h. h. pennes’ study of tissue heating in the human forearm, which appeared in volume 1, issue 2 of the*Journal of Applied Physiology* (10), is a landmark work that has the curious attribute of a higher citation frequency today than when first published fifty years ago.1It is the inception of the “Pennes bioheat transfer equation,” deeply ingrained in the literature of physiological heat transfer modeling, which accounts for the paper’s longevity. Thus Wissler’s reexamination of Pennes’ work (14) has both historical significance and current scientific relevance.

Pennes’ paper had two principal components:*1*) the report of a series of subcutaneous, deep tissue, and arterial temperature measurements in the forearms of unanesthetized subjects; and*2*) the evaluation of a mathematical model, based on an energy balance for an arbitrary tissue volume. In the Pennes model, the rate of tissue-energy increase is given by the sum of the net heat conduction into the tissue, metabolic heat generation, and heating (or cooling) effects of the arterial supply.

The experimental results are truly invaluable; such invasive measurement procedures might not be feasible today. The forearm temperature measurements appeared to validate the Pennes model. Despite some questionable assumptions about the nature of blood flow in perfused tissue, the model seemed to work quite well.

However, Wissler’s scrutiny (14) points out some fairly significant problems with Pennes’ analysis (10). One potentially serious shortcoming is the use of substantially incorrect tissue-property data. Pennes used a tissue thermal conductivity value, which is only one-third that of water. The range of tissue perfusion rates examined (1.2–1.8 g ⋅ min^{−1} ⋅ 100 ml^{−1}) is also low; the values probably should be twice as high. Thus Pennes apparently underestimated the magnitudes of the conduction and blood flow terms in the energy balance by comparable amounts (one-half to two-thirds). For a steady-state analysis, the only remaining term is metabolic heating, which is small. Thus serendipity prevailed—the errors are essentially offsetting, and (as Wissler shows) the “correct” parametric values do not yield substantially different temperature results.

In addition, there appears to be a fundamental flaw in Pennes’ presentation of the temperature data. Subcutaneous and deep-tissue temperatures were measured by using guided thermocouples to traverse the medial-lateral axis in the forearms of nine male subjects. Wissler (14) presents convincing evidence that Pennes (10) did not normalize the radial coordinates of the measurements but simply plotted all temperature data collectively as a function of absolute distance (cm) from the presumed center axis of the limb. This is an astounding error in a series of experiments that are otherwise quite well conceived and apparently carefully performed. The failure to normalize the data resulted in comparing subcutaneous temperatures in some subjects with surface temperatures in others while implying that the measurements were from corresponding locations. However, Wissler’s analysis (14) demonstrates that—again fortuitously—a “proper” scaling of Pennes’ temperature data still yields quite good agreement with the model.

Beyond these analytic problems, some basic conceptual difficulties with the Pennes model have been identified. These issues have been discussed extensively in the literature and motivated numerous alternative models. Reviews and summaries of the relevant modeling issues are found in Refs. 1 and 2. The fundamental criticism of the Pennes model is that the treatment of the blood flow term as a distributed heat source/sink mistakenly presumes that the capillary bed is the principal site of heat exchange. In fact, analytic evidence strongly indicates that the temperature equilibrates by the time the blood reaches the arterioles (4, 5), and the heat exchange in the capillaries is therefore small.

Pennes wrote the arterial-tissue heat transfer term in the form ρ_{b}C_{b}V˙ (κ − 1) ⋅ (T − T_{a}), where ρ_{b} and C_{b} are the mass density and specific heat of blood, respectively, and V˙ is the (local) volume flow rate of blood, per unit volume of tissue. The quantity T_{a} is the arterial temperature (presumably, the core temperature), and T is the local tissue temperature. The parameter κ expresses the extent to which the arterial blood thermally equilibrates with the tissue. Formally, it is defined as κ = (T_{v}− T)/(T_{a} − T), where T_{v} is “the temperature of the venous blood leaving the tissue” (10). Thus a value of κ = 0 is consistent with the view of the capillary bed as a heat source (or sink) for the tissue. In that scenario, heat loss from the arterial blood is negligible before it enters the capillaries. Complete thermal equilibration with the surrounding tissue occurs in the capillary bed, and the exiting blood temperature T_{v} equals the local tissue temperature.

The other extreme value, κ = 1, corresponds to zero heat transfer between the arterial blood and the perfused tissue; i.e., the venous temperature equals the deep arterial temperature. This would preclude heat transfer between arteriovenous vessel pairs and between the capillary bed and surrounding tissue. [Note: Pennes incorrectly states (see p. 6 in this issue of the Journal) that a value of κ approaching unity implies “complete...equilibration between capillary blood and tissue,” although the subsequent analysis is consistent with the correct interpretation.]

Pennes assumed “...the physical conditions of the capillary circulation favor almost complete equilibration [between tissue and capillary blood].” This was the justification for assuming a uniform value of κ = 0 throughout the tissue, eliminating consideration of direct arteriovenous heat exchange. However, a nonzero value of κ would account for some level of countercurrent exchange between small vessel pairs. The difficulty, of course, is determination of an appropriate value for the equilibration constant. Ultimately, we have to resort to empiricism.

Given the problems with the Pennes model, what accounts for its widespread acceptance and use? One advantage is its ultimate simplicity. If one assumes, as Pennes did, that the equilibration parameter κ is zero everywhere, then a very simple field problem results. Given the relevant properties and perfusion rates, it becomes fairly easy to solve for tissue temperature as a function of spatial location and time. The alternative to the Pennes equation is to employ a decidedly more complex model that explicitly describes heat exchange between vessel pairs. Keller and Seiler (8) proposed such a model, which includes both countercurrent heat exchange between vessel pairs and thermal equilibration in the capillary bed. An alternative model developed by Chen and Holmes (5) replaces the single-perfusion term in the Pennes equation with three terms, requiring substantially more detailed anatomic knowledge. Weinbaum et al. (12, 13) incorporated incomplete countercurrent exchange in conjunction with a vessel “bleed-off” term, which is mathematically similar to the Pennes perfusion term. Other models are much more rigorous and do not introduce some of the gross approximations of the Pennes model but may require significantly greater computing resources (6).

Of course, the virtue of simplicity loses much of its luster if it introduces great inaccuracies. It is on this point, perhaps, where Wissler’s analysis (14) is most valuable. His calculations, when compared with Pennes’ measurements (10), demonstrate good agreement. This validation of the Pennes model is confirmed by other studies; the model has shown consistency with observations when applied to perfused phantoms (3) and to simulating temperature fields in the human brain (9, 11).

This consistency between predictions and measurements is truly remarkable when viewed in the context of the gross simplifications inherent to the model. For many practical applications, the simplicity of the Pennes model is appropriate to the required accuracy and the level of detailed anatomic knowledge available.

## Footnotes

- Copyright © 1998 the American Physiological Society