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.     

Electrical communication in branching arterial networks

Electrical communication and its role in blood flow regulation are built on an examination of charge movement in single, isolated vessels. How this process behaves in broader arterial networks remains unclear. This study examined the nature of electrical communication in arterial structures where vessel length and branching were varied. Analysis began with the deployment of an existing computational model expanded to form a variable range of vessel structures. Initial simulations revealed that focal endothelial stimulation generated electrical responses that conducted robustly along short unbranched vessels and to a lesser degree lengthened arteries or branching structures retaining a single branch point. These predictions matched functional observations from hamster mesenteric arteries and support the idea that an increased number of vascular cells attenuate conduction by augmenting electrical load. Expanding the virtual network to 31 branches revealed that electrical responses increasingly ascended from fifth- to first-order arteries when the number of stimulated distal vessels rose. This property enabled the vascular network to grade vasodilation and network perfusion as revealed through blood flow modeling. An elevation in endothelial-endothelial coupling resistance, akin to those in sepsis models, compromised this ascension of vasomotor/perfusion responses. A comparable change was not observed when the endothelium was focally disrupted to mimic disease states including atherosclerosis. In closing, this study highlights that vessel length and branching play a role in setting the conduction of electrical phenomenon along resistance arteries and within networks. It also emphasizes that modest changes in endothelial function can, under certain scenarios, impinge on network responsiveness and blood flow control.     

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.     

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.     

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.     

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.     

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.     

Inhibition of Notch signaling by Dll4-Fc promotes reperfusion of acutely ischemic tissues

Notch pathway regulates vessel development and maturation. Dll4, a high-affinity ligand for Notch, is expressed predominantly in the arterial endothelium and is induced by hypoxia among other factors. Inhibition of Dll4 has paradoxical effects of reducing the maturation and perfusion in newly forming vessels while increasing the density of vessels. We hypothesized that partial and/or intermittent inhibition of Dll4 may lead to increased vascular response and still allow vascular maturation to occur. Thus tissue perfusion can be restored rapidly, allowing quicker recovery from ischemia or tissue injury. Our studies in two different models (hindlimb ischemia and skin flap) show that inhibition of Dll4 at low dose allows faster recovery from vascular and tissue injury. This opens a new possibility for Dll4 blockade's therapeutic application in promoting recovery from vascular injury and restoring blood supply to ischemic tissues.     

An advanced computational bioheat transfer model for a human body with an embedded systemic circulation

In the present work, an elaborate one-dimensional thermofluid model for a human body is presented. By contrast to the existing pure conduction-/perfusion-based models, the proposed methodology couples the arterial fluid dynamics of a human body with a multi-segmental bioheat model of surrounding solid tissues. In the present configuration, arterial flow is included through a network of elastic vessels. More than a dozen solid segments are employed to represent the heat conduction in the surrounding tissues, and each segment is constituted by a multilayered circular cylinder. Such multi-layers allow flexible delineation of the geometry and incorporation of properties of different tissue types. The coupling of solid tissue and fluid models requires subdivision of the arterial circulation into large and small arteries. The heat exchange between tissues and arterial wall occurs by convection in large vessels and by perfusion in small arteries. The core region, including the heart, provides the inlet conditions for the fluid equations. In the proposed model, shivering, sweating, and perfusion changes constitute the basis of the thermoregulatory system. The equations governing flow and heat transfer in the circulatory system are solved using a locally conservative Galerkin approach, and the heat conduction in the surrounding tissues is solved using a standard implicit backward Euler method. To investigate the effectiveness of the proposed model, temperature field evolutions are monitored at different points of the arterial tree and in the surrounding tissue layers. To study the differences due to flow-induced convection effects on thermal balance, the results of the current model are compared against those of the widely used modelling methodologies. The results show that the convection significantly influences the temperature distribution of the solid tissues in the vicinity of the arteries. Thus, the inner convection has a more predominant role in the human body heat balance than previously thought. To demonstrate its capabilities, the proposed new model is used to study different scenarios, including thermoregulation inactivity and variation in surrounding atmospheric conditions.     

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.     

Radio-frequency ablation in a realistic reconstructed hepatic tissue

This study uses a reconstructed vascular geometry to evaluate the thermal response of tissue during a three-dimensional radiofrequency (rf) tumor ablation. MRI images of a sectioned liver tissue containing arterial vessels are processed and converted into a finite-element mesh. A rf heat source in the form of a spherically symmetric Gaussian distribution, fit from a previously computed profile, is employed. Convective cooling within large blood vessels is treated using direct physical modeling of the heat and momentum transfer within the vessel. Calculations of temperature rise and thermal dose are performed for transient rf procedures in cases where the tumor is located at three different locations near the bifurcation point of a reconstructed artery. Results demonstrate a significant dependence of tissue temperature profile on the reconstructed vasculature and the tumor location. Heat convection through the arteries reduced the steady-state temperature rise, relative to the no-flow case, by up to 70% in the targeted volume. Blood flow also reduced the thermal dose value, which quantifies the extent of cell damage, from approximately 3600 min, for the no-flow condition, to 10 min for basal flow (13.8 cms). Reduction of thermal dose below the threshold value of 240 min indicates ablation procedures that may inadequately elevate the temperature in some regions, thereby permitting possible tumor recursion. These variations are caused by vasculature tortuosity that are patient specific and can be captured only by the reconstruction of the realistic geometry.     

The neurogenic constrictor response of isolated rat saphenous artery is reduced after 2-week tail suspension

Under real or simulated microgravity conditions the control of arterial vascular tone is greatly disturbed. The low arterial vessel reactivity to sympathetic influences may be the cause of an increase in flow in hind limb skeletal muscles in tail-suspended (TS) rats. Our previous experiments with constant pressure perfusion of rat hind limb demonstrated the reduced vasoconstrictor responses to sympathetic nerve stimulation in TS rats. Responses to exogenous noradrenaline depended on the perfusion conditions. It is known that the vessels of various branching orders noticeably differ in nerve density and in sensitivity to vasoconstrictor agonists. So under neurogenic or exogenous noradrenaline influences the vascular resistance may be increased at different levels of vascular bed, thus making the data analysis seriously complicated. This uncertainty may be overcome by investigation of a single vessel isolated from hind limb vascular bed. The saphenous artery, a resistance artery with dense innervation, is a very convenient object for this purpose. Thus, this study was aimed at comparing the effects of 2-week tail suspension upon the constrictor responses of isolated saphenous artery to neurogenic and exogenous noradrenaline stimuli in rats.     

Incorporating an immersed boundary method to study thermal effects of vascular systems during tissue cryo-freezing

In this paper, the three-dimensional thermal effects of a clinically-extracted vascular tissue undergoing cryo-freezing are numerically investigated. Based on the measured experimental temperature field, the numerical results of the Pennes bioheat model combined with the boundary condition-enforced immersed boundary method (IBM) agreed well with experimental data with a maximum temperature discrepancy of 2.9 °C. For simulating the temperature profile of a tumor sited in a dominantly vascularized tissue, our model is able to capture with ease the thermal effects at specified junctions of the blood vessels. The vascular complexity and the ice-ball shape irregularity which cannot be easily quantified via clinical experiments are also analyzed and compared for both two-dimensional and three-dimensional settings with different vessel configurations and developments. For the three-dimensional numerical simulations, a n-furcated liver vessels model from a three-dimensional segmented volume using hole-making and subdivision methods is applied. A specific study revealed that the structure and complexity of the vascular network can markedly affect the tissue's freezing configuration with increasing ice-ball irregularity for greater blood vessel complexity.     

Temperature difference between the body core and arterial blood supplied to the brain during hyperthermia or hypothermia in humans

 A vascular heat transfer model is developed to simulate temperature decay along the carotid arteries in humans, and thus, to evaluate temperature differences between the body core and arterial blood supplied to the brain. Included are several factors, including the local blood perfusion rate, blood vessel bifurcation in the neck, and blood vessel pairs on both sides of the neck. The potential for cooling blood in the carotid artery by countercurrent heat exchange with the jugular veins and by radial heat conduction to the neck surface was estimated. Cooling along the common and internal carotid arteries was calculated to be up to 0.87 °C during hyperthermia by high environmental temperatures or muscular exercise. This model was also used to evaluate the feasibility of lowering the brain temperature effectively by placing ice pads on the neck and head surface or by wearing cooling garments during hypothermia treatment for brain injury or other medical conditions. It was found that a 1.1 °C temperature drop along the carotid arteries is possible when the neck surface is cooled to 0 °C. Thus, the body core temperature may not be a good indication of the brain temperature during hyperthermia or hypothermia. Received: 10 January 2002 / Accepted: 7 May 2002 This research was supported by a UMBC Summer Faculty Fellowship.     

A mathematical model of diffusional shunting of oxygen from arteries to veins in the kidney

To understand how arterial-to-venous (AV) oxygen shunting influences kidney oxygenation, a mathematical model of oxygen transport in the renal cortex was created. The model consists of a multiscale hierarchy of 11 countercurrent systems representing the various branch levels of the cortical vasculature. At each level, equations describing the reactive-advection-diffusion of oxygen are solved. Factors critical in renal oxygen transport incorporated into the model include the parallel geometry of arteries and veins and their respective sizes, variation in blood velocity in each vessel, oxygen transport (along the vessels, between the vessels and between vessel and parenchyma), nonlinear binding of oxygen to hemoglobin, and the consumption of oxygen by renal tissue. The model is calibrated using published measurements of cortical vascular geometry and microvascular Po(2). The model predicts that AV oxygen shunting is quantitatively significant and estimates how much kidney Vo(2) must change, in the face of altered renal blood flow, to maintain cortical tissue Po(2) at a stable level. It is demonstrated that oxygen shunting increases as renal Vo(2) or arterial Po(2) increases. Oxygen shunting also increases as renal blood flow is reduced within the physiological range or during mild hemodilution. In severe ischemia or anemia, or when kidney Vo(2) increases, AV oxygen shunting in proximal vascular elements may reduce the oxygen content of blood destined for the medullary circulation, thereby exacerbating the development of tissue hypoxia. That is, cortical ischemia could cause medullary hypoxia even when medullary perfusion is maintained. Cortical AV oxygen shunting limits the change in oxygen delivery to cortical tissue and stabilizes tissue Po(2) when arterial Po(2) changes, but renders the cortex and perhaps also the medulla susceptible to hypoxia when oxygen delivery falls or consumption increases.     

A new approach to blood flow simulation in vascular networks

A proper analysis of blood flow is contingent upon accurate modelling of the branching pattern and vascular geometry of the network of interest. It is challenging to reconstruct the entire vascular network of any organ experimentally, in particular the pulmonary vasculature, because of its very high number of vessels, complexity of the branching pattern and poor accessibility in vivo. The objective of our research is to develop an innovative approach for the reconstruction of the full pulmonary vascular tree from available morphometric data. Our method consists of the use of morphometric data on those parts of the pulmonary vascular tree that are too small to reconstruct by medical imaging methods. This method is a three-step technique that reconstructs the entire pulmonary arterial tree down to the capillary bed. Vessels greater than 2 mm are reconstructed from direct volume and surface analysis using contrast-enhanced computed tomography. Vessels smaller than 2 mm are reconstructed from available morphometric and distensibility data and rearranged by applying Murray's laws. Implementation of morphometric data to reconstruct the branching pattern and applying Murray's laws to every vessel bifurcation simultaneously leads to an accurate vascular tree reconstruction. The reconstruction algorithm generates full arterial tree topography down to the ?rst capillary bifurcation. Geometry of each order of the vascular tree is generated separately to minimize the construction and simulation time. The node-to-node connectivity along with the diameter and length of every vessel segment is established and order numbers, according to the diameter-de?ned Strahler system, are assigned. In conclusion, the present model provides a morphological foundation for future analysis of blood flow in the pulmonary circulation     

Mathematical modeling of the response of a resistive vessel to pressure

The response of small arterial vessels to internal pressure makes an essential contribution to autoregulation in the vascular bed. It is believed that free cytosolic Ca²⁺ concentration plays a pivotal role in the regulation of smooth muscle contractility and hence of the vascular lumen. A simple mathematical model of blood flow in a resistive vessel is suggested. The model is based on the experimental data obtained for cerebral arteries, but may be used for any other resistive vessel. The model not only describes the regulation of the vascular lumen by transmural pressure but also shows realistic behavior of the vessel radius and cytosolic [Ca²⁺] at different rates of pressure change. Possible variations in the radius along the vessel due to the Bayliss effect are considered.     

The effects of large blood vessels on temperature distributions during simulated hyperthermia.

Several three-dimensional vascular models have been developed to study the effects of adding equations for large blood vessels to the traditional bioheat transfer equation of Pennes when simulating tissue temperature distributions. These vascular models include "transiting" vessels, "supplying" arteries, and "draining" veins, for all of which the mean temperature of the blood in the vessels is calculated along their lengths. For the supplying arteries this spatially variable temperature is then used as the arterial temperature in the bioheat transfer equation. The different vascular models produce significantly different locations for both the maximum tumor and the maximum normal tissue temperatures for a given power deposition pattern. However, all of the vascular models predict essentially the same cold regions in the same locations in tumors: one set at the tumors' corners and another around the inlets of the large blood vessels to the tumor. Several different power deposition patterns have been simulated in an attempt to eliminate these cold regions; uniform power in the tumor, annular power in the tumor, preheating of the blood in the vessels while they are traversing the normal tissue, and an "optimal" power pattern which combines the best features of the above approaches. Although the calculations indicate that optimal power deposition patterns (which improve the temperature distributions) exist for all of the vascular models, none of the heating patterns studied eliminated all of the cold regions. Vasodilation in the normal tissue is also simulated to see its effects on the temperature fields.(ABSTRACT TRUNCATED AT 250 WORDS)     

Dominance of geometric overelastic factors in pulse transmission through arterial branching

The branching characteristic of the arterial system is such that blood pressure pulses propagate with minimum loss. This characteristic depends on the geometric and elastic properties of branching vessels. In the current investigation, mathematical relations of branching geometry and elastic properties are formulated and their relative contributions to pulse reflection at an arterial junction are analyzed. Results show that alteration of pulse transmission through the junction is more significantly affected by changes in branching vessel radii and wall thickness than by corresponding percentage changes in vessel wall elastic moduli.