Application of the Method of Fundamental Solutions to Potential-based Inverse Electrocardiography

Abstract

Potential-based inverse electrocardiography is a method for the noninvasive computation of epicardial potentials from measured body surface electrocardiographic data. From the computed epicardial potentials, epicardial electrograms and isochrones (activation sequences), as well as repolarization patterns can be constructed. We term this noninvasive procedure Electrocardiographic Imaging (ECGI). The method of choice for computing epicardial potentials has been the Boundary Element Method (BEM) which requires meshing the heart and torso surfaces and optimizing the mesh, a very time-consuming operation that requires manual editing. Moreover, it can introduce mesh-related artifacts in the reconstructed epicardial images. Here we introduce the application of a meshless method, the Method of Fundamental Solutions (MFS) to ECGI. This new approach that does not require meshing is evaluated on data from animal experiments and human studies, and compared to BEM. Results demonstrate similar accuracy, with the following advantages: 1. Elimination of meshing and manual mesh optimization processes, thereby enhancing automation and speeding the ECGI procedure. 2. Elimination of mesh-induced artifacts. 3. Elimination of complex singular integrals that must be carefully computed in BEM. 4. Simpler implementation. These properties of MFS enhance the practical application of ECGI as a clinical diagnostic tool.

APPENDIX: FORMULATION OF MFS FOR LAPLACIAN OPERATOR

MFS has evolved from traditional boundary integral methods. Without forming a mesh, MFS uses a set of points to solve numerically partial differential equations. Details of this method can be found in the review article.37 The following Dirichlet boundary value problem is used to describe the theoretical formulation of MFS for the Laplacian operator:

$$\nabla ^2 u(x) = 0,\quad x \in \Omega$$

(a1)

$$u(x) = b(x),\quad \quad x \in \Gamma ,\quad \Gamma = \partial \Omega$$

(a2)

where $\nabla ^2$ is the Laplace differential operator with a known fundamental solution $f(r)$ in 3D space. $u(x)$ is a potential function in a source-free domain $\Omega$ and $b(x)$ is the Dirichlet boundary condition. According to the definition of fundamental solution,27 the fundamental solution of the Laplace operator can be obtained by solving the following equation for $f(r)$:

$$\nabla ^2 f(r) = \delta (r)$$

(a3)

where $\delta (r)$ is the delta function, $r = \left\| {x - y} \right\|$ is the 3D Euclidean distance between point x and point y, $x,y \in \Omega$. $f(r)$ in two dimensions (2D) and three dimensions (3D) is:53

$$f(r) = \left\{ {\begin{array}{*{20}l}{ - \displaystyle\frac{1}{{2\pi }}\ln \,r,\quad 2D} \\{\displaystyle\frac{1}{{4\pi r}},\quad 3D} \end{array}} \right.$$

(a4)

Both BEM and MFS use the same Green's function $f(r) = \frac{1}{{4\pi r}}$ in 3D space. However BEM integrates this function locally over elements of the real surface and requires computation of complex singular integrals. By placing the fictitious source points outside the domain of interest, MFS employs global integration and avoids the need to compute complex singular integrals.

The traditional boundary integral approach is to represent the solution $u(x)$ in term of a double layer potential:36,63

$$u(x) = \int_\Gamma {\frac{{\partial f(\left\| {x - y} \right\|)}}{{\partial n}}} e(y)dy,\quad x \in \Omega ,\quad y \in \Gamma$$

(a5)

where, n is the outward pointing normal at point y, $e(y)$ is an unknown density function. Equivalently a single layer potential representation of $u(x)$ can be used19,36

$$u(x) = \int_\Gamma {f(\left\| {x - y} \right\|)\,e(y)\,dy},\quad x \in \Omega ,\quad y \in \Gamma$$

(a6)

The source density distribution $e(y)$ can be determined by solving the following equation under the assumption of a double layer:

$$\int_\Gamma {\frac{{\partial f\left( {\left\| {x - y} \right\|} \right)}}{{\partial n}}\,} e(y)\,dy = b(x),\quad x \in \Gamma ,\quad y \in \Gamma$$

(a7)

or under the assumption of a single layer:

$$\int_\Gamma {f\left( {\left\| {x - y} \right\|} \right)\,} e(y)\,dy = b(x),\quad x \in \Gamma ,\quad y \in \Gamma$$

(a8)

However, singular integrals are involved in both cases. To alleviate this difficulty, the following formulation, similar to the single layer potential in (a6), has been used:50

$$u(x) = \int_{\hat \Gamma } {f\left( {\left\| {x - y} \right\|} \right)\,} e(y)\,dy,\quad x \in \Omega ,\quad y \in \hat \Gamma$$

(a9)

where the auxiliary boundary $\hat \Gamma$ is the surface of the auxiliary domain $\hat \Omega$ containing the domain $\Omega$ (Fig. 1).

Two different approaches for selecting $\hat \Gamma$ and its fictitious source points y are described in the literature:36 static configuration and dynamic configuration. In static configuration, the fictitious boundaries are fixed and pre-selected. The method is easy to implement and use in practical applications. For dynamic configuration, the location of fictitious boundaries is determined together with the solution50 by a complex, time-consuming nonlinear optimization procedure, which greatly limits its practical application. Since the geometry of the 3D domain between the torso surface and the heart surface is similar for all humans, the static approach is the method of choice for ECGI application.

Because $f\left( {\left\| {x - y} \right\|} \right)$ is the fundamental solution of the Laplace operator [Eq. (a3)], (a9) satisfies the differential Eq. (a1). Therefore we need only to apply the boundary condition (a2):

$$\int_{\hat \Gamma } {f\left( {\left\| {x - y} \right\|} \right)\,} e(y)\,dy = b(x),\quad x \in \Gamma ,\quad y \in \hat \Gamma \quad$$

(a10)

where the source density distribution $e(y)$, $y \in \hat \Gamma$, is to be determined. Once the source density is determined, Eq. (a1) subject to (a2) is solved. The analytic integral representation of (a10) implies that there is an infinite number of source density points on $\hat \Gamma$. In order to apply numerical methods to the solution, it is necessary to discretize $e(y)$. Assume $\psi _i (y), i = 1,2, \ldots \infty$ is a complete set of functions on $\hat \Gamma$, $e(y)$ can be approximated by:

$$e(y) = \sum\limits_{i = 1}^\infty {c_i \psi _i (y)} ,\quad y \in \hat \Gamma$$

(a11)

Substituting (a11) into (a10) and satisfying the boundary conditions at the N boundary points $x_k \in \Gamma ,\quad k = 1,2, \ldots N;$ we have

$$\sum\limits_{i = 1}^\infty {c_i \int_{\hat \Gamma } {f\left( {\left\| {x_k - y} \right\|} \right)\psi _i (y)\,dy} } = b(x_k ),\\ 1 \le k \le N,\quad y \in \hat \Gamma$$

(a12)

Since the fictitious boundary $\hat \Gamma$ is located outside the physical domain (Fig. 1), the integrand $f\left( {\left\| {x_k - y} \right\|} \right)$ is nonsingular and standard quadrature rules can be applied giving

$$\int_{\hat \Gamma } {f( {\| {x_k - y} \|} )\psi _i (y)\,dy} \approx \sum\limits_{j = 1}^M {w_j f( {\| {x_k - y_j } \|} )\psi _i (y_j )} ,\\ y_j \in \hat \Gamma ,\quad j = 1,2, \ldots ,M$$

(a13)

where $w_j$ is a weight factor and M is the number of fictitious nodes on the fictitious boundary $\hat \Gamma$.37

From (a12) and (a13), we obtain:

$$\sum\limits_{i = 1}^\infty {c_i \sum\limits_{j = 1}^M {w_j f( {\| {x_k - y_j } \|} )\psi _i (y_j )} }\\ = \sum\limits_{j = 1}^M {w_j \left[ {\sum\limits_{i = 1}^\infty {c_i } \psi _i (y_j )} \right]} f( {\| {x_k - y_j } \|} ) = b(x_k ),\\ 1 \le k \le N.$$

(a14)

Then:

$$\sum\limits_{j = 1}^M {a_j f( {\| {x_k - y_j } \|} ) = b(x_k )} ,\quad 1 \le k \le N.$$

(a15)

where:

$$a_j = w_j \sum\limits_{i = 1}^\infty {c_i } \psi _i (y_j )$$

(a16)

For completeness,11 a constant $a_0$ is added to (a15):

$$a_0 + \sum\limits_{j = 1}^M {a_j f( {\| {x_k - y_j } \|} ) = b(x_k )} ,\quad 1 \le k \le N.$$

(a17)

After Eq. (a17) is solved for $a_0$ and $a_j (j = 1,2, \ldots ,M)$, the solution to (a1) can be approximated by:

$$u_a (x) = a_0 + \sum\limits_{j = 1}^M {a_j f( {\| {x - y_j } \|} ),\quad x \in \Omega ,\quad y_j \in \hat \Gamma }$$

(a18)

The approximate solution $u_a$ to Eq. (a1) is represented by a linear combination of fundamental solutions of the governing equation with the singularities $y_j ,j = 1,2, \ldots ,M$ placed outside the domain of the problem.

MFS is applicable not only to the above boundary value problem, but also to the Cauchy problem42 that underlies ECGI. In this problem, both Dirichlet and Neumann boundary conditions are given only on portion of the boundary:42

$$\nabla ^2 u(x) = 0,\quad x \in \Omega\vspace*{-12pt}$$

(a1)

$${\rm Dirichlet}\;{\rm conditions}:\; u(x) = b(x),\quad x \in \Gamma _1 ,\\ \Gamma _1 \subset \Gamma = \partial \Omega$$

(a19)

$${\rm Neumann}\;{\rm conditions}: \frac{\partial }{{\partial n}}u(x) = i(x),\quad x \in \Gamma _1 ,\\ \Gamma _1 \subset \Gamma = \partial \Omega$$

(a20)

For the Neumann condition (a20), the gradient at point $x$is along the outward normal to the boundary at that point. Similar to Eq. (a17), MFS can be used to discretize the Dirichlet and Neumann boundary conditions (Eqs. (a19) and (a20)) as follows:

$$a_0 + \sum\limits_{j = 1}^M {a_j f( {\| {x_k - y_j } \|} ) = b(x_k )} ,\quad x_k \in \Gamma _1 ,\\ k = 1,2, \ldots ,N,\quad y_j \in \hat \Gamma .$$

(a21)

$$\sum\limits_{j = 1}^M {a_j \frac{\partial }{{\partial n}}f( {\| {x_k - y_j } \|} ) = i(x_k )} ,\quad x_k \in \Gamma _1 ,\\ k = 1,2, \ldots ,N,\quad y_j \in \hat \Gamma .$$

(a22)

After solving for the coefficients $(a_0 ,a_j ,j = 1,2, \ldots ,M)$, subject to the boundary conditions (a19) and (a20), the solution to (a1) can be approximated using Eq. (a18).

Convergence analysis of MFS for Laplace's equation was conducted by Cheng.21 When the problem boundary and boundary conditions are smooth functions, MFS converges exponentially to the solution of the problem. This analysis was conducted for 2D; Golberg and Chen36 provided arguments that similar convergence properties exist in 3D.