Fractal analysis of the effective thermal conductivity of biological media embedded with randomly distributed vascular trees

摘要

The biological tissue can be treated as a porous medium consisting of randomly distributed vascular trees and extra-vascular tissue matrix. In this paper, a fractal model for the effective thermal conductivity of biological media is derived based on the assumptions that the mother channel diameters of vascular trees follow the fractal scaling law. It is found that the analytical expression for the effective thermal conductivity of biological media is a function of the thermal conductivity ratio of extra-vascular tissue matrix to blood, porosity and structural parameters of the vascular trees, and there is no any empirical constant in the proposed model. Moreover, the relationships among them are also discussed. The results indicate that structural parameters of the vascular trees and porosity have significant effects on the effective thermal conductivity. It is found that the optimal diameter ratio β_m = 0.707≈ 2~(-1/2) in biological media embedded with randomly distributed symmetric two branching vascular trees also exactly satisfies the expression d~Δ = d_1~Δ +d_2~Δ with Δ = 2, but is different from Murray's law. A good agreement between the present model predictions and available experimental data is obtained. The proposed fractal model may have the potential in prediction of temperature distributions in therapy region in tumor and enhancement of efficacy in hyperthermia.