Mathematical Modeling of Physiological Systems
Although mathematical modeling has a long and very rich tradition in physiology, the recent explosion of biological, biomedical, and clinical data from the cellular level all the way to the organismic level promises to require a renewed emphasis on comput
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Mathematical Modeling of Physiological Systems Thomas Heldt, George C. Verghese, and Roger G. Mark
Abstract Although mathematical modeling has a long and very rich tradition in physiology, the recent explosion of biological, biomedical, and clinical data from the cellular level all the way to the organismic level promises to require a renewed emphasis on computational physiology, to enable integration and analysis of vast amounts of life-science data. In this introductory chapter, we touch upon four modeling-related themes that are central to a computational approach to physiology, namely simulation, exploration of hypotheses, parameter estimation, and modelorder reduction. In illustrating these themes, we will make reference to the work of others contained in this volume, but will also give examples from our own work on cardiovascular modeling at the systems-physiology level.
2.1 Introduction Mathematical modeling has a long and very rich history in physiology. Otto Frank’s mathematical analysis of the arterial pulse, for example, dates back to the late nineteenth century [12]. Similar mathematical approaches to understanding the mechanical properties of the circulation have continued over the ensuing decades, as recently reviewed by Bunberg and colleagues [5]. By the middle of the last century,
T. Heldt () G.C. Verghese Computational Physiology and Clinical Inference Group, Research Laboratory of Electronics, Massachusetts Institute of Technology, 10-140L, 77 Massachusetts Avenue, Cambridge, MA 02139, USA e-mail: [email protected]; [email protected] R.G. Mark Laboratory for Computational Physiology, Harvard-MIT Division of Health Sciences and Technology, Massachusetts Institute of Technology, E25-505, 77 Massachusetts Avenue, Cambridge, MA 02139, USA e-mail: [email protected] J.J. Batzel et al. (eds.), Mathematical Modeling and Validation in Physiology, Lecture Notes in Mathematics 2064, DOI 10.1007/978-3-642-32882-4 2, © Springer-Verlag Berlin Heidelberg 2013
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Hodgkin and Huxley had published their seminal work on neuronal action-potential initiation and propagation [25], from which models of cardiac electrophysiology readily emerged and proliferated [33]. To harness the emergent power of first analog and later digital computers, mathematical modeling in physiology soon shifted from analytical approaches to computational implementations of governing equations and their simulation. This development allowed for an increase in the scale of the problems addressed and analyzed. In the late 1960s, Arthur Guyton and his associates, for example, developed an elaborate model of fluid-electrolyte balance that still impresses today for the breadth of the physiology it represents [16]. Since the days of Guyton’s initial work, the widespread availability of relatively low-cost, high-performance computer power and storage capacity has enabled physiological modeling to move from dedicated—and oftentimes single-purpose— computers to the researcher’s desktop, as even small-scale computer clusters can be
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