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.     

Formation of the branching pattern of blood vessels in the wall of the avian yolk sac studied by a computer simulation

The present study was performed to provide data to support the notion previously believed but not proved experimentally or theoretically, that blood vessels are formed by the selection of capillaries in the network. In an attempt to understand the mechanism of formation of blood vessel branching structures, the transformation of a capillary network to a branching system in the wall of quail yolk sac was successively recorded by a series of photographs, and a computer simulation was carried out for the process of in vivo vascularization based on the photographs. The simulation demonstrated that a positive feedback system participated in the formation of a branching structure. That is, vessels which had been much used were enlarged, whereas less used vessels were reduced in their size and finally extinguished. The enlarged vessels became major components of the branching system. As the body of an embryo grew, it was observed that polygonal capillary networks enlarged, which led each polygon of the network to divide into a few finer polygons. Then, some of the capillary vessels were again selected and formed a branching system. This process repeated during the body growth, indicating that the vascular system developed adaptively to the body growth. A region where the growth was fast, received much blood flow and produced finer networks of capillaries. Thus, it was experimentally demonstrated for the first time that capillaries in the network are successively selected by a positive feedback mechanism and form blood vessels.     

Bifurcation asymmetry of the porcine coronary vasculature and its implications on coronary flow heterogeneity

The branching pattern of the coronary arteries and veins is asymmetric, i.e., many small vessels branch off of a large trunk such that the two daughter vessels at a bifurcation are of unequal diameters and lengths. One important implication of the geometric vascular asymmetry is the dispersion of blood flow at a bifurcation, which leads to large spatial heterogeneity of myocardial blood flow. To document the asymmetric branching pattern of the coronary vessels, we computed an asymmetry ratio for the diameters and lengths of all vessels, defined as the ratio of the daughter diameters and lengths, respectively. Previous data from silicone elastomer cast of the entire coronary vasculature including arteries, arterioles, venules, and veins were analyzed. Data on smaller vessels were obtained from histological specimens by optical sectioning, whereas data on larger vessels were obtained from vascular casts. Asymmetry ratios for vascular areas, volumes, resistances, and flows of the various daughter vessels were computed from the asymmetry ratios of diameters and lengths for every order of mother vessel. The results show that the largest orders of arterial and venous vessels are most asymmetric and the degree of asymmetry decreases toward the smaller vessels. Furthermore, the diameter asymmetry at a bifurcation is significantly larger for the coronary veins (1.7-6.8 for sinus veins) than the corresponding arteries (1.5-5.8 for left anterior descending coronary artery) for orders 2-10, respectively. The reported diameter asymmetry at a bifurcation leads to significant heterogeneity of blood flow at a bifurcation. Hence, the present data quantify the dispersion of blood flow at a bifurcation and are essential for understanding flow heterogeneity in the coronary circulation.     

Anatomically based finite element models of the human pulmonary arterial and venous trees including supernumerary vessels.

Studies of the origin of pulmonary blood flow heterogeneity have highlighted the significant role of vessel branching structure on flow distribution. To enable more detailed investigation of structure-function relationships in the pulmonary circulation, an anatomically based finite element model of the arterial and venous networks has been developed to more accurately reflect the geometry found in vivo. Geometric models of the arterial and venous tree structures are created using a combination of multidetector row X-ray computed tomography imaging to define around 2,500 vessels from each tree, a volume-filling branching algorithm to generate the remaining accompanying conducting vessels, and an empirically based algorithm to generate the supernumerary vessel geometry. The explicit generation of supernumerary vessels is a unique feature of the computational model. Analysis of branching properties and geometric parameters demonstrates close correlation between the model geometry and anatomical measures of human pulmonary blood vessels. A total of 12 Strahler orders for the arterial system and 10 Strahler orders for the venous system are generated, down to the equivalent level of the terminal bronchioles in the bronchial tree. A simple Poiseuille flow solution, assuming rigid vessels, is obtained within the arterial geometry of the left lung, demonstrating a large amount of heterogeneity in the flow distribution, especially with inclusion of supernumerary vessels. This model has been constructed to accurately represent available morphometric data derived from the complex asymmetric branching structure of the human pulmonary vasculature in a form that will be suitable for application in functional simulations.     

Vessel tortuousity and reduced vascularization in the fetoplacental arterial tree after maternal exposure to polycyclic aromatic hydrocarbons

Polycyclic aromatic hydrocarbons (PAHs) are ubiquitous environmental pollutants and the main toxicants found in cigarettes. Women are often exposed to PAHs before pregnancy, typically via prepregnancy smoking. To determine how prepregnancy exposure affects the fetoplacental vasculature of the placenta, we exposed female mice to PAHs before conception, perfused the fetoplacental arterial trees with X-ray contrast agent, and imaged the vasculature ex vivo by microcomputed tomography (micro-CT) at embryonic day 15.5. Automated vascular segmentation and flow calculations revealed that in control trees, <40 chorionic plate vessels (diameter>180 μm) gave rise to ～1,300 intraplacental arteries (50-180 μm), predicting an arterial vascular resistance of 0.37±0.04 mmHg·s·μl(-1). PAH exposure increased vessel curvature of chorionic plate vessels and significantly increased the tortuousity ratio of the tree. Intraplacental arteries were reduced by 17%, primarily due to a 27% decrease in the number of arteriole-sized (50-100 μm) vessels. There were no changes in the number of chorionic vessels, the depth or span of the tree, the diameter scaling coefficient, or the segment length-to-diameter ratio. PAH exposure resulted in a tree with a similar size and dichotomous branching structure, but one that was comparatively sparse so that arterial vascular resistance was increased by 30%. Assuming the same pressure gradient, blood flow would be 19% lower. Low flow may contribute to the 23% reduction observed in fetal weight. New insights into the specific effects of PAH exposure on a developing arterial tree were achieved using micro-CT imaging and automated vascular segmentation analysis.     

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 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.     

Endothelial mediators and communication through vascular gap junctions

Cellular interaction in vessels is achieved by multiple communication pathways, including gap junctions (GJs). They provide intercellular channels, allowing direct interaction of endothelial and smooth muscle cells and the coordination of cellular behaviour along the vessel. The latter is a prerequisite for large flow increases because an adaptation of resistance along the vessel length is required. Longitudinal communication is studied by confined local stimulation of arterioles and the observation of responses at distant locations. Certain vascular stimuli induce local and concomitant remote responses of a similar type, verifying rapid longitudinal conduction of vasomotor signals, most likely changes in membrane potential. This is achieved for dilatory responses via the endothelium, possibly by an endothelium-derived hyperpolarising factor (EDHF) that induces local hyperpolarisation, which is then transferred to remote sites through GJs. In vessels, GJs are composed of different connexins (Cx), but Cx40 is of special importance because its lack impairs longitudinal conduction of vasodilations. Interestingly, Cx40-deficient mice are hypertensive, suggesting that Cx40-dependent coupling is necessary to regulate vascular behaviour and peripheral resistance. While the role of other connexins is less well established, an abundance of data has proven the necessity of GJ communication to coordinate vascular behaviour during blood flow regulation.     

Analysis of blood flow in the entire coronary arterial tree

A hemodynamic analysis of coronary blood flow must be based on the measured branching pattern and vascular geometry of the coronary vasculature. We recently developed a computer reconstruction of the entire coronary arterial tree of the porcine heart based on previously measured morphometric data. In the present study, we carried out an analysis of blood flow distribution through a network of millions of vessels that includes the entire coronary arterial tree down to the first capillary branch. The pressure and flow are computed throughout the coronary arterial tree based on conservation of mass and momentum and appropriate pressure boundary conditions. We found a power law relationship between the diameter and flow of each vessel branch. The exponent is approximately 2.2, which deviates from Murray's prediction of 3.0. Furthermore, we found the total arterial equivalent resistance to be 0.93, 0.77, and 1.28 mmHg.ml(-1).s(-1).g(-1) for the right coronary artery, left anterior descending coronary artery, and left circumflex artery, respectively. The significance of the present study is that it yields a predictive model that incorporates some of the factors controlling coronary blood flow. The model of normal hearts will serve as a physiological reference state. Pathological states can then be studied in relation to changes in model parameters that alter coronary perfusion.     

Sensory nerves determine the pattern of arterial differentiation and blood vessel branching in the skin

Nerves and blood vessels are branched structures, but whether their branching patterns are established independently or coordinately is not clear. Here we show that arteries, but not veins, are specifically aligned with peripheral nerves in embryonic mouse limb skin. Mutations that eliminate peripheral sensory nerves or Schwann cells prevent proper arteriogenesis, while those that disorganize the nerves maintain the alignment of arteries with misrouted axons. In vitro, sensory neurons or Schwann cells can induce arterial marker expression in isolated embryonic endothelial cells, and VEGF(164/120) is necessary and sufficient to mediate this induction. These data suggest that peripheral nerves provide a template that determines the organotypic pattern of blood vessel branching and arterial differentiation in the skin, via local secretion of VEGF.     

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     

Blood flow regulation in the cerebral microvasculature with an arcadal network: a numerical simulation

Blood flow regulation in the cerebral microvasculature with an arcadal network was investigated using a numerical simulation. A mathematical model for blood flow in the arcadal network, based on in vivo data of cat cerebral microvasculature and flow velocity was developed. The network model consists of 45 vessel segments and 25 branching points. To simulate microvascular response to blood flow, non-reactive (solid), cerebral arteriole-like, or skeletal muscle arteriole-like responses to wall shear stress were taken into account. Numerical calculation was carried out in the flow condition where the inlet (arterial) pressure was changed from 60 to 120 mmHg. Flow-rate in each efferent vessel and the mean flow-rate over all efferent vessels were evaluated for assessment of blood supply to the local area of cerebral tissue. The simulation demonstrated the wall shear stress-induced vasodilation in the arcadal network worked to maintain the blood flow at a constant level with pressure variable in a wide range. It is suggested that an individual microvessel (segment) should join in the regulatory process of flow, interacting with other microvessels (cooperative regulation).     

Investigation of the structure of human coronary vasculature

Some results of a morphometric study of the parameters of coronary arteries are presented. The parameters that characterize the structure of the arterial vasculature as an optimal branching system have been calculated. Statistically reliable correlations between the diameter of the bigger of two daughter vessels in a bifurcation with the diameter of the parent vessel as well as between the diameter of the smaller daughter vessel and the asymmetry coefficient have been obtained. Differences in the structural parameters of the two types of coronary arteries that provide blood delivery and distribution have been revealed. The relationships between the lengths and diameters of the arteries of different subsystems have been obtained. It is shown that asymmetrical branching is characteristic of the coronary vasculature, and self-similar asymmetric tree-like systems may be used for its modeling.     

Tissue-growth-based synthetic tree generation and perfusion simulation

Biological tissues receive oxygen and nutrients from blood vessels by developing an indispensable supply and demand relationship with the blood vessels. We implemented a synthetic tree generation algorithm by considering the interactions between the tissues and blood vessels. We first segment major arteries using medical image data and synthetic trees are generated originating from these segmented arteries. They grow into extensive networks of small vessels to fill the supplied tissues and satisfy the metabolic demand of them. Further, the algorithm is optimized to be executed in parallel without affecting the generated tree volumes. The generated vascular trees are used to simulate blood perfusion in the tissues by performing multiscale blood flow simulations. One-dimensional blood flow equations were used to solve for blood flow and pressure in the generated vascular trees and Darcy flow equations were solved for blood perfusion in the tissues using a porous model assumption. Both equations are coupled at terminal segments explicitly. The proposed methods were applied to idealized models with different tree resolutions and metabolic demands for validation. The methods demonstrated that realistic synthetic trees were generated with significantly less computational expense compared to that of a constrained constructive optimization method. The methods were then applied to cerebrovascular arteries supplying a human brain and coronary arteries supplying the left and right ventricles to demonstrate the capabilities of the proposed methods. The proposed methods can be utilized to quantify tissue perfusion and predict areas prone to ischemia in patient-specific geometries.

     

Comparative characterization of reactions of skeletal muscle arterial and venous vessels to combined adrenergic effects

In experiments with the constant blood flow perfusion of the cat calf muscle and combined actions of adrenalin and noradrenaline were tested as to the blood flow resistance changes of the arterial and venous blood vessels. Separately applied the catecholamines evoked vascular resistance changes practically similar in value; combined effects of catecholamines realized in greater increase of arterial than venous resistance. In contrast to arterial vessels supramaximal stimuli resulted in much lesser constrictive effect as compared with reaction of intramural veins to separately applied catecholamines. Greater doses of catecholamines being combined, stability of effector system of skeletal muscle veins is decreased as compared to arteries.     

Simulation of 3D Solid Tumour Angiogenesis Including Arteriole,Capillary and Venule

In this paper, a 3D mathematical model of tumour angiogenesis is developed, to generate a functional tumour vasculature for blood microcirculation. The model follows that of Anderson and Chaplain (1998) ^[1] with three exceptions: (a) extending the model from 2D to 3D, one arteriole and one venule is induced as two parent vessels to form an intact circulation network for blood flow; (b) generating networks able to penetrate into the tumour interior rather than the exterior only; (c) considering branching generations with different diameters, based on which three groups of vessels, such as arterioles, venules and capillaries are classified. The present study contains four steps: 1. Generation of 3D angiogenic vasculature induced from one arteriole and one venule, with branching generations considered. 2. Examination of vessel connectivity among each other to construct a functional network for blood circulation, investigation of sensitivity of network architectures to changes in some model parameters. 3. Simulation of blood flow in the developed vasculatures. 4. Comparisons of blood flow calculated on the networks induced from an arteriole-venule system and from a single parent vessel.
The networks from simulations could present basic geometric and morphological features of tumour vasculatures. The sensitivity analysis indicates the controllability of the created networks, which could construct architectures of some specific geometric features to suit different types of tumours. The comparisons of blood flow mentioned above demonstrate the validity of the present vasculature, which could be served as a more realistic network structure for research of microcirculation, drug delivery in solid tumors.     

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.     

Theoretical comparison of wall-derived and erythrocyte-derived mechanisms for metabolic flow regulation in heterogeneous microvascular networks

The objective of this study is to compare the effectiveness of metabolic signals derived from erythrocytes and derived from the vessel wall for regulating blood flow in heterogeneous microvascular networks. A theoretical model is used to simulate blood flow, mass transport, and vascular responses. The model accounts for myogenic, shear-dependent, and metabolic flow regulation. Metabolic signals are assumed to be propagated upstream along vessel walls via a conducted response. Arteriolar tone is assumed to depend on the conducted metabolic signal as well as local wall shear stress and wall tension, and arteriolar diameters are calculated based on vascular smooth muscle mechanics. The model shows that under certain conditions metabolic regulation based on wall-derived signals can be more effective in matching perfusion to local oxygen demand relative to regulation based on erythrocyte-derived signals, resulting in higher extraction and lower oxygen deficit. The lower effectiveness of the erythrocyte-derived signal is shown to result in part from the unequal partition of hematocrit at diverging bifurcations, such that low-flow vessels tend to receive a reduced hematocrit and thereby experience a reduced erythrocyte-derived metabolic signal. The model simulations predict that metabolic signals independent of erythrocytes may play an important role in local metabolic regulation of vascular tone and flow distribution in heterogeneous microvessel networks.     

Flow and pressure distributions in vascular networks consisting of distensible vessels

We examine the influence of vessel distensibility on the fraction of the total network flow passing through each vessel of a model vascular network. An exact computational methodology is developed yielding an analytic proof. For a class of structurally heterogeneous asymmetric vascular networks, if all the individual vessels share a common distensibility relation when the total network flow is changed, this methodology proves that each vessel will continue to receive the same fraction of the total network flow. This constant flow partitioning occurs despite a redistribution of pressures, which may result in a decrease in the diameter of one and an increase in the diameter of the other of two vessels having a common diameter at a common pressure. This theoretical observation, taken along with published experimental observations on pulmonary vessel distensibilities, suggests that vessel diameter-independent distensibility in the pulmonary vasculature may be an evolutionary adaptation for preserving the spatial distribution of pulmonary blood flow in the face of large variations in cardiac output.