Constitutive Models of Coronary Vasculature

Abstract

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.

Appendices

Appendix 1: Analysis of Shear Modulus (Lu et al., 2003)

The coronary artery is assumed to be a two-layer cylinder (Fig. 4.18). Since the intima is relatively thin in the normal swine artery, the intima-media is considered as the inner layer and the adventitia as the outer layer. Initially, the arterial wall is assumed to be a homogenous material so that an “apparent” shear modulus could be obtained. The artery is subjected to a transmural pressure (P_i), a longitudinal force (F_λ), and a torque (T). The radii of the inner boundary of the intima, the interface of the media and adventitia, and the outer boundary of the adventitia are denoted by r_i, r_m, and r_a, respectively, as shown in Fig. 4.18. A vessel segment of length L undergoing a twist angle θ and an angle of twist per unit length (θ/L) can be considered. The average shear stress in the vessel wall is σ_zθ for the intact artery. It is found that the T is linearly proportional to θ/L in the range of interest. Hence, σ_zθ is linearly proportional to θ/L in this range.

Fig. 4.18

Two-layer model of coronary artery that is exposed to transmural pressure (P), axial tension force (F_λ), and twist torque (T). Reproduced from Lu et al. (2003) by permission

Since the shear strain (e_zθ) is, by definition, ${e}_{z\theta}=\frac{r\;\theta }{2L}$, the linearity implies that

$$ {\overline{\sigma}}_{z\theta}=2{Ge}_{z\theta}=\frac{Gr\theta}{L} $$

(4.1)

where G is the shear modulus of elasticity of the intact arterial wall. The torque is given by an integral of the product of the average shear stress and the moment arm and the wall area as:

$$ T=2\pi {\int}_{r_{\mathrm{i}}}^{r_{\mathrm{a}}}\frac{G r\theta}{L}{r}^2 dr=\frac{\pi }{2}\frac{G\theta}{L}\left({r}_{\mathrm{a}}^4-{r}_{\mathrm{i}}^4\right)= GJ\frac{\theta }{L} $$

(4.2a)

where J is the polar moment of inertia of the intact vessel given by:

$$ J=\frac{\pi }{2}\left({r}_{\mathrm{a}}^4-{r}_{\mathrm{i}}^4\right) $$

(4.2b)

Finally, the desired relationship between the torque and the shear modulus is obtained as:

$$ T= GJ\frac{\theta }{L} $$

(4.3)

To extend this analysis to a two-layer model, it is assumed that there is no slip between the medial and adventitial layers during torsion. This implies that the two layers have the same twist angle (θ) during torsion. Let ${\overline{\sigma}}_{z\theta}^m$ and ${\overline{\sigma}}_{z\theta}^a$ denote the average shear stress in the intima-medial and adventitial layer, respectively. Hence, the relationship between the average shear stress and shear modulus takes on a similar form to Eq. (4.1) where G is replaced by G_m or G_a for the intima-medial or adventitial layer, respectively. The vessel wall force resultant is equal to the sum of the forces in the two layers:

$$ {F}_{\theta }={\int}_{r_{\mathrm{i}}}^{{\mathrm{r}}_{\mathrm{m}}}\frac{G_{\mathrm{m}} r\theta}{L}2\pi r\; dr+{\int}_{r_{\mathrm{m}}}^{r_{\mathrm{a}}}\frac{G_{\mathrm{a}} r\theta}{L}2\pi r\; dr $$

(4.4a)

or

$$ {F}_{\theta }=\frac{2}{3}\frac{\pi \theta}{L}\left[{G}_{\mathrm{m}}\left({r}_{\mathrm{m}}^3-{r}_{\mathrm{i}}^3\right)+{G}_{\mathrm{a}}\left({r}_{\mathrm{a}}^3-{r}_{\mathrm{m}}^3\right)\right] $$

(4.4b)

The total force is calculated from the measured torque as follows:

$$ {F}_{\theta }=\frac{T}{\left({r}_{\mathrm{i}}+{r}_{\mathrm{a}}\right)/2} $$

(4.5)

The desired relationship between the shear modulus of the intact vessel and that of its two layers can be obtained by combining Eqs. (4.3), (4.4a, 4.4b) and (4.5) to yield:

$$ G=\frac{\pi }{3}\frac{r_{\mathrm{i}}+{r}_{\mathrm{a}}}{J}\left[{G}_{\mathrm{m}}\left({r}_{\mathrm{m}}^3-{r}_{\mathrm{i}}^3\right)+{G}_{\mathrm{a}}\left({r}_{\mathrm{a}}^3-{r}_{\mathrm{m}}^3\right)\right] $$

(4.6)

For a thin walled vessel where r_i ~ r_m and r_m ~ r_a, Eq. (4.6) can be simplified to the form:

$$ GJ={J}_{\mathrm{m}}{G}_{\mathrm{m}}+{J}_{\mathrm{a}}{G}_{\mathrm{a}} $$

(4.7)

Incidentally, Eq. (4.7) can be directly derived from considerations of membrane torque resultant. Since the intima-medial layer is tested mechanically after the adventitia is dissected, the shear modulus of adventitia can be computed using Eq. (4.6) or vice versa for the media.

The mean stresses in the circumferential, σ_θ, and longitudinal, σ_z, directions are given by

$$ {\sigma}_{\theta }=\frac{\Pr_{\mathrm{i}}}{h} $$

(4.8)

and

$$ {\sigma}_z=\frac{F_{\lambda }}{\pi \left({r}_{\mathrm{o}}^2-{r}_{\mathrm{i}}^2\right)}+\frac{\Pr_{\mathrm{i}}^2}{h\left({r}_{\mathrm{o}}+{r}_{\mathrm{i}}\right)} $$

(4.9)

where F_λ and h are the longitudinal force and wall thickness, respectively. The inner radius, r_i, of the vessel can be computed from the incompressibility condition for a cylindrical vessel as:

$$ {r}_{\mathrm{i}}=\sqrt{r_{\mathrm{o}}^2-\frac{A_0}{{\pi \lambda}_z}} $$

(4.10)

where r_o, A₀, and λ_z are outer radii at the loaded state, the wall area in the no-load state and the axial stretch ratio, respectively. Since all the quantities on the right-hand side of Eq. (4.10) are measured, the loaded inner radius can be computed. The linear regression data between shear modulus and circumferential stress are summarized in Table 4.1 for various axial stretches.

Table 4.1 The linear regression parameters and R² for the relationship between shear modulus (G) and circumferential stress (σ_θ) as given by G = α + βσ_θ

Appendix 2: Formulation of Incremental Moduli (Lu et al., 2004)

Mathematically, a blood vessel is assumed as a thin shell. The homeostatic in vivo state (i.e., a state of stable static equilibrium) of a blood vessel is assumed to be a circular cylinder. The distributions of the homeostatic axial and circumferential strains, referred to the zero-stress state, are nearly uniform and there is no torsion (Guo & Kassab, 2004). The circumferential deformation of an artery may be described by the mid-wall circumferential Green strain, which is defined as follows:

$$ {E}_{\theta }=\frac{1}{2}\left[{\lambda}_{\theta}^2-1\right] $$

(4.11a)

where λ_θ is the mid-wall stretch ratio (λ_θ = c/C); c refers to the mid-wall circumference of the vessel in the loaded state and C refers to the corresponding mid-wall circumference in the zero-stress state. Similarly, the longitudinal Green strain is given by:

$$ {E}_z=\frac{1}{2}\left[{\lambda}_z^2-1\right]\kern0.5em $$

(4.11b)

where λ_z is the local longitudinal stretch ratio as defined above.

The mid-wall circumference in the loaded state was computed from the average of inner and outer radius. The inner radius, r_i, of the vessel can be computed from the incompressibility condition for a cylindrical vessel as:

$$ {r}_{\mathrm{i}}=\sqrt{r_{\mathrm{o}}^2-\frac{A_0}{{\pi \lambda}_z}} $$

(4.12)

where r_o and A₀ are outer radii at the loaded state and the wall area in the no-load state, respectively. The total wall thickness, h, was computed as h = r_o − r_i. 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_θ, and longitudinal, S_z, directions are given by:

$$ {S}_{\theta }=\frac{\Pr_{\mathrm{i}}}{h{\lambda}_{\theta}^2} $$

(4.13)

and

$$ {S}_z=\frac{1}{\lambda_z^2}\left[\frac{F}{\pi \left({r}_{\mathrm{o}}^2-{r}_{\mathrm{i}}^2\right)}+\frac{\Pr_{\mathrm{i}}^2}{h\left({r}_{\mathrm{o}}+{r}_{\mathrm{i}}\right)}\right] $$

(4.14)

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_m and r_a, respectively, the two components of the mean stresses are computed according to Eqs. (4.13) and (4.14) using the respective radii and wall thicknesses. 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.

It is well known that a constitutive stress–strain relationship for an artery can be reduced from a strain energy function, ρ_oW, which represents stored deformation energy per unit volume of arterial wall. ρ_o denotes the density of the material in the 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:

$$ {S}_{ij}=\frac{\partial \left({\rho}_{\mathrm{o}}W\right)}{\partial {E}_{ij}} $$

(4.15)

where S_ij and E_ij are components of second Piola–Kirchhoff stress and Green strain, 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_ij and strains E_ij), then the perturbations may be written as:

$$ {S}_{ij}={S}_{ij}^0+\delta {S}_{ij},\kern1em {E}_{ij}={E}_{ij}^0+\delta {E}_{ij} $$

(4.16)

in which δS_ij and δE_ij are infinitesimal and quantities 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:

$$ \delta {S}_{ij}=\frac{\partial^2{\rho}_0W}{\partial {E}_{km}\partial {E}_{ij}}\delta {E}_{km}={C}_{ij km}\delta {E}_{km} $$

(4.17)

C_ijkm are the values of the second partial derivatives of ρ₀W evaluated at the 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 (circumferential direction) and 2 (longitudinal direction). If ${E}_{ij}^o$ are uniform at the homeostatic state, then C_ijkm are constants in each layer. Equation (4.17) is a linear incremental stress–strain relationship. The result can be written in the following form to introduce the definitions of the incremental elastic moduli:

$$ \delta {\mathrm{S}}_{11}={Y}_{11}\delta {E}_{11}+{Y}_{12}\delta {E}_{22},\kern1em \delta {S}_{22}={Y}_{21}\delta {E}_{11}+{Y}_{22}\delta {E}_{22},\kern1em \delta {S}_{12}=2 G\delta {E}_{12} $$

(4.18)

Y₁₁ and Y₂₂ are the classical incremental Young’s modulus in the circumferential and longitudinal directions, respectively, G is the incremental shear modulus, Y₁₂ and Y₂₁ have no equivalents in classical mechanics and have been denoted as cross-modulus by Fung and Liu (1995). The existence of strain energy function requires that Y₁₂ = Y₂₁. These equations are Hookean but not isotropic. When Eqs. (4.15)–(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.

Least Squares Method for Determination of Elastic Moduli

Since the loading does not involve shear, three elastic moduli must be determined: Y₁₁, Y₁₂, and Y₂₂. A least squares method is used to minimize the error between the theoretical stresses given by Eq. (4.18) and the experimental measurements. The result is a 3 × 3 matrix whose solution is the 3 × 1 matrix of elastic moduli:

$$ \left[\begin{array}{ccc}{A}_{11}& 0& {A}_{12}\\ {}0& {A}_{22}& {A}_{21}\\ {}{A}_{12}& {A}_{21}& {A}_{11}+{A}_{22}\end{array}\right]\left[\begin{array}{l}{Y}_{11}\\ {}{Y}_{22}\\ {}{Y}_{12}\end{array}\right]=\left[\begin{array}{l}{B}_{11}\\ {}{B}_{22}\\ {}{B}_{12}\end{array}\right] $$

(4.19)

in which

$$ {\displaystyle \begin{array}{ll}{B}_{11}=\sum \limits_n\Delta {S}_{11}^n\Delta {E}_{11}^n& {A}_{11}=\sum \limits_n\Delta {E}_{11}^n\Delta {E}_{11}^n\\ {}{B}_{12}=\sum \limits_n\Delta {S}_{22}^n\Delta {E}_{11}^n+\sum \limits_n\Delta {S}_{11}^n\Delta {E}_{22}^n& {A}_{12}={A}_{21}=\sum \limits_n\Delta {E}_{11}^n\Delta {E}_{22}^n\\ {}{B}_{22}=\sum \limits_n\Delta {S}_{22}^n\Delta {S}_{22}^n& {A}_{22}=\sum \limits_n\Delta {E}_{11}^n\Delta {E}_{22}^n\end{array}} $$

where

$$ \Delta {S}_{11}^n={S}_{11}^n-{S}_{11}^0,\Delta {S}_{22}^n={S}_{22}^n-{S}_{22}^0,\Delta {E}_{11}^n={E}_{11}^n-{E}_{11}^0,\Delta {E}_{22}^n={E}_{22}^n-{E}_{22}^0,n=\mathrm{0,1,2},\dots $$

Obviously, the in vivo state is denoted by n = 0 and consequently $\Delta {S}_{11}^0=\Delta {E}_{22}^0=\Delta {E}_{11}^0=\Delta {E}_{22}^0=0$. The symbol “δ” is eliminated for convenience but it should 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 Composite Model

Since the artery cannot be separated into two layers without damage 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.,

$$ {T}_{11}={T}_{11}^{im}+{T}_{11}^{ad} $$

(4.20a)

and

$$ {T}_{22}={T}_{22}^{im}+{T}_{22}^{ad} $$

(4.20b)

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

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:

$$ {S}_{11}=\frac{T_{11}}{h}={Y}_{11}{E}_{11} $$

(4.21a)

and

$$ {S}_{22}=\frac{T_{22}}{h}={Y}_{22}{E}_{22} $$

(4.21b)

Equations (4.21a) and (4.21b) can be substituted into Eqs. (4.20a, 4.20b) for each of the layers to yield

$$ {Y}_{11}{hE}_{11}={Y}_{11}^{\mathrm{im}}{h}^{\mathrm{im}}{E}_{11}^{\mathrm{im}}+{Y}_{11}^{\mathrm{ad}}{h}^{\mathrm{ad}}{E}_{11}^{\mathrm{ad}} $$

(4.22a)

and

$$ {Y}_{22}{hE}_{22}={Y}_{22}^{\mathrm{im}}{h}^{\mathrm{im}}{E}_{22}^{\mathrm{im}}+{Y}_{22}^{\mathrm{ad}}{h}^{\mathrm{ad}}{E}_{22}^{\mathrm{ad}} $$

(4.22b)

Since the deformation or strain, E, is the same for each of the layers in the parallel model, i.e., E = E^im = E^ad, then:

$$ {Y}_{11}=\left(\frac{h^{\mathrm{im}}}{h}\right){Y}_{11}^{\mathrm{im}}+\left(\frac{h^{\mathrm{ad}}}{h}\right){Y}_{11}^{\mathrm{ad}} $$

(4.23a)

and

$$ {Y}_{22}=\left(\frac{h^{\mathrm{im}}}{h}\right){Y}_{22}^{\mathrm{im}}+\left(\frac{h^{\mathrm{ad}}}{h}\right){Y}_{22}^{\mathrm{ad}} $$

(4.23b)

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, Y₁₁

Table 4.3 Left anterior descending (LAD)—axial incremental moduli, Y₂₂

Table 4.4 Right coronary artery (RCA)—circumferential incremental moduli, Y₁₁

Table 4.5 Right coronary artery (RCA)—axial incremental moduli, Y₂₂

Table 4.6 Left anterior descending (LAD)—cross incremental moduli, Y₁₂ = Y₂₁

Table 4.7 Right coronary artery (RCA)—cross incremental moduli, Y₁₂ = Y₂₁

Appendix 3: 2D Strain Energy Function (Pandit et al., 2005)

A well-known approach to elasticity of bodies capable of finite deformation is to postulate the form of an elastic potential or strain energy function (SEF; Green & Adkins, 1960). Following the arguments made by Fung (1993), the following form of strain energy function is used:

$$ {\rho}_0{W}^{(2)}=\frac{C}{2}\left(\exp Q-1\right) $$

(4.24a)

$$ Q={a}_1\left({E}_{\theta \theta}^2-{E}_{\theta \theta}^{\ast 2}\right)+{a}_2\left({E}_{zz}^2-{E}_{zz}^{\ast 2}\right)+2{a}_4\left({E}_{\theta \theta}{E}_{zz}-{E}_{\theta \theta}^{\ast }{E}_{zz}^{\ast}\right) $$

(4.24b)

where C, a₁, a₂, and a₄ are 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_z = 1.4). Here, the superscript “2” over ρ_oW⁽²⁾ signifies that this is a 2D approximation, treating the arterial wall as a membrane and ignoring the radial stress. The symbol W represents the strain energy per unit mass of the material and ρ_ο is the mass density at zero stress. C has the units of stress (force/area); a_l, a₂, and a₄ are dimensionless constants. Although 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:

$$ {S}_{ij}=\frac{\partial \left({\rho}_{\mathrm{o}}W\right)}{\partial {E}_{ij}},\kern1em \left(i,j=\theta, z\right) $$

(4.25)

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_θθ, E_zz are normal strains, e_θz = e_zθ 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).

4.1.1 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:

$$ {\displaystyle \begin{array}{ll}{S}_{\theta \theta}=& \frac{C}{2}\exp \left[{a}_1\left({E}_{\theta \theta}^2-{E}_{\theta \theta}^{\ast 2}\right)+{a}_2\left({E}_{zz}^2-{E}_{zz}^{\ast 2}\right)+2{a}_4\left({E}_{\theta \theta}{E}_{zz}-{E}_{\theta \theta}^{\ast }{E}_{zz}^{\ast}\right)\right]\\ {}& \times \left(2{a}_1{E}_{\theta \theta}+2{a}_4{E}_{zz}\right)\end{array}} $$

(4.26a)

and

$$ {\displaystyle \begin{array}{ll}{S}_{zz}=& \frac{C}{2}\exp \left[{a}_1\left({E}_{\theta \theta}^2-{E}_{\theta \theta}^{\ast 2}\right)+{a}_2\left({E}_{zz}^2-{E}_{zz}^{\ast 2}\right)+2{a}_4\left({E}_{\theta \theta}{E}_{zz}-{E}_{\theta \theta}^{\ast }{E}_{zz}^{\ast}\right)\right]\\ {}& \times \left(2{a}_2{E}_{zz}+2{a}_4{E}_{\theta \theta}\right)\end{array}} $$

(4.26b)

The goal of an algorithm to determine the material constants C, a₁, a₂, and a₄ is to minimize the square of the difference between theoretical (proposed function, Eqs. (4.26a, 4.26b)) and experimental values of circumferential, ${S}_{\theta \theta}^e$, and axial, ${S}_{zz}^e$, stresses as:

$$ {\displaystyle \begin{array}{ll}& \mathrm{Error}\\ {}& =\sum \limits_{n=1}^N{\left[{\left(\frac{C}{2}\exp \left[{a}_1\left({E}_{\theta \theta}^2-{E}_{\theta \theta}^{\ast 2}\right)+{a}_2\left({E}_{zz}^2-{E}_{zz}^{\ast 2}\right)+2{a}_4\left({E}_{\theta \theta}{E}_{zz}-{E}_{\theta \theta}^{\ast }{E}_{zz}^{\ast}\right)\right]\left(2{a}_1{E}_{\theta \theta}+2{a}_4{E}_{zz}\right)\right)}_n-{\left({S}_{\theta \theta}^e\right)}_n\right]}^2\\ {}& \kern1em \\ {}& +\sum \limits_{n=1}^N{\left[{\left(\frac{C}{2}\exp \left[{a}_1\left({E}_{\theta \theta}^2-{E}_{\theta \theta}^{\ast 2}\right)+{a}_2\left({E}_{zz}^2-{E}_{zz}^{\ast 2}\right)+2{a}_4\left({E}_{\theta \theta}{E}_{zz}-{E}_{\theta \theta}^{\ast }{E}_{zz}^{\ast}\right)\right]\left(2{a}_2{E}_{zz}+2{a}_4{E}_{\theta \theta}\right)\right)}_n-{\left({S}_{zz}^e\right)}_n\right]}^2\end{array}} $$

(4.27)

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.

4.1.2 Marquardt-Levenberg Method

In this approach, the determination of the material 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_1, a₂, and a₄) are evaluated. The value of a₄ is then fixed at the predetermined mean for the intact wall and each respective layer and the three parameters (C, a₁ and a₂) are re-evaluated.

4.1.3 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 approximations 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 MATLAB 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 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 a₄ is then fixed and the values of C, a₁, and a₂ are redetermined using the GA.

Table 4.8 Material constants of strain energy function obtained from experimental stress–strain data of intact left anterior descending (LAD) artery

Table 4.9 Material constants of strain energy function obtained from experimental stress–strain data of medial layer of LAD artery

Table 4.10 Material constants of strain energy function obtained from experimental stress–strain data of adventitial layer of LAD artery

Table 4.11 Material constants of strain energy function obtained from experimental stress–strain data of intact right coronary artery (RCA)

Table 4.12 Material constants of strain energy function obtained from experimental stress–strain data of medial layer of right coronary artery (RCA)

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:

$$ {E}_{\theta }=\frac{1}{2}\left[{\lambda}_{\theta}^2-1\right] $$

(4.28)

where λ_θ is the stretch ratio (λ_θ = c/C, c and C refer to the circumference in the loaded and zero-stress state, respectively). Similarly, the axial and radial Green strains are given by:

$$ {E}_z=\frac{1}{2}\left[{\lambda}_z^2-1\right],\kern1em {E}_r=\frac{1}{2}\left[{\lambda}_r^2-1\right] $$

(4.29)

where λ_z and λ_r are the local axial stretch ratio (change in axial length between loaded and no-load state) and radial stretch ratio (change in wall thickness 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):

$$ {\lambda}_r{\lambda}_z{\lambda}_{\theta }=1 $$

and

$$ {\lambda}_r=\frac{R}{\lambda_z\chi r},\kern1em {\lambda}_z=\frac{z}{Z},\kern1em {\lambda}_{\theta }=\frac{\chi r}{R} $$

(4.30)

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 = 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:

$$ {r}_{\mathrm{i}}=\sqrt{r_{\mathrm{e}}^2-\frac{A_0}{\lambda_z\pi }} $$

(4.31)

where A₀ is the cross-sectional area of vessel wall in the no-load state. The reference radius R in the zero-stress state can also be determined by the incompressibility condition as:

$$ R=\sqrt{\chi \left({r}^2-{r}_{\mathrm{i}}^2\right){\lambda}_z+{R}_{\mathrm{i}}^2} $$

(4.32)

The stretch ratios in Eq. (4.30) can be written as a function of radius r. Equations (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.

4.1.1 Strain Energy Function

Similar to Appendix 2, the following exponential form of strain energy function is:

$$ W=\frac{C}{2}\left(\exp Q-1\right) $$

(4.33)

where

$$ Q={b}_1{E}_{\theta}^2+{b}_2{E}_z^2+{b}_3{E}_r^2+2\left({b}_4{E}_{\theta }{E}_z+{b}_5{E}_z{E}_r+{b}_6{E}_{\theta }{E}_r\right) $$

where W represents the pseudo-strain energy per unit volume. C has the units of stress (force/area), b₁, b₂, b₃, b₄, b₅, and b₆ are dimensionless constants.

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:

$$ {W}^{\ast }=W+\frac{H}{2}\left[\left(1+2{E}_{\theta}\right)\left(1+2{E}_z\right)\left(1+2{E}_r\right)-1\right] $$

(4.34)

It is known that −H has the significance of a hydrostatic pressure. The Cauchy stresses can be related to the strain energy function as:

$$ {\sigma}_{ij}=\frac{\rho }{\rho_0}\frac{\partial {x}_j}{\partial {X}_{\alpha }}\frac{\partial {x}_i}{\partial {X}_{\beta }}\frac{\partial }{\partial {E}_{\beta \alpha}}{W}^{\ast}\kern1em \left(i,j,\alpha, \beta =r,z,\theta \right) $$

(4.35)

where x_i and X_α denote coordinates, and ρ and ρ₀ represent densities, in the deformed 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:

$$ {\displaystyle \begin{array}{c}{\sigma}_{\theta }=C\left(1+2{E}_{\theta}\right)\left[{b}_1{E}_{\theta }+{b}_4{E}_z+{b}_6{E}_r\right]{\mathrm{e}}^Q+H\\ {}{\sigma}_z=C\left(1+2{E}_z\right)\left[{b}_4{E}_{\theta }+{b}_2{E}_z+{b}_5{E}_r\right]{\mathrm{e}}^Q+H\\ {}{\sigma}_r=C\left(1+2{E}_r\right)\left[{b}_6{E}_{\theta }+{b}_5{E}_z+{b}_3{E}_r\right]{\mathrm{e}}^Q+H\end{array}} $$

(4.36)

Equation (4.36) will be subsequently incorporated in the equilibrium equation to obtain the desired results.

4.1.2 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):

$$ \frac{\partial {\sigma}_r}{\partial r}+\frac{\sigma_r-{\sigma}_{\theta }}{r}=0 $$

(4.37)

The following boundary conditions are considered that simulate the experimental protocol:

• 1. On the external surface r = r_e, the pressure is zero. On the inner surface r = r_i, an internal pressure p_i is imposed. Integration of Eq. (4.37) along with this boundary condition yields:

$$ {p}_{\mathrm{i}}={\int}_{r_{\mathrm{e}}}^{r_{\mathrm{i}}}C\left\{\left(1+2{E}_r\right)\left[{b}_6{E}_{\theta }+{b}_5{E}_z+{b}_3{E}_r\right]-\left(1+2{E}_{\theta}\right)\left[{b}_1{E}_{\theta }+{b}_4{E}_z+{b}_6{E}_r\right]\right\}{\mathrm{e}}^Q\frac{dr}{r} $$

(4.38)

• 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 $F+{p}_{\mathrm{i}}\pi {r}_{\mathrm{i}}^2$ equals the integral of axial stress over the vessel wall cross section, namely:

$$ F+{p}_{\mathrm{i}}\pi {r}_{\mathrm{i}}^2=2\pi {\int}_{r_{\mathrm{i}}}^{r_{\mathrm{e}}}{\sigma}_zr\; dr $$

(4.39)

Use of Eqs. (4.36) and (4.38) in (4.39) yields:

$$ {\displaystyle \begin{array}{ll}F=& 2\pi C{\int}_{r_{\mathrm{i}}}^{r_{\mathrm{e}}}r{\mathrm{e}}^Q\Big[\left(1+2{E}_z\right)\left[{b}_4{E}_{\theta }+{b}_2{E}_z+{b}_5{E}_r\right]\\ {}& \kern1em -\frac{1}{2}\left(1+2{E}_r\right)\left({b}_6{E}_{\theta }+{b}_5{E}_z+{b}_3{E}_r\right)\kern0.5em -\frac{1}{2}\left(1+2{E}_{\theta}\right)\left({b}_1{E}_{\theta }+{b}_4{E}_z+{b}_6{E}_r\right)\Big] dr\end{array}} $$

(4.40)

This equation represents a force–displacement relation.

4.1.3 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_i), axial stretch force (F), external radius (r_e), internal radius (r_i), and distribution of Green strain components (E_θ, E_z and E_r). Transmural pressure, axial stretch force, and external 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_θ(r) between the two surfaces is a function of the radius, Eqs. (4.38) and (4.40) can be simplified if a linear distribution of E_θ(r) can be assumed between two surfaces along the radial direction as:

$$ {E}_{\theta }(r)={E}_{\theta}\left({r}_{\mathrm{i}}\right)+\frac{r-{r}_{\mathrm{i}}}{r_{\mathrm{e}}-{r}_{\mathrm{i}}}\left({E}_{\theta}\left({r}_{\mathrm{e}}\right)-{E}_{\theta}\left({r}_{\mathrm{i}}\right)\right) $$

(4.41)

E_z(r) and E_r(r) can be computed using Eqs. (4.29)–(4.32). Guo, Xiao, and Kassab (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 material constants C, b₁, b₂, b₃, b₄, b₅, and b₆ is to minimize the square of the difference between theoretical (Eqs. 4.38 and 4.40) and experimental values of internal pressure ${p}_{\mathrm{i}}^e$, and axial Force F^e as:

$$ {\displaystyle \begin{array}{ll}\mathrm{Error}=& \sum \limits_{n=1}^N{\left[{\left(\begin{array}{l}{\int}_{r_{\mathrm{e}}}^{r_{\mathrm{i}}}C\Big\{\left(1+{E}_r\right)\left[{b}_6{E}_{\theta }+{b}_5{E}_z+{b}_3{E}_r\right]\\ {}\kern1em -\left(1+{E}_{\theta}\right)\left[{b}_1{E}_{\theta }+{b}_4{E}_z+{b}_6{E}_r\right]\Big\}{\mathrm{e}}^Q\frac{dr}{r}\end{array}\right)}_n-{\left({p}_{\mathrm{i}}^e\right)}_n\right]}^2\kern1em \\ {}& +\sum \limits_{n=1}^N{\left[{\left(\begin{array}{l}2\pi C{\int}_{r_i}^{r_e}{re}^Q\Big[\left(1+{E}_z\right)\kern0.5em \left[{b}_4{E}_{\theta }+{b}_2{E}_z+{b}_5{E}_r\right]\\ {}\kern3em -\frac{1}{2}\left(1+{E}_r\right)\kern0.5em \left({b}_6{E}_{\theta }+{b}_5{E}_z+{b}_3{E}_r\right)\\ {}\kern3em -\frac{1}{2}\left(1+{E}_{\theta}\right)\kern0.5em \left({b}_1{E}_{\theta }+{b}_4{E}_z+{b}_6{E}_r\right)\Big] dr\end{array}\right)}_n-{\left({F}^e\right)}_n\right]}^2\end{array}} $$

(4.42)

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.

4.1.4 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_I, is assumed to be the sum of the strain energy of the intima-media, V_IM and adventitia layer, V_A. Furthermore, the total axial force of intact vessel, ${F}_{\mathrm{I}}^{\mathrm{Total}}$, must be the sum of the axial force of the intima-media layer, F_IM, and the adventitia layer, F_A. The two assumptions can be mathematically stated as:

$$ {V}_{\mathrm{I}}={V}_{\mathrm{I}\mathrm{M}}+{V}_{\mathrm{A}} $$

(4.43a)

$$ {F}_{\mathrm{I}}^{\mathrm{total}}={F}_{\mathrm{I}\mathrm{M}}+{F}_{\mathrm{A}} $$

(4.43b)

The energy V_j (j = 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:

$$ {\displaystyle \begin{array}{c}{V}_{\mathrm{I}}={\int}_{r_{\mathrm{i}}}^{r_{\mathrm{e}}}\frac{C_{\mathrm{I}}}{2}\left(\exp {Q}_{\mathrm{I}}-1\right)2\pi rh\; dr\\ {}={\int}_{r_{\mathrm{i}}}^{r_{\mathrm{m}}}\frac{C_{\mathrm{I}\mathrm{M}}}{2}\left(\exp {Q}_{\mathrm{I}\mathrm{M}}-1\right)2\pi rh\; dr\\ {}\kern1em +{\int}_{r_{\mathrm{m}}}^{r_{\mathrm{e}}}\frac{C_{\mathrm{A}}}{2}\left(\exp {Q}_{\mathrm{A}}-1\right)2\pi rh\; dr\end{array}} $$

(4.44)

where

$$ {\displaystyle \begin{array}{ll}{Q}_j=& {b}_{1,j}{E}_{\theta}^2(r)+{b}_{2,j}{E}_z^2+{b}_{3,j}{E}_r^2\kern1em +2\left[{b}_{4,j}{E}_{\theta }(r){E}_z+{b}_{5,j}{E}_z{E}_r+{b}_{6,j}{E}_{\theta }(r){E}_r\right]\\ {}& \kern1em \left(j=I,\mathrm{IM},A\right)\end{array}} $$

r_i, r_m, and r_e are radii for inner, middle (the interface between two layers), and outer vessel wall; h is the deformed length of the arterial specimen; and E_r can be determined by E_θ(r) and E_z based on the incompressibility condition. The total force ${F}_j^{\mathrm{total}}\left(j=I,\mathrm{IM},A\right)$ is the integral of axial Cauchy stress σ_z over the wall area expressed as:

$$ {\displaystyle \begin{array}{c}{F}_{\mathrm{total}}=2\pi {\int}_{r_{\mathrm{i}}}^{r_{\mathrm{e}}}{\sigma}_{z,I}r\; dr=2\pi {\int}_{r_{\mathrm{i}}}^{r_{\mathrm{m}}}{\sigma}_{z,\mathrm{IM}}r\; dr+2\pi {\int}_{r_{\mathrm{m}}}^{r_{\mathrm{e}}}{\sigma}_{z,A}r\; dr\\ {}{\sigma}_{z,j}=C\left(1+2{E}_z\right)\left[{b}_{4,j}{E}_{\theta }(r)+{b}_{2,j}{E}_z+{b}_{5,j}{E}_r\right]{\mathrm{e}}^Q\kern1em \left(j=I,\mathrm{IM},A\right)\end{array}} $$

(4.45)

Equations (4.44) and (4.45) are also simplified by assuming a linear distribution of E_θ(r) along the radial direction expressed in Eq. (4.41). Material constants b_{i, j} (i = 1–7; j = 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:

$$ {\displaystyle \begin{array}{ll}\mathrm{Error}=& \sum \limits_{n=1}^N{\left[\begin{array}{l}{\left(\begin{array}{l}{\int}_{r_{\mathrm{i}}}^{r_{\mathrm{e}}}\frac{C_{\mathrm{I}}}{2}\left(\exp {Q}_{\mathrm{I}}-1\right)2\pi r\; dr\\ {}\kern1em -{\int}_{r_{\mathrm{m}}}^{r_{\mathrm{e}}}\frac{C_{\mathrm{A}}}{2}\left(\exp {Q}_{\mathrm{A}}-1\right)2\pi r\; dr\end{array}\right)}_n\\ {}\kern1em -{\left({\int}_{r_{\mathrm{i}}}^{r_{\mathrm{m}}}\frac{C_{\mathrm{I}\mathrm{M}}}{2}\left(\exp {Q}_{\mathrm{I}\mathrm{M}}-1\right)2\pi r\; dr\right)}_n\end{array}\right]}^2\\ {}& \kern1em +\sum \limits_{n=1}^N{\left[{\left({\int}_{r_{\mathrm{i}}}^{r_{\mathrm{e}}}{\sigma}_{z,\mathrm{I}}r\; dr-{\int}_{r_{\mathrm{m}}}^{r_{\mathrm{e}}}{\sigma}_{z,A}r\; dr\right)}_n-{\left({\int}_{r_{\mathrm{i}}}^{r_{\mathrm{m}}}{\sigma}_{z,\mathrm{I}\mathrm{M}}r\; dr\right)}_n\right]}^2\end{array}} $$

(4.46)

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 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 similar to Appendix 3 for the determination of material constants.

4.1.5 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 $\left(\begin{array}{ccc}{b}_1& {b}_4& {b}_6\\ {}{b}_4& {b}_2& {b}_5\\ {}{b}_6& {b}_5& {b}_3\end{array}\right)$ are positive. The eigenvalues are confirmed to be positive for all 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)

Table 4.14 Material constants of exponential strain energy function obtained from experimental data of right coronary artery (RCA)

Table 4.15 Material constants of exponential strain energy function obtained from experimental data of left anterior descending (LAD) artery

Table 4.16 Material constants of exponential strain energy function obtained from experimental data of left anterior descending (LAD) artery

Appendix 5: 2D Linearization of Fung’s Exponential Strain Energy Function (SEF) (Zhang & Kassab, 2007)

To linearize the constitutive relation, the major task is to find a strain measure that can best approximate the nonlinear curvature of load versus stretch ratio plots obtained in experiments. Since Fung’s model represents the experimental measurements well, the only task is to examine if Fung’s model can be linearized. In the 1D strain case where E_zz = 0 (${E}_{\theta \theta}^{\ast }={E}_{zz}^{\ast }=0$ can be assumed as well), the second Piola–Kirchhoff stress in circumferential direction is given by (Appendix 3):

$$ {S}_{\theta \theta}=\frac{\partial W}{\partial {E}_{\theta \theta}}={Ca}_1{E}_{\theta \theta}\exp \left({a}_1{E}_{\theta \theta}^2\right)={c}_{11}{D}_{\theta \theta}, $$

(4.47)

where c₁₁ = Ca₁ and ${D}_{\theta \theta}={E}_{\theta \theta}\exp \left({a}_1{E}_{\theta \theta}^2\right)$. Equation (4.47) shows that Hooke’s law can be obtained in the 1D case if D_θθ is viewed as a strain measure. A similar relation exists in the axial direction.

The expression of Eq. (4.47) in a linearized form does not highlight the advantages in the 1D model, but it certainly requires less material constants than a polynomial form (Sokolis, Boudoulas, & Karayannacos, 2002). In 2D, the linearization of the stress–strain relationship is preferred since it may be difficult to experimentally determine material constants due to the nonlinearity of the constitutive law (Pandit et al., 2005).

4.1.1 A Generalized Strain Measure

Although Green strain is the best choice for Fung’s model since it directly yields second Piola–Kirchhoff stress from the differentiation of strain energy function, other strain measures can also be selected to establish a stress–strain relationship. In the spirit of Seth-Hill strain family (Farahani & Naghdabadi, 2000), a new strain measure is proposed that takes into account the exponential increase of stress with Green strain to absorb the nonlinearity and thus result in a simpler stress–strain relation. In the circumferential (θ) and axial (z) directions, the new strains are defined as:

$$ {D}_{\theta \theta}={E}_{\theta \theta}\exp \left({nE}_{\theta \theta}^2\right),\kern1em {D}_{zz}={E}_{zz}\exp \left({nE}_{zz}^2\right), $$

(4.48)

where n is a material constant characterizing the strain nonlinearity with respect to Green strain E_θθ and E_zz (for n = 0, Eq. (4.48) reduces to Eqs. (4.28) and (4.29), Appendix 4). These generalized strains (Green-exponential strains) are products of Green strains and their exponential functions.

4.1.2 A Bilinear Stress–Strain Relation

Given the coupling between circumferential and axial directions in a biaxial deformation, a simple bilinear constitutive relation is postulated between second Piola–Kirchhoff stress and the new strain measure as:

$$ \left\{\begin{array}{c}{S}_{\theta \theta}={c}_{11}{D}_{\theta \theta}+{c}_{12}{D}_{zz}+{b}_{12}{D}_{\theta \theta}{D}_{zz}\\ {}{S}_{zz}={c}_{21}{D}_{\theta \theta}+{c}_{22}{D}_{zz}+{b}_{21}{D}_{\theta \theta}{D}_{zz}\end{array}\right., $$

(4.49)

in which there are six constants having the unit of stress and one constant n that characterizes the nonlinearity of strain. The constant c₁₁ is the elastic stiffness in the circumferential direction when the strain in the axial direction is zero, c₁₂ reflects the elastic modulus of S_θθ versus D_zz when there is no deformation in the circumferential direction, and b₁₂ is the bilinear coupling constant for circumferential stress. The other three parameters have similar meanings in the axial direction.

In general, the material constants in Eq. (4.49) do not have symmetric properties, e.g., c₁₂ ≠ c₂₁ since the hyperelasticity assumption (the existence of strain energy potential (Fung, 1993)) cannot be invoked which can reduce the number of constants.

4.1.3 Evaluation of the Bilinear Model

The nonlinearity parameter n in the strain defined in Eq. (4.48) can be obtained with experimental data using a least squares method. If the stretch ratio in the axial direction is kept constant (e.g., E_zz = 0), the first equation in Eq. (4.49) will reduce to Eq. (4.47), where a₁ = n is a constant to be determined. Of course, if n is extracted from data in the axial direction, it may differ from that obtained from the circumferential direction. Fortunately, n is not very sensitive to the direction chosen and thus either circumferential or axial data (or an average value) can be used for the determination of n. In practice, one set of “the most reliable” (or the most representative) experimental data can be used to determine n. Once n is determined, the values of the elastic constants can be easily obtained using least squares method for the bilinear model.

Appendix 6: 3D Linearization of Fung’s Exponential Strain Energy Function (SEF) (Zhang, Wang, & Kassab, 2007)

This appendix extends the strain linearization notion to the 3D stress–strain relationship. By introducing a more complicated but more rigorous strain measure in tensor form, the constitutive equation can be written in the form of a generalized Hooke’s law which connects the second Piola–Kirchhoff stresses with the new strains. In general, the deformation gradient tensor in a cylindrical coordinate system can be written as:

$$ \mathbf{F}=\left[\begin{array}{ccc}\frac{\partial r}{\partial R}& \frac{\partial r}{R\mathrm{\partial \Theta }}& \frac{\partial r}{\partial Z}\\ {}\frac{r\partial \theta }{\partial R}& \frac{r\partial \theta }{R\mathrm{\partial \Theta }}& \frac{r\partial \theta }{\partial Z}\\ {}\frac{\partial z}{\partial R}& \frac{\partial z}{R\mathrm{\partial \Theta }}& \frac{\partial z}{\partial Z}\end{array}\right], $$

(4.50)

where (R, Θ, Z) and (r, θ, z) are spatial coordinates in the undeformed and deformed configurations, respectively.

Since the Lagrangian strain tensors are of interest, the following unique polar decomposition of the deformation gradient is considered (Ogden, 1984):

$$ \mathbf{F}=\mathbf{RU}, $$

(4.51)

where R is an orthogonal second-order tensor which reflects the rigid rotation and U is a positive definite symmetric second-order tensor (the right stretch tensor).

The general form of Seth-Hill (Doyle-Ericksen) strain tensor family is defined as (Mahnken, 2005; Miehe & Lambrecht, 2001; Ogden, 1984):

$$ {\mathbf{E}}^{(m)}=\frac{1}{m}\left({\mathbf{U}}^m-\mathbf{I}\right), $$

(4.52)

where I is the identity matrix (second-order tensor) and m is a real number. An important example is the Green strain tensor (classically denoted by E) which corresponds to m = 2, namely:

$$ \mathbf{E}={\mathbf{E}}^{(2)}=\frac{1}{2}\left({\mathbf{U}}^2-\mathbf{I}\right)=\frac{1}{2}\left({\mathbf{F}}^T\mathbf{F}-\mathbf{I}\right). $$

(4.53)

It is noted that U² = F^TF because R^TR = I, where superscript T denotes the tensor (or matrix) transpose.

Hencky (natural, logarithmic, true) strain tensor is a special case of the Seth-Hill strain family when m → 0:

$$ {\mathbf{E}}^{(0)}=\frac{1}{2}\ln {\mathbf{U}}^2=\ln \mathbf{U}. $$

(4.54)

The Hencky strain has many good features such as additively separable dilatation and distortion (Criscione, Humphrey, Douglas, & Hunter, 2000). To take advantage of these properties, a generalized Hencky (logarithmic-exponential) strain tensor is defined as:

$$ \mathbf{D}=\ln \mathbf{U}\exp \left[n\left({J}_1-3\right)\right], $$

(4.55)

where J₁ is the first invariant of the right Cauchy-Green deformation tensor F^TF = U² (Ogden, 1984) and the nondimensional n characterizes the nonlinear variation of strain D with respect to the Hencky strain measure and is thus called the nonlinearity parameter. The objectivity (i.e., independence on observer (Ogden, 1984) of strain measure D is the same as the Hencky strain since J₁ is an invariant in an observer transformation.

For simplicity, it is assumed that the blood vessel is a cylindrically orthotropic material (Fung, 1993). In an inflation–stretch test without shear deformation, the principal stretches (λ_θ, λ_z, λ_r) are aligned in the circumferential (θ), axial (z), and radial (r) directions. In such cases, the principal Green strains (Eq. 4.53) can be written as:

$$ {E}_{ii}=\frac{1}{2}\left({\lambda}_i^2-1\right)\kern1em \left(i=\theta, z,r\right), $$

(4.56)

and the principal logarithmic-exponential (log-exp) strains in Eq. (4.55) becomes:

$$ {D}_{ii}=\ln {\lambda}_i\exp \left[n\left({J}_1-3\right)\right]\kern1em \left(i=\theta, z,r\right), $$

(4.57)

where

$$ {J}_1={\lambda}_{\theta}^2+{\lambda}_z^2+{\lambda}_r^2. $$

(4.58)

No summation is assumed in Eqs. (4.56) and (4.57). Similar to the Hencky strain, a spectral decomposition method (Ogden, 1984) can be used to compute the general log-exp strain tensor when the principal stretches are known.

To gain a better understanding of the log-exp strain measure, it will be compared with the Green and Hencky strains (note that D reduces to Hencky strain when n = 0). n is found to typically vary in a range of 0 < n < 2 for coronary vessels.

4.1.1 Strain Potential

For a hyperelastic material, it is assumed that a strain energy density function exists and the stress components can be derived by differentiating the strain potential with respect to the corresponding strains. For example, the second Piola–Kirchhoff stress can be obtained by Fung (1993) and Ogden (1984):

$$ {\widehat{S}}_{ij}={S}_{ij}+q\frac{\partial V}{\partial {E}_{ij}}=\frac{\partial W}{\partial {E}_{ij}}+q\frac{\partial V}{\partial {E}_{ij}}\kern1em \left(i,j=\theta, z,r\right), $$

(4.59)

where W(E_ij) is the strain potential, q is an arbitrary scalar, V(E_ij) = 0 is a possible internal constraint (e.g., incompressibility), and S_ij = ∂W/∂E_ij is the stress with no constraint.

The Fung’s 3D strain energy per unit volume is given by:

$$ W=\frac{C}{2}\left[\exp (Q)-1\right], $$

(4.60)

where

$$ {\displaystyle \begin{array}{llll}Q={b}_1{E}_{\theta \theta}^2+{b}_2{E}_{z z}^2+{b}_3{E}_{r r}^2+2\left({b}_4{E}_{\theta \theta}{E}_{z z}+{b}_5{E}_{z z}{E}_{r r}+{b}_6{E}_{\theta \theta}{E}_{r r}\right)\\ \kern1em +{b}_7\left({E}_{\theta z}^2+{E}_{z\theta}^2\right)+{b}_8\left({E}_{z r}^2+{E}_{r z}^2\right)+{b}_9\left({E}_{\theta r}^2+{E}_{r\theta}^2\right).\end{array}} $$

(4.61)

Note that the shear strains E_ij = E_ji (i, j = θ, z, r) are written separately for the convenience of deriving stress.

In a special case without shear deformation, the second Piola–Kirchhoff stresses obtained from the Fung model are:

$$ \left(\begin{array}{l}{S}_{\theta \theta}\\ {}{S}_{zz}\\ {}{S}_{rr}\end{array}\right)=C\left[\begin{array}{ccc}{b}_1& {b}_4& {b}_6\\ {}{b}_4& {b}_2& {b}_5\\ {}{b}_6& {b}_5& {b}_3\end{array}\right]\left(\begin{array}{l}{E}_{\theta \theta}\\ {}{E}_{zz}\\ {}{E}_{rr}\end{array}\right)\exp (Q). $$

(4.62)

It is noted that only stresses resulting from the strain energy function W(E_ij) are considered, while the contribution of the internal constraint V(E_ij) can be addressed separately.

Analogous to the classical theory of elasticity, a nominal strain energy W_n(D_ij) is proposed from which the second Piola–Kirchhoff stress can be symbolically derived as:

$$ {\widehat{S}}_{ij}={S}_{ij}+p\frac{\partial {V}_n}{\partial {D}_{ij}}=\frac{\partial {W}_n}{\partial {D}_{ij}}+p\frac{\partial {V}_n}{\partial {D}_{ij}}\kern1em \left(i,j=\theta, z,r\right), $$

(4.63)

where p is an arbitrary scalar, and V_n(D_ij) = 0 is a possible internal constraint.

The nominal strain potential defined in Eq. (4.63) merely serves the purpose of mathematical derivation of stresses where the physical meaning is different from the conventional definition where strain energy function connects the conjugate stress and strain pairs (Ogden, 1984). Since the strain tensor in Eq. (4.55) is designed to absorb the nonlinearity of the stress–strain relationship and the nonlinear coupling between different directions, a quadratic nominal strain potential is proposed as:

$$ {\displaystyle \begin{array}{ll}{W}_n=& \frac{1}{2}\left({c}_{11}{D}_{\theta \theta}^2+{c}_{22}{D}_{z z}^2+{c}_{33}{D}_{r r}^2\right)+{c}_{12}{D}_{\theta \theta}{D}_{z z}\\ {}& +{c}_{13}{D}_{\theta \theta}{D}_{r r}+{c}_{23}{D}_{z z}{D}_{r r}\kern1em +{c}_{44}\left({D}_{z r}^2+{D}_{r z}^2\right)+{c}_{55}\left({D}_{\theta r}^2+{D}_{r\theta}^2\right)\\ {}& +{c}_{66}\left({D}_{\theta z}^2+{D}_{z\theta}^2\right).\end{array}} $$

(4.64)

According to Eq. (4.63), the resulting stress–strain relation is linear. Note the contribution of constraint V_n(D_ij) are discarded for brevity.

4.1.2 Generalized Hooke’s Law

With the definitions of Eqs. (4.63) and (4.64), the second Piola–Kirchhoff stresses and the log-exp strains (without shear components) are connected by:

$$ \left(\begin{array}{l}{S}_{\theta \theta}\\ {}{S}_{zz}\\ {}{S}_{rr}\end{array}\right)=\left[\begin{array}{ccc}{c}_{11}& {c}_{12}& {c}_{13}\\ {}{c}_{12}& {c}_{22}& {c}_{23}\\ {}{c}_{13}& {c}_{23}& {c}_{33}\end{array}\right]\left(\begin{array}{l}{D}_{\theta \theta}\\ {}{D}_{zz}\\ {}{D}_{rr}\end{array}\right), $$

(4.65)

which, including n in Eq. (4.55), requires 7 model parameters (same as in the Fung model).

Equation (4.65) is in the form of the generalized Hooke’s law, where the constants c’s (with the unit of stress) can be interpreted as the elastic moduli with respect to the log-exp strains. For the convenience of illustration, it will be further assumed that the vessel wall is volumetrically incompressible. As a result, the third invariant of the right Cauchy-Green deformation tensor U² must be unity:

$$ {J}_3={\lambda}_{\theta}^2{\lambda}_z^2{\lambda}_r^2=1, $$

(4.66)

which implies that the principal log-exp strains in Eq. (4.57) are linearly related as:

$$ {D}_{\theta \theta}+{D}_{zz}+{D}_{rr}=0. $$

(4.67)

This result can be used to reduce the 3D Hooke’s law (Eq. 4.65) to a 2D form.

Substitution of the D_rr solved from Eq. (4.67) and λ_r = 1/(λ_θλ_z) from Eq. (4.66) into Eqs. (4.65), (4.57) and (4.58) yields the following equation:

$$ \left(\begin{array}{l}{S}_{\theta \theta}\\ {}{S}_{zz}\\ {}{S}_{rr}\end{array}\right)=\left[\begin{array}{cc}{d}_{11}& {d}_{12}\\ {}{d}_{21}& {d}_{22}\\ {}{d}_{31}& {d}_{32}\end{array}\right]\left(\begin{array}{l}{D}_{\theta \theta}\\ {}{D}_{zz}\end{array}\right), $$

(4.68)

where

$$ {\displaystyle \begin{array}{c}{d}_{11}={c}_{11}-{c}_{13},\kern1em {d}_{12}={c}_{12}-{c}_{13},\\ {}{d}_{21}={c}_{12}-{c}_{23},\kern1em {d}_{22}={c}_{22}-{c}_{23},\\ {}{d}_{31}={c}_{13}-{c}_{33},\kern1em {d}_{32}={c}_{23}-{c}_{33}.\end{array}} $$

(4.69)

Equation (4.68) is actually a 2D formulation in which stresses depend on D_θθ and D_zz.

Under the assumption of incompressibility, the 3D Fung model (Eq. 4.62) also reduces to a 2D form. Based on Eqs. (4.56) and (4.66), one can obtain:

$$ {E}_{rr}=-\frac{2{E}_{\theta \theta}{E}_{zz}+{E}_{\theta \theta}+{E}_{zz}}{\left(2{E}_{\theta \theta}+1\right)\left(2{E}_{zz}+1\right)}. $$

(4.70)

Substitution of Eq. (4.70) into Eq. (4.62), the second Piola–Kirchhoff stresses in the Fung model can be expressed as functions of the Green strains E_θθ and E_zz. In the following sections, the material constants of the generalized Hooke’s law can be determined from the Fung model for blood vessels.

The experimental data are fitted with the Hooke’s law in Eq. (4.68) to determine the nonlinearity parameter n which defines the log-exp strains (Eq. 4.57) that best linearize the Fung model. In other words, an optimal n that makes the linear model best represent the “experimental data” must be found. Specifically, it is required that n results in a minimum relative least squares error (RLSE) defined as:

$$ \mathrm{RLSE}=\frac{{\mathrm{LSE}}_{\theta }+{\mathrm{LSE}}_z+w{\mathrm{LSE}}_r}{\sum \limits_N\left|{S}_{\theta \theta}^e\right|+\left|{S}_{zz}^e\right|+w\left|{S}_{rr}^e\right|}, $$

(4.71)

where ${S}_{\theta \theta}^e$, ${S}_{zz}^e$, and ${S}_{rr}^e$ are the experimental data, N is the total number of sampled points, w is the weight of least squares error in the radial direction (w = 1 for the 3D and w = 0 for the 2D model) and

$$ {\mathrm{LSE}}_i=\sqrt{\sum \limits_N{\left({S}_{ii}^e-{S}_{ii}^f\right)}^2}\kern1em \left(i=\theta, z,r\right) $$

(4.72)

denotes the error contribution from θ, z, and r directions, respectively. The superscript f in Eq. (4.72) means “fitted” value.

For comparison purpose, please see Appendix 4 for the Fung’s exponential model while the bi-phasic model is introduced below. The strain energy function of the bi-phasic model is proposed by Holzapfel et al. (2005) which can be written as:

$$ W=\mu \left({I}_1-3\right)+\frac{k_1}{k_2}\left(\exp \left\{{k}_2\left[\left(1-\rho \right){\left({I}_1-3\right)}^2+\rho {\left({I}_4-1\right)}^2\right]\right\}-1\right), $$

(4.73a)

where ${I}_1={\lambda}_{\theta}^2+{\lambda}_z^2+{\lambda}_r^2$ and I₄ > 1 are invariants, μ > 0 and k₁ > 0 have the units of stress, and k₂ > 0 and ρ ∈ [0, 1] are dimensionless parameters. μ is associated with the non-collagenous matrix of the material, which describes the isotropic part of the overall response of the tissue (Holzapfel et al., 2000). The constants k₁ and k₂ are associated with the anisotropic contribution of collagen to the overall response (Holzapfel et al., 2000). Since there are no shear loadings in the experiments (Wang et al., 2006) and the assumption that all the fibers are embedded in the tangential surface of the tissue (no components in the radial direction), I₄ in Eq. (4.73a) can be expressed as:

$$ {I}_4={\lambda}_{\theta}^2{\cos}^2\varphi +{\lambda}_z^2{\sin}^2\varphi, $$

(4.73b)

The collagen fibers in this model are assumed to be arranged in helical structures, and φ is the angle of the fibers with respect to circumferential direction. The second term in Eq. (4.73a) contributes to W only when I₄ > 1 (Holzapfel et al., 2005; Holzapfel, Gasser, & Ogden, 2004).

4.1.3 Determination of Material Constants

Experimental data are provided by a previous study on the passive mechanical properties of porcine coronary arteries (Chap. 3). Briefly, a series of inflation tests are done on cannulated vessels under different axial stretch ratios (Wang, Zhang, & Kassab, 2008). Outer radius r_o, internal pressure p_i and axial force F are measured. Vessel rings are taken from the specimen and a radial cut is made to the vessel ring to reveal the zero-stress state. Inner circumference C_i, outer circumference C_o, and wall area A are recorded. There are two steps in determining the material constants for Hooke’s law. The first step is to derive the equations that express the external loadings (internal pressure p_i and axial force F) as functions of strains and material constants. The second step is to use a separable nonlinear least squares method to determine the material constants by minimizing the differences between theoretical and measured values of external loadings. Material constants for the bi-phasic model are determined by the standard nonlinear Levenberg-Marquardt method. The results are summarized in Tables 4.17 and 4.18 for the RCA and LAD arteries, respectively.

Table 4.17 Material constants of the constitutive equation obtained from experimental data of right coronary artery (RCA)

Table 4.18 Material constants of the constitutive equation obtained from experimental data of LAD (left anterior descending) artery

(b) Fung’s exponential model. Material parameter C has the units of stress (kPa), b1, b2, b3, b4, b5, and b6 are dimensionless constants
	C	b 1	b 2	b 3	b 4	b 5	b 6	R2 (FT)	R2 (pi)	RMS% (FT)	RMS% (pi)
Heart 7	11.64	1.03	2.09	0.38	0.37	0.02	0.06	0.98	0.97	26.3	21.1
Heart 8	5.94	1.27	2.78	0.62	0.45	0.13	0.13	0.97	0.99	25.1	21.4
Heart 9	4.16	1.58	2.66	0.73	0.30	0.03	0.08	0.96	0.98	34.5	18.4
Heart 10	11.39	1.39	2.62	0.51	0.32	0.06	0.06	0.99	0.99	28.0	18.5
Heart 11	5.48	1.54	4.29	1.31	0.05	0.22	0.21	0.98	0.99	12.6	13.0
Heart 12	4.43	1.48	5.15	0.24	0.25	0.46	0.01	0.88	0.97	33.2	22.1
Mean	7.17	1.38	3.27	0.63	0.29	0.15	0.09	0.96	0.98	26.6	19.1
SD	3.43	0.21	1.18	0.37	0.13	0.17	0.07	0.04	0.01	7.8	3.3

(c) Bi-phasic model. Material parameter μ > 0 and k1 > 0 have the units of stress, k2 > 0 and ρ ∈ [0, 1] are dimensionless parameters, and the unit of φ is degree
	μ	k 1	k 2	φ	ρ	R2 (FT)	R2 (pi)	RMS% (FT)	RMS% (pi)
Heart 7	10.87	0.47	0.65	37.89	0.26	0.99	0.98	10.6	19.5
Heart 8	11.62	0.83	0.79	89.71	0.47	0.97	0.98	15.7	19.8
Heart 9	10.14	0.01	1.39	6.97	0.51	0.98	0.99	9.3	19.3
Heart 10	13.80	0.27	1.18	39.13	0.56	0.99	0.99	10.4	14.0
Heart 11	9.72	0.51	0.96	1.15	0.41	0.98	0.98	12.7	18.5
Heart 12	17.97	0.60	2.17	90.00	0.91	0.98	0.91	24.3	35.6
Mean	12.35	0.45	1.19	44.14	0.52	0.98	0.97	13.8	21.1
SD	3.10	0.28	0.55	38.66	0.22	0.01	0.03	5.6	7.4

1. R² is correlation coefficient; RMS% is the percentage of the root mean square error of the fit compared to the mean value. Both were computed for the total axial force (F_T) and inner pressure (p_i). Reproduced from Zhang, Wang, et al. (2007) with permission

Table 4.19 Results of the sensitivity analysis for generalized Hooke’s law (a), Fung’s exponential model (b), and bi-phasic model (c)

Appendix 7: Shear Modulus in Reference to New Strain Measure Zhang, Wang, et al. (2007)

The coronary arteries are assumed to be homogeneously elastic, cylindrically orthotropic, and volumetrically incompressible solids. The arterial wall is also assumed to remain as an axisymmetric tube with a uniform cross section at various combinations of inflation pressure P_i, axial stretch λ_z, and torque T, as illustrated in Fig. 4.20. Therefore, the cylindrical coordinates of a vessel segment in the deformed state (r, θ, z) are related to those in the zero-stress state (R, Θ, Z) by:

$$ r=r(R),\kern1em \theta =\chi \Theta +\frac{\alpha Z}{L_0},\kern1em z={\lambda}_zZ, $$

(4.74)

where χ = π/(π − Φ) with Φ being the opening angle that characterizes the residual strain, α is the twist angle, L₀ the length at zero-stress state, λ_z = L/L₀ the axial stretch ratio with L being the length at deformed state. The origins of the coordinate systems are assumed to be at one end of the vessel segment. The change of axial stretch from zero-stress to no-load state is neglected.

Fig. 4.20

(a) The zero-stress state, (b) the deformed cross section, and (c) the deformed coronary artery under transmural pressure P_i, axial stretch λ_z, and twist torque T. R_i and R_o, r_i and r_o, are the inner and outer radii in the (R, Θ, Z) and (r, θ, z) coordinate systems, respectively. Φ is opening angle, L is vessel segment length, α is twist angle. Reproduced from Zhang, Wang, et al. (2007) with permission

The deformation gradient matrix F with respect to the zero-stress state is given by:

$$ \left[{F}_{jk}\right]=\left[\begin{array}{ccc}{\lambda}_r& 0& 0\\ {}0& {\lambda}_{\theta }& {\xi}_{\theta}\\ {}0& 0& {\lambda}_z\end{array}\right], $$

(4.75)

where j, k = r, θ, z and

$$ {\lambda}_r=\frac{dr}{dR},\kern1em {\lambda}_{\theta }=\frac{\chi r}{R},\kern1em {\xi}_{\theta }=\frac{\alpha r}{L_0}=\frac{\lambda_z\alpha r}{L} $$

(4.76)

The incompressibility condition, det[F_jk] = 1 in Eq. (4.75), yields

$$ {\lambda}_r=\frac{R}{{\chi \lambda}_zr},\kern1em r=\sqrt{r_{\mathrm{i}}^2+\frac{R^2-{R}_{\mathrm{i}}^2}{{\chi \lambda}_z}} $$

(4.77)

Therefore, the right Cauchy-Green deformation tensor C (Ogden, 1984) is given by:

$$ \mathbf{C}={\mathbf{F}}^T\mathbf{F}=\left[\begin{array}{ccc}{\lambda}_r^2& 0& 0\\ {}0& {\lambda}_{\theta}^2& {\lambda}_{\theta }{\xi}_{\theta}\\ {}0& {\lambda}_{\theta }{\xi}_{\theta }& {\lambda}_z^2+{\xi}_{\theta}^2\end{array}\right] $$

(4.78)

Lagrangian strains can be defined by the symmetric tensor C.

4.1.1 Generalized Hooke’s Law

To simplify the constitutive relation of blood vessels, a logarithmic-exponential (log-exp) strain tensor is defined as (Appendix 6):

$$ \mathbf{D}=\frac{1}{2}\ln \mathbf{C}\exp \left[n\left({J}_1-3\right)\right] $$

(4.79)

where n is called the nonlinearity parameter and J₁ denotes the first invariant of the right Cauchy-Green deformation tensor C. The D reduces to the Hencky (logarithmic) strain (ln C)/2 when n = 0. In the specific case of Eq. (4.78), one finds:

$$ {J}_1=\mathrm{tr}\left(\mathbf{C}\right)={\lambda}_{\theta}^2+{\lambda}_z^2+{\lambda}_r^2+{\xi}_{\theta}^2 $$

(4.80)

It is postulated that the second Piola–Kirchhoff stress and the log-exp strain are connected by a generalized Hooke’s law (Appendix 6), namely,

$$ \left(\begin{array}{l}{S}_{\theta \theta}\\ {}{S}_{zz}\\ {}{S}_{rr}\\ {}{S}_{zr}\\ {}{S}_{\theta r}\\ {}{S}_{\theta z}\end{array}\right)=\left[\begin{array}{cccccc}{c}_{11}& {c}_{12}& {c}_{13}& 0& 0& 0\\ {}{c}_{12}& {c}_{22}& {c}_{23}& 0& 0& 0\\ {}{c}_{13}& {c}_{23}& {c}_{33}& 0& 0& 0\\ {}0& 0& 0& {c}_{44}& 0& 0\\ {}0& 0& 0& 0& {c}_{55}& 0\\ {}0& 0& 0& 0& 0& {c}_{66}\end{array}\right]\left(\begin{array}{l}{D}_{\theta \theta}\\ {}{D}_{zz}\\ {}{D}_{rr}\\ {}2{D}_{zr}\\ {}2{D}_{\theta r}\\ {}2{D}_{\theta z}\end{array}\right) $$

(4.81)

where c’s (with unit of stress) can be interpreted as the elastic moduli with respect to the log-exp strains. In particular, c₆₆ is the shear modulus for the torsion test in Fig. 4.20.

To compute the log-exp strains, the logarithm of tensor C (Eqs. 4.78 and 4.79) must be calculated. In terms of spectral decomposition (Ogden, 1984), if u⁽ⁱ⁾ (i = 1, 2, 3) are the unit vectors (eigenvectors) along the Lagrangian principal axes of C, then:

$$ \mathbf{C}=\sum \limits_{i=1}^3{\lambda}_i^2{\mathbf{u}}^{(i)}\otimes {\mathbf{u}}^{(i)},\kern1em \ln \mathbf{C}=\sum \limits_{i=1}^3\ln \left({\lambda}_i^2\right){\mathbf{u}}^{(i)}\otimes {\mathbf{u}}^{(i)} $$

(4.82)

The eigenvalues and eigenvectors of tensor C (the first equation) must be determined, and subsequently the logarithm of those eigenvalues must be calculated to project the result back to the original coordinates.

4.1.2 Shear Modulus

In the experiments of Lu et al. (2003), a typical angle of twist is α = 0.4 radian (23°). The representative radius and length of porcine left anterior descending artery are r = 2.5 mm and L = 20 mm, respectively, and the typical value of ξ_θ is ξ_θ = λ_zαr/L = 0.05λ_z which is much smaller than λ_θ and λ_z (on the order of unity). If only the first-order terms of ξ_θ is retained, the logarithm of tensor C in Eq. (4.78) can be approximated by the following Taylor expansion (which is obtained from Eq. (4.82) and then expanded with respect to ξ_θ):

$$ \ln \mathbf{C}\approx \left[\begin{array}{ccc}\ln {\lambda}_r^2& 0& 0\\ {}0& \ln {\lambda}_{\theta}^2& 2{e}_{\theta z}\\ {}0& 2{e}_{\theta z}& \ln {\lambda}_z^2\end{array}\right] $$

(4.83)

where

$$ {e}_{\theta z}=\frac{\lambda_{\theta }{\xi}_{\theta}\ln \left({\lambda}_{\theta }/{\lambda}_z\right)}{\lambda_{\theta}^2-{\lambda}_z^2} $$

(4.84)

Note that e_θz ≈ ξ_θ/(2λ_z) = ar/(2L) when λ_θ ≈ λ_z, which is the small shear strain used in Lu et al. (2003).

Since it is found that the measured torque (T) and the angle of twist per unit length (α/L) are linearly proportional in the range of interest, Cauchy shear stress has been used to compute shear modulus with respect to shear strain (αr/(2L)) (Lu et al., 2003). According to the theory of continuum mechanics (Humphrey & Na, 2002), the Cauchy stress of incompressible materials can be derived from the second Piola–Kirchhoff stress as:

$$ {\sigma}_{jk}={F}_{jl}{F}_{km}{S}_{lm}+H{\delta}_{jk}\kern1em \left(j,k,l,m=r,\theta, z\right) $$

(4.85)

where δ_jk is the Kronecker delta and H is an arbitrary scalar that needs to be determined from boundary conditions. Einstein summation has been assumed in Eq. (4.85). Equations (4.75) and (4.85) yield the Cauchy shear stress in the inflation–stretch–torsion test as:

$$ {\sigma}_{\theta z}=\left({\lambda}_{\theta }{S}_{\theta z}+{\xi}_{\theta }{S}_{zz}\right){\lambda}_z $$

(4.86)

Equation (4.86) shows that Cauchy shear stress depends on both shear and axial second Piola–Kirchhoff stresses. Substituting the incompressibility condition D_θθ + D_zz + D_rr = 0 (i.e., λ_θλ_zλ_r = 1, also see Eqs. (4.79) and (4.83)) into Eq. (4.81), the following holds:

$$ {S}_{\theta z}=2{c}_{66}{D}_{\theta z},\kern1em {S}_{zz}={d}_{21}{D}_{\theta \theta}+{d}_{22}{D}_{zz} $$

(4.87)

where d₂₁ = c₁₂ − c₂₃ and d₂₂ = c₂₂ − c₂₃. Equation (4.83) can be substituted into Eq. (4.79) to obtain the following log-exp strains needed in Eq. (4.87):

$$ {D}_{\theta z}={e}_{\theta z}\exp \left[n\left({J}_1-3\right)\right] $$

(4.88)

$$ {D}_{\theta \theta}=\ln {\lambda}_{\theta}\exp \left[n\left({J}_1-3\right)\right] $$

(4.89)

$$ {D}_{zz}=\ln {\lambda}_z\exp \left[n\left({J}_1-3\right)\right] $$

(4.90)

If Eqs. (4.87)–(4.90) are substituted into Eq. (4.86), the result is:

$$ {\sigma}_{\theta z}=\left[\frac{2{c}_{66}{\lambda}_{\theta }{e}_{\theta z}}{\xi_{\theta }}+{d}_{21}\ln {\lambda}_{\theta }+{d}_{22}\ln {\lambda}_z\right]{\xi}_{\theta }{\lambda}_z\exp \left[n\left({J}_1-3\right)\right]. $$

(4.91)

The shear modulus of Cauchy stress with respect to Cauchy small strain can be evaluated as (Lu et al., 2003):

$$ G=\frac{\sigma_{\theta z}}{\left(\alpha r/L\right)}=\frac{\lambda_z{\sigma}_{\theta z}}{\xi_{\theta }} $$

(4.92)

The substitution of Eqs. (4.84) and (4.91) into Eq. (4.92) yields

$$ G=\left[\frac{2{c}_{66}{\lambda}_{\theta}^2\ln \left({\lambda}_{\theta }/{\lambda}_z\right)}{\lambda_{\theta}^2-{\lambda}_z^2}+{d}_{21}\ln {\lambda}_{\theta }+{d}_{22}\ln {\lambda}_z\right]{\lambda}_z^2\exp \left[n\left({J}_1-3\right)\right] $$

(4.93)

Since λ_r = 1/(λ_θλ_z) and ξ_θ is a small value in Eq. (4.80), i.e., ${J}_1\approx {\lambda}_{\theta}^2+{\lambda}_z^2+1/\left({\lambda}_{\theta}^2{\lambda}_z^2\right)$, one can readily conclude from Eq. (4.93) that the incremental shear modulus G depends on the circumferential and axial stretches nonlinearly. Providing that c₆₆, d₂₁ and d₂₂ are material constants, G will be constant for given λ_θ and λ_z, independent of Cauchy small shear strain ξ_θ/(2λ_z) = αr/(2L), consistent with the findings in (Lu et al., 2003). The small shear strain assumption is still valid in the above analysis although the strains in the circumferential and axial directions are large in the inflation–stretch–torsion test.

Appendix 8: Incompressibility in the Generalized Hooke’s Law (Liu, Zhang, et al., 2011)

A cylindrically orthotropic elastic model that accounts for large deformation of arteries is necessary to capture the mechanical response of the vessel. For simplicity, a linearized stress–strain relation (constitutive model) is used that makes the analysis easier and more amenable to interpretation as outlined Appendix 6.

As per standard notations in continuum mechanics, the deformation gradient F can be decomposed as tensor product of a rigid rotation tensor R (R^TR = I) and right stretch tensor U; i.e., F = RU. The right Cauchy-Green deformation tensor C = F^TF = U². Following the approach in Appendix 6, a logarithmic-exponential (log-exp) strain tensor D that absorbs the material nonlinearity is defined as:

$$ \mathbf{D}=\frac{1}{2}Q\ln \mathbf{C}=\exp \left[n\left({J}_1-3\right)\right]\ln \mathbf{U} $$

(4.94)

with

$$ Q=\exp \left[n\left({J}_1-3\right)\right],\kern1em {J}_1=\mathrm{tr}\left(\mathbf{C}\right)=\mathrm{tr}\left({\mathbf{U}}^2\right), $$

(4.95)

where n is a nonlinearity parameter, i.e., the tensor D is the product of the logarithmic (natural, Hencky) strain and a scalar Q which is proposed to account for the strong exponential strain–stress relation found in vascular tissues (Fung, 1993). It can be shown, under the principal-axis representation, that D satisfies all the mathematical requirements for a strain tensor (Ogden, 1997). It can also be proven that tr(D)= ln(J)Q, where J = det(F) = det(U), i.e., tr(D) = 0 for incompressible deformation where J ≡ 1. In fact, this property arises from the classical Hencky strain. Furthermore, D is a Lagrangian strain similar to the Green strain E.

Deformation of blood vessels is considered incompressible under physiological loading. The Cauchy stress (σ) for incompressible material is decomposed as σ = σ^e − pI, where p is a Lagrange multiplier that accounts for the hydrostatic pressure and σ^e is the “extra stress” due to deformation of the material. σ^e is also interpreted as the deviatoric stress, i.e., tr(σ^e) = 0. It is noted for incompressible materials that hydrostatic stress σ =  − pI does not lead to any deformation. In other words, the hydrostatic pressure p must be determined from the boundary condition rather than the stress–strain relation. The second Piola–Kirchhoff stress (S), however, cannot be physically decomposed into hydrostatic stress and “extra stress” since S = F⁻¹σF^−T = F⁻¹σ^eF^−T − pC⁻¹ (note J = 1). This presents some complexity for modeling anisotropic incompressible materials in Lagrangian frame, e.g., the linear relationship between D and S becomes problematic since it is difficult to ensure tr(D) = 0 under arbitrary stress S.

A modified formulation is presented to directly address the hydrostatic pressure and material incompressibility, with the introduction of a co-rotational Cauchy stress T (Ogden, 1984) as:

$$ \mathbf{T}=\mathbf{USU}={\mathbf{R}}^T\boldsymbol{\upsigma} \mathbf{R}={\mathbf{R}}^T{\boldsymbol{\upsigma}}^e\mathbf{R}-p\mathbf{I}={\mathbf{T}}^e-p\mathbf{I} $$

(4.96)

Although T is in Lagrangian frame (because both U and S are in Lagrangian frame), it preserves the decomposition of σ, i.e., T^e is a deviatoric “extra stress” generated by material deformation:

$$ \mathrm{tr}\left({\mathbf{T}}^{\mathrm{e}}\right)=\mathrm{tr}\left({\mathbf{R}}^T{\boldsymbol{\upsigma}}^e\mathbf{R}\right)=\mathrm{tr}\left({\boldsymbol{\upsigma}}^e\right)=0 $$

(4.97)

and the trace tr(T) is exclusively contributed by the hydrostatic stress −pI which is independent of deformation.

4.1.1 Incompressibility Condition

It is proposed that the stress T and the log-exp strain D are connected by a constant fourth-order compliance tensor M as:

$$ \mathbf{D}=\mathbf{M}:\mathbf{T}=\mathbf{M}:\left({\mathbf{T}}^e-p\mathbf{I}\right). $$

(4.98)

The tensor M has major and minor symmetries, and must satisfy certain conditions such as M : I = 0 (Mehrabadi & Cowin, 1999), i.e., to ensure zero deformation (D = 0) when the stress is hydrostatic (T =  − pI). If the orthotropic symmetry of vessel tissue is considered, Eq. (4.98) can be expressed in a matrix form (generalized Hooke’s law):

$$ {\displaystyle \begin{array}{ll}\left(\begin{array}{c}{D}_{\theta \theta}\\ {}{D}_{zz}\\ {}{D}_{rr}\\ {}\sqrt{2}{D}_{zr}\\ {}\sqrt{2}{D}_{\theta r}\\ {}\sqrt{2}{D}_{\theta z}\end{array}\right)=& \left[\begin{array}{cccccc}1/{E}_{\theta }& -{\nu}_{\theta z}/{E}_{\theta }& -{\nu}_{\theta r}/{E}_{\theta }& 0& 0& 0\\ {}-{\nu}_{\theta z}/{E}_{\theta }& 1/{E}_z& -{\nu}_{zr}/{E}_z& 0& 0& 0\\ {}-{\nu}_{\theta r}/{E}_{\theta }& -{\nu}_{zr}/{E}_z& 1/{E}_r& 0& 0& 0\\ {}0& 0& 0& 1/2{G}_{zr}& 0& 0\\ {}0& 0& 0& 0& 1/2{G}_{\theta r}& 0\\ {}0& 0& 0& 0& 0& 1/2{G}_{\theta z}\end{array}\right]\\ {}& \left(\begin{array}{c}{T}_{\theta \theta}^e\\ {}{T}_{zz}^e\\ {}{T}_{rr}^e\\ {}\sqrt{2}{T}_{zr}^e\\ {}\sqrt{2}{T}_{\theta r}^e\\ {}\sqrt{2}{T}_{\theta z}^e\end{array}\right),\end{array}} $$

(4.99a)

or

$$ \left\{D\right\}=\left[M\right]\left\{{T}^e\right\} $$

(4.99b)

where θ, z, and r denote circumferential, axial, and radial directions. The parameters E_θ, E_z, and E_r can be interpreted as Young’s moduli in reference to the log-exp strains, G_θz, G_θr, and G_zr are shear moduli, and v_θz, v_θr, and v_zr are Poisson’s ratios. Except for the definition of strain and stress, this constitutive relation is identical to the classical generalized Hooke’s law for orthotropic elastic materials. Note that the use of factor $\sqrt{2}$ in Eqs. (4.99a, 4.99b) preserves the tensorial properties in the matrix presentation (Mehrabadi & Cowin, 1999).

Unlike the general orthotropic materials where the six parameters E’s and ν’s in Eqs. (4.99a, 4.99b) are independent, the incompressibility condition M : I = 0 imposes restrictions (Itskov & Aksel, 2002; Loredo & Klocker, 1997), namely:

$$ {v}_{ij}=\frac{E_i}{2}\left(\frac{1}{E_i}+\frac{1}{E_j}-\frac{1}{E_k}\right)\kern1em \left( ij=\theta z,\theta r, zr;k\ne i,j\right) $$

(4.100)

such that Eqs. (4.99a, 4.99b) is rewritten for vessel tissue as:

$$ {\displaystyle \begin{array}{ll}& \left(\begin{array}{c}\frac{2{D}_{rr}-{D}_{\theta \theta}-{D}_{zz}}{\sqrt{6}}\\ {}\frac{D_{zz}-{D}_{\theta \theta}}{\sqrt{2}}\\ {}\sqrt{2}{D}_{zr}\\ {}\sqrt{2}{D}_{\theta r}\\ {}\sqrt{2}{D}_{\theta z}\end{array}\right)=\left[\begin{array}{ccccc}\frac{3}{2{E}_r}& \frac{\sqrt{3}}{2}\left(\frac{1}{E_{\theta }}-\frac{1}{E_z}\right)& 0& 0& 0\\ {}\frac{\sqrt{3}}{2}\left(\frac{1}{E_{\theta }}-\frac{1}{E_z}\right)& \frac{1}{E_{\theta }}+\frac{1}{E_z}-\frac{1}{2{E}_r}& 0& 0& 0\\ {}0& 0& 1/2{G}_{zr}& 0& 0\\ {}0& 0& 0& 1/2{G}_{\theta r}& 0\\ {}0& 0& 0& 0& 1/2{G}_{\theta z}\end{array}\right]\\ {}& \left(\begin{array}{c}\frac{2{T}_{rr}^e-{T}_{\theta \theta}^e-{T}_{zz}^e}{\sqrt{6}}\\ {}\frac{T_{zz}^e-{T}_{\theta \theta}^e}{\sqrt{2}}\\ {}\sqrt{2}{T}_{zr}^e\\ {}\sqrt{2}{T}_{\theta r}^e\\ {}\sqrt{2}{T}_{\theta z}^e\end{array}\right)\end{array}} $$

(4.101a)

or

$$ \left\{\widehat{D}\right\}=\left[\widehat{M}\right]\left\{{\widehat{T}}^e\right\} $$

(4.101b)

which involves six, instead of nine, material parameters: {E_θ, E_z, E_r, G_θz, G_θr, G_zr}. Together with tr(D) = D_θθ + D_zz + D_rr = 0 and $\mathrm{tr}\left({\mathbf{T}}^e\right)={T}_{\theta \theta}^e+{T}_{zz}^e+{T}_{rr}^e=0$, Eqs. (4.101a, 4.101b) is identical to Eqs. (4.99a, 4.99b) with the advantage that the relation is invertible (note: matrix [M] in Eqs. (4.99a, 4.99b) is singular), i.e.,

$$ \left\{{\widehat{T}}^e\right\}={\left[\widehat{M}\right]}^{-1}\left\{\widehat{D}\right\} $$

(4.102)

which is convenient for strain-based computations.

Equilibrium equations and boundary conditions for finite strain deformations are conventionally formulated in terms of second Piola–Kirchhoff stress S or Cauchy stress σ. Given a deformation gradient F, the log-exp strain D is calculated by Eqs. (4.94) and (4.95), then the deviatoric co-rotational Cauchy stress T^e is calculated by Eq. (4.102) and condition tr(T^e) = 0, and finally S or σ is obtained by Eq. (4.96). For some deformation modes such as the inflation–stretch experiment on vessels, the shear stress vanish, and the stress–strain relation (Eqs. 4.101a, 4.101b and 4.102) can be further simplified as in Eq. (4.104).

As formulated in (Appendix 7), the right Cauchy-Green deformation tensor C has the following non-zero components:

$$ {C}_{rr}={\lambda}_r^2,\kern1em {C}_{\theta \theta}={\lambda}_{\theta}^2,\kern1em {C}_{z z}={\lambda}_z^2+{\xi}_{\theta}^2,\kern1em {C}_{\theta z}={C}_{z\theta}={\lambda}_{\theta }{\xi}_{\theta }, $$

(4.103)

where λ_θ, λ_z, and λ_r are the local material stretches in the circumferential, axial, and radial directions, respectively, computed from the applied axial stretch (λ_z), the deformed radius $\left({r}_{\mathrm{o}}^e\right)$, and the zero-stress data (R_i, R_o, and Φ) (Wang et al., 2006). The torsion term (ξ_θ) is calculated by ξ_θ(r) = αr/L₀, where r is the current radius and L₀ is the initial (no-load) length of the specimen. In the log-exp strain (Eqs (4.94) and (4.95) ${J}_1=\mathrm{tr}\left(\mathbf{C}\right)={C}_{rr}+{C}_{\theta \theta}+{C}_{zz}={\lambda}_{\theta}^2+{\lambda}_z^2+{\lambda}_r^2+{\xi}_{\theta}^2$ for axial torsion deformation according to Eq. (4.103). The stress–strain relation is given by:

$$ {\displaystyle \begin{array}{ll}\left(\begin{array}{c}\frac{2{D}_{rr}-{D}_{\theta \theta}-{D}_{zz}}{\sqrt{6}}\\ {}\frac{D_{zz}-{D}_{\theta \theta}}{\sqrt{2}}\\ {}\sqrt{2}{D}_{\theta z}\end{array}\right)=& \left[\begin{array}{ccc}\frac{3}{2{E}_r}& \frac{\sqrt{3}}{2}\left(\frac{1}{E_{\theta }}-\frac{1}{E_z}\right)& 0\\ {}\frac{\sqrt{3}}{2}\left(\frac{1}{E_{\theta }}-\frac{1}{E_z}\right)& \frac{1}{E_{\theta }}+\frac{1}{E_z}-\frac{1}{2{E}_r}& 0\\ {}0& 0& 1/2{G}_{\theta z}\end{array}\right]\\ {}& \left(\begin{array}{c}\frac{2{T}_{rr}^e-{T}_{\theta \theta}^e-{T}_{zz}^e}{\sqrt{6}}\\ {}\frac{T_{zz}^e-{T}_{\theta \theta}^e}{\sqrt{2}}\\ {}\sqrt{2}{T}_{\theta z}^e\end{array}\right).\end{array}} $$

(4.104)

4.1.2 Identification of Material Parameters

The material parameters involved in the stress–strain relation Eq. (4.104), {n, E_θ, E_z, E_r}, are first fitted from stretch–inflation data where the axial twist angle is α = 0. As formulated in Appendix 6, the complete axisymmetric Cauchy stress field σ (=T^e − pI noting that R = I here) is solved by integrating the Eulerian equilibrium equation in radial direction and taking into account the stress-free condition on the outer surface of the vessel. The calculated (denoted by superscript c) internal pressure ${P}^c\left({r}_{\mathrm{o}}^e,{\lambda}_z\right)$ and axial force ${F}_z^c\left({r}_{\mathrm{o}}^e,{\lambda}_z\right)$ are obtained by integration across the vessel wall as:

$$ {P}^c={\int}_{r_{\mathrm{i}}^e}^{r_{\mathrm{o}}^e}\frac{\sigma_{rr}^e-{\sigma}_{\theta \theta}^e}{r} dr\kern1em \mathrm{and}\kern1em {F}_z^c={\int}_{r_{\mathrm{i}}^e}^{r_{\mathrm{o}}^e}2{\pi \sigma}_{zz}r\; dr+\pi {P}^e{\left({r}_{\mathrm{i}}^e\right)}^2, $$

(4.105)

where ${\sigma}_{rr}^e$ and ${\sigma}_{\theta \theta}^e$ are the components of the “extra stress” σ^e calculated by σ^e = RT^eR^T as detailed below. The second relation is because the axial force measured by force transducer includes contribution from both the tissue and the pressure (Wang et al., 2006). Similar to previous works (Wang et al., 2006; Zhang, Wang, et al., 2007) or Appendix 7), a search for {n, E_θ, E_z, E_r} is made that minimizes an error function given by:

$$ \mathrm{err}1\left(n,{E}_{\theta },{E}_z,{E}_r\right)=\frac{\sum \left[{\left({P}^c-{P}^e\right)}^2+{w}_{\mathrm{F}}{\left({F}_z^c-{F}_z^e\right)}^2/{A}_{\mathrm{l}}^2\right]}{\sum \left({P}^{e2}+{w}_{\mathrm{F}}{F}^{e2}/{A}_{\mathrm{l}}^2\right)}, $$

(4.106)

where summation is conducted over all stretch–inflation data points, A_l is the luminal area (calculated from ${r}_{\mathrm{o}}^e$ and the zero-stress geometry using incompressible condition). It is found that w_F = 0.4 gives overall good fit for the data. In all measurements and calculations, the unit is kPa for pressure, mN for the axial force, and mm² for the luminal area. It should be noted that a combined fitting error is used, while in Wang et al. (2008) the fitting errors for pressure and force are treated separately. A limited-memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS) quasi-Newton minimization method (Liu & Nocedal, 1989) is used for the data curve fitting.

4.1.3 Identification of Shear Parameters

The shear parameter G_θz is identified from torsion data with twist angle α ≠ 0. Given a trial G_θz, the deviatoric stress T^e is calculated via Eq. (4.104), and the deviatoric Cauchy stress σ^e is obtained as σ^e = RT^eR^T. The total torque M^c is then calculated as:

$$ {M}^c=2\pi {\int}_{r_{\mathrm{i}}^e}^{r_{\mathrm{o}}^e}{\sigma}_{\theta z}{r}^2 dr $$

(4.107)

where ${r}_{\mathrm{o}}^e$ and ${r}_{\mathrm{i}}^e$ are the outer and inner radius of the deformed vessel. It is noted that the shear stress σ_θz is coupled with the normal stresses as indicated by the definition of D and that R ≠ I in torsion. Therefore, σ^e depends on normal parameters {n, E_θ, E_z, E_r}. Finally,G_θz is identified by minimizing an error function:

$$ \mathrm{err}2\left({G}_{\theta z}\right)=\frac{\sum {\left({M}^c-{M}^e\right)}^2}{\sum {M}^{e2}} $$

(4.108)

4.1.4 Deformation

As detailed in Appendix 7, the deformation gradient F and the right Cauchy-Green deformation tensor C of the vessel with respect to the zero-stress state are given by:

$$ \mathbf{F}=\left(\begin{array}{ccc}{\lambda}_r& 0& 0\\ {}0& {\lambda}_{\theta }& {\xi}_{\theta}\\ {}0& 0& {\lambda}_z\end{array}\right),\kern1em \mathbf{C}=\left(\begin{array}{ccc}{\lambda}_r^2& 0& 0\\ {}0& {\lambda}_{\theta}^2& {\lambda}_{\theta }{\xi}_{\theta}\\ {}0& {\lambda}_{\theta }{\xi}_{\theta }& {\lambda}_{\theta}^2+{\xi}_{\theta}^2\end{array}\right)\kern1em \mathrm{in}\kern0.17em \mathrm{coordinate}\kern1em \left\{{\mathbf{e}}_r,{\mathbf{e}}_{\theta },{\mathbf{e}}_z\right\} $$

(4.109)

where ξ_θ = λ_zαr ⋅ L⁻¹, with r being the current radius of the material point, and L the current length of the vessel. The eigenvalues (Λ_i) and eigenvectors {u_i} (i = 1,2,3) of C are calculated numerically as C = ∑_{i = 1, 3}Λ_iu_i ⊗ u_i. Then, spectral decomposition F = RU is conducted as:

$$ \mathbf{U}=\sum \limits_{i=1,3}\sqrt{\Lambda_i}{\mathbf{u}}_i\otimes {\mathbf{u}}_i\kern1em \mathrm{and}\kern1em \mathbf{R}={\mathbf{FU}}^{-1}=\mathbf{F}\cdot \left(\sum \limits_{i=1,3}{\left(\sqrt{\Lambda_i}\right)}^{-1}{\mathbf{u}}_i\otimes {\mathbf{u}}_i\right) $$

(4.110)

Finally, the logarithmic-exponential strain tensor D defined in Eqs. (4.94) and (4.95) is:

$$ \mathbf{D}=\frac{1}{2}Q\ln \mathbf{C}=\exp \left[n\left({\Lambda}_1+{\Lambda}_2+{\Lambda}_3-3\right)\right]\left(\sum \limits_{i=1,3}\ln {\Lambda}_i{\mathbf{u}}_i\otimes {\mathbf{u}}_i\right) $$

(4.111)

4.1.5 Stress

It follows from Eq. (4.104) together with the condition tr(σ^e) = 0 that:

$$ {\displaystyle \begin{array}{ll}\left(\begin{array}{c}{T}_{\theta \theta}^e\\ {}{T}_{zz}^e\\ {}{T}_{rr}^e\\ {}\sqrt{2}{T}_{\theta z}^e\end{array}\right)=& \left(\begin{array}{ccc}\sqrt{\frac{2}{3}}& 0& 0\\ {}-\frac{1}{\sqrt{6}}& \frac{1}{\sqrt{2}}& 0\\ {}-\frac{1}{\sqrt{6}}& -\frac{1}{\sqrt{2}}& 0\\ {}0& 0& 1\end{array}\right){\left[\begin{array}{ccc}\frac{3}{2{E}_r}& \frac{\sqrt{3}}{2}\left(\frac{1}{E_{\theta }}-\frac{1}{E_z}\right)& 0\\ {}\frac{\sqrt{3}}{2}\left(\frac{1}{E_{\theta }}-\frac{1}{E_z}\right)& \frac{1}{E_{\theta }}+\frac{1}{E_z}-\frac{1}{2{E}_r}& 0\\ {}0& 0& 1/2{G}_{\theta z}\end{array}\right]}^{-1}\\ {}& \left(\begin{array}{c}\frac{2{D}_{rr}-{D}_{\theta \theta}-{D}_{zz}}{\sqrt{6}}\\ {}\frac{D_{zz}-{D}_{\theta \theta}}{\sqrt{2}}\\ {}\sqrt{2}{D}_{\theta z}\end{array}\right)\end{array}} $$

(4.112)

4.1.6 Stress Components in Axisymmetric Deformation (Eq. 4.105)

In axisymmetric deformation, R = I such that σ^e = T^e. Therefore,

$$ \left(\begin{array}{l}{\sigma}_{rr}^e\\ {}{\sigma}_{\theta \theta}^e\\ {}{\sigma}_{zz}^e\end{array}\right)=\left(\begin{array}{cc}\sqrt{\frac{2}{3}}& 0\\ {}-\frac{1}{\sqrt{6}}& -\frac{1}{\sqrt{2}}\\ {}-\frac{1}{\sqrt{6}}& \frac{1}{\sqrt{2}}\end{array}\right){\left[\begin{array}{cc}\frac{3}{2{E}_r}& \frac{\sqrt{3}}{2}\left(\frac{1}{E_{\theta }}-\frac{1}{E_z}\right)\\ {}\frac{\sqrt{3}}{2}\left(\frac{1}{E_{\theta }}-\frac{1}{E_z}\right)& \frac{1}{E_{\theta }}+\frac{1}{E_z}-\frac{1}{2{E}_r}\end{array}\right]}^{-1}\left(\begin{array}{l}\frac{2{D}_{rr}-{D}_{\theta \theta}-{D}_{zz}}{\sqrt{6}}\\ {}\frac{D_{zz}-{D}_{\theta \theta}}{\sqrt{2}}\end{array}\right) $$

(4.113)

The integration of the radial equilibrium dσ_rr/dr + (σ_rr − σ_θθ)/r = 0, subject to boundary condition σ_rr = 0 on the external surface of the vessel (r = r_out), gives

$$ {\sigma}_{rr}(r)={\int}_{r_{\mathrm{out}}}^r\frac{1}{s}\left({\sigma}_{rr}^e(s)-{\sigma}_{\theta \theta}^e(s)\right) ds $$

(4.114)

with ${\sigma}_{rr}-{\sigma}_{\theta \theta}={\sigma}_{rr}^e-{\sigma}_{\theta \theta}^e$. In addition, the axial stress σ_zz can be derived as:

$$ {\sigma}_{zz}={\sigma}_{rr}+{\sigma}_{zz}^e-{\sigma}_{rr}^e $$

(4.115)

4.1.7 Stress Components in Axial Torsion (Eq. 4.107)

In axial torsion, the rigid rotation tensor R ≠ I. Therefore, the shear stress σ_θz involves all components in T^e through the relation σ^e = RT^eR^T. Specifically, the shear stress in Eq. (4.107) is given by:

$$ {\sigma}_{\theta z}={\sigma}_{\theta z}^e={R}_{\theta \theta}{R}_{z\theta}{T}_{\theta \theta}^e+\left({R}_{\theta z}{R}_{z\theta}+{R}_{\theta \theta}{R}_{z z}\right){T}_{\theta z}^e+{R}_{\theta r}{R}_{z r}{T}_{rr}^e+{R}_{\theta z}{R}_{z z}{T}_{z z}^e $$

(4.116)

Table 4.20 Material parameters of porcine left anterior descending arteries (LAD)

Appendix 9: Viscoelasticity

4.1.1 Coronary Arteries (Zhang, Chen, et al., 2007)

Figure 4.21 is a schematic diagram of the generalized Maxwell model, where a linear spring with the elastic modulus (stiffness constant) μ₀ is connected with m Maxwell elements in parallel. In the i-th Maxwell body, the spring has an elastic modulus μ_i and the dashpot has a viscous coefficient η_i (i = 1, …, m). The stress in the i-th Maxwell body is σ_i = μ_iε_i = η_id(ε − ε_i)/dt.

Fig. 4.21

A generalized Maxwell viscoelastic model

Since the components are connected in parallel in Fig. 4.21, all elements have the same strain (deformation) equal to the overall strain ε(t). The total stress of the whole system is the sum of the stresses in each element as given by Fung (1993):

$$ \sigma (t)=\sum \limits_{i=0}^m{\sigma}_i(t)=\left({\mu}_0+\sum \limits_{i=1}^m\frac{D}{D/{\mu}_i+1/{\eta}_i}\right)\varepsilon (t), $$

(4.117)

where D = d/dt denotes differentiation with respect to time.

To reduce the number of model parameters, ω_i is denoted as the characteristic frequency of i-th Maxwell element and assume that the characteristic frequencies form a geometric series, namely:

$$ {\omega}_i=\frac{\mu_i}{\eta_i}={\rho}^{i-1}/\tau \kern1em \left(i=1,\dots, m\right), $$

(4.118)

where ρ is a nondimensional real constant characterizing the “gap” between successive frequencies, τ represents the characteristic relaxation time of the first Maxwell element (inverse of the characteristic frequency ω₁).

Substituting Eq. (4.118) into Eq. (4.117), the following differential form of the constitutive equation can be obtained:

$$ \sigma (t)=\left({\mu}_0+\sum \limits_{i=1}^m\frac{\mu_iD}{D+{\rho}^{i-1}/\tau}\right)\varepsilon (t). $$

(4.119)

The constitutive relation in Eq. (4.119) still has the main shortcoming of a generalized Maxwell model, i.e., the number of material constants increases with the number of Maxwell elements. To acquire a series of μ_i that allow Eq. (4.119) to capture rate-insensitive hysteresis behavior, the cyclic loading condition must be considered.

4.1.2 Response to Oscillatory Loading

The response of a linear viscoelastic material to an oscillating load is typically studied with complex variable functions (Fung, 1993). If the applied load is assumed to be an oscillatory strain containing a single angular frequency ω (the real part corresponds to the actual load), the oscillating strain can be written as:

$$ \varepsilon (t)={\varepsilon}_0{\mathrm{e}}^{j\omega t}, $$

(4.120)

where ε₀ is the amplitude of strain and $j=\sqrt{-1}$ denotes the imaginary number. Under the strain loading of Eq. (4.120), the stress response is also an oscillation at the same frequency with a leading phase angle δ (Findley, Lai, & Onaran, 1989):

$$ \sigma (t)={\sigma}_0{\mathrm{e}}^{j\left(\omega t+\delta \right)}=\left({\sigma}_0{\mathrm{e}}^{j\delta}\right){\mathrm{e}}^{j\omega t}, $$

(4.121)

where σ₀ is the amplitude of stress.

According to Eq. (4.120), the differentiation of strain with respect to time can be simply written as (Fung, 1993):

$$ D\varepsilon (t)= j\omega \varepsilon (t), $$

(4.122)

which reveals that the differentiation D is equivalent to a multiplication by jω.

Given Eqs. (4.120)–(4.122), the differential constitutive equation (Eq. 4.119) can be rewritten as:

$$ \left({\sigma}_0{\mathrm{e}}^{j\delta}\right){\mathrm{e}}^{j\omega t}=\left({\mu}_0+\sum \limits_{i=1}^m\frac{j{\omega \mu}_i}{j\omega +{\rho}^{i-1}/\tau}\right){\varepsilon}_0{\mathrm{e}}^{j\omega t}. $$

(4.123)

From Eq. (4.123), the complex modulus (or dynamic modulus) (4.118, 4.122, 4.123) of the generalized Maxwell body can be obtained as:

$$ {E}^{\ast}\left(\omega \right)=\frac{\sigma (t)}{\varepsilon (t)}=\frac{\sigma_0}{\varepsilon_0}{\mathrm{e}}^{j\delta}={\mu}_0+\sum \limits_{i=1}^m\frac{j{\omega \mu}_i}{j\omega +{\rho}^{i-1}/\tau }. $$

(4.124)

The mechanical loss (a measure of the internal friction) is defined as the tangent of the phase angle δ (Fung, 1993):

$$ \tan \left(\delta \right)=\frac{\operatorname{Im}\left({E}^{\ast}\left(\omega \right)\right)}{\operatorname{Re}\left({E}^{\ast}\left(\omega \right)\right)}, $$

(4.125)

where Re and Im represent the real and imaginary parts of the complex variable, respectively. The nondimensional tan(δ) in Eq. (4.125) is a function of frequency ω and is proportional to the ratio of dissipated energy to stored energy in a dynamic loading cycle (Lakes, 1999). Thus, rate insensitivity of the generalized Maxwell model can be realized by selecting an appropriate elastic modulus for each Maxwell element to make tan(δ) nearly independent of ω. The following series of elastic moduli fits the criterion:

$$ {\mu}_i=\beta {\left(1+\beta \right)}^{i-1}{\mu}_0\kern1em \left(i=1,\dots, m\right), $$

(4.126)

where β = μ₁/μ₀ is the ratio of the elastic modulus of the first Maxwell body (which has a characteristic relaxation time τ as shown in Eq. (4.118)) to that of the first spring (see Fig. 4.21). Using Eq. (4.126), the differential constitutive model (Eq. 4.119) can be written as:

$$ \sigma (t)={\mu}_0\left(1+\beta \sum \limits_{i=1}^m\frac{{\left(1+\beta \right)}^{i-1}D}{D+{\rho}^{i-1}/\tau}\right)\varepsilon (t), $$

(4.127)

and the complex modulus (Eq. 4.124) becomes:

$$ {E}^{\ast}\left(\omega \right)={\mu}_0\left(1+\beta \sum \limits_{i=1}^m\frac{j\omega {\left(1+\beta \right)}^{i-1}}{j\omega +{\rho}^{i-1}/\tau}\right). $$

(4.128)

There are five material constants in the model: μ₀, τ, m, ρ, and β, where μ₀ has the unit of stress, τ has the unit of time, and the other three parameters are nondimensional. For given model parameters, the internal friction or the normalized energy dissipation can be examined by plotting tan(δ) against frequency ω (the normalized energy dissipation differs from tan(δ) by a multiplication factor (Lakes, 1999)).

The real part of the applied complex strain load in Eq. (4.120) is a cosine function containing a single angular frequency ω:

$$ \varepsilon (t)={\varepsilon}_0\cos \left(\omega t\right). $$

(4.129)

The corresponding stress response is the real part of Eq. (4.121), which is another cosine function with the same frequency (Lakes, 1999):

$$ \sigma (t)={\sigma}_0\cos \left(\omega t+\delta \right). $$

(4.130)

The hysteresis loop can be obtained by plotting σ(t) versus ε(t), which should be nearly independent of angular frequency ω in a wide range because the internal friction tan(δ) is insensitive to the loading frequency.

4.1.2.1 Relaxation and Creep Functions

Under a given constant strain load ε₀, the relaxation function can be expressed by Fung (1993):

$$ G(t)=\frac{\sigma (t)}{\varepsilon_0}={\mu}_0+\sum \limits_{i=1}^m{\mu}_i{\mathrm{e}}^{-{\omega}_it}. $$

(4.131)

Using Eqs. (4.118) and (4.126), the reduced relaxation function (Fung, 1993) can be obtained from Eq. (4.131) as:

$$ g(t)=\frac{G(t)}{G(0)}=\frac{1+\beta {\sum}_{i=1}^m{\left(1+\beta \right)}^{i-1}{\mathrm{e}}^{-{\rho}^{i-1}t/\tau }}{1+\beta {\sum}_{i=1}^m{\left(1+\beta \right)}^{i-1}}, $$

(4.132)

from which one can easily show the following holds:

$$ g(0)=1,\kern1em g\left(\infty \right)=\frac{1}{{\left(1+\beta \right)}^m}. $$

(4.133)

Equation (4.133) indicates that the lower bound of the reduced relaxation function, g(∞), depends on m and β only.

Under a constant stress load σ₀, the strain of a viscoelastic body varies with time. The creep function (compliance) of a linear viscoelastic material can be defined by Fung (1993):

$$ J(t)=\frac{\varepsilon (t)}{\sigma_0}. $$

(4.134)

According to the linear viscoelastic theory, the creep function and the relaxation function are related by the following convolution (Lakes, 1999):

$$ {\int}_0^tJ\left(t-\xi \right)E\left(\xi \right) d\xi =t. $$

(4.135)

Equation (4.135) can be used to obtain the creep function if the relaxation function is known, or vice versa. Findley et al. (1989) provided a partial fraction expansion method to solve Eq. (4.135) using the Laplace transform.

4.1.3 Opening Angle (Zhang et al., 2008)

To simulate the viscoelasticity of opening angle, a generalized Maxwell body is considered where a spring with elastic modulus μ₀ is connected serially with m Voigt elements (springs in parallel with dashpots, Fig. 4.22). In the i-th Voigt body, the spring has an elastic modulus μ_i and the dashpot has a viscous coefficient η_i (i = 1, …, m). Under a constant stress load σ₀, the creep function J(t) is the responsive strain ε(t) divided by σ₀ (Fung, 1993; Findley et al. 1989):

$$ J(t)=\frac{\varepsilon (t)}{\sigma_0}=\frac{1}{\mu_0}+\sum \limits_{i=1}^m\frac{1}{\mu_i}\left(1-{\mathrm{e}}^{-t/{\tau}_i}\right), $$

(4.136)

where t denotes time.

Fig. 4.22

A generalized Maxwell viscoelastic model (a linear spring in serial with m Voigt elements)

To reduce model parameters, it is assumed that τ_i = η_i/μ_i = ρⁱ⁻¹τ (where i − 1 indicates a power) and μ_i = μ₀/[β(1 + β)ⁱ⁻¹] which imply that the characteristic frequencies form a geometric series and the elastic moduli of each Voigt element are interrelated (see Appendix “Coronary Arteries (Zhang, Chen, et al., 2007)”). As a result, the reduced creep function can be obtained as:

$$ j(t)=\frac{J(t)}{J\left(\infty \right)}=\frac{1+\beta {\sum}_{i=1}^m{\left(1+\beta \right)}^{i-1}\left(1-{\mathrm{e}}^{-{\rho}^{1-i}t/\tau}\right)}{{\left(1+\beta \right)}^m}, $$

(4.137)

where τ is a characteristic relaxation time, ρ characterizes the gap between successive relaxation times, β = μ₀/μ₁, and J(∞) = (1 + β)^m/μ₀.

For a ring cut from loaded state, the strain decreases first elastically as a step, followed by a creep process. The creep recovery can be described by the superposition principal (Findley et al., 1989). Suppose the loaded artery is under a stress σ₀ in the circumferential direction and the viscous stress has been fully relaxed (the corresponding strain is (Eqs. 4.136 and 4.137):

$$ {\varepsilon}_0={\sigma}_0J\left(\infty \right)={\sigma}_0{\left(1+\beta \right)}^m/{\mu}_0. $$

(4.138)

Starting from t = 0, the ring is cut open and a stress −σ₀ induces a new strain (using Eqs. (4.136) and (4.137)) given by:

$$ {\varepsilon}_0^{\prime }(t)=-{\sigma}_0J(t)=-\frac{\sigma_0J\left(\infty \right)}{{\left(1+\beta \right)}^m}\left[1+\beta \sum \limits_{i=1}^m{\left(1+\beta \right)}^{i-1}\left(1-{\mathrm{e}}^{-{\rho}^{1-i}t/\tau}\right)\right]. $$

(4.139)

The total strain recovery ε_r(t) after the radial cut (stress is completely released) is:

$$ {\varepsilon}_r(t)={\varepsilon}_0+{\varepsilon}_0^{\prime }(t)=\frac{\varepsilon_0\beta }{{\left(1+\beta \right)}^m}\sum \limits_{i=1}^m{\left(1+\beta \right)}^{i-1}{\mathrm{e}}^{-{\rho}^{1-i}t/\tau }. $$

(4.140)

Note that for m = 1, the model in Fig. 4.22 is mathematically equivalent to a spring in parallel with a Maxwell body (Fung, 1993; Orosz, Molnarka, & Monos, 1997). Therefore, the Kelvin model considered by Rehal et al. (2006) is a special case of the generalized Maxwell model in Eq. (4.140).

For a vessel ring cut from no-load state, the analysis assumes the following three steps: (1) Loaded artery is fully relaxed (stress is σ₀ at t = 0⁻); (2) No-load state is obtained after elastomer removal (stress is σ₁ from t = 0⁺ to t = t₁ = 1800 s); (3) Vessel ring is cut open at t = t₁ and all stress is released. It is noted that the second step is an approximation (i.e., not strictly under constant stress).

According to the superposition principal (Findley et al., 1989), the strain recovery in the artery after the radial cut can be written as:

$$ {\varepsilon}_n(t)={\sigma}_0J\left(\infty \right)+\left({\sigma}_1-{\sigma}_0\right)J(t)-{\sigma}_1J\left(t-{t}_1\right)\kern1em \left(t>{t}_1\right), $$

(4.141)

where ε_n(t) → 0 when t → ∞ (fully recovered zero-stress state). The substitution of Eqs. (4.137)–(4.140) into Eq. (4.141) results in

$$ {\varepsilon}_n(t)={\varepsilon}_r(t)+\frac{\varepsilon_1\beta {\sum}_{i=1}^m{\left(1+\beta \right)}^{i-1}\left({\mathrm{e}}^{\rho^{1-i}{t}_1/\tau }-1\right){\mathrm{e}}^{-{\rho}^{1-i}t/\tau }}{{\left(1+\beta \right)}^m}\kern1em \left(t>{t}_1\right), $$

(4.142)

where ε₁ = σ₁J(∞) is the residual strain due to residual stress σ₁.

The artery is assumed to be incompressible. The loaded state is a tube with a circular cross section. The circumferential stretch ratio at the inner surface is:

$$ {\lambda}_{\theta }=\frac{2\pi {r}_{\mathrm{i}}}{C_{\mathrm{i}}}, $$

(4.143)

and at the outer surface is:

$$ {\lambda}_{\theta }=\frac{2\pi {r}_{\mathrm{o}}}{C_{\mathrm{o}}}, $$

(4.144)

where r_i and r_o are the inner and outer radii at the loaded state, C_i and C_o are the inner and outer circumferences at the fully relaxed zero-stress state, respectively.

In the generalized Maxwell model, a linear stress–strain relation is expected. Although arteries are known to exhibit nonlinear constitutive behavior, Zhang and Kassab (2007) proposed to absorb the material nonlinearity with a new strain measure in the two-dimensional case (Appendix 5). The same idea is then extended to the three-dimensional Hooke’s law (Appendix 6) where the logarithmic-exponential (log-exp) strain is defined as (no shear deformation):

$$ {D}_{ii}=\ln {\lambda}_i\exp \left[n\left({J}_1-3\right)\right]\kern1em \left(i=\theta, z,r\right), $$

(4.145)

where λ_i are stretch ratios, n is a constant that characterizes the material nonlinearity, and J₁ is the first invariant of the right Cauchy-Green deformation tensor:

$$ {J}_1={\lambda}_{\theta}^2+{\lambda}_z^2+{\lambda}_r^2. $$

(4.146)

The second Piola–Kirchhoff stress and the log-exp strain in the circumferential direction can be written as (Zhang, Wang, et al., 2007):

$$ {S}_{\theta \theta}={c}_{11}{D}_{\theta \theta}+{c}_{12}{D}_{zz}+{c}_{13}{D}_{rr}, $$

(4.147)

where c’s are elastic moduli with respect to the log-exp strains.

As an estimate, it is assumed that D_zz ≈ 0 (λ_z ≈ 1) which implies that the artery is axially relaxed. In such a case, D_rr = −D_θθ (λ_r = 1/λ_θ) according to Eq. (4.145). The circumferential strain becomes:

$$ \varepsilon ={D}_{\theta \theta}=\ln {\lambda}_{\theta}\exp \left[n\left({\lambda}_{\theta}^2+1/{\lambda}_{\theta}^2-2\right)\right]. $$

(4.148)

It is noted that Eq. (4.147) can naturally reduce to a 1D linear model (S_θθ ∝ ε). Other models using Green strain measure (e.g., Fung model) can be used approximately since the strain is small (thus stress–strain relation can be linearized) during the creep process.

The open sector is characterized by an opening angle Φ. Considering the volumetric incompressibility condition λ_θλ_zλ_r = 1, the cross-sectional wall area A₀ can be calculated by Chuong and Fung (1986):

$$ {A}_0={\pi \lambda}_z\left({r}_{\mathrm{o}}^2-{r}_{\mathrm{i}}^2\right)=\frac{L_{\mathrm{o}}^2(t)-{L}_{\mathrm{i}}^2(t)}{4\left(\pi -\varPhi (t)\right)}, $$

(4.149)

where L_i and L_o are inner and outer circumferences of the sector (not fully relaxed). Equation (4.149) yields the opening angle expressed by:

$$ \varPhi (t)=\pi -\frac{L_{\mathrm{o}}^2(t)-{L}_{\mathrm{i}}^2(t)}{4{A}_0}. $$

(4.150)

For a given A₀, L_i, and L_o, Φ can be computed from Eq. (4.150). It is found that the change of these measures is negligible 2 h after the radial cut. Thereafter, the artery is assumed to be fully relaxed, i.e., L_i = C_i, L_o = C_o, and Φ = Φ₀ at t = 7200 s. The stretch ratio of the open sector λ_θ(t) = L(t)/C is equal to the circumference at a given time over that at the fully relaxed state, which differs from that at the loaded state (Eqs. 4.143 and 4.144). The strain is computed with Eq. (4.148).

Appendix 10: Active Mechanical Properties (Huo et al., 2012)

The blood vessel is assumed to be a thin-walled elastic tube deformed in the circumferential (θ) and axial (z) directions. Green strains are defined as:

$$ {E}_{\theta \theta}=\frac{1}{2}\left[{\lambda}_{\theta}^2-1\right]\kern1em \mathrm{and}\kern1em {E}_{zz}=\frac{1}{2}\left[{\lambda}_z^2-1\right] $$

(4.151)

where ${\lambda}_{\theta }=\frac{D}{D_0}$ and ${\lambda}_z=\frac{L}{L_0}$ are the circumferential and axial stretch ratios, respectively; D and D₀ are diameters in the loaded and zero-stress states, respectively; L and L₀ are axial lengths in the loaded and no-load states, respectively. If the densities of vessel wall are identical in the loaded and no-load state, first Piola–Kirchhoff and Cauchy stresses can be written as:

$$ \left\{\begin{array}{l}{T}_{\theta \theta}={\lambda}_{\theta }{S}_{\theta \theta}\\ {}{T}_{zz}={\lambda}_z{S}_{zz}\end{array}\right.\kern1em \mathrm{and}\kern1em \left\{\begin{array}{l}{\sigma}_{\theta \theta}={\lambda}_{\theta}^2{S}_{\theta \theta}\\ {}{\sigma}_{zz}={\lambda}_z^2{S}_{zz}\end{array}\right. $$

(4.152)

where S_θθ and S_ZZ are the circumferential and axial second Piola–Kirchhoff stresses, respectively.

The total strain energy function consists of passive and active components as:

$$ {W}_{\mathrm{total}}={W}_{\mathrm{passive}}+{W}_{\mathrm{active}} $$

(4.153)

where W_passive is the strain energy of passive vessel; W_active is the active strain energy caused by the K⁺-induced smooth muscle contraction; and W_total is the total strain energy of the K⁺-induced active vessel.

4.1.1 Passive Strain Energy Function

In Fung’s 2D model without shear deformation, the strain energy per unit volume (W_passive) is given by:

$$ {W}_{\mathrm{passive}}=\frac{C_1}{2}\left[\exp (Q)-1\right] $$

(4.154a)

and

$$ Q={a}_1{E}_{\theta \theta}^2+{a}_2{E}_{zz}^2+2{a}_4{E}_{\theta \theta}{E}_{zz} $$

(4.154b)

where C₁, a₁, a₂, and a₄ are constants. The passive second Piola–Kirchhoff stresses are obtained by differentiating the strain energy with respect to the corresponding Green strains as:

$$ \left\{\begin{array}{c}{S}_{{\theta \theta}_{\mathrm{passive}}}=\frac{\partial {W}_{\mathrm{passive}}}{\partial {E}_{\theta \theta}}={C}_1\left({a}_1{E}_{\theta \theta}+{a}_4{E}_{zz}\right)\exp (Q)\\ {}{S}_{zz_{\mathrm{passive}}}=\frac{\partial {W}_{\mathrm{passive}}}{\partial {E}_{zz}}={C}_1\left({a}_2{E}_{zz}+{a}_4{E}_{\theta \theta}\right)\exp (Q)\end{array}\right. $$

(4.155)

The passive first Piola–Kirchhoff stresses can be written as:

$$ \left\{\begin{array}{c}{T}_{{\theta \theta}_{\mathrm{passive}}}={C}_1\left({a}_1{E}_{\theta \theta}+{a}_4{E}_{zz}\right)\sqrt{2{E}_{\theta \theta}+1}\exp (Q)\\ {}{T}_{zz_{\mathrm{passive}}}={C}_1\left({a}_2{E}_{zz}+{a}_4{E}_{\theta \theta}\right)\sqrt{2{E}_{zz}+1}\exp (Q)\end{array}\right. $$

(4.156)

where C₁, a₁, a₂, and a₄ for coronary arteries are determined by experimental measurements.

4.1.2 Active Strain Energy Function

The active strain energy function caused by the K⁺-induced SMC contraction is proposed as:

$$ {\displaystyle \begin{array}{ll}{W}_{\mathrm{active}}& ={C}_2\left(\mathrm{Erf}\left(\frac{\sqrt{2{E}_{\theta \theta}+1}-{b}_3}{b_1}+\frac{\sqrt{2{E}_{zz}+1}-{b}_4}{b_2}\right)-1\right)\\ {}& ={C}_2\left(\mathrm{Erf}\left(\frac{\lambda_{\theta }-{b}_3}{b_1}+\frac{\lambda_z-{b}_4}{b_2}\right)-1\right)\end{array}} $$

(4.157)

where C₂, b₁, b₂, b₃, and b₄ are constants and Erf(X) is the Gauss error function. If ${b}^{\prime }=\frac{b_3}{b_1}+\frac{b_4}{b_2}$, Eq. (4.157) can be simplified as:

$$ {W}_{\mathrm{active}}={C}_2\left(\mathrm{Erf}\left(\frac{\lambda_{\theta }}{b_1}+\frac{\lambda_z}{b_2}-{b}^{\prime}\right)-1\right) $$

(4.158)

Equation (4.159) has 4 unknown parameters instead of 5 in Eq. (4.157). The active second Piola–Kirchhoff stresses are obtained by differentiating the strain energy with respect to the corresponding Green strains as:

$$ \left\{\begin{array}{c}{S}_{{\theta \theta}_{\mathrm{active}}}=\frac{\partial {W}_{\mathrm{active}}}{\partial {E}_{\theta \theta}}=\frac{2{C}_2}{b_1\sqrt{\pi }}\frac{1}{\sqrt{2{E}_{\theta \theta}+1}}\exp \left(-{\left(\frac{\sqrt{2{E}_{\theta \theta}+1}}{b_1}+\frac{\sqrt{2{E}_{zz}+1}}{b_2}-{b}^{\prime}\right)}^2\right)\\ {}{S}_{zz_{\mathrm{active}}}=\frac{\partial {W}_{\mathrm{active}}}{\partial {E}_{zz}}=\frac{2{C}_2}{b_2\sqrt{\pi }}\frac{1}{\sqrt{2{E}_{zz}+1}}\exp \left(-{\left(\frac{\sqrt{2{E}_{\theta \theta}+1}}{b_1}+\frac{\sqrt{2{E}_{zz}+1}}{b_2}-{b}^{\prime}\right)}^2\right)\end{array}\right. $$

(4.159)

The active first Piola–Kirchhoff stresses are obtained by differentiating the strain energy with respect to the corresponding stretch ratios as:

$$ {\displaystyle \begin{array}{l}\left\{\begin{array}{c}{T}_{{\theta \theta}_{\mathrm{active}}}=\frac{2{C}_2}{b_1\sqrt{\pi }}\exp \left(-{Q}^{\prime}\right)\\ {}{T}_{zz_{\mathrm{active}}}=\frac{2{C}_2}{b_2\sqrt{\pi }}\exp \left(-{Q}^{\prime}\right)\end{array}\right.\\ {}\mathrm{with}\\ {}{Q}^{\prime }={\left(\frac{\lambda_{\theta }}{b_1}+\frac{\lambda_z}{b_2}-{b}^{\prime}\right)}^2={\left(\frac{\lambda_{\theta }}{b_1}\right)}^2+{\left(\frac{\lambda_z}{b_2}\right)}^2+2\frac{\lambda_{\theta }}{b_1}\frac{\lambda_z}{b_2}-2\left(\frac{\lambda_{\theta }}{b_1}+\frac{\lambda_z}{b_2}\right){b}^{\prime }+{\left({b}^{\prime}\right)}^2\end{array}} $$

(4.160)

The material constants in Eq. (4.160) are determined by biaxial experimental measurements for intact and intima-media layers of coronary artery wall. The material parameters for six RCA coronary arteries are listed below in Tables 4.21 and 4.22 for passive and active states of intact wall and in Tables 4.23 and 4.24 for the intima-media layer, respectively. Reproduced from Huo et al. (2012) with permission.

Table 4.21 Material constants of Fung’s passive strain energy function of intact coronary artery

Table 4.22 Material constants of K⁺-induced active strain energy function (Eq. 4.160) in the pressure range of 60–200 mmHg for intact coronary artery wall as corresponding to Table 4.21

Table 4.23 Material constants of Fung’s passive strain energy function for intima-media layer of right coronary artery (RCA)

Table 4.24 Material constants of K⁺-induced active strain energy function for intima-media layer in the pressure range of 30–110 mmHg

Appendix 11: Micromechanics of Heterogeneous Materials (Chen, Zhao, Lu, & Kassab, 2013)

The idea of homogenization of heterogeneous nonbiological materials has been used to predict the macroscopic or effective mechanical properties of composites (Milton, 2002). Rigorous and reliable methods are well established and widely applied to linear elastic composites, including the classical Voigt and Reuss bounds (Hill, 1952), the variational principles of Hashin-Shtrikman (Hashin & Shtrikman, 1962, 1963), and the general self-consistent approximations (Hershey, 1954; Hutchinson, 1976). For nonlinear composites, however, rigorous methods have become available only more recently because of difficulties in addressing both strong material nonlinearity and heterogeneity. Earlier efforts are made to predict the effective constitutive behaviors of composites by extending linear methods to nonlinear materials, such as the extensions of the self-consistent procedures (Hill, 1965), or the Hashin-Shtrikman variational principles for linear composites (Talbot & Willis, 1985; Willis, 1983).

A general variational procedure for estimating the effective behavior of nonlinear composites is proposed by Ponte Castañeda (1991). He introduced a “linear elastic comparison composite” (LCC) with the same microstructure of the nonlinear composite, which allows for the use of numerous bounds and estimates for linear composites. Ponte Castañeda further proposed an alternative approach with more sophisticated LCC to generate estimates that are exact to the second order in the phase contrast (Ponte Castañeda, 1996; Ponte Castañeda & Willis, 1999). This is the so-called second-order estimate (SOE) homogenization, which yields significant improvements over previous micromechanics models. During the past decade, successive developments have been made to the SOE approach by Ponte Castañeda and his colleagues to better predict the nonlinear macroscopic properties of heterogeneous materials (Agoras, Lopez-Pamies, & Ponte Castañeda, 2009; Chen, Liu, Zhao, et al., 2011; Kailasam, Ponte Castañeda, & Willis, 1997; Liu, Gilormini, & Ponte Castañeda, 2003; Liu & Ponte Castañeda, 2004; Lopez-Pamies & Ponte Castañeda, 2004b; Ponte Castañeda, 2002). Applications have been extended to various nonlinear materials with randomly or periodically distributed microstructure, including viscoplastic polycrystal, porous, or reinforced rubbers and fiber-reinforced elastomers.

4.1.1 Framework of Nonlinear Micromechanics

Finite strain micromechanics can be used to determine the macroscopic constitutive response of biological soft tissues based on structural and mechanical properties of the microstructure. Based on the principle of minimum strain energy, the framework can provide multiple approximate solutions for the macroscopic strain energy function (SEF) of soft tissues. The “upper bound” and a “second-order” estimate models are discussed below.

4.1.2 Hyperelastic Heterogeneous Material

A heterogeneous material, such as soft tissue, is made up of N + 1 (N ≥ 1) different phases, which are distributed (randomly or with a certain distribution of orientation) in a specimen with a volume Ω and boundary ∂Ω in the reference configuration. The constitutive behavior of each inclusion is characterized by a respective SEF W^(r)(F), so that the local SEF W(X, F)of composites is written as:

$$ W\left(\mathbf{X},\mathbf{F}\right)=\sum \limits_{r=0}^N{\chi}^{(r)}\left(\mathbf{X}\right){W}^{(r)}\left(\mathbf{F}\right) $$

(4.161)

where F is defined as the deformation gradient tensor, and χ^(r) = 1 when X ∈ Ω^(r) (the volume occupied by phase r), or 0 otherwise to describe the distribution of the microstructure. The local stress field (i.e., the microscopic constitutive behavior) of the inhomogeneous material is expressed, based on thermodynamics as:

$$ \mathbf{S}\left(\mathbf{X}\right)=\frac{\partial W\left(\mathbf{X},\mathbf{F}\right)}{\partial \mathbf{F}}-p{\mathbf{F}}^{-T} $$

(4.162)

S(X) is the first Piola–Kirchhoff stress tensor, which is related to the Cauchy stress tensor σ by S = σ ⋅ F^−T, and the hydrostatic stress p is due to incompressibility (det(F) = 1) of the soft tissue.

According to the principle of minimum strain energy, the effective SEF of a microscopically inhomogeneous hyperelastic material is defined as:

$$ \overline{W}\left(\overline{\mathbf{F}}\right)=\underset{\mathbf{F}\in \kappa \left(\overline{\mathbf{F}}\right)}{\min}\left\langle W\left(\mathbf{X},\mathbf{F}\right)\right\rangle =\underset{\mathbf{F}\in \kappa \left(\overline{\mathbf{F}}\right)}{\min}\sum \limits_{r=0}^N{c}^{(r)}{\left\langle {W}^{(r)}\left(\mathbf{F}\right)\right\rangle}^{(r)}, $$

(4.163)

where the brackets 〈⋅〉and 〈⋅〉^(r) denote volume averages over the composite Ω and over the r-th phase Ω^(r), respectively, so that c^(r) = 〈χ^(r)〉 represents the volume fraction of the r-th phase. $\kappa \left(\overline{\mathbf{F}}\right)=\left\{\left.\mathbf{F}\right|\mathbf{F}\left(\mathbf{X}\right)=\mathbf{I}+{\nabla}_{\mathbf{X}}\mathbf{u}\left(\mathbf{X}\right)\;\mathrm{in}\;\Omega, \kern1em \mathrm{and}\kern1em \mathbf{u}\left(\mathbf{X}\right)=\left(\overline{\mathbf{F}}-\mathbf{I}\right)\cdot \mathbf{X}\;\mathrm{in}\;\mathrm{\partial \Omega}\right\}$ denotes all the admissible deformation gradient field in the representative volume element (RVE) of the composite with an affine displacement boundary condition u(X). In analogy with the local expression of Eq. (4.162), assuming sufficient smoothness for $\overline{W}\left(\overline{\mathbf{F}}\right)$, the macroscopic stress of the composite is defined by:

$$ \overline{\mathbf{S}}=\frac{\partial \overline{W}\left(\overline{\mathbf{F}}\right)}{\partial \overline{\mathbf{F}}}-p{\overline{\mathbf{F}}}^{-T} $$

(4.164)

where $\overline{\mathbf{S}}=\left\langle \mathbf{S}\right\rangle$ and $\overline{\mathbf{F}}=\left\langle \mathbf{F}\right\rangle$, which are also called the average stress and average deformation gradient, respectively. The effective SEF $\overline{W}$ physically represents the average elastic energy stored in the composite. In general, the rigorous minimization solution for Eq. (4.163) is difficult to obtain for a heterogeneous material with complex microstructure since a set of highly nonlinear partial differential equations must be solved. Consequently, approximate solutions or estimates have been pursued (Hill, 1965; Lopez-Pamies & Ponte Castañeda, 2006; Ponte Castañeda, 1991, 2002; Ponte Castañeda & Willis, 1999; Talbot & Willis, 1985). The efforts to seek an approximate minimizing field F(X) have resulted in several classes of finite strain micromechanics models including the uniform-field upper bound model and the second-order estimate model.

4.1.3 Uniform-Field Upper Bound Model

The simplest trial field for Eq. (4.163) is $\mathbf{F}\left(\mathbf{X}\right)=\overline{\mathbf{F}}$, which assumes a uniform deformation field in the composite. Hence, the macroscopic SEF $\overline{W}\left(\overline{\mathbf{F}}\right)$ is simply the volumetric sum of the constituents,

$$ \overline{W}\left(\overline{\mathbf{F}}\right)\approx {\overline{W}}_U\left(\overline{\mathbf{F}}\right)=\sum \limits_{r=0}^N{c}^{(r)}{W}^{(r)}\left(\overline{\mathbf{F}}\right) $$

(4.165)

This solution leads to an upper bound of the exact SEF $\overline{W}$ as $\overline{\mathbf{F}}$ is an admissible deformation field that does not minimize the total strain energy. It is usually referred to as the Voigt upper bound in linear micromechanics. Consequently, the effective stress ${\overline{\mathbf{S}}}_U=\partial {\overline{W}}_U/\partial \overline{\mathbf{F}}-p{\overline{\mathbf{F}}}^{-T}$ is an upper bound of the exact macroscopic stress for a given $\overline{\mathbf{F}}$. This approximation requires that all phases must deform with the same deformation regardless of their different material properties (e.g., stiffness). This assumption is not true for composites that have widely different phases. In addition, the affine deformation assumption only includes information on the phase volume fraction c^(r) (as indicated by Eq. (4.165)) but neglects the interactions between phases that may affect the overall mechanical response of a composite.

4.1.4 Second-Order Estimate Approach

A second-order estimate (SOE) homogenization approach, utilizing the concept of an LCC, is proposed by Ponte Castañeda (Ponte Castañeda, 1996; Ponte Castañeda & Willis, 1999). This method has been widely applied to nonlinear elastomeric composites, including reinforced and porous elastomers, as well as other heterogeneous elastomeric systems (Agoras et al., 2009;Lopez-Pamies & Ponte Castañeda, 2004a; Ponte Castañeda, 2002). This approach accounts for the statistical microstructure beyond the volume fraction of the phases and incorporates interactions among different phases. Hence, it provides a more accurate prediction of the constitutive behavior of nonlinear composites than the uniform-field upper bound.

Following previous derivations (Lopez-Pamies & Ponte Castañeda, 2004a; Ponte Castañeda, 2002), an LCC is introduced with effective SEF ${\overline{W}}_T\left(\overline{\mathbf{F}}\right)={\sum}_{r=0}^N{\chi}^{(r)}\left(\mathbf{X}\right){W}_T^{(r)}\left(\mathbf{F}\right)$ to denote that it has the same microstructure as the nonlinear composites, but each of the phases is linearly elastic with the SEF ${W}_T^{(r)}$ (i.e., the second-order Taylor approximation to the nonlinear SEF W^(r)). According to generalized Legendre transform of the strain energy (Lopez-Pamies & Ponte Castañeda, 2004a; Ponte Castañeda, 2002), the exact macroscopic SEF expressed in Eq. (4.163), can be approximated by:

$$ \overline{W}\left(\overline{\mathbf{F}}\right)\le \underset{\left\{{\mathbf{F}}^{\ast (s)},{\mathbf{L}}^{(s)}\right\}}{\min}\left\{{\overline{W}}_T\Big(\overline{\mathbf{F}};\left\{{\mathbf{F}}^{\ast (s)},{\mathbf{L}}^{(s)}\right\}\left)+\sum \limits_{r=0}^N{c}^{(r)}{V}^{(r)}\right({\mathbf{F}}^{\ast (r)},{\mathbf{L}}^{(r)}\Big)\right\}, $$

(4.166)

where L^(r) is the unknown reference modulus and F^∗(r) is the virtual residual deformation gradient of the r-th phase of LCC, both of which will be determined by the minimization procedure of Eq. (4.166). ${V}^{(r)}\left({\mathbf{F}}^{\ast (r)},{\mathbf{L}}^{(r)}\right)=\underset{\widehat{\mathbf{F}}}{\sup}\left\{{W}^{(r)}\left(\widehat{\mathbf{F}}\right)-{W}_T^{(r)}\left(\widehat{\mathbf{F}}\right)\right\}$ is a “corrector” function, i.e., the error induced by approximating the nonlinear composite SEF W^(r) with the SEF of LCC ${W}_T^{(r)}$. This approximation translates the optimization over continuous deformation field $\mathbf{F}\in \kappa \left(\overline{\mathbf{F}}\right)$ to the modulus tensor, L^(r)_, and the deformation gradient tensor, F^∗(r)_, of the r-th phase of composite (r varies from 0 to N). In theory, the field $\mathbf{F}\in \kappa \left(\overline{\mathbf{F}}\right)$ must satisfy the continuity and compatibility conditions, while L^(r) and F^∗(r) can be arbitrary within the physically meaningful ranges. Although L^(r) and F^∗(r) are allowed to change from point to point in the material, it is assumed that they are uniform for the r-th phase based on standard nonlinear micromechanics (Lopez-Pamies & Ponte Castañeda, 2004a; Ponte Castañeda, 2002; Willis, 1977). The work of Ponte Castañeda and Willis (1999) suggested that replacing the minimization over the variables L^(r) and F^∗(r) in Eq. (4.166) by the corresponding stationary points will yield a stationary estimate, and thus generate the so-called second-order estimate for the exact SEF of the nonlinear composites. The stationary procedures involved in the above derivation (maximization of V^(r) and the minimization in Eq. (4.166)) yield a set of nonlinear tensorial equations for the determination of the unknown reference modulus L^(r) and the reference deformation gradient F^∗(r)(Lopez-Pamies & Ponte Castañeda, 2004a, 2006; Ponte Castañeda, 2002). These equations have multiple solutions that lead to various estimations of the macroscopic SEF $\overline{W}\left(\overline{\mathbf{F}}\right)$. A tangent solution is widely used for composites with complex microstructure, in which the reference deformation gradient is taken to be the average deformation gradient, i.e., ${\mathbf{F}}^{\ast (r)}={\overline{\mathbf{F}}}^{(r)}$ in the r-th phase of the LCC, and L^(r) is the tangent stiffness tensor evaluated at ${\overline{\mathbf{F}}}^{(r)}$ (Ponte Castañeda and Tiberio, 2000). This solution leads to V^(r)(F^∗(r), L^(r)) = 0 and an estimate of $\overline{W}\left(\overline{\mathbf{F}}\right)$ as:

$$ \overline{W}\left(\overline{\mathbf{F}}\right)\approx {\overline{W}}_{\mathrm{S}}\left(\overline{\mathbf{F}}\right)=\sum \limits_{r=0}^N{c}^{(r)}\left\{{W}^{(r)}\left({\overline{\mathbf{F}}}^{(r)}\right)+\frac{1}{2}{\boldsymbol{\uprho}}^{(r)}\left({\overline{\mathbf{F}}}^{(r)}\right)\left(\overline{\mathbf{F}}-{\overline{\mathbf{F}}}^{(r)}\right)\right\}. $$

(4.167)

where ρ^(r) is used to denote the first derivation of the phase potential W^(r), i.e., ${\boldsymbol{\uprho}}^{(r)}\left({\overline{\mathbf{F}}}^{(r)}\right)={\left(\partial {W}^{(r)}/\partial \mathbf{F}\right)}_{\mathbf{F}={\overline{\mathbf{F}}}^{(r)}}$. In order to compute this estimate, the effective strain energy ${\overline{W}}_T\left(\overline{\mathbf{F}}\right)$ and the average deformation gradient of phases ${\overline{\mathbf{F}}}^{(r)}$ of the LCC are determined by extended finite strain Hashin-Shtrikman theory (Kailasam et al., 1997; Liu, 2003) which takes into account the stiffness L^(r) as well as the shape and distribution of the composite phases (Chen, Liu, Zhao, et al., 2011; Lopez-Pamies & Ponte Castañeda, 2006; Ponte Castañeda, 2002).

The advantage of this method is that it can be used for any type of nonlinear composite and can consider the strongly nonlinear constraint of material incompressibility, which is relevant for biological soft tissues. The SOE model statistically accounts for the heterogeneous deformation in composites (i.e., ${\overline{\mathbf{F}}}^{(r)}$ varies in phases), which is due to the heterogeneities of microstructural geometries and material properties, as well as interactions among constituents. The deformation field is therefore more realistic than that of the upper bound solution. Consequently, SOE leads to more accurate estimates of the macroscopic SEF and stress for the composites, as well as microscopic stress for microstructure, as compared with the upper bound solution.

4.1.5 Micromechanical Models for Soft Tissues

Numerous microstructural constitutive models of soft tissues have been proposed in the past several decades that proved to be more accurate than previous phenomenological models. Three major types of these micromechanical models are classified based on various assumptions and approximations in nonlinear micromechanics as described below.

4.1.5.1 Uniform-Field Models Based on a Solid-Like Matrix

The first class of micromechanical models is motivated by anisotropic mechanical behaviors of soft tissues. Previous studies showed that the elastin becomes straightened and starts to take load in the early deformation of the tissue as an isotropic material. For instance, the experimental study of Gundiah et al. (2007) suggested that elastin can be described by an isotropic neo-Hookean constitutive model. Meanwhile, collagen fibers, with preferred orientation, are largely associated with the anisotropic response of soft tissues (Holzapfel & Weizsäcker, 1998). Based on these observations, the tissue is considered as a collagen fiber-reinforced composite with a solid-like matrix that can bear load. The SEF of non-collagenous matrix material, including elastin fibers, cells, and ground substance (GS) (matrix defined here is not the same as ECM), W_M, is associated with the isotropic deformation, and the SEF of collagen W_C is anisotropic due to the deformation of two families of collagen fibers (Holzapfel et al., 2000). Hence, the effective SEF of tissue is the sum of these two functions:

$$ \overline{W}\left(\overline{\mathbf{C}}\right)={W}_{\mathrm{M}}\left(\overline{\mathbf{C}}\right)+{W}_{\mathrm{C}}\left(\overline{\mathbf{C}},{\mathbf{N}}_1,{\mathbf{N}}_2\right), $$

(4.168a)

$$ {\displaystyle \begin{array}{c}{W}_{\mathrm{M}}\left(\overline{\mathbf{C}}\right)={A}_1\left({\overline{I}}_1-3\right),\kern0.5em {W}_{\mathrm{C}}\left(\overline{\mathbf{C}},{\mathbf{N}}_1,{\mathbf{N}}_2\right)\\ {}=\frac{k_1}{2{k}_2}\sum \limits_{i=4,6}\left\{\exp \left[{k}_2{\left({\overline{I}}_i-1\right)}^2\right]-1\right\},\end{array}} $$

(4.168b)

where N₁ and N₂ are the direction vectors of the two families of collagen fibers, A₁ is a material parameter associated with elastin fiber, k₁ is a material parameter associated with collagen fiber and k₂ is a dimensionless parameter (Holzapfel et al., 2000). The right Cauchy-Green deformation tensor $\overline{\mathbf{C}}$ is related to the deformation gradient by $\overline{\mathbf{C}}={\overline{\mathbf{F}}}^T\cdot \overline{\mathbf{F}}$. The first invariant of $\overline{\mathbf{C}}$ is ${\overline{I}}_1=\mathrm{tr}\left(\overline{\mathbf{C}}\right)$, and the fourth and sixth invariants are ${\overline{I}}_4={\mathbf{N}}_1\cdot \overline{\mathbf{C}}\cdot {\mathbf{N}}_1$ and ${\overline{I}}_6={\mathbf{N}}_2\cdot \overline{\mathbf{C}}\cdot {\mathbf{N}}_2$, respectively.

This model, where the uniform-field approximation $\mathbf{F}\left(\mathbf{X}\right)=\overline{\mathbf{F}}$ is employed (as in Eqs. (4.168a) and (4.168b)), presents an upper bound of the exact effective SEF of soft tissue. In addition, the direction vectors N₁ and N₂ of fibers are determined by empirical curve fitting rather than based on histological observations (Holzapfel et al., 2000). Given that fiber orientation follows a certain continuous distribution in biological tissues (Chen, Liu, Zhao, et al., 2011; Sacks, 2003), these two parameters are phenomenological variables rather than structural parameters. Moreover, the engagement of undulated collagen fibers (characterized by the fiber waviness distribution) is described by an exponential SEF W_C, and thus is also phenomenological. To obtain more accurate predictions, this model has been subsequently revised.

Zulliger, Fridez, et al. (2004) refined the model by accounting for not only the wavy nature of collagen fibers but also the volume fraction of both elastin and collagen, based on different SEFs of the matrix and collagen fibers. Kroon and Holzapfel (2008) later incorporated this model into multi-layered structures with the mean fiber alignments in various layers. Li and Robertson (2009) also extended this model to account for either a finite number of fiber orientations or a fiber distribution function. In summary, the mechanical predictions of these uniform-field models based on a solid-like matrix are more accurate than those of phenomenological models because they reflect the heterogeneity of material properties and some of the geometrical features of tissues. These models, however, cannot accurately predict the microenvironment (strain and stress of individual fiber or cell) of soft tissues because they assume affine deformation in tissue and use non-histological-based microstructure.

4.1.5.2 Uniform-Field Models Based on a Fluid-Like Matrix

The second class of micromechanical models are structurally motivated and are proposed based on the following assumptions:

• 1. Fibers are thin and flexible with only tensile strength (i.e., fibers cannot resist compressive load).

• 2. Fibers are embedded in a fluid-like matrix, of which the mechanical contribution is only via hydrostatic pressure.

• 3. The second assumption leads to a simplification that all the microstructures deform identically to the macroscopic deformation of the tissue (i.e., uniform deformation) since no fiber–fiber interactions are considered.

On the basis of these assumptions and thermodynamic consideration, Lanir developed a general multi-axial theory for the constitutive relations in fibrous connective tissues (Lanir, 1979, 1980, 1983). In his model (Lanir, 1983), an important structural feature is the density distribution function of the fiber orientation R_i(N) where N is a unit vector tangent to the fiber. Thus, R_i(N)ΔΘ is the volumetric fraction of fibers of type i (classified by waviness and fiber type) which are oriented in direction N and occupy a spatial angle ΔΘ. According to previous derivation for uniform-field models, the macroscopic SEF in this model is the volumetric sum of the SEF of fibers in all directions:

$$ \overline{W}\left(\overline{\mathbf{F}}\right)=\sum \limits_i\sum \limits_{\mathbf{N}}{c}^{(i)}{W}^{(i)}\left({\lambda}_{\mathrm{f}}\right)\cdot {R}_i\left(\mathbf{N}\right)\cdot \Delta \Theta, $$

(4.169)

where c⁽ⁱ⁾ is the volumetric fraction of unstrained fibers of type i, and W⁽ⁱ⁾(λ_f) is the fiber SEF of type i, which depends on fiber stretch ratio ${\lambda}_{\mathrm{f}}=\sqrt{\mathbf{N}\cdot \left({\overline{\mathbf{F}}}^T\cdot \overline{\mathbf{F}}\right)\cdot \mathbf{N}}$. This equation is identical to Eq. (4.165) as ∑_Nc⁽ⁱ⁾R_i(N)ΔΘ denotes the volumetric fraction of a particular fiber phase with the type i SEF and orientation N in soft tissues. The SEF of fiber W⁽ⁱ⁾(λ_f) and the density distribution function R_i(N) are specific to different tissue types, of which microstructural geometries and material properties may be determined by histological and mechanical measurements. This model can account for geometrical distributions (e.g., orientation, waviness) as well as the mechanical response of single fibers, but it assumes affine deformation and neglects inter-fiber interactions due to the assumption of a fluid-like matrix.

The fluid-like matrix assumption is also employed by Decraemer et al. (1980) to develop a parallel wavy fibers model for soft biological tissues in uniaxial tension, assuming a normal distribution for initial length of fibers (i.e., fiber waviness). Wuyts et al. (1995) extended this model by utilizing a Lorentz distribution function for the initial fiber length. In summary, the fluid-like matrix models are more conceptual than the solid-like matrix models because they involve structural and constitutive behaviors of both the functional constituents of soft tissues: collagen and elastin. The basic fluid-like matrix assumption may be an accurate description for certain tissues where the non-fibrous constituents are mainly fibroblasts, macrophages, and amorphous gel-like GS that do not take up significant non-hydrostatic loading. These micromechanical models with a fluid-like matrix have been widely applied to different soft tissues, including tendon (Sverdlik & Lanir, 2002), skin (Lokshin & Lanir, 2009a, 2009b), myocardium (Horowitz et al., 1988), and blood vessels (Hollander et al., 2011b)

4.1.5.3 SOE Models Based on Solid-Like Matrix

Chen, Liu, Zhao, et al. (2011) developed a finite strain micromechanical model (based on the aforementioned SOE approach) to predict the macroscopic stress–strain relation and microstructural deformation of soft fibrous tissue. This model shows significant improvements over previous microstructure models when compared to finite element method (FEM) simulations. In this model, the tissue is assumed as a composite with reinforcing fibers and soft solid matrix (Fig. 4.23). The orientation of the r-th fiber is described by θ^(r), the shape (dimension) is described by a geometric tensor Z^(r), and the spatial distribution of the r-th fiber is characterized by another geometric tensor ${\mathbf{Z}}_{\mathrm{d}}^{(r)}$ while the waviness is included in the SEF of the r-th fiber to reflect fiber recruitment under macroscopic tissue deformation. Specifically, a piecewise function is used to describe the constitutive behavior of a single fiber:

$$ {W}^{(r)}\left(\mathbf{F}\right)=\left\{\begin{array}{ll}{W}^{(0)}\left(\mathbf{F}\right)& {\lambda}_{\mathrm{f}}<{\lambda}_0^{(r)},\\ {}{W}^{(0)}\left(\mathbf{F}\right)+{W}_{\mathrm{f}\mathrm{iber}}^{(r)}\left({\lambda}_{\mathrm{f}}\right)& {\lambda}_{\mathrm{f}}\ge {\lambda}_0^{(r)},\end{array}\right. $$

(4.170)

where λ_f is the fiber stretch ratio, ${\lambda}_0^{(r)}$ is the waviness of the r-th fiber, and W⁽⁰⁾ denotes the SEF of the matrix. This function implies that a fiber deforms the same as the soft matrix before straightening and becomes stiffer with additional SEF of fiber ${W}_{\mathrm{fiber}}^{(r)}$ after straightening (as shown in Fig. 4.23b). The matrix is described by a neo-Hookean SEF, while the anisotropic term ${W}_{\mathrm{fiber}}^{(r)}\left(\lambda \right)$ in the SEF of fibers is selected as:

$$ {W}_{\mathrm{f}\mathrm{iber}}^{(r)}\left({\lambda}_{\mathrm{f}}\right)={E}_1{\left(\ {\lambda}_{\mathrm{f}}-{\lambda}_0^{(r)}\right)}^2/2+{E}_2{\left({\lambda}_{\mathrm{f}}-{\lambda}_0^{(r)}\right)}^3/3 $$

(4.171)

where E₁, E₂ are material parameters of individual fibers. This is a generalization of a linear model (Decraemer et al., 1980; Lanir, 1979, 1983; Wuyts et al., 1995) where only the first term ${\left({\lambda}_{\mathrm{f}}-{\lambda}_0^{(r)}\right)}^2$ is included. In principle, the homogenization model can use any well-defined constitutive models of the matrix and fibers.

Fig. 4.23

Conceptual demonstration of the SOE method. (a) A representative volume element of a fibrous tissue at reference state. All the fibers are undulated and exhibit the same property as the matrix with strain energy function (SEF) W₀; (b) When subjected to a macroscopic $\overline{\mathbf{F}}$, some fibers are straightened and show stiffer property with SEF W^(r) as in Eq. (4.170). (c) The effectively homogeneous material with macroscopic SEF $\overline{W}$. Reproduced from Chen, Zhao, et al. (2013) with permission

By substituting fiber geometrical features ${\mathbf{G}}^{(r)}\left({\mathbf{Z}}^{(r)},{\mathbf{Z}}_{\mathrm{d}}^{(r)},{\theta}^{(r)},{\lambda}_0^{(r)}\right)$ and mechanical properties (Eqs. 4.170 and 4.171) into Eqs. (4.166) and (4.167), and utilizing the finite strain Hashin-Shtrikman theory (Chen, Liu, Zhao, et al., 2011; Kailasam et al., 1997), the macroscopic SEF and stress of soft tissue as well as microscopic deformation ${\overline{\mathbf{F}}}^{(r)}$ of every component can be obtained through solving multiple nonlinear equations (Chen, Liu, Zhao, et al. 2011; Lopez-Pamies & Ponte Castañeda, 2004a). It should be noted that the microscopic deformation ${\overline{\mathbf{F}}}^{(r)}$ of the r-th fiber is not identical to either the macroscopic tissue deformation or other fiber phases since the admissible deformation field F(X) is not $\overline{\mathbf{F}}$ (as in the uniform-field models) and is determined by tissue inhomogeneity. As compared with the first two model types, this SOE micromechanical model not only considers realistic geometrical features and material properties of tissue constituents and their interactions, but also allows flexible deformation in each constituent. Hence, the model is an actual estimate rather than an upper bound of the exact effective SEF and provides a more accurate prediction of the macroscopic and microscopic mechanical behavior of the soft tissue.

Appendix 12: A 3D Microstructure-Based Model of Coronary Adventitia (Chen, Guo, et al., 2016)

The adventitia is considered to be a cylindrical tube, with the following kinematic assumptions:

• 1. Incompressible.

• 2. Deformations are axis-symmetric and independent of axial position.

• 3. Transverse sections remain planar.

• 4. There is a unique undeformed reference configuration (i.e., ZSS).

A cylindrical coordinate system is used with circumferential direction g₁, radial direction g₂ and axial direction g₃ as principal directions, the corresponding stretches λ_θ, λ_r, and λ_z are determined, respectively,

$$ {\lambda}_{\theta }=\left(\frac{\pi }{\pi -{\Theta}_0}\right)\frac{r}{R},\kern1em {\lambda}_r=\frac{\partial r}{\partial R},\kern1em {\lambda}_z=\frac{l}{L} $$

(4.172)

where Θ₀ is opening angle measured at ZSS, R is radius to a point at ZZS and r is the radius to the same point in the current configuration. L is the axial length of the segment at ZSS and l is the loaded axial length. According to material incompressibility: J = λ_θλ_rλ_z = 1, for the mapping between ZSS and loaded state, loaded radius r is determined as a function of unloaded R:

$$ r(R)=\sqrt{r_{\mathrm{o}}^2-\left({R}_{\mathrm{o}}^2-{R}^2\right)\frac{\pi -{\Theta}_0}{\lambda_z\pi }} $$

(4.173)

where r_o is the outer radius in the loaded state while R_o is that at ZSS.

The radial component of the force equilibrium equation imposed on the loaded configuration is given by:

$$ \frac{\partial {\sigma}_{rr}}{\partial r}+\frac{\sigma_{rr}-{\sigma}_{\theta \theta}}{r}=0 $$

(4.174)

where σ_ij as Cauchy stress tensor. According to boundary conditions, ${\left.{\sigma}_{rr}\right|}_{r_{\mathrm{i}}}=-{p}_{\mathrm{i}},\kern0.5em {\left.{\sigma}_{rr}\right|}_{r_{\mathrm{o}}}=0$, the luminal pressure p_i can be written as:

$$ {p}_{\mathrm{i}}={\int}_{r_{\mathrm{i}}}^{r_{\mathrm{o}}}\left({\sigma}_{\theta \theta}-{\sigma}_{rr}\right)\frac{1}{r} dr $$

(4.175)

The axial force required to maintain the vessel axial stretch is given by:

$$ F=\pi {\int}_{r_{\mathrm{i}}}^{r_{\mathrm{o}}}\left(2{\sigma}_{zz}-{\sigma}_{\theta \theta}-{\sigma}_{rr}\right)r\; dr $$

(4.176)

4.1.1 Strain Energy Function

The coronary adventitia is considered as an incompressible hyperelastic solid and characterized by a strain energy function W(E) as a function of the Green-Lagrange strain tensor $\mathbf{E}=\frac{1}{2}\left({\mathbf{F}}^{\mathrm{T}}\cdot \mathbf{F}-\mathbf{I}\right)$. The Cauchy stress tensor σ is given by:

$$ \boldsymbol{\upsigma} =\mathbf{F}\frac{\partial W}{\partial \mathbf{E}}{\mathbf{F}}^{\mathrm{T}}-p\mathbf{I}=\mathbf{F}\cdot \mathbf{S}\cdot {\mathbf{F}}^{\mathrm{T}}-p\mathbf{I} $$

(4.177)

where F is the deformation gradient tensor and S is the second Piola–Kirchhoff stress tensor. I is the second-order identity tensor, scalar p is hydrostatic pressure, which acts as a Lagrange multiplier and must be determined from equilibrium and boundary conditions.

The strain energy function (SEF) W(E) of a microstructural model involves structural features. Experimental studies show that coronary adventitia is divided into an outer and inner adventitia as seen in Fig. 4.24a–c (Fig. 4.24a, b are lateral sections as denoted in Fig. 4.24d). The outer adventitia, consisting of thicker and wavier collagen bundles and few elastin fibers, supports the vessel and connects with the surrounding tissue rather than significantly resisting the transmural pressure (Fig. 4.24e, f), while the inner adventitia is a layered structure with concentric densely packed fiber sheets and has few radial fiber bundles distributing between sheets (Fig. 4.24b, e, f are cross sections as denoted in Fig. 4.24d). At low pressures, elastin fibers bear the loads and collagen fibers are still wavy in the inner adventitia. At high pressures, stretched collagen fibers in the inner adventitia are recruited to withstand stresses. Therefore, the adventitia wall is modeled as a composite containing two mechanical components: collagen and elastin fibers, while ground substance is found to have a negligible mechanical function (Fig. 4.24) and is treated as a fluid that sustains hydrostatic pressure. Both types of fibers are only resistant to tensile load, undulated collagen fibers are recruited to bear loads only after they become straightened, and there is no interaction between collagen and elastin fibers.

Fig. 4.24

(a) Outer adventitia (OA) consists of thicker collagen bundles and few elastin fibers; (b) Inner adventitia (IA) is a layered structure with entangled elastin and collagen fibers in each sublayer; (c) The cross section of a coronary artery at no-load state; (e, f) OA and IA deformed under elevated pressures: 100 and 200 mmHg, respectively, showing fibers in IA are stretched to take up loads while most of collagen bundles in OA are still undulated and unengaged; (d) A schematic diagram demonstrates the cross and lateral sections of a vessel segment. (a, c) are the lateral sections and (b, e, f) are the cross sections. Scale bar denotes 100μ. Reproduced from Chen, Guo, et al. (2016) with permission

A fluid-like matrix implies the tissue undergoes affine deformations, i.e., deformation of the fibers is the same as that of ground substance. Based on this assumption, the SEF of adventitia wall can be represented by the volume-weighted summation of individual SEF of elastin fiber W_E and collagen fiber W_C (Lanir, 1983):

$$ W\left(\boldsymbol{E}\right)={f}_{\mathrm{E}}{W}_{\mathrm{E}}+{f}_{\mathrm{C}}{W}_{\mathrm{C}} $$

(4.178)

where f_i (i = E, C) is the volume fraction of each type of fiber i. Generally, the orientation of fibers follows a continuous distribution density function, and the volume-weighted SEF of fibers is given by Lanir (1983):

$$ {W}_i={\int}_0^{\pi }{\mathrm{\mathcal{R}}}_i\left(\theta \right){w}_i(e) d\theta $$

(4.179)

where w_i(e) is the SEF of individual fibers as a function of fiber strain e. It should be noted that radially oriented elastin and collagen fibers are not engaged under inflation–extension condition where they are compressed and bear no loads, and only planar fibers contribute to mechanical behavior of the adventitia. Thus, ℛ_i(θ) is a planar orientation distribution density function of fiber i, and θ is the angle between the fiber orientation and the circumferential direction of the vessel g₁. ℛ_i(θ) satisfies the normalization criterion ${\int}_0^{\pi }{\mathrm{\mathcal{R}}}_i\left(\theta \right) d\theta =1$. The uniaxial fiber strain e(θ) is determined by the local strain tensor E and the reference fiber direction N = (Cos θ, Sin θ) as:

$$ e\left(\mathbf{E},\mathbf{N}\right)=\mathbf{E}:\mathbf{N}\otimes \mathbf{N} $$

(4.180)

Although elastin and collagen fibers distributed in each sublayer with transmural variation of fiber orientation, a mixture of two normal distribution of fiber orientation through the adventitia wall is found to describe the experimental data as (Chen, Liu, Slipchenko, et al., 2011):

$$ {W}_i=\left\{{\omega}_{i1}{\int}_0^{\pi }{\mathrm{\mathcal{R}}}_{i1}\left(\theta \right){w}_i(e) d\theta +{\omega}_{i2}{\int}_0^{\pi }{\mathrm{\mathcal{R}}}_{i2}\left(\theta \right){w}_i(e)d{\theta}_i\right\} $$

(4.181)

where ${\mathrm{ℛ}}_{\mathit{ij}}\left(\theta \right)=\frac{1}{{К}_j}\frac{1}{{\sigma }_j\surd 2\pi }\mathrm{Exp}\left[-\frac{{\left(\theta -{\mu }_j\right)}^2}{2{\sigma }_j^2}\right],\left(i=E,C;j=1,2\right)$ is a truncated normal distribution density function with µ_j and σ_j as the mean and standard deviation, respectively. К_j is a truncated parameter ${К}_j=\Phi \left(\frac{\pi -{\mu }_j}{{\sigma }_j}\right)-\Phi \left(\frac{-{\mu }_j}{{\sigma }_j}\right)$ (Φ is the cumulative distribution function of a normal distribution), and ω_ij is the weight of each normal distribution $\left({\sum}_{j=1}^2{\omega}_{ij}=1\right)$. The second Piola–Kirchhoff stress of each type of fiber is derived as:

$$ {\boldsymbol{S}}_i=\sum \limits_j{\omega}_{ij}{\int}_0^{\pi }{\mathrm{\mathcal{R}}}_{ij}\left(\theta \right)\frac{\partial {w}_i}{\partial e}\frac{\partial e}{\partial \mathbf{E}} d\theta $$

(4.182)

According to microscopic responses of individual elastin fibers under mechanical loads, the elastic properties are assumed to be linear:

$$ \frac{\partial {w}_{\mathrm{E}}}{\partial e}=\left\{\begin{array}{cc}0& e<0\\ {}{k}_{\mathrm{E}}e& e>0\end{array}\right. $$

(4.183)

where k_E is the stiffness parameter of elastin fiber. Because of the wavy nature of collagen fibers, the nonlinear constitutive relation is considered to account for the nonlinear elastic behavior (Hollander et al., 2011a):

$$ \frac{\partial {w}_{\mathrm{C}}}{\partial e}=\left\{\begin{array}{cc}0& e\le {e}_0\\ {}{k}_{\mathrm{C}}{\left(e-{e}_0\right)}^{M_{\mathrm{C}}}& e>{e}_0\end{array}\right. $$

(4.184)

where k_C and M_C are parameters characterizing the nonlinear stress–strain response of collagen, and e₀ denotes the strain beyond which the collagen can withstand tension, which are found to follow a beta distribution for the coronary adventitia (Chen, Slipchenko, et al., 2013):

$$ D\left({e}_0\right)=\frac{1}{B\left({\alpha}_1,{\alpha}_2\right)}\frac{{\left({e}_0-a\right)}^{\alpha_1-1}{\left(b-{e}_0\right)}^{\alpha_2-1}}{{\left(b-a\right)}^{\alpha_1+{\alpha}_2-2}}\kern0.75em $$

(4.185)

where B(α₁, α₂) is a beta function, and a and b the lower and upper bounds of the straightening strain e₀. The constitutive law of collagen fiber thus can be written as:

$$ \frac{\partial {w}_{\mathrm{C}}}{\partial e}=\left\{\begin{array}{cc}0& e\le {e}_0\\ {}{k}_{\mathrm{C}}{\int}_a^bD\left({e}_0\right)\ {\left(e-{e}_0\right)}^{M_C}{de}_0& e>{e}_0\end{array}\right. $$

(4.186)

Although the microstructural approach can employ any well-defined constitutive model for the fibers, a linear function (Eq. 4.183) and a power function (Eq. 4.184) are used for elastin and collagen fibers, respectively. If the constitutive laws for individual elastin and collagen fibers (Eqs. 4.183–4.186) are substituted into Eqs. (4.177) and (4.182), the Cauchy stress components of the vessel will be obtained. Given the geometrical parameters (f_E, f_C) and the measured distribution functions (ℛ_ij(θ), ω_ij, D(e₀)) (summarized in Table 4.25 for convenience) (Chen, Slipchenko, et al., 2013) there are only three unknown material parameters: k_E, k_C, and M_C that needed be determined by the boundary condition (Eqs. 4.175 and 4.176). Since the integrals of the above equations do not have analytical expressions, numerical approaches are used.

4.1.2 Parameter Estimation

Parameters are optimized by least squares fit to the experimental data by minimizing an objective function based on the sum of squared residuals (SSE) between model predictions and experimental data. The objective function is defined as follows:

$$ \mathrm{SSE}=\frac{1}{2}\frac{1}{nm}\sum \limits_{i,j}^{n,m}\left[{\left(\frac{r_{{\mathrm{o}}_{ij}}-\widehat{r_{{\mathrm{o}}_{ij}}}}{\sigma_{{\widehat{r}}_{\mathrm{o}}}}\right)}^2+{\left(\frac{F_{ij}-{\widehat{F}}_{ij}}{\sigma_{\widehat{F}}}\right)}^2\right] $$

(4.187)

where i and j denote distension and axial loads at which the corresponding outer radius ${r}_{{\mathrm{o}}_{ij}}$ and axial force F_ij are measured; n is the number of different pressures and m the number of different axial stretch ratios used; ${\sigma}_{\widehat{\cdot}}$ is the standard deviation of experimental measurement; and $\widehat{r_{{\mathrm{o}}_{ij}}}$, ${\widehat{F}}_{ij}$ are the corresponding model predicted outer radius and axial force, which are determined by numerically solving Eqs. (4.175) and (4.176). A numerical nonlinear optimization function NMinimize in Mathematica (WOLFRAM, US) is used to find a global minimum of the objective function (Eq. 4.187) and to determine three unknown parameters (k_E, k_C, M_C) with the constraints: (k_E > 0, k_C > 0, M_C > 0).

Table 4.25 Geometrical parameters and distributions

Table 4.26 Parameter estimates of individual elastin and collagen fibers based on full distension–extension experimental data (SEM denotes standard error of the mean)

Table 4.27 Parameter estimates of individual elastin and collagen fibers based on experimental data of λ_z = 1.3 and 1.5

Table 4.28 Parameter estimates of individual elastin and collagen fibers based on an idealized average model in comparison with the full continuous distributions of fiber orientation and waviness

Appendix 13: Microstructure-Based Model of Coronary Media Including Vascular Smooth Muscle Cell (SMC) Contraction (Chen, Luo, et al., 2013)

The media segment is considered as a thin-walled elastic tube deformed in the circumferential and axial directions (Chen, Slipchenko, et al., 2013). Green strains are defined as ${E}_{\theta \theta}=\left({\lambda}_{\theta}^2-1\right)/2$ and ${E}_{zz}=\left({\lambda}_z^2-1\right)/2$, where λ_θ = τ/Γ is the circumferential stretch ratio (τ refers to the midwall circumference of the loaded vessel and Γ refers to that at ZSS), and λ_z = L/L_o is the axial stretch ratios with L and L_o being axial lengths in loaded and no-load states, respectively. A structural constitutive model is used to describe the mechanical response of passive coronary media (Hollander et al., 2011a, 2011b), which contains isotropic inter-lamellar (IL) elastin networks and helically oriented collagen fibers:

$$ {W}_{\mathrm{E}}\left(\mathbf{E}\right)=\frac{f_{\mathrm{E}}}{\pi }{\int}_{-\frac{\pi }{2}}^{\frac{\pi }{2}}\ {w}_{\mathrm{E}}\left[{e}_{\mathrm{E}}\left({\theta}_{\mathrm{E}}\right)\right]\ d{\theta}_{\mathrm{E}}, $$

(4.188)

$$ {W}_{\mathrm{C}}\left(\mathbf{E}\right)=\frac{f_{\mathrm{C}}}{2}\left\{{w}_{\mathrm{C}}\left[{e}_{\mathrm{C}}\left({\theta}_{\mathrm{C}}\right)\right]+{w}_{\mathrm{C}}\left[{e}_{\mathrm{C}}\left(-{\theta}_{\mathrm{C}}\right)\right]\right\}, $$

(4.189)

where w_E and w_C are the strain energy of elastin struts and collagen fibers, depending on uniaxial fiber strain e_s (s = E, C, denoting elastin and collagen, respectively). θ_s is the fiber orientation angle (corresponding unit vector denoted by ${\overrightarrow{\mathbf{n}}}^s$) and f_s is the volume fraction. The fiber strain is determined by ${e}_{\mathrm{s}}={\overrightarrow{\mathbf{n}}}^s\cdot \mathbf{E}\cdot {\overrightarrow{\mathbf{n}}}^s$ with assuming affine deformation (i.e., a fiber is assumed to rotate and stretch in the same way as the bulk tissue). The linear stress–strain relation of elastin is considered to be ∂w_E/∂e_E = k_Ee_E while the nonlinear relation of collagen is considered to be $\partial {w}_{\mathrm{C}}/\partial {e}_{\mathrm{C}}={k}_{\mathrm{C}}{e}_{\mathrm{C}}^{N_{\mathrm{C}}}$, with the material parameter k_E representing the stiffness of the elastin, and k_C and N_C representing the stiffness and nonlinear parameter of collagen. The passive strain energy of the coronary media W_passive is calculated by taking the sum of the strain energies of the elastin and collagen networks, i.e., W_passive(E) = W_E(E) + W_C(E), and the second Piola–Kirchhoff stress is determined by S_passive = ∂W_passive(E)/∂E, and given by:

$$ {S}_{\theta \theta\;\mathrm{passive}}=\frac{k_{\mathrm{E}}}{8}\left(3{E}_{\theta \theta}+{E}_{zz}\right)+{k}_{\mathrm{C}}{\cos}^2{\theta}_{\mathrm{C}}{\left({\cos}^2{\theta}_{\mathrm{C}}{E}_{\theta \theta}+{\sin}^2{\theta}_{\mathrm{C}}{E}_{zz}\right)}^{N_{\mathrm{C}}}, $$

(4.190a)

$$ {S}_{zz\;\mathrm{passive}}=\frac{k_{\mathrm{E}}}{8}\left({E}_{\theta \theta}+3{E}_{zz}\right)+{k}_{\mathrm{C}}{\sin}^2{\theta}_{\mathrm{C}}{\left({\cos}^2{\theta}_{\mathrm{C}}{E}_{\theta \theta}+{\sin}^2{\theta}_{\mathrm{C}}{E}_{zz}\right)}^{N_{\mathrm{C}}}. $$

(4.190b)

where S_{θθ passive} and S_zz passive are passive circumferential and axial stresses, respectively.

The total strain energy of the media is the sum of the active and passive contributions, i.e., W_total = W_passive + W_active. The function W_active is the active strain energy of vessels as contributed by active vascular smooth muscle cells (SMC). Taking into consideration helical arrangement of SMC, the active strain energy can be given as:

$$ {W}_{\mathrm{active}}=\frac{f_{\mathrm{VSMC}}}{2}\left\{{\int}_0^{\frac{\pi }{2}}\Re \left({\theta}_{\mathrm{VSMC}}\right){w}_{\mathrm{VSMC}}\left({\theta}_{\mathrm{VSMC}}\right)d{\theta}_{\mathrm{VSMC}}\right){\displaystyle \begin{array}{l}+\left({\int}_{-\frac{\pi }{2}}^0,\Re, \left(-{\theta}_{\mathrm{VSMC}}\right),{w}_{\mathrm{VSMC}},\left(-{\theta}_{\mathrm{VSMC}}\right),d,{\theta}_{\mathrm{VSMC}}\right\}\end{array}}, $$

(4.191)

where θ_VSMC is the orientation angle of SMC, ℜ(θ_VSMC)is the two-dimensional orientation distribution of SMC, w_VSMC is multi-axial strain energy of a single SMC and f_VSMC is the volume fraction. A two-dimensional generalization of the uniaxial length–tension relation of active SMC [20] is used to account for the multi-axial active response of SMC (Huo et al., 2012, 2013). Two families of helical SMC with symmetrical polar angles, ±θ_VSMC, are considered for simplicity (i.e., span of orientational distribution of SMC is zero). Thus, the active strain energy of SMC can be written as follows:

$$ {W}_{\mathrm{active}}\left({\lambda}_{\mathrm{VSMC}},{\lambda}_{\mathrm{VSMC}}^{\perp}\right)=\frac{f_{\mathrm{VSMC}}}{2}\left\{\begin{array}{l}{w}_{\mathrm{VSMC}}\left\{{\lambda}_{\mathrm{VSMC}},{\lambda}_{\mathrm{VSMC}}^{\perp}\right\}\left({\theta}_{\mathrm{VSMC}}\right)\\ {}+{w}_{\mathrm{VSMC}}\left\{{\lambda}_{\mathrm{VSMC}},{\lambda}_{\mathrm{VSMC}}^{\perp}\right\}\left(-{\theta}_{\mathrm{VSMC}}\right)\end{array}\right\}, $$

(4.192)

with

$$ {w}_{\mathrm{VSMC}}\left({\theta}_{\mathrm{VSMC}}\right)=A\ {C}_{\mathrm{act}}\left\{\mathrm{Erf}\left(\frac{\lambda_{\mathrm{VSMC}}\left({\theta}_{\mathrm{VSMC}}\right)-{b}_3}{b_1}+\frac{\lambda_{\mathrm{VSMC}}^{\perp}\left({\theta}_{\mathrm{VSMC}}\right)-{b}_4}{b_2}\right)+1\right\}, $$

(4.193)

where ${\lambda}_{\mathrm{VSMC}}=\sqrt{{\overrightarrow{\mathbf{n}}}^{\mathrm{VSMC}}\cdot \left({\mathbf{F}}^T\cdot \mathbf{F}\right)\cdot {\overrightarrow{\mathbf{n}}}^{\mathrm{VSMC}}}$ is the longitudinal stretch ratio of a SMC (i.e., cell stretch), and ${\lambda}_{\mathrm{SMC}}^{\perp }=\sqrt{{\overrightarrow{\mathbf{n}}}^{\mathrm{VSMC}\prime}\cdot \left({\mathbf{F}}^T\cdot \mathbf{F}\right)\cdot {\overrightarrow{\mathbf{n}}}^{\mathrm{VSMC}\prime }}$ is the transversal stretch ratio (F is deformation gradient). ${\overrightarrow{\mathbf{n}}}^{\mathrm{VSMC}}$and ${\overrightarrow{\mathbf{n}}}^{\mathrm{VSMC}\prime }$ are the longitudinal and transversal vectors, respectively. A is the level of activation (0 is passive state and 1 is fully active), C_act, b₁, b₂, b₃ and b₄ are material constants, and Erf() is the Gauss error function. Accordingly, the active stress of the coronary media is determined as the derivatives of the strain energy function, i.e., S_active = ∂W_active(E)/∂E and given by:

$$ {S}_{\theta \theta\;\mathrm{active}}=\frac{2A\ {C}_{\mathrm{act}}}{\sqrt{\pi }}\left(\frac{\cos^2{\theta}_{\mathrm{VSMC}}}{b_1{\lambda}_{\mathrm{VSMC}}}+\frac{\sin^2{\theta}_{\mathrm{VSMC}}}{b_2{\lambda}_{\mathrm{VSMC}}^{\perp }}\right)\mathrm{Exp}\left[-Q\right], $$

(4.194a)

$$ {S}_{zz\;\mathrm{active}}=\frac{2{AC}_{\mathrm{act}}}{\sqrt{\pi }}\left(\frac{\sin^2{\theta}_{\mathrm{VSMC}}}{b_1{\lambda}_{\mathrm{VSMC}}}+\frac{\cos^2{\theta}_{\mathrm{VSMC}}}{b_2{\lambda}_{\mathrm{VSMC}}^{\perp }}\right)\mathrm{Exp}\left[-Q\right]. $$

(4.194b)

where S_{θθ active} and S_zz active are the active circumferential and axial stresses of SMC contraction, and $Q={\left(\frac{\lambda_{\mathrm{VSMC}}-{b}_3}{b_1}+\frac{\lambda_{\mathrm{VSMC}}^{\perp }-{b}_4}{b_2}\right)}^2$. Therefore, the total stress is the sum of passive (Eqs. 4.190a, 4.190b) and active stresses (Eqs. 4.194a, 4.194b): S_total = S_passive + S_active.

The biaxial data of coronary media (Huo et al., 2013) is used to determine the nine material parameters (the volume fraction f_E, f_C, f_VSMC are incorporated into the estimations of k_E, k_C, C_act, respectively). The media is considered as a thin cylindrical shell and the second Piola–Kirchhoff circumferential stress obtained by experimental measurement is determined by ${S}_{\theta \theta}^{\mathrm{exp}}=\frac{{\mathit{\Pr}}_{\mathrm{i}}}{h{\lambda}_{\theta}^2}$, where P is distension pressure, ${r}_{\mathrm{i}}=\sqrt{r_{\mathrm{o}}^2-\frac{A_{\mathrm{o}}}{{\pi \lambda}_z}}$ is the inner radius in the loaded state, r_o is the outer radius in the loaded state, A_o is the wall area in a no-load state, and h = r_o − r_i is the wall thickness in the loaded state. The axial stress is computed by ${S}_{zz}^{\mathrm{exp}}=\frac{1}{\lambda_z^2}\left(\frac{{\mathit{\Pr}}_{\mathrm{i}}^2}{h\left({r}_{\mathrm{o}}+{r}_{\mathrm{i}}\right)}+\frac{F}{\pi \left({r}_{\mathrm{o}}^2-{r}_{\mathrm{i}}^2\right)}\right)$ with F presenting the axial force. The material parameters are determined by minimizing the square of the difference between the theoretical and experimental passive circumferential and axial second Piola–Kirchhoff stresses. The four passive parameters {k_E, k_C, N_C, θ_C} are calculated by:

$$ {\mathrm{Error}}_1=\sum \limits_{n=1}^N\left[{\left({S}_{\theta \theta\;\mathrm{passive}}^{\mathrm{exp}}-{S}_{\theta \theta\;\mathrm{passive}}\right)}^2+{\left({S}_{zz\;\mathrm{passive}}^{\mathrm{exp}}-{S}_{zz\;\mathrm{passive}}\right)}^2\right] $$

(4.195)

where ${S}_{\theta \theta\;\mathrm{passive}}^{\mathrm{exp}}$ and ${S}_{zz\;\mathrm{passive}}^{\mathrm{exp}}$ are the experimentally measured circumferential and axial stresses of the passive coronary media. The five active parameters {C_act, b₁, b₂, b₃, b₄} are then determined by:

$$ {\mathrm{Error}}_2=\sum \limits_{n=1}^N\left[{\left({S}_{\theta \theta\;\mathrm{total}}^{\mathrm{exp}}-{S}_{\theta \theta\;\mathrm{passive}}^{\mathrm{exp}}-{S}_{\theta \theta\;\mathrm{active}}\right)}^2+{\left({S}_{zz\;\mathrm{total}}^{\mathrm{exp}}-{S}_{zz\;\mathrm{passive}}^{\mathrm{exp}}-{S}_{zz\;\mathrm{active}}\right)}^2\right] $$

(4.196)

where ${S}_{\theta \theta\;\mathrm{total}}^{\mathrm{exp}}$ and ${S}_{zz\mathrm{total}}^{\mathrm{exp}}$ are the total circumferential and axial stresses of K⁺-induced SMC contraction. A limited-memory quasi-Newton method for large-scale optimization (L-BFGS method) is employed to solve the above minimizations (Eqs. 4.195 and 4.196).

Table 4.29 Material parameters of the microstructural model (Eqs. 4.190a, 4.190b) of passive coronary media

Table 4.30 Material parameters of two-dimensional strain energy function (Eq. 4.193) of active vascular smooth muscle cells (SMC) of coronary media

Appendix 14: 3D Microstructure-Based Model of Active Coronary Artery (Chen & Kassab, 2017)

The following general kinematic assumptions are considered for a 3D vessel mechanical model: (1) Vessel wall is incompressible; (2) Deformations are axis-symmetric and independent of axial position; (3) Transverse sections remain planar; and (4) There is a unique un-deformed reference configuration (i.e., zero-stress state, ZSS). The vessel is assumed as a cylindrical tube using a cylindrical coordinate system, of which three principal directions are circumferential direction $\overrightarrow{\boldsymbol{\theta}}$, radial direction $\overrightarrow{\boldsymbol{r},}$ and axial direction $\overrightarrow{\boldsymbol{z}}$. The corresponding stretches (λ_θ, λ_r and λ_z) are given as below:

$$ {\lambda}_{\theta }=\left(\frac{\pi }{\pi -\Theta}\right)\frac{r}{R},\kern1em {\lambda}_r=\frac{\partial r}{\partial R},\kern1em {\lambda}_z=\frac{l}{L} $$

(4.197)

where Θ is opening angle and R is radius to a point measured at ZZS configuration of a vessel, and r is the radius to the same point in the current configuration, i.e., loaded vessel, as shown in the top row of Fig. 4.25. L is the axial length of the segment at ZSS and l is the loaded axial length. According to material incompressibility, J = λ_θλ_rλ_z = 1, for the mapping between ZSS and loaded state, loaded radius r is determined as a function of unloaded R:

$$ r(R)=\sqrt{r_{\mathrm{o}}^2-\left({R}_{\mathrm{o}}^2-{R}^2\right)\frac{\pi -\Theta}{\lambda_z\pi }} $$

(4.198)

where r_o is the outer radius in the loaded state while R_o is that at ZSS. Analogously, r_i and R_i are the inner radii in the loaded state and ZSS, respectively.

Fig. 4.25

Top: A schematic diagram of vessel configurations and geometrical parameters. The left panel is the coordinate system of vessels with three principal directions: circumferential direction θ, radial direction r and axial direction z; ZSS is zero-stress state of vessel, and loaded state means vessel under pressure P. Deformation gradient tensor F describes tissue deformation from ZSS to loaded configuration. Parameters (R_o, R_i) are the outer and inner radii of stress-free vessel and R is radius to a point measured at ZSS, while θ is opening angle of vessel at ZSS. Parameters (r_o, r_i) are the outer and inner radii of loaded vessel and r is radius to the point measured at the loaded state. Bottom: Images of coronary artery and microstructure. Left: multiphoton microscopic (MPM) image of arterial cross section shows coronary artery layered-structure (Red: collagen; Green: elastin); Middle: MPM images of longitudinal-circumferential sections of collagen and elastin in adventitia, respectively; Right: Confocal images of SMCs in media (Green: F-actin; Blue: cellular nucleus). Reproduced from Chen and Kassab (2017) with permission

The radial component of the force equilibrium equation imposed on the loaded configuration is given by:

$$ \frac{\partial {\sigma}_{rr}}{\partial r}+\frac{\sigma_{rr}-{\sigma}_{\theta \theta}}{r}=0 $$

(4.199)

where σ_rr, σ_θθ are the radial and circumferential components of Cauchy stress, respectively. According to boundary conditions, ${\left.{\sigma}_{rr}\right|}_{r_{\mathrm{i}}}=-{p}_{\mathrm{i}},\kern0.5em {\left.{\sigma}_{rr}\right|}_{r_{\mathrm{o}}}=0$, the luminal pressure p_i can be written as:

$$ {p}_{\mathrm{i}}={\int}_{r_{\mathrm{i}}}^{r_{\mathrm{o}}}\left({\sigma}_{\theta \theta}-{\sigma}_{rr}\right)\frac{1}{r} dr $$

(4.200)

The axial tension can be determined by the integration of the axial components of Cauchy stress, σ_zz, over the cross-sectional area:

$$ N=2\pi {\int}_{r_{\mathrm{i}}}^{r_{\mathrm{o}}}{\sigma}_{zz}r\; dr=F+{p}_{\mathrm{i}}\pi\ {r}_{\mathrm{i}}^2 $$

(4.201)

where axial force is determined as $F=\pi {\int}_{r_{\mathrm{i}}}^{r_{\mathrm{o}}}\left(2{\sigma}_{z z}-{\sigma}_{z\theta}-{\sigma}_{rr}\right)r\; dr$, and ${p}_{\mathrm{i}}\pi\ {r}_{\mathrm{i}}^2$ accounts for the internal pressure in the closed tube during the test. The circumferential and axial equilibrium equations under distension–extension loading yield all shear stress components σ_rθ = σ_zθ = σ_zr = 0. Therefore, three components of Cauchy stress are provided by the 3D model, of which the radial component σ_rr cannot be predicted by a 2D model (Chen, Luo, et al., 2013; Huo et al., 2012).

The coronary artery is considered as an incompressible hyperelastic solid and characterized by a strain energy function (SEF) W(E) as a function of the Green-Lagrange strain tensor $\mathbf{E}=\frac{1}{2}\left({\mathbf{F}}^{\mathrm{T}}\cdot \mathbf{F}-\mathbf{I}\right)$. The Cauchy stress tensor σ is given by Gurtin (1982):

$$ \boldsymbol{\upsigma} =\mathbf{F}\frac{\partial W}{\partial \mathbf{E}}{\mathbf{F}}^{\mathrm{T}}-p\mathbf{I}=\mathbf{F}\cdot \mathbf{S}\cdot {\mathbf{F}}^{\mathrm{T}}-p\mathbf{I} $$

(4.202)

where F is the deformation gradient tensor and S is the second Piola–Kirchhoff stress tensor. I is the second-order identity tensor, scalar p is hydrostatic pressure, which acts as a Lagrange multiplier and must be determined from equilibrium and boundary conditions.

4.1.1 Microstructural Features of Active Coronary Arteries

Based on previous microstructural studies (Chap. 3, Fig. 3.29), the following axioms have been integrated to synthesize a realistic microstructural constitutive model: (1) A fluid-like matrix is employed and suggests the tissue undergoes affine deformations. Therefore, the SEF of the vessel wall can be represented by the volume-weighted summation of individual SEFs of every constituent. (2) All fibers are only resistant to tensile load and have negligible compressive and bending rigidities. Elastin fibers take up most of load at low pressures, while collagen fibers gradually become straightened and are recruited with an increase of pressure. (3) In adventitia, many collagen fibers oriented towards the axial direction and the others aligned nearly in the circumferential direction following a mixture of two normal distributions. The orientation of elastin fibers also follows two normal distributions where the minor orientation of elastin is approximately orthometric to the major orientation. (4) The straightening strain of adventitia collagen is found to follow a beta distribution with a mean and standard deviation of (0.35, 0.051). (5) In media, most SMCs arrange in θ − z (circumferential-axial) plane and slightly aligned off circumferential direction of blood vessels with symmetrical polar angles, and the axial active response of blood vessels is associated with SMC biaxial contraction. (6) The majority of media fiber bundles are also planar and their orientations are consistent with but not exactly identical to that of SMCs (O’Connell et al., 2008). Finally, there exist isotropic inter-lamellar (IL) elastin networks in media, which disperse over orientation space including the radial direction (Hollander et al., 2011a; O’Connell et al., 2008).

Based on the above axioms, the total SEF of vessel wall is a sum of volume-weighted SEFs of fibers and SMCs:

$$ W\left(\boldsymbol{E}\right)={f}_{\mathrm{IL}}{W}_{\mathrm{IL}}+{f}_{\mathrm{E}}{W}_{\mathrm{E}}+{f}_{\mathrm{C}}{W}_{\mathrm{C}}+{f}_{\mathrm{SMC}}{W}_{\mathrm{SMC}} $$

(4.203)

where f_E, f_C, and f_SMC are volume fractions of elastin, collagen, and SMC, respectively.

4.1.2 Passive SEF of Coronary Artery

Although the SMCs are the predominate contributors of active behavior, they provide negligible contributions to the passive properties of blood vessels (Matsumoto & Nagayama, 2012; Roach & Burton, 1957; Wolinsky & Glagov, 1964). The passive SEF is thus only determined by SEFs of elastin and collagen fibers:

$$ {W}_{\mathrm{P}}\left(\boldsymbol{E}\right)={f}_{\mathrm{IL}}{W}_{\mathrm{IL}}+{f}_{\mathrm{E}}{W}_{\mathrm{E}}+{f}_{\mathrm{C}}{W}_{\mathrm{C}} $$

(4.204)

The SEF of a fiber family is a function of fiber orientation θ as below (i = IL, E, C):

$$ {W}_i={\int}_0^{\pi }{\mathrm{\mathcal{R}}}_i\left(\theta \right){w}_i(e) d\theta $$

(4.205)

where ℛ_i(θ) is the orientation distribution density function of fiber i, and w_i(e) is the SEF of a single fiber. The uniaxial fiber strain e(θ) is determined by the local strain tensor E and the reference fiber direction N as: e(E, N) = E : N ⊗ N. It should be noted that ℛ_i(θ) satisfies the normalization criterion ${\int}_0^{\pi }{\mathrm{\mathcal{R}}}_i\left(\theta \right) d\theta =1$.

In adventitia, orientations of elastin and collagen fibers follow a mixture of two normal distribution of fiber (i = E, C):

$$ {W}_{Ai}=\sum \limits_j^2{\omega}_{ij}{\int}_0^{\pi }{\mathrm{\mathcal{R}}}_{ij}\left(\theta \right){w}_i(e) d\theta $$

(4.206)

where ${\mathrm{ℛ}}_{\mathit{ij}}\left(\theta \right)=\frac{1}{{К}_{\mathit{ij}}}\frac{1}{{\sigma }_{\mathit{ij}}\surd 2\pi }\mathrm{Exp}\left[-\frac{{\left(\theta -{\mu }_{\mathit{ij}}\right)}^2}{2{\sigma }_{\mathit{ij}}^2}\right],\left(i=E,C;j=1,2\right)$ is a truncated normal distribution density function with µ_ij and σ_ij as the mean and standard deviation, respectively. К_ij is the total weight of the truncated normal distribution ${К}_{\mathit{ij}}=\Phi \left(\frac{\pi -{\mu }_{\mathit{ij}}}{{\sigma }_{\mathit{ij}}}\right)-\Phi \left(\frac{-{\mu }_{\mathit{ij}}}{{\sigma }_{\mathit{ij}}}\right)$ (Φ is the cumulative distribution function of a normal distribution), and ω_ij is the weight of each normal distribution $\left({\sum}_j^2{\omega}_{ij}=1\right)$.

In media, elastin and collagen fibers are aligned off of circumferential direction of blood vessels, following two symmetric normal distributions are analogous to SMCs. It is assumed that both fibers have the same distributions ℛ_E(θ) = ℛ_C(θ) = ℛ_M(θ), which is a normal distribution density function with mean and standard deviation (µ_M, σ_M). Moreover, media IL elastin is an isotropic network, the overall SEF of the passive media can be thus written as (i = E, C):

$$ {\displaystyle \begin{array}{c}{W}_{\mathrm{M}i}=\left\{\frac{1}{2}{\int}_0^{\frac{\pi }{2}}{\mathrm{\mathcal{R}}}_i\left(\theta \right){w}_i(e) d\theta +\frac{1}{2}{\int}_{-\frac{\pi }{2}}^0{\mathrm{\mathcal{R}}}_i\left(-\theta \right){w}_i(e) d\theta \right\}\\ {}\kern1em +\frac{1}{\pi }{\int}_{-\frac{\pi }{2}}^{\frac{\pi }{2}}{w}_{\mathrm{IL}}(e) d\theta \end{array}} $$

(4.207)

According to microscopic responses of individual elastin fibers under mechanical loads (Chap. 3), the elastic properties is assumed to be linear, thus the SEF of elastin is given by:

$$ {w}_{\mathrm{E}}=\frac{1}{2}\ {k}_{\mathrm{E}}{e}^2 $$

(4.208)

where fiber strain e is larger than zero (fiber is only resistant to tensile load), and k_E is stiffness parameter of elastin fiber. IL elastin has a similar function but with a different stiffness k_IL:

$$ {w}_{\mathrm{IL}}=\frac{1}{2}\ {k}_{\mathrm{IL}}{e}^2 $$

(4.209)

Because of the wavy nature of collagen fibers, the SEF is considered to account for the nonlinear elastic behavior (Hollander et al., 2011a):

$$ {w}_{\mathrm{C}}=\frac{1}{1+{M}_{\mathrm{C}}}\ {k}_{\mathrm{C}}{\left(e-{e}_0\right)}^{1+{M}_{\mathrm{C}}} $$

(4.210)

where fiber strain e is larger than collagen straightening strain e₀, beyond which the collagen can withstand tension (which also denotes fiber waviness). k_C and M_C are parameters characterizing the nonlinear stress–strain response of collagen. It is found that collagen straightening strain e₀ follows a beta distribution in LAD adventitia (Chen, Slipchenko, et al., 2013):

$$ D\left({e}_0\right)=\frac{1}{B\left({\alpha}_1,{\alpha}_2\right)}\frac{{\left({e}_0-a\right)}^{\alpha_1-1}{\left(b-{e}_0\right)}^{\alpha_2-1}}{{\left(b-a\right)}^{\alpha_1+{\alpha}_2-1}}\kern0.75em $$

(4.211)

where B(α₁, α₂) is a beta function, and a and b the lower and upper bounds of the straightening strain e₀. A uniform distribution of straightening strain is assumed for media collagen since collagen bundles are thinner and have a lower volume fraction as compared to adventitia. The material properties of elastin and collagen fibers are assumed to remain constant throughout the vessel wall.

4.1.3 Active Stresses of Coronary Artery with SMC Contraction

Since the force of SMC contraction is generated by active energy consuming processes, a stored elastic energy (i.e., SEF), which is a function of strain state, is not appropriate. An empirical length–tension relationship is typically employed in this case. Here, a 3D model is proposed to account for triaxial responses of a single cell to better predict the overall behavior of blood vessels. A phenomenological stress–strain law is employed in the cell direction (i.e., major axis of a cell) as Schmitz and Böl (2011):

$$ {\sigma}_{\mathrm{SMC}}=A\ \left[\frac{\ {\rho}_1}{2}\left({\lambda}_{\mathrm{max}}^{\rho_2}-{\lambda}_{\mathrm{SMC}}^{\rho_2}-2\right){\left({\lambda}_{\mathrm{SMC}}-{\lambda}_{\mathrm{max}}\right)}^2+{\sigma}_{\mathrm{max}}\right] $$

(4.212)

where A is the level of activation (0 is passive state and 1 is fully active, A = 1 for the present study), ${\lambda}_{\mathrm{SMC}}=\sqrt{\boldsymbol{n}\cdot \left({\mathbf{F}}^{\boldsymbol{T}}\cdot \mathbf{F}\right)\cdot \boldsymbol{n}}$ is the longitudinal stretch of a SMC (i.e., cell stretch), n and n^′ are the longitudinal and transversal vector, respectively. λ_max denotes an optimal stretch ratio at which a SMC generated maximum stress σ_max. ρ₁ and ρ₂ determine the curvature and the skewness of the curve, respectively. When ρ₂ > 0, the absolute slope of lower stretch region (λ_SMC ≤ λ_max) is smaller than that of high stretch region (λ_SMC > λ_max), while ρ₂ < 0 leads an inverse behavior; and ρ₂ = 0 provides a symmetric curve as employed in previous models.

An analogous relation is used to account for SMC transverse stress in one of minor axes (there are two minor axes of a single cell: transverse and radial axes) as:

$$ \kern0.5em {\sigma}_{\mathrm{SMC}}^{\prime }=\tau A\left[\frac{\rho_1}{2}\left({\lambda}_{\mathrm{max}}^{\rho_2}-\lambda {\hbox{'}}_{\mathrm{SMC}}^{\rho_2}-2\right){\left({\lambda}_{\mathrm{SMC}}^{\prime }-{\lambda}_{\mathrm{max}}\right)}^2+{\sigma}_{\mathrm{max}}\right] $$

(4.213)

of which ${\lambda}_{\mathrm{SMC}}^{\prime }=\sqrt{{\boldsymbol{n}}^{\prime}\cdot \left({\mathbf{F}}^{\boldsymbol{T}}\cdot \mathbf{F}\right)\cdot {\boldsymbol{n}}^{\prime }}$ is SMC stretch in transverse direction, and τ is a dimensionless parameter, that determines the contractile properties in this direction, which can be regarded as the ratio of active axial stress to circumferential stress.

In the radial direction of a cell, a similar stress–strain law should be considered. SMCs, however, are largely compressed in the radial direction and stretched in other two directions under current distention-extension loading conditions. Moreover, Eq. (4.212) is a strongly nonlinear and non-monotonic function, of which a large span in stretch variable may lead to divergence of the solutions. A linear stress–strain law is thus considered as a simplification of the stiffness of active SMCs in the radial direction as:

$$ \kern0.5em {\sigma}_{\mathrm{SMC}}^{{\prime\prime} }={k}_{\mathrm{SMC}}\kern1em {\lambda}_{\mathrm{SMC}}^{{\prime\prime} } $$

(4.214)

of which ${\lambda}_{\mathrm{SMC}}^{{\prime\prime} }$ is the radial stretch of SMCs which is equivalent to the radial stretch ratio of the vessel wall as SMCs arrange in θ − z plane, and k_SMC is the stiffness in radial direction during SMC contraction.

The second Piola–Kirchhoff stress of each constituent is thus derived as:

$$ {\boldsymbol{S}}_i=\frac{\partial W\left(\boldsymbol{E}\right)}{\partial \mathbf{E}}=\sum \limits_i^{E,C,\mathrm{SMC}}{\omega}_{ij}{\int}_0^{\pi }{\mathrm{\mathcal{R}}}_{ij}\left(\theta \right)\frac{\partial {w}_i}{\partial e}\frac{\partial e}{\partial \mathbf{E}} d\theta $$

(4.215)

and the total stress is the volume-weighted sum as given by:

$$ \boldsymbol{S}={f}_{\mathrm{IL}}{\boldsymbol{S}}_{\mathrm{IL}}+{f}_{\mathrm{E}}{\boldsymbol{S}}_{\mathrm{E}}+{f}_{\mathrm{C}}{\boldsymbol{S}}_{\mathrm{C}}+{f}_{\mathrm{SMC}}{\boldsymbol{S}}_{\mathrm{SMC}} $$

(4.216)

of which f_ES_E + f_CS_C presents passive second Piola–Kirchhoff stress of vessel wall. f_SMCS_SMC is the active stress generated by SMC contraction, of which the components can be written as:

$$ {S}_{\mathrm{SMC}\_ ij}={\sigma}_{\mathrm{SMC}}\frac{\partial {\lambda}_{\mathrm{SMC}}}{\partial {E}_{ij}}+\kern0.5em {\sigma}_{\mathrm{SMC}}^{\prime}\frac{\partial {\lambda}_{\mathrm{SMC}}^{\prime }}{\partial {E}_{ij}}+\kern0.5em {\sigma}_{\mathrm{SMC}}^{{\prime\prime}}\frac{\partial {\lambda}_{\mathrm{SMC}}^{{\prime\prime} }}{\partial {E}_{ij}} $$

(4.217)

The Cauchy stress components (i.e., Eq. 4.202) of the vessel will then be obtained by substituting the constitutive laws for individual fibers and cells (Eqs. 4.209–4.214, and 4.217) into Eq. (4.216).

The full microstructural model has 15 geometrical parameters and requires 9 material parameters of individual fibers and cells. The orientation and undulation distribution parameters of fibers and cells and obtained statistical distributions based on two groups (one for adventitia and another for media) of coronary artery specimens have been measured (Chap. 3). These statistical measured parameters are directly integrated into the model to predict mechanical responses of additional porcine LAD arteries (n = 5). The 9 material parameters are determined by optimization with appropriate boundary condition (Eqs. 4.200 and 4.201). Moreover, microstructural geometrical parameters are refined for each sample by imposing restrictions to ensure fiber orientation and waviness still follow statistic distributions measured to obtain a better agreement between model predictions and experimental data.

4.1.4 Parameter Estimation

Parameters are optimized by least squares fit to the experimental data by minimizing an objective function based on the sum of squared residuals (SSE) between model predictions and experimental data. In general, passive and active parameters are determined separately. The passive material parameters of fibers are first determined by pressure–radius and pressure–force relations, and subsequently integrated into the active model to determine active parameters of SMCs. These passive parameters, however, are determined in a limited loading range, i.e., range of only passive loading conditions. The passive parameters, determined under axial stretch ratio λ_z = 1.3 and circumferential stretch ratio λ_θ from 0.9 to 1.8 (corresponding pressure varying from 20mmHg to 160 mmHg), are integrated into the active model to determined active parameters under λ_z = 1.3 and lower λ_θ from 0.6 to 1.7. Thus, the active parameters are under- or overestimated somewhat as passive parameters are not validated in the range of λ_θ from 0.6 to 0.9, which is an important region of vessel active response (i.e., pressure varying from 20 to 60 mmHg). Therefore, an objective function simultaneously accounting for both passive and active behaviors is defined as follows:

$$ \mathrm{SSE}=\frac{1}{nm}\sum \limits_{i,j}^{n,m}\left[{\left(\frac{{r_{\mathrm{o}}}_{ij}^P-{\widehat{r_{\mathrm{o}}}}_{ij}^P}{\sigma_{{\widehat{r}}_{\mathrm{o}}}^P}\right)}^2+{\left(\frac{F_{ij}^P-{\widehat{F}}_{ij}^P}{\sigma_{\widehat{F}}^P}\right)}^2+{\left(\frac{{r_{\mathrm{o}}}_{ij}^F-{\widehat{r_{\mathrm{o}}}}_{ij}^F}{\sigma_{{\widehat{r}}_{\mathrm{o}}}^F}\right)}^2+{\left(\frac{F_{ij}^F-{\widehat{F}}_{ij}^F}{\sigma_{\widehat{F}}^F}\right)}^2\right] $$

(4.218)

where i and j denote distension and axial loads at which the corresponding outer radius (${r_o}_{ij}^P$, ${r_o}_{ij}^F$) and axial force $\left({F}_{ij}^P,{F}_{ij}^F\right)$ are measured. P denotes passive responses and F denotes full responses (including both passive and active) of coronary arteries, n is the number of different pressures and m is the number of different axial stretch ratios used. ${\sigma}_{\widehat{\cdot}}$ is the standard deviation of experimental measurement, and ${\widehat{r_o}}_{ij}$, ${\widehat{F}}_{ij}$ are corresponding model predicted outer radius and axial force. The objective function with more restrictions (i.e., typically separated into two objective functions) leads to a better identification of material parameters; especially, for the active parameters. A genetic algorithm method is employed to search optimal parameter sets (summarized in Tables 4.31, 4.32, and 4.33 below) using Fortran language executed in Linux.

Table 4.31 Material parameter estimates of fibers and smooth muscle cells (SMC) determined from grouped statistical geometrical distributions of fibers and cells into the model, based on both passive and full distension–extension experimental data of axial stretch ratios λ_z = 1.3 and 1.5

Table 4.32 Material and geometrical parameter estimates of individual elastin, collagen fibers, and smooth muscle cells (SMC) with refined microstructural geometrical parameters that still follow statistic distributions

Table 4.33 Material and geometrical parameter estimates of fibers and smooth muscle cells (SMC) determined by a mean-value approach to eliminate continuous distribution of microstructure