Mathematical models of emergence of drug resistance and growth in cancer

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(1)U NIVERSIDAD DE C ASTILLA L A M ANCHA D OCTORAL T HESIS. Mathematical models of emergence of drug resistance and growth in cancer. Author: Arturo Á LVAREZ -A RENAS. Supervisors: Dr. Gabriel F. C ALVO Dr. Juan B ELMONTE -B EITIA Dr. Víctor M. P ÉREZ -G ARCÍA. A thesis submitted in fulfillment of the requirements for the degree of Doctor of Philosophy in the Research Group Name MOLAB. July 26, 2019.

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(3) iii. Declaration of Authorship I, Arturo Á LVAREZ -A RENAS, declare that this thesis titled, “Mathematical models of emergence of drug resistance and growth in cancer” and the work presented in it are my own. I confirm that: • This work was done wholly or mainly while in candidature for a research degree at this University. • Where any part of this thesis has previously been submitted for a degree or any other qualification at this University or any other institution, this has been clearly stated. • Where I have consulted the published work of others, this is always clearly attributed. • Where I have quoted from the work of others, the source is always given. With the exception of such quotations, this thesis is entirely my own work. • I have acknowledged all main sources of help. • Where the thesis is based on work done by myself jointly with others, I have made clear exactly what was done by others and what I have contributed myself.. Signed: Date:.

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(5) v. “The book of nature is written in the language of Mathematic” Galileo Galilei.

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(7) vii. UNIVERSIDAD DE CASTILLA LA MANCHA. Abstract Faculty Name MOLAB Doctor of Philosophy Mathematical models of emergence of drug resistance and growth in cancer by Arturo Á LVAREZ -A RENAS This Thesis explores a set of key features of one of the most important group of diseases: cancer. Making use of the tools provided by applied mathematics, several aspects of this complex disease are modeled, validated and used to answer relevant biological questions. Two main subjects of relevance in patient’s prognosis are addressed during this work: tumor growth and drug resistance. Tumor growth is responsible for the progression of the disease, ultimately causing the death of the patient. The possibility of predicting and managing tumor growth may improve patient’s quality of life. To that end, we explore the dynamics governing tumor growth and we study the effect of different therapies in the control of the disease. In this thesis we also study the problem of chemotherapy resistance. Cancer cells adopt different strategies to escape from the action of chemotherapeutic agents. To provide new insight in this field, several mathematical analyses are proposed with the aim of understanding this complex issue and proposing alternatives to current treatments..

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(9) ix. Agradecimientos Primero de todo quiero agradecer a mis directores de tesis por su trabajo y por su dedicación. A Víctor, por su visión de la matemática aplicada, por su constante trabajo y por su visión innovadora. A Juan, por hacer de la sencillez una virtud, por su proximidad y por su constante buena disposición. A Gabriel, por su excelencia, por su sabiduría y por su bondad. Aparte de la ciencia, de ellos he aprendido muchas cosas que me llevo como recuerdo de este doctorado y que me acompañarán durante mi vida. En segundo lugar, quiero agradecer a todos los miembros del grupo de investigación MOLAB. Especialmente, a María Jesús, por darme la mejor bienvenidad a una ciudad que recuerdo, y por apoyarme en las decisiones importantes de mi vida. A Araceli, porque su compañía hacía los congresos y el día a día mucho más divertidos. A Julián, por su simpatía como compañero de despacho y por arreglar los problemas técnicos antes de que yo me encontrara con ellos. A Juan y Javi, por ser parte de los tres mosqueteros y por llenar los pausas del café y los días de trabajo con risas y bromas. A Alicia, por ser una excelente investigadora y ser un perfecto modelo a seguir para todos los estudiantes de doctorado del grupo. Y a muchos otros, a toda la gente de la universidad con la que he compartido parte de mi tiempo durante este periodo. En tercer lugar, quería agradecer a Culipardisk, el equipo de Ultimate Frisbee que ha sido la mejor experiencia deportiva de mi vida. Nunca olvidaré como con pasión y entusiasmo construimos este equipo desde la nada. Los primeros entrenamientos, los primeros torneos y nuestras participaciones en los Campeonatos de España son momentos que siempre recordaré con una sonrisa. Por hacer de Ciudad Real un sitio del que no me quiero ir. Estoy convencido de que solo hemos puesto la primera piedra de un proyecto tan bonito como duradero. También quiero agraceder a mis amigos. A aquellos que conozco desde la infancia y al resto que he ido conociendo en diferentes etapas de mi vida. Ir a Madrid nunca se hizo un viaje largo si al llegar estaban ellos para tomar una cerveza o hacer cualquier plan. Especial agradecimiento a aquellos que vinieron a visitarme a Ciudad Real. Gracias a mi familia, por todos los momentos de felicidad que hemos pasado juntos y por ser un ejemplo de familia unida que se quiere. A mi madre, por motivarme para continuar con mi carrera académica. Soy muy consciente de que este documento nunca se habría escrito si no fuera por su esfuerzo y apoyo. A mi padre, por su simpatía y buen sentido del humor, y por cuidarme. A mi hermana, por ser un ejemplo de constancia y esfuerzo, y por estar siempre a mi lado acompañandome en todo momento. Todos ellos son ejemplos en los que me fijo diaramente y de los que me siento extremadamente orgulloso. Para acabar, mi más grande y sincero agradecimiento va para Odelaisy. Por ser un ejemplo de todas las cosas buenas. For ser la primera persona con la que puedo hablar sobre matemáticas, frisbee, chirigotas y todas las cosas raras que me gustan. Por ser la sonrisa de los días buenos y el apoyo de los días malos. Por hacerme feliz. Por todo. Este trabajo ha sido apoyado por el Ministerio de Economía y Competititivdad/FEDER, España [MTM2015-71200-R], la Consejería de Educación Cultura y Deporte de la Junta de Comunidades de Castilla-La Mancha, España, [PEII-2014-031-P],.

(10) x la James S. Mc. Donnell Foundation (USA) 21st Century Science Initiative in Mathematical and Complex Systems Approaches for Brain Cancer [220020450] y por los contratos predoctorales de la UCLM [2015/4062].

(11) xi. Acknowledgements I would firstly like to thank my thesis directors for their work and for his dedication. To Víctor, for his vision of applied mathematics, for his constant work and for his innovative vision. To Juan, for making simplicity a virtue, for its proximity and for its constant willingness. To Gabriel, for his excellence, for his wisdom and for his kindness. Beside science, from them I have learned many things that I take as a memory of this PhD and that will accompany me throughout my life. Secondly, I would like to thank to all the members in my research group MOLAB. Special thanks to María Jesús, for giving me the best welcome to a city that I remember and for supporting me in the important decisions of my life. To Araceli, because your company made congresses and day to day much more fun. To Julian, for his sympathy as an office partner and for fixing the technical problems before I found them. To Juan y Javi, for being part of the three musketeers and filling coffee times and working days with laughter and jokes. To Alicia, for being a great researcher and a perfect role model to all PhD students in our group. And to many others, all the people in the university with whom I shared part of my time during this period. Thirdly, I would like to thank Culipardisk, the Ultimate Frisbee team that has been the best experience of my sporting life. I will never forget how with passion and enthusiasm we have built this team from nothing. The first practices, the first tournaments and our participation in the Spanish Championships are moments that I will always remember with a smile. For making a Ciudad Real a place I don’t want to leave. I am convinced that we have only laid the foundation stone for a very beautiful and lasting project. I want also to thank all my friends. Those that I know since childhood and the rest I have known at different stages of my life. The trip to Madrid has never been long if you were there to have a beer or make a plan. Specially thanks to those friends who came to visit me in Ciudad Real. Thanks to my family, for all the moments of happiness we have spent together and for being an example of a united family that loves each other. To my mother, to encorauge me to continue with my academic career. I am very aware that this document would not be written if it were not for your efforts and your support. To my father, for his sympathy and his great sense of humor, and for taking care of me. To my sister, for being a perfect example of of constancy and effort, and for always being by my side accompanying me in the different moments of my life. They are all people that I look closely at and I feel extremely proud of. Finally, the biggest and most sincere thanks go to Odelaisy. For being an example of all good things. For being the first person I can talk about maths, frisbee, chirigotas and all the weird things that I like. For being the smile of good days and the support of bad days. For making me happy. For everything. This work was supported by Ministerio de Economía y Competitividad/FEDER, Spain [grant number MTM2015-71200-R], Consejería de Educación Cultura y Deporte from Junta de Comunidades de Castilla-La Mancha (Spain) [grant number PEII-2014-031-P], James S. Mc. Donnell Foundation (USA) 21st Century Science Initiative in Mathematical and Complex Systems Approaches for Brain Cancer (Collaborative award 220020450) and UCLM PhD Fellowship [2015/4062]..

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(13) xiii. Contents Declaration of Authorship. iii. Abstract. vii. Agradecimientos. ix. Acknowledgements. xi. 1. Introduction 1.1 Cancer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Classification . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 The hallmarks of cancer . . . . . . . . . . . . . . . . . . . . 1.1.3 Cancer treatments . . . . . . . . . . . . . . . . . . . . . . . 1.2 Chemotherapy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Types of chemotherapy drugs . . . . . . . . . . . . . . . . . 1.3 Chemotherapy resistance . . . . . . . . . . . . . . . . . . . . . . . 1.4 Mathematics and biology/cancer . . . . . . . . . . . . . . . . . . . 1.5 Mathematical models of tumor growth . . . . . . . . . . . . . . . . 1.6 Mathematical models of development of chemotherapy resistance 1.6.1 Pre-existence of a resistant clone . . . . . . . . . . . . . . . 1.6.2 Induced Resistance . . . . . . . . . . . . . . . . . . . . . . . 1.6.3 Location resistance . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . .. . . . . . . . . . . . . .. . . . . . . . . . . . . .. 1 1 2 2 3 3 5 6 7 9 11 11 12 14. 2. Objectives of the thesis 17 2.1 Glioblastoma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.2 Emergence of cancer resistance . . . . . . . . . . . . . . . . . . . . . . . 19. I. Glioblastomas. 21. 3. Introduction. 23. 4. Analysis of glioblastoma growth in 3D cultures 4.1 Introduction . . . . . . . . . . . . . . . . . . . . 4.2 Methods and experimental data . . . . . . . . . 4.2.1 Cell Culture . . . . . . . . . . . . . . . . 4.2.2 Biosphere formation . . . . . . . . . . . 4.2.3 Cell growth . . . . . . . . . . . . . . . . 4.2.4 Image data . . . . . . . . . . . . . . . . . 4.2.5 Quantitative data obtained from images 4.3 Mathematical model . . . . . . . . . . . . . . . 4.4 Results . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 PDCs proliferation rates . . . . . . . . . 4.4.2 Neurospheres compactness . . . . . . .. 25 25 26 26 26 27 28 29 31 32 32 35. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . ..

(14) xiv. 4.5 5. 6. II 7. 4.4.3 Circularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38. Effects of tumor-activated stroma cells in glioblastoma cells 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Methods and Experimental Results . . . . . . . . . . . . . . . . . . . . 5.2.1 Cell culture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 TASCs preparation . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.3 Obtention of Rho 0 cells . . . . . . . . . . . . . . . . . . . . . . 5.2.4 Analysis of mitochondria transfer between TASCs and GBM cells in 3D scenarios . . . . . . . . . . . . . . . . . . . . . . . . 5.2.5 Analysis of mitochondria transfer between TASCs and GBM cells in 2D scenarios . . . . . . . . . . . . . . . . . . . . . . . . 5.2.6 Analysis of isolated mitochondria transfer . . . . . . . . . . . 5.3 Mathematical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Effects of co-cultured in 3D scenarios . . . . . . . . . . . . . . 5.4.2 Mitochondria can be transferred to GBM cells via tunneling nanotubes and extracellular vesicles . . . . . . . . . . . . . . . 5.4.3 Effect of TASCs in co-cultures with GBM or ρ0 GBM cells . . . 5.4.4 Mitochondria purified from MSCs uptaked by GBM cells . . . 5.5 Conclusions and discussion . . . . . . . . . . . . . . . . . . . . . . . .. . . . . .. 43 43 43 45 45. . . . .. 45 47 48 51. Mathematical analysis of a simple model of high-grade glioma 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Analysis of travelling waves . . . . . . . . . . . . . . . . . . . 6.3.1 Fast solitons . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2 Finite speed solutions . . . . . . . . . . . . . . . . . . 6.4 Bright solitons . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1 Solitary wave solutions . . . . . . . . . . . . . . . . . 6.4.2 Minimum speed of positive solutions . . . . . . . . . 6.5 Linear Stability . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .. 53 53 53 55 55 58 61 62 63 64 67. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . .. . 43. Cancer Resistance Mathematical Models of Transfer of Resistance Characteristics 7.1 Transfer of resistance characteristics. An analysis with ODE 7.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 7.1.2 Experimental data and methods . . . . . . . . . . . . Cell Culture . . . . . . . . . . . . . . . . . . . . . . . . Assessment of cell proliferation in real time . . . . . . Analysis of P-gp expression . . . . . . . . . . . . . . . 7.1.3 Mathematical Analysis . . . . . . . . . . . . . . . . . . Cell-cell direct contact P-gp exchange model . . . . . Basic properties of the model . . . . . . . . . . . . . . MVs mediated P-gp exchange model . . . . . . . . . Basic properties of the model . . . . . . . . . . . . . . 7.1.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . .. 41 41 42 42 42 42. 69 . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . .. 71 71 71 72 72 72 73 74 74 75 76 78 79.

(15) xv. 7.2. 8. 9. 7.1.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Transfer of resistance characteristics. An analysis with an integrodifferential kinetic transport model . . . . . . . . . . . . . . . . . . . . 7.2.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.3 Methods and experimental data . . . . . . . . . . . . . . . . . Assessment of cell proliferation in real time . . . . . . . . . . . Analysis of P-gp expression . . . . . . . . . . . . . . . . . . . . Detection of P-gp transfer . . . . . . . . . . . . . . . . . . . . . Sorting of double stained cells: duration of P-gp changes . . . Statistical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.4 Mathematical model . . . . . . . . . . . . . . . . . . . . . . . . Cell-number conservative interactions . . . . . . . . . . . . . . Cell-number non-conservative interactions . . . . . . . . . . . Kinetic transport equations . . . . . . . . . . . . . . . . . . . . Numerical simulations . . . . . . . . . . . . . . . . . . . . . . . 7.2.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . P-gp expresison kinetics . . . . . . . . . . . . . . . . . . . . . . Cell growth dynamics . . . . . . . . . . . . . . . . . . . . . . . Effect of P-gp transfer in sensitive cells . . . . . . . . . . . . . Duration of transfer resistance . . . . . . . . . . . . . . . . . . Response to different treatment protocols . . . . . . . . . . . . Effect of non-genetic processes on tumor growth . . . . . . . 7.2.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. Transient resistance. An intermediate estate 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Methods and experimental data . . . . . . . . . . . . . . . 8.2.1 Agents and Cell Culture . . . . . . . . . . . . . . . 8.2.2 Cytotoxicity assay and cell counts . . . . . . . . . 8.2.3 Gene expression assay . . . . . . . . . . . . . . . . 8.2.4 Mice and ethics statement . . . . . . . . . . . . . . 8.2.5 Orthotopic injections of cells in NSG mice . . . . . 8.3 Mathematical models . . . . . . . . . . . . . . . . . . . . . 8.3.1 Pre-existence of a resistant population . . . . . . . 8.3.2 Induction of a resistant population . . . . . . . . . 8.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.1 Acquisition of resistance in U251 cells . . . . . . . 8.4.2 Clonal selection model results . . . . . . . . . . . Acquired resistance model results . . . . . . . . . 8.4.3 The effect of TSA in the intermediate populations 8.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .. Treatment schedule. An analysis with optimal control techniques 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.1 Existence and local stability of solutions of the model . . . . . 9.3 Optimal control problem . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.1 A general formulation of the optimal control problem . . . . . 9.3.2 Pontryagin’s Minimum Principle: necessary optimality conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . 85 . . . . . . . . . . . . . . . . . . . . . .. 86 86 86 87 87 87 87 87 88 88 89 92 93 94 94 94 97 98 99 100 103 104. . . . . . . . . . . . . . . . .. 107 107 108 108 108 108 109 109 109 109 111 113 113 115 118 120 122. . . . . .. 125 125 126 127 128 128. . 129.

(16) xvi 9.3.3. 9.4. 9.5. Chemotherapy as an optimal control problem and singular controls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Numerical solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.1 Optimal control for the functional J0,1 (u): Short and long times. 9.4.2 Optimal control for the functional J1,0 (u): Short and long times. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 130 133 133 136 139. 10 Conclusions 141 10.1 Conclusions of mathematical models in glioblastoma . . . . . . . . . . 141 10.2 Conclusions of mathematical models of development of resistance . . 142 11 Publications, Congress contributions and Honors 11.1 Full Publication List . . . . . . . . . . . . . . . . Publications in ISI-indexed journals . . Other publications . . . . . . . . . . . . 11.2 Congress contributions . . . . . . . . . . . . . . Invited talks . . . . . . . . . . . . . . . . Oral Presentations . . . . . . . . . . . . Poster Contributions . . . . . . . . . . . 11.3 Honors . . . . . . . . . . . . . . . . . . . . . . . Bibliography. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. 145 145 145 145 146 146 146 146 147 149.

(17) xvii. List of Figures 1.1 1.2. 1.3. 1.4 1.5 1.6 1.7 2.1 2.2. 2.3. The hallmarks of cancer. Figure adapted from Hanahan and Weinberg, 2011. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 a) Typical side effects of chemotherapy treatment. Figure adapted from Medical News Today. b) Cell cycle phase in which different chemotherapeutical agents are active. Figure adapted from https://chemoth.com/cellcycle. Basic scheme of a mathematical model of a biological process. The development of a mathematical process is iterative. Figure adapted from Fischer, 2008. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Basic scheme of Darwinian Selection interpretation. . . . . . . . . . . . 12 Basic scheme of Lamarckian induction interpretation. . . . . . . . . . . 13 Basic scheme of resistant transition in a model of Lamarckian induction interpretation with three different resistant phenotype. . . . . . . 14 Blood vessels within the tumor microenvironment contribute to produce spatial gradients of the drug concentration. . . . . . . . . . . . . 15 Role played by 3D experimental modelling approaches in the sequence to improve therapy decision-making. . . . . . . . . . . . . . . . . . . . 18 a) Schematic representation of the different areas of glioblastoma in vivo. Figure adapted from Pérez-García et al., 2011. b) Kaplan-Meier curve for the spherical rim width (denoted by δs ). Figure adapted from Pérez-Beteta et al., 2017. . . . . . . . . . . . . . . . . . . . . . . . . 19 Possible scenarios displaying the response of sensitive and resistant cancer cell subpopulations to the absence/presence of a specific drug and intercellular communication. (Left panel) Selection and/or induction drive the cell number and P-gp expression kinetics when no communication exists between these two subpopulations. (Right panel) In addition to selection and/or induction, a transfer mechanism may arise when both subpopulations are in contact via an extracellular medium through which they can exchange microvesicles (MVs). Color code: blue and red for constitutively sensitive and resistant cells, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20. 3.1. Gliomas classification. Figure adapted from Louis et al., 2016. . . . . . 23. 4.1 4.2. Protocol for the experimental growth of 3D biospheres. . . . . . . . . . 27 Cell number growth examples. a) Examples of GBM8 growth. Blue line represents the growth of only GBM8 cells, red line the growth of GBM8 cells with ASC, and yellow line the growth of GBM8 cells with activated ASC (TASCs) b) Examples of GBM22 growth. Blue line represents the growth of only GBM22 cells, red line the growth of GBM22 cells in combination with fibroblasts and yellow line in combination with activated fibroblast (TASCs). . . . . . . . . . . . . . . . . . . . . . 28 Example of images obtained with the confocal microscope. . . . . . . 29. 4.3. 5.

(18) xviii 4.4 4.5 4.6 4.7 4.8. 4.9. 4.10. 4.11. 4.12. 4.13. 5.1. 5.2 5.3 5.4 5.5 5.6 5.7. 2D geometrical measurements of individual neurospheres obtained directly from the photographs. . . . . . . . . . . . . . . . . . . . . . . . Histogram evolution of the neurospheres area of GBM8. The number n is the number of neurospheres measured at each time. . . . . . . . . Cell number fits for the primary culture GBM3, in a serum medium without fibroblasts/MSCs/ASCs. . . . . . . . . . . . . . . . . . . . . . Cell number fits for the primary culture GBM8. In a) GBM cells where cultured in a serum medium, in b) in a defined medium with fibroblasts. Cell number fits for the primary culture GBM22. PDCs cells are cultured in a) in a medium with collagen, in b) in a medium with gelatin, in c) in the presence of fibroblasts in a defined medium and in d) with fibroblasts in a serum medium. . . . . . . . . . . . . . . . . . . . . . . . Cell number fits for the primary culture GBMA1. PDCs cells are cultured in a) in a medium with collagen, in b) in a medium with gelatin. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Boxplots showing a representation of the doubling times of the different primary cultures. The number of experiments n used for each primary culture is shown on top of each boxplot. . . . . . . . . . . . . Boxplot comparison for the messenchymal primary culture GBM22. The experimental results (E) are compared to the theoretically spherical results (S). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Boxplot comparison for the proneural primary culture GBM8. The experimental results (E) are compared to the theoretically spherical results (S). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Levene quadratic test for the different primary cultures. Spheroids are classified in groups, according to their area. The 1st and 2nd group comprises the spheroids with an area below or above the area median respectively. This analysis was done with primary cultures with the same environmental initial conditions, and circularity is measured between days 14 and 16, to ensure a sufficient size of the spheroids. Subfigures represent the different primary cultures, a) GBMA1, b) GBM22, c) GBM3 and d) GBM8. . . . . . . . . . . . . . . . . . . . . . . Diagram of the different experiments performed in this study. Samples from glioma cells were extracted for subsequent flow cytometry analysis to measure their mitochondria level. Tumor cells were cultured in the absence/presence of a medium enriched with mitochondria. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . a) Incorporation of TASCs into the neurospheres, b) Neurospheres communication via TNTs. . . . . . . . . . . . . . . . . . . . . . . . . . Mitochondria transfer from TASCs to GBM cells through TNTs. . . . a) EVs containing labeled mitchondria, b) Electron microcospy of on the EVs containing mitochondria. . . . . . . . . . . . . . . . . . . . . a) Cell number results of irradiated and non-irradiated cellsl, b) Viability of the different populations with and without radiotherapy. . . a) Cell number of the populations in the different scenarios, b) Viability of the populations in the different scenarios. . . . . . . . . . . . . Mitochondria uptake by ρ0 GBM cells. . . . . . . . . . . . . . . . . . .. 30 31 32 33. 33. 34. 34. 36. 36. 38. . 44 . 45 . 46 . 47 . 48 . 48 . 49.

(19) xix 5.8. Simulation of GBM3 in a medium enriched with mitochondria from MSC. Dashed curves correspond to the experimental results, and solid lines to the numerical simulations. Red, dark red, blue and dark blue colors correspond to experiments 1, 2, 3 and 4 respectively. . . . . . . . 50 5.9 Simulation of A1 in a medium enriched with mitochondria from ASC. Dashed curves correspond to the experimental results, and solid lines to the numerical simulations. Red, dark red, blue and dark blue colors correspond to experiments 1, 2, 3 and 4 respectively. . . . . . . . . . . 50 5.10 Simulation of GBMG5 in a medium enriched with mitochondria from MSC. In the experiments with cancer cells without mitochondria (right panel), there are two subpopulations, i.e. i = 2. Dashed curves correspond to the experimental results, and solid lines to the numerical simulations. Red, dark red, blue and dark blue colors correspond to experiments 1, 2, 3 and 4 respectively. . . . . . . . . . . . . . . . . . . . 51 6.1. 6.2. 6.3 6.4. 7.1 7.2 7.3. 7.4. 7.5. (a) Heteroclinic orbits of Eqs. (6.10)) for β = 0.1. (b) Kink solution for β = 0.1 corresponding to the marked heteroclinic orbit in (a). (c) The bright soliton solution U corresponds to the kink obtained in (b). . . Bright solitary waves for β = 0.3, φ0 = 0.5 and c = 2.5. (a) Profiles from Eq. (6.45) (solid curve) and the explicit solution given by Eq. (6.46) (dashed curve). (b) Homoclinic orbits from Eq. (6.45) (solid curve) and Eq. (6.46) (dashed curve). . . . . . . . . . . . . . . . . . . . Characteristic curves of the minimum speed c0 for positive bright solitary solutions, in terms of (a) β and (b) φ0 , calculated from Eq. (6.53). Plot of the depth of resection b versus the recurrence time t for different values of the proliferation rate: ρ = 1/5 (blue solid line), ρ = 1/4 (green dashed line) and ρ = 1/3 (red dotted-dashed line). . . . . . . . Basic scheme of the experiments. See text for details. . . . . . . . . . Basic scheme of the two theories of P-gp exchange, a) Cell-cell direct contact b) MVs mediated. . . . . . . . . . . . . . . . . . . . . . . . . . (Left) Phase portrait of the orbits of Eqs. (7.1). Black curve represents the set of equilibrium points P2 (λ), corresponding to the center manifold. Red arrows show the flow of the orbits, which are given by blue lines. (Right) Plot of eigenvalues corresponding to the equilibrium points E2 (α) as a function of α. . . . . . . . . . . . . . . . . . . . . . . Cell index measurements when 2000 (left panel), or 4000 (right panel) sensitive cells are seeded per well (dotted red line). The best least squares fits using the logistic models given by Eqs. (7.1)) or (7.6) are indicated by solid blue lines. Vertical gray lines represent the standard deviations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cell index measurements when 2000 (left panel), or 4000 (right panel) sensitive cells are seeded per well (dotted red line). The best least squares fits using the logistic models given by Eqs. (7.1)) (7.6) are indicated by solid blue lines. Vertical gray lines represent the standard deviations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . 57. . 62 . 64. . 67 . 74 . 77. . 79. . 79. . 80.

(20) xx 7.6. 7.7. 7.8. 7.9. 7.10. 7.11. 7.12. 7.13. Experimental growth data for an initial population of 4000 cells per well (dotted red line) and best fit using the logistic model (7.1) (solid blue line). Both curves represent the total cell index (S + R + SR ). Initial data are S(0) = 0.0065, R(0) = 0.0065, SR (0) = 0 for the 1:1 mixture (left panel), S(0) = 0.012, R(0) = 0.004, SR (0) = 0 for the 3:1 mixture (center panel) and S(0) = 0.00875, R(0) = 0.00125, SR (0) = 0 for the 7:1 mixture (right panel). Vertical gray lines represent the standard deviations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . Experimental growth data for an initial population of 4000 cells per well (dotted red line) and best fit using the logistic model (7.6) (solid blue line). Both curves represent the total cell index (S + R + SR ). Initial data are S(0) = 0.06, R(0) = 0.06, SR (0) = 0 1:1 mixture (left panel), S(0) = 0.14, R(0) = 0.048, SR (0) = 0 for the 3:1 mixture (medium panel) and S(0) = 0.09, R(0) = 0.013, SR (0) = 0 for the 7:1 mixture (right panel). Vertical gray lines represent the standard deviations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Comparison of the percentages of sensitive (left panel) and resistant cells (right panel) in initial mixtures of 1 : 1 sensitive to resistant cells. Solid blue lines represent the model Eqs. (7.1) predictions and dashed red lines experimental results. . . . . . . . . . . . . . . . . . . . . . . . Comparison of the percentages of sensitive (left panel) and resistant cells (right panel) in initial mixtures of 3 : 1 sensitive to resistant cells. Solid blue lines represent the model Eqs. (7.1) predictions and dashed red lines experimental results. . . . . . . . . . . . . . . . . . . . . . . . Comparison of the percentages of sensitive (left panel) and resistant cells (right panel) in initial mixtures of 7 : 1 sensitive to resistant cells. Solid blue lines represent the model Eqs. (7.1) predictions and dashed red lines experimental results. . . . . . . . . . . . . . . . . . . . . . . . Comparison of the percentages of sensitive (left panel) and resistant cells (right panel) in initial mixtures of 1 : 1 sensitive to resistant cells. Solid blue lines represent the model Eqs. (7.6) predictions and dashed red lines experimental results. . . . . . . . . . . . . . . . . . . . . . . Comparison of the percentages of sensitive (left panel) and resistant cells (right panel) in initial mixtures of 3 : 1 sensitive to resistant cells. Solid blue lines represent the model Eqs. (7.6) predictions and dashed red lines experimental results. . . . . . . . . . . . . . . . . . . . . . . . Comparison of the percentages of sensitive (left panel) and resistant cells (right panel) in initial mixtures of 7 : 1 sensitive to resistant cells. Solid blue lines represent the model Eqs. (7.6)) predictions and dashed red lines experimental results. . . . . . . . . . . . . . . . . . . . . . . . Continous versus discrete interpretation. . . . . . . . . . . . . . . . .. . 80. . 81. . 81. . 82. . 82. . 83. . 83. . 84 7.14 . 86 7.15 In the absence of exogenous agents and interactions, isolated subpopulations with constitutively low or high expression levels of P-gp will exhibit (basal). (basal). small fluctuations around their characteristic basal values x1 and x2 , respectively. However, if an interaction between these cell subpopulations exists, those with constitutively low expression levels may transiently display higher values of P-gp for a time duration of the order of T. Such higher values will persist if the interaction between these subpopulations is sustained for sufficiently long times. . . . . . . . . . . . . . . . . . . . . . . . . 93.

(21) xxi 7.17 P-gp distribution changes after 72 h under (a) 0 nM and (b) 50 nM of DOX for initial sensitive, resistant and mixed cells (sensitive:resistant fractions 1:1, 3:1 and 7:1). Insets: Cell number versus time for initial populations of sensitive, resistant and mixed cells with 0 and 50 nM of DOX. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.16 Evolution of P-gp expression in the different scenarios. Left/right column show experiments in the absence/presence of DRUG. Cells were seeded at t = 0 h and measured at subsequent intervals of 24 h. Dashed and solid lines represent the experimental results and the numerical simulations of our model, respectively. . . . . . . . . . . . 7.18 Fit of our model to NCI-H460 cell line growth assessment measured by a impedance-based (xCELLigence) system with 4000 initial cells and different drug conditions. Solid and dashed lines represent both the numerical simulation of our model and the experimental data, respectively. The higher the drug concentration, the lower the cell growth curve. Cell growth curves are displayed for 0 nM, 10 nM, 50 nM and 100 nM of DOX concentrations. The abrupt transient drops in the experimental curves were caused by voltage fluctuations on the measuring equipment. Inset: Therapy function versus P-gp expression for 0, 10, 50 and 100 nM of DOX concentrations. . . . . . . . . . 7.19 Detection of P-gp transfer during 48 h in different culture media with sensitive cells (' 2 × 104 ). Upper/lower rows represent the experimental results and the model predictions, with the corresponding calculated MV transfer function (inset). The culture media comprised only the H460 cells (blue solid curves), H460 cells grown in a conditioned medium (CM) exchanged from the H460/R cells to the H460 cells medium (red dashed curves), H460 cells grown in the presence of 50 nM of DOX (yellow dotted curves) and H460 cells grown both in the presence of 50 nM of DOX and CM (violet dashed-dotted curves). Asterisks in the upper row denote the p-values for pairwise comparisons: p < 0.005 (**) and p < 0.0005 (***). . . . . . . . . . . . . . . . . 7.20 Duration of P-gp changes. The upper four rows correspond to the numerical simulations over a time period of 240 h, while the last row shows the experimental results (at t = 240 h). . . . . . . . . . . . . . 7.21 Drug administration scheduling in the three protocols considered. . 7.22 Treatment response for the control (no drug) and the three considered protocols for H460 (first row), Mix 7:1 (second row), Mix 3:1 (third row), Mix 1:1 (fourth row) and RH460 (fifth row) initial populations when administering three different concentrations of DOX. The initial cell number was 20000 cells in all cases. . . . . . . . . . . . . . . . . . 7.23 Accumulated deviation in cell number for a period of 240 h when all MDR processes are present versus the case when Lamarkian induction is absent. The curves have been calculated using the L2 norm of the difference between the curves shown in Fig. 7.22 with those in which induction has been inactivated. All parameters correspond to Fig. 7.22. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.24 Accumulated deviation in cell number for a period of 240 h when all MDR processes are present versus the case when MV transfer is absent. The curves have been calculated using the L2 norm of the difference between the curves shown in Fig. 7.22 with those in which transfer has been inactivated. All parameters correspond to Fig. 7.22.. . 95. . 96. . 98. . 99. . 100 . 101. . 102. . 103. . 104.

(22) xxii 8.1 8.2. 8.3. 8.4 8.5. 8.6 8.7. 8.8. 8.9 8.10. 8.11 8.12. 8.13. 8.14 9.1. 9.2. 9.3. See text for details. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . a) Basic scheme of the clonal selection theory. b) Basic dynamics of the different subpopulations (upper row) and the total population (lower row). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . a) Basic scheme of the acquired resistance theory. b) Basic dynamics of the different subpopulations (upper row) and the total population (lower row). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Functions used to represent the drug delay effect on cancer cells. . . Cell number results. Each color represent ta different replicate of the experiment. Solid blue line represent the mean of the four replicates. Vertical blue lines represent the standard deviation of the four replicates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . MGMT expression dynamics, represented with boxplots. The inset in the figure it is an amplification of MGMT expression from day 0 to 6. a) Obtaining process for U251S ,U251R ,U251R3 ,U251R6 . The population U251R/WO is obtained after U251R was left untreated for 3 months b) Viability of the different subpopulations obtained. . . . . . . . . . Four possible fits of the experimental data. Pink solid lines represent the results coming from Eqs. (8.1) and dashed blue lines represent the experimental results (mean ± sd). . . . . . . . . . . . . . . . . . . . . MGMT variability for the fits shown in Fig. 8.8, with different values of α. MGMT expression is calculated with Eq. (8.5). . . . . . . . . . . Four possible fits of the experimental data. Pink solid lines represent the results coming from Eqs. (8.6) and dashed blue lines represent the experimental results (mean ± sd). . . . . . . . . . . . . . . . . . . . . MGMT variability for the fits shown in Fig. 8.10, with different values of α. MGMT expression is calculated with Eq. (8.10). . . . . . . . . . Effect of TSA in three subpopulations. Significant changes are observed in the populations U251TR− D3 (medium) and U251R (right), but no significant changes in U251S (left). . . . . . . . . . . . . . . . . Effect of TSA and TMZ in U251TR . Black bar represents the experiments with TMZ and gray bars with TMZ and TSA. Statistical significant changes are observed between the two experiments at day 9 and 12. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Effect of TSA in vivo. Statistical significant changes are only observed in the U251TR− D9 population between normal and TSA experiments.. . 108. . 110. . 111 . 112. . 114 . 115. . 115. . 116 . 118. . 119 . 120. . 121. . 121 . 122. Phase portrait of the orbits of system (9.2). Examples of convergent trajectories to P2 for x0 + y0 + z0 ≤ K. Parameters used to calculate the phase portrait are given in Table 9.1. . . . . . . . . . . . . . . . . . 128 Optimal solutions for the objective J0,1 (u) and M = 1/6. a) Optimal control u∗ (t) (blue curve) and switching function φ (dotted red line) satisfying the bang-bang control law (9.9). b) Sensitive cells x (t) c) Damaged cells y(t) d) Resistant cells z(t). The time horizon was fixed to T = 1 month. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 Optimal solutions for the objective J0,1 (u) and M = 5. a) Optimal control u∗ (t) (blue curve) and switching function φ (dotted red line) satisfying the bang-singular-bang control law (9.21). b) Sensitive cells x (t) c) Damaged cells y(t) d) Resistant cells z(t). The time horizon was fixed to T = 30 months. . . . . . . . . . . . . . . . . . . . . . . . . 135.

(23) xxiii 9.4. 9.5. 9.6. 9.7. 9.8. Suboptimal protocol for the objective J0,1 (u) and M = 5. a) Suboptimal bang-bang control. b) Sensitive cells x (t) c) Damaged cells y(t) d) Resistant cells z(t). The time horizon was fixed to T = 30 months. . Optimal solutions for the objective J1,0 (u) and M = 1/6. a) Optimal control u∗ (t) (blue curve) and switching function φ (dotted red line) satisfying the bang-bang control law (9.9). b) Sensitive cells x (t) c) Damaged cells y(t) d) Resistant cells z(t). The time horizon was fixed to T = 1 month. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Optimal solutions for the objective J1,0 (u) and M = 5. a) Optimal control u∗ (t) (blue curve) and switching function φ (dotted red line) satisfying the bang-singular-bang control law (9.23). b) Sensitive cells x (t) c) Damaged cells y(t) d) Resistant cells z(t). The time horizon was fixed to T = 30 months. . . . . . . . . . . . . . . . . . . . . . . . Suboptimal protocol for the objective J1,0 (u) and M = 5. a) Suboptimal bang-bang control. b) Sensitive cells x (t) c) Damaged cells y(t) d) Resistant cells z(t). The time horizon was fixed to T = 30 months. . Suboptimal protocol for the objective J1,0 (u) and M = 5. a) Suboptimal bang-bang control. b) Sensitive cells x (t) c) Damaged cells y(t) d) Resistant cells z(t). The time horizon was fixed to T = 30 months. .. . 136. . 137. . 138. . 139. . 140.

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(25) xxv. List of Tables 7.1. 7.2. 7.3 7.4. 7.5. 8.1. Data for the percentages of sensitive and resistant cells obtained by flow-cytometry and predictions from the mathematical model given by Eqs. (7.1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Data for the percentages of sensitive and resistant cells obtained with flow-cytometry and predictions from the mathematical model given by Eqs. (7.6). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Residual errors for both mathematical models against the experimental data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Estimated parameters’ values for P-gp expression analysis. Induction function v I ( x ) = a(1 − tanh(b( x − c))). Therapy function T ( x ) = d(1 − tanh(e( x − f ))). Transfer function v Ti ( x ) = − gd (tanh(h( x − x0 ))) + gd (2 − i ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Statistical analysis (p-values) of the experimental results obtained to study the effect of MVs on the P-gp level of sensitive cells under various culture conditions after 48h. Here, a comparison of the P-gp levels of sensitive cells with and without medium exchange (ME in the table) and in the presence/absence of drug (DOX) is made. The experimental results are shown in Fig. 5 of the main manuscript. Here, *p < 5 · 10−2 , **p < 5 · 10−3 , ***p < 5 · 10−4 . . . . . . . . . . . . . . . .. 8.4. Estimated parameters’ values for clonal selection analysis. Parameters K and a were fixed before the analysis. T (0) = S(0) + R(0). . . . Estimated range of values for β assuming the clonal selection hypothesis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Estimated parameters’ values for acquired resistance analysis. T (0) = S(0) + TR(0) + R(0). . . . . . . . . . . . . . . . . . . . . . . . . . . . Estimated parameters’ values for acquired resistance analysis. . . . .. 9.1. Values of the biological parameters for the system (9.2).. 8.2 8.3. . 83. . 84 . 85. . 97. . 99 . 117 . 118 . 119 . 120. . . . . . . . . 127.

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(27) xxvii. List of Abbreviations ASC CNS DNA DOX ECM EVs GBM HGG LGG MDR MGMT MSC MV PDC P-gp RNA ROS TASCs TMZ TNT. Mesenchymal stem cells (fat tissue) Central nervous system Deoxyribonucleic acid Doxorubicin Extracellular matrix Extracellular vesicles Glioblastoma High-grade gliomas Low-grade gliomas Multidrug resistance O6-alkylguanine DNA alkyltransferase Mesenchymal stem cells (bone marrow) Microvesicle Patient-derived cell P-glycoprotein Ribonucleic acid Reactive oxygen species Tumor-activated stromal cells Temozolomide Tunneling nanotubes.

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(29) xxix. For/Dedicated to/To my. . ..

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(31) 1. Chapter 1. Introduction 1.1. Cancer. Cancer is one of the major health problems worldwide. In the industrialized world, it is the second leading cause of death, only surpassed by heart diseases (Siegel, Miller, and Jemal, 2016). Only in 2018, there were 18.1 million new cancers diagnosed and 9.6 million cancer deaths (Bray et al., 2018). Cancer is not a single disease, but a group of more than 100 diseases. Nowadays most of the processes that lead to the formation of cancer are well-known. This set of processes is named as carcinogenesis, and can be basically summarized as follows. Firstly, and according to the standard dogma of cancer, a cell mutates and instead of proliferating in a regulated fashion, it starts to divide in a haphazard manner. There are two categories of gene mutations. The first type are mutations affecting genes involved in cell proliferation, called oncogenes. Cells with some of these genes mutated receive proliferation signals, even in the absence of growth factors. The second general type of mutations affects genes involved in the repair or destruction of defective cells. These genes are called anti-oncogenes or tumor suppressor genes. When some of these genes are mutated, cell division is no longer under control. Abnormal cell division can happen either when oncogenes or anti-oncogenes are mutated, independently or simultaneously (Ewald and Ewald, 2012). These mutations are transmitted to the offspring which continues to proliferate at a faster rhythm. This fact causes the accumulation of a mass of cells, called a primary tumor (except for cancer affecting the blood or bone marrow). In this initial stage tumoral cells present an avascular growth. Secondly, the tumor recruits vessels. The process of tumor neovascularization is mediated by genes and signaling pathways that are activated in the presence of inflammation and hypoxia. This process is known as angiogenesis. The new blood vessels formed in the tumor differ from the vessels in normal tissues, as they are typically abnormal, fragile, and hyperpermeable. The recruitment of new vasculature allows the tumor to continue expanding, in a new phase called vascular growth (Burrell and Zadeh, 2012) Thirdly, tumoral cells escape from their original tissue through the blood vessels, lymphatic system or regional spread, and are able to set up a new mass of cells in a different tissue. This process is known as metastasis, and the new tumors are refered to as secondary tumors (Liu et al., 2017). When this process happens, we refer to that tumor as a cancer..

(32) 2. Chapter 1. Introduction. 1.1.1. Classification. Cancers are usually named according to the location of the body in which they originate. Lung cancer, breast cancer or prostate cancer are some examples of this nomenclature. Cancer can also be classified by the type of tissue in which it originates. This classification consists of 6 main groups. • Carcinomas: cancers originated in epithelial tissues. They can arise in the skin or in the lining of the internal organs. They represent the 80 − 90% of all cancers. • Sarcomas: cancers originating in tissues of mesodermal origin, such as supportive and connective tissues (bone, cartilage, muscle). • Myelomas: cancers originated in the plasma cells of bone marrow. • Leukemias: cancers of the bone marrow. They are usually named as "liquid cancer" or "blood cancers" because they usually affect white or red blood cells. • Lymphoms: cancers originated in the glands or nodes of the lymphatic system. • Mixed types: cancers presenting two or more components of the previous types.. 1.1.2. The hallmarks of cancer. Every cancer is a different disease. However, the vast catalog of cancers presents some common traits, which are essential alterations in cell physiology that collectively dictate the behavior of cancer. In Hanahan and Weinberg, 2000, six common traits were suggested. A subsequent review by the same authors ( Hanahan and Weinberg, 2011) extended this list to ten, summarized in Fig. 1.1.. F IGURE 1.1: The hallmarks of cancer. Figure adapted from Hanahan and Weinberg, 2011..

(33) 1.2. Chemotherapy. 3. Understanding these processes is essential for the development of efficient anticancer therapeutic strategies. In Fig. 1.1 specific treatments targeting the different hallmarks are also suggested.. 1.1.3. Cancer treatments. Nowadays, it is expected that 1 in 2 men and 1 in 3 women will be diagnosed with cancer over the course of his or her lifetime ( Ahmad, Ormiston-Smith, and Sasieni, 2015). This scaring prognostic would be terrible if it were not for the large amount of possible therapies available, which allows many cancers to be controlled. The objectives of cancer therapies are to prevent proliferation (cytostatic effect) and to kill the cancer cells (cytotoxic effect). Some of the most common treatments are: • Surgery: It is a process by which a tumor mass is mechanically removed from the body. The tumor can be completely or partially ressected. • Chemotherapy: conventional chemotherapy uses chemicals that target deoxyribonucleic acid (DNA), ribonucleic acid (RNA), and protein to disrupt the cell cycle in rapidly dividing cancer cells and thus has broad specificity. The ultimate goal of cytotoxic chemotherapy is to cause severe DNA damage and to trigger apoptosis in the rapidly dividing cancer cells. • Radiotherapy: It is the use of focalized ionizing radiation to inflict a damage into the DNA of cancer cells. These cells, upon mitotic catastrophe undergo cell death in subsequent replication attempts. • Immunotherapy: A promising area in cancer treatment. Its main is to stimulate and contribute to the activation of cell groups of the immune system to eliminate cancer cells. Other possible treatments are targeted therapies, hormone therapies and stem cell transplantations. These treatments constitute the unique alternative in some specific cancers. However, to maximize the possible benefits, therapies are often combined.. 1.2. Chemotherapy. Chemotherapy is widely used in cancer treatment. The use of chemotherapy can have different goals: to cure the cancer, to control the disease or to palliate symptoms. It can be employed as a unique therapy, but it is normally combined with other types of treatment. If chemotherapy is used to shrink the tumor before surgery or radiotherapy, it receives the name of neoadjuvant therapy. On the other hand, if it is used after surgery or radiation to target the remaining cancer cells it is called adjuvant chemotherapy. Chemotherapy can be administered by different techniques, being the most common intravenously or orally when the drug comes as a pill, capsule or liquid. To better understand how chemotherapy works, the series of events that take place in an eukaryotic cell during the cell-division cycle (or cell cycle) are detailed in what follows. During the cell cycle, the DNA is duplicated, and the organelles and the cytoplasm are split into two daughter cells. Four different phases are distinguished in the cell cycle ( Fig. 1.2(a))..

(34) 4. Chapter 1. Introduction • G1 phase: During this phase, the cell grows in size. Protein supply and the number of organelles (such as mitochondria, ribosomes) increases. • S phase: In this phase, the DNA synthesis takes place. From one original DNA molecule, the cell produces two identical replicas of DNA. • G2 phase: After DNA replication, the cell prepares itself for division. Proteins are synthetized and the cell continues to growth in size. • Mitotic phase: In this phase, the original cell divides into two new daughter cells. Firstly, the nucleus of the mother cell is divided. Secondly, the cytoplasm, the organelles, the nuclei and cell membrane are separated, forming two new cells with approximately equal proportion of these components. This second step is referred to as cytokinesis.. Chemotherapy drugs produce damages in the cell by altering some of the processes that take place during the cell cycle. The specific cell cycle phase in which damages are produced varies between drugs. As chemotherapy affects cells that are proliferating, cancer cells are directly targeted by this therapy. One of the particularities of chemotherapy is its global effect. While surgery or radiotherapy have mostly a local action in a specific area, chemotherapy has systemic effects. Therefore, chemotherapy drugs attack also metastatic cells. This characteristic is defined as systemic treatment. The bad news is, however, that chemotherapy drugs that travel through the vascular system and reach cells all over the body barely distinguish between cancer cells and normal cells. For that reason, fast-growing healthy cells such like those of the skin, hair, intestines, and bone marrow are also damaged by chemotherapy. Consequently, illness, easy bruising or bleeding, hair loss, nausea and vomiting are some of the common side effects of chemotherapy treatments. These side effects do not normally last. We are able to replace or to repair the damaged normal cells, henceforth most side effects disappear after the treatment is halted. Some of the most common side effects of chemotherapy are displayed in Fig. 1.2(a)..

(35) 1.2. Chemotherapy. 5. Antibiotics Anti-metabolities. S G2. Vinca alkaloids M. Mitotic inhibitors Taxoids. Alkylating agents. G1. (a). (b). F IGURE 1.2: a) Typical side effects of chemotherapy treatment. Figure adapted from Medical News Today. b) Cell cycle phase in which different chemotherapeutical agents are active. Figure adapted from https://chemoth.com/cellcycle.. 1.2.1. Types of chemotherapy drugs. Chemotherapy drugs can be characterized in groups according to their functionality (ALDRED, 2009). This helps to understand how the drugs work, their possible side effects and the potentiality of combination with other drugs. Some of this groups are: • Alkylating agents: These cell-cycle non-specific drugs can act in any of the phases of the cell cycle. They arrest cell division by damaging its DNA. Temozolomide, a drug considered in some of the studies of this thesis, belongs to this group. • Antimetabolites: Their effects are produced in the S phase, specifically when chromosomes are being copied. They work by interfering with DNA and RNA growth, stopping normal development and cell division. • Antitumor antibiotics: These drugs are considered cell-cycle specific but their action can occur during the G1, S and G2 phase. DNA synthesis and replication are affected by these agents by inserting into DNA strands or by producing superoxide conditions causing the DNA strands to break. To this group belongs the drug doxorubicin (DOX), also studied in this thesis. • Topoisomerase inhibitors: Topoisomerase enzymes participate both in the overwinding and/or underwinding of DNA. These enzymes are necessary for replication, and its blockage prevents the cell to divide. There are two types, named as topoisomerase I and topoisomerase II. • Mitotic inhibitors: Their action is reserved to the M phase. They prevent the cell to divide by disrupting microtubule polymerization, structures involved in the separation of chromosomes during mitosis. Some specific mitotic inhibitors are vinca akaloids and taxoids..

(36) 6. Chapter 1. Introduction. The effect on the cell cycle of the different types of chemotherapies is summarized in Fig. 1.2(b). In many cases, more than one drug is used during a chemotherapy treatment. This allows to target cells at various stages of their cell cycle, i.e. attacking different cells at the same time, and hopefully enhancing therapy benefits.. 1.3. Chemotherapy resistance. One of the major problems affecting chemotherapy treatment is drug resistance. Cells can escape from the total action of chemotherapy agents through different mechanisms ( Mansoori et al., 2017; Housman et al., 2014). • Drug inactivation: Clinical efficacy of many anti-cancer drugs depends on metabolic activation. This activation depends on the interaction with proteins. Depending on the drug, this interaction could cause the modification of the drug, its partial degradation or its coupling with other proteins or molecules. All these different mechanisms are responsible for drug activation. Cancer cells can produce drug inactivation by the down-regulation or mutation of the genes involved in the pathways responsible of drug activation. This fact produces drug resistance in cancer cells, as inactivated chemotherapeutic drugs reduce their clinical efficacy. • Alteration of drug targets: Chemotherapy drugs normally target specific objectives involved in cell division. If those targets are modified or mutated, drug resistance might occur. For example, the topoisomerase inhibitors directly target the topoisomerase enzymes, as previously explained. Mutations in the topoisomerase II gene make ineffective some of these drugs. • Drug efflux: One of the most important and studied mechanisms in drug resistance. A group of proteins of the ATP-binding cassette transporter family proteins can be found in normal cells. These proteins are located in the cellular membrane and are responsible to efflux toxins out of the cells. Apart from being involved in normal physiological processes, these proteins can also be present in cancer cells and are able to reduce the drug accumulation inside the cells. By enhancing drug efflux cells are able to decrease the clinical efficacy of chemo-drugs. The over-expression of these proteins do not confer resistance to a one specific drug, but to several different drugs. Three transporters are implicated in many drug resistance cancers; multi drug resistance protein 1 (MDR1), multidrug resistance-associated protein 1 and breast cancer resistance protein. These transporters are able to pump out of the cells vinka alkaloids and taxanes among other drugs, protecting cancer cells from many first-line chemotherapy. P-glycoprotein (P-gp), a protein produced by the MDR1 gene, is studied during this thesis as a mechanism of drug resistance. • DNA damage repair: As previously explained, many cancer drugs produce DNA damage, by an action that can be directly or indirectly. However, cancer cells have mechanisms to fix the damage caused. This can be made thanks to DNA damage repair mechanisms. The final damage produced to the cells is a combination between the drug-induced damaged and the cell ability to repair that damage. The protein O6-alkylguanine DNA alkyltransferase (MGMT), which is able to repair the damage produced in the guanine base before a mismatch is occurred, it is considered during this thesis..

(37) 1.4. Mathematics and biology/cancer. 7. • Cell death inhibition: Cell death can be mediated by apoptosis, necrosis and autophagy (Galluzzi et al., 2018). These processes are carried out by means of internal or external pathways. Mutations in genes involved in those pathways might altered the final processes. DNA damaged caused by chemotherapeutic agents cannot produce cell death if some of those pathways are modified, resulting in drug resistance. All these possible mechanisms of drug resistance considerably complicate the landscape of cancer treatment. In addition, these mechanisms are not mutually exclusive, so there is a possibility that the cancers may become unresponsive against multiple drugs. All of these considerations must be taken into account when designing a treatment protocol.. 1.4. Mathematics and biology/cancer. Biology has attracted the attention of mathematicians for a long time. In the 12th century, Fibonacci described the growth of a rabbit population with the famous Fibonacci series. However, from the 1960s onwards the interest of mathematicians in biology and medicine has grown rapidly. Some of the reasons explaining this growth are the increasing volume of data collected (due to the genomic revolution), the recent development of mathematical tools that allow to understand complex non-linear mechanisms in biology, the growth in computing power has allowed a significant speed-up of simulations, and the interest on in-silico approaches to perform large scale simulations and to avoid ethical problems in human and animal research. Mathematical models need a precise description of the biological processes involved. Biological systems can be very complex, in the sense that many factors can be operating at the same time. However, when studying specific aspects of the biological systems not all the processes should be included. Modeling all the processes that take place can be as tedious as unnecessary. The first question that a modeler has to ask is what the important processes are. Selecting the relevant features and neglecting the irrelevant ones is the first step to develop a useful model. The simplicity of some models is actually their strength, even if sometimes they seem gross simplifications. Simple models are useful to show the implications of the basic assumptions made (Allman and Rhodes, 2003; Byrne, 2010; Gerlee, 2013; Altrock, Liu, and Michor, 2015; Barbolosi et al., 2015; Karolak et al., 2018) Mathematical models (also called in-silico models) can help biologists and medical doctors in their research. Data sets and biological results can be complemented with analyses coming from mathematical models. One of the main characteristics of these models is their ability to describe the observed phenomenology. When there are doubts about the processes that are behind of the obtained results, mathematical models might be useful to clarify the situation. A model based on a possible theory explaining the biological process can be contrasted with experimental biological data. If there exist discrepancies, analysis may point out some components are missing in the mathematical model, or some processes are unnecessary or not correctly included. Mathematical models can thereby assist to develop a more comprehensive picture of biological process. Sometimes mathematical models do not provide the answer of what processes are incorrectly included or excluded but might help to design in a new set of experiments that could clarify this issue. In Fig. 1.3 it is possible to see an scheme of the development of a mathematical model, with several interactions between experiments and simulations..

(38) 8. Chapter 1. Introduction. F IGURE 1.3: Basic scheme of a mathematical model of a biological process. The development of a mathematical process is iterative. Figure adapted from Fischer, 2008.. Another characteristic that makes mathematical models powerful, is to establish the relative importance of different processes in the final result. When there is agreement between the behaviors predicted by a mathematical model and actual behaviors observed in experiments, this means that the relevant biological processes may be correctly understood. In that case, it is possible to quantify the different processes included in the model. Not all biological processes are equally important for the overall behavior of the system. A small variation in some of the relevant processes can result in large differences in the final result, while there could be other processes that have only a slightly influence in the dynamics. The relevance of each process can be estimated performing computational simulations with different parameters values and comparing the results. Performing laboratory experiments to study the relevance of the different process is usually much more difficult. In addition, mathematical simulations are faster and cheaper than laboratory experiments. Once a reasonable model is available, computer simulations can be quickly carried out to analyze the outcome of the system under different initial conditions. For example, if we know how different drug concentrations affect the volume of a tumor, and we want to find a good protocol to reduce the volume, a good first approach would be to perform many simulations and compare the final results of the different protocols. Afterwards, it could be possible to select the best protocols computationally and perform a set of experiments to confirm the hypothesis. Many of the biological processes are dynamic, meaning that they change in time. If a biological system seems to be stable, it is normally the balance of different processes pushing in different directions. Mathematical models in biology aim to describe biological processes, which can be grouped according to similar characteristics. Two of these groups are:.

(39) 1.5. Mathematical models of tumor growth. 9. • Deterministic processes: These processes are characteristic of dynamical systems. Initial conditions determine the future states of the system, no randomness is involved. From a given initial condition, the same output is always obtained. Deterministic processes can be modeled with difference equations, ordinal differential equations, partial differential equations or deterministic cellular automata. • Stochastic processes: These processes are based on random dynamical systems. Initial conditions do not fully determine the future state of the system. Each state of the system is considered a random variable with a certain probability distribution. From a given initial condition, many different outputs can be obtained. Non-Markovian processes, jump Markovian processes and continuous Markovian processes belongs to this group, which are mathematically modeled with generalized master equations, master equations and stochasticdifferential equations respectively.. 1.5. Mathematical models of tumor growth. As explained in Section 1.1, there are three phases of tumor expansion; avascular growth, vascular growth and metastasis. Mathematical models of tumor growth will be different according to tumor growth phase. For the avascular initial phase, we presented the following models. The simplest model is based in the following hypothesis. Let P(t) denote the number of a certain population, let’s say a number of cancer cells, at time t. The population can increase by means of individual proliferation, with a rate b. Individuals can also die, with a rate d. Therefore, the changes in the population over time are given by the following ordinal differential equation Eq. (1.2). dP(t) = bP(t) − dP(t) = (b − d) P(t) = rP(t), dt P(0) = P0 .. (1.1) (1.2). This equation has the following analytical solution P(t) = P0 ert ,. (1.3). and is commonly referred to as simple exponential growth model o Malthusian growth model, in honor of its creator. If the birth processes are more notable than the death processes (meaning r > 0), the population will always increase in number with time ( lim P(t) = ∞). On the other hand, if the death processes are more relet→∞. vant (r < 0) the population number will decrease until its extinction ( lim P(t) = 0). t→∞. In population cancer dynamics, a continuous exponential growth is rarely observed. While it can be a good model to represent the initial growth of tumors, biological constraints such as competition for nutrients or space limits this kind of growth to a short period of time. For that reason, when a population increases it limits its growth. This behavior can be modeled with Eq. (1.4). dP(t) = P(t) f ( P(t)), P(0) = P0 , dt. (1.4).

(40) 10. Chapter 1. Introduction. where f (·) is a continuous decreasing function that allows exponential growth to small vales of p and limits the growth as P(t) increases. If we assume that there is a maximum number of individuals that the population can reach, and we name this maximum as Pmax , then f (0) = 1 and lim f ( P) = 0. P→ Pmax. Different structures of the function f (·) results in different models of limited tumor growth. For example, if the function takes the following form . P. f ( P) = r 1 −. . Pmax. ,. (1.5). then Eq. (1.4) becomes P(t) dP(t) = rP(t) 1 − , P(0) = P0 . dt Pmax. (1.6). Named as logistic model, this differential equation has the following exact solution. P(t) =. P0 ert 1+. P0 rt Pmax e. .. (1.7). In contrast to the exponential model, if r > 0 the model does not tend to infinity, as lim P(t) = Pmax t→∞. These equations can be easily extended to additional populations. Let’s imagine that there are two subpopulations growing at different velocities. In the case of exponential growth, one equation of the form (1.3) with a different value of the parameter r is set for each population. In the case of logistic growth, it is easy to adapt Eq. (1.6) into Eqs. (1.8)-(1.9). The non-linear system of ordinal differential equation are coupled by means of the logistic term. dP1 (t) P1 (t) + P2 (t) = r1 P1 (t) 1 − , P1 (0) = P10 , dt Pmax P1 (t) + P2 (t) dP2 (t) = r2 P2 (t) 1 − , P2 (0) = P20 . dt Pmax. (1.8) (1.9). With these models, we are able to reproduce basic dynamics of tumor growth. In the previous model, P(t) can stands for the number of cells, the tumor volume or cell density. In all those cases, tumor is considered as a global mass. This simple approach can be useful in many situations. However, cancer cells not only proliferate but also migrate. If this property wants to be included in the model, it is necessary to take into account the space, which can be considered as a new variable. The simplest case considers cell migration as a diffusive process, in which cells moves from higher cell densities to lower cell densities zones. This migration can be modeled in the following form. ∂P(x, t) = D∆P(x, t), P(x, 0) = P0 (x). ∂t. (1.10).

(41) 1.6. Mathematical models of development of chemotherapy resistance. 11. In this case, we can consider that P(x, t) represents the cell density in the position x at time t. The spatial variable x is actually a vector of variables of Rn , x = ( x1 , ..., xn ) The coefficient diffusion D must be positive D > 0. In addition, ∆P(x, t) represents 2 the Laplacian of P respect to the space variable x, ∆P = ∑in=1 ∂∂xP2 . If Eq. (1.10) holds i. ∀x ∈ Rn , the Cauchy problem is fully specified. In the case Eq. (1.10) is only defined in a bounded domain Ω ∈ Rn , proper boundary conditions must be defined. Modeling tumor growth it is not a choice between a proliferative or a diffusive model. Some models combine the proliferative and migration behavior of cancer cells. To that end, reaction-diffusion systems are normally used. In the general form, this models present the form of Eq. (1.11). ∂P(x, t) = D∆P(x, t) + F ( P), ∂t. (1.11). where F ( P) is a function that can adopt many different forms, especially in physics. In cancer, one of the most extended models (especially in gliomas) is the FisherP Kolmogorov equation, which is obtained when F ( P) = rP(1 − Pmax ). In the case of one dimensional space, Eq. (1.11) becomes (1.12). P ∂P ∂2 P = D 2 + rP 1 − . ∂t ∂x Pmax. (1.12). This equation presents solitary wave solutions, fronts, as shown in Murray, 2002 and Kyrychko and Blyuss, 2009. Extension of Eq. (1.12) to the 2-dimensional scenario does not display any kind of exact travelling wave solution. Henceforth, its analysis must be resort to numerical methods.. 1.6. Mathematical models of development of chemotherapy resistance. In Section 1.3 the different mechanisms of cancer resistance have been explained. However, the emergence of resistance deserves a separately analysis. Some tumors that have reduced its volume thanks to a chemotherapy treatment do not longer responde in more advances stages of treatment. Some different mechanisms of emergence of resistance have been studied.. 1.6.1. Pre-existence of a resistant clone. To explain this situation a Darwinian-like evolutionary dynamics interpretation has often been invoked. The pre-existence of a population exhibiting advantageous features due to random mutations in the initial tumor would be the responsible of tumor resistance. Under different stress conditions, such as the presence of chemotherapeutic agents, only the fittest clones survive. When these clones are within the tumor, the drug efficacy decreases considerably. These selective dynamics can be modeled by means of discrete models. A first simple approach defines two initial populations, the first one which would be considerably affected by chemotherapy, called Sensitive, and the second one representing the fittest clones, which are expected to not being importantly affected by the presence of drugs, called Resistant. A basic simple scheme can be seen in Fig. 1.4(b)..