Kwong T. Ng

A new three-dimensional finite-difference bidomain formulation for inhomogeneous anisotropic cardiac tissues

Bidomain modeling of cardiac tissues provides important information about various complex cardiac activities. The cardiac tissue consists of interconnected cells which form fiber-like structures. The fibers are arranged in different orientations within discrete layers or sheets in the tissue, i.e., the fibers within the tissue are rotated. From a mathematical point of view, this rotation corresponds to a general anisotropy in the tissues conductivity tensors. Since the rotation angle is different at each point, the anisotropic conductivities also vary spatially. Thus, the cardiac tissue should be viewed as an inhomogeneous anisotropic structure. In most of the previous bidomain studies, the fiber rotation has not been considered, i.e., the tissue has been modeled as a homogeneous orthotropic medium. Here, the authors describe a new finite-difference bidomain formulation which accounts for the fiber rotation in the cardiac tissue and hence allows a more realistic modeling of the cardiac tissue. The formulation has been implemented on the data-parallel CM-5 which provides the computational power and the memory required for solving large bidomain problems. Details of the numerical formulation are presented together with its validation by comparing numerical and analytical results. Some computational performance results are also shown. In addition, an application of this new formulation to provide activation patterns within a tissue slab with a realistic fiber rotation is demonstrated.

New finite difference formulations for general inhomogeneous anisotropic bioelectric problems

Due to its low computational complexity, finite difference modeling offers a viable tool for studying bioelectric problems, allowing the field behaviour to be observed easily as different system parameters are varied. Previous finite difference formulations, however, have been limited mainly to systems in which the conductivity is orthotropic, i.e., a strictly diagonal conductivity tensor. This in turn has limited the effectiveness of the finite difference technique in modeling complex anatomies with arbitrarily anisotropic conductivities, e.g., detailed fiber structures of muscles where the fiber can lie in any arbitrary direction. Here, the authors present both two-dimensional and three dimensional finite difference formulations that are valid for structures with an inhomogeneous and nondiagonal conductivity tensor. A data parallel computer, the connection machine CM-5, is used in the finite difference implementation to provide the computational power and memory for solving large problems. The finite difference grid is mapped effectively to the CM-5 by associating a group of nodes with one processor. Details on the new approach and its data parallel implementation are presented together with validation and computational performance results. In addition, an application of the new formulation in providing the potential distribution inside a canine torso during electrical defibrillation is demonstrated.

Anatomically constrained electrical impedance tomography for anisotropic bodies via a two-step approach

Discusses the inclusion of anatomical constraints and anisotropy in static Electrical Impedance Tomography (EIT) using a two-step approach to EIT. In the first step, the boundaries between regions of different conductivities are anatomically constrained using Magnetic Resonance Imaging (MRI) data. In the second step, the conductivity values in different regions are determined. Anisotropic conductivity regions are included to allow better modeling of the muscle regions (e.g., skeletal muscle) which exhibit a greater conductivity in the direction parallel to the muscle fiber. This two-step approach is used to reconstruct the conductivity profile of a canine torso, illustrating its potential application in extracting conductivity values for bioelectric modeling.

Three-dimensional finite-difference bidomain modeling of homogeneous cardiac tissue on a data-parallel computer

A data-parallel computer is used to provide the memory and reduction in computer time for solving large finite-difference bidomain problems. The finite-difference grid is mapped effectively to the processors of the parallel computer, simply by mapping one node to one (virtual) processor. Implemented on the connection machines (CMs) CM-200 and CM-5, the data-parallel finite-difference algorithm has allowed the solution of finite-difference bidomain problems with over 2 million nodes. Details on the algorithm are presented together with computational performance results.

Three-dimensional modeling of electrical defibrillation on a massively parallel computer

A finite-element solver for electrical defibrillation analysis has been developed on a massively parallel computer, Thinking Machines Corporations Connection Machine 2 (CM-2), which allows a high degree of parallelism and the solution of large problems. The solver uses a nodal assembly technique where each node in the finite-element grid is mapped to a virtual processor in the computer. Using this solver, potential and current density distributions during transthoracic defibrillation have been calculated for different anatomic models, including a realistic 3-D finite-element model constructed from a series of cross-sectional magnetic resonance imaging (MRI) images of a mongrel dog. Numerical results obtained with this model are presented together with computational performance data for the algorithm.<<ETX>>

Numerical analysis of electrical defibrillation: The parallel approach

Numerical modeling offers a viable tool for studying electrical defibrillation, allowing the behavior of field quantities to be observed easily as the different system parameters are varied. One numerical technique, namely the finite-element method, has been found particularly effective for modeling complex thoracic anatomies. However, an accurate finite-element model of the thorax often requires a large number of elements and nodes, leading to a large set of equations that cannot be solved effectively with the computational power of conventional computers. This is especially true if many finite-element solutions need to be achieved within a reasonable time period (eg, electrode configuration optimization). In this study, the use of massively parallel computers to provide the memory and reduction in solution time for solving these large finite-element problems is discussed. Both the uniform and unstructured grid approaches are considered. Algorithms that allow efficient mapping of uniform and unstructured grids to data-parallel and message-passing parallel computers are discussed. An automatic iterative procedure for electrode configuration optimization is presented. The procedure is based on the minimization of an objective function using the parallel direct search technique. Computational performance results are presented together with simulation results.

Two-dimensional Chebyshev pseudospectral modelling of cardiac propagation

Bidomain or monodomain modelling has been used widely to study various issues related to action potential propagation in cardiac tissue. In most of these previous studies, the finite difference method is used to solve the partial differential equations associated with the model. Though the finite difference approach has provided useful insight in many cases, adequate discretisation of cardiac tissue with realistic dimensions often requires a large number of nodes, making the numerical solution process difficult or impossible with available computer resources. Here, a Chebyshev pseudospectral method is presented that allows a significant reduction in the number of nodes required for a given solution accuracy. The new method is used to solve the governing nonlinear partial differential equation for the monodomain model representing a two-dimensional homogeneous sheet of cardiac tissue. The unknown transmembrane potential is expanded in terms of Chebyshev polynomial trial functions and the equation is enforced at the Gauss-Lobatto grid points. Spatial derivatives are obtained using the fast Fourier transform and the solution is advanced in time using an explicit technique. Numerical results indicate that the pseudospectral approach allows the number of nodes to be reduced by a factor of sixteen, while still maintaining the same error performance. This makes it possible to perform simulations with the same accuracy using about twelve times less CPU time and memory.

THREE-DIMENSIONAL UNIFORM GRID MODELING OF ELECTRICAL DEFIBRILLATION ON A DATA PARALLEL COMPUTER

Finite element modeling has played an increasingly important role in the study of defibrillation. In order to model well the complex anatomical details, a large number of elements are required in the finite element grid, leading to a large set of equations that often cannot be solved effectively with the computational power of conventional computers. In this paper, we describe the use of a data parallel computer to provide the memory and reduction in solution time for solving these large finite element problems. Using a uniform grid and a nodal assembly technique, the discretized problem domain can be mapped efficiently to the parallel computer, allowing the solution of problems with over two million unknowns. The finite element algorithm for a three-dimensional inhomogeneous anisotropic body is described together with its parallel implementation. Test results for a canine torso model constructed from CT images are also presented.

Three-dimensional bidomain simulation of electrical propagation in cardiac tissue

Though cardiac tissues are inherently three-dimensional in nature, three-dimensional bidomain models have been used only in a limited number of studies due to the constraints on computer processing power. Here, the authors describe the study of three-dimensional propagation effects in cardiac tissues using a data parallel finite difference bidomain model. The data parallel approach allows the use of a massively parallel computer to provide the solutions for large problems. Electrical propagation results are presented for different stimulation schemes, various ionic current models and different conductivities.

Control of high common mode voltage during transthoracic defibrillation

A high common mode voltage (V/sub cm/) relative to earth ground is produced on the myocardium during the delivery of a defibrillator pulse and can generate a differential error signal when potential gradients are recorded with bipolar electrodes and isolation amplifiers. The error signal is proportional to V/sub cm/, and therefore, a reduction in V/sub cm/ improves the accuracy of the potential gradient data. Experiments were conducted on 5 dogs to determine whether V/sub cm/ can be controlled using a bridge circuit. The bridge circuit consisted of a 5 k/spl Omega/ power rheostat in parallel with the transthoracic resistance of the dog. The variable contact of the rheostat was connected to earth ground, and by adjusting the rheostat, V/sub cm/ on the myocardium could be varied. In each dog, 20 A shocks were delivered through stainless steel transthoracic electrodes. Point contact electrodes sutured to the epicardium were used to measure V/sub cm/. It was determined that V/sub cm/ could be reduced to approximately zero at a given electrode on the heart. In addition, for the 5 dogs studied, the maximum measured V/sub cm/ on the heart was only 10% of the transthoracic voltage when the bridge circuit was balanced for an interior point in the heart.