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Mathematical biomedicine is a rapidly developing interdisciplinary field of research that connects the natural and exact sciences in an attempt to respond to the modeling and simulation challenges raised by biology and medicine. There exist a large number of mathematical methods and procedures that can be brought in to meet these challenges and this book presents a palette of such tools ranging from discrete cellular automata to cell population based models described by ordinary differential equations to nonlinear partial differential equations representing complex time- and space-dependent continuous processes. Both stochastic and deterministic methods are employed to analyze biological phenomena in various temporal and spatial settings. This book illustrates the breadth and depth of research opportunities that exist in the general field of mathematical biomedicine by highlighting some of the fascinating interactions that continue to develop between the mathematical and biomedical sciences. It consists of five parts that can be read independently, but are arranged to give the reader a broader picture of specific research topics and the mathematical tools that are being applied in its modeling and analysis. The main areas covered include immune system modeling, blood vessel dynamics, cancer modeling and treatment, and epidemiology. The chapters address topics that are at the forefront of current biomedical research such as cancer stem cells, immunodominance and viral epitopes, aggressive forms of brain cancer, or gene therapy. The presentations highlight how mathematical modeling can enhance biomedical understanding and will be of interest to both the mathematical and the biomedical communities including researchers already working in the field as well as those who might consider entering it. Much of the material is presented in a way that gives graduate students and young researchers a starting point for their own work.



Immune System Modeling


Spatial Aspects of HIV Infection

Human immunodeficiency virus type 1 (HIV-1) is one of the most and intensely studied viral pathogens in the history of science. However, despite the huge scientific effort, many aspects of HIV infection dynamics and disease pathogenesis within a host are still not understood. Mathematical modeling has helped to improve our understanding of the infection as well as the dynamics of the immune response. Fitting models to clinical data has provided estimates for the turnover rate of target cells [82, 83,111], the lifetime of infected cells and viral particles [104, 109], as well as for the rate of viral production by infected cells [21, 44]. Most mathematical models applied to experimental data on viral infections have been formulated as systems of ordinary differential equations (ODE) [91, 101, 104].
Frederik Graw, Alan S. Perelson

Basic Principles in Modeling Adaptive Regulation and Immunodominance

In this chapter we overview our recent work on mathematical models for the regulation of the primary immune response to viral infections and immunodominance. The primary immune response to a viral infection can be very rapid, yet transient. Prior to such a response, potentially reactive T cells wait in lymph nodes until stimulated. Upon stimulation, these cells proliferate for a limited duration and then undergo apoptosis or enter dormancy as memory cells. The mechanisms that trigger the contraction of the T cell population are not well understood. Immunodominance refers to the phenomenon in which simultaneous T cell responses against multiple target epitopes organize themselves into distinct and reproducible hierarchies. In many cases, eliminating the response to the most dominant epitope allows responses to subdominant epitopes to expand more fully. Likewise, if the two most dominant epitopes are removed, then the third most dominant response may expand. The mechanisms that drive immunodominance are also not well understood.
Peter S. Kim, Peter P. Lee, Doron Levy

Evolutionary Principles in Viral Epitopes

The infection of a cell by a virus elicits a Cytotoxic T Lymphocyte (CTL) response to viral peptides presented by the Major Histocompatibility Complex class I molecules [6, 20]. Such a CTL response plays a critical role in the host’s anti-viral immune response [39]. This role is suggested by studies indicating a drop of viral loads and the relief of the acute infection symptoms following the emergence of virus-specific CTLs [8], as well as by data from CTL depleted animal models [33, 41]. The CTL response is also associated with a rapid selection of viral CTL escape variants [23, 34]. In the last few years we have applied an immunomic methodology combining genomic data and multiple bioinformatic tools to study the anti-viral CTL response [5, 19, 28, 35, 38, 55–57] and found that viruses selectively mutate proteins inducing the highest danger to their survival. We here summarize these results, and propose some general conclusions regarding the selective forces affecting viruses and their human host.
Yaakov Maman, Alexandra Agranovich, Tal Vider Shalit, Yoram Louzoun

Blood Vessel Dynamics


A Multiscale Approach Leading to Hybrid Mathematical Models for Angiogenesis: The Role of Randomness

In biology and medicine we may observe a wide spectrum of formation of patterns, usually due to self-organization phenomena. This may happen at any scale; from the cellular scale of embryonic tissue formation, wound healing or tumor growth, and angiogenesis to the much larger scale of animal grouping. Patterns are usually explained in terms of a collective behavior driven by “forces,” either external and/or internal, acting upon individuals (cells or organisms). In most of these organization phenomena, randomness plays a major role; here we wish to address the issue of the relevance of randomness as a key feature for producing nontrivial geometric patterns in biological structures. As working examples we offer a review of two important case studies involving angiogenesis, i.e., tumor-driven angiogenesis [7] and retina angiogenesis [8]. In both cases the reactants responsible for pattern formation are the cells organizing as a capillary network of vessels, and a family of underlying fields driving the organization, such as nutrients, growth factors, and alike [18, 19].
Vincenzo Capasso, Daniela Morale

Modeling Tumor Blood Vessel Dynamics

Tumor blood vessels are structurally abnormal and functionally inefficient, resulting in incomplete perfusion of tumor vessel networks and nonuniform delivery of chemotherapeutics to the tumor cells. Excessive production of the angiogenic growth factor VEGF (vascular endothelial growth factor) contributes to tumor vessel abnormalities, and many anti-VEGF therapies can cause remodeling or stabilization of tumor blood vessels. This remodeling resembles the process of angioadaptation previously studied in the context of normal physiology and ischemia. During angioadaptation (also know as adaptive remodeling), endothelial cells respond to blood forces to alter blood flow. Some segments dilate, while others contract, eventually producing an efficient network. Although not well understood, it is likely that adaptive remodeling depends on blood shear forces, transvascular pressure, upstream signals transmitted along the endothelium as well as growth factors such as VEGF. To provide an analytical framework for understanding these processes in the context of tumor vasculature, we have developed a mathematical model, supported by multiparameter imaging methodology, which incorporates the necessary elements for predicting the transport of nutrients and drugs throughout tumor vessels and tissue, as well as the adaptive remodeling of the blood vessel network. A better understanding of the mechanisms responsible for the network dynamics may lead to novel approaches for treating tumors or other diseases involving vascular pathologies.
Lance L. Munn, Christian Kunert, J. Alex Tyrrell

Influence of Blood Rheology and Outflow Boundary Conditions in Numerical Simulations of Cerebral Aneurysms

Disease in human physiology is often related to cardiovascular mechanics. Impressively, strokes are one of the leading causes of death in developed countries, and they might occur as a result of an aneurysm rupture, which is a sudden event in the majority of cases. On the basis of several autopsy and angiography series, it is estimated that 0.4–6 % of the general population harbors one or more intracranial aneurysms, and on average the incidence of an aneurysmal rupture is of 10 per 100,000 population per year, with tendency to increase in patients with multiple aneurysms [14, 20].
Susana Ramalho, Alexandra B. Moura, Alberto M. Gambaruto, Adélia Sequeira

Cancer Modeling


The Steady State of Multicellular Tumour Spheroids: A Modelling Challenge

Cells from different tumour cell lines can be grown in vitro to form spheroidal masses, called multicellular tumour spheroids, currently considered valuable experimental models of avascular tumours [35, 37, 50, 51, 58]. Multicellular tumour spheroids have been extensively investigated in that they provide a useful model to assess the effects of oxygenation and nutrition on growth, as well as the effects of treatments with drugs and radiation.
Antonio Fasano, Alberto Gandolfi

Deciphering Fate Decision in Normal and Cancer Stem Cells: Mathematical Models and Their Experimental Verification

All tissues in the body are derived from stem cells (SCs). SCs are undifferentiated cells with two essential properties: unlimited replication capacity and the ability to differentiate into one or more specialized cell types. Embryonic SCs are pluripotent, meaning that they can give rise to nearly all cell types. Non-embryonic, adult SCs are found in various tissues and are capable of generating a limited set of tissue-specific cell types. The first discovered and most extensively studied type of adult SC is the hematopoietic SC, found in the bone marrow, which can give rise to all lineages of mature blood cells [12, 84]. Organ-specific SCs have been identified in many other tissues, including the liver, skin, brain, and mammary gland (see [19] for review).
Gili Hochman, Zvia Agur

Data Assimilation in Brain Tumor Models

A typical problem in applied mathematics and science is to estimate the future state of a dynamical system given its current state. One approach aimed at understanding one or more aspects determining the behavior of the system is mathematical modeling. This method frequently entails formulation of a set of equations, usually a system of partial or ordinary differential equations. Model parameters are then measured from experimental data or estimated from computer simulation or other methods, for example chi-squared parameter optimization as done in[26] or genetic algorithms which are frequently used in neuroscience [33]. Solutions to the model are then studied through mathematical analysis and numerical simulation usually for qualitative fit to the dynamical system of interest and any relative time-series data that is available. While mathematical modeling can provide meaningful insight, it may have limited predictive value due to idealized assumptions underlying the model, measurement error in experimental data and parameters, and chaotic behavior in the system. In this chapter we explore a different approach focused on optimal state estimation given a model and observational data of a biological process, while accounting for the relative uncertainty in both. The case explored here is the growth and spread of glioblastoma multiforme (GBM), a very aggressive form of glioma brain tumor which remains extremely difficult to manage clinically. The method employed is different from other approaches used in biology in that it is independent of the mathematical model and seeks an optimal initial condition. This is in contrast to other techniques such as those discussed in [21], which are model dependent and seek to find an optimal model parameterization given the observations and uncertainties in the system of interest.
Joshua McDaniel, Eric Kostelich, Yang Kuang, John Nagy, Mark C. Preul, Nina Z. Moore, Nikolay L. Matirosyan

Cancer Treatment


Optimisation of Cancer Drug Treatments Using Cell Population Dynamics

Cancer is primarily a disease of the physiological control on cell population proliferation. Tissue proliferation relies on the cell division cycle: one cell becomes two after a sequence of molecular events that are physiologically controlled at each step of the cycle at so-called checkpoints, in particular at transitions between phases of the cycle [105]. Tissue proliferation is the main physiological process occurring in development and later in maintaining the permanence of the organism in adults, at that late stage mainly in fast renewing tissues such as bone marrow, gut and skin.
Frédérique Billy, Jean Clairambault, Olivier Fercoq

Tumor Development Under Combination Treatments with Anti-angiogenic Therapies

Tumors are a family of high-mortality diseases, each differing from the other, but all exhibiting a derangement of cellular proliferation and characterized by a remarkable lack of symptoms [52] and by time courses that, in a broad sense, may be classified as nonlinear. As a consequence, despite the enormous strides in prevention and, to a certain extent, cure, cancer is one of the leading causes of death worldwide, and, unfortunately, is likely to remain so for many years to come [4, 53].
Urszula Ledzewicz, Alberto d’Onofrio, Heinz Schättler

Saturable Fractal Pharmacokinetics and Its Applications

In this chapter we discuss an application of fractal kinetics under steady state conditions to model the enzymatic elimination of a drug from the body. A one-compartment model following fractal Michaelis–Menten kinetics under a steady state is developed and applied to concentration-time data for the cardiac drug mibefradil in dogs. The model predicts a fractal reaction order and a power law asymptotic time-dependence of the drug concentration. A mathematical relationship between the fractal reaction order and the power law exponent is derived. The goodness-of-fit of the model is assessed and compared to that of four other models suggested in the literature. The proposed model provides the best fit to the data. In addition, it correctly predicts the power law shape of the tail of the concentration-time curve. The new fractal reaction order can be explained in terms of the complex geometry of the liver, the organ responsible for eliminating the drug. Furthermore, we investigate the potential for identifying global characteristics in the pharmacokinetics of the anticancer drug paclitaxel. An analysis of data in the literature yields both clearance curves and dose-dependency curves that are accurately described by power laws with similar exponents.
Rebeccah E. Marsh, Jack A. Tuszyński

A Mathematical Model of Gene Therapy for the Treatment of Cancer

Cancer is a major cause of death worldwide, resulting from the uncontrolled growth of abnormal cells in the body. Cells are the body’s building blocks, and cancer starts from normal cells. Normal cells divide to grow in order to maintain cell population equilibrium, balancing cell death. Cancer occurs when unbounded growth of cells in the body happens fast. It can also occur when cells lose their ability to die. There are many different kinds of cancers, which can develop in almost any organ or tissue, such as lung, colon, breast, skin, bones, or nerve tissue. There are many known causes of cancers that have been documented to date including exposure to chemicals, drinking excess alcohol, excessive sunlight exposure, and genetic differences, to name a few [38]. However, the cause of many cancers still remains unknown. The most common cause of cancer-related death is lung cancer. Some cancers are more common in certain parts of the world. For example, in Japan, there are many cases of stomach cancer, but in the USA, this type of cancer is pretty rare [49]. Differences in diet may play a role. Another hypothesis is that these different populations could have different genetic backgrounds predisposing them to cancer. Some cancers also prey on individuals who are either missing or have altered genes as compared to the mainstream population. Unfortunately, treatment of cancer is still in its infancy, although there are some successes when the cancer is detected early enough. To begin to address these important issues, in this work we will focus solely on genetic issues related to cancer so that we can explore a new treatment area known as gene therapy as a viable approach to treatment of cancer.
Alexei Tsygvintsev, Simeone Marino, Denise E. Kirschner

Epidemiological Models


Epidemiological Models with Seasonality

Epidemiology is the branch of medicine that deals with incidence, distribution, and control of diseases in a population. At the basic level the population is divided into susceptible, exposed, infected, and recovered compartments. However, often infection is caused not only by exposed or infected individuals but also by other species, such as mosquitos in the case of malaria, or waste water in the case of cholera. In attempting to model the transmission of the disease one has to take into account the facts that infection rates may vary among different populations (due, for instance, to those who underwent vaccination and those who did not), as well as from one season to another. In this chapter we focus on seasonality-dependent diseases and ask the question whether initial infection of one or a small number of individuals will cause the disease to spread or whether the disease will die out. To answer this question we invoke the concept of the basic reproduction number, a number which is easy to compute in the case of seasonality-independent diseases, but difficult to compute in the case of diseases with seasonality.
Avner Friedman

Periodic Incidence in a Discrete-Time SIS Epidemic Model

Mathematical models have continued to increase our understanding of the spread of infectious diseases and their control in both humans and animals. In most infectious diseases, the incidence coefficient or contact rate (the rate of new infections) plays a key role in ensuring that the model gives a reasonable qualitative description of the real disease dynamics. To accurately gauge the impact of infectious diseases prevention efforts, it is important to understand the relation between disease transmission and the host population dynamics. In [8–11], Castillo-Chavez and Yakubu introduced a framework for studying infectious disease dynamics in strongly fluctuating populations. In their model framework, Castillo-Chavez and Yakubu assumed that the host demographics is governed by the Ricker model and the contact rate is constant. However, periodicity in infectious disease incidence is known to occur in chickenpox, measles, pertussis, gonorrhea, mumps, influenza, and other infectious diseases.
Najat Ziyadi, Abdul-Aziz Yakubu
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