Vol. 282, Issue 2, H622-H629, February 2002
Regional arterial stress-strain distributions referenced to
the zero-stress state in the rat
Jingbo
Zhao1,2,
Judd
Day1,
Zhuang Feng
Yuan2, and
Hans
Gregersen1,3
1 Institute of Experimental Clinical Research, Aarhus
University, DK-8200 Aarhus; 2 Institute of Clinical-Medicine
Science, China-Japan Friendship Hospital, 100013 Beijing, China;
3 Center of Sensory-Motor Interaction, Aalborg University,
DK-9220 Aalborg; and Department of Abdominal Surgery, Aalborg
Hospital, DK-9000 Aalborg, Denmark
 |
ABSTRACT |
Morphometric and stress-strain
properties were studied in isolated segments of the thoracic aorta,
abdominal aorta, left common carotid artery, left femoral artery, and
the left pulmonary artery in 16 male Wistar rats. The mechanical test
was performed as a distension experiment where the proximal end of the
arterial segment was connected via a tube to the container used for
applying pressures to the segment and the distal end was left free.
Outer wall dimensions were obtained from digitized images of the
arterial segments at different pressures as well as at no-load and
zero-stress states. The results showed that the morphometric data, such
as inner and outer circumference, wall and lumen area, wall thickness,
wall thickness-to-inner radius ratio, and normalized outer diameter, as
a function of the applied pressures, differed between the five arteries
(P < 0.01). The opening angle was largest in the
pulmonary artery and smallest in thoracic aorta (P < 0.01). The absolute value of both the inner and outer residual strain
and the residual strain gradient were largest in the femoral artery and
smallest in the thoracic aorta (P < 0.01). In the
circumferential and longitudinal direction, the arterial wall was
stiffest in the femoral artery and in the thoracic aorta, respectively,
and most compliant in the pulmonary artery. These results show that the
morphometric and biomechanical properties referenced to the zero-stress
state differed between the five arterial segments.
morphometry; distension
| |
INTRODUCTION |
BLOOD VESSEL
ELASTICITY is important in physiological and pathophysiological
problems involving surgery, remodeling, engineering, and angioplasty
(11). Extensive data (1-30) have been
published on stress-strain distributions of the major vessels. Most
biomechanical studies before 1983 used the zero-pressure condition
(no-load state) as reference for the analysis. Vaishnav and Vossoughi
(31) and Fung et al. (12) found that vascular
tissue experiences residual strain in the no-load state, i.e., the
vascular tissue is prestressed. This can be demonstrated by making a
radial cut in a ring of tissue. In many circumstances, the ring will
spring open. This new geometry is referred to as the zero-stress state. To quantify this phenomenon, the opening angle is defined as the angle
subtended by two radii drawn from the midpoint of the inner wall to the
tips of the inner wall of the open sectors.
Because prestressing reduces the concentration of circumferential
stress at the inner vessel wall under homeostatic conditions, Liu and
Fung (19) emphasized that any mechanical analysis must begin with the zero-stress state. Hence, the zero-stress state provides
the reference state for calculation of strain. It also provides a
standard morphological state to describe vascular tissue because the
tissue is not affected by internal or external forces in the
zero-stress state. The opening angle varies considerably along the
vascular tree (19). Liu and Fung (20) studied
the stress-strain distribution referenced to the zero-stress state in
the thoracic aorta of normal and diabetic rats. Liu and Fung (21) also used inflation and deflation tests to study the
effect of cigarette smoking on the stress-strain relationship of
pulmonary arteries in 2- and 3-mo-old smoke-exposed rats. Matsumoto and Hayashi (25) studied the effects of hypertension on the
stress and strain distribution through the wall thickness of the
thoracic aorta. To the best of our knowledge, no data exist on the
axial variation of arterial stress-strain relations referenced to the zero-stress state. The purpose of this study is to compare the elastic
properties, defined with respect to the zero-stress state, from vessels
of different size, structure and composition. We studied the
stress-strain distribution including residual strains of the thoracic
aorta, abdominal aorta, femoral artery, common carotid artery, and
pulmonary artery.
| |
MATERIALS AND METHODS |
Experimental Procedures
Sixteen 4-mo-old male Wistar rats (354 ± 34 g body
wt) were used in this study. The rats were anesthetized with an
intraperitoneal injection of pentobarbital sodium (50 mg/kg).
The right femoral artery was cannulated for systemic blood pressure
measurement. The blood pressure was 109 ± 4 mmHg. The thoracic
aorta, abdominal aorta, left femoral artery, left common carotid
artery, and the left pulmonary artery were dissected, excised, and
placed immediately into a calcium-free Krebs solution with 6% dextran
and EGTA and aerated with 95% O2-5% CO2. At
the time of testing, the vessels were taken out of the bath and dried
gently with a piece of absorbent paper. The surface was sprayed with
microbeads (60-125 µm stainless steel spheres, Duke Scientific)
that easily adhere to the tissue for determination of changes in
length. For the distension experiments, the proximal end of the
arterial segment was connected to a water column. The other end was
ligated and left free in the axial direction. Branches were ligated.
Stepwise pressurization was carried out by increasing the level of the
column to induce pressures of 20, 40, 60, 80, 100, 120, and 140 mmHg
for systemic arteries and 5, 10, 15, 20, 25, and 30 mmHg for the
pulmonary arteries. Each step lasted at least 2 min until steady state
was reached (no change in outer diameter over a 20-s period). The outer
diameter and position of the microbeads were recorded with the use of a
Sony charge-coupled device camera.
To obtain data on the no-load state and zero-stress state, one
arterial ring was excised from the middle region of the various arterial segments and placed in Krebs solution. A photograph was taken
of the cross-section of the ring in the no-load state. A radial
cut was then made in the ring, which opened into a sector (22). Photographs were taken after ~20 min to allow
viscoelastic creep to take place (11).
Mechanical Data Analysis
The morphometric data were measured from the digitized images of
the arterial segments at the preselected pressures and at no-load and
zero-stress states (Fig. 1). The no-load
state was defined as that with no transmural pressure or axial
loads. The zero-stress state was the stress-free configuration obtained
by cutting vessel rings into sectors. The following parameters were measured with the use of Optimas image analysis software: 1)
outer diameter and microbead displacement at different pressures,
2) inner and outer circumferential length of arterial rings
in no-load and zero-stress state, and 3) opening angle of
the cut-open sectors. These measures were used for computation of
the biomechanical parameters.
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Fig. 1.
Diagram of distended, no-load, and zero-stress states of arterial
segments. The outer diameter and microbead (*) displacements at
different pressures (P), inner (Ln-i) and outer (Ln-o)
circumference length, thickness, and area in no-load and zero-stress
state, and the opening angle of the arterial rings could be directly
measured from the digitized images.
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|
Residual strain of inner and outer wall.
Residual strain (E) was computed according to Green's
formula using the inner (i) and outer (o) wall circumferences,
respectively, measured at zero-stress state
(Lz-i and Lz-o) and
no-load state (Ln-i and
Ln-o). This was computed at the inner wall
and at the outer wall
|
(1)
|
where L is the midwall length.
Circumferential and longitudinal wall stress.
Average wall stresses were computed as Kirchhoff stresses assuming a
circular cylindrical vessel geometry and a homogenous wall. The
circumferential Kirchhoff stress was computed as
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(2)
|
and the longitudinal stress was computed as
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(3)
|
where 
P was the transmural pressure, i.e., the difference
between the height of the pressure column and the fluid level in the
organ bath in our experiments. The luminal radius
ri at various distension pressures was computed
assuming circular geometry. The wall volume was computed from the wall
area and length of arterial segment in no-load state. By assuming
incompressibility, h as the arterial wall thickness could be
calculated at various pressures. 
is the stretch ratio referenced to
the zero-stress state. Radial stress components were ignored in this study.
Circumferential and longitudinal strains.
Circumferential strain was computed from the midwall circumference
referenced to the zero-stress state. The midwall circumferential Green
strain was computed as
|
(4)
|
where Cp was the midwall circumference at
any of the pressure loads and Cz was the midwall
circumference at the zero-stress state.
The longitudinal Green strain was computed from the midwall lengths at
each pressure (L) and at the no-load state
(L0) as
|
(5)
|
The no-load state was used as reference in the longitudinal
direction because midwall residual strains in longitudinal direction were negligible (H. Gregersen, unpublished data). L
and L0 were measured directly from the
displacement of two microbeads on the surface of the arterial segments.
We adopted the exponential strain energy function by Fung
(11) for analyzing the data further. This function has the
form
|
(6)
|
where a1,
a2, and a4 are
nondimensional material constants and C is a material
constant with the unit of stress. 
and 
refer to the
longitudinal and circumferential direction.
E*
and
E*
are strains corresponding to an
arbitrarily selected pair of stresses. The meaning of the constants was
discussed previously (11).
Statistics
The data were representative of a normal distribution and
accordingly the results are expressed as means ± SE. Analysis of variance was used for statistical analysis. The results were considered significant if P < 0.05.
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RESULTS |
Morphometric Data
Figure 2 shows the basic
morphometric parameters of the arteries. The inner and outer
circumferential length, wall and lumen area, wall thickness, and wall
thickness-to-inner radius ratio differed between the arteries
(P < 0.01). The inner and outer circumference at
no-load state and zero-stress state, wall area, and lumen area in
no-load state were largest in the thoracic aorta and smallest in the
femoral artery. The wall thickness at no-load state was greatest in the
thoracic aorta and thinnest in the pulmonary aorta. The wall
thickness-to-inner radius ratio was largest in the femoral artery and
smallest in the abdominal aorta.
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Fig. 2.
Means ± SE of the morphometric parameters of different
arteries. T, thoracic aorta; A, abdominal aorta; F, femoral artery; C,
common carotid artery; P, pulmonary artery. The sample size was 16. The
inner (A) and outer (B) circumferential length of
no-load state and zero-stress state, wall area and lumen area
(C-E) were largest in thoracic aorta and
smallest in femoral artery (P < 0.01). The wall was
thickest in thoracic aorta and thinnest in pulmonary aorta
(P < 0.01). The wall thickness-to-inner radius ratio
(F) was largest in the femoral artery and smallest in the
abdominal aorta (P < 0.01).
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The outer diameter and normalized diameter (diameter at a given
pressure minus the diameter at zero pressure) as a function of pressure
are shown in Fig. 3. The no-load diameter
was largest in the thoracic aorta and smallest in the femoral artery.
The normalized diameter was largest in the pulmonary artery and
smallest in femoral artery. Significant differences were found between all arteries (P < 0.01).
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Fig. 3.
Means ± SE of the outer diameter (A) and the
normalized outer diameter (B; the diameter at 0 pressure
subtracted from the diameter at pressurized conditions) of the thoracic
aorta (T), abdominal aorta (A), femoral artery (F), common carotid
artery (C) and pulmonary artery (P) at different pressures. The sample
size was 16. Outer diameter of arteries before applying pressure was
largest in thoracic aorta and smallest in the femoral artery
(P < 0.01). The normalized outer diameter was biggest
in the pulmonary artery and smallest in femoral artery
(P < 0.01).
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Biomechanical Data
The opening angle and residual strain of the arteries are shown in
Fig. 4. A significant difference was
found in both opening angle and residual strain between the vessels
(P < 0.01). The opening angle was largest in the
pulmonary artery and smallest in thoracic aorta. The absolute value of
both the inner and outer residual strain was largest in the femoral
artery and smallest in the thoracic aorta. The residual strain gradient
and residual strain gradient normalized with the wall thickness are
also shown in Fig. 4. Both measures were biggest in the femoral artery
and smallest in the thoracic aorta (P < 0.01).
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Fig. 4.
Means ± SE of the opening angle (A), residual
strain (B), residual strain gradient (C), and
residual strain gradient-to-wall thickness ratio (D). The
opening angle was largest in the pulmonary artery and smallest in
thoracic aorta (P > F > C > A > T;
P < 0.01). The absolute values of both the inner and
outer residual strain were largest in the femoral artery and smallest
in the thoracic aorta (P < 0.01). The residual strain
gradient and gradient-to-wall thickness were largest in the femoral
artery and smallest in the thoracic aorta (P < 0.01).
The sample size was 16.
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The relations between stress and strain are shown in Fig.
5 for the circumferential (A)
and longitudinal (B) direction. Stress in the femoral artery
was higher at a given strain than in the others and that in the
pulmonary artery was lower whereas in the longitudinal direction, the
thoracic aorta had the higher stress at a given strain, and again, the
pulmonary artery had the lower. The constants from the analysis of the
stress-strain curves are presented in Table
1. The constants differed between
the segments (P < 0.05 for each segment). The
correlation between the material constants and the residual strain
gradient was studied to evaluate whether these variables were
associated. No significant association was found. The highest
determination coefficient (R2) was found for the
association between the a1 constant and the residual strain (R2 = 0.27).
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Fig. 5.
Means ± SE of the relation between circumferential
(A) and longitudinal (B) stress and strain in
different arteries. The sample size was 16.
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Figure 6 demonstrates the effect of
considering the zero-stress state rather than the no-load state on the
pressure-strain relation. The midwall strain referenced to the
zero-stress state did not differ from that referenced to the no-load
state, whereas the inner and outer surface strains were lower and
higher when referenced to the zero-stress state compared with the
no-load state as reference. Thus, when the midwall strain is computed, the no-load state and the zero-stress states can be used as reference with the same result. However, if strains at the inner and outer surfaces are computed, then it is important to know the true
zero-stress state. The data in the figure are obtained from the
thoracic aorta, but a similar effect was observed in all segments.
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Fig. 6.
Means ± SE of the relation between strain and pressure in the
thoracic aorta referenced to the no-load state (n) and the zero-stress
state (z). Strains were computed at the inner surface (inner), midwall
(mid), and outer surface (outer). The midwall strain referenced to the
zero-stress state did not differ from that referenced to the no-load
state, whereas the inner and outer surface strains were lower and
higher when referenced to the zero-stress state compared with the
no-load state. A similar effect was found in all five segments. The
sample size was 16.
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DISCUSSION |
The mechanical properties of arteries are important determinants
of hemodynamics. The aorta, pulmonary artery, and large distributing arteries are distended rapidly during ventricular ejection, transiently accommodating 50% or more of the stroke volume (8). These
vessels then retract during diastole. Because of these dimensional
changes, the viscoelastic properties of the walls of these large
vessels are factors determining instantaneous arterial pressure. The
large arteries also transmit the pressure pulse and contribute the
dynamic resistance to the oscillatory components of blood flow
(8). Finally, certain baroreceptor areas of the arterial
tree monitor blood pressure by distension, relaying this information to
the central nervous system. All of these functions are modulated by the
mechanical properties of the arterial wall.
Many studies have been conducted on stress and strain in the aorta
(1, 6, 16-21, 24-25), femoral artery (2,
11), carotid artery (4, 8, 9), and pulmonary artery
(3, 21, 22). Most studies before 1983 used the no-load
state as reference for the strain analysis. However, it is now well
documented that the vascular system expresses residual strain in the
no-load state. This effect can be demonstrated by making a radial cut in a ring of tissue. It was demonstrated that this behavior is a
mechanism that prestresses the vessel, thus reducing the concentration of circumferential stress at the inner wall. The zero-stress state provides the reference state for calculation of strain and the standard
morphological state to describe vascular tissue. The zero-stress state
is sensitive to tissue remodeling because the tissue is not deformed by
stress. Fung and colleagues (11, 19, 20) studied the zero-stress state of the aorta and its
main branches, small blood vessels and the pulmonary artery of
rats. They found that the opening angle varied along the rat
aorta and its branches. The opening angle of the pulmonary artery
showed axial variation (21, 22). We measured the opening
angle of the arterial segments of thoracic aorta, abdominal aorta,
common carotid artery, femoral artery, and pulmonary artery. The
results in our experiment are concordant with those obtained by Fung
and colleagues. We also computed the residual strain distribution on
the inner and outer wall of the arterial segments. We demonstrate that
the absolute value of residual strain both at inner and outer surface
was largest in the femoral artery and smallest in the thoracic aorta.
For the residual strain distribution in the thoracic aorta, the common
carotid artery and the femoral artery, the results of our experiment
were similar to the results reported by Li and Hayashi
(18) in the same arterial segments in rabbits. The
residual strain was largest in the femoral artery and smallest in the
thoracic aorta among three arteries (15). Thus the same
trend seems to exist between different species. We further computed the
residual strain gradient and residual strain gradient-to-wall thickness ratio. The result showed that the residual strain gradient and residual
strain gradient-to-wall thickness ratio were largest in the femoral
segment and smallest in the thoracic aortic segment. These results
suggest structural differences between the various arteries. However,
these were not studied in greater detail in this study, and, in
general, correlations between structure and residual strains have been
inconclusive. This is likely due to the structural complexity of the
arterial wall.
A volume-pressure method based on a step test was used to generate the
data for computation of the stress and strain distribution of the five
arteries. In the analysis, it was assumed that the arterial wall was
homogenous, incompressible, and elastic. Furthermore, stress and strain
gradients throughout the arterial wall thickness were ignored. The
analysis resulted in determination of material constants for the
arterial segments. The exact meaning of these constants is discussed
(11). The results (see Fig. 5 and Table 1) showed that in
the circumferential direction, the arterial wall was stiffest in the
femoral artery and softest in the pulmonary artery. In the longitudinal
direction, the arterial wall was stiffest in the thoracic aorta and
most compliant in the pulmonary artery. Biaxial testing of arteries has
been done before (8, 10, 17), though the former studies
did not consider the zero-stress state. Qualitatively, however, the
same differences in elasticity as shown in this study were found previously.
The main structural components of blood vessels are elastin fibers,
collagen fibers, and smooth muscle cells. The proportion of those
components varies throughout the circulation system. The passive
elements are prevalent in the large vessels whereas in small vessels at
least 50% of the vessel wall may consist of smooth muscle cells. The
mechanical characteristics of blood vessels are determined by both
passive and active tissue components. When the vascular smooth muscle
cells are inactivated, the elastic modulus is determined by connective
tissue, primarily elastin and collagen fibers. Despite the complexity
of the connections among elastin fibers, collagen fibers, and smooth
muscle cells, the properties of the vessel are represented by an
element (the parallel elastic element) in parallel with the series
combination of elements representing the force generators in the smooth
muscle cells (the contractile element) and the structures connecting them (the series elastic element). Because the structure and the proportion of the elements in the thoracic aorta, abdominal aorta, common carotid artery, femoral artery, and pulmonary artery differ, the
stress-strain distribution in these arterial segments will also differ.
Despite the fact that the thoracic aorta is an elastic artery, it was
stiffest in the longitudinal direction when pressurized. Furthermore,
the femoral artery is a muscular artery, but it was stiffest in the
circumferential direction when pressurized. The components determining
the circumferential or longitudinal vascular stress-strain
distributions and the physiological significance of these findings
still need further study.
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ACKNOWLEDGEMENTS |
We acknowledge the Karen Elise Jensens Foundation and The NOVO
Nordic Center of Growth and Regeneration for financial support.
| |
FOOTNOTES |
Address for reprint requests and other correspondence: H. Gregersen, Biomechanics Lab, Center for Sensory-Motor Interaction, Aalborg Univ., Fredrik Bajers Vej 7D-3, DK-9220 Aalborg, Denmark (E-mail: hag{at}smi.auc.dk).
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.
10.1152/ajpheart.00620.2000
Received 6 July 2000; accepted in final form 15 October 2001.
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Am J Physiol Heart Circ Physiol 282(2):H622-H629
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TOP
ABSTRACT
INTRODUCTION
![<IT>+</IT>2<IT>a</IT><SUB>4</SUB>(<IT>E<SUB>&thgr;&thgr;</SUB>E</IT><SUB>&phgr;&phgr;</SUB><IT>−E<SUP>*</SUP><SUB>&thgr;&thgr;</SUB>E<SUP>*</SUP></IT><SUB>&phgr;&phgr;</SUB>)]](/content/vol282/issue2/fulltext/H622/img008.gif)
