Heat is continuously generated in the human body by metabolic processes and exchanged with the environment and among internal organs by conduction, convection, evaporation and radiation. Transport of heat by the circulatory system makes heat transfer in the body — or bioheat transfer — a specific branch of this general science. As in all entities, the principle of conservation of energy yields:
where the terms denote, from left to right, the rate of heat generation due to metabolic processes; rate of heat stored in body tissues and fluids; heat lost to the environment and adjacent tissues; and the rate of work performed by the tissue. This latter quantity is usually negligible at the tissue level.
In the tissue element shown in Figure 1, heat due to metabolic processes (5–10,000 W/m3) is generated at a variable rate, which, when integrated over the entire control volume, obtains:
where denotes the spatial coordinate and t is time.
Under unsteady conditions, part of the heat flow will be stored in the control volume:
where ρ is tissue density (900–1600) kg/m3); c is tissue specific heat (2.1–3.8 kJ/kgK); and T is tissue temperature.
The term representing heat lost to adjacent tissues and to the environment contains a number of components, one of which is heat exchanged by diffusion (Fourier’s Law of conduction):
where k is the thermal conductivity of the tissue (0.29–1.06 W/mK); T is tissue temperature gradient; is outward-pointing unit vector; and A is control volume surface area.
A second component of the heat lost to adjacent tissues is due to blood perfusion. Blood circulates in a variety of vessels ranging in lumen diameter from the 2.5 cm aorta to the 6–10 mm capillaries. Due to this four-fold size distribution, heat transport effects of blood are coupled to the specific group of vessels under consideration. A common approach to modeling this effect is to assume that the rate of heat taken up by the circulating blood at the capillary level equals the difference between the venous and arterial temperatures times the flow rate (Fick’s Law):
where rbcb is blood heat capacity (≈4000 kJ/m3K) and wb is volumetric blood perfusion rate (0.17–50 kg/m3s). At the capillary level, blood flow velocity is very slow and thermal equilibration with surrounding tissue occurs. Thus, Eq. (5) may be modified by setting Tv = T (Pennes, 1948). When all the terms are substituted into Eq. (1) and integrated over the entire volume and surface area, the well-known bioheat equation is obtained:
Equation (6) has been very useful in the analysis of heat transfer in various body organs and tissues characterized by a dense capillary bed. Other thermal effects due to blood flow are not adequately accounted for by Eq. (6), including: 1) countercurrent heat transfer between adjacent vessels; 2) directionality effects due to the presence of larger blood vessels; and 3) heat exchange with larger vessels in which complete thermal equilibrium may not be assumed. These issues have been analyzed by Chen and Holmes (1980) and by Weinbaum et al. (1984).
Heat is exchanged with the environment through a complex combination of conduction, convection, radiation and evaporation. Clothing worn by humans and natural integuments also play a role. As a good approximation, these effects may be calculated by an equation of the form:
where is the amount of heat exchanged and hi is the heat exchange coefficient (2.3–2.7 W/m2K for free convection, 7.4V0.67 for forced convection, where V is wind velocity, m/s, and 3.8–5.1 for radiation); is average body surface temperature; To is environmental temperature; and AD is Dubois’ body surface given by:
where m is body mass in kg and h is height in m.
Chen, M. M. and Holmes, K. R. (1980) Microvasculature Contributions in Tissue Heat Transfer, In: Thermal Characteristics of Tumors: Applications in Detection and Treatment, R. K. Jain and P. M. Gullino Eds., Ann. N.Y. Acad. Sci., 335:137–150.
Pennes, H. H. (1948) Analysis of Tissue and Arterial Blood Temperatures in the Resting Human Forearm, J. Appl. Physiol., 1:93–122.
Shitzer, A. and Eberhart, R. C., Eds. (1985) Heat transfer in medicine and biology — analysis and applications, Plenum Press, New York.
Weinbaum, S., Jiji, L. M., and Lemons, D. (1984) Theory and Experiment for the Effect of Vascular Microstructure on Surface Heat Transfer, ASME J. Biomech. Eng., 106:321–330 (Pt. 1); 331–341 (Pt. 2).