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Appendix 2: Formulation of Incremental Moduli (Lu et al., 2004) 215
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θ
) as given by G ¼ α + βσ
a
0.877 0.325
θ
a
0.501 0.185
a
d
15.3 3.49 17.9 5.99
c
16.5 4.08
b,d
27.2 5.68
a,d
1.44 0.384
a
0.824 0.233
a,c
1.23 0.271
b,d
0.951 0.193
d
c
b,d
18.8 5.92 22.1 3.36
19.8 3.37
35.4 7.29
a
2.45 0.787
a
1.12 0.207
a
1.95 0.642
b
2.02 0.559
a
37.9 11.7
a
24.7 6.79
a
32.9 7.61
b
45.1 16.4
b
for the relationship between shear modulus (G) and circumferential stress (σ
2
11.1 2.56
b,c
LAD RCA
Measured intact wall Measured media Computed adventitia Measured intact wall Measured media Computed adventitia
α (kPa) 19.4 2.87
Table 4.1 The linear regression parameters and R
λ ¼ 1.2 β 0.553 0.101 0.522 0.189 0.586 0.247 0.732 0.191
0.974 0.030 0.990 0.007 0.932 0.076 0.964 0.041 0.985 0.025 0.953 0.057
2
R
b
0.706 0.242
c
λ ¼ 1.3 β 0.825 0.172
b
b,c
13.5 3.15
α (kPa) 24.7 3.86
0.990 0.011 0.990 0.007 0.964 0.050 0.958 0.033 0.983 0.016 0.929 0.051
2
R
b
0.916 0.287
b
λ ¼ 1.4 β 1.47 0.369
b
b
20.7 4.01
α (kPa) 33.1 7.98
0.993 0.004 0.998 0.002 0.986 0.008 0.973 0.032 0.993 0.006 0.961 0.023
2
R
The data were obtained for the left anterior descending (LAD) and right coronary artery (RCA) from 8 animals where the intact vessel and media were measured
and the adventitial properties computed according to Eq. (4.7) from Appendix 1. Reproduced from Lu et al. (2003 ) with permission
Note: The parameters are presented as mean SD
Denotes a statistically significant difference ( p < 0.05) between media and intact wall or adventitia of the RCA
a
Denotes a statistically significant difference ( p < 0.05) between media and intact wall or adventitia of the LAD
b
Denotes a statistically significant difference ( p < 0.05) between the intact wall of LAD and RCA
c
Denotes a statistically significant difference ( p < 0.05) between the adventitia of LAD and RCA
d
216 4 Constitutive Models of Coronary Vasculature
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circumference in the zero-stress state. Similarly, the longitudinal Green strain is
given by:
1
2
E
λ
¼
z
1
z
2
ð4:11bÞ
where λ
is the local longitudinal stretch ratio as defined above.
z
The mid-wall circumference in the loaded state was computed from the average
of inner and outer radius. The inner radius, r
, of the vessel can be computed from the
i
incompressibility condition for a cylindrical vessel as:
s
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
A
0
2
¼
r
o
πλ
z
where r
r
i
and A0are outer radii at the loaded state and the wall area in the no-load
o
state, respectively. The total wall thickness, h, was computed as h ¼ r
o
ð4:12Þ
ri. Since all
the quantities on the right-hand side of Eq. (4.12) are measured, the loaded inner
radius can be computed.
The mean second Piola–Kirchhoff stresses in the circumferential, S
tudinal, S
, directions are given by:
z
Pr
S
i
¼
θ
2
hλ
θ
, and longi-
θ
ð4:13Þ
and
"#
1
¼
S
z
2
λ
z
F
2
π r
o
r
þ
2
i
2
Pr
i
hroþ r
ðÞ
i
ð4:14Þ
where F and h are the longitudinal force and wall thickness, respectively.
Equations (4.11a)–(4.14) are also applied individually to each separate layer. If
the radii of the interface of the media and adventitia and the outer boundary of the
adventitia are denoted by r
stresses are computed according to Eqs. (4.13) and (4.14) using the respective radii
and wall thicknesse s. Similarly, the strain is computed with the respective circumference as given by Eqs. (4.11a, 4.11b). Finally, the wall thickness of each individual
layer is similarly determined by Eq. (4.12) with the appropriate radius and wall area.
Incremental Moduli The foregoing analysis is based on several assumptions. The
material of each layer of the vessel wall is assumed to be homogeneous, incompressible, orthotropic, and assumed to obey linear elasticity law with distinct moduli.
Therefore, the classical theory of thin-walled elastic shells is applicable to each
cylindrical layer. The major simplification is to ignore the radial stress, and radial
shear, so that each layer is treated as a two-dimensional shell.
and ra, respectively, the two components of the mean
m
Appendix 2: Formulation of Incremental Moduli (Lu et al., 2004) 217
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It is well known that a constitutive stress–strain relationship for an artery can be
reduced from a strain energy function, ρ
energy per unit volume of arterial wall. ρ
W, which represents stored deformation
o
denotes the density of the material in the
o
unstressed state and W is the strain energy per unit mass. Under the assumptions that
the arterial wall is homogeneous and pseudoelastic, the strain–strain relationship can
be expressed as follows:
∂ ρoWðÞ
S
¼
ij
∂E
ij
ð4:15Þ
where S
and Eijare components of second Piola–Kirchhoff stress and Green strain,
ij
respectively.
The incremental theory is developed under the assumption of linear elasticity. If a
small perturbation of stress and strain from a homeostatic in vivo state are considered
(defined by stress S
in which δS
ij
and strains Eij), then the perturbations may be written as:
ij
0
S
¼ S
þ δSij, Eij¼ E
ij
ij
0
þ δE
ij
ij
ð4:16Þ
and δEijare infinitesimal and quantiti es with a superscript “o” are
homeostatic values. On substituting Eq. (4.16) into Eq. (4.15), the following result
(after omitting higher order terms) was obtained:
δS
C
are the values of the second partial derivatives of ρ0W evaluated at the
ijkm
∂2ρ0W
¼
ij
∂E
km∂Eij
δEkm¼ C
ijkmδEkm
ð4:17Þ
homeostatic state. The summation convention is used such that a repetition of an
index in a single term means a summation over the range of the index, 1 (cir cumferential direction) and 2 (longitudinal direction). If E
state, then C
are constants in each layer. Equation (4.17) is a linear incremental
ijkm
o
are uniform at the homeostatic
ij
stress–strain relationship. The result can be written in the following form to introduce the definitions of the incremental elastic moduli:
¼ Y11δE11þ Y12δE22, δS22¼ Y21δE11þ Y22δE22, δS
δS
11
¼ 2GδE
12
12
ð4:18Þ
Y
and Y22are the classical incremental Young’s modulus in the circumferential and
11
longitudinal directions, respectively, G is the incremental shear modulus, Y
Y
have no equivalents in classical mechanics and have been denoted as cross-
21
modulus by Fung and Liu (1995). The existence of strain energy function requires
that Y
¼ Y21. These equations are Hookean but not isotropic. When Eqs. (4.15)–
12
(4.18) are applied to the intima-medial and adventitial layer of the blood vessel,
every symbol should have a superscript “im” and “ad,” respectively.
and
12
218 4 Constitutive Models of Coronary Vasculature
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Least Squares Method for Determination of Elastic Moduli Since the loading
does not involve shear, three elastic moduli must be determined: Y
, Y12, and Y22.A
11
least squares method is used to minimize the error between the theoretical stresses
given by Eq. (4.18) and the experimental measurem ents. The result is a 3 3 matrix
whose solution is the 3 1 matrix of elastic moduli:
2
A
0 A
11
4
0 A
22
A12A21A11þ A
3
243
12
A
21
Y
5
Y
Y
22
243
11
22
12
B
5
11
¼
5
B
22
B
12
ð4:19Þ
in which
B
11
B12¼
B22¼
X
n
¼
ΔS
n
X
ΔS
n
X
ΔS
n
n
ΔE
11
11
n
ΔE
22
n
ΔS
22
X
n
þ
11
n
22
n
ΔS
n
n
ΔE
11
22
X
A11¼
ΔE
n
A12¼ A21¼
X
A22¼
ΔE
n
n
11
X
n
11
n
ΔE
ΔE
ΔE
n
11
n
n
ΔE
11
22
n
22
where
n
n
0
n
n
0
n
n
0
n
n
ΔS
¼ S
S
,ΔS
¼ S
S
,ΔE
¼ E
E
,ΔE
11
11
11
22
22
22
11
11
11
¼ E
22
0
E
22
,n
22
¼ 0; 1; 2, ...
Obviously, the in vivo state is denoted by n ¼ 0 and consequen tly
ΔS
0
11
¼ ΔE
0
22
¼ Δ E
0
11
¼ Δ E
0
¼ 0. The symbol “δ” is eliminated for convenience
22
but it shoul d be recalled that the quantities of stress and strain are all defined in the
incremental equation (Eq. 4.18). The constants A and B are determined from the
n experiments and the matrix is solved for the three elastic moduli.
A Linear Comp osite Model Since the artery cannot be separated into two layers
without damag e to one of the layers, it is useful to have a model where the
incremental modulus of one of the layers can be computed from the moduli of the
other layer and the intact vessel. To develop a simple model, two springs are
considered in parallel representing the two layers as shown in Fig. 4.19. The total
tension, T, is equal to the sum of the tensions in each layer for the circumferential and
longitudinal directions, i.e.,
im
T
¼ T
11
ad
þ T
11
11
ð4:20aÞ
and
im
¼ T
T
22
ad
þ T
22
22
ð4:20bÞ
Appendix 2: Formulation of Incremental Moduli (Lu et al., 2004) 219
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For an incremental analysis, a linear stress–strain relation is assumed as given by
Eq. (4.18 ). If the cross-modulus is assumed to be significantly smaller than the
circumferential and longitudinal moduli, then Eq. (4.18) is reduced to:
T
11
S
11
¼ Y
¼
11E11
h
ð4:21aÞ
and
T
22
¼ Y
¼
S
22
22E22
h
ð4:21bÞ
Equations (4.21a) and (4.21b) can be substituted into Eqs. (4.20a, 4.20b) for each
of the layers to yield
im
im
ad
Y
11hE11
¼ Y
11
himE
þ Y
11
11
hadE
ad
11
ð4:22aÞ
and
im
im
ad
Y
22hE22
¼ Y
22
himE
þ Y
22
22
hadE
ad
22
ð4:22bÞ
Since the deformation or strain, E, is the same for each of the layers in the parallel
model, i.e., E ¼ E
im
¼ Ead, then:
Y
¼
11
im
h
Y
h
im
11
h
þ
h
ad
ad
Y
11
ð4:23aÞ
and
Fig. 4.19 Schematic of a two-layer linear model of coronary artery. Y, T, E represent modulus,
tension, and strain, respectively. Superscript “im” and “ad” denote intima-media and adventitia
layers, respectively. Reproduced from Lu et al. (2004) with permission
220 4 Constitutive Models of Coronary Vasculature
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Y22¼
im
h
h
Y
im
22
h
þ
h
ad
ad
Y
22
ð4:23bÞ
Hence, the composite modulus can be obtained from the moduli of the two
individual layers and their respective wall thicknesses. Equations (4.23a) and
(4.23b) correspond to the circumferential and longitudinal directions, respectively.
The incremental moduli are considered at the same level of stress in the vicinity of
in vivo loading conditions.
Table 4.2 Left anterior descending (LAD)—circumferential incremental moduli, Y
Mean stress: 45–48 kPa
LAD measured and calculated circumferential incremental moduli (Y
Measured intact Measured media Calculated adventitia
Heart No. 1 184 340 94.9
Heart No. 2 143 313 23.8
Heart No. 3 142 330 60.1
Heart No. 4 180 214 152
Heart No. 5 164 188 145
Mean SD 16322.9 29957.9 82.754.6
LAD measured and calculated circumferential incremental moduli (Y
Measured intact Calculated media Measured adventitia
Heart No. 6 224 261 193
Heart No. 7 180 199 157
Heart No. 8 225 329 137
Heart No. 9 105 123 84.1
Heart No. 10 144 188 90.3
Mean SD 176 51.7 220 77.9 132 45.6
Comparison of circumferential incremental moduli, Y
(kPa), for intact LAD artery and medial and
11
adventitial layers. Reproduced from Lu et al. (2004) with permission
11
)
11
)
11
Table 4.3 Left anterior descending (LAD)—axial incremental moduli, Y
Mean stress: 44–48 kPa
Heart No. 1 219 81.6 259
Heart No. 2 129 20.8 246
Heart No. 3 110 41.1 178
Heart No. 4 74.1 36.9 106
Heart No. 5 164 71.7 158
Mean SD 133 61.7 45.1 25.9 197 70.4
Heart No. 6 129 36.2 208
22
LAD measured and calculated axial incremental moduli (Y
)
22
Measured intact Measured media Calculated adventitia
LAD measured and calculated axial incremental moduli (Y
)
22
Measured intact Calculated media Measured adventitia
(continued)
Appendix 2: Formulation of Incremental Moduli (Lu et al., 2004) 221
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LAD measured and calculated axial incremental moduli (Y22)
Measured intact Calculated media Measured adventitia
Heart No. 7 62.4 42.9 91.2
Heart No. 8 69.9 55.4 87.3
Heart No. 9 118 123 113
Heart No. 10 109 34.3 190
Mean SD 97.5 29.6 58.4 37.1 138 56.6
Axial incremental moduli, Y
(kPa), for intact LAD and medial and adventitial layers. Reproduced
22
from Lu et al. (2004) with permission
Table 4.4 Right coronary artery (RCA)—circumferential incremental moduli, Y
11
Mean stress: 36–38 kPa
RCA measured and calculated circumferential incremental moduli (Y
Measured intact Measured media Calculated adventitia
Heart No. 1 119 206 63.5
Heart No. 2 142 209 82.3
Heart No. 3 147 282 35.8
Heart No. 4 158 225 82.9
Heart No. 5 174 202 146
Mean SD 148 20.6 226 33.1 81.9 40.8
Table 4.5 Right coronary artery (RCA)—axial incremental moduli, Y
22
Mean stress: 36–38 kPa
RCA calculated and measured axial incremental moduli (Y
)
22
Measured intact Measured media Calculated adventitia
Heart No.1 156 89.1 198
Heart No.2 106 104 110
Heart No.3 105 46.6 153
Heart No.4 106 99.1 113
Heart No.5 142 109 212
Mean SD 123 24.3 89.4 25.0 157 46.9
Axial incremental moduli, Y
(kPa), for intact right coronary artery (RCA) and its medial and
22
adventitial layers. Reproduced from Lu et al. (2004) with permission
)
11
Table 4.6 Left anterior descending (LAD)—cross incremental moduli, Y
Mean stress: 45–48 kPa
Heart No. 1 49.6 66.4
Heart No. 2 98.6 10.2
Heart No. 3 43.9 8.5
Heart No. 4 4.76 46.0
Heart No. 5 94.6 107
¼ Y
12
21
LAD cross incremental moduli (Y
12
¼ Y21)
Measured intact Measured media
(continued)
222 4 Constitutive Models of Coronary Vasculature
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Mean stress: 45–48 kPa
LAD cross incremental moduli (Y
12
¼ Y21)
Measured intact Measured media
Mean SD 58.3 39.0 47.6 41.3
LAD cross incremental moduli (Y
12
¼ Y21)
Measured intact Measured adventitia
Heart No. 6 150 183
Heart No. 7 23.5 68.4
Heart No. 8 86.2 72.5
Heart No. 9 35.0 120
Heart No. 10 99.6 74.3
Mean SD 78.8 51.2 104 49.3
Cross incremental moduli, Y
¼ Y21(kPa), for intact LAD and its medial and adventitial layers.
12
Reproduced from Lu et al. (2004) with permission
Table 4.7 Right coronary artery (RCA)—cross incremental moduli, Y
¼ Y
12
21
Mean stress: 36–38 kPa
RCA measured cross incremental moduli (Y
12
¼ Y21)
Measured intact Measured media
Heart No. 1 22.3 119
Heart No. 2 60.3 85.4
Heart No. 3 95.8 36.0
Heart No. 4 34.6 73.7
Heart No. 5 97.7 56.9
Mean SD 62.1 34.4 74.1 31.1
Cross incremental moduli, Y
¼ Y21(kPa), for intact right coronary artery (RCA) and its adventitial
12
layers. Reproduced from Lu et al. (2004) with permission
Appendix 3: 2D Strain Energy Function (Pandit et al., 2005)
A well-known approach to elasticity of bodies capabl e of finite deformation is to
postulate the form of an elastic potential or strain energy function (SEF; Green &
Adkins, 1960). Fo llowing the arguments made by Fung (1993), the following form
of strain energy function is used:
Q ¼ a
1
ρ
W
0
2
E
2
E
θθ
θθ
2
þ a
E
2
zz
expQ 1ðÞ ð4:24aÞ
¼
E
2
2
zz
þ 2a
4EθθEzz
E
E
θθ
zz
ð4:24bÞ
C
2ðÞ
Appendix 3: 2D Strain Energy Function (Pandit et al., 2005) 223
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where C, a1, a2, and a4are constants and starred quantities are Green strains
(Appendix 2) corresponding to a reference pair of stresses at the homeostatic state
(physiological pressure, 80 mmHg, and axial stretch, λ
“2” over ρ
(2)
W
signifies that this is a 2D approximation, treating the arterial wall as a
o
¼ 1.4). Here, the superscript
z
membrane and ignoring the radial stress. The symbol W represents the strain energy
per unit mass of the material and ρ
of stress (force/area); a
, a2, and a4are dimensionless constants. Although
l
is the mass density at zero stress. C has the units
o
Eqs. (4.24a, 4.24b) applies either to the loading or the unloading curve with different
set of constants, the former is the focus. The differentiation of the strain energy
equation leads to the stresses as:
∂ ρoWðÞ
S
¼
ij
, i; j ¼ θ; zðÞ ð4:25Þ
∂E
ij
In these formulas, (θ, z ) is a set of local right-handed cylindrical coordinates with
an origin lying on the neutral surface of the blood vessel wall, the axis θ pointing in
the circumferential direction and z in the axial direction. The strains are finite and
referred to the zero-stress state; E
, Ezzare normal strains, e
θθ
θz ¼ezθ
are shear strains
taken as zero because of the axisymmetric loading conditions. The subscripts i and
j range over 1, 2; with 1 referring to θ; 2 referring to z. When Eqs. (4.24a, 4.24b)–
(4.25) are applied to intima-media, every symbol should have a superscript (im).
Similarly, when the equations are applied to the adventitial layer every symbol
should have a superscript (ad). Note that the quadratic form of Q is written for two
dimensions in the spirit of the theory of thin shells in classical mechanics. The blood
vessel material is incompressible (Carew, Vaishnav, & Patel, 1968; Chuong & Fung,
1984). In two dimensions, however, it is not incompressible (Chuong & Fung,
1986).
Determination of Elastic Constants
If Eqs. (4.24a, 4.24b) and (4.25) are combined, a constitutive relation that relates the
circumferential and axial stresses to strains can be obtained as:
C
S
θθ
exp a
¼
2
2a
ðÞ
and
C
exp a
¼
S
zz
2
2a
ðÞ
2
2
E
E
1
θθ
þ 2a4E
1Eθθ
2
E
E
1
θθ
þ 2a4E
2Ezz
þ a
θθ
zz
2
þ a
θθ
θθ
2
2
E
E
2
zz
2
E
E
2
zz
þ 2a
zz
2
þ 2a
zz
4EθθEzz
4EθθEzz
E
E
E
θθ
zz
ð4:26aÞ
E
θθ
zz
ð4:26bÞ
224 4 Constitutive Models of Coronary Vasculature
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The goal of an algorithm to determine the material constants C, a1, a2, and a4is to
minimize the square of the difference between theoretical (proposed function,
Eqs. (4.26a, 4.26b)) and experimental values of circumferential, S
e
, and axial, S
θθ
stresses as:
Error
N
X
¼
n¼1
N
X
þ
n¼1
C
exp a
E
1
2
C
exp a
2
E
1
2
E
θθ
2
E
θθ
2
E
þ a
2
θθ
2
E
þ a
2
θθ
2
2
E
zz
zz
2
2
E
zz
zz
þ 2a
4EθθEzz
þ 2a
4EθθEzz
E
E
E
θθ
zz
E
θθ
zz
þ 2a4E
ðÞ
2a
1Eθθ
þ 2a4E
ðÞ
2a
2Ezz
e
S
zz
θθ
n
n
S
θθ
e
zz
n
ð4:27Þ
where N represents the total number of experimental points used to determine the
material constants of each curve. Two approaches are used to minimize the error
expressed by Eq. (4.27) as outlined below.
Marquardt-Levenberg Method
In this approach, the determination of the mat erial constants of the strain energy
function is carried out using Mathematica which utilizes the Marquardt-Levenberg
(M-L) method for nonlinear optimization. The optimized cost function is expressed
by Eq. (4.27). Initially, all four parameters (C, a
value of a
respective layer and the three parameters (C, a
is then fixed at the predetermined mean for the intact wall and each
4
1
, and a4) are evaluated. The
1,a2
and a2) are re-evaluated.
e
,
zz
2
n
2
Genetic Algorithm Method
The genetic algorithm (GA) is a stochastic global search that attempts to mimic
biological evolution. GA operates on a population of potential solutions applying the
principle of survival of the fittest to produce the best approximation to a solution. At
each generation, a new set of approximat ions is created by the process of selecting
individuals according to their level of fitness in the problem domain and breeding
them together using operators analogous to biological genetics. This process leads to
the evolution of populations of individuals that are better suited to their environment
than their parents, similar to natural selection. The objective is to minimize the cost
function in Eq. (4.27) based on a probabilistic rather than numerical approach
(Coley, 1999; Sverdlik & Lanir, 2002).
Briefly, a simple code is developed to implement the GA using the MATLA B
Genetic Algorithm Toolbox (The MATLAB Genetic Algorithm Toolbox, 1995). A
number of parameters are selected including the size of the population; probability of
crossover and mutation; scale for mutation and Tournament probability; and initial
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