Originally published as Biophys J. BioFAST on February 18, 2005.
doi:10.1529/biophysj.104.058537
Biophysical Journal 88:3731-3733 (2005)
© 2005 The Biophysical Society
Interpreting Biomagnetic Fields of Planar Wave Fronts in Cardiac Muscle
Rodrigo Weber dos Santos and
Hans Koch
Division of Medical Physics and Metrological, Information Technology, Physikalisch-Technische, Bundesanstalt, Berlin, Germany
Correspondence: Address reprint requests to Rodrigo Weber dos Santos, Division of Medical Physics and Metrological Information Technology, Physikalisch-Technische Bundesanstalt, D-10587 Berlin, Germany. Tel.: 49-30-3481511; Fax: 49-30-3481361; E-mail: rwdsantos{at}yahoo.com.
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ABSTRACT
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The recent results of Holzer and co-workers reveal the existence of net currents that flow along the front of a planar wave propagating through cardiac tissue. This is an important contribution toward the better understanding of the physics of biomagnetic fields. However, although the authors claim their results reveal particular bidomain properties, we show in this short letter that the results allow multiple interpretations. For instance, cardiac anisotropy by itself may also explain the existence of a net current along the wave front. Based on our calculations, we suggest additional experiments that would allow distinguishing between these two explanations and thus provide further evidence on the basic physics behind cardiac biomagnetism.
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INTRODUCTION
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The magnetic field generated during the electric propagation on cardiac tissue is still a subject of research. The recent article of Holzer et al. (2004)
is an important contribution toward the understanding of the basic physics behind such biomagnetic fields. Using a unique combination of transmembrane potential and Bz magnetic component maps, the authors' experiment reveals the existence of net currents that flow along the front of a propagating planar wave (Jy, as illustrated in Fig. 1). These findings are in contrast to the traditional assumption of Frank (1953), where a layer of current dipoles travels parallel to the propagation direction (Jx, as in Fig. 1). The new results of Holzer et al. (2004) show that a more sophisticated mechanism is taking place.
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FIGURE 1 A planar wave propagating toward the x direction on an infinite plane. Cardiac fibers have an inclination  . Current density (J) flows mainly along the cardiac fibers. Only the projection of J into y (Jy) generates the z component of the magnetic field, Bz. The Jx contribution to Bz cancels out due to the symmetry of the plane wave (see text for details).
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A possible explanation for the experimental findings is based on the theoretical work of Roth and Woods (1999) and on the bidomain model (Henriquez, 1993): the magnetic field would be generated by a net current that accounts for the sum of intra- and extracellular currents of cardiac tissue. Due to the unequal anisotropies in the intra- and extracellular domains, the bidomain net current has a very distinct property—it flows along the front of a propagating planar wave, i.e., there is no net current orthogonal to the wave front (Jx = 0; see Fig. 1). This is a very distinct source mechanism from the one accepted to explain cardiac electric fields.
Although Holzer and co-workers claim their recent results reveal such distinct bidomain properties, we show next in this short letter that the results allow multiple interpretations by the fact that Jx does not contribute to Bz. For instance, a current density J that flows mainly along the cardiac fibers (see Fig. 1) generates the same Bz magnetic component as the one generated by the Jy current component. Therefore, the experimental results presented in (Holzer et al., 2004) do not support for bidomain properties.
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THE INFORMATION CONTENT OF BZ IS INSUFFICIENT
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The experimental work of Corbin and Scher (1977) has shown the importance of the cardiac tissue anisotropy in the generation of electrocardiograms. We next calculate the magnetic field vector generated by a propagating planar wave, assuming that the net current (J) flows preferentially along the cardiac fibers (Jx, Jy 
0 for 0°, 90°), as a result of the cardiac tissue bulk anisotropy. Our calculations show that the Bz magnetic component depends only on Jy. Therefore, the cardiac tissue anisotropy is also a possible explanation for the experimental results of Holzer et al. (2004).
We consider a planar wave of transmembrane potentials (
Vm/y = 0) propagating toward the x direction on an infinite plane of small width h at z = 0. Cardiac fibers have an inclination . The anisotropy of the tissue can be modeled by a conductivity tensor that in some arbitrary direction is given by
where the matrix R reflects the change of coordinate from the cardiac fiber direction to the direction of propagation, and 
l and t are the conductivity values along and transversal to the fibers, respectively. Since all cardiac fibers have an inclination of , we have
The net current can be expressed as J = 
Vm, and thus
 | (1) |
From Eq. 1, we see that due to the tissue anisotropy, even with Vm/y = 0, we have Jy 0.
The magnetic field can be calculated with the Biot-Savart equation
where r = (x,y,0) and r' = (x',y',z').
Under the above assumptions, since the current J varies only with x, we can rewrite the Biot-Savart equation and find
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 | (3) |
where we have used that
and
Equation 3 shows that Jx does not contribute to Bz due to the symmetry of the planar wave propagation, i.e., Jx does not vary with y (Jx/y = 0). Bz reflects only the y component of the current distribution. Therefore, the experimental results of Holzer et al. (2004), by revealing the existence of currents along the wave front, i.e., Jy 0, do not distinguish between the distinct bidomain properties (which predicts Jx = 0 and Jy 0) and the influence of the tissue anisotropy (Jx 0 and Jy 0).
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PROPOSAL OF ADDITIONAL EXPERIMENTS
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One way to distinguish between the two mechanisms, bidomain and tissue anisotropy, is via the -dependence of the maximum amplitude of Bz. Bidomain theory predicts the following dependence on for Jy (Roth and Woods, 1999):
The dependence on for Jy due to the tissue anisotropy is given by Eq. 1. Thus, by initiating planar waves with different inclinations with respect to the cardiac fiber, one could try to verify which model best fits the experimental data. Unfortunately, the large error for estimating the fiber angle (12°) and the Bz amplitude variation in the order of 1.0 nT, as reported in Holzer et al. (2004), could still disturb the experimental data interpretation.
Equation 2 shows another and perhaps easier way how the two described mechanisms could be distinguished. The horizontal component By of the magnetic field depends on Jx, and should be thus always equal to zero, as predicted by the bidomain theory. In contrast, if only the anisotropy of the tissue is playing the role, Jx should always be different than zero, as described by Eq. 1. A SQUID microscope design with a vertical pickup coil (Matthews et al., 2003) would be able to measure the horizontal By and Bx components.
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CONCLUSION
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So far, the relation between bidomain theory and cardiac biomagnetism was mainly investigated by theoretical and modeling works and still lack sound experimental evidence. The confirmation of this theory would support a novel tool based on magnetic sensors, extremely promising for basic investigation on cardiac electrophysiology. The combination of electric and magnetic sensors during in vitro experiments would provide one with a better understanding of cardiac intracellular currents, intracellular conductivity, and gap junction effects during normal and abnormal cardiac electric propagation. We believe the additional experiments described here would be a beneficial supplement to the valuable approach of Holzer et al. (2004) and provide further evidence for the establishment and proof of a theory for cardiac biomagnetism.
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ACKNOWLEDGEMENTS
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We acknowledge the support provided by 13N8125 BMFT German Ministry of Research and Technology.
Submitted on December 22, 2004;
accepted for publication February 7, 2005.
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REFERENCES
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Corbin 2nd, L. V., and A. M. Scher. 1977. The canine heart as an electrocardiographic generator. Dependence on cardiac cell orientation. Circ. Res. 41:58–67.[Abstract/Free Full Text]
Frank, E. 1953. A comparative analysis of the eccentric double-layer representation of the human heart. Am. Heart J. 46:364–378.[CrossRef][Medline]
Henriquez, C. S. 1993. Simulating the electrical behavior of cardiac tissue using the bidomain model. Crit. Rev. Biomed. Eng. 21:1–77.[Medline]
Holzer, J. R., L. E. Fong, V. Y. Sidorov, J. P. Wikswo, and F. Baudenbacher. 2004. High resolution magnetic images of planar wave fronts reveal bidomain properties of cardiac tissue. Biophys. J. 87:4326–4332.[Abstract/Free Full Text]
Matthews, J., S. Y. Lee, F. C. Wellstood, A. F. Gilbertson, G. E. Moore, and S. Chatraphorn. 2003. Multi-channel high Tc scanning SQUID microscope. IEEE Trans. Appl. Supercond. 13:219–222.[CrossRef]
Roth, B. J., and M. C. Woods. 1999. The magnetic field associated with a plane wave front propagating through cardiac tissue. IEEE Trans. Biomed. Eng. 46:1288–1292.[CrossRef][Medline]
Copyright © 2005 by the Biophysical Society.
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