## 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

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.

## 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 (ΔP) 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 (η)
Equation 1Modulation 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 (τ) is the frictional force acting on the endothelium due to the flow of blood. It can be calculated from known values of *r*, η, and Q, given the same assumptions required for Poiseuille's Law (16)
Equation 2Equivalently, endothelial shear stress can be expressed in terms of radius and pressure gradient (11)
Equation 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 ΔP 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.1
*A*). 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 ΔP. The pressure source (P_{in}) in conjunction with a source resistance (*R*
_{s}) 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. 1
*A*. 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 (ΔP_{tot} = P_{in} − P_{out}, where ΔP_{tot} is the total pressure and P_{out} is the pressure at the end of the vessel) by the sum of*R*
_{s} and *R*(*r*)
Equation 4Substituting *Eq. 4
* and *1* into *Eq.2
* yields the endothelium shear stress
Equation 5which depends on the vessel properties (radius and length), blood properties (viscosity), and properties of the system in which the vessel is embedded (*R*
_{s} and ΔP_{tot}). 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 (τ_{max}) at a particular radius (*r*
_{max}). These values can be determined by differentiating *Eq. 5
* with respect to *r*, setting the result equal to 0, and solving for τ and *r*
Equation 6Because it is not clear from inspection of *Eq. 5
* how particular values of η, *R*
_{s}, *L*, and ΔP_{tot} influence *τ(r*), *r *and τ are normalized (yielding *r*′ and τ′, respectively)
Equation 7τ′ (plotted in Fig. 1
*B*) has a maximum value at*r*′ equal to 1. Nondimensionalization allows representation of a function independent of particular parameter values (i.e., η,*R*
_{s}, *L*, and ΔP_{tot}) (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” (τ*). 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 (Δ*r*) is set proportional to the difference between the actual shear stress (τ) and τ*. A proportionality constant (*K*) represents the sensitivity
Equation 8When τ = τ*, then a vessel is in equilibrium and there will be no adaptation. If τ < τ*, Δ*r* is positive. If τ > τ*, Δ*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, τ* = 0.2τ_{max} is plotted on the same graph as τ′ (Fig.2
*A*). As revealed by Fig. 2, long-term adaptation introduces a difficulty; there are two possible radii that can result in the same τ*. One radii (*r _{a}
*) is small; the other radii (

*r*) is large.

_{b}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 *r _{b}
*, the vessel radius would be stimulated to grow larger or smaller until the final radius settled at

*r*. Any perturbation of this equilibrium radius would cause the radius to return to

_{b}*r*. The equilibrium radius

_{b}*r*is thus recognized to be stable. However, if the radius were initially at

_{b}*r*, a small decrease in radius would initiate a growth in radius from

_{a}*r*to zero. In contrast, a small increase in radius would initiate growth in radius from

_{a}*r*to

_{a}*r*. Thus

_{b}*r*is recognized to be unstable. All radii <

_{a}*r*

_{max}are similarly unstable (as illustrated by the dashed portion of the curve in Fig. 2

*A*). 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.,

*R*

_{s}≠ 0) (4).

Another type of instability arises when two vessels in parallel adapt concurrently (4). In this case, the shear stress in*vessel 1* (τ_{1}) depends on its own radius (*r*
_{1}) as well as the radius of *vessel 2* (*r*
_{2}). As in *Eq. 4
*,*τ*
_{1} and the shear stress of *vessel 2*(*τ*
_{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 *r*
_{max} and τ_{max}(compare with *Eq. 7
*)
Equation 9From *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 τ_{1} and τ_{2} are initially equal to τ*, then the system would be in equilibrium. However, if*r*
_{1} were initially slightly larger than*r*
_{2}, then τ_{1} would be greater than τ_{2}. This would stimulate *r*
_{1} to grow larger in an attempt to lower τ. In response, τ_{2}decreases, stimulating *r*
_{2} to decrease. This further increases τ_{1}, and so on. The result of this process is depicted in Fig. 2
*B* (arrows). Note that there are only two cases where τ_{1} = τ_{2} = τ*. These two equilibria (*a*′ and *b*′) correspond to the two equilibria identified in Fig. 2
*A* [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 Q_{AR}) 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 (*R*
_{MVG}) is nonlinear and depends on the value of ΔP
Equation 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 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*r*
_{max}, 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 *r*
_{max} (Fig.4
*A*), because they are assumed to be influenced by metabolic stimuli. The small conducting vessels, without direct metabolic stimuli, are larger than *r*
_{max} and do not exhibit the instability illustrated in Fig. 2
*A*.

Furthermore, the addition of autoregulating MVGs makes two vessels in parallel conditionally stable. The derivation of shear stress in the system (shown in Fig. 4
*B*) follows that of *Eq. 9
*, with flow through each branch modified by*R*
_{MVG}(ΔP). 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: P_{in} = 100 mmHg, P_{out} = 0 mmHg, *L* = 0.031 cm, μLLA = 17 mmHg, μULA = 117 mmHg, Q_{AR} = 7.8 μl/min, and *R*
_{s} = 1.01 · 10^{9}g cm^{−4} s^{−1}. In particular, *R*
_{s} was chosen such that the combination of the vessel of interest, *R*
_{s} and P_{in}, behave like the more complicated model in Fig. 5. As in Fig. 2
*B*, Fig. 4
*B* (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.5
*B*, *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. 2
*B*.

#### 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. 5
*A* 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 (*L _{n}
*) and radii (

*r*). This structural similarity results in hemodynamic similarity, wherein the pressure (P

_{n}_{n}), resistance (

*R*), and shear stress (τ

_{n}_{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 (Q

_{o}). The values of

*R*and the flow within vessels of a common generation

_{n}*n*(Q

_{n}) can be calculated from

*R*(

*r*) and Q

_{o}. To simplify,

*L*is halved in each generation Equation 11The value of the radius of vessels of a common generation

_{n}*n*(

*r*) can be calculated from

_{n}*Eqs. 2*and

*11*. Equation 12The pressure drop across each generation can then be calculated from

*Eqs. 1*and

*12*. Equation 13Terminating 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 A

_{0}… A

_{N-1}and V

_{0}… V

_{N-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* (τ_{n}) must be specified. When fully adapted, τ_{n} will equal τ*. In general, τ* in the high-pressure arteries is higher than τ* 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, τ*(P), to this data (20, 21)
Equation 14For the present purposes, τ*(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 τ*(P) was originally fit, it is nonetheless expected to capture the essential behavior of the small conductance vessels.

## 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: P_{in} = 100 mmHg, P_{out} = 0 mmHg, input length (*L*
_{o}) = 1 cm, μLLA = 17 mmHg, μULA = 117 mmHg, and Q_{o} = 250 μl/min. The values of Q_{o} and*L*
_{o} 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 5
*B* 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 *r*
_{max}, 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. 4
*A*.

Because all the vessels in a generation are assumed identical, the second type of instability (illustrated in Fig. 4
*B*) is not directly investigated. However, the value of *R*
_{s}in Fig. 4
*B* was chosen so that the reduced model mimics the spatially distributed model in Fig. 5. Figure 4
*B* illustrates how two parallel vessels in the last arterial generation (*generation A _{5}
* 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. 5

*B*.

#### Structural adaptation to chronic hypotension.

To explore the process of adaptation in chronic hypotension, the model in Fig. 5
*B* is allowed to adapt to an input pressure of P_{in} = 35 mmHg. It is assumed that the μLLA and μULA of the MVGs remain fixed. As illustrated in Fig.6
*A*, the process of adaptation does not alter venous radii. However, hypotension causes arterial radii to dilate appreciably (>29% in *generation A _{0}
*and >50% in

*generation A*). The cause of this dilation can be identified by considering the shear stress before and after adaptation (Fig. 6

_{5}*B*). 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 P_{in} = 100 mmHg at the entrance of the vascular bed to P_{in} = 69 mmHg at the MVG (Fig. 7
*A*). If the model with the structure described in Fig. 5
*B* 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.

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. 7
*A*). This structural adaptation raises the perfusion pressure of the MVGs to 24 mmHg (above the assumed μLLA) (Fig. 7
*B*) 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. 7
*C*. The LLA is decreased 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 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.

## Acknowledgments

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.

- Copyright © 2000 the American Physiological Society