Forward and Inverse Solutions of Electrocardiography Problem Using an Adaptive BEM Method

F.B. Sachse and G. Seemann (Eds.): FIMH 2007, LNCS 4466, pp. 290–299, 2007. © Springer-Verlag Berlin Heidelberg 2007

Forward and Inverse Solutions of Electrocardiography Problem Using an Adaptive BEM Method

Guofa Shou1, Ling Xia1, Mingfeng Jiang1,2, Feng Liu3, and Stuart Crozier3

1 Department of Biomedical Engineering, Zhejiang University, Hangzhou, P.R. China, 310027

[email protected] 2 The College of Electronics and Informatics, Zhejiang Sci-Tech University, Hangzhou,

P.R. China, 310018 3 The School of Information Technology and Electrical Engineering,

The University of Queensland, St. Lucia, Brisbane, Queensland 4072, Australia

Abstract. The construction of geometry models of heart-torso is critical for solving the forward and inverse problems of magneto- and electro-cardiography (MCG/ECG). Boundary element method (BEM) is a commonly used numerical approach for the solution of these problems and it requires the modeling of interfaces between various tissue regions. In this study, a new BEM (h-adaptive type) has been applied to the ECG forward/inverse problems. Compared with those traditional BEMs, the adaptive BEM can self-adjust the number and size of the boundary element (BE) meshes according to an error indicator, and thus can save a lot of computational time and also improve the accuracy of the forward and inverse solutions. In this paper, the procedure of the adaptive triangular mesh generation is detailed and the algorithm is tested using a concentric sphere model and a realistic heart-torso model. For the realistic torso model, to improve the numerical accuracy, a number of new nodes are added on the basis of initial torso BE meshes, and the corresponding node coordinates are determined using an approach called Parametric Fourier Representation (PFR) of closed polygons. The simulation results show that the adaptive BEM is more accurate and efficient than traditional BEMs and therefore it is a very promising numerical scheme for ECG forward/inverse problems.

Keywords: h-adaptive BEM, ECG, Forward Problem, Inverse problem.

1 Introduction

The main task of Electrocardiography (ECG) is to interpret the electrical activities of the heart from the recorded body surface potentials (BSPs), which involves the so-called “forward” and “inverse” problems [1]. The former deals with the modeling of the potential distribution on the body surface from equivalent cardiac sources or epicardial potentials (EPs), and the latter estimates the cardiac equivalent sources or reconstructs the EP distribution, myocardial activation sequences, etc.[1,2]. Similar to the electromagnetic imaging simulation study for electro- and magneto-encephalography (EEG/MEG) [3], the ECG forward/inverse problems require the construction of geometry models including torso, heart, and lung, etc. Once the geometry models are available, a numerical approach is required to solve the involved

Forward and Inverse Solutions of Electrocardiography Problem 291

field problem. The widely used numerical methods for electromagnetic problem can be classified into two categories [4]:

1. Volume approach, it is based on differential equations techniques, and typical algorithms include finite element method (FEM) [5, 6], finite difference method (FDM) [7] and finite volume method (FVM) [8].

2. Surface approach, it is based on integral function technique and the typical algorithm is the boundary element method (BEM) [5, 9, 10, 11].

For the surface approach, the interfaces of different conductivity regions are merely required for modeling. Therefore this approach can significantly reduce the number of unknown variables, however, it is very difficult to handle anisotropic regions; while the volume approach is very suitable to model the inhomogeneous volume conductor problems but it will lead to a large amount of unknown variables. These two approaches have their own advantages and can be applied to different problems. Seger [4] and Martin [12] even integrated these two methods for ECG field calculation.

In the BEM, the BE mesh is critical and directly influences the solution accuracy. In the field of computational bio-electromagnetic field, previous research on BEM can be summarized into two classes: construction of the realistic BE mesh model using a number of elements [13, 14], or sparse mesh model using higher-order elements [9, 11]. For ECG problem, to our knowledge, the adaptive method has only been used in FEM modeling of cardiac field [6]. In this study, we employ the adaptive BEM for this problem and attempt to find an accurate and efficient numerical scheme for forward/inverse solution.

Since the introduction of the adaptive BEM in the 1980s, it has been applied in various fields [15, 16]. In this paper, we used the h-adaptive BEM, in which the number of element increases by adding new nodes and the order of interpolation function is unchanged during refinement. In the h-adaptive BEM, it is important to determine the coordinates of those new added nodes. For regular model such as sphere, it is very easy to find the coordinates. For realistic torso model, it will be obtained indirectly, for example, through medical image reconstruction methods [17, 18]. In this study, the Parametric Fourier Representations (PFR) [18] of closed polygon is applied to estimate both the in-slices and inter-slices nodes.

In this paper, we firstly describe the modeling process of h-adaptive BEM for ECG problem, and then test the proposed algorithm with a concentric sphere model and a realistic human model. The inverse solutions using initial BE meshes and adaptive BE meshes are also compared.

2 Adaptive BEM Analysis for ECG Forward Problem

2.1 The Forward Problem

In terms of the EPs, the ECG forward problem can be formulated as Laplace’s equation with Cauchy boundary conditions [1]:

0σ∇ ⋅ ∇Φ = in Ω (1)

0Φ = Φ on T⊆ Γ∑ and 0nσ∇Φ ⋅ = on TΓ (2)

292 G. Shou et al.

where Φ is the quasi-electrostatic potentials, σ is the conductivity tensor, ΓT and Ω represent the surface and volume of the thorax, respectively.

2.2 Boundary Integral Equation

Using the Green second identity

2 2( ) ( )S V

A B B A ndS A B B A d∇ − ∇ ⋅ = ∇ − ∇ Ω∫ ∫ (3)

and taking the fundamental solution u＊ which satisfies

2 0u δ∗∇ + = (4)

where δ denotes Dirac’s delta function, we have

( ) ( ) ( ) 0S

c p p u q dS∗ ∗Φ − ∇Φ − Φ ⋅ =∫ (5)

where p is an arbitrary field point, q＊ denotes 1

r∇ , while r is the distance between

the source point and field point p, and c(p) depends on the location of p: c(p)=4π when p is inside Ω; c(p)=0 when p is outside Ω; c(p)=2π when p is on the smooth boundary.

2.3 Discretization Error and Residual

After the discretization of the boundary S with M triangles and defining q as the charge and displace Φ in Eq. (5), we have

1

( ) ( ) ( ) 0m

M

i i Sm

c p p u q q d∗ ∗

=

Φ − −Φ ⋅ Γ =∑∫ (6)

The potential and flux charge values on the jth element are approximated by a linear combination of their nodal values and the interpolation functions are as follows.

( ) ( )j i j ii

p N pΦ = ⋅Φ∑ and ( ) ( )j i j ii

q p N p q= ⋅∑ (7)

The approximated solutions of Φ and q are Φ and q , respectively, and the errors

of potential and charge, Φe and qe , are

ˆeΦ = Φ − Φ , ˆqe q q= − (8)

In Eq. (6), if we substitute Φ and q for Φ and q, Eq. (6) is not satisfied perfectly

and therefore, the residual R is


1

ˆ ˆˆ( ) ( ) ( ) ( )m

M

j j j Sm

R p c p p u q q d∗ ∗

== Φ − − Φ ⋅ Γ∑∫ (9)

In the collocation formulation, R is forced to be zero when the boundary points are taken as the collocation points.

Finally, the system equation with matrix-form can be written as

Ax b= (10)

where A is the transfer matrix, x and b are the potential vectors on epicardial surface and body surface.

2.4 The Error Indicator from the Boundary Integral Equation

In the adaptive BEM, the error estimation is the most important component, which is the factor to determine the accuracy of the approximated solutions and drives the whole discrete meshes’ adaptive refinement procedure. The error estimation schemes are classified into the residual-type, the integral equation-type, the node sensitivity-type and the solution difference type [15]. In this paper, we used residual-type [16].

The approximated solution by BEM does not necessarily satisfy Eq. (5) at positions other than collocation nodes. In this work, the square of residual is used as the error indicator (see Eq. (11)). Since the collocation node is positioned at each end of the line element and the maximum discrepancy between the interpolated boundary potential and the value obtained by Eq. (5) can be expected at the points furthest from these nodes, i.e., at the centroid of that triangle element. The local error indicator at the element center is used directly as the error indicator for the entire element. In order to reduce the computational effort, the boundary element error indicator is obtained from the weighted summation of local error indicators (see Eq. (12)).

2 2

1

ˆ ˆ ˆˆ( ) ( ( ) ( ) ( ) )m

M

j j j Sm

R p c p p u q q d∗ ∗

== Φ − − Φ ⋅ Γ∑∫ (11)

2 2ˆ ˆ ( )m

j jj

R R pωΓ

= ⋅∑ and ˆ ˆm

m

R R mΓ Γ=∑ (12)

where ωj is the weight and is used to compute the element error indicator, ˆm

RΓ is the

error indicator of the elementΓm and RΓ is the mean error indicator of the whole

element. The selection of ωj and pj is the same as that in Bächtold’s [16].

2.5 h-Adaptive BE Mesh Generation

The flowchart of adaptive mesh generation is shown in Fig. 1. The calculation of electric potential starts from an initial triangular mesh and the triangular mesh is partially divided into small triangles by a refinement procedure and new node is

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Fig. 1. A. The flowchart of the adaptive triangular mesh generator. B. The process of refine-ment for a specific element.

added on the center point of the refined element. Then the triangles are rearranged using the joint quality factors of triangular pair [19]. The criterion for the mesh refinement is an error indicator expressed in Eq. (13). In this equation, the parameter k is a constant which controls the degree of refinement in each iterative procedure. The iteration is stopped when the error estimation RΓ

is less than a predetermined

value (See Fig.1 (A)).

ˆ ˆm

R k RΓ Γ≥ × (13)

3 The Parametric Fourier Representation of Realistic Torso Model

In the h-adaptive BEM described above, new nodes need to be added to the initial mesh. For the regular geometry model such as sphere, it is very easy to find the parameter function. However, additional efforts have to be taken to determine the nodes in the realistic torso model and we use PFR for this problem [18]. In general, there are very sparse nodes in an initial torso model; with PFR approach [18], dense nodes can be obtained not only in in-slices but also within inter-slices. Here, the approximation quality can be easily controlled by the alternation of number of harmonics.

At first, the PFR for each closed polygon is determined. In the next step, a new set of equiangular points is generated by PFR for each closed polygon. For the realistic torso model, two PFR sets are used. The first set is for the transversal slice and it can generate more nodes in one slice. After the nodes in each transversal slice are determined, a closed curve in the sagittal direction is then formed. The second set of PFR is for the closed curve in the sagittal direction. The dense nodes can be gotten determined from the second set. Fig.2 shows the meshing results for a human torso using the above described PFR method. In this case, the number of harmonics is 10, and the nodes of the torso boundary are 412(A), 1322(B) and 5402(C), respectively.


4 The Inverse Calculation of the ECG Problem

The geometry model generated by the proposed h-adaptive BEM is used for the ECG inverse calculation. The ECG inverse problem is ill-posed and it requires the regularized numerical procedure [1, 2]. In the paper, the zero-order Tikhonov regularization and the GCV are applied [20, 21]. With respect to assumed ‘exact’ solution from forward simulation, the inverse solution is evaluated using relative error (RE) and correlation coefficient (CC).

5 Results

Two simulation protocols with increased complexity are used to validate the proposed h-adaptive BEM for ECG problem. The first one is the concentric sphere heart-torso model which is often used in ECG simulation, with known analytical BSP distribution induced by a limited number of dipoles placed within the inner sphere. In this model, the radii of the two spheres are 4 and 10 centimeter, respectively. The number of nodes and elements of ‘heart’ sphere are 122 and 240, and the ‘torso’ sphere is 38 and 72. The second one is the realistic heart and torso geometry models based on CT images with less nodes on the torso compared with previous application [22]. The initial numbers of nodes and elements are 165 and 315 for the heart, and 210 and 416 for the torso, respectively.

Fig. 2. The BE model of a realistic torso. A, the initial torso model (412 nodes); B, the torso model generated with PFR for transversal slices (1322 nodes); C, the torso model using PFR for both the transversal slices and sagittal direction (5402 nodes).

(A) (B)

Fig. 3. The refined results of sphere mesh for torso. (A) Model I, dipole orientation: zero degree. (B) Model II, dipole orientation: 30 degrees.

296 G. Shou et al.

In this study, we only refine the torso meshes and keep BE meshes of the heart model unchanged. For the sphere model, two different dipole orientations are used for the forward ECG calculations, i.e., the single dipole is located at the center of the sphere with a zero degree of azimuth angle in the first simulation (Model I), and with a 30 degree of azimuth angle in the second one (Model II). In the realistic torso-heart model based simulation, a single dipole is located inside the heart model and its orientagtion: both altitude and azimuth angles are 100 degree.

For the concentric sphere model, the BE meshes are generated with two-iteration refinement. Fig.3 shows the mesh profile and Fig.3 (A) is for the Model I and Fig.3 (B) is for Model II. It is noted that for the first-iteration refinement, the mesh results of these two models are quite similar to each other, while for the second-iteration the forward calculation of BSPs modify the mesh profiles. The RE and CC of those forward calculated BSPs using different meshes are summarized in Table 1. From this table, it can be seen that the RE of forward calculation is decreased while the CC is increased after mesh refinement.

A B

Fig. 4. The refinement of the realistic torso model. A. The initial mesh; B. The refined mesh.

(A) (B)

Fig. 5. Epicardial and body surface potential (BSP) distributions. A. Eepicardial potential distribution; B. BSP distribution simulated using initial BE meshes; C. BSP distribution simuated using refined BE meshes.

Fig.4 shows the results for the realistic heart-torso BEM model and Fig.4 (A) is the initial mesh model and (B) is the refined mesh model. For the realistic torso model, single-iteration refinement is enough and the parameter k is 0.3 (see Eq.15). The


corresponding BSP distributions in the realistic-model based simulation are shown in Fig.5. From this figure, it can be seen that the potential distribution is much smoother using the refined meshes.

For all these three models with various meshes, the EPs are inversely calculated. Fig.6 shows the EP distribution and Fig.6 (A) is the ‘exact’ solution obtained from forward modeling, Fig.6(B) and (C) are the inverse solutions using initial and refined torso meshes. RE and CC of the inverse calculations are summarized in Table.1. From this table, it can be seen that the RE and CC values in Model I and Model II are not constant, the RE in Model I increases and CC in Model I decreases, and the RE in Model II decreases and CC increases. These variations in the inverse solutions implied that the dense mesh structure might not provide the improved solution for ECG problem. In the realistic torso-heart model, the RE decreases almost 10% and the CC increases as well, this demonstrates that the h-adaptive BEM provide high-quality BE meshes and leads to better solution. This is clearer in Fig.6, in which the inversely recovered EP distribution using the refined torso meshes is more similar to the ‘exact’ profile, especially in the back part of the heart model. Therefore, the application of the h-adaptive BEM can improve the ECG inverse solution.

Fig. 6. The ‘exact’ (forward solution) and the inversely calculated EPs using the initial and refined torso mesh models. Left panel: ‘exact’ solution; Middle panel: inverse solution using initial mesh; Right panel: inverse solution using refined mesh model.

Table 1. CC and RE results of the application of h-adaptive BEM in three simulations

Iteration node

mesh

CC forward

RE forward

CC inverse

RE inverse

0 38 72 0.9998 0.1872 0.9994 0.0345 1 110 216 1.0000 0.0776 / /

Model I

2 326 648 1.0000 0.0471 0.9988 0.0501 0 38 72 0.9968 0.1670 0.9995 0.0316 1 110 216 0.9971 0.0874 / /

Model II

2 304 604 0.9974 0.0743 1.0000 0.0026 0 210 416 / / 0.9769 0.2001 Realistic

model 1 421 838 / / 0.9901 0.1296

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6 Discussion and Conclusion

This study describes an adaptive BEM implementation and an approach for creating a more detailed realistic BEM models from the initial coarse torso meshes. The proposed algorithm is tested with a concentric sphere model and realistic human data model with a dipole source inside the heart model. Compared to the standard BEM, the accuracy of the forward and inverse solution for this specific ECG problem is improved by using the new adaptive BEM. For example, for the concentric sphere model, the RE of the forward solution is decreased by near 10%; for the realistic human torso model, the RE of the inverse solution is also decreased by 10%. Interestingly, the results show that denser mesh structure does not always lead to better numerical solution. From the sphere model simulation, it is found that the dipole orientation influences the accuracy of the adaptive BEM solution. Therefore, to obtain robust and accurate inverse solution, more work need be done for the realistic model, in which the dipole orientations can be varied constantly during the activation course.

In conclusion, the application of adaptive BEM in the ECG problem can avoid complicated mesh construction procedure and thus save a lot of computational time. Using this adaptive technique, for patient-specific ECG/EEG studies, it is feasible to build BEM models with automatic mesh generation.

Acknowledgments. This project is supported by the 973 National Key Basic Research & Development Program (2003CB716106), the 863 High-tech Research & Development Program (2006AA02Z307), the National Natural Science Foundation of China (30570484) and the Program for New Century Excellent Talents in University (NCET-04-0550).

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Forward and Inverse Solutions of Electrocardiography Problem Using an Adaptive BEM Method

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