The anatomical model, that forms the foundation of this study, results from the MEET Man project (Models for Simulation of Electromagnetic, Elastomechanic and Thermal Behaviour of Man), a project of the Institute of Biomedical Engineering, University of Karlsruhe (Germany). The purpose of this project is the creation of models for simulating the physical behaviour of man. The model originates from computertomographic scans and thin-section photos of the Visible Man Data Set [1], which is provided by the National Library of Medicine, Bethesda, Maryland (USA). The model was created using advanced strategies of digital image processing ([2], [4], [5], [6]).

      The anatomical model (Figure 1) is stored in a three dimensional dataset, which consists of approximately 400 millions cubic voxels. Each voxel has a size of 1 mm x 1 mm x 1 mm and is assigned to one out of forty different tissue classes. The model includes the orientation of skeletal and cardial muscle fibres (Figure 2). Therefore, an additional orientation dataset is used. Two angles were assigned to every voxel that belongs to the tissue classes skeletal or cardiac muscle. Each of these angles has a range from 0 to 180 degrees (Figure 2).
 
      This model was used to derive anatomical models with lower resolution of the human thorax by determining the occurrence of tissue class in every single voxel. The most frequent occurring tissue class was assigned to each voxel. Conductivity models were derived from these anatomical models by assigning an electrical conductivity to each tissue class. These partly anisotropic, frequency dependend electrical conductivities were taken from in vitro measurement ([7]).

      The selection of appropiate conductivity models is a task of great importance. This selection can be supported by analysis and comparison of solutions obtained from numerical calculations based on a set of examplary models. In this way different properties of models, e. g. inhomogeneities and anisotropy of tissue, spatial resolution of models and their influence on solutions have been examined in previous studies ([8], [9]).

Finite Difference Method

      The forward problem in electrophysiology can be solved employing the generalized Poisson's equation for electrical conduction [3]:

      In this study the finite difference method was chosen to solve the forward problem. This method leads to a system of linear equations. The dimension of this system is determined by the number of mesh points. The solution of the linear system was determined using an iterative solver based on the Full Multigrid Algorithm ([10]). The implementation of the finite difference method allows the efficient solving of large problems on shared memory, multiprocessor compute servers.

      An appropriate initial solution for the iterative solving can accelerate the calculations. An acceleration of the calculation of a complete heart cycle was achieved by using the technique. The calculation of a complete heart cycle consists of determining potential distributions at equidistant chosen times. As initial solution serves the potential distribution, which was calculated in the predecessor time step.

Cardiac Sources
      A cardiac source model was applied to determine the current density source   ([11]), which is needed to compute the electric field distribution in the body. The model describes the electrical excitation propagation by means of a finite automaton, which bases on the anatomical data set. The model calculates the potential across the cell membrane and the impressed current per volume unit for each voxel (Figure 3).