Am J Physiol Heart Circ Physiol
279: H1645–H1653, 2000.
Model of structural and functional adaptation of small
conductance vessels to arterial hypotension
CHRISTOPHER M. QUICK,1 WILLIAM L. YOUNG,1,2 EDWARD F. LEONARD,3
SHAILENDRA JOSHI,5 ERZHEN GAO,4 AND TOMOKI HASHIMOTO1
1
Department of Anesthesia and Perioperative Care and 2Departments of Neurosurgery and
Neurology, University of California San Francisco, San Francisco, California 94110; and
3
Departments of Biomedical Engineering and Chemical Engineering, 4Department of Electrical
Engineering, and 5Department of Anesthesiology, Columbia University, New York, New York 10032
Received 29 December 1999; accepted in final form 14 April 2000
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
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.
mathematical modeling; autoregulation; hemodynamics; instability
EVIDENCE EXISTS THAT
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H1645
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Quick, Christopher M., William L. Young, Edward F.
Leonard, Shailendra Joshi, Erzhen Gao, and Tomoki
Hashimoto. Model of structural and functional adaptation of
small conductance vessels to arterial hypotension. Am J Physiol
Heart Circ Physiol 279: H1645–H1653, 2000.—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.
H1646
ADAPTATION OF SMALL CONDUCTANCE VESSELS
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.
THEORY AND METHODS
R共r兲 ⫽
⌬P 8␩L
⫽
Q
␲r 4
(1)
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)
␶共r, Q兲 ⫽
4␩
䡠Q
␲r 3
Q⫽
⌬P tot
R s ⫹ R共r兲
(4)
Substituting Eq. 4 and 1 into Eq. 2 yields the endothelium
shear stress
␶⫽
⌬P tot
4␩
䡠
␲r 3 R s ⫹ 共8␩L/␲r 4 兲
(5)
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
(2)
Equivalently, endothelial shear stress can be expressed in
terms of radius and pressure gradient (11)
␶共r, ⌬P兲 ⫽
r
䡠 ⌬P
2L
(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
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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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 (␩)
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).
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)
ADAPTATION OF SMALL CONDUCTANCE VESSELS
solving for ␶ and r
r max ⫽
冉 冊
8␩L
3␲R s
1/4
␶ max ⫽ ⌬P tot
冉
27␩
512␲L 3 R s
冊
1/4
(6)
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)
r⬘ ⫽
r
r max
␶⬘ ⫽
␶
␶ max
⫽
4r⬘
3 ⫹ r⬘ 4
(7)
⌬r ⫽ K共␶ ⫺ ␶*兲
(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, ␶* ⫽ 0.2␶max 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.
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)
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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␶⬘ (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
H1647
H1648
␶⬘1 ⫽
ADAPTATION OF SMALL CONDUCTANCE VESSELS
␶1
4r⬘1
⫽
␶max 3 ⫹ r⬘14 ⫹ r⬘24
␶⬘2 ⫽
␶2
4r⬘2
⫽
␶max 3 ⫹ r⬘14 ⫹ r⬘24
(9)
R MVG共⌬P兲 ⫽
冦
␮LLA/Q AR, ⌬P ⬍ ␮LLA
⌬P/Q AR, ␮LLA ⱕ ⌬P ⱕ ␮ULA
␮LLA 䡠 ⌬P/关Q AR共⌬P ⫹ ␮LLA
⫺ ␮ULA兲兴, ⌬P ⬎ ␮ULA
(10)
Equation 10 corresponds to a “Type 3” empirical description
of autoregulation delineated in Gao et al. (3).
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 con-
Q n ⫽ Q o/2 n
Fig. 3. Representation of simple functional model of a microvascular
group (MVG). Between the microvascular lower limit of autoregulation (␮LLA) and the upper limit of autoregulation (␮ULA), flow is
maintained at a value QAR. Above and below these limits, the system
becomes passive, and flow increases linearly with pressure.
R n ⫽ R共r兲/2 n
L n ⫽ L o/2 n
(11)
The value of the radius of vessels of a common generation n
(rn) can be calculated from Eqs. 2 and 11.
1/3
4␩Q o
(12)
rn ⫽
n
␲␶ n 2
冉 冊
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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 (␮LLA) and the
upper limit of microvascular autoregulation (␮ULA). Below
the ␮LLA and above the ␮ULA, the system becomes passive.
The resistance of the MVG (RMVG) is nonlinear and depends
on the value of ⌬P
ductance arteries, which introduce a pressure drop between
the point of observation and the microcirculation. Thus ␮LLA
and ␮ULA, 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 ␮LLA and ␮ULA 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.
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, ␮LLA ⫽ 17 mmHg, ␮ULA ⫽ 117 mmHg, QAR ⫽ 7.8
␮l/min, and Rs ⫽ 1.01 䡠 109g cm⫺4 s⫺1. 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.
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
ADAPTATION OF SMALL CONDUCTANCE VESSELS
H1649
The pressure drop across each generation can then be calculated from Eqs. 1 and 12.
⌬P n ⫽ L n ␶ n
冉
2 n ⫹ 1 ␲␶ n
␩Q o
冊
1/3
(13)
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 ␮m. They fit an empirical
equation, ␶*(P), to this data (20, 21)
␶*共P兲 ⫽
再
14,
P ⱕ 10 mmHg
100 ⫺ 86 䡠 exp共⫺5,000 䡠 兵log 关log 共P兲兴其 5.4 兲, (14)
P ⱖ 10 mmHg
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.
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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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.
H1650
ADAPTATION OF SMALL CONDUCTANCE VESSELS
RESULTS
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.
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 ␮LLA).
According to Eq. 10, autoregulation would no longer
function, and flow would fall from a regulated value of
250 ␮l/min to a value of 175 ␮l/min.
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
␮LLA) (Fig. 7B) and raises the flow through the MVGs
back to 250 ␮l/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
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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, ␮LLA ⫽ 17 mmHg,
␮ULA ⫽ 117 mmHg, and Qo ⫽ 250 ␮l/min. The values
of Qo and Lo were chosen to illustrate a vascular network branching off a small artery. The values of ␮LLA
and ␮ULA 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
␮LLA and ␮ULA 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).
Functional adaptation to chronic hypotension. In the
normotensive case, pressure falls from Pin ⫽ 100
ADAPTATION OF SMALL CONDUCTANCE VESSELS
in chronic hypotension, shifting the autoregulation
curve to the left.
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 lessexplored 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 delin-
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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 ␮LLA (dashed line). Adaptation raises microvascular pressure above ␮LLA. C: resulting shift in the autoregulation
curve due to adaptation to chronic hypotension.
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H1652
ADAPTATION OF SMALL CONDUCTANCE VESSELS
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.
The authors thank Joyce Ouchi for assistance with preparation of
the manuscript.
Portions of this work were supported by National Institutes of
Health Grants RO1 NS-37921, NS-27713, K24 NS02091 and 5-T32GM08464.
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