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Appendix 12: A 3D Microstructure-Based Model of Coronary Adventitia (C... 285
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"#
SSE ¼
121
nm
n, m
X
r
cr
o
ij
σ
^r
,j
i
o
o
2
þ
Fij^F
σ
ij
2
ij
^
F
ð4:187Þ
where i and j denote distension and axial loads at which the corresponding outer
radius r
m the number of different axial stretch ratios used; σ
experimental measurement; andcr
and axial force Fijare measured; n is the number of different pressures and
o
ij
is the standard deviation of
^
,^Fijare the corresponding model predi cted outer
o
ij
radius and axial force, which are determined by numerically solving Eqs. (4.175)
and (4.176). A numerical nonlinear optimization function NMinimize in
Mathematica (WOLFRAM, US) is used to find a global minimum of the objective
function (Eq. 4.187) and to determine three unknown parameters (k
the constraints: (k
Table 4.25 Geometrical parameters and distributions
Geometries
Orientation
angle
Waviness Collagen Beta distribution α1¼ 5.01, α2¼ 52.67, a ¼ 0.0,
Volume fraction Collagen f
Reproduced from Chen, Guo, et al. (2016) and Chen, Slipchenko, et al. (2013) with permission
Note: (μ
of fiber i (i ¼C,E and j¼ 1,2) where C, E denote collagen and elastin fiber, respectively; W
weight of the j th normal distribution for fiber i orientation angle; f
i; (α
lower and upper bounds
, σij) are the mean and standard deviation of the jth normal distribution of orientation angle
ij
, α2) are parameters of the beta distribution of collagen straightening strain and (a, b) are its
1
> 0, kC> 0, MC> 0).
E
Fiber
type Distribution type Parameters
Collagen Bimodal normal
distribution
Elastin Bimodal normal
distribution
¼ 0.37, σC1¼ 0.20, wC1¼ 0.22
μ
C1
μ
¼ 1.91, σC2¼ 0.50,
C2
¼ 1.0 w
w
C2
μE1¼ 0.33, σE1¼ 0.20, wE1¼ 0.4
μ
¼ 1.99, σE2¼ 0.57,
E2
¼ 1.0 w
w
E2
b ¼ 4.0
¼ 33%
C
Elastin f
¼ 22%
E
is the volume fraction of fi ber
i
, kC, MC) with
E
C1
E1
ij
is the
Table 4.26 Parameter estimates of individual elastin and collagen fibers based on full distension–
extension experimental data (SEM denotes standard error of the mean)
Sample no.
Parameters
k
(kPa) 29.37 20.01 171.2 446.8 293.7 192.2 72.6 100.0
E
K
(MPa) 45.8 19.8 15.3 20.10 35.1 27.2 5.12 18.8
C
M
C
SSE for r
12345
5.22 4.88 4.00 7.57 5.17 5.37 0.53 4.74
0.19 0.22 0.31 0.11 0.31 0.23 0.03 0.11
o
SSE for F 0.30 0.29 0.07 0.22 0.08 0.19 0.04 0.06
Total SSE 0.50 0.51 0.38 0.33 0.39 0.42 0.03 0.17
Reproduced from Chen, Slipchenko, et al. (2013) with permission
is stiffness parameter of elastin fiber; kCand MCare parameters characterizing the stiffness
Note: k
E
and nonlinear parameter of collagen
Average SEM Sample mean
286 4 Constitutive Models of Coronary Vasculature
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Table 4.27 Parameter estimates of individual elastin and collagen fibers based on experimental
data of λ
¼ 1.3 and 1.5
z
Sample no.
Parameters
k
(kPa) 35.7 15.3 29.0 172.0 445.6 139.5 73.0
E
K
(MPa) 90.5 22.5 56.0 83.4 36.7 57.8 11.7
C
M
C
SSE for partial database of λ
¼1.3 and
z
1234 5
Average SEM
6.51 5.08 5.10 7.52 5.39 5.92 0.43
0.17 0.28 0.64 0.43 0.17 0.34 0.08
1.5
SSE for full database 0.29 0.53 0.99 1.18 0.42 0.68 0.15
Reproduced from Chen, Slipchenko, et al. (2013) with permission
Note:k
is stiffness parameter of elastin fiber; kCand MCare parameters characterizing the stiffness
E
and nonlinear parameter of collagen
Table 4.28 Parameter estimates of individual elastin and collagen fibers based on an idealized
average model in comparison with the full continuous distributions of fiber orientation and
waviness
Sample no.
Parameters
k
(kPa) 153.2 37.7 219.8 247.5 259.4 183.5 36.5
E
K
(MPa) 106.8 20.3 28.9 28.6 73.3 51.6 14.9
C
M
C
SSE for r
12345
4.97 4.08 4.00 5.03 4.83 4.58 0.2
o
0.20 0.14 0.27 0.53 0.20 0.27 0.06
Average SEM
SSE for F 0.38 0.25 0.08 0.07 0.15 0.19 0.05
Total SSE 0.58 0.39 0.35 0.60 0.35 0.45 0.05
Reproduced from Chen, Slipchenko, et al. (2013)
is stiffness parameter of elastin fiber; kCand MCare parameters characterizing the stiffness
Note: k
E
and nonlinear parameter of collagen
Appendix 13: Microstructure-Based Model of Coronary
Media Including Vascular Smooth Muscle Cell (SMC)
Contraction (Chen, Luo, et al., 2013)
The media segment is considered as a thin-walled elastic tube deformed in the
circumferential and axial directions (Chen, Slipchenko, et al., 2013). Green strains
are defined as E
circumferential stretch ratio (τ refers to the midwall circumference of the loaded
vessel and Γ refers to that at ZSS), and λ
L
being axial lengths in loaded and no-load states, respectively. A structural
o
constitutive model is used to describe the mechanical response of passive coronary
media (Hollander et al., 2011a, 2011b), which contains isotropic inter-lamellar
(IL) elastin networks and helically oriented collagen fibers:
2
¼ λ
θθ
θ
1
=2 and Ezz¼ λ
2
1
=2, where λθ¼ τ/Γ is the
z
¼ L/Lois the axial stretch ratios with L and
z
Appendix 13: Microstructure-Based Model of Coronary Media Including... 287
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Z
f
E
π
π
π
2
wEeEθEðÞ½dθ
2
C
w
ðÞ½þwCeCθCðÞ½
fg
CeCθC
2
,
E
, ð4:189Þ
ð4:188Þ
W
C
WEEðÞ¼
f
EðÞ¼
where w
depending on uniaxial fiber strain e
respectively). θ
!
s
by n
) and fsis the volume fraction. The fiber strain is determined by es¼ n
and wCare the strain energy of elastin struts and collagen fibers,
E
(s ¼ E, C, denoting elastin and collagen,
s
is the fiber orientation angle (corresponding unit vector denoted
s
!
s
E n
!
with assuming affine deformation (i.e., a fiber is assumed to rotate and stretch in the
same way as the bulk tissue). The linear stress–strain relation of elastin is considered
to be ∂w
∂w
elastin, and k
The passive strain energy of the coronary media W
/∂eE¼ kEeEwhile the nonlinear relation of collagen is considered to be
E
=∂eC¼ kCe
C
N
C
, with the material parameter kErepresenting the stiffness of the
C
and NCrepresenting the stiffness and nonlinear parameter of collagen.
C
is calculated by taking the
passive
sum of the strain energies of the elastin and collagen networks, i.e.,
W
(E) ¼ WE(E)+WC(E), and the second Piola–Kirchhoff stress is determined
passive
by S
S
θθ passive
S
where S
passive
zz passive
¼ ∂W
k
¼
k
¼
θθ passive
(E)/∂E, and given by:
passive
E
3E
þ E
ðÞþkCcos2θCcos2θCEθθþ sin2θCE
θθ
þ 3E
zz passive
zz
zz
are passive circumferential and axial stresses,
8
E
E
ðÞþkCsin2θCcos2θCEθθþ sin2θCE
θθ
8
and S
N
C
zz
N
C
: ð4:190bÞ
zz
, ð4:190aÞ
respectively.
The total strain energy of the media is the sum of the active and passive
contributions, i.e., W
total
¼ W
passive
+ W
. The function W
active
is the active strain
active
energy of vessels as contributed by active vascular smooth muscle cells (SMC).
Taking into consideration helical arrangement of SMC, the active stra in energy can
be given as:
s
W
¼
active
where θ
VSMC
orientation distribution of SMC, w
and f
is the volume fraction. A two-dimens ional gen eralization of the uniaxial
VSMC
length–tension relation of active SMC [20] is used to account for the multi-axial
active response of SMC (Huo et al., 2012, 2013). Two families of helical SMC with
Z
π
f
VSMC
2
2
ℜ θ
0
Z
þ
ðÞw
VSMC
0
ℜ θ
ðÞw
π
2
ðÞdθ
VSMCθVSMC
VSMC
VSMCθVSMC
ðÞ; dθ
is the orientation angle of SMC, ℜ(θ
is multi-axial strain energy of a single SMC
VSMC
:
VSMC
VSMC
)is the two-dimensional
VSMC
)
, ð4:191Þ
288 4 Constitutive Models of Coronary Vasculature
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symmetrical polar angles, θ
, are considered for simplicity (i.e., span of
VSMC
orientational distribution of SMC is zero). Thus, the active strain energy of SMC
can be written as follows:
W
activeλVSMC
; λ
⊥
VSMC
f
VSMC
¼
2
w
VSMCλVSMC
þw
VSMCλVSMC
; λ
⊥
VSMC
⊥
; λ
VSMC
ðÞ
θ
VSMC
ðÞ
θ
VSMC
, ð4:192Þ
with
ðÞ¼AC
w
VSMCθVSMC
λ
ðÞb
VSMCθVSMC
Erf
act
b
1
⊥
λ
θ
3
þ
ðÞb
VSMC
VSMC
b
2
4
þ 1
ð4:193Þ
q
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
!
¼
where λ
VSMC
n
SMC (i.e., cell stretch), and λ
stretch ratio (F is deformation gradient). n
VSMC
FT F
⊥
SMC
!
VSMC
n
q
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
!
¼
n
is the longitudinal stretch ratio of a
VSMC0
!
VSMC
FT F
and n
!
!
VSMC0
n
VSMC0
is the transversal
are the longitudinal and
transversal vectors, respectively. A is the level of activation (0 is passive state and
1 is fully active), C
, b1, b2, b3and b4are material constants, and Erf() is the Gauss
act
error function. Accordingly, the active stress of the coronary media is determined as
the derivatives of the strain energy function, i.e., S
active
¼ ∂W
(E)/∂E and given
active
by:
cos2θ
act
ffiffiffiπp
act
ffiffiffiπp
b1λ
sin2θ
b1λ
VSMC
VSMC
VSMC
VSMC
þ
þ
sin2θ
b2λ
cos2θ
b2λ
VSMC
⊥
VSMC
VSMC
⊥
VSMC
Exp Q½, ð4:194aÞ
Exp Q½: ð4:194bÞ
S
θθ active
S
zz active
¼
¼
2AC
2AC
,
where S
θθ active
contraction, and Q ¼
passive (Eqs. 4.190a, 4.190b) and active stresses (Eqs. 4.194a, 4.194b):
¼ S
S
total
passive
The biaxial data of coronary media (Huo et al., 2013) is used to determine the nine
material parameters (the volume fraction f
estimations of k
cal shell and the second Piola–Kirchhoff circumferential stress obtained by experimental measurement is determined by S
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
q
¼
r
i
A
o
2
r
o
πλ
z
loaded state, A
thickness in the loaded state. The axial stress is computed by
and S
+ S
active
, kC, C
E
is the inner radius in the loaded state, r
is the wall area in a no-load state, and h ¼ ro riis the wall
o
are the active circumferential and axial stresses of SMC
zz active
λ
VSMCb3
b
1
⊥
λ
VSMC
þ
2
b
4
. Therefore, the total stress is the sum of
b
2
.
, fC, f
E
, respectively). The media is considered as a thin cylindri-
act
exp
Pr
i
¼
, where P is distension pressure,
θθ
2
hλ
θ
are incorporated into the
VSMC
is the outer radius in the
o
Appendix 13: Microstructure-Based Model of Coronary Media Including... 289
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2
1
2
hroþriðÞ
λ
z
Pr
i
þ
2
π r
ðÞ
o
F
with F presenting the axial force. The material param-
2
r
i
exp
S
¼
zz
eters are determined by minimizing the square of the difference between the theoretical and experimental passive circumferential and axial second Piola–Kirchhoff
stresses. The four passive parameters {k
N
Error
X
¼
1
n¼1
exp
S
θθ passive
S
θθ passive
, kC, NC, θC} are calculated by:
E
2
þ S
exp
zz passive
S
zz passive
2
ð4:195Þ
where S
exp
θθ passive
axial stresses of the passi ve coronary media. The five active parameters {C
b
, b4} are then determined by:
3
N
Error
X
¼
2
n¼1
and S
exp
S
θθ total
exp
zz passive
S
are the experimentally measured circumferential and
act
exp
θθ passive
S
θθ active
2
þ S
exp
zz total
S
exp
zz passive
S
zz active
, b1, b2,
2
ð4:196Þ
where S
exp
θθ total
and S
exp
are the total circumferential and axial stresses of K+-induced
zztotal
SMC contraction. A limited-memory quasi-Newton method for large-scale optimization (L-BFGS method) is employed to solve the above minimizations (Eqs. 4.195
and 4.196).
Table 4.29 Material parameters of the microstructural model (Eqs. 4.190a, 4.190b) of passive
coronary media
2
R
Animal no. k
(kPa) kC(MPa) N
E
C
θC()
S
θθ passive
Sample 1 43.2 6.4 5.6 36.4 0.99 0.97
Sample 2 40.1 2.2 5.3 50.5 0.99 0.91
Sample 3 68.7 1.2 4.2 38.4 0.99 0.93
Sample 4 64.7 4.2 5.5 31.2 1.00 0.98
Sample 5 53.3 1.8 4.0 38.5 0.99 0.99
Mean SD 54.0 11.3 3.1 1.9 4.9 0.7 9.0 6.3
for
Fit of all data of
40.5 5.4 5.6 0.7 1.00 1.00
Sample 1–5
Reproduced from Chen, Luo, et al. (2013) with permission
Note: k
is stiffness parameter of elastin fiber; kCand MCare parameters characterizing the stiffness
E
and nonlinear parameter of collagen. θ
is orientation angle of smooth muscle cell
C
R2for
S
zz passive
290 4 Constitutive Models of Coronary Vasculature
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1
b
for
2
zz active
S
R
for
2
θθ active
S
R
2
/b
1
b
0
SMC1:3
λ
0
SMC1:2
λ
3
¼ b
¼ 1.3
z
0
SMC1:2
is SMC optimal stretch ratio at λ
2
4
1:3 b
ðÞ=b
1
b
3
¼ b
4
b
0
SMC1:3
Table 4.30 Material parameters of two-dimensional strain energy function (Eq. 4.193) of active vascular smooth muscle cells (SMC) of coronary media
¼ 1.2, λ
3
b
2
b
1
(kPa) b
act
Sample 1 9.7 0.2 0.5 1.4 1.3 1.41 1.36 0.4 0.80 0.83
Sample 2 5.1 0.3 0.8 1.4 1.4 1.44 1.41 0.3 0.87 0.78
Sample 3 4.2 0.2 0.4 1.4 1.2 1.39 1.35 0.4 0.89 0.93
Animal no. C
Sample 4 15.7 0.2 0.4 1.4 1.5 1.53 1.47 0.5 0.97 0.73
Sample 5 16.8 0.3 0.6 1.3 1.3 1.37 1.32 0.5 0.81 0.82
Mean SD 10.3 5.2 0.2 0.04 0.5 0.2 1.4 0.02 1.3 0.1 1.43 0.06 1.38 0.05 0.4 0.06
8.1 0.2 0.5 1.3 1.4 1.41 1.36 0.5 0.91 0.62
Fit of mean of Sam-
ple 1–5
z
) are material constants of a phenomenological active strain energy function for smooth muscle cell (SMC); λ
4
, b
3
, b
2
is SMC optimal stretch ratio at axial stretch ratio λ
, b
1
2
, b
act
4
1:2 b
Reproduced from Chen, Luo, et al. (2013) with permission
Note: (C
ðÞ=b
Appendix 14: 3D Microstructure-Based Model of Active Coronary Artery (... 291
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Appendix 14: 3D Microstructure-Based Model of Active
Coronary Artery (Chen & Kassab, 2017)
The following general kinematic assumptions are considered for a 3D vessel
mechanical model: (1) Vessel wall is incompressible; (2) Deformations are axissymmetric and independent of axial position; (3) Transverse sections remain planar;
and (4) There is a unique un-deformed reference configuration (i.e., zero-stress state,
ZSS). The vessel is assumed as a cylindrical tube using a cylindrical coordinate
system, of which three principal directions are circumferential direction θ
direction r,
!
and axial direction z!. The corresponding stretches (λθ, λrand λz) are
given as below:
π
r
λ
¼
θ
π Θ
R
, λ
∂r
, λ
¼
r
∂R
l
¼
z
L
where Θ is opening angle and R is radius to a point measured at ZZS configuration of
a vessel, and r is the radius to the same point in the current configuration, i.e., loaded
vessel, as shown in the top row of Fig. 4.25. 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 mappi ng between ZSS and loaded state, loaded radius r is
θλrλz
determined as a function of unloaded R:
!
, radial
ð4:197Þ
s
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
where r
2
2
rRðÞ¼
is the outer radius in the loaded state while Rois that at ZSS. Analogously, r
o
r
R
o
o
R
2
π Θ
λ
π
z
and Riare the inner radii in the loaded state and ZSS, respectively.
The radial component of the force equilibrium equation imposed on the loaded
configuration is given by:
∂σ
σrr σ
rr
þ
∂r
where σ
, σθθare the radial and circumferential components of Cauchy stress,
rr
respectively. According to boundary conditions, σ
nal pressure p
can be written as:
i
Z
r
p
o
¼
i
r
i
θθ
¼ 0 ð4:199Þ
r
¼pi, σ
j
rr
r
i
1
σθθ σ
ðÞ
dr ð4:200Þ
rr
r
¼ 0, the lumi-
j
rr
r
o
The axial tension can be determined by the integration of the axial components of
Cauchy stress, σ
, over the cross-sectional area:
zz
ð4:198Þ
i
292 4 Constitutive Models of Coronary Vasculature
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Fig. 4.25 Top: A schematic diagram of vessel configurations and geometrical parameters. The left
panel is the coordinate system of vessels with three principal directions: circumferential direction θ,
radial direction r and axial direction z; ZSS is zero-stress state of vessel, and loaded state means
vessel under pressure P. Deformation gradient tensor F describes tissue deformation from ZSS to
loaded configuration. Parameters (R
radius to a point measured at ZSS, while θ is opening angle of vessel at ZSS. Parameters (r
, Ri) are the outer and inner radii of stress-free vessel and R is
o
, ri) are
o
the outer and inner radii of loaded vessel and r is radius to the point measured at the loaded state.
Bottom: Images of coronary artery and microstructure. Left: multiphoton microscopic (MPM)
image of arterial cross section shows coronary artery layered-structure (Red: collagen; Green:
elastin); Middle: MPM images of longitudinal-circumferential sections of collagen and elastin in
adventitia, respectively; Right: Confocal images of SMCs in media (Green: F-actin; Blue: cellular
nucleus). Reproduced from Chen and Kassab (2017) with permission
where axial force is determined as F ¼ π
accounts for the internal pressure in the closed tube during the test. The circumfer-
ential and axial equilibrium equations under distension–extension loading yield all
shear stress components σ
stress are provided by the 3D model, of which the radial component σ
predicted by a 2D model (Chen, Luo, et al., 2013; Huo et al., 2012).
The coronary artery is considered as an incompressible hyperelastic solid and
characterized by a strain energy function (SEF) W(E) as a function of the GreenLagrange strain tensor E ¼
Gurtin (1982):
Z
r
N ¼ 2π
rθ¼σzθ¼σzr
1
2
o
σzzrdr ¼ F þ piπ r
r
i
¼0. Therefore, three components of Cauchy
FT F I
2
i
R
r
o
2σzz σzθ σ
ðÞrdr, and piπ r
r
i
rr
ð4:201Þ
cannot be
rr
. The Cauchy stress tensor σ is given by
2
i
Appendix 14: 3D Microstructure-Based Model of Active Coronary Artery (... 293
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∂W
T
σ ¼ F
F
pI ¼ F S FT pI ð4:202Þ
∂E
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.
Microstructural Features of Active Coronary Arteries
Based on previous microstructural studies (Chap. 3, Fig. 3.29), the following axioms
have been integrated to synthesize a realistic microstructural constitutive model:
(1) A fluid-like matrix is employed and suggests the tissue undergoes affine deformations. Therefore, the SEF of the vessel wall can be represented by the volumeweighted summation of individual SEFs of every constituent. (2) All fibers are only
resistant to tensile load and have negligible compressive and bending rigidities.
Elastin fibers take up most of load at low pressures, while collagen fibers gradually
become straightened and are recruited with an increase of pressure. (3) In adventitia,
many collagen fibers oriented towards the axial direction and the others aligned
nearly in the circumferential direction following a mixture of two normal distributions. The orientation of elastin fibers also follows two normal distributions where
the minor orientation of elastin is approximately orthometric to the major orientation. (4) The straightening strain of adventitia coll agen is found to follow a beta
distribution with a mean and standard deviation of (0.35, 0.051). (5) In media, most
SMCs arrange in θ z (circumferential-axial) plane and slightly aligned off circumferential direction of blood vessels with symmetrical polar angles, and the axial
active response of blood vessels is associated with SMC biaxial contraction. (6) The
majority of media fiber bundles are also planar and their orientations are consistent
with but not exactly identical to that of SMCs (O’Connell et al., 2008). Finally, there
exist isotropic inter-lamellar (IL) elastin networks in media, which disperse over
orientation space including the radial direction (Hollander et al., 2011a;O’Connell
et al., 2008 ).
Based on the above axioms, the total SEF of vessel wall is a sum of volumeweighted SEFs of fibers and SMCs:
where f
, fC, and f
E
respectively.
W EðÞ¼f
SMC
WILþ fEWEþ fCWCþ f
IL
SMC
W
SMC
ð4:203Þ
are volume fractions of elastin, collagen, and SMC ,
294 4 Constitutive Models of Coronary Vasculature
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Passive SEF of Coronary Artery
Although the SMCs are the predominate contributors of active behavior, they
provide negligible contributions to the passive properties of blood vessels
(Matsumoto & Nagayama, 2012; Roach & Burton, 1957; Wolinsky & Glagov,
1964). The passive SEF is thus only determined by SEFs of elastin and collagen
fibers:
W
EðÞ¼fILWILþ fEWEþ fCW
P
C
The SEF of a fiber family is a function of fiber orientation θ as below (i ¼ IL, E,
C):
Z
π
W
¼
i
ℛiθðÞwieðÞdθ ð4:205Þ
0
ð4:204Þ
where ℛ
(θ) is the orientation distribution density function of fiber i, and wi(e) is the
i
SEF of a single fiber. The uniaxial fiber strain e(θ) is determined by the local strain
tensor E and the reference fiber direction N as: e(E, N) ¼ E : N N. It should be
noted that ℛ
(θ) satisfies the normalization criterion
i
R
π
ℛiθðÞdθ ¼ 1.
0
In adventitia, orientations of elastin and collagen fibers follow a mixture of two
normal distribution of fiber (i ¼ E, C):
Z
2
X
W
¼
Ai
j
θμ
where ℛ
ij
θðÞ¼
1
1
К
ij
√2π
σ
ij
Exp
ðÞ
2σ
distribution density function with μ
respectively. К
πμ
ij
Φ
σ
ij
tion), and ω
is the total weight of the truncated normal distribution Кij¼ Φ
ij
μ
ij
(Φ is the cumulative distribution function of a normal distribu-
σ
ij
is the wei ght of each normal distribution
ij
π
ω
ℛijθðÞwieðÞdθ ð4:206Þ
ij
0
2
ij
, i ¼ E; C; j ¼ 1; 2ðÞis a truncated normal
2
ij
and σijas the mean and standard deviation,
ij
P
2
ωij¼ 1
j
.
In media, elastin and collagen fibers are aligned off of circumferential direction of
blood vessels, following two symmetric normal distributions are analogous to
SMCs. It is assumed that both fibers have the same distributions
(θ) ¼ ℛC(θ) ¼ ℛM(θ), which is a normal distribution density function with
ℛ
E
mean and standard deviation (μ
, σM). Moreover, media IL elastin is an isotropic
M
network, the overall SEF of the passive media can be thus written as (i ¼ E, C):
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