Over the last two decades, computer modeling and simulation have become increasingly important to the fields of Bioengineering and Medicine. The reasons for this growing importance are manyfold. First, mathematical modeling has been shown to be a substantial tool for the investigation of complex biophysical phenomena. Secondly, since the level of complexity one can model parallels existing hardware configurations, primarily tuned to memory, disk capacity, and CPU speed advances in computer technology have made it feasible to apply the computational modeling paradigm to complex living systems.
For these reasons, biological complexity still outstrips the capabilities of even the largest computational systems, and will for some time to come, the computational methodology has taken hold in biology and medicine and has been used successfully to suggest physiologically and clinically important scenarios and results.
No matter the size of the system to be modeled, the goal remains the same: to understand which of the system's characteristics and interactions are essential in order to quantify and represent its behavior. One always seeks biologically useful solutions and interpretations of the mathematical and computational results. Such results may help describe known behavior as well as predict unknown responses and suggest new representations. While simple to state, the aforementioned goals are difficult to realize. Once the task of defining the problem is accomplished, the modeler must be able to bring the proper tools to bear on solving the problem. While the requisite toolset will depend upon the application, it is sure to include techniques from a wide variety of disciplines, including computer science, mathematics, physics, and engineering.
In this case study, we will describe the techniques of modeling and simulation which can be applied to a class of bioelectric field problems. Bioelectric field problems can be found in a wide variety of biomedical applications which range from single cells [1], to organs [2], up to models which incorporate partial to full human structures [6][5][4][3]. We will describe some general modeling techniques which will be applicable, in part, to all the aforementioned applications. We will focus our study on a class of direct and inverse volume conductor problems which arise in electrocardiography and electroencephalography. The solutions to these problems have applications to defibrillation studies, detection and location of arrhythmias, impedence imaging techniques, and localization and analysis of spontaneous brain activity in epileptic patients; furthermore, they can, in general, be used to estimate the electrical activity inside a volume conductor, either from potential measurements at an outer surface, or directly from the interior bioelectric sources.
Specifically, we first describe the physics and physiology of a bioelectric volume conductor, continue by describing the model construction process, emphasizing mesh generation, and follow with mathematical formulations, numerical solutions, and computational considerations of a set of direct and inverse problems. Throughout the chapter, we provide opportunities for you, the reader, to investigate various techniques in computational science yourself. We conclude with a section in which we interpret the results in terms of an error analysis and briefly introduce adaptive techniques to reduce the error due to discretization.