Quantification of Stochastic Behavior in Cardiac Electrophysiological
Models
Mathematical models of intrinsically stochastic systems have
typically been reduced to deterministic forms to avoid the
computational expense and cumbersome implementation commonly
associated with solutions to stochastic systems. The generalized
polynomial chaos theory provides an efficient means of representing
stochastic processes within random systems. In combination with either
the stochastic Galerkin or stochastic collocation numerical
techniques, one wields an extremely powerful tool for efficiently
computing the stochastic solutions of random systems. Stochastic
modeling has proven useful for accomplishing a number of
goals. Firstly, intrinsically stochastic physiological phenomena are
often modeled with deterministic mathematical systems to reduce
complexity. In focusing on mean behaviors only, important aspects of a
system's behavior may go unnoticed. With the gPC representations
coupled with either the stochastic Galerkin or stochastic collocation
method, it is now possible to quantitatively examine ranges of
behavior outputs to distributions of model parameters or
inputs. Secondly, these techniques can be applied to quantitatively
determine the impact uncertainty in directly measured model inputs and
parameters has upon model outputs. A third utility of the method
applies to systems where the model parameters cannot be measured
directly. In this case, parameter values must instead be fit to
recorded response (output) data. Typically, modelers utilize
statistics for the response, yet seek only to fit the mean parameter
values. The excellent computational efficiency of the generalized
polynomial chaos-stochastic collocation gPC-SC methods allows one to
seek to fit not only mean model parameter values, but higher
statistical moments as well, thus quantifying the uncertainty inherent
in such parameters.
In this thesis, we apply both the gPC-SG and gPC-SC techniques to
cardiac electrophysiological models for: (1) characterization of
simulated behavior response, (2) uncertainty quantification and
sensitivity analysis, and (3) determination of higher stochastic
moments in fitted model parameters for cardiac electrophysiological
systems of interest within the bioengineering community. For breadth,
we focus on both microscopic level Markovian ion channel models as
well as a high level physiological system- that of the forward problem
of electrocardiography. During the process, we explore and expand the
mathematical and computational aspects of the stochastic Galerkin and
collocation methods.