Investigation of the source spaces of ECG electrode arrangements

M. Nalbach, O. Skipa and O. Dössel
Institute of Biomedical Engineering, Universität Karlsruhe (TH),
Kaiserstrasse 12, D-76128 Karlsruhe, Germany

Abstract: Non invasive source reconstruction from Electrocardiography (ECG) and Magnetocardiography (MCG) data is a highly discussed research field. In this work we investigate mathematically the source space of the Inverse Problem of Electrocardiography with respect to the information content. Starting from the modeled source space of distributed epicardial potentials we compare several ECG electrode configurations, i.e. a 32-, 64-optimized channel electrode arrangement, that we determined for our individual torso model, the 256-lead ECG configuration which was recorded at the Ragnar Granit Instiute, Technical University of Tampere, Finland and the 128 channel system of the BioMag Laboratory, Helsinki University Central Hospital, Finland.

INTRODUCTION

In the research field of the Inverse Problem of Electrocardiography and Magnetocardiography an important issue is the question which source patterns are detectable and which cannot be seen. In this work an approach is presented to compare the source subspaces of different sensor systems using orthognal projections. Four different ECG-sensor arrangements are compared.

METHODS

In order to solve the Inverse Problem of non invasive source imaging we first have to solve the Forward Problem to compile the lead field matrix A which maps linearly the source distribution x onto the measurement vector b.

Ax = b                                                             (1)

For the compilation of the lead field matrix A 300 forward calculations corresponding to 300 epicardial potential source nodes on the heart are carried out in a volume conductor model of the human body. The torso model was constructed from a 4D-MRI-dataset of a healthy volunteer which was recorded at the Institute of Biomedical Engineering, ETH and University Zurich. For this work the geometric changes of the heart during the cardiac cycle are not taken into account because they have no effect on the characteristics of the lead field matrix and the corresponding source space [1]. So we use a static enddiastolic isotropic torso model that consists of skin, fat, skeletal muscles, lungs, liver, kidneys in the thorax. The heart model contains left and right ventricles, left and right atria and blood (40000 nodes, 280000 tetrahedrons).

The forward calculations are carried out using finite element method. The source space is constructed using 300 equidistant epicardial source nodes placed on the ventricles and atria.

The Inverse Problem is solved using a Singular-Value-Decomposition (SVD) approach followed by a regularization (Regularizationmatrix R). In order to investigate the source subspaces we need to study the SVD of the transfer matrix A.

A = U S V^T                                                 (2a)

U = {u_i}, S = diag{s_i}, V = {v_i}                (2b)

A^*-1 = V R S^–1 U^T                                       (3)

The matrix V of the SVD contains the orthonormale base vectors v_i of the source space of the epicardial potential distributions. The singular values, the diagonal elements of the matrix S, describe the weights between the source and measurement space. In a first step we investigate the slope of the singular values which yields important information about the base vectors that can be reconstructed [1, 2].

In a second step we study the matrix V itself. Two different sensor arrangments are taken and the corresponding lead field matrices A₁, A₂ respective the orthonormal bases V₁ and V₂ are determined. We extract the main base vectors [1, 2] and define the main subspace V^m₁. P^m₁ (P₂) is the unique, orthogonal projection onto V^m₁ (V₂) if ran(P^m₁)=V^m₁ (ran(P₂)=V₂), (P^m₁)²=P^m₁ ((P₂)²=P₂) and P^m₁^T=P^m₁ (P₂^T=P₂) [4]. We project the base vectors of V₂ in the subspace of the main vectors of the source subspace of V^m₁ using P^m₁.

V¹₂ = P^m₁ V2                                           (4)

V¹₂ includes only parts of V₂ which are registered by V^m₁ too, but in the subspace of V^m₁. We have to project V¹₂ onto the original subspace of V₂ using P₂.

V^1,2₂ = P₂V¹₂ = P₂P^m₁V₂                         (5)

The norm of each base vector ||v_i^1,2₂||₂ shows how this vector is represented in the source subspace V^m₁

We study four different electrode configurations. The first with 128 channels is used at the BioMag Laboratory, Helsinki, Finland (Fig. 1a). The second one, a 256-lead ECG, was installed at the Ragnar Granit Institute, Technical University of Tampere, Tampere, Finland (Fig. 1b). The 3^th and 4^th were a 32-channel and 64-channel electrode arrangement (Fig. 1c,1d) which were developed for the used individual thorax dataset at the Institute of Biomedcial Engineering, Universität Karlsruhe (TH), Germany [4].

   (a)                            (b)                       (c)                          (d)

Figure 1. Investigated electrode configurations: (a) 128-lead arrangement BioMag Laboratory, Helsinki, (b) 256-lead arrangement Ragnar Granite Instiute, Tampere, (c) 32-lead system IBT, Karlsruhe, (d) 64-lead system IBT, Karlsruhe

RESULTS

Investigation of the singular values

Figure 2 shows the slope of the singular values of the four different electrode configurations. The slopes of the 128-lead and 256-lead system are similar because the electrodes are distributed in a similar way. The slopes of the optimized arrangements (32 and 64 channels) are not so steep. This corresponds to former investigations [1, 4].

Figure 2. Singular values of the investigated sensor configurations

Investigation of the source subspaces

The norm of the projected base vectors for each configuration in comparison to one of the others was calculated. Significant results are presented in figure 3.

Figure 3. Norm ||v^1,2_i2||₂ of the projected source base vectors of T256

Figure 4. Norm ||v^1,2_i2||₂ of the projected source base vectors in comparison to T256

DISCUSSION

This investigation shows how the source subspace of one arrangement can be represented by another. The subspace of the 256-lead configuration can acquire the largest number of base vectors of any other configuration over a large range of indices (Fig. 3, 4). It is able to represent satisfactory all other configurations for source subspace purposes. Our optimized configuration can detect the source subspaces of the other arrangements but not with this large number of base vectors (Fig. 4). For the first 35 base vectors we achieve the highest accuracy. Base vectors which are not represented in a configuration deliver the source patterns that are not detectable with this sensor configuration.

Acknowledgments: We thank Oliver Weber, Institute of Biomedical Engineering, ETH and University Zurich, for the recording of the MRI dataset. We are indebted to Jukka Nenonen for the recording of the 128-lead BSPM. Furthermore we thank Pasi Kauppinen, Jari Hyttinen and Jaakko Malmivuo for the good collaboration in recording of the 256-lead ECG. This work is supported by Deutsche Forschungsgemeinschaft DFG, Graduate Course GK329.

REFERENCES

[1]    M. Nalbach, J. Nenonen, O. Weber, O. Dössel, “MCG and ECG Source Reconstruction using a 4D-Model of the Human Body,” Biomedizinische Technik, vol. 46-2, pp. 57-59, 2001.

[2]    F.R. Schneider, Das Inverse Problem der Elektro- und Magnetokardiographie, Dissertation, Institute of Biomedical Engineering, Universität Karlsruhe (TH), 1999.

[3]    G.H. Golub, C.F. van Loan, Matrix Computations, London : The John Hopkins University Press, 1996

[4]    M. Nalbach, O. Dössel, “Comparison of MCG and ECG sensor arrangements with respect to the information content“, Physica C, in press, 2002