Development of a general method for designing microvascular networks using distribution of wall shear stress

In the present study, theoretical formulations for calculation of optimal bifurcation angle and relationship between the diameters of mother and daughter vessels using the power law model for non-Newtonian fluids are developed. The method is based on the distribution of wall shear stress in the mother and daughter vessels. Also, the effect of distribution of wall shear stress on the minimization of energy loss and flow resistance is considered. It is shown that constant wall shear stress in the mother and daughter vessels provides the minimum flow resistance and energy loss of biological flows. Moreover, the effects of different wall shear stresses in the mother and daughter branches, different lengths of daughter branches in the asymmetric bifurcations and non-Newtonian effect of biological fluid flows on the bifurcation angle and the relationship between the diameters of mother and daughter branches are considered. Using numerical simulations for non-Newtonian models such as power law and Carreau models, the effects of optimal bifurcation angle on the pressure drop and flow resistance of blood flow in the symmetric bifurcation are investigated. Numerical simulations show that optimal bifurcation angle decreases the pressure drop and flow resistance especially for bifurcations at large Reynolds number.     

Numerical investigation of the non-Newtonian pulsatile blood flow in a bifurcation model with a non-planar branch

The pulsatile flow of non-Newtonian fluid in a bifurcation model with a non-planar daughter branch is investigated numerically by using the Carreau-Yasuda model to take into account the shear thinning behavior of the analog blood fluid. The objective of this study is to deal with the influence of the non-Newtonian property of fluid and of out-of-plane curvature in the non-planar daughter vessel on wall shear stress (WSS), oscillatory shear index (OSI), and flow phenomena during the pulse cycle. The non-Newtonian property in the daughter vessels induces a flattened axial velocity profile due to its shear thinning behavior. The non-planarity deflects flow from the inner wall of the vessel to the outer wall and changes the distribution of WSS along the vessel, in particular in systole phase. Downstream of the bifurcation, the velocity profiles are shifted toward the flow divider, and low WSS and high shear stress temporal oscillations characterized by OSI occur on the outer wall region of the daughter vessels close to the bifurcation. Secondary motions become stronger with the addition of the out-of-plane curvature induced by the bending of the vessel, and the secondary flow patterns swirl along the non-planar daughter vessel. A significant difference between the non-Newtonian and the Newtonian pulsatile flow is revealed during the pulse cycle; however, reasonable agreement between the non-Newtonian and the rescaled Newtonian flow is found. Calculated results for the pulsatile flow support the view that the non-planarity of blood vessels and the non-Newtonian properties of blood are an important factor in hemodynamics and may play a significant role in vascular biology and pathophysiology.     

Pressure drop and flow rate measurements in a human aortic bifurcation cast for steady and pulsatile flow

Pressure drop and flow rate measurements in a rigid cast of a human aortic bifurcation under both steady and physiological pulsatile flow conditions are reported. Integral momentum and mechanical energy balances are used to calculate impedance, spatially averaged wall shear stress and viscous dissipation rate from the data. In the daughter branches, steady flow impedance is within 30% of the Poiseuille flow prediction, while pulsatile flow impedance is within a factor of 2 of fully developed, oscillatory, straight tube flow theory (Womersley theory). Estimates of wall shear stress are in accord with measurements obtained from velocity profiles. Mean pressure drop and viscous dissipation rate are elevated in pulsatile flow relative to steady flow at the mean flow rate, and the exponents of their Reynolds number dependence are in accord with available theory.     

Branching exponent heterogeneity and wall shear stress distribution in vascular trees

A bifurcating arterial system with Poiseuille flow can function at minimum cost and with uniform wall shear stress if the branching exponent (z) = 3 [where z is defined by (D(1))(z) = (D(2))(z) + (D(3))(z); D(1) is the parent vessel diameter and D(2) and D(3) are the two daughter vessel diameters at a bifurcation]. Because wall shear stress is a physiologically transducible force, shear stress-dependent control over vessel diameter would appear to provide a means for preserving this optimal structure through maintenance of uniform shear stress. A mean z of 3 has been considered confirmation of such a control mechanism. The objective of the present study was to evaluate the consequences of a heterogeneous distribution of z values about the mean with regard to this uniform shear stress hypothesis. Simulations were carried out on model structures otherwise conforming to the criteria consistent with uniform shear stress when z = 3 but with varying distributions of z. The result was that when there was significant heterogeneity in z approaching that found in a real arterial tree, the coefficient of variation in shear stress was comparable to the coefficient of variation in z and nearly independent of the mean value of z. A systematic increase in mean shear stress with decreasing vessel diameter was one component of the variation in shear stress even when the mean z = 3. The conclusion is that the influence of shear stress in determining vessel diameters is not, per se, manifested in a mean value of z. In a vascular tree having a heterogeneous distribution in z values, a particular mean value of z (e.g., z = 3) apparently has little bearing on the uniform shear stress hypothesis.     

Healthy and diseased coronary bifurcation geometries influence near-wall and intravascular flow: A computational exploration of the hemodynamic risk

Local hemodynamics has been identified as one main determinant in the onset and progression of atherosclerotic lesions at coronary bifurcations. Starting from the observation that atherosensitive hemodynamic conditions in arterial bifurcation are majorly determined by the underlying anatomy, the aim of the present study is to investigate how peculiar coronary bifurcation anatomical features influence near-wall and intravascular flow patterns. Different bifurcation angles and cardiac curvatures were varied in population-based, idealized models of both stenosed and unstenosed bifurcations, representing the left anterior descending (LAD) coronary artery with its diagonal branch. Local hemodynamics was analyzed in terms of helical flow and exposure to low/oscillatory shear stress by performing computational fluid dynamics simulations.Results show that bifurcation angle impacts lowly hemodynamics in both stenosed and unstenosed cases. Instead, curvature radius influences the generation and transport of helical flow structures, with smaller cardiac curvature radius associated to higher helicity intensity. Stenosed bifurcation models exhibit helicity intensity values one order of magnitude higher than the corresponding unstenosed cases. Cardiac curvature radius moderately affects near-wall hemodynamics of the stenosed cases, with smaller curvature radius leading to higher exposure to low shear stress and lower exposure to oscillatory shear stress. In conclusion, the proposed controlled benchmark allows investigating the effect of various geometrical features on local hemodynamics at the LAD/diagonal bifurcation, highlighting that cardiac curvature influences near wall and intravascular hemodynamics, while bifurcation angle has a minor effect.     

Modeling the effect of blood vessel bifurcation ratio on occlusive thrombus formation

Vascular geometry is a major determinant of the hemodynamics that promote or prevent unnecessary vessel occlusion from thrombus formation. Bifurcations in the vascular geometry are repeating structures that introduce flow separation between parent and daughter vessels. We modelled the blood flow and shear rate in a bifurcation during thrombus formation and show that blood vessel bifurcation ratios determine the maximum shear rate on the surface of a growing thrombus. We built an analytical model that may aid in predicting microvascular bifurcation ratios that are prone to occlusive thrombus formation. We also observed that bifurcation ratios that adhere to Murray’s law of bifurcations may be protected from occlusive thrombus formation. These results may be useful in the rational design of diagnostic microfluidic devices and microfluidic blood oxygenators.     

Vertex analysis of Purkinje cell dendritic trees in the cerebellum of the rat

All networks are made up of vertices (points interconnected by segments), which include terminals interconnected by terminal segments, nodes interconnected by link segments and the root point connected to the tree by the root segment. All nodes may be classified into unique types according to the number of terminal and link segments they drain. For example, there are three distinct dichotomous nodes, a 'primary' node draining two terminal segments, a 'secondary' node draining one terminal segment and a link segment, and a 'tertiary' node draining two link segments. The numbers of primary and tertiary nodes approximate to equality in large networks and thus the ratio of primary to secondary nodes defines topology. All higher order nodes ( trichotomous and beyond) may be resolved into dichotomous forms and incorporated into the analysis. Different forms of growth may thus be analysed by comparing the frequency distributions of nodes with those generated by computer simulated growth models. Moreover, all vertices can be ordered so that metrical parameters are easily incorporated and the hierarchical arrangements of vertices of different order discerned. The dendritic trees of 48 Purkinje cells, taken from folia along the primary fissure, were analysed using vertex analysis. The mean number of segments in Purkinje cell trees was 881 +/- 23 (s.e.) and mean total dendritic length 7959 +/- 233 (s.e.) micrometers. Segment lengths were longest over proximal segments but over most of the tree segment lengths were constant at 10 +/- 0.2 (s.e.) micrometers. Vertex, segment and terminal frequency distributions of equivalent orders were all normal with a slight positive skew. Peak frequencies were recorded at the 12th equivalent order. The mean primary/secondary nodal vertex ratio was 0.93 and the proportion of trichotomous branch points in the tree was 5%. Comparison of the frequency distribution of all vertices with computer generated models showed that growth of the Purkinje cell was most closely simulated by a random terminal growth model, incorporating 5% trichotomy , in which the branching of high order terminals was more likely than low order terminals. It was concluded that growth of the Purkinje cell tree could proceed by random terminal branching with growth occurring preferentially over a front composed of terminals that are ascending through a corridor in the molecular layer whose margins are defined by neighbouring trees.     

Shear-stress dependence of dinoflagellate bioluminescence

Fluid flow stimulates bioluminescence in dinoflagellates. However, many aspects of the cellular mechanotransduction are incompletely known. The objective of our study was to formally test the hypothesis that flow-stimulated dinoflagellate bioluminescence is dependent on shear stress, signifying that organisms are responding to the applied fluid force. The dinoflagellate Lingulodinium polyedrum was exposed to steady shear using simple Couette flow in which fluid viscosity was manipulated to alter shear stress. At a constant shear rate, a higher shear stress due to increased viscosity increased both bioluminescence intensity and decay rate, supporting our hypothesis that bioluminescence is shear-stress dependent. Although the flow response of non-marine attached cells is known to be mediated through shear stress, our results indicate that suspended cells such as dinoflagellates also sense and respond to shear stress. Shear-stress dependence of flow-stimulated bioluminescence in dinoflagellates is consistent with mechanical stimulation due to direct predator handling in the context of predator-prey interactions.     

Effect of fluid force on vascular cell function

Endothelial cells (ECs) that line the inner surface of blood vessels are continuously exposed to fluid frictional force (shear stress) induced by blood flow, and shear stress affects the intracellular calcium ([Ca2+]i), which initiates cellular responses. Here, we studied the effect of long-term exposure of shear stress on [Ca2+]i responses in cultured ECs by using a confocal laser microscope and calcium indicator. At the initiation of shear stress of 20 dyn/cm2 (0 hr), 27% of the cells exhibited [Ca2+]i responses. This percentage gradually decreased with increasing exposure time, reaching about 4% after 24 hr of exposure. These data indicate that long-term shear-stress exposure affects [Ca2+]i responses in cultured ECs. Furthermore, we studied the effect of magnitude of shear stress on macromolecule uptake. For the low shear-stress, the uptake was enhanced, whereas the uptake was inhibited for higher shear-stress.     

Terminal and intermediate segment lenghts in neuronal trees with finite length

A basic but neglected property of neuronal trees is their finite length. This finite length restricts the length of a segment to a certain maximum. The implications of the finite length of the tree with respect to the segment length distributions of terminal and intermediate segments are shown by means of a stochastic model. In the model it is assumed that branching is governed by a Poisson process. The model shows that terminal segments are expected to be longer than intermediate segments. Terminal and intermediate segments are expected to decrease in length with incrasing centrifugal order. The results are compared with data fromin vivo pyramidal cells from rat brain and tissue cultured ganglion cells from chicken. A good agreement between data and model was found.     

Analysis of particle trajectories in aortic artery bifurcations with stenosis

The fluid-particle dynamics in a two-dimensonal symmetric branching channel with local occlusions representing a diseased segment of an aortic artery bifurcation has been analyzed. The validated finite element model simulates the trajectories and landing or impact sites of spherical particles for laminar flow in bifurcation channels with generalized wall conditions. Two hypotheses relating critical wall shear stress levels and plaque formation, previously postulated by Kleinstreuer et al. (1988) and Nazemi et al. (1989), have been confirmed. Low shear stress may contribute to the onset of atherosclerotic lesions and areas of critically low and higher shear stresses are susceptible to accelerated growth of plaque.     

Numerical investigation of the non-Newtonian blood flow in a bifurcation model with a non-planar branch

The non-Newtonian fluid flow in a bifurcation model with a non-planar daughter branch is investigated by using finite element method to solve the three-dimensional Navier–Stokes equations coupled with a non-Newtonian constitutive model, in which the shear thinning behavior of the blood fluid is incorporated by the Carreau–Yasuda model. The objective of this study is to investigate the influence of the non-Newtonian property of fluid as well as of curvature and out-of-plane geometry in the non-planar daughter vessel on wall shear stress (WSS) and flow phenomena. In the non-planar daughter vessel, the flows are typified by the skewing of the velocity profile towards the outer wall, creating a relatively low WSS at the inner wall. In the downstream of the bifurcation, the velocity profiles are shifted towards the flow divider. The low WSS is found at the inner walls of the curvature and the lateral walls of the bifurcation. Secondary flow patterns that swirl fluid from the inner wall of curvature to the outer wall in the middle of the vessel are also well documented for the curved and bifurcating vessels. The numerical results for the non-Newtonian fluid and the Newtonian fluid with original Reynolds number and the corresponding rescaled Reynolds number are presented. Significant difference between the non-Newtonian flow and the Newtonian flow is revealed; however, reasonable agreement between the non-Newtonian flow and the rescaled Newtonian flow is found. Results of this study support the view that the non-planarity of blood vessels and the non-Newtonian properties of blood are an important factor in hemodynamics and may play a significant role in vascular biology and pathophysiology.     

3D reconstruction techniques of human coronary bifurcations for shear stress computations

Background

Heterogeneity in plaque composition in human coronary artery bifurcations is associated with blood flow induced shear stress. Shear stress is generally determined by combing 3D lumen data and computational fluid dynamics (CFD). We investigated two new procedures to generate 3D lumen reconstructions of coronary artery bifurcations for shear stress computations.

Methods

We imaged 10 patients with multislice computer tomography (MSCT) and intravascular ultrasound (IVUS). The 3D reconstruction of the main branch was based on the fusion of MSCT and IVUS. The proximal part of side branch was reconstructed using IVUS data or MSCT data, resulting in two different reconstructions of the bifurcation region. The distal part of the side branch was based on MSCT data alone. The reconstructed lumen was combined with CFD to determine the shear stress. Low and high shear stress regions were defined and shear stress patterns in the bifurcation regions were investigated.

Results

The 3D coronary bifurcations were successfully generated with both reconstruction procedures. The geometrical features of the bifurcation region for the two reconstruction procedures did not reveal appreciable differences. The shear stress maps showed a qualitative agreement, and the low and high shear stress regions were similar in size and average shear stress values were identical. The low and high shear stress regions showed an overlap of approximately 75%.

Conclusion

Reconstruction of the side branch with MSCT data alone is an adequate technique to study shear stress and wall thickness in the bifurcation region. The reconstruction procedure can be applied to further investigate the effect of shear stress on atherosclerosis in coronary bifurcations.     

Basal EDRF activity helps to keep the geometrical configuration of arterial bifurcations close to the Murray optimum

We have used X-ray microangiography to investigate the hypothesis that the potent endogenous vasodilator endothelium-derived relaxing factor (EDRF) contributes to the maintenance of "optimality" in vascular branching by modulating the diameters of the parent (D0) and daughter (D1 and D2) arteries at bifurcations. Five anatomically different types of bifurcation were studied in buffer-perfused rabbit ear preparations both under resting conditions and after pharmacological constriction by 5-hydroxytryptamine (5HT). A range of flow rates (1-5 ml min-1) was employed as release of EDRF from endothelial cells is stimulated by shear stress. Experimental data obtained in the presence and absence of EDRF activity were compared with theoretical predictions in three ways. (1) Junction exponents (x) were determined at each bifurcation from the equation Dx1 + Dx2 = Dx0, and their frequency distributions constructed. Murray (1926a, Proc. natn. Acad. Sci., U.S.A. 12, 207-214; 1926b, J. gen. Physiol. 9, 835-841.) proposed that x will be exactly 3 if power losses and intravascular volume are minimized simultaneously. In unconstricted preparations, either in the presence or absence of EDRF activity, and in preparations constricted by 0.1 microM 5HT in the presence of EDRF activity, the modes and medians of the frequency distributions of x were found to be close to 3 at all flow rates. In contrast, in 0.1 microM 5HT-constricted preparations in the absence of EDRF activity, no single mode common to all flow rates was apparent and medians were significantly larger at all flow rates. (2) Theoretically "optimal" branching angles were derived from experimental diameter measurements using four mathematical models which minimize respectively the total surface area, total volume, total drag (shear stress) and total power losses at bifurcations (Murray, 1926b). These calculated branching angles were then compared with actual branching angles. EDRF activity was found to be necessary for accurate prediction of branching angles by the minimum volume and power loss models in 5HT-constricted but not in resting preparations. (3) For each model or "minimization principle", there is an optimal mathematical relationship between the junction exponent, x, and the angle between daughter arteries, psi 12, at a bifurcation (Roy & Woldenberg, 1982, Bull. math. Biol. 44, 349-360.) Experimentally determined values of x and psi 12 agreed closely with those predicted both by the minimum volume and the minimum power loss principles, except again in 5HT-constricted preparations in the absence of EDRF activity.(ABSTRACT TRUNCATED AT 250 WORDS)     

Heterogeneous perfusion is a consequence of uniform shear stress in optimized arterial tree models

Using optimized computer models of arterial trees we demonstrate that flow heterogeneity is a necessary consequence of a uniform shear stress distribution. Model trees are generated and optimized under different modes of boundary conditions. In one mode flow is delivered to the tissue as homogeneously as possible. Although this primary goal can be achieved, resulting shear stresses between blood and the vessel walls show very large spread. In a second mode, models are optimized under the condition of uniform shear stress in all segments which in turn renders flow distribution heterogeneous. Both homogeneous perfusion and uniform shear stress are desirable goals in real arterial trees but each of these goals can only be approached at the expense of the other. While the present paper refers only to optimized models, we assume that this dual relation between the heterogeneities in flow and shear stress may represent a more general principle of vascular systems.     

Topological properties of binary trees grown with order-dependent branching probabilities

This paper describes a growth model for binary topological trees. The model defines the branching probability of all segments in the tree. The branching probability of a segment is formulated as a function of two variables, one indicating its type (intermediate or terminal), the other representing its order, i.e. the topological distance to the root segment. The function is determined by two parameters, namely the ratio of branching probabilities of intermediate and terminal segments and the strength of the order dependency, implemented in an exponential form. Expressions are derived for the calculation of symmetry properties of the partitions and it is indicated which part of the parameter domain results in predominantly symmetrical trees.     

Stochastic continuous time neurite branching models with tree and segment dependent rates

In this paper we introduce a continuous time stochastic neurite branching model closely related to the discrete time stochastic BES-model. The discrete time BES-model is underlying current attempts to simulate cortical development, but is difficult to analyze. The new continuous time formulation facilitates analytical treatment thus allowing us to examine the structure of the model more closely. We derive explicit expressions for the time dependent probabilities p(γ,t) for finding a tree γ at time t, valid for arbitrary continuous time branching models with tree and segment dependent branching rates. We show, for the specific case of the continuous time BES-model, that as expected from our model formulation, the sums needed to evaluate expectation values of functions of the terminal segment number μ(f(n),t) do not depend on the distribution of the total branching probability over the terminal segments. In addition, we derive a system of differential equations for the probabilities p(n,t) of finding n terminal segments at time t. For the continuous BES-model, this system of differential equations gives direct numerical access to functions only depending on the number of terminal segments, and we use this to evaluate the development of the mean and standard deviation of the number of terminal segments at a time t. For comparison we discuss two cases where mean and variance of the number of terminal segments are exactly solvable. Then we discuss the numerical evaluation of the S-dependence of the solutions for the continuous time BES-model. The numerical results show clearly that higher S values, i.e. values such that more proximal terminal segments have higher branching rates than more distal terminal segments, lead to more symmetrical trees as measured by three tree symmetry indicators.     

Human coronary arteries. II. Branching anatomical pattern and arterial wall microarchitecture

A study on 22 pairs, including subjects of similar age, sex and cause of death but with different branching anatomical patterns, shows, that the amount and structural organization of elastic and muscular tissue of the thickened intima are strongly influenced by (a) the size, length, mode of ramification of the parent vessels of the coronary tree; (b) the number of branch orifices in a given arterial segment, and (c) the collateral or terminal character of certain daughter branches.     

Remodelling during development of the Purkinje cell dendritic tree in the mouse

The development of the Purkinje cells in normal C57 mice was studied from 7-100 d post natum . The growth of the dendritic trees was analysed both metrically and topologically using the method of vertex analysis (Berry & Flinn 1983 a). Granule and PUrkinje cell counts were made so that Purkinje cell segment production could be correlated with the number of parallel fibres deposited. Both topological and metrical results indicate that from 7 to 30 d post natum the Purkinje cell dendritic trees expand massively; accounting for 87% of total segment elaboration, reaching their lateral boundaries by 12-15 d post natum and then advancing towards the pial surface. Continued lateral expansion is constrained by the proximity of dendrites from neighbouring trees. Growth proceeds upwards through the neuropil as a front of prolific random terminal branching with inhibitory forces acting at the edges of the growth corridor and behind the growth front to prevent overlapping of dendrites. By 30 d post natum all boundaries are reached and the size of the dendritic field is fixed. Trees averaged 711.2 segments +/- 21.45 with a mean distance from root to terminal segment of 133.5 +/- 2.9 micrometers. The Va/Vb vertex ratios and the levels of trichotomy during this period indicate that branching patterns deviate from pure random terminal additions in a dichotomous tree. There is opportunity for non-random growth at the areas of inhibitory action. Beyond 30 d post natum remodelling occurs within the arbor which involves segment loss in the subpial region (orders above 16) and segment elaboration within the tree (orders 8-16) causing increased density of dendrites and overlapping of segments. The frequencies of segments and terminals are restored to symmetrical distributions through the orders of the trees from the skewed distributions associated with the frontal advance in earlier growth. During remodelling the Va/Vb vertex ratios and percentage of trichotomous nodes are consistent with growth through dichotomous random terminal branching. Path lengths of 8 micrometers between each order are seen as regular increments throughout entire trees at 100 d post natum . The final tree produced is indistinguishable from a network grown entirely by random terminal dichotomous branching with some 6% trichotomy and a Va/Vb vertex ratios of 0.92. Granule cell number within the granular layer increases rapidly up to 15 d post natum after which cell death causes a decrease to stable levels beyond 30 d post natum .(ABSTRACT TRUNCATED AT 400 WORDS)