Engineering, 2013, 5, 341-346
http://dx.doi.org/10.4236/eng.2013.510B069 Published Online Octob er 2013 (http://www.scirp.org/journal/eng)
Biaxial Constitutive Model of Active Coronary Media
Based on Microstructural Information
Huan Chen1*, Yunlong Huo2, Ghassan S. Kassab1,3
1Department of Biomedical Engineer ing, Indiana University Purdue University Indianapolis, Indianapolis, USA
2Mechan ics and Engineering Sci ence, State Key Laboratory for Turbulence and Complex Systems,
College of Engineering, Peking University, Beijing, China
3Department of Surgery, Cellular and Integrative Physiology, Indiana University Purdue
University Indianapolis, Indianapolis, USA
Received June 2013
ABSTRACT
Detailed morphological data of vascular smooth muscle cells (VSMC) of coronary arteries were limited. The present
study was to quantify dimensions and orientation of swine coronary VSMC and to develop a micro-structura l constitu-
tive model of active media. I t was found that geo metrical pa rameters of VSMC (length, width, spa tial aspect ratio, and
orientation) follow normal distributions, and VSMCs orientate towards the circumferential direction of vessels with
oblique and symmetrical angles. A micro-structural model of media layer was developed to accurate → accurately pre-
dict biaxial active responses o f coronary arterial media, based on experimental measurements. The present morphologi-
cal data base and micro-structural model lead to a better understanding of bio mechanics of muscular vessels.
Keywords: Biaxial Constitutive Model; Cell Deformation; Morphology; Vascular Smooth Mu scle Cell
1. Introduction
Many mechanical models of blood vessels suggested
uniaxial active constitutive relations, based on one-di-
mensional length-tension relationships of vascular smooth
muscle cells (VSMC s) in the circumferential direction of
vessels. Some studies, however, observed significant
biaxial responses (circumferential and axial) of blood
vessels during vasoconstriction [1-5], which cannot be
explained by previous uniaxial active models. The sig-
nificant axial active response may be induced by VSMC
helical orientation [6-9] and/or multi-axial active res-
ponses of individual VSMCs [2,3,10,11]. The arrange-
ment of VSMCs, however, has been debated in literature.
VSMCs were fou nd to align in the circu mferential d irec-
tion in some studies [ 6,12-14], while they were observed
to be obliq ue in other s [ 6-9].
The present study aimed to describe biaxial vasoactiv-
ity by a micro-structural constitutive model of coronary
artery, which accounts for geometrical and mechanical
properties of individual collagen fibers, elastin fiber and
VSMCs, respectively [11]. The morphology of VSMCs
must be measured by using confocal microscopy, and
four geometrical parameters (length, width, sp atial aspect
ratio, and orientation) were then quantified, and the de-
formation of individual VSMCs under various pressure
distensions was also measured. Based on measured geo-
metrical and deformation features of VSMC, a micro-
structural constitutive model, including muscle contrac-
tion was proposed to describe the biaxial contraction of
coronary arterial media.
2. Method s
2.1. Preparation of Samples
6 hearts of healthy pigs were used in present study. The
coronary arteries were dissected from hearts and cleaned
carefully. 6 segments of ~2 cm in length of each heart
were prepared for 6 different distension pressures. A
custom-made excess surface-area balloon tip catheter
was inserted into each segment and distended to fully
transmit pres sure to the vessel lumen [15]. T he distended
segments were immersion-fixed in 0.8% methanol-free
paraformaldehyde solution, the osmolarity of which was
adjusted to 292 ± 11 mOsm) with pH of 7.4, mimicking
the osmolarit y of nor mal extrac ellular fluids to avoid cell
shri nkage a nd swell ing [16]. The segment was then fixe d
at room temperature for 48 hrs for later preparing of his-
tological sections. 5 × 5 mm2 cr oss-sections were then
sectioned from the segment and the circumferential di-
rection was marked after distention-fixation. The adven-
*Corresponding author.

H. CHEN ET AL.
342
titia was peeled off carefully from the vessel to expose
media layer. The main protein type of the cellular cy-
toskeleton, F -actin, and the cellular nucleus were labeled
(by Alexa Fluor 488 phalloidin and DAPI) to track geo-
metries of VSMC under different distension loads. The
sections were wet mounted on microscope slides using a
glycerin-water mixture and then viewed under a FV1000-
MPE confocal microscope with a 60 × 1.1NA water im-
mersion objective. Each segment was scanned at 3 dif-
ferent locations for each loading condition. T he six load-
ing conditions were considered: 1) zero-stress st ate (ZSS);
2) no-load state (0 mmHg distension); 3) 40 mmHg dis-
tension; 4) 80 mmHg distension; 5) 120 mmHg disten-
sion; and 6) 160 mmHg distension.
2.2. Measurement of VSMC Geometry
The geometry of VSMCs was determined by four para-
meters: length, width, spatial a spect ratio, and orientatio n
angles (the circumferential direction of vessel was taken
as 0o and the axial direction as 90˚). The orientation of
the VSMC was measured using an automated algorithm,
and dimensions of VSMCs were directly measured on
confocal images [11]. The cell length and width were
determined by the major and minor axes of each cell, and
the aspect ratio, featuring the cell shape, was identified
by the ratio of length to width of a given cell. The RGB
image of the nucleus was first converted to a binary im-
age, where the nucleus-containing pixels were clearly
distinguished from the background by median filtering.
The geometrical parameters of the nucleus were calcu-
lated automatically based on this binary image [11]. The
measured result was expressed as the mean, mean ± SD
(standard deviation), of all measured cells in the images.
The significa nce of the d ifference between the parameter
under various loads was evaluated by a one-way ANO-
VA test (SigmaStat 3.5), while the results were consi-
dered statistically different when P < 0.05.
2.3. Microstructure-Based Constitutive Model of
Active Coronary Media
T he me dia segment was considered as a thin-walled elas-
tic tube deformed in the circumferential and axial direc-
tions. A previously proposed structural constitutive mod-
el was used to describe the mechanical response of pas-
sive coronary media [17,18], which contains isotropic
inter-lamellar ( IL) ela stin net works a nd hel icall y orie nted
collagen fibers:
{ }
2
2
[( ( )]
( )[()]
)
[ ()]
2
EEE EE
C
CC
E
C CCCC
fwW ed
f
Wwe we
π
π
θθ
π
θθ
−
=
= +






−
∫
E
E
(1)
where E w and C w are the strain energy of elastin
struts and collagen fibers, depending on uniaxial fiber
strain
s
e
(
,s EC=
denoting elastin and collagen, re-
spectively).
s
θ
is the fiber orientation angle (corres-
ponding unit vector denoted by
s
n

) and
s
f
is the vo-
lume fraction. The fiber strain was determined by
ss
s
e= ⋅⋅n En

with assuming affine deformation (i.e., a
fiber is assumed to rotate and stretch in the same way as
the bulk ti s sue) . T he li ne ar stress -strai n relatio n o f elasti n
was considered as
E EEE
w eke∂ ∂=
while the nonlinear
relation of collagen was considered as
C
N
C CCC
w eke∂ ∂=
,
with the material parameter
E
k
representing the stiff-
ness of the elastin, and
C
k
and
C
N
representing the
stiffness and nonlinear parameter of collagen. The pas-
sive strain energy of the coronary media
passive
W
was
calc ulated b y taking t he sum of the str ain ener gies of t he
elastin and collagen networks; i.e.,
() ()
passive E
WW=EE
()
C
W+E
, and the second Piola-Kirchhoff stress was ac-
cordingly determined by
()
passive passive
W=∂∂S EE
.
The total strain energy of media is the sum of the ac-
tive and passive contributions, and the active strain
energy is dominantly contributed by active VSMC. Tak-
ing into co ns ideration two families of helical V SM C wi th
symmetrical angles, it can be given as:
{ }
( )
{
{ }
( )
(,)
,
2
,
VSMC VSMC
VSMC V
acti ve
VSMC VSMC VSMC
VSMC
SMC
VSMC VSMVSCMC
W
fw
w
λλ
λλ
λλ
θ
θ
⊥
⊥
⊥
=
+−
(2)
where
VSMC
θ
is the orientation angle of VSMC,
VSMC
w
is multi-axial strain energy o f a single VSMC and
VSMC
f
is the volume fraction. A two-dimensional generalization
of the uniaxial length-tension relation of active VSMC
[19] was used to account for the multi-axial active re-
sponse of VSMC [2,3]:
( )
( )
( )
3
1
4
2
Erf
+1
VSMC VSMC
VSMC
VSMC
VSMC
act
VSMC
b
AC b
b
b
w
θλ
θ
λ θ
⊥
−

=





−
+


(3)
where
( )
VSMC TVSMC
VSMC
λ =⋅ ⋅⋅n FFn

is the longitu-
dinal stretch ratio of a VSMC (i.e., cell stretch), and
( )
VS
SM MC C TVSMC
λ
⊥
′′
=⋅ ⋅⋅n FFn

is the transversal stretch
ratio (
F
is deformation gradient).
VSMC
n

and
VSMC′
n

are the longitudinal and transversal vectors, respectively.
A
is the level of activation (0 is passive state and 1 is
fully active),
123
,,,
act
C bbb
and
4
b
are material con-
stants, and Erf() is the Gauss error function. Accordingly,

H. CHEN ET AL.
343
the active stress of coronary media is determined as the
derivatives of the strain energy function; i.e.,
acti ve
=S
()
acti ve
W∂∂EE
. Therefore, the total stress is the sum of
passive and ac t i ve stresses.
We used biaxial data of coronary media obtained in
our previous study [3] to determine material parameters
in the structural model. The media was considered as a
thin cylindrical shell and the 2nd Piola-Kirchhoff cir-
cumferential stress obtained by experimental measure-
ment was determined by
2
/
exp i
SPrh
θθ θ
λ =
, where P is
distension pressure,
2
/π
i ooz
r rA λ = −
is the inner ra-
dius in the loaded state,
o
r
is the outer radius in the
loaded state,
o
A
is the wall area in a no-load state, and
oi
hr r= −
is the wall thickness in the loaded state. The
axial stress was computed by
222 2
(/( ())/(π())) /
exp
zzio ioiz
SPrhrrFrr λ =++ −
with F pr esenting the axial force. The material parame-
ters were determined by minimizing the square of the
difference between the theoretical and experimental pas-
sive and active circumferential and axial 2nd Piola-Kir -
chhoff stresses [11]. The predicted strain-stress curves
were thus determined by Equations (1)-(3).
3. Resul ts
The pr obability distribution funct ions of cell geo metrica l
parameters were found to follow to continuous normal or
bimodal normal distributions (Table 1) at ZSS. The
lengt h of indi vidual VSMC was 56.0 ± 10.3 µm which is
larger than that of the nuclei (15.0 ± 4.7 µm) while their
widths were similar (3.9 ± 0.7 µm vs. 3.4 ± 0.8 µm). T he
means of aspect ratio of the VSMC and the nuclei were
14.7 ± 3.5 and 4.6 ± 1.7, respectively. VSMC aligned off
the circumferential direction of the vessel with a bimodal
distribution with a mean an gle of ± 18.7˚ ± 10.9˚, co nsis -
tent with the angl e of the nucleus ± 19.9˚ ± 10.7˚.
The VSMC of passive coronary media deformed sig-
nificantly with an increase in distention pressure (Figure
1). The cells gradually shifted in the circumferential di-
rection at elevated pre ssure. T he VSMC wer e sig nif ica ntl y
Table 1. The distribution of geometrical parameters of
VSMC and the nucleus were fitted to a continuous normal
distribution (or a bimodal normal distribution).
µ is the
mean of the distribution,
σ
is standard deviation and R2
presents goodness of fit.
Nucleus VSMC
Parameter s µ
σ
R2
µ
σ
R2
Length (µm) 15.0 4.7 0.96 56.0 10.3 0.98
Width (µm) 3.4 0.8 0.85 3.9 0 .7 0.93
Aspect ratio 4.6 1 .7 0.80 14.7 3.5 0.88
Orientation (o) ±19 .9 10.7 0.98 ±18.7 10. 9 0.92
stretched in the axial direction and became morespindled
with longer tails, while the nuclei did not significantly
deform. Changes of geometries of VSMC and nuclei
were fitted to a logarithmic function in Table 2. The
lengt h o f V SM C ra ise d sig ni fi ca ntl y a nd no nl ine ar l y ( P <
0.05), with increase of distension pressure, while the
length of the nuclei increased relatively slightly. The
change of VSMC length increased sharply from the
no-load state to 80 mmHg distention and became pla-
teaued at higher pressure. The widths of VSMC and the
nucle i d id no t c ha n ge si g ni fic a ntl y u nd e r a ll p r es s ure s ( P >
0.05) (insignificant changes were not listed in Table 2).
Accordingly, the aspect ratio of VSMC increased nonli-
nearly, while that of the nuclei did not change signifi-
cantly (P > 0.05). The orientation angles of VSMC and
the nuclei decreased nonlinearly from the no-load state to
160 mmHg distension (P < 0.05).
The predicted reorientation of VSMC, determined by
2222
arccos(cos/ cossin)
VSMC
VSMCVSMCVSMC z θθ
θ
θλθλ θλ
′=
+
(
1.0
z
λ =
in experimental study), was plotted in Figure
2(a), according to the affine deformation assumption.
Correspondingly, the VSMC stretch ratio, determined by
Figure 1. The deformed VSMC of coronary media under
various distention pressures: The loading pressures in (a) -
(e) are: 0 mmHg, 40 mmHg, 80 mmHg, 120 mmHg, 160
mmHg, res pectively.
Table 2. Non-linear relations between geometrical para-
meter s a nd dis te nsio n pres sur e s obt ain ed by curve fit ting to
a logarithmic function:
12
y= a+ aLog(x)
.
Parameter s
1
a
2
a
R2
VSMC
Length (µm) 60.2 7.7 0.95
Aspect Ratio 17.4 1.8 0.90
Orientation (o) 18.2 -2.9 0.99
Stre tch Ratio 1.1 0.1 0.95
Nucleus Length (µm) 16.1 0.9 0.81
Orientation (o) 17.2 -2.5 0.99

H. CHEN ET AL.
344
Figure 2. (a) Measured and predicted changes of the orien-
tation angle of the VSMC and nucleus; (b) Measured and
predicted VSMC and tissue deformation.
the ratio of cell le ngth in the lo aded state r elative to ZSS,
was predicted as
2222
cos sin
VSMCVSMCVSMC z θ
λθλ θλ = +
,
and plotted in Figure 2(b). The results showed that af-
fine-model predictions of VSMC deformation agreed
with experiment al mea surements (P > 0.05).
The total, passive and active stresses of media were
plotted in Figure 3 for axial stretch ratio
1.2
z
λ =
and
1.3
z
λ =
. The experimental data averaged over 5 sam-
ples were presented by symbols (diamond, circle and
solid triangle), and model predictions were presented by
solid lines. The mean ratio of axial to circumferential
active stresses of the media was 0.63 ± 0.02 fo r
1.2
z
λ
=
and 0.59 ± 0.02 for
1.3
z
λ =
, while the mean ratio of a
single VSMC (i.e.,
12
bb
) was 0.4 ± 0.06, smaller than
that of the media. The peak active circumferential stress
of media slightly preceded the axial stress, while the
peak active circumferential and axial stresses of individ-
ual VSMC occurred at the same stretch level. The larger
axial stretch ratio
1.3
z
λ =
advanced VSMC contraction
as compared with
1.2
z
λ =
.
4. Discussion and Summary
Detailed morphological data of coronary media was col-
lected and then was used to construct a micro-structure-
Figure 3 . The t otal , pass i ve a n d act ive stre s ses of c or onar y med ia. (a, b) T he ci rc u mfere nti al a n d ax i al s tres ses at
z
= 1.2λ
; (c,
d) The circumferential and axial s t res s es at
z= 1.3λ
.

H. CHEN ET AL.
345
based model of active coronary media, which accounts
for material properties of individual collagen and elastin
fibers, and VSMCs. The measured VSMC orientation
and distribution implies that VSMC constriction gene-
rates not only circumferential active stress, but also axial
active responses in coronary arteries [11].
The nonlinear geometrical parameter-pressure rela-
tions of VSMC and nuclei, consistent with that of the
outer diameter of vessels, suggest that recruited collagen
fibers either in media or adventitia prevent the intact
vessel as well as VSMC from overstretch at high pres-
sure [12,15,20,21]. Additionally, the deformation of in-
dividual VSMC was found to be affine. VSMC connect
with the extracellular matrix (ECM) via focal adhesion
and the deformation thus strongly depends on ECM de-
formation as well as the macroscopic deformation of
blood vessels. It is likely that flexible actin filaments
deform with ECM through dense bodies in passive tissue
such that collagen and elastin fibers follow affine defor-
mation [22,23]. VSMC become stiffer during vasocon-
striction due to the forces generated by actin-myosin in-
teraction and tensile properties of cytoskeletal filaments
increase significantly in contraction. Hence, the affine
deformation assumption needs to be directly tested in
active VSMC. Moreover, the elongation of the nuclei
suggested that the tension developed by the cytoskeleton
is transferred to the nuclei which may influence gene
transcription and cellular phenotypes [14,24]. Conse-
quently, the determination of strain and stress on indi-
vidual VSMC is essential for better understanding of
VSMC functions in normal and diseased arteries. This
requires the development of microstructure based models
to accurately predict the micro-environment of cells and
nuclei.
The biaxial vasoactiviy of blood vessels were found in
many st udies. Huo et al. found that the porcine coronary
artery displayed significant biaxial vasoconstriction in-
duced by a K+ physiological s aline solut ion [2,3]. Lu and
Kassab [1] observed that there were significant axial
force changes during vasomotions of carotid and femoral
arteries, and Hayman et al showed that VSMC vasocon-
striction reduced artery buckling as compared with re-
laxed conditions [5], indicating that vasoactivity may
shorten the artery in the axial direction. These studies
suggest that the biaxial vasoactivity of arteries is related
to the helical structure of VSMC in muscular arteries.
When a ssuming a simple o ne-dimension constit utive la w
for active VSMCs, the ratio of active axial to circumfe-
rential stresse s was pr edicte d as 0 .12 for an y axial stre tc h
ratio, which was significantly lower than experimental
measurements (≈0.6). It suggests that there exists mul -
ti-axial VSMC vasoconstriction in coronary media. The
axial active response was principall y induced by the mul-
ti-axial contraction of VSMC, denoted by the mean ratio
of transversal to axial active stress of a single muscle
fiber (0.4), while the oblique VSMC arrangement con-
tributed ab out 30 %. W ith the influe nce o f he lical o rienta -
tion of VSMC, the larger axial stretch ratio
1.3
z
λ =
further stretches the oblique VSMC and the peak active
stresses, thus occurs earlier than that of
1.2
z
λ =
. The
present micro-structural model, based on a structural pas-
sive model and a two-dimensional model of active
VSMC [2,3], can accurately predict the biaxial vasoac-
tivity of coronary media based on the measured micro-
structure [11].
Some limitations need to be mentioned. First, a three-
dimensional constitutive model should be developed for
individual active VSMCs, and a three-dimensional mi-
cro-structural model for vessels can be further proposed.
Second, the present model does not account for the
VSMCs in the lamella adjacent to intima or adventitia,
where VSMC aligned towards the axial direction [14,25]
and may contribute partially to the biaxial active re-
sponse of blood vessels. Finally, a microstructure-based
model of the entire vessel (adventitia and med ia), sho uld
be integrated to investigate macro- and micro-scopic me-
chanical behaviors of active vessels [3,26].
In sum mar y, the p res ent s tud y p rovi ded mor phol ogic al
data of VSMCs of coronary media, based on which an
active biaxial microstructure-based constitutive model
was developed. The micro-structural model can accu-
rately predict the biaxial mechanical responses of coro-
nary media, and provides a more accurate framework for
the biomechanics of blood vessels.
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