A rigorous derivation of the bioheat equation for local tissue heat transfer based on a volume averaging theory

A general three-dimensional bioheat equation for local tissue heat transfer has been derived with less assumptions, exploiting a volume averaging theory commonly used in fluid-saturated porous media. The volume averaged energy equations obtained for the arterial blood, venous blood and tissue were combined together to form a single energy equation in terms of the tissue temperature alone. The resulting energy equation turns out to be remarkably simple as we define the effective thermal conductivity tensor, which accounts not only for the countercurrent heat exchange mechanism but also for the thermal dispersion mechanism. The present equation for local tissue heat transfer naturally reduces to the Weinbaum-Jiji equation for the unidirectional case. List of symbols Aint Interface between the blood vessel and tissue (m ) aa,v Specific surface area (1/m) Cs Tissue specific heat (J/kgK) cpf Blood specific heat at constant pressure (J/kgK) d Blood vessel diameter (m) e Spacing between centers of neighboring vessels (m) ha,v Interfacial heat transfer coefficient (W/m K) k Thermal conductivity (W/mK) li Local unit vector along the vessel axis (-) nj Unit vector pointing outward from the fluid side to solid side (-) Sh Metabolic reaction rate (W/m ) t Time (s) T Temperature (K) uB Bulk mean blood velocity (m/s) ui Blood velocity vector (m/s) V Representative elementary volume (m) xi Cartesian coordinates (m) ea,v Porosity (-), volume fractions of arterial blood phase and venous blood phase qf,s Density (kg/m ) x0 Perfusion bleed-off rate (1/s) Special symbols ~ / Deviation from intrinsic average / h i Volume average / h i ;s;a;v Intrinsic average Subscripts and superscripts a Artery dis Dispersion f Fluid s Tissue stag Stagnant v Vein