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Am J Physiol Heart Circ Physiol 281: H1015-H1025, 2001;
0363-6135/01 $5.00
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Vol. 281, Issue 3, H1015-H1025, September 2001

Structural adaptation of microvascular networks: functional roles of adaptive responses

A. R. Pries1,2, B. Reglin1, and T. W. Secomb3

1 Department of Physiology, Freie Universität Berlin, D-14195 Berlin; 2 Deutsches Herzzentrum Berlin, D-13353 Berlin, Germany; and 3 Department of Physiology, University of Arizona, Tucson, Arizona 85724


    ABSTRACT
TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
REFERENCES

Terminal vascular beds continually adapt to changing demands. A theoretical model is used to simulate structural diameter changes in response to hemodynamic and metabolic stimuli in microvascular networks. Increased wall shear stress and decreased intravascular pressure are assumed to stimulate diameter increase. Intravascular partial pressure of oxygen (PO2) is estimated for each segment. Decreasing PO2 is assumed to generate a metabolic stimulus for diameter increase, which acts locally, upstream via conduction along vessel walls, and downstream via metabolite convection. By adjusting the sensitivities to these stimuli, good agreement is achieved between predicted network characteristics and experimental data from microvascular networks in rat mesentery. Reduced pressure sensitivity leads to increased capillary pressure with reduced viscous energy dissipation and little change in tissue oxygenation. Dissipation decreases strongly with decreased metabolic response. Below a threshold level of metabolic response flow shifts to shorter pathways through the network, and oxygen supply efficiency decreases sharply. In summary, the distribution of vessel diameters generated by the simulated adaptive process allows the network to meet the functional demands of tissue while avoiding excessive viscous energy dissipation.

shear stress; pressure; conducted response; oxygen transport; mathematical modeling


    INTRODUCTION
TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
REFERENCES

TERMINAL VASCULAR BEDS in normal tissues must meet the functional needs of the tissue, including the supply of oxygen and other metabolites and the maintenance of a relatively low capillary pressure without demanding excessive shares of the total blood volume or the pumping capacity of the heart. Furthermore, they must possess the ability to adapt structurally to long-term changes in functional needs. Given the inverse fourth-power dependence of flow resistance on vessel diameter, the ability of the vascular beds to meet these demands implies the existence of relatively sensitive adaptive control systems in which the diameter of each segment in a network of microvessels adapts structurally according to the physiological status of the network and the tissue it supplies.

Each vascular segment in a network experiences a number of stimuli that depend on the flow and metabolic conditions in the segment itself and in other segments in the network. The wall shear stress and the intravascular pressure in each segment depend on the distribution of flow throughout the network. Structural responses of vessel diameters to chronic changes in these hemodynamic variables have been demonstrated in many studies. Increased wall shear stress generally leads to increase in diameter, whereas increased intravascular pressure causes vessel diameter to decrease (8, 20, 42). The roles of pressure and shear stress in adaptation of vascular networks have also been investigated in model simulations (11, 12, 25). Metabolites such as oxygen or adenosine are transported by convection in the blood. Their concentrations in each segment depend on uptake or release rates in upstream segments, and they can influence vascular growth or regression (24, 32, 47). In addition, investigations of acute vasoactive responses have shown that signals are propagated along vessel walls (6, 36, 38, 45) probably by conduction of changes in membrane potential. A similar conduction mechanism may also play a role in structural adaptation.

Pries et al. (29) used a theoretical model to investigate structural adaptation and stability of microvascular networks. They argued that vascular responses to wall shear stress, intravascular pressure, and metabolic conditions including a conducted metabolic response form a minimal set of requirements for stable network structures with realistic distributions of vessel diameters and flow velocities.

These requirements are summarized in Fig. 1. In the absence of any adaptive response (Fig. 1A), vessel diameters and flow resistance of segments coupled in series would vary randomly. Harmonizing of diameters along flow pathways can be achieved by invoking a response to wall shear stress (15, 16) in which segment diameters grow or shrink so that wall shear stress achieves a target level. The resulting network structures show an orderly decrease in diameter from proximal to distal segments (Fig. 1B) and lead to minimal viscous energy dissipation for a given flow rate and blood volume (22). However, the resulting symmetric networks are unrealistic in that capillary pressure is too high, being the mean of arterial and venous pressures. Furthermore, they are unstable and would decay by loss of segments to structures in which all flow passes along a single pathway (11). Introduction of a response to intravascular pressure, such that the target level of wall shear stress depends on pressure (28), leads to asymmetric networks with higher flow resistance on the arteriolar side and lower capillary pressure (Fig. 1C). However, such networks are still unstable.


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  		Fig. 1.   Schematic representation of roles of adaptive responses in a hypothetical small microvascular network. A: without adaptive responses, random distribution of vessel diameters gives inefficient blood supply. B: with response to shear stress diameters are coordinated along flow pathways. C: with response to pressure, arterioles are smaller than corresponding venules. D: parallel pathways are maintained by responses to metabolic stimuli. E: information transfer from small segments upstream to their feeding and downstream to draining vessels is needed to ensure that flow is balanced between long and short flow pathways. *Adaptation in response to shear stress alone or to shear and pressure is not stable; the network would collapse to a single flow pathway.

Instability can be avoided by including a metabolic response (29) so that segments with insufficient supply generate a signal stimulating diameter increase (Fig. 1D). If sufficiently strong, such a response has the important property of stabilizing network structure by preventing excessive shrinking of segments. Finally, realistic network structures can only be achieved if additional means of information transfer, both in the upstream and downstream directions, from distal to proximal segments of the network are included (29) (Fig. 1E). In the absence of such transmitted responses, short pathways through the network, such as proximal shunts, tend to increase in diameter and carry an unduly large proportion of the total flow in the network.

Assumptions regarding generation and conduction of metabolic responses in the model of Pries et al. (29) were made for simplicity and were unrealistic in some significant respects. In particular, the metabolic signal for a given segment was based only on blood flow in that segment. Furthermore, responses transmitted both upstream and downstream originated only in capillary segments, an assumption not supported by experimental observations.

In the present study, a model is developed to simulate structural adaptation of vessel diameters in microvascular networks while overcoming deficiencies of the previous model (29). Oxygen is chosen as a key metabolite for the generation of metabolic stimuli, and oxygen exchange occurring in upstream segments is taken into account. Signals are transmitted upstream from any segment in the network via conduction along vessel walls. Information transfer in the downstream direction is assumed to occur via convection of a metabolite generated in regions of low PO2. This model is used to analyze the functional roles of the assumed vascular responses to hemodynamic and metabolic stimuli by considering the consequences of altering their strengths for functional parameters of terminal vascular beds such as the distributions of wall shear stress and intravascular pressure, the adequacy of oxygen supply, the viscous energy dissipation in the network, and the mean path length for blood flow through the network.


    METHODS
TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
REFERENCES

Experimental observations of microvessel network structure. Details of the animal preparation and intravital microscopy setup have been given elsewhere (28, 30, 31). All procedures were approved by the local and state authorities for animal welfare. Male Wistar rats (n = 6, 300-450 g body wt) were premedicated (0.1 mg/kg im atropine, 20 mg/kg im pentobarbital sodium), anesthetized (100 mg/kg im ketamine) and placed on a special stage. During 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. After cannulation of trachea, jugular vein, and carotid artery, the animals were transferred with the stage to an intravital microscope (Leitz). The small bowel was exteriorized through an abdominal midline incision and superfused continuously with a thermostated (36.5°C) bicarbonate-buffered saline. In this preparation of the exposed mesentery, vessels did not exhibit smooth muscle tone and there were no indications of changes in vessel diameters or blood flow velocities during the recording periods. As additional precaution to prevent the development of tone and variation of vessel diameters during the experiments, papaverine (10-4 M) was added to the superfusate. Heart rate and arterial blood pressure (range 105 to 140 mmHg) were continuously monitored via the catheter in the carotid artery.

Microvascular networks in the fat-free portions of the mesentery were visualized with a ×25/numerical aperture 0.6 salt-water immersion objective (Leitz). In each experiment, one vascular network in the fat-free portion of the mesenteric membrane with an area between 25 and 80 mm2 and good visibility of all vessels was selected. This network was scanned and recorded both on videotape and on photographic film. The complete scan took ~30 min and consisted of ~300 individual fields of view (300 × 400 µm). The inner diameters of the arterioles feeding and the venules draining the networks ranged from ~25 to 35 and 45 to 65 µm, respectively. The volume flow rate through the networks as determined from blood flow velocities and diameters in the feeding arterioles varied between ~500 and 1,200 nl/min.

In two networks, an additional scan using strobe asynchronous illumination was made to allow the determination of blood flow velocity (Vb) in each segment with a digital image analysis system (31). From the video recordings, light intensity patterns along the centerline of the image of a microvessel were determined from the two fields of each video frame. Because of the double flash system used for illumination, these fields represented consecutive intravital images with a time delay of only 0.5 ms. 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 obtained as the length of this spatial shift divided by the time delay between the two recordings (spatial correlation principle). Centerline velocities (Vcl) determined by spatial correlation were averaged over ~4 s (100 individual measurements) and then converted into mean Vb using a procedure described previously (31).

The topological structure of the networks (connection matrix) and the lengths of all vessel segments between branch points were determined from photomontages of the photographs obtained during the scanning procedure. The diameters of all vessel segments were determined from the video recordings with the flash illumination in two networks and from the photographic negatives for the remaining networks. The number of vessel segments in a given network varied between 389 and 913.

Procedure for simulation of network blood flow and adaptation. The overall approach used in the theoretical simulations has been described previously (29) and is summarized in Fig. 2. It involves four nested levels of iterative computational procedures. At the innermost level (loop 1), the volume flow rate in each vessel segment and the pressure at each node (junction of segments) were calculated for prescribed values of the diameter, length, and apparent viscosity of blood in each segment. This procedure uses information on the network structure derived from experimental observations as already described (29) including the lengths and topological connections of the segments. Also, the volume flow rates and hematocrits in all segments feeding the network and the volume flow rates for those segments leaving the network were prescribed, with the exception of the main venular draining segment, whose pressure is prescribed. The conditions for conservation of blood flow at nodes may be expressed as a system of linear equations solved iteratively to obtain the nodal pressures and the flows.


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  		Fig. 2.   Overall organization of the simulation procedure. Each loop (1 to 4) is repeated until a preset level of convergence is achieved before another iteration of the next higher loop is initiated, starting with the innermost loop (1).

At the next level (loop 2, Fig. 2), hematocrit and viscosity in each segment were calculated. Given the distribution of flows, the discharge hematocrit (HD) in each segment was calculated based on a parametric description of phase separation (i.e., nonproportional partition of red blood cell and plasma flows) at diverging bifurcations (27) and the apparent viscosity of blood corresponding to the segment diameter and hematocrit was computed using an empirical equation previously determined for this tissue (31). The segment flows were then recalculated as just described, and this procedure was continued until all parameters approached constant values, usually within 20-30 iterations. From the resulting distributions of pressure, blood flow, and hematocrit, values for shear stress, oxygen saturation, and partial pressure were estimated for all vessel segments (see below).

Adaptive changes in vessel diameters were introduced at the third level (loop 3, Fig. 2). For a given distribution of segment diameters and flows, the wall shear stress, intravascular pressure, and PO2 were computed. These quantities were used to estimate a net signal (Stot) for diameter change (Delta D) in each segment as described in the next section. The segment diameters were then incremented by an amount Delta D = Stot · D where D is the diameter, and the calculations at the inner two levels were repeated. This procedure was repeated iteratively until the diameters approached equilibrium values. Because the goal is to predict stable equilibrium states resulting from adaptive processes, information on the time needed to reach equilibrium and on the absolute time constants of different mechanisms is not required and is not included. Simulated diameter adaptation was started either using the diameter values determined experimentally, identical diameters (10 µm) for all segments, or randomized diameter sets based on the experimental diameters.

As described below, the calculation of Stot involves several unknown parameters (Table 1). Values of these parameters for each network were optimized by minimizing deviations between predicted and measured segment flow velocities [for networks with measured velocity values available, velocity error (EV)] or diameters [diameter error (ED)], i.e., by aiming for the closest possible fit between model results and experimental network data. The best-fit parameter values were obtained by a multidimensional optimization procedure minimizing the EV or ED (Fig. 2, loop 4). For each trial set of parameter values, the calculations at the inner three levels were repeated. The complete procedure was programmed in the pascal language and typically took 10-30 h to complete on a personal computer.

                              
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  		Table 1.   Mean parameter values and standard deviations for the 6 experimental networks obtained by minimizing the flow velocity error or segment diameter error

Rules for adaptive diameter changes. As described earlier, adaptive changes in segment diameters were assumed to occur in response to hemodynamic and metabolic stimuli. This was represented mathematically by expressing the Stot as the sum of terms representing the assumed stimuli. The responses to wall shear stress and intravascular pressure were assumed to have a similar form as in the previous model (29). Stau is the growth stimulus derived from shear stress according to
S<SUB><IT>&tgr;</IT></SUB><IT>=</IT>log (<IT>&tgr;</IT><SUB>w</SUB><IT>+&tgr;</IT><SUB>ref</SUB>)
where tau w is the actual wall shear stress in a vessel segment calculated from pressure drop, vessel diameter, and vessel length, and tau ref is a small constant included to avoid singular behavior at low wall shear rates. The (negative) sensitivity to intravascular pressure was described by
S<SUB>p</SUB><IT>=</IT>−log<IT> &tgr;</IT><SUB>e</SUB>(P)
where tau e(P) is the level of wall shear stress expected from the actual intravascular pressure (P) according to a parametric description of experimental data obtained in the rat mesentery (28) exhibiting a sigmoidal increase of wall shear stress with increasing pressure1
&tgr;<SUB>e</SUB>(P)<IT>=</IT>100<IT>−</IT>86<IT>·</IT>exp{−5,000<IT>·</IT>[log (log P)]<SUP>5.4</SUP>}
In the present model, the assumptions regarding the vascular response to metabolic conditions and its transmission upstream and downstream in the network were based on known physiological processes in the microcirculation. The metabolic signal was estimated based on a consideration of oxygen transport and consumption in the network. Oxygen supply is generally the most critical requirement that must be satisfied by the vascular system, and hypoxia is known to stimulate both acute and structural changes in the vasculature (14, 24, 26, 43, 46, 47). The aim of the present model was to use oxygen availability as a representative index of metabolic conditions in the tissue surrounding the vascular network rather than to provide accurate quantitative predictions of tissue or vascular oxygen levels. Therefore, a highly simplified model for oxygen transport was used in which oxygen is assumed to diffuse out of each vessel segment at a fixed rate per unit vessel length until it is exhausted, independent of the diameters of the segments and their geometrical configuration. This approximate representation of transmural oxygen exchange does not take into account the effect of irregularity of vessel spacing and arrangement on diffusive fluxes. Inclusion of such effects would require a much more complex model (35) incorporating detailed information about the spatial distribution of vessels and leading to prohibitively long computation times.

It was assumed that each vessel supplies a tissue region 200 µm wide and 20 µm thick according to the average spacing of vessels in the mesenteric membrane and the membrane thickness. Using a typical consumption rate for connective tissue of 0.01 cm3 O2/(cm3 × min) results in a consumption per unit vessel length and time of 4 × 10-11 cm3 O2/(µm × min). Blood was assumed to enter the network with a PO2 of 95 mmHg. The oxygen flux of flowing blood was calculated as Q × CO × HD × S(PO2), where Q is the flow rate, CO = 0.5 cm3 O2/cm3 is the oxygen binding capacity of red blood cells and the saturation is given by the Hill equation S(PO2) = (PO2/P50)N/ [1 + (PO2/P50)N]. Values for P50 (38 mmHg) and N (3), typical for rat blood, were chosen.

Downstream transmission of information about the metabolic state of the tissue was assumed to occur via the convection of a metabolic signal substance added to flowing blood at a rate that depends on the PO2 in each segment. Collins et al. (6) suggested that ATP released from red blood cells in response to hypoxia functions as a vasoactive agent. ATP is therefore a candidate metabolite, but other substances, e.g., nitrosohemoglobin (43), have been proposed. However, the identity of the metabolite is not a crucial feature of the model. The convective flux (Jm) of the metabolite (measured in arbitrary units) is assumed to increase as blood flows from upstream to downstream by additional input of each segment, according to J<UP><SUB>m</SUB><SUP>down</SUP></UP> = J<UP><SUB>m</SUB><SUP>up</SUP></UP> + Ls (1 - PO2/PO2,ref) whenever the intravascular PO2 falls below a reference level for PO2 (PO2,ref) where Ls is the length of the segment. The metabolic signal for diameter increase is assumed to vary approximately logarithmically with the intravascular concentration of the metabolite (Jm/Q), and is given by Sm = log[1 + Jm/(Q + Qref)] where Qref, the reference value for blood flow, is a small constant included to avoid singular behavior at low blood flow values.

For upstream transmission of information, signal conduction along vessel walls is assumed (37), probably via electrotonic spread of changes in membrane potential through gap junctions coupling smooth muscle cells and/or endothelial cells (1, 45). Most intravital studies have used local application of substances such as acetylcholine to elicit conducted responses. However, the role of acetylcholine in regulatory processes in vivo is still unclear. Recent experimental studies investigating effects of adenosine, prostaglandins, or muscle contraction suggest a relation between local PO2 or metabolic state and conducted responses (2, 6, 34, 44). In the present model, the conducted signal is assumed to originate in each segment in proportion to the local value of the metabolic signal (Sm), to be conducted only in the upstream direction with summation or equal partition at each bifurcation, and to decay exponentially with length constant (L). In each segment, therefore, the conducted signal Jc at the upstream end is related to that at the downstream end by J<UP><SUB>c</SUB><SUP>up</SUP></UP> = (J<UP><SUB>c</SUB><SUP>down</SUP></UP> + Sm) · exp(-Ls/L). The conducted signal is assumed to depend on the value of Jc evaluated at the midpoint of each segment, with a saturable response, i.e., Sc = Jc/(Jc + J0).

The total signal for diameter change is represented in the model by the following equation
S<SUB>tot</SUB><IT>=</IT>S<SUB><IT>&tgr;</IT></SUB><IT>+</IT><IT>k</IT><SUB>p</SUB>S<SUB>p</SUB><IT>+</IT><IT>k</IT><SUB>m</SUB>[S<SUB>m</SUB><IT>+</IT><IT>k</IT><SUB>c</SUB>S<SUB>c</SUB>]<IT>−</IT><IT>k</IT><SUB>s</SUB>
Here, the unknown parameters kp, km, and kc are introduced to allow variation of the sensitivity with respect to pressure, the metabolic, and the conducted signal relative to the sensitivity to shear stress. The additional parameter, shrinking tendency (ks) represents a basal tendency of vessels to shrink in the absence of positive growth stimuli. To establish baseline values for the unknown parameters of the model (kp, km, kc, ks, L, J0, PO2,ref and Qref), their values were optimized to minimize the root mean square deviations between predicted segment diameters (ED) and flow velocities (EV) and the corresponding measured values (29) using a multidimensional optimization procedure [downhill simplex method (23)].

Further simulations were performed to examine the functional roles of the assumed adaptive responses by changing the sensitivity to different stimuli. In these simulations, the parameters kp and km were systematically varied from their baseline values. The other parameters were held fixed with the exception of ks. This parameter was adjusted so as to preserve the total blood volume in each network, permitting investigation of functional changes due to redistribution of volume within the networks. Furthermore, the energy consumption for maintenance of blood and vessel mass is held constant by this assumption, and only changes in viscous energy dissipation occur. The overall flow rate through a given network is also constant in these simulations (as in previous simulations), but the overall driving pressure is allowed to vary.

Functional network parameters. To characterize the functional state of the simulated networks under a given set of conditions, several additional variables were computed. The total viscous energy dissipation of blood flow in the network is equal to the product of segment flow with pressure drop, summed over all the segments in the network. The global oxygen deficit (O2,def) of the network is defined as
O<SUB>2,def</SUB><IT>=</IT><LIM><OP>∑</OP></LIM> (O<SUB>2,dem</SUB><IT>−</IT>O<SUB>2,del</SUB>)<IT>/</IT><LIM><OP>∑</OP></LIM> O<SUB>2,dem</SUB>
where O2,dem is oxygen demand and O2,del is oxygen delivery, and the sums are taken over all segments of a given network. An oxygen deficit occurs if the local PO2 in individual segments reaches zero. The flow path length of the network is equal to the flow-weighted mean of the lengths of all pathways for flow through the network. The root mean square variability of wall shear stress is evaluated with each segment weighted according to its flow rate and represents a measure of the departure of the network from the optimality condition defined by Murray (22). Relative average capillary pressure is defined as (capillary pressure - outflow pressure)/(inflow pressure - outflow pressure).


    RESULTS
TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
REFERENCES

Distributions of hemodynamic and metabolic variables. The parameters kp, km, kc, ks, L, J0, PO2,ref, and Qref were chosen to minimize the EV in two networks and ED in four networks. The averages of the final values are given in Table 1. Simulated diameter adaptation with these optimal parameter values yielded distributions of hemodynamic and metabolic parameters analogous to those obtained using vessel diameter values determined experimentally during intravital microscopy. Furthermore, the results obtained did not change if the adaptation process was started with identical diameters for all vessel segments or with randomized data sets instead of using experimentally determined diameters.

Examples are given in Figs. 3 and 4 that compare results after diameter adaptation with the initial values obtained with measured vessel diameters. In each case, the main features of the distributions are preserved although the predicted values show a somewhat reduced scatter. The distributions of wall shear stress and intravascular PO2 are given as functions of intravascular pressure in Fig. 3. The decline of average wall shear stress and PO2 with intravascular pressure is very similar for initial and adapted vessel diameters. However, the number of vessels in an intermediate pressure range with very low shear stress and PO2 is reduced substantially in the adapted state. Given that the initial state includes scatter resulting from both measurement errors (EV and ED) and physiological effects not included in the adaptation model, such a reduction in scatter in the adapted state is to be expected. The agreement between initial and final adapted states is better than that obtained using the simpler model of Pries et al. (29).


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  		Fig. 3.   Distributions of shear stress (top) and oxygen partial pressure (bottom) as functions of intravascular pressure for a network with 546 segments (162 arterioles, 167 capillaries, and 217 venules). Left: values calculated with vessel diameters measured by intravital microscopy. Right: values obtained after simulated vessel adaptation.



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  		Fig. 4.   Color-coded maps of calculated blood flow velocity for a network with 389 segments. Left: values based on experimentally determined vessel diameters. Right: values based on diameters obtained after simulated adaptation. Vessel diameters are doubled relative to scale for better visibility. Main feeding arteriole and draining vein are indicated by arrows. Area shown is ~5 mm × 5.5 mm.

Distributions of flow velocities estimated with the measured diameter values and with diameters obtained by the simulated adaptation are compared in Fig. 4 in a computer-generated map of a microvascular network with 389 vessel segments. Close similarity is seen between the measured diameters and those obtained after diameter adaptation and among the resulting flow velocity distributions. Thus simulated diameter adaptation according to the present model can lead to stable networks with realistic properties, suggesting that the model provides an adequate representation of the processes of controlling structural diameter changes in vivo.

A similar color-coded map is used to represent the distributions of wall shear stress, intravascular pressure, PO2, and the derived adaptation signals (combined hemodynamic signal, metabolic signal, and conducted signal) (Fig. 5). Intravascular pressure exhibits a continuous decline from the arteriolar inflow to the venular outflow, according to fundamental principles of fluid mechanics with low inertia. The highest shear stress values are not seen in the feeding arterioles but in short segments connecting high pressure arterioles with low pressure venules. The lowest values, in contrast, are found in longer segments connecting branch points at similar pressure levels. These segments also tend to exhibit low PO2 levels. However, balanced flow distribution to proximal and distal regions of the network appears to prevent an overall disparity in oxygen supply. The heterogeneity in the distributions of hemodynamic and metabolic parameters is reflected in the distributions of the derived adaptation signals, which taken together, have to balance ks in the fully adapted state.


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  		Fig. 5.   Maps of calculated parameters for the network shown in Fig. 4. Left: In the pressure-coded map, larger arterioles are red and larger venules blue; the small terminal segments, especially capillaries, are mostly green. Right: total hemodynamic signal elicited by shear stress and intravascular pressure, metabolic signal, and conducted signal.

Functional roles of adaptive responses. The functional effects of pressure response and conduction sensitivity are indicated in Fig. 6 in which the parameters kp and kc are varied, keeping total intravascular volume and blood flow constant. As expected (28), weakening kp (<100% of baseline level) causes an increase in mean capillary pressure. At the same time, energy dissipation in the network decreases markedly. This reflects the energy cost of requiring an asymmetric distribution of shear stress between venous and arteriolar parts of the network. The oxygen deficit remains low and is essentially unaffected by variation in kp. With variation of kc, capillary pressure and energy dissipation show an opposite behavior, which is that decreased conduction increases network asymmetry (decreased relative capillary pressure) and energy dissipation. At low levels of kc (<90% of baseline level) an oxygen deficit develops.


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  		Fig. 6.   Variations of functional network parameters in a network (546 segments) with changes of the sensitivity to intravascular pressure (top) and conduction sensitivity (bottom) from its optimized value (100%).

Figure 7 shows the effects of varying the strength of km. With decreasing km (<100%), the number of poorly oxygenated segments (oxygen deficit) increases steeply while energy dissipation decreases. The oxygen deficiency is also evident from the graphical representation of the same network in Fig. 8. This underperfusion of nutritional segments occurs despite maintained overall blood volume and bulk flow, and results from a redistribution of blood flow to shorter arteriovenous pathways that have developed into shunts. Reducing km also leads to a decrease in network asymmetry as evidenced by low relative capillary pressures (data not shown).


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  		Fig. 7.   Variations of functional parameters in a network (546 segments) with changes of metabolic sensitivity from its optimized value (100%). Top: oxygen deficit, viscous energy dissipation and capillary pressure as in Fig. 6. Bottom: flow-weighted mean path length through the network. Shear stress variability is defined as root mean square variation of wall shear stress.



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  		Fig. 8.   Map of PO2 in a mesenteric network. Left: results for optimized parameter values; very few segments have low PO2. Right: results with metabolic sensitivity reduced to 55% of the optimized value. Most of the arterial blood supply is shunted via short pathways through the network and large tissue areas contain only segments with low or near zero PO2.

Some insight into changes in network properties with decreasing sensitivity to metabolic signals is obtained by considering the variation of flow-weighted mean path length and shear stress variability (Fig. 7). With decreasing km, the path length shows an abrupt decline, representing a substantial redistribution of blood flow from longer to shorter pathways through the network. The variability in shear stress also declines, implying a closer approach to the condition of minimum energy dissipation [Murray's law (22)]. Conversely, increasing km leads to an increase in the variability in shear stress and in the energy dissipation in the network. However, no functional advantage is predicted for values of km above baseline, because mean path length does not increase significantly and nearly no further reduction in the oxygen deficit is seen.


    DISCUSSION
TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
REFERENCES

The present study shows that networks with stable, functionally adequate distributions of segment diameters can be generated if each segment adapts its diameter in response to local shear stress and pressure and to a metabolic stimulus derived from local oxygen availability. A key requirement is that information on the metabolic stimulus can be transmitted from one segment to other segments upstream and downstream of it in a given flow pathway. In a previous model (29), a metabolic signal was assumed that depended only on flow rate in the segments, and the transmitted response originated only in capillary segments. In the present model, more realistic assumptions are made, namely that the metabolic signal is directly related to a metabolite concentration (O2) (2, 6, 34, 44) and that the signal can be transmitted both upstream and downstream from any segment. Transmission of information upstream and downstream is postulated to occur by distinct mechanisms [conducted responses in vessel walls in the upstream direction (1, 37) and convective metabolite transport in the downstream direction (6, 43)] that are plausible based on observed mechanisms in microvascular networks, even though their role in vascular adaptation has not been directly demonstrated.

The network structures predicted by the model exhibit distributions of vessel diameters and hemodynamic parameters very similar to those obtained experimentally that can be assumed to represent a comparatively stable state generated by action of adaptive processes over an extended time period. Although this similarity does not prove the involvement of specific mechanisms in vivo, it suggests that the simulated adaptive reactions mimic biological processes occurring in living microvascular networks. Furthermore, the theoretical approach makes it possible to define basic requirements with respect to adaptive reactions to hemodynamic and metabolic stimuli and their quantitative relationships. Such information is difficult to obtain with standard experimental methods of vascular biology. For example, the model predicts that upstream and downstream information transfer along vessels plays an essential role in long-term adaptation of vessels to their environment, a possibility that has not been considered extensively in this context. Such findings suggest possible future directions for experimental investigations in vivo.

Nevertheless, this approach still has a number of inherent limitations. Although acute adaptive responses to wall shear stress and pressure (3, 4, 10 18), to metabolic state, and to conducted stimuli (6, 7, 33, 36, 38, 45) are well documented, less is known about structural adaptive responses to shear stress, pressure, and metabolic state (9, 13, 20, 41, 42), and structural adaptation to conducted stimuli has not yet been demonstrated experimentally. However, strong relationships between mechanisms and mediators regulating vascular tone and those involved in vessel growth or regression (28) support the assumption that acute conducted responses, if sustained, lead to structural adaptation (28).

In the model, a number of assumptions have to be made, for instance, regarding the mathematical forms of the vascular responses to the assumed stimuli. However, the main predictions of the model do not depend greatly on the specific forms chosen. For example, the formulation of the present model differs substantially from that of Pries et al. (29), e.g., by deriving metabolic signals from local PO2 and by implementing conduction and convection as mechanisms for upstream and downstream information transfer, respectively. Yet it leads to similar conclusions with respect to the fundamental requirements of structural adaptation. Metabolites other than oxygen may play a role in generating the metabolic signal, and mechanisms for its transmission upstream and downstream may be different from those assumed here. Further development of the model is likely to depend on the availability of additional experimental information about these aspects of the control systems in vivo. Such experiments may include perturbation of a quasi-stable microvascular network in vivo, generating stimuli for adaptation to changed conditions. This would make it possible to test the validity of the model and to include information on time dependence of adaptive processes.

For many years, discussions of the design of the vascular system have been strongly influenced by the concept of optimality with respect to low, overall "cost" (5, 19, 22, 39). In this context, the total energy cost of a vascular system was defined by Murray (22) as the sum of the viscous energy dissipation in the network and a term proportional to the total blood volume in the network. However, the functional requirements of vascular systems must, in principle, have priority over energy cost (28). These requirements include adequate supply of oxygen and substrates to the tissue and a low capillary pressure level for maintenance of fluid balance, and are described in the model by the oxygen deficiency and the relative capillary pressure, respectively. Meeting such requirements almost inevitably leads to total energy costs above the theoretical minimum.

The simulations in which the strengths of the adaptive responses were varied give some insight into the consequences for energy dissipation of the assumed responses (Figs. 6 and 7). In these simulations, the blood volume was held constant, and therefore, changes in total energy dissipation are equivalent to changes in total energy cost. The results show that minimization of energy dissipation in functional vascular beds is possible only within the constraints imposed by functional demands. In Fig. 6, reduction of the pressure response (kp) or strengthening the conducted response (kc) leads to decreased dissipation but at the expense of increased intracapillary pressure. Such an increase would lead to excessive fluid filtration and compromise the fluid balance of the tissue (21, 28). However, varying the pressure response has little effect on the oxygen deficit in the network.

Conversely, as shown in Fig. 7, reduction of the km also leads to a decrease in dissipation, but with a rapid increase in the number of segments receiving inadequate oxygen supply (Fig. 8), and a decrease in mean path length through the network, reflecting a loss of flow to longer flow pathways. It is striking that the km value that gives optimal agreement with the experimental data is only just large enough to avoid significant hypoxia and that further increases in this parameter lead to a rapid increase in energy dissipation. In this sense, the system does appear to exhibit optimal behavior, by setting the strength of the adaptive response at a level such that energy dissipation is minimized but functional demands are met. This suggests that the strength of the response may itself be controlled by a feedback mechanism. For instance, the expression of hypoxia-sensitive genes (17, 40) that promote vascular growth may be upregulated by hypoxia.

In summary, a theoretical model has been used to simulate structural diameter adaptation in microvascular networks. Reactions to the local signals of shear stress and pressure and to a metabolic signal derived from PO2 as well as to information transmitted both upstream (by conduction) and downstream (by convection) are included. When the strengths of these reactions are set appropriately, stable distributions of structural and hemodynamic parameters are predicted similar to those observed in vivo. Altering these strengths results in networks that either fail to meet the crucial functional requirements of oxygen supply to the tissue and low capillary pressure level allowing maintenance of fluid balance, or meet these requirements but with increased viscous energy dissipation. Thus not only are adaptive responses of the types assumed here necessary, but their relative strengths must be controlled to provide adequate functionality without excessive energy cost.


    ACKNOWLEDGEMENTS

The assistance of A. Scheuermann in preparing the manuscript for this article is gratefully acknowledged.


    FOOTNOTES

This study was supported by the Deutsche Forschungsgemeinschaft Grants Pr 271/5-4 and FOR 341/TP1 and by National Heart, Lung, and Blood Institute Grant HL-34555.

Address for reprint requests and other correspondence: A. R. Pries, Freie Universität Berlin, Dept. of Physiology, Arnimallee 22, D-14195 Berlin, Germany (E-mail: pries{at}zedat.fu-berlin.de).

1 A typographical error in the previously stated (see Ref. 29) formula has been corrected here.

The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.

Received 24 August 2000; accepted in final form 30 April 2001.


    REFERENCES
TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
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Am J Physiol Heart Circ Physiol 281(3):H1015-H1025
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