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(Circulation Research. 1995;77:1017.)
© 1995 American Heart Association, Inc.

Articles

Design Principles of Vascular Beds

Axel R. Pries, Timothy W. Secomb, Peter Gaehtgens

From the Department of Physiology (A.R.P., P.G.), Freie Universität Berlin (Germany), and the Department of Physiology (T.W.S.), University of Arizona, Tucson.

Correspondence to A.R. Pries, MD, Freie Universität Berlin, Department of Physiology, Arnimallee 22, D-14195 Berlin, Germany.

	Abstract

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Abstract 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 simulations. 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 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.

Key Words: shear stress • intravascular pressure • growth • optimal design • vascular remodeling

	Introduction

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Like other fluid transport systems in biology, the mammalian circulation must comply with a number of physical design principles¹ to operate efficiently. A small number of large conduit vessels (macrocirculation) provide convective transport over large distances, whereas a large number of narrow vessels (microcirculation) permit diffusive exchange between blood and tissue. Vessels form hierarchical branching networks, the architecture of which continually adapts to functional demands and tissue growth. This adaptive capability implies that the details of network structure are not exclusively prescribed by genetic information but also result from local action of general principles governing responses to biochemical stimuli (including oxygen/metabolite levels and growth factors) and/or mechanical stimuli (including wall shear stress and transmural pressure). Understanding these principles, which appear to be particularly dominant in the periphery of the vascular system, where resistance to flow and exchange efficiency are determined, can provide insight into the normal functioning of the circulation and into related disorders.

Murray² hypothesized that the design of the vascular system is such that "operating costs" of the circulatory system are minimized. The operating costs consist of the cardiac work incurred in generating the pressure that drives the flow of blood and the metabolic work needed to make and maintain the blood and the blood vessels. Murray concluded that in an optimal vessel system, volume flow is proportional to vessel diameter cubed and that shear stress at the vessel wall (approximately proportional to shear rate) is uniform throughout the network. Shear stress at the endothelial vessel surface is known to influence both smooth muscle tone (acutely) and vessel wall structure (chronically).^{3 4 5} Such mechanisms have provided the basis for the concept of vascular shear stress autoregulation,⁶ and fairly constant shear rates were reported for arterial vessel trees in several tissues.^{7 8 9 10} However, these data relate to arteries and arterioles only; shear stress levels are much lower in veins and venules,^{11 12} contradicting the general applicability of the "minimum-cost" principle.

Blood vessels are exposed not only to shear stress but also to the distending intravascular or transmural pressure. Like shear stress, intravascular pressure affects vessel diameter both acutely (through the myogenic response to vessel distension) and chronically (via remodeling of the vessel wall),^{13 14 15} and intravascular pressure is thus likely to modulate the shear-dependent responses of blood vessels.^{16 17 18} Therefore, the present study was designed to analyze the hydrodynamic design of a terminal vascular bed, ie, the rat mesentery, with respect to both pressure and shear. The mesenteric microvasculature was chosen for this study because of its accessibility to intravital microscopy, its two-dimensional architecture, and its low vasomotor activity, which provides stable measurement conditions over extended periods of time. To avoid possible artifacts associated with sampling selected vessel populations, an approach was used that provides sets of hemodynamic data for complete microvascular networks by combining experimental measurements with model simulations of pressure and flow distribution.

	Materials and Methods

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Male Wistar rats (body weight, 300 to 450 g) were prepared for intravital microscopy of the mesenteric microcirculation after premedication (0.1 mg/kg IM atropine and 20 mg/kg IM sodium pentobarbital), anesthesia (100 mg/kg IM ketamine), cannulation of trachea, jugular vein, and carotid artery, and abdominal midline incision as described in detail previously.¹⁹ Anesthetic level and fluid balance were maintained by intravenous infusion of physiological saline (24 mL · kg^-1 · h^-1) containing sodium pentobarbital (0.3 mg/mL). Heart rate and arterial blood pressure (range, 105 to 140 mm Hg) were monitored. The small bowel was exteriorized, and fat-free portions of the mesenteric membrane were selected for investigation. Control experiments showed that in this preparation, all investigated vessels generally exhibit no spontaneous (resting) smooth muscle tone. As a precaution to prevent the development of tone and thus temporal variation of flow and pressure distributions during the period of time required for measurements (40 minutes) papaverine (10^-4 mol/L) was continuously superfused.

Six microvascular networks were scanned photographically and videotaped by using monochromatic illumination (448 nm) for densitometric hematocrit measurement.¹⁹ In three animals, a second scan was performed with asynchronous strobe illumination (model 11360-1, Chadwick-Helmuth) to allow off-line determination of flow velocity in every segment by use of a spatial correlation technique.²⁰ A single scan was completed in 30 minutes and consisted of 300 individual fields of view. Photographs were assembled in photomontages of the complete networks for determination of topological network structure and segment length between branch points. Vessel segments were classified as arterioles (connecting divergent bifurcations), venules (connecting convergent bifurcations), or capillaries (connecting arteriolar to venular trees, ie, divergent to convergent bifurcations). The area covered by the networks ranged between 25 and 80 mm² and contained 432±102 (mean±SD) vessel segments. The mean blood supply to these networks was 727±302 nL/min, and the diameters of the feeding arterioles and draining venules averaged 32±13 and 54±23 µm, respectively.

From the video recordings, vessel diameter and flow velocity were measured in all segments between branch points by using a digital image analysis system.²⁰ The spatial correlation analysis of video recordings obtained with asynchronous illumination allowed measurement of velocities up to 40 mm/s. The resulting centerline velocity was converted into mean blood flow velocity by using a parametric description of the Fahraeus effect for vessel diameters of <5 µm and a spatial averaging model for vessel diameters of >40 µm.²⁰ A linear transition between the predictions obtained by the two methods was used in the intermediate diameter range.

Since the direct measurement of pressure in all vessel segments of a microvascular network is not technically feasible, a mathematical model simulation described previously was used to obtain pressure values.²⁰ An iterative algorithm based on a continuum approach was used to compute the flows and hematocrits in all segments and the pressures at all branch points.²¹ The simulations are based on the experimentally recorded network structures for the six networks analyzed, information on blood rheology, and the hemodynamic parameters in boundary segments. Volume flow rates in all vessel segments feeding and draining the networks (boundary segments) were calculated from measured velocities where available (three networks). In the remaining three networks, mean blood flow velocity for the main feeding segments was estimated from the following empirical relation: v=0.4 · D-1.9, where v is flow velocity in millimeters per second and D is vessel diameter in micrometers. Flow rates in smaller vessel segments were derived from those in the main feeding segments normalized with respect to the number of capillaries supplied. An inflow hematocrit of 0.45 was assumed for all networks.

In the mathematical flow simulations, parametric descriptions of three rheological phenomena, as previously described,²⁰ were used: (1) the dynamic reduction of hematocrit in blood flowing through narrow vessels (Fahraeus effect), (2) the nonproportional distribution of red blood cells and plasma at arteriolar branch points (phase-separation effect), and (3) effective blood viscosity as a function of vessel diameter and hematocrit (Fahraeus-Lindqvist effect). The parametric relation between blood viscosity, hematocrit, and diameter used in the present study was developed previously to characterize effective viscosities in microvascular networks in vivo.²⁰ In the model calculations, venous outflow pressure of the networks was set to a value obtained in venules draining networks of comparable size in an earlier study²⁰ by direct measurements using micropipettes and a servo-nulling system (model 5, IPM). Intravascular pressure in these venules, which were similar in diameter and location to the main draining venules of the networks studied here, averaged 13.8±3.3 mm Hg (n=20). After convergence of the iterative simulation algorithm used, all pressures provided by the model represent pressure differences between a given segment and this main outflow pressure. Wall shear stresses in all segments were deduced from the computed pressure differences between upstream and downstream branch points and segment geometry.

	Results

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To assess the validity of the mathematical simulations, comparisons were made, wherever possible, with the experimentally obtained measurements. As an example, Fig 1 shows flow velocities in the three categories of vessel segments as a function of segment diameter. Whereas rather uniform distributions are seen in venules and capillaries, a strong decrease of velocity with diameter is seen in arterioles. More important, good agreement between measurement and model prediction is evident, thus confirming earlier evaluations of the model in terms of segment hematocrit, velocity, and pressure.^{19 20 22}

View larger version (22K): [in this window] [in a new window]   Figure 1. Comparison of flow velocity in arterioles, capillaries, and venules measured from video recordings (top) and predicted by mathematical flow simulation (bottom). For each of three networks, segments were grouped according to their diameter (class width, 2 µm). For each parameter and class, mean values were determined and then averaged across networks. These averages are shown for arterioles, capillaries, and venules with standard errors between networks.

Fig 2 shows the relation between wall shear stress and segment diameter for all vessel segments of the six mesenteric microvascular networks. Shear stress in arterioles varies little with diameter down to 15 µm but increases substantially in even smaller arterioles. Wall shear stresses in venules and capillaries are significantly lower than in arterioles and increase with decreasing diameter below 30 µm.

View larger version (19K): [in this window] [in a new window]   Figure 2. Wall shear stress in vessel segments of six networks of the rat mesentery as a function of vessel diameter and class (arteriole, capillary, and venule). Data representation is as described for Fig 1.

Fig 3 shows the relation between wall shear stress and intravascular pressure in arterioles, capillaries, and venules of the same mesenteric microvascular networks as in Fig 2, along with literature data for larger arterioles/arteries (100 dyne/cm²) and venules/veins (5 dyne/cm²). All three types of segments exhibit an essentially identical variation of shear stress with pressure: A monotonic transition from high to low shear stresses is seen as intravascular pressure falls from 50 to 15 mm Hg. At pressures exceeding 50 mm Hg, the relation between shear stress and pressure observed in the mesenteric microvessels levels off and approaches the values reported by previous investigators for larger arteries.

View larger version (23K): [in this window] [in a new window]   Figure 3. Wall shear stress in arterioles, capillaries, and venules versus midsegment pressure (class width, 2.4 mm Hg). Numbers of segments and data representation are as in Figs 2 and 1, respectively. In addition, wall shear stress for larger arterioles/venules and arteries/veins calculated from literature reports are given. Values given by Zamir23 are derived from anatomic examinations12 24 ; two other studies10 25 report results of intravital microscopy.

	Discussion

Top Abstract Introduction Materials and Methods Results Discussion References

 
According to Murray’s minimum-cost hypothesis, volume flow should be proportional to vessel diameter cubed, and wall shear stress should be approximately uniform throughout vascular networks. This hypothesis represents the only approach to date that provides quantitative predictions of parameters related to vascular network design based on universal simple assumptions. Although the structure and hemodynamics of arterial vessel trees in a number of tissues are consistent with these predictions,^{7 8 9 10} the large difference between average wall shear stresses in arteries and veins^{12 23} demonstrates that Murray’s law does not describe the structure of the entire vascular system. The present data obtained from complete mesenteric microvascular networks show that a transition from the high arterial to the low venous shear stresses occurs within the microcirculation.

Furthermore, uniformity of shear stress throughout the circulation would not be compatible with the different physiological functions of the series-coupled vascular compartments of the circulatory system, as discussed in more detail below. Optimization of vascular beds according to the minimum-cost principle is therefore possible only within functionally uniform sections of the circulation and only within the limits imposed by functional requirements.

Pressure-Shear Hypothesis
Although present data and previous studies demonstrate that the design of the complete vascular system is not in accordance with Murray’s minimum-cost principle, the concept of vascular design being dynamically controlled through local responses of vessels to the shear stress acting at their endothelial surface⁶ is plausible. However, vessels are exposed not only to wall shear stress but also to the mechanical forces generated by transmural pressure. Since intravascular pressure must decline monotonically along any arteriovenous pathway, it can serve as indicator of the location of each vessel segment within this pathway, and local pressure could potentially modulate the response of a vessel segment to wall shear stress. Therefore, it is appealing to consider whether the observed distributions of hemodynamic parameters in vascular beds may result from the combined effects of local vascular responses to wall shear stress and pressure.

The data shown in Fig 3 support this line of argument by showing that all three types of vessel segments exhibit an essentially identical decrease of wall shear stress with decreasing intravascular pressure. Even though literature data on shear stress for larger vessels exhibit substantial variability,²⁶ average wall shear stress levels in larger arteries and veins agree with extrapolations of the present data. Since the present data were obtained in a tissue with no smooth muscle tone, the observed relation between wall shear stress and intravascular pressure must reflect structural network properties resulting from long-term adaptations. This suggests the following "pressure-shear" hypothesis for the design of vascular beds: Vascular systems grow and adapt in response to hemodynamic conditions so as to maintain local wall shear stress at a set point that is a function of local transmural pressure.

According to the pressure-shear hypothesis, equilibrium states, in which the effects of pressure and flow on the adaptive response of individual vessels are in balance, are given by a characteristic relation between wall shear stress and pressure. This is illustrated schematically in Fig 4, which is based on the present experimental observations. The regions to the lower right or upper left, respectively, of the sigmoidal equilibrium curve correspond to nonequilibrium conditions brought about by sustained changes of pressure and/or flow in a given vessel segment (Fig 4, upper panel). Such hemodynamic changes would trigger structural adaptations of the vessel, as has been described by previous investigators.^{26 27} Increased pressure and/or decreased wall shear stress (brought about by changes of blood flow) would lead to a structural reduction of internal vessel diameter. This adaptation would cause an increase of wall shear stress until the equilibrium curve is reached again (Fig 4, lower panel). Changes of vessel structure in the opposite direction are expected when vascular pressure is chronically decreased and/or shear stress is increased. It is noteworthy that parallel changes of pressure and flow resulting in a shift along the equilibrium line might not lead to vascular adaptations.

View larger version (23K): [in this window] [in a new window]   Figure 4. Schematic drawings explaining the concept of a maintained relation between wall shear stress and transmural pressure. The upper panel defines the equilibrium curve as given by a sigmoidal relation according to Fig 3, which balances the combined effects of wall shear stress and pressure on vessel structure. The regions on both sides of the equilibrium curve are reached by changes of pressure and/or blood flow. The lower panel shows the adaptive mechanisms available to an individual vessel in order to reestablish the equilibrium condition in the face of long-term hydrodynamic perturbations.

Mechanisms
Endothelial cells are known to respond to changes of wall shear stress, which act as a direct stimulus.^{5 28} By contrast, the nature of the mechanical signal sensed at the cellular level upon changes of transmural pressure is not known. Most probably, elements of the vessel wall respond to pressure-induced changes in circumferential stress. However, circumferential stress levels may vary greatly among the different structural components composing vascular walls.²⁹ Furthermore, the radial distribution of circumferential stress depends markedly on the degree of vessel constriction or dilation and the distending transmural pressure.³⁰ Therefore, the mean value of circumferential stress (), estimated as =p · D/2h, where p is the transmural pressure difference, D is the vessel diameter, and h is the wall thickness, is unlikely to represent the signal at the cellular level corresponding to changes of distending pressure. Identification of the potential pressure-sensing mechanisms in the vessel wall will require consideration of inhomogeneous wall structure and thus nonuniform circumferential stress distribution.

As pointed out above, the pressure-shear relation observed in the present study reflects structural features of the vascular networks investigated. However, blood vessels are also known to react acutely to changes in shear and pressure.^{5 13 31} The resulting short-term alterations of vessel diameter correspond to those seen during long-term adaptation of vessel structure. Typically, vessels dilate acutely in response to increased shear stress and constrict in response to increased pressure, in parallel with the chronic responses. Furthermore, wall mass is evidently conserved during acute responses, as is the case in a number of chronic vascular adaptations to sustained changes in blood flow or pressure.^{32 33 34} Therefore, it is conceivable that mechanisms involved in acute adjustment of smooth muscle tone also play a role in establishing the proposed pressure-shear relation (Fig 4, lower panel).

Cells of the vessel wall exhibit a broad range of responses to mechanical stimuli, with typical response times varying from milliseconds up to hours.²⁸ For example, K⁺ channels are involved in both the short-term regulation of endothelial autacoids³⁵ and in the long-term regulation of growth factors³⁶ upon changes of shear stress and distension.³⁷ A close relation between the short-term regulation of vascular tone and the long-term adaptation of vascular structure is also suggested by observations that a number of growth factors act as mediators of vascular tone and vice versa.^{38 39 40 41} Furthermore, the "shear stress–responsive element" (SSRE) is found in growth factor genes (eg, genes coding for platelet-derived growth factor-B and transforming growth factor-ß1) as well as in genes related to the production of mediators involved in the acute adjustment of vessel tone such as, for example, endothelial cell nitric oxide synthase.^{38 42}

Physiological Implications
In contrast to Murray’s minimum-cost concept, the pressure-shear hypothesis provides an explanation for the physiologically important arteriovenous asymmetry of the vascular system as a whole. This asymmetry is evident in (1) the high arteriolar flow resistance that is required for the role of arterioles in regulating organ perfusion, (2) the large pressure drop across the arterial system that is necessary to control, via the low average capillary pressure, the fluid balance between intravascular and extravascular space, (3) the low venular shear rates that support leukocyte margination and interaction with venular walls, and (4) the large venous compliance that allows effective volume buffering. Despite variations in the design of vascular systems between different tissues, all of these functions depend on a systematic difference between the geometric and hemodynamic properties of arterial and venous vessel trees and therefore, a priori, contradict the general applicability of Murray’s law.

According to the present hypothesis, the shape of the preset relation between shear stress and pressure determines, for a given network topology, the details of the hemodynamic arteriovenous asymmetry, including mean capillary pressure: The decrease in intravascular pressure along arteriovenous pathways results in a decrease in wall shear stress by way of dynamic adjustments of vascular diameters. Consequently, vessel diameters are smaller and the pressure drop is larger on the arterial compared with the venous side.^{11 12} This is reflected by the pressure profiles shown in Fig 5, which demonstrate that 70% of the pressure drop is located in arterioles with diameters below 100 µm, irrespective of the tissue or species investigated, even in tissues with low or even absent vascular smooth muscle tone (as in the present data set).

View larger version (23K): [in this window] [in a new window]   Figure 5. Intravascular pressure as a function of vessel diameter in different tissues and species. Present data obtained from mathematical model calculations for six mesenteric networks are compared with literature reports based on direct micropipette pressure measurements.43 44 45 46

Obviously, the pressure-shear principle proposed in the present study does not uniquely define the design of vascular networks, and it must therefore act in conjunction with vascular responses to other stimuli (Fig 6). Topological network structure, as well as the density of vessels per unit tissue volume, depends strongly on the action of growth factors and metabolic stimuli, which are, in turn, regulated by tissue supply conditions and diffusion distances.⁴⁷ This establishes a biochemical feedback control responding to the metabolic conditions of the tissue. In conjunction with this control mechanism, the adaptation of vessel diameter and wall thickness to mechanical stimuli (pressure and shear stress) will govern the overall design of vascular networks. In a scenario that involves the development of new vessels, eg, wound healing, biochemical stimuli cause vessel formation and topological arrangement of vessels, while local vascular reactions to the mechanics of blood flow would generate the appropriate hemodynamic conditions. Both feedback mechanisms contribute to the control of overall network parameters such as flow resistance and total vascular surface area.

View larger version (24K): [in this window] [in a new window]   Figure 6. Schematic representation of the hypothesized biochemical (upper loop) and mechanical (lower loop) feedback mechanisms controlling the design of vascular networks, indicating how they may operate in parallel to regulate hemodynamic and mass transport characteristics of the vasculature.

The proposed pressure-shear hypothesis has been developed on the basis of data from a single tissue preparation, and general arguments supporting the hypothesis have been presented. Clearly, its validity must be examined in other tissues and in physiological as well as pathophysiological conditions. Since it has been shown that the average shear stress level in large arteries changes with body size,²⁶ the dependence of the relation between pressure and shear stress should also be investigated for different body sizes and developmental stages.

In summary, our studies of complete microvascular networks in the rat mesentery have revealed a unified relation between wall shear stress and intravascular pressure for arterioles, capillaries, and venules. We hypothesize that this relation represents a general characteristic of the design of vascular networks brought about by local vascular adaptations to both distending transmural pressure and wall shear stress. Such a mechanism, operating in parallel with vascular responses to biochemical (metabolic) stimuli, would control hemodynamic features and arteriovenous asymmetry of vascular beds.

	Acknowledgments

 
This study was supported by the Deutsche Forschungsgemeinschaft (Pr 271/1-1, 1-2, and 5-1) and by National Institutes of Health grant HL-34555.

Received February 21, 1995; accepted July 18, 1995.

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