Origination of Foucault Cardiogram by Impedance Redistribution: Transfer Coefficient Approach

Konstantin Skaburskas, Jüri Vedru

Institute of Experimental Physics and Technology, University of Tartu, Tartu, Estonia

Correspondence: K Skaburskas, Institute of Experimental Physics and Technology, University of Tartu, Tähe 4-101, Tartu, 51010, Estonia.
E-mail: konstan@ut.ee, phone +372 7 375530, fax +372 7 375858

1.  Introduction

Tracking heart mechanical activity by the measurement of the variation of high frequency electromagnetic energy absorption in the heart region, caused by the induction of Foucault eddy currents in it, is denominated as Foucault cardiography (FCG). We consider here FCG as the measurement of power losses taking place in single- or multiple-inductor arrangements placed onto the thorax in the vicinity of the heart. Two mechanisms of FCG signal (Foucault cardiogram) origination can be distinguished. The measured signal originates from: 1) the variation of the specific impedance of tissues and organs; 2) the spatial redistributions of impedance due to contraction and dilatation of the heart. In this work the second mechanism only is considered. At FCG signal interpretation it is necessary to know, which parts of the contracting and dilatating heart cause the signal. To measure their contributions, one should have quantities indicating the sensitivity of the signal to the variation of the situation in the regions under consideration.

The FCG signal simulation has been reduced to the solving of forward problem of FCG for the time-determined series of static configurations [Vedru et al., 2001]. The forward problem of FCG is a task to calculate the power losses due to eddy currents, induced in a volume conductor having complex, human-like geometry and conductivity distribution. This problem is a sort of the known eddy current problem, with appended calculation of the power absorbed by the conductor.

On the basis of a set of ultra-fast CT scans, a model of the heart shape variation has been developed [Malchenko et al., 2003]. The inner structure of the heart has been omitted, replacing it by a homogeneous medium. This model was used as well to determine the region occupied by the heart in the calculation domain as to find its variations for 18 time moments during a single cardiac cycle.

As a result of simulation, an FCG signal waveform and a corresponding set of power absorption distributions were obtained. The waveform expresses the variation of the integral power absorption in the model during the cardiac cycle. On the basis of the distributions of absorbed power, study of the transformation of impedance redistribution into FCG signal was performed.

The FCG signal comprises the time-variable component of the power losses. Therefore, the magnitude comparison of the analyzed quantities, i.e., local mechanical movements and local power losses, was performed for their variable components only. Thus, we excluded the constant components of the analyzed quantities by the subtraction of their values at the end-diastolic (ED) state. This was done for all the spatially distributed values of the analyzed quantities at each of the 18 time moments.

Admitting that the system under consideration is non-linear, but noticing the fact that the variations in it are small, we have tried the approach of linear systems analysis. We denote the local movements as distributed Inputs of the system and the corresponding variable component of local power absorption as distributed Outputs. The transfer coefficient (TC), showing the efficiency of transformation of the movements (local heart volume increments) into the Outputs (local absorption increments) was introduced:

Ti(x) = ∆Pi(x) / ∆Li (x),           i = 1,…,n.

(1)

Here T_i(x) - transfer coefficient; ∆P_i(x) = P_i(x) - P₁(x) - the local power absorption increments = Outputs; ∆L_i(x) = L_i(x) - L₁(x) - local heart volume increments = Inputs; i = 1,…,n; n – number of time moments, i = 1 corresponds to the ED state. Vector x denotes the spatial coordinates.

For each time moment, spatial reduction of the Inputs and Outputs onto the heart ED surface was done. The significant magnitudes of the variable components of the Inputs and Outputs were localized in the layer of 1.5 cm thickness on either side of the ED surface. Thus, the cubic region with edge size 3 cm was chosen for reduction volume. Reduction was performed onto the shell of the voxels representing the ED surface, by a passage of the reduction cell through all the shell voxels. Reduction comprised the summation of all the magnitudes enclosed in the reduction cell, while its centre was po­sitioned into the current voxel of the shell. Thus, the TC was determined as distribution on the ED surface.

To study the introduced approach, a test simulation was performed with the forward problem solved on the rectangular finite difference grid having the step size of 1/3 cm. Test consisted in stimulating of all Inputs by Test Input and examining Output and TC. In our case the Test Input was a dilatation of the heart model by a layer of one voxel thickness. This gave almost uniform stimulation to the Inputs. The resulting map of TC − transfer map − showed nonzero values (which were positive) only on its side turned to the inductor. The transfer map did not change noticeably if Input of two voxels thickness was applied instead. Thus, we may conclude that for such kind of Inputs (if the shape of the heart does not change) the system behaves almost linearly.

Using the heart model, the real FCG signal was simulated. The Outputs were reduced. After the reduction of Inputs, regions without movements appeared for every time point. It led to the impossibility of TC calculation for such regions. It was found, that certain delocalization had led to non-zero values of Outputs even for the regions with zero Inputs. This was just a symptom of the presence of cross-connections between adjacent Inputs and Outputs. Another issue was the appearance of the places with high values of TC, localized on the ED surface near the boundaries between the regions with zero and nonzero Inputs. Division of delocalized Outputs to small local Inputs had caused this effect. Although this is a natural property of the system, in some cases (e.g., at visualization) it merely hinders. Such peaks should be discriminated by applying appropriate thresholds. It is important, that there were also regions where the TC was negative.

2.  Results and Discussion

As a part of ongoing study of the mechanisms of the FCG signal origination, the technique for the efficiency estimation of the transformation of the impedance redistribution into the FCG signal has been introduced. Quantitative results have been obtained, but the main result of the work is testing of the proposed approach to estimation of sensitivities for such kind of impedance measurements.

We have outlined an approach to dimensional reduction of the analysis of FCG origination from 3D to 2D. After the reduction, the particularity of the considered system lies in Inputs and Output having the forms of maps - the distributions over 2D surface. A transfer map shall represent the TC also. Non-linearity of the system would appear in the dependence of the transfer map on the shape of the Input. Generally, non-linearity demands introduction of some constraints to limit the volume of the problem. A pragmatic constraint would be limiting the study to only those transfer maps that can be got by real Inputs. Of the entire transfer map, only the regions with Output greater than some threshold shall be studied. Further, using the common methods, it must be tried to reduce the analysis procedure to calculation of a small number of numerical characteristics of the TC maps.

Acknowledgements

The authors thank the Estonian Science Foundation for the grants No. 3914, 5530.

References

Malchenko S, Vedru J. Model of heart shape cyclic variation for Foucault cardiography simulations. In this volume.

Vedru J, Skaburskas K, Malchenko S. Solving simply the forward problem for Foucault cardiography. In proc. of XI Int. Conf. Electrical Bio-impedance, Oslo, 2001, 565-568.