Problem of Electrocardiography: Estimating the Location of Cardiac Ischemia in a 3D Realistic Geometry

(1)

where $J_{stim}$ represents the initial stimulus, $J_{in}$ and $J_{out}$ denotes the sum of all inward and outward currents, respectively, which are defined as

$\begin{aligned} J_{in}(v,h)=\frac{h(1-v)(v-v_{rest})^2}{\tau _{in}}, \end{aligned}$

(2)

$\begin{aligned} J_{out}(v,h)=- \frac{v-v_{rest}}{\tau _{out}}. \end{aligned}$

(3)

where $v_{rest}$ is the resting potential. This parameter was not considered in the original formulation. It was incorporated in [1] to simulate the increase of the resting potential during ischemia.

The gating variable h(t) is dimensionless and varies between zero and one. This variable regulates inward current flows and obeys the following equation

$\begin{aligned} \frac{dh}{dt}=\left\{ \begin{array}{c c} (1-h)/\tau _{open}, &{} v<v_{crit} \\ -h/\tau _{close}, &{} v\ge v_{crit} \\ \end{array}\right. \end{aligned}$

(4)

where $v_{crit}$ is the change-over voltage. This model contains four time constants ( $\tau _{in}$ , $\tau _{out}$ , $\tau _{open}$ and $\tau _{close}$ ) which correspond to the four phases of the cardiac action potential: initiation, plateau, decay and recovery (see [1] for details).

The effects of ischemia are simulated by modifying the values of $\tau _{in}$ and $v_{rest}$ [4]. For ischemic cells, we set the parameter $\tau _{in}$ and $v_{rest}$ equal to $0.8\,\text {ms}$ and 0.1, respectively. For healthy cells $\tau _{in}$ and $v_{rest}$ are set to $0.2\,\text {ms}$ and 0, respectively.

2.2 Cardiac Tissue Model

Let ${\varOmega }_H\in \mathcal {R}^3$ be the cardiac tissue, and $v=v(\varvec{r_H},t)$ the membrane potential with $\varvec{r_H}\in {\varOmega }_H$ . The propagation of v is described according to the monodomain formalism,

$\begin{aligned} \frac{\partial v}{\partial t} = \nabla D \cdot (\nabla v ) +J_{TC} \end{aligned}$

(5)

where $J_{TC}$ is the ion current term current provided by TC model, and D is the intracellular conductivity tensor (assumed constant $D = 1.4 \,\mathcal {I} \,(\text {mm}^2/\text {ms})$ ). Equation (5) is solved by imposing the initial conditions $v=v_{rest}$ at

, and no-flux boundary conditions.

To simulate the effects of ischemia at the tissue level, we consider a regional model of ischemia where the parameters $\tau _{in}(\varvec{r_H})$ , $v_{rest}(\varvec{r_H})$ vary linearly between healthy and ischemic values [1, 21].

2.3 Model of Remote Recordings

According to the Volumen Conductor Theory [13, 16], the resulting potential distribution at position $\varvec{r_T}\in {\varOmega }_{T}$ outside the cardiac tissue ${\varOmega }_H$ , is calculated as

$\begin{aligned} \upvarphi (\varvec{r_T},t)= \frac{1}{4 \pi \sigma _{0}} \int _{{\varOmega }_{H}} \frac{\nabla D \cdot (\nabla v(\varvec{r_H},t))}{R(\varvec{r_H},\varvec{r_T})} \, d {\varOmega }_{H} \end{aligned}$

(6)

where $R(\varvec{r_H}, \varvec{r_T})=||\varvec{r_T}-\varvec{r_H}||$ represents the distance from the source location point $\varvec{r_H}$ to the observation point $\varvec{r_T}$ , $\sigma _0$ is the medium conductivity (assumed homogeneous and set to 1 $\mathrm{Sm}^{-1}$ ), and $v(\varvec{r_H},t)$ is solution of (5). Equations (1)–(6) comprise a complete description of the forward problem.

3 Inverse Procedure

Let ${\varOmega }_H$ be a synthetic cardiac tissue which contains an ischemic region, and $\upvarphi ^i_R(t)$ be a set of associated remote measurements. Note that $\upvarphi ^i_R(t)=\upvarphi (\varvec{r^i_T},t)$ with $t\in [0,T]$ and $i=1,2,\ldots ,N$ , being T the total recording time, and N the number of recording points. Hereinafter, we assume that the regional ischemia constitute our true ischemic configuration, and $\upvarphi ^i_{R}(t)$ is our observed data.

The aim of the inverse procedure is to estimate the shape and the location of the ischemic areas by adjusting the spatial distribution of $\tau _{in}$ and $v_{rest}$ . This is achieved by minimizing the misfit between the observed data $\upvarphi ^i_R(t)$ , and the data associated to a guess configuration: $\upvarphi ^i_S(t;\tau _{in},v_{rest})=\upvarphi (\varvec{r^i_T},t;\tau _{in},v_{rest})$ . We use an iterative scheme in which the following cost functional

$\begin{aligned} {\mathcal {J}}(\tau _{in},v_{rest})= & {} \frac{1}{2} \int _0^T\sum _{i=1}^N \left| \left| \upvarphi _{R}^i(t)-\upvarphi _{S}^i(t;\tau _{in},v_{rest}) \right| \right| ^2 \,\, dt \end{aligned}$

(7)

is reduced at each step of the reconstruction process. We use an adjoint formulation to find a direction in the parameter space $(\tau _{in},v_{rest})$ such that the cost functional decreases. The ischemic region is defined by a level set function [1].

Since both parameters, $\tau _{in}$ and $v_{rest}$ , define the same region, we only consider the variation of $\tau _{in}$ for the gradient computation. Following [1], the gradient of the function ${\mathcal {J}}$ over the parameter $\tau _{in}$ is given by

$\begin{aligned} grad_{\tau _{in}} {\mathcal {J}} = \int _{0}^T w \frac{h \left( v-v_{rest}\right) ^2\left( v-1\right) }{\tau ^2_{in}} \, dt \end{aligned}$