Comparison of Two Formulations for Computing Body Surface Potential Maps

TL;DR

This work compares F1 and F2, to determine which formulation is the most relevant to use, and shows that the Balance Formulation F2 is robust to the input data ($\tilde{v}$ and $\psi$).

Abstract

In order to establish a link between the activation map and the body surface signals, we want to compute body potentials from activation maps and a frontlike approximation of the activation of the heart. To do so, two formulations that map the extracardiac potential from the transmembrane voltage naturally emerge from the bidomain model. Either the extracellular/extracardiac potential solves a Laplace equation with discontinuous conductivity coefficient and ionic current as a source (Source Formulation F1); or the quasi-stationary electrical balance between the intra- and extracellular fields (Balance Formulation F2). In this work, we compare F1 and F2, to determine which formulation is the most relevant to use. We compute reference activation map $ψ$ , transmembrane voltage $v$ and body surface map $u$ with a bidomain 2D code. We design two alternative shapes $\tilde{v}$ for a frontlike approximation of $v$ . Afterwards, two extracellular / extracardiac potentials are computed from the activation time $ψ$ and $\tilde{v}$ , using the two different formulations. Then the extracardiac potentials solutions of F1 and F2 are compared respectively to the solution of the bidomain $u$ . Results show that the Balance Formulation F2 is robust to the input data ($\tilde{v}$ and $ψ$). On the contrary, the Source Formulation F1 is very unstable and generates very large errors on the body surface map.

Figures (5)

• Figure 1: Comparison between $f_{\text{MS}}$ in green and $f_{\text{int}}$ in red, as functions of $v$. The stars are the reference values $f_\text{ref}(v)$ obtained by the bidomain propagation.
• Figure 2: Bidomain solution $u_\text{ref}$ on the left, and $u_1$ solution of \ref{['SF']} on the right for $\partial_t v = \partial_t v_\text{ref}$ and $f(v) = f_\text{int}\left (v_\text{ref}(t^n) \right )$ in \ref{['sbdf2']} (line 2 of Table \ref{['table:1']}).
• Figure 3: Activation times computed from the bidomain propagation.
• Figure 4: Relative differences between $u_1$ (red), $u_2$ (blue) and $u_\text{ref}$ respectively, and $\tilde{v}$ and $v_\text{ref}$ (black). $\tilde{v}$ may be the MS0D model (horizontal dashed lines), or $v_\varepsilon$ with varying duration parameter $\varepsilon$ (points).
• Figure 5: Relative difference between $u_2$ and $u_\text{ref}$ on the torso at time $t=35$ ms for $5000$ realisations of white noise $w$. $\varepsilon=2.5$.