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Файл:Ординатура / Хирургия / Библиотека им академика М.И. Перельмана / Книга_3787_Библиотеки_им_академика_М_И_Перельмана
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4.4 Microstructural Models of Coronary Artery 205
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The total, passive, and active circumferential stresses of media are shown in
Fig. 4.14a, c for axial stretch ratio λ
¼ 1.2 and λz¼ 1.3, respectively, while the
z
corresponding axial stresses are shown in Fig. 4.14b, d. The experimental data
averaged over 5 vessels are presented by symbols (diamond, circle, and solid
triangle), and model predictions are presented by solid lines. The predicted mean
ratio of axial to circumferential active stresses of the media is 0.63 0.02 for
λ
¼ 1.2 and 0.59 0.02 for λz¼ 1.3 which are nearly identical to experimental
z
measurements (0.64 0.09 and 0.58 0.04, respectively). Furthermore, the peak
active circumferential stress of media (referred to as optimal stretch ratio λ
0
) slightly
θ
precedes the axial stress, while the peak active circumferential and axial stresses of
individual SMC occur at the same stretch level. Finally, the maximal active circumferential and axial stresses occur at λ
as compared to λ
0
¼ 1:39 and λ
θ
0
¼ 1:34 and λ
θ
0
¼ 1:41 (λz¼ 1.2).
θ
0
¼ 1:36 (λz¼ 1.3), respectively,
θ
Smooth muscle cells (SMC) become stiffer during vasoconstriction due to the
forces generated by actin–myosin interaction and tensile properties of cytoskeletal
filaments increase significantly in contraction. Hence, the affine deformation
assumption needs to be directly tested in active SMC. Moreo ver, the elongation of
the nuclei suggests that the tension developed by the cytoskeleton is transferr ed to
the nuclei which may influence gene transcription and cellular phenotypes (Ingber,
2006;O’Connell et al., 2008). Consequently, the determination of strain and stress
on individual SMC is essential for better understanding of SMC function in health
and disease. This requires the development of microstructure-based models to
accurately predict the microenvironment of cells and nuclei.
The biaxial vasoactivity of coronary arteries has also been found in other experimental studies (Gaballa et al., 1998; Hayman, Zhang, Liu, Xiao, & Han, 2013;Lu&
Kassab, 2007). Lu and Kassab (2007) found that there are significant axial force
changes during vasomotion of carotid and femoral arteries. Another study showed
that SMC vasoconstriction reduced artery buckling as compared with relaxed conditions (Gaballa et al., 1998; Hayman et al., 2013; Lu & Kassab, 2007), indicating
that vasoactivity may shorten the artery in the axial direction (i.e., vessel become
much stiffer in both circumferential and axial directions). These studies suggest that
the biaxial vasoactivity of arteries is related to the helical structure of SMC in
muscular arteries. The ratio of active axial to circumferential stresses is predicted
sin2θ
VSMC
as
cos2θ
constitutive law for active SMC (b
¼ 0:12 for any axial stretch ratio λz, assuming a simple one-dimension
VSMC
! 0 in Eqs. (4.194a, 4.194b), Appendix 13).
2
This value, however, is significantly lower than experimental measurements. It
suggests that there exists multi-axial SMC vasoconstriction in the coronary media,
and that the biaxial vasoactivity is induced by oblique SMC arrangement as well as
multi-axial muscle vasoconstriction.
The axial active response is principally induced by the multi-axial contraction of
SMC, denoted by the mean ratio of transverse to axial active stress of a single muscle
fiber (
(θ
of SMC, the larger axial stretch ratio λ
b
1
¼ 0:4 0:06 ; Appendix 13), while the oblique SMC arrangement
b
2
¼ 18.7) contributes about 30%. With the influence of helical orientation
VSMC
¼ 1.3 further stretches the oblique SMC and
z
206 4 Constitutive Models of Coronary Vasculature
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Fig. 4.14 The second Piola–Kirchhoff total, passive and active stresses of coronary media. (a) The
circumferential stresses at λ
stresses at λ
and solid lines present the predicted values from theory. Reproduced from Chen, Luo, et al. (2013)
with permission
¼ 1.3; (d) The axial stresses at λz¼ 1.3. Symbols present experimental measurements,
z
the peak active stresses, thus occurs earlier than that of λz¼ 1.2 (Table 4.30,
Appendix 13).
There is a need to extend the current phenomenological 2D model to 3D
microstructural constitutive model for active SMC, where the material parameters
have physical significance. The actin–myosin interaction is the fundamental mechanism of tension development in contractile cells and involves several signal transduction pathways. The constitutive law of SMC must be based on a combination of
molecular biology and nonlinear mechanics (Gestrelius & Borgst röm, 1986;
Stålhand et al., 2011; Yang, Clark, Bryan, & Robertson, 2003). The Chen, Luo,
et al. (2013) model shoul d also be extended to account for the SMC in the lamella
adjacent to intima or adventitia, where SMC align towards the axial direction
(O’Connell et al., 2008; Timmins, Wu, Yeh, Moore, & Greenwald, 2010) and may
contribute to the biaxial active response of blood vessels. Previous experimental
¼ 1.2; (b) The axial stresses at λz¼ 1.2; (c) The circumferential
z
4.4 Microstructural Models of Coronary Artery 207
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studies (Matsumoto & Nagayama, 2012; Nagayama & Matsumoto, 2004) showed
that actin filaments largely align in cell direction with a slightly oblique arrangement
in isolated SMC, suggesting that active stress generated by actin–myosin crossbridge may not only apply in the cell direction (i.e., the major axis of a cell), but also
in the transverse direction (minor axes). Moreover, the tensile properties of SMC
freshly isolated from rat aortas measured in both major and minor axes showed that
under relaxation, both major and minor axes present very low stiffness of
13.6 10.8 KPa and 1.8 3.0 KPa, respectively, which can be neglected as
compared to those of elastin and collagen fibers (with a magnitude of MPa). The
stiffness increases significantly in both axes, however, in contracted cells (stimulated
by Serotonin) as 92.4 30.1 KPa and 38.3 25.8 KPa, respectively (Matsumoto &
Nagayama, 2012; Nagayama & Matsumoto, 2004). These observations suggest that
contracted cells can present significant stiffness in their mino r axes, implying that a
1D active model may not be accurate for contracted cells in a 3D analysis of blood
vessels. Moreover, previous phenomenological models suggest symmetric active
stress–strain curves due to a symmetric simplification of SMC responses (Chen, Luo,
et al., 2013 ; Huo et al., 2012; Zhou, Rachev, & Shazly, 2015). Some studies dispute
this symmetry as they show tension gradually increases prior to a peak and then
declines more steeply in a non-symmetric fashion (Herlihy & Murphy, 1973 ;
Schmitz & Böl, 2011; Winters, Takahashi, Lieber, & Ward, 2011). Clearly, a triaxial
asymmetric constitutive law of SMC in the microstructural model that accounts for
active responses of SMC is warranted.
4.4.3 Integrated 3D Model of Coronary Artery Wall
The discussions above suggest a need for a fully integrated 3D microstructure model
of the entire coronary wall, which accounts for all microstructure constituents
including both passive and active responses. A unified fully integrated 3D model
should p redict the arterial biaxial vasoactivity and stress distribution in the vessel
wall as well as provide reliable parameter estimations of individual fibers and SMC
(Chen & Kassab, 2017). In this section, we introduce a 3D microstructural model of
coronary artery that integrates individual fibers and cells, as well as their orientation
and undulation distributions in adventitia and media along with an asymmetric
constitutive model to describe biaxial active properties of a single SMC.
The 3D model formulations of the full coronary artery wall including active
properties are outlined in Appendix 14 along with the method of material parameter
determination. This is the first 3D microstructural active model of coronary artery
with SMC contraction that accounts for microstructure, including collagen waviness
distribution, elastin and collagen orientation distributions in individual media and
adventitia, biaxial vasoactivity and symmetrical helical arrangement of media SMC,
as well as isotropic inter-lamellar elastin distribution in media (Chen & Kassab,
2017). The 3D model of coronary artery is considered under animal-specific
mechanical data (Chen & Kassab, 2017) and three sets of input geometrical data
208 4 Constitutive Models of Coronary Vasculature
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measured from separate set of animals (Chen, Slipchenko, et al., 2013; Case I)
Population-based statistical measurements grouped together from all animals into
single distributions for various geometric parameters (Chen, Slipchenko, et al., 2013;
Case II) “Animal-specific” statistical measurements for each sample; and Case III)
Mean model based on average values for each animal.
Case I The material parameters of population of fibers and cells estimated by
pooling of all measured microstructural data (from all animals) into the model are
provided in Table 4.31 (Appendix 14), and the corresponding model predictions of
pressure–radius and pressure–force relations are shown in Fig. 4.15. The measurements of total (passive and active) and passive arterial responses are compared with
model predictions under two different axial stretch ratios (λ
¼ 1.3 and 1.5, respec-
z
tively). The predictions are in good agreement with experimental measurements
which capture the nonlinear passive nonlinear behavior as well as the biaxial active
responses of coronary arteries. The larger axial stretch λ
¼ 1.5 leads to smaller
z
diameters and higher axial force.
Case II Since there are microstructural variations among different animals, the
geometrical parameters of fibers and cells for each animal are used during the
process of parameter optimization. The estimated material and geometrical parameters are provided in Table 4.32 (Appendix 14). Figure 4.16 shows the model
incorporating individual animal distributions of microstructure provides better predictions of vessel responses with improved R
stiffness parameter of IL elastin fibrils is k
planer elastin fibers k
k
¼ 29.5 22.1 MPa and MC¼ 5.23 0.96 (see Appendix 14 for definition and
C
¼ 0.27 0.07 MPa, while parameters of collagen fiber are
E
2
(as listed in Table 4.32 ). The mean
¼ 0.18 0.14 MPa, similar to that of
IL
physical meaning of parameters). These estimated mat erial parameters of elastin and
collagen based on experimental data of the full vessel response are consistent with
those estimated on passive individual adventitia, i.e., k
k
¼ 27.2 5.1 MPa, and MC¼ 5.37 0.53 (Chen, Guo, et al., 2016). The active
C
¼ 0.19 0.07 MPa,
E
material parameters are summarized in Table 4.32. The optimal stretch ratio for
SMC is λ
¼ 1.34 0.12, at which the maximum stre ss σ
max
¼ 0.09 0.01 MPa
max
is generated. The ratio of axial to circumferential stresses τ ¼ 0.23 0.08, indicating
that axial active response is significant and cannot be ignored.
Case III We performed a microstructural sensitivity analysis by replacing the
continuous spatial distributions of fiber orientation and waviness with a uniform
mean orientation angle. We considered two families of elastin with two mean
orientation angles (μ
angles (μ
and μC2) with a straightening strain e01in adventitia (see Appendix 14
C1
for definition of parameters). In media, there are two symmetrical families of SMC
and elastin fibers, while collagen fibers are parallel to elastin fibers and have a
different straightening strain e
in Table 4.33. The mean-value approach achieved lower R
of full statistical distributions.
Figure 4.16 shows the stress–strain curves of individual fibers of all samples
using material parameters based on full fiber distributions (Case I: Full model) and
and μE2), and two families of collagen with mean orien tation
E1
. Parameter estimates of fibers and cells are shown
02
2
as compared with those
4.4 Microstructural Models of Coronary Artery 209
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Fig. 4.15 Model predictions with statistical geometric data from all animals grouped together.
Solid lines denote model predictions of passive responses, and dashed lines denote model predictions of full responses of coronary arteries with K
experiment measurements and error bars denote standard deviation. (a and b) show the pressure–
radius relationships at two axial stretch ratios of λ
corresponding pressure–force relationships. Reproduced from Chen and Kassab ( 2017) with
permission
+
-induced SMC contraction. Symbols indicate
¼ 1.3 and 1.5, respectively, and (c and d) show
z
uniform distributions (Case III: Mean-value model). In general, elastin fibers take up
most of loads at low strain e < 0.5, and collagen is then gradually recruited to carry
loads at high strain levels. When SMC contract, they work with elastin to resist loads
at low strain, and SMC contraction declines where collagen is then engaged to
support the vessel wall. It is found that elastin stiffness determined by the full model
(Case I) is larger than that of mean-value model (Case III). For collagen fibers, the
stress–strain curve determined by mean-value approach enable earlier fiber recruitment than that of full model, while the active stress–strain of SMC do not show a
shift of the curve.
For coronary arteries, the majority of collagen fibers orient towards the longitudinal
direction and the other fibers align nearly in the circumferential direction, following
two normal distributions. This is in contrast with previous assumption that fibers are
210 4 Constitutive Models of Coronary Vasculature
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Fig. 4.16 Model predictions with individual samples (“animal-specific”) microstructural distribu-
tions showing better agreements with experimental data. Solid lines denote model predictions of
passive responses, and dashed lines denote predictions of full responses of coronary arteries with
+
-induced SMC contraction. Symbols indicate experiment measurements. (a and b) show the
K
pressure–radius relationships at two axial stretch ratios of λ
d) show corresponding pressure–force relationships. Reproduced from Chen and Kassab (2017)
with permission
¼ 1.3 and 1.5, respectively, and (c and
z
symmetrically disposed with respect to the circumferential direction of the vessel
(Hollander et al., 2011a; Holzapfel et al., 2000; Zulliger, Fridez, et al., 2004). The
outer radius of passive arteries rapidly increased at low pressure and began to plateau
at higher pressures as a result of circumferentially oriented collagen fibers engaged to
withstand loads with increase of pressure (Figs. 4.15 and 4.16). The axial force is
found to increase significantly from λ
are recruited to provoke a rapid increase of axial force at λ
¼ 1.3 to 1.5 as most longitudinal-oriented fibers
z
¼ 1.5, which is beyond
z
straightening strains of most collagen fibers. Other geometrical parameters, including
collagen straightening strain distribution, elastin and SMC orientations in adventitia
and media, respectively, and inter-lamellar elastin distribution, are also engaged to
achieve a full microstructure-based model of coronary arteries.
When the focus is on macroscopic behaviors rather than stresses of individual
cells and fibers, the mean-value approach model (Case III) c an predict nonlinear
responses and stress distribution of the vessel wall, which is the sum of stresses of
Appendix 1: Analysis of Shear Modulus (Lu et al., 2003) 211
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every constituent, with much higher computational efficiency. Moreover, the meanvalue approach model offers reliable predictions for tissues with uniformly distributed fibers (Chen, Zhao, et al., 2016). Some other microstructure models, such as
“four fiber-family” model (Achille, Celi, Puccio, & Forte, 2011; Baek, Gleason,
Rajagopal, & Humphrey, 2007; Hansen, Wan, & Gleason, 2009; Liu, Wen,
Mottahedi, & Han, 2014), can also provide good macroscopic predictions under
certain conditions. The models that lack realistic microstructure basis, however, are
not able to provide reliable material parameters or to predict micro-stresses on
individual fibers and cells (Fig. 4.17). In summary, the mean-value app roach (Case
III), with lower computational expense, provides good predictions of macroscopic
mechanical behaviors of blood vessel but it cannot obtain reliable material parameters of individual constituents, while a full microstructural model (Cases I and II)
can achieve both goals with including detailed microstructural distributions and thus
requires more computational cost.
Appendix 1: Analysis of Shear Modulus (Lu et al., 2003)
The coronary artery is assumed to be a two-layer cylinder (Fig. 4.18). Since the
intima is relatively thin in the normal swine artery, the intima-media is considered as
the inner layer and the adventitia as the outer layer. Initially, the arterial wall is
assumed to be a homogenous material so that an “apparent” shear modulus could be
obtained. The artery is subjected to a transmural pressure (P
(F
), and a torque (T ). The radii of the inner boundary of the intima, the interface of
λ
the media and adventitia, and the outer boundary of the adventitia are denoted by r
r
,andra, respectively, as shown in Fig. 4.18. A vessel segm ent of length
m
L undergoing a twist angle θ and an angle of twist per unit length (θ/L) can be
considered. The average shear stress in the vessel wall is σ
found that the T is linearly proportional to θ/L in the range of interest. Hence, σ
linearly proportional to θ/L in this range.
Since the shear strain (e
) is, by definition, ezθ¼
zθ
r θ
2L
), a longitudinal force
i
for the intact artery. It is
zθ
, the linearity implies that
zθ
,
i
is
where G is the shear modulus of elasticity of the intact arterial wall. The torque is
given by an integral of the product of the average shear stress and the moment arm
and the wall area as:
where J is the polar moment of inertia of the intact vessel given by:
T ¼ 2π
σzθ¼ 2Gezθ¼
Z
r
a
Grθ
L
r
i
r
2
dr ¼
Grθ
L
π2Gθ
4
r
r
a
L
4
i
¼ GJ
ð4:1Þ
θ
L
ð4:2aÞ
212 4 Constitutive Models of Coronary Vasculature
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Fig. 4.17 The stress–strain curves of individual fibers and SMC of all samples. Comparisons are
made between the full microstructure model and the mean-value approach. Solid lines denote the
stresses of fibers and cells predicted by the full model, and dashed lines are predictions of meanvalue approach. Reproduced from Chen and Kassab (2017) with permission
Fig. 4.18 Two-layer model
of coronary artery that is
exposed to transmural
pressure (P), axial tension
), and twist torque
force (F
λ
(T ). Reproduced from Lu
et al. (2003) by permission
Appendix 1: Analysis of Shear Modulus (Lu et al., 2003) 213
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π
4
4
J ¼
r
r
a
2
i
ð4:2bÞ
Finally, the desired relationship between the torque and the shear modulus is
obtained as:
T ¼ GJ
θ
L
ð4:3Þ
To extend this analysis to a two-layer model, it is assumed that there is no slip
between the medial and adventitial layers during torsion. This implies that the two
m
layers have the same twist angle (θ) during torsion. Let
σ
zθ
a
and σ
denote the average
zθ
shear stress in the intima-medial and adventitial layer, respectively. Hence, the
relationship between the average shear stress and shear modulus takes on a similar
form to Eq. (4.1) where G is replaced by G
or Gafor the intima-medial or
m
adventitial layer, respectively. The vessel wall force resultant is equal to the sum
of the forces in the two layers:
Z
r
m
Gmrθ
F
¼
θ
r
i
L
2πrdr þ
Z
r
a
Garθ
2πrdr ð4:4aÞ
L
r
m
or
23πθ
F
¼
θ
G
r
m
L
3
3
r
m
i
3
3
r
þ G
r
a
a
m
ð4:4bÞ
The total force is calculated from the measured torque as follows:
¼
F
θ
T
r
þ r
ðÞ=2
i
a
ð4:5Þ
The desired relationship between the shear modulus of the intact vessel and that
of its two layers can be obtained by combining Eqs. (4.3), (4.4a, 4.4b ) and (4.5)to
yield:
For a thin walled vessel where r
the form:
Incidentally, Eq. (4.7) can be directly derived from considerations of membrane
torque resultant. Since the intima-medial layer is tested mechanically after the
G ¼
π3riþ r
J
a
GJ ¼ J
3
r
m
mGm
r
3
i
þ JaG
G
m
~ rmand rm~ ra, Eq. (4.6) can be simplified to
i
3
þ G
a
a
3
r
r
a
m
ð4:6Þ
ð4:7Þ
214 4 Constitutive Models of Coronary Vasculature
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adventitia is dissected, the shear modulus of adventitia can be compu ted using
Eq. (4.6) or vice versa for the media.
The mean stresses in the circumferential, σ
, and longitudinal, σz, directions are
θ
given by
Pr
σ
i
¼
θ
h
ð4:8Þ
and
2
Pr
i
hroþ r
ðÞ
i
ð4:9Þ
where F
radius, r
F
λ
σ
¼
z
and h are the longitudinal force and wall thickness, respectively. The inner
λ
, of the vessel can be computed from the incompressibility condition for a
i
π r
2
o
r
þ
2
i
cylindrical vessel as:
s
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
A
0
where r
2
¼
r
r
i
, A0, and λzare outer radii at the loaded state, the wall area in the no-load
o
o
πλ
z
ð4:10Þ
state and the axial stretch ratio, respectively. Since all the quantities on the right-hand
side of Eq. (4.10) are measured, the loaded inner radius can be computed. The linear
regression data between shear modulus and circumferenti al stress are summarized in
Table 4.1 for various axial stretches .
Appendix 2: Formulation of Incremental Moduli (Lu et al.,
2004)
Mathematically, a blood vessel is assumed as a thin shell. The homeostatic in vivo
state (i.e., a state of stable static equilibrium) of a blood vessel is assumed to be a
circular cylinder. The distributions of the homeostatic axial and circumferential
strains, referred to the zero-stress stat e, are nearly uniform and there is no torsion
(Guo & Kassab, 2004). The circumferential deformation of an artery may be
described by the mid-wall circumferential Green strain, which is defined as follows:
where λ
is the mid-wall stretch ratio (λθ¼ c/C); c refers to the mid-wall circumfer-
θ
ence of the vessel in the loaded state and C refers to the corresponding mid-wall
1
E
θ
2
λ
¼
1
θ
2
ð4:11aÞ
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