An analytical study of 'Poisson conduction shape factors' for two thermally significant vessels in a finite, heated tissue

To conveniently and properly account for the vessel to vessel and vessel to tissue heat transfer rates to predict in vivo tissue temperature distributions, this paper analyses two different types of Poisson conduction shape factors (PCSFs) for unheated and/or uniformly heated, non-insulated, finite tissue domains. One is related to the heat transfer rate from one vessel to another (vessel-vessel PCSF (VVPCSF)) and the other is related to the vessel to tissue heat transfer rates (vessel-tissue PCSF (VTPCSF)). Two alternative formulations for the VTPCSFs are studied; one is based on the difference between the vessel wall and tissue boundary temperatures, and the other on the difference between the vessel wall and the average tissue temperatures. The effects of a uniform source term and of the diameters and locations of the two vessels on the PCSFs are studied for two different cases: one, when the vessel wall temperatures are lower than the tissue boundary temperature, i.e., the vessels cool the tissue, and vice versa. Results show that, first, the VVPCSFs are only geometry dependent and they do not depend on the applied source term and the vessel wall and tissue boundary temperatures. Conversely, the VTPCSFs are strong functions of the source term and of the temperatures of the vessel walls and tissue boundary. These results suggest that to account for the vessel to vessel heat transfer rates, the VVPCSFs can be evaluated solely based on the vessel network geometry. However, to account for the vessel to tissue heat transfer rates, the VTPCSFs should be used iteratively while solving for the tissue temperature distributions. Second, unlike the tissue boundary temperature-based VTPCSFs which may become singular only in heated tissues, the average tissue temperature-based VTPCSFs have the potential to become singular in both unheated and heated tissues. These results suggest that caution should be exercised in the use of the VTPCSFs since they may approach singularity by virtue of their definition and thus may introduce large errors in the evaluation of tissue temperature distribution. Presented results are new and complementary to the previous shape factor results since these include the effect of (1) source term and (2) unequal vessel-tissue heat transfer rates from the two vessels to the tissue