Awada K A, Jackson D R, Williams J T, Wilton D R, Baumann S B, Papanicolaou A C
Department of Neurological Surgery, University of Pittsburgh, PA 15213, USA.
IEEE Trans Biomed Eng. 1997 Aug;44(8):736-52. doi: 10.1109/10.605431.
A comparison is made of two different implementations of the finite element method (FEM) for calculating the potential due to dipole sources in electroencephalography (EEG). In one formulation (the direct method) the total potential is the unknown that is solved for and the dipole source is directly incorporated into the model. In the second formulation (the subtraction method) the unknown is the difference between the total potential and the potential due to the same dipole in an infinite region of homogeneous conductivity, corresponding to the region where the dipole is located. Both methods have the same FEM system matrix. However, the subtraction method requires an additional calculation of flux integrations along the edges of the elements in the computation of the right-hand side (RHS) vector. It is shown that the subtraction method is usually more accurate in the forward modeling, provided the flux integrations are computed accurately. Errors in calculating the flux integrations may result in large errors in the forward solution due to the ill-conditioned nature of the FEM system matrix caused by the Neumann boundary condition. To minimize the errors, closed-form expressions for the flux integrations are used for both linear and quadratic triangular elements. It is also found that FEM forward modeling errors may cause false extrema in the least-square objective function obtained from the boundary potential, near boundaries between media of differing conductivity. Multiple initial guesses help eliminate the possibility of the solution getting trapped in these false extrema.
本文对用于计算脑电图（EEG）中偶极子源所产生电势的有限元法（FEM）的两种不同实现方式进行了比较。在一种公式化方法（直接法）中，总电势是待求解的未知量，偶极子源被直接纳入模型。在第二种公式化方法（减法）中，未知量是总电势与在均匀电导率的无限区域中同一偶极子所产生电势的差值，该无限区域对应于偶极子所在的区域。两种方法具有相同的有限元系统矩阵。然而，减法在计算右侧（RHS）向量时需要额外计算沿单元边缘的通量积分。结果表明，在正向建模中，减法通常更准确，前提是通量积分计算准确。由于诺伊曼边界条件导致有限元系统矩阵的病态性质，通量积分计算中的误差可能会导致正向解出现较大误差。为了最小化误差，对于线性和二次三角形单元，通量积分均使用闭式表达式。还发现，有限元正向建模误差可能会在不同电导率介质之间的边界附近，由边界电势得到的最小二乘目标函数中导致虚假极值。多个初始猜测有助于消除解陷入这些虚假极值的可能性。
