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164 3 Mechanical Properties and Microstructure of the Coronary Vasculature
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Chapter 4
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Constitutive Models of Coronary
Vasculature
4.1 Introduction
The significance of mechanical stresses and strains in biology, physiology, and
pathology is well recognized. Although deformations or strains can be measured,
there is no instrument or method to measure stresses. Stresses must be calculated
from the constitutive equation, i.e., stress–strain relationship. The constitutive relation of the vessel wall is seminal to hemodynamics, wave propagation, distensibility
of arteries, plaque stability and rupture, as well as to vascular growth and
remodeling.
In this chapter, constitutive elastic and viscoelastic models are provided. Finite
strain micromechanics is introduced, and two homogenization methods are provided. Three types of microstructure-based models of soft tissues are compared in
relation to their assumptions and micromechanical structural basis, i.e., (1) uniformfield models with fluid-like matrix, (2) uniform-field models with solid-like matrix,
and (3) Second-order estimate models. Final ly, microstructural models of coronary
artery layers (media and adventitia) are highlighted.
4.2 Phenomenological Constitutive Models
4.2.1 Shear Modulus
On the epicardial surface of the heart, the large coronary arteries experience significant extensions and torsion in the axial direction due to changes in the shape and
size of the heart during the cardiac cycle (Dobrin, 1978; Pao, Lu, & Ritman, 1992;
Waldman, Fung, & Covell, 1985). Therefore, the study of the shear properties of
these vessels is important for understanding coronary physiology and pathophysiology. Lu, Yang, Zhao, Gregersen, and Kassab (2003) determined the
© Springer Science+Business Media, LLC, part of Springer Nature 2019
G. S. Kassab, Coronary Circulation, https://doi.org/10.1007/978-3-030-14819-5_4
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dependence of the shear modulus on the pressure, circumferential stress, longitudinal
stretch and stress. The intact coronary segment is mounted on the cannulae of the
triaxial machine as shown in Fig. 3.15 (Chap. 3). In the protocol of triaxial measurements (Chap. 3), the longitudinal stretch ratio (λ
) is varied from 1 to 1.4 in
z
increments of 0.1. The transmural pressure (P) is set at various pressures at each λ
At each λ
and pressure, a ramp of twist is performed from 0to 25, and 0to 25.
z
The direct measurements show that the shear stress is linearly related to the shear
strain, i.e., the shear modulus does not depend on the shear stress or strain (Lu et al.,
2003). Furthermore, the relation between torque and twist rate is linear at the
inflation and longitudinal extensions examined for the intact wall, media, and
adventitia. The dependence of the shear modulus on the circumferential strains,
however, is nonlinear. Table 4.1 (Appendix 1) summarizes the data on the linear
regressions between shear modulus and circumferential stress for the measured intact
left anterior descending (LAD) and right coronary artery (RCA) arteries and their
media. The shear modulus of the adventitia is found to be greater than that of the
intact vessel which is greater than that of the media. This relationship is expressed by
Eq. (4.7) (Appendix 1) which states that the product of the polar moment of inertia
and shear modulus of the intact vessel is equal to the sum of the products of the polar
moment of inertia and shear modulus of each of the two layers as validated by the
experimental data.
Although the coronary arterial vascular smooth muscle is reported as a helical
structure in the media (Rhodin, 1980), the direction of twist (clockwise versus
counterclockwise) is not found to be statistically significant for the range of pressures and longitudinal stretch ratios examined. Furthermore, there is no statistically
significant differences in the values of shear moduli between RCA and LAD artery at
λ
¼ 1.4 (physiological longitudinal stretch).
z
The above data on coronary torsion or medial–adventitial shear have important
implications for physiology and pathology of the coronary arterial wall. For example, the two layers are innervated by baroreceptors which are stimulated by deformation and perfused by small blood vessels called vasa vasorum. Shear forces at the
media–adventitia border may stimulate the mechanoreceptors by mechanical deformation as well as affect the vascular geometry of the vasa vasorum and hence the
blood perfusion of the vessel wall. Kwon et al. (1998) used micro-CT to study the
anatomy of the vasa vasorum in the porcine coronary arteries. They demonstrated
that the vasa vasorum originates from the lumen of the coronary artery and runs
longitudinally along the medial–adventitial border. Hence, if the transmural forces at
the border become abnormally high, they may collapse the vasa vasorum and cause
ischemia of the vessel wall which may lead to coronary artery disease or arteriosclerosis (Ritman & Lerman, 2007). Furthermore, significant differential torsional
stresses at the media–adventitia border during percutaneous coronary intervention
may lead to coronary artery dissection.
.
z
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