Am J Physiol Heart Circ Physiol
281: H1015–H1025, 2001.
Structural adaptation of microvascular networks:
functional roles of adaptive responses
A. R. PRIES,1,2 B. REGLIN,1 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
Received 24 August 2000; accepted in final form 30 April 2001
shear stress; pressure; conducted response; oxygen transport; mathematical modeling
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 longterm 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 structur-
Address for reprint requests and other correspondence: A. R. Pries,
Freie Universität Berlin, Dept. of Physiology, Arnimallee 22,
D-14195 Berlin, Germany (E-mail: [email protected]).
http://www.ajpheart.org
ally 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 strucThe 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.
0363-6135/01 $5.00 Copyright © 2001 the American Physiological Society
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Pries, A. R., B. Reglin, and T. W. Secomb. Structural
adaptation of microvascular networks: functional roles of
adaptive responses. Am J Physiol Heart Circ Physiol 281:
H1015–H1025, 2001.—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.
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VASCULAR ADAPTATION AND NETWORK FUNCTION
METHODS
tures 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
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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)
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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.
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.
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.
VASCULAR ADAPTATION AND NETWORK FUNCTION
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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).
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.
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
(⌬D) in each segment as described in the next section. The
segment diameters were then incremented by an amount
⌬D ⫽ Stot 䡠 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
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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 䡠 kg⫺1 䡠
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 ␮m). The inner
diameters of the arterioles feeding and the venules draining
the networks ranged from ⬃25 to 35 and 45 to 65 ␮m,
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
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VASCULAR ADAPTATION AND NETWORK FUNCTION
Table 1. Mean parameter values and standard deviations for the 6 experimental networks obtained by
minimizing the flow velocity error or segment diameter error
Means
SD
kp
km
kc
ks
L, ␮m
J0
␶ref,
dyn/cm2
Qref,
nl/min
0.68
0.04
0.70
0.06
2.45
0.26
1.72
0.15
17,300
2,020
27.9
7.4
0.103
0.004
0.198
0.030
For all networks PO2,ref was set to 93.2 mmHg. kp, km, and kc, scaling parameters for sensitivity to pressure, metabolic signal, and
conducted signal, respectively; ks, basal shrinking tendency of vessels; L, length constant; J0, saturation constant for conducted signal; ␶ref
and Qref, reference values for shear stress and blood flow, respectively.
S ␶ ⫽ log 共␶w ⫹ ␶ref兲
where ␶w is the actual wall shear stress in a vessel segment
calculated from pressure drop, vessel diameter, and vessel
length, and ␶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 p ⫽ ⫺log ␶e(P)
where ␶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
␶ e(P) ⫽ 100 ⫺ 86 䡠 exp{⫺5,000 䡠 关log (log P)]5.4}
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
1
A typographical error in the previously stated (see Ref. 29) formula has been corrected here.
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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 ␮m wide and 20 ␮m 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 /(␮m ⫻ min). Blood was assumed to enter the network
with a PO2 of 95 mmHg. The oxygen flux of flowing blood was
calculated as Q̇ ⫻ CO ⫻ HD ⫻ S(PO2), where Q̇ 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 segdown
up
ment, according to J m
⫽ Jm
⫹ 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/Q̇), and is given by Sm ⫽
log[1 ⫹ Jm/(Q̇ ⫹ Q̇ref)] where Q̇ref, 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
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using the diameter values determined experimentally, identical diameters (10 ␮m) 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.
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). S␶ is the growth stimulus derived from shear stress
according to
VASCULAR ADAPTATION AND NETWORK FUNCTION
S tot ⫽ S␶ ⫹ kpSp ⫹ km[Sm ⫹ kcSc] ⫺ ks
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 Q̇ref), 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 2,def ⫽
兺 (O
2,dem
兺O
⫺ O2,del)/
2,dem
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
AJP-Heart Circ Physiol • VOL
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
Distributions of hemodynamic and metabolic variables. The parameters kp, km, kc, ks, L, J0, PO2,ref , and
Q̇ref 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).
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
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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
down
is related to that at the downstream end by J up
⫹
c ⫽ (J c
Sm) 䡠 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
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VASCULAR ADAPTATION AND NETWORK FUNCTION
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.
Functional roles of adaptive responses. The functional effects of pressure response and conduction sen-
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.
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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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VASCULAR ADAPTATION AND NETWORK FUNCTION
sitivity 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
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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.
Figure 7 shows the effects of varying the strength of
km. With decreasing km (⬍100%), the number of poorly
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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.
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VASCULAR ADAPTATION AND NETWORK FUNCTION
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
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%).
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).
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
AJP-Heart Circ Physiol • VOL
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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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
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VASCULAR ADAPTATION AND NETWORK FUNCTION
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.
AJP-Heart Circ Physiol • VOL
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
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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
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VASCULAR ADAPTATION AND NETWORK FUNCTION
The assistance of A. Scheuermann in preparing the manuscript
for this article is gratefully acknowledged.
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.
AJP-Heart Circ Physiol • VOL
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