Abstract
A theoretical model was developed to simulate longterm changes of vessel diameters during structural adaptation of microvascular networks in response to tissue needs. The diameter of each vascular segment was assumed to change with time in response to four local stimuli: endothelial wall shear stress (τ_{w}), intravascular pressure (P), a flowdependent metabolic stimulus (M), and a stimulus conducted from distal to proximal segments along vascular walls (C). Increases in τ_{w}, M, or C or decreases in P were assumed to stimulate diameter increases. Hemodynamic quantities were estimated using a mathematical model of network flow. Simulations were continued until equilibrium states were reached in which the stimuli were in balance. Predictions were compared with data from intravital microscopy of the rat mesentery, including topological position, diameter, length, and flow velocity for each segment of complete networks. Stable equilibrium states, with realistic distributions of velocities and diameters, were achieved only when all four stimuli were included. According to the model, responses to τ_{w} and P ensure that diameters are smaller in peripheral than in proximal segments and are larger in venules than in corresponding arterioles, whereas M prevents collapse of networks to single pathways and C suppresses generation of large proximal shunts.
 shear stress
 pressure
 conducted response
 mathematical modeling
vascular beds are capable of continuous adjustment in response to the functional needs of the tissues that they supply. The inverse fourthpower dependence of flow resistance on luminal diameter implies that blood flow in vascular networks is very sensitive to vessel diameters. Acute regulation of flow is achieved largely by contraction or relaxation of smooth muscle in vessel walls. However, accommodation to chronic changes in tissue needs involves longterm structural adaptation of vessel diameters, in which each vessel segment responds to the local mechanical and biochemical stimuli that it experiences (27).
Many studies have shown that vessels respond structurally to the mechanical forces exerted by flowing blood, i.e., transmural pressure and shear stress at the endothelial surface (2, 11, 12, 1416, 34,35, 37, 38). In response to sustained increases of shear stress, vessels generally exhibit a structural increase of luminal diameter, and at constant volume flow, this reduces shear stress. Therefore, it was proposed that shear stress is regulated by structural adaptation of vessel diameters (13, 41). However, model studies (8, 11) showed that adaptation of individual segments in response to local shear stress alone cannot produce realistic hemodynamic conditions in vascular networks.
In addition to wall shear stress, circumferential wall stress produced by transmural pressure is the main hemodynamic force acting on vessel walls. Chronic increase of transmural pressure results in changes of vascular morphology (10, 17) and network structure of terminal vascular beds (7, 9, 20). Simulations of the hemodynamics of microvascular networks suggest that circumferential wall stress influences arteriolar proliferation and rarefaction (21, 22).
Studies of terminal vascular beds of the rat mesentery have yielded information on the interaction between effects of transmural pressure and wall shear stress (23). A characteristic relationship was observed between intravascular pressure and wall shear stress, independent of vessel type (arteriole, capillary, or venule). On the basis of these data, a “pressureshear” hypothesis was proposed stating that vascular adaptation in response to hemodynamic conditions tends to maintain wall shear stress at a set point that is a function of local transmural pressure.
Structural adaptation of vascular beds involves responses not only to mechanical stimuli but also to the metabolic state of the tissue (33). Furthermore, the signals causing vascular adaptation may be transmitted from one segment to another in a manner analogous to the conduction of acute vasoactive responses (30). Responses to hemodynamic and metabolic stimuli may be difficult to distinguish experimentally, because reactions of one segment affect flow and pressure in other segments and thus contribute to adaptive responses throughout the network.
In the present study, a theoretical model was developed to simulate the changes that occur with time in vessel diameters during adaptation of microvascular networks. Each segment in the networks considered was assumed to adjust its diameter in response to the local stimuli described earlier. The following main criteria were used in developing the model. It should predict stable network structures, i.e., segment diameters should approach finite equilibrium values if the simulation is continued over a sufficient time, in the absence of externally imposed timedependent effects. Furthermore, the equilibrium network structures should be consistent with experimentally observed networks in terms of total flow resistance, distribution of pressure along arteriovenous flow pathways, and distribution of flow among pathways.
The model was developed in two stages. First, analyses of simplified hypothetical networks (networks 1,2, and3) were used to establish a minimal set of adaptive responses that must be included in the model for it to predict stable, realistic network structures. The model was then applied to microvascular networks (networks I, II, andIII) of the rat mesentery to test whether the assumed adaptive responses can lead to equilibrium distributions of morphological and hemodynamic parameters in quantitative agreement with experimental observations.
METHODS
Experimental Observations of Microvessel Network Structure
The animal preparation and the setup used for intravital microscopy have been described in detail elsewhere (23, 25, 26). Male Wistar rats (n = 3, 300–450 g body wt) were prepared for intravital microscopy of the mesenteric microcirculation following premedication (atropine 0.1 mg/kg im and pentobarbital sodium 20 mg/kg im); anesthesia (ketamine 100 mg/kg im); cannulation of trachea, jugular vein, and carotid artery; and abdominal midline incision. The animals were then transferred to a special stage mounted on an intravital microscope. The small bowel was exteriorized, and fatfree portions of the mesentery were selected for investigation with a ×25/NA 0.6 saltwater immersion objective (Leitz). During the experiments, the level of anesthesia and fluid balance were maintained by intravenous infusion of physiological saline (24 ml ⋅ kg^{−1} ⋅ h^{−1}) containing 0.3 mg/ml pentobarbital sodium. In this preparation of the exposed mesentery, vessels generally exhibit no spontaneous smooth muscle tone. However, as a precaution to prevent the development of tone and thus temporal variation of vessel diameters and flow resistance during the measurement period, papaverine (10^{−4} M) was continuously superfused. Heart rate and arterial blood pressure (ranging from 105 to 140 mmHg) were continuously monitored via the catheter in the carotid artery.
The networks selected for this study were supplied by feeding arterioles with inner diameters of ∼30 μm and drained by venules of ∼45 μm. The volume flow rate through the networks varied between ∼200 and 1,000 nl/min. A selected area of the mesenteric membrane (ranging from 35 to 80 mm^{2}) was scanned with an SW 25/0.6 saltwater immersion objective (Leitz) and recorded on both videotape and blackandwhite film. The complete scan took ∼30 min and consisted of ∼300 individual fields of view (300 × 400 μm). There were no indications of changes in vessel diameters or blood flow velocities during the recording period.
In networks I andII, the flow velocity in each vessel segment was determined with a digitizedimage analysis system (26) from an additional second scan using a strobed asynchronous illumination. From the original video recordings, the lightintensity pattern of a line along the centerline of the image of a microvessel was repeatedly determined at two closely spaced time instances. The intensity patterns corresponding to moving blood cells were shifted in position between the two successive recordings. A measure of the centerline flow velocity was calculated as the length of this spatial shift divided by the time delay between the two recordings (spatial correlation principle) (6). Centerline velocities determined by spatial correlation were averaged over ∼4 s, corresponding to 100 individual measurements, and then converted into mean blood velocity according to a procedure described previously (26).
The photographs exposed during the scanning procedure were used to assemble photomontages of the complete microvascular networks that were then used to determine network topological structure (connection matrix) and the lengths of all vessel segments between branch points. The diameters of all vessel segments were determined from the video recordings obtained with the strobed flash illumination innetworks I andII and from the photonegatives fornetwork III. The number of vessel segments per networks I,II, andIII was 546, 383, and 913, respectively.
Simulation of Network Blood Flow
Mathematical simulations of blood flow in microvascular networks with prescribed structures were used to estimate mean wall shear stress, pressure, and volume flow rate in each vessel segment. Details of the simulation have been described earlier (26). For any given network, the volume flow rate in each segment and the pressure at each branch point were calculated using an iterative algorithm. The following information and assumptions were required for these calculations.1) Network structure data was necessary, including topology (connection matrix of vessel segments) and geometry (diameters and lengths of each segment).2) Boundary conditions were required, including the volume flow rates and hematocrits in all vessel segments feeding the network, and the volume flow rates for those segments leaving the network, with the exception of the main venular draining segment. This segment was assigned a pressure of 13.8 mmHg according to previous measurements in similarsized venules in the same tissue (26). For two of the rat mesentery networks (networks I and II), volume flow rates in the boundary segments were derived from the measured flow velocities. For the third experimental network (network III) and the simple hypothetical networks (networks 1,2, and3), volume flow rates were assigned to match measured values for corresponding vessel diameters.3) Equations were necessary to describe rheological phenomena in the microcirculation: the phaseseparation effect (nonproportional partition of red cell and plasma flows) at diverging bifurcations, and the effective viscosity of blood flowing through microvessels. The parametric description of phase separation was based on experimental data obtained previously in arteriolar bifurcations of the rat mesentery (25) and describes the distribution of blood and red cell flow at individual bifurcations. The flow resistance in microvessels as a function of vessel diameter and hematocrit was derived previously for the same tissue (26).
At each step in the iteration, current values of segment hematocrits were used to estimate the flow resistance of each segment, from which updated nodal pressures and segment flows were computed. The rheological equations were then used to calculate updated values of segment hematocrits and flow resistance. This process was repeated until convergence was achieved, usually within 20–30 iterations. The simulation yielded predicted values of pressure, volume flow rate, flow resistance, and hematocrit in all segments, with the exception of the pressure in the main draining venule and the flows in all other boundary segments, which were prescribed.
Simulation of Adaptive Diameter Changes
For each segment in the network, the change of its diameter (ΔD) for a time step Δt was assumed to be proportional to the sum of terms representing different adaptive stimuli (S_{tot}) and to the vessel diameter (D) as
Adaptation of microvascular networks was simulated as follows. An initial set of segment diameters was chosen for each of the hypothetical networks. The flow and hematocrit in each segment were computed using the network flow simulation and were used to estimate S_{tot} (seeresults). The diameters were updated according to Eq. 1 . This process was repeated until the set of segment diameters reached either an equilibrium steady state or unrealistic values.
Optimization of Parameter Values
The model used to simulate adaptive diameter changes involves several unknown parameters. Comparisons between experimentally observed network states and model predictions were used to estimate the values of these parameters. Measured values of segment diameters, together with flow, resistance, and hematocrit values resulting from the network flow simulation, represented the initial (observed) state. To simulate adaptation for a given set of parameter values, vessel diameters were changed according to the combined adaptation stimulus, S_{tot}. The final distributions of hemodynamic parameters, when all vessel segments reached equilibrium (S_{tot} = 0), depended on the assumed parameter values.
The experimentally observed network states themselves reflect the result of the adaptation occurring in the tissue. A model that best represents this in vivo adaptation process should therefore minimize the differences between the initial (observed) state and the final (predicted) state. In principal, differences in either flow velocities or segment diameters can be used to assess these differences. Fornetworks I andII, in which velocities were measured, differences in velocities were used because they more sensitively reflect changes in flow resistance distribution within the networks. For these networks, the unknown parameters in the adaptation model were adjusted to minimize the velocity error (E_{V}), i.e., the root mean square deviation of the predicted segment flow velocities (V
_{p}) from the measured velocities (V
_{m})
RESULTS
Theory for Effects of Adaptation on Network Properties
Wall shear stress.
Several previous studies have assumed that vessels adapt so as to maintain a preset level of wall shear stress. This assumption was used as a starting point by setting
Transmural pressure.
Observations of blood flow in vascular networks show that wall shear stress is higher on the arterial side than on the venous side. Pries et al. (23) proposed that structural adaptation tends to maintain a preset relationship between wall shear stress and local transmural pressure. This was introduced in the present calculations by setting
Metabolic stimulus.
Network 1 is a special case in which parallel segments with the same generation numbers have identical lengths and diameters. In vivo, parallel pathways exhibit unequal lengths and diameters, as in network 2. Such a network is unstable when wall shear stress alone acts as the stimulus for adaptation, as shown by Hacking et al. (8). This may be seen by considering two segments connected in parallel, which initially experience the same pressure drop. Whichever segment has the larger wall shear stress tends to dilate, receiving more flow and still higher wall shear stress, whereas the parallel segment tends to shrink. This process continues until only a single pathway through the network remains (Fig.2). The same instability is found if both shear stress and pressure are used as adaptive stimuli (appendix ). Therefore, the model including only hemodynamic stimuli cannot adequately represent network adaptation.
In reality, structural adaptation of vessel networks must also respond to the metabolic needs of the tissue that they supply. If, at a given metabolic demand of the tissue, flow in a segment drops so that the surrounding tissue is poorly supplied with oxygen or other metabolic materials, then the segment must be stimulated to increase its diameter to enhance perfusion. Therefore, an additional contribution to S_{tot} was included in the model, dependent on the volume flux of red blood cells passing through the segment (represented byQ˙H_{D}, whereQ˙ is the blood volume flow and H_{D} is the discharge hematocrit) and increasing with decreasing flux. The resulting total stimulus was represented in the model by
Inclusion of the metabolic stimulus in the model tends to stabilize network structure. When flow in a segment drops to a low level, the effect of declining τ_{w} is compensated by the increasing metabolic stimulus. The analysis inappendix shows that a single segment connected to a fixed pressure source has a stable diameter if the metabolic stimulus is sufficiently strong. Such a segment is denoted as “pressure stable.” In terms of Eq.3 , pressure stability requires thatk _{m} > ¼. For a pair of segments in parallel, supplied by a constantpressure source in series with a fixed resistance, the configuration is stable if the segments are pressure stable, but not otherwise. For a more complex network, pressure stability of individual segments is sufficient to ensure stability of the entire network under the conditions stated in appendix . The behavior according to this model of a network with irregular morphology is illustrated in Fig. 1 C, with k _{m} = 0.7. A stable equilibrium state is reached, with flow in all segments. Because of the different metabolic stimulus acting on each segment, the levels of wall shear stress for individual segments are scattered around the single functional relationship of shear stress with pressure according to Eq. 2 .
Conducted stimulus.
Inclusion of a sufficiently strong local metabolic stimulus ensures the stability of segment diameters, but the resulting diameters do not necessarily agree with observations in the mesentery. Observed networks contain lowgeneration capillaries that provide short pathways between the major feeding arterioles and draining venules. These lowgeneration capillaries have larger pressure gradients but smaller diameters and flows than their parent vessels. In contrast, the model including only shear stress, pressure, and local metabolic stimuli leads to structures in which segments with larger pressure gradients have higher flows, as proven in appendix . In this model, lowgeneration capillaries increase in diameter and flow at the expense of the more distal segments, as illustrated in Fig.3. A realistic model of structural adaptation must predict that segments that feed a large, dependent network have larger diameters and flows than nearby segments that experience similar pressures and metabolic environments but that supply relatively short and nonramified “shunt” pathways through the network. To satisfy this requirement, it is necessary to introduce a further stimulus that reflects the topological position of a segment in the network, i.e., the number of dependent segments fed or drained.
Regulatory signals can be propagated along microvessels (30, 32). In the arteriolar network, such signals can be propagated upstream so that a parent vessel receives a signal that is the result of signals originating in its dependent branches (28). Conducted signals reaching a given segment provide an additional stimulus to which the segment may respond by longterm adaptation, the “conducted stimulus” (C). The effect of including such a stimulus in the model is shown in Fig.1 D. In accordance with experimental data, arteriolar and venular diameters are larger than those of the capillary vessels fed, and capillary diameters increase with generation number (24). In the corresponding model, it was assumed that metabolic stimuli generated in individual segments are propagated upstream in the arteriolar vessel tree and downstream in the venular vessel tree.
The calculation of the assumed conducted stimulus started at the most distal branch points of the arteriolar tree and proceeded proximally until the main feeding vessel was reached. The conducted stimulus at each junction (S_{c}) was assumed to be the sum of the metabolic stimuli (Table1) of the two downstream vessel segments (segments a andb) together with conducted stimuli from the two downstream bifurcations. The conducted signals were assumed to decay exponentially with distance traveled as exp (−x/L), where x is the length of the vessel segment between bifurcations and L is a length constant. Therefore, at each bifurcation
Clearly, this involves a number of assumptions that cannot be fully justified on the basis of available physiological data. However, the exact form of the response is not crucial. A number of modifications of the model, all of which included the conduction of a signal generated in peripheral vessel segments along vessel walls to feeding or draining vessels, were found to produce realistic distributions of vessel diameters and hemodynamic parameters. For example, this was true for a model assuming generation of a conducted signal only in capillary segments (and not in arteriolar or venular segments) and also for a model assuming generation of a conducted signal of equal strength in all vessel segments independent of their blood flow. In contrast, diffusion of metabolites across the tissue, from capillary vessels to the feeding and draining segments, cannot replace the conductive mechanism. Such a process would affect both the main feeding vessels and lowgeneration capillaries in a given area and could not prevent the development of lowgeneration capillaries into large arteriovenous shunts.
The inclusion of responses to these four types of stimuli (τ_{w}, P, M, and C; Table 1) is thus a minimal requirement for a realistic description of diameter adaptation. To test the adequacy of the resulting model, its predictions were compared with observed network structures in the rat mesentery.
Simulation of Observed Network Structures
Estimation of parameters in adaptation model.
Preliminary simulations using data obtained from the experimental networks were carried out in which the adaptation model was used with various combinations of only some of the full set of stimuli. These tests confirmed that all four stimuli were needed to prevent strong deviations of the model results from observed data in terms of segment diameters or flow velocities. For the complete model, the parametersk
_{m},k
_{c}, andk
_{s} were chosen to minimize E_{V} innetworks I andII and E_{D} in network III. Their final values are given in Table2. A further optimization was performed with respect to the function τ_{e}(P) describing the effect of pressure on the set point for τ_{w}. The experimentally observed variation of τ_{w} with P (23) reflects the combined effects of all adaptation stimuli and does not necessarily represent the equilibrium relationship τ_{e}(P) that would be found in the absence of metabolic stimuli. Therefore, the functional form of τ_{e}(P) was chosen empirically to reflect the observed dependence (23), but the parameters were adjusted to minimize E_{V}, yielding
Distributions of hemodynamic variables.
Detailed results are presented for one of the three experimental networks, network I containing 546 segments. Experimental values of segment wall shear stress and diameter and values obtained after the adaptation model was applied are shown in Fig. 4 as functions of intravascular pressure. Results for the two other networks were very similar. Despite significant changes for individual segments, the distributions of morphological and hemodynamic parameters were largely conserved. E_{D} ranged between 0.31 and 0.36 (Table 2), indicating that typical diameter changes during adaptation were below ∼30%. Although diameter changes of up to ±30% can strongly influence the flow resistance of individual vessel segments, these changes were distributed within the networks in such a way as to produce only minimal changes in the overall distributions of pressure and wall shear stress (Fig. 4).
For networks I andII, in which velocity was measured, the initial relative E_{V} were 0.63 and 0.79. These errors reflect both measurement uncertainties, particularly those of diameters, and limitations of the hemodynamic model. During the simulated adaptation, vessel diameters were free to adjust independent of their initial values. The final errors were similar (0.65 and 0.75) to the initial values, i.e., the segment velocities predicted from the diameters obtained after adaptation were no less accurate than the velocities predicted from measured diameters. These results show that the adaptation model can predict stable, realistic distributions of diameters and hemodynamic parameters.
Distributions of stimuli.
Figure 5 shows the initial and final strengths of the adaptive stimuli for network I. Results for networks II and III were nearly identical. On average, the two hydrodynamic stimuli balance each other over the whole pressure range. The strength of both the pressure (P) and shear stress (τ_{w}) stimuli increased with increasing pressure, reflecting the assumed effects of shear stress and pressure on vascular growth. A similar balance is seen in the metabolic (M) and conducted (C) stimuli. According to its mathematical formulation, the metabolic stimulus was larger in vessels with lower blood flow, whereas the conducted stimulus showed the opposite behavior. The coupling of the conducted stimulus to flow is indirect: vessels with high flow generally feed largervessel trees containing more capillaries, which generate the conducted stimuli. The distributions of the individual stimuli are very similar in the initial and final states. The combined stimulus (S_{tot}, not shown) is distributed around zero in the initial state, with a value of 0.08 ± 0.44 (mean ± SD) for network I, shown in Fig.5. In the final state, S_{tot} is zero in every segment, as required by convergence of the model.
Sensitivity analysis.
To test the sensitivity of the results to the most important parameters (k _{s},k _{m}, andk _{c}), these parameters were varied around the values obtained by minimizing E_{V}. Results fornetworks I andII (Fig.6) showed that E_{V} is highly dependent on the shrinking constant,k _{s}, with a narrow, welldefined minimum. Whenk _{s} was varied over the range shown, the pressure drop across the network increased strongly from 9 and 7 mmHg to 323 and 217 mmHg for the two networks, respectively. However, at the optimal values ofk _{s}, yielding minimal E_{V}, pressure drops in the realistic range were predicted: 69 mmHg for the larger network, and 37 mmHg for the smaller. In further sensitivity tests,k _{s} was optimized with respect to minimal E_{V} for each value of k _{m}and k _{c} tested. The level of both the metabolic and conducted stimuli depends onk _{m}, because it was assumed in the model that the conducted response originates from the metabolic stimuli of individual segments. Therefore,k _{m} determines the balance between these two stimuli and the hydrodynamic stimuli. The balance between metabolic and conducted stimuli is set byk _{c}. Figure 6shows results of varyingk _{m} andk _{c}. The resulting minima of E_{V} were much broader than those fork _{s}. Reasonable agreement between model predictions and observations was obtained over a range of about ±20–30% in these parameters.
DISCUSSION
Growth and adaptation of vascular beds are complex processes, involving many interacting stimuli and responses. A number of potential mechanisms have been identified in experiments at the vascular and cellular level (1, 3, 4, 15). Such experiments could provide a basis for an integrated, quantitative description of adaptation in complete networks, but this has not been achieved. A different approach was used here: a theoretical model for network adaptation was combined with measurements of network properties representing a single instant in time to deduce information about the principles governing the system dynamics. Previous studies (8, 2123) have shown the value of theoretical models for gaining insight into the processes of network growth and adaptation. The approach used here additionally takes advantage of extensive data sets on in vivo network structure and flow derived from previous studies. It also provides a framework for interpreting results of experiments at the cellular and singlevessel levels and suggests future directions for experimentation. The mesenteric preparation is especially suited for the purpose of analyzing the longterm structural adaptation of vessel diameters. The vessels constituting the analyzed networks exhibit no spontaneous tone, so the experimental results are not confounded by interactions between shortterm regulation of vessel diameter via changes in tone and structural diameter adaptation.
The approach used here has some inherent limitations. Because the observations are made at a single instant, no information is provided about the time scale of the process. Also, to obtain useful results, some a priori assumptions are needed. The key assumptions of the present study are that each segment responds autonomously to the local stimuli that it receives, including those transmitted along the vessel wall, and that the response characteristics as defined by the equations (Table 1) are identical for all vessel segments. These assumptions are reasonable, because it is difficult to imagine a mechanism that could provide centralized control over the diameters of very many segments and yet retain their responsiveness to local conditions and need. In addition, the assumptions are supported by the adaptive responses seen in vessels grafted to positions with different hemodynamic conditions (5, 18, 19). Further assumptions had to be made about the specific forms of the responses to these stimuli so that they could be expressed in terms of a reasonable number of unknown parameters. Here, information from other experimental studies has been incorporated where possible. Also, the arguments used to develop the model for adaptation, including the analyses in appendixes and , are largely independent of the specific functional forms used. Therefore, although the present model for adaptation is not the only one possible, other models capable of predicting the observed structures will probably be at least as complex and are likely to incorporate, in some form, the types of response included in the present model.
The present analysis leads to the following conclusion: continuous diameter adaptation, which leads to stable networks with structures and hemodynamic properties consistent with observations, can be explained by responses of each segment to a set of four stimuli. These are the shear stress τ_{w} at the endothelial surface, the transmural pressure P (corresponding to circumferential wall stress), a metabolic stimulus M dependent on blood flow in the segment, and a conducted stimulus C dependent on the number and also, possibly, the metabolic state of the exchange vessels supplied by the segment. Adaptation in response to pressure and shear stress is necessary to obtain the existing and functionally significant arteriovenous asymmetry with respect to pressure, diameter, and flow velocity (23). A local metabolic stimulus is needed to prevent the collapse of networks to single arteriovenous pathways, and a conducted metabolic stimulus suppresses the generation of large, short arteriovenous shunts.
Because the exact definitions of these stimuli and the corresponding vascular responses cannot be deduced from available experimental measurements, their implementation in the adaptive model was arbitrary to some extent. A number of different assumptions and formulations of the model were tested and found to yield similar results, provided the parameter values were optimally chosen. This was true for the expressions used to describe the effects of shear and pressure and particularly for those of the metabolic and conducted stimuli. The assumption that the local metabolic stimulus depends only on the red cell flux in a segment is clearly a gross simplification. In reality, this stimulus should reflect the balance between oxygen supply and demand in the region surrounding each segment and should depend on the size of the region supplied by the segment, the oxygen saturation of the blood flowing in the segment, and the flows in neighboring segments. Metabolites other than oxygen may also play a role. Inclusion of these effects would complicate the model and involve further arbitrary assumptions because the mechanisms by which metabolic needs influence adaptation are not established. Therefore, only the crucial dependence on red cell flux is included here.
The relevance of wall shear stress, intravascular pressure, and local metabolic conditions to vascular adaptation was already evident from numerous previous studies using a variety of experimental approaches, as described in the introduction (2, 7, 912, 14, 16, 17, 20,3335, 37, 38). The necessity for a fourth, distinct stimulus is an important conclusion of the present work. Again, the precise form of this stimulus is not known, and different assumptions could lead to similar predicted behavior. For example, if the conducted stimulus was assumed to depend only on the number of capillaries that a given segment feeds or drains (“magnitude”) but not on metabolic stimuli or local blood flow, then similar results were obtained. On the other hand, the conducted stimulus cannot be interpreted as a purely diffusive process from capillaries to feeding and draining segments. Such a process would lead to a continuous spatial distribution of the stimulus, with similar signals in different types of vessels in a given tissue area. In particular, this would apply to neighboring arterioles feeding large capillary networks and short arteriovenous connections. Thus a diffusive mechanism would not be able to prevent the growth of the vessels constituting such arteriovenous connections and would not inhibit generation of short, largediameter shortcuts.
In the present model, the conducted stimulus is assumed to originate in individual segments and to be conducted from daughter to parent vessels in the vascular tree, assuming summation at junctions and exponential decay with distance traveled, with a length constant of 1,500 μm. This assumption is consistent with many studies showing that vessel walls conduct information created locally, most likely by a mechanism involving changes of the membrane potential and transmission via gap junctions (28, 31, 39, 40). Although these studies were concerned with acute changes of vascular tone, the longterm adaptive responses analyzed in the present work require structural changes of the vessel wall. Investigation of the local reactions of cells and vessels to mechanical stimuli, however, suggests that short and longterm reactions are closely linked (4). An essential feature of the model is that stimuli are transmitted unidirectionally upstream (for arteriolar segments) and downstream (for venular segments). Without this assumption, the conducted stimulus would enter small segments branching from larger feeding vessels, causing formation of short, largediameter arteriovenous shunts. Although this aspect of the model is at present not supported by direct experimental evidence in the microcirculation, it may be linked to unidirectional coupling and rectification observed at gap junctions (29, 36).
As shown in this study, vessel diameter adaptation in response to a set of four stimuli can lead to realistic network structures. Many further questions remain, however. The present model was used to simulate the approach of a network to an equilibrium state, assuming constant functional demands. In principle, a similar approach could be used to model the dynamic adaptation of a network subject to varying demands. Adaptation may involve responses to stimuli other than the four considered here. Although some experimental information is available on adaptive responses at the singlevessel level, it is not clear how such information can be incorporated into the present model. The existence of a conducted stimulus for adaptation, predicted from the model, remains to be verified experimentally. Finally, the addition or loss of segments from a network is a crucial aspect of the adaptation process for which satisfactory theoretical models remain to be developed.
Acknowledgments
The expert technical help of B. Giesicke and M. Ehrlich as well as the assistance of A. Scheuermann in preparing the manuscript is gratefully acknowledged.
Footnotes

Address for reprint requests: A. R. Pries, Freie Universität Berlin, Dept. of Physiology, Arnimallee 22, D14195 Berlin, Germany.

This study was supported by Deutsche Forschungsgemeinschaft Pr 271/11, 12, 51, and 52 and by National Heart, Lung, and Blood Institute Grant HL34555.
 Copyright © 1998 the American Physiological Society
Appendix
Stabilization of Segment Diameters by a FlowDependent Metabolic Stimulus
Adaptation of vascular diameters in response to wall shear stress alone leads to instability (8). Here, the goal is to establish conditions under which the inclusion of a flowdependent metabolic stimulus stabilizes diameter. Responses to pressure and conducted stimuli are neglected here, and constant H_{D}and apparent viscosity (η) are assumed. Suppose that segmenti adjusts its diameter (D_{i}
) with time (t) in response to its wall shear stress (τ_{i}) and flow (Q˙_{i}). If continuous variation with time is assumed, Eq.1
can be expressed as
First, consider one active segment in series with a fixed resistance (i = 1) so that ΔP_{1} = P_{0}/(1 + ρ_{1}). At equilibrium,f(τ_{1}) +g(Q˙_{1}) = 0. For stability, the derivative of this quantity with respect toD
_{1} must be negative at equilibrium so thatD
_{1} returns to equilibrium after a perturbation. From Eq.EA2
, we find
Next, consider two parallel active segments in series with a fixed resistance (i = 1, 2) so that P_{1} = P_{2} = P_{0}/(1 + ρ_{1} + ρ_{2}). From the theory of ordinary differential equations, a pair of equilibrium diameters (D
_{1},D
_{2}) is stable if and only if the eigenvalues of the matrix
The analysis of larger networks is more complex. However, such networks are likely to contain pairs of parallel segments, and such segments must be pressure stable to ensure the stability of the whole network. In fact, it can be shown that if all segments in the network are pressure stable, then the network as a whole is stable, subject to the additional condition that τ_{i} f′(τ_{i}) +Q˙_{i} g′(Q˙_{i}) > 0 in all segments. In the simulations, this condition is satisfied if k _{m} < 1. The main steps in the proof are as follows. The general form of the matrix (A) is shown to have the same eigenvalues as a real symmetric matrix. Therefore, the eigenvalues are real, and the system is stable if they are all negative. Suppose, on the contrary, that there is a positive eigenvalue. The corresponding eigenvector represents a disturbance to the system in which all segments whose diameters increase experience an increased pressure gradient, and vice versa. However, such a disturbance can be ruled out using a minimumdissipation argument, completing the proof.
The introduction of a pressuredependent stimulus for adaptation complicates the above analysis and could produce different behavior in some cases. However, the dominant mode of instability in networks with shear response occurs when two segments are connected in parallel, with one shrinking while the other grows. In this case, both segments are subject to the same pressures, so a pressure stimulus cannot provide a differential stimulus to stabilize the pair. This suggests that inclusion of a flowdependent metabolic stimulus as described here leads to stability even in the presence of a pressuredependent response.
Appendix
Stability of Networks With LowGeneration Shunts
Consider a network of segments whose adaptation is governed byEq. EA1
. In each segment,Q˙ = πD
^{4}G/128η and τ = DG/4, where G = ΔP/L is the pressure gradient. Therefore, τ = (ηQ˙G^{3}/2π)^{¼}. The condition for equilibrium,F(Q˙,G) =f[τ(Q˙,G)] + g(Q˙) = 0, implicitly defines a functional relationship betweenQ˙ and G. From wellknown properties of partial derivatives