Constitutive Modeling of Coronary Arterial Media—Comparison of Three Model Classes

Abstract

Accurate modeling of arterial elasticity is imperative for predicting pulsatile blood flow and transport to the periphery, and for evaluating the mechanical microenvironment of the vessel wall. The goal of the present study is to compare a recently developed structural model of porcine left anterior descending artery media to two commonly used typical representatives of phenomenological and structure-motivated invariant-based models, in terms of the number of model parameters, model descriptive and predictive powers, and requisite different test protocols for reliable parameter estimation. The three models were compared against 3D data of radial inflation, axial extension, and twist tests. Also checked are the models predictive capabilities to response data not used for estimation, including both tests outside the range of estimation database, as well as protocols of a different nature. The results show that the descriptive estimation error (model fit to estimation database), measured by the sum of squared residuals (SSE) between full 3D data and model predictions, was about twice as low for the structural (4.58%) model compared to the other two (9.71 and 8.99% for the phenomenological and structure-motivated models, respectively). Similar SSE ratios were obtained for the predictive capabilities. Prediction SSE at high stretch based on estimation of two low stretches yielded an SSE value of 2.81% for the structural model, and 10.54% and 7.87% for the phenomenological and structure-motivated models, respectively. For the prediction of twist from inflation-extension data, SSE values for the torsional stiffness was 1.76% for the structural model and 39.62 and 2.77% for the phenomenological and structure-motivated models. The required number of model parameters for the structural model is four, whereas the phenomenological model requires six to nine and the structure-motivated has four parameters. These results suggest that modeling based on the tissue structural features improves model reliability in describing given data and in predicting the tissue general response.

1. Introduction

Arteries function as conduits responsible for transport of blood to respective tissues and organs. The mechanical properties of arteries are of central importance in propagation of pulsatile blood along the arterial tree and to quantify the load imposed on the vessel wall cells. The cells can induce vessel remodeling which may occur under such pathologies as hypertension, atherosclerosis, and aneurysms. Reliable modeling of the mechanical behavior of arteries is of essential importance in understanding the disease progression and in the design of prevention and treatment strategies. The artery wall is mechanically nonlinear, anisotropic and heterogeneous [1–5] and nearly incompressible [6,7].

Three major types of constitutive modeling have been adopted in the research literature: phenomenological models [8–20] which provide specific mathematical expressions (exponential relations are most prevalent) to fit mechanical (stress-strain) response data. As such, their parameters are not related to the vessel wall micro-structure and are often difficult to interpret physically. Due to their phenomenological nature, they represent average properties of the wall tissue constituent, and therefore are likely to inaccurately predict the loads applied on the SMCs since these cells are known [21–23] to be in direct contact with just some of the tissue constituents (the fluid matrix and the elastin fibers). A second major class of models is the structure-motivated constitutive relations in which modeling of some wall constituents is structural while that of others is phenomenological. For instance, Holzapfel and co-workers [2, 24–27] considered the contributions of two anisotropic 2D families of helical wavy collagen fibers and an isotropic noncollagenous neo-Hookean matrix (for elastin, smooth muscle cells and ground substance). Although these models have a structural basis, they consider only the structure of collagen. This type of structure-motivated model accounts for the possibiility of collagen fiber buckling. However, the use of a neo-Hookean isotropic matrix which can sustain compressive loads does not account for the consequences of possible interlamellar fiber buckling which occurs as the vessel is inflated. The third class of models is the purely structural—they are entirely based on the tissue fiber networks and on the fluidlike matrix, and are free of phenomenological representation. This structural approach to tissue characterization [28–30] was previously applied to the tendon [31], the skin [32, 33], the passive [34] and active [35] LV myocardium, as well as to the heart valves, pericardium, and nonwoven tissue engineered scaffold [36–39]. In the present study it was developed for the coronary aortic media. Since structural models sum the contribution of the tissue microstructural elements, they hold a better potential to predict stresses applied on cells.

The mechanostructure link in arteries was first proposed by Roach and Burton[40]. Oka and co-workers [41–43] examined the rheological properties of arteries and veins in terms of idealized networks of collagen, elastin, and smooth muscle cells. These models; however, used highly simplified wall structures, neglecting some essential features (e.g., fibers orientation distribution, anisotropy of the interlamellar strut system, etc.). Recently, a model of the coronary artery media that incorporates the realistic 3D structure was developed within the framework of hyperelastic theory [44]. Based on histological reports [22,23,45–47], the structures include the media 3D elastin scaffold consisting of lamellae of helical elastin fibers interconnected by a network of thin and short inter-lamellar struts. The media also contains bundles of thick, helically oriented wavy collagen fibers dispersed between the concentric laminae.

The purpose of the present study is to compare the reliability of typical representatives of commonly used phenomenological and structure-motivated invariant-based (referred to as partial structure) models, with a recently validated structural model [44]. The comparison was made in terms of the model reliability in prediction of mechanical response patterns of LAD coronary media layer under 3D protocols of extension, inflation, and twist. Twist data is important since although arteries are mainly subjected to internal pressure and axial stretch, some vessels (such as the coronaries) may also twist under normal contraction of the heart. More generally, inclusion of twist is important for model validation over a wider range of deformation patterns.

2. Methods

2.1. The Models.

The passive media layer is considered as a hyperelastic incompressible solid with a strain energy function (SEF) W (E), where E = (F^TF ‒ I)/ 2 is the Green–Lagrange strain tensor, with F as the deformation gradient tensor, and I as the identity tensor. The Cauchy stress σ is obtained from W by the relation σ = F. ∂W/∂E. F^T ‒ pI, where p is a Lagrange multiplier needed to ensure incompressibility. Three models are investigated here, of which two have been extensively used in archival literature of arterial constitutive models. The two models are compared with a structural model recently validated for the arterial media [44].

2.1.1. Fung-Type Exponential Phenomenological Model.

Based on uniaxial experimental data of soft tissues, Fung [48] proposed an exponential stress-strain law which was found to be highly suitable for many types of tissues. Fung generalized [10] this model to a 2D exponential SEF for arteries and later generalized it into a 3D form [8]. Deng et al. [49] extended the 2D Fung model to incorporate shear deformations. A general form of Fung-type SEF was given by Humphrey [13]. The exponential-type constitutive models are commonly used for soft tissues, particularly for arteries, in both their strain component-based and invariant-based forms. In general form, Fung-type exponential SEF model is expressed as

$$\begin{array}{c}W=\frac{1}{2}C\left[\text{exp}\left(Q\right)-1\right]\\ Q=b_1{E}_{\mathit{R}\mathit{R}}^2+b_2{E}_{\Theta \Theta }^2+b_3{E}_{\mathit{Z}\mathit{Z}}^2\\ +2b_4{E}_{\mathit{R}\mathit{R}}{E}_{\Theta \Theta }+2b_5{E}_{\mathit{Z}\mathit{Z}}{E}_{\Theta \Theta }\\ +2b_6{E}_{\mathit{R}\mathit{R}}{E}_{\mathit{Z}\mathit{Z}}+b_7{E}_{R\Theta }^2+b_8{E}_{\Theta Z}^2+b_9{E}_{R2}^2\end{array}$$ (1)

where E_ij (i, j = R, Θ, Z) are the Green strain components, (R, Θ, Z) are the cylindrical coordinates in the radial, circumferential, and axial reference directions, respectively, and C, b₁,….,b₉ are material parameters. Although this model was proposed for the entire arterial response, it will be used here for the media. In Eq. (1), the shear terms are completely decoupled from the normal strain components. The coupling can be readily incorporated by means of mixed terms of normal and shear strain components, but this would significantly increase the number of model parameters.

2.1.2. Invariant-Based Partial Structure Model.

Holzapfel and co-workers [25, 26] adopted a fiber-reinforced structure model for blood vessels and media [2]. The model considers the contributions of an isotropic noncollagenous neo-Hookean matrix (elastin, SMCs) reinforced by two 2D families of helical wavy collagen fibers, symmetrically oriented with respect to the axial direction. Holzapfel's model is expressed by

$$\begin{array}{c}W=C_{I_1}(I_1-3)+\frac{k_1}{2k_2}\sum _{\alpha =4,6}\left\{\hspace{0.17em}\text{exp}\left[k_2{(I_{\alpha }-1)}^2\right]-1\right\}\\ I_1=\text{trC},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{I}_4=\mathrm{C}\hspace{0.17em}:\hspace{0.17em}\text{MM},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{I}_6=\mathrm{C}\hspace{0.17em}:\hspace{0.17em}\mathrm{M}\mathrm{\prime }\mathrm{M}\mathrm{\prime }\end{array}$$ (2)

with C as the right Cauchy–Green tensor, M and M^′ as the unit vector reference directions of the two symmetric collagen helices, and C_I1, k₁, k₂, and φ as material parameters where the latter is the angle of inclination of the two helical collagen fiber families relative to the axis. The product MM represents dyadic product of vector M with itself. In order to account for possible collagen fiber buckling, the anisotropic contribution of energy function W is active only when either I₄ ≥ 0or I₆ ≥ 0.

2.1.3. Structural Model.

In the microstructural approach [28], W reflects the strain energy of the tissue fibers with the basic assumptions that: (I) the passive tissue total strain energy is the sum of the strain energies of the fibrous constituents, elastin and collagen; (II) the fibers deform as if they are parts of a continuum (affine deformation); and (III) the fluidlike matrix (the ground substance) in which the fibers are embedded contributes hydrostatic pressure. For the present case of arterial media, the following additional specific assumptions were made in developing the fully microstructural model [44]: (i) the elastin network forms a 3D continuous scaffold made of concentric thick lamellae that are interconnected by a network of thin elastin fibrils (struts); (ii) the thin interlamellar struts are dispersed over orientation space with a mean radial direction [23]; (iii) the concentric lamellae are composed of two families of symmetric helical fibers with a mean pitch angle [45] (the fibers' orientations are distributed around the mean angle where one family consists of straight elastin fibers, while the other, of wavy collagen fibers arranged in parallel with the elastin fibers); (iv) all fibers are thin and long and have negligible compressive and bending rigidities compared to their tensile rigidity; and (v) as the tissue is stretched, wavy collagen fibers become gradually straight and mechanically active, a process which increases the tissue stiffness and induces nonlinear stress-strain relationship even if the mechanical response of individual fiber is linear.

A main microstructural feature is the fibers orientation density distribution function ${\mathcal{R}}_i^{*}(\psi ,\hspace{0.17em}\theta )$ [34] for each type of fiber i such that ${\mathcal{R}}_i^{*}(\psi ,\hspace{0.17em}\theta )$ sin ψdψdθ is the area fraction over a unit sphere surface of i-fibers oriented between the azimuthal and polar angles (ψ, θ) and (ψ + dψ, θ + dθ). For convenience, we will use a modified version of ℛ* (ψ, θ), namely, ${\mathcal{R}}_i\hspace{0.17em}\mathrm{=}{\mathcal{R}}_i^{*}\hspace{0.17em}\text{sin}(\psi )$. Each parallel fiber bundle has its own strain energy w_i(e), which depends on the uniaxial fiber strain e. Following the affine deformation assumption, e can be derived from the global strain tensor E using the projection e = E: NN, where N is the fiber reference orientation unit vector. With this approach, W (E) is expressed as the sum of strain energies of fibers in all directions

$$W(\mathbf{E})=\sum _i{\varphi }_i^0{\mathrm{\iint }}^{}{\mathcal{R}}_i(\psi ,\theta )w_i(e)d\psi \hspace{0.17em}d\theta$$ (3)

Where ${\varphi }_i^0$ is the reference (stress-free) volume fraction of type i fibers. If we differentiate W in Eq. (3) with respect to E, and use the aforementioned relation for e (E, N), we obtain the volume averaged fiber stress tensor as

$$\frac{\partial W}{\partial \mathbf{E}}=\sum _i{\varphi }_i^0{\mathrm{\iint }}^{}{\mathcal{R}}_i(\psi ,\theta )\frac{\partial w_i\left(e\right)}{\partial e}\mathbf{N}\mathbf{N}d\psi \hspace{0.17em}d\theta$$ (4)

In realizing the structural model, specific expressions must be assigned to the fibers mechanical properties and their density orientation distributions [44]. The choices made are general enough to incorporate wide ranges of structures and properties, based on histological and mechanical observations.

The interlamellar (IL) 3D elastin strut network was taken to be a combination of an isotropic and a 3D β function ℛ_il (α_il, β_il) over the azimuthal and polar angles α_il and β_il given by

$${\mathcal{R}}_{\mathit{i}\mathit{l}}({\alpha }_{\mathit{i}\mathit{l}},{\beta }_{\mathit{i}\mathit{l}})=\frac{1}{{\pi }^2(1+c_0)}\left[{\widehat{\mathcal{R}}}_m\right({\alpha }_{\mathit{i}\mathit{l}}\left){\widehat{\mathcal{R}}}_n\right({\beta }_{\mathit{i}\mathit{l}})+c_0]$$ (5)

Where ${\widehat{\mathcal{R}}}_k(\delta )$ (k = m, n and δ = α_il, β_il) is defined as

$${\widehat{\mathcal{R}}}_k(\delta )=\{\frac{1}{B\left(k\right)}{\left(\frac{1}{2}+\frac{\delta }{\pi }\right)}_{0,}^{k-1}{\left(\frac{1}{2}-\frac{\delta }{\pi }\right)}^{k-1},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\underset{\text{otherwise}}{\delta \in \left[-\frac{\pi }{2},\frac{\pi }{2}\right]}$$ (6)

with B(k) = [Γ(k)]²/ Γ(2k) (k > 1), and m, n, and c₀ as material constants. The distribution in Eqs. (5) and (6) is appropriately bounded and satisfies the normalization criterion. For the interlamellar network, a linear elastin fiber stress response was assumed, namely, ∂w_il/∂e_il = k_ile_il, with k_il as a material parameter (subscript “il” is the label index for this network). To account for the negligible compressive fiber rigidity, the condition ∂w_il(e_il) / ∂e_il = 0 for e_il ≤ 0 was imposed.

The families of helical elastin and collagen fibers are assumed to be aligned in two groups with symmetrical polar angles. The polar orientation of fibers in each group was allowed to be dispersed around the mean ±β_h by using a standard symmetric β distribution function; namely,

$${\mathcal{R}}_h\left({\beta }_h^{\mathrm{\prime }}\right)=\{\underset{0,}{\frac{1}{2\Delta {\beta }_{h}B\left(m_h\right)}{\left\{\frac{1}{4}\left[1-{\left(\frac{{\beta }_h^{\mathrm{\prime }}-{\beta }_h}{\Delta {\beta }_h}\right)}^2\right]\right\}}^{m_h-1}},\underset{\text{otherwise}}{{\beta }_h^{\mathrm{\prime }}\in \left[{\beta }_h-\Delta {\beta }_h,{\beta }_h+\Delta {\beta }_h\right]}$$ (7)

which satisfies the normalization criterion ${\int }_0^{\pi /2}{\mathcal{R}}_h({\beta }_h^{\mathrm{\prime }})d{\beta }_h^{\mathrm{\prime }}=1$. In Eq. (7) the symmetric β function exponent is denoted by m_h and the range of orientations is denoted by Δβ_h (both are additional material parameters). An equivalent orientational distribution of medial helical fibers was considered by Gasser [24], who used the von-Mises distribution function. For the helical fibers, a combined law of linear and power law branches is assumed for the fiber stress response, namely,

$$\begin{array}{c}\frac{\partial w_h\left(e_h\right)}{\partial e_h}\\ \begin{array}{cc}& =\{\begin{array}{cc}0\hfill & e<0\hfill \\ k_h^0{e}_h\hfill & 0<e\le e_{01}\hfill \\ k_h^0{e}_h+k_h{\left(e-e_{01}\right)}^{N_h}\hfill & e_{01}<e\le e_{02}\hfill \\ k_h^0{e}_h+k_h{\left(e_{02}-e_{01}\right)}^{N_h}\left[\frac{N_h\left(e-e_{02}\right)}{e_{02}-e_{01}}+1\right]\hfill & e>e_{02}\hfill \end{array}\end{array}\end{array}$$ (8)

Where $k_h^0$ is the stiffness related to the helical elastin fibers, k_h and N_h are related to the collagen fiber bundle stress-strain response which is nonlinear due to the fiber gradual recruitment with strain, and e₀₁ and e₀₂ are bounds of span of the collagen straightening strain, such that bellow e₀₁ all collagen fibers are undulated, while above e₀₂ all collagen fibers are stretched. To summarize, the number of model parameters in the most general case is 12. They are listed in Table 1 together with their associated physical significance.

Table 1.

The twelve parameters of the general structural model and their physical significance: Those underlined are of significant importance for the LAD media model

Parameter	Physical meaning
$k_h^0$	Helical elastin stiffness
$\underset{¯}{k_h}$	Helical collagen stiffness
Nh	Power law term representing gradual recruitment of collagen fibers
e01, e02	Collagen lowest and highest straightening strains
mh	Exponent of symmetric beta function related with helical fiber distribution
$\underset{¯}{{\beta }_h^0}$	Mean polar angle of helical fibers
Δβh	Range of polar angle dispersion of helical fibers
kil	Inter-lamellar struts stiffness
m, n, c0	Anisotropic inter-lamellar network

3. Residual Stress

Unloaded arteries are not stress-free. In rings cut from blood vessels, Fung and co-workers [51–54] showed that following a radial cut the ring opens to a circular section with an opening angle (OA) which is a measure of residual stress. Although it was earlier thought that residual stress in tissues stems solely from growth and remodeling of solid constituents, recent studies indicate that osmolarity-induced swelling is likely to affect the zero-stress state as well [50,55]. These findings are supported by similar observations of myocardial swelling effects in the rat left ventricle [56] and in aorta [57].

Only some of the previous phenomenological and structure-motivated models incorporated residual stress. Also, previous models did not include the effect of tissue swelling. In contrast, both swelling and residual stress are incorporated in the recently developed structural model [44]. It is assumed, for this model, that the open sector obtained by cutting the unloaded vessel is not stress-free, but swelled by a volume ratio J_SW compared to the true unswollen stress-free configuration. In the present study, all models compared have been introduced with identical measured residual stress data. In terms of swelling, two different comparisons were carried out. One was done with each model evaluated under the conditions in which it was developed, namely, without swelling for the phenomenological and partial structure models, and with swelling, prescribed by J_SW = 1.35, for the structural model. The value of J_SW = 1.35 was taken from Ref. [58] for a blood vessel which resembles geometrically pig's LADs. In addition, runs were performed (results not shown), in which the value of J_SW was perturbed (ranged between 1.00 and 1.35). It was found that the estimations for values of J_SW lower than 1.35 were poorer than those achieved for J_SW = 1.35. For the phenomenological and partial structure models, additional comparison were done with swelling of J_SW = 1.35 incorporated into these models. The results (not shown) indicate that although the introduction of swelling into these models has some impact on the estimated parameters, it has little effect on the results of comparison between the three models. Hence in the subsequent investigation, each model was presented under the conditions in which it was developed (i.e., no swelling in the phenomenological and structure-motivated models).

4. Experimental Database

Models were tested against triaxial mechanical data [19, 60] of porcine coronary left anterior descending (LAD) media. Briefly, the database includes five specimens. The cylindrical media was separated from the adventitia and cannulated to a triaxial machine and preconditioned prior to mechanical testing. Data was measured for the outer vessel radius, the axial force, and the torque, in response to a series of applied stepwise luminal pressures (0 to 130 mm Hg), axial stretches (1.2 to 1.4), and twist (‒25 ° to 25°, which reflects a range of twist angle per unit length γ = ±0.04 rad/mm). In addition, the stress-free geometric data were recorded for each specimen to account for residual stress.

5. Kinematics

The media tube was loaded by internal pressure P_i, external axial force F, and external torque M, and responded with inflation, axial stretch, and twist. For these protocols, the following kinematical assumptions were adopted: (i) deformations are axis-symmetric and independent of axial position; (ii) incompressible vessel media; (iii) transverse sections remain planar (no warping); (iv) both the twist angle and axial displacement are independent of radial position; (v) quasistatic and elastic response; (vi) no luminal flow and associated shear stress; and (vii) there is a unique stress-free reference configuration, which can be obtained through a combination of cutting the tissue radially and unswelling it by immersion in a hyperosmotic solution [50, 56]. Following these assumptions, the deformation field is

$$r=r(R),\hspace{0.17em}\theta =\theta (\Theta ,Z),\hspace{0.17em}Z=Z(\mathrm{Z})$$ (9)

where (r, θ, z) and (R, Θ, Z) are the radial, circumferential, and axial cylindrical coordinates in the deformed and reference configurations, respectively. Four distinct configurations were considered as displayed in Fig. 1.

Fig. 1.

Planar schematics of vessel mappings. Left: In the true reference state the vessel is open, unswollen, and stress-free (SF) with inner and outer radii, R_i and R_o, length L, and an opening angle Θ₀. In the swollen state (SW) it is open but not stress free, with corresponding dimensions ${\widehat{R}}_i$ ${\widehat{R}}_0$, Λ₀ L, with Λ₀ as the SF-SW transformation stretch ratio, and an opening angle Θ₁. The closed unloaded (UL) configuration is not subjected to further swelling, and with dimensions ρ_i, ρ_o, and Λ₀ Λ₁ L, with Λ₁ as the SW-UL stretch ratio. The closed loaded (L) configuration has dimensions: inner and outer radii r_i and r_o, length λΛ₀Λ₁ L, with λ as the UL-L stretch ratio, and twist per unit length γ (not shown in figure). Right: Force and bending moment resultants of circumferential stress acting on the swollen open sector free edges; both vanish, yielding a boundary constraint on the circumferential stress distribution.

The combined mapping between stress-free and loaded configurations (SF-L) is given by Ref. [3] as

$$\begin{array}{ccc}r=r\left(R\right),& \theta =\frac{\pi }{{\Theta }_0}\Theta +\gamma {\Lambda }_0{\Lambda }_{1}Z,& Z=\lambda {\Lambda }_0{\Lambda }_{1}Z,\end{array}$$ (10)

By incompressibility, det F = 1 for the intermediate mappings SW-UL and UL-L, where F is the deformation gradient. For the mapping SF-SW an input volume ratio J_SW is subscribed.

6. Equilibrium Equations and Boundary Conditions

The radial component of the vectorial equilibrium equation (∇ · σ = 0) written in the loaded configuration has the form

$$\frac{d{\mathbf{\sigma }}_{\mathit{r}\mathit{r}}}{\mathit{d}\mathit{r}}+\frac{{\sigma }_{\mathit{r}\mathit{r}}-{\sigma }_{\theta \theta }}{r}=0$$ (11)

with σ_rr and σ_θθ as the Cauchy stress radial and circumferential components. The boundary conditions associated with Eq. (11) are as follows

$$\begin{array}{cc}{\mathbf{\sigma }}_{\mathit{r}\mathit{r}}(r=r_i)=-P_i,& {\mathbf{\sigma }}_{\mathit{r}\mathit{r}}\end{array}(r=r_0)=0$$ (12)

The total axial force F and torque M required to maintain equilibrium in the loaded state are related to the stress components by

$$\begin{array}{cc}F=2\pi {\int }_{r_i}^{r_0}{\mathbf{\sigma }}_{\mathit{z}\mathit{z}}rdr\hspace{0.17em}\mathrm{-}{P}_i\pi r_i^2,\hspace{0.17em}& M=2\pi \end{array}\hspace{0.17em}{\int }_{r_i}^{r_0}{\mathbf{\sigma }}_{\theta z}{r}^{2}dr\hspace{0.17em}$$ (13)

The circumferential and axial equilibrium equations, in absence of blood flow, yield σ_rθ = σ_rz = 0 for all r_i ≤ r ≤ r_o.

Radial equilibrium equations similar to Eq. (11) were solved for the SW and UL configurations as well, with associated boundary conditions of zero external loads. For the SW configuration both the resultant force f and bending moment m on the cut edges (associated with the circumferential stress component) vanish for every radial cut (see right part of Fig. 1). The first condition (f = 0) is identically satisfied from the radial equilibrium equation. The condition m = 0 implies the independent boundary condition [56]

$$m={\int }_{{\widehat{R}}_i}^{{\widehat{R}}_o}{\mathbf{\sigma }}_{\widehat{\Theta }\widehat{\Theta }}\widehat{R}d\widehat{R}=0$$ (14)

where ${\mathbf{\sigma }}_{\widehat{\Theta }\widehat{\Theta }}$ is the circumferential stress component acting on the swollen sector free edges. Solution of equilibrium equations for each configuration is subjected to the relevant boundary conditions: the luminal pressure P_i, the loaded stretch ratio λ, the twist angle per unit length γ, and the SW configuration geometry characterized by ${\widehat{R}}_i$, ${\widehat{R}}_0$, and Θ₀. The unknowns to be estimated are the SF-SW and SW-UL stretch ratios Λ₀ and Λ₁, and the stress-free opening angle Θ₀. This was done based on the fit of the loaded vessel outer radius r_o, the axial force F, and torque M to experimental data.

7. The Torsional Stiffness

For the experimentally given twist range, recorded data shows [49,60] that in response to twist, the torque M and twist angle per unit length γ are linearly proportional. We denote the proportional ratio (torsional stiffness) by μ*(P_i, γ), which depends on the inner luminal pressure P_i and axial stretch ratio λ.

Under inflation, the media tube sample were observed to bulge, primarily along short transition zones near the rigidly cannulaefixed ends. Along most of its intermediate length, the tube remains cylindrical. Hence, the tube kinematics, while axially independent along the long cylindrical mid region, admits axial gradients in the short transition zones. The main effect of this noncylindrical vessel shape is the existence of a nonvanishing shear stress component σ_rz along the transition zones which should be incorporated into the radial equilibrium equation. To this end, a computational method was developed [44] which provides the apparent (measured) torsional stiffness μ and axial force F, which can be compared with the data. This procedure was applied for all models considered here.

8. Parameter Estimation

Parameters of each of the tested models were optimized to fit the triaxal data by least squares minimization of an objective function consisting of the sum of squared residuals (SSE) between model predictions and experimental data. The objective function incorporates the averaged normalized residuals of all three protocols and was defined as

$$\text{SSE}=\frac{1}{n}\sum _i\left[\left(\frac{r_o^i-{\widehat{r}}_o^i}{{\mathbf{\sigma }}_{{\widehat{r}}_o}}\right)+{\left(\frac{F^i-{\widehat{F}}^i}{{\mathbf{\sigma }}_{\widehat{F}}}\right)}^2+{\left(\frac{{\mu }^i-{\widehat{\mu }}^i}{{\mathbf{\sigma }}_{\widehat{\mu }}}\right)}^2\right]$$ (15)

where n is the number of data points, ${(•)}^i$ and ${(\widehat{•})}^i$ are the ith point model prediction and measured data, respectively, and ${\sigma }_{\widehat{•}}$ are the standard deviations of the experimental data for the three protocols. The search for optimal parameter set was carried out using the Genetic Algorithm (GA) method employing an MPI parallel computation based version of the C-code GAUL package [59]. The measure of model reliability was the value of SSE, where the lower its value the more reliable the estimation is.

9. Aspects of Model Comparison

The following attributes were considered in comparing the three models: (a) the number of model parameters which should be as low as possible to reduce the likelihood of ill-conditioning due to interaction between parameters. (b) The model reliability which determines the model utility. This was evaluated based on both the model descriptive and predictive powers. The descriptive power relates to the goodness of fit to data used for estimating the model parameters. This test was based on both the model-to-data sum of squared residuals [SSE, Eq. (15)], and on graphical inspection of the resemblance between model and data response. The model predictive power was examined from two aspects. In the first, the 3D response predictive power was evaluated where parameters estimated from 3D data (inflation, extension, and torsion) under two axial stretches (1.2 and 1.3) were used to predict response under axial stretch of 1.4 and to compare this prediction with data. The second aspect is the 2D-to-3D response prediction power where the model parameters were estimated using only 2D inflation-extension data and the model prediction of twist response was compared to data. In both tests, parameters estimates were compared as well. (c) The requisite data dimensionality which relates to the dimensions (2D versus 3D) of the protocols required to obtain reliable estimates of model parameters and reliable response prediction. Experimentally, 2D (e.g., inflation-extension) are preferred over 3D protocols (e.g., inflation-extension-twist) since they are simpler to perform.

10. Results

10.1. Number of Model Parameters.

The exponential Fung-type phenomenological model (1) [8, 19] requires seven parameters in the 2D inflation-extension tests (no shear components), eight parameters in the 3D inflation-extension-twist tests (nonzero E_{Θ Z} shear), and nine parameters that account for noncylindircal deformation (transition near the cannulae which involves E_RZ shear component as well). The invariant-based model [2, 25, 61] requires only four parameters (2). The largest number of parameters in the structural model is twelve. However, model structure analysis (parsimony test) revealed [44] that for the porcine LAD media in the physiological range of loadings, most structural features considered have only very small effects on the predicted response and are thus not needed. Hence, a reliable response prediction of the LAD media can be obtained with a reduced form having only four parameters (Table 1, the underlined parameters). One is the helical angle of the collagen fibers, two parameters represent the fibers stress-strain relationship and the fourth parameter is the IL elastin fibers stiffness. In the following comparisons, the four parameters version will be used for further analysis.

For convenient, the reduced four parameter structural model equations are summarized below.

• Interlamellar fiber energy function [using Eqs. (3) and (5) with m = 1, n = 1, c₀ = 1]

  $$W_{\mathit{i}\mathit{l}}\left(\mathbf{E}\right)={\varphi }_{\mathit{i}\mathit{l}}^0{\mathrm{\iint }}^{}\frac{1}{\pi }{w}_{\mathit{i}\mathit{l}}\left[e_{\mathit{i}\mathit{l}}\right({\alpha }_{\mathit{i}\mathit{l}},{\beta }_{\mathit{i}\mathit{l}}\left)\right]d{\alpha }_{\mathit{i}\mathit{l}}d{\beta }_{\mathit{i}\mathit{l}},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\frac{\partial w_{\mathit{i}\mathit{l}}}{\partial e_{\mathit{i}\mathit{l}}}=k_{\mathit{i}\mathit{l}}{e}_{\mathit{i}\mathit{l}}$$ (16)
• Helical fiber energy function [using Eqs. (7) and (8) with m_h = 0, Δβ_h ≪ 1, $k_h^0=0$, e₀₁ = 0, e₀₂ ≫ 1]

  $$W_h\left(\mathbf{E}\right)=\frac{{\varphi }_h^0}{2}\left\{w_h\left[e_h\right({\beta }_h^0\left)\right]+w_h\left[e_h\right(-{\beta }_h^0\left)\right]\right\},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\frac{\partial w_h}{\partial e_h}=k_h{e}_h^{N_h}$$ (17)

In Eqs. (16) and (17) the values of the volume fractions ${\varphi }_{\mathit{i}\mathit{l}}^0$ and ${\varphi }_h^0$ cannot be estimated separately from the other parameters and are thus incorporated into the estimations of k_il and k_h.

10.2. Model Descriptive Power.

A list of the parameters estimates for the three compared models is given in the upper panels of Tables 2–4. The fit to the data of the three models for the combined 3D protocols (inflation-extension-twist) is plotted in Figs. 2–4. The corresponding SSE values for each of the measured variables (external sample radius r_o, axial force F, and torsion apparent stiffness μ) are listed in the lower panels of Tables 2–4 for each of the models, and compared between the models in Table 5.

Table 2.

Descriptive power of the Fung-type nine parameters phenomenological model [Eq. (1)]. Upper panel: Parameters estimates for the five samples. Lower panel: The corresponding SSE values for the measured variables (external sample radius r_o, axial force F, and torsion apparent stiffness μ).

Sample		1	2	3	4	5	Mean ± SEM
C	(KPa)	28.66	34.53	23.43	19.79	31.66	27.61 ± 1.50
b1	(‒)	2.88	3.56	2.57	2.77	2.79	2.91 ± 0.09
b2	(‒)	2.89	3.38	3.51	3.39	3.22	3.28 ± 0.06
b3	(‒)	7.44	6.72	5.96	5.36	6.03	6.30 ± 0.20
b4	(‒)	1.67	1.23	1.28	0.56	0.96	1.14 ± 0.10
b5	(‒)	2.59	2.29	2.45	1.21	1.87	2.08 ± 0.14
b6	(‒)	2.32	2.08	1.85	1.55	2.16	1.99 ± 0.07
b8	(‒)	2.15	2.22	2.54	2.37	1.79	2.21 ± 0.07
b9	(‒)	4.27	4.90	3.97	4.12	4.92	4.43 ± 0.11
SSE
ro	(10‒2)	2.96	3.25	2.38	1.48	3.64	2.74 ± 0.21
F	(10‒2)	4.34	4.65	2.50	3.50	5.82	4.16 ± 0.31
μ	(10‒2)	2.38	3.07	4.61	2.34	1.65	2.81 ± 0.28
Total	(10‒2)	9.68	10.97	9.49	7.32	11.11	9.71 ± 0.38

Table 4.

Descriptive power of the structural model [Eq. (4)]. Upper panel: Parameters estimates for the five samples. Lower panel: The corresponding SSE values for the measured variables (external sample radius r_o, axial force F, and torsion apparent stiffness μ).

Sample		1	2	3	4	5	Mean ± SEM
kh	(102 KPa)	4.64	5.69	3.40	2.51	2.34	3.72 ± 0.36
Nh	(‒)	6.72	6.09	5.85	7.05	9.68	7.08 ± 0.38
βh	(rad)	0.63	0.56	0.70	0.68	0.59	0.63 ± 0.01
kil	(101 KPa)	7.99	11.06	4.73	8.05	12.02	8.77 ± 0.72
SSE
ro	(10‒2)	1.93	2.59	2.80	1.23	3.08	2.33 ± 0.19
F	(10‒2)	1.63	0.79	2.03	0.75	0.58	1.16 ± 0.16
μ	(10‒2)	1.03	1.81	1.36	0.77	0.50	1.09 ± 0.13
Total	(10‒2)	4.59	5.20	6.19	2.74	4.16	4.58 ± 0.32

Fig. 2.

Fung-type phenomenological model (1) descriptive power (sample No. 1). Predictions (lines) and experimental data (symbols) of (a) outer radius r_o, (b) axial force F, and (c) torsional apparent stiffness μ, versus inner luminal pressure P_i, under three axial stretch ratios (λ) of 1.2 (○), 1.3 (□), and 1.4 (+).

Fig. 4.

Fully structural model (4) descriptive power (sample No. 1). Predictions (lines) and experimental data (symbols) of (a) outer radius r_o, (b) axial force F, and (c) torsional apparent stiffness μ, all versus inner luminal pressure P_i, under three axial stretch ratios (λ) of 1.2 (○), 1.3 (□), and 1.4 (+).

Table 5.

Comparison of the models descriptive power: SSE levels (mean ± SEM) for each model and for each of the three measured features (external sample radius r_o, axial force F, and torsion apparent stiffness μ) and for the combined data (total)

SSE		Fung Type	Structurally motivated	Structural
ro	(10‒2)	2.74 ± 0.21	3.94 ± 0.27	2.33 ± 0.19
F	(10‒2)	4.16 ± 0.31	3.57 ± 0.41	1.16 ± 0.16
μ	(10‒2)	2.81 ± 0.28	1.47 ± 0.15	1.09 ± 0.13
Total	(10‒2)	9.71 ± 0.38	8.99 ± 0.64	4.58 ± 0.32

10.3. Model Predictive Power

10.3.1. Predictive Power.

Tables 6–8 list the three model parameters for each sample, as estimated from 3D inflation-extension-twist data under extension ratios of 1.2 and 1.3, together with the corresponding SSE levels of both the estimation data and of the predicted response under extension ratio of λ = 1.4, for each of the three measured outputs (external sample radius r_o, axial force F, and torsion apparent stiffness μ). Tables 6–8 also contain the ratios of partial (λ = 1.2 and 1.3) to full (λ = 1.2, 1.3, 1.4) data SSEs. From comparing SSE values of the prediction at stretch ratio 1.4 it is apparent that the SSE level for the structural model is the lowest among the three (2.81 ± 0.49 versus 10.54 ± 0.93 and 7.87 ± 1.09 for the Fung-type and invariant-based models, respectively). Figures 5–7 show a graphical illustration of model predictions at each stretch ratio, from which it can be clearly seen that the prediction for the highest stretch ratio is better for the structural model.

Table 6.

Axial stretch predictive power of the Fung-type nine parameters phenomenological model [Eq. (1)]. Upper panel: Parameters estimates for the five samples using only data of stretch ratios 1.2 and 1.3. Lower panel: The corresponding SSE values for the measured data, where simulation was performed for stretch ratio 1.4 (external sample radius r_o, axial force F, and torsion apparent stiffness μ).

Sample		1	2	3	4	5	Mean ± SEM
C	(KPa)	18.92	37.20	37.05	20.60	41.59	31.07 ± 2.63
b1	(‒)	3.00	4.17	2.12	2.36	2.25	2.78 ± 0.21
b2	(‒)	3.43	3.63	2.84	3.94	3.72	3.51 ± 0.10
b3	(‒)	4.88	5.16	3.45	8.26	4.87	5.33 ± 0.44
b4	(‒)	1.70	1.59	0.97	0.90	0.86	1.20 ± 0.10
b5	(‒)	1.93	1.97	1.69	2.34	1.87	1.96 ± 0.06
b6	(‒)	1.35	2.57	1.28	1.74	1.70	1.73 ± 0.13
b8	(‒)	2.30	1.81	1.68	2.16	1.33	1.85 ± 0.10
b9		2.45	3.44	2.97	3.07	4.88	3.36 ± 0.23
SSE
ro	(10‒2)	3.29	3.50	1.71	1.74	3.24	2.70 ± 0.22
F	(10‒2)	10.64	13.82	3.06	8.16	14.01	9.94 ± 1.14
μ	(10‒2)	1.98	3.32	4.96	1.56	1.31	2.62 ± 0.38
Total	(10‒2)	15.91	20.64	9.72	11.4%	18.56	15.26 ± 1.16
Partial/ Full data SSE	(‒)	1.64	1.88	1.02	1.57	1.67	1.56 ± 0.08
SSE λ = 1.2	(10‒2)	2.45	2.94	2.06	1.24	2.22	2.18 ± 0.16
SSE λ = 1.3	(10‒2)	2.80	2.75	2.25	1.53	2.34	2.33 ± 0.13
SSE λ = 1.4	(10‒2)	10.67	14.94	5.41	8.69	13.00	10.54 ± 0.93

Table 8.

Axial stretch predictive power of the structural model [Eq. (4)]. Upper panel: Parameters estimates for the five samples using only data of stretch ratios 1.2 and 1.3. Lower panel: The corresponding SSE values for the measured variables, where simulation was performed for stretch ratio 1.4 (external sample radius r_o, axial force F, and torsion apparent stiffness μ).

Sample		1	2	3	4	5	Mean ± SEM
kh	(102 KPa)	6.16	4.47	1.32	1.65	1.58	3.04 ± 0.54
Nh	(‒)	7.51	6.06	3.97	5.89	9.40	6.57 ± 0.51
βh	(rad)	0.61	0.52	0.66	0.66	0.57	0.60 ± 0.02
kil	(101 KPa)	7.48	10.20	3.70	6.47	10.90	7.75 ± 0.73
SSE
ro	(10‒2)	2.22	2.61	2.46	1.25	3.17	2.34 ± 0.18
F	(10‒2)	1.71	1.23	4.64	0.95	0.54	1.81 ± 0.41
μ	(10‒2)	0.95	2.09	2.74	1.33	0.68	1.56 ± 0.21
Total	(10‒2)	4.88	5.93	9.83	3.53	4.39	5.71 ± 0.62
Partial/ Full data SSE	(‒)	1.07	1.15	1.59	1.29	1.05	1.23 ± 0.06
SSE λ = 1.2	(10‒2)	1.84	1.47	2.22	0.38	1.85	1.55 ± 0.18
SSE λ = 1.3	(10‒2)	1.70	1.61	1.48	0.72	1.24	1.35 ± 0.10
SSE λ = 1.4	(10‒2)	1.34	2.86	6.13	2.42	1.30	2.81 ± 0.49

Fig. 5.

Fung-type model (1) stretch predictive power (sample No. 1): predictions (lines) compared with experimental data (symbols) of (a) outer radius r_o, (b) axial force F, and (c) torsional apparent stiffness μ, all versus inner luminal pressure P_i, under three axial stretch ratios (λ) of 1.2 (○), 1.3 (□), and 1.4 (+). Parameters are estimated from data at stretch ratios 1.2 and 1.3, and predictions are then simulated for stretch ratio 1.4 (bold dashed-dotted lines).

Fig. 7.

Structural model (4) stretch predictive power (sample No. 1). Predictions (lines) compared with experimental data (symbols) of (a) outer radius r_o, (b) axial force F, and (c) torsional apparent stiffness μ, all versus inner luminal pressure P_i, under 3 axial stretch ratios (λ) of 1.2 (○), 1.3 (□), and 1.4 (+). Parameters are estimated from data at stretch ratios 1.2 and 1.3, and predictions are then simulated for stretch ratio 1.4 (bold dashed-dotted lines).

10.3.2. 2D-to-3D Predictive Power.

Values of parameters for the three models, as estimated from 2D inflation-extension data, together with the corresponding SSE levels are given in Tables 9–11. These parameters were then used to predict the twist response in comparison with the data. A graphical presentation of the model-to-data fit in the twist response of the invariant-based and the structural models is shown in Fig. 8. The predicted twist response of the Fung-type exponential model is not shown since twist prediction in that model must incorporate the shear term E_ZΘ whose parameter (b₈) can only be estimated from twist data itself. In Fig. 8(a) it is seen that the 2D-based prediction grossly overestimates the torsion apparent stiffness in the partial structure model, while in the structural model [Fig. 8(b)], the 2D-based prediction fits the twist data well. From Tables 3 and 4, and 10 and 11, the SSE levels of the twist apparent stiffness μ for the invariant-based model are 2.77 ± 0.37 and 1.47 ± 0.15 using the 2D-based versus 3D-based estimation respectively, while for the structural model they are 1.76 ± 0.33 and 1.09 ± 0.13, respectively. These figures indicate that not only are the torsion apparent stiffness SSE levels of the structural model lower than those of the invariant-based one, but they are less sensitive to application of 2D versus 3D data for parameter estimation.

Table 9.

2D-to-3D predictive power of the Fung-type model [Eq. (1)]. Upper panel: Parameters estimated from 2D inflation-extension data. Lower panel: The corresponding SSE values for each of the measured variables (external sample radius r_o, axial force F, and torsion apparent stiffness μ). The torsion apparent stiffness is predicted based on the 2D measured data. Note that the twist associate parameters b8 and b9 are absent since twist data was not used in the estimation.

Sample		1	2	3	4	5	Mean ± SEM
C	(KPa)	43.32	57.01	48.11	66.31	58.97	54.74 ± 2.28
b1	(‒)	2.49	3.09	1.97	2.18	2.08	2.36 ± 0.11
b2	(‒)	2.16	2.73	2.29	2.20	2.34	2.34 ± 0.06
b3	(‒)	7.27	8.31	7.01	7.75	8.85	7.84 ± 0.19
b4	(‒)	1.54	1.41	1.28	1.32	0.94	1.30 ± 0.06
b5	(‒)	2.37	2.66	2.48	2.50	2.20	2.44 ± 0.04
b6	(‒)	2.41	2.83	2.07	2.91	2.57	2.56 ± 0.08
SSE
ro	(10‒2)	3.56	5.18	3.36	1.71	5.51	3.86 ± 0.38
F	(10‒2)	4.35	3.05	1.63	2.15	2.80	2.80 ± 0.26
μ	(10‒2)	34.55	40.68	46.89	34.83	41.17	39.62 ± 1.28
Total	(10‒2)	42.46	48.92	51.88	38.69	49.48	46.28 ± 1.37

Table 11.

2D-to-3D predictive power of the structural model [Eq. (4)]. Upper panel: Parameters estimated from 2D inflation-extension data. Lower panel: The corresponding SSE values for each of the measured variables (external sample radius r_o, axial force F, and torsion apparent stiffness μ). The torsion apparent stiffness is predicted based on the 2D measured data.

Sample		1	2	3	4	5	Mean ± SEM
kh	(102 KPa)	4.18	5.27	2.68	1.47	2.52	3.22 ± 0.37
Nh	(‒)	5.94	5.52	5.89	5.12	8.91	6.28 ± 0.38
βh	(rad)	0.64	0.58	0.67	0.68	0.60	0.63 ± 0.01
kil	(101 KPa)	6.93	10.49	5.67	8.09	12.46	8.73 ± 0.69
SSE
ro	(10‒2)	2.10	2.65	2.60	1.23	3.30	2.37 ± 0.19
F	(10‒2)	1.36	0.74	1.89	1.08	0.70	1.16 ± 0.12
μ	(10‒2)	1.56	2.01	3.89	0.81	0.54	1.76 ± 0.33
Total	(10‒2)	5.01	5.40	8.37	3.12	4.54	5.29 ± 0.48

Fig. 8.

2D-to-3D predictive power (sample No. 1). Comparison between predicted twist apparent stiffness (lines) and measured data (symbols) at three axial stretch ratios (λ) 1.2 (○), 1.3 (□), and 1.4 (+) for (a) the invariant-based and (b) the structural models.

Table 3.

Descriptive power of the invariant-based model [Eq. (2)]: Upper panel: Parameters estimates for the five samples. Lower panel: The corresponding SSE values for the measured variables (external sample radius r_o, axial force F, and torsion apparent stiffness μ).

Sample		1	2	3	4	5	Mean ± SEM
k1	(102 KPa)	2.03	1.59	1.22	2.10	0.81	1.55 ± 0.14
k2	(‒)	1.65	2.47	1.76	1.03	1.85	1.75 ± 0.13
φ	(rad)	0.65	0.62	0.70	0.72	0.63	0.66 ± 0.01
CI1	(102 KPa)	2.53	3.69	2.36	2.38	3.43	2.88 ± 0.16
SSE
ro	(10‒2)	3.60	4.98	2.35	3.92	4.87	3.94 ± 0.27
F	(10‒2)	2.62	2.62	2.80	6.49	3.33	3.57 ± 0.41
μ	(10‒2)	1.20	1.12	1.00	2.46	1.58	1.47 ± 0.15
Total	(10‒2)	7.41	8.72	6.15	12.87	9.78	8.99 ± 0.64

Table 10.

2D-to-3D predictive power of the invariant-based model [Eq. (2)]. Upper panel: Parameters estimated from 2D inflation-extension data. Lower panel: The corresponding SSE values for each of the measured variables (external sample radius r_o, axial force F, and torsion apparent stiffness μ). The torsion apparent stiffness is predicted based on the 2D measured data.

Sample		1	2	3	4	5	Mean ± SEM
k1	(102 KPa)	0.50	0.29	1.02	0.35	0.20	0.47 ± 0.08
k2	(‒)	3.01	4.74	1.77	2.55	3.06	3.03 ± 0.27
φ	(rad)	0.64	0.60	0.68	0.70	0.64	0.65 ± 0.01
CI1	(102 KPa)	2.72	3.88	2.24	2.80	3.59	3.04 ± 0.17
SSE
ro	(10‒2)	1.76	3.25	2.39	1.78	3.44	2.52 ± 0.20
F	(10‒2)	2.59	3.01	4.84	7.05	3.12	4.12 ± 0.46
μ	(10‒2)	4.40	1.62	0.91	3.00	3.92	2.77 ± 0.37
Total	(10‒2)	8.75	7.87	8.13	11.84	10.48	9.41 ± 0.42

Table 7.

Axial stretch predictive power of the invariant-based model [Eq. (2)]. Upper panel: Parameters estimates for the five samples using only data of stretch ratios 1.2 and 1.3. Lower panel: The corresponding SSE values for the measured variables, where simulation was performed for stretch ratio 1.4 (external sample radius r_o, axial force F, and torsion apparent stiffness μ).

Sample		1	2	3	4	5	Mean ± SEM
k1	(102 KPa)	2.08	1.53	2.34	1.81	0.78	1.71 ± 0.15
k2	(‒)	1.61	2.63	1.10	1.22	1.84	1.68 ± 0.15
φ	(‒)	0.64	0.58	0.67	0.67	0.62	0.63 ± 0.01
CI1	(102 KPa)	2.39	3.39	1.81	2.05	3.24	2.58 ± 0.18
SSE
ro	(10‒2)	3.59	4.22	2.59	2.37	4.51	3.45 ± 0.24
F	(10‒2)	3.60	5.37	9.30	12.90	4.47	7.12 ± 0.97
μ	(10‒2)	0.75	0.82	1.48	1.79	1.26	1.22 ± 0.11
Total	(10‒2)	7.93	10.40	13.36	17.05	10.24	11.80 ± 0.88
Partial/Full data SSE	(‒)	1.07	1.19	2.17	1.32	1.05	1.36 ± 0.12
SSE λ = 1.2	(10‒2)	1.81	1.87	1.64	1.49	3.37	2.04 ± 0.19
SSE λ = 1.3	(10‒2)	2.67	1.60	1.02	1.72	2.43	1.89 ± 0.17
SSE λ = 1.4	(10‒2)	3.46	6.94	10.70	13.84	4.44	7.87 ± 1.09

10.4. Requisite Data Dimensionality.

The requisite data dimensionality has two aspects. The first relates to reliability (closeness) of parameters estimated from 2D data compared to those estimated from the 3D data. From Tables 2–4 and 9–11, comparison of each of the Fung-type model parameters estimated from the 3D inflation-extension-twist data versus their estimates based on the 2D inflation-extension data (excluding the shear parameters) yields mean values of (27.61, 2.91, 3.28, 6.30, 1.14, 2.08, 1.99) versus (54.74, 2.36, 2.34, 7.84, 1.30, 2.44, 2.56) for the parameters C, b₁, b₂, b₃, b₄, b_{5 ,} and b₆ (1), respectively_. For the structure-motivated invariant-based model, the corresponding estimates are (1.55, 1.75, 0.66, 2.88) versus (0.47, 3.03, 0.65, 3.04) for the parameters k₁, k₂, φ and C_I1 respectively (2). For the structural model (4) the estimates are (3.72, 7.08, 0.63, 8.77) versus (3.22, 6.28, 0.63, 8.73) for the parameters k_h, N_h, β_h, and k_il, respectively. Robustness of parameter estimates to 2D versus 3D based estimations can be measured by the highest and lowest levels of the parameters ratios between the two estimations over all model parameters. Ideally these ratios should be unity and close to each other. The above estimates show that these ratios are 1.40 and 0.50 for the Fung-type model, 3.30 and 0.58 for the structure-based model, and 1.16 and 1.00 for the structural one. Hence, the parameters of the structural model are the most robust with respect to 2D versus 3D-based estimations.

The other aspect of requisite data dimensionality is closely related to the 2D-to-3D predictive power of the twist response based on inflation and extension data. A comparison is shown in Fig. 8. The associated SSE levels for the predicted torsion apparent stiffness based on the 3D versus the 2D data (Tables 3 and 4, 10 and 11) are 1.47 ± 0.15 and 2.77 ± 0.37, respectively, for the invariant-based model and 1.09 ± 0.13 and 1.76 ± 0.33 for the structural model. The change in SSE levels between the two estimations is lower (61%) in the structural model than in the invariant-based model (88%).

11. Discussion

The two major classes of arterial constitutive models (phenomenological and partial structure invariant-based) were compared with a recently developed structural model. A comparison was made in terms of the number of model parameters, model descriptive and predictive potentials, and requisite data dimensionality for reliable parameter estimation. Model predictions were compared with 3D data of radial inflation, axial extension, and twist responses of the porcine LAD media layer. The results show that the structural model performed best followed by the partial structure and then the phenomenological models. These results suggest that inclusion of partial structure improves model reliability whereas inclusion of all structural tissue features provides the best performance.

11.1. Model Structure

11.1.1. Number of Model Parameters.

For the present case of the pig LAD media, the Fung-type model requires between eight to ten parameters (depending on the protocol) while both the invariant-based and structural models, require only four. The number of parameters is important since higher numbers pose the risk of significant interactions between parameters which may lead to ill-conditioning and decrease reliability.

11.1.2. Model Convexity.

The strain energy function should be convex under all combinations of model parameters, to ensure both physical response under any deformation pattern, and stability. As previously shown [62], structural models based on tissues fibrous networks have an important advantage as they are inherently strongly elliptic since the fibers stress-strain relationship are convex. In contrast, convexity of phenomenological models such as the exponential ones is parameter dependent [61] and must be checked for each set of parameters.

11.1.3. Physical Significance of Parameters.

Parameters of the phenomenological model (1) have no direct physical significance. The collagen parameters in the structure-motivated model (2) are based on similar structural considerations as in the structural model. However, the parameter of the phenomenological neo-Hookean elastin/cells/ground substance contribution cannot be assigned by a specific physical significance. In structural models parameters represent specific structural features, or specific properties of the tissue constituents. The advantage of this attribute is threefold. First, parameter estimates can be obtained not only from data of the tissue mechanical response, but also from morphometric data (the structural features) and from well controlled uniaxial tests of isolated constituents (such as the tissue fibers). This provides an independent manner of model validation. Second, the physical significance of parameters provides inherent constraints on permissible ranges of their estimates (e.g., fibers orientation must be in the range 0 ‒ 2π) thereby reducing their dispersion. Finally, structural models can be used to diagnose and monitor the progress of pathologies which affect specific tissue constituents.

11.2. Model Descriptive Power.

The model-to-data fit with parameters estimated from the full 3D response data was evaluated qualitatively based on inspection of response pattern, and quantitatively based on the respective SSE measure of deviation.

11.2.1. Pattern Analysis.

Fung-type model [8,10,13,48] provides good fit to the inflation data [Fig. 2(a)], and reasonable fit to the axial force [Fig. 2(b)] and torsional apparent stiffness data [Fig. 2(c)] under extension ratios of 1.2 and 1.3. The fit under extension ratio of 1.4 is less good as the model underestimates the torsional apparent stiffness and over estimates the axial force in the medium to high pressure range, while it provides an under estimation in the low pressure range. In addition, the predicted axial force-pressure relationship indicates leveling of the curve and a changing trend (maxima under low pressures), which is not supported by the data (Fig. 2). The structure-based model [2, 25, 61] provides good fit to the inflation data [Fig. 3(a)] and to the axial force data [Fig. 3(b)] at extensions 1.2 and 1.3, while it slightly underestimates the 1.4 axial force data. For the torsional apparent stiffness [Fig. 3(c)], the model is seen to slightly overestimate the data at the medium to high pressure ranges. In addition, the model predicts leveling of axial force data at low pressures, a feature which is not supported by experimental data (Fig. 3). The structural model provides good fit to the inflation, axial force and torsional apparent stiffness data under all three extension ratios.

Fig. 3.

Structure-motivated Invariant-based model (2) descriptive power (sample No. 1). Predictions (lines) and experimental data (symbols) of (a) outer radius r_o, (b) axial force F, and (c) torsional apparent stiffness μ, all versus inner luminal pressure P_i, under three axial stretch ratios (λ) of 1.2 (○), 1.3 (□), and 1.4 (+).

Fig. 6.

Invariant-based model (2) stretch predictive power (sample No. 1). Predictions (lines) compared with experimental data (symbols) of (a) outer radius r_o, (b) axial force F, and (c) torsional apparent stiffness μ, all versus inner luminal pressure P_i, under three axial stretch ratios (λ) of 1.2 (○), 1.3 (□), and 1.4 (+). Parameters are estimated from data at stretch ratios 1.2 and 1.3, and predictions are then simulated for stretch ratio 1.4 (bold dashed-dotted lines).

11.2.2. Quantitative Analysis.

The SSEs of the Fung-type model (Table 2) for the inflation and torsional twist data are low (2.74 ± 0.21 and 2.81 ± 0.28, respectively), while for the axial force it is higher (4.16 ± 0.31). The combined (total) SSE equals 9.71 ± 0.38. The corresponding SSEs for the partial structure model (Table 3) are, respectively, 3.94 ± 0.27, 3.57 ± 0.41, 1.47 ± 0.15, and total SSE of 8.99 ± 0.64. For the structural model (Table 4), the SSE levels are 2.33 ± 0.19, 1.16 ± 0.16, 1.09 ± 0.13, with a total SSE of 4.58 ± 0.32. These figures show that the structural model agrees with the data better than the other two models for every measured feature and for the combined response as well.

Both these qualitative and quantitative comparisons show a clear advantage of the structural model in terms of the model descriptive power. In fact, it is the only model that provides good fit to the 3D data under all conditions tested.

It is instructive to reflect on the possible reason for the differences in performance between the structure-based and structural models. This is an intriguing issue since both models rely on similar structural features, namely, helical fibers embedded in an isotropic matrix. The performance differences may stem from the fact that the structure-based model treats the matrix as a neo-Hookean isotropic solid, which can sustain compressive loads. In actuality, the tissue response to tension is carried by the fibers, which buckle under compression. The compressive loads are resisted by a fluidlike ground substance matrix, which contributes hydrostatic pressure. Another potential reason for the observed performance differences stems from the structure of the tensor ∂W(E)/∂E, which for the neo-Hookean matrix equals the constant diagonal tensor 2C_I1 I. This is a rather restrictive form of the matrix response. In the structural model on the other hand, ∂W(E)/∂E depends on the Green-Lagrange strain tensor and has in general both mutually unequal diagonal as well as deviatory terms. It thereby allows for more general response features of the elastin network compared to the neo-Hookean model.

11.3. Model Predictive Power

11.3.1. 3D Predictive Power.

An extensive examination was made on the three models 3D predictive power under increasing stretch. Parameter estimation was performed with data of the two low axial stretches while extrapolative simulation was made for stretch 1.4. An extrapolative rather than interpolative prediction was made since the former is a more challenging test for a model realism. For all models the SSE for stretch 1.4 was higher compared to the 1.2 and 1.3 SSEs, but for the structural model it was the lowest of the three. This result is also illustrated in Figs. 5–7. A possible explanation for this performance difference between the models is that an adequate characterization of fiber properties can be obtained from data at low stretches. Thus, a model which includes only fibers is more likely to be valid at stretches above those used for its estimation.

11.3.2. 2D-to-3D Predictive Power.

As indicated above, the twist response of Fung-type model cannot be predicted from 2D inflation-extension data. For the other two models, Fig. 8 shows that while the partial structure invariant-based model overestimates the torsion rigidity under all three extension ratios, predictions of the structural model agree well with the torsion apparent stiffness data. Tables 10 and 11 reveal a similar picture: the model-to-data comparison SSE level of the apparent torsion stiffness for the invariant-based model is 2.77 ± 0.37 while for the structural model it is 1.76 ± 0.33. There is a physical reason for the superior predictive potential of the latter model: under twist, some IL fibers buckle. These fibers are represented in the partial structure models by a phenomenological invariant-based matrix model which unlike the fibers, can sustain and resist compressive strain, thereby contributing to unrealistic higher apparent torsion stiffness than the one induced by buckled fibers.

11.3.3. Requisite Data Dimensionality.

The SSE figures regarding the estimation reliability based on 2D data versus those SSE based on 3D estimation indicate that estimates of parameters of the structural model vary little between 3D and 2D based estimation (the parameter which changed the most is N_h from 7.08 to 6.28—less than 12% change). This implies that for this model 2D based estimation is adequate for reliable characterization of the tissue 3D response. In the Fung-type model six of the parameters estimates vary little but the estimate of the prefactor C increases by nearly twofold (from 27.61 to 54.74) between 3D and 2D based estimation. In the invariant-based model, estimates of two of the four model parameters (k₁, k₂) vary substantially (from 1.55 and 1.75 to 0.47 and 3.03 for the two parameters). These results imply that only in the structural model, reliable parameter estimation can be carried out based on 2D inflation-extension data. The other two models require 3D (inflation-extension-twist) data.

11.4. Computational Issues and Physical Realization.

The realization of the structural model involves a significant computational effort which stems from the need to perform a 3D integration of the interlamellar network at each integration point as a part of the solution of the radial equilibrium equation. In contrast, the computation of the phenomenological and partial structure strain energy functions is simpler. The computational run time differences are problem dependent. On an average, the realization of the structural model is about five times slower for a typical estimation problem than that of the two others. These differences are related to the fact that the structural model aims to realize the real nature of the fibrous tissue structure, especially the morphology of the inter-lamellar strut system. However, there are good reasons why it is advantageous to use structural models in spite of the aforementioned difficulty: (a) it was demonstrated in the present study that this model out-performs the phenomenological and invariant-based models in terms of descriptive and predictive capabilities to experimental data; (b) structural models allow physical insight and understanding of the contribution of each of the tissue components to its global mechanical response; (c) these models can be readily generalized to include more complex features such as active properties induced by smooth muscle cells; and (d) as computational platforms get stronger and more efficient, the numerical effort in solving structural models will be substantially reduced.

11.5. Summary of Models' Comparisons.

The results show that for the pig LAD media, the Fung-type phenomenological model (1) requires 7–9 parameters while the partial structure (2) and structural (4) models require only four. Model convexity is a priori assured for the structural and partial structure models [61, 62]. As for the phenomenological model, convexity must be checked for each parameter set. Both the phenomenological and partial structure model provide reasonable fit to data under certain deformation regions, but deviate from the data in other deformation regions in terms of both response patterns and closeness to the data. The structural model provides good fit under all conditions tested. In terms of predictive potential, the structural model outperforms the other two models, in predicting response in deformation regions not used for parameter estimation, as well as for twist response when parameters were estimated from 2D inflation-extension data. Reliable parameter estimates can be obtained from 2D inflation-extension data only with the structural model, while for the other two models such estimates are less reliable.