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4.2 Phenomenological Constitutive Models 185
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of energy form may also make it difficult to investigate the material stability of the
model, such as polyconvexit y or rank-one convexity. On the other hand, the model
has been found to be stable in finite element simulations (Liu, Zhao, et al., 2011)
where the arteries deform within the typical in vivo range. In summary, due to the
need for fewer independent parameters, greater ease of interpretation, the stability of
the model in finite element situations, and that this model has been embedded in
certain simulation tools, the generalized Hooke’s model may be a simpler alternative
to existing models of coronary arteries.
4.2.7 Linear Viscoelasticity and Maxwell’s Model
4.2.7.1 Artery
It is well known that the mechanical properties of blood vessels exhibit viscoelastic
properties such as creep (time-dependent increase in length while the force is
constant), relaxation (time-dependent decrease in force while the length is constant),
and hysteresis (differences in the loading and unloading curves) (Fung, 1993). The
viscoelastic mechanical behavior is relatively insensitive to strain rate within several
decades of time (Fung, 1993). The classical linear viscoelastic models (i.e., Maxwell, Voigt, or Kelvin, see de finitions below) have a single characteristic frequency
(Fung, 1993), and hence are unable to account for the rate-insensitive feature of
biological tissues. Moreover, stress–strain relationships of soft tissues are usually
highly nonlinear and strongly anisotropic (Fung, 1993). To this end, the quasi-linear
viscoelastic theory developed by Fung (1993) has been extensively used to describe
biological constitutive behaviors.
In the quasi-linear viscoelastic formulation, the dependence of stress on time
(loading history) and strain (total deformation) is separated, and the stre ss relaxation
response (dependence on time) is linear (Fung, 1993). The stress–strain relationship,
however, remains nonlinear. Since the stress–strain relationship of coronary arteries
in the full range of elasticity can be linearized by defining a new strain measure to
absorb the nonlinearity, the stress in quasi-linear viscoelastic materials can be
linearly dependent on both time and strain. In this regard, a linear viscoelastic
model that is insensitive to the loading frequency can sufficiently capture the
constitutive behavior of biological soft tissues.
There are three kinds of classical linear viscoelastic models: (1) Maxwell model
which is composed of a linear spring and a linear viscous dashpot connected in series;
(2) Voigt model is formed by a line ar spring and a linear dashpot in parallel; and
(3) Kelvin model (standard linear solid) consists of a linear spring in parallel with a
Maxwell body (Fung, 1993). A common feature of these models is that they have a
single relaxation time (or characteristic frequency) and their hysteresis loops notably
depend on loading rate. To capture the rate-insensitive hysteresis behavior of soft
tissues, continuous spectrum relaxation functions must be considered (Fung, 1993).
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In principle, many of the classic linear viscoelastic models with various characteristic frequencies can be used to approximate the continuous spectrum functions of
blood vessels. The parameter determination, howe ver, can be problematic because of
the increased number of material constants. On the other hand, a continuous spectrum function may need fewer parameters albeit it is usually hard to obtain closedform solutions. Zhang, Chen, and Kassab (2007) considered a generalized Maxwell
model with a finite number of characteristic frequencies that form a geometric series
and cover multiple time scales (Appendix “Coronary Arteries (Zhang, Chen, et al.,
2007)”). As a result, the rate-insensitive feature of biological tissues can be captured
with significantly reduced number of model parameters as outlined below.
4.2.8 Opening Angle
The generalized Maxwell model can also be used to describe the viscoelasticity of
the opening angle as described in Sect. 3.4 where Rehal, Guo, Lu, and Kassab (2006)
showed that the opening angle decreases with time if the artery is cut from the loaded
state, while it increases if the cut is made from the no-load state due to viscoelasticity
(Fig. 3.13). In both cases, the opening angle approaches the same value in 3 h which
implies that the characteristic relaxation time is about 10,000 s. The creep function of
a generalized Maxwell model (a spring in series with six Voigt bodies) is used to
predict the temporal change of opening angle in multiple time scales (Appendix
“Opening Angle Zhang, Guo, & Kassab, 2008”). It is demonstrated that the generalized Maxwell model captures the salient features of the experimental results
(Zhang et al., 2008).
4.2.9 Active Properties
Although some multi-axial active models have been proposed, they have been of
theoretical forms not rooted in experimental measurements (Stålhand, Klarbring, &
Holzapfel, 2011; Zulliger, Rachev, & Stergiopulos, 2004). Huo, Cheng, Lu, Liu, and
Kassab (2012) provided experimental data on the passive and active properties of
both intact coronary artery wall and the intima-media layer. Figure 4.4a, b show the
data for the circumferential stress as a function of the circumferential stretch ratios at
axial stretches of 1.3 and 1.5, respectively. Figure 4.5a, b show the axial stress as a
function of circumferential stretch ratios at the same axial stretches in Fig. 4.4a, b.
The biomechanical formulation for the determination of active material properties of
coronary arteries is outlined in Appendix 10. Table 4.21 (Appendix 10) summarizes
the material constants of the passive energy function (from Eqs. (4.154a, 4.154b),
Appendix 10) while Table 4.22 lists material constants of the K
strain energy function (from Eq. (4.157), Appendix 10) for six RCAs. At axial
stretch ratio of 1.3 and pressure of 80 mmHg, the mean circumferential stress and
+
-induced active
4.2 Phenomenological Constitutive Models 187
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(kPa)
θθ
T
λ
θ
(A)
(kPa)
θθ
T
Fig. 4.4 Circumferential first Piola–Kirchhoff stress (Tθθ) as a function of circumferential stretch
)atλzof (a) 1.3 and (b) 1.5 measured at K+-induced vasoconstriction and Ca2+-free-induced
ratio (λ
θ
vasodilation. Total Stress at vasoconstriction (square marker); Passive Stress at vasodilation
λ
(B)
θ
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strain (averaged through the wall) are 40.3 kPa and 0.80 in passive state and 35.9 kPa
and 0.58 in active state, respectively. Given the decreased inner diameter and
increased wall thickness, the isobaric constriction significantly decreased mean
circumferential stress and strain.
Results from Huo et al. (2012) illustrate that as the transmural pressure increases
in the physiological pressure range, the active stress–strain relationship resembles a
Gaussian (normal) distribution, which has a maximal value when the passi ve and
active pressures are in the range of 80–100 mmHg and 140–160 mmHg, respectively, as shown in Figs. 4.4 and 4.5. This relationship is consistent with the sliding
actin–myosin molecules, where the number of active cross-bridges is proportional to
the active stress (Fung, 1993; Mulvany & Warshaw, 1979). Although vascular
smooth muscle cell (SMC) contraction shows an in vitro optimal point at the passive
pressure of 80–100 mmHg and at the active pressure of 140–160 mmHg, the in vivo
optimal point for SMC contraction in coronary arteries needs further study.
Huo, Zhao, Che ng, Lu, and Kassab (2013) extended the measurements and
analysis to a two-layer model where the intima-media layer is mechanically tested
under both passive and active conditions using a method similar to that described
above. Figure 4.6a, b show the theoretical a nd experi mental data for the circumferential stress as a function of circumferential stretch ratios at axial stretch ratio of 1.2
and 1.3, respectively. Similarly, Fig. 4.7a, b show the axial stress as a function of
circumferential stretch ratio at axial stretch ratio of 1.2 and 1.3 in correspondence
with Figs. 4.6a, b, respectively. Table 4.23 summarizes the passive material constants while Table 4.24 lists the material constants for six RCA intima-media layers.
The radius, circumferential length, and opening angle are increased in the intimamedia layer but decreased in adventitia layer in zero-stress state in comparison with
those of an intact vessel wall. Furthermore, the active stress–stretch relationship
(intima-media layer) shows less curvature and lower stress than that in the intact
vessel. The implications of these findings on circumferential stress and stretch will
be examined in Chap. 8.
4.3 Microstructure-Based Constitutive Models
The idea of relating the macroscopic mechanical properties of arteries to the arterial
microstructures, including elastin and collagen fibers and cells, is first demonstrated
by Burton and Yamada (1951). Roach and Burton ( 1957) made a quantitative study
by differential digestion of elastin or collagen and measured the mechanical properties of the partially digested artery. Oka (1972) proposed a theoretical analysis of the
arterial wall, culminating in several well-known papers (Azuma & Hasegawa, 1971;
Fig. 4.4 (continued) (triangle marker); and Active Stress (thin solid line with standard error bars of
SE value). Thick dotted line (theoretical total stress), dashed line (theoretical passive stress), and
solid line (theoretical active stress). Reproduced from Huo et al. (2012) with permission
4.3 Microstructure-Based Constitutive Models 189
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(kPa)
zz
T
λ
θ
(A)
(kPa)
zz
T
Fig. 4.5 Axial first Piola–Kirchhoff stress (Tzz) as a function of circumferential stretch ratio (λθ)at
of (a) 1.3 and (b) 1.5 measured at K+-induced vasoconstriction and Ca2+-free-induced vasodi-
λ
z
lation. Total Stress at vasoconstriction (square marker), Passive Stress at vasodilation (triangle
λ
(B)
θ
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Azuma & Oka, 1971; Oka & Azuma, 1970). Azuma and Hasegawa (1971) discussed
the rheological properties of arteries and veins in terms of the networks of collagen,
elastin, and smooth muscle cells. Their works showed that the mechanical properties
of the vessels are intimately associated with microstructural components such as
elastin and collagen fibers, cells, and ground substance. Based on this idea, much
effort has been made to derive the constitutive model of the soft tissue from the
geometry, distribution, and the mechanical properties of the individual
microstructures.
The determination of microstructural stress requires a constitutive model based on
the ultrastructure and the corresponding material properties. Compl ex microstructure
and strong nonlinear mechanical behaviors of soft tissue, however, present significant challenges in const itutive modeling. Despite the complexity, the current trend in
biomechanics is to move from phenomenological to microstructural models to
predict the overall nonlinear and micromechanical responses of inhomogeneous
soft tissues. New developments in imaging and biochemistry will continue to
provide more details of the constitutive properties of soft tissues and continue to
advance the development of structure-based models.
Based on standard approximations in nonlinear mechanics, the microstructural
models can be classified into three categories: (1) Uniform-field models with solidlike matrix (Holzapfel, Gasser, & Ogden, 2000), (2) Uniform-field models with
fluid-like matrix (Hollander, Durban, Lu, Kassab, & Lanir, 2011b; Holzapfel et al.,
2000), and (3) Second-order estimate (SOE) models (Ponte Castañeda, 1996; Ponte
Castañeda & Willis, 1999). The first two categories assume an affine deformation
field where the deformation of microstructure is the same as that of the tissue,
regardless of material heterogeneities, i.e., uniform-field (UF) model which represents the upper bounds of the effective strain energy and stress of soft tissues. In
addition, the first type is structurally idealized and not necessarily consistent with
ultrastructure and hence cannot accurately predict the microscopic mechanical
behaviors of soft tissues. The third category considers realistic geometrical features,
material properties of microstructure, and interactions among them and allows for
flexible deformation in each constituent. The UF model with fluid-like matrix and
the SOE model are microstructure-based and can be applied to different tissues based
on microstructural features as described in Appendix 11 and below.
The two major classes of UF micromechanical models are described in Appendix
11. Briefly, the first class of UF models proposed by Lanir (1979, 1983) considers
the tissue as a composite of elastin and collagen fibers embedded in a fluid-like
matrix. Thus, the fibers are the only constituent phases that sustain non-hydrostatic
loading such as tension and shear, while the contribution of the fluid-like matrix is a
hydrostatic pressure. The UF assumption leads to a simplification that all the
microstructures deform identically to the macroscopic deformation of the tissue
Fig. 4.5 (continued) marker), and Active Stress (thin solid line with error bars of SE value). Thick
dotted line (theoretical total stress), dashed line (theoretical passive stress), and solid lines (theoretical active stress). Reproduced from Huo et al. (2012) with permission
4.3 Microstructure-Based Constitutive Models 191
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(kPa)
θθ
T
λ
θ
(A)
(kPa)
θθ
T
Fig. 4.6 Circumferential first Piola–Kirchhoff stress (Tθθ) as a function of circumferential stretch
)atλzof (a) 1.2 and (b) 1.3 measured at K+-induced contraction and Ca2+-free-induced
ratio (λ
θ
vasodilation for intima-media. Total Stress at contraction (square marker), Passive Stress at
λ
(B)
θ
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since no fiber interactions are considered, such that the macroscopic SEF is the
volumetric sum of the individual fibers’ SEF. On the basis of this assumption and
thermodynamic consideration, Lanir developed a general multi-axial theory for the
constitutive relations in fibrous connective tissues Lanir (1979, 1983). Similarly,
Decraemer, Maes, and Vanhuyse (1980) proposed a parallel wavy fibers model for
soft biological tissues in uniaxial tension, followed by Wuyts et al. (1995). More
recent developments of fluid-l ike matrix-based models can be referred to in Humphrey and Yin (1987), Dahl et al. (2008), and Lokshin and Lanir (2009a).
A second class of UF micromechanical models assume the tissue as a collagen
fiber-reinforced composite, whose matrix is a solid-like material that can take up
loading. This assumption is motivated by the fact that the elastin, which is part of the
matrix, becomes straightened and starts to take the load in the early deformation of
the tissue. For example, the experimental study of Gundiah, Ratcliffe, and Pruitt
(2007) suggested that the elastin is described with a neo-H ookean constitutive
model. Based on this solid-like matrix assumption, Holzapfel and Weizsäcker
(1998) and Holzapfel et al. (2000) modeled the arterial wall as a two-layer fiberreinforced composite where the macroscopic SEF of soft tissue stems from two
sources: (1) An isotropic part associated with the mechanical response of the
non-collagenous matrix material (elastin fibers, cells, and ground substance), and
(2) an anisotropic part due to the deformation of two classes of collagen fibers
symmetrically disposed with respect to the axis of the vessel. Successive developments of this model can be found in Zulliger, Fridez, Hayashi, and Stergiopulos
(2004), Kroon and Holzapfel (2008), and Li and Robertson (2009). Specifically,
Zulliger, Fridez, et al. (2004) made further refinement to account for the distribution
of the waviness of collagen fibers and different SEF of the matrix and collagen fibers.
Mechanical predictions of these models are more accurate than those of phenomenological models as they account for heterogeneity of material properties and
geometrical features of vessel components. These models, however, cannot accurately predict microenvironments of the vessel, i.e., strain and stress of individual
fiber or cell since they all assume affine deformation in tissue, i.e., the deformati on of
the collagen fibers and the matrix are identical to the macroscopic deformation of the
tissue, and the microstructure is not based on histological measurements in arteries
(Chen, Liu, Zhao, Lanir, & Kassab, 2011).
UF models assume the deformation of all fibers to be the same and do not account
for the heterogeneity of the fibers or fiber–fiber interactions. To account for the
complexity of the microstructural components of the blood vessel, Chen, Liu, Zhao,
et al. (
2011) developed a finite strain homogenization approach based on the second-
order estimate (SOE) theory (Appendix 11) to predict the macroscopic stress –strain
relation and microstructural deformation of vascular tissue. This micromechanical
model considers measured histological geometrical features and material properties
Fig. 4.6 (continued) vasodilation (triangle marker), and Active Stress (thin solid line with error bars
of SE value). Thick dotted line (theoretical total stress), dashed line (theoretical passive stress), and
solid line (theoretical active stress). Reproduced from Huo et al. (2013) with permission
4.3 Microstructure-Based Constitutive Models 193
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of vessel constituents and allows flexible deformation in each component. A comparison of the various model predictions of macro- and microscopic mechanical
behavior of vascular tissue is presented below.
4.3.1 Comparison of Microstructural Models
Both the uniform-field (UF) approaches (solid-like matrix and fluid-like matrix) and
tangent second-order (SOE) micromechanics model are applied to 2D material with
distributed collagen fibers, respectively, in comparison with finite element
(FE) simulations (Chen, Liu, Zhao, et al., 2011). The planar tissue stretches along
the principal direction of the fibers but shrinks in the transverse direction to maintain
material incompressibility. The material property of the matrix is directly taken from
Holzapfel et al. (2000). To simplify the simulations, all the fibers are assumed to be
isotropically distributed in space and have the same under-formed cross-sectional
geometry. The fiber orientation is assumed to follow a beta distribution (Sacks,
2003), and the waviness is also assumed to be beta distributed but direction depen-
dent in line with tissue ultrastructure (Brown, 1973; Sverdlik & Lanir, 2002). The
orientation distribution is discretized into 13 regions and the waviness is discretized
into 5 regions, so that the collagen fibers are categorized into 65 phases according to
their orientation and waviness. The total volume fraction of collagen fibers is taken
as 20% and that of the matrix is 80%. The volume fraction of every fiber phase is
determined by the cumulative probabilities of the orientation and waviness. FE
simulations are used for the purpose of model validation. The FE model contained
~200 randomly distributed fibers whose orientation angle and waviness are randomly assigned according to a beta distribution.
The model predictions of the macroscopic SEF and the tensile Cauchy stress of
fibrous tissue are plotted in Fig. 4.8a, b, in comparison with FE computational
results. At low stretch level
been straightened and deform the same as the matrix so that effective macroscopic
SEF and Cauchy stress of tissue predicted by UF and SOE models are very similar
and approximate to that of pure matrix material. When the macroscopic deformation
λ is >1.6, predictions of the two models become different. The UF predictions of
SEF and Cauchy stress increase rapidly with higher macroscopic stretch ratio while
SOE results increase slowly and predicted Cauchy stress is consistent with FE
simulation as shown in Fig. 4.8b. At this loading level, collagen fibers become
straighten completely to take up loads and deform less than the matrix. The UF
model, however, overestimates the deformation field of stiffer fibers (assuming it is
identical to that of matrix) and hence approaches upper bounds of the exact SEF and
Cauchy stress. The SOE model statistically accounts for the heterogeneous deformation of the matrix and fibers, which is due to the heterogeneities of microstructural
geometries and material properties as well as matrix–fiber and fiber–fiber interactions. Thus, the deformation field employed by SOE is more realistic and leads to
lower estimates of the macroscopic SEF and stress than the UF upper bounds.
λ < 1:6, most fibers are still undulated or have just
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(kPa)
zz
T
λ
θ
(A)
(kPa)
zz
T
Fig. 4.7 Axial first Piola–Kirchhoff stress (Tzz) as a function of circumferential stretch ratio (λθ)at
of (a) 1.2 and (b) 1.3 measured at K+-induced contraction and Ca2+-free-induced vasodilation in
λ
z
the intima-media layer. Total Stress at contraction (square marker), Passive Stress at vasodilation
λ
(B)
θ
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