regression is proportional to growth rate (linear t in red), with higher regression rates associated with high growth rates of early stage tumors. (D) Tumor regression, , can also be calculated as the slope of a dose response curve (red), where therapy is administered for a range of dose densities (0:7 c 1:0) for a single timepoint, 8000 cell divisions (i.e. single growth rate). 134 9 APPENDIX II: AN EVOLUTIONARY MODEL OF TUMOR CELL KINETICS AND THE EMERGENCE OF MOLECULAR HETEROGENEITY DRIVING GOMPERTZIAN GROWTH 9 Appendix II: An evolutionary model of tumor cell kinetics and the emergence of molecular heterogeneity driving Gompertzian growth 9.1 Abstract We describe a cell-molecular based evolutionary mathematical model of tumor development driven by a stochastic Moran birth-death process. The cells in the tumor carry molecular information in the form of a numerical genome which we represent as a four-digit binary string used to dierentiate cells into 16 molecular types. The binary string is able to undergo stochastic point mutations that are passed to a daughter cell after each birth event. The value of the binary string determines the cell tness, with lower t cells (e.g. 0000) dened as healthy phenotypes, and higher t cells (e.g. 1111) dened as malignant phenotypes. At each step of the birth-death process, the two phenotypic sub-populations compete in a prisoner's dilemma evolutionary game with the healthy cells playing the role of cooperators, and the cancer cells playing the role of defectors. Fitness, birth-death rates of the cell populations, and overall tumor tness are dened via the prisoner's dilemma payo matrix. Mutation parameters include passenger mutations (mutations conferring no tness advantage) and driver mutations (mutations which increase cell tness). The model is used to explore key emergent features associated with tumor development, including tumor growth rates as it relates to intratumor molecular heterogeneity. The tumor growth equation states that the growth rate is proportional to the logarithm of cellular diversity/heterogeneity. The Shannon entropy from information theory is used as a quantitative measure of heterogeneity and tumor complexity based on the distribution of the 4-digit binary sequences produced by the cell population. To track the development of heterogeneity from an initial population of healthy cells (0000), we use dynamic phylogenetic trees which show clonal and sub-clonal expansions of cancer cell sub-populations from an initial malignant cell. We show tumor growth rates are not constant throughout tumor development, and are generally much higher in the subclinical range than in later stages of development, which leads to a Gompertzian growth curve. We explain the early exponential growth of the tumor and the later saturation associated with the Gompertzian curve which results from our evolutionary simulations using simple statistical mechanics principles related to the degree of functional coupling of the cell states. We then compare dosing strategies at early stage development, mid-stage (clinical stage), and late 135 9 APPENDIX II: AN EVOLUTIONARY MODEL OF TUMOR CELL KINETICS AND THE EMERGENCE OF MOLECULAR HETEROGENEITY DRIVING GOMPERTZIAN GROWTH stage development of the tumor. If used early during tumor development in the subclinical stage, well before the cancer cell population is selected for growth, therapy is most eective at disrupting key emergent features of tumor development. 9.2 Introduction At the molecular and cellular levels, cancer is an evolutionary process [126, 138, 138, 138] driven by random mutational events [138, 104, 192, 166] responsible for genetic diversication which typically arises via waves of clonal and sub-clonal expansions [75, 146], operating over an adaptive tness landscape in which Darwinian selection favors highly proliferative cell phenotypes which in turn drive rapid tumor growth [111, 65, 17]. The tumor environment should be viewed as a complex Darwinian adaptive eco-system consisting of cell types which have evolved over many years [126]. As a result, all but the most well designed and tailored therapeutic strategies often deliver disappointing outcomes and potentially introduce a potent new source of selective pressure for the proliferation of variant cells which develop an enhanced ability to resist future therapeutic assaults [93, 54, 106, 105, 74]. The prospects for in uencing and controlling such a system are likeliest at the emerging early stages of tumor development when the cell population has not yet been selected for growth and survival, and the tumor size is small. But by the time a typical tumor becomes clinically detectable (often after several years of growth), it already contains O(10 8 ) or more malignant cells (and potentially occupies a volume of 1 2 mm 3 ), some of which may have entered the blood circulation [65]. Since there is very little human data available in this early subclinical stage of tumor development, computational models can serve as a useful surrogate in this critical developmental stage which clearly in uences and determines many important emergent features of the tumor at later stages. Our goals in this paper are to describe a mathematical model for stochastic cell kinetics in the beginning stages of tumor development (from a single malignant cell) that includes cell reproduction and death, muta- tions, evolution, and the subsequent emergence of genetic heterogeneity well documented in many soft-tissue tumors [133, 123, 55, 2, 85, 173, 71, 186, 86]. The model is a computational one, driven by a stochastic Moran (birth-death) process with a nite cell population, in which birth-death rates are functions of cell tness. The tness is determined by the cell's numerical genome in the form of a four-digit binary string capable of undergoing point mutational dynamics with one digit in the string ipping values stochastically. The corresponding numerical value of the binary string determines whether the cell is healthy (low-tness) or cancerous (high tness). These two classes of cells compete against each other at each birth-death event, with tness calculated according to the payo matrix associated with the prisoner's dilemma evolutionary zero-sum game [179, 143, 201, 10]. The healthy cells play the role of cooperators, while the cancer cells play 136 9 APPENDIX II: AN EVOLUTIONARY MODEL OF TUMOR CELL KINETICS AND THE EMERGENCE OF MOLECULAR HETEROGENEITY DRIVING GOMPERTZIAN GROWTH the role of defectors [143, 10]. Our goal is to understand how the model parameters: passenger (m p ) and driver mutation rates (m d ), selection strength (w), birth and death rates, aect tumor growth characteristics, such as tumor growth rates, xation probabilities of malignant and healthy cell types, saturation rates of cancer cells, and the emergence of genetic heterogeneity in a tumor at later stages of development when the tumor is clinically detectable. An important outcome of the model is that growth of the cancer cell population is directly in uenced by the intratumor heterogeneity (represented as the distribution of the 4-digit binary strings throughout the cell population), with high heterogeneity driving more rapid growth. The connection between heterogeneity and growth has been discussed in the literature [100, 112, 3, 209, 7, 86]. We quantify heterogeneity in a tumor using tools from information theory [39, 134], as well as quantitative analysis of phylogenetic trees associated with clonal and sub-clonal expansions [170, 186] in the developing tumor. Because our numerical simulations are carried out from initial conditions corresponding to a homogeneous population of healthy cells (0000) all the way to a saturated population of cancer cells, we can use the model to test basic dose and scheduling strategies [195, 1] at the very early stages of tumor development in the subclinical range, well before a tumor would be clinically detectable by current technology. Our point of view is that this emerging subclinical tumor should be more amenable (and potentially vulnerable) to a well planned therapeutic assault than a more mature tumor comprised (on average) of larger numbers of cells with more aggressive proliferative capabilities (having undergone generations of selection), that are potentially in the early stages of migration to other organs. More complex features that might in uence early stage dynamics, like human-immune response [3] and the tumor microenvironment [117] are not included in this model in order to keep things as simple and clear as possible. 9.3 Description of the model The ingredients in our model includes a stochastic birth-death process that is the engine which drives tumor growth, with heritable mutations operating over a tness landscape so that natural selection can play out over many cell division timescales. Genetic mutations (point mutations) are modeled using a four-digit binary string of information that each cell carries with it. 2 This simple sequence divides the cells into 16 dierent \genotypes", ranging from 0000 up to 1111, and this information is passed on to the daughter cell during a birth event. The birth-death replacement process is based on a tness function dened in terms of interactions quantied by the prisoner's dilemma payo matrix which operates on two general classes of cells: healthy (the cooperators), and cancerous (the defectors). Natural selection acts on each generation of the 2 To be clear, the four digit sequence is not meant as a bare-bones representation of the full human genome, but as a simple representation of the relevant dierences in genetic information contained in dierent cells, allowing us to course-grain the cells into 16 dierent categories based on their genetic/epigenetic proles. 137 9 APPENDIX II: AN EVOLUTIONARY MODEL OF TUMOR CELL KINETICS AND THE EMERGENCE OF MOLECULAR HETEROGENEITY DRIVING GOMPERTZIAN GROWTH cell population as the computational simulation proceeds on a cell division timescale. In this version of the model we typically simulate up to O(10 11 ) cell divisions, so our mutation rates are chosen to be relatively high to accommodate these somewhat modest timescales. See [192] for discussions on mutation rates in cancer. 9.3.1 The Moran birth-death process The stochastic engine [165] that drives tumor growth in our model is a nite cell-based Moran process consisting of a population of N cells, divided into two sub-populations consisting of i cancer cells, and Ni healthy cells. In all of our simulations, N is large enough so that there is not a signicant dierence between the results from our nite-cell model and the (deterministic) replicator equation approach for innite populations, a connection that is discussed in detail in [196]. At each time-step in the simulation, one cell is chosen for reproduction and one cell is chosen for elimination. The cells are chosen randomly, based on their prevalence in the population pool which, in turn, is weighted by the tness function based on a chosen payo matrix. The probability of choosing a cancer cell at any given step is i=N, while the probability of choosing a healthy cell is (Ni)=N. As it unfolds, the process is a stochastic birth-death process where the total population size,N, stays constant and the number of cancer cells in the population, i, is the stochastic state variable. At any given step, the probability of transitioning from i cancer cells to j cancer cells is denoted P ij , with j = i + 1 or j = i 1. These probabilities are determined by the birth/death rates associated with the cancer cell population, which in turn are determined by a cell population tness function. Each cell carries with it a binary string in the form of a four digit binary sequence from 0000 up to 1111. This denes 16 dierent cell types, which are course-grained into two groups: healthy cells (0000 - 1010), and cancer cells (1011-1111). These two sub-populations interact at each birth-death time-step with tness dened in terms of the prisoner's dilemma payo matrix. The algorithmic details are shown in the appendix Figure 67. To set the stage for more complex simulations, Figure 58 shows the result of a stochastic simulation (depicting i) driven by the Moran process alone, with no mutations, and no selection. Figure 58 shows three dierent simulations, one leading to the elimination of all cancer cells via random drift (red), another uctuates between a mixed cell population after 10,000 cell divisions (yellow), and a third leading to xation of the cancer cell population (blue) after around 5000 cell divisions. The average of 25 stochastic simulations is also plotted (note that the average will converge to half cancer cells and half healthy cells by the law of large numbers). The mean time to xation of the cancer cell population which starts with `i' cells in this simple setting (no mutations, no selection) is given by 138 9 APPENDIX II: AN EVOLUTIONARY MODEL OF TUMOR CELL KINETICS AND THE EMERGENCE OF MOLECULAR HETEROGENEITY DRIVING GOMPERTZIAN GROWTH k =N 2 4 i X j=1 Ni Nj + N1 X j=i+1 i j 3 5 : (125) With no mechanism for natural selection, there is no shape to the growth curves. Figure 58: Stochastic Moran birth-death process | Cancer cell population,i(t), during three stochastic simulations of the Moran birth-death process in a population of 100 cells and an initial condition of i = 50 cells. The blue curve leads to xation of the cancer cell population, the red curve leads to elimination of the cancer cell population, and the yellow curve remains uctuating in a mixed population of cells after 10,000 cell divisions. An average of 25 stochastic simulations (black dashed line) is also plotted. 9.3.2 The prisoner's dilemma payo matrix To introduce the eect of selection which will regulate cell birth and death rates, we use the prisoner's dilemma evolutionary game in which two players compete against each other for the best payo. Each has to decide whether to cooperate (healthy cell) or defect (cancer cell) and each receives a payo determined from the prisoner's dilemma payo matrix 3 , A: A = 0 B @ a b c d 1 C A = 0 B @ 3 0 5 1 1 C A: (126) 3 What denes a prisoner's dilemma matrix are the inequalities c > a > d > b. The chosen values in (126) are relatively standard, but not unique. More discussion of why the prisoner's dilemma matrix, which models the evolution of defection, is a useful paradigm for cancer can be found in [211] and some of the references therein. 139 9 APPENDIX II: AN EVOLUTIONARY MODEL OF TUMOR CELL KINETICS AND THE EMERGENCE OF MOLECULAR HETEROGENEITY DRIVING GOMPERTZIAN GROWTH The essence of the prisoner's dilemma game is the two players compete against each other, and each has to decide what best strategy to adopt in order to maximize their payo. This 2 x 2 matrix assigns the payo (e.g. reward) to each player on each interaction. My options, as a strategy or, equivalently, as a cell type, are listed along the rows, with row 1 associated with my possible choice to cooperate, or equivalently my cell type being healthy, and row 2 associated with my possible choice to defect, or equivalently my cell type being cancerous. Your options are listed down the columns, with column 1 associated with your choice to cooperate (or you being a healthy cell), and column 2 associated with your choice to defect (or you being a cancer cell). The analysis of a rational player in a prisoner's dilemma game runs as follows. I do not know what strategy you will choose, but suppose you choose to cooperate (column 1). In that case, I am better o defecting (row 2) since I receive a payo of 5 instead of 3 (if I also cooperate). Suppose instead you choose to defect (column 2). In that case, I am also better o defecting (row 2) since I receive a payo of 1 instead of 0 (if I were to have cooperated). Therefore, no matter what you choose, I am better o (from a pure payo point of view) if I defect. What makes this game such a useful paradigm for strategic interactions ranging from economics, political science, biology [211], and even psychology [143] is the following additional observation. You will analyze the game in exactly the same way I did (just switch the roles of me and you in the previous rational analysis), so you will also decide to defect no matter what I do. The upshot if we both defect is that we will each receive a payo of 1, instead of each receiving a payo of 3 if we had both chosen to cooperate. The defect-defect combination is a Nash equilibrium [179], and yet it is sub-optimal for both players and for the system as a whole. Rational thought rules out the cooperate-cooperate combination which would be better for each player (3 points each) and for both players combined (6 points). In fact, the Nash equilibrium strategy of defect-defect is the worst possible system wide choice, yielding a total payo of 2 points, compared to the cooperate-defect or defect-cooperate combination, which yields a total payo of 5 points, or the best system-wide strategy of cooperate-cooperate yielding a total payo of 6 points. The game becomes even more interesting if it is played repeatedly [179, 143, 201, 10], with each player allowed to decide what strategy to use on each interaction so as to accumulate a higher payo than the competition over a sequence of N games. In order to analyze this kind of an evolving set-up, a tness function must be introduced based on the payo matrix A. Let us now switch our terminology so that the relevance to tumor cell kinetics becomes clear. In this case, we randomly select pairs of cells out of the total population at each step, and subject them to a birth-death process, basing our birth rates and death rates on the prisoner's dilemma payo matrix. Thus, in our context, it is not the strategies that evolve, as cells cannot change type based on strategy (only based on mutations), but the prevalence of each cell type in the population is evolving, with the winner identied as the sub-type that rst reaches xation in the population. As the populations evolve, the tness of the two competing sub-populations can be tracked, as 140 9 APPENDIX II: AN EVOLUTIONARY MODEL OF TUMOR CELL KINETICS AND THE EMERGENCE OF MOLECULAR HETEROGENEITY DRIVING GOMPERTZIAN GROWTH well as the overall tness of the combined total population of cells. 9.3.3 The tness landscape Let us start by laying out the various probabilities of pairs of cells interacting and clearly dening payos when there are i cancer cells, and Ni healthy cells in the population. The probability that a healthy cell interacts with another healthy cell is given by (Ni 1)=(N 1), whereas the probability that a healthy cell interacts with a cancer cell is i=(N 1). The probability that a cancer cell interacts with a healthy cell is (Ni)=(N 1), whereas the probability that a cancer cell interacts with another cancer cell is (i 1)=(N 1). The payos associated with the healthy cells and cancer cells, obtained by weighting the payo matrix values with appropriate probabilities, are given by (following notation in [196]): H = 3(Ni 1) + 0i N 1 ; (127) C = 5(Ni) + 1(i 1) N 1 : (128) This gives rise to the average payo associated with the population of cells: hi = H (Ni) + C (i) N : (129) Based on these formulas, we dene the tness of the healthy cells as: f H = 1w H +w H H ; (130) and the tness of the cancer cells as: f C = 1w C +w C C : (131) Here, (w H ;w C ) are `selection strength' parameters, 0 w H 1; 0 w C 1, that measure the strength of selection pressure on each of the population of cells. If w H = 0, there is no natural selection acting on the healthy cell population and the dynamics is driven purely by the Moran process. When w H = 1, the selection pressure on the healthy cell population is strongest and the prisoner's dilemma payo matrix has maximum eect. Likewise for the parameter w C and how it controls selection pressure in the cancer cell population. Since therapy imposes selection pressure on dierent sub-populations of cells, w H and w C are the two parameters we alter to administer simulated therapeutic responses. We discuss this in sectionx3.5. 141 9 APPENDIX II: AN EVOLUTIONARY MODEL OF TUMOR CELL KINETICS AND THE EMERGENCE OF MOLECULAR HETEROGENEITY DRIVING GOMPERTZIAN GROWTH The expected tness of each of the sub-populations are: H = Ni N f H ; (132) C = i N f C ; (133) with total expected tness: = H i + C : (134) From these formulas, we can dene the transition probability of going from i toi + 1 cancer cells on a given step: P i;i+1 = if C if C + (Ni)f H Ni N : (135) The rst term represents that probability that a cancer cell is selected for reproduction (weighted by tness), and a healthy cell is selected for death. Likewise, the transition probability of going from i to i 1 cancer cells on a given step is: P i;i1 = (Ni)f H if C + (Ni)f H i N : (136) Here, the rst term is the probability healthy cell is selected for reproduction (weighted by tness), and a cancer cell is selected for death. The remaining transition probabilities are as follows: P i;i = 1P i;i+1 P i;i1 ; P 0;0 = 1; P N;N = 1: (137) It is these simple formulas that drive the subsequent dynamics of the competing populations of cells and determine the emergent features of the forming tumor (cancer cell population). A typical set of simulations of the evolving tness of the healthy cell population, H , the cancer cell population C , and the total tness, , is shown in Figure 59 as the selection parameter varies from 0 to 1 (w H =w C w). As the population evolves, the tness of the healthy cell population decreases, the tness of the cancer cell population increases (sometimes reaching a maximum point), while the total population tness decreases. 142 9 APPENDIX II: AN EVOLUTIONARY MODEL OF TUMOR CELL KINETICS AND THE EMERGENCE OF MOLECULAR HETEROGENEITY DRIVING GOMPERTZIAN GROWTH Figure 59: Fitness as a function of the selection parameter w w H w C | (A) Monotonically decreasing tness of healthy cell sub-population H ; (B) Fitness of cancer cell sub-population C . Note that C has a maximum at i = N 2 + (N1) 8w , which is between 0 and N for w > 1 4 (1 1=N); (C) Monotonically decreasing tness of the total population, . 9.3.4 Passenger and driver mutations Two remaining parameters in our model are the passenger mutation rate, m p and the driver mutation rate, m d [192]. Passenger mutations confer no tness advantage, hence m p controls point mutations that act on the digit strings that dene the 11 levels of healthy cells 0000-1010, and the 5 levels of cancer cells 1011-1111. A mutation diagram is shown in Figure 60 depicting all of the possible point mutation transitions at each step. Mutations that stay within either of those two ranges do not alter the cell tness. On the other hand, the driver mutation parameter controls mutations that take a binary string from a healthy cell and, via a point mutation, alter it so that the string becomes a cancer cell 4 . A simple example would be a mutation that alters 1010 (healthy) to 1011 (cancer) by stochastically ipping the rst digit from 0 to 1. The interested reader can consult the ow diagram in Figure 67 of the Appendix for more details of the algorithm. The full code is available from the authors upon request. 9.4 Results Gompertzian growth arising from multicellular systems occurs in many settings with dierent physical and biological constraints acting in concert. Hence it appears as if this universal growth curve does not depend on specic physical mechanisms (e.g. oxygen diusion, blood supply, tumor microenvironment, etc.) but more on multi-cellularity and the ability for populations of cells to assume a heterogeneous distribution of functional states, as was described most clearly in Kendal's 1985 paper [100] and documented clinically in 4 In our simulations, we assume that driver mutations cannot revert to passenger mutations, i.e. once a cancer cell is born, it stays in that category. We do not know of any evidence in the literature that shows the reversion of a cancer cell to a healthy cell, nor is this particularly a focus of this manuscript. 143 9 APPENDIX II: AN EVOLUTIONARY MODEL OF TUMOR CELL KINETICS AND THE EMERGENCE OF MOLECULAR HETEROGENEITY DRIVING GOMPERTZIAN GROWTH Figure 60: Markov Point Mutation Diagram | Left: diagram shows 16 genetic cell types based on 4-digit binary string and the eect of a point mutation on each cell type. Blue indicates healthy cell type (0000 | 1010), red indicates cancerous cell type (1011 | 1111). Black arrows indicate passenger mutations (healthy to healthy or cancer to cancer), red arrows indicate driver mutations (healthy to cancer). Top right: 3 scenarios may occur during the reproduction process: no mutation, passenger mutation, or driver mutation. breast [1] and other tumor types. Alternative bio-mechanistic models of tumor growth at the cellular level have been developed (see [163, 42, 213, 117, 66, 102]) although do not generally include molecular information or evolutionary eects. Features of the Gompertzian growth curve dened by eqns (138), (139) allow us to clearly describe three distinct growth regimes, the earliest being subclinical and the most critical regime in which to in uence future tumor kinetics, the second being the clinical regime where growth measurements are typically obtained [65], and the third being the lethal burden phase where growth saturates. The growth equation, (146), relates tumor heterogeneity to growth rates, and we quantify heterogeneity via the Shannon entropy [39, 134] of the cellular population. One of the main features of our evolutionary simulations is to show how it (i) leads to Gompertzian growth, (ii) how growth is driven by heterogeneity quantitated via Shannon entropy, (iii) how the initiation of heterogeneity and tness can be tracked via dynamic phylogenetic trees, and (iv) how tumor kinetics can be in uenced via therapeutic strategies that target heterogeneity best in earlier growth regimes. In keeping consistent with the notation of the Gompeterzian growth curve, we now represent the tumor growth as the proportion of cancer cells in the population, n G (t). 144 9 APPENDIX II: AN EVOLUTIONARY MODEL OF TUMOR CELL KINETICS AND THE EMERGENCE OF MOLECULAR HETEROGENEITY DRIVING GOMPERTZIAN GROWTH 9.4.1 Gompertzian tumor growth and three growth regimes The basic (top-down) equations giving rise to pure Gompertzian growth [63, 109, 194, 118] are the coupled equations: dn G dt = n G ; (138) d dt = : (139) Here, is the proportion of growing cancer cells in the mixed population, which grows exponentially according to (138), but with a time-dependent growth rate which is exponentially decaying according to (139). It is straightforward to integrate (138) to obtain: n G (t) =N 0 exp 1 t Z t 0 dt t : (140) Then, (139) is solved with: (t) = 0 exp(t): (141) Plugging (141) into (140) and integrating yields the Gompertzian curve: n G (t) =N 0 exp h 0 (1 exp(t)) i ; (142) where in the long-time limit , the population saturates to the value n 1 =N 0 exp( 0 =); (143) which we normalize to one (without loss of generality). The key features of Gompertzian growth are shown in Figure 61. As the cancer cell proportion n G (t) increases (Figure 61A), there are three distinct growth regimes dened by the in ection point on the n G growth curve (maximum of _ n G and d 2 n G =dt 2 = 0), and the two in ection points on the growth-velocity curve _ n G (maximum/minimum of n G and d 3 n G =dt 3 = 0). As shown in Figure 3.1(a), there are three points that divide the growth curve into four distinct regions. For convenience, and symmetry, we lump the second and third regions together and dene three basic growth regimes: Regime 1 (Subclinical): Increasing velocity _ n G , increasing acceleration d 2 n G =dt 2 . Cell population 145 9 APPENDIX II: AN EVOLUTIONARY MODEL OF TUMOR CELL KINETICS AND THE EMERGENCE OF MOLECULAR HETEROGENEITY DRIVING GOMPERTZIAN GROWTH and tumor volume grows at an exponential rate; Regime 2 (Clinical): In this regime, _ n G reaches its maximum value. In the early part of the regime, _ n G is increasing whiled 2 n G =dt 2 decreases. In the later part of the regime, _ n G is decreasing andd 2 n G =dt 2 becomes negative (deceleration). Growth rates are clinically typically measured as linear; Regime 3 (Saturation/Lethal): Decreasing tumor velocity _ n G with decreasing deceleration. Growth rate rapidly slows towards full saturation of the cancer cell population. Figure 61: Gompertzian equation | Numerical simulation of the Gompertzian equation (138), (139) with parametersN 0 = 0:001, 0 = 10, and = 0:2895. The three regimes of tumor growth are demarcated by the blue dots in each subgure, representing the maximum and minimum of the second-derivative. (A) Cancer cell proportion, n(t), over time; (B) First- and second-derivatives of the tumor growth curve; (C) Growth rate, (t), over time, with the average growth rate in regimes 1, 2, 3 plotted in red; (D) First derivative of growth rate. Regime 1, generally speaking, is the subclinical growth regime where the developing tumor has substan- tially fewer than 10 8 malignant cells with a tumor size smaller than 1 or 2 mm 3 . Typically, the clinically measurable regime is Regime 2, while the lethal stage when the tumor saturates is associated with Regime 3. In reality, the boundaries of these regimes are, of course, not sharp and depend on tumor type and location which in uence detectability. But the clarity of the pure Gompertzian curve gives a useful framework which delineates the three distinct growth regimes based on clear principles associated with growth, velocity, and 146 9 APPENDIX II: AN EVOLUTIONARY MODEL OF TUMOR CELL KINETICS AND THE EMERGENCE OF MOLECULAR HETEROGENEITY DRIVING GOMPERTZIAN GROWTH acceleration. The growth rate curve is shown in Figure 61C, with its derivatives shown in Figure 61D. It is most instructive to show the average growth rates dened in each of the three regimes, also shown in the Figure 61C. The average growth rate in the time interval from t 1 to t 2 is dened as: ave = 1 t 2 t 1 Z t2 t1 (t)dt (144) The subclinical regime 1 has the highest average growth, whereas regime 2, where tumor growth is typically measured, average growth rates are lower, followed by the lowest average growth in the clinically lethal regime 3. This implies that clinically measured growth rates typically underestimate growth rates that preceded it in the subclinical stage. It also implies that linear extrapolation back from clinically measured growth rates to estimate tumor initiation times (see [111, 65, 17]) will systematically overestimate the amount of time the tumor has been developing before being measured. While this might generally be seen as good news (since the cancer initiation event was more recent than estimated via linear extrapolation), it also gives the clinician a shorter window of time in which to act. 9.4.2 Heterogeneity and growth via statistical mechanics Kendal [100] lays out a clear argument of how this growth curve arises from a purely statistical mechanics point of view. In a nutshell, his argument can be explained by considering a population of n cells, let the jth cell (j = 1; 2; 3;:::;n) have the potential to assume one of q j possible states. The number of combinations of states possible within the population, P , can be thought of as a measure of intra-neoplastic diversity: P =q 1 q 2 q 3 :::q n ; (145) and is related to the growth rate of a tumor via the equation: dn dt = logP; (146) where n(t) is the number of cells capable of proliferation at a given time t and is a parameter that sets the timescale of growth 5 . There are two basic cases to consider. First, suppose the cells have no interaction at all, say in the earliest stages of tumor development, and let each of the n cells have the ability to assume 5 Kendal's formulation [100] assumes a cell population made up of three sub-groups: (1) proliferative cells; (2) nonproliferative and nonclonogentic cells; (3) nonproliferative but clonogenic cells, with an assumption that the neoplasm's growth rate is in uenced by the proportion of proliferating to nonproliferating cells and an expression of each clone's growth potential. The log is chosen based on the fact that heterogeneity is measured as the multiplicative combination of achievable states in the tumor, and the requirement thatG(P 1 P 2 ) =G(P 1 ) +G(P 2 ) for any two sub-populationsP 1 ,P 2 and growth functionG. The discussion of the relationship between tumor heterogeneity and growth is an ongoing topic in the current literature [133, 123, 85, 71, 186]. 147 9 APPENDIX II: AN EVOLUTIONARY MODEL OF TUMOR CELL KINETICS AND THE EMERGENCE OF MOLECULAR HETEROGENEITY DRIVING GOMPERTZIAN GROWTH m d m p t Emax t SAT n d n p ave;1 ave;2 ave;3 0.4 0.1 5.50e+5 1.830e+6 1.289e+4 4.68e+4 3.14e-5 3.68e-6 1.448e-7 0.3 0.2 4.88e+5 1.753e+6 1.682e+4 8.26e+4 4.04e-5 4.31e-6 1.677e-7 0.2 0.3 4.85e+5 1.761e+6 1.715e+4 1.230e+4 3.86e-5 4.41e-6 1.729e-7 0.1 0.4 5.40e+5 1.426e+6 1.362e+4 1.836e+4 3.04e-5 3.81e-6 1.658e-7 Table 22: m d : driver mutation rate;m p : passenger mutation rate;t Emax : time to maximum entropy;t SAT : time to saturation; n d : number of driver mutations; n p : number of passenger mutations; ave;1 : average growth rate in regime 1; ave;2 : average growth rate in regime 2; ave;3 : average growth rate in regime 3. one of m possible states. Then, P =m n , and the growth equation becomes dn dt =n logm = ( logm)n: (147) The solution to this equation is the exponentially growing population: n(t) =N 0 exp(( logm)t): (148) Thus, early stage development is characterized by exponential growth (regime 1), with a growth rate proportional to the log of the number of assumable states of the cells comprising the tumor population. This stage is characterized by the Gompertzian curve shown in Figure 61A to the left of the rst blue dot, in regime 1. Contrast this with later stages of tumor growth, when the sub-populations of cells communicate and in uence each other's growth characteristics, either via competition, or cooperation (regime 3) within the tumor microenvironment. In eect, this will constrain (reduce) the number of assumable states of each cell, since the population is eectively coupled. In the extreme, suppose P = m n =n n . In other words, suppose P is now inversely related to the total number of possible intercellular interactions. Inserting this into (146) yields dn dt = log ( m n ) n =n [logm logn]: (149) The solution to this equation is exactly the Gompertzian growth curve (142) and accounts for regimes 2 and 3 previously discussed in which tumor growth slows down. The growth equation (146) which relates cancer cell population growth to tumor heterogeneity is capable of producing a family of growth curves, depending on details of intercellular coupling, which of course is in uenced by details of the biological and physical constraints in uencing the tumor microenvironment. Thus, the growth equation (146) has the ability to produce dierent detailed shapes based on assumptions associated with intercellular coupling. Table 22 shows the average growth rates in the three regimes as a function of the key parameters in the model. 148 9 APPENDIX II: AN EVOLUTIONARY MODEL OF TUMOR CELL KINETICS AND THE EMERGENCE OF MOLECULAR HETEROGENEITY DRIVING GOMPERTZIAN GROWTH 9.4.3 Quantitative measures of tumor heterogeneity and growth For our purposes, we measure heterogeneity using the Shannon entropy from information theory [39]: E(t) = N X i=1 p i log 2 p i ; (150) (here, log is dened as base 2). The probability p i measures the proportion of cells of type i, with i = 1;:::; 16 representing the distribution of binary strings ranging from 0000 to 1111. We then course-grain this distribution further so that cells having strings ranging from 0000 up to 1010 are called \healthy", while those ranging from 1011 to 1111 are \cancerous" 6 . The growth equation (146) then becomes dn E dt =E(t): (151) It follows from (151) that the cancer cell proportion n E (t) can be written in terms of entropy as: n E (t) = Z t 0 E(t)dt: (152) The panel in Figure 62 shows the results from our cell-based simulations. Figure 62A shows the Gompertzian curve associated with the proportion of cancer cells in the population, while Figure 62B shows the velocity and accelerations associated with growth, and can be compared with Figure 61B. In Figure 62C we show the entropy during a typical simulation, marking the maximum entropy point which peaks relatively early in the simulation before the entropy returns back down to zero, re ecting the fact that cancer cells have reached xation and have saturated the population. Figure 62D shows the tness of the cancer cell sub-population, healthy cell sub-population, and the overall tumor tness (w H = w C w = 0:5). As a typical simulation proceeds, the cancer cell sub-population tness increases, the healthy cell sub-population tness decreases, while the overall tumor tness decreases. Figure 62E, 62F shows the Gompertzian growth curves as the selection pressure increases (Figure 62E) and as the mutation rate increases (Figure 62F). High values for either of these parameters leads to a very steep growth curve, as is expected. Figure 63 shows the growth curves linearly extrapolated back to give a prediction of when the rst driver mutation occurred that initiated tumor growth. The growth rates from regime 2 (linear regime) are used to extrapolate back to the initiation event. Since the actual growth rate in regime 1 is much faster than linear, the linear extrapolation extends the event too far back in time as compared to when the event actually occurred. The inset of Figure 63 shows histograms of the average growth rates in each of the three regimes as a function of the mutation rate m (here, we take m p =m d =m). 6 Our results are relatively insensitive to where we draw the dividing line between healthy and cancerous. 149 9 APPENDIX II: AN EVOLUTIONARY MODEL OF TUMOR CELL KINETICS AND THE EMERGENCE OF MOLECULAR HETEROGENEITY DRIVING GOMPERTZIAN GROWTH Figure 62: Moran birth-death process with selection | (A) Cancer cell population, i(t) (w = 0:5, m = 0:2,N = 10 10 ) plotted with a spline curve connecting 200 data points from a single stochastic simulation; (B) First- and second-derivatives of the tumor growth curve in (A) are plotted with maximum and minimum of second-derivative indicated (blue); (C) Entropy of the cell population from eqn. (150) as it relates to the growth equation (151); (D) Fitness of healthy cell population and cancer cell population and total tness as dened by eqns. (132), (133), (134); (E) Simulations of cancer cell population, i(t), for a range of selection parameter values; (F) Simulations of cancer cell population, i(t), for a range of mutation rate values. A typical stochastic simulation showing the evolution of all 16 possible cell types is shown in Figure 64. We also showE(t), where entropy is computed using the most extreme course-grained two-state system comprised of the two sub-populations of healthy cells and cancer cells. We compare in Figure 64 the Gompertzian growth curve (eqn. (142)) and the corresponding curve obtained from eqn. (152) to the stochastic simulation and the agreement is excellent. Likewise, we also show a comparison of dn=dt with eqn. (151) and eqn. (138) with E(t) normalized so that limiting values match the stochastic simulation, 150 9 APPENDIX II: AN EVOLUTIONARY MODEL OF TUMOR CELL KINETICS AND THE EMERGENCE OF MOLECULAR HETEROGENEITY DRIVING GOMPERTZIAN GROWTH Figure 63: Tumor initiation prediction | Five sample stochastic simulations of tumor growth (N = 10 10 cells,w = 0:5,m = 0:1; 0:2; 0:3; 0:4; 0:5) plotted on a log-linear graph where the model output (i(t), solid lines) is t in the clinical regime (greater than 10 8 cells) using an exponential growth equation and extrapolated backwards in simulation time (dashed lines). The inset bar graph shows the average growth rate in each regime. and the agreement is also excellent. In the beginning, entropy is zero, since the population consists purely of healthy cells, and in the end of the simulation, entropy is again zero as the population consists purely of cancer cells. Entropy peaks somewhere early in the simulation when the mixture of cell types is equally distributed over cancer and healthy types. It is this intermediate but important heterogeneously distributed state that is the key driver of growth, as is clear from eqn. (151). 9.4.4 Dynamic phylogenetic trees and evolution of tness To track the initiation of cellular heterogeneity from an initially homogeneous state, we follow all of the mutations that take place during the course of a simulation, and organize this in the form of a phylogenetic tree in Figure 65 showing the typical size of the genotypic space and the evolution of the genotypic landscape. As the simulation proceeds, the phylogenetic tree dynamically branches out into an increasingly complex structure, with tness characteristics color coded in Figure 65A. We also show the bins associated with each of the 16 cell types, the number of cancer cells i(t), and the entropy associated with the sub-population of cell types as a simulation proceeds, in Figure 65B. Knowing exactly the types of cells comprising the tumor at any given time allows us to target cell distributions for simulated therapies to test dierent strategies, which we describe next. 151 9 APPENDIX II: AN EVOLUTIONARY MODEL OF TUMOR CELL KINETICS AND THE EMERGENCE OF MOLECULAR HETEROGENEITY DRIVING GOMPERTZIAN GROWTH Figure 64: Comparison of stochastic Moran birth-death process, Gompertzian, and Shannon entropy growth curves | (a) A single stochastic simulation (N = 10 10 cells, m = 0:5, w = 0:5, m p = m d = 0:25) growth curve, n(t), compared with the Gompertzian growth curve, n G (t), eqn. (142), and Shannon entropy growth curve, n E (t), eqn. (152). Growth curves n G (t) and n E (t) are normalized to equal one in the limit; (b) Comparison of rst-derivatives of n(t), n E (t), n G (t); (c) Comparison of growth rates associated with n(t), n E (t), n G (t), with average growth rates of n(t) plotted for each regime, eqn. (144). 9.4.5 A comparison of early vs. late therapy In Figure 66 we show the results from asking the simple question of how early therapy (administered in regime 1) compares with therapy in the middle stages of tumor development (regime 2), or in the later 152 9 APPENDIX II: AN EVOLUTIONARY MODEL OF TUMOR CELL KINETICS AND THE EMERGENCE OF MOLECULAR HETEROGENEITY DRIVING GOMPERTZIAN GROWTH Figure 65: Emergence of genetic heterogeneity | (A) Left: sample dendritic phylogenetic tree tracking point mutations as time extends radially. Right: three snapshots in time of a dendritic tree in a simulation of 30 cells with strong selection (w = 1, m p = 0:1, m d = 0:2). Pathways are color coded to indicate genetic cell type; (B) Linear phylogenetic tree of the same stochastic simulation shown in (a) along with histogram plots of the distribution of genetic cell types and a plot of the cancer cell population i(t) and entropy. stages of development (regime 3). Eqns (135), (136) are the governing equations controlling birth/death rates of the cancer cell, healthy cell sub-populations as natural selection plays out. Since the proliferation of cancer cells can be thought of as an imbalance of selection pressures on the competing sub-populations in favor of the cancer sub-population, the goal of any therapeutic intervention is to alter this complex imbalance 153 9 APPENDIX II: AN EVOLUTIONARY MODEL OF TUMOR CELL KINETICS AND THE EMERGENCE OF MOLECULAR HETEROGENEITY DRIVING GOMPERTZIAN GROWTH in favor of the healthy cell sub-population. We implement this by adjusting the selection pressure parameters (w H ;w C ) in the formulas (130), (131). In particular, when therapy is `on', we choose w C = 0, and w H = 1, tilting the selection pressure in favor of the healthy cell sub-population. When therapy is `o', the two parameters return to their original baseline values, which here we take as w H = 0:1, w C = 0:1. Figure 66 depicts the proportion of cancer cells in the population both in the absence of therapy, and when therapy is administered. As a comparative tool, in each case, we administer the therapy until a xed number of cancer cells remains (in each case, we take this threshold number to be 25 cancer cells), and we compare the amount of time, t, it takes to achieve this low level. The gure clearly shows t 1 < t 2 < t 3 < t 4 , while if therapy is administered too late, as in t 5 , the low threshold is never achieved. The simulations show that a shorter therapeutic time-period is needed if administered earlier to gain the same level of success. The topic of how best to optimize computational therapies is complex, and these simulations are only meant as a conrmation and quantication of how early stage therapy is more eective than late stage therapy. Figure 66: Simulated therapy | An average of 25 stochastic simulations (N = 10 3 cells,w = 0:5,m = 0:1) where therapy (w H = 1,w C = 0) is administered at dierent time points (t = 6000; 8000; 10000; 12000; 14000 cell divisions) until all cancer cells are eliminated below a small threshold value (25 cells). Time required (t) for tumor elimination increases as the tumor volume increases (i.e. t 1 < t 2 < t 3 < t 4 , blue, red, yellow, purple arrows respectively), until, at later simulation time points, therapy is unable to regress tumor size (t 5 , green arrow). 9.5 Discussion To summarize the main points forming the framework of our model: 154 9 APPENDIX II: AN EVOLUTIONARY MODEL OF TUMOR CELL KINETICS AND THE EMERGENCE OF MOLECULAR HETEROGENEITY DRIVING GOMPERTZIAN GROWTH (i) A tumor is a complex Darwinian ecosystem of competing cells operating on an adaptive tness landscape driven by mutational dynamics and shaped by evolutionary pressures; (ii) The basic competitors in an evolutionary game theory model of tumor development are cell populations with a broad distribution of tness characteristics course grained into two types: healthy cells (cooperators) and cancer cells (defectors). Each of these cell sub-populations attempts to maximize its own tness; (iii) Cell tness is associated with reproductive prowess and in this respect, healthy cells are less t than cancer cells; (iv) Primary tumors initiate from a single malignant cell that has undergone the appropriate mutational steps and subsequently undergoes clonal and sub-clonal expansion. Polyclonality and heterogeneity are thus seen as emergent features of tumor development; (v) Parameters and distributions measured in the detectable range of tumor growth, such as tumor growth rates and xation probabilities, are emergent features that have developed from a monoclonal state via cell kinetics and evolutionary development taking place in the subclinical regime; (vi) Tumor growth is driven by molecular heterogeneity of the cell population comprising the tumor and re ected in the growth equation (146); (vii) Tumor cell populations are more amenable to therapeutic strategies in the early stages of development, before selection for growth and survival have shaped the environment. We believe the simple evolutionary model described in this paper, driven by a Moran process and shaped by heritable mutations with a tness landscape based on the prisoner's dilemma evolutionary game is useful in helping to understand early stage tumor growth and how it is in uenced by the interplay of a few select small number of key parameters. When a malignant tumor cell population has already exceededO(10 8 10 10 ) cells, some of which may have entered the circulation or lymphatics and migrated to other sites, the opportunity to control or even shape future events may be limited. Attacking tumor heterogeneity as soon as it develops seems to be a useful strategy, particularly if heterogeneity is the driver of growth, as in eqn. (146). Whether these concepts can be developed in the more general context when cell dissemination to other sites is included in the model, and then translated into actionable clinical strategies is a challenge for the future. 155 9 APPENDIX II: AN EVOLUTIONARY MODEL OF TUMOR CELL KINETICS AND THE EMERGENCE OF MOLECULAR HETEROGENEITY DRIVING GOMPERTZIAN GROWTH 9.6 Appendix Figure 67: A ow chart of the Moran process with selection and mutation algorithm | Box 1: mutation rate m (wherem =m p +m d ), selection pressure w and the initial state vectorx containingN total cells are the inputs for a simulation. Box 2: the prisoner's dilemma game (a = 3;b = 0;c = 5;d = 1) is used to calculate the tness of each healthy and cancer cell type, which is a function of the payo values and the state vector,x. Box 3, 4: a single cell is chosen for death according to the relative proportion of the cell type in the cell population. Simultaneously, a single cell is selected for birth according to the relative proportion, weighted by cell tness. Box 5: During the replication process, the daughter cell inherits a replica of the parent cell's genetic string, with errors occurring at a rate of m. A single bit of the daughter cell's genetic string may ip during each cell division. The possible mutations can be thought of as a single step random walk on the Markov diagram shown in Figure 60. 156 10 PUBLICATIONS 10 Publications In-progress Submitted Completed 10.1 First author publications Journal article: Low-dimensional characterization of human performance of cancer patients using motion data [81] Conference paper: Mining Human Mobility to Quantify Performance Status [136] Journal article: Machine learning models for long-term outcome prediction in bladder cancer Journal article: Feature and model based characterization of spatial and temporal organoid dynamics Journal article: Kinematics from clinical exercises dierentiate patients by unexpected hospitalizations and physical activity Journal article: Temporal dierentiation of phenotypes using a Hidden Markov model of organoid shape dynamics 10.2 Second author publications Journal article: The prisoner's dilemma as a cancer model [211] Journal article: An evolutionary model of tumor cell kinetics and the emergence of molecular heterogeneity driving Gompertzian growth [210] Journal article: Pathways of metastatic bladder cancer from a longitudinal patient data set 157 11 REFERENCES 11 References [1] LH Abbott and F Michor. \Mathematical models of targeted cancer therapy". 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Abstract (if available)

Abstract Progression of cancer occurs across multiple length and time scales and encompasses biodynamical processes with intrinsically unique implications for the host patient. The compartmentalized study of these processes has not only revealed scale-specific biomedical questions but also led to an equally diverse set of quantitative approaches derived from various fields in the applied spirit of engineering. Due to advances in imaging, experimental, and patient engagement technologies, a growing subset of cancer progression studies are empirical in nature and yield data which is often spatio-temporal. The focus of this thesis is to explore a general design approach for developing quantitative analysis tools which leverage data from these studies to provide quantified answers to biomedical questions across time and length scales, and to apply the methodology to three case studies representing three different scales. ❧ Biodynamical processes or systems observed via experiments, clinical studies, and longitudinal medical assessments span the physical spectrum from nanometer scale molecular phenomena, human body locomotion at the meter scale, to population level disease trends. These processes may be characterized by features from statistics, mechanics, signal processing, or any domain-specific life science. The direct use of these features, optionally in combination with dimensionality reduction and distance metrics, forms the first arm of biodynamical analytical study design called the feature based approach. This is the more direct approach and leads to an intuitive understanding of a process' role in cancer progression, for instance, by comparing, correlating, and ranking samples or features themselves. The second arm of biodynamical analytical study design, the model based approach, uses stochastic, physics, or statistical modeling techniques to create models of the biodynamical processes or system features. This is the more complex design arm and requires a greater amount of data, but is distinct from the feature based approach due the more advanced questions the models are able to answer. Both approaches' ultimate goal is to provide a quantified output in the form of one or a mixture of the following: i) clustering or differentiating samples, ii) classifying samples as a type, iii) forecasting or predicting a numerical sample characteristic, and iv) characterizing the behavior of a given sample. ❧ The first case study of morphogenesis of 40 organoids focuses on extracting geometric and signal processing based features from time series of images at the micrometer scale. The feature set is used to cluster organoids into groups of similar phenotypes to separate invasive and normal behaving organoids. Secondly, two model based approaches are used to understand the dynamics of growth: i) a statistical model of shape is created to corroborate the feature based clustering, and ii) a hidden Markov model of organoid shape is used to discover unmeasured genomic states and their corresponding shape dynamics. A standardized code is provided in order to apply the feature based clustering to image data from organoid experiments. ❧ The second case study is a human performance assessment of 37 cancer patients where a non-invasive motion capture camera is used to record patients performing clinically supervised exercises. The goal of the study is to identify patients with higher risk of deteriorating health while undergoing a chemotherapy regimen. A kinematic feature based approach is used to cluster patients into groups of low and high risk of unexpected hospitalization and limited physical activity. Secondly, a machine learning model based on kinematic and demographic features is used to classify individual patient fitness levels based on a single exercise instance. This work serves as a proof-of-concept of a non-invasive human performance based risk calculator of clinical utility. ❧ The final case study uses a feature based approach to understand the importance of clinical measures in a longitudinal bladder cancer dataset of 3500 patients, and machine learning models to stratify patient survival and disease recurrence post-surgery. The combination of feature and model based approaches developed here can be applied to most clinical datasets which contain survival data, without additional domain specific knowledge of the cancer type. The accuracy and temporal resolution of predictions made with the developed models can dramatically change post-cystectomy patient care.

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Creator Hasnain, Zaki (author)

Core Title Feature and model based biomedical system characterization of cancer

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Mechanical Engineering

Publication Date 12/03/2018

Defense Date 08/01/2018

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

Tag Applied Mathematics,biomedical data,cancer,cancer progression,clinical data,clinical prediction,data analysis,dynamics,health forecasting,human motion,Human Performance,machine learning,mathematical modeling,OAI-PMH Harvest,organoids, bladder cancer,patient clustering,time series analysis

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Advisor Newton, Paul K. (committee chair), Ghanem, Roger Georges (committee member), Kanso, Eva (committee member)

Creator Email syedzakihasnain@gmail.com,zhasnain@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c89-109733

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