Am J Physiol Heart Circ Physiol 288: H881–H886, 2005;
doi:10.1152/ajpheart.00607.2004.
Transmural strain distribution in the blood vessel wall
Xiaomei Guo, Xiao Lu, and Ghassan S. Kassab
Department of Biomedical Engineering, University of California, Irvine, California
Submitted 18 June 2004; accepted in final form 22 September 2004
opening angle; aorta; coronary arteries; stress
of circumferential residual strain by Fung (5)
and Vaishnav and Vossoughi (30) has placed vascular biomechanics on firmer grounds. Previous to what has now become
axiomatic, it was thought that the circumferential strain (and
consequently stress) are greatest at the inner wall and decrease
towards the outer wall at the in vivo state. In a computational
study, Chuong and Fung (3) showed that the existence of
circumferential residual strain reduces the transmural gradients
of stress and strain, i.e., the inner (intima) and outer (adventitia) circumferential stresses and strains are similar at the in
vivo state. Their computational approach was based on the
existence of a strain energy density function whose constants
were determined experimentally. The stresses and strains used
in the strain energy function were based on the zero-stress
state, which was characterized by an opening angle (␪; defined
as the angle subtended by two radii connecting the midpoint of
the inner wall). Takamizawa and Hayashi (28, 29) solved the
inverse problem, i.e., they showed that under the uniform strain
THE RECOGNITION
Address for reprint requests and other correspondence: G. S. Kassab, Dept.
of Biomedical Engineering, Univ. of California, 204 Rockwell Engineering
Center, Irvine, CA 92697-2715 (E-mail: [email protected]).
http://www.ajpheart.org
hypothesis, the thin-wall theory can be used to predict the
material constants in the strain energy density function.
The first direct experimental evidence for the uniform transmural strain hypothesis at the in vivo state was provided by
Fung and Liu (10) on small vessels where they measured the
circumferences in the loaded and zero-stress state and computed the corresponding strains at the inner and outer wall. In
both the computational and experimental studies, the vessels
studied had ␪ ⬍ 180°. Although the majority of vessels fall into
this category, there are regions of the rat and human aorta, rat
pulmonary artery, porcine coronary artery, and rat ileal arterioles that have ␪ ⬎ 180° (4, 6, 9, 10, 18, 27). Furthermore, ␪
is known to increase beyond 180° in hypertension-, cigarette
smoke-, and diabetes-induced remodeling (8, 9, 11, 13). Finally, other tubular organs such as the dog trachea and guinea
pig small intestines are known to have ␪ well in excess of 180°
(14, 16).
The objective of the present study is to examine the validity
of the uniform transmural strain hypothesis at the in vivo state
along the aorta and coronary arterial tree. Our hypothesis is
that the uniform transmural strain hypothesis cannot apply to
those vessels that turn inside out (␪ ⬎ 180°). In those cases, we
will show that the loaded circumferential strain on the inner
wall will become smaller than that at the outer wall, which is
the converse of the case where the residual strain is ignored.
The mechanical and physiological implication of these observations will be discussed.
METHODS
Existing Data
The method of animal preparation and the measurement of strain in
the aorta and coronary arterial tree have been recently described in
detail (15). In that study, the distribution of mean stress and strain
throughout the porcine coronary arterial tree and in the aorta and its
branches were presented. The same data were used in the present
study to determine the inner and outer strain, and the discussion herein
is focused on the transmural variation of loaded strain. In total, 572
aortic (186 thoracic and 136 abdominal rings) and secondary arterial
branches (250 rings from iliac, femoral, and other branches) were
analyzed. Additionally, 387 vessel rings from the coronary arterial
tree were considered with diameters ⬎50 ␮m.
Aorta and its Primary Branches
The surgical preparations of the 10 pigs used for the aorta have
been described in Guo and Kassab (15). Briefly, the ascending aorta
was cannulated and perfused with catalyzed silicone elastomer at 100
mmHg after the animal was euthanized (18). The aorta, with the
solidified elastomer in its lumen, was dissected and cut transversely
into rings. All rings were photographed transverse to the long axis of
the vessel in the loaded state with the hardened elastomer maintained
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.
0363-6135/05 $8.00 Copyright © 2005 the American Physiological Society
H881
Downloaded from http://ajpheart.physiology.org/ by 10.220.32.247 on June 18, 2017
Guo, Xiaomei, Xiao Lu, and Ghassan S. Kassab. Transmural
strain distribution in the blood vessel wall. Am J Physiol Heart Circ
Physiol 288: H881–H886, 2005; doi:10.1152/ajpheart.00607.2004.—
The transmural distributions of stress and strain at the in vivo state
have important implications for the physiology and pathology of the
vessel wall. The uniform transmural strain hypothesis was proposed
by Takamyzawa and Hayashi (Takamizawa K and Hayashi K. J
Biomech 20: 7–17, 1987; Biorheology 25: 555–565, 1988) as describing the state of arteries in vivo. From this hypothesis, they derived the
residual stress and strain at the no-load condition and the opening
angle at the zero-stress state. However, the experimental evidence
cited by Takamyzawa and Hayashi (J Biomech 20: 7–17, 1987; and
Biorheology 25: 555–565, 1988) to support this hypothesis was
limited to arteries whose opening angles (␪) are ⬍180°. It is well
known, however, that ␪ ⬎ 180° do exist in the cardiovascular system.
Our hypothesis is that the transmural strain distribution cannot be
uniform when ␪ is ⬎180°. We present both theoretical and experimental evidence for this hypothesis. Theoretically, we show that the
circumferential stretch ratio cannot physically be uniform across the
vessel wall when ␪ exceeds 180° and the deviation from uniformity
will increase with an increase in ␪ beyond 180°. Experimentally, we
present data on the transmural strain distribution in segments of the
porcine aorta and coronary arterial tree. Our data validate the theoretical prediction that the outer strain will exceed the inner strain when
␪ ⬎ 180°. This is the converse of the gradient observed when the
residual strain is not taken into account. Although the strain distribution may not be uniform when ␪ exceeds 180°, the uniformity of stress
distribution is still possible because of the composite nature of the
blood vessel wall, i.e., the intima-medial layer is stiffer than the
adventitial layer. Hence, the larger strain at the adventitia can result in
a smaller stress because the adventitia is softer at physiological loading.
H882
TRANSMURAL STRAIN HYPOTHESIS
in the lumen. The elastomer was then pushed out of each ring, and a
radial cut was made at the anterior position. This process causes the
ring to open up into a sector. The cross section of each sector was
photographed 60 min after the radial cut. The morphological measurement of inner and outer circumference, wall thickness, and ␪ in
the loaded and zero-stress states were made from the images using an
image-analysis system (SigmaScan). ␪ was defined as the angle
subtended by two radii connecting the midpoint of the inner wall.
Because the aorta and coronary arteries have different wall thicknesses, it is useful to define a normalized parameter for the transmural
strain. As such, the strain distribution in the arterial wall can be
characterized in analogy to a coefficient for stress distribution (vε) as
proposed by Rachev et al. (25), namely,
Coronary Arterial Vessels
This parameter will be used an index of nonuniformity of strain across
the wall thickness.
Classification of aortic data. The aorta was subdivided into thoracic (descending aorta, just below the arch, to the diaphragm),
abdominal (from diaphragm to the common iliac artery), and secondary branches ⬎1.5 mm in diameter (femoral, renal, etc.). The data
were grouped together for the thoracic, abdominal, and secondary
branches, respectively.
Ordering of coronary arterial branches. We have previously developed and implemented an ordering system to classify various size
vessels into orders based on a diameter-defined Strahler system (18).
This has resulted in a unique relationship between diameter and order
number for the entire coronary arterial tree. The relationship between
the diameter range and order number obtained from the previous study
is as follows: order 5 (48.1–101 ␮m), order 6 (102–217 ␮m), order
7 (218 –384 ␮m), order 8 (385–554 ␮m), order 9 (555–986 ␮m),
order 10 (987–2,189 ␮m), and order 11 (2,190 – 4,500 ␮m). In the
present study, we determined the relationship between inner and outer
strains and diameter for coronary vessels ⬎50 ␮m in diameter. Hence,
by using the relation between diameter and order number from the
previous study, we determined the relationship inner and outer strains
and the order number.
Biomechanical Analysis
Let the circumference of a deformed vessel in the loaded state be
designated by C and that of the undeformed vessel in the zero-stress
state be designated by Czs. Hence, the circumferential deformation of
a cylindrical can be described by Green’s strain (ε), which is defined
as follows:
1 2
ε i,o ⫽ 共␭i,o
⫺ 1兲
2
(1)
where ␭i,o ⫽ Ci,o/C and is the stretch ratio; Ci,o refers to the inner or
i,o
refers to
outer circumference of the vessel in the loaded state, and Czs
the corresponding inner or outer circumference in the zero-stress state.
To assess the degree of nonuniformity of transmural strain, we can
evaluate the ratio of outer to inner strain as
i,o
zs
冉
εo
C ⫺C
⫽
εi
C ⫺C
2
o
2
i
冊冉 冊
zs2
o
zs2
i
C
C
zs 2
i
zs
o
(2)
Hence, the product of the first and second terms of Eq. 2 gives the
ratio of outer to inner Green strain. Equation 2 can be simplified if we
consider the quotient in terms of ␭, as follows:
冉 冊冉 冊
Co
␭o
⫽
␭i
Ci
C zs
i
C ozs
(3)
The first and second terms become linearized and are easier to
interrupt physically, as discussed later.
Fig. 1. Photos of coronary rings cut radially to reveal sectors with opening
angles (␪) ⬍180° (left) and ⬎180° (right).
AJP-Heart Circ Physiol • VOL
ε o ⫺ εi
εo ⫹ εi
(4)
Statistical Analysis
A Student’s t-test was used to compare inner with outer strain for
coronary and aortic vessels. A P value of ⬍0.05 was indicative of
statistical significance.
RESULTS
The variation of the first term on the righthand side of Eq. 2
with ␪ is shown in Fig. 2A. It can be seen that this ratio is
always greater than one for all values of ␪. Figure 2B shows
data for the second term on the righthand side of Eq. 2. When
␪ is equal to 180°, the inner and outer circumferences are equal
and hence the ratio is one. When ␪ ⬍ 180°, the inner circumference is smaller than the outer circumference, and the ratio is
⬍1. The converse is true when ␪ ⬎ 180°, as shown in Fig. 2B.
The ratio of outer to inner strain, i.e., product of the two ratios
shown in Fig. 2, A and B, is demonstrated in Fig. 2C. An
interesting pattern is revealed where the nonuniformity of
strain increases with an increase in ␪, especially when ␪ ⬎
180°. Figure 3 shows the corresponding data for the aorta and
its primary branches. Because the wall thickness for the aorta
is significantly larger than that for the coronary vessels, the
ratio of inner to outer circumference was significantly smaller
for the smaller ␪. This gave rise to a greater nonuniformity of
strain for the aorta, i.e., the ratio of outer to inner strain was
smaller for the smaller angles (e.g., ␪ ⬍ 45°). It should be
noted that the largest angle for the aortic vessels did not exceed
180°, whereas significantly smaller angles were found for the
aorta compared with the coronary vessels.
To assess the nonuniformity of strain distribution for the
coronary artery and aorta that have very different wall thicknesses, we defined a strain parameter as per Eq. 4. The data are
shown for the coronary and aortic vessels in Fig. 4, A and B,
288 • FEBRUARY 2005 •
www.ajpheart.org
Downloaded from http://ajpheart.physiology.org/ by 10.220.32.247 on June 18, 2017
Ten hearts were obtained from a local slaughterhouse on the
morning of the experiment (15). A cast of the left anterior descending
(LAD) arterial tree at 100 mmHg was made with silicone elastomer
(18), and the vessels were carefully dissected down to small branches
with diameters of ⬃50 ␮m. Each vessel segment along the LAD
arterial tree was cut perpendicular to the longitudinal axis into rings.
A photograph of every transverse section was taken, and the inner and
outer dimensions (and hence wall thickness) were measured. The
elastomer was then pushed out, and each ring was cut radially to
obtain the zero-stress state, as shown in Fig. 1. The morphological
data of the coronary vessels in the loaded and zero-stress states were
obtained with the same method described for the aorta.
vε ⫽
TRANSMURAL STRAIN HYPOTHESIS
respectively. It is apparent that the transmural strain is nonuniform for the coronary vessels when ␪ ⬎ 180° and for the aorta
when ␪ ⬍ 45°.
The inner and outer strains are listed in Tables 1 and 2 for
the coronary arterial tree and aorta, respectively. In Table 1, the
data are classified according to order number and range of ␪ (in
increments of 45°) for the coronary arterial tree. The outer
strain is significantly larger than the inner strain for orders
6 –11 by 7– 45%, respectively. When the outer strain was
compared with respect to ␪, it was significantly larger than the
inner strain for ␪ ⬎ 135°. In Table 2, the data are classified for
different segments of the aorta (thoracic, abdominal, and primary branches) and range of ␪. The inner and outer strains are
not statistically different for the abdominal aorta and branches.
In the thoracic aorta, the inner strain is 27% larger than the
outer strain. In relation to ␪, the inner strain is larger than the
outer strain for ␪ ⬍ 90°, whereas the converse is true for ␪ ⬎ 135°.
Fig. 3. A: relationship between the ratio of Co to Ci in the loaded state and ␪.
B: relationship between the ratio of Cizs to Czs
o in the zero-stress state and ␪. C:
relationship between the ratio of εo to εi and ␪. The data correspond to the aorta
and its branches.
the inner wall was a direct consequence of the starting assumption that the unloaded (zero transmural pressure) blood vessel
or ventricle is at the zero-stress state. Knowledge of the
zero-stress state is of vital significance in mechanics because
all calculations of stress and strain are made in reference to
such state. Simultaneously and independently, Fung (5) and
Vaishnav and Vossoughi (30) challenged the starting assump-
DISCUSSION
Circumferential Residual Strain
Before 1983, every study pointed to the existence of a stress
concentration at the intima of the blood vessel and the subendocardium of ventricle, to the extent that the circumferential
tension at the inner wall was much higher than that at the outer
wall (2). The stress concentration implied high local energy
consumption by the vessel or ventricle and consequently a high
oxygen demand at the inner wall. The stress concentration at
AJP-Heart Circ Physiol • VOL
Fig. 4. Relationship between the coefficient for strain distribution as defined
by Eq. 4 and ␪ for the data corresponding to the coronary arterial tree (A) and
to the aorta and its branches (B).
288 • FEBRUARY 2005 •
www.ajpheart.org
Downloaded from http://ajpheart.physiology.org/ by 10.220.32.247 on June 18, 2017
Fig. 2. A: relationship between the ratio of outer (Co) to inner circumferences
(Ci) in the loaded state and ␪. B: relationship between the ratio of inner (Cizs)
to outer circumferences (Czs
o ) in the zero-stress state and ␪. C: relationship
between the ratio of outer (εo) to inner stretch (εi) and ␪. The data correspond
to the left anterior descending arterial tree.
H883
H884
TRANSMURAL STRAIN HYPOTHESIS
Table 1. Comparison of ␧i and ␧o for the coronary arterial
tree in accordance to different order numbers and
different ranges of opening angles
εo
n
P Value
0.37⫾0.05
0.41⫾0.06
0.48⫾0.07
0.55⫾0.11
0.62⫾0.13
0.60⫾0.14
0.62⫾0.12
0.39⫾0.05
0.44⫾0.06
0.53⫾0.07
0.62⫾0.10
0.72⫾0.14
0.78⫾0.13
0.90⫾0.13
21
64
46
46
64
66
80
0.252
0.006
0.007
0.004
⬍0.001
⬍0.001
⬍0.001
0.49⫾0.12
0.49⫾0.11
0.58⫾0.16
0.59⫾0.13
0.55⫾0.12
0.52⫾0.09
0.47⫾0.12
0.51⫾0.11
0.68⫾0.17
0.85⫾0.15
0.85⫾0.16
0.85⫾0.12
36
82
125
59
51
14
0.539
0.067
⬍0.001
⬍0.001
⬍0.001
⬍0.001
Values are means ⫾ SD; n, no. of arterial trees. εi and εo, inner and outer
Green’s strains, respectively.
tion that the unloaded blood vessel is at the zero-stress state. A
radial cut of a vessel ring relieved the residual stress and strain
and changed the no-load circular geometry into an open sector
(5, 30). The open sector was quantified by ␪. The recognition
of residual stress and strain reduced the stress concentration
problem and simplified the stress-strain relation because it
referred to a well-defined state. Rachev and Greenwald (24)
provide a thorough review of the literature on residual strain of
blood vessels.
Uniform Transmural Strain Hypothesis
The physiological implication of ␪ was investigated theoretically by Chuong and Fung (3) based on an experimentally
determined constitutive equation, the geometry of the zerostress sector, and a ␪ of 108.7°. They showed that the undesired
stress concentration at the physiological loading conditions
was significantly reduced (3). Fung and Liu (10) later reached
a similar conclusion experimentally in regard to strain distribution in small arteries. The intima and adventitia strains were
computed in reference to the zero-stress state in small arteries,
and it was found that the transmural strain was uniform.
Although ␪ was not specified, the calculations of inner and
outer circumferences were based on vessels whose inner circumferences were smaller than the outer and suggested ␪ ⬍ 180°.
Fung and Liu (10) examined the transmural variation of
stress and strain at the homeostatic condition of 120 mmHg in
the saphenous artery in response to diabetes. The control vessel
had ␪ ⬍ 180° but increased well above 180° during diabetogenesis. They showed that the transmural strain and stress were
fairly uniform for the control vessel but were higher at the
outer wall for the remodeled vessel with ␪ ⬎ 180°. These are
similar observations to our normal vessels with ␪ ⬎ 180°.
In the present study, our contention is that the intimal strain
cannot equal to the adventitial strain when ␪ ⬎ 180°. This point
can be simply illustrated if we consider the deformation in
terms of ␭ as given by Eq. 3. The first term, the ratio of outer
to inner circumference in the loaded state, is physically always
⬎1. The second term, the ratio of inner to outer circumference
in the zero-stress state, is ⬍1 if ␪ ⬍ 180° and ⬎1 if ␪ ⬎ 180°.
AJP-Heart Circ Physiol • VOL
Experimental Evidence for the Nonuniformity of
Transmural Strain
Experimentally, the nonuniformity in strain occurs well
below the 180° angle for the coronary arteries. Table 1 shows
that the outer strain is greater than the inner strain for ␪ ⬎ 135°.
This occurs because the second term in Eq. 2 (Fig. 2B) is
disproportionately smaller than 1 compared with the first term
in Eq. 2 being ⬎1 (Fig. 2A). Hence, the product is smaller than
1 (Fig. 2C). Although the results for the aorta were similar to
those of coronary arteries, there were some interesting differences. Similar to the coronary vessels, the first term was ⬎1
(Fig. 3A), and the second term was ⬍1 for ␪ ⬍ 180° (Fig. 3B).
The ratio of outer to inner strain (product of Fig. 3, A and B)
was smaller than 1 for ␪ ⬍ 90° and ⬎1 for ␪ ⬎ 90°. Below 90°,
the ratio was ⬍1, implying that inner strain is greater than
outer strain, whereas above 90°, the converse was true, as
shown in Table 2. The differences between the aorta and
coronary data are in part due to the thickness difference.
Because the aorta is significantly thicker than the coronary
arteries, the ratio of inner to outer circumference in the zerostress state is reduced more than the increase in the ratio of
outer to inner circumference in the loaded state and hence the
product is ⬍1. Furthermore, the ␪ for the aorta and branches
reach smaller values than those of the coronary vessels, which
amplifies the ratios described.
In addition to examining the difference between inner and
outer strain, it is important to consider the gradient of strain, vε,
which vanishes when the strain distribution is uniform across
the arterial wall and approaches 1 as the circumferential strain
at the outer wall becomes much larger than the strain at the
inner wall. Conversely, the parameter will approach ⫺1 as the
inner strain becomes much larger than the outer strain. Despite
the differences in wall thicknesses, we find that the nondimensionalized gradient parameter has similar values for the coroTable 2. Comparison of ␧i and ␧o for the aorta and its
branches in accordance to different segments of aorta and
different range of opening angles
Aortic segment
Thoracic
Abdominal
Branches
Opening angle, °
0–45
45.1–90
90.1–135
135.1–180
εi
εo
n
P Value
0.33⫾0.11
0.48⫾0.11
0.49⫾0.11
0.26⫾0.09
0.48⫾0.12
0.49⫾0.09
186
136
250
⬍0.001
0.605
0.748
0.45⫾0.15
0.41⫾0.13
0.47⫾0.12
0.46⫾0.11
0.26⫾0.15
0.38⫾0.14
0.50⫾0.12
0.55⫾0.13
145
259
109
58
⬍0.001
0.002
0.051
⬍0.001
Values are means ⫾ SD; n, no. of vessels.
288 • FEBRUARY 2005 •
www.ajpheart.org
Downloaded from http://ajpheart.physiology.org/ by 10.220.32.247 on June 18, 2017
Order number
5
6
7
8
9
10
11
Opening angle, °
45.1–90
90.1–135
135.1–180
180.1–225
225.1–270
270.1–315
εi
Hence, when ␪ ⬍ 180°, the product of the two terms (the first
term is ⬎1 and the second is ⬍1) can be approximately equal
to 1 and hence implies uniformity of strain. On the other hand,
when ␪ ⬎ 180°, both terms are ⬎1 and hence their product
must further deviate from unity. The same argument applies to
ε, as shown in Fig. 2, A–C, respectively. The degree of
deviation from unity increases with an increase in ␪. Hence, we
find that theoretically and experimentally, the strain cannot be
transmurally uniform when ␪ ⬎ 180°. We shall explore the
experimental evidence in more detail below.
TRANSMURAL STRAIN HYPOTHESIS
nary arteries and aorta (Fig. 4, A and B). For the coronary
vessels with ␪ ⬎ 180°, it is clear that the adventitial strain is
significantly larger than the intimal strain.
Differences Between Elastic and Muscular Arteries
Implications of Nonuniformity of Transmural Strain on
Stress Distribution
Physiologically, it may not be disconcerting that the outer
strain (at the adventitia) is larger than the inner strain (at the
intima). Our group has recently determined the incremental
moduli of intima-media and adventitial layers for the proximal
LAD artery (20). We found that the modulus of the intimiamedia is greater than that of the adventitial layer, which is true
for other vessels as well (31, 33, 34). Hence, it may be
advantageous that the outer strain is larger than the inner strain
because of the composite nature of the vessel wall. For example, if the strain distribution was uniform, the stress would be
significantly higher at the inner wall because the intima-medial
layer is significantly stiffer than the adventitial layer under in
vivo loading conditions. A larger strain at the adventitia can
translate into a lower stress because of the smaller modulus.
We can demonstrate this quantitatively for the incremental,
linear elasticity case where the circumferential stress (S11) can
be expressed as follows:
im
im im
im im
⫽ E 11
ε11 ⫹ E 12
ε 22
S 11
(5a)
a
a a
a
a
S 11
⫽ E 11
ε11 ⫹ E 12
ε 22
(5b)
and
where E11 is the incremental Young’s modulus in the circumferential direction and E12 denotes the cross-modulus (12). The
superscripts im and a represent intima-media and adventitial
AJP-Heart Circ Physiol • VOL
layers, respectively. Lu et al. (20) obtained data from LAD
arteries of 10 porcine hearts. In five hearts, the biaxial incremental moduli of intact wall and intima-media layer were
measured, and those of the adventitia layers were computed. In
five additional hearts, the biaxial incremental moduli of intact
wall and adventitia were measured, and those of intima-media
were computed. The mean circumferential modulus of 10
intima-media and adventitia layers (5 measured and 5 comim
a
puted) were 259 (E11
) and 107 kPa (E11
), respectively (20). The
mean cross-moduli for the intima-media and adventitia were
im
a
47.6 (E12
) and 104 kPa (E12
), respectively. The mean ε at in
vivo loading for the LAD artery is 0.62 (inner) and 0.90
(outer), as listed in Table 1 for order 11 vessels. The axial
stretch ratio for the LAD artery is ⬃1.4, which gives a ε of 0.48
for both the inner and outer layers. Given these parameters, we
can compute the ratio of outer (Eq. 5b) to inner (Eq. 5a) mean
circumferential stress, which yields a value of 0.80. Hence,
although the ratio of outer to inner strain is 1.45, the stress
gradient is significantly smaller. If the outer and inner strains
were uniform at 0.62, similar calculations would yield a stress
ratio of 0.63 and hence a less uniform transmural stress
distribution.
Critique of Method
We have previously observed ␪ in the proximal coronary
arteries that exceed 180° (4, 17). In the present study, we
obtained ␪ that were significantly larger than those of the
previous studies. The reason for this discrepancy was that in
the present study, the vessel rings were distended with elastomer to obtain strain at the loaded condition. The ring was cut
open 1 or 2 min after the elastomer was pushed out of the
vessel. In the previous study, the vessel was maintained at
no-load state for longer duration before the radial cut. Hence,
we found that the initial state of stress in the vessel wall and its
time history affects ␪. This reflects the viscoelastic properties
of the vessel wall. These interesting features will be described
in a future publication and are beyond the scope of the present
study. The main issue here is that ␪ ⬎ 180° have been
documented not only for the coronary arteries but also for the
aorta, pulmonary artery, and ileal arterioles (6, 9, 10, 27). For
those vessels, the strain distribution cannot possibly be uniform
theoretically or experimentally.
The definition of ␪ as described in Fig. 1 assumes a circular
geometry for the vessel sector. This is, of course, only an
approximation because the majority of vessel sectors are not
exactly circular. The utility of this definition lies in its simplicity. More complex approaches can be found in the literature. Matsumoto et al. (21) proposed a method by which the
vessel sector is cut into numerous segments to eliminate the
noncircularity of the whole sector, which is particularly important for diseased vessels. The residual strain of the whole sector
was then estimated from the curvature and dimensions of each
segment. Alternative measures of ␪ based on measurement of
edge angle and tangent rotation angle have also been introduced in the literature (14, 19).
A question arises as to whether a single radial cut of vessel
ring is sufficient to ensure a zero-stress and zero-moment state
of the vessel sector. In theory, this can only occur if the tissue
is cut into infinite surfaces, each of which has zero traction. In
reality, Fung and Liu (7) have demonstrated that subsequent
additional radial cuts do not affect the vessel sector configu-
288 • FEBRUARY 2005 •
www.ajpheart.org
Downloaded from http://ajpheart.physiology.org/ by 10.220.32.247 on June 18, 2017
Arteries are generally subdivided into two types: elastic
(e.g., aorta, carotids, and pulmonary arterial vessels) and muscular (e.g., coronary, femoral, and cerebral arteries) (1). The
wall structure of both types of arteries consists of intima,
media, and adventitia. The aortic media contains many elastic
laminae and relatively few smooth muscle cells. There is a
gradual transition in structure and function between the elastic
and muscular arteries. The amount of elastic tissue decreases as
the vessels become smaller and the smooth muscle component
becomes more prominent. Hence, the proportion of smooth
muscle cells is much greater in the coronary arteries than in the
aorta. Consequently, the strain in the coronary artery can be
regulated by muscle contraction, which can affect the transmural deformation reported in the present study. Indeed,
Matsumoto et al. (22) have shown that ␪ increases with
contraction and decreases with relaxation in the rat thoracic
aorta. Contrary to these findings, Zeller and Skalak (35) reported an increase in ␪ due to vasodilatation in small rat
arteries. The discrepancy may be due to the difference in
location of the neutral axis relative to the smooth muscle cells
in large and small vessels. If the neutral axis is closer to the
intima, smooth muscle contraction will increase ␪. Conversely,
if the neutral axis is closer to the adventitia, muscle contraction
should decrease ␪. Finally, if the neutral axis coincides with the
smooth muscle layer, contraction will not affect ␪. The location
of the neutral axis is unknown for the coronary arteries and
should be determined in future studies.
H885
H886
TRANSMURAL STRAIN HYPOTHESIS
Significance of Study
The uniform transmural strain hypothesis has been generally
accepted for all blood vessels. The significance of the present
study is to provide a clarification that the strain cannot be
transmurally uniform for vessels whose zero-stress state reveals ␪ ⬎ 180°. For the coronary vessels, the transmural strain
was uniform only for ␪ ⬍ 135°, whereas for the aorta and its
primary branches, the transmural strain was uniform in the
range of 135 ⬎ ␪ ⬎ 90°. The nonuniformity of transmural
strain, however, may lead to uniformity of transmural stress
because the vessel is a composite structure with different
material properties in each layer as discussed above. Indeed, if
the vessel were homogenous with uniform material properties,
the nonuniform strain would lead to a very nonuniform stress
because of the highly nonlinear stress-strain relation. This underscores the importance of considering the composite nature of
blood vessels for a realistic mechanical analysis. A full mechanical analysis of the coronary artery that accounts for the composite
nature of the vessel remains a worthwhile task for the future.
ACKNOWLEDGMENTS
We thank Professor Y. C. Fung for the kind review of the manuscript.
GRANTS
This research was supported in part by National Heart, Lung, and Blood
Institute Grant 2 R01 HL-055554-06. G. S. Kassab was the recipient of an
American Heart Association Established Investigator Award.
REFERENCES
1. Burkitt GH, Young B, and Health JW. Wheater’s Functional Histology:
A Text and Colour Atlas. New York: Churchill Livingstone, 1993.
2. Chuong CT and Fung YC. Three-dimensional stress distribution in
arteries. J Biomech Eng 105: 268 –274, 1983.
3. Chuong CT and Fung YC. On residual stresses in arteries. J Biomech
Eng 108: 189 –192, 1986.
4. Frobert O, Gregersen H, Bjerre J, Bagger JP, and Kassab GS.
Relation between the zero-stress state and the branching orders of the
porcine left coronary arterial tree. Am J Physiol Heart Circ Physiol 275:
H2283–H2290, 1998.
5. Fung YC. What principle governs the stress distribution in living organs?
In: Biomechanics in China, Japan and USA, edited by Fung YC, Fukada
E, and Wang J. Beijing, China: Science, 1983, p. 1–13.
6. Fung YC and Liu SQ. Zero-stress states of arteries. J Biomech Eng 110:
1525–1527, 1988.
7. Fung YC and Liu SQ. Change of residual strains in arteries due to
hypertrophy caused by aortic constriction. Circ Res 65: 1340 –1349, 1989.
8. Fung YC and Liu SQ. Changes of zero-stress state of rat pulmonary
arteries in hypoxic hypertension. J Appl Physiol 70: 2455–2470, 1991.
AJP-Heart Circ Physiol • VOL
9. Fung YC and Liu SQ. Influence of STZ-diabetes on zero-stress states of
rat pulmonary and systemic arteries. Diabetes 41: 136 – 46, 1992.
10. Fung YC and Liu SQ. Strain distribution in small blood vessels with
zero-stress state taken into consideration. Am J Physiol Heart Circ Physiol
262: H544 –H552, 1992.
11. Fung YC and Liu SQ. Material coefficients of the strain energy function
of pulmonary arteries in normal and cigarette smoke-exposed rats. J Biomech 26: 1261–1269, 1993.
12. Fung YC and Liu SQ. Determination of the mechanical properties of the
different layers of blood vessels in vivo. Proc Natl Acad Sci USA 92:
2169 –2173, 1995.
13. Fung YC, Liu SQ, and Zhou JB. Remodeling of the constitutive
equation while a blood vessel remodels itself under stress. J Biomech Eng
115: 1670 –1676, 1993.
14. Gregersen H, Kassab GS, Pallencaoe E, Lee S, Chien C, Skalak R, and
Fung YC. Morphometry and strain distribution in guinea pig duodenum
with reference to the zero-stress state. Am J Physiol Gastrointest Liver
Physiol 273: G865–G874, 1997.
15. Guo X and Kassab GS. Distribution of stress and strain along the porcine
aorta and coronary arterial tree. Am J Physiol Heart Circ Physiol 286:
H2361–H2368, 2004.
16. Han HC and Fung YC. Residual strains in porcine and canine trachea.
J Biomech 24: 307–315, 1991.
17. Kassab GS, Gregersen H, Nielsen SL, Lu X, Tanko L, and Falk E.
Remodeling of the coronary arteries in supra-valvular aortic stenosis.
J Hypertens 20: 2429 –2437, 2002.
18. Kassab GS, Rider CA, Tang NJ, and Fung YC. Morphometry of pig
coronary arterial trees. Am J Physiol Heart Circ Physiol 265: H350 –H365,
1993.
19. Li X and Hayashi K. Alternate method for the analysis of residual strain
in the arterial wall. Biorheology 33: 439 – 449, 1996.
20. Lu X, Pandit A, and Kassab GS. Biaxial incremental homeostatic elastic
moduli of coronary artery: two-layer model. Am J Physiol Heart Circ
Physiol 287: H1663–H1669, 2004.
21. Matsumoto T, Hayashi K, and Ide K. Residual strain and local strain
distributions in the rabbit atherosclerotic aorta. J Biomech 28: 1207–1217,
1995.
22. Matsumoto T, Tsuchida M, and Sato M. Change in intramural strain
distribution in rat aorta due to smooth muscle contraction and relaxation.
Am J Physiol Heart Circ Physiol 271: H1711–H1716, 1996.
23. Omens JH and Fung YC. Residual strain in rat left ventricle. Circ Res
66: 37– 45, 1990.
24. Rachev A and Greenwald SE. Residual strains in conduit arteries.
J Biomech 36: 661– 670, 2003.
25. Rachev A, Greenwald SE, Kane TPC, Moore JE, and Meister JJ.
Analysis of the strain and stress distribution in the wall of the developing
and mature rat aorta. Biorheology 32: 473– 485, 1995.
26. Rachev A and Hayashi K. Theoretical study of the effects of vascular
smooth muscle contraction on strain and stress distributions in arteries.
Ann Biomed Eng 27: 459 – 68, 1999.
27. Saini A, Berry C, and Greenwald S. Effect of age and sex on residual
stress in the aorta. J Vasc Res 32: 398 – 405, 1995.
28. Takamizawa K and Hayashi K. Strain energy density function and
uniform strain hypothesis for arterial mechanics. J Biomech 20: 7–17, 1987.
29. Takamizawa K and Hayashi K. Uniform strain hypothesis and thinwalled theory in arterial mechanics. Biorheology 25: 555–565, 1988.
30. Vaishnav RN and Vossoughi J. Estimation of residual strains in aortic
segments. In: Biomedical Engineering. II. Recent Developments, edited by
Hall CW. New York: Pergamon, 1983, p. 330 –333.
31. Von Maltzahn WW, Warriyar RG, and Keitzer WF. Experimental
measurements of elastic properties of media and adventitia of bovine
carotid arteries. J Biomech 17: 839 – 847, 1984.
32. Vossoughi J, Hedjazi H, and Borris FSI. Intimal residual stress and
strain in large arteries. In: ASME Bioengineering Conference. New York:
1993, p. 434 – 437.
33. Xie J, Zhou J, and Fung YC. Bending of blood vessel wall: stress-strain
laws of the intima-media and adventitial layers. J Biomech Eng 117:
136 –145, 1995.
34. Yu Q, Zhou J, and Fung YC. Neutral axis location in bending and
Young’s modulus of different layers of arterial wall. Am J Physiol Heart
Circ Physiol 265: H52–H60, 1993.
35. Zeller PJ and Skalak TC. Contribution of individual structural components in determining the zero-stress state in small arteries. J Vasc Res 35:
8 –17, 1998.
288 • FEBRUARY 2005 •
www.ajpheart.org
Downloaded from http://ajpheart.physiology.org/ by 10.220.32.247 on June 18, 2017
ration. In the myocardium, Omens and Fung (23) found that a
second radial cut produced deformations significantly smaller
than those produced by the first cut. In contrast, Vossoughi et
al. (32) found that the sector geometry changes when an
additional circumferential cut is made to separate the vessel
wall into two layers. Similarly, Lu et al. (20) found that when
the vessel is separated into intima-media and adventitial layers,
␪ of the intima-media sector increases, whereas that of the
adventitial layer decreases relative to the intact wall.
The present study does not take into account the effect of
muscle tone. The in vivo and the residual stress in arteries have
been found to be strongly dependent on muscle tone (22, 26).
Although the present results were obtained under passive state,
they provide the framework for exploring the effect of active
contraction in future studies.