[精品]边界元法求解脑电正问题初探

边界元法求解脑电正问题初探-A Modifield Boundary Element Methed for the Estimation of Potential Fields on the Scalp,赵旭 电子科技大学生命科学与技术学院 20041202,内容提要,1课题及边界元法简介2Modifield BEM 简介3Modifield BEM 的实现4初步总结以及展望,1课题及边界元法简介,电流偶极子源的假设脑电正问题（Forward Problem) 脑电逆问题 (Inverse Problem),1课题及边界元法简介,边界元法的一般原理 许多物理问题可通过不同的途径归结为不同的 数学模型（大多数没有解析解）： a 偏微分方程的边值问题有限差分法 b 区域上的变分问题有限元法 C 边界上的积分问题边界元法,1课题及边界元法简介,边界元法是将区域内微分方程通过积分定理变为边界上的积分方程再将积分方程在边界上离散为代数方程。,2Modifield BEM 简介,Let us start by formulating the BEM equations asWhere Vi and gi are vectors of potentials on the Ith surface,and Bij is a matrix.,2Modifield BEM 简介,2Modifield BEM 简介,Consider the expression for the potentials on the inner surface S3 of a three-surface head model For V1 and V2 are much more smaller than V3,we can ignore V1 and V2 entirely and still achieve a good estimate of V3 .,2Modifield BEM 简介,By Greens theorem we can get :Pxy a matrix with coefficients equal to pitimes the solid angle subtended by an element of unit area on surface y at a point on surface x;Gxy a matrix with coefficients equal to ds an areal element of surface y, r the distance between a point on surface x and element ds; the normal component of the gradient of V3,2Modifield BEM 简介,First,since the surfaces are relatively smooth ,the self matrices,P11,P33,and G33 are strongly diagonal.Also ,as already noted,we do not have to maintain the accuracy of the magnitude of the scalp field,so we can take the self matrices as identity matrity matrices.Thus,2Modifield BEM 简介,We can get:Premultiply by G13 to giveHowever ,all the terms of the matrix product G13P13 are much smaller than one and,2Modifield BEM 简介,(I-G13P31) is dominated by the identity matrix,so,3Modifield BEM 的实现,采用三层球模型并归一化半径为一。每层都离散成642个点，1280个三角形。第一层为头皮层，二层为颅骨层，三层为大脑组织层。三层的电导率分别取为：1；1/80；1,3Modifield BEM 的实现,由上面的分析可知要求头皮的电位必须求：g3,B33,P13,G13.G3D为电流偶极子极矩；r0为偶极子位置；r为场点位置。,3Modifield BEM 的实现,B33（I,j）为立体角；k=l=3的计算：a,b,c为相对于参考点的三角形的三个顶点,3Modifield BEM 的实现,P13(I,j)= piB13B13的计算方法与B33相同G13(I,j) pids(j)/r(I,j)ds为三角形的面积；r为点到三角形的距离,4初步总结以及展望,vp=0 0 1; r=0.8 0 0;(jxf),4初步总结以及展望,vp=0 0 1;vr0=0.8 0 0;(BEM),4初步总结以及展望,vp=0 0 1; r=0.5 0 0;(jxf),4初步总结以及展望,vp=0 0 1;vr0=0.5 0 0;(BEM),4初步总结以及展望,Modifield BEM直观上看比起解析解要模糊一些。这个结果不太理想，要仔细