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.     

A new simplified bioheat equation for the effect of blood flow on local average tissue temperature

A new simplified three-dimensional bioheat equation is derived to describe the effect of blood flow on blood-tissue heat transfer. In two recent theoretical and experimental studies [1, 2] the authors have demonstrated that the so-called isotropic blood perfusion term in the existing bioheat equation is negligible because of the microvascular organization, and that the primary mechanism for blood-tissue energy exchange is incomplete countercurrent exchange in the thermally significant microvessels. The new theory to describe this basic mechanism shows that the vascularization of tissue causes it to behave as an anisotropic heat transfer medium. A remarkably simple expression is derived for the tensor conductivity of the tissue as a function of the local vascular geometry and flow velocity in the thermally significant countercurrent vessels. It is also shown that directed as opposed to isotropic blood perfusion between the countercurrent vessels can have a significant influence on heat transfer in regions where the countercurrent vessels are under 70-micron diameter. The new bioheat equation also describes this mechanism.     

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.     

A three-dimensional variable geometry countercurrent model for whole limb heat transfer.

A new formulation of the combined macro and microvascular model for heat transfer in a human arm developed in Song et al. [1] is proposed using a recently developed approximate theory for the heat exchange between countercurrent vessels embedded in a tissue cylinder with surface convection [2]. The latter theory is generalized herein to treat an arm with an arbitrary variation in cross-sectional area and continuous bleed off from the axial vessels to the muscle and cutaneous tissue. The local microvascular temperature field is described by a "hybrid" model which applies the Weinbaum-Jiji [3] and Pennes [4] equations in the peripheral and deeper tissue layers, respectively. To obtain reliable end conditions at the wrist and other model input parameters, a plethysmograph-calorimeter has been used to measure the blood flow distribution between the arm and hand circulations, and hand heat loss. The predictions of the model show good agreement with measurements for the axial surface temperature distribution in the arm and confirm the minimum in the axial temperature variation first observed by Pennes [4] for an arm in a warm environment.     

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.     

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.     

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.     

Blood vessels and the occurrence of arteriovenous anastomoses in cephalic heat loss areas of mallards,Anas platyrhynchos (Aves)

Summary 1. The blood supply to cephalic heat loss areas (nasal and oropharyngeal mucosa, bill, eyelids) was studied in mallards by using plastic corrosion casts. The structure and organization of the blood vessels, as well as the occurrence of arteriovenous anastomoses (AVAs), were examined by scanning electron microscopy of vascular casts and by paraffin sections.2. Submucosal venous plexuses (cavernous tissue) are present in the nasal cavity, tongue, and lateral margins of the palate. These plexuses receive blood from post-capillary venules, but may also receive a non-nutritive component via numerous AVAs.3. High densities of AVAs were found in the eyelids and in the tip of the bill. In the tongue and nasal mucosa, the AVAs decreased in number caudally. The reason for regional differences in the density of AVAs is discussed in relation to variation in mechanical and thermal stimulation of the tissues.4. The connection of the different heat loss areas with the Rete ophthalmicum, which is a countercurrent heat exchanger important for brain cooling, is pointed out. The vascular pattern of the head suggests that sphincteric veins are involved in regulating the venous return from the evaporative surfaces of the nasal cavity and palate. One of these veins had, in addition to the normal circular smooth muscle fibres, a conspicuous component of longitudinally arranged, subendothelial, smooth muscle fibres.     

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.     

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.     

Heat transport by countercurrent blood vessels in the presence of an arbitrary temperature gradient

This paper presents a three-dimensional analysis of the temperature field around a pair of countercurrent arteries and veins embedded in an infinite tissue that has an arbitrary temperature gradient along the axes of the vessels. Asymptotic methods are used to show that such vessels are thermally similar to a highly conductive fiber in the same tissue. Expressions are developed for the effective radius and thermal conductivity of the fiber so that it conducts heat at the same rate that the artery and vein together convect heat and so that its local temperature equals the mean temperature of the vessels. This result allows vascular tissue to be viewed as a composite of conductive materials with highly conductive fibers replacing the convective effects of the vasculature. By characterizing the size and thermal conductivity of these fibers, well-established methods from the study of composites may be applied to determine when an effective conductive model is appropriate for the tissue and vasculature as a whole.     

Small-scale temperature fluctuations in perfused tissue during local hyperthermia

We develop analytical expressions (scaling laws) for the local temperature fluctuations near isolated and countercurrent blood vessels during hyperthermia. These scaling laws relate the magnitude of such fluctuations to the size of the heated region and to the thermal equilibration length of the vessels. A new equilibration length is identified for countercurrent vessels. Significant temperature differences are predicted between the vessels and the immediately adjacent tissue when the equilibration length is comparable to or longer than the size of the heated tissue region. Countercurrent vessels are shown to have shorter equilibration lengths and produce smaller temperature fluctuations than isolated vessels of the same size.     

Evaluation of temperature distribution during hyperthermic treatment in biliary tumors: a computational approach

A computational approach is adopted to predict the temperature distribution in the biliary tissue during hyperthermic treatments in biliary tumors. Two different models are developed: an axisymmetric model and a three-dimensional model. In the first model the Pennes bioheat transfer equation is applied. It is aimed at simulating the thermoregulatory effect of the capillary bed and it can also give a pressure criterion to determine whether the blood perfusion term should be included in the mathematical model. The second model is aimed at simulating the convective effect of the large hepatic vessels: A constant Nusselt number is assumed on the sides of the vessels. The simulations of the three-dimensional model have been carried out with and without capillary perfusion in the tissue, i.e., respectively in the worst case and in the best case that may occur during heating. The results show that it is possible to obtain therapeutic temperature values in the tissue for time intervals considered acceptable by physicians. Moreover, the model is able to give more precise information about the volumes of tumoral tissue heated above therapeutic temperatures with the hyperthermic technique considered.     

How well mixed is inert gas in tissues?

The washout of inert gas from tissues typically follows multiexponential curves rather than monoexponential curves as would be expected from homogeneous, well-mixed compartment. This implies that the ratio for the square root of the variance of the distribution of transit times to the mean (relative dispersion) must be greater than 1. Among the possible explanations offered for multiexponential curves are heterogeneous capillary flow, uneven capillary spacing, and countercurrent exchange in small veins and arteries. By means of computer simulations of the random walk of gas molecules across capillary beds with parameters of skeletal muscle, we find that heterogeneity involving adjacent capillaries does not suffice to give a relative dispersion greater than one. Neither heterogeneous flow, nor variations in spacing, nor countercurrent exchange between capillaries can account for the multiexponential character of experimental tissue washout curves or the large relative dispersions that have been measured. Simple diffusion calculations are used to show that many gas molecules can wander up to several millimeters away from their entry point during an average transit through a tissue bed. Analytical calculations indicate that an inert gas molecule in an arterial vessel will usually make its first vascular exit from a vessel larger than 20 micron and will wander in and out of tissue and microvessels many times before finally returning to the central circulation. The final exit from tissue will nearly always be into a vessel larger than 20 micron. We propose the hypothesis that the multiexponential character of skeletal muscle tissue inert gas washout curves must be almost entirely due to heterogeneity between tissue regions separated by 3 mm or more, or to countercurrent exchanges in vessels larger than 20 micron diam.     

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.     

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.     

Structure of the brain and eye heater tissue in marlins, sailfish, and spearfishes

Marlins, sailfish, and spearfishes have a heat-producing tissue beneath the brain and adjacent to the eyes. This tissue warms the brain and eyes while the rest of the body remains at water temperature. The heater tissue is derived from the superior rectus eye muscle. Only a portion of this eye muscle contains normal skeletal muscle tissue; the rest consists of the modified muscle tissue that is associated with heat production. The heat-producing portion is supplied with blood through a countercurrent heat exchanger that originates from the carotid artery. The vascular rate prevents the heat being produced by the tissue from being dissipated at the gill. An unusual circulatory supply to the eyes and brain is associated with the presence of the heater tissue in these fishes.     

Renal vascular anatomy of the toad (Bufo marinus)

The renal vasculature of the toad, Bufo marinus, was studied mainly by means of scanning electron microscopy of vascular corrosion casts. All arterial branches terminated in a glomerulus. Each glomerulus was supplied by only one afferent arteriole. No shunts between afferent and efferent arterioles were observed. The glomerular channels appeared to be permanent capillaries. No evidence supporting the theory of freely shifting glomerular blood channels was found. Efferent arterioles radiated out towards the dorsal surface of the kidney where they connected with peritubular vessels. The renal portal veins produced an anastomosing plexus on the dorsal surface of the kidney, giving rise to the peritubular vessels. Peritubular vessels ran radially toward the ventral surface of the kidney, where they formed the roots of the renal veins. Attention is drawn to the possibility of hairpin countercurrent exchange between the capillary-like efferent arterioles and the peritubular vessels in the dorsal kidney.