INTRODUCTION

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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

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Experimental Analysis

7

4

View larger version (33K): [in this window] [in a new window]   Fig. 1.   A: evolution of mean pressure during the experiment (solid line, hypertensive; dashed line, normotensive). B: idealized step change in pressure used as load condition for the model. C: evolution of active stress as a function of stretch ratio for vascular smooth muscle (VSM) cells under maximal contraction (bold lines) and under normal VSM tone (thin lines). D: results of the theoretical model corresponding to the curves in C. Model for active stress under maximal contraction (first term of Eq. 7, bold line) is fitted through the entire set of experimental curves (controls, and 2, 4, 8, and 56 days postsurgery). The arrows show the nonmonotonic evolution of the two different parts of the active stress during hypertension (days are in bold characters). At low stretch ratio the basal tone ratio (Sbas) accounts for total active stress. At higher stretch ratio, the myogenic tone has the major effect on the active stress. Evolution of the myogenic tone is obtained by adjusting the myogenic critical strain (cr).

Theoretical Model of Arterial Wall

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

(1)

where R₀ and H₀ are the midwall radius and thickness, respectively, at the ZSS at the beginning of the experiments (day 0) and k₁ and k₂ 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 k₁ and k₂ 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 () and longitudinal direction are the mean measures of deformation, and are defined as follows

(2)

where 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. _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

(3)

Also, because the wall material is assumed to be incompressible, the deformed wall thickness h is

(4)

Considering 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, _tot) is represented as a sum of a passive stress (_pas), which is borne by the wall material when VSM is fully relaxed, and an active stress (_act) developed by VSM when it is contracted

(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 _pas = 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

(6)

where a₁, a₂, a₃, and a₄ 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

(7)

where c₁ and c₂ 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. 1D). The term (1  S_bas)f_myo() 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() is the following sigmoid function accounting for the strain-dependence of myogenic tone

(8)

where _cr is the strain at the inflection point of the sigmoid f_myo() and it can be regarded as the middle point of the range of  values associated with the myogenic response. Thus we will refer to _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() at the inflection point. Finally, ₀ is the stretch ratio of the circumferential midwall fiber at zero pressure and is therefore entirely determined by the experimental geometry. ₀  1 because the arterial segments are stretched longitudinally and this causes the length of the circumferential midwall fiber, ₀, to contract to a value <1. Fridez et al. (unpublished observations) reported a value of ₀ = 0.86 ± 0.01 for the rat common carotid artery, which was the value used in the present analysis.

(9)

Equilibrium equation of the arterial wall. Considering the overall equilibrium of the vessel in the circumferential direction, it follows that

(10)

where P is the mean arterial pressure and r_i is the deformed inner radius which, by using Eqs. 2 and 4, is described as

(11)

Adaptation Rate Equations for Wall Thickness

Induced arterial hypertension is modeled by a step increase in blood pressure from P^N to P^H (Fig. 1B), 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

(12)

where is the current total circumferential stress in the hypertensive artery at a given time, is the corresponding total circumferential stress in the normotensive artery at the same time and _H is the characteristic time constant for the adaptation speed of wall thickness.

Adaptation Rate Equations for VSM Tone

(13)

(14)

1

2

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.

(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

(16)

	RESULTS

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Model Parameter Identification

1

2

1

2

Model Predictions of Thickening and VSM Tone Adaptation

View larger version (18K): [in this window] [in a new window]   Fig. 2.   Time evolution in response to a step change in pressure of wall thickness at zero load state (A), Sbas (B), and cr (C). Solid lines show the results of the model. Dashed lines show the evolution of the control (normotensive) as a result of natural animal growth. Symbols denote experimental results of Fridez et al. (unpublished observations).

Model Predictions of Total and Active Stress Evolution

View larger version (23K): [in this window] [in a new window]   Fig. 3.   Time evolution of total stress (A) and active stress (B). Solid lines show the predictions of the model (passive elastic properties are kept constant). Dashed lines show the evolution of the control (normotensive) as a result of the natural animal growth. Symbols denote experimental results of Fridez et al. (unpublished observations).

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 _H, _S1, _S2, ₁, and ₂ obtained by using the extended model are given in Table 3.

View larger version (22K): [in this window] [in a new window]   Fig. 4.   Time evolution of total stress (A) and active stress (B). Solid lines show the predictions of the model (adaptation of the passive elastic properties are taken into account). Dashed lines show the evolution of the control (normotensive) as a result of natural animal growth. Symbols denote experimental results of Fridez et al. (unpublished observations).

	DISCUSSION

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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

Adaptation Rate Equations

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 _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 _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 _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 characteristic time constants are an indicator of the relative speed and the prominence of different mechanisms of adaptation. Values such as 1/_H and 1/_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/_S1  70/_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 _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 _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. 1C) 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

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.