The matching of thermal fields surrounding countercurrent microvessels and the closure approximation in the Weinbaum-Jiji equation.

A second order perturbation theory is developed to show the difference between the average artery-vein temperature Tm and the local average tissue temperature theta. This theory demonstrates that the closure approximation in the Weinbaum-Jiji bioheat equation does not require that theta = Tm and that although the difference between these two temperatures is second order the magnitude of the countercurrent convection terms in the Weinbaum-Jiji equation can be of order unity. The theory also shows that to second order this new bioheat equation is the same as the simplified set of one-dimensional model equations used in Baish et al.