The knowlege of the state of the heart is usually obtained by evaluating measured data of the potential distribution on the body surface (body surface potential maps, BSPM) which arises from the electrical sources in the heart. The calculation of such sources may be done by solving the inverse problem of electrocardiology. In contrast to the inverse problem the forward problem of electrocardiology can be solved by calculating the body surface potential maps from a given electrical source distribution.

Both applications need a realistic source distribution either as boundary condition (forward problem) or for validation purposes (inverse problem). To obtain a realistic source distribution several approches are possible.

In this study a simulation of the electrical excitation propagation based on an anatomical model of the heart of the Visible Man is performed by means of a cellular automaton. Thus, it is possible to determine the electrical source distribution numerically from the results of the cellular automaton. In addition, an example of a calculation of the body surface potential map is done as well.

Anatomical Model
As an initial step to simulate the electrophysiological processes in the heart, a highly detailed model of the anatomy of the heart was created as part of the MEET Man project [5], [6], [7]. Therefore, different strategies of 3D image processing were applied to the cryosections of the Visible Man dataset provided by the National Library of Medicine, Bethesda, Maryland (USA) [1] to obtain a voxel representation of the heart. The strategies involved different filter operations, such as median, average and morphological filtering, manual use of 3D spline editors to separate anatomical structures of similar color, and 3D region growing [3]. Appling these strategies, the tissue classes left ventricle (epicardial), left ventricle (endocardial), right ventricle (epicardial), right ventricle (endocardial), left atrium, right atrium, arterious blood, venous blood, fat, and background were derived.

In addition to these classes, it was necessary to integrate the tissue classes of the cardiac conduction system manually into the anatomical model. These classes comprise sino-atrial node, atrial-ventricular node, bundle of His, right bundle branch, left bundle branch, and purkinje fibres. Figure 1 shows the classified heart in a 3D view.

The anatomical model contains about 30 million of voxels of the size of 1/3 mm x 1/3 mm x 1 mm. For performance reasons the model resolution for the simulation was reduced to a dataset containing about 3 million cubic voxels of 1 mm edge length.
To obtain simulation results which are close to reality the anisotropic properties of cardiac muscle with regard to conductivity and excitation velocity could not be neglected. Therefore the knowledge of the orientation of the muscle fibres was determined for each voxel by texture analysis.
At first, the orientation was calculated automatically for a set of anchor points within cardiac muscle applying the hotelling transform [3] to the cryosections to determine the principle axes of the fibres. Using an interactive 3D editor, these calculated orientations could be validated and manually corrected. Finally, the orientations of the anchor points were used as boundary condition for an iterative interpolation within cardiac muscle. The interpolation starts with an initialization of the entire cardiac muscle region setting the orientation for each voxel to the value of the closest anchor point. Afterwards, the interpolation continues by averaging orientations in the 6-neighbourhood. The interpolation finishes if the degree of change between two interpolation steps becomes negligible [8] (Figure 2).
Physiological Model
To simulate the electrical excitation propagation the anatomical model including the fibre orientation dataset was expanded to a physiological model by adding physiological parameters of the excitation. These parameters comprise the course of action potential, propagation velocity along and across the muscle fibre, refractory periods, autorhythmicity and possibilty of propagation depending on the tissue class.
In addition to the tissue class and fibre orientation, which are provided by the anatomical model, the action potential and refractory state depends on the location of each voxel. The excitation is simulated applying a cellular automaton [10], [11] to the physiological model. This automaton calculates the potential across the cell membrane for any time over a total heart cycle. In this simulation one cell is represented by one voxel. Thus, for this simulation the number of cardiac cells is about 350000.
It simulates the excitation propagation by passing action impulses from one cell to its neighbours if allowed by the cells' states. The states depend on the time passed since the last time the cell was excitated as well as on the refractory period related to the action potential course for the cell. If a cell is currently activated by one of its neighbours it will be called active cell further on. If there are a lot of active cells located next to each other, these cells represent a depolarizing wavefront. If these active cells pass impulses to their neighbours, the depolarizing wavefront propagates through the entire cardiac tissue. Usually such a propagation starts at the sino-atrial node, continues by passing the atria, atrio-ventricular node, left and right bundle branches, and the purkinje fibres. Finally it spreads throughout the left and right ventricles.
The repolarization of the cardiac tissue is determined by the action potential course, which is given in this study for the different tissue classes. The course of action potential also depends on the refractory state of the cell which itself depends on the time of last excitation. In addition to this the action potential course for the ventricular tissue classes depends also on the location of the cell. The course for epicardial and endocardial cells is scaled in the time domain, so that the repolarization wavefront propagates in reverse order with regard to the depolarization wavefront.

Cardiac Sources
To calculate the electrical sources in the human heart, a bidomain model [2] was applied to the transmembrane potential dataset obtained by the simulation. Thus, it is possible to calculate an impressed current density for every voxel of the entire cardiac muscle. This current density is calulated by the product of the negative gradient of the transmembrane potential with the intra cellular conductivity tensor. The ratio between conductivity along and across a single muscle fibre is averaged from values taken from literature [4]. In this study the ratio was set to 6:1.

This electrical source distribution can be taken as boundary condition when calculating body surface potential maps [9]. Therefore the following equation

(1)

Results and Conclusions
This work shows one way to simulate the electrical excitation propagation and to calculate the electrical source distribution within the heart of the Visible Man dataset. Figure 3 and Figure 4 show the transmembrane potential distribution resulting from the simulation.
The following ECG  (Figure 5 Einthoven I) results from a calculation of the body surface potentials solving equation (1) by means of a finite difference solver [9].
The presented methods may be applied to different datasets for educational and diagnostic purposes. Future work will focus on pathologies and the impact of motion of the heart on the excitation propagation as well as on the body surface potential maps.