A general bioheat transfer model based on the theory of porous media

Abstract

A volume averaging theory (VAT) established in the field of fluid saturated porous media has been successfully exploited to derive a general set of bioheat transfer equations for blood flows and its surrounding biological tissue. A closed set of macroscopic governing equations for both velocity and temperature fields in intra-and extra-vascular phases has been established, for the first time, using the theory of anisotropic porous media. Firstly, two individual macroscopic energy equations are derived for the blood flow and its surrounding tissue under the thermal non-equilibrium condition. The blood perfusion term is identified and modeled in consideration of the transvascular flow in the extravascular region, while the dispersion and interfacial heat transfer terms are modeled according to conventional porous media treatments. It is shown that the resulting two-energy equation model reduces to Pennes model, Wulff model and their modifications, under appropriate conditions. Subsequently, the two-energy equation model has been extended to the three-energy equation version, in order to account for the countercurrent heat transfer between closely spaced arteries and veins in the circulatory system and its effect on the peripheral heat transfer. This general form of three-energy equation model naturally reduces to the energy equations for the tissue, proposed by Chato, Keller and Seiler. Controversial issues on blood perfusion, dispersion and interfacial heat transfer coefficient are discussed in a rigorous mathematical manner.

The role of mathematical modelling in thermal medicine

“…Nakayama and Kuwahara in 2008 [8] also used a volume averaging approach. The assumptions included: 1) tissues are filled with interstitial blood flow as porous media, 2) local thermal nonequilibrium exists between blood and tissue, and 3) interface integrals need to be simplified.…”

1D generalized dual-phase lag equation. Sensitivity analysis with respect to the porosity

“…The generalized dual-phase lag equation describing the heat transfer processes occurring in the heated tissues is based on the theory of porous media [6,12,13]. In this approach the tissue is divided into two regions: the vascular region (blood vessels) and the extravascular region (tissue).…”

“…Under the assumption that the metabolic heat sources Q, Q b and the blood temperature T b are constant, the generalized dual-phase lag equation for 1D problem takes the form [13,15,16] ( )…”

“…The parameter τ q is the phase lag of the heat flux and parameter τ T is the phase lag of the temperature gradient. Extension of this model is the generalized dual-phase lag equation [12][13][14][15]. In this equation the phase lag times τ q and τ T are expressed in terms of the blood and tissue properties, the interphase convective heat transfer coefficient and the blood perfusion rate.…”

Operating Conditions for the Hemodialysis Treatment Based on the Volume Averaging Theory

“…Under these situations, Sano and Nakayama [17] proposed a membrane transport model based on the volume averaging theory for the analysis of hollow fiber hemodialysis systems. The concept of local volume-averaging theory, namely, VAT, widely used in the study of porous media (Cheng [18], Quintard and Whitaker [19], Nakayama [20], Vafai and Tien [21], Nakayama and Kuwahara [22], Yang and Nakayama [23]) is quite useful under these situations, in which thousands of small-scale elements exist in a large space. Subsequently, Sano et al [24] provided three-dimensional numerical computations for revealing the mass transport phenomena within the three individual phases, namely the blood (lumen), the dialysate (shell) and the membrane phases in the countercurrent hollow fiber dialyzer.…”