A theoretical
model was developed to simulate long-term 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 (
w),
intravascular pressure (P), a flow-dependent metabolic stimulus (M),
and a stimulus conducted from distal to proximal segments along
vascular walls (C). Increases in
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
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.
 |
INTRODUCTION |
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 |
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
fat-free 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 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
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 similar-sized 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, and
3), 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
phase-separation 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
(
D) for a time step
t was assumed to be proportional to
the sum of terms representing different adaptive stimuli
(Stot) and to the vessel
diameter (D) as
|
(1)
|
As outlined in RESULTS, contributions
of the adaptive stimuli to Stot
were expressed in terms of hemodynamic variables calculated in the
simulation of network blood flow. They were based on consideration of
three simple hypothetical 22-segment network structures.
Network 1 has symmetric topology as
evidenced by the equal "generation number" for all capillaries.
The generation number of a vessel segment is defined as the number of
branch points on the arterial (for arterioles and capillaries) and
venous (for venules and capillaries) pathways from the respective main
feeding vessel to the given segment. In addition,
network 1 exhibits regular morphology
(all segments with the same generation number have identical length and
diameter). Network 2 has symmetric
topology and irregular morphology, and network
3 has asymmetric topology and irregular morphology
(Fig. 1).

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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).
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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
Stot (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 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,
Stot. The final distributions of
hemodynamic parameters, when all vessel segments reached equilibrium
(Stot = 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. 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
(EV), i.e., the root mean square
deviation of the predicted segment flow velocities
(Vp) from the
measured velocities
(Vm)
Sensitivity analyses were performed to determine the dependence of
EV on model parameters. For
network III, velocities were not
measured, so the root mean square diameter deviation
(ED) between predicted diameters
(Dp) and
measured diameters
(Dm) was minimized where
 |
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
|
(2)
|
where
w is the wall shear stress in
the segment. Logarithmic dependence was introduced to achieve constant
sensitivity of Stot 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.

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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.
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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
Stot was included in the model,
dependent on the volume flux of red blood cells passing through the
segment (represented by
HD, where
is the blood volume flow and
HD is the discharge hematocrit)
and increasing with decreasing flux. The resulting total stimulus was
represented in the model by
|
(3)
|
The functional form given in Eq. 3 was
chosen so that the metabolic stimulus is always positive and increases
with decreasing red cell flux. The value of the constant
km reflects both
the sensitivity of the tissue metabolic state to blood flow and the adaptive response of vessel diameter to changes in metabolic state. Changes in the functional state of the tissue would be expected to
influence the value of
km, but a
constant value was assumed here. For flux levels below the reference
flow
(
ref),
which was assumed to be larger than actual fluxes in most segments, the
relation between the red cell flux and the resulting metabolic component of the stimulus is logarithmic. In this range,
km gives the
increase of the metabolic stimulus (M) for a decrease of the red cell
flux by one order of magnitude. Inclusion of a positive metabolic
stimulus tends to drive all diameters to large values. A further
constant, ks
("shrinking tendency"), was therefore introduced in
Eq. 3, which is subtracted from the
hydrodynamic and metabolic terms. The shrinking tendency can be
interpreted as reflecting the basal need of vessels for factors
stimulating growth to maintain or increase their cell mass and
diameter. In the present context, these factors are positive
hydrodynamic, metabolic, or conducted stimuli (see
Conducted stimulus). Atrophy or
degradation in the absence of positive tropic stimuli is generally seen
in organs, e.g., denervated muscle, and in cell cultures if no
growth/survival factors are added to the culture medium.
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
w is
compensated by the increasing metabolic stimulus. The analysis in
APPENDIX A 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 that
km > 1/4. For a pair of segments in parallel, supplied by a
constant-pressure 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 A.
The behavior according to this model of a network with irregular
morphology is illustrated in Fig. 1C,
with km = 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 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.

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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.
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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 long-term adaptation, the "conducted stimulus" (C).
The effect of including such a stimulus in the model is shown in Fig.
1D. 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 (Sc) was assumed
to be the sum of the metabolic stimuli (Table
1) of the two downstream vessel segments (segments a and
b) 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
where
subscripts a and
b denote the two daughter vessels. At
the most distal branch points, whose downstream bifurcations lie within
the venous tree, Sc = Ma + Mb. The same procedure was used
proceeding downstream from the most distal branch points in the venular
tree. Possible nonlinearity of the summation was accounted for by a
saturable response, with a reference value
(S0), giving
|
(4)
|
where
kc is a conducted
stimulus constant.
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 low-generation capillaries in a given area and could not prevent
the development of low-generation capillaries into large arteriovenous
shunts.
The inclusion of responses to these four types of stimuli
(
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 parameters
km,
kc, and
ks were chosen to
minimize EV in
networks I and
II and
ED 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 EV, yielding
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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.
ED 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).

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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.
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For networks I and
II, in which velocity was measured,
the initial relative EV 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 (
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 (Stot, 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, Stot is
zero in every segment, as required by convergence of the model.

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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.
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Sensitivity analysis.
To test the sensitivity of the results to the most important parameters
(ks,
km, and
kc), these
parameters were varied around the values obtained by minimizing
EV. Results for
networks I and II (Fig.
6) showed that
EV is highly dependent on the
shrinking constant,
ks, with a
narrow, well-defined minimum. When
ks 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 ks, yielding
minimal EV, pressure drops in the
realistic range were predicted: 69 mmHg for the larger network, and 37 mmHg for the smaller. In further sensitivity tests,
ks was optimized
with respect to minimal EV for
each value of km
and kc tested.
The level of both the metabolic and conducted stimuli depends on
km, because it
was assumed in the model that the conducted response originates from
the metabolic stimuli of individual segments. Therefore,
km determines the
balance between these two stimuli and the hydrodynamic stimuli. The
balance between metabolic and conducted stimuli is set by
kc. Figure 6
shows results of varying km and
kc. The resulting
minima of EV were much broader
than those for
ks. Reasonable
agreement between model predictions and observations was obtained over
a range of about ±20-30% in these parameters.

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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.
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|
 |
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, 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.
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
if'(
i) +
ig'(
i) > 0 in all segments. In the simulations, this condition is satisfied
if km < 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 minimum-dissipation argument, completing the proof.
The introduction of a pressure-dependent 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 flow-dependent metabolic stimulus as described here
leads to stability even in the presence of a pressure-dependent
response.
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.
This study was supported by Deutsche Forschungsgemeinschaft Pr 271/1-1,
1-2, 5-1, and 5-2 and by National Heart, Lung, and Blood Institute
Grant HL-34555.
Address for reprint requests: A. R. Pries, Freie Universität
Berlin, Dept. of Physiology, Arnimallee 22, D-14195 Berlin, Germany.
Received 1 December 1997; accepted in final form 6 April 1998.