de Munck J C, Peters M J
Low Temperature Department, University of Twente, Enschede, The Netherlands.
IEEE Trans Biomed Eng. 1993 Nov;40(11):1166-74. doi: 10.1109/10.245635.
A series expansion is derived for the potential distribution, caused by a dipole source in a multilayered sphere with piecewise constant conductivity. When the radial coordinate of the source approaches the radial coordinate of the field point the spherical harmonics expansion converges only very slowly. It is shown how the convergence can be improved by first calculating an asymptotic approximation of the potential and using the so-called addition-subtraction method. Since the asymptotic solution is an approximation of the true solution, it gives some insight on the dependence of the potential on the conductivities. The formulas will be given in Cartesian coordinates, so that difficulties with coordinate transformations are avoided. Attention will be paid to the (fast) computation of the partial derivatives of the potential, which is useful for inverse algorithms.
推导了由具有分段恒定电导率的多层球体中的偶极子源引起的电势分布的级数展开式。当源的径向坐标接近场点的径向坐标时，球谐展开的收敛非常缓慢。展示了如何通过首先计算电势的渐近近似并使用所谓的加减法来改善收敛性。由于渐近解是真实解的近似，它给出了电势对电导率依赖性的一些见解。公式将以笛卡尔坐标给出，从而避免坐标变换的困难。将关注电势偏导数的（快速）计算，这对逆算法很有用。
