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Appendix 11: Micromechanics of Heterogeneous Materials (Chen, Zhao... 275
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error induced by approximating the nonlinear composite SEF W
rðÞ
LCC W
mation field F 2 κ
tensor, F
F 2 κ
(r)
F
(r)
F
. This approximation translates the optimization over continuous defor-
T
(r)
,
Fmust satisfy the continuity and compatibility conditions, while L
Fto the modulus tensor, L
(r)
and the deformation gradient
,
of the r-th phase of composite (r varies from 0 to N). In theory, the field
can be arbitrary within the physically meaningful ranges. Although L
are allowed to change from point to point in the material, it is assumed that they
(r)
with the SEF of
(r)
and
(r)
and
are uniform for the r-th phase based on standard nonlinear micromechanics (LopezPamies & Ponte Castañeda, 2004a; Ponte Castañeda, 2002; Willis, 1977). The work
of Ponte Castañeda and Willis (1999) suggested that replacing the minimization over
the variables L
(r)
and F
(r)
in Eq. (4.166) by the corresponding stationary points will
yield a stationary estimate, and thus generate the so-called second-order estimate for
the exact SEF of the nonlinear compo sites. The stationary procedures involved in the
above derivation (maximization of V
set of nonlinear tensorial equations for the determination of the unknown reference
modulus L
(r)
and the reference deformation gradient F
Castañeda, 2004a, 2006; Ponte Castañeda, 2002). These equations have multiple
solutions that lead to various estimations of the macroscopic SEFW
(r)
and the minimization in Eq. (4.166)) yield a
(r)
(Lopez-Pamies & Ponte
F.Atangent
solution is widely used for composites with complex microstructure, in which the
reference deformation gradient is taken to be the average deformation gradient, i.e.,
rðÞ
F
evaluated atF
(r)(F(r)
V
rðÞ
¼F
in the r-th phase of the LCC, and L
rðÞ
(Ponte Castañeda and Tiberio, 2000). This solution leads to
) ¼ 0 and an estimate ofW
, L
(r)
Fas:
(r)
is the tangent stiffness tensor
FW
W
where ρ
rðÞ
rðÞ
F
ρ
energyW
S
(r)
is used to denote the first derivation of the phase potential W
¼ ∂W
Fand the average deformation gradient of phasesF
T
F¼
rðÞ
=∂F
X
r¼0
c
F¼F
rðÞWrðÞ
rðÞ
N
determined by extend ed finite strain Hashin-Shtrikman theory (Kailas am et al.,
1997; Liu, 2003) which takes into account the stiffness L
rðÞ
F
þ
1
rðÞ
ρ
2
rðÞ
F
F F
rðÞ
: ð4:167Þ
(r)
, i.e.,
. In order to compute this estimate, the effective strain
rðÞ
of the LCC are
(r)
as well as the shape
and distribution of the composite phases (Chen, Liu, Zhao, et al., 2011; LopezPamies & Ponte Castañeda, 2006; Ponte Castañeda, 2002).
The advantage of this method is that it can be used for any type of nonlinear
composite and can consider the strongly nonlinear constraint of material
incompressibility, which is relevant for biological soft tissues. The SOE model
statistically accounts for the heterogeneous deformation in composites (i.e.,F
varies in phases), which is due to the heterogeneities of microstructural geometries
and material properties, as well as interactions among constituents. The deformation
field is therefore more realistic than that of the upper bound solution. Consequently,
SOE leads to more accurate estimates of the macroscopic SEF and stress for the
composites, as well as microscopic stress for mic rostructure, as compared with the
upper bound solution.
rðÞ
276 4 Constitutive Models of Coronary Vasculature
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Micromechanical Models for Soft Tissues
Numerous microstructural constitutive models of soft tissues have been proposed in
the past several decades that proved to be more accurate than previous phenomenological models. Three major types of these micromechanical models are classified
based on various assumptions and approximations in nonlinear micromechanics as
described below.
Uniform-Field Models Based on a Solid-Like Matrix
The first class of micromechanical models is motivated by anisotropic mechanical
behaviors of soft tissues. Previous studies showed that the elastin becomes straightened and starts to take load in the early deformation of the tissue as an isotropic
material. For instance, the experimental study of Gundiah et al. (2007) suggested that
elastin can be described by an isotropic neo-Hookean constitutive model. Meanwhile, collagen fibers, with preferred orientation, are largely associated with the
anisotropic response of soft tissues (Holzapfel & Weizsäcker, 1998). Based on these
observations, the tissue is considered as a collagen fiber-reinforced composite with a
solid-like matrix that can bear load. The SEF of non-collagenous matrix material,
including elastin fibers, cells, and ground substance (GS) (matrix defined here is not
the same as ECM), W
collagen W
is anisotropic due to the deformation of two families of collagen fibers
C
(Holzapfel et al., 2000). Hence, the effective SEF of tissue is the sum of these two
functions:
W
C¼ A
M
1
, is associated with the isotropic deformation, and the SEF of
M
I
1
W
3, W
C¼ W
C
C; N
Cþ W
M
; N
1
2
C; N
C
X
k
1
¼
2k
2
i¼4,6
; N
, ð4:168aÞ
1
2
hi
no
exp k
I
1
2
i
2
1
ð4:168bÞ
,
where N
and N2are the direction vectors of the two families of collagen fibers, A1is
1
a material parameter associated with elastin fiber, k
ated with collagen fiber and k
The right Cauchy-Green deform ation tensorC is related to the deformation gradient
byC ¼F
T
F. The first invariant ofC isI1¼ tr
invariants areI
This model, where the uniform-field approximation FXðÞ¼F is employed (as in
Eqs. (4.168a) and (4.168b)), presents an upper bound of the exact effective SEF of
soft tissue. In addition, the direction vectors N
empirical curve fitting rather than based on histological observations (Holzapfel
et al., 2000). Given that fiber orientation follows a certain continuous distribution in
biological tissues (Chen, Liu, Zhao, et al., 2011; Sacks, 2003), these two parameters
is a material parameter associ-
1
is a dimensionless parameter (Holzapfel et al., 2000).
2
C, and the fourth and sixth
¼ N1C N1andI6¼ N2C N2, respectively.
4
and N2of fibers are determined by
1
Appendix 11: Micromechanics of Heterogeneous Materials (Chen, Zhao... 277
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are phenomenological variables rather than structural parameters. Moreover, the
engagement of undulated collagen fibers (characterized by the fiber waviness distribution) is described by an exponential SEF W
, and thus is also phenomenological.
C
To obtain more accurate predictions, this model has been subsequently revised.
Zulliger, Fridez, et al. (2004)refined the model by accounting for not only the
wavy nature of collagen fibers but also the volume fraction of both elastin and
collagen, based on different SEFs of the matrix and collagen fibers. Kroon and
Holzapfel (2008) later incorporated this model into multi-layered structures with the
mean fiber alignments in various layers. Li and Robertson (2009) also extended this
model to account for either a finite number of fiber orientations or a fiber distribution
function. In summary, the mechanical predictions of these uniform-field models
based on a solid-like matrix are more accurate than those of phenomenological
models because they reflect the heterogeneity of material properties and some of
the geometrical features of tissues. These models, however, cannot accurately
predict the microenvironment (strain and stress of individual fiber or cell) of soft
tissues because they assume affine deformation in tissue and use non-histologicalbased microstructure.
Uniform-Field Models Based on a Fluid-Like Matrix
The second class of micromechanical models are structurally motivated and are
proposed based on the following assumptions:
1. Fibers are thin and flexible with only tensile strength (i.e., fibers cannot resist
compressive load).
2. Fibers are embedded in a fluid-like matrix, of which the mechanical contribution
is only via hydros tatic pressure.
3. The second assumption leads to a simplification that all the microstructures
deform identically to the macroscopic deformation of the tissue (i.e., uniform
deformation) since no fiber–fiber interactions are considered.
On the basis of these assumptions and thermodynamic consideration, Lanir
developed a general multi-axial theory for the constitutive relations in fibrous
connective tissues (Lanir, 1979, 1980, 1983). In his model (Lanir, 1983), an important structural feature is the density distribution function of the fiber orientation R
where N is a unit vector tangent to the fiber. Thus, R
(N)ΔΘ is the volumetric
i
(N)
i
fraction of fibers of type i (classified by waviness and fiber type) which are oriented
in direction N and occupy a spatial angle ΔΘ. According to previous derivation for
uniform-field models, the macroscopic SEF in this model is the volumetric sum of
the SEF of fibers in all directions:
X
W
F¼
X
ðÞ
ðÞ
i
i
c
W
λfðÞRiNðÞΔΘ, ð4:169Þ
i
N
278 4 Constitutive Models of Coronary Vasculature
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where c
fiber SEF of type i, which depends on fiber stretch ratio λ
equation is identical to Eq. (4.165)as∑
of a particular fiber phase with the type i SEF and orientation N in soft tissues. The
SEF of fiber W
(i)
is the volumetric fraction of unstrained fibers of type i, and W
(i)
c
Ri(N)ΔΘ denotes the volumetric fraction
N
(i)
(λf) and the density distribution function Ri(N) are specificto
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
¼
N
f
(i)
T
F
F Nq. This
(λf) is the
different tissue types, of which microstructural geometries and material properties
may be determined by histological and mechanical measurements. This model can
account for geometrical distributions (e.g., orientation, waviness) as well as the
mechanical response of single fibers, but it assumes affine deformation and neglects
inter-fiber interactions due to the assumption of a fluid-like matrix.
The fluid-like matrix assumption is also employed by Decraemer et al. (1980)to
develop a parallel wavy fibers model for soft biological tissues in uniaxial tension,
assuming a normal distribution for initial length of fibers (i.e., fiber waviness).
Wuyts et al. (1995) extended this model by utilizing a Lorentz distribution function
for the initial fiber length. In summary, the fluid-like matrix models are more
conceptual than the solid-like matrix models because they involve structural and
constitutive behaviors of both the functional constituents of soft tissues: collagen and
elastin. The basic fluid-like matrix assumption may be an accurate description for
certain tissues where the non-fibrous constituents are mainly fibroblasts, macrophages, and amorphous gel-like GS that do not take up significant non-hydrostatic
loading. These micromechanical models with a fluid-like matrix have been widely
applied to different soft tissues, including tendon (Sverdlik & Lanir, 2002), skin
(Lokshin & Lanir, 2009a, 2009b), myocardium (Horowitz et al., 1988), and blood
vessels (Hollander et al., 2011b)
SOE Models Based on Solid-Like Matrix
Chen, Li u, Zhao, et al. (2011) developed a finite strain micromechanical model
(based on the aforementioned SOE approach) to predict the macroscopic stress–
strain relation and microstructural deformation of soft fibrous tissue. This model
shows significant improvements over previous microstructure models when compared to finite element method (FEM) simulations. In this model, the tissue is
assumed as a composite with reinforcing fibers and soft solid matrix (Fig. 4.23).
The orientation of the r-th fiber is described by θ
described by a geometric tensor Z
characterized by another geometric tensor Z
SEF of the r-th fiber to reflect fiber recruitment under macroscopic tissue deformation. Specifically, a piecewise function is used to describe the constitutive behavior
of a single fiber:
(r)
(r)
, and the spatial distribution of the r-th fiber is
rðÞ
while the waviness is included in the
d
, the shape (dimension) is
Appendix 11: Micromechanics of Heterogeneous Materials (Chen, Zhao... 279
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0
()
W F
()
0
()
W F
()
F
=
=
, WF
(c)
rr
(a) (b)
(a) (b)
=
=
0
()
W F
()
Fig. 4.23 Conceptual demonstration of the SOE method. (a) A representative volume element of a
fibrous tissue at reference state. All the fibers are undulated and exhibit the same property as the
matrix with strain energy function (SEF) W
are straightened and show stiffer property with SEF W
homogeneous material with macroscopic SEFW. Reproduced from Chen, Zhao, et al. (2013) with
permission
() ()
W FG
=
=
,
()
0
()
W F
()
;(b) When subjected to a macroscopicF, some fibers
0
(r)
as in Eq. (4.170). (c) The effectively
(c)
(
0ðÞ
FðÞ λf< λ
rðÞ
W
FðÞ¼
where λ
denotes the SEF of the matrix. This function implies that a fiber deforms the same as
the soft matrix before straightening and becomes stiffer with additional SEF of fiber
W
neo-Hookean SEF, while the anisotropic term W
selected as:
where E
linear model (Decraemer et al., 1980; Lanir, 1979, 1983; Wuyts et al., 1995) where
only the first term λ
can use any well-defined constitutive models of the mat rix and fibers.
is the fiber stretch ratio, λ
f
rðÞ
after straightening (as shown in Fig. 4.23b). The matrix is described by a
fiber
rðÞ
W
λfðÞ¼E1λf λ
fiber
, E2are material parameters of individual fibers. This is a generalization of a
1
λ
f
W
0ðÞ
W
FðÞþW
rðÞ
0
2
rðÞ
is included. In principle, the homogenization model
0
rðÞ
λfðÞ λf λ
fiber
is the waviness of the r-th fiber, and W
rðÞ
fiber
2
rðÞ
=2 þ E2λf λ
0
rðÞ
,
0
rðÞ
,
0
λðÞin the SEF of fibers is
3
rðÞ
=3 ð4:171Þ
0
ð4:170Þ
(0)
280 4 Constitutive Models of Coronary Vasculature
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By substituting fiber geometrical features G
rðÞ
rðÞ
rðÞ
Z
; Z
d
; θ
rðÞ
; λ
rðÞ
0
and mechan-
ical properties (Eqs. 4.170 and 4.171) into Eqs. (4.166) and (4.167), and utilizing the
finite strain Hashin-Shtrikman theory (Chen, Liu, Zhao, et al., 2011; Kailasam et al.,
1997), the macroscopic SEF and stress of soft tissue as well as microscopic defor-
rðÞ
mationF
of every component can be obtained through solving multiple nonlinear
equations (Chen, Liu, Zhao, et al. 2011; Lopez-Pamies & Ponte Castañeda, 2004a). It
should be noted that the microscopic deformationF
rðÞ
of the r-th fiber is not identical
to either the macroscopic tissue deformation or other fiber phases since the admissible
deformation field F(X)isnotF (as in the uniform-field models) and is determined by
tissue inhomogeneity. As compared with the first two model types, this SOE
micromechanical model not only considers realistic geometrical features and material
properties of tissue constituents and their interactions, but also allows flexible
deformation in each constituent. Hence, the model is an actual estimate rather than
an upper bound of the exact effective SEF and provides a more accurate predictio n of
the macroscopic and microscopic mechanical behavior of the soft tissue.
Appendix 12: A 3D Microstructure-Based Model
of Coronary Adventitia (Chen, Guo, et al., 2016)
The adventitia is considered to be a cylindrical tube, with the following kinematic
assumptions:
1. Incompressible.
2. Deformations are axis-symmetric and independent of axial position.
3. Transverse sections remain planar.
4. There is a unique undeformed reference configuration (i.e., ZSS).
A cylindrical coordinate system is used with circumferential direction g
direction g
λ
, λr, and λzare determined, respectively,
θ
where Θ
and axial direction g3as principal directions, the corresponding stretches
2
π
r
λ
¼
θ
π Θ
is opening angle measured at ZSS, R is radius to a point at ZZS and r is the
0
, λ
R
0
∂r
, λ
¼
r
∂R
l
¼
z
L
, radial
1
ð4:172Þ
radius to the same point in the current configuration. L is the axial length of the
segment at ZSS and l is the loaded axial length. According to material
incompressibility: J ¼ λ
¼ 1, for the mapping between ZSS and loaded state,
θλrλz
loaded radius r is determined as a function of unloaded R:
Appendix 12: A 3D Microstructure-Based Model of Coronary Adventitia (C... 281
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s
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
rRðÞ¼
2
2
r
R
o
o
R
2
π Θ
λzπ
0
ð4:173Þ
where r
is the outer radius in the loaded state while Rois that at ZSS.
o
The radial component of the force equilibrium equation imposed on the loaded
configuration is given by:
∂σ
σrr σ
where σ
σ
¼pi, σ
j
rr
r
i
rr
þ
∂r
as Cauchy stress tensor. According to boundary conditions,
ij
¼ 0, the luminal pressure pican be written as:
j
rr
r
o
Z
r
p
o
¼
i
r
i
θθ
¼ 0 ð4:174Þ
r
1
σθθ σ
ðÞ
dr ð4:175Þ
rr
r
The axial force required to maintain the vessel axial stretch is given by:
Z
r
F ¼ π
o
2σzz σθθ σ
ðÞrdr ð4:176Þ
r
i
rr
Strain Energy Function
The coronary adventitia is considered as an incompres sible hyperelastic solid and
characterized by a strain energy function W(E) as a function of the Green-Lagrange
strain tensor E ¼
1
FT F I
2
. The Cauchy stress tensor σ is given by:
where F is the deformation gradient tensor and S is the second Piola–Kirchhoff stress
tensor. I is the second-order identity tensor, scalar p is hydrostatic pressure, which
acts as a Lagrange multiplier and must be determined from equilibrium and boundary conditions.
The strain energy function (SEF) W(E) of a microstructural model involves
structural features. Experimental studies show that coronary adventitia is divided
into an outer and inner adventitia as seen in Fig. 4.24a–c (Fig. 4.24a, b are lateral
sections as denoted in Fig. 4.24d). The outer adventitia, consisting of thicker and
wavier collagen bundles and few elastin fibers, supports the vessel and connects with
the surrounding tissue rather than significantly resisting the transmural pressure
σ ¼ F
∂W
T
F
pI ¼ F S FT pI ð4:177Þ
∂E
282 4 Constitutive Models of Coronary Vasculature
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Fig. 4.24 (a) Outer adventitia (OA) consists of thicker collagen bundles and few elastin fibers; (b)
Inner adventitia (IA) is a layered structure with entangled elastin and collagen fibers in each
sublayer; (c) The cross section of a coronary artery at no-load state; (e, f) OA and IA deformed
under elevated pressures: 100 and 200 mmHg, respectively, showing fibers in IA are stretched to
take up loads while most of collagen bundles in OA are still undulated and unengaged; (d)A
schematic diagram demonstrates the cross and lateral sections of a vessel segment. (a, c) are the
lateral sections and (b, e, f) are the cross sections. Scale bar denotes 100μ. Reproduced from Chen,
Guo, et al. (2016) with permission
(Fig. 4.24e, f), while the inner adventitia is a layered structure with concentric
densely packed fiber sheets and has few radial fiber bundles distributing between
sheets (Fig. 4.24b, e, f are cross sections as denoted in Fig. 4.24d). At low pressures,
elastin fibers bear the loads and collagen fibers are still wavy in the inner adventitia.
At high pressures, stretched collagen fibers in the inner adventitia are recruited to
withstand stresses. Therefore, the adventitia wall is modeled as a composite
containing two mechanical components: collagen and elastin fibers, while ground
substance is found to have a negligible mechanical function (Fig. 4.24) and is treated
as a fluid that sustains hydrostatic pressure. Both types of fibers are only resistant to
tensile load, undulated collagen fibers are recruited to bear loads only after they
become straightened, and there is no interaction between collagen and elastin fibers.
A fluid-like matrix implies the tissue undergoes affine deformations, i.e., deformation of the fibers is the same as that of ground substance. Based on this assumption,
the SEF of adventitia wall can be represented by the volume-weighted summation of
individual SEF of elastin fiber W
and collagen fiber WC(Lanir, 1983):
E
W EðÞ¼f
WEþ fCW
E
C
ð4:178Þ
Appendix 12: A 3D Microstructure-Based Model of Coronary Adventitia (C... 283
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where fi(i ¼ E, C) is the volume fraction of each type of fiber i. Generally, the
orientation of fibers follows a continuous distribution density function, and the
volume-weighted SEF of fibers is given by Lanir (1983):
Z
π
W
¼
i
ℛiθðÞwieðÞdθ ð4:179Þ
0
where w
(e) is the SEF of individual fibers as a function of fiber strain e. It should be
i
noted that radially oriented elastin and collagen fibers are not engaged under
inflation–extension condition where they are compressed and bear no loads, and
only planar fibers contribute to mechanical behavior of the adventitia. Thus, ℛ
(θ)is
i
a planar orientation distribution density function of fiber i, and θ is the angle between
the fiber orientation and the circumferential direction of the vessel g
the normalization criterion
R
π
ℛiθðÞdθ ¼ 1 . The uniaxial fiber strain e(θ) is deter-
0
. ℛi(θ) satisfies
1
mined by the local strain tensor E and the reference fiber direction N ¼ (Cos θ, Sin θ)
as:
e E; NðÞ¼E : N N ð4:180Þ
Although elastin and collagen fibers distributed in each sublayer with transmural
variation of fiber orientation, a mixture of two normal distribution of fiber orientation
through the adventitia wall is found to describe the experimental data as (Chen, Liu,
Slipchenko, et al., 2011):
Z
π
W
¼ ω
i
where ℛ
ij
θðÞ¼
К
distribution density function with μ
respectively. К
j
cumulative distribution function of a normal distribution), and ω
each normal distribution
ℛi1θðÞwieðÞdθ þ ω
i1
0
1
1
Exp
j
√2π
σ
j
2
θμ
ðÞ
j
2
2σ
j
j
is a truncated parameter Кj¼ Φ
P
2
ωij¼ 1
j¼1
Z
π
ℛi2θðÞwieðÞdθ
i2
0
i
ð4:181Þ
, i ¼ E; C; j ¼ 1; 2ðÞis a truncated normal
and σjas the mean and standard deviation,
πμ
j
Φ
σ
j
μ
j
(Φ is the
σ
j
is the weight of
ij
. The second Piola–Kirchhoff stress of each
type of fiber is derived as:
According to microscopic responses of individual elastin fibers under mechanical
loads, the elastic properties are assumed to be linear:
Z
X
S
¼
i
j
π
ω
ij
0
ℛijθðÞ
∂w
∂e
∂e
i
dθ ð4:182Þ
∂E
284 4 Constitutive Models of Coronary Vasculature
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∂e
¼
0 e < 0
k
ee> 0
E
ð4:183Þ
E
where k
∂w
is the stiffness parameter of elastin fiber. Because of the wavy nature of
E
collagen fibers, the nonlinear constitutive relation is considered to account for the
nonlinear elastic behavior (Hollander et al., 2011a):
∂e
C
¼
0 e e
kCe e
ðÞ
0
e > e
0
0
M
C
ð4:184Þ
where k
and MCare parameters characterizing the nonlinear stress–strain response
C
of collagen, and e
∂w
denotes the strain beyond which the collagen can withstand
0
tension, which are found to follow a beta distribution for the coronary adventitia
(Chen, Slipchenko, et al., 2013):
De
where B(α
, α2) is a beta function, and a and b the lower and upper bounds of the
1
straightening strain e
∂w
∂e
ðÞ¼
0
. The constitutive law of collagen fiber thus can be written as:
0
1
B α
; α
ðÞ
1
2
8
<
:
Z
k
C
b
De0ðÞe e
a
C
¼
α11
e0 aðÞ
ðÞ
α1þα22
b aðÞ
0 e e
ðÞ
M
0
α21
b e
0
C
de0e > e
ð4:185Þ
0
0
ð4:186Þ
Although the microstructural approach can employ any well-defined constitutive
model for the fibers, a linear function (Eq. 4.183) and a power function (Eq. 4.184)
are used for elastin and collagen fibers, respectively. If the constitutive laws for
individual elastin and collagen fibers (Eqs. 4.183–4.186) are substituted into
Eqs. (4.177) and (4.182), the Cauchy stress components of the vessel will be
obtained. Given the geometrical parameters ( f
functions (ℛ
ij
Slipchenko, et al., 2013) there are only three unknown material parameters: k
and M
that needed be determined by the boundary condition (Eqs. 4.175 and
C
4.176). Since the integrals of the above equations do not have analytical expressions,
numerical approaches are used.
Parameter Estimation
Parameters are optimized by least squares fit to the experimental data by minimizing
an objective function based on the sum of squared residuals (SSE) between model
predictions and experimental data. The objective function is defined as follows:
, fC) and the measured distribution
E
(θ), ωij, D(e0)) (summarized in Table 4.25 for convenience) (Chen,
, kC,
E
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