The cost of departure from optimal radii in microvascular networks

MATHEMATICAL MODELS OF HUMAN RESPIRATORY AND BLOOD CIRCULATORY SYSTEMS

Aim. To analyze modern approaches to mathematical modeling of human respiratory and blood circulatory systems. Methods. Comprehensive review of scientific literature sources extracted from domestic and international resources databases. Results. Historical information and modern data concerning mathematical modeling of human functional respiratory and blood circulatory systems were summarized and analyzed in present ¬review; current trends in approaches to the construction of these models were revealed. Conclusions. Currently, two main approaches to the mathematical modeling of respiratory and blood circulatory systems exist. One of them is the construction of models of the mechanics of respiration and blood circulation. They are based on the models of mechanics of solid deformable body, thermomechanics, hydromechanics, and continuum mechanics. This approach uses complex mathematical apparatus, including Navier-Stokes equation, which makes it possible to obtain a number of theoretical results, but it is hardly usable for real problems solutions at present time. The second approach is based on the model of F. Grodins, who represented the process of breathing as a controlled dynamic system, described by ordinary differential equations, in which the process control is carried out according to the feedback principle. There is a significant number of modifications of this model, which made it possible to simulate various disturbing influences, such as physical activity, hypoxia and hyperemia, and to predict parameters characterizing functional respiratory system under these disturbing influences.

Retinal Vasculature Fractal and Stroke Mortality

BACKGROUND AND PURPOSE

Fractal analysis is a method of quantifying the branching complexity and density of the retinal vessels. We hypothesized that reduced fractal dimension, signifying a sparser vascular network, is associated with long-term stroke mortality.

METHODS

We examined the relationship of fractal dimension and stroke mortality in a prospective, population-based cohort of 3143 participants aged 49 years or older. Fractal dimension was measured from digitized fundus photographs using a computer-automated method. Stroke mortality was documented from Australian National Death Index records. We defined reduced fractal dimension as values in the lowest quartile.

RESULTS

Over 12 years, there were 132 (4.2%) stroke-related deaths. Stroke-related mortality was higher in participants with reduced fractal dimension (lowest quartile) compared with the highest quartile (7.7% versus 1.3%, P<0.01). After controlling for age, gender, smoking, blood pressure, history of stroke, and other factors, participants with reduced fractal dimension had higher stroke mortality (hazard ratio, 2.42 [95% CI, 1.15-5.07], lowest versus highest quartile). When modeled as a continuous variable, reduced fractal dimension was associated with increased stroke mortality (multivariable-adjusted hazard ratio, 1.26 [95% CI, 1.06-1.51], per SD decrease).

CONCLUSIONS

Reduced retinal vascular fractal dimension is independently associated with 12-year stroke mortality. Reduced fractal dimension may indicate cerebral tissue hypoxia and increased risk of stroke.

On deriving Murray's law from constrained minimization of flow resistance

Murray's law, which states that the cube of the radius of a parent vessel equals the sum of the cubes of the radii of the daughter vessels, was originally derived by minimizing the cost of operation of blood flow in a single cylindrical tube. An alternative widely cited derivation by Sherman is based upon the optimization problem of minimizing the total flow resistance subject to a material constraint, and that study claimed that "Conservation of the sum of the cubes of the radii is the condition for minimal resistance whether the parent vessel divides symmetrically or asymmetrically, and whether it divides into two, three, four, or, presumably, any number of daughter vessels." In this paper we show that Sherman's analysis is flawed, since with N daughter vessels there are 2^N-N-1 sets of vessel radii which satisfy Murray's law but which do not yield minimal total flow resistance. Moreover, we show that when there are N daughter vessels, each with the same radius, the minimal total flow resistance is an increasing function of N for N⩾1. Since N=1 corresponds to the degenerate case of no branching at all, our result implies that bifurcation (N=2) achieves the minimal total flow resistance. Our analysis thus offers an explanation for the preponderance of bifurcations (as opposed to trifurcations or higher level branchings) in many biological systems.

Gibbs point field model quantifies disorder in microvasculature of U87-glioblastoma

Microvascular proliferation in glioblastoma multiforme is a biological key mechanism to facilitate tumor growth and infiltration and a main target for treatment interventions. The vascular architecture can be obtained by Single Plane Illumination Microscopy (SPIM) to evaluate vascular heterogeneity in tumorous tissue. We make use of the Gibbs point field model to quantify the order of regularity in capillary distributions found in the U87 glioblastoma model in a murine model and to compare tumorous and healthy brain tissue. A single model parameter Γ was assigned that is linked to tissue-specific vascular topology through Monte-Carlo simulations. Distributions of the model parameter Γ differ significantly between glioblastoma tissue with mean 〈Γ_G〉=2.1±0.4, as compared to healthy brain tissue with mean 〈Γ_H〉=4.9±0.4, suggesting that the average Γ-value allows for tissue differentiation. These results may be used for diagnostic magnetic resonance imaging, where it has been shown recently that Γ is linked to tissue-inherent relaxation parameters.

Branching Pattern of the Cerebral Arterial Tree

Quantitative data on branching patterns of the human cerebral arterial tree are lacking in the 1.0–0.1 mm radius range. We aimed to collect quantitative data in this range, and to study if the cerebral artery tree complies with the principle of minimal work (Law of Murray).

To enable easy quantification of branching patterns a semi‐automatic method was employed to measure 1,294 bifurcations and 2,031 segments on 7 T‐MRI scans of two corrosion casts embedded in a gel. Additionally, to measure segments with a radius smaller than 0.1 mm, 9.4 T‐MRI was used on a small cast section to characterize 1,147 bifurcations and 1,150 segments. Besides MRI, traditional methods were employed. Seven hundred thirty‐three bifurcations were manually measured on a corrosion cast and 1,808 bifurcations and 1,799 segment lengths were manually measured on a fresh dissected cerebral arterial tree. Data showed a large variation in branching pattern parameters (asymmetry‐ratio, area‐ratio, length‐radius‐ratio, tapering). Part of the variation may be explained by the variation in measurement techniques, number of measurements and location of measurement in the vascular tree.

This study confirms that the cerebral arterial tree complies with the principle of minimum work. These data are essential in the future development of more accurate mathematical blood flow models. Anat Rec, 302:1434–1446, 2019. © 2018 The Authors. The Anatomical Record published by Wiley Periodicals, Inc. on behalf of American Association of Anatomists.

Impact of coronary bifurcation morphology on wave propagation

The branching pattern of the coronary vasculature is a key determinant of its function and plays a crucial role in shaping the pressure and velocity wave forms measured for clinical diagnosis. However, although multiple scaling laws have been proposed to characterize the branching pattern, the implications they have on wave propagation remain unassessed to date. To bridge this gap, we have developed a new theoretical framework by combining the mathematical formulation of scaling laws with the wave propagation theory in the pulsatile flow regime. This framework was then validated in multiple species using high-resolution cryomicrotome images of porcine, canine, and human coronary networks. Results demonstrate that the forward well-matchedness (no reflection for pressure/flow waves traveling from the coronary stem toward the microcirculation) is a salient feature in the coronary vasculature, and this result remains robust under many scenarios of the underlying pulse wave speed distribution assumed in the network. This result also implies a significant damping of the backward traveling waves, especially for smaller vessels (radius, <0.3 mm). Furthermore, the theoretical prediction of increasing area ratios (ratio between the area of the mother and daughter vessels) in more symmetric bifurcations found in the distal circulation was confirmed by experimental measurements. No differences were observed by clustering the vessel segments in terms of transmurality (from epicardium to endocardium) or perfusion territories (left anterior descending, left circumflex, and right coronary artery).

Structure and hemodynamics of vascular networks in the chorioallantoic membrane of the chicken

The chick chorioallantoic membrane (CAM) is extensively used as an in vivo model. Here, structure and hemodynamics of CAM vessel trees were analyzed and compared with predictions of Murray's law. CAM microvascular networks of Hamburger-Hamilton stage 40 chick embryos were scanned by videomicroscopy. Three networks with ∼3,800, 580, and 480 segments were digitally reconstructed, neglecting the capillary mesh. Vessel diameters (D) and segment lengths were measured, and generation numbers and junctional exponents at bifurcations were derived. In selected vessels, flow velocities (v) and hematocrit were measured. Hemodynamic simulations, incorporating the branching of capillaries from preterminal vessels, were used to estimate v, volume flow, shear stress (τ), and pressure for all segments of the largest network. For individual arteriovenous flow pathways, terminal arterial and venous generation numbers are negatively correlated, leading to low variability of total topological and morphological pathway lengths. Arteriolar velocity is proportional to diameter (v∝D^1.03 measured, v∝D^0.93 modeling), giving nearly uniform τ levels (τ∝D^0.05). Venular trees exhibit slightly higher exponents (v∝D^1.3, τ∝D^0.38). Junctional exponents at divergent and convergent bifurcations were 2.05 ± 1.13 and 1.97 ± 0.95 (mean ± SD) in contrast to the value 3 predicted by Murray's law. In accordance with Murray's law, τ levels are (nearly) maintained in CAM arterial (venular) trees, suggesting vascular adaptation to shear stress. Arterial and venous trees show an interdigitating arrangement providing homogeneous flow pathway properties and have preterminal capillary branches. These properties may facilitate efficient oxygen exchange in the CAM during rapid embryonic growth.

Optical Coherence Tomography Angiography Signs of Vascular Abnormalization With Antiangiogenic Therapy for Choroidal Neovascularization

PURPOSE

To investigate the vascular appearance of choroidal neovascularization (CNV) treated with recurrent intravitreous anti-vascular endothelial growth factor (VEGF) injections, which have been proposed to cause transient vascular normalization along with decreased vascularity and leakage.

DESIGN

Retrospective case series with perspective on the topic.

METHODS

Patients with treated CNV secondary to age-related macular degeneration from a community-based retinal referral practice were evaluated with optical coherence tomography angiography employing split-spectrum amplitude decorrelation. The choroidal neovascular morphology of the 17 eyes of 14 consecutive patients was described.

RESULTS

The mean age of the patients, 8 men and 6 women, was 78.4 (standard deviation ± 9.3) years. The mean greatest linear dimension of the lesion was 3600 μm. The mean number of anti-VEGF injections was 47 (±21). The vascular diameter of the vessels in the CNV appeared large even in small lesions, with feeder vessels approaching the size of the major arcade vessels of the retina. The vessels had few branch points and many vascular anastomotic connections among larger vessels. There was a paucity of capillaries visualized within the lesions.

CONCLUSIONS

The findings of this study do not support the hypothesis of vascular normalization in eyes receiving recurrent periodic antiangiogenic treatment. The observed "abnormalization" of the vessels may be explained by periodic pruning of angiogenic vascular sprouts by VEGF withdrawal in the face of unimpeded arteriogenesis. As the eye is a readily accessible VEGF laboratory, features expressed therein may also apply to neovascularization elsewhere in the body, such as in tumors.

Two-dimensional transient model for prediction of arteriolar NO/O2 modulation by spatiotemporal variations in cell-free layer width

Despite the significant roles of the cell-free layer (CFL) in balancing nitric oxide (NO) and oxygen (O2) bioavailability in arteriolar tissue, many previous numerical approaches have relied on a one-dimensional (1-D) steady-state model for simplicity. However, these models are unable to demonstrate the influence of spatiotemporal variations in the CFL on the NO/O2 transport under dynamic flow conditions. Therefore, the present study proposes a new two-dimensional (2-D) transient model capable of predicting NO/O2 transport modulated by the spatiotemporal variations in the CFL width. Our model predicted that NO bioavailability was inversely related to the CFL width as expected. The enhancement of NO production by greater wall shear stress with a thinner CFL could dominate the diffusion barrier role of the CFL. In addition, NO/O2 availability along the vascular wall was inhomogeneous and highly regulated by dynamic changes of local CFL width variation. The spatial variations of CFL widths on opposite sides of the arteriole exhibited a significant inverse relation. This asymmetric formation of CFL resulted in a significantly imbalanced NO/O2 bioavailability on opposite sides of the arteriole. The novel integrative methodology presented here substantially highlighted the significance of spatiotemporal variations of the CFL in regulating the bioavailability of NO/O2, and provided further insight about the opposing effects of the CFL on arteriolar NO production.

The influence of branch order on optimal leaf vein geometries: Murray's law and area preserving branching

Models that predict the form of hierarchical branching networks typically invoke optimization based on biomechanical similitude, the minimization of impedance to fluid flow, or construction costs. Unfortunately, due to the small size and high number of vein segments found in real biological networks, complete descriptions of networks needed to evaluate such models are rare. To help address this we report results from the analysis of the branching geometry of 349 leaf vein networks comprising over 1.5 million individual vein segments. In addition to measuring the diameters of individual veins before and after vein bifurcations, we also assign vein orders using the Horton-Strahler ordering algorithm adopted from the study of river networks. Our results demonstrate that across all leaves, both radius tapering and the ratio of daughter to parent branch areas for leaf veins are in strong agreement with the expectation from Murray’s law. However, as veins become larger, area ratios shift systematically toward values expected under area-preserving branching. Our work supports the idea that leaf vein networks differentiate roles of leaf support and hydraulic supply between hierarchical orders.

Nanocarrier Hydrodynamics and Binding in Targeted Drug Delivery: Challenges in Numerical Modeling and Experimental Validation

This review discusses current progress and future challenges in the numerical modeling of targeted drug delivery using functionalized nanocarriers (NC). Antibody coated nanocarriers of various size and shapes, also called functionalized nanocarriers, are designed to be injected in the vasculature, whereby they undergo translational and rotational motion governed by hydrodynamic interaction with blood particulates as well as adhesive interactions mediated by the surface antibody binding to target antigens/receptors on cell surfaces. We review current multiscale modeling approaches rooted in computational fluid dynamics and nonequilibrium statistical mechanics to accurately resolve fluid, thermal, as well as adhesive interactions governing nanocarrier motion and their binding to endothelial cells lining the vasculature. We also outline current challenges and unresolved issues surrounding the modeling methods. Experimental approaches in pharmacology and bioengineering are discussed briefly from the perspective of model validation.

Relating function to branching geometry: a micro-CT study of the hepatic artery, portal vein, and biliary tree

Utilizing micro-computed tomography images, the hierarchical structure, interbranch segment lengths and diameters of a hepatic artery, a portal vein, and two biliary trees from intact rat liver lobes were characterized. The data were investigated by analyzing the geometric properties of the vascular structures, such as how interbranch segment diameters change at bifurcation points. In the case of the hepatic artery and portal vein trees (in which the flow rate is high by comparison with that in the biliary tree), the vascular geometry is consistent with a fluid transport system which aims to simultaneously minimize both the power loss of laminar flow, and a cost function proportional to the total volume of material needed to maintain the system (lumenal contents). In comparison, the biliary tree (which has a low flow rate and an opposite flow direction to that of the hepatic artery and portal vein) was found to have a geometry in which the lumen cross-sectional area is maintained at bifurcations. These findings imply that the histological makeup and therefore the pathophysiology of biliary tree vasculature are likely very different from that of the vasculature within the systemic arterial tree. The extent to which the characteristic variability/scatter in the data may have resulted from imaging and/or measurement errors was examined by simulating such errors in a theoretical tree model and comparing the results with the measured data.

Theoretical models for coronary vascular biomechanics: progress & challenges

A key aim of the cardiac Physiome Project is to develop theoretical models to simulate the functional behaviour of the heart under physiological and pathophysiological conditions. Heart function is critically dependent on the delivery of an adequate blood supply to the myocardium via the coronary vasculature. Key to this critical function of the coronary vasculature is system dynamics that emerge via the interactions of the numerous constituent components at a range of spatial and temporal scales. Here, we focus on several components for which theoretical approaches can be applied, including vascular structure and mechanics, blood flow and mass transport, flow regulation, angiogenesis and vascular remodelling, and vascular cellular mechanics. For each component, we summarise the current state of the art in model development, and discuss areas requiring further research. We highlight the major challenges associated with integrating the component models to develop a computational tool that can ultimately be used to simulate the responses of the coronary vascular system to changing demands and to diseases and therapies.

Murray's law, the 'Yarrum' optimum, and the hydraulic architecture of compound leaves

There are two optima for maximizing hydraulic conductance per vasculature volume in plants. Murray's law (ML) predicts the optimal conduit taper for a fixed change in conduit number across branch ranks. The opposite, the Yarrum optimum (YO), predicts the optimal change in conduit number for a fixed taper. We derived the solution for YO and then evaluated compliance with both optima within the xylem of compound leaves, where conduits should have a minimal mechanical role. We sampled leaves from temperate ferns, and tropical and temperate angiosperms Leaf vasculature exhibited greater agreement with ML than YO. Of the 14 comparisons in 13 species, 12 conformed to ML. The clear tendency towards ML indicates that taper is optimized for a constrained conduit number. Conduit number may be constrained by leaflet number, safety requirements, and the fact that the number of conduits is established before their diameter during development. Within a leaf, ML compliance requires leaf-specific conductivity to decrease from petiole to petiolule with the decrease in leaf area supplied. A similar scaling applied across species, indicating lower leaf-specific petiole conductivity in smaller leaves. Small leaf size should offset lower conductivity, and petiole conductance (conductivity/length) may be independent of leaf size.

Minimum mass vascular networks in multifunctional materials

A biomimetic analysis is presented in which an expression for the optimum vessel diameter for the design of minimum mass branching or vascular networks in engineering applications is derived. Agreement with constructal theory is shown. A simple design case is illustrated and application to more complex cases with branching networks of several generations discussed. The analysis is also extended into the turbulent flow regime, giving an optimization tool with considerable utility in the design of fluid distribution systems. The distribution of vessel lengths in different generations was also found to be a useful design variable. Integrating a network into a structure is also discussed. Where it is necessary to adopt a non-optimum vessel diameter for structural integration, it has been shown that small deviations from the minimum mass optimum can be tolerated, but large variations could be expected to produce a punitive and rapidly increasing mass penalty.

Vascular metabolic dissipation in Murray's law

The metabolic dissipation in Murray's minimum energy hypothesis includes only the blood metabolism. The metabolic dissipation of the vascular tree, however, should also include the metabolism of passive and active components of the vessel wall. In this study, we extend the metabolic dissipation to include blood metabolism, as well as passive and active components of the vessel wall. The analysis is extended to the entire vascular arterial tree rather than a single vessel as in Murray's formulation. The calculations are based on experimentally measured morphological data of coronary artery network and the longitudinal distribution of blood pressure along the tree. Whereas the model includes multiple dissipation sources, the total metabolic consumption of a complex vascular tree is found to remain approximately proportional to the cumulative arterial volume of the unit. This implies that the previously described scaling relations for the various morphological features (volume, length, diameter, and flow) remain unchanged under the generalized condition of metabolic requirements of blood and blood vessel wall.

Design of cerebellar and nontegmental rhombencephalic microvascular bed in the sterlet, Acipenser ruthenus: a scanning electron microscope and 3D morphometry study of vascular corrosion casts

The design of the microvasculature of cerebellum and nontegmental rhombencephalic areas was studied in eight adult Acipenser ruthenus L. by scanning electron microscopy of vascular corrosion casts and three-dimensional morphometry. Gross vascularization was described and diameters and total branching angles of parent and daughter vessels of randomly selected arterial and capillary bifurcations (respectively, venous mergings) were measured. With diameters ranging from 15.9 +/- 1.9 microm (cerebellum; mean +/- S.D.) to 15.9 +/- 1.7 mm (nontegmental rhombencephalon; mean +/- S.D.) capillaries in Acipenser were significantly (p > or = .05) smaller than in cyclostomes (18-20 microm) but significantly thicker than in higher vertebrates and men (6-8 microm). With the exception of the area ratio beta (i.e., sum of squared daugther diameters divided by squared diameter of parent vessel) of the venular mergings in the nontegmental rhombencephalon, no significant differences (p > or = .05) existed between the two brain areas. Data showed that arteriolar and capillary bifurcations and venular mergings are optimally designed in respect to diameters of parent vessel to daughter vessels and to branching (merging) angles. Quantitative data are discussed both in respect to methodical pitfalls and the optimality principles possibly underlying the design of vascular bifurcations/mergings in selected brain areas of a nonteleost primitive actinopterygian fish.

Microcirculation and Hemorheology

Major experimental and theoretical studies on microcirculation and hemorheology are reviewed with the focus on mechanics of blood flow and the vascular wall. Flow of the blood formed elements (red blood cells (RBCs), white blood cells or leukocytes (WBCs) and platelets) in individual arterioles, capillaries and venules, and in microvascular networks is discussed. Mechanical and rheological properties of the formed elements and their interactions with the vascular wall are reviewed. Short-term and long-term regulation of the microvasculature is discussed; the modes of regulation include metabolic, myogenic and shear-stress-dependent mechanisms as well as vascular adaptation such as angiogenesis and vascular remodeling.

Fractal Analysis of the Vascular Tree in the Human Retina

The retinal circulation of the normal human retinal vasculature is statistically self-similar and fractal. Studies from several groups present strong evidence that the fractal dimension of the blood vessels in the normal human retina is approximately 1.7. This is the same fractal dimension that is found for a diffusion-limited growth process, and it may have implications for the embryological development of the retinal vascular system. The methods of determining the fractal dimension for branching trees are reviewed together with proposed models for the optimal formation (Murray Principle) of the branching vascular tree in the human retina and the branching pattern of the human bronchial tree. The limitations of fractal analysis of branching biological structures are evaluated. Understanding the design principles of branching vascular systems and the human bronchial tree may find applications in tissue and organ engineering, i.e., bioartificial organs for both liver and kidney.

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.

On the principles of the vascular network branching

We propose an explanation of Murray's law without applying the minimality principles. The model deals with a "delivering" artery system of an organ that is characterized, first, by the space-filling embedding into the organ tissue and, second, by the uniform distribution of the blood pressure drop over it. The latter assumption is justified using the available physiological data and the idea about conditions needed for perfect self-regulation. Based on the two statements we get Murray's law, and so, demonstrate that it can be also regarded as a direct consequence of the organism's capacity for controlling finely the blood flow redistribution over peripheral vascular networks.

Structural adaptation of microvascular networks: functional roles of adaptive responses

Terminal vascular beds continually adapt to changing demands. A theoretical model is used to simulate structural diameter changes in response to hemodynamic and metabolic stimuli in microvascular networks. Increased wall shear stress and decreased intravascular pressure are assumed to stimulate diameter increase. Intravascular partial pressure of oxygen (PO(2)) is estimated for each segment. Decreasing PO(2) is assumed to generate a metabolic stimulus for diameter increase, which acts locally, upstream via conduction along vessel walls, and downstream via metabolite convection. By adjusting the sensitivities to these stimuli, good agreement is achieved between predicted network characteristics and experimental data from microvascular networks in rat mesentery. Reduced pressure sensitivity leads to increased capillary pressure with reduced viscous energy dissipation and little change in tissue oxygenation. Dissipation decreases strongly with decreased metabolic response. Below a threshold level of metabolic response flow shifts to shorter pathways through the network, and oxygen supply efficiency decreases sharply. In summary, the distribution of vessel diameters generated by the simulated adaptive process allows the network to meet the functional demands of tissue while avoiding excessive viscous energy dissipation.

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.

Shear stress-induced release of prostaglandin H(2) in arterioles of hypertensive rats

The nitric oxide-mediated portion of shear stress-induced dilation of rat gracilis muscle arterioles was shown to be impaired in spontaneously hypertensive rats (SHR). Because shear stress-induced dilation is primarily mediated by endothelium-derived prostaglandins in rat cremasteric arterioles, we hypothesized that in the cremasteric vascular bed the mediation of shear stress-induced dilation by prostaglandins is altered in hypertension. At a constant intraluminal pressure of 80 mm Hg, the active diameters of isolated rat cremasteric arterioles of normotensive 30-week-old Wistar-Kyoto rats (WKY) and SHR were 58.0+/-3.1 and 51.7+/-3.6 microm, respectively, whereas their passive diameters were 109.4+/-4.4 and 101.9+/-6.7 microm, respectively. Dilations to increases in shear stress elicited by increases in intraluminal flow (from 0 to 25 microL/min) were significantly less (P<0.05) in cremasteric arterioles isolated from SHR than from WKY. Arachidonic acid (10(-5) mol/L) elicited constrictions in SHR arterioles but dilations in WKY arterioles. The prostaglandin H(2)/thromboxane A(2) (PGH(2)/TxA(2)) receptor antagonist SQ 29,548 (10(-6) mol/L) significantly increased basal diameter by 11% and normalized the attenuated shear stress-induced dilation in SHR, whereas it did not affect basal diameter and arteriolar responses of WKY. Furegrelate, a specific inhibitor of TxA(2) synthase, did not affect the response in SHR. Also, SQ 29,548 reversed the arachidonic acid-induced constriction to dilation in SHR arterioles, whereas it did not affect the dilator response in WKY arterioles. Constrictions of arterioles of WKY and SHR to U46,619 (a PGH(2)/TxA(2) receptor agonist) were not different. These results demonstrate that in cremasteric arterioles of hypertensive rats, shear stress elicits an enhanced release of PGH(2), resulting in a reduced shear stress-dependent dilation. Thus, augmented hemodynamic forces can alter the shear stress-induced synthesis of prostaglandins, which may contribute to the elevated vascular resistance in hypertension.

Regulatory interaction between myogenic and shear-sensitive arterial segments: conditions for stable steady states

Myogenic and shear stress-sensitive mechanisms control the caliber of a small blood vessel in this modeling study. This blood vessel in our model was composed of a pressure-sensitive (myogenic) component and a series-connected shear-sensitive component. The response of this model to imposed pressure and the conditions that result in a stable steady-state vessel diameter were investigated. The requirement that the model parameters need to satisfy for a stable steady state to exist are expressed by the numerical solution of simultaneous nonlinear equations. Also, if a vessel is put into an initial state that is not an equilibrium state, then the system must occupy a range of initial conditions to arrive at a stable equilibrium. These are described graphically for three cases. In general, the initial shear stress should be higher than the equilibrium value of shear stress, and/or the initial transmural pressure should be low, compared with the imposed feed pressure. Increasing the imposed pressure on the vessel can lead to elimination of the equilibrium state and vasospasm, according to this model. When a stable steady state is not reached, the model predicts elimination of the vessel or vasospasm. The model is in qualitative agreement with experimental observations that, during angiogenesis, vessels with low flow are often eliminated.

Design principles of vascular beds

Hemodynamic parameters were determined in each vessel segment of six complete microvascular networks in the rat mesentery by using a combination of experimental measurements and theoretical stimulations. For a total number of 2592 segments, a strong unified dependence of wall shear stress on intravascular pressure for arterioles, capillaries, and venules was obtained. All three types of segments exhibit an essentially identical variation of shear stress from high to low values (from approximately 100 to 10 dyne/cm2) as intravascular pressure falls from 70 to 15 mm Hg. On the basis of these observations, it is proposed that vascular beds grow and adapt so as to maintain the shear stress in each vessel at a level that depends on local transmural pressure. In contrast to Murray's classic 'minimum-cost' hypothesis, which implies uniformity of wall shear rate throughout the vasculature, the proposed design principle provides an explanation for the functionally important arteriovenous asymmetry of wall shear rates and flow resistance in the circulation.

Shear stress-induced dilation is attenuated in skeletal muscle arterioles of hypertensive rats

Hypertension is thought to alter many of the functions of the vascular endothelium. The present study examines whether shear stress-induced endothelium-dependent skeletal muscle arteriolar dilation is compromised in genetically hypertensive rats. Changes in the diameter of isolated, perfused arterioles (approximately 60 microns) from gracilis muscles of 12-week-old normotensive Wistar rats (NWR) and spontaneously hypertensive rats (SHR) were investigated. At a constant perfusion pressure (80 mm Hg), the active diameter of NWR and SHR arterioles was 57.1 +/- 2.0 and 50.9 +/- 3.5 microns, respectively (mean +/- SEM), while the passive diameter (in Ca(2+)-free solution) was 113.2 +/- 3.1 and 100.6 +/- 2.9 microns, respectively. Increases in wall shear stress (from 0 to 100 dyne/cm2) elicited by increases in perfusate flow (from 0 to 25 microL/min) resulted in marked increases in the diameter of NWR arterioles, but such increases produced substantially smaller dilations in SHR arterioles (43.0 versus 18.9 microns). The prostaglandin synthesis inhibitor indomethacin (10(-5) mol/L) significantly attenuated the shear stress-induced dilations in both strains of rats. In contrast, the nitric oxide synthase inhibitor N omega-nitro-L-arginine (10(-4) mol/L) significantly shifted the shear stress-diameter curve to the right in vessels from NWR (by 50 dyne/cm2) but not in those from SHR. Thus, in gracilis muscle arterioles of SHR, the reduced dilation to increases in shear stress seems to be due to the lack of nitric oxide synthesis and/or release in response to shear stress.(ABSTRACT TRUNCATED AT 250 WORDS)

Endothelial expression of thrombomodulin is reversibly regulated by fluid shear stress

The vascular endothelium, by virtue of its position at the interface between blood and the vessel wall, is known to play a critical role in the control of thrombosis and fibrinolysis. Thrombomodulin (TM) is a surface receptor that binds thrombin and is a potent activator of the protein C anticoagulant pathway. Although TM expression is known to be regulated by various cytokines, little is known about its response to ever-present biomechanical stimuli. We have explored the role of fluid shear stress, imparted on the luminal surface of the endothelial cell as a result of blood flow, on the expression of TM mRNA and protein in both bovine aortic endothelial (BAE) and bovine smooth muscle (BSM) cells in an in vitro system. We report in the present study that TM expression is regulated by flow. Subjecting BAE cells to fluid shear stress in the physiological range of magnitude of 15 (moderate shear stress) and 36 (elevated shear stress) dynes/cm2 resulted in a mild transient increase followed by a significant decrease in TM mRNA to 37% and 16% of its resting level, respectively, by 9 hours after the onset of flow. In contrast, shear stress at the low magnitude of 4 dynes/cm2 did not affect TM mRNA levels. The sensitivity of TM mRNA expression by flow was found to be specific to endothelium, since it was not observed in BSM cells exposed to steady laminar shear stress of 15 dynes/cm2. Furthermore, unlike BAE cells, BSM cells did not exhibit altered cell shape nor align in the direction of flow after 24 hours of shear stress at 15 dynes/cm2.(ABSTRACT TRUNCATED AT 250 WORDS)

Role of shear stress and endothelial prostaglandins in flow- and viscosity-induced dilation of arterioles in vitro

We have studied the effect of changes in shear stress on diameter of isolated arterioles of rat cremaster muscle. The steady-state active diameter of arterioles at a constant perfusion pressure (60 mm Hg) was 80 +/- 1.2 microns. The vessels' passive diameter (Ca(2+)-free solution) was 156 +/- 1.8 microns. Changes in shear stress were induced either by an increase in flow (velocity) or by an increase in viscosity of the perfusion solution. At a constant perfusion pressure, the stepwise increase in perfusion flow (0-80 microliters/min in 10-microliters/min steps) elicited, with a delay of approximately 20 seconds, a gradual increase in diameter up to 46%. At a constant 20-microliters/min flow rate, increases in viscosity of the perfusate (2%, 4%, and 6% dextran [molecular weight, 77,800]) caused a gradual vasodilation up to 22%. Varying flow and viscosity of the perfusate simultaneously resulted in an upward shift of the flow-diameter curve. Both flow- and viscosity-induced dilations were eliminated by the removal of the endothelium of arterioles (by air) or were inhibited by indomethacin (10(-5) M). The efficacy and specificity of these inhibitory treatments were assessed with vasoactive agents whose action, with regard to endothelial mediation, has been determined previously. The arteriolar dilation maintained calculated wall shear stress close to control values during increases in flow and/or viscosity of the perfusate, but when the dilation was inhibited by removal of the endothelium or by indomethacin, wall shear stress increased significantly in a cumulative manner.(ABSTRACT TRUNCATED AT 250 WORDS)

Blood flow and optimal vascular topography: role of the endothelium

We have used x-ray microangiography to investigate the influence of EDRF and endothelin-1 on arterial diameters (70-800 microns) at bifurcations in the isolated rabbit ear and the "optimality" of its branching geometry. The median value of the junction exponent x (which is given by d0x = d1x + d2x, where d0, d1 and d2 are parent and daughter artery diameters respectively) was close to 3 at different flow rates in unconstricted preparations. When x = 3, branching geometry is optimal in that i) power losses and intravascular volume are both minimised, and ii) fractal considerations suggest that the total surface area for metabolic exchange is maximised. Under conditions of vasoconstriction (by 5HT/histamine) the junction exponent deviated from its control value but was restored towards 3, both by basal and by acetylcholine-stimulated EDRF activity. In contrast, endothelin-1 caused a dose-dependent reduction in the junction exponent from its optimal value 3. This suggests that the endothelium helps to optimise microvascular function through EDRF but not endothelin-1 release.

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)

Principles of design of fluid transport systems in zoology

Fluid transport systems mediate the transfer of materials both within an organism and between an organism and its environment. The architecture of fluid transport systems is determined by the small distances over which transfer processes are effective and by hydrodynamic and energetic constraints. All fluid transport systems within organisms exhibit one of two geometries, a simple tube interrupted by a planar transfer region or a branched network of vessels linking widely distributed transfer regions; each is determined by different morphogenetic processes. By exploiting the signal inherent in local shear stress on the vessel walls, animals have repeatedly evolved a complex branching hierarchy of vessels approximating a globally optimal system that minimizes the costs of the construction and maintenance of the fluid transport system.