Vol. 280, Issue 6, H2752-H2760, June 2001
Model of geometrical and smooth muscle tone adaptation of
carotid artery subject to step change in pressure
P.
Fridez1,
A.
Rachev2,
J.-J.
Meister1,
K.
Hayashi3, and
N.
Stergiopulos1
1 Biomedical Engineering Laboratory, Swiss Federal Institute
of Technology, 1015 Lausanne, Switzerland; 2 Institute of
Mechanics, 113 Sofia, Bulgaria; and 3 Division of Mechanical
Science, Department of Systems and Human Science, Graduate School
of Engineering Science, Osaka University, Osaka 560, Japan
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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
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INTRODUCTION |
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.
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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.
1A). 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 104 [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. 1C).
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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).
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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 (RN and
HN, respectively) are described as a function of
time through the following relationships
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(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
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(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
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(3)
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Also, because the wall material is assumed to be
incompressible, the deformed wall thickness h is
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(4)
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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
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(5)
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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,ez)
(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
![&sfgr;<SUB>pas</SUB><IT>=&lgr;<SUP>2</SUP></IT>[<IT>a<SUB>1</SUB>+a<SUB>2</SUB>e+a<SUB>3</SUB> </IT>exp(<IT>a<SUB>4</SUB>e</IT>)]](/content/vol280/issue6/fulltext/H2752/img006.gif)
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(6)
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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
![&sfgr;<SUB>act</SUB><IT>=</IT>(<IT>c<SUB>1</SUB>+c<SUB>2</SUB>&lgr;</IT>)<IT>·</IT>[<IT>S</IT><SUB>bas</SUB><IT>+</IT>(<IT>1−S</IT><SUB>bas</SUB>)<IT>·f</IT><SUB>myo</SUB>(<IT>&lgr;</IT>)]](/content/vol280/issue6/fulltext/H2752/img007.gif)
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(7)
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where c1 and c2
are material constants representing the maximal capacity of the VSM to
contract. Sbas is the basal tone ratio, that is
the ratio of active stress at normal VSM tone to active stress under
maximal contraction. Sbas accounts for VSM tone
at lower strains and it is independent of strain (Fig. 1D).
The term (1 
Sbas)fmyo() 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
Sbas) and basal-myogenic tone (our myogenic
tone) in their study. The factor (1 Sbas) in Eq. 7 means that the
myogenic tone can only operate within the range delimited by the total
contraction (S = 1), and Sbas. In other words, it represents the remaining tone capacity of the VSM.
Finally, fmyo() is the following sigmoid
function accounting for the strain-dependence of myogenic tone
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(8)
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where cr is the strain at the inflection point of
the sigmoid fmyo() 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 fmyo() 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.
The parameters in Eq. 7 contribute to the active stress in
different ways. The constants c1, c2, q,
and 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,
c1, c2, q, and
0 are constant for a given artery and their values are
not dependent of VSM activation. The parameters
Sbas and 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
Sbas and 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. Sbas and
cr vary during the development and maturation. Within
the period of duration of our experiments, it is assumed that the time
dependence of Sbas and cr is
appropriately represented by a linear function of the form
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(9)
|
where Sbas0 and cr0 are
the basal tone ratio and myogenic critical strain values at the
beginning of the experiments and k3 and
k4 are constants accounting for the rate of
change of Sbas and cr, due to
natural animal growth.
Equilibrium equation of the arterial wall.
Considering the overall equilibrium of the vessel in the
circumferential direction, it follows that
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(10)
|
where P is the mean arterial pressure and
ri is the deformed inner radius which, by using
Eqs. 2 and 4, is described as
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(11)
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Adaptation Rate Equations for Wall Thickness
Under normotensive conditions, the midwall radius and wall
thickness at zero-load are RN and
HN. The vessel is subjected to normal blood
pressure, PN, 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 PN to PH
(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
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(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
Figure 1C 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 1C 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. 1D). 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 Sbas and 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
Sbas and cr
![<FR><NU>dS<SUP>H</SUP><SUB>bas</SUB>(<IT>t</IT>)</NU><DE><IT>dt</IT></DE></FR><IT>=</IT><FR><NU><IT>S</IT><SUP>N</SUP><SUB>bas</SUB>(<IT>t</IT>)</NU><DE><IT>&tgr;<SUB>S1</SUB></IT></DE></FR><IT>·</IT><FR><NU><IT>&sfgr;</IT><SUP>H</SUP><SUB>tot</SUB>(<IT>t</IT>)<IT>−&sfgr;</IT><SUP>N</SUP><SUB>tot</SUB>(<IT>t</IT>)</NU><DE><IT>&sfgr;</IT><SUP>N</SUP><SUB>tot</SUB>(<IT>t</IT>)</DE></FR><IT>−</IT><FR><NU><IT>1</IT></NU><DE><IT>&tgr;<SUB>S2</SUB></IT></DE></FR> [<IT>S</IT><SUP>H</SUP><SUB>bas</SUB>(<IT>t</IT>)<IT>−S</IT><SUP>N</SUP><SUB>bas</SUB>(<IT>t</IT>)]](/content/vol280/issue6/fulltext/H2752/img015.gif)
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(13)
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![<FR><NU>d&lgr;<SUP>H</SUP><SUB>cr</SUB>(<IT>t</IT>)</NU><DE><IT>dt</IT></DE></FR><IT>=</IT>−<FR><NU><IT>&lgr;</IT><SUP>N</SUP><SUB>cr</SUB>(<IT>t</IT>)</NU><DE><IT>&tgr;<SUB>&lgr;1</SUB></IT></DE></FR><IT>·</IT><FR><NU><IT>&sfgr;</IT><SUP>H</SUP><SUB>tot</SUB>(<IT>t</IT>)<IT>−&sfgr;</IT><SUP>N</SUP><SUB>tot</SUB>(<IT>t</IT>)</NU><DE><IT>&sfgr;</IT><SUP>N</SUP><SUB>tot</SUB>(<IT>t</IT>)</DE></FR><IT>+</IT><FR><NU><IT>1</IT></NU><DE><IT>&tgr;<SUB>&lgr;2</SUB></IT></DE></FR> [<IT>&lgr;</IT><SUP>H</SUP><SUB>cr</SUB>(<IT>t</IT>)<IT>−&lgr;</IT><SUP>N</SUP><SUB>cr</SUB>(<IT>t</IT>)]](/content/vol280/issue6/fulltext/H2752/img016.gif)
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(14)
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where S1,
S2, 
1, and
2 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 Sbas and cr. The
particular form of these terms suggests that the rate of change of
Sbas or cr is proportional to the
deviation of the total circumferential stress from its value under
normotensive conditions. These are on the basis that both
Sbas and cr rapidly respond to a
step increase in pressure (Fig. 1C). Note that the first
terms on the right hand side of Eqs. 13 and 14
are of opposite signs because Sbas increases,
whereas 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 Sbas and 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.
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(15)
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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
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(16)
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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 Table
1. Parameters characterizing passive
stress, active stress and VSM tone (Sbas and
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 H,
S1, S2,
1, and 2 are identified by using
evolution Eqs. 12-14 for wall thickness,
VSM, Sbas, and critical strain through matching
(best fit) theoretical and experimental results. Equations
12-14, 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
H, S1,
S2, 1, and
2 are given in Table 3.
Model Predictions of Thickening and VSM Tone Adaptation
The evolution of the wall thickness, Sbas,
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, Sbas, and myogenic critical
strain. The characteristics of these solutions obtained for wall
thickness, Sbas, and critical strain are not
sensitive to changes in parameter values within the physiological
range.
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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).
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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).
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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).
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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.
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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).
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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 (Sbas) 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 Sbas 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 Sbas 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 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/H and
1/S1 depict the characteristic speed at which
the arterial wall adapts in proportion to the baseline values of
thickness and Sbas, respectively. From this
point of view, the adaptation of Sbas is more
important than that of wall thickening: the characteristic speed of
Sbas adaptation is indeed ~70 times
"faster" than that of thickening (i.e.,
1/S1 
70/H). On
day 8 postsurgery, the significant increase in
Sbas (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 Sbas 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
Sbas 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
Sbas. 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
Sbas within the initial adaptation phase. The
operating points of the artery (in vivo mean pressure and normal VSM
tone; Fig. 1C) lie in the Sbas 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.
| |
ACKNOWLEDGEMENTS |
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).
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.
Received 27 July 2000; accepted in final form 8 January 2001.
| |
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ABSTRACT
INTRODUCTION
