Feb 2, 1994 - Department of Applied Sciences, College of Staten Island / CUNY, Staten. Department of Veterinary Biosciences and Bioengineering Faculty. Uni-. Island, NY 10301 ...... Foster at the University of Pennsylvania for suggesting the topic of this review and ..... Assistant Professor with the Bioengineering Faculty.
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IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING. VOL. 41. NO. 2, FEBRUARY 1994
Recent Developments in Modeling Heat Transfer in Blood Perfused Tissues H. Arkin, L. X. Xu and K. R. Holmes
Abstracf- Successful hyperthermia treatment of tumors requires understanding the attendant thermal processes in both diseased and healthy tissue. Accordingly, it is essential for developers and users of hyperthermia equipment to predict, measure and interpret correctly the tissue thermal and vascular response to heating. Modeling of heat transfer in living tissues is a means towards this end. Due to the complex morphology of living tissues, such modeling is a difficult task and some simplifying assumptions are needed. Some investigators have recently argued that Pennes’ interpretation of the vascular contribution to heat transfer in perfused tissues fails to account for the actual thermal equilibration process between the flowing blood and the surrounding tissue and proposed new models, presumably based on a more realistic anatomy of the perfused tissue. The present review compares and contrasts several of the new bio-heat transfer models, emphasizing the problematics of their experimental validation, in the absence of measuring equipment capable of reliable evaluation of tissue properties and their variations that occur in the spatial scale of blood vessels with diameters less than about .2 mm. For the most part, the new models still lack sound experimental grounding, and in view of their inherent complexity, the best practical approach for modeling bio-heat transfer during hyperthermia may still be the Pennes model, providing its use is based on some insights gained from the studies described here. In such cases, these models should yield a more realistic description of tissue locations and/or thermal conditions for which the Pennes’ model might not apply.
I. NOMENCLATURE
A C
d 9
k 1
L n P r! T 9
t T U
U V
cross-section of blood vessel, [m2] heat capacity, [J/kg.K] vessel diameter, [mm] bleed-off mass flow, [rn3/s] thermal conductivity, [W/m.K] length, [mm] actual vessel length, [mm] vessel number density, [#/m2] circumference, [m] volumetric heat, [W/m3] vessel radius, [mm] coordinate along path of small blood vessel. ml time, [SI temperature, [K] average volumetric blood flux density, [m/s] local overall heat transfer coefficient, [W/m‘ blood velocity within vessel, [m/s]
The National Institute for Building Research, Technion, Haifa 32000, Israel. Department of Applied Sciences, College of Staten Island / CUNY, Staten Island, NY 10301. Department of Veterinary Biosciences and Bioengineering Faculty. University of Illinois, Urbana, IL 61801.
Greek letters angle between the direction of a blood vessel y and the local tissue temperature gradient I, /l-ratio between equilibration length and actual E vessel length wave number in Fourier integral [j V Grad o shape factor p tissue density, [kg/m3] blood perfusion, [ml/(ml . s)] w Subscripts a arterial h blood c convective mode e equilibration eff effective i index variable for the spectral component in Fourier integral 711. metabolic p perfusive mode ‘11 venous * largest vessel forming part of continuum in CH model conduction between tissue and both countercurrent C vessels A conduction between two countercurrent vessels Superscripts vector; tensor 11. INTRODUCTION
Thermal models for blood perfused tissues have been used in a wide range of applications. Several simplified models quantify conditions for human thermal comfort and/or stress [I], [2], while others combine thermal modeling with empirical formulae for physiological thermoregulatory mechanisms. Among others, Stolwijk et al [3] developed a model in which the whole body is subdivided into compartments, each with a uniform temperature. Wissler [4] provided a more detailed description of body temperature using partial differential equations-an approach subsequently expanded by Arkin et al [SI into a two-dimensional model to cope with asymmetric environmental thermal loads. Gordon et a1 [6] modeled human thermal response to cold environment. For medical applications, thermal modeling has been widely used in analysis of bums [7], [8], cryosurgery [91, [IO], hypothermia [ 1 I], and in measurement of tissue thermal prop-
0018-9294/94$04.00 @ 1994 IEEE
IEEE TRANSACTIONS ON BIOMEOICAL ENGINEERING, VOL. 41. NO. 2, FEBRUARY 1994
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erties [ 121 and tissue blood flow [13]-[ 151; a comprehensive review was published by Shitzer and Eberhart [16]. Much work has been devoted to modeling of body thermal response as related to detection [ 171, [ 181 and treatment [ 191, [20] of cancer by hyperthermia [7]. Accordingly, several thermal modes of hyperthermia have been developed (including its combination with other techniques, such as chemotherapy or radiation [21], [22]) with provision for minimizing thermal damage to healthy tissues. The method of heat application depends on the type of tumor, its location and its size [23]. Elaboration on the various modes of hyperthermia is beyond the scope of the present review. Investigators and practitioners commonly accept that adequate thermal modeling is essential for effective treatment by hyperthermia. Accordingly, models have been developed for predicting and simulating thermal responses to whole-body [18], [24], 1251 and local [261-[28] hyperthermia, and for evaluating and designing various protocols for hyperthermia applicators [29]-[31]. This requires knowledge of the spatial and temporal patterns of the thermal properties and of blood perfusion within the diseased and the surrounding healthy tissue. To this end, investigators have developed techniques for quantifying thermal conductivity [32], [ 3 3 ] and blood flow 134]-[36] inside tumors and for on-line, simultaneous prediction of the temperature field and blood perfusion during hyperthermia treatments 1371. Many publications on these topics have appeared over the past few years. At present, it appears that the relatively limited success of hyperthermia in treating tumors is due, in part, to the technical difficulties of monitoring and controlling tissue temperature during treatment, and to insufficient knowledge of the heat transfer processes in blood perfused tissues. The microvascular geometry in tumors is very complicated and quantitative data on this irregular structure is sparse. As a consequence, attempts to better understanding of heat transfer mechanisms in tumors. have relied primarily upon information gained on the vascular geometry in healthy tissues. The discussion here refers to several proposed models that claim to provide more precise descriptions of heat transfer in living healthy tissues and covers the material published prior to 1991. The evaluation is based on the authors’ understanding of the hypotheses regarding heat transfer processes in living tissues, as well as on their experience with the tissue vasculature. 111. FORMULATION OF BIOHEAT TRANSFER MODELS
A. The Pennes (BHT) Model
The Pennes model (1948) for describing heat transfer in a tissue, originally designed for predicting temperature fields in the human forearm, has become well known as the “bio-heat transfer” (BHT) equation [38]:
Eqn. 1 includes a term on the right-hand side that represents the contribution of flowing blood to the overall energy balance. Pennes’ primary premise was that energy exchange between blood vessels and the surrounding tissue occurs mainly across
the wall of capillaries (blood vessels with 0.0054.015mm in diameter), where blood velocity is very low. He assumed, therefore, that the thermal contribution of blood can be modeled as if it enters an imaginary pool (the capillary bed) at the temperature of major supply vessels, Ta,and immediately equilibrates (thermally) with the surrounding tissue. Thus it exits the “pool” and enters the venous circulation at tissue temperature, T . He postulated therefore, that the total energy exchange by the flowing blood can be modeled as a nondirectional heat source, whose magnitude is proportional to the volumetric blood flow and the difference between local tissue and major supply arterial temperatures.
B . Early Models Other investigators have attempted to quantify the thermal interaction between a large blood vessel and its surrounding tissue. Among these, Mitchell and Myers [39] were perhaps the first to investigate the effect of countercurrent heat exchange between an artery and an adjacent vein. Their model incorporated equations describing the heat exchange between the vessels and between each vessel and its surrounding tissue. However, their model did not cover heat conduction within the tissue itself. In addition, the assumption of constant blood flow rate restricted the applicability of the model to the main supply/collection vessels. The separate treatment of tissue and vessels has served as a point of departure from the Pennes’ concept for most of the recent studies in tissue heat transfer. Keller and Seiler 1401 not only incorporated the countercurrent heat exchange between the major artery and vein as had Mitchell and Myers [39] when modeling peripheral heat transfer in humans, but they also added an energy conservation equation for the surrounding tissue which coupled with the artery and vein equations. As they adhered to the conception of Pennes’ model regarding the thermal significance of capillaries, the tissue equation included a “sink” term to account for the thermal contribution of the blood. Wisslcr [4]. in his whole body model, also recognized that thermal variations in large blood vessels should be treated separately from those in the tissue. His model consisted of three coupled equations; the first, similar to the BHT equation (Eqn. I), described the energy balance of the perfused tissue and the other two modeled the heat transfer within a large artery and a large vein. However, the actual distinction between “large” and “small” vessels was not based on thermal analysis, but rather was defined qualitatively, based on vessel dimensions. During the last decade several investigators have questioned the physical and physiological validity of the assumptions underlying the Pennes bioheat equation, in particular the thermal contribution of the flowing blood. The discussion seems to have been initiated by Wulff (421, who claimed that the convective effect originates from the net blood flux within the tissue. Thus the blood flow contribution must be modeled by a directional term of the form ( p c ) b T i . V T rather than the scalar perfusion term suggested by Pennes [38]. Klinger [43] subsequently emphasized the convective contribution to heat transfer caused by the blood flow inside the vessels. In place
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ARKIN et a/ RECENT DEVELOPMENTS IN MODELING HEAT TRANSFER IN BLOOD PERFUSE0 TISSUES
of Pennes' perfusion term, he suggested a convective term that takes the form ( p C ) b v ( ? , t ) . V T within the vessels.
C. Thermal Equilibration Length ( E , ) of Blood Vessels. An imperative step towards clarification of the heat transfer mechanisms in living tissues was achieved by Chen and Holmes [44]. They evaluated the thermal equilibration length of individual vessels ( E p ) defined as a length of blood vesse1 for which the temperature difference between blood and tissue is reduced to I/" of its initial value. Assuming that time-dependent variations of the blood temperature are much smaller than the spatial variations along a characteristic length of the vessel, the blood vessel temperature, T b , is govemed by the equation: -dTb A(p
