An FDTD algorithm for transient propagation in biological tissue with a Cole-Cole dispersion relation

The finite difference time domain (FDTD) method has become a popular numerical technique for analyzing the propagation of electromagnetic fields in the human body. Because of the high water content in many biological tissues, a Debye dispersion relation has often been used to describe the frequency variation in their dielectric properties. However, an accurate representation over a broad frequency range usually requires using a linear combination of several Debye functions. An alternative is to describe the frequency dependence using the Cole-Cole dispersion relation. While a number of frequency dependent FDTD formulations have been developed for Debye and Lorentz media, the Cole-Cole dispersion relation has not received nearly as much attention. In the approach presented here, the Cole-Cole dispersion relation is transformed into a time domain relation between the electric field and polarization current which involves a convolution integral. A recursive update of the convolution integral is made possible by approximating a time series by a sum of decaying exponentials.