On the generalization of the Weinbaum-Jiji bioheat equation to microvessels of unequal size; the relation between the near field and local average tissue temperatures

The extensive series of experiments reported in Lemons et al. [1] show that measureable local tissue temperature fluctuations are observed primarily in the vicinity of the 100-500 micron countercurrent vessels of the microcirculation and thus strongly support the basic hypothesis in the new bioheat equation of Weinbaum and Jiji [2] that these countercurrent microvessels are the principal determinants of local blood-tissue heat transfer. However, the detailed temperature profiles in the vicinity of these vessels indicate that large asymmetries in the local temperature field can result from the significant differences in size between the countercurrent artery and vein. Using the superposition techniques of Baish et al. [9], the paper first presents a solution to the classic problem of an unequal countercurrent heat exchanger with heat loss to the far field. This solution is then used to generalize the Weinbaum-Jiji bioheat equation and the conductivity tensor that appears in this equation to vessels of unequal size. An asymptotic analysis has also been developed to elucidate the relationship between the near field temperature of the artery-vein pair and the local average tissue temperature. This analysis is used to rigorously prove the closure approximation relating the local arterial-venous temperature difference and the mean tissue temperature gradient which had been derived in [2] using a more heuristic approach.     

Experimental measurements of the temperature variation along artery-vein pairs from 200 to 1000 microns diameter in rat hind limb

Theoretical studies have indicated that a significant fraction of all blood-tissue heat transfer occurs in artery-vein pairs whose arterial diameter varies between 200 and 1000 microns. In this study, we have developed a new in vivo technique in which it is possible to make the first direct measurements of the countercurrent thermal equilibration that occurs along thermally significant vessels of this size. Fine wire thermocouples were attached by superglue to the femoral arteries and veins and their subsequent branches in rats and the axial temperature variation in each vessel was measured under different physiological conditions. Unlike the blood vessels < 200 microns in diameter, where the blood rapidly equilibrates with the surrounding tissue, we found that the thermal equilibration length of blood vessels between 200 microns and 1000 microns in diameter is longer than or at least equivalent to the vessel length. It is shown that the axial arterial temperature decays from 44% to 76% of the total core-skin temperature difference along blood vessels of this size, and this decay depends strongly on the local blood perfusion rate and the vascular geometry. Our experimental measurements also showed that the SAV venous blood recaptured up to 41% of the total heat released from its countercurrent artery under normal conditions. The contribution of countercurrent heat exchange is significantly reduced in these larger thermally significant vessels for hyperemic conditions as predicted by previous theoretical analyses. Results from this study, when combined with previous analyses of vessel pairs less than 200 microns diameter, enable one estimate the arterial supply temperature and the correction coefficient in the modified perfusion source term developed by the authors.     

The bleed off perfusion term in the Weinbaum-Jiji bioheat equation.

The microvascular organization and thermal equilibration of the primary and secondary arteries and veins that comprise the bleed off circulation to the muscle fibers from the parent countercurrent supply artery and veins are analyzed. The blood perfusion heat source term in the tissue energy equation is shown to be related to this vascular organization and to undergo a fundamental change in behavior as one proceeds from the more peripheral tissue, where the perfusion term is proportional to the Ta--Tv difference in the parent supply vessels, to the deeper tissue layers where the bleed off vessels themselves form a branching countercurrent system for each muscle tissue cylinder and the venous return temperature can vary between the local tissue temperature and Ta. The consequences of this change in behavior are examined for the Weinbaum-Jiji bioheat equation and a modified expression for the effective conductivity of perfused tissue is derived for countercurrent bleed off exchange.     

Heat transport mechanisms in vascular tissues: a model comparison

We have conducted a parametric comparison of three different vascular models for describing heat transport in tissue. Analytical and numerical methods were used to predict the gross temperature distribution throughout the tissue and the small-scale temperature gradients associated with thermally significant blood vessels. The models are: an array of unidirectional vessels, an array of countercurrent vessels, and a set of large vessels feeding small vessels which then drain into large vessels. We show that three continuum formulations of bioheat transfer (directed perfusion, effective conductivity, and a temperature-dependent heat sink) are limiting cases of the vascular models with respect to the thermal equilibration length of the vessels. When this length is comparable to the width of the heated region of tissue, the local temperature changes near the vessels can be comparable to the gross temperature elevation. These results are important to the use of thermal techniques used to measure the blood perfusion rate and in the treatment of cancer with local hyperthermia.     

Parametric studies on the three-layer microcirculatory model for surface tissue energy exchange

The new three-layer microvascular mathematical model for surface tissue heat transfer developed in, which is based on detailed vascular casts and tissue temperature measurements in the rabbit thigh, is used to investigate the thermal characteristics of surface tissue under a wide variety of physiological conditions. Studies are carried out to examine the effects of vascular configuration, arterial blood supply rate, distribution of capillary perfusion, cutaneous blood circulation and metabolic heat production on the average tissue temperature profile, the local arterial-venous blood temperature difference in the thermally significant countercurrent vessels, and surface heat flux.     

A combined macro and microvascular model for whole limb heat transfer

A new prototype model for whole limb heat transfer is proposed wherein the countercurrent heat exchange from the large central arteries and veins in the core of the limb is coupled to microvascular models for the surrounding muscle and the cutaneous tissue layers. The local microvascular temperature field in the muscle tissue is described by the bioheat equation of Weinbaum and Jiji. The new model allows for an arbitrary axial variation of cross-sectional area and blood distribution between the muscle and cutaneous tissue, accounts for the blood flow to and heat loss from the hand and treats the venous return temperature and surface temperature distribution as unknowns that are determined as part of the solution to the overall boundary value problem. Representative solutions are presented for a wide range of environmental conditions for a limb in both the resting state and during exercise.     

A new fundamental bioheat equation for muscle tissue--part II: Temperature of SAV vessels

In this study, a new theoretical framework was developed to investigate temperature variations along countercurrent SAV blood vessels from 300 to 1000 microm diameter in skeletal muscle. Vessels of this size lie outside the range of validity of the Weinbaum-Jiji bioheat equation and, heretofore, have been treated using discrete numerical methods. A new tissue cylinder surrounding these vessel pairs is defined based on vascular anatomy, Murray's law, and the assumption of uniform perfusion. The thermal interaction between the blood vessel pair and surrounding tissue is investigated for two vascular branching patterns, pure branching and pure perfusion. It is shown that temperature variations along these large vessel pairs strongly depend on the branching pattern and the local blood perfusion rate. The arterial supply temperature in different vessel generations was evaluated to estimate the arterial inlet temperature in the modified perfusion source term for the s vessels in Part I of this study. In addition, results from the current research enable one to explore the relative contribution of the SAV vessels and the s vessels to the overall thermal equilibration between blood and tissue.     

Conventional and newly developed bioheat transport models in vascularized tissues: A review

Heat transfer in a biological system is a complex process and its analysis is difficult. Heterogeneous vascular architecture, blood flow in the complex network of arteries and veins, varying metabolic heat generation rates and dependence of tissue properties on its physiological condition contribute to this complexity. The understanding of heat transfer in human body is important for better insight of thermoregulatory mechanism and physiological conditions. Its understanding is also important for accurate prediction of thermal transport and temperature distribution during biomedical applications. During the last three decades, many attempts have been made by researchers to model the complex thermal behavior of the human body. These models, viz., blood perfusion, countercurrent, thermal phase-lag, porous-media, perturbation, radiation, etc. have their corresponding strengths and limitations. Along with their biomedical applications, this article reviews various contextual issues associated with these models. After brief discussion of early bioheat models, the newly developed bioheat models are discussed in detail. Dependence of these models on biological properties, viz., thermophysical and optical properties are also discussed.     

Local heating of human skin by millimeter waves: effect of blood flow

We investigated the influence of blood perfusion on local heating of the forearm and middle finger skin following 42.25 GHz exposure with an open ended waveguide (WG) and with a YAV mm wave therapeutic device. Both sources had bell-shaped distributions of the incident power density (IPD) with peak intensities of 208 and 55 mW/cm(2), respectively. Blood perfusion was changed in two ways: by blood flow occlusion and by externally applied vasodilator (nonivamide/nicoboxil) cream to the skin. For thermal modeling, we used the bioheat transfer equation (BHTE) and the hybrid bioheat equation (HBHE) which combines the BHTE and the scalar effective thermal conductivity equation (ETCE). Under normal conditions with the 208 mW/cm(2) exposure, the cutaneous temperature elevation (DeltaT) in the finger (2.5 +/- 0.3 degrees C) having higher blood flow was notably smaller than the cutaneous DeltaT in the forearm (4.7 +/- 0.4 degrees C). However, heating of the forearm and finger skin with blood flow occluded was the same, indicating that the thermal conductivity of tissue in the absence of blood flow at both locations was also the same. The BHTE accurately predicted local hyperthermia in the forearm only at low blood flow. The HBHE made accurate predictions at both low and high perfusion rates. The relationship between blood flow and the effective thermal conductivity (k(eff)) was found to be linear. The heat dissipating effect of higher perfusion was mostly due to an apparent increase in k(eff). It was shown that mm wave exposure could result in steady state heating of tissue layers located much deeper than the penetration depth (0.56 mm). The surface DeltaT and heat penetration into tissue increased with enlarging the irradiating beam area and with increasing exposure duration. Thus, mm waves at sufficient intensities could thermally affect thermo-sensitive structures located in the skin and underlying tissue.     

Readdressing the issue of thermally significant blood vessels using a countercurrent vessel network

A physiologically realistic arterio-venous countercurrent vessel network model consisting of ten branching vessel generations, where the diameter of each generation of vessels is smaller than the previous ones, has been created and used to determine the thermal significance of different vessel generations by investigating their ability to exchange thermal energy with the tissue. The temperature distribution in the 3D network (8178 vessels; diameters from 10 to 1000 microm) is obtained by solving the conduction equation in the tissue and the convective energy equation with a specified Nusselt number in the vessels. The sensitivity of the exchange of energy between the vessels and the tissue to changes in the network parameters is studied for two cases; a high temperature thermal therapy case when tissue is heated by a uniformly distributed source term and the network cools the tissue, and a hypothermia related case, when tissue is cooled from the surface and the blood heats the tissue. Results show that first, the relative roles of vessels of different diameters are strongly determined by the inlet temperatures to those vessels (e.g., as affected by changing mass flow rates), and the surrounding tissue temperature, but not by their diameter. Second, changes in the following do not significantly affect the heat transfer rates between tissue and vessels; (a) the ratio of arterial to venous vessel diameter, (b) the diameter reduction coefficient (the ratio of diameters of successive vessel generations), and (c) the Nusselt number. Third, both arteries and veins play significant roles in the exchange of energy between tissue and vessels, with arteries playing a more significant role. These results suggest that the determination of which diameter vessels are thermally important should be performed on a case-by-case, problem dependent basis. And, that in the development of site-specific vessel network models, reasonable predictions of the relative roles of different vessel diameters can be obtained by using any physiologically realistic values of Nusselt number and the diameter reduction coefficient.     

Temperature evolution in tissues embedded with large blood vessels during photo-thermal heating

During laser-assisted photo-thermal therapy, the temperature of the heated tissue region must rise to the therapeutic value (e.g., 43 °C) for complete ablation of the target cells. Large blood vessels (larger than 500 micron in diameter) at or near the irradiated tissues have a considerable impact on the transient temperature distribution in the tissue. In this study, the cooling effects of large blood vessels on temperature distribution in tissues during laser irradiation are predicted using finite element based simulation. A uniform flow is assumed at the entrance and three-dimensional conjugate heat transfer equations in the tissue region and the blood region are simultaneously solved for different vascular models. A volumetric heat source term based on Beer–Lambert law is introduced into the energy equation to account for laser heating. The heating pattern is taken to depend on the absorption and scattering coefficients of the tissue medium. Experiments are also conducted on tissue mimics in the presence and absence of simulated blood vessels to validate the numerical model. The coupled heat transfer between thermally significant blood vessels and their surrounding tissue for three different tissue-vascular networks are analyzed keeping the laser irradiation constant. A surface temperature map is obtained for different vascular models and for the bare tissue (without blood vessels). The transient temperature distribution is seen to differ according to the nature of the vascular network, blood vessel size, flow rate, laser spot size, laser power and tissue blood perfusion rate. The simulations suggest that the blood flow through large blood vessels in the vicinity of the photothermally heated tissue can lead to inefficient heating of the target.     

Predicting effects of blood flow rate and size of vessels in a vasculature on hyperthermia treatments using computer simulation

Background

Pennes Bio Heat Transfer Equation (PBHTE) has been widely used to approximate the overall temperature distribution in tissue using a perfusion parameter term in the equation during hyperthermia treatment. In the similar modeling, effective thermal conductivity (K_eff) model uses thermal conductivity as a parameter to predict temperatures. However the equations do not describe the thermal contribution of blood vessels. A countercurrent vascular network model which represents a more fundamental approach to modeling temperatures in tissue than do the generally used approximate equations such as the Pennes BHTE or effective thermal conductivity equations was presented in 1996. This type of model is capable of calculating the blood temperature in vessels and describing a vasculature in the tissue regions.

Methods

In this paper, a countercurrent blood vessel network (CBVN) model for calculating tissue temperatures has been developed for studying hyperthermia cancer treatment. We use a systematic approach to reveal the impact of a vasculature of blood vessels against a single vessel which most studies have presented. A vasculature illustrates branching vessels at the periphery of the tumor volume. The general trends present in this vascular model are similar to those shown for physiological systems in Green and Whitmore. The 3-D temperature distributions are obtained by solving the conduction equation in the tissue and the convective energy equation with specified Nusselt number in the vessels.

Results

This paper investigates effects of size of blood vessels in the CBVN model on total absorbed power in the treated region and blood flow rates (or perfusion rate) in the CBVN on temperature distributions during hyperthermia cancer treatment. Also, the same optimized power distribution during hyperthermia treatment is used to illustrate the differences between PBHTE and CBVN models. K_eff (effective thermal conductivity model) delivers the same difference as compared to the CBVN model. The optimization used here is adjusting power based on the local temperature in the treated region in an attempt to reach the ideal therapeutic temperature of 43°C. The scheme can be used (or adapted) in a non-invasive power supply application such as high-intensity focused ultrasound (HIFU). Results show that, for low perfusion rates in CBVN model vessels, impacts on tissue temperature becomes insignificant. Uniform temperature in the treated region is obtained.

Conclusion

Therefore, any method that could decrease or prevent blood flow rates into the tumorous region is recommended as a pre-process to hyperthermia cancer treatment. Second, the size of vessels in vasculatures does not significantly affect on total power consumption during hyperthermia therapy when the total blood flow rate is constant. It is about 0.8% decreasing in total optimized absorbed power in the heated region as γ (the ratio of diameters of successive vessel generations) increases from 0.6 to 0.7, or from 0.7 to 0.8, or from 0.8 to 0.9. Last, in hyperthermia treatments, when the heated region consists of thermally significant vessels, much of absorbed power is required to heat the region and (provided that finer spatial power deposition exists) to heat vessels which could lead to higher blood temperatures than tissue temperatures when modeled them using PBHTE.     

Finite-element solution of thermal conductivity of muscle during cold water immersion

The in vivo or effective thermal conductivity (keff) of muscle tissue of the human forearm was determined through a finite-element (FE) model solution of the bioheat equation. Data were obtained from steady-state temperatures measured in the forearm after 3 h of immersion in water at temperatures (Tw) of 15 (n = 6), 20 (n = 5), and 30 degrees C (n = 5). Temperatures were measured every 0.5 cm from the longitudinal axis of the forearm to the skin approximately 9 cm distal from the elbow. Heat flux was measured at two sites on the skin adjacent to the temperature probe. The FE model is comprised of concentric annular compartments with boundaries defined by the location of temperature measurements. Through this approach, it was possible to include both the metabolic heat production and the convective heat transfer between blood and tissue at two levels of blood flow, one perfusing the compartment and the other passing through the compartment. Without heat exchange at the passing blood flow level, the arterial blood temperature would be assumed to have a constant value everywhere in the forearm muscles, leading to a solution of the bioheat equation that greatly underpredicts keff. The extent of convective heat exchange at the passing blood flow level is estimated to be approximately 60% of the total heat exchange between blood and tissue. Concurrent with this heat exchange is a decrease in the temperature of the arterial blood as it flows radially from the axis to the skin of the forearm, and this decrease is enhanced with a lowered Tw.(ABSTRACT TRUNCATED AT 250 WORDS)     

Heat transfer analysis of skin during thermal therapy using thermal wave equation

Specifying exact geometry of vessel network and its effect on temperature distribution in living tissues is one of the most complicated problems of the bioheat field. In this paper, the effects of blood vessels on temperature distribution in a skin tissue subjected to various thermal therapy conditions are investigated. Present model consists of counter-current multilevel vessel network embedded in a three-dimensional triple-layered skin structure. Branching angles of vessels are calculated using the physiological principle of minimum work. Length and diameter ratios are specified using length doubling rule and Cube law, respectively. By solving continuity, momentum and energy equations for blood flow and Pennes and modified Pennes bioheat equations for the tissue, temperature distributions in the tissue are measured. Effects of considering modified Pennes bioheat equation are investigated, comprehensively. It is also observed that blood has an impressive role in temperature distribution of the tissue, especially at high temperatures. The effects of different parameters such as boundary conditions, relaxation time, thermal properties of skin, metabolism and pulse heat flux on temperature distribution are investigated. Tremendous effect of boundary condition type at the lower boundary is noted. It seems that neither insulation nor constant temperature at this boundary can completely describe the real physical phenomena. It is expected that real temperature at the lower levels is somewhat between two predicted values. The effect of temperature on the thermal properties of skin tissue is considered. It is shown that considering temperature dependent values for thermal conductivity is important in the temperature distribution estimation of skin tissue; however, the effect of temperature dependent values for specific heat capacity is negligible. It is seen that considering modified Pennes equation in processes with high heat flux during low times is significant.     

Pulsatile blood flow effects on temperature distribution and heat transfer in rigid vessels

The effect of blood velocity pulsations on bioheat transfer is studied. A simple model of a straight rigid blood vessel with unsteady periodic flow is considered. A numerical solution that considers the fully coupled Navier-Stokes and energy equations is used for the simulations. The influence of the pulsation rate on the temperature distribution and energy transport is studied for four typical vessel sizes: aorta, large arteries, terminal arterial branches, and arterioles. The results show that: the pulsating axial velocity produces a pulsating temperature distribution; reversal of flow occurs in the aorta and in large vessels, which produces significant time variation in the temperature profile. Change of the pulsation rate yields a change of the energy transport between the vessel wall and fluid for the large vessels. For the thermally important terminal arteries (0.04-1 mm), velocity pulsations have a small influence on temperature distribution and on the energy transport out of the vessels (8 percent for the Womersley number corresponding to a normal heart rate). Given that there is a small difference between the time-averaged unsteady heat flux due to a pulsating blood velocity and an assumed nonpulsating blood velocity, it is reasonable to assume a nonpulsating blood velocity for the purposes of estimating bioheat transfer.     

A theoretical model for peripheral tissue heat transfer using the bioheat equation of Weinbaum and Jiji

In this paper the new bioheat equation derived in Weinbaum and Jiji is applied to the three layer conceptual model of microvascular surface tissue organization proposed in. A simplified one-dimensional quantitative model of peripheral tissue energy exchange is then developed for application in limb and whole body heat transfer studies. A representative vasculature is constructed for each layer and the enhancement in the local tensor conductivity of the tissue as a function of vascular geometry and blood flow is examined. Numerical solutions for the boundary value problem coupling the three layers are presented and these results used to study the thermal behavior of peripheral tissue for a wide variety of physiological conditions from supine resting state to maximum exercise.     

A self-heated thermistor technique to measure effective thermal properties from the tissue surface

A microcomputer based instrument to measure effective thermal conductivity and diffusivity at the surface of a tissue has been developed. Self-heated spherical thermistors, partially embedded in an insulator, are used to simultaneously heat tissue and measure the resulting temperature rise. The temperature increase of the thermistor for a given applied power is a function of the combined thermal properties of the insulator, the thermistor, and the tissue. Once the probe is calibrated, the instrument accurately measures the thermal properties of tissue. Conductivity measurements are accurate to 2 percent and diffusivity measurements are accurate to 4 percent. A simplified bioheat equation is used which assumes the effective tissue thermal conductivity is a linear function of perfusion. Since tissue blood flow strongly affects heat transfer, the surface thermistor probe is quite sensitive to perfusion.     

Perfused phantom models of microwave irradiated tissue

The theoretical basis, practical design considerations, and prototype testing of a perfused model suitable for simulation studies of microwave heated tissue are presented. A parallel tube heat exchanger configuration is used to simulate the internal convection effects of blood flow. The global thermal response of the phantom, on a scale of several tube spacings, is shown theoretically to be nearly identical to that predicted by Pennes' bioheat equation, which is known to give a reasonable representation of tissue under many conditions. A parametric study is provided for the relationships between the tube size, spacing and material properties and the simulated perfusion rate. A prototype with a physiologically reasonable perfusion rate was tested using a typical hyperthermia applicator. The measured thermal response of the phantom compares favorably with the numerical solution of the bioheat equation under the same irradiation conditions. This similarity sheds light on the unexpected success of the bioheat equation for modeling the thermal response of real tissue.     

A laboratory exercise using a physical model for demonstrating countercurrent heat exchange

A physical model was used in a laboratory exercise to teach students about countercurrent exchange mechanisms. Countercurrent exchange is the transport of heat or chemicals between fluids moving in opposite directions separated by a permeable barrier (such as blood within adjacent blood vessels flowing in opposite directions). Greater exchange of heat or chemicals between the fluids occurs when the flows are in opposite directions (countercurrent) than in the same direction (concurrent). When a vessel loops back on itself, countercurrent exchange can occur between the two arms of the loop, minimizing loss or uptake at the bend of the loop. Comprehension of the physical principles underlying countercurrent exchange helps students to understand how kidneys work and how modifications of a circulatory system can influence the movement of heat or chemicals to promote or minimize exchange and reinforces the concept that heat and chemicals move down their temperature or concentration gradients, respectively. One example of a well-documented countercurrent exchanger is the close arrangement of veins and arteries inside bird legs; therefore, the setup was arranged to mimic blood vessels inside a bird leg, using water flowing inside tubing as a physical proxy for blood flow within blood vessels.