Effect of Aging on the Biaxial Mechanical Behavior of Human Descending Thoracic Aorta: Experiments and Constitutive Modeling Considering Collagen Crosslinking

Abstract

Collagen crosslinking, an important contributor to the stiffness of soft tissues, was found to increase with aging in the aortic wall. Here we investigated the mechanical properties of human descending thoracic aorta with aging and the role of collagen crosslinking through a combined experimental and modeling approach. A total of 32 samples from 17 donors were collected and divided into three age groups: <40, 40–60 and >60 years. Planar biaxial tensile tests were performed to characterize the anisotropic mechanical behavior of the aortic samples. A recently developed constitutive model incorporating collagen crosslinking into the two-fiber family model (Holzapfel and Ogden, 2020) was modified to accommodate biaxial deformation of the aorta, in which the extension and rotation kinematics of bonded fibers and crosslinks were decoupled. The mechanical testing results show that the aorta stiffens with aging with a more drastic change in the longitudinal direction, which results in altered aortic anisotropy. Our results demonstrate a good fitting capability of the constitutive model considering crosslinking for the biaxial aortic mechanics of all age groups. Furthermore, constitutive modeling results suggest an increased contribution of crosslinking and strain energy density to the biaxial stress-stretch behaviors with aging and point to excessive crosslinking as a prominent contributor to aortic stiffening.

1. Introduction

The aortic wall stiffens with age (Cecelja and Chowienczyk, 2012), which gives rise to an augmented systolic blood pressure with increased risks for the development of multiple comorbid diseases such as hypertension, stroke and chronic kidney disease (Wang et al., 2005; North and Sinclair, 2012). The microstructural mechanisms that contribute to aging-induced aortic stiffening are diverse. Within the aortic extracellular matrix (ECM), crosslinking plays an important role in determining arterial stiffness. Two types of crosslinks were identified in aortic collagen: enzyme driven crosslinking (Fujimoto, 1982; Eyre et al., 1984; Reiser et al., 1992; Carmo et al., 2002), as well as nonenzymatic advanced glycation end-products (AGEs) (Wolff et al., 1991; Hoshino et al., 1995). The density of enzymatic crosslinking increases rapidly before maturation (Brüel and Oxlund 1996; Watanabe et al., 1996). However, controversial information exists regarding the post-maturation change of enzymatic crosslinking density, as a slightly increasing trend was found in some studies (Fujimoto, 1982; Watanabe et al., 1996), whereas other researchers reported a constant (Brüel and Oxlund 1996) or decreasing trend (Hoshino et al., 1995). Nevertheless, the density of nonenzymatic crosslinking increases with age throughout the lifespan (Hoshino et al., 1995; Sims et al., 1996; Snedeker and Gautieri, 2014), and was found to have a positive correlation with aortic stiffness (Brüel and Oxlund, 1996).

Crosslinks stiffen the collagen network primarily by restraining sliding between fibers and engaging more fibers to load-bearing (Depalle et al., 2015; Žagar et al., 2015). The network stiffness is dictated by the density rather than stiffness of crosslinks as revealed by molecular and fiber level computational studies (Depalle et al., 2015; Lin and Gu, 2015; Chen et al., 2017). Besides aging, increased collagen crosslinking density has been found to accelerate with the progression of other diseases such as diabetes (Sims et al., 1996) and aneurysm (Carmo et al., 2002). Although excessive crosslinking is detrimental, it should be noted that a proper amount of enzymatic crosslinking in the ECM is necessary for a normal mechanical function of soft tissues by providing stiffness and strength necessary for maintaining the stability of the ECM (Brüel et al., 1998), facilitating stress transfer among neighboring fibers to make the tissue behave as a coherent network (Lindeman et al., 2010), as well as reducing energy loss during cyclic loading (Kim et al., 2017). In addition to collagen, crosslinks can form among other aortic ECM components including elastin and glycoproteins. Specifically, AGEs in aortic elastin increase with aging (Konova et al., 2004) while the two major enzymatic elastin crosslinks desmosine and isodesmosine decrease slightly (John and Thomas, 1972). Histidinoalanine, a crosslink between glycoproteins, and between glycoproteins and collagen and elastin, increases drastically with aging (Fujimoto, 1982).

While experimental characterization of the contribution of collagen crosslinks to arterial wall mechanics remains a challenge, in several efforts, modeling approaches have been used to incorporate collagen crosslinking at multiple length scales. Molecular dynamics models (Buehler, 2008; Depalle et al., 2015; Kwansa et al., 2016) and discrete fiber network models (Lin and Gu, 2015; Žagar et al., 2015; Chen et al., 2017; Yu and Zhang, 2022) were employed to elucidate the effect of crosslinking density, type and mechanical properties on the deformation and failure behavior of fibers and networks. At the tissue level, statistical mechanics-based model was used to describe arterial mechanics where the crosslinking density is inversely related to chain length (Zhang et al., 2005, 2007; Tian et al., 2016). Several recent continuum mechanics-based models were developed in which the contribution of collagen crosslinking to tissue behavior was decoupled from the rest of the ECM. In a study by Sacks et al. (Sacks et al., 2016), the interaction between collagen fiber families induced by crosslinking was modeled by decoupling extensional and rotational effects. In Sáez et al. (Sáez et al., 2014), the strain energy density function was extended with an additional isotropic term to incorporate the mechanical contribution of crosslinking. Recently, a constitutive model considering the stretch of crosslinks and crosslink-fiber interaction was incorporated into a two-fiber family model (Holzapfel and Ogden, 2020). These studies reported better fitting and predicting capabilities of constitutive models considering the contribution of crosslinking.

Invariant I₈ was introduced by Spencer (1984) to model the coupling interactions between two families of fibers under finite deformation, and then adopted in several studies to model fiber-fiber and crosslink-fiber interactions (Merodio and Ogden, 2006; Holzapfel and Ogden, 2020). While most biomechanical investigations of human thoracic aorta were performed with uniaxial tensile tests (Manopoulos et al., 2018; Amabili et al., 2019; Amabili et al., 2021), for anisotropic soft tissues such as the aortic wall, biaxial tensile testing allows for controlled biaxial loading ratio in uniquely determining the parameters for three-dimensional constitutive models (Sacks, 2000). However, under biaxial deformation, extension and rotation kinematics could have opposite mathematical effects on the change of the magnitude of I₈ and thus diminish the contribution of crosslink-fiber interaction to the strain energy density.

In this study, motivated by Holzapfel and Ogden (2020), collagen crosslinking was incorporated into a two-fiber family model to describe the biaxial mechanical behavior of the aorta. The extension and rotation kinematics of crosslinking was decoupled so that both mechanisms can independently contribute to biaxial deformation. Exponential functions were used in describing the crosslinking-associated strain energy density in nonlinear aortic stiffening. Also, changes of the mechanical properties of human descending thoracic aorta with aging was studied to provide new insights into the microstructural mechanisms of aortic stiffening through constitutive modeling. Biaxial tensile tests were performed on human descending thoracic aortas from 24 to 90 years old donors. The effect of aging on aortic stiffness and anisotropy as well as the mechanical role of crosslinking in the aging aorta was examined through a combined experimental and modeling approach.

2. Material and methods

2.1. Experiments

2.1.1. Sample preparation

Human descending thoracic aortas were obtained from the National Disease Research Interchange (NDRI) and transported to laboratory on dry ice and stored at −80°C before experiment. The age, gender and selected medical information of aorta donors is summarized in Table 1. The use of human tissue was reviewed and approved by the Institutional Biosafety Committee at Boston University. Before experiments, aortas were defrosted at −20°C for 5 hours followed by 3 hours at 4°C and then 1 hour at room temperature. Square samples free of visible atherosclerotic plaques with a dimension of approximately 2.5 × 2.5 cm were cut with the edges of the samples aligned with the longitudinal and circumferential directions of the aorta. Loose connective tissue was removed from the adventitial surface. Two samples were obtained from each aorta except for aortas N8 and N16 due to limited tissue size. All samples were obtained from the proximal region of the descending thoracic aorta to reduce longitudinal variation of the mechanical properties of aortic tissues (Purslow, 1983; Zou and Zhang, 2009; Rouleau et al., 2012; Kim et al., 2013; Zeinali-Davarani et al., 2015). Donors were divided into three age groups: <40 (n=8), 40–60 (n=11), and >60 (n=13) years.

Table 1.

Age, gender and selected medical information of human descending thoracic aorta donors used in this study.

Age group	Donor	Age (years)	Gender	Hypertension	Hyperlipidemia	BMI	Cause of death
<40 years	N1	24	F	Y*	N	30.27	SIGSW
	N2	26	M	N	N	26.5	MVA
	N3	32	M	N	N	64.83	CPA
	N4	39	F	N	N	24.69	CVA
40–60 years	N5	42	M	Y	N	44.3	CPA
	N6	44	F	Y	N	38.23	ABI
	N7	47	M	Y	N	47.47	CPA
	N8	49	M	Y	N	24.06	CVA
	N9	57	F	N	N	18.5	ABI
	N10	59	M	N	N	29.18	HD
>60 years	N11	62	F	Y	N	25.81	CVA
	N12	64	F	N	Y	38.6	CPA
	N13	70	M	Y	N	Unknown	CVA
	N14	72	F	N	N	72.45	CPA
	N15	74	F	N	N	14.98	COPD
	N16	87	M	Y	Y	29.53	CPA
	N17	90	F	Y	N	16.14	CPA

BMI - body mass index; F - female; M - male; Y - yes; N - no;

^{* -}during pregnancy;

SIGSW - self-inflicted gunshot wound; MVA - motor vehicle accident; CPA - cardiopulmonary arrest; CVA - cerebrovascular accident; ABI - anoxic brain injury; HD - heart disease; COPD - chronic obstructive pulmonary disease.

2.1.2. Planar biaxial tensile testing

Samples were mounted on a custom-built planar biaxial tensile testing device and submerged in 1× phosphate buffered saline (PBS) under room temperature. Biaxial tensile preload of 5±0.1 N/m was applied in both the longitudinal and circumferential directions to straighten the sutures connecting the sample to actuators. Eight preconditioning cycles of 40 N/m equibiaxial tension was applied to each sample to achieve repeatable mechanical response. Samples were then subjected to the following five biaxial loading protocols: T_L: T_C=200:200, 200:150, 200:100, 100:200, and 150:200 N/m, where T_L and T_C are tension applied to the longitudinal and circumferential directions of the sample, respectively. Each loading protocol consisted of eight cycles with 10 s half cycle time, and data of the last loading cycle were used for analysis. The longitudinal and circumferential stretches were calculated by optically tracking the displacement of four carbon marker dots attached to the center region of each sample. Cauchy stresses in the longitudinal and circumferential directions were then calculated as (Zou and Zhang, 2009):

$${\sigma }_L=\frac{F_L{\lambda }_L}{h{L}_C},\text{and}{\sigma }_C=\frac{F_C{\lambda }_C}{h{L}_L}$$ (1)

where σ is Cauchy stress, λ is the stretch, h is the thickness of the sample, and L is the edge length of the sample. Subscripts L and C denote the longitudinal and circumferential directions, respectively. Tangent modulus was obtained by taking the derivative of a fifth order polynomial fit to the Cauchy stress-stretch data. The peak stretch under 200:200 N/m equibiaxial tension were obtained in both circumferential and longitudinal directions and used to assess the degree of anisotropy (DA) as (Jadidi et al., 2020):

$$DA=\frac{{\lambda }_L-{\lambda }_C}{0.5\left({\lambda }_C+{\lambda }_L\right)}$$ (2)

2.2. Constitutive modeling

2.2.1. Model development

Two-fiber family model was chosen upon which the crosslinking mechanisms are based to represent the two prominent collagen fiber families found in the intima, media and adventitia in human thoracic aortas (Schriefl et al., 2012). In the newly developed constitutive model, deformation of collagen crosslinks and crosslink-fiber interaction were incorporated into the two-fiber family model considering the kinematics of bonded fiber and crosslink under biaxial deformation. A schematic of the model, shown in Fig. 1, contains two collagen fiber families with the angle between each collagen fiber family and the longitudinal direction denoted by θ. E₁ = {cosθ, sinθ, 0}^T and E₂ = {cosθ, −sinθ, 0}^T are unit vectors along the direction of the two collagen fibers. The crosslinks are symmetrically distributed within each collagen fiber family. As shown in Fig. 1, the angle between the orientation of crosslinks and their associated collagen fibers is denoted by α₀. $L_{1,2}^{+}$, and $L_{1,2}^{-}$ are unit vectors in the direction of crosslinks, where subscripts 1 and 2 denote the family of collagen fibers that the crosslinks are attached to, and superscripts + and – are used to distinguish between the two crosslink families, and can be written as:

$$L_1^{\pm }=\pm \mathit{\text{cos}}{\alpha }_0{E}_1+\mathit{\text{sin}}{\alpha }_0{E}_{1\mathit{R}}\text{,}\text{and}{L}_2^{\pm }=\pm \mathit{\text{cos}}{\alpha }_0{E}_2+\mathit{\text{sin}}{\alpha }_0{E}_{2\mathit{R}}$$ (3)

where E_1R = {−sinθ, cosθ, 0}^T and E_2R = {−sinθ, −cosθ, 0}^T are unit vectors normal to collagen fiber orientations (Fig. 1).

Fig. 1.

Schematic of a sample with two families of collagen fibers and crosslinks. E_1, is the unit vector in the orientation of collagen fibers; E_1,2R is the unit vector normal to the orientation of collagen fibers; $L_{1,2}^{\pm }$ is the unit vector in the orientation of crosslinks; θ denotes the angle between the longitudinal direction of the aorta and the orientation of collagen fibers; and α₀ denotes the angle between the orientations of collagen fibers and their associated crosslinks.

With crosslinks bonded to collagen fibers, under mechanical loading, both collagen fibers and crosslinks are stretched, leading to relative rotation between the bonded fibers and crosslink. The bonded fiber and crosslink may mechanically interact by extension and relative rotation, both of which contribute to the increase of the strain energy density of the aortic wall under deformation (Sacks et al., 2016; Mansouri et al., 2021). Considering the kinematics of bonded fiber and crosslink under biaxial tension (Appendix A), the extension and relative rotation of bonded fiber and crosslink was decoupled in this model. The strain energy density function of the aorta, W_2FC, is thus given by:

$$W_{2\mathit{\text{FFC}}}=W_{2\mathit{\text{FF}}}+W_c+W_{cf}$$ (4)

where W_2FF, W_c and W_cf are strain energy density functions of the two-fiber family model (Holzapfel et al., 2000), deformation of collagen crosslinks, and crosslink-fiber interaction, respectively. The strain energy density function of the two-fiber family model is:

$$W_{2FF}=\frac{\mu }{2}\left(I_1-3\right)+\frac{k_1}{2k_2}\left\{\mathit{\text{exp}}\left[k_2{\left(I_{4,1}-1\right)}^2\right]-1\right\}+\frac{k_1}{2k_2}\left\{\mathit{\text{exp}}\left[k_2{\left(I_{4,2}-1\right)}^2\right]-1\right\}$$ (5)

where the first term represents the contribution of ground substance, smooth muscle cells, and elastin with μ being the shear modulus, and I₁ is the first invariant of the right Cauchy-Green deformation tensor C. For the two collagen fiber families represented by the second and the third terms, k₁ is a stress-like parameter, and k₂ is a dimensionless parameter. I_4,1 and I_4,2 are square of collagen fiber stretch and can be obtained as E₁ · CE₁ and E₂ · CE₂, respectively.

The strain energy density function of crosslinking extension, W_c, is assumed to follow the same exponential form as that of collagen fibers to accommodate the reported increased nonlinearity of aortic stress-stretch behavior with aging (Geest et al., 2004; Jadidi et al., 2020):

$$W_c=\frac{v_1}{2v_2}\left\{\mathit{\text{exp}}\left[v_2{\left(I_1^{+}-1\right)}^2\right]-1\right\}+\frac{v_1}{2v_2}\left\{\mathit{\text{exp}}\left[v_2{\left(I_1^{-}-1\right)}^2\right]-1\right\}+\frac{v_1}{2v_2}\left\{\mathit{\text{exp}}\left[v_2{\left(I_2^{+}-1\right)}^2\right]-1\right\}+\frac{v_1}{2v_2}\left\{\mathit{\text{exp}}\left[v_2{\left(I_2^{-}-1\right)}^2\right]-1\right\}$$ (6)

where v₁ is a stress-like parameter, and v₂ is a dimensionless parameter. $I_{1,2}^{\pm }=L_{1,2}^{\pm }\cdot C{L}_{1,2}^{\pm }$ is the square of crosslink stretch.

To model the relative rotation between the bonded fiber and crosslink, a modified invariant, $I_{8*}^{1,2\pm }$, which decouples the extensional effects of bonded fiber and crosslink from its unmodified counterpart, and only incorporates the relative rotation between bonded fiber and crosslink in considering the crosslink-fiber interaction, is introduced as:

$$I_{8*}^{1,2\pm }=\frac{I_8^{1,2\pm }}{\sqrt{I_{4,1,2}{I}_{1,2}^{\pm }}}=\mathit{\text{cos}}{\alpha }^{\pm }$$ (7)

As shown in Eq. 7, $I_{8*}^{1,2\pm }$ is equal to the cosine of the angle between bonded fiber and crosslink in the deformed configuration. α⁺ and α⁻ are the angles between collagen fiber orientated in E_1,2 and associated crosslinks oriented in $L_{1,2}^{+}$ and $L_{1,2}^{-}$ directions in the deformed configuration, respectively. The invariant $I_8^{1,2\pm }$ couples the extension and relative rotation of bonded fiber and crosslink under deformation and is defined as (Merodio and Ogden, 2006; Holzapfel and Ogden, 2020):

$$I_8^{1,2\pm }=L_{1,2}^{\pm }\cdot C{E}_{1,2}$$ (8)

The strain energy density function of the crosslink-fiber interaction, W_cf, captures the relative rotation between the bonded fiber and crosslink and follows the same exponential form as those of fiber and crosslink extension:

$$W_{cf}=\frac{{\kappa }_1}{2{\kappa }_2}\left\{\mathit{\text{exp}}\left[{\kappa }_2{\left(I_{8*}^{1+}-\mathit{\text{cos}}{\alpha }_0\right)}^2\right]-1\right\}+\frac{{\kappa }_1}{2{\kappa }_2}\left\{\mathit{\text{exp}}\left[{\kappa }_2{\left(I_{8*}^{1-}+\mathit{\text{cos}}{\alpha }_0\right)}^2\right]-1\right\}+\frac{{\kappa }_1}{2{\kappa }_2}\left\{\mathit{\text{exp}}\left[{\kappa }_2{\left(I_{8*}^{2+}-\mathit{\text{cos}}{\alpha }_0\right)}^2\right]-1\right\}+\frac{{\kappa }_1}{2{\kappa }_2}\left\{\mathit{\text{exp}}\left[{\kappa }_2{\left(I_{8*}^{2-}+\mathit{\text{cos}}{\alpha }_0\right)}^2\right]-1\right\}$$ (9)

where k₁ is a stress-like parameter, and k₂ is a dimensionless parameter.

For comparison purpose, the two-fiber family model without the consideration of collagen crosslinking is also adopted to describe the aortic biaxial mechanical response. The Cauchy stress tensor, σ, can be calculated as:

$${\sigma }_{2FFC,2FF}=-pI+2F\frac{\partial W_{2FFC,2FF}}{\partial C}{F}^T$$ (10)

where I is the identity tensor and p = μ/(λ_L²λ_C²) is the Lagrange multiplier.

2.2.2. Parameter estimation

The constitutive models were implemented in MATLAB (Version R2019a, The MathWorks, Inc., Natick, MA) and fit to the experimental Cauchy stress-stretch data considering all five loading protocols simultaneously for each sample. Model parameters μ, k₁, k₂, v₁, v₂, k₁ and k₂ were set to be greater than zero, θ and α₀ varied between 0 and 90°. Parameters were estimated by minimizing the following objective function:

$$\psi ={\sum }_{i=1}^{m/2}\left[{{\left({\sigma }_L^c-{\sigma }_L^e\right)}_i}^2+{{\left({\sigma }_C^c-{\sigma }_C^e\right)}_i}^2\right]$$ (11)

where m is the total number of data points. Superscripts c and e denote Cauchy stresses from the model and experiment, respectively. The fminsearch function with embedded Nelder-Mead direct search method was used to minimize the objective function with a tolerance of 1 × 10⁻⁸. To ensure that the global minimum of the objective function is reached, and to enhance the reliability and robustness of the estimated parameters, the model was fit to five biaxial tensile loading protocols (T_L: T_C=200:200, 200:150, 200:100, 100:200, and 150:200 N/m) simultaneously. Also, the nonlinear regression with fminsearch was carried out with multiple random initial guesses. The estimation was accepted only when different initial guesses converge to the same set of parameters. The goodness of fit was measured by the coefficient of determination (R²) as well as the root mean square error (RMSE) defined as (Holzapfel et al., 2005):

$$\text{RMSE}=\frac{\sqrt{\frac{\psi }{m-q}}}{{\sigma }_{\mathit{\text{ref}}}}$$ (12)

where q is the number of parameters of the constitutive model, and σ_ref is the sum of Cauchy stresses of each data point divided by the number of data points m. Additionally, to account for the different numbers of parameters in the two constitutive models, the corrected Akaike information criterion (AIC_C) which penalizes the goodness of fit for increasing number of model parameters was also used and is defined as (Zeinali-Davarani et al., 2009; Ferruzzi et al., 2011):

$${\text{AIC}}_{\text{C}}=m\text{ln}\left(\frac{\psi }{m}\right)+2(q+1)+\frac{2(q+1)(q+2)}{m-q-2}$$ (13)

A better fit corresponds to a lower AIC_C. Further, contributions of the strain energy density due to crosslink deformation (W_c) and crosslink-fiber interaction (W_cf) to the total strain energy density (W) of all samples under 200 N/m equibiaxial tension were obtained as W_c/W and W_cf/W, respectively, which were then compared across age groups to provide a better understanding of the contribution of collagen crosslinking to the mechanical behavior of human descending thoracic aorta with aging.

2.3. Statistical analysis

Data are summarized using box plots and are also presented as mean ± standard error of the mean (SEM). Data normality of the peak stretch, tangent modulus, degree of anisotropy, R², RMSE, AIC_C, W_c/W and W_cf/W was examined with the Kolmogorov-Smirnov and the Shapiro-Wilk tests. Within each age group, paired t-test was used to compare the peak stretch and tangent modulus between the circumferential and longitudinal directions, and R², RMSE and AIC_C of the stress-stretch fittings with constitutive models with and without crosslinking. Across the three age groups, one-way analysis of variance (ANOVA) was used with Bonferroni correction for the peak stretch, tangent modulus, degree of anisotropy, R², RMSE, W_c/W and W_cf/W. AIC_C was used to compare the goodness of fit of a constitutive model for samples of matching age groups. Correlation analysis of age relevant to W_c/W and W_cf/W, as well as mechanically or structurally meaningful model parameters μ, θ and α₀, is performed. For statistical analysis involving data that are not normally distributed, the Wilcoxon signed-rank test and the Kruskal-Wallis H test were used as nonparametric counterparts of the paired t-test and one-way ANOVA. All statistical analysis was performed in SPSS (Version 27, IBM Corp., Armonk, NY). p<0.05 was considered statistically significant.

3. Results

The Cauchy stress-stretch curves of all samples under 200 N/m equibiaxial tension are shown in Fig. 2. With aging, the stiffness in both longitudinal and circumferential directions increases as the Cauchy stress-stretch curves become steeper and gradually shift towards left. The increase of stiffness with aging is also supported by the decreasing peak stretch (Fig. 3a) and increasing tangent modulus (Fig. 3b) in both directions. The longitudinal and circumferential peak stretches decrease from 1.31 ± 0.03 and 1.29 ± 0.02 for the <40 age group to 1.21 ± 0.01 and 1.26 ± 0.01 for the 40–60 age group and then to 1.10 ± 0.01 and 1.16 ± 0.02 for the >60 age group, respectively. The longitudinal tangent modulus increases from 2.08 ± 0.17 MPa to 3.31 ± 0.48 MPa and then to 5.48 ± 0.49 MPa for the <40, 40–60 and >60 age groups, respectively; while the circumferential tangent modulus was found to be 1.80 ± 0.19 MPa and 1.80 ± 0.20 MPa for the <40 and 40–60 years groups, respectively, and increases to 3.11 ± 0.30 MPa for the >60 age group. A more prominent stiffening effect was found in the >60 age group for both circumferential and longitudinal directions (p<0.05), as no significant difference of the peak stretch and tangent modulus was found between the <40 and 40–60 years age groups. However, aortic stiffening with aging is more drastic in the longitudinal than in the circumferential direction (Fig. 3), as from the <40 to 40–60 age groups the mean longitudinal peak stretch decreases by 7.63% while the mean circumferential peak stretch decreases by only 2.33%, and from 40–60 to >60 age groups the mean longitudinal peak stretch decreases by 9.09% (p<0.05) while the mean circumferential peak stretch decreases by 7.94% (p<0.05). Similarly, from the <40 age group to the >60 age group, the mean longitudinal tangent modulus increases by 163.46% (p<0.05) while the mean circumferential tangent modulus increases by 72.78% (p<0.05). Aging also alters the degree of anisotropy as the in-plane stiffness is higher in the circumferential direction than the longitudinal direction in the <40 years age group; while as the age further increases, the longitudinal direction exhibits a significantly higher stiffness than the circumferential direction for both 40–60 and >60 age groups (p<0.05), as shown in Fig. 3. The change of the degree of anisotropy with aging is given in Fig. 4, where a positive value of DA corresponds to a higher circumferential stiffness and a negative value of DA corresponds to a higher longitudinal stiffness. The degree of anisotropy decreases from 0.019 ± 0.026 to −0.042 ± 0.013 and then to −0.052 ± 0.018 for the <40, 40–60 and >60 age groups, respectively, with significant difference between the <40 and >60 age groups (p<0.05).

Fig. 2.

Cauchy stress-stretch curves in the circumferential and longitudinal directions of human descending thoracic aortas under 200 N/m equibiaxial tension test from the <40 yrs (years), 4060 yrs, and >60 yrs age groups.

Fig. 3.

Effect of aging on a peak stretch and b tangent modulus of human descending thoracic aortas under 200 N/m equibiaxial tensile test. Boxes in the plots represent the middle 50% (from the first quartile to the third quartile) of the data set; the horizontal line inside the box represents the median; the cross symbol represents the mean; and the dotted symbols represent individual data points. (*p<0.05, **p<0.01, ***p<0.001, ****p<0.0001)

Fig. 4.

Effect of aging on the degree of anisotropy (DA) of human descending thoracic aortas under 200 N/m equibiaxial tensile test. The absolute value of DA increases suggesting a more anisotropic behavior of the sample. A positive DA corresponds to a higher circumferential stiffness while a negative DA corresponds to a higher longitudinal stiffness. Boxes in the plot represent the middle 50% (from the first quartile to the third quartile) of the data set; the horizontal line inside the box represents the median; the cross symbol represents the mean; and the dotted symbols represent individual data points. (*p<0.05)

Representative model fitting results of the experimental Cauchy stress-stretch curves with the two constitutive models with and without the consideration of collagen crosslinking are shown in Figs. 5 and 6. Parameters as well as the associated R², RMSE and AIC_C of each sample are listed in Tables 2 and 3. The correlation coefficients and p-values of the correlation analysis of μ, θ, α₀ with age are shown in Table 4. Collagen orientation, θ, shows a strong negative correlation with age while the other two parameters do not. Compared to the model with crosslinking, the model without crosslinking is less capable of capturing the mechanical response at high stretch and exhibits relatively large fitting error for protocols with highly unequal biaxial tensile loads, i.e., when T_L: T_C=200:100 and 100:200 N/m, as shown in Fig. 5. The fitting capacity of the constitutive model without crosslinking decreases with aging, as the RMSE increases from 0.18 ± 0.01 to 0.20 ± 0.01 and then to 0.27 ± 0.01 for the <40, 40–60 and >60 years age groups, respectively; and R² decreases from 0.96 ± 0.002 to 0.95 ± 0.005 and then to 0.92 ± 0.005 for the <40, 40–60 and >60 years age groups, respectively. Significant difference was found between the <40 and >60 as well as between the 40–60 and >60 years age groups for both R² and RMSE (p<0.05). On the contrary, the model with crosslinking shows improved goodness of fit (Fig. 6), and the improved fitting capacity leads to significantly lower RMSE (p<0.05) and AIC_C (p<0.05) as well as higher R² (p<0.05) for all age groups than the model without crosslinking (Fig. 7). The RMSE considering crosslinking was found to be 0.07 ± 0.01, 0.06 ± 0.01 and 0.09 ± 0.01 for the <40, 40–60 and >60 years age groups, respectively; and R² considering crosslinking was found to be 0.99 ± 0.002, 0.99 ± 0.001 and 0.99 ± 0.002 for the <40, 40–60 and >60 years age groups, respectively. In addition, without crosslinking, the AIC_C was found to be 3087.1 ± 131.6, 3002.0 ± 114.4 and 3126.8 ± 72.8 for the <40, 40–60 and >60 years age groups, respectively, which was reduced to 1446.3 ± 229.4, 1086.5 ± 192.0 and 1361.5 ± 186.7 after the effects of collagen crosslinking are incorporated into the constitutive model. The goodness of fit of the model with crosslinking is not affected by aging as no significant difference was found in either RMSE (p=0.07) or R² (p=0.21) across the three age groups (Fig. 7). Compared with the <40 and 40–60 years groups, contributions of crosslinking-related strain energy density to the total strain energy density of samples under 200 N/m equibiaxial tension increased significantly in the >60 years age group (p<0.05). The ratio of the strain energy density due to crosslink deformation to the total strain energy density, W_c/W, was found to be 34.90% ± 6.32%, 30.06% ± 2.94% and 58.98% ± 6.12% for the <40, 40–60 and >60 years age groups, respectively. The ratio of the strain energy density due to crosslink-fiber interaction to the total strain energy density, W_cf/W, was found to be 0.10% ± 0.05% for the <40 years group, which increased slightly to 0.17% ± 0.07% and drastically to 2.34% ± 0.72% for the 40–60 and >60 years groups, respectively, as shown in Fig. 8. Correlation analysis also demonstrates that both W_c/W and W_cf/W are positively correlated with age (p<0.05), as shown in Table 4.

Fig. 5.

Representative curve fitting results of the two-fiber family model without considering the effect of crosslinking. Symbols represent experimental data and curves represent constitutive model fitting.

Fig. 6.

Representative curve fitting results of the two-fiber family model considering the effect of crosslinking. Symbols represent experimental data and curves represent constitutive model fitting.

Table 2.

Model parameters of the two-fiber family model without considering the effect of crosslinking, and the associated root mean square error (RMSE), coefficient of determination (R²), and corrected Akaike information criterion (AIC_C) as measures of the goodness of fit.

Sample	Constitutive model parameters				Goodness of fit
	μ (kPa)	k1 (kPa)	k 2	θ (°)	RMSE	R 2	AICC
N1(1)	40.98	31.83	0.67	49.53	0.173	0.967	3590.0
N1(2)	38.77	32.96	0.56	48.47	0.166	0.969	3524.5
N2(1)	57.96	8.78	5.32	46.70	0.154	0.971	3020.3
N2(2)	34.31	12.72	2.58	45.29	0.192	0.956	3116.7
N3(1)	54.20	8.42	7.08	51.72	0.188	0.958	3158.7
N3(2)	39.26	11.41	4.23	48.49	0.191	0.957	3103.7
N4(1)	24.17	6.83	3.17	44.44	0.180	0.964	2482.6
N4(2)	21.53	7.52	3.05	44.80	0.199	0.959	2700.2
N5(1)	39.70	9.15	4.91	46.91	0.161	0.968	2482.9
N5(2)	38.04	5.12	5.62	41.80	0.167	0.965	2630.7
N6(1)	46.24	19.84	2.57	46.46	0.194	0.959	3229.1
N6(2)	48.21	14.61	2.92	43.60	0.158	0.975	3038.0
N7(1)	26.93	15.63	2.96	46.94	0.235	0.931	3144.0
N7(2)	40.63	10.41	3.84	43.86	0.175	0.964	2872.6
N8	26.19	6.51	7.88	42.56	0.253	0.926	2756.6
N9(1)	59.08	19.24	15.23	46.95	0.242	0.940	3656.1
N9(2)	45.78	28.95	10.34	41.86	0.262	0.937	3611.8
N10(1)	30.57	8.94	7.61	44.10	0.210	0.950	2822.6
N10(2)	34.01	4.04	8.16	43.23	0.196	0.959	2778.0
N11(1)	56.18	6.23	42.27	39.20	0.251	0.930	2801.9
N11(2)	32.17	6.93	22.32	37.55	0.256	0.933	2873.1
N12(1)	49.64	4.64	55.94	35.90	0.323	0.904	3246.3
N12(2)	28.26	10.15	28.76	48.80	0.249	0.939	2795.4
N13(1)	64.94	4.92	40.19	36.75	0.268	0.928	3224.7
N13(2)	43.51	5.93	18.52	39.43	0.215	0.953	2876.3
N14(1)	63.02	14.01	50.08	42.01	0.309	0.893	3377.4
N14(2)	35.93	10.29	11.53	44.05	0.216	0.949	2892.4
N15(1)	67.47	6.75	32.87	40.06	0.295	0.910	3579.4
N15(2)	65.29	8.64	57.27	34.48	0.307	0.909	3333.1
N16	40.99	10.75	25.32	40.68	0.265	0.928	3000.0
N17(1)	74.73	7.66	112.84	35.08	0.319	0.901	3309.7
N17(2)	109.32	31.39	237.41	48.16	0.298	0.908	3338.5

Two samples were obtained from each aorta except for N8 and N16. In the column of “Sample”, numbers 1 and 2 in the parentheses denote the first and second sample obtained from the corresponding aorta.

Table 3.

Model parameters of the two-fiber family model considering the effect of crosslinking, and the associated root mean square error (RMSE), coefficient of determination (R²), and corrected Akaike information criterion (AIC_C) as measures of the goodness of fit.

Sample	Constitutive model parameters									Goodness of fit
	μ (kPa)	k1 (kPa)	k 2	θ (°)	v1 (kPa)	v 2	κ1 (kPa)	κ 2	α0 (°)	RMSE	R 2	AICC
N1(1)	18.97	22.95	2.50E-11	54.78	12.95	0.44	9.36E-09	7025.75	28.12	0.032	0.999	1061.3
N1(2)	23.93	33.59	4.87E-11	61.18	5.32	0.77	8.66	5.13E-14	90.00	0.026	0.999	715.5
N2(1)	41.73	3.32	2.62E-12	43.20	8.19	3.59	11.64	21.15	82.16	0.073	0.993	1895.7
N2(2)	29.48	5.19E-06	26.18	44.53	6.94	2.40	6473.79	3.07E-09	17.98	0.049	0.997	1036.5
N3(1)	34.20	23.28	3.80	51.38	2.10E-04	15.23	39.83	15.35	58.24	0.130	0.979	2599.2
N3(2)	30.66	12.17	3.31	59.38	2.03	3.43	6.87	7.52E-08	78.47	0.085	0.991	1879.2
N4(1)	13.84	4.15	3.81	39.26	5.19	0.48	2.03E-04	1221.35	44.42	0.096	0.989	1543.2
N4(2)	13.73	1.70	5.37	36.02	5.69	0.81	5.34E-13	6.33E-03	57.87	0.057	0.997	839.7
N5(1)	31.43	12.80	2.33	56.65	1.22	5.94	4.91	21.42	90.00	0.043	0.998	511.5
N5(2)	35.25	0.44	11.18	33.96	2.83	4.23	1598.61	2117.17	18.87	0.057	0.996	995.7
N6(1)	29.85	12.52	0.37	85.57	8.77	2.81	0.23	1133.84	54.97	0.080	0.993	1885.7
N6(2)	34.91	3.81	0.32	77.93	9.00	2.56	21.88	103.98	51.36	0.029	0.999	471.0
N7(1)	16.79	24.82	2.60E-13	54.82	0.52	9.50	8.92	5.23E-10	72.48	0.069	0.994	1286.9
N7(2)	29.07	11.07	7.81E-10	54.07	3.48	4.72	8.95	6.03E-06	73.55	0.038	0.998	539.3
N8	12.19	10.61	0.02	70.31	2.57	11.05	9.87	133.33	69.87	0.088	0.991	1164.2
N9(1)	4.19E-11	68.21	0.88	46.57	3.64	20.72	56998.71	81231.28	15.27	0.089	0.992	2135.5
N9(2)	33.98	15.35	12.91	37.01	5.73	10.06	7.78E-09	0.27	21.41	0.083	0.993	1866.3
N10(1)	28.31	5.62	4.64	60.72	1.22	13.21	6.67	5.01E-07	84.31	0.053	0.997	724.5
N10(2)	24.36	9.47	0.28	54.07	0.96	10.75	6.56	52.09	78.59	0.041	0.998	371.3
N11(1)	45.17	1.03	74.96	7.45	4.03	33.20	84.37	313.89	41.70	0.068	0.995	819.2
N11(2)	27.20	3.65	44.92	19.05	2.69	17.31	60.51	404.88	28.82	0.042	0.998	85.5
N12(1)	17.76	0.39	98.67	6.79E-08	10.73	28.16	252.29	337.07	39.69	0.167	0.972	2244.5
N12(2)	1.14E-09	2.93	2.94	10.30	14.63	18.84	71.70	168.76	54.88	0.077	0.994	1006.3
N13(1)	49.59	1.19	71.46	24.59	4.60	20.95	504.41	2.96E-09	27.17	0.070	0.995	1173.1
N13(2)	30.14	0.65	27.59	6.83	5.85	13.34	96.93	162.75	38.94	0.055	0.997	801.4
N14(1)	3.90E-14	0.98	51.46	10.97	33.77	20.53	4.74E-06	0.13	48.58	0.155	0.972	2348.1
N14(2)	32.79	2.84	8.00	11.41	3.51	12.67	43.07	1.42E-10	48.90	0.058	0.996	916.0
N15(1)	9.83	4.75	46.48	22.53	23.94	3.89	1.60E-12	0.83	54.12	0.103	0.988	1988.2
N15(2)	31.18	6.96	121.88	18.00	12.38	17.44	256.07	3.06E-12	33.30	0.092	0.991	1506.0
N16	13.73	1.95	48.38	8.20	13.96	12.96	105.05	182.50	43.01	0.074	0.994	1065.7
N17(1)	15.44	16.36	110.84	24.93	22.46	26.81	1509.29	3.72E-11	26.07	0.138	0.979	2053.2
N17(2)	9.02E-09	15.00	68.95	10.86	48.04	144.34	51.08	5.62E-09	51.18	0.101	0.988	1692.2

Two samples were obtained from each aorta except for N8 and N16. In the column of “Sample”, numbers 1 and 2 in the parentheses denote the first and second sample obtained from the corresponding aorta.

Table 4.

Correlation coefficients and p-values of the correlation between age and model parameters as well as crosslinking-related strain energy contributions.

	μ	θ	α 0	W c /W	W cf /W
Correlation coefficient	−0.308	−0.678	−0.305	0.523	0.512
p-value	0.086	2.0E-5	0.089	0.002	0.003

μ - shear modulus of the non-collagenous components, θ - the angle between collagen fiber orientation and the longitudinal direction of the aorta, α₀ - angle between orientations of bonded crosslink and fiber, W_c - strain energy density of crosslink stretch, W_cf - strain energy density of crosslink-fiber interaction, W - total strain energy density.

Fig. 7.

Comparison of a root mean square error (RMSE), b coefficient of determination (R²) and c corrected Akaike information criterion (AIC_C) as measurements of the goodness of fit for the two-fiber family model (2FF) and the two-fiber family with crosslinking model (2FFC). Boxes in the plots represent the middle 50% (from the first quartile to the third quartile) of the data set; the horizontal line inside the box represents the median; the cross symbol represents the mean; and the dotted symbols represent individual data points. (**p<0.01, ***p<0.001, ****p<0.0001)

Fig. 8.

Contributions of the strain energy density due to a crosslink deformation (W_c) and b crosslink-fiber interaction (W_cf) to the total strain energy density (W) of human descending thoracic aorta samples under 200 N/m equibiaxial tensile test. Boxes in the plots represent the middle 50% (from the first quartile to the third quartile) of the data set; the horizontal line inside the box represents the median; the cross symbol represents the mean; and the dotted symbols represent individual data points. (*p<0.05, **p<0.01)

4. Discussion

Aortic stiffening is a significant contributor and predictor of the progression of various diseases (Dernellis and Panaretou, 2005; O’Rourke and Safar, 2005; Wang et al., 2005; Riambau et al., 2017). Computational modeling of aging effects on aortic stiffening coupled with microstructural characteristics will greatly benefit our understanding of the mechanisms underlying aortic aging and pathologies, and lead to more reliable metrics for aortic function assessment and prediction. This study shows that human descending thoracic aorta stiffens with aging with a faster increase of stiffness in the longitudinal direction than the circumferential direction along with a reversed degree of anisotropy. Constitutive model considering the deformation of collagen crosslinks and crosslink-fiber interaction provides improved fitting capacity to the biaxial tensile behavior as compared to the two-fiber family model. It is found that collagen crosslinking has an increasing contribution to the stress-stretch behavior and strain energy density in human aorta senescence.

Our results demonstrate an increasing stiffness of aging human descending thoracic aorta (Figs. 2 and 3), which is consistent with earlier biomechanics studies (Langewouters et al., 1984; Sherebrin et al., 1989; Haskett et al., 2010; García-Herrera et al., 2012; Weisbecker et al., 2012; Amabili et al., 2019; Jadidi et al., 2020; Amabili et al., 2021; Franchini et al., 2021). The compromised mechanical functionality with aging dovetails with the reported increase in collagen concentration (Spina et al., 1983; Cattell et al., 1996; Wheeler et al., 2015; Cavinato et al., 2021), while the concentration of elastin decreases (Spina et al., 1983; Andreotti et al., 1985; Seyedsalehi et al., 2015). Meanwhile, similar as previously reported (Haskett et al., 2010; García-Herrera et al., 2012; Jadidi et al., 2020), our study shows faster longitudinal stiffening and reversed anisotropy (Fig. 4). Interestingly, correlation analysis shows θ decreases significantly with age (Table 4), suggesting a shift of collagen fibers towards the longitudinal direction, which explains the faster longitudinal stiffening. Furthermore, the prominent decrease of elastin mass fraction reported by Seyedsalehi et al. (Seyedsalehi et al., 2015) could largely change its interaction with collagen and the overall wall behavior. Also, elastin degradation may be direction-dependent and may play a role in preferentially promoting collagen recruitment in the descending thoracic aortic media. A recent study found that matrix metalloproteinase-12 (MMP12) preferentially stiffens the longitudinal direction of aging artery in mouse (Brankovic et al., 2019). As two closely interwoven ECM components (Wolinsky and Glagov, 1964; Chow et al., 2014), future investigations of the remodeling of collagen and elastin as well as their interactions are necessary to better understand the elevated longitudinal stiffening of human descending thoracic aorta.

Constitutive modeling results suggest a significant contribution of collagen crosslinking to aortic biaxial mechanical behavior regardless of age, as significant improvements of the goodness of fit are achieved by considering crosslinking as measured by R², RMSE and AIC_C in all age groups (Fig. 7). The strain energy density contributed by crosslink deformation is considerable in all age groups (Fig. 8a). This is corroborated by the fact that the amount of collagen crosslinks, pyridinoline and AGEs per collagen molecule, is already substantial in the thoracic aorta of young rats and the amount of AGEs keeps increasing with age after maturation (Brüel and Oxlund 1996). Even for the thoracic aorta of young rats (11 weeks old), eliminating pyridinoline significantly reduces the stiffness of the tissue (Brüel et al., 1998). On the other hand, considering collagen crosslinking is even more important in the aged group. Compared with the reduced fitting capacity of the model without crosslinking in the aged group, the model considering crosslinking showed sustained fitting capability for all age groups (Fig. 7). The positive correlation of W_c/W and W_cf/W with age (Table 4) along with the increase of W_c/W and W_cf/W in the >60 years group (Fig. 8) further confirms the essential role of collagen crosslinking in regulating aortic biaxial stress-stretch behavior as well as the capacity of the new model in describing aging induced aortic stiffening.

Exponential functions were adopted for the strain energy density of crosslink deformation and crosslink-fiber interaction (Eqs. 6 and 9), as compared to quadratic functions proposed by Holzapfel and Ogden (2020). Molecular dynamics simulations have found that the force-displacement curves of enzymatic collagen crosslinks under tension are highly nonlinear with distinct toe and knee regions (Depalle et al., 2015). In this study, increasing nonlinearity was observed in the Cauchy stress-stretch curves of human aorta with aging (Fig. 2), which calls for exponential functions with a more nonlinear nature and better fitting capability for describing the crosslinking-related mechanical responses. Although not reported here, with quadratic strain energy density functions, we found that crosslink deformation and crosslink-fiber interaction would dominate the biaxial mechanical response of samples in the <40 years age group where less nonlinear stress-stretch behaviors are exhibited.

In this study, extensions of fibers and crosslinks are decoupled from the constitutive relation of crosslink-fiber interaction (Eqs. 7 and 9), to avoid the opposite mathematical effects of the extension and rotation kinematics on I₈ in biaxial deformation configuration while maintaining the simplicity of the constitutive model with minimal parameters (Eq. 9). This is because that both fibers and crosslinks will be elongated under biaxial stretch (Eqs. A.4 and A.5) and lead to an increase of the magnitude of I₈, while the relative rotation between bonded fiber and crosslink might either increase or decrease the magnitude of I₈ depending on multiple factors such as the initial angle between the longitudinal direction of the aorta sample and fiber orientation (θ), the initial angle between fiber and crosslink (α₀), and the ratio of stretches between the circumferential and longitudinal directions, as shown in Eq. A.7. The relative rotation between bonded fiber and crosslink leads to an increase of the magnitude of I₈ only when cosα⁺⁄cosα₀ and cosα⁻⁄co(180° − α₀) are greater than 1, which is not guaranteed under biaxial stretch as shown in Figs. S1–S5 of the supplementary material.

Although our study focuses on the descending thoracic aorta, difference in the stiffening behavior between the longitudinal and circumferential directions was found along the entire length of the aorta (Haskett et al., 2010). However, the degree of aging-induced stiffening was found to increase from proximal to the distal regions along the aortic tree from both clinical reports and animal model studies (Hickson et al., 2010; Ferruzzi et al., 2018). With an increasing collagen to elastin ratio (Sokolis, 2007; Concannon et al., 2020), a decreasing undulation and alignment of collagen fibers (Haskett et al. 2010; Zeinali-Davarani et al. 2015) and a decreasing density of pyridinoline crosslinks (Whittle et al., 1987) from the proximal to the distal regions along the aortic tree, remodeling of individual ECM components and their interaction with aging is likely to be regional dependent. Structural-based constitutive modeling will be a powerful tool in understanding the regional-dependency in aging and pathophysiology of the aorta.

5. Limitations

Several limitations regarding the experimental and modeling approaches of this study need to be pointed out. For mechanical testing, due to the required size of the aortic tissue samples for planar biaxial tensile testing, the circumferential variation in stiffness and thickness of the descending thoracic aorta, as previously reported (Kim and Baek, 2011; Rouleau et al., 2012; Kim et al., 2013), was not taken into consideration. Two-fiber family model is adopted to consider collagen crosslinking. However, the modeling framework can be extended to constitutive models in which fiber dispersion and/or additional fiber families are considered (Gasser et al., 2006; Zeinali-Davarani et al., 2009; Jadidi et al., 2020; Holzapfel et al., 2015). Description of the crosslink-fiber interaction is rather phenomenological due to the lack of information on the interaction mechanisms between crosslinks and fibers. Future models with a refined structural representation of collagen crosslinks would lead to a more robust constitutive descriptor of aortic mechanics. In addition, the affine deformation assumption of the constitutive modeling approach adopted in this study could underestimate the reorientation underwent by the fiber and crosslink segments of the interconnected network (Chandran and Barocas, 2006).

Although this study focuses on collagen crosslinking, aortic stiffening results from combined effects of ECM remodeling and interactions of the constituents. The deposition, crosslinking and decreased undulation of collagen could have a combined effect on the biaxial stiffness of the aorta. As a laminated fibrous soft tissue, each aortic layer undergoes unique microstructural alterations with aging (Amabili et al., 2021; Franchini et al., 2021). Layer-specific changes in ECM constituents may play important roles in the stiffening behavior of the aorta. Elastin fragmentation, which was observed with aging (Spina and Garbin, 1976; Schlatmann and Becker, 1977; Toda et al., 1980), may release the compressive stress exerted on collagen fibers and thus promote early collagen recruitment (Chow et al., 2014). The stiffness of vascular smooth muscle cells also increases with aging in the aorta (Qiu et al., 2010). While aging is the focus of this study, future studies are needed in understanding the role of crosslinking in aortic function considering the complex interplay among multiple physiological and pathological factors, such as aging, sex, and cardiovascular diseases.

6. Conclusions

Despite the reported increasing crosslinking density with aging, the mechanistic role of collagen crosslinking in aortic stiffening is yet to be delineated. In this study, the mechanical behavior of human descending thoracic aorta with aging was studied through constitutive modeling coupled with biaxial tensile testing. Our study revealed anisotropic stiffening of the arterial wall with the longitudinal direction stiffens more rapidly than the circumferential direction with aging. A constitutive model incorporating collagen crosslinking and deformation as well as crosslink-fiber interaction was developed to study the contribution of collagen crosslinking in regulating aortic stress-stretch behavior. The modeling results demonstrated an increased contribution of collagen crosslinking to aortic mechanical properties with aging. Results from this study are important for elucidating the mechanistic mechanism of aortic stiffening with aging and the establishment of biomechanical-based metrics for vascular health assessment.

Appendix A. Kinematics of bonded fiber and crosslink under biaxial tension

Under biaxial tension, the deformation gradient, F, is given by:

$$F=\left[\begin{array}{ccc}{\lambda }_1& 0& 0\\ 0& {\lambda }_2& 0\\ 0& 0& \frac{1}{{\lambda }_1{\lambda }_2}\end{array}\right]$$ (A.1)

where λ₁ and λ₂ are stretches in the X₁ (longitudinal) and X₂ (circumferential) directions respectively, as shown in Fig. 1. The right Cauchy-Green deformation tensor, C, is given by:

$$C=F^{T}F=\left[\begin{array}{ccc}{{\lambda }_1}^2& 0& 0\\ 0& {{\lambda }_2}^2& 0\\ 0& 0& \frac{1}{{{\lambda }_1}^2{{\lambda }_2}^2}\end{array}\right]$$ (A.2)

In the undeformed configuration, the unit vectors that are parallel (E₁ and E₂) or normal (E_1R and E_2R) to the orientation of the two collagen fiber families are defined in section 2.2.1. Since the geometry and mechanical properties of bonded fiber and crosslink are symmetric about the longitudinal direction of the aortic sample as are the boundary conditions of biaxial tension (Fig. 1), only the deformation of fibers oriented in the direction of E₁ and their associated crosslinks are considered here. The unit vectors parallel to the orientation of crosslinks in the undeformed configuration are obtained using Eq. 3 as:

$$L_1^{+}=\left\{\begin{array}{c}\text{cos}\theta \text{cos}{\alpha }_0-\text{sin}\theta \text{sin}{\alpha }_0\\ \text{sin}\theta \text{cos}{\alpha }_0+\text{cos}\theta \text{sin}{\alpha }_0\\ 0\end{array}\right\}\text{,}\text{and}{L}_1^{-}=\left\{\begin{array}{c}-\text{cos}\theta \text{cos}{\alpha }_0-\text{sin}\theta \text{sin}{\alpha }_0\\ -\text{sin}\theta \text{cos}{\alpha }_0+\text{cos}\theta \text{sin}{\alpha }_0\\ 0\end{array}\right\}$$ (A.3)

The square of fiber stretch under biaxial tension, I_4,1, is obtained as:

$$I_{4,1}=E_1\cdot C{E}_1=e_1\cdot e_1={{\lambda }_1}^2{\text{cos}}^2\theta +{{\lambda }_2}^2{\text{sin}}^2\theta >1$$ (A.4)

where e₁(=FE₁) is the push-forward of the unit vector in the direction of collagen fibers in the deformed configuration.

The squares of crosslink stretch under biaxial tension, $I_1^{+}$ and $I_1^{-}$, are obtained as:

$$\begin{gathered} I_1^{+}=L_1^{+}\cdot C{L}_1^{+}=l_1^{+}\cdot l_1^{+}={{\lambda }_1}^2{\text{cos}}^2\left(\theta +{\alpha }_0\right)+{{\lambda }_2}^2{\text{sin}}^2\left(\theta +{\alpha }_0\right)>1\text{,}\text{and} \\ {I}_1^{-}=L_1^{-}\cdot C{L}_1^{-}=l_1^{-}\cdot l_1^{-}={{\lambda }_1}^2{\text{cos}}^2\left({\alpha }_0-\theta \right)+{{\lambda }_2}^2{\text{sin}}^2\left({\alpha }_0-\theta \right)>1 \end{gathered}$$ (A.5)

where $l_1^{+}\left(=F{L}_1^{+}\right)$ and $l_1^{-}\left(=F{L}_1^{-}\right)$ are push-forwards of the unit vector in the direction of crosslinks in the deformed configuration.

The invariants $I_8^{1+}$ and $I_8^{1-}$ under biaxial deformation are given by:

$$\begin{gathered} I_8^{1+}=L_1^{+}\cdot C{E}_1=l_1^{+}\cdot e_1=\Vert l_1^{+}\Vert \Vert e_1\Vert \text{cos}{\alpha }^{+}={{\lambda }_1}^2\text{cos}\theta \text{cos}\left(\theta +{\alpha }_0\right)+{{\lambda }_2}^2\text{sin}\theta \text{sin}\left(\theta +{\alpha }_0\right),\text{and} \\ {I}_8^{1-}=L_1^{-}\cdot C{E}_1=l_1^{-}\cdot e_1=\Vert l_1^{+}\Vert \Vert e_1\Vert \text{cos}{\alpha }^{-}=-{{\lambda }_1}^2\text{cos}\theta \text{cos}\left({\alpha }_0-\theta \right)+{{\lambda }_2}^2\text{sin}\theta \text{sin}\left({\alpha }_0-\theta \right) \end{gathered}$$ (A. 6)

The cosine of the angle between bonded fiber and crosslink in the deformed configuration is given by Eq. 7 as:

$$\begin{gathered} I_{8^{*}}^{1+}=\text{cos}{\alpha }^{+}=\frac{I_8^{1+}}{\sqrt{I_{4,1}{I}_1^{+}}}=\frac{\text{cos}\theta \text{cos}\left(\theta +{\alpha }_0\right)+r^2\text{sin}\theta \text{sin}\left(\theta +{\alpha }_0\right)}{\sqrt{{\text{cos}}^2\theta {\text{cos}}^2\left(\theta +{\alpha }_0\right)+r^2{\text{cos}}^2\theta {\text{sin}}^2\left(\theta +{\alpha }_0\right)+r^2{\text{sin}}^2\theta {\text{cos}}^2\left(\theta +{\alpha }_0\right)+r^4{\text{sin}}^2\theta {\text{sin}}^2\left(\theta +{\alpha }_0\right)}},\text{and} \\ {I}_{8*}^{1-}=\text{cos}{\alpha }^{-}=\frac{I_8^{1-}}{\sqrt{I_{4,1}{I}_1^{-}}}=\frac{-\text{cos}\theta \text{cos}\left({\alpha }_0-\theta \right)+r^2\text{sin}\theta \text{sin}\left({\alpha }_0-\theta \right)}{\sqrt{{\text{cos}}^2\theta {\text{cos}}^2\left({\alpha }_0-\theta \right)+r^2{\text{cos}}^2\theta {\text{sin}}^2\left({\alpha }_0-\theta \right)+r^2{\text{sin}}^2\theta {\text{cos}}^2\left({\alpha }_0-\theta \right)+r^4{\text{sin}}^2\theta {\text{sin}}^2\left({\alpha }_0-\theta \right)}} \end{gathered}$$ (A.7)

where r = λ₂/λ₁ is the ratio of stretches between the circumferential and longitudinal directions.