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<p>Engineered monolayers created by using microabrasion and micropatterning methods have provided a simplified in vitro system to study the effects of anisotropy and fiber direction on electrical propagation. Interpreting the behavior in these culture systems has often been performed using classical computer models with continuous properties. Such models, however, do not account for the effects of random cell shapes, cell orientations and cleft spaces inherent in these monolayers on the resulting wavefront conduction. Additionally when the continuous computer model is built to study impulse propagations, the intracellular conductivities of the model are commonly assigned to match impulse conduction velocity of the model to the experimental measurement. However this method can result in inaccurate intracellular conductivities considering the relationship among the conduction velocity, intracellular conductivities and ion channel properties. In this study, we present novel methods for modeling a monolayer cardiac tissue and for estimating intracellular conductivities from an optical mapping. First, in the proposed method for modeling a monolayer of cardiac tissue, the factors governing cell shape, cell-to-cell coupling and the degree of cleft space are not constant but rather are treated as spatially random with assigned distributions. This approach makes it possible to simulate wavefront propagation in a manner analogous to performing experiments on engineered monolayer tissues. Simulated results are compared to reported experimental data measured from monolayers used to investigate the role of cellular architecture on conduction velocities and anisotropy ratios. We also present an estimate for obtaining the electrical properties from these networks and demonstrate how variations in the discrete cellular architecture affect the macroscopic conductivities. The simulation results agree with the common assumption that under normal ranges of coupling strengths, tissues whose cell shapes and connectivity show relatively uniform distributions can be represented using continuous models with conductivities derived from random discrete cellular architecture using either estimates. The results also reveal that in the presence of abrupt changes in cell orientation, local estimates of tissue properties predict smoother changes in conductivities that may not adequately predict the discrete nature of propagation at the transition sites. Second, a novel approach is proposed to estimate intracellular conductivities from the optical mapping of the monolayer cardiac tissue under subthreshold stimulus. This method uses a simplified membrane model, which represents the membrane as a second order polynomial of the membrane potential. The simplified membrane model and the intracellular conductivities are estimated from the optical mapping of the monolayer tissue under the subthreshold stimulus. We showed that the proposed method provides more accurate intracellular conductivities compared to a method using a constant membrane resistance.</p> / Dissertation

Accurate numerical simulation of reaction-diffusion systems can come with a high cost. A system may be stiff, and solutions may exhibit sharp localized features that require fine grids and small time steps to properly resolve the physical phenomena they represent. The development of efficient methods is crucial to cut down the demands of computational resources. In this thesis we consider the use of adaptive space and time methods driven by a posteriori error estimation. The error estimators for the spatial discretization are built from a variety of sources: the residual of the partial differential equation (PDE) system, gradient recovery operators and interpolation estimates. The interpolation estimates are anisotropic, not relying on classical mesh regularity assumptions. The adapted mesh is therefore allowed to include elements elongated in specified directions, as dictated by the type of solution being approximated. This thesis proposes an element-based adaptation method to be used for a residual estimator. This method avoids the usual conversion of the estimator to a metric, and instead applies the estimator to directly control the local mesh modifications. We derive a new error estimator for the L^2-norm in the same anisotropic setting and adjust the element-based adaptation algorithm to the new estimator. This thesis considers two new adaptive finite element settings for reaction-diffusion problems. The first is the extension to a PDE setting of an estimator for the time discretization with the backward difference formula of order 2 (BDF2), based on an estimator for ordinary differential equation (ODE) problems. Coupled with the residual estimator, we apply a space-time adaptation method. The second is the derivation of anisotropic error estimates for the monodomain model from cardiac electrophysiology. This model couples a nonlinear parabolic PDE with an ODE and this setting presents challenges theoretically as well as numerically. In addition to theoretical considerations, numerical tests are performed throughout to assess the reliability and efficiency of the proposed error estimators and numerical methods.

In mammalian hearts, initiation of the heartbeat occurs in a region of specialised pacemaker cells known as the sinoatrial node (SAN). The SAN is a highly complex spatially distributed structure which displays considerable cellular heterogeneity and is subject to complex electrotonic interactions with the surrounding atrial tissue. In this study, biophysically detailed ionic models of central and peripheral SAN pacemaker cells are described. These models are able to accurately reproduce experimental recordings of the membrane potential from central and peripheral SAN tissue. These models are used to investigate frequency synchronisation of electrically coupled cardiac pacemaker cells. Based on simulation results presented, it is proposed that cellular heterogeneity in the SAN plays an important role in achieving rhythm coordination and possibly contributes to the efficient activation of the surrounding atrial myocardium. This represents an important, previously unexplored, mechanism underlying pacemaker synchronisation and cardiac activation in vivo. A spatial-gradient model of action potential heterogeneity within the SAN is then formulated using a large-scale least squares optimisation technique. This model accurately reproduces the smooth spatial variation in action potential characteristics observed in the SAN. One and two dimensional models of the intact SAN are then formulated and three proposed models of SAN heterogeneity are investigated: 1) the discrete-region model, in which the SAN consists of a compact central region surrounded by a region of transitional pacemaker cells, 2) the gradient model, in which cells of the SAN exhibit a smooth variation in properties from the centre to the periphery of the SAN, and 3) the mosaic model, in which SAN and atrial cells are scattered throughout the SAN region with the proportion of atrial cells increasing towards the periphery. Simulation results suggest that the gradient model achieves frequency entrainment more easily than the other models of SAN heterogeneity. The gradient model also reproduces action potential waveshapes and a site of earliest activation consistent with experimental observations in the intact SAN. It is therefore proposed that the gradient model of SAN heterogeneity represents the most plausible model of SAN organisation.