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4.3 Microstructure-Based Constitutive Models 195
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In the SOE prediction, the statistical deformation is flexible and non-affine, i.e.,
fibers with different orientation and waviness deform differently once they are
straightened and become stiffer to take up the load. Figure 4.9a describes the
influence of waviness λ
of collagen fibers that are parallel to the loading axis (θ ¼ 0
λ
¼ 1.5, 16, 1.7 and 1.8 are plotted. In the UF method, the deformation gradient of
0
on the microscopic stretch of the fibers. The axial stretch
0
) and with waviness of
the matrix and all fibers are identical to the applied macroscopic deformation; i.e.,
In the SOE model, the microscopic deformation is non-affine once some fibers are
straightened and become stiffer to take up the load. For fibers with a low waviness,
such as λ
¼ 1.5, their stretch shows a tendency of becoming straight at large
0
macroscopic deformation. As a result, the matrix phase must deform slightly more
to accommodate the applied macroscopic deformation, as indicated by the solid line
in Fig. 4.9b, which is noticeably higher than the UF prediction. Figure 4.9b plots the
stretch of fibers with orientation angle θ ¼ 0
,10and 20, and waviness λ0¼ 1.6,
showing the influence of the orientation angle θ on the statistical average microscopic stretch of the fibers. The UF predictions depend on the orientation but not the
waviness. The SOE predictions also show similar trend that larger orientation angle
θ leads to later engagement of the fiber.
The macroscopic stretch
λ
from 1.4 to 1.8, denoted as the macroscopic straightening stretch, is plotted in
0
λ that straightens the fibers with θ ¼ 0,15, and 25, and
Fig. 4.9c. The macroscopic straightening stretches predicted by SOE model are
consistently lower than the UF predictions, indicating earlier engagement of the
fibers. The difference is more significant for fibers with larger angle and/or higher
λ.
Fig. 4.8 For fibrous tissues with 20% randomly distributed collagen fibers, (a) The macroscopic
strain energy function, and (b) Cauchy stress–stretch relation. Reproduced from Chen, Liu, Zhao,
et al. (2011) with permission
⁄
Fig. 4.7 (continued) (triangle marker), and Active Stress (thin solid line with error bars of SE
value). Thick dotted line (theoretical total stress), dashed line (theoretical passive stress), and solid
lines (theoretical active stress). Reproduced from Huo et al. (2013) with permission
196 4 Constitutive Models of Coronary Vasculature
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Fig. 4.9 The influence of
waviness (λ
) and
0
orientation angle (θ) on the
statistical average
microscopic stretch of the
fibers. (a) The microscopic
stretch of matrix phase
(solid line) and collagen
fibers (symbols) with
orientation angle θ ¼ 0
and
different waviness; (b) The
microscopic stretch of
collagen fibers with
waviness λ
¼ 1.6 and
0
different orientation angle.
(c) The macroscopic stretch
that straightens the
undulated collagen fibers
with different orientation
angle. Reproduced from
Chen, Liu, Zhao, et al.
(2011) with permission
4.4 Microstructural Models of Coronary Artery 197
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waviness. As discussed above, the earlier fiber engagement is due to the heterogeneous microscopic deformation predicted by SOE method. Once a fiber is straightened, it deforms less than those undulated fibers and the macroscopic deformation,
due to the engagement of higher SEF. This mechanism prevents overstretch of the
fibers with low waviness λ
in the tissue under large macroscopic deformation
0
(Fig. 4.9). Since there is a limit for the collagen fibers to stretch beyond the straightening value (Mohanaradhakrishnan, Ramanathan, & Nayudamma, 1970), this prediction is qualitatively consistent with the protective role of fibers in the tissue.
It is important to investigate the predictive capability of UF and SOE models
under different conditions. Figure 4.10 shows model predictions of macroscopic
Cauchy stress of two tissues with the same waviness λ
or variance of the fiber orientation distribution (span Δ ¼ 0
These tissues are stretched along the principal direction of the fibers (θ ¼ 0
contracted in the transverse direction. While Δ ¼ 60
icantly higher than both SOE and FE. For Δ ¼ 0
¼ 1.6 but with different span
0
and 60, respec tively).
, the UF estimates are signif-
, the UF estimates, however, are
) but
similar to both FE and SOE results. The span or variance of fiber orientation
distribution plays a very important role in FE and SOE predictions but is much
less significant in UF model. It seems that the UF prediction of the macroscopic
stress is accurate only for tissue with very narrow distribution of fiber orientation.
In summary, the recently developed SOE model can provide the most accurate
predictions of macroscopic and microscopic mechanical behaviors for soft tissues,
while the uniform-field model with a fluid-like matrix assumes affine deformation in
soft tissue and neglects fiber–fiber interactions based on the fluid-like matrix
assumption. As such, the SOE method overcomes the shortcomings of the classical
uniform-field theory to enable more realistic predictions of the macroscopic stress
and the statistical deformation of the wavy collagen fibers and compares favorably
with direct FE numerical simulations. The shortcoming of the SOE model is the
computational cost which is significantly greater than the UF theory models (first
and second type) albeit less than the direct FE method. The choice of constitutive
models for the uniform theory depends on whether the matrix (GS and cells) is fluidlike or solid-like which is explored further in the next section. Accurate quantitative
data of microstructure are needed in the structural models (both for second and third
classes). Thus, the modeling developments point to the need for morphometric data
and constitutive behaviors of tissue components. Furthermore, with increasing
interest in microenvironment of soft tissues in physiology and pathology, the full
microstructural model with capabilities to predict heterogeneous microscopic stress
and deformation in health and vascular disease is a laudable goal.
4.4 Microstructural Models of Coronary Artery
The majority of microstructure-based constitutive models simplify the microstructure by assuming fibers to be symmetrically disposed with respect to the axis of the
vessel (with a preferred orientation) to yield a macroscopic orthotropic constitutive
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Fig. 4.10 The macroscopic
tensile stress–strain curve of
fibrous tissues with 20%
randomly distributed
collagen fibers. The beta
distribution of the fiber
orientation angle θ has mean
and span or variance of
of 0
and 60,
Δ ¼ 0
respectively. Reproduced
from Chen, Liu, Zhao, et al.
(2011) with permission
law for each layer (Holzapfel et al., 2000; Holzapfel & Weizsäcker, 1998; Zulliger,
Fridez, et al., 2004). Other models assume the geometrical features of fibers follow a
typical continuous distribution such as beta distribution with primarily planar array
(Chen, Liu, Zhao, et al., 2011; Hollander, Durban, Lu, Kassab, & Lanir, 2011a;
Hollander et al., 2011b; Lanir, 1983; Lokshin & Lanir, 2009b). The lack of accurate
quantitative data of microstructure and deformation of elastin and collagen fibers in
structural models is a barrier to accurately predict a vessel microenvironment.
Therefore, comprehensive geometrical data of tissue microstructure are fundamental
for understanding the morphology of arteries and for the development of biomechanical models. Classically, histology and more recently multi-photon microscopy
(MPM) have shed a great deal of light on vascular microstructure as described
below.
4.4.1 Adventitia
The coronary adventitia consists of three major constituents: Elastin, collagen fibers,
and ground substance. The dense and wavy collagen fibers form an interwoven
network that tangles with elastin fibers (Rhodin, 1980; Wolinsky & Glagov, 1967).
Histologically, a collagen or elastin fiber is well described as a bundle of loosely
bounded fibrils (Chen, Liu, Slipchenko, Cheng, & Kassab, 2011; Fratzl et al., 1998;
Ottani, Raspanti, & Ruggeri, 2001). In an undulating state, such a fiber can deform
with very little stress. When straightened, it can sustain a significant amount of
stress. Experimental observations showed that elastin fibers are much less undulated
4.4 Microstructural Models of Coronary Artery 199
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than collagen at zero-stress state, so they gradually extend (Chen, Liu, Slipchenko,
et al., 2011; Lu et al., 2004) to take up load together with the fibroblasts and ground
substance at very low strain level. At this level of deformation, the stress–strain
behavior of the tissue exhibits only weak nonlinearity since the stiffness of elastin
fibers is not significantly higher than the fibroblasts and ground substance. When the
artery is distended beyond a certain stretch ratio, the collagen fibers are gradually
straightened and begin to take up increasing loads in a manner, which depends on the
deformation of the network. Since the collagen fibers have much higher tensile
stiffness than elastin fibers and ground substance, full engagement of collagen leads
to the highl y nonlinear overall mechanical properties of the adventitia. The mechanical function of adventitia ground substance, an amorphous gel-like material
containing glycosaminoglyc ans (GAGs), proteoglycans, glycoproteins, and fibroblasts, has yet to be tested.
4.4.1.1 Uniform-Field Models: Behavior of Ground Substance
As noted above and in Appendix 11, two major classes of micromechanic al models,
with different assumptions for the matrix material (cells and ground substance) of the
tissue, have been developed in the past decades. The first ones considered the vessel
wall as a composite of elastin and collagen fibers embedded in a fluid-like matrix.
The fibers are the only constituent phases that sustain non-hydrostatic loading such
as tension and shear, while the contribution of the fluid-like matrix is only a
hydrostatic pressure. The second class of micromechanical models assume the tissue
as a fiber-reinforced composite, of which the non-collagenous matrix material
(including elastin fibers, cells, and ground substance) or the non-fibrous matrix
(including cells and ground substance) is a solid-like material that can take up
load. Although the computational simulations of the idealized fiber sheets in the
preceding section suggests that the fluid-like model is more consistent with the SOE
and FE models, one question is which of these two hypotheses (fluid-like or solidlike) can be validated experimentally for the adventitia ground substance?
To test this hypothesis, Chen, Guo, Luo, and Kassab (2016) performed enzymatic
degradation of the ground substance to assess whether the behavior is fluid-like
(no mechanical contribution) or solid-like. The ground substance (an amorphous
gel-like media) consists largely of GAGs and their associated proteins (i.e., proteoglycans and glycoproteins). To determine the mechanical contribution of the
ground substance, GAGs are digested by chondroitinase ABC (Chen, Guo, et al.,
2016). Intact arterial specimens are used to confirm digestion of GAGs with incu-
bation in a 0.2 U/mL solution of chondroitinase ABC. Since the anti-chondroitin
sulfate antibody, CS-56, recognizes an epitope on intact chondroitin sulfate GAGs
chains, GAGs are labeled with CS-56. After vessel specimens are incubated with the
CS-56-specific mouse monoclonal antibody, they are visible with an ABC staining
system and examined by light microscopy. Controls are obtained by using arterial
segment incubated without the CS-56 primary antibody. Mechanical testing is
performed before and after GAGs removal. The segment is stretched to λ
¼ 1.3
z
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(in vivo length) and inflated by different pressures to obtain the diameter – pressure
curve of arterial adventitia with or without GAGs.
Removal of GAGs in the adventitia is confirmed by immunohistochemical
images as shown in Fig. 4.11a–c. Intact arterial samples are stained with an ABC
kit without CS-56 antibody incubation (Fig. 4.11a, control group, purple), and some
samples are labeled with antibody CS-56 to specify GAGs followed by ABC
staining (Fig. 4.11b, brown is specific for GAGs). GAGs-digested samples labeled
with antibody CS-56 showed purple color indicating that few GAGs are left in the
adventitia after digestion (Fig. 4.11c). Figure 4.11d provides the pressure–diameter
curves of the adventitia segments before and after treatment of chondroitinase ABC
under physiological axial stretch λ
¼ 1.3. The data did not show a significant
z
difference between these two groups which suggests the mechanical contribution
of ground substance can be ignored at this load range (i.e., behaves as fluid-like).
GAGs resist vessel wall compression to prevent luminal radius collapse from smooth
muscle cell contraction but do not contribute to tension (Viidik, Danielson, &
Oxlund, 1982). Hence, collage n and elastin fibers embedded in a fluid-like matrix
satisfy an affine deformation field as a model of coronary adventitia as described
below.
4.4.1.2 3D Micro structural Model of Coronary Adventitia
Chen, Guo, et al. (2016) presented a structure-based model of coronary artery
adventitia based on microstructure of elastin and collagen nested in a fluid-like
matrix (Appendix 12). Collagen and elastin fibers embedded in fluid-like matrix
satisfy an affine deformation field as a model of coronary adventitia. The inner
adventitia is modeled as a layered structure with concentric densely packed fiber
sheets with few radially oriented fibers within the inner adventitia. The model is
informed with measurements that showed that ~80% of collagen fibers oriented
towards the longitudinal direction (with μ
¼ 1.91, σC2¼ 0.50, wC2¼ 0.78)
C2
(Appendix 12, Table 4.25) and the other fibers aligned nearly in the circumferential
direction (with μ
¼ 0.37, σC1¼ 0.20, wC1¼ 0.22), following two normal
C1
distributions (Appendix 12). This model differs from the assumption used in most
microstructural models where fibers are symmetrically disposed with respect to the
circumferential direction of the vessel (with a preferred orientation). The longitudinal arrangement of fibers accounts for the signi ficant increase of axial force at high
stretch ratio λ
¼ 1.5 (Fig. 4.12, bottom row). The other geometrical parameters,
z
including fiber waviness distribution and volume fraction, are also employed to
achieve a realistic microstructure-based model of the coronary adventitia. The
detailed formulations are presented in Appendix 12. The model presented by
Chen, Guo, et al. (2016) is validated based on biaxial stretch (inflation and axial
stretch) of swine coronary adventitia. The microstructural model is used to predict
the macro-mechanical biaxial vessel responses and reliable parameter estimations of
individual fibers.
4.4 Microstructural Models of Coronary Artery 201
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Fig. 4.11 (a–c) Immunohistochemical images of control and glycosaminoglycans (GAGs)digested vessels: (a) Control group (purple); (b) vessel labeled with antibody CS-56 (brown
specifying GAGs); (c) GAGs-digested vessel labeled with antibody CS-56 (purple confirming
GAGs removal); (d) Mechanical testing of coronary adventitia with or without GAGs under
physiological axial stretch λ
( p ¼ 0.384). Reproduced from Chen, Guo, et al. (2016) with permission
¼ 1.3. There is no significant difference between these two groups
z
The least squares estimation of the material parameters of fibers is summarized in
Table 4.26 (Appendix 12). The mean stiffness parameter of elastin fiber is
k
¼192.2 72.6 kPa, while the parameters of collagen fiber are kC¼27.2 5.1 MPa
E
and M
¼ 5.37 0.53. The parameter estimation for the sample mean (average over
C
5 experimental data sets) is also given. The measured outer radius–pressure and axial
force–pressure relations are compared with model predictions as shown in Fig. 4.12,
under three different axial stretch ratios λ
¼ 1.0, 1.3, and 1.5, respectively. The
z
predictions are in good agreement with experimental measurements which accurately capture the nonlinear responses of the coronary adventitia.
Chen, Guo, et al. (2016) model accurately predicts the nonlinear responses of the
adventitia as shown in Fig. 4.12. The outer radius of adventitia 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 the increase of pressure.
The outer radius sli ghtly declined at elevated axial stretch ratio, while the axial force
is found to increase greatly at λ
¼ 1.5. The axial force is very low under λz¼ 1.0
z
where only elastin fibers contribute and collagen fibers are undulated and
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Fig. 4.12 Comparison of model predictions (solid line) with experimental measurements (dashed
line, sample mean, error bar denotes standard deviation of experimental data). The top and lower
panels show outer radius and axial force, respectively; at three axial stretch ratios λ
1.5. Reproduced from Chen, Guo, et al. (2016) with permission
¼ 1.0, 1.3, and
z
unengaged. When λz¼ 1.3 (equivalent strain is 0.35), a few longitudinal-oriented
collagen fibers are recruited so that axial force increased moderately. When λ
reached 1.5, which is beyond the straightening strain of most collagen fibers, most
longitudinal-oriented fibers are recruited to provoke a rapid increase of axial force
and axial stress (as shown in bottom panel of Fig. 4.12).
It should be noted that the axial stretch of coronary arteries during the cardiac
cycle may be about 10% smaller or larger than the physiological axial stretch ratio of
1.3 (Ohayon et al., 2011), which is included in the range of mechanical testing
protocol: λ
¼ 1.0, 1.3, and 1.5. In systole, collagen fibers are likely unengaged and
z
the adventitia can shrink axially, while fibers become straightened to withstand
tension such that the adventitia becomes stiffer against axial overstretch in diastole.
The natural undulation of collagen fibers provides the coronary arteries less resistance to axial compression in systole, while their axial alignment prevents coronary
arteries from overstretching in diastole. The structure–function relation of the collagen orientation in coronary arteries is in contrast with the predominantly circumferential alignment of collagen in other types of arteries that do not undergo axial
dynamic deformation (Gh azanfari et al., 2012; Roveri et al., 1980 ; Sáez, García,
Peña, Gasser, & Martínez, 2016; Smith, Canham, & Starkey, 1981).
The material parameters are also estimated from the data under two axial stretches
and then used to simulate responses under the third axial stretch ratio. Model
predictive power is examined by comparing model predictions (in terms of SSE)
z
4.4 Microstructural Models of Coronary Artery 203
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based on partial and full data sets. Parameter estimates based on partial database of
axial stretch ratios of λ
The mean stiffness of elastin is k
collagen are k
¼ 57.8 11.7 MPa and MC¼ 5.92 0.43, with an increased total
C
¼ 1.3 and 1.5 are summarized in Table 4.27 (Appendix 12).
z
¼ 139.5 73.0 kPa and the parameters for
E
SSE (0.68 0.15). Material parameter estimation based on the partial database of
axial stretch ratios of λ
¼ 1.0 and 1.3 are also performed but show a lower total SSE.
z
A sensitivity analysis is performed to replace the continuous spatial distributions
of fiber orientation and waviness wi th the mean values. Two groups of elastin are
considered with two mean orientation angles (μ
groups of collagen with mean orientation angles (μ
unique straightening strain e
¼ 0.35. Parameter estimates of individual fibers are
0
¼ 0.33 and μE2¼ 1.99), and two
E1
¼ 0.37 and μC2¼ 1.91) with a
C1
shown in Table 4.28 (Appendix 12). The stiffness parameter of elastin is
k
¼ 183.5 36.5 kPa, which is similar with kEestimated by the model with full
E
fiber distributions. The mean material parameters of collagen fibers are
k
¼ 51.6 14.9 MPa and MC¼ 4.58 0.2, which are somewhat different with
C
that of full distributions. The simplified approach achieved a slightly increased mean
value of SSE as compared with that of full distributions (0.45 0.05
vs. 0.42 0.03).
Figure 4.13 shows the stress–strain curves of individual fibers using material
parameters based on full fiber distributions with all experimental data sets (E1, C1),
full distributions with partial data sets (E2, C2), and simplified mean values with full
data sets (E3, C3). The stiffness of elastin is small compared with that of collagen
fibers, and collagen shows a strong nonlinear response. The stress–strain curves of
collagen fibers based on full distributions (C1 and C2) are close to each other with
similar toe regions, while the one estimated by the simplified mean values
(C3) shows a steeper stress–strain slope. The stress–strain curves C1 and C2 are
comparable to the circumferential Cauchy stress–strain curve of the adventitia at
middle wall σ
. This sensitivity analysis to assess the effect of using mean values of
θθ
the distributions for fiber orientation and waviness as opposed to the full distributions shows that the simplified mean analysis affects the fiber stress–strain relation,
resulting in incorrect estimation of mechanical parameters. This underscores the
need for measurements of fiber distribution for a rigorous analysis of fiber
mechanics.
The stiffness parameter of elastin k
is estimated at magnitude of 102 kPa, which
E
is within the reported values of Young’s modulus: 100 kPa to 1 MPa (Gundiah et al.,
2007). The elasticity modulus of collagen fiber has a large reported range of
100–1000 MPa, depending on testing methodology, specimen dimension, animal
species, etc. (Gentleman et al., 2003; Gundiah et al., 2007; Kato et al., 1989; Yang,
2008). The estimated value of parameter k
indicates that the tangent modulus (also
C
called as Young’s modulus) of collagen fibers has a magnitude of about 10 MPa.
Based on the estimated nonlinear stress–strain curves (C1 and C2 in Fig. 4.13)of
single collagen fiber, the tangent moduli, determined at fiber strain e
¼ 1.0 (fiber
f
stretching 30% after becoming straightened), are 23.0 and 42.7 MPa for C1 and C2,
respectively. The stre ss of C1 begins to rise at a steep slope at a critical strain
(e
0.7), while the stress of C2 rapidly increased at a similar critical strain but with
f
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Fig. 4.13 The stress–strain curves of individual elastin and collagen fibers using mean material
parameters estimated by various models. Straightening strain e
Capital E and C denote the stress–strain curves of elastin and collagen fibers, respectively. E1 and
C1 are the curves estimated by full distribution model with full data sets; E2 and C2 are the curves
estimated by full distribution model with partial data sets of λ
curves estimated by the simplified mean-value model with full data sets. σ
ferential stress–strain relation of the adventitia at middle wall under axial stretch ratio λ
Reproduced from Chen, Guo, et al. (2016) with permission
is set to 0.345 for collagen fiber.
0
¼ 1.3 and 1.5; E3 and C3 are the
z
denotes the circum-
θθ
z
¼ 1.3.
a steeper slope, indicating that estimates based on database of larger axial stretch
ratios λ
¼ 1.3 and 1.5 provide a higher fiber stiffness that is closer to the reported
z
values. The curve C3 determined by the simplified mean-values model has a smaller
critical strain that is largely induced by the unique straightening strain of collagen
fibers (e
¼ 0.35). There are no collagen fibers engaged to withstand loads before
0
fiber strain reached 0.35, so the material response of individual fiber have to match
the overall mechanical response of adventitia wall, resulting in an overestimation of
k
and underestimation of MC(as shown in Tables 4.26 and 4.28 , Appendix 12).
C
This suggests that the estimated parameters are not independent of structure and
accurate measurements of microstructure are necessary for predictive models.
4.4.2 Media
A microstructure model of the media has also been developed that accounts not only
for the elastin and collagen fibers but also the passive and active SMC (Chen, Luo,
et al., 2013). The model formulations (Appendix 13) along with the morphometric
data SMC (Tables 4.29 and 4.30, Appendix 13) and material parameters of individual collagen fibers, elastin network, and SMC (Tables 4.29 and 4.30, Appendix 13)
are used to predict the total, passive and active stresses of media (citation needed).
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