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Appendix 3: 2D Strain Energy Function (Pandit et al., 2005) 225
https://t.me/med1917
guess values for lower and upper limit of C, a1, a2, and a4; and number of
generations. The error function (Eq. 4.27) is evaluated based on the initial parameters. These values are stored in an array and the search is initiated for the best
“individuals” or material constants. An elitism and recombinant individual is set as
per the Tournament Algorithm or the Roulette Algorithm based on the probability of
subsequent tournament algorithms. Two parents are selected based on Tournament
Algorithm and roulette principle, i.e., the lower the value, the better chance. The
selection of children from the two parents is made using crossover, mutations, and
elitism. This process is repeated several times and the fitness value is computed at
each cycle. The converged values represent a minima for Eq. (4.27). The parameter
a
is then fixed and the values of C, a1, and a2are redetermined using the GA.
4
Table 4.8 Material constants of strain energy function obtained from experimental stress–strain
data of intact left anterior descending (LAD) artery
Material constants for intact left anterior descending artery (LAD)
Animal # C (kPa) a
1
a
2
E
θθ
E
ZZ
Heart 1 30.7 1.4 2.3 0.82 0.47 0.99 0.98
Heart 2 38.4 1.5 2.4 0.71 0.51 0.98 0.98
Heart 3 34.7 2.6 1.1 0.47 0.50 0.99 0.98
Heart 4 41.3 1.5 1.7 0.67 0.47 0.98 0.97
Heart 5 35.2 1.0 1.8 0.93 0.49 0.98 0.98
Heart 6 69.3 0.7 1.5 0.91 0.49 0.98 0.96
Heart 7 55.9 1.5 1.1 0.56 0.47 0.98 0.98
Mean SD 43.6 13.9 1.4 0.6 1.7 0.5 0.72 0.17 0.49 0.02
The value of a
permission
C (kPa), a
axial directions, respectively. S
was fixed at 0.15 for all animals. Reproduced from Pandit et al. (2005) with
4
, a2are material parameters. E
1
and SZare second Piola–Kirchhoff stresses in the circumferential
θ
θθ
and E
are Green strains in the circumferential and
ZZ
and axial directions, respectively. Starred quantities correspond to a reference homeostatic state
(physiological pressure, 80 mmHg, and axial stretch, λ
¼ 1.4)
z
R
for
S
2
2
R
for
S
Z
θ
Table 4.9 Material constants of strain energy function obtained from experimental stress–strain
data of medial layer of LAD artery
Material constants for medial layer of left anterior descending
artery (LAD)
Animal # C (kPa) a
Heart 1 48.0 4.1 1.6 0.56 0.50 0.98 0.99
Heart 2 32.3 1.5 1.3 0.78 0.49 0.98 0.96
Heart 3 40.1 7.5 1.9 0.37 0.48 0.98 0.98
Heart 4 31.7 6.3 1.9 0.37 0.48 0.97 0.95
Heart 5 29.2 3.9 2.1 0.64 0.47 0.94 0.96
Heart 6 55.0 5.0 2.5 0.42 0.50 0.99 0.96
2
R
1
a
2
E
θθ
E
ZZ
for
S
2
R
for
S
Z
θ
(continued)
226 4 Constitutive Models of Coronary Vasculature
https://t.me/med1917
Material constants for medial layer of left anterior descending
artery (LAD)
2
2
R
for
S
Z
θ
Animal # C (kPa) a
R
1
a
2
E
θθ
E
ZZ
for
S
Heart7 45.3 3.8 1.6 0.48 0.47
Mean SD 40.2 9.7 4.6 1.9 1.8 0.4 0.52 0.15 0.49 0.01
The value of a
permission
C (kPa), a
axial directions, respectively. S
was fixed at 0.34 for all animals. Reproduced from Pandit et al. (2005) with
4
, a2are material parameters. E
1
θ
are second Piola–Kirchhoff stresses in the circumferential
Z
θθ
and E
are Green strains in the circumferential and
ZZ
and axial directions, respectively. Starred quantities correspond to a reference homeostatic state
(physiological pressure, 80 mmHg, and axial stretch, λ
¼ 1.4)
z
Table 4.10 Material constants of strain energy function obtained from experimental stress–strain
data of adventitial layer of LAD artery
Material constants for adventitial layer of left anterior
descending artery (LAD)
2
R
for S
Animal # C (kPa) a
1
a
2
E
θθ
E
ZZ
R2for
S
Z
Heart 8 53.2 1.0 4.1 0.91 0.49 0.97 0.97
Heart 9 63.9 1.6 1.1 0.96 0.48 0.96 0.97
Heart 10 78.3 1.7 1.0 0.71 0.48 0.96 0.98
Heart 11 24.4 2.3 2.5 0.80 0.48 0.98 0.99
Heart 12 34.7 2.4 2.6 0.75 0.48 0.95 0.97
Mean SD 50.9 21.7 1.8 0.6 2.3 1.3 0.82 0.10 0.48 0
The value of a
permission
C (kPa), a
axial directions, respectively. S
was fixed at 0.77 for all animals. Reproduced from Pandit et al. (2005) with
4
are material parameters. E
1,a2
θ
are second Piola–Kirchhoff stresses in the circumferential
Z
θθ
and E
are Green strains in the circumferential and
ZZ
and axial directions, respectively. Starred quantities correspond to a reference homeostatic state
(physiological pressure, 80 mmHg, and axial stretch, λ
¼ 1.4)
z
θ
Table 4.11 Material constants of strain energy function obtained from experimental stress–strain
data of intact right coronary artery (RCA)
Material constants for intact right coronary artery (RCA)
Animal# C (kPa) a
Heart 1 32.0 1.3 2.1 0.82 0.47 0.98 0.98
Heart 2 34.7 1.5 2.5 0.87 0.48 0.99 0.99
Heart 3 25.4 1.8 2.1 0.71 0.48 0.99 0.98
Heart 4 34.4 2.1 1.8 0.52 0.47 0.99 0.97
Heart 5 30.8 0.9 1.3 0.91 0.48 0.99 0.98
Mean SD 31.5 3.8 1.5 0.4 2.0 0.5 0.77 0.15 0.48 0.01
2
R
1
a
2
E
θθ
E
ZZ
for
S
2
R
for
S
Z
θ
(continued)
Appendix 4: 3D Strain Energy Function (Wang et al., 2006) 227
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Material constants for intact right coronary artery (RCA)
2
2
R
for
S
Z
θ
Animal# C (kPa) a
R
1
a
2
E
θθ
E
ZZ
for
S
The value of a4was fixed at 0.15 for all animals. Reproduced from Pandit et al. (2005) with
permission
C (kPa), a
, a2are material parameters. E
1
axial directions, respectively. S
and SZare second Piola–Kirchhoff stresses in the circumferential
θ
θθ
and E
are Green strains in the circumferential and
ZZ
and axial directions, respectively. Starred quantities correspond to a reference homeostatic state
(physiological pressure, 80 mmHg, and axial stretch, λ
¼ 1.4)
z
Table 4.12 Material constants of strain energy function obtained from experimental stress–strain
data of medial layer of right coronary artery (RCA)
Material constants for medial layer of right coronary artery (RCA)
2
2
R
for
S
Z
θ
Animal # C (kPa) a
R
1
a
2
E
θθ
E
ZZ
for
S
Heart 1 36.1 3.3 3.5 0.56 0.54 0.99 0.98
Heart 2 44.3 3.9 2.1 0.46 0.50 1.00 0.99
Heart 3 35.7 7.7 2.2 0.40 0.50 0.99 0.99
Heart 4 37.6 4.4 2.5 0.45 0.50 0.98 0.99
Heart 5 49.5 2.6 1.9 0.55 0.48 0.91 0.88
Mean SD 40.6 6.0 4.4 2.0 2.4 0.6 0.49 0.07 0.50 0.02
The value of a
permission
C (kPa), a
axial directions, respectively. S
was fixed at 0.32 for all animals. Reproduced from Pandit et al. (2005) with
4
are material parameters. E
1,a2
θ
are second Piola–Kirchhoff stresses in the circumferential
Z
θθ
and E
are Green strains in the circumferential and
ZZ
and axial directions, respectively. Starred quantities correspond to a reference homeostatic state
(physiological pressure, 80 mmHg, and axial stretch, λ
¼ 1.4)
z
Appendix 4: 3D Strain Energy Function (Wang et al., 2006)
Similar to Appendix 2, the circumferential deformation of an arbitrary point on
the artery may be described by the circumferential Lagrangian-Green strain as
defined by:
where λ
is the stretch ratio (λθ¼ c/C, c and C refer to the circumference in the
θ
loaded and zero-stress state, respectively). Similarly, the axial and radial Green
strains are given by:
1
2
E
λ
¼
θ
1
θ
2
ð4:28Þ
228 4 Constitutive Models of Coronary Vasculature
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where λ
1
2
Ez¼
and λrare the local axial stretch ratio (change in axial length between
z
λ
1
, Er¼
z
2
1
2
λ
1
r
2
ð4:29Þ
loaded and no-load state) and radial stretch ratio (change in wall thic kness between
loaded and no-load state), respectively.
The generalization of 2D analysis to 3D can be made by invoking the
incompressibility assumption which requires the following relationship between
the principal stretch ratios (Chuong & Fung, 1986):
λ
¼ 1
rλzλθ
and
R
, λ
¼
λ
r
λ
χr
z
z
, λ
¼
z
Z
χr
¼
θ
R
ð4:30Þ
where R denotes the radius of an arbitrary point at zero-stress state (an open sector)
as a reference, r is the radial coordinate in the deformed configuration, χ ¼ π/(π Φ)
is a factor that depends on the opening angle Φ defined as the angle subtended by
two radii connecting the midpoint of the inner wall of the open sector. Although the
axial and circumferential stretch ratios are measured independently, the radial stretch
ratio is computed from the incompressibility assumption (Eq. 4.30). The inner and
outer radii at no-load state and zero-stress states are obtained from the measurements
of the vessel rings to determine the residual circumferential strains at the no-load
state. The outer radius of the vessel segment in the loaded configuration is measured
directly while the inner radius is computed from the incompressibility condition for a
cylindrical vessel as:
where A
is the cross-sectional area of vessel wall in the no-load state. The reference
0
radius R in the zero-stress state can also be determined by the incompressibility
condition as:
The stretch ratios in Eq. (4.30) can be written as a function of radius r. Equ ations
(4.28)–(4.32) can be combined to give the components of Green strain tensor at any
given deformed state. Equations (4.28)–(4.32) are also applied individually to each
separate layer. The wall thickness of each individual layer is determined as the
difference between inner and outer radius.
r
¼
i
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
q
R ¼
χ r
s
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
r
e
2
r
A
0
λzπ
2
i
2
þ R
λ
z
i
ð4:31Þ
ð4:32Þ
Appendix 4: 3D Strain Energy Function (Wang et al., 2006) 229
https://t.me/med1917
Strain Energy Function
Similar to Appendix 2 , the following exponential form of strain energy function is:
C
W ¼
where
expQ 1ðÞ ð4:33Þ
2
Q ¼ b
2
E
1
θ
þ b2E
2
þ b3E
z
2
þ 2 b4EθEzþ b5EzErþ b6EθE
ðÞ
r
r
where W represents the pseudo-strain energy per unit volume. C has the units of
stress (force/area), b
, b2, b3, b4, b5, and b6are dimensionless constants.
1
The vessel wall is assumed to be incompressible. This constraint is added to the
strain energy function by the method of a Lagrangian multiplier, H, to yield:
W
¼ W þ
H
1 þ 2E
ðÞ1 þ 2E
2
ðÞ1 þ 2E
θ
ðÞ1½ð4:34Þ
z
r
It is known that H has the significance of a hydrostatic pressure. The Cauchy
stresses can be related to the strain energy function as:
ρ
∂x
∂x
∂X
∂
i
Wi; j; α; β ¼ r; z; θðÞð4:35Þ
∂E
β
βα
where x
σ
¼
ij
and Xαdenote coordinates, and ρ and ρ0represent densities, in the deformed
i
j
ρ
∂X
α
0
and reference states, respectively. The summation convention is used in these
expressions. The principle stress components can be determined from Eqs. (4.28)
to (4.30) and Eqs. (4.34)–(4 .35) to yield:
σ
¼ C 1 þ 2E
ðÞb1Eθþ b4Ezþ b6E
θ
σ
¼ C 1 þ 2E
ðÞb4Eθþ b2Ezþ b5E
z
σ
¼ C 1 þ 2E
ðÞb6Eθþ b5Ezþ b3E
r
½eQþ H
θ
½eQþ H
z
½eQþ H
r
r
r
r
ð4:36Þ
Equation (4.36) will be subsequently incorporated in the equilibrium equation to
obtain the desired results.
Equation of Equilibrium and Boundary Conditions
The problem of a pre-strained thick wall vessel under transmural pressure and axial
force can be solved by substituting Eq. (4.36) into the equation of equilibrium as
(Chuong & Fung, 1983):
230 4 Constitutive Models of Coronary Vasculature
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∂σ
σr σ
∂r
r
þ
θ
¼ 0 ð4:37Þ
r
The following boundary conditions are considered that simulate the experimental
protocol:
1. On the external surface r ¼ r
internal pressure p
is imposed. Integration of Eq. (4.37) along with this boundary
i
, the pressure is zero. On the inner surface r ¼ ri,an
e
condition yields:
Z
r
i
¼
C 1 þ 2E
p
i
ðÞb6Eθþb5Ezþb3E
fg
r
e
½1 þ 2E
r
ðÞb1Eθþb4Ezþb6E
r
½
θ
dr
Q
e
r
ð4:38Þ
2. On the two ends of a blood vessel segment, there exists an external axial force F.
For static equilibrium, the sum of the axial and pressure force F þ p
2
πr
i
i
equals the
integral of axial stress over the vessel wall cross secti on, namely:
Z
r
F þ p
2
πr
¼ 2π
i
i
e
σzrdr ð4:39Þ
r
i
Use of Eqs. (4.36) and (4.38)in(4.39) yields:
Z
r
e
F ¼ 2πC
Q
re
ðÞb4Eθþb2Ezþb5E
r
i
1
1 þ2E
ðÞb6Eθþb5Ezþb3E
2
1 þ2E
ðÞ
r
½
z
r
1
1 þ2E
ðÞb1Eθþb4Ezþb6E
r
2
ðÞ
θ
dr
r
ð4:40Þ
r
This equation represents a force–displacement relation.
Determination of Elastic Constants
Equations (4.38) and (4.40) are two integral relations from which the material
constants can be determined given the values of transmural pressure ( p
force (F ), external radius (r
components (E
radius are direct experimental measurements while internal radius is calculated from
the outer radius and no-load wall area based on the incompressibility assumption.
Green strains at inner and outer surfaces are computed using Eqs. (4.28)–(4.32). Since
the circumferential strain E
), axial stretch
i
), internal radius (ri), and distribution of Green strain
e
, Ezand Er). Transmural pressure, axial stretch force, and external
θ
(r) between the two surfaces is a function of the radius,
θ
Appendix 4: 3D Strain Energy Function (Wang et al., 2006) 231
https://t.me/med1917
Eqs. (4.38) and (4.40) can be simplified if a linear distribution of Eθ(r) can be assumed
between two surfaces along the radial direction as:
r r
E
rðÞ¼EθriðÞþ
θ
E
(r) and Er(r) can be computed using Eqs. (4.29) – (4.32). Guo, Xiao, and Kassab
z
re r
i
EθreðÞEθriðÞðÞð4:41Þ
i
(2005) verified that the computed strain distribution is very close to linear as given
by Eq. (4.28 ).
The goal of an algorithm to determine the mat erial constants C, b
and b
is to minimize the square of the difference between theoretical (Eqs. 4.38 and
6
4.40) and experimental values of internal pressure p
Z
0
2
r
Error ¼
N
X
n¼1
þ
N
X
n¼1
4
@
2
6
6
4
i
C1 þ E
ðÞb6Eθþ b5Ezþ b3E
r
e
ðÞb1Eθþ b4Ezþ b6E
1 þ E
Z
0
2πC
B
B
@
r
½
r
½
θ
r
e
Q
1 þ E
re
ðÞb4Eθþ b2Ezþ b5E
i
1
1 þ E
ðÞb6Eθþ b5Ezþ b3E
2
1
1 þ E
ðÞb1Eθþ b4Ezþ b6E
2
½
z
ðÞ
r
ðÞ
θ
e
, and axial Force Feas:
i
r
Q dr
e
r
r
1
A
r
r
p
n
r
dr
, b2, b3, b4, b5,
1
3
2
e
5
i
n
1
C
C
FeðÞ
A
n
3
7
7
n
5
ð4:42Þ
2
where N represents the total number of experimental data points used to determine
the material constants for each curve. The strain energy function must be positive
definite, which means any possible strain configuration except zero strain should
have positive strain energy. Although this condition is not applied as a constraint on
the material constants, all the material constants are verified to satisfy this condition.
Determination of Elastic Constants of the Dissected Layer
One layer has to be dissected or sacrificed to allow the testing of the other layer. A
method to determine the material constants of the dissected layer from the intact
vessel and the measured layer is proposed. The total strain energy of the intact vessel
segment, V
and adventitia layer, VA. Furthermore, the total axial force of intact vessel, F
must be the sum of the axial force of the intima-media layer, F
layer, F
, is assumed to be the sum of the strain energy of the intima-media, V
I
, and the adventitia
IM
. The two assumptions can be mathematically stat ed as:
A
V
¼ VIMþ V
I
total
F
I
¼ FIMþ F
A
A
IM
Total
I
ð4:43aÞ
ð4:43bÞ
,
232 4 Constitutive Models of Coronary Vasculature
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The energy Vj( j ¼ I, IM, A) can be computed by taking an integral of strain
energy density given by Eq. (4.33) over the respective wall volume as:
Z
r
e
C
V
¼
I
¼
I
expQ
1ðÞ2π rh dr
IM
expQ
I
1ðÞ2πrh dr þ
IM
Z
r
e
C
A
expQ
1ðÞ2πrh dr ð4:44Þ
2
r
m
A
2
r
i
Z
r
m
C
2
r
i
where
Q
¼ b
j
1, j
E
2
rðÞþb
θ
2, j
2
E
z
þ b
3, j
2
E
r
þ 2 b
4, jEθ
rðÞEzþ b
5, jEzEr
þ b
6, jEθ
rðÞE
r
j ¼ I ; IM; AðÞ
, and reare radii for inner, middle (the interface between two layers), and outer
r
i,rm
vessel wall; h is the deformed length of the arterial specimen; and E
determined by E
total
force F
j
(r) and Ezbased on the incompressibility condition. The total
θ
j ¼ I ; IM; AðÞis the integral of axial Cauchy stress σzover the wall area
can be
r
expressed as:
F
total
σ
z,j
Z
r
¼ 2π
¼ C 1 þ 2E
e
σ
r
i
ðÞb
rdr ¼ 2π
z, I
z
4, jEθ
Z
r
r
i
rðÞþb
m
σ
z, IM
2, jEz
rdr þ2π
þ b
5, jEr
Z
r
e
σ
rdr
z, A
r
m
Q
j ¼ I ; IM; AðÞ
e
ð4:45Þ
Equations (4.44) and (4.45) are also simplified by assuming a linear distribution
(r) along the radial direction expressed in Eq. (4.41). Material constants b
of E
θ
i, j
(i ¼ 1–7;j¼ I, IM, A) of the intact vessel and the measured layer are obtained by
minimization of the objective function (Eq. 4.42). Hence, the material constants of
the dissected layer can be determined by minimization of the following error
function:
Z
Error ¼
N
X
n¼1
0
2
r
e
C
I
expQ
1ðÞ2πrdr
B
6
B
6
@
6
6
6
6
4
2
r
i
Z
Z
Z
N
X
þ
n¼1
r
I
r
e
C
A
expQ
1ðÞ2πrdr
expQ
rdr
A
1ðÞ2πrdr
IM
Z
r
e
σ
r
m
z, A
2
r
m
r
m
C
IM
2
r
i
r
e
σ
z,I
i
rdr
1
3
2
C
7
C
7
A
7
7
7
n
7
5
n
Z
r
n
m
σ
rdr
z,IM
r
i
2
n
ð4:46Þ
Equation (4.45) is the square of the difference between the theoretical values of
strain energy and axial force of the sacrificed layer and the values determined by the
Appendix 4: 3D Strain Energy Function (Wang et al., 2006) 233
https://t.me/med1917
material constants of the intact and the tested layer. The above function can be used
to determine the material constants of the intima-media layer. In order to determine
the material constants of the adventitia, the terms of the two layers are interchanged.
A genetic algorithm is adopted sim ilar to Appendix 3 for the determination of
material constants.
Convexity of the Strain Energy Function
The values of the material constants in the strain energy function must be such that
the strain energy is convex, i.e., the condition that the material must be stable under
loading (Ogden, 1997). Although the convexity condition is not applied as a
constraint in the genetic algorithm, all the material constants are verified to satisfy
this condition. Because of the quadratic form of Q in Eq. (4.33), it can be verified that
the strain energy is locally strictly convex when the eigenvalues of the matrix
0
b
1b4b6
@
b4b2b
b6b5b
the material constants (Tables 4.13, 4.14, 4.15, and 4.16).
Table 4.13 Material constants of exponential strain energy function obtained from experimental
data of right coronary artery (RCA)
Animal#C(kPa) b
Heart1 7.73 1.29 2.04 0.64 0.26 0.06 0.07 0.98 0.98 13.6 16.1
Heart2 7.78 1.32 2.73 0.16 0.38 0.08 0.02 0.99 0.98 14.9 20.5
Heart3 10.13 0.95 2.69 0.82 0.30 0.05 0.15 0.99 0.97 12.5 23.5
Heart4 4.37 1.21 2.68 0.75 0.28 0.03 0.03 0.99 0.99 18.9 16.1
Heart5 7.78 1.64 2.54 0.25 0.31 0.01 0.01 0.99 0.99 16.7 20.9
Mean 7.56 1.28 2.54 0.52 0.31 0.05 0.06 0.99 0.98 15.3 19.4
SD 2.06 0.25 0.29 0.30 0.05 0.03 0.06 0.004 0.01 2.5 3.2
Heart1 8.57 2.12 3.80 0.38 0.60 0.04 0.11 0.99 0.98 13.6 28.2
Heart2 3.08 5.04 3.97 0.93 0.27 0.06 0.15 0.98 0.99 23.7 13.5
Heart3 4.36 2.27 3.81 1.76 0.55 0.06 0.14 0.97 0.99 25.1 10.9
Heart4 10.48 0.92 3.18 0.25 0.31 0.07 0.04 0.98 0.96 14.0 24.7
Heart5 6.13 2.16 3.26 0.85 0.49 0.01 0.07 0.97 0.97 20.9 20.9
Mean 6.53 2.50 3.61 0.83 0.44 0.05 0.10 0.98 0.97 19.4 19.6
SD 3.02 1.52 0.36 0.60 0.15 0.02 0.05 0.01 0.02 5.4 7.3
(a) Material constants of the five intact RCAs; (b) material constants of the five corresponding
intima-media layers. R
error of the fit compared to the mean value. Both were computed for the total axial force (F
inner pressure ( p
1
A
are positive. The eigenvalues are confirmed to be positive for all
5
3
2
b2b
1
2
is correlation coefficient; RMS% is the percentage of the root mean square
). Reproduced from Wang et al. (2006) with permission
i
b4b5b
3
(a) Intact
(b) Intima-media
R
(FT) R2( pi)
6
RMS
%(FT)
RMS
%(pi)
) and
T
234 4 Constitutive Models of Coronary Vasculature
https://t.me/med1917
Table 4.14 Material constants of exponential strain energy function obtained from experimental
data of right coronary artery (RCA)
2
2
R
Animal#C(kPa) b
( pi)
RMS%
(FT)
b2b
1
b4b5b
3
(FT)
6
R
RMS%
( pi)
(a) Intact
Heart6 29.01 0.61 1.53 0.44 0.32 0.06 0.12 0.98 0.96 10.5 14.1
Heart7 6.19 1.59 3.74 0.17 0.48 0.12 0.14 0.95 0.99 19.2 12.9
Heart8 4.35 1.30 3.57 0.90 0.26 0.09 0.03 0.99 0.96 10.9 17.4
Heart9 21.63 0.90 1.62 0.76 0.25 0.06 0.11 0.99 0.98 11.8 12.8
Heart10 20.08 0.55 3.56 1.25 0.29 0.07 0.13 0.92 0.82 22.1 31.1
Mean 16.25 0.99 2.81 0.70 0.32 0.08 0.11 0.96 0.94 14.9 17.6
SD 10.60 0.45 1.12 0.42 0.09 0.02 0.04 0.03 0.07 5.4 7.7
(b) Adventitia
Heart6 7.70 1.04 2.43 1.63 0.42 0.06 0.02 0.99 0.99 5.6 9.1
Heart7 2.03 1.65 3.01 1.99 0.27 0.04 0.01 0.93 0.91 19.7 22.1
Heart8 0.92 0.97 3.62 1.61 0.59 0.07 0.11 0.97 0.93 14.8 24.8
Heart9 2.41 1.72 3.91 0.98 0.40 0.16 0.06 0.97 0.92 13.3 21.2
Heart10 13.42 0.54 3.54 0.12 0.37 0.02 0.16 0.84 0.90 21.4 38.0
Mean 5.30 1.19 3.30 1.27 0.41 0.07 0.07 0.94 0.93 15.0 23.0
SD 5.24 0.50 0.58 0.74 0.12 0.05 0.06 0.06 0.03 6.2 10.3
(a) Material constants of the five intact RCAs; (b) material constants of the five corresponding
adventitia layers. R
error of the fit compared to the mean value. Both were computed for the total axial force (F
inner pressure ( p
2
is correlation coefficient; RMS% is the percentage of the root mean square
). Reproduced from Wang et al. (2006) with permission
i
) and
T
Table 4.15 Material constants of exponential strain energy function obtained from experimental
data of left anterior descending (LAD) artery
2
2
R
Animal#C(kPa) b
( pi)
RMS%
(FT)
b2b
1
b4b5b
3
(FT)
6
R
RMS%
( pi)
(a) Intact
Heart11 11.64 1.03 2.09 0.38 0.37 0.02 0.06 0.98 0.97 26.3 21.1
Heart12 5.94 1.27 2.78 0.62 0.45 0.13 0.13 0.97 0.99 25.1 21.4
Heart13 4.16 1.58 2.66 0.73 0.30 0.03 0.08 0.96 0.98 34.5 18.4
Heart14 11.39 1.39 2.62 0.51 0.32 0.06 0.06 0.99 0.99 28.0 18.5
Heart15 11.45 1.00 2.16 0.52 0.39 0.04 0.06 0.97 0.98 25.3 19.2
Mean 8.92 1.25 2.46 0.55 0.36 0.06 0.08 0.97 0.98 27.8 19.7
SD 3.59 0.25 0.31 0.13 0.06 0.05 0.03 0.01 0.01 3.9 1.4
(b) Intima-Media
Heart11 4.90 1.94 2.42 0.61 0.55 0.11 0.05 0.99 1.00 28.3 15.7
Heart12 6.29 1.62 2.61 1.03 0.30 0.04 0.04 0.97 0.96 16.0 25.2
Heart13 4.62 2.22 2.81 1.81 0.47 0.04 0.15 0.97 0.98 29.9 21.1
Heart14 8.48 2.41 2.82 0.35 0.42 0.10 0.11 0.97 0.98 17.5 18.1
(continued)
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