CONSTITUTIVE DESCRIPTION OF HUMAN FEMOROPOPLITEAL ARTERY AGEING

Abstract

Introduction

Femoropopliteal artery (FPA) mechanics play a paramount role in pathophysiology and the artery’s response to therapeutic interventions, but data on FPA mechanical properties are scarce. Our goal was to characterize human FPAs over a wide population to derive a constitutive description of FPA ageing to be used for computational modeling.

Methods

Fresh human FPA specimens (n=579) were obtained from n=351 predominantly male (80%) donors 54±15 years old (range 13–82 years). Morphometric characteristics including radius, wall thickness, opening angle, and longitudinal pre-stretch were recorded. Arteries were subjected to multi-ratio planar biaxial extension to determine constitutive parameters for an invariant-based model accounting for the passive contributions of ground substance, elastin, collagen and smooth muscle. Nonparametric bootstrapping was used to determine unique sets of material parameters that were used to derive age-group-specific characteristics. Physiologic stress-stretch state was calculated to capture changes with ageing.

Results

Morphometric and constitutive parameters were derived for seven age groups. Vessel radius, wall thickness, and circumferential opening angle increased with ageing, while longitudinal pre-stretch decreased (p<0.01). Age-group-specific constitutive parameters portrayed orthotropic FPA stiffening, especially in the longitudinal direction. Structural changes in artery wall elastin were associated with reduction of physiologic longitudinal and circumferential stretches and stresses with age.

Conclusions

These data and the constitutive description of FPA ageing shed new light on our understanding of peripheral arterial disease pathophysiology and arterial ageing. Application of this knowledge might improve patient selection for specific treatment modalities in personalized, precision medicine algorithms and could assist in device development for treatment of peripheral artery disease.

INTRODUCTION

Femoropopliteal artery (FPA) disease is usually due to obstructive atherosclerotic lesions that reduce blood flow to the lower limbs. This condition, frequently referred to as Peripheral Arterial Disease (PAD), is associated with high morbidity, mortality and impairment in quality of life. In addition, the total annual hospital cost for patients with PAD exceed $21 billion per year, with per-patient costs of PAD higher than those for both coronary artery disease and cerebrovascular disease(Mahoney et al. 2008). The staggering cost of PAD is primarily attributed to the high number of peripheral vascular operations and interventions that fail, resulting in poor clinical outcomes and a frequent need for repetitive intervention(Adam et al. 2005; Schillinger et al. 2006; Conte et al. 2006; Schillinger et al. 2007). Specifically, restenosis within three years after FPA bypass surgery occurs in 27% of patients(Siracuse et al. 2012). For endovascular interventions, over 45% of patients develop hemodynamically significant restenosis within two years after treatment, leading to repeated intervention in 37–54% of patients(Schillinger et al. 2007).

Though systemic risk factors for restenosis are often the same for carotid, iliac and femoropopliteal artery reconstructions, FPA interventions fail significantly more often than those in other anatomic locations. This observation suggests that local factors unique to the FPA, such as blood flow alterations(Stonebridge and C.M. 1991) and limb flexion-induced artery deformations(Iida et al. 2006; MacTaggart et al. 2014), might strongly influence disease development and contribute to reconstruction failure. FPA morphometric, mechanical, and structural characteristics influence each of these two factors, in addition to the interaction between the artery wall and reconstruction material or device.

The human FPA has a unique anisotropic structure that facilitates its function as a highly dynamic artery in a flexing limb. The medial layer is composed primarily of smooth muscle cells (SMCs). At the outer edge of the media blending into the adventitia, the external elastic lamina (EEL) demonstrates heavily concentrated elastin, oriented primarily in the longitudinal direction. This longitudinal alignment gives the FPA significant in situ longitudinal pre-stretch that facilitates energy efficient function(Kamenskiy et al. 2014b; Kamenskiy et al. 2015; Kamenskiy et al. 2016) and reduces kinking of the artery during limb flexion(MacTaggart et al. 2014). Degradation and fragmentation of the EEL has been demonstrated to influence anisotropic FPA stiffening that occurs more rapidly in the longitudinal direction with ageing(Kamenskiy et al. 2015). These structural and mechanical changes in the artery likely affect disease development and the artery’s response to therapeutic interventions.

The goal of the current study is to present human FPA morphometric and structural characteristics obtained from a robust sample (n=579) of human FPAs, and to determine constitutive parameters across a wide range of ages. Constitutive parameters were determined using biaxial testing, nonparametric bootstrapping, and constitutive modeling techniques. These data were then used to model the evolution of in vivo physiologic states during the ageing process, accounting for residual stresses and longitudinal pre-stretch.

METHODS

2.1 Materials

Fresh femoropopliteal arteries (n=579) from organ and tissue donors (n=351) were obtained from the Nebraska Organ Recovery System (NORS) within 24 hours of subject’s death after obtaining consent from the next of kin. Donors were predominantly male (80%) and on average 54±15 years old (range 13–82 years). Prior to excision from the body, the in situ length of the FPA segment was measured from the takeoff of the profunda femoris artery to the tibioperoneal trunk using an umbilical tape. The tape and the artery were cut together and while the umbilical tape maintained its length, the artery typically shortened due to in situ longitudinal pre-stretch ${\lambda }_z^{\mathit{\text{in situ}}}$. Pre-stretch was then defined as the ratio of the umbilical tape length to the excised artery length.

2.2 Mechanical testing

Arterial rings of 5mm length were cut from the FPA and photographed. The images were used to measure outer vessel radius in the load-free configuration (ρ_o). Rings were then cut radially to relieve most of the residual stress(Humphrey 2002; Holzapfel et al. 2007; Holzapfel and Ogden 2010) and allowed to equilibrate in 0.9% NaCl physiological saline solution at room temperature for 20 min along with a longitudinal 13 mm strip of adjacent artery wall. Inner and outer radii (R_i and R_o), circumferential opening angle (α), and inner radius of the longitudinal strip (R_zo) were determined by contouring images of the samples and inscribing circles which allowed calculation of the best-fit radii and angle(Kamenskiy et al. 2015).

Additional segments of the artery were then cut into 13×13mm sheets (when permitted by diameter) for planar biaxial mechanical testing. Testing was performed on the whole segment, without separating it into layers. To eliminate shear, the longitudinal and circumferential axes were aligned with the test axes. The first 38% of all experiments were performed with a custom-made planar biaxial tensile tester equipped with “250g” (max load 2.5N) load cells (Honeywell Sensotec). Maximum load for the majority of specimens was set at 1N, and nine force-controlled loading protocols that varied maximal load at 1:1, 1:2, 1:4, 1:10 loading ratios(Kamenskiy et al. 2014c; Kamenskiy et al. 2014b; Kamenskiy et al. 2015) were utilized to characterize the spectrum of stretches. Specimens were attached using hooks and loops of surgical suture. The other 62% of experiments were performed with a CellScale planar biaxial tensile tester using 21 stretch-controlled multi-ratio protocols ranging from 1:1 to 1:0.1 on each axis and performed at 0.01s⁻¹ strain rate(Kamenskiy et al. 2016). Prior to testing, an equibiaxial loading protocol with gradually increasing stretches was executed to estimate stretch limits corresponding to 1N maximum load. No sizable shear was observed when using neither hooks nor surgical loops(Kamenskiy et al. 2015), and CellScale tests were performed using a rake system which significantly simplified specimen attachment.

A deformation gradient was calculated for the center of the sample which was marked with graphite markers or aluminum oxide crystals. Principal in-plane stretches in the longitudinal (λ_z) and circumferential (λ_θ) directions were calculated and related to forces on each of the axes (P_z and P_θ respectively). Through-thickness average experimental Cauchy stresses for each direction were calculated as:

$$t_{zz}^{\mathit{\text{exp}}}={\lambda }_z\frac{P_z}{H{L}_{\theta }},t_{\theta \theta }^{\mathit{\text{exp}}}={\lambda }_{\theta \theta }\frac{P_{\theta }}{H{L}_z},$$ (1)

where H is the undeformed thickness that was measured using images of the radially cut arterial ring, and L_i are the undeformed lengths over which the applied loads P_i act (i = θ, z). All tissues were pre-loaded with a tare load of 0.01N prior to initiation of the test and pre-conditioned with 10–20 equibiaxial cycles to achieve a repeatable response. More details on specimen preparation and specific testing procedures can be found in previous works(Kamenskiy et al. 2014c; Kamenskiy et al. 2014b; Kamenskiy et al. 2015).

2.3 Constitutive modeling

The wall of the femoropopliteal artery was assumed to contain an extracellular matrix composed of randomly organized amorphous ground substance, longitudinally oriented elastin fibers concentrated primarily in the EEL and adventitia, and two symmetrical families of collagen fibers in the adventitia wrapping around the artery in a helix (Figure 1)(Kamenskiy et al. 2015; Kamenskiy et al. 2016). Most of the tunica media is populated by SMCs that have a predominantly circumferential alignment, which facilitates rapid changes in arterial diameter in response to downstream organ flow demands.

Figure 1.

Intramural structure of the FPA from a 17-year-old male in the transverse and longitudinal directions. VVG stain: elastin black, collagen red, smooth muscle brown.

Based on this histological structure, the total elastic strain energy of the artery per unit reference volume (W) may be considered as consisting of four contributors:

$$W=W_{\mathit{\text{gr}}}+W_{\mathit{\text{el}}}+W_{\mathit{\text{smc}}}+W_{\mathit{\text{col}}}.$$ (2)

The contribution of SMCs here is passive, just like the contribution of the extracellular components, so W = W (λ_i, P) where P is the set of constitutive model parameters determined from the experimental data and i = z, θ. Specifically, the following Holzapfel-Gasser-Ogden (HGO) forms of strain energy density constituents have been previously demonstrated to accurately portray the behavior of the FPA(Kamenskiy et al. 2014b; Kamenskiy et al. 2015):

$$W_{\mathit{\text{gr}}}=\frac{C_{\mathit{\text{gr}}}}{2}(I_1^{\mathit{\text{gr}}}-3),$$

$$W_{\mathit{\text{el}}}=\frac{C_1^{\mathit{\text{el}}}}{4C_2^{\mathit{\text{el}}}}(e^{C_2^{\mathit{\text{el}}}{\langle I_4^{\mathit{\text{el}}}-1\rangle }^2}-1),$$

$$W_{\mathit{\text{smc}}}=\frac{C_1^{\mathit{\text{smc}}}}{4C_2^{\mathit{\text{smc}}}}(e^{C_2^{\mathit{\text{smc}}}{\langle I_4^{\mathit{\text{smc}}}-1\rangle }^2}-1),$$ (3)

$$W_{\mathit{\text{col}}}=\frac{C_1^{\mathit{\text{col}}}}{2C_2^{\mathit{\text{col}}}}(e^{C_2^{\mathit{\text{col}}}{\langle I_4^{\mathit{\text{col}}}-1\rangle }^2}-1).$$

where the corresponding invariants of the right Cauchy-Green tensor C are defined (enforcing incompressibility) as

$$I_1^{\mathit{\text{gr}}}=\mathit{\text{tr}}\mathbf{C}={\lambda }_z^2+{\lambda }_{\theta }^2+\frac{1}{{\lambda }_z^2{\lambda }_{\theta }^2},$$

$$I_4^{\mathit{\text{el}}}={\lambda }_z^2,$$

$$I_4^{\mathit{\text{smc}}}={\lambda }_{\theta }^2,$$ (4)

$$I_4^{\mathit{\text{col}}}={\lambda }_z^2{\text{cos}}^2\gamma +{\lambda }_{\theta }^2{\text{sin}}^2\gamma .$$

Here γ is the angle between the collagen fibers and the longitudinal direction, and $\mathbf{P}=(C_{\mathit{\text{gr}}},C_1^{\mathit{\text{el}}},C_2^{\mathit{\text{el}}},C_1^{\mathit{\text{col}}},C_2^{\mathit{\text{col}}},C_1^{\mathit{\text{smc}}},C_2^{\mathit{\text{smc}}},\gamma )$ are constitutive parameters to be determined from the experimental data by minimizing the difference between principal experimental (1) and theoretical (5) isochoric Cauchy stresses:

$$\overline{t_{zz}}=c_{\mathit{\text{gr}}}({\lambda }_z^2-\frac{1}{{\lambda }_{\theta }^2{\lambda }_z^2})+c_1^{\mathit{\text{el}}}\langle {\lambda }_z^2-1\rangle e^{c_2^{\mathit{\text{el}}}{({\lambda }_z^2-1)}^2}{\lambda }_z^2+2c_1^{\mathit{\text{col}}}\langle I_4^{\mathit{\text{col}}}-1\rangle e^{c_2^{\mathit{\text{col}}}{(I_4^{\mathit{\text{col}}}-1)}^2}{\lambda }_z^2{\text{cos}}^2\gamma$$ (5)

$$\overline{t_{\theta \theta }}=c_{\mathit{\text{gr}}}({\lambda }_{\theta }^2-\frac{1}{{\lambda }_{\theta }^2{\lambda }_z^2})+c_1^{\mathit{\text{smc}}}\langle {\lambda }_{\theta }^2-1\rangle e^{c_2^{\mathit{\text{smc}}}{({\lambda }_{\theta }^2-1)}^2}{\lambda }_{\theta }^2+2c_1^{\mathit{\text{col}}}\langle I_4^{\mathit{\text{col}}}-1\rangle e^{c_2^{\mathit{\text{col}}}{(I_4^{\mathit{\text{col}}}-1)}^2}{\lambda }_{\theta }^2{\text{ sin}}^2\gamma .$$

Here Macaulay brackets $\langle (·)\rangle =\frac{1}{2}[(·)+|(·)|]$ filter positive values such that fibers only contribute to the strain energy density during tension, but not during compression. Note that (5) is mathematically equivalent to the 4-fiber family HGO model(Baek et al. 2007), but has a different interpretation of parameters in the context of the FPA.

2.4 Simulation of physiologic conditions

Constitutive parameters, longitudinal pre-stretch, and morphometric measurements of radii and opening angle allowed calculation of physiologic conditions through two-dimensional kinematic framework simulating inflation-extension testing. Equilibrium provides the relation for the internal pressure p_i and the longitudinal force F_z:

$$p_i={\int }_{r_i}^{r_o}(\overline{t_{\theta \theta }}-\overline{t_{rr}})\frac{dr}{r}$$ (6)

$$F_z=\pi {\int }_{r_i}^{r_o}(2\overline{t_{zz}}-\overline{t_{\theta \theta }}-\overline{t_{rr}})rdr.$$ (7)

where $\overline{t_{zz}},\overline{t_{\theta \theta }}$ are mean transmural stresses through the wall thickness as determined from (5), isochoric radial stress $\overline{t_{rr}}={\lambda }_r\frac{\partial W}{\partial {\lambda }_r}=\frac{c_{\mathit{\text{gr}}}}{{\lambda }_{\theta }^2{\lambda }_z^2}$, and inner and outer radii (r_i, r_o) are given by

$$r_i=\sqrt{{({\rho }_o{\lambda }_{\theta })}^2-\frac{R_o^2-R_i^2}{k{\lambda }_{z,\mathit{\text{res}}}{\lambda }_z}}\text{ and }{r}_o={\rho }_o{\lambda }_{\theta }.$$ (8)

Here the effects of residual stretches stemming from the opening of the arterial ring into a sector and curving of the longitudinal strip are taken into account, but no variation of stresses through thickness is considered. Further, $k=\frac{2\pi }{2\pi -\alpha }$ is a measure of the circumferential opening angle α in the reference configuration, ${\lambda }_{z,\mathit{\text{res}}}=\frac{2R_{zo}+H}{2(R_{zo}+H)}$ is the axial residual stretch, ρ_o is arterial outer radius in the unloaded configuration, R_o, R_i are outer and inner radii in the stress-free configuration, and R_zo is the inner radius of the curved longitudinal strip in the stress-free configuration. Note that R_zo is measured from the center of the sector to the adventitia because the FPA strip curves with the intima facing outward. Details of this kinematic framework are given in Kamenskiy et al (Kamenskiy et al. 2015).

2.5 Nonparametric bootstrapping

Constitutive parameters in (5) may not uniquely represent arterial properties, which can result in unpredictable material behavior under new loading conditions. In order to reveal nonuniqueness, a technique called nonparametric bootstrapping is used (Ferruzzi et al. 2011). Nonparametric bootstrapping involves random resampling and fitting of a dataset, followed by analysis of the probability distribution of each parameter for all fits. Say the matrix X represents stress-stretch data obtained from planar biaxial testing using n protocols. This matrix has 4n columns, and it can be divided into n matrices X_i, each representing a different testing protocol. Each of these matrices is composed of m_i arrays x_ij, where $x_{ij}=\{{\lambda }_z,\overline{t_{zz}},{\lambda }_{\theta },\overline{t_{\theta \theta }}\}$. The value of m_i may change depending on the protocol due to the varying length of the experimental data sets. For each bootstrapping iteration k, a unique data matrix X* is created with n matrices for each protocol. Each of these new matrices ${\mathit{X}}_i^{*}$ is composed of m random samples from the original matrix X_i. Here, m is the minimum protocol length. Note that during filling of the bootstrapping protocol matrix, x_ij values that have already been included are still available for addition into the dataset, meaning that the bootstrapping dataset X* represents a redistribution of weight onto different pieces of data randomly.

For each bootstrapping iteration, the new dataset is introduced into an error function (9) that aids in minimizing the differences between experimental and theoretical stresses using lsqnonlin function in MATLAB (MathWorks, Natick, Massachusetts, USA).

$$e({\mathbf{P}}_k^{*})={\Vert \overline{\mathit{t}}(\lambda ,\theta )-{\mathit{t}}^{\mathit{\text{exp}}}\Vert }_2^2={\sum }_{i=1}^n\{{[\overline{t_{zz}}{({\lambda }_z,{\lambda }_{\theta };{\mathbf{P}}_k^{*})}_i-{\left(t_{zz}^{\mathit{\text{exp}}}\right)}_i]}^2+{[\overline{t_{\theta \theta }}{({\lambda }_z,{\lambda }_{\theta };{\mathbf{P}}_k^{*})}_i-{\left(t_{\theta \theta }^{\mathit{\text{exp}}}\right)}_i]}^2\}$$ (9)

Initially, random values are assigned to constitutive parameters, and zeros are assigned to lower bounds. After minimization, output parameter estimations are extracted in the form of an eight-component array ${\mathbf{P}}_k^{*}$ of constitutive parameters, and the entire process is repeated, ultimately yielding a matrix of constitutive parameters P. Clustering of parameters around different values on the probability distribution constitutes a mode, and presence of several modes suggests local minima and non-uniqueness of the solution. Multimodal results were observed as both distributions of blended modes (where they are close together), or significantly different modes (where they do not overlap at all). One could test individual modes in a multimodal distribution for their statistical significance using Silverman’s modality test based on the kernel density estimate of the probability density function (Ferruzzi et al. 2011), but this approach was not implemented here due to large number of unimodal samples. Multimodal results can stem from a variety of factors, including poor fit and insufficient number of testing protocols. In our samples they were also generally associated with lower R² (coefficient of determination) and hence were not considered in the final analysis. A sample was selected for further analysis only if the R² obtained for bootstrapped parameters was ≥ 0.9.

2.6 Determination of constitutive parameters for age groups

Averaging of constitutive parameters to represent a group of samples is not appropriate because of the interrelation between these parameters. A better approach is to determine “representative experimental data” for this group and fit them with equations (5) to determine group-specific constitutive parameters. The challenge however is often in the difference of loading conditions for each sample that complicates the comparison of the experimental curves. In our case, 38% of samples were tested under force-controlled protocols, and the remaining 62% under stretch-controlled protocols which produce essentially different ranges of stress-stretch responses for anisotropic tissues.

In order to obtain uniform loading condition for all samples, a normalization procedure is required prior to generation of the representative stress-stretch curves. These loading conditions can either be stress-controlled or stretch-controlled, but stress-controlled approach is generally beneficial when dealing with anisotropic materials that demonstrate nonlinear stiffening. Substantial non-linearity and anisotropy of the FPA makes the artery highly sensitive to stretch, and when pushed to a set stretch value selected for the group, can result in exponential stress increase that is usually observed in the circumferential direction in the FPA.

To resolve this issue, the approach described in Figure 2 was implemented. Specifically, the raw experimental dataset consisting of n=579 samples was used to determine initial constitutive parameters P_ini using nonparametric bootstrapping. While most specimens fit well with parameters representing local minima, only 56% (n=322 samples) had unimodal distributions of constitutive parameters that produced an R² ≥ 0.9. These samples were then used to determine biaxial stretches that corresponded to 100 kPa Cauchy stress under stress-controlled conditions. This was achieved through numerically solving (5) for λ_z, λ_θ using vpasolve function of MATLAB. Resulting stretch limits were then used to generate stress-controlled protocols that were uniform for all samples in terms of Cauchy stress. Uniformity in stress allowed calculation of a representative sample response for each age group by averaging stretches corresponding to the same level of stress within the group. This resulted in seven stress-stretch protocols each representing an age group: 11–20 (n=21), 21–30 (n=15), 31–40 (n=26), 41–50 (n=39), 51–60 (n=106), 61–70 (n=83) and 71–80 (n=32) years old. These protocols were again fitted with (5) using nonparametric bootstrapping to generate constitutive values P_av for each age group (Figure 2). A representative stress-stretch curve (C) and the corresponding constitutive parameter distributions (B) are presented on the right panel of Figure 2.

Figure 2.

A) Bootstrap procedure schematics demonstrating how constitutive parameters for each age group were obtained, and B) bootstrap results demonstrating unimodal distributions of parameters, and C) curve fit obtained with these parameters. Multiple curves of the same color represent multi-ratio loading protocols in the longitudinal (blue) and circumferential (red) directions.

2.7 Histopathological changes with ageing

Structural changes associated with ageing were studied using methacarn-fixed and Verhoeff-Van Gieson (VVG) stained n=671 transverse and n=1186 longitudinal sections of the FPAs immediately adjacent to the biaxially tested samples. Multiple sections from the same artery allowed assessment of structural variation. Longitudinal sections were particularly enlightening because they allowed characterization of longitudinal elastin in the EEL, which appears to play a significant role in ageing and adaptation of muscular arteries(Kamenskiy et al. 2014a; MacTaggart et al. 2014; Kamenskiy et al. 2014c; Kamenskiy et al. 2014b; Kamenskiy et al. 2015; Kamenskiy et al. 2016). VVG stained sections were scanned at ×10 resolution and used to measure thickness of the tunica media, defined as the space between the internal and external elastic laminae. EEL thickness, density and discontinuity were also characterized, with EEL discontinuity measured on a scale 1–5 with 1 representing perfectly continuous fibers. In addition, the thickness of the individual elastic fibers, density of elastin in the EEL and the total amount of elastin in the arterial section were quantified with image analysis. All measurements were independently repeated and verified by three operators.

RESULTS

Changes in arterial morphometry with age are summarized in Figure 3. Aging resulted in outward remodeling with an increase in the load-free outer radius from 2.76 mm to 4.16 mm (p<0.01). Correlation with age was strong, with a Spearman correlation coefficient of ρ = 0.51. Spearman correlation coefficient describes how well the relationship between two variables can be described using a monotonic (but not necessarily linear) function. Opening angle also increased with age (p=0.03), from 122° to 142°, but correlation with age was weak and variation was significant (ρ = 0.09). Larger opening angles resulted in increased outer and inner radii in the stress-free configuration (p<0.01) that changed from 3.82 mm and 2.43 mm in the youngest age group to 6.66 mm and 4.88 mm in the oldest age group respectively. Interestingly, correlation with age was stronger for the radii than for the opening angle (ρ = 0.32, ρ = 0.26), and outer radius increased faster resulting in the increase in wall thickness from 1.39 mm in the youngest age group to 1.78 mm in the oldest age group (ρ = 0.29, p<0.01). The inner radius of the longitudinal strip R_zo in the stress-free configuration increased with age (p<0.01) indicating that axial strips progressively became more flat with ageing, but correlation with age was nonlinear (ρ = 0.74). Finally, in situ longitudinal pre-stretch significantly decreased with age (ρ = −0.68, p<0.01) from 1.53 for the youngest age group to 1.10 for the oldest.

Figure 3.

Changes in A) load-free outer radius (ρ_o), stress-free B) outer (R_o) and C) inner ((R_i) radii, D) opening angle (α), E) inner radius in the stress-free longitudinal strip (R_zo), and F) in situ longitudinal pre-stretch (${\lambda }_z^{\mathit{\text{in situ}}}$). Note that stress-free longitudinal strip is curved intima (I) outward. Box plots represent data between 25^th and 75^th percentiles with median and mean marked with a red line and a blue asterisk respectively.

Constitutive parameters for each age group are summarized in Table 1. Some parameters, like $C_2^{\mathit{\text{el}}},C_1^{\mathit{\text{smc}}},C_2^{\mathit{\text{smc}}},C_2^{\mathit{\text{col}}}$ and γ monotonically increased with age – a trend that can be accurately described with an exponential function. Other parameters, like $C_1^{\mathit{\text{el}}},C_1^{\mathit{\text{col}}}$ decreased until approximately 50 years of age, but rapidly increased in the sixth decade. Finally, C_gr changed irregularly, but demonstrated an overall decrease with age suggesting substitution of ground substance with other intramural constituents, such as collagen.

Table 1.

Constitutive parameters for each age group. Number of samples in each age group is signified by n.

Age group, years	n	Mean age, years	Cgr, kPa	$C_1^{el}$, kPa	$C_2^{el}$	$C_1^{smc}$, kPa	$C_2^{smc}$	$C_1^{\mathit{\text{col}}}$, kPa	$C_2^{\mathit{\text{col}}}$	γ, °
11–20	21	16.7	10.51	20.92	0.22	2.34	1.76	3.32	2.05	60.84
21–30	15	25.0	16.81	17.35	0.69	7.72	3.68	3.80	3.28	50.17
31–40	26	35.7	9.96	17.57	0.99	4.74	2.81	2.45	4.10	55.88
41–50	39	47.2	13.46	14.54	2.41	8.20	7.29	2.31	9.96	46.84
51–60	106	56.1	10.69	20.77	3.60	11.58	8.38	3.28	13.19	45.62
61–70	83	64.6	7.83	24.81	5.73	16.82	12.56	5.24	21.76	46.58
71–80	32	76.1	7.01	37.03	12.81	36.09	15.96	11.65	36.08	45.02

The interplay of constitutive parameters resulted in the overall reduction of total strain energy density with age calculated at equibiaxial stress-controlled state (ρ = −0.73, p<0.01). This signifies overall stiffening of the tissue (Figure 4). Strain energy density of the ground substance and elastin also decreased (p<0.01), while strain energies of collagen and SMCs fluctuated, demonstrating possible turnover and adaptation.

Figure 4.

Reduction of total strain energy per unit reference volume (W, kPα) with age. Strain energy density was calculated by using P_sc constitutive parameters at stretches that corresponded to 100 kPa equibiaxial Cauchy stress. Reduction of strain energy density at the equibiaxial stress state indicates overall stiffening of the FPA with age.

Piecewise cubic Hermite polynomial interpolation was performed using constitutive parameters (Table 1), allowing modeling of FPA ageing. Figure 5 illustrates changes in the stress-stretch response with age demonstrating stiffening in both longitudinal and circumferential directions. Young FPAs were substantially more compliant longitudinally than circumferentially. With ageing however the FPA stiffened faster longitudinally than circumferentially, resulting in more isotropic response of older FPAs. Circumferential stress response also appeared to have a more pronounced toe region with stiffening occurring around λ_θ~1.25 for the youngest and λ_θ~1.05 for the oldest age group, while the longitudinal stress-stretch curve appeared more linear. Stiffening with age occurred in bursts progressing rapidly from the second to third decade, slowing down during the third and fourth decade, rapidly progressing to the fifth decade, and again slowing down before accelerating into the eighth decade of life.

Figure 5.

Cauchy stress - stretch curves for each of the age groups in longitudinal (A, z) and circumferential (B, θ) directions. Legend summarizes average ages (years) for each of the groups. Figure demonstrates longitudinal and circumferential stiffening with age. Curves of the same color demonstrate multiple loading protocols for the same age group. Stress-stretch curves for all age groups are statistically different from each other (p<0.01) except 21–30 (average age 25 years) and 31–40 (average age 36 years) groups in the longitudinal direction (orange and yellow colors in the left panel).

At simulated physiologic conditions of 120 mmHg of pressure and in situ longitudinal pre-stretch, ageing FPAs demonstrated reductions of longitudinal (ρ = −1, p < 0.01) and circumferential (ρ = −0.86, p = 0.02) Cauchy stresses, but no change in radial stress (ρ = 0.14, p = 0.78) (Figure 6). Reduction of longitudinal stress can be described with a linear function $\overline{t_{zz}}=-1.63·\mathit{\text{age}}+147$ producing an R² = 0.92. Circumferential stress demonstrated some fluctuation with increases and decreases in $\overline{t_{\theta \theta }}$ resulting in R² = 0.69 for a linear fit of $\overline{t_{\theta \theta }}=-0.66·\mathit{\text{age}}+77.77$. Longitudinal stress was higher than circumferential stress in all age groups (p=0.01). In the youngest age group the difference was 1.8-fold but it reduced to less than 10% in the oldest age group highlighting the change towards more isotropic behavior. Radial stress was significantly smaller than both longitudinal and circumferential stresses in all age groups. The difference between radial, longitudinal and circumferential stresses for the youngest age group was 53-fold and 29-fold respectively, and 7-fold and 6-fold for the same stresses in the oldest age group respectively.

Figure 6.

Physiologic stresses A) $\overline{t_{zz}}$, B) $\overline{t_{\theta \theta }}$, C) $\overline{t_{rr}}$, stretches D) θ_z, E) λ_θ and tethering force F) F_z at 120 mmHg pressure. Correlation with age is measured with Spearman's rank correlation coefficient ρ, while the quality of linear fit is assessed with coefficient of determination R².

Longitudinal and circumferential physiologic stretches also decreased with age. Longitudinal physiologic stretch changed from 1.52 in the youngest to 1.10 in the oldest age group, while circumferential physiologic stretch decreased more slowly from 1.34 in the youngest to 1.09 in the oldest age group. Again, longitudinal stretch was larger than circumferential for all age groups (p=0.01). Linear functions provided a good description of both λ_z and λ_θ with λ_z = −0.006749 · age + 1.592 (R² = 0.97) and λ_θ = −0.003722 · age + 1.371 (R² = 0.83).

Finally, tethering force F_z reduced with age F_z = −0.01413 · age + 1.735 (R² = 0.85) from 1.49 N in the youngest age group to 0.72 N in the oldest age group. Reduction of physiologic longitudinal (p<0.01) and circumferential (p=0.03) stresses, stretches, and tethering force (p<0.01), and no change in radial stress with age (p=0.38) described by the ageing model was confirmed by running the analysis on all samples in addition to the seven age groups presented in Table 1.

Structural changes that occurred with age are plotted in Figure 7. Ageing resulted in increased tunica media thickness (ρ = 0.09, p = 0.03), from 490 µm in the youngest age group to 604 µm in the oldest age group. The thickness of the EEL decreased with age (ρ = −0.14, p < 0.01) from 114 µm in the youngest to 79 µm in the oldest age group. Individual elastic fiber thickness also decreased with age (ρ = −0.17, p < 0.01) from 4.4 µm to 3.2 µm. Elastin density in the EEL dropped with age (ρ = −0.31, p < 0.01) from 32% to 22% as did the total amount of elastin in the tissue (ρ = −0.28, p < 0.01), reduced from 5.3% to 3.1% from youngest to oldest age groups. Finally, elastic fibers became significantly more discontinuous with ageing (ρ = 0.35, p<0.01).

Figure 7.

Histopathological changes in the FPAs with age demonstrating increase in tunica media thickness (A) and elastic fiber discontinuity (F), and decrease in thickness of the External Elastic Lamina (EEL) (B), thickness of individual elastic fibers (C), total elastin composition (D), and elastin density in the EEL (E) with age. Box plots represent data between 25^th and 75^th percentiles with median and mean marked with a red line and a blue asterisk respectively.

DISCUSSION

Ageing is a non-modifiable risk factor for vascular disease, frequently associated with stiffening of the arteries(Learoyd and Taylor 1966; Jani and Rajkumar 2006; Greenwald 2007; O’Rourke 2007; Lakatta et al. 2009; Lee and Oh 2010). It may include a variety of biological and mechanical changes that occur at different rates affecting the artery in the circumferential, longitudinal and radial directions. In this study we provide a comprehensive quantitative description of the morphometric, structural, and mechanical changes that occur in the human FPA with ageing.

The FPA is a muscular artery and its mechanics are primarily dependent on SMCs, elastic fibers, collagen fibers and ground substance(MacTaggart et al. 2014; Kamenskiy et al. 2014b; Kamenskiy et al. 2015; Kamenskiy et al. 2016). SMCs dominate the media and are oriented primarily circumferentially to regulate rapid diameter changes in response to flow demands. Changes in cardiac output(Brandfonbrener et al. 1955) and the cumulative effects of hypertension with age(Mozaffarian et al. 2015) may affect SMCs by inducing hypertrophy(Intengan and Schiffrin 2001), hyperplasia(Ueno et al. 2000), and alterations in SMC phenotype(Qiu et al. 2014). Along with diameter and opening angle changes(Matsumoto et al. 1996), these processes have been described as adaptive responses to alterations in flow and pressure. Our data support prior experimental observations by demonstrating an increase in FPA wall thickness, diameter, and opening angle, as well as an increase in tunica media thickness with age.

Most elastin in the FPA is located in the EEL and adventitia(Kamenskiy et al. 2015; Kamenskiy et al. 2016). Its highest concentration is within the EEL where the elastin fibers are oriented primarily longitudinally(Kamenskiy et al. 2014b). This is in contrast to elastic arteries, in which elastin is organized multi-dimensionally(Kamenskiy et al. 2014a; Kamenskiy et al. 2014c) and can be assumed isotropic(Holzapfel et al. 2000; Gasser et al. 2006; Sommer 2008). Elastin is a stable, resilient protein with a high concentration of nonpolar hydrophobic amino acids and a natural turnover rate believed to be on the order of a human lifespan(Mithieux and Weiss 2005). Having formed during the perinatal period, continuous elastic fibers stretch significantly as the artery grows in length, resulting in significant longitudinal tension in maturity. This tension likely serves to reduce kinking of the FPA during locomotion(MacTaggart et al. 2014) and ensures energy efficient function(Kamenskiy et al. 2014b; Kamenskiy et al. 2016). The longitudinal nature of elastin in the FPA likely contributes to the almost linear stress-stretch response in the longitudinal direction, while the location of elastin in the EEL and adventitia is likely responsible for the curving of the longitudinal artery strip intima outward. This observation suggests that distribution of λ_z,res may be non-uniform throughout the wall thickness and additional studies, utilizing a three-dimensional rather than two-dimensional kinematic framework, are required to investigate its influence on physiologic tethering force F_z. Our histological studies demonstrate that ageing results in flattening of the longitudinal strips, elastin fiber discontinuity and thinning of the individual elastic fibers. Degradation and fragmentation of elastin in the EEL, either by means of mechanical fatigue or chemical breakdown due to local proteases, is also associated with reduction of in situ longitudinal prestretch and stiffening of the FPA in the longitudinal direction.

Degradation of elastin may be accompanied by the accumulation of collagen in an attempt by the body to maintain the homeostatic stress state (Humphrey 2008). Stiff collagen fibers dominate the adventitia and contain several symmetric fiber families that protect the artery from overstretch and aid with structural stability. Collagen fibers are undulated below physiologic pressure which results in mostly linear elastic responses due primarily to the action of elastic fibers; however as blood pressure increases, collagen fibers straighten resulting in a rapid increase in vascular stiffness. Since the major role of adventitial collagen is to protect against overstretch, collagen fibers have a preferred circumferential alignment. This alignment results in a distinctive toe region in the circumferential stress-stretch curve that is less evident in the longitudinal direction. Collagen turns over at a much higher rate than elastin, and accumulation of new collagen with age may contribute to both increased stiffening and decreased collagen fiber angle in older FPAs. This suggests reorganization or addition of new collagen in the longitudinal direction, likely to compensate for the damaged elastin. Further studies are required to validate this speculation.

The organization of elastin and collagen as well as differences in their turnover rates can explain the significant anisotropy in the FPA wall that demonstrates higher compliance in the longitudinal direction compared to the circumferential direction. The degradation and fragmentation of longitudinally oriented elastin, accompanied by deposition of new collagen, appears to influence the overall drop in compliance and anisotropy with age. This drop of compliance occurs faster longitudinally than circumferentially, thereby resulting in movement of the equibiaxial stress-stretch curves closer to each other. Degradation and fragmentation of longitudinal elastin may also be responsible for the reduction of the in situ longitudinal prestretch with age. These phenomena appear to occur independent of the most common vascular diseases such as atherosclerosis and medial calcification (Kamenskiy et al. 2016; Horný et al. 2016). This suggests that reduction of longitudinal pre-stretch may be part of an adaptive response rather than a pathologic result. Interestingly, reduction of longitudinal pre-stretch with age has been observed in other human arteries, namely the common carotid and the abdominal aorta, and results were artery-dependent (Horný et al. 2016). Comparison of our data with prior publications demonstrate that longitudinal pre-stretch and the associated axial force are higher in the FPA than in the common carotid artery and the abdominal aorta for all age groups (Horny et al. 2012; Horný et al. 2013; Horný et al. 2016).

At physiologic levels of mechanical loading, both stress and stretch demonstrated reduction with age. While reduction in stretch indicates stiffening, reduction in stress suggests that FPAs may remodel to reduce mechanical injury. Longitudinal changes with age were found to be more dramatic than those in the circumferential direction, but circumferential stress did decrease with age. This observation disagrees with previous data that suggested the artery remodels to maintain its homeostatic circumferential stress target (Humphrey et al. 2009; Kamenskiy et al. 2015). One possible explanation could be the fact that the reduction in circumferential physiologic stress observed in our study was not monotonic but rather fluctuated with age. This may be a result of the partitioning of data into age groups. Less stratified grouping could have resulted in the constant circumferential stress observed previously. Further studies are required to investigate the nature of these stress fluctuations.

Interestingly, muscular arteries were previously demonstrated to maintain their in vivo functional metrics with age (Benetos et al. 1993; Bortolotto et al. 1999). In particular, circumferential elastic modulus at physiologic pressure was not found to be affected by age in the femoral and radial arteries. This elastic modulus can be estimated by relating in vivo stress to stretch, both of which were shown to decrease with age by our current results. Similar slopes in Figure 6 B) and E) demonstrate that physiologic circumferential elastic modulus did not change with age (ρ = 0.04, p = 0.94). These results, along with findings of biaxial stiffening with age observed during mechanical testing, reveal that the FPA undergoes complex remodeling processes that involve morphologic changes as well as changes to the residual stresses that affect in vivo stress-stretch conditions.

The findings of this study should be interpreted within the context of its limitations. First and foremost, it does not differentiate between subjects with and without risk factors, such as diabetes, that likely play a significant role in arterial mechanics. Data on risk factors have been collected by our laboratory and this analysis will be the focus of future work. The second limitation is concerned with the phenomenological nature of the constitutive model used in this study. Though the model is based on the structural composition of the FPA and is straightforward for computational use, it does not explicitly consider intramural constituents, their volume fractions or interactions. More work is required to determine these additional parameters. The arterial wall in this study was assumed homogeneous, while histology obviously demonstrates otherwise. While layer-specific properties have been determined for some of our samples, separation of the layers is extremely difficult in young subjects and additional studies are required to understand whether properties obtained for individual layers can adequately describe the behavior of the composite arterial wall. Lastly, while elastin appears to play a major role in FPA mechanics and ageing, other matrix molecules such as collagen and versican, may also play significant roles. Detailed studies examining other artery wall constituents are required. While these limitations are being addressed, the results reported in this study may provide a better understanding of FPA pathophysiology and act as a stepping stone for future computational models of FPA behavior to assist in material and device development for personalized, precision medicine approaches.