Mathematical and Computational Studies on Progress, Prognosis, Prevention and Panacea of Breast Cancer (Forum for Interdisciplinary Mathematics)

This book’s aim is to study the mathematical and computational models to analyze the progress, prognosis, prevention, and panacea of breast cancer. The book discusses application of Markov chains and transient mappings, Charlie–Simpson numerical algorithm, models represented by nonlinear reaction–diffusion-type partial differential equations, and related techniques. The book also attempts to design mathematical model of targeted strategic treatments by using Skilled Killer Drugs (SKD1 and SKD2) to suggest the improvisation of future cancer treatments. Both graduate students and researchers of computational biology and oncologists will benefit by studying this book. Researchers of cancer studies and biological sciences will also find this work helpful. Dedication Describing the Dictum that Directed Me Foreword Preface Organization of Chapters Acknowledgements A Congratulatory Note from Mohit De Words of Gratitude from a Patient’s Spouse on the Success of our Modeling Introduction and Summary of the Research Model 1: The Uncontrolled Growth Model 2: A Noticeable Scenario of Uncontrolled Growth Model 3: The Most Realistic and Simplistic Model on the Growth of Cancer Model 4: A Method of Treatment of Uncontrolled Growth Necrotic Tumors Conclusion References Contents Notations 1 Introduction 1.1 Rationale 1.2 The Nature of a Cancer Cell 1.3 A Need for a Practical 3D Mathematical Modeling 1.4 The Mathematical Mapping Behind Our Computational Studies 1.4.1 Surjective Mapping 1.4.2 Some Deadly Aspects of Cancer Cells 1.5 A Fundamental Drawback of the Use of Logistic Equations 1.6 The Dynamics of Violence Started by Cancer Cells 1.7 Awakening of the Dynamics of the Immune System 1.8 The Role of Thymus [Ref. Rejuvenate Your Thymus Gland-Dr. J .E. Williams] 1.9 Role of Entropy 1.10 Few Special Proteins Released by Cancer Cells 1.11 A Short Outline of Breast Cancer 1.12 Some Tests to Detect Cancer 1.13 Some Risk Factors: Carcinogens 1.14 Free Radicals 1.15 Estrogen and Progesterone 1.16 HER2 Protein 1.17 Mutations of Genes 1.18 A Note on Surgery 1.19 The Role of Stress 1.20 Other Well-Known Causes 1.21 Some Perspectives of Our Models 1.22 Conclusion References 2 Statistics: The Background and the Basis 2.1 Rationale 2.2 Some Preliminary Concepts on Sets 2.2.1 Cardinality 2.2.2 Union and Intersection of Sets 2.2.3 Probability Using Sets 2.2.4 Conditional Probability 2.2.5 Independent Events 2.2.6 Mutually Exclusive vs. Independent Events 2.3 Bayes’ Formula 2.4 Testing Tumors: Specificity and Sensitivity and Applications of Bayes’ Formula 2.5 Confidence in the Estimation of Prognosis 2.6 Analysis of Multiple Probabilistic Tests for Tumor Detection 2.7 A Few Topics on Fundamentals of Probability Theory 2.7.1 Uniform Distribution 2.7.2 Binomial Distribution 2.7.3 Normal Distribution 2.7.4 Significance of μ in the Practical World 2.7.5 The Central Limit Theorem (CLT) 2.8 Hypotheses Testing: The p-Value 2.9 A Search For Panacea for Metastatic Breast Cancer by Applying Dey–Markov Chain 2.9.1 Modified Markov Chain or The Dey–Markov Chain 2.9.2 Definition: The Dey–Markov Chain 2.10 Conclusion References 3 Attacker and Defender Model: The Dynamics of the Immune System 3.1 Rationale 3.2 A Preliminary Model 3.3 First Line of Defense of the Body 3.4 Immune System Gets A Helper Drug 3.5 Immune System Changing Strategies to Fight Cancer 3.6 Stiff Computations 3.7 Computational Studies Applying CDey (Charlie’s) Algorithm [4, 5] 3.8 Use of an Additional Drug 3.9 Modeling with Logistic Equation 3.10 Conclusion References 4 Mathematical Modeling of Metastatic Cancer 4.1 Rationale 4.2 Mathematical Derivation of Reaction-Dispersion (Diffusion) Equation 4.2.1 The Attacker–Defender Model [11] 4.2.2 Rates of Growth of Cancer 4.2.3 Dimensional Analysis 4.2.4 A Condition for Successful Treatment 4.3 Analytical Solution Revealing How Cancer Spreads 4.4 Algorithm of CDey-Simpson. A Difference-Integro Method 4.4.1 Definition: CDey-Simpson Operator 4.5 Time-Dependent Extrapolated Boundary Conditions 4.6 Modeling General Therapy Targeting All Cancer Cells 4.7 A Fast Growing Fast Spreading Cancer. Use of a Second Chemo 4.8 Reductions of Both the Rate of Growth and Dispersion of Cancer Cells 4.9 Slow Growing Tumor with No Therapy. (Tubular Breast Cancer) 4.10 Intratumoral Cancer Treatment 4.10.1 The Entire Field Including the Boundaries is a War Zone 4.10.2 Conclusion References 5 Modeling Advanced Immunotherapy with Monoclonal Drugs 5.1 Rationale 5.2 Three Reaction–Dispersion (Diffusion) Equations: The Model 5.2.1 Numerical Solution by the CDey-Simpson 5.2.2 A Special Note 5.3 Computational Studies 5.4 Applications of Immunotherapy with Variable Dosages of Monoclonal Drugs 5.5 A Severe Case with Five Tumors 5.6 More Powerful Immunotherapy 5.7 Boundaries Are No Longer Safe Havens for Malignant Cells 5.8 More Applications of Monoclonal Drugs 5.9 A Very Severe Case 5.10 Conclusion References 6 Modeling Strategies to Win the War Against Breast Cancer 6.1 Rationale 6.2 The Burning Question on Validation of Computational Findings 6.2.1 Definition: Surjective, Injective, and Bijective Mappings 6.2.2 Surjective Transformation in Set 6.2.3 Mathematical Models on Breast Cancer Treatments. The Attacker–Defender Models 6.2.4 A Measure of Success of Treatment 6.2.5 Condition for Non-replicability of Malignant Cells 6.2.6 Conditions for the Rates of Growths of Chemicals/Biochemicals Fighting Cancer 6.2.7 Mathematical Model for Standard Breast Cancer Treatment 6.2.8 Variables Used in the Equations 6.2.9 The Equations Representing the Model 6.2.10 Measure of Aggression of Cancer 6.2.11 Immunostimulation/Immunosuppression Parameter ISN 6.2.12 Dimensional Analysis 6.2.13 Defeat of Cancer Cells 6.2.14 The Law of Physiology on the Replication of the Defenders Used in the Model 6.2.15 A Note on the Rate Constants rij and the Coefficients of Dispersions κi 6.2.16 The CDey-Simpson Method for Numerical Solution 6.2.17 A Special Note on Graphing 6.2.18 Computational Studies of the Model 6.2.19 Radiation is Administered Intravenously Using Nanoparticles as Vectors 6.2.20 The Confused T-Cells and An Inactive Chemo 6.2.21 Use of Fixed Dosages of Medications 6.3 Immunotherapy for Breast Cancer Treatment 6.4 Modeling a Stark Tragedy in Cancer Treatment 6.4.1 The Necessity for Adjuvant Therapy 6.5 More on Adjuvant Therapy 6.5.1 The Mathematical Model for Adjuvant Therapy 6.6 The Computer Visualization of Numerical Solutions 6.6.1 Graphical Results on Long Term Effects of Cancer Treatment 6.6.2 Application of Skilled Killer Drug (SKD) 6.7 The Conclusion References 7 Gene Therapy 7.1 Rationale 7.1.1 Mathematical Preliminaries [3] 7.1.2 An Application 7.1.3 A Preliminary Model 7.2 Introduction of D-Matrices [5–7] 7.3 A Model for Eliminating Multiple Proteins 7.3.1 Definition. D-Mapping [5–7] 7.3.2 Theorem on Elimination of Defective mRNA 7.4 Two Examples 7.4.1 The Generalized Form 7.4.2 Error Analysis for a Coupled System 7.5 Solution of Interacting Proteins 7.5.1 An Example 7.6 Conclusion References 8 The Smartest Fighters 8.1 Rationale 8.2 The Irresistible Fighters Against Cancer 8.2.1 The Mathematical Modeling and Computational Studies: Two Smart Combat Drugs SCD1(w) and SCD2(r) 8.2.2 Total Immunotherapy by SCD1 ( w ) and SCD2 ( r ) 8.3 Enhancement of the Immune Response 8.4 Fixed Values of SCD1 and SCD2 for Cancer Cells Dispersing Faster in Tissues 8.5 Introduction of the Skilled Killer Drug (SKD) 8.6 SKD: The Intelligent Nanoparticle 8.7 Faster Dispersion of SKD 8.8 Conclusion References 9 Nutritional Therapy 9.1 Rationale 9.2 A Mathematical Model 9.3 If a Tumor has Attained a Steady State 9.3.1 A Sheer Myth 9.4 Statistical Studies on Fat Intake and Breast Cancer [6] 9.5 A Mathematical Model on Statistical Studies 9.6 Probabilistic Analysis of Diets 9.7 Conclusion References 10 The Fateful Code and The Future Course 10.1 Rationale 10.2 The Fateful Code 10.3 Cancer Subsides, Yet It Could be Life Threatening 10.4 Growth Factor Reduction is not Enough (A Continuation of the Treatment of the Same Patient) 10.4.1 An Example: Difference Between Only Time Dependent and Both Time and Space-Dependent Models 10.5 A Deviation from the Previous Models 10.6 Mathematical Validations of All Models Through Microstatic Analysis 10.6.1 Three Models Representing the Three States of Cancer 10.6.2 Dimensional Analysis 10.6.3 Analytical Solution 10.6.4 The Treatment 10.7 Conclusion References 11 Conclusion 11.1 Rationale 11.2 An Observation on Mathematical Modeling 11.3 The Graphs 11.4 Numerical Challenges in the Models 11.4.1 An Extended Charlie Model 11.5 A Stark Reminder from Previous Studies 11.6 Breast Tumor Detection 11.7 Tumor Necrosis 11.8 Stress Reduction 11.9 Some Final Comments References Appendix A Appendix B Appendix C References