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.     

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.     

Effect of variable heat transfer coefficient on tissue temperature next to a large vessel during radiofrequency tumor ablation

Background  

One of the current shortcomings of radiofrequency (RF) tumor ablation is its limited performance in regions close to large blood vessels, resulting in high recurrence rates at these locations. Computer models have been used to determine tissue temperatures during tumor ablation procedures. To simulate large vessels, either constant wall temperature or constant convective heat transfer coefficient (h) have been assumed at the vessel surface to simulate convection. However, the actual distribution of the temperature on the vessel wall is non-uniform and time-varying, and this feature makes the convective coefficient variable.     

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.     

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.     

Investigation of the influence of blood flow rate on large vessel cooling in hepatic radiofrequency ablation.

Radiofrequency (RF) ablation using high-frequency current has become an important treatment method for patients with non-resectable liver tumors. Tumor recurrence is associated with tissue cooling in the proximity of large blood vessels. This study investigated the influence of blood flow rate on tissue temperature and lesion size during monopolar RF ablation at a distance of 10 mm from single 4- and 6-mm vessels using two different approaches: 1) an ex vivo blood perfusion circuit including an artificial vessel inserted into porcine liver tissue was developed; and 2) a finite element method (FEM) model was created using a novel simplified modeling technique for large blood vessels. Blood temperatures at the inflow/outflow of the vessel and tissue temperatures at 10 and 20 mm from the electrode tip were measured in the ex vivo set-up. Tissue temperature, blood temperature and lesion size were analyzed under physiological, increased and reduced blood-flow conditions. The results show that changes in blood flow rate in large vessels do not significantly affect tissue temperature and lesion size far away from the vessel. Monopolar ablation could not produce lesions surrounding the vessel due to the strong heat-sink effect. Simulated tissue temperatures correlated well with ex vivo measurements, supporting the FEM model.     

Numerical evaluation reveals the effect of branching morphology on vessel transport properties during angiogenesis

Blood flow governs transport of oxygen and nutrients into tissues. Hypoxic tissues secrete VEGFs to promote angiogenesis during development and in tissue homeostasis. In contrast, tumors enhance pathologic angiogenesis during growth and metastasis, suggesting suppression of tumor angiogenesis could limit tumor growth. In line with these observations, various factors have been identified to control vessel formation in the last decades. However, their impacts on the vascular transport properties of oxygen remain elusive. Here, we take a computational approach to examine the effects of vascular branching on blood flow in the growing vasculature. First of all, we reconstruct a 3D vascular model from the 2D confocal images of the growing vasculature at postnatal day 5 (P5) mouse retina, then simulate blood flow in the vasculatures, which are obtained from the gene targeting mouse models causing hypo- or hyper-branching vascular formation. Interestingly, hyper-branching morphology attenuates effective blood flow at the angiogenic front, likely promoting tissue hypoxia. In contrast, vascular hypo-branching enhances blood supply at the angiogenic front of the growing vasculature. Oxygen supply by newly formed blood vessels improves local hypoxia and decreases VEGF expression at the angiogenic front during angiogenesis. Consistent with the simulation results indicating improved blood flow in the hypo-branching vasculature, VEGF expression around the angiogenic front is reduced in those mouse retinas. Conversely, VEGF expression is enhanced in the angiogenic front of hyper-branching vasculature. Our results indicate the importance of detailed flow analysis in evaluating the vascular transport properties of branching morphology of the blood vessels.     

Synergy in cancer treatment between liposomal chemotherapeutics and thermal ablation

Minimally invasive image-guided tumor ablation using short duration heating via needle-like applicators using energies such as radiofrequency or microwave has seen increasing clinical use to treat focal liver, renal, breast, bone, and lung tumors. Potential benefits of this thermal therapy include reduced morbidity and mortality compared to standard surgical resection and ability to treat non-surgical patients. However, improvements to this technique are required as achieving complete ablation in many cases can be challenging particularly at margins of tumors>3 cm in diameter and adjacent to blood vessels. Thus, one very promising strategy has been to combine thermal tumor ablation with adjuvant nanoparticle-based chemotherapy agents to improve efficiency. Here, we will primarily review principles of thermal ablation to provide a framework for understanding the mechanisms of combination therapy, and review the studies on combination therapy, including presenting preliminary data on the role of such variables as nanoparticle size and thermal dose on improving combination therapy outcome. We will discuss how thermal ablation can also be used to improve overall intratumoral drug accumulation and nanoparticle content release. Finally, in this article we will further describe the appealing off-shoot approach of utilizing thermal ablation techniques not as the primary treatment, but rather, as a means to improve efficiency of intratumoral nanoparticle drug delivery.     

Heating of tissues by microwaves: A model analysis

We consider the thermal response times for heating of tissue subject to nonionizing (microwave or infrared) radiation. The analysis is based on a dimensionless form of the bioheat equation. The thermal response is governed by two time constants: one(τ₁) pertains to heat convection by blood flow, and is of the order of 20–30 min for physiologically normal perfusion rates; the second (τ₂) characterizes heat conduction and varies as the square of a distance that characterizes the spatial extent of the heating. Two idealized cases are examined. The first is a tissue block with an insulated surface, subject to irradiation with an exponentially decreasing specific absorption rate, which models a large surface area of tissue exposed to microwaves. The second is a hemispherical region of tissue exposed at a spatially uniform specific absorption rate, which models localized exposure. In both cases, the steady-state temperature increase can be written as the product of the incident power density and an effective time constant τ_eff, which is defined for each geometry as an appropriate function of τ₁ and τ₂. In appropriate limits of the ratio of these time constants, the local temperature rise is dominated by conductive or convective heat transport. Predictions of the block model agree well with recent data for the thresholds for perception of warmth or pain from exposure to microwave energy. Using these concepts, we developed a thermal averaging time that might be used in standards for human exposure to microwave radiation, to limit the temperature rise in tissue from radiation by pulsed sources. We compare the ANSI exposure standards for microwaves and infrared laser radiation with respect to the maximal increase in tissue temperature that would be allowed at the maximal permissible exposures. A historical appendix presents the origin of the 6-min averaging time used in the microwave standard. Bioelectromagnetics 19:420–428, 1998. © 1998 Wiley-Liss, Inc.     

Effects of bradykinin on the hemodynamics of tumor and granulating normal tissue microvasculature.

Bradykinin (BK) is an important endogenous mediator of microvascular flow modulation. Since the structure of the microcirculation is very different in tumor tissues than in normal tissues, bradykinin may elicit different responses in tumors. This study was designed to test the hypothesis that local administration of bradykinin increases blood flow preferentially in normal tissue relative to adjacent tumor tissue, resulting in a "vascular steal" phenomenon. Microvessel diameters (D), velocities (Vc), length densities, shear rates, and intermittent flow frequencies were measured every 10 min before, during, and after 40 min exposure to BK in rats with dorsal flap window chambers 9 days after tumor implantation. Separate studies were made of normal vessels outside the tumor margin, the hypervascular tumor periphery, and the tumor center. Bradykinin was administered with a suffusion medium flowing over the tissue at 1-2 ml/min with a BK concentration of 1.6 x 10(7) M. Administration of BK created five distinct changes in normal and tumor vessel function that varied over time, but coincidentally reached a maximum effect after 20 min exposure to BK. In normal vessels, increased Vc and D led to increased flow, which reached a peak 20 min after onset of suffusion with BK. In contrast, in centrally located tumor vessels, decreased D and Vc were observed in most vessels during the initial 10-20 min of suffusion. In addition, there was a significant increase in intermittent flow frequency in tumor central vessels, which peaked after 20 min of suffusion with BK. These five separate observations that coincided at 20 min of suffusion are consistent with a "vascular steal" phenomenon. The increase in normal microvessel D and Vc at 20 min suggests that BK causes vasodilation in arterioles. The coincident decrease in tumor microvessel D and Vc suggests that tumor feeding vessels are less able to respond to BK by vasodilating. The concomitant increase in intermittent flow frequency in tumor vessels suggests that a reduction in pressure drop occurred after 20 min exposure to BK, which is also consistent with "vascular steal." Since BK is also known to increase vascular permeability, it is possible that increases in interstitial fluid pressure brought on by exposure to BK contributed to the observed reduction in tumor blood flow. In normal vessels, reduced D and Vc, relative to peak values, were noted after 40 min suffusion with BK. Adherence of leukocytes to the vessel walls was prominent and microthrombi were also observed during this period. No evidence of such adhesion was seen in tumor vessels, although microthrombi were observed.(ABSTRACT TRUNCATED AT 400 WORDS)     

The enigmatic role of angiopoietin-1 in tumor angiogenesis

A tumor vasculature is highly unstable and immature, characterized by a high proliferation rate of endothelial cells, hyper-permeability, and chaotic blood flow. The dysfunctional vasculature gives rise to continual plasma leakage and hypoxia in the tumor, resulting in constant on-sets of inflammation and angiogenesis. Tumors are thus likened to wounds that will not heal. The lack of functional mural cells, including pericytes and vascular smooth muscle cells, in tumor vascular structure contributes significantly to the abnormality of tumor vessels. Angiopoietin-1 (Ang 1) is aphysiological angiogenesis promoter during embryonic development. The function of Angl is essential to endothelial cell survival, vascular branching, and pericyte recruitment. However, an increasing amount of experimental data suggest that Angl-stimulated association of mural cells with endothelial cells lead to stabilization of newly formed blood vessels. This in turn may limit the otherwise continuous angiogenesis in the tumor, and consequently give riseto inhibition of tumor growth. We discuss the enigmatic role of Angl in tumor angiogenesis in this review.     

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.     

An analytical solution countercurrent heat transfer between parallel vessels with a linear axial temperature gradient

Presented in this paper is a solution for countercurrent heat exchange between two parallel vessels embedded in an infinite medium with a linear temperature gradient along the axes of the vessels. The velocity profile within the vessel is assumed to be parabolic. This solution describes the temperature field within the vessels, as well as in the tissue, and establishes that the intravessel temperature is not uniform, as is generally assumed to be the case. An explicit expression for the intervessel thermal resistance based on the difference between cup-mixed mean temperatures is derived.     

Emergent vascular network inhomogeneities and resulting blood flow patterns in a growing tumor

Tumors acquire sufficient oxygen and nutrient supply by coopting host vessels and neovasculature created via angiogenesis, thereby transforming a highly ordered network into chaotic heterogeneous tumor specific vasculature. Vessel regression inside the tumor leads to large regions of necrotic tissue interspersed with isolated surviving vessels. We extend our recently introduced model to incorporate Fahraeus-Lindqvist- and phase separation effects, refined tissue oxygen level computation and drug flow computations. We find, unexpectedly, that collapse and regression accelerates rather than diminishes the perfusion and that a tracer substance flowing through the remodeled network reaches all parts of the tumor vasculature very well. The reason for decreased drug delivery well known in tumors should therefore be different from collapse and vessel regression. Implications for drug delivery in real tumors are discussed.     

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.     

Coupled field analysis of heat flow in the near field of a microwave applicator for tumor ablation

Microwave tumor ablation (MTA) offers a new approach for the treatment of hepatic neoplastic disease. Reliable and accurate information regarding the heat distribution inside biological tissue subjected to microwave thermal ablation is important for the efficient design of microwave applicators and for optimizing experiments, which aim to assess the effects of therapeutic treatments. Currently there are a variety of computational methods based on different vascular structures in tissue, which aim to model heat distribution during ablation. This paper presents results obtained from two such computational models for temperature distributions produced by a clinical 2.45 GHz MTA applicator immersed in unperfused ex vivo bovine liver, and compares them with measured results from a corresponding ex vivo experiment. The computational methods used to model the temperature distribution in tissue caused by the insertion of a 5.6 mm diameter "wandlike" microwave applicator are the Green's function method and the finite element method (FEM), both of which provide solutions of the heat diffusion partial differential equation. The results obtained from the coupled field simulations are shown to be in good agreement with a simplified analysis based on the bio-heat equation and with ex vivo measurements of the heat distribution produced by the clinical MTA applicator.     

Steady-state analysis of self-heated thermistors using finite elements

The purpose of this work is to validate, using numerical, finite element methods, the thermal assumptions made in the analytical analysis of a coupled thermistor probe-tissue model upon which a thermal conductivity measurement scheme has been based. Analytic, closed form temperature profiles generated by the self-heated thermistors can be found if three simplifying assumptions are made: the thermistor is spherical; heat is generated in all regions of the bead; and heat is generated uniformly in the bead. This analytic solution is used to derive a linear relationship between tissue thermal conductivity and the ratio of thermistor temperature rise over electrical power required to maintain that temperature rise. This derived, linear relationship is used to determine thermal conductivity from the observed experimental data. However, in reality, the thermistor bead is a prolate spheroid surrounded by a passive shell, and the heating pattern in the bead is highly nonuniform. In the physical system, the exact relationship between the tissue thermal conductivity and parameters measured by the thermistor is not known. The finite element method was used to calculate the steady-state temperature profiles generated by thermistor beads with realistic geometry and heating patterns. The results of the finite element analysis show that the empirical, linear relationship remains valid when all three simplified assumptions are significantly relaxed.     

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.     

Monitoring sunitinib-induced vascular effects to optimize radiotherapy combined with soy isoflavones in murine xenograft tumor

Using dynamic contrast-enhanced magnetic resonance imaging (DCE-MRI) to monitor vascular changes induced by sunitinib within a murine xenograft kidney tumor, we previously determined a dose that caused only partial destruction of blood vessels leading to "normalization" of tumor vasculature and improved blood flow. In the current study, kidney tumors were treated with this dose of sunitinib to modify the tumor microenvironment and enhance the effect of kidney tumor irradiation. The addition of soy isoflavones to this combined antiangiogenic and radiotherapy approach was investigated based on our studies demonstrating that soy isoflavones can potentiate the radiation effect on the tumors and act as antioxidants to protect normal tissues from treatment-induced toxicity. DCE-MRI was used to monitor vascular changes induced by sunitinib and schedule radiation when the uptake and washout of the contrast agent indicated regularization of blood flow. The combination of sunitinib with tumor irradiation and soy isoflavones significantly inhibited the growth and invasion of established kidney tumors and caused marked aberrations in the morphology of residual tumor cells. DCE-MRI studies demonstrated that the three modalities, sunitinib, radiation, and soy isoflavones, also exerted antiangiogenic effects resulting in increased uptake and clearance of the contrast agent. Interestingly, DCE-MRI and histologic observations of the normal contralateral kidneys suggest that soy could protect the vasculature of normal tissue from the adverse effects of sunitinib. An antiangiogenic approach that only partially destroys inefficient vessels could potentially increase the efficacy and delivery of cytotoxic therapies and radiotherapy for unresectable primary renal cell carcinoma tumors and metastatic disease.