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Mathematics and 21st Century Biology 5 Understanding Organisms The step in hierarchical scale from cells to organisms is huge. This chapter largely discusses tissues, organs, and organ systems in multicellular organisms made up of immense numbers of highly differentiated cells. In some instances, as in the discussion of locomotion, the focus is on the integrated properties of the whole organism. While much mathematical analysis directed at understanding organisms involves systems that are far removed from the cellular level, there are also levels of biological organization intermediate between cells and organisms. Biofilms formed by bacteria are one such intermediate level that has received modeling attention (Morgenroth et al., 2004), as are still more complex cellular aggregates such as the slug phase of the slime mold Dictyostelium discoideum (Umeda and Inouye, 2004) and the aggegration phase of myxobacteria (Igoshin and Oster, 2004). Analysis of systems of this type—poised as they are between cells and organisms on the scale of biological organization—offers a promising path toward improving mathematical approaches to multicellular processes. However, the committee draws its main examples in this chapter from more traditional areas of mathematical modeling, such as physiological processes. In recent years the importance of mathematical models in the study of physiological processes has become widely accepted. There are many instances of how experimentalists and theoreticians, working together, have made discoveries that would be difficult, if not impossible, for each working independently. One such discovery involves the phenomenon of electrical excitability and the propagation of action potentials in cardiac and neural tissue. How oscillations in the cell cycle lead to regular cell divi-
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Mathematics and 21st Century Biology sions; how intercellular calcium waves coordinate cellular responses over large areas; how tumors grow and respond to chemotherapy; and how the HIV virus is produced and cleared within cells: all are areas where mathematical models have played an important role. The need for mathematical models has never been greater. Much of the biological investigation of the past can be described as a compilation and categorization of the list of parts, whether as the delineation of genomic sequences, genes, proteins, or species. The past decade has seen an explosion in probing genetic or cellular defects that alter properties and behaviors at the tissue or organ level, thereby identifying the root basis for many diseases. As examples, we know the mutation to a chloride ion channel that results in cystic fibrosis and the mutations to potassium channels that lead to long QT syndrome (an abnormality of the heart’s electrical system). There have also been many striking advances in imaging and measurement of function, some due to mathematical advances that provide insight into the level and extent of functional degradation or guide clinical intervention. For example, the ability to interpret electrocardiograms has led to spectacular advances in the reliability of implantable pacemakers and defibrillators (Kenknight et al., 1996). Missing is the ability to integrate how the various components of organs work together to achieve dynamic function, and how change of specific components or combinations thereof impact function. Thus, the challenge of systems physiology is to provide an understanding of how the interactions of biological entities across spatial and temporal scales lead to observable behavior and function. Two important organizing principles need emphasis. First, an integrated understanding of systems requires mathematics and the development of theory, supplemented by simulations. One of the important lessons of the past is that there are behaviors and phenomena that are the consequences of interactions of several or many individual components that cannot occur with the components uncoupled, and the principles governing these emergent behaviors require theory for their full explanation. Secondly, theory cannot be relevant if it is not driven and inspired by experimental data. The committee illustrates these with some examples where systems physiology has great promise. CARDIAC PHYSIOLOGY Failure of the cardiac system remains the leading cause of death in the Western world. The cardiac cycle consists of two primary events: (1) a contractile, or mechanical, event, controlled by (2) an electrical event, the cardiac action potential. Failure of either of these can lead to death. Either cardiomyopathies, in which the cardiac muscle does not provide enough
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Mathematics and 21st Century Biology force or blood volume (decreased cardiac output), or a disruption of the electrical signal in an otherwise mechanically adequate heart may be the primary dysfunction. Of course, these disruptions are rarely completely independent, as diseased and damaged tissue often results in an increased likelihood of electrical malfunction. One challenge presented by the cardiac system is to understand the physiological mechanisms underlying the electrical signal, so as to understand the mechanisms of the variety of arrhythmias and to learn how to control or prevent these arrhythmias. A substantial amount of ongoing research is aimed at understanding the dynamics of cardiac cells using mathematical and computational models. There is a long history to this direction of investigation, which has its origins with the Hodgkin-Huxley equations. The Hodgkin-Huxley theory was extended to cardiac cells by Noble, Beeler-Reuter, and others. More recently, detailed cellular ionic models have been developed by, for instance, Luo and Rudy (1994), Jafri et al. (1998), and Puglisi and Bers (2001). In spite of the remarkable success of these models, they all fall short of providing an understanding of many important arrhythmias. This short-coming is illustrated by the history of antiarrhythmic drugs. Many of the so-called antiarrhythmic drugs are known to be ion channel blockers. When they were first discovered, it was thought that arrhythmias were caused by overactive ion channels and if these were blocked, then the arrhythmias could be prevented. Indeed, tests on single cells and small patches of tissue verified this conjecture. However, when drugs were tested in the CAST and SWORD clinical trials (CAST Investigators, 1989; Waldo et al., 1996), it was discovered that many of these drugs were actually proarrhythmic. The fundamental difficulty was that an understanding of how single cells or small patches of tissue behave or respond to drugs does not answer the question of how the entire spatiotemporal system will behave. (While some arrhythmias are the result of cellular automaticity and ectopic foci, which occur when a cell or small collection of cells oscillates without external stimulus and thereby takes over as the pacemaker of the heart, the most significant life-threatening arrhythmias are maintained because they are spatiotemporal patterns and cannot occur in single cells or small patches of tissue.) In the case of the CAST and SWORD studies, the lack of a suitable spatiotemporal model led people astray. They relied instead on their best guess, but their best guess was wrong. It is now recognized that almost all drugs that were previously classified as antiarrhythmic are actually proarrhythmic. It is also now recognized that the response of a single cell to ion-channel blockers does not adequately predict the response at the spatiotemporal level. Thus, the challenge is to develop mechanistic, functionally integrated, multiscale mathematical models of the heart from molecular to
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Mathematics and 21st Century Biology cellular and whole-organ scales, which would lead to a deeper understanding of the excitation and contraction of the heart. Research needs to move from understanding atrial and ventricular electrophysiology based on models of the biophysics of single-ion channels to predicting the electrocardiogram recorded at the body surface. (Of course we will ultimately want to use this understanding in an inverse way: interpreting electrocardiogram signals at the surface as indicators of the functioning of cardiac subsystems.) An overarching theme will be how mathematical models can help elucidate mechanisms, improve diagnoses, and identify therapeutic targets for cardiac arrhythmias. Simultaneously, there is a need to address the mechanical function of the heart using models of the biophysics and biochemistry of molecular motors to predict the three-dimensional mechanical performance of the whole heart (Vetter and McCulloch, 1998). Related questions are how mathematical models can help improve the diagnosis and treatment of cardiac mechanical dysfunction during disease, especially congestive heart failure, and how to elucidate the mechanisms by which mechanical factors can regulate cardiac remodeling in vivo. Integrative computational modeling of the heart has a long history dating to Laplace. Laplace’s law provides an explanation for the fact that a dilated heart must create a larger wall tension in order to create the normal pressure, giving a theoretical basis for the surgical procedure of ventricular remodeling. The first cardiac myocyte ionic models were published in Noble (1962), followed by Moe’s cellular automata model of atrial fibrillation in 1964 (Moe et al., 1964). Crossbridge and continuum models of ventricular mechanics started to appear in 1970. Today, an established multidisciplinary community of mathematicians, bioengineers, biophysicists, and physiologists is working on the experimental, theoretical, and computational challenges associated with formulating, implementing, and validating predictive models that integrate functionally across interacting cellular processes such as electrical excitation, mechanical contraction, and energy metabolism, and structurally across scales of biological organization from molecule to organ and system (McCulloch et al., 1998). Many in this community have advocated ambitious multicenter programs under banners such as the Cardiome Project, headed by A.D. McCulloch at the University of California at San Diego. Several large sponsored collaborations are under way (McCulloch et al., 1998; McCulloch and Huber, 2002). However, in spite of the growing sophistication of these integrative modeling efforts, the investigators are the first to point out the manifest weaknesses and shortcomings. While excellent progress has been made in applying cellular system models of action potentials or contractile processes to three-dimensional continuum models of impulse propagation or ventricular pumping, multiscale electromechanical models are in their
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Mathematics and 21st Century Biology infancy and will require further development before they can provide the insight that is needed. Another frontier is the development of models of the metabolic and neurohormonal (cell signaling) mechanisms that regulate excitation and contraction and their interactions (Saucerman and McCulloch, 2004). Finally, the application of integrative models to understanding the pathogenesis of genetic and acquired heart diseases and identifying new therapeutic targets is an emerging and timely field (Sussman et al., 2002). CIRCULATORY PHYSIOLOGY The function of the systemic circulatory system is to distribute and remove materials and heat as needed throughout the body. Transport is achieved by convection in the blood and diffusive exchange with surrounding tissue (Pittman, 2000). Because diffusion is effective only over short distances, blood must be brought close to every point in every tissue. To make this possible, the peripheral circulation consists of a highly branched system of blood vessels containing more than 109 segments ranging in diameter from about 1 cm down to a few microns. The set of vessels of diameter about 100 microns or less is referred to as the microcirculation. A remarkable feature of the systemic circulatory system is its ability to adjust to short- and long-term changes in local functional requirements. This is achieved by a combination of central and local mechanisms. Short-term local control of blood flow is accomplished when vessels change their diameters by contracting and relaxing vascular smooth muscle. Longer-term changes in needs are met by structural changes, including changes in wall thickness and diameter and the addition of new vessels (angiogenesis). Many of these changes are driven primarily by responses to local stimuli, without central control. The peripheral circulation can therefore be considered as a highly distributed adaptive system. Understanding this system has important implications both for normal physiological processes and for many diseases, including heart disease, hypertension, and cancer. Important areas of research are blood flow and mass transport in the microcirculation; short-term regulation of blood flow, including vascular smooth-muscle behavior; and structural adaptation of blood vessels, including angiogenesis. Mathematical and computational approaches can make important contributions in all of these areas. Continuum and multiphase models can be applied to study blood flow. Simulations of mass and heat transport also typically require solution of nonlinear partial differential equations. Consideration of network properties is also critical to understanding short- and long-term control of blood flow (Segal,
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Mathematics and 21st Century Biology 2000; Secomb and Pries, 2002). The network can be regarded as a dynamic system in which the properties of each segment (diameter, etc.) evolve with time (Segal, 2000; Ursino, 2003; Zakrzewicz et al., 2002). Simulations of angiogenesis can use a variety of approaches, including deterministic and stochastic models and cellular automata. RESPIRATORY PHYSIOLOGY As with many organ-level pathologies, the past decade has seen an explosion in probing lung pathology from the bottom up (e.g., genetic or cellular defects or manipulations initiating the processes that alter airway and tissue properties) as well as from the top down (e.g., advances in imaging and function measurements that provide insight on the level and extent of functional degradation). However, the chasm that remains between the two approaches must be bridged in a manner that can more effectively guide therapeutic targets and assessment. Past and even current experimental and modeling research focuses either on a specific level of lung structure—for example, on a single airway, the airway wall, tissue rheology, airway smooth muscle, or even airway smooth muscle and alveolar cell—or on function at a gross level—for example, whole-lung mechanical properties and indices of ventilation distribution. What is missing is the capacity to integrate how all the components in the lung work together to achieve dynamic function and how degradation in specific components or combinations of components might impact function. Examples of lung pathologies in need of a more comprehensive understanding of how integrated structures lead to function include asthma, adult respiratory distress syndrome, and emphysema. Computational modeling promises a new era in the fundamental understanding of how lung morphometry and biomechanical/biomaterial properties impact lung function. With continuous improvement in imaging modalities, it is becoming increasingly possible to establish precise physical locations and degrees of structural or functional defects in the lung during disease. Such data will provide a foundation for addressing how explicit defects of biological components, processes, and structure at specific anatomic locations alter function. Computational power now permits the development of models that are closer anatomic replicas of a real lung, while incorporating the fundamental biophysical properties and relations for each component of each airway. Rational and efficient disease management could be enhanced by understanding or predicting how alterations in the individual components of lung structure and properties impact the emergent lung function. A holy grail is a so-called in silico lung, which would reflect a personalized condition and enable simulated treatments to be performed and
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Mathematics and 21st Century Biology evaluated. Such a virtual lung would lead to the generation or rejection of specific treatment hypotheses, in turn leading to more scientific and financially cost-effective experiments or technology development. While a multiscale and personalized modeling approach has emerged for other physiological systems (e.g., cardiovascular), it remains in its infancy for the lung. INFORMATION PROCESSING A system physiology approach is also needed for information processing in the visual system. The traditional feedforward model of the visual system invokes a sequence of processing stages, beginning with the relay of retinal input to neurons in the primary visual cortex (V1) via the lateral geniculate nucleus (LGN) and subsequent higher-order processing through a hierarchy of cortical areas. According to this model, neurons at each successive stage process inputs from increasingly larger regions of space and code for increasingly more complex aspects of visual stimuli. The selectivity of a neuron to a given stimulus parameter (e.g., orientation, color, depth) is assumed to result from the ordered convergence of afferents from the lower stages. Although feedforward models can perform a surprising number of object-recognition tasks in some simple environments, they perform badly in many situations that are simple for human vision—for example, where an object might be partially masked or occluded by other objects. It has become clear that more complex forms of visual information processing require global-to-local interactions, both within a given stage and between different stages of the visual hierarchy. Long-range horizontal connections provide an anatomical substrate for the former, whereas feedforward and feedback connections provide a substrate for the latter. One of the major outstanding theoretical challenges is to bridge the gap between the systems physiology of vision, characterized by spatiotemporal dynamics at multiple scales (synapses, neurons, networks), and the computational/information theoretic aspects of vision (neural code, statistics of natural scenes, redundancy) (Barlow, 1961; Laughlin, 1981; Atick et al., 1992). In a similar way, the human auditory system from the inner ear to the auditory cortex is a complex multilevel pathway of sound information processing (Dallos et al., 1990). One of the early stages of sound processing occurs in the cochlea, where the vibration pattern of the basilar membrane encodes the acoustic characteristics of incoming sound signals. Though well-known partial differential equations in classical mechanics provide a solid foundation for describing these mechanical activities, additional nonlinearities and active behaviors must be modeled to capture nonlinear responses such as tonal suppressions and the ob-
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Mathematics and 21st Century Biology served frequency selectivity. The next level of information processing occurs in as many as 30,000 nerve fibers connecting the inner ear to the brain. Nonlinearities are associated with peripheral auditory neurons when the hair cell converts sound signals from mechanical to neural representation. It is now well known that outer hair cells of the cochlea play an active role in increasing the sensitivity and dynamic range of the ear. In addition, the frequency distribution of sound is maintained by the wave patterns on the basilar membrane and is preserved along the fibers, resulting in an organization of frequency responses in the auditory cortex of the brain whereby different tone frequencies are transmitted separately along different parts of the structure. An additional challenge regarding processing in the auditory system is to account for the extremely fast temporal resolution of hearing, which is at the timescale of microseconds rather than the typical millisecond timescale of individual neurons. Although mathematical models exist for many levels of visual and auditory processing, a better understanding of the connections within the systems will depend on progress in physiological experiments as well as theoretical advances to connect these individual levels. Moreover, many of the neural models take the form of integro-differential equations, in which the interaction kernel represents the spatial distribution of synaptic weights. Such equations are much less well understood than the more familiar partial differential equations of reaction diffusion systems. ENDOCRINE PHYSIOLOGY Mathematical modeling has led to an improved understanding of several important endocrine processes (Bertram and Sherman, 2004; Kukkonen et al., 2001; Mosekilde et al., 2001). However, there are numerous areas of endocrine physiology and associated pathologies that are in need of a more integrated approach. Consider, for instance, diabetes. It is well appreciated within the diabetes research community that diabetes is a multifactorial disease that involves the interaction over disparate spatial scales (from genes to cells to organs to the whole body) and timescales (from milliseconds to decades) (Porksen et al., 1997; Sedaghat et al., 2002; Smolen et al., 2001; Topp et al., 2000). In addition, genetic, metabolic, and ionic events all have to be integrated to achieve a workable understanding of normal and pathological regulation (Bergmann, 1989; Cobelli et al., 1998; Tornheim, 1997). Some diabetes-related questions that could be explored through a system physiology approach include these: How is weight regulated? Is there such a thing as a set point for weight? Why is it easier to gain weight than to lose it? Why is insulin resistance associated with inflammation, hypertension, and a high LDL/HDL cholesterol ratio? How do insulin resis-
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Mathematics and 21st Century Biology tance and beta-cell failure interact to produce a global failure of regulation? Can we go beyond descriptive diagrams of hormone/peptide interaction networks to predictive models? How do ionic and metabolic oscillations articulate to produce pulsatile insulin secretion? How is the renin-angiotensin system regulated genetically, and what are the genetic factors underlying high blood pressure? These questions invite mathematical modeling and simulations, some of which are currently taking place. MORPHOGENESIS AND PATTERN FORMATION The combination of developmental genetics with rapidly advancing imaging and transcriptional profiling technologies promises a golden age for the modeling and computational analysis of developing systems. The main efforts for modeling and computational analysis can be subdivided into three groups: (1) analysis and synthesis of genetic and imaging data, with the main goal of formulating realistic models, (2) formulation of models that reflect the complexity of developing tissues, and (3) analysis of these models and their testing in direct genetic experiments. Models of developmental pattern formation are necessarily spatially distributed, dynamic, and multivariable. All of these aspects can be now explored experimentally, opening the door to a mutually beneficial interplay between modelers and experimentalists. The expression of tens to thousands of genes in any given context can be visualized by multicolor in situ hybridizations; antibody stainings; spatially resolved, quantitative, real-time polymerase chain reactions (PCRs); or microarray experiments (Tomancak et al., 2002; Fraser and Marcotte, 2004). Few of the current techniques for gene-expression analysis in development are real-time, so the multivariable dynamics of a system have to be pieced together from a number of still shots taken from different embryos. Data normalization and image processing techniques, such as morphing, can be used to construct the spatiotemporal atlases of gene expression or pathway activity (Pereanu and Hartenstein, 2004). Recently, this approach was successfully used to develop a comprehensive multivariable and dynamic picture of gap gene expression in fruit fly embryogenesis (Kozlov et al., 2002; Jaeger et al., 2004). The spatiotemporal information about gene expression or pathway activity can be integrated, again through morphing, with the results of real-time microscopic analysis of the morphological changes in the developing system (Huisken et al., 2004; Kosman et al., 2004). Atlases of gene expression and pathway activity directly lead to models. At the simplest level, correlation between expression patterns of multiple genes can be used to formulate Boolean or Bayesian models of gene regulation. Systematic methods for formulating such models from data
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Mathematics and 21st Century Biology must be developed, along with the computational techniques for their analysis, with an emphasis on spatially distributed systems (Friedman, 2004; Nachman et al., 2004). At the next level of complexity, multivariable spatial data can be used to fit parameters in the continuous-time dynamic models—for example, in the form of reaction-diffusion equations. This requires robust numerical methods for parameter estimation based on spatially resolved and dynamic data. Reinitz and co-workers have used stochastic optimization to fit parameters in the reaction-diffusion model of gene regulation in the early Drosophila embryo (Jaeger et al., 2004). The estimated parameters can be used to interpret the dynamics of genetic interactions in development. Gene expression patterns in developing tissues can be very fine-grained, with the characteristic domains of gene expression spanning only a few cell diameters. Such patterns cannot be captured with the continuum models traditionally used to model developmental patterning (Murray, 1993). Models of developing tissues must account for cell-cell interactions by both localized and spreading signals and for the dynamics of gene expression mediated by extracellular signals (Monk, 2000; Shvartsman et al., 2002; Eldar et al., 2003). An important modeling direction involves incorporating cell-level models into the descriptions of multicellular systems and tissues (Pribyl et al., 2003). A major challenge for the development of truly predictive pattern formation models lies in choosing the appropriate modeling formalism for describing the regulatory patterns of gene expression. Indeed, the expression of a single gene can be a highly complex function of extracellular conditions (Yuh et al., 1998; Davidson, 2001; Setty et al., 2003). Despite this complexity, it is worthwhile to explore the utility of simple logic and switchlike models for modeling gene expression in developing tissues (Thieffry and Sanchez, 2003). The most productive approach to modeling is likely to be hybrid, with threshold functions that couple extracellular signals to gene expression in individual cells. Numerical techniques for the analysis of such models are being developed (Ghosh and Tomlin, 2001). Analysis of robustness is crucial in the evaluation of mathematical and computational models of development (von Dassow et al., 2000; Eldar et al., 2004). Indeed, developing systems can frequently tolerate gene dose reductions due to heterozygocity, so they are robust to twofold changes in the developmental parameters. This experimental observation can be used to rule out models and mechanisms that require fine-tuning of parameters. In fact, the robustness that is so common in biological systems—and which is seen as a relative insensitivity to variations in parameters—suggests that a model that requires fine-tuning may be overlooking a layer of regulation in the system. In connection with this, modeling standards
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Mathematics and 21st Century Biology for robustness analysis must be developed. Current approaches are based on random sampling of system parameters (von Dassow et al., 2000; Meir et al., 2002; Eldar et al., 2003). More sophisticated methods for parameter sampling and statistical verification of results of random parameter sampling must be developed. The amazing robustness of developing systems, e.g., the stability of the morphologies of eggshells or wings, is contrasted with large interspecies variations. Since the time of Turing, the nonlinear instabilities induced by variations of system parameters were considered one of the mechanisms for generating increasingly complex patterns and morphologies (Turing, 1952; Meinhardt and Gierer, 2000). While true in physicochemical systems, this hypothesis still awaits its experimental verification in developing systems. This verification requires the identification of appropriate experiments where system parameters can be varied and the effects of these variations on gene expression patterns and the emerging morphologies can be examined. Model organisms, such as fruit flies and worms, can be used to generate genetic backgrounds with controlled levels of gene expression. The design of such genetic experiments can be model-based in the sense that nonlinear analysis of the model can suggest the most critical genetic perturbations (Shvartsman et al., 2002; Nakamura and Matsuno, 2003). In addition, analysis of the results of genetic experiments (e.g., the effect of overexpressing a gene on the shape of a wing) can be accelerated by the development of new image analysis and pattern recognition tools for rapid phenotyping. For instance, when studying wing development, tools for the rapid detection of morphological changes in a large number of fruit fly cells, embryos, or wings would be desirable (Myasnikova et al., 2001; Houle et al., 2003; Kiger et al., 2003). LOCOMOTION An important area of mathematical analysis at both the cellular and organismal level is the study of locomotion. Much productive research has been carried out on locomotion at many scales, ranging from the mechanical repositioning of subcellular organelles to the gaits of running animals. The subject is too large to review comprehensively in this report. Instead, the committee simply cites examples that illustrate the breadth of important problems on which progress has been made. At the subcellular level, molecular motors based on actin and tubulin polymerization are of central importance in many basic cellular processes, including chromosome segregation during cell division, cell motility, muscle contraction, and intracellular transport of organelles (for a review, see Mogilner and Oster, 2003). Although bacteria themselves lack systems based on actin and tubulin, some pathogens have evolved systems for
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Mathematics and 21st Century Biology utilizing the molecular motors of their hosts for propulsion. A dramatic example is Lysteria monocytogenes: The relatively simple mechanical system through which Lysteria moves when in contact with mammalian cells lends itself well to both detailed experimental characterization and mathematical modeling (Alberts and Odell, 2004). Rotary motors, exemplified by those that drive flagellar motion in E. coli, have also received extensive attention (Coombs et al., 2002; for a review, see Oster and Wang, 2003). At an organismal level, mathematical analysis has illuminated basic mechanisms of insect flight (Combes and Daniel, 2003; Miller and Peskin, 2005). A distinctive characteristic of these studies—and those of locomotion in general—has been the close interplay between experiment and theory. This interplay has long been evident even in the analysis of locomotion at larger spatial scales. Examples include the swimming motions of lampreys (Cohen et al., 1992; Lighthill, 1995) and the gaits of quadrupeds (Buono and Golubitsky, 2001). Indeed, most biomechanical processes would be difficult to study effectively without a close connection between theory and experiment. CANCER While cancer can be studied at the genetic and cellular levels, it is not until it is understood at the tumor level that its intrinsic cancerous behavior can be recognized. It follows that the outcome of chemotherapy cannot be understood without understanding the effects of spatial organization and intercellular communication on the dynamics of tumor development. The dynamic interplay of several biological factors determines the response of a cell to therapy and, ultimately, the outcome of chemotherapy. The key issues are (1) delivery of therapy to target tumor cells, (2) mechanisms of drug action, (3) growth and differentiation of cell populations, and (4) development of resistance. Delivery of Therapy to Target Tumor Cells Over 80 percent of human cancers are solid tumors. Presentation of a drug to cells in a solid tumor and the accumulation and retention of a drug in tumor cells depend on how the drug is delivered, the ability of the drug to diffuse through the interstices, and the binding of the drug to intracellular macromolecules. Some of these factors also depend on time and drug concentration. For example, the interstices, which determine the porosity and therefore the diffusion coefficient, might be expanded as a result of drug-induced apoptosis. Mathematical models depicting how these processes affect drug delivery to tumor cells could suggest the treat-
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Mathematics and 21st Century Biology ment regimens that will result in the most effective drug concentration and residence time in the target sites. Mechanisms of Drug Action Most anticancer drugs act on specific molecular targets, often molecules that are involved in the regulation of cell growth, cell differentiation, and cell death. Mathematical models to link the effective drug concentration in the tumor cells with the molecular targets, in a time- and concentration-dependent manner, are needed to improve the understanding of drug-target interaction (see Panetta et al., 2000; McDougall et al., 2002; http://calvino.polito.it/~biomat/). Growth and Differentiation of Cell Populations Efforts here involve the modeling of growth and differentiation of laboratory cell populations, of populations of normal cells, and of cells in tumors. Precise mathematical models exist for the processes of haemopoiesis (blood cell production) and self-renewal of colon epithelium. The mathematical tools used include stochastic processes (which are useful when describing small colonies or early stages of cancer), particularly branching processes, nonlinear ordinary differential equations (which are useful for modeling feedbacks of cell-production systems), and integral equations and partial differential equations (which are useful for modeling heterogeneous populations). The challenges involve integrating newly described genetic and molecular mechanisms into the models of proliferation, mathematically modeling the geometric growth of tumors in various phases (prevascular, vascular, anoxic), and modeling the heterogeneity of tumor populations. The mathematical tools needed include partial differential equations with free boundary conditions, bifurcation in systems of many nonlinear ordinary differential equations, and branching processes with infinite-type space. Development of Resistance Cancer cells are genetically unstable and can acquire genetic and phenotypic changes that permit them to escape cytotoxic insults. Development of drug resistance is common, and it is a major problem in cancer chemotherapy. Development of drug resistance is often a function of the frequency, intensity, and duration of drug exposure, as well as the chronological age of the cells. These biological parameters can be described in mathematical terms. The modeling and optimization of chemotherapy protocols is an area
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Mathematics and 21st Century Biology of potentially great practical importance. Classical models involve populations of normal and cancer cells described as systems of ordinary differential equations with control terms representing treatment intervention. The most common classical approach involves defining a performance index that summarizes the efficiency of the therapy and the damage done to normal (noncancer) cells and using methods of control theory to find the best value of the index. These models had a good deal of appeal in the early days of chemotherapy, when the complexity of tumor cell populations was not entirely appreciated. There also exist models that take into account emerging resistance (like the Coldman-Goldie clonal resistance model) and heterogeneity (like gene amplification), but they are based on unrealistic biological hypotheses. Challenges for the field involve the development of more realistic models of drug action and cell proliferation and heterogeneity as well as new methods for parameter estimation. IN VIVO DYNAMICS OF THE HIV-1 INFECTION Mathematical models of HIV infection and treatment have provided quantitative insights into the main biological processes that underlie HIV pathogenesis and helped establish the treatment of patients with combination therapy.1 This in turn has changed HIV from a fatal disease to a treatable one. The models successfully describe the changes in viral load in patients under therapy and have yielded estimates of how rapidly HIV is produced and cleared in vivo, how long HIV-infected cells survive while producing HIV, and how fast HIV mutates and evolves drug resistance. They have also provided clues to the T-cell depletion that characterizes AIDS. The models also allow the rapid screening of antiviral drug candidates for potency in vivo, hastening the introduction of new antiretroviral therapies. HIV on average takes about 10 years to advance from initial infection to immune dysfunction (or AIDS). During this period the amount of virus measured in a person’s blood hardly changes. Because of this slow progression and the unchanging level of virus, it was initially thought that the infection was slow. It was unclear if treating the disease early, when symptoms were not apparent, was worthwhile. Recognizing that constant levels of virus meant that the rates of viral production and clearance were in balance but not necessarily slow, Alan Perelson and David Ho (Perelson et al., 1996) used experimental drug therapy to perturb the viral steady state. Mathematically modeling the response to this perturbation using a system of ordinary differential equations that kept track of the concentra- 1 The material in this section was generously contributed by Alan Perelson.
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Mathematics and 21st Century Biology tions of infected cells and HIV and fitting the experimental data to the model revealed a plethora of new features about HIV infection. After therapy was initiated, levels of HIV RNA (a surrogate for virus) fell 10- to 100-fold in the first week or two of therapy. This suggested that HIV has a half-life of 1 or 2 days, so to maintain the pretherapy constant level of virus requires enormous virus production—in fact the amount of virus in the body must double every 1 or 2 days. Detailed analysis showed that this viral decay was governed by two processes: the clearance of free virus particles and the loss of productively infected cells. From this rapid clearance of virus one could compute that at steady state, ~1010 virions are produced daily and, given the mutation rate of HIV, that each single and most double mutations of the HIV genome are produced daily. Thus, effective drug therapy would require drug combinations that can sustain at least three mutations before resistance arises, and this engendered the idea of triple combination therapy. Other analyses showed that the slope of viral decay was proportional to the drug combination’s antiviral efficacy, providing a means of comparing therapies. Following the rapid 1-2 week, first-phase loss, the rate of HIV RNA decline slows. Models of this second phase of decline, when fitted to the kinetic data, suggested that a small fraction of infected cells might live for a period of weeks while infected. Following on the success of these joint modeling and experimental efforts, many similar studies were undertaken that revealed a fourth, much longer timescale, between 6 and 44 months, for the decay of latently infected cells. Latently infected cells, which harbor the HIV genome but do not produce virus, can hide from the immune system and reignite infection when the cells become stimulated into proliferation. Clearing latently infected cells is one of the last remaining obstacles to eradicating HIV from the body. The modeling of the HIV virus is but one example of the extensive contributions of the mathematical sciences to immunology and epidemiology. Many exciting opportunities remain. FUTURE DIRECTIONS The examples described above briefly illustrate the broad challenge and opportunities for mathematical modeling and simulation in system physiology. The use of mathematical models to describe processes in system physiology will improve our understanding of the dynamic interplay between those processes and ultimately aid in the translation of basic science findings to clinical application. At the same time, these mathematical investigations will undoubtedly lead to new mathematical problems and to new mathematical and computational methods with application in many other areas of science.
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Representative terms from entire chapter: