Ouyang S, Maynard D E
Department of Electrical and Electronic Engineering, University of Hertfordshire, Hatfield, Herts, UK.
Med Eng Phys. 1997 Mar;19(2):164-70. doi: 10.1016/s1350-4533(96)00038-0.
Finite difference methods for the volume conductor problem have used a single coordinate system for the mesh and made approximations of Laplace's equation. This method is simple but has two major problems. Firstly, to deal with boundary conditions properly, the normal potential gradient at the boundary must be known. However it is complicated to compute at a curved surface point. Secondly, for an inverse solution the equation on a curved boundary is difficult to reverse since more than one inner mesh node appears in the approximation equation for each surface point. The new method developed in this paper is a dual coordinate system. One system serves as a frame mesh, the other is a sub-coordinate system in which surface points become mesh points (regular nodes). The equation at each surface point is then directly reversible since only one inner point appears in the equation. The forward solution is applied to both centric and eccentric bone models and uses the conventional successive over-relaxation (SOR) method. Noise is added to this solution for input to the inverse procedure which is a direct step-in non-iterative method. Low pass filtering was effective in reducing the effects of noise. In the examples given, only one coordinate subsystem is used but, for complex shape boundaries, multiple subsystems would be necessary.
用于容积导体问题的有限差分法采用单一坐标系划分网格，并对拉普拉斯方程进行近似。该方法简单，但存在两个主要问题。其一，为妥善处理边界条件，必须知道边界处的法向电位梯度。然而，在曲面点处计算该梯度很复杂。其二，对于逆解，曲面上的方程难以求逆，因为每个表面点的近似方程中会出现多个内部网格节点。本文提出的新方法是一种双坐标系。一个系统用作框架网格，另一个是子坐标系，其中表面点成为网格点（规则节点）。这样每个表面点处的方程就可以直接求逆，因为方程中只出现一个内部点。正解应用于同心和偏心骨模型，并使用传统的逐次超松弛（SOR）方法。向该解中添加噪声作为逆过程的输入，逆过程是一种直接的非迭代方法。低通滤波在降低噪声影响方面很有效。在给出的例子中，只使用了一个坐标子系统，但对于复杂形状的边界，需要多个子系统。
