A viscoelastic constitutive framework for aging muscular and elastic arteries

Abstract

The evolution of arterial biomechanics and microstructure with age and disease plays a critical role in understanding the health and function of the cardiovascular system. Accurately capturing these adaptative processes and their effects on the mechanical environment is critical for predicting arterial responses. This challenge is exacerbated by the significant differences between elastic and muscular arteries, which have different structural organizations and functional demands. In this study, we aim to shed light to these adaptive processes by comparing the viscoelastic mechanics of autologous thoracic aortas (TA) and femoropopliteal arteries (FPA) in different age groups. We have extended our fractional viscoelastic framework, originally developed for FPA, to both types of arteries. To evaluate this framework, we analyzed experimental mechanical data from TA and FPA specimens from 21 individuals aged 13 to 73 years. Each specimen was subjected to a multi-ratio biaxial mechanical extension and relaxation test complemented by bidirectional histology to quantify the structural density and microstructural orientations. Our new constitutive model accurately captured the mechanical responses and microstructural differences of the tissues and closely matched the experimentally measured densities. It was found that the viscoelastic properties of collagen and smooth muscle cells (SMCs) in both the FPA and TA remained consistent with age, but the viscoelasticity of the SMCs in the FPA was twice that of the TA. Additionally, changes in collagen nonlinearity with age were similar in both TA and FPA. This model provides valuable insights into arterial mechanophysiology and the effects of pathological conditions on vascular biomechanics.

Statement of significance

Developing durable treatments for arterial diseases necessitates a deeper understanding of how mechanical properties evolve with age in response to mechanical environments. In this work, we developed a generalized viscoelastic constitutive model for both elastic and muscular arteries and analyzed both the thoracic aorta (TA) and the femoropopliteal artery (FPA) from 21 donors aged 13 to 73. The derived parameters correlate well with histology, allowing further examination of how viscoelasticity evolves with age. Correlation between the TA and FPA of the same donors suggest that the viscoelasticity of the FPA may be influenced by the TA, necessitating more detailed analysis. In summary, our new model proves to be a valuable tool for studying arterial mechanophysiology and exploring pathological impacts.

1. Introduction

Both aortic aneurysm (AA) and peripheral artery disease (PAD) significantly increase cardiovascular mortality [1,2]. Thoracic AA carries an in-hospital mortality rate of 22% and surgical mortality rate of 18%. Meanwhile, an estimated 236 million people worldwide – 5% of the population – suffer from PAD [3,4]. PAD is among the most costly vascular diseases to manage based on expenses per patient, due to the high rate of repair failures and the need for repeated care [5]. Developing durable treatments for these arterial diseases necessitates a deep understanding of how arterial mechanical properties evolve with age in response to their mechanical environments. This understanding is essential for devising personalized medical strategies that consider the specific mechanical environments of individual arteries and the broader dynamics of the cardiovascular system.

Arteries can be broadly divided into two types: elastic, which are generally located closer to the heart, and muscular, found nearer to the periphery. One of the most studied elastic arteries is the aorta – the largest artery in the human body [6]. It is composed of substantial amounts of circumferentially aligned elastin lamellar sheets [7–9] and is primarily responsible for facilitating transport throughout the cardiac cycle. This allows the aorta to function as an elastic buffer chamber, expanding during systole to store elastic energy. It then recoils following the closure of the aortic valve, returning this energy during diastole [8,10–12]. This feature, known as the Windkessel effect, protects the left ventricle from pressure-related injuries, perfuses the coronary bed, and provides nearly continuous peripheral flow [13,14].

Muscular arteries, such as the femoropopliteal artery (FPA, the main artery in the lower limb), distribute blood from the elastic arteries to their respective peripheral organs. They regulate the amount of that delivery by relaxing and contracting the concentric layers of smooth muscle cells (SMCs) that populate most of their tunica media [6]. In the FPA, the most prominent elastin structure is the external elastic lamina (EEL) at the border of tunica media and adventitia, which contains longitudinally-oriented elastic fibers [15,16]. These elastic fibers facilitate longitudinal pre-stretch [17], ensure energy-efficient function [18], and aid organ motions, such as flexion of the limbs, by reducing arterial bending and kinking [19]. In addition, the SMCs in the FPA are primarily circumferentially oriented, assisting the artery with diameter regulation in response to downstream oxygen demands and easing arterial bending during limb flexion.

Comparative analysis of aging in these two arteries can elucidate the structural and mechanical adaptations related to their different functional roles and varying mechanical environments. This approach offers more insights than examining each artery individually, providing a deeper understanding of their responses to aging. For example, the study by Laucyte-Cibulskiene et al. has shown that differences in aortic and femoral artery stiffness correlate with vascular calcification [20]. Female type I diabetic patients were demonstrated to have increased elastic but not muscular artery stiffness [21], and the health of peripheral muscular arteries was closely linked to aortic health and complications of transcatheter aortic valve procedures [22–25].

The TA and the FPA demonstrate different morphological, mechanical, physiological, and structural adaptations to aging [15,26–29]. Although both arteries increase in diameter and thickness with age, the TA thickens and widens significantly faster than the FPA [29–34]. The residual strain in these tissues also adapts differently. The circumferential opening angle in the TA does not change with age but increases 2.4-fold in the FPA [29]. Young TAs are mostly isotropic, but become more anisotropic as they age due to longitudinal stiffening. The FPAs also stiffen with age but remain 51% more compliant longitudinally than circumferentially. In both arteries, the densities of elastin and SMCs decrease with age, collagen remains constant due to medial thickening, and the amount of glycosaminoglycans increases [29,35–38]. Mechanical testing shows significantly more hysteresis in the FPA than in the TA, with a strong preference for the circumferential direction.

In addition, viscoelasticity is seldom incorporated into arterial modeling, especially for organ-level simulations. Although studies have shown that viscoelasticity reduces the magnitude and temporal variation of circumferential stress and strain and causes reduced radial wall movement [39]. Incorporating viscoelasticity can result in 11% greater cross-sectional area throughout the cardiac cycle and a 25% decrease in wall shear stress [40]. It has also been shown to play a role in maintaining the Windkessel effect [41–43]. Burattini et al. [44] postulate that the viscoelastic properties may help to modulate the resonance frequency of arterial flow. Other works have shown that incorporating viscoelasticity resolves the differences in compliance between the Windkessel model and the true compliance of the arterial wall [43,45]. In addition, viscoelasticity can also affect wave reflection in arteries [46,47], which is correlated with cardiovascular risk and diseases [48–50]. The difference in viscoelasticity in these arteries, particularly in the context of aging, is still an area that is not well understood.

When combined with computational analysis, structural constitutive modeling leverages microstructural information to predict functional responses at the organ scale. This approach provides insights into how the mechanical behavior of tissue correlates with its microstructure and facilitates the derivation of intrinsic tissue properties that are challenging to measure experimentally. Most constitutive analysis of arteries focuses on hyperelasticity. Traditional viscoelastic approaches [51,52] mainly integrate rate effects using differential equations based on spring and dashpot elements. These approaches generally require many material parameters and extensive experimental data. In recent studies, fractional viscoelasticity has been used to model the behavior of soft tissue [53–55] by representing the material with a continuous viscoelastic relaxation spectrum. This approach is practical and extends the existing hyperelastic models. It introduces only one or two additional parameters with intuitive physical meaning, i.e., the degree of viscoelasticity, which varies from no rate dependence (elastic) to pure stress rate dependence. We have demonstrated this in some examples such as the liver, myocardium, and arteries [56–59].

Our present work aims to develop a generalized viscoelastic constitutive formulation that is not limited to a specific artery. This formulation identifies the stresses from the main contributors to the extracellular matrix (ECM), including the viscoelasticity of collagen and SMCs. We extended and generalized the constitutive model formulation and optimization framework developed previously for the FPA to characterize both the FPA and the TA. This framework also utilizes viscoelasticity to enrich the parameter estimation approach by integrating multiple data sources to improve material parameter estimation and was optimized for high throughput and automation. The new formulation was tested on autologous TA and FPA specimens obtained from 21 human subjects between the ages of 13 and 73 years. The results for the various ECM components were compared against experimental measures of their respective microstructure to validate the findings. The aging of additional material characteristics that are challenging to measure directly (i.e., nonlinearity and viscoelasticity) was examined for both the TA and the FPA. A more comprehensive picture, including the effects of aging on the mechanical response and viscoelasticity of separated components, opens the door for future analyses of how various risk factors differ in their effects on aging. This may provide additional insights into how to improve the predictive capabilities of computational analysis. We also examined the correlation between the parameters across artery types and studied their changes with age. Overall, these findings illuminate the mechanical adaptations of elastic and muscular arteries and explore the potential interrelations in their development.

Below, we will first compare the TA and FPA microstructures and discuss the experimental data used for this study. We will then review the viscoelastic constitutive model developed for the FPA and propose its extension to elastic arteries. Next, we will discuss the challenges of performing parameter estimation for the TA to obtain structurally accurate parameters and present a multi-stage approach that is robust and reflective of the TA microstructure. Parameter estimation results will then be compared against experimental measurements, and the changes in the mechanical properties of the different structural components of the TA and FPA with age will be examined.

2. Materials and methods

2.1. Arterial microstructure

The TA ensures consistent blood flow from the heart to various organs. Its mechanical properties are optimized to reduce the load on the left ventricle and enhance diastolic flow to the peripheral arteries. This functionality is supported by a dense network of elastic lamellae and elastin fibers within the medial layer [60–62]. The microstructure of the aorta is well described [63]. Given that the elastic lamellae are 3D elastic sheets, elastin generally exhibits little mechanical directionality. Meanwhile, the collagen fibers are organized into double-helical spirals with a pitch angle of approximately 45° in the adventitia. The SMCs in the tunica media also have a pitch angle of between 40° and 60° and are aligned with collagen [64]. Histological images of the TA ECM components are shown in Fig. 1A and C, and the structure is schematically illustrated in Fig. 2B. Mechanically, healthy young TAs are relatively isotropic and have similar circumferential and longitudinal physiologic stresses (Fig. 1E and G).

Fig. 1.

Microstructure of the Thoracic Aorta (TA) and the Femoropopliteal artery (FPA). The VVG stain of transverse (left) and longitudinal (right) slices of zoomed-in regions is shown in the middle, demonstrating elastin in black color for the A) TA and B) FPA. MTC stain of transverse (left) and longitudinal (right) slices depicting collagen (blue color) and smooth muscle cells (red color) for the C) TA and D) FPA. The longitudinal (left) and circumferential (right) Cauchy stresses plotted versus stretches for arteries in different age groups representing E) AT and F) FPA. Solid curves depict average specimens in each age group, while semi-transparent regions bound 25th and 75th percentile ranges. Scale bars in histological images have a length of 250 μm for the larger images and 50 μm for the zoomed-in images (adapted from [29,123,124]). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 2.

An illustration of the material axes of smooth muscle cells (${\mathbf{m}}_s$), elastin (${\mathbf{m}}_e$), and collagen (${\mathbf{m}}_c$) for the TA in A)-C) and FPA in D)-F); $m_4$ and $m_6$ illustrate the orientation of the two families of collagen fibers. A) and D) show layer differences from histology (adapted from [124]), B) and E) illustrate the components, and C) and F) show the axes relative to the Cartesian coordinate for mechanical testing $\stackrel{\mathrm{ˆ}}{\text{e}}$. The color and subscript correspond to the components: collagen (green, $c$), elastin (blue, $e$), and SMCs (red, $s$) when applicable. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

The FPA needs to support two essential physiological functions: cardiac cycle pulsation and the flexion-induced deformations of the lower limbs [19,65–67]. The former is facilitated by the SMCs of the tunica media, while the latter is supported by the elastic fibers in the external elastic lamina (EEL) located at the border of the tunica media and the adventitia [17]. These roles have led to distinct differences in the microstructure compared to the TA, notably in the cellular and fiber patterns (Fig. 1B and D). The SMCs are adapted to be predominantly circumferentially oriented, allowing efficient regulation of the vessel diameter depending on downstream flow demands, and assisting arterial deformations during limb flexion. On the other hand, the elastic fibers in the EEL are oriented longitudinally to help maintain axial pre-stretch and avoid buckling during limb flexion-induced axial compression [66,68] (Fig. 2E). External support of the tunica adventitia prevents the FPA from overstretching, primarily through the families of undulated type I collagen fibers mixed with dispersed elastic fibers. This unique structure gives rise to more viscoelastic anisotropic behaviors and distinct differences between the mechanical responses of the FPA along its main two axes.

2.2. Mechanical testing

The FPAs and TAs were obtained from 21 donors between the ages of 13 and 73 years as a part of the Live On Nebraska procurement that occurred within 24 hours of death and with the consent from the next of kin to use tissues for research [29]. Donor characteristics are summarized in Table 1. Fresh tissues were transported in 0.9% phosphate-buffered saline (PBS) at 4°C and tested within 4 hours of procurement. Mechanical testing was conducted using a CellScale BioTester (CellScale, Waterloo, ON, Canada) with 2.5 N load cells. The tests were performed on 13 × 13 mm specimens immersed in 0.9% PBS at 37°C. The longitudinal and circumferential directions of the arteries were aligned with the test axes (Fig. 2C and F). All specimens were preconditioned for 20 loading-unloading cycles at the maximum a priori estimated strain to ensure reproducible results [57,69–71]. The data set included 21 biaxial loading cycles with different longitudinal to circumferential stretch ratios ranging from 1:1 to 1:0.1 with a step size of 0.1, then 1:1 to 0.1:1, and one more 1:1 (Fig. 3). The interspersed equibiaxial strain protocols were used to verify that the tissues did not accumulate damage and maintained the same viscoelastic behavior. The experiment was completed with an equibiaxial relaxation protocol where the decay in longitudinal and circumferential forces was recorded over 600 s. All testing was done with a maximum strain rate of 0.01 s⁻¹ relative to the maximum strain, and the deformation gradient was measured by tracking the movements of graphite markers using a top-mounted camera. The resulting mechanical data were filtered to remove outliers and high-frequency noise (see Appendix A of [59]).

Table 1.

Subject age, sex (M = male, F = female), body mass index (BMI), cause of death (SI: Self-inflicted; GSW: gunshot wound; ICB: intracerebral bleeding; ICH: intracerebral hemorrhage; CVA: cerebral vascular accident), hypertension (HTN), diabetes mellitus (DM), Dyslipidemia (DLD), coronary artery disease (CAD), and smoking (packs per year × years).

ID	Age	Sex	BMI	Cause of Death	HTN	DM	DLD	CAD	Smoking
0	13	F	19.7	Trauma					0
1	19	M	40.6	SI SW					0
2	22	M	29.5	Trauma					0
3	26	M	22.9	SI hanging					0
4	28	M	28.2	Trauma					0
5	31	M	28.3	Electrocution					7.5 × 15
6	38	F	53.8	Cardiac Arrest	✓	✓		✓	0
7	39	M	32.1	Anoxia					22 × 22
8	43	M	21.9	Motorcycle accident					0
9	44	M	33.2	Cardiac Arrest	✓				0
10	47	F	46.6	ICB/ICH				✓	4.5 × 30
11	51	M	26.9	Cardiopulmonary Arrest	✓		✓		52.5 × 35
12	51	M	29.1	ICB/ICH		✓		✓	30 × 30
13	54	M	40.2	Cardiac Arrest	✓		✓	✓	0
14	55	M	40.1	Cardiac Arrest	✓		✓	✓	10 × 20
15	57	F	44.4	Cardiac Arrest					73 × 42
16	59	M	38.5	Cardiac Arrest	✓	✓			30 × 15
17	59	M	25.4	Trauma	✓				0
18	64	M	19.5	Cardiac Arrest	✓		✓		0
19	69	M	31.3	CVA	✓		✓		20 × 20
20	73	M	28.1	Cardiac Arrest	✓	✓			2.5 × 5

Fig. 3.

A) Representative loading paths for biaxial testing using the 1:1 protocol are shown in red. B) Graphical representation of the loading protocols showing the Green-Lagrange strain $E_l$ in the longitudinal direction and $E_c$ in the circumferential direction of each cycle. There are four phases: preconditioning, decreasing ratios of circumferential stretch, decreasing ratios of longitudinal stretch, and stress relaxation. Black curves are simulated, but only the blue curves are used in parameter estimation. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

2.3. Kinematics and kinetics

Motion can be characterized by the mapping of material points in the reference configuration, $\mathbf{X}\in {\mathrm{\Omega }}_0,{\mathrm{\Omega }}_0\subset {\mathbb{R}}^3$, to the spatial points in the physical configurations at time $t\in [0,T]$, $\mathbf{x}\in {\mathrm{\Omega }}_t$, ${\mathrm{\Omega }}_t\subset {\mathbb{R}}^3$ such that $\mathbf{x}(\mathbf{X},t)=\mathbf{u}(\mathbf{X},t)+\mathbf{X}$ [72–74]. The deformation gradient tensor, $\mathbf{F}={\nabla }_{\mathbf{X}}\mathbf{u}+\mathbf{I}$, characterizes the stretch and shear of the material and its determinant, $J=\mathrm{d}\mathrm{e}\mathrm{t}\mathbf{F}>0$, characterizes the volume change [74].³ Although it is debated in the literature [75–78], arteries are often assumed to be incompressible [79], i.e., $J=1$. The invariants of the right Cauchy–Green tensor, $\mathbf{C}={\mathbf{F}}^{\mathrm{T}}\mathbf{F}$, often provide the basis for characterizing deformations in constitutive relations (e.g., [72]), i.e.,

$$I_{\mathbf{C}}=\mathbf{C}:\mathbf{I},I{I}_{\mathbf{C}}=\mathbf{C}:\mathbf{C},II{I}_{\mathbf{C}}=\mathrm{d}\mathrm{e}\mathrm{t}\mathbf{C}.$$ (1)

Pseudo-invariants are used to model anisotropy in arteries resulting from the organized SMCs and embedded collagen and elastin fibers [74], i.e.,

$$I_4=\mathbf{C}:\mathbf{m}\otimes \mathbf{m}=\mathbf{m}\cdot \left(\mathbf{C}\mathbf{m}\right)=\left(\mathbf{F}\mathbf{m}\right)\cdot \left(\mathbf{F}\mathbf{m}\right)$$ (2)

where $\mathbf{m}$ is a unit normal vector describing the fiber orientation in the reference configuration and $I_4$ denotes the square of the stretch in the direction $\mathbf{m}$. A more generalized way of representing the distribution of fiber orientations is to use structure tensors [80–82], e.g.,

$$\mathbf{H}=\frac{1}{4\pi }{\int }_S\rho (\mathbf{m})\mathbf{m}\otimes \mathbf{m}\mathbf{d}S,\mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h}\mathrm{t}\mathrm{r}\mathbf{H}=1,$$ (3)

where $\mathbf{H}$ is a structure tensor involving the fiber dispersion via $\rho \left(\mathbf{m}\right)$, i.e., the probability density of the fiber orientation $\mathbf{m}$ in the reference configuration, and $D$ denotes the unit sphere $S$ [81].

The relation between applied traction and the stress is given by the Cauchy stress formula $\mathbf{t}=\sigma \cdot \mathbf{n}$, which is expressed in the deformed domain [72–74].

Alternatively, the first and second Piola-Kirchhoff stress tensors $\mathbf{P}$ and $\mathbf{S}$ are expressed with respect to the reference domain and are related to $\sigma$ by the Piola transformation [74]

$$\mathbf{P}=J\sigma {\mathbf{F}}^{-\mathrm{T}},\mathbf{S}=J{\mathbf{F}}^{-1}\sigma {\mathbf{F}}^{-\mathrm{T}}.$$ (4)

$\mathbf{S}$, together with its strain conjugate $\mathbf{E}=(\mathbf{C}-\mathbf{I})/2$, is suitable for constitutive modeling because it is completely defined in the reference configuration.

2.4. Fractional viscoelasticity

Traditional methods [51,52] for modeling arterial viscoelasticity mainly integrate rate effects using differential equations based on spring and dashpot elements. These approaches generally require more material parameters and extensive experimental data to describe the directional dependence and the nonlinearity of soft tissues as well as variable viscoelastic effects. Studies have shown that fractional viscoelasticity captures some of the key viscoelastic behaviors of soft tissues [58,83–85]. Conceptually, viscoelasticity in soft biological tissues arises from a hierarchical distribution of relaxation mechanisms [86,87]. Instead of modeling the resulting viscoelastic relaxation spectrum using discrete independent elements, e.g., in a generalized Maxwell model, the fractional model represents the entire continuous spectrum using one constant, the order of the fractional derivative, $v$. Each $v$ characterizes a spectrum ranging from $v=0$, purely elastic, to $v=1$, purely viscous. In general, fractional viscoelastic models compare more favorably to classical approaches for modeling viscoelasticity in cardiovascular tissues despite introducing only one addition parameter [88]. Only having one additional parameter is important for parameter estimation uniqueness. Also, the resulting fractional model response shows more physical responses across different loading frequency ranges and mechanical testing regimes. A frequently used definition is the Caputo derivative [89], i.e.

$$D_t^{v}g=\frac{1}{\mathrm{\Gamma }(1-v)}{\int }_0^t(t-s{)}^{-v}\dot{g}(s)ds,v\in [0,1],$$ (5)

which has the advantage that the Caputo derivative of any constant is 0. This approach was used to model viscoelasticity in a variety of soft tissues [55,90–93], including the heart [57,94], lung [95], liver [56], breast [96], heart valves [97], and arteries [54,58,98]. The numerical implementation can be found in [99] and [100].

2.5. Generalized constitutive model for arteries

There is a long history of developing constitutive models for aortic tissues. The most common TA models are extensions of the work by Holzapfel et al. [52]. For the FPA, our team has developed a nonlinear viscoelastic model [59] that enables the separate characterization of collagen, elastin, and SMC properties. The current models for the TA do not differentiate between the contributions of collagen and SMCs, and the variations in their constitutive forms complicate comparisons of properties between the two arteries and their aging processes. We aimed to develop a constitutive model that bridges this gap, accompanied by a parameter estimation approach to accurately determine these properties.

Our prior FPA model [59] has four components, of which two are viscoelastic:

$$\mathrm{\Psi }={\mathrm{\Psi }}_g+{\mathrm{\Psi }}_e+D_t^{{\alpha }_s}{\mathrm{\Psi }}_s+D_t^{{\alpha }_c}{\mathrm{\Psi }}_c-p(J-1),$$ (6)

where $p$ serves as a hydrostatic pressure and $p(J-1)$ is a penalty function. The subindices, $\{g,c,e,s\}$, are indices for ground matrix, collagen, elastin, and SMCs. The ground matrix is represented by a neo-Hookean material, where the stress is given by

$${\mathrm{\Psi }}_g=\frac{{\mu }_g}{2}\left(I_{\mathbf{C}}-3\right),{\mathbf{S}}_g=2\frac{\partial {\mathrm{\Psi }}_g}{\partial \mathbf{C}}={\mu }_g\mathbf{I},$$ (7)

where ${\mu }_g$ is the modulus. Collagen, elastin, and SMCs all share a similar Holzapfel-type fiber form, i.e.,

$$\begin{array}{l}{\mathrm{\Psi }}_k=\frac{{\mu }_k}{4b_k}\left(e^{b_k{I}_k^2}-1\right),{\mathbf{S}}_k=2\frac{\partial {\mathrm{\Psi }}_k}{\partial \mathbf{C}}={\mu }_k{I}_k{e}^{b_k{I}_k^2}{\mathbf{H}}_k,\\ I_k=\mathbf{C}:{\mathbf{H}}_k-1,\end{array}$$ (8)

but different structure tensors $\mathbf{H}$.

Here, we extend this framework for modeling the TA, focusing on the microstructural differences between the two arteries. We will assume that collagen fibers and SMCs lie in the circumferential-longitudinal plane, which was flattened. Here, the radial direction is ${\mathbf{e}}_r={\mathbf{e}}_3$, longitudinal direction is ${\mathbf{e}}_z=\mathrm{c}\mathrm{o}\mathrm{s}\theta {\mathrm{e}}_1+\mathrm{s}\mathrm{i}\mathrm{n}\theta {\mathrm{e}}_2$, and the circumferential direction ${\mathbf{e}}_{\theta }$ is the remaining orthonormal vector, when expressed with respect to the cartesian basis, i.e., the testing reference configuration. Elastin is modeled by an in-plane distribution defined by the structure tensor

$${\mathbf{H}}_e={\kappa }_e{\mathbf{I}}_{\text{plane}}+\left(1-2{\kappa }_e\right){\mathbf{e}}_z\otimes {\mathbf{e}}_z,{\mathbf{I}}_{\text{plane}}={\mathbf{e}}_z\otimes {\mathbf{e}}_z+{\mathbf{e}}_{\theta }\otimes {\mathbf{e}}_{\theta }$$ (9)

where ${\kappa }_e$ is the dispersion of elastin [81], i.e., ${\kappa }_e=0$ is longitudinally aligned and ${\kappa }_e=0.5$ is planer isotropic.

Since the exact 3D distribution of the SMCs was not known, SMCs were modeled as two families oriented with a pitch angle of $\beta$, i.e., ${\mathbf{m}}_{s,4}=\mathrm{c}\mathrm{o}\mathrm{s}\beta {\mathbf{e}}_z+\mathrm{s}\mathrm{i}\mathrm{n}\beta {\mathbf{e}}_{\theta }$ and ${\mathbf{m}}_{s,6}=\mathrm{c}\mathrm{o}\mathrm{s}(-\beta ){\mathbf{e}}_z+\mathrm{s}\mathrm{i}\mathrm{n}(-\beta ){\mathbf{e}}_{\theta }$. The corresponding structure tensors are

$${\mathbf{H}}_{s,i}={\mathbf{m}}_{s,i}\otimes {\mathbf{m}}_{s,i},i\in \{4,6\}.$$ (10)

Collagen is modeled according to the previously established bifamily form [59,81], i.e.

$${\mathbf{H}}_{c,i}=A\mathbf{I}+B{\mathbf{m}}_{c,i}\otimes {\mathbf{m}}_{c,i}+(1-3A-B){\mathbf{e}}_r\otimes {\mathbf{e}}_r,i\in \{4,6\},$$ (11)

where ${\mathbf{m}}_{c,4}=\mathrm{c}\mathrm{o}\mathrm{s}(\alpha ){\mathbf{e}}_z+\mathrm{s}\mathrm{i}\mathrm{n}(\alpha ){\mathbf{e}}_{\theta }$ and ${\mathbf{m}}_{c,6}=\mathrm{c}\mathrm{o}\mathrm{s}(-\alpha ){\mathbf{e}}_z+\mathrm{s}\mathrm{i}\mathrm{n}(-\alpha ){\mathbf{e}}_{\theta }$ are the mean orientation of each fiber family and $\alpha$ is the pitch angle (Fig. 2). The shape parameters $A=0.132,B=0.586$, and $C=0.02$ are typically derived by experimentally measuring $\rho \left(\mathbf{m}\right)$ [81,101] and fitting the resulting data. Due to difficulty measuring these parameters for all specimens and the consistency of these values in previously evaluated human arteries [101], we set these as constants.

As described in Section 2.1, these microstructural components are organized differently for TA and FPA to facilitate their function. For the TA, ${\kappa }_e\to 1/2$ and $\beta \to \alpha$ match the orientation of SMCs to collagen for the TA (Fig. 2 top row), following assumptions from the results of Cordoba and Daly [64]. On the other hand, for the FPA, ${\kappa }_e\to 0,\beta \to \pi /2$ gives the FPA microstructure (Fig. 2 bottom row).

Combining the above, the full model in terms of the second Piola-Kirchhoff stress tensor is given by

$$\mathbf{S}={\mathbf{S}}_g+{\mathbf{S}}_e+D_t^{v_c}\left[{\mathbf{S}}_{c,4}+{\mathbf{S}}_{c,6}\right]+D_t^{v_s}\left[{\mathbf{S}}_s\right]-p{\mathbf{C}}^{-1}.$$ (12)

The parameters that are needed to characterize this form are $\xi =\left\{{\mu }_g,{\mu }_e^{\mathrm{*}},{\mu }_c^{\mathrm{*}},b_c,{\mu }_s^{\mathrm{*}},b_s,\theta ,\alpha ,v_c,v_s\right\}$. Here ${\mu }_k$ are the modulus parameters of each structural component of the tissue, and $b_k,k\in \{c,s\}$ are the corresponding exponents, see Eq. (8). The parameter $\alpha$ is the angle of the collagen fiber families in Eq. (11) relative to the specimen orientation $\theta$. Lastly, $v_c$ and $v_s$ are the viscoelasticity constants for the fractional derivative of the collagen and SMCs.

2.6. Parameter optimization

Structural parameter estimation is often challenging due to the need to distinguish between the contributions of different ECM components. This generally needs to be done through known structural characteristics of the major contributors to the mechanical response. Such characteristics can include information about fiber alignment and dispersion, linearity vs nonlinearity of the different components, or, as we have demonstrated for the FPA [59], their viscoelastic response. The TA demands a different approach due to its behavior being more linear, more planer isotropic, and less viscoelastic. In the following subsections, we will summarize the optimization approach for the FPA and then present a new step-wise optimization approach for the TA, with each step focused on a specific model component.

2.6.1. Optimization for the FPA

The parameter optimization strategies for the FPA were presented in [59]. To summarize, the objective function has three parts,

$$ℱ(\xi )=[\chi (\xi )+\varphi (\xi )](1+𝒫(\xi )),$$ (13)

where $\chi$ is the residual function, $𝒫$ is a penalty/regularization term for the parameters, and $\varphi$ is additional fitted data – in this case, the hysteresis. Including the hysteresis improves the ability to distinguish the different constituents (see [59] Appendix B for a more thorough exploration of these terms and their impacts). For the stress residuals, we used the weighted $L^2$ norm, i.e.

$$\chi (\xi )={\sum }_{i\in \mathrm{1,2}}{\sum }_{p=1}^{n_p}{𝒲}_{ii}^p{\sum }_{k=N_0^p}^{N_e^p}{\left(S_{ii}^k(\xi )-{\hat{S}}_{ii}^k\right)}^2.$$ (14)

Only the normal stresses were considered due to the nature of the performed testing which did not include shear [16,102–104]. Here, ${𝒲}_{ii}^p$ are the weights for each loading cycle $p$ and the stress tensor component $S_{ii}.N_0^p$ and $N_e^p$ denote each loading cycle’s initial and final index, respectively. The weights are defined as

$${𝒲}_{ii}^p=\left[1+\frac{1}{4}{\left(\frac{p}{n_p}\right)}^2\right]{\left({\sum }_{k=N_0^p}^{N_e^p}{\hat{S}}_{ii}^k+\frac{1}{n_p}{\sum }_{k=0}^{N_e^{n_p}}{\hat{S}}_{ii}^k\right)}^{-1}.$$ (15)

The first term represents an increase in confidence for the later cycles, with values in the range [1,1.25]. The second term balances the weight of each loading cycle during optimization. This increases the robustness of the parameter estimation by balancing the importance of each protocol to characterize the material behavior and susceptibility of each loading ratio to noise.

The hysteresis term is calculated by the midpoint rules, i.e., for loading cycle $p$, the hysteresis ${\mathscr{H}}_{ii}^p$ is given by

$${\mathscr{H}}_{ii}^p(\mathbf{S},\mathbf{C})={\sum }_{k=N_0^p}^{N_e^p}\frac{1}{2}\left(S_{ii}^k+S_{ii}^{k-1}\right)\left(C_{ii}^k-C_{ii}^{k-1}\right).$$ (16)

The full contribution to the objective function is given by

$$\varphi (\xi )={\sum }_{i\in \mathrm{1,2}}{\sum }_{p=1}^{n_p}{𝒲}_{ii}^p{\left({\mathscr{H}}_{ii}^p(\mathbf{S}(\xi ),\stackrel{\mathbf{ˆ}}{\mathbf{C}})-{\mathscr{H}}_{ii}^p(\stackrel{\mathbf{ˆ}}{\mathbf{S}},\stackrel{\mathbf{ˆ}}{\mathbf{C}})\right)}^2.$$ (17)

In the previous study, we found that hysteresis was necessary for the unique determination of the SMC and collagen fiber orientations. This was due to the lack of shear stresses, resulting in less data for coupling between the material axes. To remedy this, we added two minor penalties on $\theta$ and $\alpha$ to account for these effects, i.e.,

$$\begin{array}{l}𝒫(\xi )=0.001{\theta }^2+0.01{\left(\alpha -\frac{\pi }{4}\right)}^2+{\left(v_c-v_{\text{long}}\right)}^2\\ +{\left[\frac{1}{2}\left(v_c+v_s\right)-v_{\text{circ}}\right]}^2.\end{array}$$ (18)

The scaling constants for these two terms are deliberately small, so they only affect the results significantly when the minimum is ambiguous. The precomputed viscoelastic constants $v_{\text{long}}$ and $v_{\text{circ}}$ were obtained by fitting the relaxation data ${\hat{S}}_{i,k}$ along each axis, i.e.

$$\begin{array}{l}{v}_{\text{long}}=\mathrm{a}\mathrm{r}\mathrm{g}{\mathrm{m}\mathrm{i}\mathrm{n}}_v{\sum }_k{\left(\frac{{\hat{S}}_{11}^k}{{\hat{S}}_{11}^0}-{\left(t^k\right)}^{-v}\right)}^2,\\ & v_{\text{circ}}=\mathrm{a}\mathrm{r}\mathrm{g}{\mathrm{m}\mathrm{i}\mathrm{n}}_v{\sum }_k{\left(\frac{{\hat{S}}_{22}^k}{{\hat{S}}_{22}^0}-{\left(t^k\right)}^{-v}\right)}^2.\end{array}$$ (19)

Due to the structural orientation of collagen and SMCs, only collagen affects the stress decay in the longitudinal direction, while both collagen and SMCs can affect stress decay circumferentially.

2.6.2. Optimization of the elastin phase in the TA

For the TA, we designed a stepwise approach. The first step focuses on elastin, utilizing the toe region of the complete biaxial data set. The second step focuses on the SMCs, utilizing the difference between the ensemble stress curve and the elastin in the toe region. The third and last step focuses on the collagen and the structure parameters using the complete data set.

Based on experimental evidence, the mechanical response of elastin can be reasonably approximated by a linear curve in $\mathbf{S}$ and $\mathbf{E}$ [105–107]. We observed a similar response in our previous results on the FPA [59], where the parameter $b_e$ was zero or close to zero. The contribution of elastin can be determined from the toe region of the mechanical response, where elastin is the most dominant [58,108–110]. Generally, this toe region is estimated from the lower bound of the collagen recruitment, i.e., when collagen fibers first begin straightening. The measurement of collagen fiber crimp distribution is difficult to acquire, especially for a large data set. Thus, another common approach is to approximate this as the point where the mechanical response deviates from linearity.

In the TA, elastin appears to be isotropically distributed inplane (see Section 2.1), which we model in Eq. (9) with ${\kappa }_e\to 0.5$. Following previous approaches, the mechanical response of elastin can be approximated by the steepest linear surface approaching the data from below (Fig. 4A and B). To perform the parameter estimation, we used the biased objective function,

$$\chi (\xi )=\left\{\begin{array}{ll}{\sum }_{i\in \left\{\mathrm{1,2}\right\}}{\sum }_{p=1}^{n_p}{𝒲}_{ii}^p{\sum }_{k=N_0^p}^{N_e^p}{\left(S_{ii}^k(\xi )-{\hat{S}}_{ii}^k\right)}^4& \text{if}{S}_{ii}^k(\xi )>{\hat{S}}_{ii}^k,\\ {\sum }_{i\in \left\{\mathrm{1,2}\right\}}{\sum }_{p=1}^{n_p}{𝒲}_{ii}^p{\sum }_{k=N_0^p}^{N_e^p}{\left(S_{ii}^k(\xi )-{\hat{S}}_{ii}^k\right)}^2& \text{if}{S}_{ii}^k(\xi )\le {\hat{S}}_{ii}^k.\end{array}\right.$$ (20)

Fig. 4.

Best fit of the elastin demonstrating A) the longitudinal stress surface and B) the circumferential stress surface. Lower panels show the remaining difference between the data and elastin in C) the longitudinal direction and D) the circumferential direction.

This objective function penalizes the model quartically when it exceeds the data. The per-protocol weighting $𝒲$ from Eq. (15) is retained. The low ratio protocols proved to be especially important due to fewer collagen fibers and SMCs being engaged, establishing the upper limit of the elastin response. In this step, ${\mu }_s={\mu }_c=0$, i.e., SMCs and collagen do not contribute to the stress response. The only parameters being fit are ${\mu }_g$ and ${\mu }_e$. There remains a gap between the data and model, even in the toe region for high ratio protocols (Fig. 4C and D). This is assumed to be due to the SMCs and collagen, which have a directional dependence due to fiber rotation and shear.

2.6.3. SMC Optimization: An ensemble stress approach

Another approach to reduce the optimization problem is to consider only the ensemble stress. This approach considers planar soft tissues under biaxial mechanical testing, i.e., when the fiber microstructures are in-plane. For incompressible tissues, the ensemble stress is defined as the trace of the second Piola-Kirchhoff stress tensor under equibiaxial strain, i.e.,

$$\begin{array}{l}{S}_{\text{ens}}(t)=S_{11}(t)+S_{22}(t),C_{\text{ens}}(t)=\text{λ}{(t)}^2,\\ \text{given}[\mathbf{C}(t)]=\left[\begin{array}{ccc}\text{λ}{(t)}^2& 0& 0\\ 0& \text{λ}{(t)}^2& 0\\ 0& 0& 1/{\text{λ}}_z{(t)}^4\end{array}\right]\text{.}\end{array}$$ (21)

This expression is useful because there is no fiber rotation under equibiaxial deformation, e.g., the pseudo-invariant given by Eq. (2) leads to $\mathbf{m}\cdot \mathbf{C}\mathbf{m}={\mathrm{\lambda }}^2{\mathrm{c}\mathrm{o}\mathrm{s}}^2\theta +{\mathrm{\lambda }}^2{\mathrm{s}\mathrm{i}\mathrm{n}}^2\theta ={\mathrm{\lambda }}^2$. This also holds if the given structure tensors are used to describe the fiber dispersion; since $\mathrm{t}\mathrm{r}\mathbf{H}=1,\mathbf{H}:\mathbf{C}={\mathrm{\lambda }}^2$. Fitting the ensemble stress is independent of the microstructural orientations of the fibrous components. This approach has been used for parameter estimation of valvular and pericardial tissues for other structural models [109,110].

Although three 1:1 ratio protocols were utilized, they do not lie precisely on the equibiaxial line (Fig. 3A and Fig. 5A). This gives the illusion of increased extensibility, affecting the estimated exponent parameters $b_s$ and $b_c$. To get a better approximation of the ensemble stress curve, we first determine the maximum strain from the convex hull of the strain (Fig. 5A), then interpolate the loading and unloading surfaces separately (Fig. 5B) to derive a better estimate of the ensemble response (Fig. 5C).

Fig. 5.

Illustration of the fitting procedure for the thoracic aorta. A) Loading paths for a representative specimen with a poor ratio for the equibiaxial path (blue) due to preconditioning effects, plotted with the ideal equibiaxial path (red). B) Interpolation of the loading (blue) and unloading (red) paths. C) Corrected equibiaxial response (red) compared to the experimental 1:1 ratio protocol (blue). An example of an ensemble model fit for D) a 19 year old and E) a 59 year old artery. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fixing the ground matrix and elastin parameters, ${\mu }_g$ and ${\mu }_e$, and ignoring the structural parameters $\theta$ and $\alpha$, the only parameters being optimized at this stage are ${\mu }_s,b_s,v_s,{\mu }_c,b_c$, and $v_c$. We begin by focusing on the SMCs. Here, we apply the constraint $b_c>b_s$ as described in our previous work [59]. This constraint is to enforce a more nonlinear collagen fiber response (which is a result of fiber crimping) to better fulfill their functional role. Applying the biased objective function, Eq. (20), to the ensemble stress gives preference to the SMCs in the toe region, which has the smaller exponent. The response of the collagen naturally fills the gap in the high-stress region. A modification to the penalty function (18) is also made to take into account that the SMCs are aligned with the collagen in the TA, i.e.,

$$\begin{array}{l}𝒫(\xi )=0.01{\theta }^2+0.01{\left(\alpha -\frac{\pi }{4}\right)}^2+{\left(\frac{1}{2}\left(v_c+v_s\right){\mathrm{c}\mathrm{o}\mathrm{s}}^2(\alpha )-v_{\text{long}}\right)}^2\\ +{\left(\frac{1}{2}\left(v_c+v_s\right){\mathrm{s}\mathrm{i}\mathrm{n}}^2(\alpha )-v_{\text{circ}}\right)}^2.\end{array}$$ (22)

2.6.4. Collagen and microstructure optimization

The collagen parameters estimated in the previous step tend to be overestimated. Collagen fiber typically exists in a crimped state and does not bear significant stress until straightened. This type of behavior, which involves a zero stress region followed by a sharp exponential-like increase, is difficult to replicate with an exponential model. Overestimation of the exponent parameter $b_c$ is common. This is exacerbated by penalizing the model for exceeding the data in the toe region, which is unavoidable in an exponential model. To obtain a better estimate, a second fit to the ensemble stress curve is done for the collagen. In this step, the parameters ${\mu }_s,b_s$, and $v_s$ are also fixed, and the parameters ${\mu }_c,b_c$, and $v_c$ are refitted using the standard residual function (14). These updated parameters are then used as the initial guess for the final optimization step. In the final step, the parameters ${\mu }_g,{\mu }_e,{\mu }_s,b_s$ and $v_s$ are fixed to the values from the previous steps while ${\mu }_c,b_c,v_c,\theta$ and $\alpha$ are optimized using the full objective function (13).

2.6.5. Potential for multi-stage optimization for the FPA

An interesting question is whether this (or a similar) approach can be applied to the FPA. The isotropic distribution of the elastin fibers was crucial for the TA and allowed the elastin phase to be fitted independently. In the FPA, elastin would rely on the accurate knowledge of the circumferential axis. An ensemble stress approach can be used, but separating the contributions of the elastin and SMCs in the toe region becomes an issue. Extending this approach may have benefits, especially for the accuracy of the parameter determination for older individuals, but it is better saved for future considerations.

2.6.6. Optimization algorithm

For optimization, we used the differential evolution algorithm in the SciPy optimization and root-finding package [111]. Differential evolution attempts to find the global minimum by mutating a population of randomly seeded parameter vectors. We enabled the following options for the algorithm: polishing, which refines the best seed of each item using the L-BFGS-B gradient method, 64 seeds per parameter, a mutation factor of [0.7, 1.7], and a recombination ratio of 0.7. The constitutive models and objective functions were coded entirely in C++ and compiled as a Python module using the Cython framework [112].

To improve optimization performance, we can utilize the scaled modulus approach from the previous study [59]. This approach transforms the objective function surface to be more elliptic while not changing the best-fit parameters. This is done by replacing the constituent stresses ${\mathbf{S}}_k$ by ${\mathbf{S}}_k^{\mathrm{*}}$, where

$${\mathbf{S}}_k^{\mathrm{*}}={\gamma }_k{\mathbf{S}}_k,{\gamma }_k=\frac{1}{I_{\max ,k}}{e}^{-b_k{I}_{\max ,k}^2},I_{\max ,k}=\underset{t}{\max }{I}_k\left(t\right).$$ (23)

It can reduce the number of iterations to convergence by a factor of 2 or better and improve consistency of reaching the minimum [59]. In addition, this allowed setting the bounds for the modulus parameters to ${\mu }_c^{\mathrm{*}},{\mu }_e^{\mathrm{*}},{\mu }_s^{\mathrm{*}}\in \left[{10}^{-4},300\right]\mathrm{k}\mathrm{P}\mathrm{a}$. No specimens were loaded with stresses greater than 300 kPa for our experimental protocol. A modulus of zero leads to ambiguity of the exponents, requiring a non-zero lower bound. The modulus of the ground matrix was set to ${\mu }_g\in [0,3]\mathrm{k}\mathrm{P}\mathrm{a}$. A more adaptive bound was required for the exponent parameters. Following the observed trend, the bounds were set to $b_c\in \left[0.0,30.0/\left(C_{\mathrm{m}\mathrm{a}\mathrm{x}}-1.0\right)\right],b_e\in \left[0.0,15.0/\left(C_{\mathrm{m}\mathrm{a}\mathrm{x}}-1.0\right)\right]$, and $b_s\in \left[0.0,15.0/\left(C_{\mathrm{m}\mathrm{a}\mathrm{x}}-1.0\right)\right]$. The viscoelastic constants were in the range of [0,0.5]. In addition, we required that $b_c>b_s$ as mentioned above. Finally, $\theta \in [-\pi /4,\pi /4]$ and $\alpha \in [0,\pi /2]$ represented the entire search space.

3. Results

3.1. Parameter estimation

The constitutive parameters that describe the biomechanical responses of all 21 TAs and FPAs are summarized in Tables 2 and 3. Variations in the health and risk factors among donors likely contributed to the observed differences; however, the sample size was too small to analyze the impact of specific diseases and risk factors comprehensively. Nevertheless, general trends related to aging were still observable.

Table 2.

Best-fit model parameters for the TAs from all 21 subjects. The subscripts are $g$ for ground matrix, $c$ for collagen, $e$ for elastin, and $s$ for SMCs. The $\mu$’s are the scaled moduli (Eq. (23)), $b$’s are the exponents where applicable, $\alpha$ denotes the orientation of the collagen fiber families around the longitudinal axis, and $\theta$ is the overall orientation of the specimen in radians. $R^2$ is the coefficient of determination.

	GM	Elas.	SMC			Collagen			Structure
ID	${\mu }_g$ (kPa)	${\mu }_e^{\mathrm{*}}$ (kPa)	${\mu }_s^{\mathrm{*}}$ (kPa)	$b_s$	$v_s$	${\mu }_c^{\mathrm{*}}$ (kPa)	$b_c$	$v_c$	$\theta$ (Rad)	$\alpha$ (Rad)	$R^2$
0	3.00	64.53	192.93	0.00	0.05	1.38	11.46	0.01	−0.41	0.98	0.981
1	3.00	108.17	106.63	0.00	0.01	43.93	1.22	0.03	0.60	0.96	0.994
2	3.00	138.34	218.41	0.55	0.05	29.23	0.00	0.01	0.52	0.89	0.992
3	3.00	82.21	53.86	0.69	0.01	5.16	0.00	0.01	0.58	0.98	0.996
4	3.00	95.61	204.25	0.33	0.01	5.13	3.05	0.01	0.62	0.79	0.995
5	3.00	80.58	58.01	0.00	0.02	157.98	2.21	0.01	0.44	0.41	0.943
6	3.00	93.88	1.14	3.73	0.04	86.92	1.03	0.01	0.50	0.67	0.992
7	3.00	69.38	23.65	0.00	0.07	80.84	4.66	0.01	0.33	0.58	0.982
8	3.00	77.87	62.53	0.00	0.02	60.29	3.49	0.01	−0.54	0.54	0.989
9	3.00	117.48	23.89	0.00	0.03	117.79	8.92	0.02	0.42	0.42	0.986
10	3.00	68.16	16.01	0.00	0.03	91.28	4.16	0.06	0.57	0.32	0.982
11	3.00	48.50	1.48	6.88	0.01	6.02	100.00	0.02	0.58	0.58	0.989
12	3.00	154.30	0.20	35.99	0.02	62.74	20.89	0.01	−0.57	0.29	0.994
13	3.00	104.98	38.05	0.00	0.05	143.28	9.99	0.02	−0.41	0.72	0.991
14	3.00	62.10	17.47	0.00	0.04	47.38	14.12	0.01	−0.60	0.49	0.989
15	3.00	94.04	38.21	0.00	0.08	69.55	13.43	0.01	−0.44	0.82	0.989
16	3.00	95.09	17.48	0.00	0.03	101.16	7.10	0.02	0.46	0.57	0.993
17	3.00	175.59	7.50	0.00	0.09	123.70	46.54	0.02	−0.44	0.32	0.991
18	3.00	91.69	7.76	0.00	0.05	48.37	17.13	0.02	−0.56	0.69	0.996
19	3.00	76.82	11.13	0.00	0.06	85.63	19.09	0.01	−0.41	0.60	0.990
20	3.00	119.52	5.23	0.00	0.01	117.66	42.61	0.04	−0.31	0.55	0.992

Table 3.

Best-fit model parameters for the FPAs from all 21 subjects. The subscripts are $g$ for ground matrix, $c$ for collagen, $e$ for elastin, and $s$ for SMCs. The $\mu$’s are the scaled moduli (Eq. (23)), $b$’s are the exponents where applicable, $\alpha$ denotes the orientation of the collagen fiber families around the longitudinal axis, and $\theta$ is the overall orientation of the specimen in radians. $R^2$ is the coefficient of determination.

	GM	Elas.	SMC			Collagen			Structure
ID	${\mu }_g$ (kPa)	${\mu }_e^{\mathrm{*}}$ (kPa)	${\mu }_s^{\mathrm{*}}$ (kPa)	$b_s$	$v_{\text{s}}$	${\mu }_c^{\mathrm{*}}$ (kPa)	$b_c$	$v_c$	$\theta$ (Rad)	$\alpha$ (Rad)	$R^2$
0	1.60	55.52	45.10	0.74	0.06	38.05	1.26	0.03	0.04	0.84	0.992
1	3.00	29.27	105.57	2.50	0.04	31.40	2.50	0.04	0.23	0.58	0.993
2	3.00	25.31	117.18	1.39	0.04	54.36	1.39	0.03	0.12	0.57	0.991
3	2.96	18.48	141.06	0.97	0.05	108.32	1.01	0.03	0.17	0.49	0.948
4	3.00	17.38	63.28	3.37	0.05	42.57	3.37	0.03	−0.14	0.61	0.995
5	0.95	30.35	89.28	1.64	0.06	25.34	1.64	0.08	−0.27	0.28	0.986
6	3.00	22.60	146.08	10.05	0.07	4.48	10.05	0.06	−0.14	0.51	0.983
7	3.00	10.16	91.18	3.62	0.05	51.99	3.62	0.02	0.11	0.53	0.995
8	3.00	25.34	67.55	1.23	0.05	93.07	4.73	0.02	0.27	0.40	0.997
9	3.00	14.00	181.32	16.84	0.03	58.74	23.44	0.02	0.06	0.92	0.995
10	3.00	24.77	104.35	4.49	0.03	89.47	18.61	0.02	−0.32	0.49	0.997
11	2.74	24.60	48.36	1.63	0.06	66.64	10.26	0.02	0.23	0.27	0.998
12	3.00	48.63	155.33	21.45	0.06	149.08	38.53	0.05	−0.08	0.63	0.948
13	3.00	6.05	50.65	28.15	0.03	17.35	28.15	0.02	0.02	0.87	0.997
14	3.00	9.07	69.64	31.39	0.06	33.44	31.61	0.01	0.11	0.93	0.994
15	3.00	5.20	36.16	2.50	0.04	20.70	4.26	0.01	0.19	0.51	0.999
16	3.00	21.91	57.16	4.89	0.04	48.26	20.68	0.04	−0.26	0.46	0.997
17	3.00	18.01	129.35	25.68	0.02	196.32	25.68	0.04	0.16	0.57	0.998
18	3.00	19.88	22.34	0.87	0.07	22.28	20.80	0.06	−0.26	0.19	0.995
19	3.00	7.72	70.34	38.97	0.02	24.84	39.18	0.01	−0.10	0.71	0.997
20	3.00	8.65	143.73	23.08	0.04	104.62	30.84	0.04	−0.21	0.92	0.996

The quality of fit for the FPA was previously discussed in [59]. The new step-wise optimization approach proposed here for the TA was more robust than that for the FPA, optimizing fewer and less covariant parameters at each stage. Despite exhibiting less nonlinearity and hysteresis, and more coupled microstructures compared to the FPA, all specimens achieved reasonable convergence. This resulted in a minor improvement and produced the $R^2$ of 0.995 ± 0.005 for the TA and 0.994 ± 0.007 for the FPA, despite the restrictive nature of the step-wise process. The better fitting quality in the TA is likely due to its more linear elastin phase, which is typically easier to fit and leads to higher $R^2$ values.

This approach leveraged the isotropic elastin structure. An example of the fit to the elastin phase is demonstrated in Fig. 4. The 0.1:1 and 1:0.1 ratio protocols limited the elastin stress surface (Fig. 4A and B). Under low stretch, i.e., the short stretched axis of the 0.1:1 ratio protocols, the mechanical response reduced to a more linear behavior, similar to the longitudinal direction of the FPA [59], indicating a more elastin-driven response. On the other hand, the equibiaxial strain had slight nonlinearity in the toe region and left a gap (Fig. 4C and D). This is indicative of the existence of a nonlinear anisotropic phase that was not previously reported in less muscular tissues such as heart valves [110] and pericardium [109]. It was attributed to the SMCs (Fig. 5). The remaining highly nonlinear response at high strain was attributed to collagen. The collagen response was small in young arteries (e.g., 19-year-old (Fig. 5D)), but increased in older vessels (e.g., 59-yearold (Fig. 5E)). This approach was effective for both linear and nonlinear specimens.

3.2. Differences in the TA and FPA responses

The FPAs were significantly more nonlinear than the TAs (Fig. 6). This was partially due to the more significant contribution of collagen to the FPA behavior. In young individuals, the mechanical response of the TA was dominated by elastin, and collagen did not bear a significant load in the physiological range. With the advance in age, the mechanical response of the TA became more nonlinear in both the toe region and the high-stress region. The collagen began to bear more total stress at comparable stress levels. In general, the circumferential direction of the TA and the FPA shows relatively similar changes with age. For the TA, the mechanical response remained more similar between axes than for the FPA, whose circumferential direction became significantly more nonlinear/stiff. These differences were consistent with the histological assessment of the microstructure (Section 2.1).

Fig. 6.

Representative model fits for A),B) autologous TAs and C),D) FPAs depicting the longitudinal (left) and circumferential (right) responses of the 1:1 protocol and the contribution of the main ECM components in A),C) a 19 year old and B),D) a 59 year old artery. The best E),F) TA and G),H) FPA fits of the longitudinal response of all protocols in phase 2 (left) and the circumferential response of all protocols in phase 3 (right) for E),G) a 19 year old and F),H) a 59 year old artery.

3.3. Comparison vs experimentally measured density

The degradation of elastin represents a significant microstructural change associated with aging in both the TA and the FPA [29]. Accurately capturing this phenomenon through structural models is crucial for validating the physiological accuracy of these models. The moduli ${\mu }_e$, Eq. (8), corresponded well with the experimentally-measured changes in elastin density with age for both the TA and the FPA (Fig. 7A). The confidence errors shown for the fitted parameters are calculated following the method described in Appendix A. Despite the noise and variations between specimens, a clear linear correlation was observed between the experimentally measured density and model-fitted parameters when the datasets for the TA and FPA were combined (Fig. 8A). This suggests that the elastin modulus may be consistent between the arteries, scaling only with quantity and orientation.

Fig. 7.

A) Changes in the measured elastin (left), SMCs (middle), and collagen (right) densities with age. B) Maximum tangent modulus (MTM) of the same ECM components estimated from best-fit model parameters. The error bars show the parameter confidence error from a local linear regression at the optimum (Appendix A).

Fig. 8.

A) Experimentally measured elastin density vs ${\mu }_e$ from parameter estimation. B) Experimentally measured SMC density vs maximum tangent modulus (MTM) of SMCs. Changes in C) the nonlinearity parameter $b_s$ and D) the viscoelasticity $v_s$ of the SMCs with age. Changes in E) the nonlinearity parameter $b_c$ and F) the viscoelasticity $v_c$ for collagen with age. The error bars show the parameter confidence error from a local linear regression at the optimum (Appendix A).

The relationship between measured elastin density and modulus is straightforward due to the linear behavior of elastin. However, this simplicity does not extend to nonlinear materials. The source of nonlinearity in SMCs and the reason for its variation between specimens remains unclear. For collagen, nonlinearity is primarily attributed to recruitment and the gradual straightening of collagen fibers [110]. Once all collagen fibers are straightened, collagenous tissues exhibit a linear response. SMCs may demonstrate similar recruitment-like behavior, suggesting that the maximum tangent modulus could be a more comparable metric to SMC density. Unfortunately, measuring the maximum tangent modulus requires overextending the tissue, which is not possible within the physiological range. We approximated the maximum tangent modulus using the tangent modulus at maximum equibiaxial strain. Using this metric, we observed a similar age-related trend in SMC maximum tangent modulus and SMC density (Fig. 7). Differences are likely due to the fact that the true maximum tangent modulus was not directly measured.

We did not observe a consistent correlation between collagen density and the fitted collagen maximum tangent modulus. This lack of correlation is likely due to the recruitment effect, where the maximum tangent modulus largely depends on the number of collagen fibers that have already straightened. Unlike SMCs, collagen fibers do not respond in the toe region and only begin to recruit at higher strains. In collagenous tissues, there is a substantial structural reserve of collagen fibers, with only a small fraction of which become straight under physiological load. The collagen response was rarely observed in younger arteries. Furthermore, the long recruitment stretches associated with collagen are challenging to model with exponential constitutive models. The extended flat zero-stress response that occurs when all collagen fibers are crimped typically results in an overestimation of the collagen exponent $b_c$. Shorter recruitment stretches can reduce this bias, which partially explains the observed results.

3.4. Differences between nonlinearity in aging

The most pronounced difference between the TA and the FPA lay in the SMCs, particularly in their orientation and nonlinearity. In the FPA, the nonlinearity of SMCs was gauged from the toe region in the circumferential direction, whereas in the TA, it was determined from the toe region of the ensemble stress curve. The nonlinearity exponent of the SMCs in the TA remained consistent with age, while in the FPA, it increased significantly by the age of 70 years (Fig. 8C). Conversely, the nonlinearity of collagen fibers in both the TA and FPA appeared similar and increased with age, becoming more pronounced (Fig. 8E).

3.5. Differences between viscoelasticity in aging

The viscoelasticity of SMCs and collagen in both the TA and the FPA remained constant with age. The viscoelastic parameter for collagen, $v_c$, was similar in both arteries. However, the viscoelastic parameter for the SMCs, $v_s$, was notably higher in the FPA, being twice as high as in the TA. Visually, the mechanical response appeared to show an increase in hysteresis with age. This change did not appear to stem from intrinsic changes in the viscoelasticity of collagen and SMCs. Rather, it was likely due to two factors: (i) a reduction in arterial extensibility, making the hysteresis more prominent relative to the stress-strain curve, and (ii) the viscoelastic components – the SMCs and collagen – accounting for a larger relative proportion of the total stress, as a result of the elastin degradation with age.

3.6. Parameter correlation

Identifying correlations between different material parameters was challenging, primarily due to specimen heterogeneity and noise, as observed in Fig. 7. Aging compounds this challenge by inducing further variations in the parameters. Notably, the exponents of SMCs and collagen increased nonlinearly and varied more widely with age, as shown in Fig. 8. A significant correlation existed between the exponents of collagen fibers and SMCs in the FPA, with an $r^2=0.9279$ (Fig. 9A right), indicating distinct aging processes in the FPA compared to the TA. There were observable trends among other parameters, particularly in the exponent parameters related to elastin (Fig. 9 first and second column). However, a closer inspection suggested that these may not imply a causal relationship but rather that both parameters correlated with aging (Fig. 9 black curves). When examining parameters less associated with aging, such trends disappeared, e.g., in Fig. 2 third and fourth columns. This pattern was also evident when comparing parameters between the TA and FPA (Fig. 9D).

Fig. 9.

Selection of parameter pairs in the TA and FPA. A) Correlation between the nonlinearity of SMCs and collagen in the TA and FPA. Correlation of the elastin modulus and the nonlinearity and viscoelasticity of SMCs and collagen for B) the TA, C) FPA, and D) between the TA and FPA.

Interestingly, the lack of expected correlation, or the discovery of correlations where none should exist, proves more intriguing. We did not observe a strong correlation between the viscoelastic parameters and age, as illustrated in Fig. 8. Similarly, the parameters $v_s$ and $v_c$ did not show correlation (Fig. 10). However, when comparing the viscoelasticity of different arteries, more remarkable behaviors emerged. Specifically, the viscoelasticity of the SMCs, $v_s$, appeared to exhibit almost an inverse relationship, a finding not paralleled in the viscoelasticity of collagen, $v_c$ (Fig. 10). More extensive experimental data and analysis are required to determine if this suggests a coupled development between the two arteries.

Fig. 10.

Correlation between the viscoelastic parameters for collagen and SMCs in the TA and FPA from the same individuals. Panel A) depicts how these two parameters change between individual artery specimens, while panel B) illustrates their relationships between arteries.

4. Discussion

4.1. Generalized constitutive model form for arterial tissues

In this study, we developed a generalized viscoelastic constitutive model for both elastic and muscular arteries, designed to accurately capture the contributions of key microstructural components. By adapting the structure tensor representing the microstructures of elastin, SMCs, and collagen, we demonstrated that the same model form can accurately depict both the TA and the FPA. These structure tensors are based on experimental histological examinations. Likewise, this modeling approach can be applied to a broad range of other arteries. Using the same model form facilitates meaningful comparisons of material parameters between the TA and FPA for the different constituents. By contrasting how these arteries age, we gain insights into their adaptation to different mechanical environments and functional roles. The most significant differences in the mechanical responses of the TA and FPA, as well as their adaptations to aging, can be attributed to three factors: (i) the relative composition of collagen, elastin, and SMCs; (ii) the structural orientation of fibrous components, as represented by structure tensors, i.e. Eqs. (9)–(11), and (iii) the nonlinearity of collagen fibers and SMCs. Despite variations in nonlinearity, the correlation between maximum tangent modulus and density fraction may suggest that the intrinsic properties of individual SMCs and collagen fibers are relatively consistent. Instead, orientation and density likely account for the more significant differences.

4.2. Structural and functional changes with age

The degradation of elastin in arteries with age is well-documented [103,103,113–116], and our findings align with these observations. More notably, we investigated how other arterial components, such as SMCs and collagen, adapt alongside elastin, and whether changes occur in their stiffness, nonlinearity, and viscoelasticity. Our constitutive model demonstrates that the moduli of elastin and SMCs correlate well with experimental data on their respective densities (Fig. 7). Both the TA and the FPA exhibited a decrease in SMC density and, consequently, SMC modulus with age. The changes in collagen fiber density and modulus were less pronounced. However, in both the TA and FPA, SMCs and collagen increasingly bear the mechanical loads as they age. The most significant change observed was an increase in collagen nonlinearity in both arteries and SMCs nonlinearity in the FPA. These changes in nonlinearity likely indicate alterations in the reference configuration of the tissue, such as the reduction of collagen fiber crimp. This adjustment may relate to the degradation of elastin, which complicates the maintenance of large pre-strain during cyclic loading.

Quantifying collagen modulus through constitutive modeling is inherently challenging. Experimental and modeling studies suggest that collagen fibers exhibit a linear response in the first Piola-Kirchhoff stress and stretch, i.e., force-displacement, within the functional range [110,117–119]. The effective response of collagen fibers is highly dependent on the microstructure of the tissue. Collagen fibers are very stiff, with moduli in the range of hundreds of MPa [110]. To increase extensibility, collagenous tissues employ two mechanisms: collagen fiber crimp and collagen fiber rotation. Both mechanisms alter how tissue-level deformations affect collagen fiber-level stretch, effectively softening the fiber response [110]. To accurately quantify the mechanical properties of collagen, the number of fibers in the straightened state must be estimated. This typically requires either multiphoton imaging to quantify collagen fiber crimp or high mechanical loading during the test to straighten all fibers. However, both methodologies are complex and not available for the current dataset.

In collagenous tissues, only a subset of collagen fibers is engaged under physiological loading, particularly in younger individuals (Fig. 5). These additional fibers serve as a structural reserve, protecting the arteries against over-extension. For example, mitral valves have a structural reserve of 60% [110]. However, the size of this reserve, its variation with age, and whether the increased load borne by collagen fibers leads to a decrease in this reserve – potentially increasing the risk of mechanical failure in older individuals – remain uncertain. Consequently, direct comparisons of collagen modulus and density remain challenging at this stage.

4.3. Differences in viscoelasticity vs functional roles

For the TA, lower viscoelasticity means reduced energy loss under consistent and stable loading conditions. However, the role of viscoelasticity in supporting the Windkessel effect, which is crucial for dampening the pulsatile output of the heart, has been well-studied [41–43]. In contrast, the FPA faces complex and unpredictable loading environments due to limb flexion, necessitating additional considerations. An increase in viscoelasticity could help stabilize the artery and reduce wall shear stress [40]. Visually, the mechanical response in the FPA shows an increase in the size of the hysteresis loop, indicating more energy dissipation during loading cycles. However, the specific causes and effects of this increased hysteresis remain unclear.

We have found that viscoelastic constants for both collagen and SMC do not change with age in the TA and FPA. Furthermore, there appears to be no difference in the viscoelasticity of collagen between the TA and FPA. However, the viscoelasticity of SMCs was twice as high in the FPA as in the TA. The nonlinearity of collagen fibers also showed nearly identical changes with age in both arteries, indicating similar mechanical responses of collagen in the TA and FPA. Conversely, the nonlinearity of SMCs in the FPA increased exponentially with age, whereas it remained relatively linear in the TA. This suggests that SMCs are key factors distinguishing the viscoelastic properties of TA and FPA. The circumferential alignment of SMCs contributes to the increased hysteresis observed in the circumferential axis of the FPA [59]. Increased relative mass fraction and nonlinearity both contribute to this hysteresis. Nonlinearity increases the stress rate, amplifying the viscoelastic response. The balance of these factors is important in determining the overall viscoelastic response, suggesting that the linearity of SMCs in the TA plays a stabilizing role.

4.4. Correlations between TA and FPA development

Specimen heterogeneity posed significant challenges for a comprehensive statistical analysis. This was further complicated by increased variation in parameter values, particularly in the exponent parameters, as age advanced. More data points are necessary; however, certain trends can still be discerned when parameters are more closely correlated with age (Fig. 9). In some instances, these trends appeared weaker when comparing the TA and FPA from the same individuals, making it difficult to determine if this is due to specimen heterogeneity or insufficient data sizes. Notably, observations related to the SMCs of the arteries are particularly intriguing. In the FPA, the correlation between $b_s$ and $b_c$ was strong (Fig. 9), a pattern less evident in the TA, where the correlation between $b_s$ and the elastin modulus $k_e$ was more pronounced. The viscoelastic parameters present even more fascinating aspects; despite the lack of correlation between $v_s,v_c$, and age in both the TA or FPA, the comparison of $v_s$ between these arteries suggests a positive inverse relationship. More data are needed, but investigating whether functional properties of the TA influence peripheral artery development, particularly with respect to SMCs, represents a compelling question.

4.5. Limitations

Our study comprised 21 donors aged between 13 and 73 years, which can be considered a sizable sample for constitutive modeling studies. However, the distribution of ages within this range is sparse, contributing to notable variance in the parameters due to specimen-to-specimen variations. The computed parameter confidence intervals are reasonable but are much smaller than the variance observed relative to the aging trends. This parameter error analysis can only account for the confidence errors associated with the parameter estimation procedure. It cannot account for additional variability attributable to unavoidable experimental measurement errors, including marker tearing, preconditioning, and voltage drifts, as well as differences in subject health, vascular pathology, and the accompanying risk factors (Table 1). To draw more definitive conclusions on the statistical trends with age, larger sample sizes are necessary. Nonetheless, our current methodologies for the constitutive modeling of the TA and FPA have successfully captured physiologically realistic parameters that correlate well with experimental measurements. This model has provided new insights into the mechanical response and development of these arteries in relation to their microstructure, laying a foundation for further research in this area.

We are modeling the elastic and muscular arteries at the constituent level, i.e., an entire family of fibers or SMCs. For example, this does not distinguish between the response of individual collagen fibers and crosslinking between collagen fibers. Constitutive models for crosslinking do exist [109,120–122], but additional parameters will make optimization more difficult and the presented approach will no longer be sufficient. The remaining crosslinks that are unaccounted for, e.g., between the explicit constituents, are assumed to be accounted for in the ground matrix. The ground matrix, when considered as a separate constituent to elastin, is small in native tissues. For example, modeling of the exogenously crosslinked bovine pericardium illustrates the differences in this ground matrix response before and after the application of glutaraldehyde. The mechanical response of bovine pericardium is almost entirely due to collagen fibers, making such analysis easier. In our previous work [57], we assumed that the distribution of the relaxation time constant in the fractional viscoelasticity arises from the interactions and friction of hierarchical structures at the scales. The differences in crosslinking may be a factor affecting the viscoelastic constants between individuals.

5. Conclusions

We developed a generalized viscoelastic constitutive model applicable to both elastic and muscular arteries and outlined a methodology for accurately estimating the parameters of their different microstructural components. Using human arteries from 21 tissue donors ranging from 13 to 73 years of age, we demonstrated that the derived parameters correlate well with histological measures of microstructural composition. Our analysis demonstrates that the viscoelasticity of collagen and SMCs remains constant with age and that the viscoelasticity of SMCs in the FPA was double that of the TA. We also observed that collagen nonlinearity aged similarly in both arteries, but the nonlinearity of TA SMCs did not increase significantly with age. Our results also suggest that the viscoelasticity of the FPA could be influenced by that of the TA, although further data are required for a more detailed analysis. Overall, our new model proves to be a valuable tool for studying arterial mechanophysiology and exploring the impact of pathological factors on human artery data sets.

Nomenclature

TA

Thoracic aorta

FPA

Femoropopliteal artery

SMC

Smooth muscle cell

ECM

Extracellular matrix

EEL

External elastic lamina

${\square }_k$

Subscript $k\in \{g,e,s,c\}$ indicate the corresponding constituent ground matrix ($g$), elastin ($e$), SMC ($s$), and collagen ($c$)

${\stackrel{\mathrm{ˆ}}{\square }}_{ij}^{(k)}$

Data point with subscript $ij$ indicates the component tensor, and superscript $k$ indicates time at $t=t^{(k)}$

$D_t^v$

Fractional derivative operator with order $v$

$\mathrm{\Gamma }$

Gamma function

$\rho$

Orientation distribution function

${\mathbf{e}}_1,{\mathbf{e}}_2,{\mathbf{e}}_3$

Cartesian basis in testing configuration

${\mathbf{e}}_r,{\mathbf{e}}_{\theta },{\mathbf{e}}_z$

Cylindrical basis, radial, circumferential, and longitudinal, relative to the intact arteries

$\mathbf{m}$

Arbitrary unit vector represents fiber orientation

$\mathbf{H}$

Structure tensor involving the fiber dispersion

${\mathrm{\Omega }}_0$

Reference domain

${\mathrm{\Omega }}_t$

Deformed domain at time $t$

$\mathbf{X}$

Material position

$\mathbf{x}$

Spatial position

$\mathbf{u}$

Displacement vector

$\mathbf{I}$

Identity tensor

${\mathbf{I}}_{\text{plane}}$

In-plane identity tensor

$\mathbf{F}$

Deformation gradient tensor

$\mathbf{C}$

Right Cauchy-Green tensor

$\mathbf{E}$

Green-Lagrange strain tensor

$J$

Volume ratio or Jacobian determinant

$\mathrm{\lambda }$

Stretch

$I_{\mathbf{C}}$

First invariant of $\mathbf{C}$

${II}_{\mathbf{C}}$

Second invariant of $\mathbf{C}$

${III}_{\mathbf{C}}$

Third invariant of $\mathbf{C}$

${\mathrm{I}}_4$

Pseudo-invariant represents the square of the stretch in the fiber direction

$\mathrm{\Psi }$

Strain-energy function

$p$

Hydrostatic pressure

$\sigma$

Cauchy stress tensor

$\mathbf{P}$

First Piola-Kirchhoff stress tensor

$\mathbf{S}$

Second Piola-Kirchhoff stress tensor

${\mathbf{C}}_{\text{ens}}$

Ensemble strain under equibiaxial extension

$S_{\text{ens}}$

Ensemble stress under equibiaxial extension

${\mathbf{C}}_{\text{max}}$

Maximum strain of the 1:1 loading ratio protocol

$I_{\text{k,max}}$

Maximum invariant of constituent $k\in e,s,c$ of the 1:1 loading ratio protocol

$n_p$

Number of loading protocols for biaxial testing

$N_p^0$

Starting index of the $p^{\text{th}}$ protocol

$N_p^e$

Ending index of the $p^{\text{th}}$ protocol

${\mu }_k,{\mu }_k^{\mathrm{*}}$

Modulus parameter $k\in \{g,e,s,c\}$; $\mathrm{*}$ denotes the scaled form of Eq. (23) used for optimization

$b_k$

Exponent parameter, $k\in \{e,s,c\}$

$v_k$

Viscoelastic parameter, $k\in \{s,c\}$, for the order of the applied fractional derivative

$\theta$

Specimen orientation, i.e., the angle between the longitudinal direction ${\mathbf{e}}_z$ and the ${\mathbf{e}}_1$ axis of the testing configuration

$\alpha$

Collagen fiber pitch angle

$\beta$

SMC pitch angle

$\gamma$

Constant multiplier for the scaled model form

${\mathbf{S}}_k^{\mathrm{*}}$

Scaled constituent model form

$v_{\text{long}}$

Estimated fractional order from the longitudinal relaxation curve

$v_{\text{circ}}$

Estimated fractional order from the circumferential relaxation curve

$\xi$

Parameter vector

$ℱ$

Objective function

${𝒲}_{ij}^p$

Weights applied on protocol $p$ for the $i{j}^{\text{th}}$ component

$\chi$

Residual part of the objective function

$\varphi$

Hysteresis part of the objective function

${\mathscr{H}}_{ij}^p$

Function for calculating the hysteresis for the loading protocol $p$

$𝒫$

Penalty on the objective function value based on the parameter values

${\sigma }^2$

Residual distribution variance

${\hat{s}}^2$

Sample variance

$\mathscr{H}$

Hessian matrix of the objective function

$\mathbf{J}$

Local Jacobian matrix for the objective function

$\mathbf{\Sigma },{\mathbf{\Sigma }}_r,{\mathbf{\Sigma }}_h$

A priori estimate of the variance of the residual terms, i.e., a diagonal matrix containing the multiplicative weights; contains the stress-strain part ${\mathbf{\Sigma }}_r$ and the hysteresis part ${\mathbf{\Sigma }}_h$

$ϵ,{ϵ}_r,{ϵ}_h$

Residual vector containing the stress-strain ${ϵ}_r$ and the hysteresis part ${ϵ}_h$

$v$

Residual degree of freedom

$n_{\xi }$

Number of parameters

${\tau }_{\alpha /2,v}t$

-statistic for degree of freedom $v$ and a critical value of $\alpha$

Appendix A. Confidence Errors of Parameter Estimation

When analyzing the quality of the best-fitting parameters, two aspects must be taken into account: how consistently is the global optimum achieved and what is the confidence interval of the parameters? Nonlinear optimization typically involves multiple local minima, where the optimum is assumed to be the global minimum. The consistency of finding the global minimum is heavily influenced by the choice of the optimization algorithm. We tailored our approach to increase the frequency of finding the global optimum by using the differential evolution algorithm and variable protocol weights. The results for the FPA are shown in Appendix B of [59] and are also analyzed for the current study (Fig. A.11. The global minimum is defined as the run with the best objective function value out of 10. A run is defined as converged if all parameters are within a tolerance of 10⁻⁶ of the global minimum. Most specimens consistently reached the global minimum. Results can be improved at the cost of increasing the density of the randomized seeds or running parameter estimation multiple times.

The standard approach for determining the confidence error on nonlinear optimization is to perform a local linear regression at the optimum, assuming that $ϵ~𝒩\left(0,{\sigma }^2\mathbf{I}\right)$, i.e., the residuals are independent and normally distributed with a variance of ${\sigma }^2$. This is equivalent to the standard sum of squares for the objective function, i.e., the negative log-likelihood [125]. In our work, we apply the weights through penalties, which are a priori assumptions about the confidence of each protocol [59] (Appendix B). This includes the assumption that

$${ϵ}_r~𝒩\left(0,{\sigma }^2{\mathbf{\Sigma }}_r\right),{ϵ}_h~𝒩\left(0,{\sigma }^2{\mathbf{\Sigma }}_h\right)$$ (A.1)

for the stress-strain residuals and the hysteresis part of the objective functions, respectively. Here ${\mathbf{\Sigma }}_r$ and ${\mathbf{\Sigma }}_h$ are the diagonalized weight matrices of the residual and hysteresis terms, respectively, i.e., ${\left((1+𝒫(\xi ))W_{ii}^p\right)}^{-1}$ for the $k^{\text{th}}$ data point in the protocol $p\left(k\in \left[N_0^p,N_e^p\right]\right)$ or simply the $p^{\text{th}}$ protocol for the hysteresis. When these assumptions are incorporated into the objective function form, the confidence error for the $i^{\text{th}}$ parameter is given by

$${\xi }_i\pm {\tau }_{0.025,v}\sqrt{{\hat{s}}^2{\mathscr{H}}_{ii}^{-1}},$$ (A.2)

where ${\tau }_{0.025,v}$ is the coefficient from $t$-distribution for a critical value of 0.05 and degree of freedom $v,{\hat{s}}^2{\mathscr{H}}^{-1}$ is the approximation of the covariance matrix from the sample variance ${\hat{s}}^2$ and the Hessian matrix $\mathscr{H}$, which is defined as

$${\mathscr{H}}_{ij}=\frac{{\partial }^2ℱ(\xi )}{\partial {\xi }_i\partial {\xi }_j}.$$ (A.3)

The sample variance for weighted regression and the corresponding nonlinear form approximated by reduced chi-squared as

$${\hat{s}}^2\approx \frac{{ϵ}^T{\stackrel{\mathbf{ˆ}}{\mathbf{\Sigma }}}^{-1}ϵ}{v}=\frac{ℱ}{v}.$$ (A.4)

Here, $\stackrel{\mathbf{ˆ}}{\mathbf{\Sigma }}$ is the combined diagonal matrix from ${\mathbf{\Sigma }}_r$ and ${\mathbf{\Sigma }}_h,ϵ$ is the appended residual vector combining ${ϵ}_r$ and ${ϵ}_h$, and of the residual vector $ϵ$. The residual degree of freedom $v$ is given by the trace of the residual projection/smoother matrix for the local linear regression problem [126,127], $\mathbf{I}-\mathbf{J}\left({\mathbf{J}}^{\mathit{T}}{\mathbf{\Sigma }}^{-1}\mathbf{J}\right)\mathbf{J}{\mathbf{\Sigma }}^{-1}(\mathbf{J}$ is the Jacobian matrix). This projection matrix is idempotent, so the trace is equal to the rank. Because the objective function (Eq. (13)) also contains no regularization terms, this is equal to

$$\nu =2\left({\sum }_{p=1}^{n_p}\left(N_p^e-N_p^0+1\right)+n_p\right)-n_{\xi },$$ (A.5)

where $n_{\xi }$ is the number of model parameters and $\mathbf{\Sigma }$ is the weight matrix for each parameter. These confidence errors are presented in Figs. 9 and 10.

Limitations

We have made standard assumptions related to the use of the sum of squares, i.e., that the covariance matrix of the residual distribution is diagonal. In general, this may not always be the case. The data are not random observations, but rather a continuous sequence in time. The imperfection of data fitting to the model leads to biased errors. Errors in residuals are often correlated if they are close in time. This can be addressed by an a priori estimate of the covariance between residual terms. This exists in theory, but applications to real biomechanical data are difficult to find in the literature. Thus, the current approach remains standard.

Fig. A1.

The number of optimizations that converged to the global optimum (out of 10) for the TA and FPA.