Biomedical Engineering Reference
In-Depth Information
of parameter space, and one region is found to exhibit behavior quite similar to
that seen in the real system. In many cases, the model behavior is rather sensi-
tive to parameter variations, so if the model parameters can be measured in the
real system the model shows realistic behavior at those values, and one can have
some confidence that the model has captured the essential features of the sys-
tem. Third, there are cases in which model equations are constructed based on
detailed descriptions of known (bio)chemistry or biophysics. Numerical experi-
ments can then generate information about variables inaccessible to physical
experiments.
In many cases, all three approaches are applied in parallel to the same sys-
tem. Consider, for example, the problem of fibrillation in cardiac tissue. At the
cellular level, the physics of the propagation of an electrical signal involves
complicated physicochemical processes. Models involving increasingly realistic
descriptions of the interior of the cell, its membrane, and the intercellular me-
dium are being developed in attempts to include all the features that may give
rise to macroscopic properties implicated in fibrillation. (See, for example, arti-
cles in (5).) At the same time, recognizing the general phenomenon of action
potential propagation as similar to chemical waves in reaction-diffusion systems
allows one to construct plausible, though idealized, mathematical models in
which phenomena quite similar to fibrillation can be observed and understood
(10,11)). These models can then be refined using numerical simulations that
incorporate more complicated features of the tissue physiology. In parallel with
these theoretical efforts, experiments on fibrillation or alternans in real cardiac
tissue yield time series data that must be analyzed on its own (with as little mod-
eling bias as possible) to determine whether the proposed models really do cap-
ture the relevant physics (13).
Almost all mathematical modeling of biomedical processes involves a sig-
nificant computational component. This is less a statement about the complexity
of biomedical systems than a reflection of the mathematical structure of nonlin-
ear systems in general, even simple ones. Indeed, in large measure the rise of
nonlinear dynamics as a discipline can be attributed to the development of the
computer as a theoretical tool. Though one can often prove theorems about gen-
eral features of solutions to a set of nonlinear equations, it is rarely possible to
exhibit those solutions in detail except through numerical computation. More-
over, it is often the case that the numerical simulation has to be done first in or-
der to give some direction to theoretical studies. Though the catalogue of well-
characterized, generic behaviors of deterministic nonlinear systems is large and
continues to grow, there is no
a priori
method for classifying the expected be-
havior of a particular nonlinear dynamical system unless it can be directly
mapped to a previously studied example.
Rather than attempting a review of the state of the art in time-series analy-
sis, numerical methods, and theoretical characterization of nonlinear dynamical
systems, this chapter presents some of the essential concepts using a few exam-
