Am J Physiol Heart Circ Physiol 293: H3130-H3139, 2007.
First published September 7, 2007; doi:10.1152/ajpheart.00684.2007
0363-6135/07 $8.00
Arterial remodeling in response to hypertension using a constituent-based model
Alkiviadis Tsamis and
Nikos Stergiopulos
Laboratory of Hemodynamics and Cardiovascular Technology, École Polytechnique Fédérale de Lausanne, Lausanne, Switzerland
Submitted 13 June 2007
; accepted in final form 5 September 2007
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ABSTRACT
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Hypertension-induced arterial remodeling has been previously modeled using stress-driven remodeling rate equations in terms of global geometrical adaptation (Rachev A, Stergiopulos N, Meister JJ. Theoretical study of dynamics of arterial wall remodeling in response to changes in blood pressure. J Biomech 29: 635–642, 1996) and was extended later to include adaptation of material properties (Rachev A, Stergiopulos N, Meister JJ. A model for geometric and mechanical adaptation of arteries to sustained hypertension. J Biomech Eng 120: 9–17, 1998). These models, however, used a phenomenological strain energy function (SEF), the parameters of which do not bear a clear physiological meaning. Here, we extend the work of Rachev et al. (1998) by applying similar remodeling rate equations to a constituent-based SEF. The new SEF includes a statistical description for collagen engagement, and remodeling now affects material properties only through changes in the collagen engagement probability density function. The model predicts asymptotic wall thickening and unchanged deformed inner radius as to conserve hoop stress and intimal shear stress, respectively, at the final adapted hypertensive state. Mechanical adaptation serves to restore arterial compliance to control levels. Average circumferential stress-strain curves show that the material at the final adapted hypertensive state is softer than its normotensive counterpart. These findings as well as the predicted pressure-diameter curves are in good qualitative agreement with experimental data. The novelty in our findings is that biomechanical adaptation leading to maintenance of compliance at the hypertensive state can be perfectly achieved by appropriate readjustment of the collagen engagement profile alone.
vascular adaptation; material properties; collagen engagement
SUSTAINED CHANGES in pressure, flow, or axial stretch cause a long-term transformation of an artery, a phenomenon often termed remodeling. Remodeling is a dynamic process, manifested as changes in both the geometrical dimensions and material properties of an artery, and is dependent on mechanical stresses and strains. In certain pathological cases with altered mechanical loads, such as in hypertension or changes in blood flow, it has been experimentally shown (17, 20) that remodeling serves to restore hoop stress and intimal shear stress, respectively, to control levels.
There have been many earlier attempts to provide for physiologically relevant theoretical models of vascular remodeling. Hypertension-induced arterial remodeling has been modeled by means of stress-driven evolution laws applied to the global geometrical characteristics of the zero-stress state (ZSS) (23) and was extended (24) to include adaptation of material properties. Temporal changes in stress, geometry, and opening angle of arteries were computed by Taber (27) during development and following the onset of sudden hypertension based on the theory of local volumetric growth (25). A similar model was also employed by Taber and Humphrey (28), who inferred that opening angles depend strongly on the heterogeneity of material properties of the vessel wall, and thus a combination of radial and circumferential cuts may be needed to produce a stress-free configuration. A fundamentally different approach was followed by Gleason and Humphrey (11, 12), who simulated the responses of common carotid arteries to sustained changes in pressure by means of a constrained mixture model, wherein individual constituents turnover at different rates and have different natural configurations.
These models have been based on the phenomenological SEF of Chuong and Fung (3) to describe the elastic properties of the wall. The use of this phenomenological SEF, however, results in two major shortcomings: first, its elastic constants do not bear a clear physiological meaning and second, and most important, it is assumed that the wall is a homogeneous medium, and thus it is impossible to ascribe different remodeling laws to the different wall components [i.e., elastin, collagen, and vascular smooth muscle (VSM) cells] that are affected differently by the remodeling process.
To overcome the aforementioned shortcomings, we revisit and extend the theoretical model for arterial remodeling under sustained hypertension of Rachev et al. (24), whereby the phenomenological SEF of Chuong and Fung (3) is replaced by the constituent-based SEF proposed by Zulliger and Stergiopulos (35). The new SEF, although still pertaining to a homogeneous averaged one-component material, accounts for the main wall constituents and their structural properties, giving thus a clear physiological meaning to the constants defining the elastic properties of the arterial wall. The theoretical model is applied to simulate arterial wall remodeling under sustained hypertension, and the model predictions are compared with existing experimental data to validate, in a qualitative sense, the predictive power of the model and assess the physiological relevance of the results.
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METHODS
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Mathematical Model of the Arterial Wall
The artery is modeled as a thick-walled tube made of nonlinear, elastic, anisotropic, and incompressible material. In the ZSS, the vessel cross section is considered to be a circular sector with the following dimensions: inner radius of curvature Ri, outer radius of curvature Ro, and opening angle 
(Fig. 1A). The inner arc length (Li), outer arc length (Lo), and thickness (H) are calculated using the following formulas:
 | (1) |
Under physiological loading, arteries are in a finite axisymmetric plane strain state. The deformed arterial cross section is shown in Fig. 1B. The governing equations and the equations used to derive the Cauchy stress field can be found in the original article (24) and are given for completeness in APPENDIX A.
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Fig. 1. Global geometrical adaptation. A: schematic diagram of an artery cross section in the zero-stress state (ZSS). , Opening angle;  , angle; R, radius; Ri, inner radius of curvature; Ro, outer radius of curvature; H, thickness of the artery. B: schematic diagram of an artery cross section in the loaded state.  , circumference in the loaded state; r, radius in the loaded state; ri, deformed inner radius; ro, deformed outer radius.
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Strain Energy Function
We base our analysis on a constituent-based SEF (W), which accounts for the constituents and structural properties of the arterial wall. This SEF was proposed by Zulliger and Stergiopulos (35), and we recapitulate its important features here. The general approach is a separation of the SEF into isotropic and anisotropic parts (15). We assume that the three main constituents of the arterial wall (elastin, collagen, and VSM cells), each one represented by an individual SEF (Welast, Wcoll, and WVSM), coexist in one homogeneous layer and act in parallel. For the sake of simplicity, we hypothesize that VSM cells are totally passive. The contribution of WVSM is then neglected on the basis that the passive elastic modulus of VSM has been reported to be about one order of magnitude less than that of elastin (1, 30). If we multiply Welast and Wcoll by the respective mass fraction occupied by each constituent (felast and fcoll), then Eq. A3 results in effective stresses for each component. Hence, adding the products algebraically yields the total SEF:
 | (2) |
However, VSM still exists in the arterial wall, occupying a mass fraction (fVSM), and for this reason felast + fcoll + fVSM = 1 or felast + fcoll < 1.
Elastin is a soft nonlinearly elastic material (21) and is assumed here to be isotropic. It was found (34) that a good description for the SEF Welast is the following convex relation:
 | (3) |
where celast > 0 is a parameter describing the elastic properties of elastin; I1 is the first invariant of the Cauchy-Green strain tensor (26); and Er, E, and Ez are radial, circumferential, and axial strain, respectively. As can be seen from Eqs. 2 and 3, felast and celast are always multiplied together, and they act as a single parameter (felast x celast). So, effectively, a lower value of felast is compensated for by a higher value of celast. The same holds for collagen, which is discussed later. According to Zulliger et al. (34), the attempt to use a simpler Neo-Hookean SEF for elastin fails to reproduce the initial phase of both the pressure-radius and pressure-axial force experiments with similar precision as the here-proposed SEF.
Contrary to elastin, collagen fiber mesh is stiff, nonlinear, and anisotropic. Moreover, collagen fibers appear to be coiled and wavy in their unloaded state (4, 5). We assume that collagen fibers are oriented circumferentially (5, 32). We further hypothesize that collagen engagement as a function of E is distributed in some statistical manner (32, 33). Following Zulliger and Stergiopulos (35), we assume that the probability density function (PDF) describing E at which collagen fibers engage in bearing load is the log-logistic PDF, 
fiber (Fig. 2):
 | (4) |
where E0 
0 is a lower bound E, below which all collagen fibers do not bear load; b > 0 is a scaling parameter and k > 0 is a parameter defining the shape of the PDF. Strain E0 cannot be negative, because in that case a percentage of collagen fibers would be load bearing at E = 0 and this would have meant that the ZSS does not hold true.
In Eq. 4, when parameter k increases, PDF variance decreases and its height increases, restricting the strain domain over which collagen engages and thus making collagen fiber engagement more abrupt (Fig. 2A). Parameter b is associated with the E in which 50% of collagen fibers are engaged. When parameter b increases, PDF shifts to the right (Fig. 2B), making collagen fiber engagement occur at higher E. At the same time, PDF height decreases and PDF variance increases, widening the strain domain over which collagen engages and thus making collagen fiber engagement less abrupt (Fig. 2B).
An individual collagen fiber is assumed to behave as a linear isotropic material described by the fiber SEF (Wfiber) as follows:
 | (5) |
where ccoll > 0 is a parameter describing the elastic properties of collagen and E is the local strain in the direction of the fiber. To describe the ensemble of circumferentially oriented collagen fibers, we can convolute Wfiber with fiber:
 | (6) |
Remodeling Rate Equations
Geometrical adaptation.
The remodeling rate equations describing the evolution of the geometry of the ZSS are identical to those of Rachev et al. (24). In brief, changes in geometry are driven by the differences in wall stress between the hypertensive and normotensive state. Inner and outer arc lengths are remodeled according to their local circumferential stress, wall thickness is remodeled according to average circumferential stress (
,av), and arterial length is remodeled according to average axial stress (z,av). The deformed inner radius (ri) is remodeled to restore intimal shear stress. Details on the formulation of the remodeling laws can be found in the original article (24) and are given in succinct format in APPENDIX B.
Remodeling effects on material properties (collagen turnover).
Arterial compliance depends on material parameters as well as geometry and loading state. At physiological pressures, an increase in blood pressure results in a decrease in arterial compliance. In the absence of changes in material properties, hypertension-induced remodeling would not restore arterial compliance to control levels, because geometrical adaptation results in wall thickening. Changes in compliance mean that the amplitude of the pulsatile E experienced by the arterial wall will be different assuming that the pulse pressure and heart rate remain unchanged. This would then affect the production of structural components of the arterial wall by VSM cells (18). It is therefore postulated that remodeling of material properties serves to restore arterial compliance at operating arterial pressure. So, the range within which the pulsatile strain varies is restored to baseline values. Since the vascular compliance is related to the pulse wave velocity, this implies that the pulse wave velocity returns to control values, and, in general, the loading on the heart remains unchanged (31).
Area compliance (CA), given by
 | (7) |
depends on all SEF parameters (felast, celast, fcoll, ccoll, k, and b). In Eq. 7, A is the lumen area of the vessel at pressure P and in situ length, and 
A is the increment of the lumen area for the pressure increment P. felast and fcoll do not undergo significant changes under sustained hypertension (8), despite an arterial thickness increase. Moreover, celast and ccoll remain constant, because remodeling does not alter the inherent material characteristics of these fibers. The choice of parameters is thus restricted to k and b, which define the collagen engagement characteristics. Because CA seems to be more sensitive to changes in parameter b rather than k, and changes in k alone do not suffice for the full restoration of compliance, we hypothesize that adaptation of CA results primarily from changes in parameter b. It is therefore assumed that the rate of change of parameter b is driven by CA. Thus, the following remodeling rate equation is postulated to account for mechanical adaptation:
 | (8) |
where 
(t) = bH(t)/bN is a material growth parameter and tM is a characteristic time constant. Superscripts N and H denote values under normotensive and hypertensive conditions, respectively. A positive mismatch between compliances [CAN – CAH(t)] leads to an increase in parameter b. This causes collagen to engage at higher E, thereby reducing the percentage of collagen fibers being engaged under hypertension and increasing arterial compliance.
Integration Procedure
The integration procedure of the governing equations comprises six stages, which are shown and explained in Fig. 3. Details on a similar integration procedure can be found in Ref. 24. In brief, stage 1 gives the initial values of deformed inner radius ri and stresses under normotensive conditions. Stage 2 calculates deformed geometry and stresses immediately following the onset of hypertension (t = 0). In stage 3, remodeling rate equations are integrated in time to yield the estimates of ZSS and elastic constant b at the next time step. In stage 4, growth parameters 
, 
, and 
are adjusted so that they are consistent with the shear-stress-driven adaptation of ri (Eq. B4). In stage 5, the new ZSS geometry is obtained, based on the growth parameters, and the new stress and strain field is calculated. In stage 6, the convergence criterion is applied, according to which the remodeling process terminates if compliance is restored.
Arterial Model
The theoretical study of typical arterial remodeling in response to hypertension was based on data available in the literature. Data for the ZSS of a rabbit thoracic aorta (Lo = 11.25 mm, Li = 9.75 mm, and = 108.6°) were taken from Ref. 3, with normotensive pressure (PN) = 13.33 kPa (100 mmHg) and in situ axial stretch ratio (
zN) = 1.6, respectively; felast was assumed to equal 0.3 and fcoll was assumed to equal 0.2, respectively (8). The celast was set to 43 kPa (34), and ccoll = 130 MPa (2, 21). Parameters k and b of the PDF were chosen as k = 30 and bN = 0.98, to yield a physiological pressure-diameter curve for the normotensive rabbit thoracic aorta. For simplicity, and based on unpublished pressure-diameter data from pig carotids we have taken, E0 = 0, which means that collagen fibers are gradually involved in load bearing when E > 0. Finally, for the purposes of qualitative analysis, induced hypertension was simulated by a step increase to PH = 21.33 kPa (160 mmHg).
Time was nondimensionalized with respect to characteristic time T, t* = t/T, and dimensionless time constants were chosen (t*L = 10, t*H = 1, t*Z = 10, t*Q = 1, and t*M = 70 in Eqs. 8 and B2–B4) such that all aspects of geometrical remodeling take place at the same pace and mechanical adaptation is the slowest procedure (20). Table 1 shows model predictions for the geometrical and mechanical characteristics of the adapted hypertensive aorta compared with their values under PN.
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Table 1. Geometrical and mechanical characteristics of the adapted hypertensive artery as predicted by the model compared with their normotensive counterparts
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RESULTS
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The model predicts a rapid asymptotic increase of ZSS thickness H toward a new steady state (Fig. 4A). The time course of is nonmonotonic, and the final adapted value is close to control (Fig. 4B). Li exhibits a transient upward peak followed by a gradual decrease toward an asymptotic value close to and below its normotensive counterpart (Fig. 4C). Lo shows a transient downward peak followed by a relatively rapid increase toward a new homeostatic value close to and above control (Fig. 4D). An abrupt jump of ri is initially caused by the step increase in pressure (Fig. 4E); ri is restored thereafter to its initial normotensive value. The axial in situ stretch ratio z stays fairly constant during the remodeling process (Fig. 4F).
Initially, average circumferential stress ,av and average axial stress z,av increase abruptly following the pressure increase (Fig. 5, A and B); ,av decreases thereafter monotonically toward control. After its initial increase, z,av varies nonmonotonically toward normotensive levels.
Material property b exhibits a gradual monotonic increase toward a new steady state (Fig. 6A). CA, which is a structural property and varies with the loading state, initially drops as a result of the increase in pressure and is asymptotically augmented thereafter toward control (Fig. 6B). The step increase in pressure causes an initial abrupt increase of pressure-strain elastic modulus Ep (Fig. 6C), as the latter is not a material constant and depends on the loading conditions. Then, Ep is asymptotically reduced toward a new steady state that is close to and above its baseline counterpart. The pressure jump results in an abrupt increase of incremental elastic modulus H (Fig. 6D). Then, changes in geometry and material property b lead to a monotonic decrease in H toward a new homeostatic state that is close to and below its normotensive counterpart.
The normalized outer radius (ro) versus pressure curve (Fig. 7A) under normotensive conditions (dashed line) has shifted to the left in the final adapted hypertensive state (solid line). In the final adapted hypertensive state (solid line), Ep is higher than its normotensive counterpart (dashed line) for P < 12 kPa (90 mmHg), whereas Ep becomes lower than control for higher values of pressure (Fig. 7B). The ,av – average circumferential stretch (,av; defined at the midwall radius) curve remains relatively unchanged for ,av < 1.6 (Fig. 7C), whereas for ,av > 1.6, the curve for the adapted hypertensive artery (solid line) is below its normotensive counterpart (dashed line).
The changes in elastic parameter b (Fig. 6A) impact on the PDF of collagen fiber engagement in a way similar to that shown in Fig. 2B, with k = 30 and b = 0.98 
1.08. The increase in b shifts PDF to higher E (solid line) compared with control (dashed line). At the same time, the PDF height decreases and its variance increases.
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DISCUSSION
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Hypertension-induced arterial remodeling has been modeled earlier (23) using stress-driven remodeling rate equations in terms of global geometrical adaptation (Fig. 1) and was extended (24) to include adaptation of material properties. Models in both cases, however, used the phenomenological SEF of Ref. 3, the parameters of which do not bear a clear physiological meaning. We here extend the work of Rachev et al. (24), where the SEF of Ref. 3 is replaced by the constituent-based SEF proposed by Zulliger and Stergiopulos (35). The new SEF accounts for the constituents and structural properties of the wall, including a PDF for collagen fiber engagement. Remodeling rate equations are similar to those of Ref. 24, with the exception that compliance affects the properties of collagen engagement PDF, and this leads to a simpler set of remodeling rate equations with a better description of the collagen turnover process, which is central to remodeling phenomena under hypertension.
Remodeling Using a Constituent-Based SEF
The approach taken in this study was to extend and improve the previous theoretical model of global growth and remodeling of Ref. 24, which gave qualitatively interesting results for the arterial response to induced hypertension. Rachev et al. (24) employed the following phenomenological SEF of Chuong and Fung (3):
 | (9) |
to describe the elastic properties of the arterial wall. As discussed earlier, elastic constants c and b1–b6 do not bear a clear physiological meaning and cannot be assigned exclusively to the contribution of individual wall constituents. Mechanical adaptation was simulated in terms of changes in parameters c, b1, and b2 only. The rate of change of c was driven by the altered CA. b1 and b2 varied in a manner to maintain the current level of ,av and ,av. Here, mechanical adaptation is simulated by means of varying only one constant (b) as opposed to three constants (c, b1, and b2), yet we obtained results similar to those of Rachev et al. (24). Furthermore, parameter b has a clear physiological meaning: it defines the engagement characteristics of the collagen fiber network. Since we know that in hypertension-induced remodeling collagen is primarily turned over and elastin remains relatively unaffected (14), the choice of parameter is physiologically justified. Thus, our model yields qualitatively similar results to the one of Ref. 24 and allows monitoring of mechanical adaptation via a single and physiologically relevant parameter (b).
Geometrical Adaptation
With the chosen values of time constants, the predicted time course of geometrical remodeling precedes mechanical adaptation. Our results show that the time wall thickness reaches 80% of its asymptotic value (Fig. 4A) and the time at which compliance is approximately restored to baseline levels (Fig. 6B) is 1:10. It was found that 80% of the wall thickness increase takes place within 10 days from the induction of hypertension (9) and mechanical adaptation of the aorta lasts 
100 days (20), which gives the same order of time ratio between the two events. Furthermore, the model predicts that wall thickness reaches 80% of its asymptotic value in about t* = 2, or the real time is t = 2T, which according to experimental data (9) is equal to 10 days. Therefore, 2T 
10 days or the characteristic time T 5 days.
The time courses of ZSS thickness H (Fig. 4A) and opening angle (Fig. 4B) are in good qualitative agreement with those of Liu and Fung (19), who observed that the wall thickens fast, reaching an asymptote in 1 mo, whereas increases rapidly, reaching a maximum at the second to fifth day and decreases gradually thereafter toward an asymptote above or below control, depending on the aortic location. The time course of deformed inner radius ri (Fig. 4E) agrees qualitatively with the results of Langille (17), who concluded that the time period for the first inner radius adjustments ranges from few days to 2 wk. As in our previous model study (24), and in contrast to experimental data (29), no significant changes in axial stretch ratio z are predicted by the model (Fig. 4F).
We observe that average circumferential stress ,av (Fig. 5A) is restored at the time when thickness H reaches its new homeostatic value (Fig. 4A). This is a result of Laplace's law and of the fact that ri adapts at a faster rate than wall thickness (compare Fig. 4, A–E). Restoration of average axial stress z,av is, however, slower than that of ,av, because time constant t*Z=10 was chosen to be larger than time constant t*H = 1 (Fig. 4F).
Mechanical Adaptation
Area compliance CA is restored (Fig. 6B) at the time when material parameter b reaches its new homeostatic state (Fig. 6A). According to Eq. A10, pressure-strain elastic modulus Ep is not entirely restored (Fig. 6C) because deformed outer radius ro at the adapted hypertensive artery is bigger than control due to eccentric wall thickening. Higher Ep has been observed in the rat aorta 16 wk after the induction of hypertension (20), a time at which it is accepted that mechanical adaptation has already been completed.
The abrupt increase of Hudetz incremental elastic modulus H after the onset of sudden hypertension (Fig. 6D) is mainly due to collagen fiber engagement. The asymptotic decrease of H below control implies a softer material at the final adapted hypertensive state than that under normal pressure. The tendency of H to decrease with time has been also observed by Matsumoto and Hayashi (20).
P versus ro curves (Fig. 7A) and Ep versus P curves (Fig. 7B) express the structural response of the artery and are in good qualitative agreement with the data of Matsumoto and Hayashi (20). Ep versus P curves (Fig. 7B) predict that the arterial wall of the adapted hypertensive artery is less compliant (higher Ep) than control for P < 12 kPa (90 mmHg) and becomes more compliant (lower Ep) than control for higher values of P. Although the hypertensive aorta grows thicker, which increases structural stiffness, the material softening discussed above leads to increased compliance (reduced Ep) when compared at equal P values. This result is in agreement with the data of Matsumoto and Hayashi (20).
Softening of wall material can also be concluded by the slope, H, of the ,av versus ,av curves shown in Fig. 7C. We observe that the curves under normotensive and adapted hypertensive conditions almost coincide for ,av < 1.6. This means that for this range of ,av, in which elastin [unaffected by remodeling (14)] dominates, material properties of the normotensive and adapted hypertensive artery are similar. For higher values of ,av, in which collagen (affected by remodeling) dominates, the slope of the solid curve is lower than that of the dashed curve (Fig. 7C), implying that the material has become softer than its normotensive counterpart when compared at equal ,av. This is in good qualitative agreement with the data of Vaishnav et al. (29), who found that, after induced hypertension, stress-strain curves were similar to controls at low strains.
Mechanical adaptation increases the value of material parameter b, shifting the PDF to higher E (as in Fig. 2B, with k = 30 and b = 0.98 1.08). At the same time, PDF height decreases and its variance increases. In consequence, collagen fibers engage at higher E and more gradually, resulting in a softer material at the final adapted hypertensive state. Material softening implies that, during mechanical adaptation and collagen turnover, newly produced collagen is laid down in a matrix subjected to higher levels of circumferential stretch. Hence, when brought to the ZSS, these newly produced collagen fibers would be wavier, and thus it would take higher strains to attain comparable levels of collagen engagement compared with normotensive arteries. This conclusion is in agreement with the constrained mixture model of Gleason and Humphrey (11, 12), whereby it was postulated that each newly produced constituent is laid down at homeostatic stretch or over a distribution of stretches around homeostatic stretch levels. Therefore, hypertension-induced remodeling seems to be associated with a decrease in the stretch of collagen fibers, resulting in material softening and compliance restoration.
Limitations
The present model ignores a number of potentially important effects. First, stress-driven processes at the molecular, cellular, and tissue levels are all lumped into a "black box" expressed by phenomenological remodeling rate equations linking the cause (stress) and the effect (remodeling), disregarding other important growth-related factors, such as angiotensin I and II, platelet-derived growth factor, etc. Future models will need to take into account the different biological mechanisms involved in the remodeling process. Second, a more realistic study should consider the distribution of collagen fiber orientations (10) as well as different types of collagen separately, because hypertension may provoke different effects on the composition and structure of the different collagen types. Third, we suggested the hypothesis that all constituents coexist within one homogeneous layer, disregarding the heterogeneous multilayer structure of the arterial wall. Fourth, the contribution of VSM is neglected by assuming that VSM is totally passive, which is clearly not the case, especially in the early phase of wall remodeling in response to acute hypertension (6, 7). Finally, a better validation might be necessary for the assumption we made about the unchanged felast and fcoll (8).
Conclusions
In summary, we proposed a constituent-based model for the evolution of geometrical and mechanical adaptation of the arterial wall after a step increase in blood pressure. We suggested that geometrical adaptation aims at maintaining wall stress distribution and flow-induced intimal shear stress, whereas mechanical adaptation, principally through collagen turnover and reorganization, restores arterial function expressed in terms of CA. The model results agree qualitatively with reported experimental data for hypertension-induced remodeling. We conclude that a constituent-based SEF, based on which compliance is restored by means of changes in collagen properties only, seems an appropriate choice for modeling the arterial wall adaptation in response to hypertension. The proposed model, being a good approximation of reality as it includes the major mechanical characteristics of arteries and the remodeling process, can serve as a theoretical basis for a better understanding of arterial remodeling under a variety of physiological and pathological conditions.
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APPENDIX A: GOVERNING EQUATIONS
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Considering the deformation process to be axisymmetric, the principal stretch ratios in the radial, circumferential and longitudinal direction (r, , and z), respectively, are given by the following:
 | (A1) |
where (R, , Z) are the cylindrical coordinates of an arbitrary point in the ZSS (Fig. 1A); (r, , z) are the coordinates of the same point in the loaded state (Fig. 1B); 
= 
/( – ), where is the opening angle; and 
is the axial stretch per unit unloaded length. For the incompressible arterial wall, the condition rz = 1 or rdr = RdR must be enforced. Integration of the resultant differential equation yields the following relation between r and R:
 | (A2) |
where subscripts i and o denote inner and outer radii, respectively.
Local radial, circumferential, and longitudinal Cauchy stresses (r, , and z, respectively) in the arterial wall are as follows:
 | (A3) |
where p is a Lagrange multiplier, Ej = (j2 – 1)/2 are the nonvanishing components of Green strain tensor E (26), and W(Er, E, Ez) is the SEF. Integration of the differential equations of equilibrium across the entire wall yields the following expressions for r, , and z:
 | (A4) |
 | (A5) |
 | (A6) |
Blood pressure (P) and axial force (Fz) relate to wall stresses as follows:
 | (A7) |
where z is axial stress. Average circumferential stress (,av) and average z (z,av) are given by:
 | (A8) |
Measures of the linearized mechanical behavior of the artery around certain deformed states are the pressure-strain elastic modulus Ep (22), which is defined as:
 | (A9) |
and area compliance CA (13), which is given by Eq. 7. Ep and CA are related through:
 | (A10) |
by virtue of the incompressibility condition. Ep and CA are not material parameters. They are structural properties and depend on material parameters as well as the geometry and loading state.
An average measure of the inherent elastic properties of the wall material in the neighborhood of a known deformed state is the incremental elastic modulus (H), which was introduced by Hudetz (16) as follows:
 | (A11) |
The mean shear stress (
) at the inner arterial surface is given by Poiseuille's law:
 | (A12) |
where Q is blood flow and 
is blood viscosity.
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APPENDIX B: GEOMETRICAL ADAPTATION
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The artery is first considered under normal blood pressure (PN). The ZSS dimensions are LiN, LoN, and HN; the axial stretch ratio is zN. Induced arterial hypertension is modeled by a step increase in blood pressure from PN to PH. Superscripts N and H denote values under normotensive and hypertensive conditions, respectively.
To monitor the remodeling of ZSS, the following growth parameters (, , , and 
) are defined:
 | (B1) |
and the following remodeling rate equations are postulated:
 | (B2) |
 | (B3) |
where ,iH (t), ,oH (t), ,avH (t), and z,avH (t) are current circumferential stress at the inner surface, current circumferential stress at the outer surface, average circumferential stress, and average axial stress of the hypertensive artery, respectively; ,iN (t), ,oN (t), ,avN (t), and z,avN (t) are the corresponding stresses in the normotensive artery; and tL, tH, and tZ are characteristic time constants.
The following remodeling rate equation is postulated for the evolution of the deformed inner radius of the hypertensive artery (riH):
 | (B4) |
where shear stress is given by Eq. A12 and tQ is a characteristic time constant. The term in brackets on the righthand side of Eq. B4 is the deviation of flow-induced shear stress from its value under normotensive conditions. Assuming constant blood flow Q during hypertension-induced remodeling, substitution of Eq. A12 into Eq. B4 yields the following:
 | (B5) |
ri can also be calculated from Eqs. A2–A7 based on the current ZSS and loading. To make the calculated value of ri compatible with the value resulting from Eq. B4, the following hypothesis is introduced: at any moment during geometrical adaptation of the hypertensive artery, its deformed inner radius ri follows the compensatory adjustment driven by the flow-induced shear stress.
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GRANTS
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This work was supported by Swiss National Scientific Research Fund Grant 3100A0-100423.
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FOOTNOTES
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Address for reprint requests and other correspondence: A. Tsamis, Laboratory of Hemodynamics and Cardiovascular Technology, École Polytechnique Fédérale de Lausanne, AI 1140, Station 15, Lausanne CH-1015, Switzerland (e-mail: alkiviadis.tsamis{at}epfl.ch)
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.
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