Macroscale thermal models have been developed for biological tissues either by the mixture theory of continuum mechanics or by the porous-media theory. The former uses scaling-down from the global scale; the latter applies scaling-up from the microscale by the volume averaging. The used constitutive relations for heat flux density vector include the Fourier law, the Cattaneo–Vernotte (Cattaneo, C., 1958, “A Form of Heat Conduction Equation Which Eliminates the Paradox of Instantaneous Propagation,” Compt. Rend., 247, pp. 431–433; Vernotte, P., 1958, “Les Paradoxes de la Théorie Continue de I’equation de la Chaleur,” Compt. Rend., 246, pp. 3154–3155) theory, and the dual-phase-lagging theory. The developed models contain, for example, the Pennes (1948, “Analysis of Tissue and Arterial Blood Temperature in the Resting Human Forearm,” J. Appl. Physiol., 1, pp. 93–122), Wulff (1974, “The Energy Conservation Equation for Living Tissues,” IEEE Trans. Biomed. Eng., BME-21, pp. 494–495), Klinger (1974, “Heat Transfer in Perfused Tissue I: General Theory,” Bull. Math. Biol., 36, pp. 403–415), and Chen and Holmes (1980, “Microvascular Contributions in Tissue Heat Transfer,” Ann. N.Y. Acad. Sci., 335, pp. 137–150), thermal wave bioheat, dual-phase-lagging (DPL) bioheat, two-energy-equations, blood DPL bioheat, and tissue DPL bioheat models. We analyze the methodologies involved in these two approaches, the used constitutive theories for heat flux density vector and the developed models. The analysis shows the simplicity of the mixture theory approach and the powerful capacity of the porous-media approach for effectively developing accurate macroscale thermal models for biological tissues. Future research is in great demand to materialize the promising potential of the porous-media approach by developing a rigorous closure theory. The heterogeneous and nonisotropic nature of biological tissue yields normally a strong noninstantaneous response between heat flux and temperature gradient in nonequilibrium heat transport. Both blood and tissue macroscale temperatures satisfy the DPL-type energy equations with the same values of the phase lags of heat flux and temperature gradient that can be computed in terms of blood and tissue properties, blood-tissue interfacial convective heat transfer coefficient, and blood perfusion rate. The blood-tissue interaction leads to very sophisticated effect of the interfacial convective heat transfer, the blood velocity, the perfusion, and the metabolic reaction on blood and tissue macroscale temperature fields such as the spreading of tissue metabolic heating effect into the blood DPL bioheat equation and the appearance of the convection term in the tissue DPL bioheat equation due to the blood velocity.
A Review of Mixture Theory for Deformable Porous Media and Applications