Analytical solutions of Laplace equations have given the electrical characteristics of membranes and interiors of spherical, ellipsoidal, and cylindrical cells in suspensions and tissues from impedance measurements, but the underlying assumptions may be invalid above 50% volume concentrations. However, resistance measurements on several nonconducting, close-packing forms in two and three dimensions closely predicted volume concentrations up to 100% by equations derived from Maxwell and Rayleigh. Calculations of membrane capacities of cells in suspensions and tissues from extensions of theory, as developed by Fricke and by Cole, have been useful but of unknown validity at high concentrations. A resistor analogue has been used to solve the finite difference approximation to the Laplace equation for the resistance and capacity of a square array of square cylindrical cells with surface capacity. An 11 x 11 array of resistors, simulating a quarter of the unit structure, was separated into intra- and extra-cellular regions by rows of capacitors corresponding to surface membrane areas from 3 x 3 to 11 x 11 or 7.5% to 100%. The extended Rayleigh equation predicted the cell concentrations and membrane capacities to within a few percent from boundary resistance and capacity measurements at low frequencies. This single example suggests that analytical solutions for other, similar two- and three-dimensional problems may be approximated up to near 100% concentrations and that there may be analytical justifications for such analogue solutions of Laplace equations.
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机译:Laplace方程的解析解通过阻抗测量给出了悬浮液和组织中球形,椭圆形和圆柱状细胞的膜和内部的电学特性,但高于50%的体积浓度,基本假设可能无效。但是,通过从麦克斯韦和瑞利派生的方程式,在二维和三维的几种不导电,紧密堆积的形式上进行的电阻测量可紧密预测体积浓度高达100%。由Fricke和Cole开发的理论扩展计算悬浮液和组织中细胞的膜容量是有用的,但在高浓度下有效性尚不清楚。电阻器类似物已被用来求解拉普拉斯方程的有限差分近似值,以计算具有表面电容的方形圆柱单元的方形阵列的电阻和电容。一个11 x 11的电阻器阵列模拟了四分之一的单元结构,并通过对应于3 x 3至11 x 11或7.5%至100%的表面膜面积的电容器行分成了细胞内和细胞外区域。扩展的瑞利方程式预测细胞浓度和膜容量在低频下的边界电阻和容量测量值在百分之几以内。这个单一的例子表明,针对其他类似的二维和三维问题的解析解可能近似于接近100%的浓度,并且对于拉普拉斯方程式的此类类似解可能存在解析依据。
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