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     

Vessel distensibility and flow distribution in vascular trees

 In a class of model vascular trees having distensible blood vessels, we prove that flow partitioning throughout the tree remains constant, independent of the nonzero driving flow (or nonzero inlet to terminal outlet pressure difference). Underlying assumptions are: (1) every vessel in the tree exhibits the same distensibility relationship given by $D/D_0 = f(P)$ where $D$ is the diameter which results from distending pressure $P$ and $D_0$ is the diameter of the individual vessel at zero pressure (each vessel may have its own individual $D_0$). The choice of $f(P)$ includes distensibilities often used in vessel biomechanics modeling, e.g., $f(P) = 1 + \alpha P$ or $f(P) = b + (1-b) \exp(-c P)$, as well as $f(P)$ which exhibit autoregulatory behavior. (2) Every terminal vessel in the tree is subjected to the same terminal outlet pressure. (3) Bernoulli effects are ignored. (4) Flow is nonpulsatile. (5) Blood viscosity within any individual vessel is constant. The results imply that for a vascular tree consistent with assumptions 2–5, the flow distribution calculations based on a rigid geometry, e.g., $D=D_0$, also gives the flow distribution when assuming the common distensibility relationships. Received: 30 October 2001 / Published online: 14 March 2002     

Heterogeneous mechanics of the mouse pulmonary arterial network

Individualized modeling and simulation of blood flow mechanics find applications in both animal research and patient care. Individual animal or patient models for blood vessel mechanics are based on combining measured vascular geometry with a fluid structure model coupling formulations describing dynamics of the fluid and mechanics of the wall. For example, one-dimensional fluid flow modeling requires a constitutive law relating vessel cross-sectional deformation to pressure in the lumen. To investigate means of identifying appropriate constitutive relationships, an automated segmentation algorithm was applied to micro-computerized tomography images from a mouse lung obtained at four different static pressures to identify the static pressure–radius relationship for four generations of vessels in the pulmonary arterial network. A shape-fitting function was parameterized for each vessel in the network to characterize the nonlinear and heterogeneous nature of vessel distensibility in the pulmonary arteries. These data on morphometric and mechanical properties were used to simulate pressure and flow velocity propagation in the network using one-dimensional representations of fluid and vessel wall mechanics. Moreover, wave intensity analysis was used to study effects of wall mechanics on generation and propagation of pressure wave reflections. Simulations were conducted to investigate the role of linear versus nonlinear formulations of wall elasticity and homogeneous versus heterogeneous treatments of vessel wall properties. Accounting for heterogeneity, by parameterizing the pressure/distention equation of state individually for each vessel segment, was found to have little effect on the predicted pressure profiles and wave propagation compared to a homogeneous parameterization based on average behavior. However, substantially different results were obtained using a linear elastic thin-shell model than were obtained using a nonlinear model that has a more physiologically realistic pressure versus radius relationship.     

Network thermodynamic analysis of vasomotion in a microvascular network.

The modulation of microvascular blood flow by vasomotion in the individual vessels of a simple vascular network was simulated by means of a network thermodynamic model. The flow is driven under a pulsating pressure through two arcades of branching vasoactive arterioles into a passive resistance representing the capillary and venular beds. Each vessel was assumed to have the capability of decreasing rhythmically the local diameter over a short section by a specified fraction of the maximum value and to change the average diameter along its total length in response to alterations in intraluminal pressure. Blood was assumed to exhibit a simple linear viscous flow resistance. Alterations in flow rate and distribution through the network were determined as a function of the magnitude and frequency of vasomotion within the individual arterioles supplying blood to the microvascular bed. Specific cases are shown to illustrate how blood flow can be influenced by the patterns of vasomotion within the network.     

Distensibility of small arteries of the dog lung.

To obtain in situ measurements of the distensibility of small (100- to 1,000-microns-diam) pulmonary arterial vessels of the dog lung, X-ray angiograms were obtained from isolated lung lobes with the vascular pressure adjusted to various levels. The in situ diameter-pressure relationships were compared with the diameter-pressure relationships for small arteries that were dissected free from the lungs and cannulated with small glass pipettes for the measurement of diameter and transmural pressure. The diameter-vascular or diameter-transmural pressure curves from both in situ and cannulated vessels were sufficiently linear in the pressure range studied (0-30 Torr) that they could be characterized by linear regression to obtain estimates of D0, the diameter at zero vascular pressure, and beta, the change in diameter (micron) per Torr change in pressure. The vessel distensibility coefficient (alpha) was defined as alpha = beta/D0. The mean values of alpha were approximately 2.0 +/- 0.8%/Torr (SD) for the in situ vessels and 1.7 +/- 0.6%/Torr for the cannulated vessels, with no statistically significant difference between the two methods. The influence of vasoconstriction elicited by serotonin was evaluated in the in situ vessels. Serotonin-induced vasoconstriction caused a decrease in D0 and little change in alpha.     

Microfocal X-ray CT imaging and pulmonary arterial distensibility in excised rat lungs

The objective of this study was to develop an X-ray computed tomographic method for measuring pulmonary arterial dimensions and locations within the intact rat lung. Lungs were removed from rats and their pulmonary arterial trees were filled with perfluorooctyl bromide to enhance X-ray absorbance. The lungs were rotated within the cone of the X-ray beam projected from a microfocal X-ray source onto an image intensifier, and 360 images were obtained at 1 degrees increments. The three-dimensional image volumes were reconstructed with isotropic resolution using a cone beam reconstruction algorithm. The vessel diameters were obtained by fitting a functional form to the image of the vessel circular cross section. The functional form was chosen to take into account the point spread function of the image acquisition and reconstruction system. The diameter measurements obtained over a range of vascular pressures were used to characterize the distensibility of the rat pulmonary arteries. The distensibility coefficient alpha [defined by D(P) = D(0)(1 + alphaP), where D(P) is the diameter at intravascular pressure (P)] was approximately 2.8% mmHg and independent of vessel diameter in the diameter range (about 100 to 2,000 mm) studied.     

Growth and vascular structure of human melanoma xenografts

The growth and the vascular structure of five human melanomas grown in athymic nude mice were studied. Four growth parameters (tumour volume doubling time, fraction of cells in S-phase, growth fraction, cell-loss factor) were analysed against each of four vascular parameters (length of vessels with diameters in the range 5-15 micron, total vessel length, total vessel surface, total vessel volume--all per unit of histologically intact tumour volume). Statistically significant linear correlations between the parameters were found for any of the combinations. However, there was a consistent trend in the data: the tumour volume doubling time and the cell-loss factor tended to decrease while the fraction of cells in S-phase and the growth fraction tended to increase with increasing vascular density, whichever vascular parameter was considered. This finding indicates that the vascular density is among the factors which are decisive for the growth rate of tumours. However, the present work does not exclude the possibility that intrinsic properties of the tumour cells may also be important.     

Growth and Vascular Structure of Human Melanoma Xenografts

Abstract The growth and the vascular structure of five human melanomas grown in athymic nude mice were studied. Four growth parameters (tumour volume doubling time, fraction of cells in S-phase, growth fraction, cell-loss factor) were analysed against each of four vascular parameters (length of vessels with diameters in the range 5–15 μm, total vessel length, total vessel surface, total vessel volume-all per unit of histologically intact tumour volume). Statistically significant linear correlations between the parameters were found for any of the combinations. However, there was a consistent trend in the data: the tumour volume doubling time and the cell-loss factor tended to decrease while the fraction of cells in S-phase and the growth fraction tended to increase with increasing vascular density, whichever vascular parameter was considered. This finding indicates that the vascular density is among the factors which are decisive for the growth rate of tumours. However, the present work does not exclude the possibility that intrinsic properties of the tumour cells may also be important.     

Cardiovascular mechanics in the early stages of pulmonary hypertension: a computational study

We formulate and study a new mathematical model of pulmonary hypertension. Based on principles of fluid and elastic dynamics, we introduce a model that quantifies the stiffening of pulmonary vasculature (arteries and arterioles) to reproduce the hemodynamics of the pulmonary system, including physiologically consistent dependence between compliance and resistance. This pulmonary model is embedded in a closed-loop network of the major vessels in the body, approximated as one-dimensional elastic tubes, and zero-dimensional models for the heart and other organs. Increasingly severe pulmonary hypertension is modeled in the context of two extreme scenarios: (1) no cardiac compensation and (2) compensation to achieve constant cardiac output. Simulations from the computational model are used to estimate cardiac workload, as well as pressure and flow traces at several locations. We also quantify the sensitivity of several diagnostic indicators to the progression of pulmonary arterial stiffening. Simulation results indicate that pulmonary pulse pressure, pulmonary vascular compliance, pulmonary RC time, luminal distensibility of the pulmonary artery, and pulmonary vascular impedance are much better suited to detect the early stages of pulmonary hypertension than mean pulmonary arterial pressure and pulmonary vascular resistance, which are conventionally employed as diagnostic indicators for this disease.     

Diameter asymmetry of porcine coronary arterial trees: structural and functional implications

The coronary vasculature is characterized by highly asymmetric diameters at bifurcations, which may be an important determinant of flow distribution. To facilitate accurate reconstruction of the coronary network for hemodynamic analysis, we introduce a statistical data set of the diameter asymmetry at bifurcations based on morphometric data of the porcine coronary arterial and venous trees. The bifurcation asymmetry data were represented by the diameter ratio of the daughters relative to mother vessel and by an area expansion ratio (AER) at each bifurcation. A novel asymmetry ratio matrix was introduced to describe the diameter asymmetry of daughters to mother vessels. The relations between AER and flow velocity, and asymmetry ratio matrix and flow distribution, were considered. The results indicate that the ratio of large daughter to mother vessel has a minimum value at order 5 (mean diameter of approximately 70 microm), whereas the ratio of small daughter to mother vessel decreases monotonically with increase in order number. The AER was found to be fairly uniform for larger vessels and to increase from order 5 toward the capillaries. At order 5, we observe a transition in asymmetric bifurcation pattern that may mark a hemodynamic transition from transmural to perfusion subnetworks. The functional implications of these structural transitions are considered.     

Recruitment in networks of pulmonary capillaries.

To improve our understanding of the pressure-flow characteristics of pulmonary capillaries, we analyzed by means of computer stimulation a theoretical model composed of 50 interconnected nonlinear elements. Each element required a critical pressure across it before flow occurred and there was a subsequent linear pressure-flow region whose slope, or resistance, could be related to the transmural pressure of the element ("distensibility"). The critical pressures and resistances of each element of the network were randomly chosen from distributions. We found that recruitment (i.e., onset of flow) occurred over a large range of network upstream or "arterial" pressures, and that relatively high arterial pressures were required before all elements had no distensibility. Intermittent and reverse flow were commonly seen in some elements as the arterial pressure was raised in steps. These flow reversals were particularly common when the critical pressures and resistances of the elements were inversely related. The critical pressures required for such behavior in the capillary segments of the pulmonary microcirculation were calculated to be extremely small, of the order of 0.02 cmH2O. Pressures of this magnitude might result from sticking of red cells to capillary walls or to each other. The properties of such a network may explain the patchiness of flow in the pulmonary microcirculation and the large range of arterial pressures over which recruitment is observed to occur.     

A simple distensible vessel model for interpreting pulmonary vascular pressure-flow curves.

A simple distensible vessel model was developed for the purpose of interpreting the vascular pressure-flow curve in the zone 3 lung. The model-governing equation has two parameters: R0, representing the hemodynamic resistance of the undistended pulmonary vascular bed, and alpha, representing the distensibility of the resistance vessels. To evaluate the model, the governing equation was used in a nonlinear regression analysis of the pressure-flow data from isolated dog lung lobes. The dependency of the estimates of the model parameters in response to changes in perfusate viscosity (hematocrit) was determined. The distensible vessel model provided reasonable fits to the data, and, as predicted, R0, but not alpha, was hematocrit dependent. On the other hand, the traditional linear ohmic-Starling resistor model fit to the same pressure-flow data generally provided fits approaching those of the distensibility model only if the pressure intercept (the mean "critical closing pressure") was allowed to increase with hematocrit. Because the ohmic-Starling resistor concept does not predict a hematocrit dependence of the critical closing pressure, this latter observation is evidence that the distensible vessel model offers an alternative conceptualization of the pulmonary circulation worthy of additional study with respect to the interpretation of experimental pressure-flow data.     

Angiogenesis and vascular remodelling in normal and cancerous tissues

Vascular development and homeostasis are underpinned by two fundamental features: the generation of new vessels to meet the metabolic demands of under-perfused regions and the elimination of vessels that do not sustain flow. In this paper we develop the first multiscale model of vascular tissue growth that combines blood flow, angiogenesis, vascular remodelling and the subcellular and tissue scale dynamics of multiple cell populations. Simulations show that vessel pruning, due to low wall shear stress, is highly sensitive to the pressure drop across a vascular network, the degree of pruning increasing as the pressure drop increases. In the model, low tissue oxygen levels alter the internal dynamics of normal cells, causing them to release vascular endothelial growth factor (VEGF), which stimulates angiogenic sprouting. Consequently, the level of blood oxygenation regulates the extent of angiogenesis, with higher oxygenation leading to fewer vessels. Simulations show that network remodelling (and de novo network formation) is best achieved via an appropriate balance between pruning and angiogenesis. An important factor is the strength of endothelial tip cell chemotaxis in response to VEGF. When a cluster of tumour cells is introduced into normal tissue, as the tumour grows hypoxic regions form, producing high levels of VEGF that stimulate angiogenesis and cause the vascular density to exceed that for normal tissue. If the original vessel network is sufficiently sparse then the tumour may remain localised near its parent vessel until new vessels bridge the gap to an adjacent vessel. This can lead to metastable periods, during which the tumour burden is approximately constant, followed by periods of rapid growth.     

Influence of size of emboli on extravascular lung water

We examined the influence of the size of emboli on the vascular volume (QL) and extravascular volume (Qev) accessible to 3HOH during a single pass through an isolated dog lung lobe using the double indicator-dilution method with 125I-human serum albumin as the vascular indicator. As successively more beads of a given diameter (58, 548, or 3,175 microns) were introduced into a lung lobe, a linear relationship between QL and Qev was obtained as they both decreased. The slope of the graph of QL vs. Qev with progressive embolism was directly proportional to the bead diameter. This suggested an approach for estimating the total vascular volume in vessels smaller than the diameter of the beads before embolization, referred to as Qm. If it is assumed that most of the transvascular diffusional exchange of 3HOH occurs in vessels smaller than the smallest beads (mainly capillaries) and that vessel obstruction does not change the ratio of Qev to the perfused capillary volume, the slope of the plot of QL vs. Qev is an estimate of the fraction, Qm/QL, of the total vascular volume in vessels smaller than the bead diameter. In the dog lung lobes studied, Qm/QL was approximately 0.64 for 58-microns vessels, 0.75 for 548-microns vessels, and 0.82 for 3,175-microns vessels. The results suggest that, with occlusion of vessels greater than or equal to 58 microns, 3HOH does not diffuse significantly into unperfused regions.(ABSTRACT TRUNCATED AT 250 WORDS)     

In vivo two-photon excited fluorescence microscopy reveals cardiac- and respiration-dependent pulsatile blood flow in cortical blood vessels in mice

Subtle alterations in cerebral blood flow can impact the health and function of brain cells and are linked to cognitive decline and dementia. To understand hemodynamics in the three-dimensional vascular network of the cerebral cortex, we applied two-photon excited fluorescence microscopy to measure the motion of red blood cells (RBCs) in individual microvessels throughout the vascular hierarchy in anesthetized mice. To resolve heartbeat- and respiration-dependent flow dynamics, we simultaneously recorded the electrocardiogram and respiratory waveform. We found that centerline RBC speed decreased with decreasing vessel diameter in arterioles, slowed further through the capillary bed, and then increased with increasing vessel diameter in venules. RBC flow was pulsatile in nearly all cortical vessels, including capillaries and venules. Heartbeat-induced speed modulation decreased through the vascular network, while the delay between heartbeat and the time of maximum speed increased. Capillary tube hematocrit was 0.21 and did not vary with centerline RBC speed or topological position. Spatial RBC flow profiles in surface vessels were blunted compared with a parabola and could be measured at vascular junctions. Finally, we observed a transient decrease in RBC speed in surface vessels before inspiration. In conclusion, we developed an approach to study detailed characteristics of RBC flow in the three-dimensional cortical vasculature, including quantification of fluctuations in centerline RBC speed due to cardiac and respiratory rhythms and flow profile measurements. These methods and the quantitative data on basal cerebral hemodynamics open the door to studies of the normal and diseased-state cerebral microcirculation.     

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.     

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.     

Neurogenic-nitric oxide interactions affecting brachial artery mechanics in humans: roles of vessel distensibility vs. diameter

The purpose of this investigation was to assess the interactive influence of sympathetic activation and supplemental nitric oxide (NO) on brachial artery distensibility vs. its diameter. It was hypothesized that 1) sympathetic activation and NO competitively impact muscular conduit artery (brachial artery) mechanics, and 2) neurogenic constrictor input affects conduit vessel stiffness independently of outright changes in conduit vessel diastolic diameter. Lower body negative pressure (LBNP) and a cold pressor stress (CPT) were used to study the changes in conduit vessel mechanics when the increased sympathetic outflow occurred with and without changes in heart rate (LBNP -40 vs. -15 mmHg) and blood pressure (CPT vs. LBNP). These maneuvers were performed in the absence and presence of nitroglycerin. Neither LBNP nor CPT altered brachial artery diastolic diameter; however, distensibility was reduced by 25 to 54% in each reflex (all P < 0.05). This impact of sympathetic activation on brachial artery distensibility was not altered by nitroglycerin supplementation (21-54%; P < 0.05), although baseline diameter was increased by the exogenous NO (P < 0.05). The results indicate that sympathetic excitation can reduce the distensibility of the brachial artery independently of concurrent changes in diastolic diameter, heart rate, and blood pressure. However, exogenous NO did not minimize or reverse brachial stiffening during sympathetic activation. Therefore, sympathetic outflow appears to impact the stiffness of this conduit vessel rather than its diastolic diameter or, by inference, its local resistance to flow.     

Numerical simulation of blood flow and pressure drop in the pulmonary arterial and venous circulation

A novel multiscale mathematical and computational model of the pulmonary circulation is presented and used to analyse both arterial and venous pressure and flow. This work is a major advance over previous studies by Olufsen et al. (Ann Biomed Eng 28:1281–1299, 2012) which only considered the arterial circulation. For the first three generations of vessels within the pulmonary circulation, geometry is specified from patient-specific measurements obtained using magnetic resonance imaging (MRI). Blood flow and pressure in the larger arteries and veins are predicted using a nonlinear, cross-sectional-area-averaged system of equations for a Newtonian fluid in an elastic tube. Inflow into the main pulmonary artery is obtained from MRI measurements, while pressure entering the left atrium from the main pulmonary vein is kept constant at the normal mean value of 2 mmHg. Each terminal vessel in the network of ‘large’ arteries is connected to its corresponding terminal vein via a network of vessels representing the vascular bed of smaller arteries and veins. We develop and implement an algorithm to calculate the admittance of each vascular bed, using bifurcating structured trees and recursion. The structured-tree models take into account the geometry and material properties of the ‘smaller’ arteries and veins of radii \(\ge \) 50  \(\upmu \) m. We study the effects on flow and pressure associated with three classes of pulmonary hypertension expressed via stiffening of larger and smaller vessels, and vascular rarefaction. The results of simulating these pathological conditions are in agreement with clinical observations, showing that the model has potential for assisting with diagnosis and treatment for circulatory diseases within the lung.