Mathematical Problems From Cryobiology A Dissertation presented to the Faculty of the Graduate School University of Missouri In Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy by James Dale Benson Professor Carmen Chicone and Professor John Critser, Dissertation Supervisors JULY 2009 c Copyright by James Dale Benson 2009 All Rights Reserved The undersigned, appointed by the Dean of the Graduate School, have examined the dissertation entitled Mathematical Problems from Cryobiology presented by James Dale Benson, a candidate for the degree of Doctor of Philosophy and hereby certify that in their opinion it is worthy of acceptance. Professor Carmen Chicone Professor John Critser Professor Tanya Christiansen Professor Yuri Latushkin Professor Stephen Montgomery-Smith To Jack and (.) ACKNOWLEDGEMENTS Writing this Ph. D. has given me an opportunity to learn about myself, about my friends and about my family. I’ve learned that there is a world of knowledge whose history spans centuries of careful thought and consideration, and that my own small contributions are only small a mark on the pages of academic pursuit. These last seven years my life have revolved around problems presented in this dissertation and elsewhere, and though I’ve been focused on one problem in general, the interactions with friends, family, and colleagues from all disciplines have inspired me to believe that there must be more to academics than a single discipline, more to a discipline than academy, and more to life than an academic pursuit, and for these reasons, I’m indebted and overwhelmingly grateful to the many people who made it possible: To my parents, Dale and Barb, thank you for allowing me to follow my own path, explore the arts, humanities, and the sciences at my leisure (probably to your financial dismay). Your delicate balance of support and distance has been amazing, and now that I’m a father, I have become more proud to be your son. Thank you for being great role models, great friends, and great support. I love you both. To my siblings Chris, Chuck, and Dave. Thank you, Chris for reminding me how important it is to have the vision in mind while trudging through the details and for reminding me that there’s way more to life than science and scientists. Thank you, Chuck for continually inspiring me to love science, for teaching me how to be a scientist, for giving me my first real lab gig, and for challenging me at every step of the way. Thank you, Dave for exposing me to computers and programming, for philosophic conversations on everything, for a spectacular summer job, for continual ii positive reinforcement, and for biannual visits that bookended my summers. To my advisor John, thank you for giving me my first job, my first lab work, my first cryobiology conference, my first published paper, and for convincing me to come to Missouri to work in this truly fascinating and interdisciplinary field. Thank you for keeping me funded, keeping me published, and for introducing me to so many fantastic collaborators. Thank you for reminding me that there is a Philosophy of Science that all scientists should keep in touch with, and that I am earning a doctor of Philosophy. To my advisor Carmen, thank you for tolerating a graduate student who has more applications than time and more breadth of knowledge than depth. Thank you for keeping me grounded, reminding me of what a mathematician does, and reminding me of why I wanted to become a mathematician. Your involvement in diverse projects has been an inspiration for my future research aspirations. Thank you for your interest in my research and my life beyond my research and for your limitless support and enthusiasm, To my friends from the Mathematics and Veterinary Pathobiology departments: I am indebted for Tuesday night trivia and the “dirty deal,” Thursday, Friday, and Saturday 7 hour “happy hours,” banal conversations about hamsters in wet suits, golf, being the weakest player on an otherwise respectable (well, some years) basketball team, Scrabble, and pitchers of Boulevard Wheat at a row of tables 30 feet long at the Heidelburg. At Vet Path, thank you for making me the “math guy” who should know more about statistics, for daily lunch conversations, for letting me be the utility infielder on champion softball teams, for keeping me in touch with biology and the iii life sciences, and for giving me a desk with too much room to pile papers on. To my colleagues Yuksel Agca, Xu Han, Steve Mullen, Hongsheng Men, Gary Solbrekken, and Erik Woods, thank you for your patience, your love for science and for your inspiration. I am indebted to you for actually carrying out experiments, reminding a mathematician of the bounds of physics and biology, and giving me new problems to think about. To my mentees Claire, Sarah, and Jason, thank you for reminding me why I wanted to go into academia in the first place. You guys were all great, and great sports. I learned as much if not more from you than you from me. To Trina, Idelle, and DeShayda, thank you for being a second family for Jack. It has made life easier for me to know that Jack spends his days with people that truly care for and love him. To my mother-in-law Anna, thank you for your support, for your willingness to spend weeks at a time making my and Corinna’s lives easier. It would have been hard to make it through a few times without your love and support. To my son, Jack, who has given me so much joy these last 30 months that I am inspired to be a better man, a deeper thinker, and a more considerate person. May your beautiful inquisitiveness and love for life never falter. Finally, to my wife, Corinna, thank you for supporting me, inspiring me, prodding me, making me a better person, widening my thoughts and perspectives on a daily basis, bringing me down to reality, keeping me awake at nights, and making me laugh. I love you. iv Table of Contents ACKNOWLEDGEMENTS ii List of Figures ix List of Tables xi 1 Introduction 1 1.1 History and Fundamentals of Cryobiology . .1 1.1.1 Fundamentals of Equilibrium Freezing . .4 1.1.2 Rapid cooling approaches . .9 1.1.3 CPA addition and removal . 10 1.1.4 Archetypical experimental design for fundamental cryobiology 12 1.1.5 Empirical versus mathematical approaches to cryobiological op- timization . 14 1.2 This thesis and fundamental cryobiology . 15 2 Exact solutions of a two parameter flux model and cryobiological applications 18 2.1 Introduction . 18 2.2 Methodology . 21 v 2.2.1 A reparameterized solution to the Jacobs model . 21 2.2.2 The inverse of q .......................... 25 2.3 Results and Discussion . 27 2.3.1 Finding cell volume and intracellular solute concentration max- ima and minima . 27 2.3.2 Curve fitting . 30 2.3.3 Finite element models . 32 2.4 Conclusions . 33 3 A general model for the dynamics of cell volume, global stability, and optimal control 34 3.1 Introduction . 34 3.2 Dynamics for M(t) M ......................... 37 ≡ 3.2.1 Stability . 37 3.2.2 Rate of approach to the rest point . 41 3.3 Optimal Control . 42 3.3.1 Controllability . 42 3.3.2 Existence of an optimal control . 45 3.3.3 A control problem . 45 3.3.4 Synthesis of the optimal control in the case n = 2........ 47 3.4 Conclusions . 60 4 An optimal method for the addition and removal of cryoprotective agents 62 4.1 Introduction . 62 vi 4.2 Problem . 63 4.3 Theory . 66 4.4 Discussion . 72 4.5 Application of optimal control to human oocyte CPA addition. 80 4.6 Problems with application of this model . 82 5 The impact of diffusion and extracellular velocity fields on cell, tis- sue, and organ mass transport 88 5.1 Introduction . 88 5.1.1 Characteristic quantities of the system . 91 5.2 Model Construction . 94 5.2.1 The basic model . 96 5.2.2 Basic model after a change of coordinates . 100 5.2.3 Nondimensionalization of our model . 106 5.2.4 Exact solutions of the fluid and concentration models . 110 5.3 Numerical analysis . 110 5.4 Results . 113 5.5 Discussion and Conclusions . 122 Appendices 126 A-1 Derivation of solute-solvent flux model . 127 A-2 Pontryagin Maximum Principle . 129 A-3 Boltayanskii Sufficiency Theorem . 130 A-4 Design of an apparatus to measure the effects of fluid velocity fields and diffusion on cell permeability . 133 vii A-5 Exact solutions of fluid and concentration models . 135 A-5.1 Solutions for the φ component of u................ 135 A-5.2 Solutions for the r component of u ............... 136 A-5.3 An exact solution of the concentration model in a special case. 139 A-6 Numerical analysis of a curved boundary condition: a general approach to Dirichlet and Neumann conditions and uneven grid spacing . 140 A-6.1 A simple example: Laplace’s Equation . 142 A-6.2 A more complicated example: Laplace’s Equation on a non uniform grid . 143 A-6.3 An example with Neumann boundary conditions . 145 VITA 164 viii List of Figures 1.1 Plot of the interaction between cooling rate, “solution effects” injury, ice formation injury and cell survival. .7 1.2 Survival rates for various cell types cooled at different rates. .8 1.3 Osmotic Tolerance of Spermatozoa from Rat, Boar, and Mouse. 11 1.4 Mouse spermatozoa volume during cryopreservation.