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4.2 Phenomenological Constitutive Models 175
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4.2.2 Incremental Moduli
In general, the stress–strain–history relationships of arteries are highly nonlinear
(Fung, 1993). A well-accepted approach to nonlinear elasticity uses the incremental
formulation. In this approach, a linearized relationship between the incremental
stresses and strains is obtained by subjecting the vessel to a small perturbation
about the in vivo condition. Using this approach, one can determine the relation
between stress and strain (and consequently incremental elastic modulus) within the
physiological regime. In conjunction with the incremental modulus, the measurement of strain under in vivo conditions will yield a value of stress. Hence, the
incremental approach allows the elucidation of the full mechanical status (stress,
strain, and incremental modulus) in the vicinity of the in vivo state. Furthermore, it
simplifies the characterization of the mechanical status of the vessel wall to a single
parameter in the respective direction (e.g., incremental moduli in the circumferential,
axial, and cross direction).
Lu, Pandit, and Kassab (2004) determined the incremental moduli of the coronary
arteries as a two-layer structure: (1) Intima-media layer (endothelial cells and
vascular smooth muscles, including elastin and some collagen); and (2) Adventitia
layer (collagen, fibroblasts, and elastin). The vessel wall is initially mechanically
tested intact and subsequently as intima-medial or adventitial layer. Two experiments are done for each layer which includes inflation and axial stretching. The
longitudinal stretch ratio (λ
transmural pressure (P) is varied from 110 to 163 cm H
H
O a t every λz(using a Ca+2-free Krebs solution to prevent vessel tone). The
2
incremental elastic moduli in the individual layers at in vivo (homeostatic) conditions are computed from the stress–strain relation and zero-stress state of the whole
tube and individual layers with the method of analysis presented in Appendix 2.A
simple biomechanical model is proposed to compute the incremental modulus of
adventitia from that of the intact vessel and media or that of the media from the intact
vessel and adventitia (Appendix 2).
The opening angles for the intact RCA and LAD artery are 140 30.3
134 35.5
, respectively. When the adventitia is dissected away, additional com-
pressive residual strain is relieved and the opening angle of the media for the RCA
and LAD artery increased to 210 38.6
media is dissected away, additional tensile residual strain is relieved as reflected by
the decrease of opening angle of the adventitia for the RCA and LAD artery to
98.1 36.5
and 108 35.8, respectively.
The data on the circumferential incremental moduli for the intact LAD artery and
medial and adventitial layers are summarized in Table 4.2 (Appendix 2) for each
individual animal. In five animals, the intact LAD artery and its media are measured
while the adventitia is calculated from Eqs. (4.23a , 4.23b) at a mean stress level of
45–48 kPa. In five additional animals, the intact LAD artery and adventitia layers are
directly measured while the media is simil arly computed. In order of increasing
moduli, it is found that media > intact > adventitia as seen in Table 4.2.
) is varied from 1.3 to 1.5 in increments of 0.05 and the
z
O in increments of 13.3 cm
2
and 198 36.5, respec tively. When the
and
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Furthermore, the computed value of adventitia in the first group of animals (top
portion of Table 4.2) is not statistically different than the measured adventitia in the
second group of animals (lower portion of Table 4.2). Similarly, there is no statistically significant difference between the computed and measured media. Table 4.3
(Appendix 2) shows equivalent data for the axial direction in the same vessels. In the
axial direction, the moduli are found to increase in the order of adventitia > intact >
media. Furthermore, the vessel is non-isotropic mechanically in that the circumferential and axial material properties are different. Interestingly, the intact vessel and
the media are both stiffer in the circumferential direction, while the adventitia is
stiffer in the longitudinal direction (although not statistically significant, p ¼ 0.075)
at the in vivo loading (45–48 kPa). The differences in the calculated and measured
values of longitudinal incremental moduli are also not statistically significant.
Hence, the linear model (Fig. 4.19, Appendix 2) provides a reasonable approximation for prediction of incremental moduli.
Tables 4.4 and 4.5 (Appendix 2) show the incremental moduli in the circumferential and axial direction, respectively, for the RCA. The conclusions are similar to
those of the LAD artery in relation to the relative stiffness of the two layers and the
intact vessel. Furthermore, there are no statistically significant differences between
the RCA and LAD artery for the intact vessel, media, and adventitia. The comparison of the LAD artery and RCA is made at different homeostatic range of stress
(LAD artery: 44–48 kPa; RCA: 36–38 kPa) for the same distension pressure and
longitudinal stretch. This is due to the differences in the diameter and wall thickness
of the two vessels. Hence, the coronary arterial wall must be considered as a
composite, anisotropic structure.
The incremental cross-moduli for the intact LAD artery and the intima-medial
and adventitial layers are shown in Table 4.6 (Appendix 2). For the intact segment,
the circumferenti al and longitudinal moduli are significantly larger than the crossmodulus. For the media, the circumferential moduli are significantly larger than the
cross-modulus alth ough there are no statistically significant differences between the
longitudinal and cross-modulus. The converse is true for the adventitia where the
circumferential modulus is not significantly different, while the longitudinal modulus is significantly larger than the cross-modulus. Table 4.7 (Appendix 2) shows
similar data on the cross-modulus for the RCA artery.
The above findings suggest that the media bears 62 13% of the circumferential
tension and the adventitia bears 38 13% of the circumferential tension. Under the
same loading, the media carries 24 15% of the longitudinal tension and adventitia
layers carry 76 15% of the longitudinal tension. These results support the notion
that most of the circumferential tension lies in the media under homeostatic conditions and predicts that the converse is true for the longitudinal direction. Under
increased loading, such as in hypertension or angioplasty, the moduli in the adventitia increase significantly and the burden of tension bearing may shift to the
adventitia.
4.2 Phenomenological Constitutive Models 177
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4.2.3 Strain Energy Function (SEF)
The constitutive equation of the blood vessel wall is a debated subject in earlier years
(see Review in Fung (1993)). Eventually, the following observation by Fung (1993)
is generally accepted: Soft tissues have a special kind of viscoelasticity where the
percentage loss of energy by hysteresis per cycle in a cyclic loading and unloading
process is only a few percent, and this percentage does not vary more than a factor of
two over a frequency range of five orders of magnitude. At any frequency, however,
it is not easy to predict the exact value of the loss per cycle. Fung (1993) and others
showed that this is consistent with a model of viscoelasticity with a continuous
relaxation spectrum. In cyclic loading and unloading, the stress–strain relationship is
unique in the loading stroke and is also unique in the unloading stroke (although the
two are somewhat different because of hysteresis). Fung (1993) called such a
material pseudoelastic, a term which is now widely used. In this framework, the
viscoelastic hysteresis behavior of soft tissue can be approximated within the more
tractable framework of elasticity, provided that the elastic properties are different for
loading and unloading.
A well-known approach to the study of elasticity of bodies capable of finite
deformation is to postulate the form of a strain energy function (SEF) (Green &
Adkins, 1960 ) as outlined in Appendix 3 (2D) and Appendix 4 (3D). The SEF relates
stress to strain in a hyperelastic material which arises from changes in internal energy
or entropy with loading. The partial derivatives of the SEF with respect to Green
strain components are related to the second Piola–Kirchhoff stresses. The determination of SEF is an inverse problem. A form of the SEF is assumed and based on
computed values of stress and strain from loading and deformation experiments; the
material properties are determined such that they provide good agreement between
the theorized form and the experimental data. This procedure involves nonlinear
least squares fit using the classical Marquardt-Levenberg (Fung, Fronek, & Patitucci,
1979; Marquardt, 1963) or genetic algorithm methods (Coley, 1999; Sverdlik &
Lanir, 2002). Genetic algorithms may be advantageous because they can determine
the global minimum in highly nonlinear problems with multiple local minima
(Pandit, Lu, Wang, & Kassab, 2005; Vigdergauz, 2001). The major advantages of
this approach are that it explores the solution space by testing parameter combinations simultaneously to avoid local minimum and does not require derivative
information (Goldberg, 1989).
4.2.3.1 2D and 3D SEF Fung Model
A 2D exponential strain energy function (SEF) is introduced by Fung et al. (1979)to
describe the highly nonlinear mechanical behavior of arteries, and later is generalized into a 3D form (Chuong & Fung, 1983), in which the formulation applied to
axisymmetric deformation of the vessel where the principal directions of the stress
and strain tensors coincide with the radial, circumferential, and axial directions, i.e.,
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assuming zero shear deformation. An extension of this model is proposed by Deng,
Tomioka, Debes, and Fung (1994) by adding a radial–circumferential strain term to
determine the respective shear parameter by axial torsion experiments. Humphrey
(1995) subsequently developed a more general form of Fung-type SEF for arbitrary
3D deformations. Other forms of hyperelastic constitutive models include the fourparameter logarithmic SEF of Takamizawa and Hayashi (1987), and the polynomial
SEFs developed by Vaishnav, Young, and Patel (1973) with three, seven, or even
twelve parameters.
Pandit et al. (2005) determined the material properties of a 2D SEF as outlined in
Appendix 3. The mean circumferential stress– strain relations averaged over all
experimental data for the intact wall, media, and adventitia are shown in
Figs. 4.1a, b, c, respectively, for the LAD artery at various axial stretch ratios. The
experimental stress–strain data from all hearts are fitted to Eqs. (4.24a, 4.24b) and
(4.25) (Appendix 3) using Marquardt-Levenberg (M-L) nonlinear least squares fit
and genetic algorithm (GA) method. The initial fit of the data involved the determination of four material elastic constants for the 2D Fung SEF, i.e., C, a
, a2, and a
1
(Appendix 3). Once a4is determined for the entire set of data (intact wall, media, or
adventitia), the mean value is fixed and the remaining constants are re-evaluated.
Table 4.8 (Appendix 3) summarizes the elastic constants for seven intact LAD
arteries for the genetic algorithm method. Also listed are values of circumferential
and axial strains at homeostatic values (80 mmHg, λ
¼ 1.4). These seven LAD
z
arteries had their adventitia dissected away and their media tested. The material
constants for those hearts are listed in Table 4.9 (Appendix 3). Five other LAD
vessels are tested for their adventitia once their media is dissected away and their
material constants are shown in Table 4.10 (Appendix 3 ).
The RCA of five hearts are tested first as a whole (Table 4.11, Appendix 3)
followed by dissection of adventitia and hence tested the media as shown in
Table 4.12 (Appendix 3). For the intact vessels, all constants are statistically similar.
A comparison is made between intact and media of RCA and LAD arteries with and
without a constitutive model. There are no statistically significant differences
between RCA and LAD artery for any of the material constants (C, a
, a2and a4)
1
for the intact wall or media. There are also no significant differences in the modelindependent stress–strain data for any axial stretch.
Wang, Garcia, Lu, Lanir, and Kassab (2006) generalized the 2D model to 3D and
derived the respective material properties based on inflation–extension experimental
data. The material constants for 10 RCA arteries are listed in Tables 4.13 and 4.14
(Appendix 4). Table 4.13 shows five intact vessels and their intima-media layers.
There are statistically significant differences between the intact wall and intimamedia layer. Table 4.14 summarizes the other five intact RCA vessels and their
adventitia layers. There are statistically significant differences between the intact
wall and adventitia layer. There are no statistical ly significant differences, however,
between the two groups of five hearts (Tables 4.13 and 4.14) for the intact RCA.
Similarly, the material constants for the 10 LAD arteries are listed in Tables 4.15
and 4.16 (Appendix 4). There are statistically significant differences between the
intact LAD artery and intima-media (T able 4.15), as wel l as between the intact LAD
4
4.2 Phenomenological Constitutive Models 179
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120
100
80
60
40
20
0
Circumferential Stress(Kpa)
0 0.2 0.4 0.6 0.8 1
Circumferential Strain
(A)
120
100
80
60
40
20
Circumferential Stress(Kpa)
120
100
80
λ=1.4
λ=1.3
λ=1.2
λ=1.1
0
0 0.2 0.4 0.6 0.8 1
Circumferential Strain
60
40
20
0
Circumferential Stress(Kpa)
0 0.2 0.4 0.6 0.8 1
(C)
λ=1.4
λ=1.3
λ=1.2
λ=1.1
Circumferential Strain
(B)
λ=1.4
λ=1.3
λ=1.2
λ=1.1
Fig. 4.1 Relationship between the circumferential Kirchhoff’s stress and Green’s strain for various
axial stretch ratios for (a) intact left anterior descending (LAD) artery, (b) media of LAD artery, and
(c) adventitia of LAD artery. Reproduced from Pandit et al. (2005) with permission
and adventitia (Table 4.16 ). The differences between the two groups of intact LAD
arteries in Tables 4.15 and 4.16, however, are not statistically significant.
Figure 4.2a shows the circumferential stress–strain curves of the intact RCA and
media layer of the RCA for the first five hearts (Appendix 4, Table 4.13) obtained
from the mean material constants. Similarly, Fig. 4.2b shows the circumferential
strain–stress curves of the intact RCA and adventitia layer of the RCA for the second
five hearts (Appendix 4, Table 4.14) obtained from the mean material constants. It is
apparent that the media is stiffer and will sustain most of the circumferential force
while the adventitia is softer in the normal physiological loading state (100 mmHg,
λ
¼ 1.4).
z
Figure 4.3a, b show a comparison of the strain–stress behavior of the predicted
material constants of RCA and LAD intima-media layer and those determined from
experimental data, respectively. The differences between theoretical predictions and
experimental data are not statistically significant (all p values >0.48). This analysis
demonstrates that the material constants of the dissected layer can be determined
based on the experimental data of the intact vessel and the intact layer which
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Fig. 4.2 Stress–strain
relation of the right coronary
artery (RCA) in the
circumferential direction.
(a) Solid and dotted lines
correspond to intact vessel
and intima-media layer,
respectively. (b) Solid and
dotted lines correspond to
intact vessel and adventitia
layer, respectively. Data
corresponding to axial
stretch ratios of 1.4 and 1.3
are shown. Reproduced
from Wang et al. (2006)
with permission
150
120
90
60
30
Circumferential Stress (kPa)
0
Circumferential Strain
(A)
150
120
90
60
30
Circumferential Stress (kPa)
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
Circumferential Strain
λ=1.4
λ=1.3
λ=1.4
λ=1.3
λ=1.4
λ=1.3
λ=1.4
λ=1.3
validates the feasibility of the proposed two-layer model and the reliability of the
material constants.
4.2.4 Bilinear Model: Generalized Hooke’s Law
As demonstrated above, the stress–strain relation of blood vessels is highly
nonlinear, and there is strong coupling between the circumferential and axial directions. A good constitutive model is expected to describe these mechanical behaviors
with good fits to the experimental data and to have stable parameters, i.e., variations
between material parameters are relatively small for a group of mechanically similar
data. The logarithmic form has a limited ability to describe this anisotropic behavior
of blood vessels and may generate “physically unrealistic predictions” (Humphrey,
1999). The polynomial form is less capable of distinguishing differences between
various data sets since it has larger variations in its material constants compared to
(B)
4.2 Phenomenological Constitutive Models 181
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250
Experimental
200
150
100
50
Circumferential Stress (kPa)
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
Predicted
Circumferential Strain
(A)
200
Experimental
150
100
Predicted
50
Circumferential Stress (kPa)
0
0 0.2 0.4 0.6 0.8
Circumferential Strain
(B)
Fig. 4.3 Comparison between predicted and experimental strain–stress curves for the media layers
of (a) RCA and (b) LAD (axial stretch ratio of 1.4 for both RCA and LAD). Reproduced from Wang
et al. (2006) with permission
the exponential form (Fung et al., 1979). Compared to the logarithmic and polynomial forms, Fung’s exponential model captures the anisotropic behavior of blood
vessels relatively well with seven parameters (one linear parameter and six nonlinear
parameters). The fit can still be inadequate, however, when the data include multiple
loading curves from different axial stretch ratios. Generally, the curve fitting for
nonlinear models is difficult because of possible multiple minimum locales of the
strain energy. An additional issue is the oversensitivity of material constants, i.e.,
large variations of material constants can still be deduced for mechanically similar
blood vessels. This sensitivity to the small change in experimental data is due to the
non-uniqueness of the nonlinear optimization (Fung, 1993). As a result, some model
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parameters may be arbitrarily chosen. Finally, the empirical material constants lack
clear physical meanings.
To address some of these shortcomings, linearized 2D and 3D stress–strain
relations (generalized Hooke’s law) for coronary arteries have been proposed
where the stress is linearly dependent on a new strain tensor that is a nonlinear
function of stretch ratios (Zhang & Kassab, 2007). In an inflation–stretch test
without shear deformation, there are seven material constants. One is used in the
definition of strains (parameter n) and it represents the nonlinearity of the material,
and the other six constants are linear parameters which can be interpreted as elastic
constants with respect to the new strain measure (Appendices 5 and 6 for 2D and 3D
cases, respectively). The number of material constants is comparable to other
popular models where there are six nonlinear parameters and one linear parameter
in Fung’s exponential model (Chuong & Fung, 1983; Fung et al., 1979), and there
are two linear parameters and three nonlinear parameters in the bi-phasic model
(Holzapfel, Sommer, Gasser, & Regitnig, 2005).
The key premise to the generalized Hooke’s law is that the nonlinearity parameter
n is introduced in strain in order to linearize the Fung SEF, such that an optimal
n must be determined that makes the linear model best represent the “experimental
data.” Specifically, it is required that n results in a minimum relative least squares
error, RLSE (Appendix 6). When the parameter n is systematically varied, the value
of RLSE in Eq. (4.71) (Appendix 6) is examined for a minimum.
Tables 4.17 and 4.18 (Appendix 6) summarize the material parameters for the
generalized Hooke’s model (linear), exponential, and bi-phasic models for the RCA
and LAD porcine coronary arteries, respectively. When compared with exponential
and bi-phasic models, RMS errors are largely smaller for the linearized model. A
sensitivity analysis is done to evaluate the stability of material constants. For a set of
LAD data, the zero-stress inner circumference C
and outer circumference Coare
i
both hypothetically varied from 80 to 120% of their measured values in 10% steps to
simulate softer to stiffer vessel, respectively, as compared to actual data. The
generalized Hooke’s model (linear), the exponential model, and the bi-phasic
model are used, and the results are listed in Table 4.19 (Appendix 6). When
compared to the exponential model and the bi-phasic model, the material constants
of the generalized Hooke’s model have smaller coefficients of variation, i.e., the ratio
of the standard deviation to the mean. Therefore, the generalized Hooke’s model
may be more suitable to detect the changes in the blood vessels due to growth or
remodeling. Since the model results in less variability, when two groups are compared, small variations create less overlap. Therefore, the small changes are more
likely to be statistically significant. The goodness-of-fit is maintained throughout all
data sets with the new generalized Hooke’s model. In the exponential model, the
RMS errors of the internal pressure are increased with perturbations of data. For the
bi-phasic model, the RMS errors are significantly affected when the lengths are
elongated to 120% of the original.
It is easier to experimentally determine the material constants in a linearized
generalized Hooke’ model. The nonlinearity parameter n can be first estimated from
a single stress–strain curve. Then all the model parameters are determined by the
4.2 Phenomenological Constitutive Models 183
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least squares method for a value of n that minimizes the least squares error. Once a
criterion is reached, (e.g., the minimum of RLSE from Appendix 5 for 2D case, or
Eq. (4.71) from Appendix 6 for 3D case), the material constants are obtained.
Furthermore, the curve fitting process with biaxial test data for the linear model is
expected to be more reliable than for the highly nonlinear models. Finally, parameters in the linearized model have clearer physical meanings, i.e., n is nonlinearity of
strain, other constan ts are elastic moduli (expected to be positive in simple tests) with
respect to the strain defined by n. The variation of the moduli with direction, for
instance, the difference between c
and c22(e.g., see Appendix 5)reflects the
11
anisotropy of the material.
4.2.5 Shear Modulus
Since the shear modulus depends on the loading state (e.g., internal pressure and axial
stretch, as shown above), it may be complex to use and difficult to interpret. The
actual force or deformation under which the shear modulus is measured must be
known, and this measure can only apply at a given loading condition. As described
above, Lu et al. (2003) found a linear correlation between the torsional shear modulus
of porcine coronary artery and the circumferential or axial stress. This result may not
be convenient to use in mechanical modeling since the stress must be first computed
from the constitutive law for a given deformed state, and then the shear modulus can
be calculated. An explicit expression between the modulus and the stretch ratios
(or strains which are directly measurable), however, has not been attempted.
Zhang, Lu, and Kassab (2007) formulated a torsional shear modulus in terms of
the new strain measure used in the generalized Hooke’s law (see Appendix 7 for
formulation). In general, the linearized shear modulus c
viewed as a material constant, in contrast with the classical incremental shear
modulus G that depends on the deformation. In the generalized Hooke’s law
(Eq. (4.81), Appendix 7), the nonlinearity parameter n can be determined from
inflation–stretch experiments without shear deformation. The shear moduli can then
be obtained from torsion and other mechanical tests invoking shear deformation. This
decoupling makes the parameter determination process easier and more
deterministic.
(Appendix 7) can be
66
4.2.6 Incompressibility Condition
The incompressibility condition for the vessel wall can be invoked in the
logarithmic-exponential strain-based generalized Hooke’s law to reduce the number
of material parameters (Liu, Zhang, Wang, & Kassab, 2011). The existence of three
intrinsic relations between the elastic constants reduces the number of model parameters from ten to seven. The basic premise is to employ a constitutive model that
results in simple relation between model parameters when the incompressibility
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assumption is enforced (Appendix 8). The fitted material parameters are listed in
Table 4.20 (Appendix 8) for five pairs of experimentally measured LAD intact
vessels and the respective media layers. The outer radius and internal pressure of
the intact vessel and media from experimental measurements and theoretical predictions that are obtained by solving boundary value problems (Appendix 8). The
agreement between theory and experiment is excellent (Liu, Zhang, et al., 2011).
Liu, Zhang, et al. (2011) found the media layer has a larger nonlinearity parameter
n than the corresponding intact vessel, i.e., 1.93 0.38 for the media compared to
1.44 0.18 for the intact vessel. This implies that the load in the media will increase
faster than that in the overall vessel in the tested range. The Young’s moduli E
and E
are all found to be positive. The fitting errors defined in Eq. (4.106)of
r
, Ez,
θ
Appendix 8 are generally less than 5% which is excellent given the variance of
biological data. For both intact vessel and the media layer, the Poisson’s ratio v
v
are positive, while vθzis negative for all specimens (Table 4.20, Appendix 8)in
zr
and
θr
the range of 0.1 to 0.25. This negative Poisson’s ratio is consistent with experimental observation that the vessel tends to stretch axially with an increase of
pressure (T
dix 8), the shear parameter G
higher variation among the vessels in comparison to the Young’s moduli. A comparison is also made between the measured torque M
> 0) and can buckle at high pressure. As given in Table 4.20 (Appen-
θθ
is fitted with slightly higher error and the value show
θz
e
and prediction Mcas function
of Jα/L (Lu et al., 2003), where J is the polar moment of inertia of the deformed
vessel, α is the torsion angle (in rad), and L is the deformed length. Under various
axial stretch and pressure loading combinations, the general trend of the experimental data is well predicted.
The major advantage of the generalized Hooke’s law model lies in the fact that it
requires fewer independent parameters (only four without shear moduli). The volumetric incompressibility constraint is used to eliminate three redundant elastic
constants, which typically cannot be done in a nonlinear constitutive law due to
mathematical complexity. For example, Fung ’ s exponent ial model of blood vessels
requires eight material parameters to describe the triaxial torsion stress–strain relation, compared to four in the generalized Hooke’s model.
A limitation of the generalized Hooke’s model is that unconventional stress and
strain measures are used which may require additional computations to convert these
measures to convent ional ones, or vice versa. Furthermore, a strain energy
(or hype relastic potential) may not exist for this type of stress–strain relationship,
i.e., the hyperelasticity assumption is discarded. Regardless, a nominal strain potential can be utilized to conveniently express the constitutive law in a scalar form
(Zhang, Lu, et al., 2007). The lack of an explicit form for the strain energy may be a
limitation for application in computational methods that are developed based on
energy principles. On the other hand, many finite element computational software
have been derived based on the principle of virtual work that only requires a usersupplied stress–strain relation and the tangent stiffness tensor, which are all provided
for the generalized Hooke’s model. For example, the proposed model has been
embedded into ANSYS and FEAP and conducted simulation of angioplasty in LAD
artery as described in Chap. 7 (Liu, Zhao, Zhang, Wang, & Kassab, 2011). The lack
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