Cardiac anisotropy in boundary-element models for the electrocardiogram

Mark Potse 0 1 2 3 Bruno Dube 0 1 2 3 Alain Vinet 0 1 2 3 0 M. Potse (&) B. Dube A. Vinet Research Center, Sacre-Coeur Hospital , 5400 Boulevard Gouin Ouest, Montreal , QC H4J 1C5, Canada 1 Computational resources for this work were provided by the Reseau quebecois de calcul de haute performance (RQCHP). M. Potse was supported by the Research Center of Sacre-Coeur Hospital , Montreal, QC, Canada 2 M. Potse Laboratory for Experimental Cardiology, Heart Failure Research Center, Academic Medical Center , Amsterdam, The Netherlands 3 M. Potse Interuniversity Cardiology Institute of The Netherlands , Utrecht, The Netherlands The boundary-element method (BEM) is widely used for electrocardiogram (ECG) simulation. Its major disadvantage is its perceived inability to deal with the anisotropic electric conductivity of the myocardial interstitium, which led researchers to represent only intracellular anisotropy or neglect anisotropy altogether. We computed ECGs with a BEM model based on dipole sources that accounted for a ''compound'' anisotropy ratio. The ECGs were compared with those computed by a finitedifference model, in which intracellular and interstitial anisotropy could be represented without compromise. For a given set of conductivities, we always found a compound anisotropy value that led to acceptable differences between BEM and finite-difference results. In contrast, a fully isotropic model produced unacceptably large differences. A model that accounted only for intracellular anisotropy showed intermediate performance. We conclude that using a compound anisotropy ratio allows BEM-based ECG models to more accurately represent both anisotropies. 1 Introduction The electrocardiogram (ECG) is arguably the most important diagnostic tool in cardiology. Although it has been around for more than a century, many aspects of the ECG are still poorly understood. Computer models of the ECG play an important role in filling these knowledge gaps. Whole-heart reaction-diffusion models, which can simulate the ECG directly from processes on the membrane level, have only just begun to appear [21, 24, 25, 51]. These models, combined with patient-specific anatomic models, can predict subtle electrocardiographic effects of ion-channel malfunctions, provided that the ECG simulation is accurate enough. The boundary-element method (BEM) has been used for ECG simulation for more than four decades [1, 3, 9, 16, 19, 21, 28, 32, 47, 55]. Its attractiveness comes from the small number of surface elements necessary to describe the torso and its major inhomogeneities. The torso, skeletal muscle layer, lungs, and ventricular blood masses can be modeled with a few thousand triangles [16]. Originally the small footprint of the BEM model made it the only candidate for ECG simulation [16, 19, 32]. The continuing popularity of the method is mainly due to its speed, which makes it useful for low-end computers and interactive applications [37]. The BEM is used to model the conductivity of the torso components. It is combined with a source model, which represents the cardiac electrical activity. The source model can be a small number of dipole sources inside the myocardium [16, 19, 32, 52], which can be computed from membrane potentials simulated by a reaction-diffusion model [51] or by simpler models [28]. Other source models are the uniform- or oblique dipole layer on the activation front [6, 7, 42] and the equivalent double layer on the surface of the myocardium [11, 35, 36]. We will discuss only dipole sources. The major disadvantage of the BEM model for ECG simulation is its inability to represent the anisotropy of the extracellular space in the cardiac muscle. Both intracellular and extracellular anisotropy affect the ECG. Intracellular anisotropy can be treated straightforwardly, as has been done in several studies [20, 53]. However, when extracellular anisotropy is neglected, the effect of intracellular anisotropy in the model is exaggerated. Because of this, previous authors have expressed doubt as to whether such models should represent intracellular anisotropy [14, 50]. Many models neglected anisotropy completely. Anisotropy has important effects on the precordial ECG leads. For example, when subendocardial ischemia is modeled, the effect of anisotropy can make the difference between a positive and a negative ECG deflection [29], with important consequences for diagnosis. Thus, anisotropic ECG simulation can be important and the question is whether BEM models can reliably account for it. The purpose of this study is to demonstrate that a good approximative treatment of extracellular anisotropy in a BEM model is possible, and that accounting for both anisotropies improves the simulated ECG. We compared ECGs computed by a BEM model with those computed by a finite-difference model, in which anisotropy could be represented without compromise. 2 Methods Our methods are based on the bidomain model of cardiac tissue [14, 18, 32], which treats the myocardium as two continuous co-located media called the intracellular and extracellular domain, which are separated everywhere by the cell membrane. The conductivity in each domain is greater along than across the muscle fibers. We denote the fiber direction by a field of normalized row vectors ^a ax; ay; az: The conductivity of each domain can then be characterized by a tensor field, generated by the function GrL; rT rT1 rL where 1 is a unit tensor, and rL and rT are the conductivities parallel and perpendicular to the fiber axis, respectively [8]. Let riL and riT be the intracellular conductivities parallel and perpendicular to the fibers, respectively, and reL and reT their extracellular equivalents. We define the intracellular and extracellular conductivity tensors fields as Gi GriL; riT and Ge GreL; reT: The anisotropy ratios of the two domains are Ri riL=riT and Re reL=reT: An overview of all conductivity values and anisotropy ratios is given in Table 1. Potential fields /i and /e in the two domains are related to current density fields Ji Gir/i in the intracellular domain and Je Ger/e in the extracellular domain [14]. The divergence of each current density field equals the current that flows through the cellular membrane; this current must have equal magnitude and opposite sign in the two domains. Thus, the bidomain model can be summarized with the following equation [14, 18, 32]: It is convenient to use the transmembrane potential Vm = /i - /e to eliminate /i from Eq. 2; after rearranging terms we obtain an implicit equation for /e in terms of Vm: r Gi Ger /e r In this study ECGs were simulated from given membrane potentials Vm by a BEM model and by a finite-difference (FD) model of the human torso. The FD model solved the extracellular potential /e from Eq. 3. The BEM model is conceptually more complicated. Its source model is an equivalent current density Gc fcGRcriT; riT with Gc a proposed compound conductivity tensor (...truncated)