First, deriving the integral equations. The potential ϕ o at point O can be expressed as Equation (1) using Green’s second law, and the point O is assumed as an observation point which inside the volume but very close to surface,

ϕ o = − 1 4 π ∫ S H ϕ H r ¯ ⋅ n ¯ r 2 d S H − 1 4 π ∫ S H ∇ ϕ H ⋅ n ¯ r d S H + 1 4 π ∫ S B ϕ B r ¯ ⋅ n ¯ r 2 d S B . (1)

where S B is the body surface, S H is the epicardial surface. r is the distance that extends from the observer to elements of integration dS. ϕ B is the potential distribution on body surface, ϕ H is the potential distribution on epicardial surface. r ¯ is an unit vector in the direction of r. n ¯ is an outward pointing vector of unit magnitude normal to surface element dS.

So, two equations can be obtained from Equation (1) for observation positions located on body and epicardial surface, respectively,

− ϕ B + 1 4 π ∫ S B ϕ B r ¯ ⋅ n ¯ r 2 d S B − 1 4 π ∫ S H ϕ H r ¯ ⋅ n ¯ r 2 d S H − 1 4 π ∫ S H ∇ ϕ H ⋅ n ¯ r d S H = 0 , (2)

1 4 π ∫ S B ϕ B r ¯ ⋅ n ¯ r 2 d S B − ϕ H − 1 4 π ∫ S H ϕ H r ¯ ⋅ n ¯ r 2 d S H − 1 4 π ∫ S H ∇ ϕ H ⋅ n ¯ r d S H = 0 . (3)

To solve the Equation (2) and Equation (3) numerically, these equations should convert to simultaneous linear equations. Therefore, the surface S, consist of S B and S H , need to subdivide in small triangles, which means the integration area will be partitioned into triangle elements. And Equations (2) and Equation (3) can be simplified by the element of solid angle d Ω e f i which is subtended at an observation point of the i-th location on surface e by an area element on surface f:

d Ω e f i = r ¯ ⋅ n ¯ r 2 d S f . (4)

As it can be seen here, if the direction of n ¯ is wrong, the sign of r ¯ ⋅ n ¯ wouldn’t be correct, therefore, it will have influence on the solid angle, which means the result of potential distribution may not be right.

Substituting Equation (4) in Equation (2) and Equation (3), and calculating the surface solid angle integration approximately by discretized triangle solid angle summation as the way shown in Equation (5). The coefficients of the components of gradients term are noted in Equation (6).

1 4 π ∫ S Y ϕ Y d Ω X Y = ∑ 1 N Y P X Y Φ Y , (5)

1 4 π ∫ S Y ∇ ϕ Y ⋅ n ¯ r d S Y = ∑ 1 N Y G X Y Γ Y . (6)

The different rows of matrices P X Y and G X Y corresponds to different locations of the observer on surface X. As the theory showing, each of the P’s and G’s is a matrix of coefficients depending entirely on the geometry. In all cases the number of columns corresponds to the number of locations on the surface Y of integration [1]. Based on the derivation above, the equation about the potential distributions on epicardial surface and body surface can be derived as ϕ B = Z B H ϕ H , where Z B H is the transfer matrix described in Equation (7).

Z B H = ( P B B − G B H G H H − 1 P H B ) − 1 ( G B H G H H − 1 P H H − P B H ) . (7)

As it has been shown in the theory, the direction of n ¯ is very important, the method of Barnard et al. requires the additional computation of the normal to the plane triangle to establish the correct sign. So the direction of n ¯ must be confirmed before any further calculation.

A new method is proposed to ensure the direction of n ¯ of each triangle element to be outward. The basic idea of this method comes from the application of Kirchhoff’s laws. There are two applications of Kirchhoff’s laws and Ohm’s law, the Branch Current method and the Mesh Current method. The Mesh Current method can solve a circuit with less variables and less equations by simply determine unknown currents of each mesh in a network. Like its name, mesh current, means a current that loops around the essential mesh. It encompasses all the component of the mesh. First, there are some loops need to be identified within the circuit. As can be seen in Figure 1(a), one loop formed by V D C , R 1 and R 2 while the other loop formed by R 2 , R 3 . These current meshes are connected with each other by loops. I 1 and I 2 is the current of each mesh respectively. The direction of each current is chosen arbitrarily, and the intersecting component will be “went through” by two currents based on the assumption of the method. Second, get all the voltage drops though each resistor according to the direction of mesh currents. Then use the Kirchhoff’s Voltage laws, get equations state as ∑ k = 1 n V k = 0 . Solve a set of equations can get respective mesh current.

In the theory of Mesh Current method, if all mesh currents are chosen in a same direction, either clockwise or counterclockwise, the direction of two mesh currents that go through the intersecting side of two meshes must be opposite. So, that kind of law can be used in determining the direction of n ¯ in turn. In other words, if applying the theory of Mesh Current method to determine the direction of n ¯ , some principles should be followed: the direction of n ¯ gives the direction of rotation according to the right-hand rule, the direction of rotation of each triangle element shows a direction of each side of the triangle. If two triangles have contrary direction on their common side, it means the directions of n ¯ of them are at the same side. This can be seen in Figure 1(b).

In that case, if the direction of n ¯ of a triangle could be determined, the direction of n ¯ of the neighborhood of this triangle also could be determined. Therefore, in this new method, it needs to figure out a reference normal vector (RNV) at first, then determine the others’ direction according to the direction of

the RNV. The triangle element of which n ¯ is ensured correct and is referred as reference triangle (RT). Check the n ¯ of triangle elements in the neighborhood, which share one of RT’s sides, and then check the neighborhood of these three triangles, and so on. Eventually, we can finish examination on all triangle elements of the model.

The simulation on simple model shows in Figure 3. Figure 3(a) gives the potential distribution on epicardial surface which has been set as a known distribution. Figure 3(b) shows the potential distribution on body surface, which is a reference result here. And the potential distribution on body surface calculated by a commercial program and the program with new method is shown in Figure 3(c) and Figure 3(d), respectively.

As it can be seen in Figure 3(a), the potential on heart surface is set to increase from the bottom to the top monotonously. Consequently, Figure 3(b) and Figure 3(d) show a reasonable monotonous potential distribution on the body surface. However, the bottom part of Figure 3(c) have significant differences with Figure 3(b) and Figure 3(d), it shows an unreasonable reverse below the middle area, which means it is inaccurate.

To verify the feasibility of the new method, a more genuine heart model is used for compares. The simulation results on comparatively complex model are shown in Figure 4. Figure 4(a) gives the potential distribution on epicardial surface which has been set as a known distribution. Figure 4(b) shows the potential distribution on body surface, which is a reference result here. And the

potential distribution of body surface calculated by a commercial program and program of the new method are show in Figure 4(c) and Figure 4(d) respectively.

The results shown in Figure 4 are analogous to the results in Figure 3. The potential on heart surface also increases from the bottom to the top monotonously. Figure 4(b) and Figure 4(d) still keep this property. But Figure 4(c) appears some abnormal distribution on the shoulder area, which exhibits its inaccuracy.