Effect of Collagen Fiber Tortuosity Distribution on the Mechanical Response of Arterial Tissues

Abstract

This study investigated the effect of collagen fiber tortuosity distribution on the biomechanical failure and prefailure properties of arterial wall tissue. An in-silico model of the arterial wall was developed using data obtained from combined multiphoton microscopy imaging and uni-axial tensile testing. Layer-dependent properties were prescribed for collagen, elastin, and ground substance. Collagen fibers were modeled as discrete anisotropic elements, while elastin and ground substance were modeled as homogeneous isotropic components. Our parametric analysis, using a finite element approach, revealed that different parameters of collagen fibers tortuosity distribution significantly influence both prefailure and failure biomechanical properties. Increased fiber tortuosity improved the tissue strength whereas the dispersion in the tortuosity distribution reduced it. This study provides novel insights into the structural-mechanical interdependencies in arterial walls, offering potential targets for clinical assessments and therapeutic interventions aimed at mitigating rupture risks.

1 Introduction

Structural integrity and functional efficiency of arterial walls are crucial for maintaining hemodynamic stability of the blood vessels and ensuring effective blood flow throughout the body. However, vascular diseases, such as aneurysms and atherosclerosis, alter the components of the arterial wall as well as their organization. As a result, the maximum load carrying capability of the tissue, designated as tissue strength, can be compromised, potentially leading to wall rupture and associated major cardiovascular events [1,2]. Mechanical damage to the arterial wall due to medical treatments, such as angioplasty or due to physical trauma, such as during high impact accidents can also increase the chance of rupture [3,4]. Arterial rupture is a catastrophic event that can result in internal bleeding and, frequently, death or permanent disabilities [5]. In the case of cerebral aneurysms, rupture is fatal for approximately 45% of the patients [6].

The relationship between rupture risk and the organization of wall constituents is still poorly understood, impacting the ability of clinicians to effectively select between risky treatment and careful observation [7,8]. For example, current clinical protocols for preventing cerebral aneurysm rupture include clipping, endovascular coiling, and stenting, all of which carry substantial medical risks that can be greater than those of aneurysm ruptures [9,10]. Thus, there is a pressing need to understand arterial wall rupture from a mechanistic viewpoint to improve upon current clinical practice of rupture risk assessment. Moreover, such knowledge could provide targets for new pharmacological treatments aimed at enhancing the structural integrity of the artery wall.

From a biomechanical standpoint, rupture can occur whenever arterial wall stress focally exceeds the wall strength. Thus, to develop a mechanistic understanding of the artery rupture, elucidation of the relationship between tissue structure and its biomechanical failure properties is needed. As elastin and collagen fiber networks are the primary load bearing components of the vascular tissue, structural abnormalities of these networks are expected to directly modulate tissue biomechanical behavior [11–13]. Different features of the fiber network, such as fiber density, fiber tortuosity, and orientation distribution have been variously implicated toward vascular tissue strength alteration. While the role of fiber density and orientation distribution on tissue failure is fairly well understood due to vigorous research activities on these topics [14–16], the effect of fiber tortuosity on the biomechanical properties has yet to be fully elucidated. Furthermore, multiple imaging analysis studies have revealed aging and disease mediated alterations to fiber tortuosity [17–22], motivating further investigation into this feature to explore the impact of this feature on tissue biomechanical behavior [15,23,24]. Literature reports in this area have highlighted the significance of collagen fiber tortuosity distribution regarding the nonlinear J-shaped stress–stretch curve behavior but considered fibers without damage, and thus did not investigate the role of tortuosity on tissue failure properties [25–29]. In contrast, other studies have examined the effect of fiber damage behavior to the tissue properties, but they did not incorporate the fiber tortuosity distribution [30–35]. Thus, there is a pressing need to elucidate the role of fiber tortuosity distribution on the biomechanical behavior of cardiovascular tissues, especially its failure properties.

We hypothesized that the collagen fiber tortuosity distribution plays a pivotal role in influencing both failure and prefailure properties of arterial wall tissues. The workflow used to test our hypothesis is shown in Fig. 1. We employed a combined imaging, experimental and computational approach to create an image-based in silico model of arterial wall tissue. The computational model featured a representative volume element (RVE) that incorporated layer-specific collagen fiber architecture. The constitutive model for the wall was developed using data from multiphoton microscopy imaging of artery samples and uni-axial tensile tests. Subsequently, a comprehensive multivariable parametric study was undertaken to find the correlation between fiber tortuosity distribution and the failure properties of the arterial wall tissue under uni-axial loading in the circumferential direction. The parametric study involved finite element (FE) analysis on the RVE models with different tortuosity distribution parameters. The prefailure and failure properties of arterial wall tissue were obtained from stress–stretch (SS) curves arising from FE simulations of uni-axial loading. To evaluate our hypothesis, a correlation study was performed with the failure properties of the SS curves and the tortuosity distribution parameters. The results demonstrated that variations in fiber tortuosity distribution indeed have a significant impact on the biomechanical failure properties of arteries, a finding that has the potential to explain the relationship between pathological changes to the arterial wall and deleterious changes to tissue strength.

Fig. 1.

The computational workflow for representative arterial wall tissue model construction, model properties calibration using experimental tensile test data, model validation using confidence interval (CI) values, and parametric study on tortuosity distribution parameters. The tissue representative volume element (RVE) consists of three layers (i.e., internal elastic lamina (IEL), media, and adventitia). Fiber and matrix composition are also layer specific.

2 Methods

To investigate the biomechanical behavior of arterial wall tissue, we developed a computational model integrating bio-imaging data. The model was then calibrated and validated using experimental uni-axial tensile test data. A schematic of the workflow is presented in Fig. 1, with the different components briefly described below.

2.1 Biomechanical Testing Coupled With Bio-Imaging.

Biomechanical testing data and bio-imaging data were used from a prior study from our research group as detailed in Ref. [25]. That study employed our custom uni-axial mechanical testing device compatible with simultaneous multiphoton microscopy (MPM). Using this system, it was possible to perform non-destructive imaging of collagen fibers and elastin at different stretch levels (λ = 1, 1.4, 1.6, 1.7, 1.8, etc.) along the stress–stretch curve. Briefly, the experimental data of six carotid artery specimens (out of eight total) of New Zealand white rabbits with average weight of 4.7 ± 0.1 Kg were obtained from a previous study [25]. Strain controlled uni-axial testing in the circumferential direction was performed to obtain the stress–stretch curve. Bio-imaging data for collagen and elastin fibers were obtained for each specimen using an Olympus FV1000 MPE (Tokyo, Japan) equipped with a Spectra-Physics DeepSee Mai Tai Ti-Sapphire laser (Newport, Mountain View, CA). As elaborated on below, the MPM data sets were then postprocessed to obtain collagen fiber tortuosity distribution as a function of applied stretch.

2.2 Layer Specific Structurally Motivated Models of the Arterial Tissue.

The RVE model of the arterial wall was constructed in-silico for computational studies of the biomechanical behavior of the arterial tissue in both elastic and inelastic regimes. The RVE incorporated three wall layers: adventitia, media, and internal elastic lamina (IEL) composed of collagen, elastin, and ground substance. Geometric parameters for the RVE were inferred from MPM image analyses as shown in Tables 1 and 2.

Table 1.

Fixed parameters for all RVE samples

Input parameters for all samples	Media	Adventitia	IEL
Elastin volumetric density	44%	0%	100%
Collagen volumetric density	35.5%	39%	0%
Ground substance volumetric density	20.5%	61%	0%
Fiber dispersion	17.92 deg	23.84 deg	—
Fiber diameter	1.28 μm	1.85 μm	—
Peak fiber stretch (λp)	1.2	1.2	—
Maximum fiber stretch (λmax)	1.3	1.3	—

Table 2.

The sample specific input parameters for modeling each RVE tissue specimen. In all six samples, the adventitia MPM layers are assumed equal to that of media. The tortuosity distribution parameters are obtained by fitting the percentage of recruited fibers at different stretches into the cumulative Beta distribution function.

Sample specific input parameters	Sample 1	Sample 2	Sample 3	Sample 4	Sample 5	Sample 6
Adventitia thickness (μm)	66	74	46	76	76	70
Media thickness (μm)	66	74	46	76	76	70
IEL thickness (μm)	6	6	8	8	8	12
Total thickness (μm)	138	154	100	160	160	152
Media activation stretch λa1	1.7	1.45	1.4	1.5	1.8	1.8
Media distribution range Ra	0.5	0.69	0.3	0.37	0.62	0.55
Adventitia activation stretch λa1	1.8	1.54	1.48	1.59	1.91	1.91
Adventitia distribution range Ra	0.93	1.01	0.65	0.69	1.12	1.03
Beta PDF parameter: α	1.732	2.52	4.745	2.123	1.145	0.783
Beta PDF parameter: β	1.633	2.383	4.714	1.449	1.511	1.016

2.2.1 Areal Fraction of Wall Layers and Components of Extra Cellular Matrix.

In the prior study [25], MPM scanning of the tissue was performed from the lumenal side and generated stacks of 508 × 508 μm images spaced at 2 μm intervals, providing information about thickness of wall layers. In particular, in the present work, the areal dimension of the RVE was set to the MPM scanning window (i.e., 508 × 508 μm), while the total thickness of each wall layer was derived from the count of MPM images and the spacing between consecutive images. The average MPM imaging depth for carotid tissue was around 113 ± 20 μm, depending on the collagen density and therefore, the lumen scans stopped partially into the adventitial layer [25]. The medial thickness could therefore be determined directly from the MPM stacks. The end of the medial layer was defined by the cessation of elastin signal, which also corresponded with a change in collagen fiber architecture. The adventitial thickness had to be estimated and was set equal to that of the medial layer for this study based on a previous study [36].

The volumetric densities of elastin and collagen in each wall layer were determined using a previously developed protocol based on image thresholding with imagej software [37]. Briefly, the MPM scans from each layer were first converted to grayscale images and then an intensity threshold (60-100) was applied to obtain the areal density of elastin and collagen. The areal density for the adventitia layer was obtained from averaging the values across the partial MPM stacks (10–30) and prescribed as constant across the entire adventitia region. Representative MPM images of the media and adventitia regions, along with their thresholded counterparts, are shown in Fig. 1 available in the Supplemental Materials on the ASME Digital Collection. The measured collagen areal density was 35.5% (±2.97%, n = 6) in the media and 39% (±5.1%, n = 6) in the adventitia. The measured elastin areal density was 44% (±5.95%, n = 6) in the media, whereas the elastin content in the adventitia was negligible. Volumetric densities were assumed equal to areal densities obtained from MPM scans [28]. The residual volumetric density value was attributed to the ground substance. The mean areal densities for each material component in each layer were used in the modeling of the RVE.

Manual tracing of collagen fibers at different depths was performed to assess collagen fiber diameter for both media and adventitia regions. Width of the fibers in terms of number of pixels was recorded along the traced paths, yielding an average diameter of 1.28 and 1.85 μm for the media and adventitia collagen fibers, respectively which is consistent with findings of Ref. [38].

2.2.2 Fiber Orientation Distribution.

The orientation distribution of collagen fibers was measured from the image analysis of the MPM images using ct-fire software (Laboratory for Optical and Computational Instrumentation, University of Wisconsin-Madison, WI) [39]. The fiber dispersion for each layer (media and adventitia) were obtained using CT-Fire which traces prominently visible fibers in MPM scans and calculates the fiber orientation dispersion about the circumferential direction in form of standard deviation of the fiber angles. This process was applied to both adventitia and media layers individually for all samples. The average fiber dispersion calculated from all six samples for media and adventitia was 17.92 deg (±3.49 deg, n = 6) and 23.84 deg (±2.55 deg, n = 6), respectively.

2.2.3 Fiber Tortuosity Distribution

2.2.3.1 Medial Layer.

As previously reported in Ref. [25], medial collagen fibers show a distribution of tortuosity, resulting in a distribution of the probability density function of the activation (recruitment) stretch distribution d( ${\lambda }_a$). Prior work has demonstrated that arterial collagen fibers start to be recruited at a finite strain [25]. We denoted the recruitment stretch of the first recruited fiber as ${\lambda }_{a1}$, and used $R_a$ to denote the range between first and last fiber activation stretches Eq. (1). We represent the shape function of the fiber recruitment distribution using a Beta probability distribution function (PDF). The recruitment distribution was mathematically modeled by shifting and scaling the Beta PDF using ${\lambda }_{a1}$ and $R_a$ parameters, respectively, generating the typical fiber recruitment distribution of arterial tissue as shown in Fig. 2

$$d\left({\lambda }_a\right)={\lambda }_{a1}+R_a\times \mathrm{Beta}\left(\alpha ,\beta \right)$$ (1)

Fig. 2.

(a) Beta distribution function with parameters (α, β) along with average and full width at half maximum (FWHM) properties. (b) Beta distribution shifted and scaled to fit collagen fiber tortuosity distribution. Original Beta distribution properties are denoted with an “*” which is removed in the corresponding mapped parameters.

where α and β are two positive parameters that control the shape of the Beta distribution. Both parameters form a conjugate pair, influencing the distribution symmetrically. Increasing α tilts the distribution curve to the right, while increasing β tilts it to the left. Exchanging α and β produces a mirror image of the distribution curve, reflecting around the center of the range [26]. The average and the dispersion of Beta PDF were also obtained as these parameters will be useful for interpreting the impact of tortuosity distribution on biomechanical behavior. There are different ways of calculating average and the dispersion of a distribution function. In this study, the arithmetic mean $({\lambda }_{a\_m}^{*}=\frac{\alpha }{\alpha +\beta })$ and full width at half maximum (FWHM) ( $W_a^{*}$) was utilized for calculating the average and dispersion of Beta PDF, respectively. These parameters correspond to ${\lambda }_{a\_m}$ and $W_a$ in the shifted and scaled physical curve as per Eqs. (2) and (3) (Fig. 2).

$${\lambda }_{a\_m}={\lambda }_{a1}+R_a\times \left(\frac{\alpha }{\alpha +\beta }\right)$$ (2)

$$W_a=R_a\times {\mathrm{W}}_a^{\mathrm{*}}$$ (3)

For this study, the probability distribution of recruitment stretch was measured from the previously obtained MPM data sets obtained at different stretch levels [25]. Briefly, the tortuosity was defined as the ratio of the arc length and minimum distance between the endpoints of a fiber. The tortuosity values for individual fibers were measured in stacks of two-dimensional MPM images using the automated filament function in imaris software (Bitplane, Switzerland). A fiber was considered recruited (or activated) when the tortuosity was less than 1.02. The procedure for obtaining the experimental tortuosity data was explained in detail in Refs. [25,40]. The fraction of recruited fibers at each stretch value was used to generate a cumulative distribution function for recruitment stretch. Therefore, ${\lambda }_{a1}$ and $R_a$ could be determined directly from the tortuosity data, while the values of α and β were obtained by fitting the experimental cumulative distribution function (CDF) data points to the CDF of Beta distribution for each sample using the Levenberg-Marquardt nonlinear least squares method in matlab (R2022a, MathWorks Inc., Natick, MA) as shown in Fig. 2 available in the Supplemental Materials on the ASME Digital Collection.

2.2.3.2 Adventitial Layer.

The uni-axial testing of the tissue specimens was not performed to strains at which adventitial collagen fibers were recruited, precluding assessment of the recruitment distribution function for adventitial fibers in the same manner used for medial fibers. Consequently, in this work we posited the adventitia and medial fibers share identical beta distribution function parameters. We then estimated the values for ${\lambda }_{a1}$ and $R_a$ in the adventitia collagen fibers as follows. The tortuosity for a given adventitial fiber was mapped back to the reference configuration under the uni-axial deformation and then mapped forward under the uni-axial deformation to estimate the stretch where it was recruited, ${\lambda }_{a1}$. Then ${\lambda }_{a1}$ and ${\lambda }_{a\_\max }$ ( ${\lambda }_{a\_\max }={\lambda }_{a1}+R_a$) were determined as the minimum and maximum values of ${\lambda }_{a1}$ over the specimen. It was found that, for adventitial collagen, ${\lambda }_{a1}$ and ${\lambda }_{a\_\max }$ are 1.06 and 1.25 times those of media fibers, respectively. Due to the limited number of MPM scans that covered the adventitia region, these ratios were evaluated in Sample 1 and applied across all specimens.

2.3 Finite Element Analysis

2.3.1 Meshing.

After creating the RVE volume in Cubit (Coreform LLC, Orem, UT), it was meshed with hexahedral elements. The mesh was segmented with a 10 × 10 grid in the areal direction, and five segmentations were assigned in the thickness direction: two each for the adventitia and media regions and one for the IEL region. Consequently, the total number of nodes and elements in the continuum component of the model were 726 and 500, respectively. The ground substance and elastin composition were varied across the five subdivisions in the thickness direction, applying the rule of mixtures to ascertain the composite material properties for each subdivision.

The collagen fibers for each tissue specimen were modeled as one-dimensional rod elements. These fibers were modeled and oriented circumferentially with the axial dispersion defined in Sec. 2.2.2. Collagen fibers were modeled across each layer in the RVE uniformly by selecting a random point within the layer and an angle from the fiber orientation distribution to extend the fiber to the boundary. The number of fibers in each wall layer was calculated to be consistent with the RVE dimensions, fiber diameter, and fiber density. For both layers, the fiber tortuosity values were generated from the measured tortuosity distribution function (Eq. (1)) and then those values were assigned to discrete fibers of its respective layer.

2.3.2 Fiber-Reinforced Finite Element Model.

In this study, a fiber-reinforced finite element method was implemented for all simulations that is extensively described in our earlier works [28,30,33]. In this approach, the microstructure of fibrous material is represented as discrete one-dimensional rod elements network that are embedded within a three-dimensional continuum at the subtissue level. At the specimen level, the uni-axial tensile testing of tissue was simulated by upscaling the information obtained at the lower length scale. Affine deformation was ensured in the model throughout the domain with no-slip condition at the fiber-continuum interface. The continuum non-fibrous component was modeled with a two-parameter isotropic homogeneous nearly incompressible neo-Hookean material (Eq. (4)).

$$\Psi =0.5\mu \left(J^{-\frac{2}{3}}{I}_1-3\right)+0.5K{\left(J-1\right)}^2$$ (4)

where Ψ denotes the strain energy per unit volume, μ represents the shear modulus, J denotes the determinant of the deformation gradient, K is a penalty parameter that enforces the incompressibility condition, and I₁ refers to the first invariant of the right Cauchy-Green deformation tensor. The constitutive model for the collagen fibers was defined such that fiber stress ( $P_f$) increased linearly with fiber stretch ( ${\lambda }_f$) beyond its assigned tortuosity value ${\lambda }_a$ with elastic modulus of E_f up to the peak fiber stretch value ( ${\lambda }_p$) at which the fiber has the maximum strength (Eq. (5)), after which it underwent a subsequent linear decline to zero due to fiber damage as shown in Fig. 3. The constitutive relationship of an individual fiber is linearly dependent on stretch; however, the overall material model remains nonlinear. This approach is consistent with previous studies that have demonstrated a nearly linear stress–stretch relationship for individual collagen fibers [27,33,41]. The individual fiber will only start bearing load when its fiber stretch ${\lambda }_f$ is greater than its tortuosity value ( ${\lambda }_a)$ irrespective of applied tissue stretch λ. The relationship between fiber stretch and applied tissue stretch is dependent on the fiber angle (in the unloaded configuration) from the loading direction ( ${\lambda }_f=\sqrt{{\lambda }^2{\text{cos}}^2\mathrm{\theta }+{{\lambda }^{-1}\mathrm{s}\mathrm{in}}^2\mathrm{\theta }}$). In this model, a multiplicative decomposition of fiber stretch ( ${\lambda }_f$) has been applied between the true fiber stretch ( ${\lambda }_t$) and activation stretch ( ${\lambda }_a$) (i.e., ${\lambda }_f={\lambda }_a{\lambda }_t$) [23,25]. The fibers had no load-bearing capability below their respective activation stretch.

$$P_f=\left\{\begin{array}{cc}0& \mathrm{i}\mathrm{f}\hspace{0.17em}{\lambda }_f<{\lambda }_a\Rightarrow {\lambda }_t<1\\ E_f\left({\lambda }_f/{\lambda }_a-1\right)S& \mathrm{i}\mathrm{f}\hspace{0.17em}{\lambda }_f\ge {\lambda }_a\Rightarrow {\lambda }_t\ge 1\end{array}\right.$$ (5)

Fig. 3.

Stress-stretch constitutive relation of a single collagen fiber. Fiber will bear load for stretches greater than λ_a where λ_a denotes the activation stretch of a given fiber. The slope of elastic region is the fiber stiffness (E_f). In this relation, fibers start to damage when the fiber stretch surpasses λ_p and full fail at λ_max.

In Eq. (5), S ∈ [0,1] is an internal damage parameter for the collagen fibers that enforces the postpeak decline in fiber stress to zero as the fiber stretches beyond ${\lambda }_p$ and experiences subfailure damage and eventually fails completely. S takes the value S_init = 1 up to ${\lambda }_p$ and if ${\lambda }_f$ is increased beyond ${\lambda }_p$ up to the failure stretch of the fiber ( ${\lambda }_{\max }$), S decreases to zero. Beyond ${\lambda }_{\max }$, the fiber is fully damaged (S = 0) and cannot bear any load afterwards, even under potential unloading and reloading. The evolution of S is described based on our previous work [28,30] in the following equation:

$$S=\min \left[S_{\min },〈1-\frac{{\lambda }_t-{\lambda }_p}{{\lambda }_{\max }-{\lambda }_p}〉\right]$$ (6)

where S_min ≥ 0 is the minimum value of S during the simulation and the operator 〈x〉 is defined in the following equation:

$$〈x〉=\left\{\begin{array}{cc}x& \mathrm{i}\mathrm{f}\hspace{0.17em}x>0\\ 0& \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\mathrm{otherwise}\end{array}\right.$$ (7)

In all FE simulations ${\lambda }_p$ was set to 1.2 based on the experimental study [41] and ${\lambda }_{\max }$ was set 0.1 higher than that of ${\lambda }_p$ (i.e., ${\lambda }_{\max }=1.3$) [30]. Both fiber damage parameters were defined relative to the true fiber stretch ${(\lambda }_t)$.

2.3.3 Analysis and Post-Processing.

To simulate uni-axial tensile testing, finite element mesh nodes on the left face of the specimen were fixed in the x-direction (i.e., circumferential direction), while nodes on the right face were prescribed with a uniform stretch of 2.3 over 2000 load steps. The number of load steps was chosen to ensure temporal convergence of the simulation. Similarly, nodes on the side and bottom faces were fixed in the longitudinal and radial directions, respectively. Reaction forces were recorded at all load steps to compute the average Cauchy stress which will finally output the stress–stretch curve. The current cross-sectional area was approximated by dividing the initial cross-sectional area of the sample by the current stretch, as performed in processing the experimental data. All of the FE results were analyzed and visualized in Paraview (v5.12, Kitware Inc., Clifton Park, NY).

2.4 Model Calibration and Validation.

The material properties of the RVE constituents were the unknown parameters of our model and needed to be calibrated. The finite element generated stress–stretch curve was fitted against its experimental uni-axial tensile test data to regress the material properties of each sample. Qualitatively, the elastic regime of the tensile testing stress–stretch curve of each specimen can be divided into three regimes: low stiffness, transition, and high stiffness regimes. The low stiffness regime is known to be dominated by the non-collagenous material as collagen fibers are yet to be recruited, and a high stiffness regime where a substantial recruitment of medial fibers leads to an elevated stress response [42]. Therefore, through regression analysis, two material properties were calibrated: the elastin shear modulus from the low stiffness regime and medial collagen fiber stiffness from the slope of high stiffness regime. The shear modulus of ground substance was set to 16.07 kPa for all specimens [20]. The resulting calibrated properties for elastin and collagen for each specimen are tabulated in Table 3 along with literature reported values [41,43–47], and the correlated SS curves with experimental data for all samples are depicted in Fig. 3 available in the Supplemental Materials on the ASME Digital Collection.

Table 3.

The calibrated parameters of each tissue specimen with literature reported values

Regressed parameters	Sample 1	Sample 2	Sample 3	Sample 4	Sample 5	Sample 6	Mean ± standard dev	Literature reported values
Elastin shear modulus μ (kPa)	300	200	300	300	300	350	292 ± 49.16	290–410 [43,44]
Collagen fiber stiffness Ef (MPa)	120	130	130	300	150	140	162 ± 68.53	23–400 [41,45–47]

After the regression of material properties of elastin and collagen, the RVE model underwent validation to ascertain its capacity to replicate the biomechanical behavior of arterial wall tissue, especially the mechanical response of the fiber network. For this purpose, one random specimen was selected for the validation set and the remaining samples were classified as a training data set. The 95% confidence interval of the regressed material parameters of the training set specimens were calculated and the finite element analysis was performed on the RVE model of the testing specimen with lower and upper limits of the material properties achieved by the confidence interval analysis. RVE model validation was achieved as the regressed SS curve of the testing sample fell within the bounds produced by these confidence interval curves. Figure 4 illustrates both the experimental and FE generated SS curve from the confidence interval extreme property responses of Sample 3. This process was repeated for all other samples as well as shown in Fig. 4 available in the Supplemental Materials on the ASME Digital Collection.

Fig. 4.

Stress-stretch plot of Sample 3 with experimental and confidence interval curves. The RVE model was validated by first calculating the confidence interval (CI) properties of Sample 3 based on elastin and collagen of the training dataset to obtain the confidence interval region. The experimental stress–stretch curve of Sample 3 aligns within the bounds of the confidence interval region, validating our RVE model.

2.5 Statistical Analysis.

In the postprocessing of FE results of the parametric study, the relationship between tortuosity parameters and the prefailure and failure properties of the tissue was determined using the Pearson's correlation coefficient (r), Spearman's rank correlation coefficient (ρ), and the sensitivity coefficient (or slope) (m). All statistical analyses and data sampling of fiber tortuosity distribution for the parametric study were conducted using matlab (R2022a, MathWorks Inc., Natick, MA).

3 Results

A comprehensive parametric study was conducted to investigate the impact of individual tortuosity parameters on the stress–stretch behavior of the arterial tissue. Our analysis focused on three key attributes of the stress–stretch curve shown in Fig. 5(a): high stiffness slope or tangent modulus, failure stress or strength, and the corresponding tissue failure stretch. Here, we define the failure stress as the peak or ultimate stress for the Cauchy stress–stretch curve.

Fig. 5.

(a) The stress–stretch curve for the representative volume element of the tissue with different applied stretch (λ) points selected in different loading regimes, denoted from (i) to (v). (b) Visualization of the evolution of matrix stress, fiber stress, and fiber damage. Point (i) lies in the low stiffness regime. The matrix exhibits uniformly low stress. Fibers are not yet loaded in this regime and therefore there is zero fiber stress and damage. Point (ii) lies in the transition regime. The matrix shows moderately increased stress level, and some fibers are recruited and have non-zero stress. Point (iii) lies in the high stiffness regime. There is a substantial increase in matrix and fiber stress as more fibers are recruited, leading to heterogeneity in the matrix stress. Point (iv) occurs at the peak (ultimate) stress level where some fiber damage is seen. Complete fiber failure is also visible in some fibers. Point (v) lies in the postfailure regime where fiber damage and failure are more pronounced.

3.1 Stress and Damage Evolution in the Tissue.

We examined the progression of Cauchy stress and damage within the tissue's collagen and non-collagenous components (or Matrix) at various stretch points along the SS curve, as demonstrated in Fig. 5. Figure 5(a) presents the SS curve of the tissue obtained using the mean values of the tortuosity distribution parameters, marked with five distinct deformation points (i–v) across the curve. Point (i) lies in the low stiffness regime before the activation stretch. Point (ii) lies in the transition regime of the curve; Point (iii) falls within the region of high stiffness slope. Points (iv) and (v) are situated at the peak and postpeak locations of the stress–stretch curve, respectively.

Figure 5(b) displays the evolution of matrix and fiber stress and fiber damage at those stretch points. At point (i), the tissue matrix stress in the media region remains homogenous and smoothly varies with tissue stretch up to activation stretch of the tortuosity distribution after which it starts showing some heterogeneity due to the matrix-fiber interaction. The second column of Fig. 5(b) illustrates the evolution of stress in the media collagen fibers. At Point (i), there is no fiber stress because the tissue stretch is below the activation point; at Point (ii), some fibers begin to bear the load; by point (iii), a significant number of fibers are engaged and maximum stress is observed at Point (iv), the peak of the SS curve. Finally, the third column of Fig. 5(b) visualizes fiber damage progression, indicating minimal fiber damage up to Point (iii). However, from Point (iv) to (v), there is a marked increase in fiber damage, culminating in a postpeak decline in the tissue's overall Cauchy stress.

3.2 Effect of Activation Stretch on Stress–Stretch Curve.

The probability density function of the fiber tortuosity distribution in the parametric study of activation stretch ( ${\lambda }_{a1}$) is shown in Fig. 6(a), where all other tortuosity parameters were kept constant for this set of simulations. The effect of activation stretch on the overall constitutive behavior of the carotid tissues is presented in Fig. 6(b). This figure shows the simulated stress–stretch curves maintained the J-shape typical to soft tissues for the entire range of activation stretch (1.35–1.90) considered in this study. Increasing ${\lambda }_{a1}$ resulted in a rise of both the failure stretch and strength of the tissue, while the high stiffness slope seems to be largely unaffected. There was an increase in tissue failure stretch and strength from 1.85 ± 0.01 to 2.49 ± 0.005 and 4.64 ± 0.14 to 7.03 ± 0.23 MPa respectively as ${\lambda }_{a1}$ was increased. The high stiffness slope changed from 17.02 ± 0.10 to 20.46 ± 0.29 MPa for the range of ${\lambda }_{a1}$ considered. Figure 6(c) plots the variation of failure stretch with the activation stretch and reveals a linear relationship between these two parameters (r = 0.999, ρ = 1, m = 1.17). Figure 6(d) reveals the strength of the tissue also exhibited a linear relationship with the ${\lambda }_{a1}$ (r = 0.999, ρ = 1, m = 4.26 MPa). Figure 6(e) shows the monotonic relationship between the tangent modulus and ${\lambda }_{a1}$ (r = 0.993, ρ = 1, m = 6.15 MPa).

Fig. 6.

Parametric study of the role of activation stretch of the first fiber (λ_a1) (initiation of recruitment): (a) The tortuosity distribution plots for varying λ_a1. (b) Dependence of stress–stretch plots on λ_a1. (c)–(e) Dependence of failure stretch, failure stress, and high stretch slope on λ_a1.

3.3 Effect of Range of Tortuosity Distribution on Stress–Stretch Curve.

The probability density function of fiber tortuosity is shown in Fig. 7(a) where only the range of the recruitment stretch distribution ( $R_a$) is varied, while other tortuosity parameters were kept constant. The effect of the $R_a$ on the constitutive behavior of the tissue is presented in Fig. 7(b). Similar to the case of changing ${\lambda }_{a1}$, the SS curves presented in the Fig. 7(b) maintained the J-shape curve typical to soft tissues for the entire span of the $R_a$ (0.3–0.7) considered. Increasing $R_a$ led to a higher failure stretch and a marginal reduction in failure stress, along with a diminishing high stiffness slope (Fig. 7(c)). Figure 7(d) shows that there was an increase in failure stretch of tissue from 2.035 ± 0.01 to 2.27 ± 0.01 as $R_a$ increases. It further reveals a linear variation of failure stretch with the $R_a$ (r = 0.999, ρ = 1, m = 0.55). There was a change in tissue strength from 5.89 ± 0.15 to 5.61 ± 0.21 as $R_a$ increases. $R_a$ showed a negative linear dependency with the strength of the tissue but exhibited poor sensitivity compared to ${\lambda }_{a1}$ (r = −0.890, ρ = −0.8, m = −0.75 MPa). Similarly, Fig. 6(e) shows high stiffness slope changed from 22.06 ± 0.30 to 15.03 ± 0.18 MPa for the range of recruitment stretch distribution with inverse linear relationship (r = −0.997, ρ = −1, m = −17.70 MPa).

Fig. 7.

Parametric study of the role of range of recruitment stretch (R_a). (a) The tortuosity distribution plots for varying R_a. (b) Dependence of stress–stretch plots on R_a. (c)–(e) Dependence of failure stretch, failure stress, and high stretch slope on R_a.

3.4 Effect of Shape Function of Tortuosity Distribution.

Figures 8(a) and 9(a) show the probability density function of tortuosity distribution with varying α and β parameter, respectively. As illustrated in Fig. 8(b), an increase in the α parameter of the recruitment distribution resulted in a rightward shift in the SS curve, indicating an increment in the failure stretch value. Figure 8(c) shows that there was a change in tissue failure stretch from 1.96 ± 0.005 to 2.23 ± 0.003 with a monotonic relationship as α parameter increased (r = 0.970, ρ = 1, m = 0.066). Interestingly in Fig. 8(d), the failure stress exhibited a non-monotonic relationship with α; it initially decreased from 5.98 ± 0.19 to 5.77 ± 0.18 MPa and then increased to 5.89 ± 0.09 MPa with higher α values. Concurrently, there was a noticeable decline in the slope of the high stiffness regime from 23.84 ± 0.32 to 19.09 ± 0.17 MPa (r = −0.773, ρ = −0.6, m = −1.039) as shown in Fig. 8(e). Figure 9(b) presents the SS curves resulting from the parametric study on the β parameter of the recruitment distribution. In contrast to the α parameter, β displayed an inverse effect; the high stiffness slope intensified as β increased, whereas the failure stretches decreased. Figure 9(c) shows tissue failure stretch changed from 2.28 ± 0.005 to 2.05 ± 0.006 with linear relationship like α but in the inverse direction (r = −0.970, ρ = −1, m = −0.0594). Moreover, the failure stress once again exhibited a non-monotonic trend; it initially decreased from 5.82 ± 0.09 to 5.67 ± 0.10 MPa and then increased to 5.93 ± 0.22 MPa (Fig. 9(d)). There is also a significant increment in the slope of high stiffness regime from 17.11 ± 0.17 to 21.83 ± 0.38 MPa as β increases (r = 0.995, ρ = 1, m = 1.338).

Fig. 8.

Parametric study of the role of the Beta distribution parameter α. (a) The tortuosity distribution plots for varying α parameter. (b) Dependence of stress–stretch plots on α. (c)–(e) Dependence of failure stretch, failure stress, and high stretch slope on α.

Fig. 9.

Parametric study of the role of Beta distribution parameter β. (a) The tortuosity distribution plots for varying β parameter. (b) Dependence of stress–stretch plots on β. (c)–(e) Dependence of failure stretch, failure stress, and high stretch slope on β.

4 Discussion

Our study elucidated for the first time the role of collagen fiber tortuosity distribution on the failure properties of arterial wall tissue. By integrating results from MPM imaging, uni-axial testing, and finite element modeling, we have demonstrated that variations in the tortuosity distribution parameters significantly affect the stress–stretch behavior of the arterial tissue (Figs. 6–9). The tortuosity of collagen fibers is a fundamental structural feature of collagen network that can be altered by the aging and progression of disease [17,20,21]. Hill et al. [25] have demonstrated the importance of modeling collagen fiber tortuosity distribution rather than a single fiber tortuosity value. Prior studies have shown that importance of fiber tortuosity distribution in terms of nonlinear stiffening behavior and the shifting of characteristic J-shaped stress–stretch curve along the stretch axis of various soft tissues in addition to arteries [26,27], including brain matter [23], bladder [42,48] and cornea [24]. However, the impact of collagen tortuosity on the prefailure and particularly failure biomechanics of these tissues was not studied due to the absence of fiber failure considerations in their FE models.

4.1 Salient Features of Tortuosity Distribution Curve.

When interpreting how the form of the probability distribution function drives the mechanical response, it is useful to consider two particular features of tortuosity distribution function: the mean fiber activation stretch ( ${\lambda }_{a\_m}$), and the measure of the variability or dispersion (full width at half maximum) of the recruitment stretch distribution ( $W_a$) because those features can be obtained irrespective of chosen probability distribution function. The relationship between these two parameters and ( ${\lambda }_{a1}{,\hspace{0.17em}R}_a,\hspace{0.17em}\alpha ,\hspace{0.17em}\beta$) is stated in Eqs. (2) and (3) and illustrated in Fig. 2. An increase in ${\lambda }_{a1}$ enhances only mean recruitment stretch, ( ${\lambda }_{a\_m}$), (Fig. 6(a)), while an increase in $R_a$ elevates both the ${\lambda }_{a\_m}$ and $W_a$, (Fig. 7(a)). Increasing the α parameter in the Beta distribution causes a skew of ${\lambda }_{a\_m}$ toward higher values, with an initial rise in $W_a$ when α approaches β, that decreases as α surpasses β (Fig. 8(a)). In contrast, increasing the β parameter skews the distribution toward lower stretch values, following a similar dispersion trend as observed with changes in α (Fig. 9(a)).

We now consider the observed trends for tissue failure and prefailure properties (Figs. 6–9) in the context of parameters ${\lambda }_{a\_m}$ and $W_a$. In doing so, it is useful to consider the overall tissue stress as an additive decomposition of stresses in non-collagenous component and collagen fibers. An individual fiber starts bearing load at its activation stretch and subsequently bears load up to the fiber failure stretch (at the maximum fiber stress) where it will start failing. With an increase in the mean activation stretch ( ${\lambda }_{a\_m}$), fibers recruit later and therefore fail at higher values of tissue stretch. The maximum stress contribution from an individual fiber will remain unchanged, irrespective of its tortuosity. However, the stress contribution from the matrix at these higher tissue stretches will be elevated, which will lead to higher overall stress at failure point. Moreover, the dispersion of tortuosity distribution ( $W_a$) among fibers affects the synchronization of fiber recruitment. If all fibers have identical recruitment stretch, they all will engage and bear load simultaneously leading to higher total fiber stress. Conversely, a higher dispersion in fiber recruitment distribution in the tissue leads to a scenario where, for some ranges of stretch, some fibers will be load-bearing while other fibers are either not yet recruited or have already failed, resulting in a lower overall stress born by the fibers.

4.2 Effect of Tortuosity Parameters on Failure Properties.

In this work, it was shown that increasing activation stretch ( ${\lambda }_{a1}$) alone led to a rise in both the failure stretch and stress of the tissue by 34.59% and 51.50%, respectively (Figs. 6(b)–6(d)). Notably, this increase in tissue failure stress was due exclusively to an increase in activation stretch without increasing fiber density or fiber strength, properties already known to increase tissue strength [14,49,50]. The observed increase in failure stress can be attributed to the higher stress contribution of the matrix to the overall stress at higher stretches. This effect is reflected in the Cauchy stress ( ${\sigma }_{\mathrm{matrix}}$) of the Neo-Hookean matrix material under uni-axial tensile stretch ( $\lambda )$ described by equation: ${\sigma }_{\mathrm{matrix}}={\mu }_{\mathrm{matrix}}({\lambda }^2-1/\lambda )$.

In a second series of simulations, the range parameter ( $R_a$) of the recruitment distribution was varied keeping other parameters constant, which caused less synchronized engagement of collagen fibers, resulting in a rise in the tissue failure stretch by 11.55% and fall in tissue strength by 4.75% (Fig. 7). Here, even though ${\lambda }_{a1}$ was held fixed, both ${\lambda }_{a\_m}$ and $W_a$ increased (Eqs. (2) and (3)). As a result, two competing effects occurred. As just noted, increasing ${\lambda }_{a\_m}$ tends to increase the failure stretch and stress. However, with diminished synchronization of the recruitment of collagen (increasing $W_a)$, the collective stress born by the fibers will be diminished when a fraction of the fibers are either not recruited or have already failed. Therefore, despite an increase in matrix stress associated with the increased failure stretch, Fig. 7(b), the failure stress decreased modestly.

The Beta distribution shape parameters (α, β) play a crucial role in modulating both the average fiber tortuosity ( ${\lambda }_{a\_m}$) and the tortuosity dispersion ( $W_a$), which in turn impact tissue failure properties (Figs. 8 and 9). A closer examination of these results, through the lens of ${\lambda }_{a\_m}$ and $W_a$, provides deeper physical insights into the mechanical behavior of the tissue. For instance, increasing α from 1.774 to 4.745 (Fig. 8(a)) leads to an increase in the ${\lambda }_{a\_m}$, while simultaneously varying $W_a$. This change explains the increase in failure stretch and non-monotonic response of the failure stress of the tissue (Fig. 8). Conversely, increasing β from 1.941 to 4.714 results in a monotonic decrease in ${\lambda }_{a\_m}$ with varying $W_a$, leading to the expected reduction in failure stretch, as well as a non-monotonic behavior of failure stress (Fig. 9). This non-monotonic behavior in failure stress occurs due to the competing effects of ${\lambda }_{a\_m}$ and $W_a$ of the recruitment distribution.

These results reinforce the importance of average fiber tortuosity ( ${\lambda }_{a\_m}$) and its dispersion ( $W_a$) as key factors in understanding how the recruitment distribution influences the tissue's failure characteristics. The interplay between these parameters and the tortuosity distribution parameters ( ${\lambda }_{a1},\hspace{0.17em}{R}_a,\hspace{0.17em}$α, β) provides a novel framework for explaining the mechanical behavior of collagen fibers under loading. Importantly, these findings represent the first report that highlights the significance of ${\lambda }_{a\_m}$ and $W_a$ in predicting tissue failure based on fiber recruitment characteristics.

4.3 Effect of Tortuosity Parameters on Prefailure Properties.

Our study reveals the parameters governing tortuosity distribution significantly influenced the prefailure characteristics of arterial wall tissue, notably the high stiffness slope. This slope increased sharply when a substantial proportion of fibers became activated, greatly enhancing the tissue's stress-bearing capacity. It was found that increasing the ${\lambda }_{a1}$ parameter enhanced the high stiffness slope, while the $R_a$ parameter lowered it (Figs. 6(e) and 7(e)). Similarly, the shape parameters of the Beta distribution impacted the high stiffness slope; the α parameter reduced it, whereas the β parameter had the opposite effect (Figs. 8(e) and 9(e)).

Weisbecker et al. [26] examined the effect of tortuosity distribution parameters under the assumption of no fiber failure which resulted in same tissue stress and the high stiffness slope beyond the stretch points where all fibers had already recruited. That study also revealed that the interval of distribution function—similar to range parameter ( $R_a$) in our study—affects the slenderness of the distribution (analogous to dispersion in tortuosity distribution in our study) and resulted in a lower initial stiffness of the tissue which echoes with our findings of high stiffness slope.

Additionally, our observations indicated that stress distribution within the arterial matrix and fibers during prefailure states changes with tortuosity of collagen fibers. Initially, matrix stress was uniform until the tissue stretch reached the activation stretch point of the tortuosity distribution ( ${\lambda }_{a1}$), after which stress heterogeneity started to emerge (Fig. 5). This local stress variation can be attributed to the local differences in the geometric arrangement of the collagen fiber network, including variations in fiber orientation and tortuosity. At the microscopic level, fibers with different orientations and tortuosity engage in the load-bearing process at different stretch levels, contributing to local stress heterogeneity. For instance, fibers with a larger orientation angle or higher tortuosity may not participate in load bearing until higher applied stretches, while more aligned fibers with lower tortuosity begin bearing load at lower stretches. This difference in recruitment behavior across fibers leads to localized stress concentrations, resulting in heterogeneity in the overall stress distribution.

This observation is consistent with findings from previous studies [27,30,51], which also noted that local variations in fiber properties, such as orientation and tortuosity, can significantly influence stress heterogeneity at the tissue level. The discrete fiber-reinforced model used in this study captures these effects, as demonstrated in Fig. 5(b), where the local stress variations due to microscopic fiber properties are illustrated, further supporting the importance of local fiber mechanics in contributing to the overall mechanical behavior of the tissue.

4.4 Limitations of Current Study and Future Directions.

Our study was subject to certain limitations. The model calibration was constrained by the scope of the MPM image dataset from a prior work in Ref. [25]. That study provided extensive data on medial fiber recruitment in the elastic regime. However, as that study was focused on medial collagen fibers, imaging was performed from the lumenal side. Due to optical constraints, the lumenal side MPM imaging did not reach the adventitial layer for most specimens. Consequently, we lacked comprehensive MPM images for the adventitia region. To address this, the RVE was modeled by assuming that the adventitia thickness was equivalent to that of the media thickness. While this choice was supported by cross-sectional imaging studies for select samples, it was not quantified on a case specific basis.

Additionally, while the combined MPM-UT technique enabled the extraction of fiber recruitment distributions at different stretch levels, the uni-axial tensile tests from the existing study were not carried to the point of adventitial fiber recruitment. Rather, the minimum and maximum recruitment stretches for the adventitial layer were inferred from tortuosity measurements and we assumed the adventitial tortuosity distribution shape profile was the same as in the medial layer. The results in our work provide a great motivation not only for future experimental studies on adventitial collagen fiber recruitment distribution, but also studies exploring the relationship between adventitial recruitment distribution parameters and tissue failure properties.

Moreover, the uni-axial tensile tests were only performed in the circumferential direction. As the collagen fibers are predominantly aligned circumferentially, they will not engage significantly even under higher applied stretches in the axial direction. Therefore, the mechanical response in the axial direction would primarily be governed by the non-collagenous components, rather than the collagen fibers. Future investigations could include tensile testing in the axial direction, to further explore this point and complement our findings in the circumferential direction. Furthermore, the experimental uni-axial tensile tests were only performed in the elastic regime, and we do not have experimental data extending to tissue failure. Having experimental data that captures the failure response would significantly strengthen our RVE model calibration and validation, providing further justification for the choice of failure parameters, such as ${\lambda }_p$ & ${\lambda }_{\max }$. Future studies could conduct uni-axial tensile tests up to failure to extract these parameters and generate patient-specific data in the failure regime.

In this work, we further assumed that fiber density and orientation distribution were homogeneous across each layer of the arterial wall. In future studies, it could be of interest to explore how variations in fiber architecture across each layer influence the biomechanical behavior of the tissue. The average fiber orientation considered in this work was circumferentially aligned with a relatively narrow distribution of fiber angle. With increasing fiber angle distribution, the interpretation of the role of fiber recruitment will become more complex. For example, even if the fibers had a single tortuosity in the unloaded configuration, a dispersion in angle would alter the synchronization of the fiber recruitment. Moreover, we have used the average collagen fiber density across all samples in both layers.

In this study, we also assumed affine deformation between the matrix and collagen fibers. While previous studies have shown that the assumption of affine deformation reasonably predicts macroscopic behavior, it may not fully capture microscopic behavior, such as the potential slippage between fibers and the ground substance [52,53]. Non-affine deformations could play a role in the mechanical response, particularly in failure scenarios. Future studies will aim to develop a high-fidelity model that incorporates potential slippage between the matrix and fibers to provide a more accurate representation of tissue mechanics, especially under extreme loading conditions.

4.5 Importance for Future Clinical Directions.

This study findings on impact of fiber tortuosity distribution on failure properties of arterial wall have potential importance for developing therapeutic strategies targeting collagen fiber tortuosity to enhance tissue strength and prevent arterial wall rupture. Prior studies have demonstrated age-related changes in collagen tortuosity distribution in the layers of the bladder wall that result in a reduced toe region [42]. In a related study, purine nucleoside phosphorylase inhibition with 8-aminoguanine (8-AG) was shown to increase the fiber tortuosity in aged bladders, recovering the extended toe region to that of young bladders [54]. Future research could investigate the effects of such treatments on fiber tortuosity distribution in arterial tissues. Furthermore, one of the challenges in developing vascular grafts and tissue engineered blood vessels is replicating the mechanical properties of the native vessels [55,56]. The present study provides insights on how to tune the fiber tortuosity distributions to enhance the mechanical performance and longevity of these synthetic tissues.

5 Conclusions

In conclusion, this study developed an in-silico model to investigate the effect of collagen fiber tortuosity distribution on the biomechanical failure of arterial wall tissue. Our findings reveal for the first time that variations of two important features of collagen fiber tortuosity (average and dispersion) significantly influence the tissue's mechanical behavior, with increased fiber tortuosity enhancing the tissue strength and, in contrast, increased dispersion in tortuosity distribution reducing the strength. The insights gained from this study have important implications for understanding vascular diseases such as aneurysms, where structural alterations in tissue can lead to catastrophic failures. Our results suggest that modifying fiber tortuosity through therapeutic interventions could strengthen arterial walls and reduce rupture risks. Additionally, our findings indicate the potential for drugs that can alter fiber tortuosity of diseased arterial tissues.

Supplementary Material

Supplementary Material

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Funding Data

• National Institutes of Health (NIH) (Grant No. 2R01NS097457; Funder ID: 10.13039/100000002).