INTRODUCTION

Top Abstract Introduction Methods Results Discussion Appendix A Appendix B References

VASCULAR BEDS are capable of continuous adjustment in response to the functional needs of the tissues that they supply. The inverse fourth-power 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 long-term 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, 14-16, 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 "pressure-shear" 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 time-dependent 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, and 3) 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, and III) 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

Top Abstract Introduction Methods Results Discussion Appendix A Appendix B References

Experimental Observations of Microvessel Network Structure

1

1

4

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 mm²) was scanned with an SW 25/0.6 saltwater immersion objective (Leitz) and recorded on both videotape and black-and-white 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 and II, the flow velocity in each vessel segment was determined with a digitized-image analysis system (26) from an additional second scan using a strobed asynchronous illumination. From the original video recordings, the light-intensity 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 in networks I and II and from the photonegatives for network III. The number of vessel segments per networks I, II, and III was 546, 383, and 913, respectively.

Simulation of Network Blood Flow

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

(1)

View larger version (29K): [in this window] [in a new window]   Fig. 1.   Results of simulated adaptation for small (22 segments) networks. A-D represent results after successive incorporation of stimuli for adaptation in different network structures: network 1 (A and B); network 2 (C); and network 3 (D). Schematic drawings (top) indicate resulting diameter distributions, with arteriolar inflow indicated by arrows. Graphs provide predicted distributions of wall shear stress (middle) and diameter (bottom) for arteriolar segments (Art), capillaries (Cap), and venular segments (Ven).

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 (see RESULTS). 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 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. For networks I and II, 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) Sensitivity analyses were performed to determine the dependence of E_V on model parameters. For network III, velocities were not measured, so the root mean square diameter deviation (E_D) between predicted diameters (D_p) and measured diameters (D_m) was minimized where

	RESULTS

Top Abstract Introduction Methods Results Discussion Appendix A Appendix B References

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

(2)

where _w is the wall shear stress in the segment. Logarithmic dependence was introduced to achieve constant sensitivity of S_tot to a given proportional change in _w over a wide range of _w values. The constant ₀ defines the set point of the adaptation process and was set at ₀ = 100 dyn/cm² as found in larger arteriolar vessels of the rat. Values above this level lead to increasing vessel diameters. When this adaptation model was applied to the symmetric network 1, the results shown in Fig. 1A were obtained. In Fig. 1A, vessel diameters and _w are plotted as a function of intravascular pressure, the only parameter that varies unidirectionally with distance from the arteriolar inflow into the network. Segment diameters in the resulting equilibrium structure decreased with increasing generation, as expected. In contrast to the experimental findings, however, corresponding vessels on the arterial and venous sides of the network had equal diameters. This unrealistic behavior was a result of assuming identical responses to wall shear stress on both the arterial and venous sides of the network.

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 where P is transmural pressure and _e is the corresponding expected level of wall shear stress, which follows a sigmoidally increasing function of pressure, according to experimental data obtained in the rat mesentery. When applied to the symmetric network 1, this adaptation model generated a structure with strong arteriovenous asymmetry of shear stress, pressure, and diameter, as observed in vivo (Fig. 1B). In each segment, wall shear stress corresponded to pressure according to the assumed pressure-shear relationship.

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 A). Therefore, the model including only hemodynamic stimuli cannot adequately represent network adaptation.

View larger version (19K): [in this window] [in a new window]   Fig. 2.   Schematic representation of adaptation process for a network with topologically symmetric but morphologically irregular structure (A) with consideration of only hemodynamic stimuli. Flow pathways with initially higher wall shear stress (w) experience a larger total stimulus than parallel flow pathways with lower w. The resulting adaptive response leads to even greater imbalance of w (B, C). In the final state (D), all flow passes through a single arteriovenous pathway, and all other vessel segments are eliminated.

(3)

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 low-generation capillaries that provide short pathways between the major feeding arterioles and draining venules. These low-generation 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 B. In this model, low-generation 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.

View larger version (41K): [in this window] [in a new window]   Fig. 3.   Schematic representation of adaptation process for a topologically asymmetric and morphologically irregular network structure. Inclusion of a metabolic stimulus in addition to wall shear stress and pressure does not lead to realistic geometry. Proximal capillaries should have smaller diameters than feeding arterioles or draining venules, as in assumed initial conditions (A). However, high pressure drops and shear stresses in such vessels would cause them to enlarge to the diameters of their feeding and draining vessels following adaptation (B). Such vessels would become arteriovenous shunts, carrying most of the blood flow in the network.

(4)

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 parameters k_m, k_c, and k_s were chosen to minimize E_V in networks I and II and E_D in network III. Their final values are given in Table 2. A further optimization was performed with respect to the function _e(P) describing the effect of pressure on the set point for _w. The experimentally observed variation of _w with P (23) reflects the combined effects of all adaptation stimuli and does not necessarily represent the equilibrium relationship _e(P) that would be found in the absence of metabolic stimuli. Therefore, the functional form of _e(P) was chosen empirically to reflect the observed dependence (23), but the parameters were adjusted to minimize E_V, yielding Resulting _e values start at 14 dyn/cm² for pressures of 10 mmHg, increase sigmoidally with increasing pressure, and reach 100 dyn/cm² at 90 mmHg. In the intermediate range of P, the curve shows a slight rightward shift compared with the observed dependence (23).

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).

View larger version (37K): [in this window] [in a new window]   Fig. 4.   Distribution of characteristic hemodynamic and morphological parameters for network I with 546 segments (162 arterioles, 167 capillaries, and 217 venules in rat mesentery). Hemodynamic parameters were calculated using the network flow model based on measured network morphology and topology. Values obtained with experimentally determined vessel diameters (A) are compared with those obtained using diameters resulting from simulated adaptation (B). Parameter values employed for adaptation are given in Table 1.

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 (_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 larger-vessel 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.

View larger version (36K): [in this window] [in a new window]   Fig. 5.   Strengths of adaptive stimuli (according to Eq. 4 and Table 1) in initial state of network I (546 segments), based on measured vessel diameters (A) and simulated diameter adaptation (B). Hydrodynamic stimuli derived from w and pressure (P) are plotted as functions of pressure, and metabolic (M) and conducted stimuli (C) are plotted as functions of flow rate, to show functional dependence of these stimuli as clearly as possible.

Sensitivity analysis. To test the sensitivity of the results to the most important parameters (k_s, k_m, and k_c), these parameters were varied around the values obtained by minimizing E_V. Results for networks I and II (Fig. 6) showed that E_V is highly dependent on the shrinking constant, k_s, with a narrow, well-defined minimum. When k_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 of k_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 on k_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 by k_c. Figure 6 shows results of varying k_m and k_c. The resulting minima of E_V were much broader than those for k_s. Reasonable agreement between model predictions and observations was obtained over a range of about ±20-30% in these parameters.

View larger version (26K): [in this window] [in a new window]   Fig. 6.   Results of sensitivity analysis for networks I (A) and II (B) for which flow velocities were experimentally determined. Shown are changes of velocity error (EV) for variations of constants for shrinking tendency (ks), local metabolic stimulus (km), and conducted metabolic stimulus (kc). For each parameter, a value of 100% on abscissa corresponds to optimal value given in Table 2 for each respective network. Parameter values for which no data points are shown yielded EV outside ordinate range. Note different parameter ranges between ks and km or kc data.

	DISCUSSION

Top Abstract Introduction Methods Results Discussion Appendix A Appendix B References

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, 21-23) 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 single-vessel levels and suggests future directions for experimentation. The mesenteric preparation is especially suited for the purpose of analyzing the long-term 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 short-term 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 A AND B, 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 _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, 9-12, 14, 16, 17, 20, 33-35, 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, large-diameter 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 long-term 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 long-term 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, large-diameter 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 single-vessel 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.