© 1995 American Heart Association, Inc.
Articles |
Arterial Wall Mechanics in Conscious Dogs
Assessment of Viscous, Inertial, and Elastic Moduli to Characterize Aortic Wall Behavior
From the Basic Sciences Research Institute (R.L.A., J.G.B., R.H.P.), Favaloro Foundation, Buenos Aires, Argentina, and the Centre de Médicine Preventive Cardiovasculaire (J.L., A.S.), Hôpital Broussais, Paris, France.
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Abstract |
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Abstract To evaluate arterial physiopathology, complete arterial wall mechanical characterization is necessary. This study presents a model for determining the elastic response of elastin (
E, where is stress), collagen
(C), and smooth muscle (SM) fibers and
viscous (
) and inertial (M) aortic
wall behaviors. Our work assumes that the total stress developed by the
wall to resist stretching is governed by the elastic modulus of elastin
fibers (EE), the elastic modulus of collagen
(EC) affected by the fraction of collagen fibers
(fC) recruited to support wall stress, and the elastic
modulus of the maximally contracted vascular smooth muscle
(ESM) affected by an activation function (fA).
We constructed the constitutive equation of the aortic wall on the
basis of three different hookean materials and two nonlinear functions,
fA and fC:
 |
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is strain and 0E is strain at zero
stress. Stress-strain relations in the control state and during
activation of smooth muscle (phenylephrine, 5
µg · kg-1 · min-1 IV) were obtained
by transient occlusions of the descending aorta and the inferior vena
cava in 15 conscious dogs by using descending thoracic aortic pressure
(microtransducer) and diameter (sonomicrometry) measurements. The
fC was not linear with strain, and at the onset of
significant collagen participation in the elastic response (break point
of the stress-strain relation), 6.02±2.6% collagen fibers were
recruited at 23% of stretching of the unstressed diameter. The
fA exhibited a skewed unimodal curve with a maximum level
of activation at 28.3±7.9% of stretching. The aortic wall dynamic
behavior was modified by activation increasing viscous () and
inertial (M) moduli from the control to active state (viscous,
3.8±1.3x104 to 7.8±1.1x104
dyne · s · cm-2, P<.0005;
inertial, 61±42 to 91±23
dyne · s2 · cm-2,
P<.05). Finally, the purely elastic stress-strain
relation was assessed by subtracting the viscous and inertial
behaviors.
Key Words: aortic mechanical properties • constitutive equation • stress-strain relation • collagen recruitment function • vascular smooth muscle activation function
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Introduction |
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TopAbstractIntroductionMaterials and MethodsResultsDiscussionAppendix 1References |
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The arterial load can be separated into three components: systemic vascular resistance, total arterial elasticity, and wave reflection.1 Although arterial resistance exerts the greatest effect on ventricular stroke output, an increase of arterial elasticity might result in less optimal coupling of the heart to the arteries with a concomitant less efficient energy utilization by the left ventricle.2 In particular, a decrease in compliance might increase systolic blood pressure3 and create extra cardiac load.4 An important feature of atherogenesis is the focal smooth muscle cell proliferation accompanied by matrix fiber accumulation. Characterization of the elastic behavior of arterial wall constituents in normal and diseased vessels may provide insight into the mechanical factors that could modulate the morphogenesis of atherosclerotic plaques or contribute to the pathogenesis of medial sclerosis.5
The individual elastic behavior of structural constituents of the aortic wall has been assessed in response to vitamin D3–induced accelerated calcinosis6 and under chronic converting enzyme inhibition on aortic stiffening induced by renovascular hypertension in conscious dogs.7 In the first report,6 the accelerated, severe, experimental calcinosis–induced calcium deposition inside the wall of large arteries was shown to be accompanied by a clear-cut paradoxical reduction in arterial rigidity that was mainly due to functional and structural modification of collagen elasticity. The other report7 showed that arterial elasticity in hypertensive dogs was probably altered by changes in vascular smooth muscle (VSM) activity induced by angiotensin-mediated renovascular hypertension. Chronic converting enzyme inhibition decreased the aortic stiffness induced by renovascular hypertension, specifically changing the elastic behavior of elastin and smooth muscle fibers.7
The individual contribution of elastin and collagen fibers to whole aortic elasticity has been previously assessed in normal and diseased conscious dogs.8 In that study, the elastic modulus of the whole aortic wall was decomposed into the elastic modulus of elastin fibers (EE), the elastic modulus of collagen fibers (EC), and the recruitment of collagen fibers (fC, the strain recruitment function) supporting wall stress at a given transmural pressure. The elastic contribution of VSM in conscious animals has been recently assessed by using a modified Maxwell model consisting of a contractile element (CE), which behaves as a simple viscous element in the resting muscle and offers no permanent resistance to stretching, an elastic spring (SEC) coupled in series with the CE, and a parallel elastic component (PEC), which represents the elastic behavior of the aortic wall when the smooth muscle is relaxed or under normal vasomotor tone with negligible elastic effects.9
However, viscous and inertial contributions to aortic wall mechanics, evidenced by the hysteresis seen in the stress-strain relation, have not been previously studied.10 11 12 13 Besides, Fung14 has pointed out the lack (until the present study) of constitutive equations of smooth muscle, which are necessary to analyze the function of different organs, and the urgent need for research to fill this gap.
To our knowledge, no study regarding a complete specification of the mechanical properties of the aortic wall has been reported in conscious animals. The study of the mechanical properties of the aortic wall using conscious animal preparations was pointed out by Dobrin,15 who stated that constriction of large arteries is not usually seen in anesthetized animals. Moreover, the in vivo characterization is a striking procedure that allows a complete description of the arterial mechanical properties under a conceptual framework destined to evaluate the arterial physiopathology both in clinical and experimental research. A possible reason that this analysis has not been performed in conscious animals could be the difficulty in characterizing the complex structure of the aortic wall, whose constituents have very different mechanical properties that produce nonlinear stress-strain relations when pressure and diameter are examined over a wide range. Moreover, the nonlinearity between stress and strain signals can hinder the quantification of the viscous and inertial moduli in the frequency domain. To solve this problem, it is crucial to develop algorithms in the time domain, where nonlinearity does not cause any difficulties.
The goal of the present study is to give some insight into the complete characterization of the mechanical properties of the aortic wall. The individual elastic behaviors of elastin, collagen, and smooth muscle fibers are mathematically quantified. The hysteresis loop is characterized, in the time domain, by means of its viscous and inertial moduli; thus, the purely elastic behavior can be obtained. Finally, we suggest the possible constitutive mechanical equation of the aortic wall in conscious dogs and use this equation to show the individual contributions of the elastic, viscous, and inertial properties in the beat-to-beat hysteresis stress-strain relations.
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Materials and Methods |
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Surgical Preparation
Fifteen male mongrel dogs aged 4.9±1.9 years and weighing 22.2±2.9 kg were prepared for the present study. On arrival at the animal facility, the dogs were vaccinated against common canine diseases and were treated for skin and intestinal parasites. For 20 days before surgery, they were appropriately fed and watered and assessed for adequate clinical status. The experiments were performed in accordance with the guiding principles of the American Physiological Society concerning the use of experimental animals.
Anesthesia was induced with intravenous thiopental sodium (20 mg/kg) and, after intubation, maintained with 2% enflurane carried in pure oxygen (4 L/min) through a Bain tube connected to a Bird Mark VIII respirator. A sterile thoracotomy was made at the left fifth intercostal space. A pressure microtransducer (Konigsberg P7, 1200-Hz frequency response) and a fluid-filled polyvinyl chloride catheter (outer diameter, 2.8 mm; for later calibration of the microtransducer) were implanted in the descending thoracic aorta through a stab wound in the left brachial artery. A pair of ultrasonic crystals (5 MHz, 4-mm diameter) was sutured onto the adventitia of the aorta, after minimal dissection, to measure external aortic diameter. The transit time of the ultrasonic signal (1580 m/s) was converted into distance by using a sonomicrometer (Triton Technology Inc, 100-Hz frequency response) and observed on the screen of an oscilloscope (Tektronix 465B) to confirm optimal signal quality. A polyvinyl chloride catheter (outer diameter, 2.3 mm) was advanced through the left mammary vein to lie in the superior vena cava or right atrium for drug administration. Two hydraulic cuff occluders made from silicon rubber were implanted around the descending thoracic aorta and inferior vena cava. The aortic cuff occluder was implanted at least 30 to 40 mm distal to the ultrasonic crystals to ensure that no artifacts appeared in diameter measurements during aortic occlusions. Before repairing the thoracotomy, all cables and catheters were tunneled subcutaneously to emerge at the interscapular space. All animals were allowed to recover under veterinary care. Ampicillin (20 mg · kg-1 · d-1 per os) was given for 7 days after surgery. The catheters were flushed daily with heparinized saline.
Experimental Protocol
Experiments were performed starting on the seventh postoperative
day. Each study was performed with the dog resting quietly on its right
side in the conscious unsedated state.
The aortic pressure was measured with the pressure microtransducer, which had been calibrated against a Statham-P23 transducer connected to the aortic fluid-filled catheter. The zero reference point was set at the level of the right atrium. The Statham-P23 transducer had been previously calibrated with a mercury manometer. The external aortic diameter signal was calibrated in millimeters by using the step calibration facility of the sonomicrometer. Aortic pressure and diameter signals were stored on an FM tape recorder (Hewlett-Packard 3968-A) for later digital analysis, registered on a six-channel chart recorder (Gould 2600), and displayed on the screen of a four-channel monitor (Gould 51-2341). Instantaneous pressure-diameter loops were displayed on-line on a computer (PC-386) by using an analog-to-digital converter (National Instruments Lab PC). A specific program, developed in our laboratory,16 was modified for this purpose.
Steady States
A 5% dextrose drip (0.25 mL/min) was started through the
mammary vein catheter. At each steady state, the recordings of aortic
pressure and external diameter were performed under the control
condition and during phenylephrine administration (5
µg · kg-1 · min-1) infused in
parallel with the intravenous dextrose drip.
Assessment of Passive Elasticity
Recordings of aortic pressure and external diameter were made
under basal steady state (PEbasal, where PE is
passive elasticity) and during mechanical cuff occlusions of the
descending aorta and vena cava (PEmech). The occlusions
were made to obtain a wide range of pressure-diameter relations. Also,
aortic pressure and diameter were modified by means of intravenous
bolus doses of angiotensin II (0.1 µg/kg) and nitroglycerin (25
mg/kg) to compare mechanical and pharmacological methods in the
obtainment of changes in the aortic signals (PEangio and
PEnitro, respectively). The assembly of
PEangio and PEnitro data was called
PEpharm.
Assessment of VSM Elasticity
After PE recordings, VSM was activated by phenylephrine
infusion. The instantaneous pressure-diameter loops were monitored
until stabilization was achieved. We waited 15 to 20 minutes to ensure
steady state under phenylephrine infusion and confirmed by visual
inspection that the pressure-diameter loops shifted toward a higher
pressure level than that found in PEbasal and to the left
of the PEmech condition. Similar to PE recordings, aortic
pressure and external diameter were recorded under basal steady state
(SMEbasal, where SME is smooth muscle elasticity)
and during aortic and vena caval occlusions (SMEmech).
Two days later, the dogs were killed with an intravenous overdose of thiopental sodium followed by potassium chloride. The correct positioning of the dimension gauges in all dogs was verified at necropsy.
Data Collection
Aortic pressure and diameter signals were sampled and analyzed
off-line on a Compaq Deskpro 25e computer equipped with a Data
Translation 2801-A analog-to-digital converter. Sample frequency was
set at 200 Hz. Approximately 20 consecutive beats during
PEbasal and SMEbasal conditions were averaged
to obtain mean, systolic, diastolic, and pulse aortic pressures,
diameter, and heart rate. The onset of diastolic aortic values was
calculated as the point of maximum pressure occurring between the
negative peak and the onset of the rapid upstroke of the first
derivative of aortic pressure.
During the transient states (PEmech, PEangio, PEnitro, PEpharm, and SMEmech), all beats were digitized, starting from the beat before the onset of variation of the pressure and diameter signals until the beat before the maximal effect produced by the mechanical or pharmacological intervention.
Aortic wall thickness was calculated as the difference between the
external aortic radius (re) and the internal aortic radius
(ri). To estimate ri the following equation was
used:
 | (1) |
Strain () was obtained from the ratio of midwall radius
[R=(re+ri)/2] to the nonstressed midwall
radius (R0) measured at 
25 mm Hg of aortic
pressure17 during necropsy:
 | (2) |
) and the incremental elastic modulus
(Einc) were assessed by using a linear elastic theory and
assuming an isotropic homogeneous elastic material for the aortic
wall.18 19 Einc was calculated as the slope of
the stress-strain curve that theoretically describes the inherent
stiffness of a vessel independent of its geometry:
 | (3) |
 | (4) |
The computation of stress requires that opposing forces are at equilibrium. For this reason, we accepted this condition when the second derivative of strain was near zero, indicating the absence of acceleration.
It should be noted that the P, R, re,
ri, , and variables are functions of time and
are incorporated into the equations developed in the
"Appendix."
Constitutive Equation of the Aortic Wall
To obtain a complete mechanical characterization, we proposed a
model to assess the elastic response of elastin (E),
collagen (C), and smooth muscle (SM)
fibers and the viscous () and inertial
(M) behavior of the aortic wall. This model assumes that
total stress developed by the wall to resist stretching is governed by
the following equations:
 | (5) |
 | (6) |
0E is strain at zero stress,
ESM is the elastic modulus of the maximally contracted VSM,
fA is the activation function, is the viscous modulus,
and M is the inertial modulus. The first term in Equation 5
characterizes the linear elastic behavior of elastin fibers (see
"Appendix," Equation 8
). The second term represents the
nonlinear behavior of collagen fibers (Equations 9 to 11). The first
and second terms constitute the passive elastic stress-strain relation.
The third term exists only under VSM activation and characterizes the
elastic response of the VSM (Equations 12 to 16). The remaining terms
represent the dynamic behavior of the aortic wall and are
responsible for the hysteresis of the stress-strain relation (Equation 17).
Statistical Analysis
All measurements and calculated values were expressed as
mean±SD. Linear regression was analyzed by the least-squares method.
The presence of significant differences was assessed by paired
t test or Tukey-B and Dunnett tests following ANOVA for
repeated measures. Values of t and F ratio with
P<.05 were considered statistically
significant.20 The nonlinear curve fittings were performed
by using the Gauss-Newton iterative algorithm included in a scientific
system (Asyst Software Technologies, Inc). To evaluate the curve fit
performance, the normalized standard error of the estimate (NSE) was
calculated in each animal according to the following expression:
 | (7) |
is the mean stress, 
is curve fit
estimation of , N is the number of processed points, and 
is the
number of constants involved in the corresponding equations (Equations
11 and 15). |
Results |
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Steady State
Table 1
shows the hemodynamic parameters during
control steady state and during steady state VSM activation by
phenylephrine infusion. Heart rate was the only variable that
presented no statistical differences with respect to the control
condition.
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Assessment of Viscous and Inertial Moduli (Hysteresis
Characterization)
In Fig 1, top, recordings of pressure and diameter
waveforms of the canine aorta are shown. Both signals present the
same morphology, but the aortic diameter shows a time delay with
respect to the aortic pressure. In Fig 1, bottom, the stress-strain
hysteresis loop (open circles) calculated from these signals is
plotted. A first approximation in the elimination of the hysteresis
loop is shown when only the viscous stress has been subtracted (open
squares). Once both viscous and inertial components have been removed,
the actual stress-strain relation represents only the elastic
properties of the aortic wall (filled circles). In the control
condition, the viscous stress was 445±292% larger than inertial
stress. Similarly, with phenylephrine infusion, the dynamic arterial
wall behavior was mainly affected by the viscous properties
yielding a viscous stress 453±141% larger than inertial stress. Both
viscous and inertial moduli were significantly increased from the
control state (3.8±1.3x104 dyne
· s · cm-2 and 61±42
dyne · s2 · cm-2) to the active state
(7.8±1.1x104 dyne · s · cm-2
[P<.0005] and 91±23
dyne · s2 · cm-2
[P<.05]), respectively.
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) involves the elastic, viscous, and
inertial properties conforming a hysteresis loop. By increasing values
of the viscous modulus, the hysteresis area was reduced to a minimum
(
). Ulterior increase of the inertial modulus produced the total
disappearance of the hysteresis loop. The purely elastic stress-strain
relation (
) is coincident with the stress-strain diastolic
phase.
Transient State
Transient states were performed by using mechanical and
pharmacological maneuvers producing marked variations in diameter and
pressure waveforms (Fig 2, left). In Fig 2, right, the
corresponding stress-strain relation obtained from the individual
measured samples was depicted in the control condition and under
phenylephrine administration. In the interval marked "diastole"
(Fig 2, left), the second derivative of pressure (or stress) and
diameter (or strain) is near zero, indicating the absence of
acceleration and therefore the high degree of equilibrium required for
the proper use of the stress-strain equations. In this interval, the
purely elastic stress-strain curve was identified in the steady
state.
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Assessment of PE
During pharmacological maneuvers, systolic and diastolic pressures
were respectively increased by 67±31% (P<.0005)
(angiotensin bolus) and decreased by 35±23% (P<.01)
(nitroglycerin bolus) with respect to the control steady state.
Systolic and diastolic aortic diameters were respectively increased by
3.8±4.0% (P<.025) (angiotensin bolus) and decreased by
8.6±4.8% (P<.01) (nitroglycerin bolus) from the control
steady state.
Mechanical interventions (descending thoracic aortic and vena caval occlusions) during the control condition increased systolic pressure (80±12% [P<.00001]) and decreased diastolic pressure (20±17% [P<.0005]) and increased systolic diameter (4±3% [P<.0005]) and decreased diastolic diameter (5±3% [P<.00001]) with respect to the basal steady state.
Descending thoracic aortic occlusion during phenylephrine infusion
increased systolic pressure by 32±7% (P<.00001) from the
steady state and increased systolic diameter by 2±0.5%
(P<.0005) with respect to the activation steady state.
Mechanical vena caval occlusion during phenylephrine infusion decreased
both diastolic pressure by 33±17% (P<.0005) and diastolic
diameter by 8±3% (P<.00001) with respect to the
phenylephrine steady state. Fig 2, right, shows the stress-strain
relations during control and activation conditions converted from
pressure and diameter signals in a typical dog.
Elastic Behavior of the Elastin Fibers
Table 2 shows the totality of elastic parameters
calculated from the whole population (15 dogs) assessed during
PEmech and compares these parameters with those obtained
under PEangio, PEnitro, and
PEpharm in the dogs in which comparisons between occlusions
and angiotensin and nitroglycerin bolus doses were performed. In all
the studied parameters, no significant differences were found when
ANOVA for repeated measures was made.
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To find in which part of the cardiac cycle the break point of the
stress-strain relation was placed, we performed comparisons between
systolic, mean, diastolic, and maximum diastolic phase values of
diameter, pressure, stress, and strain versus the break-point values in
all dogs (Table 3). In this table, it can be observed
that the level at which all parameters present no differences with
respect to those corresponding to the break point is the maximum value
assessed during the diastolic phase of the cycle.
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The use of a linear mathematical model to characterize the elastic response of elastin fibers is supported by the highly linear correlation coefficients (0.972±0.022 [P<.0001]) found in the analysis of the first portion of the stress-strain relations obtained with PEangio (0.956±0.045 [P<.0001]), PEnitro (0.977±0.023 [P<.0001]), PEpharm (0.975±0.023 [P<.0001]), and PEmech (0.977±0.031 [P<.0001]).
Elastic Behavior of Collagen Fibers (Assessment of Strain
Recruitment Function)
The elastic modulus of the collagen fibers was
1102.5±532.2x106 dyne/cm2. Fig 3 shows the nonlinear function representing the
total aortic stress-strain relation and the corresponding linear
function due to the elastic behavior of the elastin fibers (top left).
Subtraction of both functions results in the collagen stress-strain
relation (bottom left) of the entire population of dogs. Fig 3, top
right, shows the mean values of measured and estimated (Equation 11)
collagen stress-strain relations. The percentage of collagen fibers
recruited from fC (Equation 10) at each level of stretching
is depicted in Fig 3, bottom right. It can be seen that the percentage
of collagen fibers recruited at the deformation level observed at the
break point of the stress-strain relation was 6.09±2.6%. The equation
used to model the collagen stress response was successfully fitted in
all dogs (NSE=0.243±0.098). The values of constants
c1, c2, and c3
obtained from the fit of the mean collagen stress-strain relation were
1.1452, 0.1117, and 0.0001, respectively (NSE=0.188).
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Assessment of VSM Elasticity
Elastic behavior of smooth muscle fibers (assessment of
activation function). Fig 4 shows the mean active
stress-strain curves in control and VSM activation (top left)
corresponding to eight dogs. Subtraction of stress values during the
control condition from those obtained after excitation with
phenylephrine results in the elastic behavior of the VSM (bottom left).
As shown by these data, VSM exhibits a unimodal strain–active stress
curve, with a maximum active stress of
0.949±0.57x106 dyne/cm2 corresponding
to a strain level of 1.299±0.083. The maximum value of the VSM elastic
modulus assessed by the maximum value of the derivative of the VSM
stress-strain curve was 8.345±7.56x106
dyne/cm2, occurring at a strain value of
1.283±0.079. Fig 4, top right, shows the mean values of measured and
estimated (Equation 15) VSM stress-strain relations. The percent VSM
activation function derived from Equation 16 is depicted in Fig 4,
bottom right. The equation used to model the VSM muscle response was
successfully fitted in all dogs (NSE=0.031±0.011). The values of
constants m1, m2,
m3, m4, and m5
obtained from the fit of the mean VSM stress-strain relation were
-71.23, 93.45, -0.49, 997.98, and 1.337, respectively
(NSE=0.0125).
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Constitutive Equation of the Aortic Wall
To reproduce the individual contribution of the elastic, viscous,
and inertial behaviors to the total stress-strain relation, we applied
the model proposed in Equation 6. Fig 5 depicts the
elastic contribution of the passive components (elastin and collagen)
to aortic stress-strain loops. These panels show that elastin fibers
mainly govern the resistance to stretch. Fig 5 also shows the viscous
and inertial components that are responsible for the hysteresis of the
stress-strain loop. It can be seen that the wall viscosity is the most
important dynamic contribution. Thus, the arterial wall is essentially
viscoelastic in the beat-to-beat stress-strain relation, and the
viscous effect can probably be attributed to smooth muscle in systole
and the elastic effect to elastin fibers in diastole. In the control
state, the VSM activation function is close to zero, since we assume
that in this condition the VSM is not significantly activated; ie, it
develops a small degree of activation whose elastic effects are
negligible. Under activation of VSM (Fig 6), there was
an increase in the collagen elastic contribution in the viscous stress
and in the VSM elastic contribution. Similar to the previous case, the
mechanical properties remain essentially viscoelastic. Fig 7 compares in a typical dog the measured and modeled
(Equation 6) stress-strain relations and the stress time course during
the control condition (top panels) and under activation (bottom
panels), with the corresponding strain measured in each condition used
as an input.
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Discussion |
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TopAbstractIntroductionMaterials and MethodsResultsDiscussionAppendix 1References |
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The aim of the present study was to obtain a complete characterization of the arterial wall mechanics in conscious, unsedated, chronically instrumented dogs. In this work, a method to obtain this relation, called the constitutive equation,11 12 14 was presented, and to our knowledge, no study regarding the complete specification of the mechanical properties of the aortic wall has previously been reported in conscious animals. To build the constitutive equation, we used second-order differential equations, where constant coefficients are the total elastic modulus, the viscous modulus, and the inertial modulus. The total elastic modulus was decomposed, quantifying the individual contributions to this modulus of the principal components of the arterial wall (elastin, collagen, and smooth muscle) and describing them by a mechanical model. The viscous and inertial moduli were assessed by using the criterion of disappearance of the hysteresis loop to obtain the purely elastic behavior. Finally, the individual contributions of the elastic, viscous, and inertial properties to the hysteresis loop could be achieved with the methods presented in the present study.
The instrumentation techniques used to obtain the pressure-diameter relation in the conscious animal18 have been reproduced in our laboratory,6 7 8 9 allowing accurate and reproducible measurements.10 21
Aortic signals were converted into stress and strain by using a
thick-walled cylindrical tube model and linear elastic
theory.18 Stress was calculated instantaneously in the
overall cardiac cycle, considering the nonvariability of wall volume
segments.17 Even though it might appear that the condition
of equilibrium was absent, the analysis of pressure-diameter
relations shows that most of the sampled data corresponded to the
diastolic phase of the cardiac cycle (Fig 1). In this phase, the second
derivative of diameter (or strain) was near zero, indicating the
absence of acceleration and therefore a high degree of equilibrium.
This fact could have been a major limitation in this approach, since we
needed the instantaneous beat-to-beat stress-strain relation to apply
the procedure that gives the purely elastic behavior. To overcome this
problem, we defined a threshold in the second derivative of diameter,
below which we assumed that it was impossible to separate signal from
noise. The points beyond this threshold were withdrawn from the
stress-strain relation (Fig 2). In all cases, only few points were
found in this condition because of the chosen acquisition sample
rate.
The first step in the present study was the evaluation of the system parameters (coefficients of the second-order equation), and the second step was the use of these parameters to predict the behavior of the system.22 In the present work, it was possible to assess the purely elastic aortic behavior in conscious dogs. This behavior was close to the diastolic stress-strain relation, demonstrating that during this phase of the beat, the very low harmonic contents yield a negligible viscous-inertial behavior. However, the diastolic phase represents only half of the cardiac cycle, whereas our procedure involves all the cardiac cycle in the assessment of the purely elastic stress-strain relation. With the same procedure, it was possible to quantify the inertial and viscous moduli. In the descending thoracic aorta, the inertial modulus was negligible compared with the viscous and elastic behaviors, in agreement with Peterson et al,23 who stated that the behavior of arteries was mainly viscoelastic. Our results showed a mean viscous modulus of 3.8±1.3x104 dyne · s · cm-2 and a mean elastic modulus of 4.99±1.58x106 dyne/cm2 in the control condition. From the data reported by Peterson et al in thoracic aorta, we calculated a mean viscous modulus of 22±104 dyne · s · cm-2 and a mean elastic modulus of 5.3x106 dyne/cm2, suggesting an understimation in our calculated viscous modulus, whereas the elastic modulus is very similar to their results. A possible explanation could be that the different experimental conditions and technology used by these authors in the pressure-diameter measurements could have altered the real time lag between signals, thereby increasing viscocity.
In the beat-to-beat steady state, the elastic deformation is
proportional to the potential energy stored during systole that will be
yielded to the system during diastole, whereas the viscous loop (Figs 5 and 6) quantifies the absorption of energy by the vessel wall. The
increment of the viscous modulus under activation signifies that this
coefficient is dependent on pressure and on the level of smooth muscle
activation, material responsible for the viscous behavior of the aortic
wall. The viscous loss of energy is small, but it is an interesting
aspect of hemodynamics that has been little explored.13
The larger viscous modulus when the muscle is active indicates a
greater expenditure of energy in the pulsatile expansion of the vessel
with each heartbeat.10
Rigorously speaking, the inertial modulus is a proportionality constant
between force and the acceleration developed by a given material and
quantifies the resistance to acceleration presented by the body. In
systemic arterial wall dynamics, inertial forces might develop at the
beginning of systole concomitant with the very fast increase in
diameter.12 In our case and in agreement with Peterson et
al,23 the inertial term is negligible in the relation
between stress and strain in the arterial wall under physiological
conditions; ie, the inertial stress is too small to influence the
stress-strain loop (Figs 5 and 6), indicating that the artery is
essentially viscoelastic. During phenylephrine administration, the
inertial modulus was increased with respect to the control condition.
This result implies that (according to the use of mass as a
quantitative measure of inertia) the aortic wall mass, defined as the
addition of the individual mass of each structural constituent, should
also be increased under activation. It is obvious that factors other
than the wall mass should offset the magnitude of the inertial modulus.
This modulus can be codetermined by the level of aortic pressure and/or
the stiffness of the aortic wall, because at greater pulse pressures,
marked nonlinearity of the inertial behavior can occur.12
Among the possible factors producing nonlinear inertial behavior,
frequency could play an important role, as was reported for the viscous
modulus of the arterial wall.24 In this case, the inertial
modulus must be regarded as a mathematical abstraction that mimics the
inertial behavior of the aortic wall and not as a physical constant.
Other effects such as the radial acceleration of blood could influence
this phenomenon, and for these reasons, further studies would be
necessary to evaluate whether the alterations that could be found in
the inertial modulus under different physiopathological states were due
exclusively to intrinsic changes in the arterial wall.
There are two theories that explain the viscosity of the vessel. Passive theories assume that viscosity is simply a property of the aortic wall constituents, and it is generally assumed that the vascular smooth muscle is mainly responsible for this behavior.25 26 Bulbring et al27 have shown that viscosity is greater in muscular than in elastic arteries, suggesting that this point of view is possible. The location of viscous elements in the infrastructure of the muscle cell has not been identified. Among several possibilities, it has been suggested that viscosity could be ascribed to (1) the protoplasm itself, (2) the viscous resistance of the cell matrix encountered by the sliding of actin and myosin filaments during muscle contraction, and finally (3) the cell membrane that could act as a restraint presenting a viscoelastic nature.10 These explanations must be taken carefully, since some stretch relaxation was still observed when the muscle was inactivated by prolonged immersion in calcium-free solutions or iodoacetic acid,13 28 suggesting that smooth muscle cannot be the only source of these effects. This observation could be a possible limitation in the present model because collagen, elastic, and ground substance might also present viscous behavior. In collagen fibers, both elasticity and viscosity probably depend to some extent on the orientation of the fibers.29 Another theory (active theory) to explain the cause of hysteresis takes into account the force-generating mechanism of the muscle30 as well as the myogenic response to stretching.10 31 32 These possible explanations are not mutually exclusive, and hysteresis might be the result of several factors. Viscous behavior may affect the velocity of the pulse wave. Any system that presents elastic and viscous behavior acts on the size and shape as well as the velocity of the wave. For these reasons, the pulse-wave velocity measured in clinical practice must be carried out by identifying the points next to the foot of the wave where the effects of viscosity and the reflected wave are negligible.33
In brief, VSM exhibits two kinds of response: one represented by the viscous behavior present in the beat-to-beat steady state, which affects mainly the systolic stress-strain relation, and the other related to pharmocological activation, which can modify the elastic behavior of the muscle through the elastic contribution of the contracted muscle.
The elastic modulus of elastin fibers and the coordinates of the break point are the result of the arterial wall structure and for this reason can be evaluated by mechanical or pharmacological maneuvers. The diameter and pressure values at the break point did not change with the different maneuvers, since the rapid variations in the stress-strain relations reflect solely the intrinsic properties of the aortic wall.
We found that the break point was close to the maximum diastolic value and that this might represent the value at which the reflected wave reaches its maximum. Beyond this point until the onset of the next beat, the stress-strain relation reflects the purely elastic behavior of the aortic wall. This suggests that the mean diastolic pressure used by O'Rourke et al34 as an index of coronary perfusion is determined exclusively by the elastic components and not by the transient effect of viscosity and inertia, which depends mainly on heart rate. Beyond the break point, collagen participation is increased, and at high levels of deformation, the totality of the collagen fibers resists stretching. Thus, the hysteresis loop is diminished because of the double effect of a collagen predominance and a decrease in heart rate produced by the high pressure level. In contrast, below the break point, the viscous behavior is predominantly accounted for by smooth muscle cells.
The recruitment of collagen fibers as a function of deformation showed
that with a 30% stretching of the arterial wall with respect to the
unstressed diameter, 100% of the collagen fibers resisted deformation,
whereas at the break point (23% of deformation with respect to the
unstressed diameter), only 6% of the total collagen fibers was
recruited (Fig 3, bottom right). The recruited collagen fibers were
increased from 6% to 100% with only a 7% increase in deformation
(from 23% to 30%), evidencing their highly nonlineal behavior. Thus,
in control conditions the nonlinearity of the stress-strain relation is
due exclusively to the collagen recruitment function. To model this
function, we used a trigonometric tangent function used by others with
excellent results.35 36
The VSM activation function depends on the degree of smooth muscle
activation. We assumed that under phenylephrine administration the
smooth muscle is maximally contracted and thus the term ESM
represents the maximum value of the VSM elastic modulus. This
skewed function was represented as a function of the
percent deformation from the unstressed values and represents
the level of activation at each step of arterial deformation. Fig 4,
bottom right, shows that its maximum value corresponded to 28% of
stretching and that beyond this point, stress fell abruptly. This curve
does not reflect the exact overlapping of actin-myosin bridges because
the series arrangement of the series elastic component and the
contractile element transforms a symmetric curve into a skewed one. For
this reason, we used a modified lorentzian function with five constants
with excellent curve fit performance.
We identified the constants of the constitutive equation for the
aortic wall (Equation 6) in conscious dogs by using the individual
characterization of its structural constituents and exploring their
contribution to stress. The stress-strain relations estimated with
Equation 6 in control resting conditions and under activation of the
smooth muscle were similar to the respective measured ones. In both
cases, the hysteresis involved in the estimated stress-strain relations
were lower than the measured ones. These differences were found at the
highest level of stress and strain near the systolic point. At this
stage, it is possible that the hysteresis of the aortic wall is
complicated by the myogenic response,10 31 32 consisting
of a small rapid stretching of the vessel immediately followed by an
active contraction of the muscle. However, to our knowledge, no study
regarding this effect in the aortic wall has been reported.
Characterization of the mechanical properties of the individual structural constituents could provide insight into the physiological and pathological processes that can affect the arterial wall, such as hypertension, atherosclerosis, aging, and diabetes.37 With this approach, it would be possible to know the consequences of structural alterations of the arterial wall, and by assessing its individual mechanical behaviors before and after treatment, it could be determined whether these alterations could be reversed.
In conclusion, the accurate assessment of the mechanical properties of the individual components of the aortic wall in conscious dogs allows the construction of a possible constitutive equation. The purely elastic stress-strain relation can be obtained by extracting the viscous and inertial arterial behaviors; thus, the arterial wall can be characterized on the basis of three different hookean materials (EE, EC, and ESM, with the latter only under activation) and two nonlinear functions, fA and fC. Finally, we suggest that another pathway (probably the myogenic response of vascular smooth muscle) in addition to the passive viscosity of the smooth muscle might play a role in the hysteresis of the stress-strain loop.
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Acknowledgments |
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The authors wish to thank Elena Lascano, BSc, for her help in the revision of the manuscript.
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Footnotes |
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Reprint requests to Ricardo Armentano, EEng, PhD, Basic Sciences Research Institute, Favaloro Foundation, Solís 453, (1078) Buenos Aires, Argentina.
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Appendix 1 |
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TopAbstractIntroductionMaterials and MethodsResultsDiscussionAppendix 1References |
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Elastic Modulus of Elastin Fibers (EE)
The first part of the stress-strain relation shows a linear behavior. Beyond a certain value of stress, the "break point," the relation curves upward as collagen fibers are recruited.7 8 Up to this critical point, the linear response of the aortic wall can be fitted with a linear model where EE is the slope of the linear relation. Therefore, the stress-strain relation for this linear portion can be written as follows:
 | (8) |
0E is the strain-axis intercept.
Elastic Modulus of Collagen Fibers (EC)
To describe the elastic response of collagen fibers, it is
necessary to separate the stress-strain relation corresponding to
elastin from the overall stress-strain relation. According to this
approach, the collagen behavior is given by the following relation:
 | (9) |
at each strain value.
Recruitment Function of Collagen Fibers as a Function of Strain
(fC)
To model the collagen elastic response represented
by the stress developed by collagen fibers resisting stretch, we
multiplied the elastic modulus of the collagen fibers,
EC, by a normalized morphology function named the
strain recruitment function, fC, expressed as
follows:
 | (10) |
into Equation 9 gives the following:
 | (11) |
Mechanics of VSM
To describe the mechanics of VSM, a modified three-element
Maxwell model was used.9 According to this model, total
stress can be written as follows:
 | (12) |
SM is the active stress supported by
the assembly of an elastic spring coupled in series with the
contractile element (CE-SEC). The passive stress developed by the
parallel elastic component (PE) could be evaluated by
using the elastic passive behavior of elastin (Equation 8) and collagen
(Equation 9) obtained during control conditions:
 | (13) |
 | (14) |
Assessment of Activation Function as a Function of Strain
(fA)
The smooth muscle active stress-strain relation obtained from in
vitro studies has been characterized in several
works.38 39 40 Recently, this was obtained in conscious
animals: the smooth muscle elastic modulus versus strain was shown as a
skewed unimodal curve with a maximum value (ESM) that
represents the highest level of elasticity that can be reached
for this level of activation.9 To model the smooth muscle
active stress-strain relation, we proposed the following skewed
function composed by a modified lorentzian function multiplied by the
maximum value of the smooth muscle modulus, ESM:
 | (15) |
 | (16) |
Assessment of the Purely Elastic Stress-Strain Relation
The total stress generated by the wall to oppose stretching is
commonly attributed to the combined effects of wall elasticity, wall
viscosity, and inertia. Bauer and colleagues11 12 have
developed a procedure that subdivides the wall stress into three terms,
the first of which depends on , the second on the first derivative
of (velocity), and the third on the second derivative of
(acceleration):
 | (17) |
, and M are the elastic, viscous, and inertial
moduli. The first term is the elastic stress; the second, the viscous
stress; and the last, the inertial stress. By definition, the purely
elastic stress-strain relation courses along the same curve for
increasing and decreasing radius; therefore, in this diagram no
hysteresis loop appears. To separate the purely elastic wall
properties, one must subtract the viscous and the inertial stress from
the total aortic stress, finding the optimal value through the
criterion of disappearance of the hysteresis loop. The input for
Equation 17 was . In a first step, M was considered to be equal to
zero, and increasing values of were given by analyzing by visual
inspection the reduction of the hysteresis loop area. When the area
reached a minimum (considered to be the value that maintained the
clockwise course of the loop), M values were incremented in steps to
obtain the total disappearance of the hysteresis loop. Units used were
dynes, centimeters, and seconds. Received July 7, 1994; accepted November 15, 1994.
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References |
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TopAbstractIntroductionMaterials and MethodsResultsDiscussionAppendix 1References |
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