Unruled Notebook

It can be argued that one of the most influential articles ever published in the Journal of Applied Physiology is the Analysis of tissue and arterial blood temperatures in the resting human forearm by Harry H. Pennes, which appeared in Volume 1, No. 2, published in August, 1948. Thus begins Prof. Wissler, his 1998 revisit to this classic paper by H. H. Pennes. In that 1948 paper, he proposed what can be identified today as the first analytical Bioheat transfer model with experimental validation from temperature variation data in human forearm. Many later models have refined what he proposed but his basic insight that blood is a carrier of heat, adding a distinct perfusion term to the the standard heat equation, remains a major contribution.

The schematic of the experiment is shown in the accompanying figure. After anesthetizing the right forearm of each human subject, Pennes measured the temperature variation from the surface skin into the deep muscle. During experiment, as Pennes reports in an understatement, ’phlegmatic subjects occasionally reported no unusual pain’.

For all the subjects, temperature measurements were done by placing their right forearm in a fixture that also controlled the exact location of the Y-type thermocouple locations inside the forearm. For the sake of those of us who pause at antique and wonder about proper infrastructure to perform any fruitful experiment, I have given below a low resolution version of the fixture that Pennes used.

Click on the above image for a larger picture with legend. In short, between the upright clamps is where you place your forearm with your elbow resting on this side while your fingers are on the far side. The thermocouple wire is inserted across your forearm and held taut between the clamps marked K. If the wire is not taut, it got deformed in the tissue inside the arm causing minor error in the measured temperatures, as Pennes observes.

The room temperature in all the experiments of Pennes varied between 26.1^∘C and 27.4^∘C (variation within ~ 1.5^∘C) and the thermocouple accuracy was within 0.01^∘C. The initial part of the paper discusses in detail, measurement of the forearm skin surface temperatures. He also measured the rectal and brachial arterial temperatures. Thermocouple readings were recorded after 3 to 4 hours as otherwise the alcohol used on the skin surface before piercing caused 4 to 5^∘C temperature drop.

In the later sections Pennes describes his extensive experiments to measure temperature data along the interior of the forearm. Typical steady state temperature measurement are as shown in the picture below. Data for three subjects are shown in the picture. In principle, the thermocouple recorded the tissue (muscle) temperatures. The trough in the middle of the curve for the atypical subject is attributed by Pennes to the possible presence of an artery near the thermocouple location.

In order to explain his forearm tissue temperature data, Pennes was able to suggest a suitable modification to the standard Heat Equation through the introduction of a blood perfusion term. Further, the Pennes bioheat equation was developed specifically for the arm treated as an axi-symmetric cylindrical medium with uniform thermo-physical properties of blood and tissue assumed along with uniform metabolic heating in arm tissue. In revised form, what is now called the Pennes Bioheat Equation is written as

The terms in the above equation are explained in the legend below.

In particular, the _P ^′′′ is a lumped parameter called the blood perfusion source term that determines the heat transfer between the blood and the tissue. Interestingly, apart from T_t, the unknown tissue temperature present in the bio-heat equation, the blood perfusion term introduces two more unknowns, T_a, the arterial inlet blood temperature and T_v, the venous blood temperature. Pennes devised an equilibrating parameter through which he related T_v and T_t in the form T_v = T_t + λ(T_a – T_t). Here, λ is the degree of thermal equilibrium between venous blood in tissue and tissue itself. From this relation it is evident when λ = 0, venous blood and tissue are in thermal equilibrium and T_v = T_t; when when λ = 1, blood is not exchanging heat with the tissue and T_v = T_a.

Pennes further assumed Ta uniform within domain and equal to arterial blood temperature at inlet. He proposed to relate T_a to the temperature he measured at the inlet brachial artery (T_a0) of each forearm. In principle, blood can heat or cool the tissue, depending on the difference between the incoming arterial blood and the tissue temperature.

The steady state form of the above bioheat equation can be solved analytically in cylindrical coordinates for boundary conditions:

where h is a combined convection/radiation heat transfer coefficient between the skin surface and the surroundings.

The solution of the Pennes bioheat equation in these radial coordinates is

where

and

Here, I₀ and I₁ are the modified Bessel functions.

Because blood perfusion rate ^′′′ could not be measured by Pennes, he varied parameters to match the analytical solution with his experimental data. For T_a0 = 36.25^∘C, assuming λ = 0 (tissue-blood thermal equilibrium), best agreement between experimental data and analytical prediction resulted for 1.2 < ^′′′∕ρ _t < 1.8. The comparison of the analytical curves with the data is shown below.

Importantly, it can be observed that when the blood perfusion is set as zero (bottom-most curve in the figure), the analytical results markedly under-predicts the actual tissue temperature data. This conclusively proves the blood flow in and out of tissue is a major heat transport mechanism in living tissues. This is a major contribution of H. H. Pennes and his bioheat equation.

For completion, there have been enough criticisms and modifications to the Pennes Bioheat equation, some grounded on later date experiments, others on first principles. Even Pennes enunciated some of the shortcoming of his model like the assumption of uniform metabolic heating, perfusion rate and thermal conductivity. But the introduction of the concept of blood perfusion by Pennes as a carrier of heat in living tissues until date remains undisputed and original.

References

1. Introduction to Bioheat transfer
2. E. H. Wissler, Pennes 1948 paper revisited, J. Appl. Physiol. 85, 35-41, 1998. [link]
3. PENNES HH (1948). Analysis of tissue and arterial blood temperatures in the resting human forearm. Journal of applied physiology, 1 (2), 93-122 PMID: 18887578 [link]
4. Mathematical Models of Bioheat Transfer by Caleb K . Charny, pp. 19 – 156, in Bioengineering Heat Transfer, Advances in Heat Transfer, v. 22, Eds. Cho et al., Academic Press, 1992.