Vol. 279, Issue 4, H1645-H1653, October 2000
Model of structural and functional adaptation of
small conductance vessels to arterial hypotension
Christopher M.
Quick1,
William L.
Young1,2,
Edward
F.
Leonard3,
Shailendra
Joshi5,
Erzhen
Gao4, and
Tomoki
Hashimoto1
1 Department of Anesthesia and Perioperative Care and
2 Departments of Neurosurgery and Neurology, University of
California San Francisco, San Francisco, California 94110; and
3 Departments of Biomedical Engineering and Chemical
Engineering, 4 Department of Electrical Engineering, and
5 Department of Anesthesiology, Columbia University, New York,
New York 10032
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ABSTRACT |
Vascular networks
adapt structurally in response to local pressure and flow and
functionally in response to the changing needs of tissue. Whereas most
research has either focused on adaptation of the macrocirculation,
which primarily transports blood, or the microcirculation, which
primarily controls flow, the present work addresses adaptation of the
small conductance vessels in between, which both conduct blood and
resist flow. A simple hemodynamic model is introduced consisting of
three parts: 1) bifurcating arterial and venous trees,
2) an empirical description of the microvasculature, and
3) a target shear stress depending on pressure. This simple
model has the minimum requirements to explain qualitatively the
observed structure in normotensive conditions. It illustrates that flow
regulation in the microvasculature makes adaptation in the larger
conductance vessels stable. Furthermore, it suggests that structural
changes in response to hypotension can account for the observed
decrease in the lower limit of autoregulation in chronically
hypotensive vasculature. Independent adaptation to local conditions
thus yields a coordinated set of structural changes that ultimately
adapts supply to demand.
mathematical modeling; autoregulation; hemodynamics; instability
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INTRODUCTION |
EVIDENCE
EXISTS THAT in the long term, shear stress is regulated in both
the microcirculation and in the conductance vessels (7, 10, 14,
25, 27). This apparent regulation raises two problems (4,
20). First, when vessel lumens grow smaller in response to
decreased shear stress, an instability may arise wherein a vessel with
too small a lumen might obstruct its own flow. In this case, a decrease
in vessel radius leads to a vicious cycle of ever-decreasing radius and
shear stress until the vessel completely closes. Second, local changes
in the vessel lumen affect not only the local endothelial shear stress
but also the pressures, flows, and shears in neighboring vessels. When
two vessels are in parallel, a small imbalance could signal one to
start increasing its lumen and the other to start decreasing its lumen.
The larger vessel "steals" blood flow from the smaller, thus
increasing its own shear stress and stimulating its own growth.
Logically, this process would continue until the smaller vessel closes
completely. Because of these two instabilities, it has been concluded
that shear stress is not sufficient to control the growth of vascular networks (4, 20).
The theoretical work of Pries et al. (20) has
helped resolve this issue for the microvasculature. They used a
mathematical model to explain how a complex interaction of various
stimuli can lead to vascular growth and adaptation. Four separate
stimuli that increase vessel lumen were necessary to yield a stable
vascular network with recognizable structural and functional
properties. First, to set shear stress at an appropriate value, a
stimulus was assumed that increases with shear stress. Second, to
prevent the instabilities mentioned above, a metabolic stimulus was
assumed that increases when blood flow is inadequate. Third, to produce arteriovenous assymetry (veins larger than arteries), a
pressure-dependent stimulus was assumed to increase in response to low
pressure. All three proposed stimuli have been observed to affect
actual vessels. A final stimulus was hypothesized that prevents the
formation of large proximal shunts, but the mechanism has yet to be
identified (20).
The knowledge gained from this powerful approach is of limited
use for interpreting the structural and functional adaptation of
conductance vessels. First, few metabolism-related stimuli are known to
act directly on the conductance vessels (5, 17). Thus this
mechanism may be insufficient to ensure stability. Adaptation to
chronic hypotension presents an additional problem. A decrease in
perfusion pressure below the lower limit of autoregulation (LLA) is
expected initially to decrease flow (9). From current knowledge of adapting vessels, a decrease in flow is expected to
decrease shear stress and ultimately lead to a smaller radius (7,
10, 14, 25). This mechanism would thus theoretically increase
resistance of the small conductance vessels in direct opposition to the
observed decrease in resistance in vessels rendered chronically
hypotensive by a nearby shunt (30).
This work proposes a simple model containing the minimum attributes
necessary to explain the structural and functional adaptation of the
small conductance vessels.
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THEORY AND METHODS |
Model criteria.
Given the problems described above, the properties of an ideal model
can be enumerated. An ideal model should 1) be consistent with the fundamental principles governing blood flow, 2) be
grounded on identifiable physiological mechanisms, 3) yield
networks predicted to be stable over time, 4) generate a
structure with dimensions consistent with those observed, and
5) explain functional adaptation to chronic arterial hypotension.
Hemodynamics of a single vessel.
Vessel radius determines not only the quantity of blood flowing through
a vessel but also its axial pressure gradient. The profound influence
of radius is best expressed by the effect of the radius on vascular
resistance [R(r)], which is the ratio of pressure gradient (
P) to flow (Q). Although limited by a number of
assumptions (16), Poiseuille's Law can predict resistance from the vessel radius (r), length (L), and
viscosity (
)
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(1)
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Modulation of resistance is accomplished through changes in
arterial radius; smooth muscle contraction and relaxation provides acute control, and vascular growth and remodeling provides long-term adaptation.
Besides determining vascular resistance, radius also determines shear
stress. Shear stress (
) is the frictional force acting on the
endothelium due to the flow of blood. It can be calculated from known
values of r, , and Q, given the same assumptions required for Poiseuille's Law (16)
|
(2)
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Equivalently, endothelial shear stress can be expressed in terms
of radius and pressure gradient (11)
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(3)
|
Equation 3 directly follows from substituting
Eq. 1 into Eq. 2. As long as Poiseuille's Law is
followed in a vessel of specified radius, Eqs. 2 and 3 are equivalent, and both must be satisfied. Equations 2 and 3 are adequate for calculating
the shear stress from a measured radius when either P or Q are
artificially kept constant in vitro. However, in vivo, changes in
radius affect both pressure and flow.
Adaptation of vessels in a vascular network.
To address this issue, the adaptation of a vessel in a vascular network
can be explored. To simplify, an entire vascular network surrounding a
particular vessel of interest is considered passive, linear, and
assumed not to adapt to changes in pressure and flow. Instead of
including the entire complexity of the network, a simplified model can
be used to functionally mimic a network. Following the procedure of linear circuit analysis (12), a complex
vascular network can be reduced to three elements: the vessel of
interest, a pressure source, and a source resistance (Fig.
1A). The vessel of interest
has resistance R(r), governed by Eq. 1. The flow and pressure gradient across the vessel of interest
are represented by Q and P. The pressure source (Pin) in
conjunction with a source resistance (Rs)
functionally represents all the vessels proximal, distal, and parallel
to the vessel of interest. As described in texts describing linear
circuit analysis, any linear network can be reduced to these elements
(12).
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Fig. 1.
A: representation of a vessel supplied by a
vascular network. Any passive, time-invariant network can be
represented by a Thevenin Equivalent (12) consisting of a
pressure source (Pin) and an internal resistance
(Rs). P, pressure gradient; Pout,
pressure at end of vessel; Q, flow. B: normalized shear
stress (') in the vessel as a function of normalized radius
(r'). Shear stress is bimodal, having a maximum shear stress
(max) at the maximum radius
(rmax). Dashed lines represent Eq. 2
(right dashed line) and Eq. 3 (left dashed line) normalized
by max.
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Shear stress in a vessel in vivo can be calculated from the reduced
model shown in Fig. 1A. For simplicity, it is first assumed that r does not influence the radii of other vessels
(4). Flow is calculated by dividing the total pressure
(Ptot = Pin 
Pout, where Ptot is the total pressure and Pout is
the pressure at the end of the vessel) by the sum of
Rs and R(r)
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(4)
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Substituting Eq. 4 and 1 into Eq. 2 yields the endothelium shear stress
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(5)
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which depends on the vessel properties (radius and length),
blood properties (viscosity), and properties of the system in which the
vessel is embedded (Rs and Ptot).
The latter is what makes shear stress in vivo different from shear
stress measured in vitro (Eq. 2) (4).
In this model, shear stress is a bimodal function of radius; shear
stress first increases and then decreases with radius. Shear stress has
a maximum value (max) at a particular radius (rmax). These values can be determined by
differentiating Eq. 5 with respect to r, setting
the result equal to 0, and solving for and r
|
(6)
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Because it is not clear from inspection of Eq. 5 how
particular values of , Rs, L, and
Ptot influence (r), r and are normalized (yielding r' and ', respectively)
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(7)
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' (plotted in Fig. 1B) has a maximum value at
r' equal to 1. Nondimensionalization allows representation
of a function independent of particular parameter values (i.e., ,
Rs, L, and Ptot)
(28). Equation 3 approximates the shear-radius
relationship for small r', whereas Eq. 2
approximates the model behavior for large r'. The model is
necessary to describe the transition between these asymptotes.
Adaptation to shear stress.
Numerous investigators (7, 10, 14, 19, 26) suggest that
vessel radii adapt chronically so that shear stress maintains a value
within a particular range to prevent atherosclerosis or endothelial
damage. In other words, shear stress is regulated at a particular
value, referred to here as a "target shear stress" (*). If shear
stress is greater than the target value, the vessel lumen increases. If
shear stress is less than the target value, the lumen decreases.
Although the particular feedback mechanism can take many forms,
analysis of the simplest case is instructive; the change in lumen
radius (r) is set proportional to the difference between
the actual shear stress () and *. A proportionality constant
(K) represents the sensitivity
|
(8)
|
When = *, then a vessel is in equilibrium and there
will be no adaptation. If < *, r is
positive. If > *, r is negative. Using these
two conditions as a guide, two types of instability can be explored
(4, 20).
Vessel instability.
First, there is an instability that arises in a vessel when no other
vessels adapt (all other radii are kept constant) (4). For
illustrative purposes, * = 0.2max is plotted on
the same graph as ' (Fig.
2A). As revealed by Fig. 2,
long-term adaptation introduces a difficulty; there are two possible
radii that can result in the same *. One radii
(ra) is small; the other radii (rb) is large.
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Fig. 2.
A: stability of a single vessel in a passive,
nonadapting vascular network. Arrows indicate how radii change when
system in disequilibrium. Dashed portion of curve represents unstable
radii. Point a is an unstable equilibrium; point
b is a stable equilibrium. B: stability of two
adapting vessels in parallel. The resulting radii are indicated by the
solid squares. Points a' and b'
represent equilibria where the shear stress in vessel 1 = shear stress in vessel 2 = target shear stress
(*). The arrows point away from both a' and b'
(b' is a saddle point), indicating that neither is stable.
Q1, flow in vessel 1; Q2, flow in
vessel 2; r'1, normalized
radius of vessel 1; r'2,
normalized radius of vessel 2;  ,
Degeneration to zero when radius is too small.
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The control mechanism described by Eq. 8 implies behavior
that is sensitive to the initial radius of the vessel. Following a
previously described convention (4, 22), the behavior of the vessel in disequilibrium (in the process of adapting) is indicated by the direction of the arrows in Fig. 2. For instance, if the initial
radius were in the neighborhood of rb, the
vessel radius would be stimulated to grow larger or smaller until the
final radius settled at rb. Any perturbation of
this equilibrium radius would cause the radius to return to
rb. The equilibrium radius rb is thus recognized to be stable. However, if
the radius were initially at ra, a small
decrease in radius would initiate a growth in radius from
ra to zero. In contrast, a small increase in
radius would initiate growth in radius from ra
to rb. Thus ra is
recognized to be unstable. All radii < rmax are similarly unstable (as illustrated by
the dashed portion of the curve in Fig. 2A). This type of
instability arises when the radius of the vessel effects the axial
pressure gradient of the vessel or blood flow through the vessel (i.e., Rs 
0) (4).
Another type of instability arises when two vessels in parallel adapt
concurrently (4). In this case, the shear stress in
vessel 1 (1) depends on its own radius
(r1) as well as the radius of vessel
2 (r2). As in Eq. 4,
1 and the shear stress of vessel 2 (2) can be derived by applying
Eqs. 1 and 2 and setting the total
inflow equal to the sum of flows through the two vessels (16). The resulting shear stress then can be
normalized by rmax and max
(compare with Eq. 7)
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(9)
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From Eq. 9, it can be shown that when the lumen of
one vessel increases, the shear stress in the other vessel decreases. If the feedback mechanism described by Eq. 8 is operative,
then a sequence of reactions can become a vicious cycle. For instance, if 1 and 2 are initially equal to *,
then the system would be in equilibrium. However, if
r1 were initially slightly larger than
r2, then 1 would be greater than
2. This would stimulate r1 to
grow larger in an attempt to lower . In response, 2
decreases, stimulating r2 to decrease. This
further increases 1, and so on. The result of this
process is depicted in Fig. 2B (arrows). Note that there are
only two cases where 1 = 2 = *. These two equilibria (a' and b') correspond
to the two equilibria identified in Fig. 2A [i.e., the
stable (a) and unstable (b)
equibilibria]. However, neither a' nor
b' are stable. This system comes to rest only when one of
the two radii degenerates to zero.
These instabilities are inconsistent with the observed stability of
extant vascular networks. However, to describe these instabilities, it
was assumed that either one or two vessels adapt and that all other
vessels remain constant. In actual arterial beds, the microvasculature adapts, modulating pressure and flow.
Adaptation of the microvasculature.
Although the structure of the microvasculature is quite complicated, it
is functionally simple. As in Gao et al. (3), it will be
treated as a "black box" and referred to as a microvascular group
(MVG). The relationship of flow to perfusion pressure is assumed to
have the form shown in Fig. 3. Flow is
regulated (at a value defined as QAR) when perfusion
pressures are between the lower limit of microvascular autoregulation
(µLLA) and the upper limit of microvascular autoregulation (µULA).
Below the µLLA and above the µULA, the system becomes passive. The
resistance of the MVG (RMVG) is nonlinear and
depends on the value of P
![R<SUB>MVG</SUB>(<IT>&Dgr;</IT>P)<IT>=</IT><FENCE><AR><R><C><IT>&mgr;</IT>LLA<IT>/</IT>Q<SUB>AR</SUB><IT>, &Dgr;</IT>P<IT><&mgr;</IT>LLA</C></R><R><C><IT>&Dgr;</IT>P<IT>/</IT>Q<SUB>AR</SUB><IT>, &mgr;</IT>LLA<IT>≤&Dgr;</IT>P<IT>≤&mgr;</IT>ULA</C></R><R><C><IT>&mgr;</IT>LLA<IT>·&Dgr;</IT>P<IT>/</IT>[Q<SUB>AR</SUB>(<IT>&Dgr;</IT>P<IT>+&mgr;</IT>LLA<IT>−&mgr;</IT>ULA)]<IT>, &Dgr;</IT>P<IT>>&mgr;</IT>ULA</C></R></AR></FENCE>](/content/vol279/issue4/fulltext/H1645/img010.gif)
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(10)
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Equation 10 corresponds to a "Type 3"
empirical description of autoregulation delineated in Gao et al.
(3).
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Fig. 3.
Representation of simple functional model of a
microvascular group (MVG). Between the microvascular lower limit of
autoregulation (µLLA) and the upper limit of autoregulation (µULA),
flow is maintained at a value QAR. Above and below these
limits, the system becomes passive, and flow increases linearly with
pressure.
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In conventional experimental settings, autoregulation is characterized
in a vascular bed by measuring perfusion pressure and the tissue
perfusion in regions served by relatively large arteries (17). From such measurements, the LLA and the upper limit
of autoregulation (ULA) are characterized. However, pressure is
measured proximal to the small conductance arteries, which introduce a pressure drop between the point of observation and the
microcirculation. Thus µLLA and µULA, describing autoregulatory
limits in the distal microvasculature, are less than LLA and ULA,
measured in proximal arteries. If the values of LLA and ULA are known
and the structure of the small conductance vessels are specified,
the values of µLLA and µULA can be calculated.
Stability of vessels with autoregulatory MVGs.
The effect of the MVGs on adaptation of conductance vessels is explored
in Fig. 4. An MVG is placed in series
with the conductance vessels shown in Fig. 2. If all of the radii
of the small conductance vessels are greater than
rmax, then they are stable. Pries et al.
(20) explained how metabolic stimuli result in the stable adaptation of microvascular networks themselves. The MVGs may therefore
contain vessels smaller than rmax (Fig.
4A), because they are assumed to be influenced by metabolic
stimuli. The small conducting vessels, without direct metabolic
stimuli, are larger than rmax and do not exhibit
the instability illustrated in Fig. 2A.
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Fig. 4.
Illustration of enhanced stability when vessels are in series with
a MVG. A: very small radii are described by MVG; larger
radii are stable. B: with two vessels in parallel, MVGs
produce one stable radius at b'. As indicated by the arrows,
a' is unstable and b' is stable. If either radius
is too small, it degenerates to zero (indicated by ).
Compare with Fig. 2.
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Furthermore, the addition of autoregulating MVGs makes two vessels in
parallel conditionally stable. The derivation of shear stress in the
system (shown in Fig. 4B) follows that of Eq. 9, with flow through each branch modified by
RMVG(P). For illustrative purposes, it is not
critical which parameter values are chosen for this simple model.
However, to permit a convenient comparison with a more complicated
model presented below (Fig. 5), the
following parameter values were chosen: Pin = 100 mmHg, Pout = 0 mmHg, L = 0.031 cm,
µLLA = 17 mmHg, µULA = 117 mmHg, QAR = 7.8 µl/min, and Rs = 1.01 · 109g cm
4 s1. In
particular, Rs was chosen such that the
combination of the vessel of interest, Rs and
Pin, behave like the more complicated model in Fig. 5. As
in Fig. 2B, Fig. 4B (arrows) indicates how radii
change when the system is in disequilibrium (in the process of
adapting). The addition of MVGs makes one of the two equilibria (Fig.
5B, point b') stable. Thus, if both radii are in
the neighborhood of b', they both will converge on
b'. However, if either radius is too small, it degenerates
to zero, as in Fig. 2B.
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Fig. 5.
A: simple model of vascular network. Arteries and veins
bifurcate in N = 6 generations, where N is
the number of generations of an arterial tree. Dashed portions
represent MVGs described in Fig. 3. B: shear-pressure
relationship given by Eq. 14 results in structural
asymmetry. Radii and lengths are proportional to numerical values
generated by the model, but the branching angles are arbitrary.
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Simple model of a vascular network consisting of small
conductance vessels.
To construct a model that delineates the minimum set of attributes for
a viable vascular network, several conditions must be met. First, the
model must be include arteries in series and parallel. Second, it must
have a limited set of parameters, ensuring that the behavior of the
model can be readily related to the constitutive properties of the
model. Third, the model must reduce mathematically, so that a network
with a large number of vessels can be described by a small number of equations.
The model illustrated in Fig. 5A was designed to fulfill
these criteria. It consists of a bifurcating arterial tree with
N generations. Within a particular generation
(n), all arteries have the same lengths
(Ln) and radii (rn). This
structural similarity results in hemodynamic similarity, wherein the
pressure (Pn), resistance
(Rn), and shear stress
(n) are the same in all vessels of a common
generation n (n = 0 ... N-1). The total flow through each generation is
the same as the input flow (Qo). The values of
Rn and the flow within vessels of a common
generation n (Qn) can be
calculated from R(r) and Qo. To
simplify, Ln is halved in each generation
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(11)
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The value of the radius of vessels of a common
generation n (rn) can be
calculated from Eqs. 2 and 11.
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(12)
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The pressure drop across each generation can then be calculated
from Eqs. 1 and 12.
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(13)
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Terminating the vessels of the arterial tree are the MVGs
described by Eq. 10. The MVGs form the entrance to a
symmetrical, bifurcating venous tree. To distinguish between arteries
(A) and veins (V), generations are denoted as A0
... AN-1 and V0 ... VN-1.
Shear-pressure relationship.
To calculate the resistance, radii, and pressures in the distributed
vascular network described above, the shear stress of vessels in a
common generation n (n)
must be specified. When fully adapted, n will
equal *. In general, * in the high-pressure arteries is higher
than * in the low-pressure veins. Pries et al. (19)
found a sigmoidal relationship of shear stress and pressure in vessels
with a radius of 5 to 55 µm. They fit an empirical equation, *(P),
to this data (20, 21)
![&tgr;*(P)<IT>=</IT><FENCE><AR><R><C><IT>14, </IT>P<IT>≤10 </IT>mmHg</C></R><R><C><IT>100−86·</IT>exp(−<IT>5,000·</IT>{log [log (P)]}<SUP><IT>5.4</IT></SUP>)<IT>,</IT></C></R><R><C>P<IT>≥10 </IT>mmHg</C></R></AR></FENCE>](/content/vol279/issue4/fulltext/H1645/img014.gif)
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(14)
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For the present purposes, *(P) is assumed applicable for the
small conductance vessels leading to (and from) the microvasculature. P
is taken to be the average of input and output pressures of each
individual vessel. Although Eq. 14 is employed to describe vessels larger than those to which *(P) was originally fit, it is
nonetheless expected to capture the essential behavior of the small
conductance vessels.
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RESULTS |
Structure of small conductance vessels at normotensive pressures.
Four basic equations, which were introduced above, were used to
construct a simple model of a vascular network. Equations 11 and 12 specified the structure of the adapted arterial
and venous trees. Equation 10 represented the
microvasculature. Equation 14 specified the target shear
stress to which each vessel adapts. Equation 13 specified
the resulting pressures. Equation 8 suggested a method for
the vessels to adapt, although Eqs. 10-14 can be solved directly without assuming a particular adaptive mechanism.
The resulting structure of the model depends on the particular
parameter values. In an actual network, there is a large variation in
lengths and radii of vessels within a single generation. It would
therefore be misleading to try to assign specific physiological values
to them. For illustrative purposes, the following parameter values were
chosen: Pin = 100 mmHg, Pout = 0 mmHg, input length (Lo) = 1 cm, µLLA = 17 mmHg, µULA = 117 mmHg, and Qo = 250 µl/min. The values of Qo and
Lo were chosen to illustrate a vascular network branching off a small artery. The values of µLLA and µULA were chosen to yield values of LLA and ULA of 50 and 150 mmHg
(3). Figure 5B represents the resulting
structure of a network presented with an assumed normal pressure of 100 mmHg.
As can be expected from the present theoretical development, the total
number of generations described by this model is limited. When
N is set too large, the radii of the smallest vessels become less than rmax, resulting in unstable
adaptation. However, in the present model, vessels with r'
<1 are described by MVGs (Eq. 10). This boundary is
illustrated in Fig. 4A.
Because all the vessels in a generation are assumed identical, the
second type of instability (illustrated in Fig. 4B) is not
directly investigated. However, the value of Rs
in Fig. 4B was chosen so that the reduced model mimics the
spatially distributed model in Fig. 5. Figure 4B illustrates
how two parallel vessels in the last arterial generation
(generation A5 in Fig. 5) are expected to adapt.
Unless one of the radii is initially very small, they will exhibit
stable adaptation, yielding the network in Fig. 5B.
Structural adaptation to chronic hypotension.
To explore the process of adaptation in chronic hypotension, the model
in Fig. 5B is allowed to adapt to an input pressure of
Pin = 35 mmHg. It is assumed that the µLLA and
µULA of the MVGs remain fixed. As illustrated in Fig.
6A, the process of adaptation does not alter venous radii. However, hypotension causes arterial radii
to dilate appreciably (>29% in generation A0
and >50% in generation A5). The cause of this
dilation can be identified by considering the shear stress before and
after adaptation (Fig. 6B). As arterial pressure falls, the
target shear given by Eq. 14 decreases. Confronted with a
lower target shear stress, the arteries are stimulated to dilate
(Eq. 12).
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Fig. 6.
A: structural adaptation of bifurcating a
vascular network to chronic hypotension. Shear-pressure relationship
given by Eq. 14 causes small arteries to dilate despite the
decreased shear stress (Eq. 12). B: shear stress
in normal and hypotensive vascular networks after adaptation.
An and Vn, arteries and
veins, respectively, where n is the number of a given
generation.
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Functional adaptation to chronic hypotension.
In the normotensive case, pressure falls from Pin = 100 mmHg at the entrance of the vascular bed to Pin = 69 mmHg at the MVG (Fig. 7A).
If the model with the structure described in Fig. 5B is
perfused at 35 mmHg, then the pressure into the tree would fall 65%.
In acute hypotension, the MVGs would be perfused with a pressure of 12 mmHg (which is below the assumed µLLA). According to Eq. 10, autoregulation would no longer function, and flow would fall
from a regulated value of 250 µl/min to a value of 175 µl/min.
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Fig. 7.
Representation of functional adaptation to chronic
hypotension. A: pressure plotted as a function of vessel
generation for normotensive, acute hypotensive, and chronic hypotensive
cases. Dilation of arteries increases perfusion pressure of the
microcirculation. B: resulting pressures in the MVGs. Acute
hypotension yields pressures below µLLA (dashed line). Adaptation
raises microvascular pressure above µLLA. C: resulting
shift in the autoregulation curve due to adaptation to chronic
hypotension.
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Allowing the system to structurally adapt causes the pressure in the
smallest vessels to increase. This is because the larger radii,
according to Eq. 1, cause less of a pressure drop across the
conductance vessels (Fig. 7A). This structural adaptation raises the perfusion pressure of the MVGs to 24 mmHg (above the assumed
µLLA) (Fig. 7B) and raises the flow through the MVGs back to 250 µl/min.
Structural adaptation to chronic hypotension leads to functional
adaptation. The global pressure-flow relationship as viewed from the
entrance of the vascular network is illustrated in Fig. 7C.
The LLA is decreased in chronic hypotension, shifting the autoregulation curve to the left.
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DISCUSSION |
The present work is the first demonstration that changes in vessel
caliber stimulated by shear stress can account for structural and
functional adaptation of small conductance vessels. To explain how
radii adapt, a simple model is developed from basic physical principles. The limited set of assumptions is based on identifiable physiological mechanisms. Stability in the adaptation process is
provided by recognized mechanisms of flow regulation in the microvasculature. The difference in arterial and venous dimensions results from the modulating effect of a pressure stimulus. The observed
functional adaptation in response to chronic hypotension is explained
by arterial dilation. This dilation increases pressure in the
microvasculature, allowing the resistance vessels in the microvasculature to adequately control flow. The simple model therefore
satisfies the five criteria enumerated in the beginning of THEORY
AND METHODS.
The present theoretical study focused on the less-explored vasculature
bridging the low-resistance macrovasculature, which primarily conducts
blood to the tissue, and the high-resistance microvasculature, which
primarily regulates blood flow. The active regulation of flow is not
necessarily confined to the microvasculature. Kontos et al.
(8) showed a continuum of participation between vessels
traditionally considered in the macrocirculation and microcirculation. This "mesocirculation" consists of vessels that, although primarily acting to conduct blood, are small enough to contribute to total peripheral resistance.
Explanation of how the small conductance vessels adapt required the
assumption of local shear stress and pressure stimuli. However, to
ensure structural stability for the model illustrated in Fig. 5,
flow-regulating MVGs were assumed. By maintaining a constant flow in
the microvasculature, the MVGs prevented the small conductance arteries
from degenerating (i.e., autoregulation in the microvasculature
prevents degeneration of the arterial and venous conductance
vessels). This allows the conductance vessels to be stable
despite the lack of a direct metabolic stimuli found necessary to keep
the microvasculature structurally stable (20).
Limitations of results.
The proposed model of the mesocirculation was intentionally made
simple. The goal was not to predict particular radii and/or lengths of
vessels in a particular vascular network. Instead, the goal was to
determine the minimum set of rules that explains structural and
functional adaptation of small conductance vessels. This simple model
required a very small set of unknown parameters, which are embedded in
Eqs. 10-14. The present work extends the work of Pries
et al. (20), who determined the minimum set of rules that
explains chronic adaptation of the microvasculature.
The assumptions necessary to specify the complex model are illustrated
in Fig. 5 and manifested in Eqs. 1, 2, and
10-14. Several important phenomena were intentionally
excluded. For instance, acute regulation of vessel radius in response
to shear stress (1, 9) was disregarded. Also absent is the
myogenic response, which adjusts the radius in response to changes in
pressure. The simple topology of the model also excludes proximal
anastamoses interconnecting the smaller arteries. This structural
complexity was explored via a model of the microvasculature by Pries et
al. (20). Furthermore, structural adaptation of the
microvasculature itself, explored in detail by Pries et al.
(20), was not addressed in the present work. Microvascular
beds were treated as black boxes with the functional characteristics
illustrated in Fig. 3. Numerous phenomena could have been added to the
model, undoubtedly increasing the predictive capabilities of the model.
However, the imposed simplicity allows delineation of the phenomena
that are not required to explain the structural and functional
adaptation of the small conductance vessels.
The present methodology has not been used to explore the adaptation of
the small conductance vessels to hypertension. The simple model does
not have the flexibility to do so; the stimuli for growth (described by
Eqs. 8 and 14) are insensitive to pressures above
100 mmHg. Pries et al. (21) explored the adaptation of the
microvasculature to hypertension. They found that an initial increase
in cardiac output increases pressure in the microcirculation, resulting
in structural changes that increase resistance. Because the conductance
vessels, responding to hydrostatic pressure and shear stress, are
indirectly altered by changes in flow, it can be speculated that the
structural changes in the conductance vessels may also be involved. To
fully model the adaptation associated with hypertension, it would be
necessary to integrate adaptation of the conductance vessels with
adaptation of the microvasculature.
Local competition and global adaptation.
For the model of the small conductance vessels, growth of the vessel
lumen was assumed to depend on only two stimuli, sensed locally by the
endothelium: hydrostatic pressure and shear stress (2).
These stimuli result in global structural and functional adaptations
that are similar to that of actual vascular networks. The MVGs of the
model, assumed to autoregulate, were assumed to regulate flow
independent of hydrostatic pressure and shear stress. Two striking
aspects of the model bear further discussion. First, the response of
the conductance vessels to shear stress and pressure stimuli is assumed
to be independent of the locations of the conductance vessels in the
vascular tree. Local conditions determine the target shear stress, and
thus the radius each vessel ultimately attains. The vessels take on the
characteristics of either an artery or vein depending on local requirements.
The second striking aspect is that this model suggests a surprising
level of coordination despite the lack of an overarching control
mechanism. Local control is effective because the microvasculature acts
to balance the competing interests of each vessel. For instance, autoregulation prevents vessels from stealing blood flow from their
neighbors. Furthermore, unless the vessels fail to fulfill their
primary responsibility of conducting blood to a MVG, conductance vessels are prevented from degenerating. The apparently coordinated adaptation of supply to demand is the result of individual units acting
in their own self-interest. This mechanism is hypothesized to
result in a system that distributes resources with the greatest efficiency (15, 23).
Implications relevant to pathological states.
With a minimum set of principles governing normal adaptation, it
becomes instructive to consider the effects of eliminating a particular
control mechanism. For instance, the effect of eliminating the
autoregulation in a MVG can be considered. It is shown above that
autoregulating MVGs are necessary to ensure that the adaptation of
conductance vessels remains stable. Without autoregulating microvasculature, some of the small vessels would greatly dilate, dramatically decreasing resistance and increasing flow. This process would theoretically lead to an arteriovenous shunt (4,
20). This is similar to cerebral arteriovenous malformations,
which are characterized by an absence of autoregulation, large
conductance vessels, low resistance, and high flow (18,
29).
Furthermore, the response of the model to chronic hypotension is
similar to a related clinical condition; high-flow arteriovenous malformations cause profound chronic hypotension in adjacent vascular beds that are structurally and functionally normal. Typically, the
shear stress in hypotensive conductance (feeding) arteries is similar
to that in the normotensive contralateral vessels (24). Furthermore, Young et al. (30) found that, despite
pressures well below the normal lower limits of autoregulation, the
vasculature in the adjacent hypotensive regions were still able to
autoregulate. From the preceding theoretical development, it can be
speculated that this functional adaptation is primarily due to the
structural adaptation of the small conductance vessels. The low
hydrostatic pressure, possibly through increasing endothelial nitric
oxide synthase expression (13), sets the target shear
stress to a lower value (Eq. 14). This stimulates the very
small conductance vessels to dilate, and thus increases perfusion of
the microvasculature. Higher pressure in the microvasculature allows
the resistance arteries to operate effectively. In terms of tissue
perfusion, this adaptation manifests as a shift in the autoregulation
curve to the left. Notably, endothelial nitric oxide synthase knockout mice have autoregulation curves that are shifted to the right (6).
In conclusion, the present work delineates the essential criteria
determining the long-term radii of small conductance vessels. Chronic
changes in global conditions, such as the development of arterial
hypotension, affect local endothelial shear stress, pressure, and flow.
The independent adaptation of vessels to local conditions yields a
coordinated set of structural changes that ultimately adapts global
supply to demand.
| |
ACKNOWLEDGEMENTS |
The authors thank Joyce Ouchi for assistance with preparation of
the manuscript.
| |
FOOTNOTES |
Portions of this work were supported by National Institutes of Health
Grants RO1 NS-37921, NS-27713, K24 NS02091 and 5-T32-GM08464.
Address for reprint requests and other correspondence: W. L. Young, Dept. of Anesthesia and Perioperative Care, Univ. of
California San Francisco, 1001 Portrero Ave., Rm. 3C-38, San Francisco,
CA 94110.
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 29 December 1999; accepted in final form 14 April 2000.
| |
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ABSTRACT
INTRODUCTION
