transmembrane potential reconstruction
in anisotropic heart model

O. Skipa, M. Nalbach, F. Sachse, C. Werner, O. Dössel
Institute of Biomedical Engineering, Universität Karlsruhe (TH)
Kaiserstrasse 12, 76128 Karlsruhe, Germany

Abstract: A new approach to the reconstruction of transmembrane potentials (TMP) in anisotropic finite element heart model is presented. The solution is sought in the form of 3D patches constructed by the interpolation of TMP distributions. The method is evaluated using TMP distributions generated with a cellular automaton.

INTRODUCTION

The distribution of transmembrane potential (TMP) describes directly the state of electrical excitation in the heart. Thus, the reconstruction of transmembrane potentials is an important issue in noninvasive cardiac source imaging. Existing approaches either apply constraints on the activation function and concentrate on isochrone computation [1] or use a boundary element model to reconstruct TMP on epicardium and endocardium [2].

This work presents a volume-based approach for TMP reconstruction. We employ a detailed anisotropic finite element model to construct a transfer matrix describing the relation between the TMP distribution in the heart volume and the potentials on the torso surface. The method is tested by computing the inverse solutions for source distributions simulated with a cellular automaton [3]. It is of special interest to reconstruct the TMP distributions in the repolarization phase when the isochrone approach does not work .

METHODS

Forward problem

The relation between the body surface potentials and TMP in the heart is derived from bidomain model equations [4]

(1)

where and are the intracellular and extracellular conductivity tensors; and are the extracellular and transmembrane potentials.

Eq. (1) is solved using the finite element method (FEM). Our finite element model comprised 50000 nodes corresponding to 327000 tetrahedrons. About 17000 nodes were placed regularly in the heart to allow for the simulation of electrocardiograms (ECG) and for the construction of the lead-field matrix. The fiber orientation in the myocardium as well as in the skeletal muscles was also available [5].

The simulation of “realistic” TMP distributions during the complete cardiac cycle was performed with a cellular automaton. It is based on a highly-detailed voxel heart model having the resolution of 1´1´1 mm.

The inverse solution was sought in the form of 3D TMP patches. Every patch corresponds to one node of a simplified coarse finite element heart model (955 nodes). The sequence of patches was generated by setting the TMP at the chosen node of the coarse grid to 1 and the rest to 0 and interpolating the TMP distribution onto the nodes of the fine finite element mesh using first-order interpolation functions for tetrahedrons. The correspondent ECG signal is then copied to the corresponding column of the lead-field matrix.

Inverse problem

Zero- and second-order Tikhonov regularization were used to solve the inverse problem [6]. Tikhonov zero-order regularization is based on the singular value decomposition of the lead-field matrix A and the inverse solution is written as

(2)

where are the singular values, and are left and right singular vectors of A, and is the regularization parameter. The optimal regularization parameter was found by the method of L-curve [6].

For second-order Tikhonov regularization, the iterative “preconditioned” conjugate gradient algorithm from [7] was applied. The L-curve was used to find the optimal number of iterations.

RESULTS

The TMP distributions were modeled with a cellular automaton and forward computations were performed to get the ECG.

Fig. 1 Transmembrane potential distribution simulated with a cellular automaton at one instant of time. Four horizontal slices through the heart volume are shown as seen from the top of the torso.

An example of a computed TMP corresponding approximately to the middle of the T-wave is presented in Fig. 1. Four slices through the heart volume are shown as seen from the top of the torso.

Fig. 2 demonstrates the results of Tikhonov zero-order reconstruction of the TMP from the simulated ECG signal with 0.5% Gaussian noise added. The range of transmembrane potentials corresponds approximately to that of Fig. 1 although the values are somewhat shifted upwards. It should be noticed that according to Eq. 1 two TMP distributions which differ only by a constant value are considered to be equivalent.

Fig. 2 Transmembrane potentials reconstructed from the simulated measurement data with 0.5% noise using Tikhonov zero-order regularization. The same slices and the time instant as in Fig. 1 are shown.

An example of a TMP distribution reconstructed with Tikhonov second-order regularization is shown in Fig. 3. For both approaches it is seen that only low spatial frequency components of source distributions could be reconstructed.

Fig. 3 Transmembrane potentials reconstructed from the simulated measurement data with 0.5% noise by Tikhonov second-order regularization. The same slices and the time instant as in Fig. 1 are shown.

DISCUSSION

The work demonstrates a new volume-oriented approach for inverse TMP computations. The method allows for taking into account the anisotropy of the heart muscle which was not possible in the inverse procedures developed so far. Tikhonov second-order regularization delivered generally smoother and less noisy solutions as the zero-order regularization. The simulated TMP distributions could be reconstructed better during the repolarization phase. This can be explained by the relatively low content of high spatial frequencies in the source distributions during this stage.

REFERENCES

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