## Abstract

Recent experimental studies have shown significant alterations of the vascular smooth muscle (VSM) tone when an artery is subjected to an elevation in pressure. Therefore, the VSM participates in the adaptation process not only by means of its synthetic activity (fibronectins and collagen) or proliferative activity (hypertrophy and hyperplasia) but also by adjusting its contractile properties and its tone level. In previous theoretical models describing the time evolution of the arterial wall adaptation in response to induced hypertension, the contribution of VSM tone has been neglected. In this study, we propose a new biomechanical model for the wall adaptation to induced hypertension, including changes in VSM tone. On the basis of Hill's model, total circumferential stress is separated into its passive and active components, the active part being the stress developed by the VSM. Adaptation rate equations describe the geometrical adaptation (wall thickening) and the adaptation of active stress (VSM tone). The evolution curves that are derived from the theoretical model fit well the experimental data describing the adaptation of the rat common carotid subjected to a step increase in pressure. This leads to the identification of the model parameters and time constants by characterizing the rapidity of the adaptation processes. The agreement between the results of this simple theoretical model and the experimental data suggests that the theoretical approach used here may appropriately account for the biomechanics underlying the arterial wall adaptation.

- arterial wall
- remodeling
- myogenic response
- theoretical model
- Hill's model
- biomechanics

hypertension is one of the major risk factors associated with the development of many cardiovascular diseases. It is related to changes that occur in geometry, structure, and composition of blood vessels, affecting their mechanical function to transport blood and distribute it according to metabolic demands. In addition to factors of genetic and humeral origin, arterial geometry and structure are strongly influenced by their mechanical environment, namely the arterial pressure and blood flow rate. Changes in the mechanical environment often elicit an arterial response directed to maintain certain mechanical characteristics such as medial stress, flow-induced shear stress and arterial compliance at their baseline values (13).

The character of the arterial response depends significantly on the duration of changes in the mechanical environment. For example, a short-term change in pressure results predominately in arterial constriction, a phenomenon known as the Bayliss effect or myogenic response (3). The myogenic response is typical for small muscular arteries, but it is also observed in large arteries (2,16, 23). When changes in blood pressure persist for a longer period (from hours to weeks), large arteries respond by altering their geometrical dimensions, principally by eccentric arterial wall media hypertrophy or hypertrophic remodeling that could not be classified as “inward” or “outward” (17). This is often termed geometrical adaptation. Several experimental studies (10, 11,15) showed that when a conduit artery is subjected to an increased arterial pressure while blood flow rate is maintained, the wall thickens monotonically to keep the deformed inner radius constant and to restore the circumferential stress under normal conditions. The geometrical adaptation phase is accompanied by a change in elastic properties of the arterial wall such as incremental modulus or pressure-radius modulus (structural adaptation) (4, 14). On the basis of these studies, it has been postulated that wall adaptation aims to restore an “optimal” biomechanical environment for the arterial wall. Understanding the mechanisms underlying the arterial wall adaptation and the factors that control arterial response necessitates better understanding of the normal arterial function and the genesis of certain pathologies.

Recent studies (6, 7) showed that the vascular smooth muscle (VSM) tone, estimated in terms of the active stress borne by the VSM, varies during the acute phase of the adaptation process of the rat common carotid artery to induced hypertension. The VSM tone rapidly increases after the step increase in pressure, and then slowly decreases towards control values as geometrical adaptation reaches asymptotic levels. These studies show that VSM plays an important role in acute arterial adaptation to hypertension. Rachev and Hayashi (19) suggested that the VSM tone affects the stress distribution through the arterial wall and, consequently, its geometrical and structural adaptation.

All of the existing models dealing with geometrical adaptation of an artery in response to sustained hypertension are on the basis of the assumption that changes in the arterial wall thickness are related to changes in the circumferential wall stress, following the idea proposed by Fung et al. (12) of the stress-growth law. This law states that the artery thickens as a result of medial stress-induced mass growth rate. Taber and Eggers (24-26) considered the arterial wall as a growing continuum and used the theory of the finite volumetric growth developed by Rodriguez et al. (22). A different approach was used by Rachev et al. (20, 21) to model the dynamics of geometrical and structural adaptation in response to sustained changes in blood pressure. They assumed that the geometry of the arterial cross section and the mechanical properties of arterial tissue change in a manner to restore the normal baseline values of the flow-induced shear stress at the intima, the normal stress distribution across the arterial wall, and the normal arterial compliance. However, none of the above models account for the changes in VSM tone.

In this study, we develop a theoretical model describing the evolution of arterial wall adaptation subjected to a step increase in pressure. The novel aspect of this work, with respect to previous theoretical models of arterial adaptation, is the consideration of VSM tone. The key assumption is that the synthetic and proliferate activity of VSM, leading to arterial wall adaptation, is associated with changes in both the contractile state of VSM and changes in total circumferential wall stress. Therefore, the model developed here describes not only the time course of the geometrical adaptation but also the associated changes and contribution of VSM tone during the adaptation process.

## METHODS

### Experimental Analysis

The theoretical model developed here is on the basis of our experimental data (6, 7). Hypertension was induced in 8-wk Wistar rats by total ligation of the aorta between the two kidneys. This procedure induced a step increase in mean blood pressure (means ± SE) from an initial level of 92 ± 2 to 145 ± 4 mmHg for the entire postsurgery period (Fig.1
*A*). Rats were euthanized 2, 4, 8, and 56 days after surgery, and the left common carotid artery was excised for mechanical investigation. In vitro measurements of pressure-diameter relationships were obtained for arterial segments extended to their own in situ length and inflated by internal pressure at a rate of ∼1.3 mmHg/s between 0 and 200 mmHg. The pressure-diameter curves were obtained by the following methods:*1*) under normal VSM tone in Krebs-Ringer solution, i.e., when the contractile state of VSM was kept close to physiological conditions, *2*) when the muscle was stimulated maximally to contract by administration of 5 × 10^{−7} [M] norepinephrine, and *3*) when the VSM was completely relaxed by administration of 10^{−4} [M] papaverine. The pressure-diameter data were used to calculate the mean circumferential wall stress viewing the artery as a thin-walled tube. At any given diameter, the associated stress at complete relaxation was subtracted from the stress under physiological conditions or maximal contraction (Hill's model). The obtained value was termed “active stress” (see Fig. 1
*C*).

### Theoretical Model of Arterial Wall

An artery was considered to be a circular membrane made of nonlinear elastic and incompressible material. For the first approximation, we assumed that mechanical properties of the arterial wall do not change significantly during the acute adaptation process and therefore are considered invariable. The state of no load, that is, when pressure and longitudinal force are zero and the VSM is fully relaxed, was taken as a reference, i.e., zero stress state (ZSS), for the strain measurements at any deformed state.

The experimental findings obtained by Fridez et al. (6) show that the process of natural growth of the rats is not completed during the period while the artery is subjected to sustained hypertension. To account for this fact, the midwall radius and wall thickness of the artery at the ZSS (*R*
^{N} and*H*
^{N}, respectively) are described as a function of time through the following relationships
Equation 1where *R*
_{0} and *H*
_{0}are the midwall radius and thickness, respectively, at the ZSS at the beginning of the experiments (*day 0*) and*k*
_{1} and *k*
_{2} are positive constants accounting for the rate of change of the geometrical dimensions of the vessel due to natural animal growth. N denotes values under normotensive conditions. It was assumed that the growth process is not affected by the induced hypertension, i.e., the rate constants*k*
_{1} and *k*
_{2} do not depend on pressure.

Under applied load the artery undergoes an axisymmetric finite deformation. The stretch ratios of the midwall surface in the circumferential (λ) and longitudinal direction are the mean measures of deformation, and are defined as follows
Equation 2where *R* and *r* are the midwall radii of the artery at the ZSS and the deformed state, respectively, and*L* and *l* are the length of an arterial segment at the ZSS and at in situ conditions, respectively. λ_{z}is the longitudinal stretch ratio of the vessel. Following the theory of finite elastic deformation (9), the Green strains corresponding to these stretch ratios are
Equation 3Also, because the wall material is assumed to be incompressible, the deformed wall thickness *h* is
Equation 4Considering the vessel as an elastic membrane, the axial and circumferential stresses are assumed to be uniformly distributed across the arterial thickness, whereas the radial stress is considered to be zero. On the basis of Hill's model, the total circumferential stress per unit deformed area (Cauchy stress, ς_{tot}) is represented as a sum of a passive stress (ς_{pas}), which is borne by the wall material when VSM is fully relaxed, and an active stress (ς_{act}) developed by VSM when it is contracted
Equation 5

#### Passive behavior of arterial wall.

Considering that arterial material is elastic and orthotropic, the most general form of the constitutive relation between the passive circumferential stress and the membrane strains (*Eq. 3
*) is ς_{pas} = λ^{2}(*e*,*e _{z}
*) (9). On the basis of experimental findings presented by Fridez et al. (7), we assumed that the axial stretch ratio does not change significantly over time nor it is affected by the induced increase in pressure. With these assumptions the above constitutive relation can be reduced to a one-dimensional circumferential stress-strain relationship. A suitable form of the one-dimensional constitutive relation describing the passive mechanical properties of a rat common carotid artery is
Equation 6where

*a*

_{1},

*a*

_{2},

*a*

_{3}, and

*a*

_{4}are material constants to be determined from experimental data on pressure-radius relationship at fixed axial stretch ratio.

#### Active behavior of arterial wall.

The active stress developed by the VSM when it is contracted depends on several factors. It is well recognized that the magnitude of the active stress developed at isometric constriction and constant stimulus depends on the actual radius of the artery after the length-tension relationship (5). On the other hand, at fixed radius and variable stimulation, the magnitude of the active stress follows the dose-tension relationship (5). Both of these factors affect the active stress in a complex manner and reflect the intercellular ionic, diffusive, and mechanical processes involved in VSM contraction. Moreover, the contribution of VSM to load bearing depends on the orientation and amount of VSM in the wall tissue. Here we use the following phenomenological description of the circumferential active stress
Equation 7where *c*
_{1} and *c*
_{2}are material constants representing the maximal capacity of the VSM to contract. *S*
_{bas} is the basal tone ratio, that is the ratio of active stress at normal VSM tone to active stress under maximal contraction. *S*
_{bas} accounts for VSM tone at lower strains and it is independent of strain (Fig. 1
*D*). The term (1 −*S*
_{bas})*f*
_{myo}(λ) refers to myogenic tone. The separate representation of basal and myogenic tone originates from the biomechanical analysis (macroscopic level) (P. Fridez et al., unpublished observations). Although other studies asserted this separation, it remains speculative at the level of mechanisms. Osol et al. (18) also considered separate representation of intrinsic tone (our*S*
_{bas}) and basal-myogenic tone (our myogenic tone) in their study. The factor (1 −*S*
_{bas}) in *Eq. 7
* means that the myogenic tone can only operate within the range delimited by the total contraction (*S* = 1), and *S*
_{bas}. In other words, it represents the remaining tone capacity of the VSM. Finally, *f*
_{myo}(λ) is the following sigmoid function accounting for the strain-dependence of myogenic tone
Equation 8where λ_{cr} is the strain at the inflection point of the sigmoid *f*
_{myo}(λ) and it can be regarded as the middle point of the range of λ values associated with the myogenic response. Thus we will refer to λ_{cr} as the critical strain for the myogenic response (myogenic critical strain). The parameter *q* is proportional to the maximal slope of the VSM tone ratio as a function of strain, i.e., proportional to the slope of *f*
_{myo}(λ) at the inflection point. Finally, λ_{0} is the stretch ratio of the circumferential midwall fiber at zero pressure and is therefore entirely determined by the experimental geometry. λ_{0} ≠ 1 because the arterial segments are stretched longitudinally and this causes the length of the circumferential midwall fiber, λ_{0}, to contract to a value <1. Fridez et al. (unpublished observations) reported a value of λ_{0} = 0.86 ± 0.01 for the rat common carotid artery, which was the value used in the present analysis.

The parameters in *Eq. 7
* contribute to the active stress in different ways. The constants *c _{1}, c_{2}, q,*and λ

_{0}reflect inherent characteristics of the VSM apparatus and the structure and composition of the specific artery studied. Because the active stress is defined per unit as a deformed area over the whole cross section, it is not the partial stress borne by VSM itself but represents an average stress measure. Therefore, the active stress varies if the ratio between the area occupied by VSM and the other structural components changes. This may happen when an artery grows during development and maturation or undergoes adaptation under hypertensive conditions. At a given time, however,

*c*

_{1},

*c*

_{2}, q, and λ

_{0}are constant for a given artery and their values are not dependent of VSM activation. The parameters

*S*

_{bas}and λ

_{cr}are also phenomenological VSM tone parameters, but they are affected by the intensity of stimulation. Fridez et al. (unpublished observations) showed that by appropriately varying the values of

*S*

_{bas}and λ

_{cr}, while keeping the other parameters constant, it is possible to describe the variation of the active stress over the entire range of stimulation and deformation of physiological interest.

*S*

_{bas}and λ

_{cr}vary during the development and maturation. Within the period of duration of our experiments, it is assumed that the time dependence of

*S*

_{bas}and λ

_{cr}is appropriately represented by a linear function of the form Equation 9where

*S*

_{bas0}and λ

_{cr0}are the basal tone ratio and myogenic critical strain values at the beginning of the experiments and

*k*

_{3}and

*k*

_{4}are constants accounting for the rate of change of

*S*

_{bas}and λ

_{cr}, due to natural animal growth.

### Adaptation Rate Equations for Wall Thickness

Under normotensive conditions, the midwall radius and wall thickness at zero-load are *R*
^{N} and*H*
^{N}. The vessel is subjected to normal blood pressure, *P*
^{N}, and is kept at constant deformed length. By using the formulas given in the previous section, stress and strain distributions in the arterial wall are calculated. Because wall dimensions change slowly over time according to *Eq. 1
*, the strain and stress measures are also time dependent.

Induced arterial hypertension is modeled by a step increase in blood pressure from *P*
^{N} to *P*
^{H}(Fig. 1
*B*), where superscript H denotes values under hypertensive condition. Considering the vascular material as an elastic solid, the stresses in the arterial wall also undergo a step increase. The magnitude of the blood flow is kept constant. Assuming that intimal wall shear stress is kept at control levels (via endothelium-mediated mechanisms), the internal diameter at perfusion pressure stays constant. This is in agreement with the experimental observation by Fridez et al. (7) where the internal diameter at mean pressure remains fairly constant.

By following the approach used by Rachev et al. (20, 21), we assumed that the rate of change in wall thickness is driven by the deviation of the mean total circumferential stress from its value under normal condition
Equation 12where ς
is the current total circumferential stress in the hypertensive artery at a given time, ς
is the corresponding total circumferential stress in the normotensive artery at the same time and τ_{H} is the characteristic time constant for the adaptation speed of wall thickness.

### Adaptation Rate Equations for VSM Tone

Figure 1
*C* shows that as the arterial wall undergoes remodeling in response to hypertension, the magnitude of the active stress developed by VSM deviates from its baseline value corresponding to normotensive conditions. Figure 1
*C* also shows that the maximum active stress developed by VSM in response to hypertension follows practically the same stress-strain relationship at various phases of the adaptation process (2, 4, 8, and 56 days). Therefore, the time variation of the active stress is assumed to depend solely upon the contractile state of VSM cells and the current deformed configuration of the vessel, while the maximum contraction capacity of the VSM remains constant (Fig. 1
*D*). This assumption is equivalent to considering that the ratio between the area occupied by the VSM cells and the area occupied by the passive constituents of the wall remain fairly constant during the whole process of remodeling. Thus the magnitude of the active stress is fully determined by*S*
_{bas} and *λ*
_{cr}. Also, experimental observations (1) indicate that VSM is sensitive to total stress rather than total stretch ratio. Taking the above facts into consideration as well as the form of the adaptation rate equation for wall thickness (*Eq. 12
*), we propose the following adaptation rate equations for the evolution of*S*
_{bas} and λ_{cr}
Equation 13
Equation 14where τ_{S1}, τ_{S2}, τ_{λ1}, and τ_{λ2} are characteristic time constants. The first terms on the right-hand side of *Eqs. 13
* and *
14
* account for the effect of changes in total stress on the active stress parameters *S*
_{bas} and λ_{cr}. The particular form of these terms suggests that the rate of change of*S*
_{bas} or λ_{cr} is proportional to the deviation of the total circumferential stress from its value under normotensive conditions. These are on the basis that both*S*
_{bas} and λ_{cr} rapidly respond to a step increase in pressure (Fig. 1
*C*). Note that the first terms on the right hand side of *Eqs. 13
* and *
14
*are of opposite signs because *S*
_{bas} increases, whereas λ_{cr} decreases in response to instantaneous increase in stress. The second terms reflect the tendency of the active stress, developed by VSM to restore its tone by pulling*S*
_{bas} and λ_{cr} back to normotensive levels.

On the basis of the experimental observation that pressure-induced arterial adaptation occurs mainly through a change in wall thickness rather than a change in arterial radius (7, 15), we assumed that the evolution of the undeformed radius of a hypertensive artery is the same as that of a normotensive one, i.e.
Equation 15
*Equations 12-14
* form a nonautonomous system of first-order differential equations describing the evolution of the wall thickness and circumferential stress after a step increase in pressure. They are coupled with the equation of equilibrium (*Eq. 10
*) and the equations describing the deformation and rheology of the arterial wall, and have to be solved simultaneously.

Because the wall thickness and the active stress are continuous functions of time, at the moment of increase in pressure (*t = 0)* the following initial conditions hold true
Equation 16

## RESULTS

### Model Parameter Identification

Parameters concerning the geometry of hypertensive and normotensive arteries (including variation due to natural growth) are obtained directly from the experimental results of Fridez et al. (6, 7). These values are given in Table1. Parameters characterizing passive stress, active stress and VSM tone (*S*
_{bas} and λ_{cr}) were obtained by fitting *Eqs. 2-11
*to the pressure-diameter data of the hypertensive and control group.*Equations 2-11
* describe the mechanical properties and the geometry of the arterial wall. The values of these parameters are given in Table 2. Finally, the characteristic time constants τ_{H}, τ_{S1}, τ_{S2}, τ_{λ1}, and τ_{λ2} are identified by using evolution *Eqs. 12-14
* for wall thickness, VSM, *S*
_{bas}, and critical strain through matching (best fit) theoretical and experimental results. *Equations12-14
*, together with the boundary and initial conditions (*Eqs. 15
* and *
16
*), were solved by using a standard procedure on the basis of Fehlberg order 4–5, the Runge-Kutta method, and the software Mathematica. A standard Levenberg-Marquardt fit was used to identify the set of time constants providing the best agreement between theoretical results and experimental data. The resulting values of characteristic time constants τ_{H}, τ_{S1}, τ_{S2}, τ_{λ1}, and τ_{λ2} are given in Table 3.

### Model Predictions of Thickening and VSM Tone Adaptation

The evolution of the wall thickness, *S*
_{bas}, and myogenic critical strain obtained for normotensive and hypertensive rats are shown in Fig. 2. The experimental data are also shown in Fig 2, *A*-*C*. The theoretical results are in good agreement with experimental data for wall thickness, *S*
_{bas}, and myogenic critical strain. The characteristics of these solutions obtained for wall thickness, *S*
_{bas}, and critical strain are not sensitive to changes in parameter values within the physiological range.

### Model Predictions of Total and Active Stress Evolution

The evolution of total and active circumferential stresses is shown in Fig. 3. The theoretical results are in good agreement with experimental data for the early stage of postsurgery period (∼10 days). However, the model predictions at the*day 56* differ considerably from the experimental data. These discrepancies were thought to be because that the proposed theoretical model does not take into account the changes in passive mechanical properties of the arterial wall that come into play at later stages of arterial response (long-term response) to hypertension (4,15).

To verify the above hypothesis, we extended the model to take into account the alteration of the passive mechanical properties which occurs during arterial adaptation. We used the pressure-radius relationships under total relaxation from the inflation tests performed at 0, 2, 4, 8, and 56 days postsurgery to calculate a smooth numerical interpolation between these experimental values. We let the passive properties change continuously over time according to experimental data, and reconsidered the dynamics of the active stress and muscle parameters. The results of these calculations are shown in Fig.4. The results of the extended model are in good agreement with experimental data both for the early and late stages of arterial wall adaptation. The new set of characteristic time constants τ_{H}, τ_{S1}, τ_{S2}, τ_{λ1}, and*τ*
_{λ2} obtained by using the extended model are given in Table 3.

## DISCUSSION

We have developed a theoretical model for arterial wall adaptation to induced hypertension taking into account changes in arterial geometry as well as alteration and contribution of VSM tone during the adaptation process. The necessity to include VSM tone into a model of arterial adaptation to hypertension arises from the recent experimental work showing that VSM tone responds rapidly and undergoes significant alterations when an artery is subjected to an acute elevation in pressure (7, 8). These studies show that VSM contributes to the adaptation process not only by means of its synthetic or proliferative activity, but also by adjusting its own degree of contraction. These changes at the level of VSM tone alter the stress distribution in arterial wall and, in particular, the balance between the stress borne by the extracellular matrix and the stress borne by VSM.

### Theoretical Models on Arterial Adaptation

Earlier theoretical models (20, 21, 24-26) describing the time evolution of the arterial wall adaptation in response to induced hypertension have ignored the contribution of VSM tone. Furthermore, because of the lack of a complete set of data to identify the values of model parameters, previous studies (20,21) have provided only qualitative predictions for the arterial adaptation in response to hypertension. In that respect, the present theoretical study is noteworthy in two ways. First, all model parameters were identified or derived from a single, comprehensive, and complete experimental data set. Second, the present study is, to our knowledge, the first theoretical model accounting for all aspects of VSM contribution into the adaptation process, namely its synthetic, proliferative, and contractile activities.

### Adaptation Rate Equations

*Equation 12
* describes the evolution of arterial thickness, which is assumed to be driven by the deviation of the average circumferential stress at the hypertensive conditions from that at normotensive state. There are experimental observations in support of this assumption. For example, Matsumoto and Hayashi (14,15) observed that an increase in pressure mainly causes a transversal adaptation due to VSM hypertrophy and production of extracellular matrix towards an eventual restoration of baseline circumferential stress values. The same assumption has been used by Rachev et al. (20, 21), Taber (24, 25), and Taber and Eggers (26) to model arterial wall adaptation under sustained hypertension and yielded results in agreement with experimental observations.

With the use of Hill's model, the total circumferential stress is separated into its passive and active components, the active part being the stress developed by VSM. Adaptation rate equations are written for both geometrical adaptation (wall thickening) and adaptation of active stress (VSM tone). The model proposed here is a phenomenological one, therefore it does not account for particular cellular and intracellular mechanisms involved in arterial wall adaptation. The proposed wall thickening rate (*Eq. 12
*) and VSM tone adaptation (*Eqs.13
* and *
14
*) are proportional to the deviation of the mean total stress in hypertensive conditions from that under normal conditions, and indirectly related to VSM parameters in a highly complex manner. The model also makes the distinction between VSM tone at lower strains (*S*
_{bas}) and the VSM tone at higher strains, the latter being termed the myogenic mechanism. This distinction was on the basis of the biomechanical analysis of Fridez et al. (unpublished observations); however, it remains somewhat controversial at the level of mechanisms (18). The effect of λ_{cr} is integrated in the model through the definition of the active stress curve (*Eqs. 7
* and *
8
*) imposing an important increase in VSM tone in response to pressure-induced distension. The similarity in the adaptation rate equations for *S*
_{bas} and the λ_{cr} is for simplicity, and it does not necessarily reflect nor imply a similarity in nature and complexity of the underlying physiological mechanisms.

The proposed descriptions for the wall thickening rate (*Eq.12
*) and the VSM tone adaptation (*Eqs. 13
* and *
14
*) do not predetermine the time course of wall thickening nor that of active stress. Geometrical parameters, such as radius and thickness, and mechanical parameters, such as active stress and muscle parameters *S*
_{bas} and λ_{cr}, are coupled with total circumferential stress through the equations describing the stress and strain state of the artery and the equation of equilibrium in a complex and highly nonlinear manner. For example, if the circumferential stretch ratio or the active stress is used in*Eqs. 12-14
* as “driving stimulus” rather than the total circumferential stress, the model does not predict the appropriate characteristic time for thickening and circumferential stress (results not shown).

### Relative Rapidity of Adaptation Events

The experimental results of Fridez et al. (unpublished observations) show that the wall thickening and active stress dynamics are much faster and more significant processes than changes in the arterial geometry and parameters of the active response due to natural growth (see Fig. 2). Consequently, the characteristic time constants identified here could unequivocally be related to the adaptation process. This is confirmed by the fact that the time constants found for the model neglecting the changes of the passive mechanical properties and those found for the model, including these changes (structural remodeling) are of the same order.

The characteristic time constants are an indicator of the relative speed and the prominence of different mechanisms of adaptation. Values such as 1/τ_{H} and 1/τ_{S1} depict the characteristic speed at which the arterial wall adapts in proportion to the baseline values of thickness and *S*
_{bas}, respectively. From this point of view, the adaptation of *S*
_{bas} is more important than that of wall thickening: the characteristic speed of*S*
_{bas} adaptation is indeed ∼70 times “faster” than that of thickening (i.e., 1/τ_{S1} ≅ 70/τ_{H}). On*day 8* postsurgery, the significant increase in*S*
_{bas} (716%) compared with 18% increase in wall thickness stems from this 70-fold difference in speed. The time scale and the characteristic speed or rapidity of nonmonotonic adaptation mechanisms can also be seen by looking at their “return time,” the time when the adaptation switches direction, i.e., the time corresponding to the extrema of the *S*
_{bas} or λ_{cr} curves in Fig. 2, *B* and *C*. These figures show that the myogenic tone adaptation begins returning back to control values at *day 13.5* for*S*
_{bas} and at *day 6.2* for λ_{cr}. This indicates that the myogenic critical strain terminates its acute phase of adaptation twice as fast as that of the*S*
_{bas}. This is in agreement with the biomechanical analysis by Fridez et al. (unpublished observations) where it is proposed that, from a physiological point of view, the adaptation of the myogenic critical strain dominates the adaptation of*S*
_{bas} within the initial adaptation phase. The operating points of the artery (in vivo mean pressure and normal VSM tone; Fig. 1
*C*) lie in the *S*
_{bas} range for the control groups (at *day 0* and *day 56*). However, in the acute hypertension phase, they lie in the myogenic response range. This indicates that during the initial phase of adaptation, the vessel works in the myogenic response range, which dominates the early stages of the adaptation process.

### Limitations of the Study

The assumptions of the model define its limitations and perspectives for future investigations. The simple Hill's model used to separate the active and the passive stresses intrinsically neglects the mechanical coupling between VSM and the extracellular matrix. Also, the long-term remodeling of mechanical properties (passive stress) is not included in this model by means of appropriate remodeling rate equations. Considering an artery as a thin-walled membrane disregards the existence of residual strains in the arterial wall when the load is removed. This simplification may be of minor importance because the residual strains in the artery tend to bring the stress through the wall under working conditions to a uniform level. However, further verification of this would be necessary for the model to be applied to a relatively thicker muscular artery (the artery viewed as a thick-walled tube and the exact stress and strain distribution across the thickness calculated considering the residual strain).

In humans, hypertension develops, with a few exceptions, gradually and over several years. One may argue that if the rate of increase in pressure is low, the VSM response may not even be required and thus may never occur. Therefore, the step change in pressure modeled in this study is from a pathological and clinical perspective of limited relevance and any extrapolation to hypertension in humans should be made with caution.

In conclusion, experiments have shown that the VSM tone adaptation is a key point in understanding arterial adaptation in hypertension. Here we propose a simple phenomenological model describing geometrical and VSM tone adaptation in large arteries exposed to hypertension. The model has the merit to be the first one on the basis of a single comprehensive data set, it is robust to changes in parameter values within the physiological range, and yields quantitative information of the adaptation such as the characteristic time constants for the geometrical and VSM tone adaptation properties. This is also the first theoretical framework for arterial adaptation including the contribution of VSM tone.

## Acknowledgments

The work presented in this paper is partly funded by the Swiss National Science Foundation (Grant No. 2100-04321.94/2).

## Footnotes

Address for reprint requests and other correspondence: N. Stergiopulos, Biomedical Engineering Laboratory, Swiss Federal Institute of Technology, PSE-A Ecublens, 1015 Lausanne, Switzerland (E-mail: nikolaos.stergiopulos{at}epfl.ch).

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- Copyright © 2001 the American Physiological Society