AbstractAcknowledgmentsIntroductionBiology BackgroundThe Healthy ColonOverview of the Immune SystemColon Cancer and TreatmentsPrior Mathematical WorkODE Tumor ModelsPDE Tumor ModelsEquationsSystem of EquationsJustifications of New TermsParametersInitial ConditionsEquation 4.1 ParametersEquation 4.2 ParametersEquation 4.3 ParametersEquation 4.4 ParametersEquation 4.5 ParametersEquation 4.6 ParametersEquation 4.7 ParametersTreatmentsResults and AnalysisCurrent MonotherapiesCurrent Combination TreatmentsTumor Growth With Experimental TreatmentsSensitivity to ParametersDiscussionBiology GlossaryReference TablesEquation Term DescriptionsParameter DescriptionsParameter ValuesBibliographyA Nonlinear ODE Model of Tumor Growthand Effect of Immunotherapy andChemotherapy Treatment in Colorectal CancerHannah SavageLisette de Pillis, AdvisorAmi Radunskaya, ReaderMay, 2010Department of MathematicsCopyrightc 2010 Hannah Savage.The author grants Harvey Mudd College the nonexclusive right to make this workavailable for noncommercial, educational purposes, provided that this copyrightstatement appears on the reproduced materials and notice is given that the copy-ing is by permission of the author. To disseminate otherwise or to republish re-quires written permission from the author.AbstractColorectal cancer will kill approximately 50,000 people in the United Statesthis year (ACS, 2008). Current treatment options, including surgery, chemo-therapy, and radiation, are often able to force the cancer into remission, butbetter treatments are needed to help those who don’t respond to currenttreatments. A new and promising treatment option, monoclonal-antibodytherapy, has the potential to help reduce the deaths caused by colorec-tal cancer, but most monoclonal-antibody drugs are currently still in trialphases, and the variations in the dosing schedule of those currently ap-proved for use have not been heavily explored. We have modified a non-linear ODE tumor/treatment model by de Pillis and colleagues (2009) to in-clude monoclonal antibody treatments. Parameter values have been mod-ified for colorectal cancer, with irinotecan as the chemotherapy agent andthe two monoclonal antibodies cetuximab, which is FDA approved, andpanitumumab, which is still undergoing clinical trials. We have run Mat-lab simulations of current treatment options, and have found new dosingschedules which, in our simulations, reduce tumor size more effectivelythat the current schedules. Equilibria of the system and its sensitivity toparameters have also been examined.ContentsAbstract iiiAcknowledgments xi1 Introduction 12 Biology Background 32.1 The Healthy Colon . . . . . . . . . . . . . . . . . . . . . . . . 32.2 Overview of the Immune System . . . . . . . . . . . . . . . . 32.3 Colon Cancer and Treatments . . . . . . . . . . . . . . . . . . 73 Prior Mathematical Work 113.1 ODE Tumor Models . . . . . . . . . . . . . . . . . . . . . . . . 113.2 PDE Tumor Models . . . . . . . . . . . . . . . . . . . . . . . . 124 Equations 154.1 System of Equations . . . . . . . . . . . . . . . . . . . . . . . 164.2 Justifications of New Terms . . . . . . . . . . . . . . . . . . . 175 Parameters 215.1 Initial Conditions . . . . . . . . . . . . . . . . . . . . . . . . . 215.2 Equation 4.1 Parameters . . . . . . . . . . . . . . . . . . . . . 255.3 Equation 4.2 Parameters . . . . . . . . . . . . . . . . . . . . . 295.4 Equation 4.3 Parameters . . . . . . . . . . . . . . . . . . . . . 315.5 Equation 4.4 Parameters . . . . . . . . . . . . . . . . . . . . . 335.6 Equation 4.5 Parameters . . . . . . . . . . . . . . . . . . . . . 345.7 Equation 4.6 Parameters . . . . . . . . . . . . . . . . . . . . . 345.8 Equation 4.7 Parameters . . . . . . . . . . . . . . . . . . . . . 355.9 Treatments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37vi Contents6 Results and Analysis 416.1 Current Monotherapies . . . . . . . . . . . . . . . . . . . . . . 426.2 Current Combination Treatments . . . . . . . . . . . . . . . . 426.3 Tumor Growth With Experimental Treatments . . . . . . . . 486.4 Sensitivity to Parameters . . . . . . . . . . . . . . . . . . . . . 517 Discussion 55A Biology Glossary 59B Reference Tables 63B.1 Equation Term Descriptions . . . . . . . . . . . . . . . . . . . 63B.2 Parameter Descriptions . . . . . . . . . . . . . . . . . . . . . . 65B.3 Parameter Values . . . . . . . . . . . . . . . . . . . . . . . . . 67Bibliography 69List of Figures2.1 Normal and cancerous colon crypts . . . . . . . . . . . . . . . 42.2 Types of blood cells . . . . . . . . . . . . . . . . . . . . . . . . 54.1 Possible MAB pathways . . . . . . . . . . . . . . . . . . . . . 175.1 Stable Equilibria. . . . . . . . . . . . . . . . . . . . . . . . . . 225.2 Small and large initial tumors. . . . . . . . . . . . . . . . . . . 245.3 The probability density function of x and y. . . . . . . . . . . 275.4 Activated CD8+T-cell treatment schedule. . . . . . . . . . . . 385.5 Irinotecan dosing schedule. . . . . . . . . . . . . . . . . . . . 385.6 Monoclonal antibody dosing schedules. . . . . . . . . . . . . 406.1 Irinotecan Monotherapies. . . . . . . . . . . . . . . . . . . . . 436.2 Cetuximab Monotherapies. . . . . . . . . . . . . . . . . . . . 436.3 Panitumumab Monotherapies. . . . . . . . . . . . . . . . . . 446.4 Combination Therapy: Weak tumor response. . . . . . . . . 456.5 Combination Therapy: Mixed tumor response. . . . . …