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Mathematical models of tumor growth and therapy

Dissertation
Author: Mark Robertson-Tessi
Abstract:
A number of mathematical models of cancer growth and treatment are presented. The most significant model presented is of the interactions between a growing tumor and the immune system. The equations and parameters of the model are based on experimental and clinical results from published studies. The model includes the primary cell populations involved in effector-T-cell-mediated tumor killing: regulatory T cells, helper T cells, and dendritic cells. A key feature is the inclusion of multiple mechanisms of immunosuppression through the main cytokines and growth factors mediating the interactions between the cell populations. Decreased access of effector cells to the tumor interior with increasing tumor size is accounted for. The model is applied to tumors of different growth rates and antigenicities to gauge the relative importance of the various immunosuppressive mechanisms in a tumor. The results suggest that there is an optimum antigenicity for maximal immune system effect. The immunosuppressive effects of further increases in antigenicity out-weigh the increase in tumor cell control due to larger populations of tumor-killing effector T cells. The model is applied to situations involving cytoreductive treatment, specifically chemotherapy and a number of immunotherapies. The results show that for some types of tumors, the immune system is able to remove any tumor cells remaining after the therapy is finished. In other cases, the immune system acts to prolong remission periods. A number of immunotherapies are found to be ineffective at removing a tumor burden alone, but offer significant improvement on therapeutic outcome when used in combination with chemotherapy. Two simplified classes of cancer models are also presented. A model of cellular metabolism is formulated. The goal of the model is to understand the differences between normal cell and tumor cell metabolism. Several theories explaining the Crabtree Effect, whereby tumor cells reduce their aerobic respiration in the presence of glucose, have been put forth in the literature; the models test some of these theories, and examine their plausibility. A model of elastic tissue mechanics for a cylindrical tumor growing within a ductal membrane is used to determine the buildup of residual stress due to growth. These results can have possible implications for tumor growth rates and morphology.

6 Table of Contents List of Figures..................................9 List of Tables..................................11 Astract......................................12 Chapter 1 Introduction...........................14 1.1 Biology of cancer.............................17 1.2 Outline of the Dissertation........................23 Part I A Mathematical Model of Tumor-Immune Interactions..27 Chapter 2 The Immune System and Interactions with a Tumor..28 2.1 Biology of the Immune System......................29 2.1.1 T-cell types............................29 2.1.2 Antigen Presentation and T-cell Activation...........30 2.1.3 T-cell proliferation........................31 2.1.4 Cytotoxicity............................32 2.2 Interactions between Tumor Cells and the Immune System......33 2.2.1 Evidence for Immune System Efficacy against Tumor Cells..33 2.2.2 Immune Escape Mechanisms of Tumors.............34 2.2.3 The Effect of Regulatory T-cells.................36 2.2.4 Summary.............................40 Chapter 3 A Model of Tumor-Immune Interactions.........41 3.1 A Mathematical Model..........................43 3.1.1 Variables..............................43 3.1.2 Equations.............................48 Chapter 4 Parameter Estimation.....................60 4.1 Parameter estimation...........................60 4.1.1 Parameter values for tumor progression.............60 4.1.2 Parameter values for dendritic cell expansion..........63 4.1.3 Parameter values for T-cell expansion..............65 4.1.4 Parameter values for IL-2,TGF-β and IL-10 concentrations.67 4.1.5 Summary.............................69

Table of Contents – Continued 7 Chapter 5 Results...............................71 5.1 Results...................................71 5.1.1 Main Qualitative Behaviors...................71 5.1.2 Effects of TGF-β,Converted Tregs,and IL-10.........75 5.1.3 Fixed point analysis.......................84 5.1.4 Behavior of small tumors in the limiting case..........88 5.2 Discussion.................................91 5.2.1 Comparison with other models..................91 5.2.2 Conclusions............................92 Chapter 6 Tumor-Immune System Interactions with Treatment.94 6.1 Chemotherapy...............................94 6.1.1 Biology of chemotherapy.....................94 6.1.2 Inclusion of chemotherapy in the model.............97 6.1.3 Results and discussion for chemotherapy............102 6.1.4 Conclusions for chemotherapy..................122 6.2 Immunotherapy..............................125 6.2.1 Adoptive T cell immunotherapy.................126 6.2.2 Dendritic cell therapy.......................128 6.2.3 IL-2 therapy............................137 6.2.4 TGF-β blockade.........................140 6.2.5 Treg depletion with ONTAK...................141 6.2.6 Conclusions for immunotherapy.................143 6.3 Combination therapies..........................144 6.3.1 Effect of Treg depletion during chemotherapy.........144 6.3.2 Effect of blocking TGF-β during chemotherapy........146 6.3.3 Combination of dendritic cell therapy and chemotherapy...146 6.4 Conclusions................................148 Part II Simplified Models of Tumor Growth..............150 Chapter 7 A Model of Tumor Metabolism and Crabtree Effect.151 7.1 Biological assumptions..........................152 7.2 Variables..................................155 7.3 Metabolic reactions............................156 7.4 Mathematical models...........................162 7.4.1 ADP competition model.....................164 7.4.2 Metabolic model including pH inhibition............175 7.4.3 Additional metabolic models and future work.........180

Table of Contents – Continued 8 7.5 Conclusions................................183 Chapter 8 Induced Stress in Solid Tumor Growth..........185 8.1 Mechanics of elastic deformation.....................186 8.2 Applications................................190 8.3 Results and discussion..........................191 8.4 Conclusions................................198 Chapter 9 Conclusions............................199 9.1 Acknowledgements............................201 References....................................202

9 List of Figures 2.1 Basic tumor-immune interactions....................29 3.1 The tumor-immune interactions used in the mathematical model...47 3.2 Hybrid growth law for tumor cells....................51 5.1 Long-term tumor behavior of the model.................73 5.2 Long-term tumor behavior of the model without TGF-β.......76 5.3 Long-term tumor behavior of the model without Treg conversion...77 5.4 Long-term tumor behavior of the model without TGF-β suppression.78 5.5 Growth simulation for a tumor with low antigenicity..........80 5.6 Growth simulation for a tumor with high antigenicity..........83 5.7 Bifurcation diagrams for the mathematical model............85 5.8 Solutions of Eq.(5.2) for fixed values of R/E..............90 6.1 Effect of chemotherapy on immunosuppression.............105 6.2 Effect of the immune system on remission time.............108 6.3 Tumor response to the immune system,under chemotherapy......110 6.4 Nadir of tumor size after chemotherapy.................112 6.5 Effect of enhanced cytotoxicity on tumor outcome...........113 6.6 Effect of the cytostatic duration on tumor outcome..........115 6.7 Effect of celluar kill fraction on tumor outcome............116 6.8 Effect of early detection on chemotherapy outcome..........118 6.9 Optimization of the chemotherapy regimen...............121 6.10 Simulation of adoptive T cell transfer therapy.............127 6.11 Effect of adoptive T cell transfer therapy on tumor outcome.....129 6.12 Simulation of dendritic cell therapy...................132 6.13 Effect of dendritic cell therapy on tumor outcome...........134 6.14 Effect of dendritic cell counts on therapy................136 6.15 Effect of IL-2 therapy...........................139 6.16 Outcome of ONTAK immunotherapy..................142 6.17 Combination of chemotherapy and ONTAK immunotherapy.....145 6.18 Effect of TGF-β suppression during chemotherapy...........147 6.19 Effect of dendritic cell therapy during chemotherapy..........148 7.1 Effect of glucose level on the basic metabolic model..........169 7.2 Effect of oxygen levels on the basic metabolic model..........172 7.3 Comparison with data on the Crabtree effect..............174

List of Figures – Continued 10 7.4 Effect of pH on aerobic metabolism...................177 7.5 Effect of glucose level on the pH inhibition model...........179 8.1 Stress as a function of strain for a neo-Hookean material........192 8.2 Stress as a function of strain for a Fung material............192 8.3 Strain and deformation for radial growth.................193 8.4 Residual stress due to radial growth in a neo-Hookean material....194 8.5 Residual stress due to radial growth in a Fung material.........194 8.6 Strain and deformation for circumferential growth............195 8.7 Residual stress due to circumf.growth in a neo-Hookean material...196 8.8 Strain and deformation for nested materials...............197 8.9 Residual stress due to growth in nested materials............197

11 List of Tables 4.1 Parameter values for tumor-immune interactions:Tumor cells....61 4.2 Parameter values for tumor-immune interactions:Dendritic cells...64 4.3 Parameter values for tumor-immune interactions:T cells.......66 4.4 Parameter values for tumor-immune interactions:Molecules.....67 6.1 Parameter values for chemotherapy...................103 6.2 Percentage of cells killed as a function of chemotherapy dose.....120 7.1 Variables used in the metabolic models.................157 7.2 Basic metabolic model:nominal concentrations and parameter values 166 7.3 pH inhibition model:nominal concentrations and parameter values.178

12 ABSTRA CT A number of mathematical models of cancer growth and treatment are presented. The most significant model presented is of the interactions between a growing tumor and the immune system.The equations and parameters of the model are based on experimental and clinical results from published studies.The model includes the primary cell populations involved in effector-T-cell-mediated tumor killing:regula- tory T cells,helper T cells,and dendritic cells.A key feature is the inclusion of multiple mechanisms of immunosuppression through the main cytokines and growth factors mediating the interactions between the cell populations.Decreased access of effector cells to the tumor interior with increasing tumor size is accounted for. The model is applied to tumors of different growth rates and antigenicities to gauge the relative importance of the various immunosuppressive mechanisms in a tu- mor.The results suggest that there is an optimumantigenicity for maximal immune systemeffect.The immunosuppressive effects of further increases in antigenicity out- weigh the increase in tumor cell control due to larger populations of tumor-killing effector T cells.The model is applied to situations involving cytoreductive treat- ment,specifically chemotherapy and a number of immunotherapies.The results show that for some types of tumors,the immune system is able to remove any tu-

13 mor cells remaining after the therapy is finished.In other cases,the immune system acts to prolong remission periods.A number of immunotherapies are found to be ineffective at removing a tumor burden alone,but offer significant improvement on therapeutic outcome when used in combination with chemotherapy. Two simplified classes of cancer models are also presented.A model of cellular metabolism is formulated.The goal of the model is to understand the differences between normal cell and tumor cell metabolism.Several theories explaining the Crabtree Effect,whereby tumor cells reduce their aerobic respiration in the presence of glucose,have been put forth in the literature;the models test some of these theories,and examine their plausibility. A model of elastic tissue mechanics for a cylindrical tumor growing within a ductal membrane is used to determine the buildup of residual stress due to growth. These results can have possible implications for tumor growth rates and morphol- ogy.

14 Chapter 1 Intr oduction A significant amount of research has been performed to understand the causes of cancer and find treatments for this ubiquitous disease.Despite this effort,cancer remains one of the leading causes of death within the human population,accounting for over 23 percent of deaths in the Unites States in 2006 [1].Incidence and mortality rates have decreased somewhat since 1990,although this is primarily attributed to the effort to reduce smoking,and the development of improved early detection technologies for some cancers.The effort to find the cure has been difficult,and with a few notable exceptions,broadly successful treatments have been elusive. The reasons for this relative failure are varied,and a detailed examination is beyond the scope of this work;however,it is clear that this is a complex disease, and new tools are needed to study the origins,progression,and treatment options for the various types of cancer. A recurring theme in cancer research is the announcement of a promising treat- ment that shows great results in the petri dish or laboratory murine animal,only to have minimal clinical success in the human population [2,3,4].Even treatments which work at one stage of cancer growth may not have any effect at a different stage for the same disease [5].In other cases,treatments will only work in a small

15 fraction of patients,all of whomostensibly have the same disease [6].These puzzling failures speak to the need for deeper understanding of the interactions between a tumor and its host at all levels,not just at the molecular biochemical level which dominates current cancer research. Indeed,one of the aspects of cancer research which makes it both frustrating and fascinating is the wide range of spatial and timescales over which the disease operates.Spatially,there are interactions on the atomic and molecular scale which determine metabolic rates and genetic mutations;cell signaling,angiogenesis,and immune system interactions operate on a cellular scale;and nutrient diffusion,me- chanical stresses,and systemic molecular concentrations are based on tissue- and host-sized scales.Temporally,there are processes such as enzymatic reactions which operate on sub-second timescales;genetic regulation occurs on the order of minutes; cell-to-cell interactions can last for hours;and cellular growth rates are on the order of days or weeks.There is increasing evidence that these multiple scales affect each other in ways yet to be understood [7].Therefore,a singular chain of enzymatic reactions that appears critical when isolated in a laboratory may in fact be insignif- icant when examined in the tumor milieu.It is likely that there exists emergent behavior in cancer that cannot be predicted from the basic underlying biochemical processes.A holistic understanding of cancer at multiple scales may be necessary to give context to the role of a specific process.In addition,the interactions between these differing scales may themselves reveal possibilities for therapeutic influence,

16 something that cannot be deduced from examination of a single biochemical path- way. One tool which is becoming increasingly useful for gaining insight into these systems is mathematical modeling [8].The complex feedback loops which manifest themselves in the tumor-host system often lead to unintuitive dynamics.A good mathematical model can be used to determine which features of the system play the most significant roles,and provide insight into why a particular treatment may not work,despite appearing to be intuitively sound from a pharmaceutical perspective. Furthermore,a model which incorporates treatment may illuminate methods for improving the treatment,either by adjusting the regimen or by adding a combination treatment for example. In this dissertation,a number of mathematical models of cancer are presented and analyzed.Often,mathematical models of biological systems have been used to explore novel mathematical techniques in their own right,with the biological impli- cations sometimes becoming secondary to the mathematical analysis.In this work, the opposite tack is chosen:the primary focus is the development of mathematical models that attempt to answer a set of biological questions about an aspect of tumor growth.Mathematical results which fall far outside of a reasonable biological regime are not examined,since they have little bearing on the treatment of the disease. Prior to introducing the models,a brief general description of cancer biology is presented in this introduction as a basis for understanding the models which followin

17 the body of the paper.Additional biological aspects which are specific to individual models are presented in the section dedicated to that particular model. 1.1 Biology of cancer The word cancer refers to a collection of diseases.The specific mechanisms at work in any given type of cancer are varied and quite complex,but a few basic commonalities are shared by all cancers.First,a group of cells must exhibit uncontrolled growth within the diseased host.This growth is unchecked by normal host cell-to-cell signaling and internal cellular control mechanisms.Second,all cancers are ultimately invasive;this is in contrast to a benign growth that remains localized and will not disrupt the adjacent tissue [9]. Cancer primarily results from mutations in cellular DNA.The causes for muta- tion are numerous [10].They can be caused by external carcinogens,such as smoke and various chemicals;they can be caused by exposure to radiation;they can be inherited from a parent that has a malformed gene;they can result from genetic manipulation by pathogens;they can be caused by reactive oxygen species within a cell;and they can be caused by DNA transcription errors when a cell is divided. These mutations accumulate over time in all humans.Given the large numbers of DNA base pairs that can be mutated,the likelihood of any particular cell receiving a series of mutations leading to carcinogenesis is exceedingly small;most mutations either cause fatal dysfunction in that cell,or cause little to no change in cellular

18 function. However,given the large number of cells in the body,approximately 10 14 , and enough time,there is a significant chance that the mutations leading to can- cer may accumulate in a cell.Exposure to carcinogens or inherited mutations will naturally accelerate the process.Still,incidence rates for cancer are correlated with age,indicating that the carcinogenesis process occurs in a series of steps acquired over time [10]. Regardless of the specific origin,the mutations cause various proteins to be malformed,leading to changes in the way the cell responds to its environment. In the case of cancer,these changes lead to uncontrolled growth and subsequent invasion.Given the complexity of signaling and metabolic pathways within a cell, the routes by which a host’s normal tissue can become cancerous are numerous, even within the same type of cancer.This complexity is one reason why universal treatments for a given type of cancer are so elusive.What works on one fraction of patients may completely fail in the remaining fraction because of subtle differences in the disease. Although the amount of genetic variation of the disease across patients with a given type of cancer is not known,it is possible that each cancer patient has their own version of the disease,dictated by the particular genetic mutations incurred and the state of the host’s body during cancer growth.It is therefore plausible that each patient may best be treated by individualized therapy.Certainly this idea has led to the development of personalized treatments [11],when these subtle genetic

19 v ariations can in fact be detected [12].However,not all genetic markers for cancer are known,and the interplay between a group of mutations is not well understood. Furthermore,even if a complete genetic profile of the tumor can be acquired,a corresponding treatment is not necessarily going to be easy to deduce. Without question,understanding the role of dysfunctional proteins caused by genetic mutation is of fundamental importance in the fight against cancer.Bio- chemical research on the function of these mutated pathways has been critical to the understanding of the steps involved in the transformation from a normal tissue cell to a cancerous cell.A number of treatments are reliant on the existence of a par- ticular genetic mutation.For example,the antibody trastuzumab (Herceptin) only has an effect on breast cancers that have increased expression of Human Epidermal Growth Factor Receptor-2 (HER2) [13]. At the same time,the context in which these mutations occurs is likely to be as important as the mutations themselves.Evidence has shown that the micro- and macro-environment in which the mutated cells exists has a significant bearing on the future of the mutated cell [14].It is clear that the proper series of oncogenic mutations must be accompanied by,or perhaps even preceded by,a degradation of tissue in the area of the abnormal cell. Once the mutations and local conditions are conducive for tumorigenesis,the cells will proliferate in an unchecked manner.In the early stages of tumor growth, the tumor is usually avascular.This lack of penetrating blood vessels means that

20 the tumor must depend on diffusion for nutrients to reach the interior of the tumor. This diffusion-limited growth can persist until the tumor reaches a size of about 1 mm [15].At this point,a spherical tumor may be stratified into three levels. The outermost layer will be comprised of viable,proliferating tumor cells.Further towards the center,the tumor cells will be alive,but minimally proliferative.The central core of the spheroid will be necrotic material,cells that have died due to lack of nutrient.The quiescent and necrotic layers have been theorized to be the result of nutrient limitation,although the exact mechanisms at work are not clear [16]. At a size of one millimeter,oxygen likely does not permeate to the center of the tumor [15].However,many tumors significantly upregulate their anaerobic metabolism [17],enabling them to process glucose in the absence of oxygen.It is not immediately clear why a lack of oxygen should lead to necrosis or quiescence. Glucose penetration has not been accurately measured in an experimental setting.A few mathematical models predicting the glucose gradient have been formulated [18, 19],but questions remain regarding the metabolic equations used in these models. The metabolic model presented in Chapter 7 takes a step towards addressing these questions;however,much work remains to be done in this area. Once the tumor reaches a size of about 1 mm,it is hypothesized that the growth of the outer layer compensates for the loss of cellular material in the central core, and therefore the tumor can grow no larger while depending on nutrient diffusion. When cells find themselves in lownutrient environments,they will secrete angiogenic

21 factors which cause blood vessels to grow into the tumor mass,towards the nutrient- deprived areas [16].In normal tissue,this angiogenesis is a controlled process,given that the growth of tissue is regulated.The structure and function of blood vessels in normal tissue is different than that seen in cancerous tissue.In tumors,the angiogenic factors are not subject to regulation,and therefore the vessels may follow tortuous paths,be leaky due to gaps in the endothelial cells which form the vessels, and generally inefficient [20]. However unregulated the blood vessels may be,they do provide a way for nu- trients to reach areas of the tumor which do not receive sufficient nutrient through diffusion alone.Due to the uneven distribution of vessels,it is likely that pockets of hypoxia remain,but overall the infusion of blood vessels allows a tumor to grow indefinitely. The final stage of tumor progression is invasion into the surrounding tissue and subsequent metastasis to distal sites.A number of mechanisms have been offered as explanation for metastasis [21].First,the tumor cells may become motile.Normally, cells are unwilling to leave their place in the tissue,and detachment can cause the cell to go into apoptosis,in some cases.Invasive tumor cells are under no such restriction, and therefore can dislodge from their position and penetrate the surrounding tissue. Second,the tumor cells produce enzymes which break down the surrounding tissue, allowing the tumor to expand more easily.This enzyme production can also break down membranes,in the case of ductal carcinoma in situ of the breast,for example.

22 Third, mechanical forces may be at work.The pressure formed by unrestricted growth may cause the tumor to push out of its initial growth area and into the area of least mechanical resistance. Once cell motility is gained,the tumor cells will enter nearby blood vessels,and travel through the bloodstream.These individual cells or small clumps of cells are mostly removed by the immune system,but some of the cells will be trapped in capillaries or otherwise exit the bloodstream at a distant site.Once the cell has entered the new tissue,it will begin to proliferate rapidly,and form a secondary tumor mass.The process of avascular to vascular growth repeats itself. Over time,multiple lesions can develop.Growth of all of these cancerous masses will continue until the tumors are detected and treated,until the immune system removes them,or until they kill the host. This basic outline of tumor biology glosses over many important aspects of tu- mor growth which are beyond the scope of this work.The critical aspect to consider is that the tumor operates on many different spatial and time scales at once.The disease is not defined by any single property or genetic mutation.Rather the dis- ease is the result of an accumulation of mutations,and a tissue terrain where that particular genotype of tumor cell can grow in an uninhibited fashion.When consid- ering that the progression of the disease is affected by everything froma single DNA base-pair mutation all the way up to the overall state of the host’s body,it is clear that cancer is a complex system which can benefit from the use of mathematical

23 mo deling. 1.2 Outline of the Dissertation The dissertation is divided in two parts.Part I describes a mathematical model of the interactions between a solid tumor and the host immune system.This model is examined alone,and in the context of several existing cancer treatments.Two additional classes of model are considered in Part II,dealing with tumor metabolism and mechanical growth. Part I begins with a background on the biology of the mammalian immune system and its interactions with a growing tumor,presented in Chapter 2.Many types of immune cells are capable of interacting with a tumor;in this work T cells are examined,since they are considered to be the primary subset of immune cells capable of tumor cytotoxicity [22].Of particular interest is the inclusion of regulatory Tcells, a class of T cell that suppresses the host immune system when cancer is present. These cells have recently come to the forefront in the biological literature as a significant barrier for T-cell mediated tumor killing [23],and their presence has been cited as a possible explanation for the failure of T cell based immunotherapies [24, 25].The inclusion of these cells adds an interesting feedback mechanism to the model. In Chapter 3,the mathematical model of tumor-immune interactions is pre- sented.This model is a set of 12 ordinary differential equations developed from the

24 exp erimental and clinical literature.Chapter 4 details the methods used to acquire the parameters for the model.Accurate parametrization was a critical component of the development of the model,ensuring that the results were within a biologically reasonable regime. The results of the model are presented in Chapter 5.Tumors are characterized by two characteristic parameters.The growth rate is a measure of how aggressively the tumor cells divide.The antigenicity is a measure of how much the tumor primes the immune system.A tumor with very low antigenicity will be invisible to the immune system,since the cells which patrol the body will not see the tumor as a threat. Tumors with high antigenicity will cause a large immune response to be mounted. Although intuitively it would seem that the greater the antigenicity,the greater the immune response and therefore the greater chance of removing the tumor with the immune system,the inclusion of immunosuppressive factors such as the regulatory T cells described above shows a different result.Depending on the aggressiveness of the tumor,there is an optimum antigenicity at which the immune system effect is maximized.This result can have bearing on treatments which effectively increase the antigenicity of the tumor. The tumor-immune model is extended to include therapies in Chapter 6.It has been suggested that chemotherapy may have an immunomodulatory effect on the recipient,and this immune effect may in fact be what leads to the removal of tu- mors when chemotherapy is successful [26].The results of the model suggest that

25 c hemotherapy offers short termbenefit to the immune system,and that the regimen of chemotherapy can be optimized to maximize this benefit.This results in pro- longed remission times,or even in tumor removal.In addition to chemotherapy,a few immunotherapies are examined,both alone and in combination with chemother- apy.The results of combination therapy show that the application of two treatments is often greater than the sum of its parts. Part II describes two simplified classes of models.In Chapter 7,a model of cellu- lar metabolismis presented.The goal of the model is to investigate which aspects of cellular metabolismin tumor cells lead to the metabolic differences observed between normal tissue and tumor tissue.Specifically,the Crabtree effect is the phenomenon where tumor cells will decrease their aerobic respiration rate in the presence of high glucose levels,an effect not seen in normal tissue.A basic model of metabolism is presented,and a possible mechanism for the Crabtree effect is demonstrated.Since a number of theories exist in the literature to explain the Crabtree effect,the basic model is extended to examine the plausibility of some of these other theories. Finally,a model of tumor tissue mechanics is presented in Chapter 8.Using a cylindrical geometry to simulate ductal carcinoma in situ,the effects of growth and mechanical stress in this geometry are described,following the work of Goriely and Ben Amar [27].Using tensor analysis,the mechanical response of an elastic tissue is derived.Depending on the mode of growth,tensile or compressive stress can develop within the tissue.This stress in turn affects the growth.Fung demonstrated

26 that residual stress within arteries can have a great effect on the function [28]. Although the ramifications of this mechanical effect on tumor growth in vivo are not well understood,there is in vitro evidence that mechanical stress can limit tumor growth [29].Theories of tumor invasion include the possibility that mechanical pressure may be a factor [21]. Chapter 9 will summarize the main results of the models presented in the body of the dissertation,and discuss future extensions of the work.

27 Par t I A Mathematical Model of Tumor-Immune Interactions

28 Chapter 2 The Immune System and Interactions with a Tumor The human immune system is very complex.The need to be able to engage an enormous variety of pathogens while maintaining tolerance to the body’s own cells necessitates a highly developed and heterogeneous collection of cells and signaling molecules.The mechanisms of action of the immune system upon encountering a pathogenic cell are highly varied. In this work,the focus is primarily on T cells,which are one subset of the cells that comprise the immune system and have the ability to combat tumors.Other immune system cells,such as natural killer cells,macrophages,and B cells can also have an effect against tumors.However,the literature suggests that T cells may have the most significant anti-tumor immune effect within the body [22,30].A diagramof the interactions between a cancer and the T-cell immune systemis shown in Figure 2.1.The tumor provokes an immune response from T cells,which can then attack the tumor.At the same time,the tumor promotes immunosuppressive factors,which damp the immune response.The strength of this response depends on the tumor’s antigenicity,which describes how much antigen a tumor presents, and also how sensitive the immune system is to this antigen. In the following sections,the biology of T-cell activation,proliferation and cyto-

29 Figure 2.1:The interactions between a tumor and the immune system.The cy- totoxic effects of the immune system are counteracted by the immunosuppressive effects created by the tumor and the autoimmune protections within the body. toxicity is described in further detail. 2.1 Biology of the Immune System 2.1.1 T-cell types Three major types of T-cells are involved in an immune response.The T cells which kill the target cells,called effector T cells or cytotoxic lymphocytes (CTLs),are distinguished by the surface marker CD8,which is highly expressed on cells of this type.A second type of T cell is known as the helper T cell,and it is distinguished by the cell surface marker CD4.While not primarily involved in the killing of the target cells,this helper cell phenotype performs several important supporting roles, including the licensing of dendritic cells and the production of important proliferative

30 cytokines. The third T-cell type is the regulatory T cell (Treg) which suppresses the function of the other T cells under certain conditions.Tregs share the CD4 marker with helper cells,but display two other markers to distinguish them from helper cells.The surface marker CD25 and the intracellular protein FoxP3 are the current paradigm for describing Tregs,although this is confounded by the fact that activated effector and helper cells can also show elevated CD25 levels [31,32]. 2.1.2 Antigen Presentation and T-cell Activation Antigens are small sections of proteins,on the order of 5 to 20 amino acids in length [33].Most cells have mechanisms for presenting antigen on their cell surfaces. These antigens can be recognized by certain types of immune systemcells,including antigen presenting cells (APCs) such as dendritic cells (DCs),and T cells.The antigens presented by a cell on its surface are related to the proteins produced by the cell;therefore,different cell types will produce different antigens.This allows the immune system to differentiate between cell types.A foreign bacteria will present a specific set of antigens different than that produced by the normal cells in the body. Research has shown that the level of antigen correlates with the proliferation of T cells and the strength of T cell response [34].The link between antigens and the T cell population is provided by dendritic cells,a class of APC.These cells,which are considered to be the primary antigen presenting cells in the body,are normally in an immature state.Upon encountering the antigens on a cell,the dendritic cell will

Full document contains 218 pages
Abstract: A number of mathematical models of cancer growth and treatment are presented. The most significant model presented is of the interactions between a growing tumor and the immune system. The equations and parameters of the model are based on experimental and clinical results from published studies. The model includes the primary cell populations involved in effector-T-cell-mediated tumor killing: regulatory T cells, helper T cells, and dendritic cells. A key feature is the inclusion of multiple mechanisms of immunosuppression through the main cytokines and growth factors mediating the interactions between the cell populations. Decreased access of effector cells to the tumor interior with increasing tumor size is accounted for. The model is applied to tumors of different growth rates and antigenicities to gauge the relative importance of the various immunosuppressive mechanisms in a tumor. The results suggest that there is an optimum antigenicity for maximal immune system effect. The immunosuppressive effects of further increases in antigenicity out-weigh the increase in tumor cell control due to larger populations of tumor-killing effector T cells. The model is applied to situations involving cytoreductive treatment, specifically chemotherapy and a number of immunotherapies. The results show that for some types of tumors, the immune system is able to remove any tumor cells remaining after the therapy is finished. In other cases, the immune system acts to prolong remission periods. A number of immunotherapies are found to be ineffective at removing a tumor burden alone, but offer significant improvement on therapeutic outcome when used in combination with chemotherapy. Two simplified classes of cancer models are also presented. A model of cellular metabolism is formulated. The goal of the model is to understand the differences between normal cell and tumor cell metabolism. Several theories explaining the Crabtree Effect, whereby tumor cells reduce their aerobic respiration in the presence of glucose, have been put forth in the literature; the models test some of these theories, and examine their plausibility. A model of elastic tissue mechanics for a cylindrical tumor growing within a ductal membrane is used to determine the buildup of residual stress due to growth. These results can have possible implications for tumor growth rates and morphology.