Computational modeling reveals the relationship between intrinsic failure properties and uniaxial biomechanical behavior of arterial tissue

Abstract

Biomechanical failure of the artery wall can lead to rupture, a catastrophic event with a high rate of mortality. Thus, there is a pressing need to understand failure behavior of the arterial wall. Uniaxial testing remains the most common experimental technique to assess tissue failure properties. However, the relationship between intrinsic failure parameters of the tissue and measured uniaxial failure properties is not fully established. Furthermore, the effect of the experimental variables, such as specimen shape and boundary conditions, on the measured failure properties is not well understood. We developed a finite element model capable of recapitulating pre-failure and post-failure uniaxial biomechanical response of the arterial tissue specimen. Intrinsic stiffness, strength and fracture toughness of the vessel wall tissue were used as the input material parameters to the model. Two uniaxial testing protocols were considered: a conventional setup with a rectangular specimen held at the grips by cardboard inserts, the other used a dogbone specimen with soft foam inserts at the grips. Our computational study indicated negligible differences in the peak stress and post-peak mechanical behavior between these two testing protocols. It was also found that the tissue experienced only modest localized failure until higher levels of applied stretch beyond the peak stress. A robust cohesive model was capable of modeling the post-peak biomechanical response, which was primarily governed by tissue fracture toughness. Our results suggest that the post-peak region, in conjunction with the peak stress, must be considered to evaluate the complete biomechanical failure behavior of the soft tissue.

1. Introduction

Normal function of arteries under physiological conditions is influenced by the biomechanical behavior of the vessel wall. Biomechanical properties of the artery wall tissue can be altered by the progression of disease, such as aneurysm. Compromised biomechanics of the diseased tissue can lead to the loss of mechanical integrity culminating in vessel wall failure, a catastrophic clinical event. For example, in the case of intracranial aneurysm rupture, the fatality rate is 45% and long term disability rate is 64% (Kelly et al., 2001; Ropper and Zervas, 1984). Another example of a life threatening arterial disease, acute dissection of the ascending thoracic aorta, is associated with a mortality rate of 1-1.4% per hour for the first 48 hours (Mészáros et al., 2000), escalating to > 60% by two weeks (Masuda et al., 1991) if left untreated. Thus, there is a pressing need to understand the failure biomechanics of the arterial wall tissue.

The first step towards understanding wall tissue failure mechanics is the identification of the material parameters that govern the biomechanical failure process. In the cardiovascular biomechanics literature, the primary focus has been the estimation of tissue strength, defined as the maximum stress the tissue can bear before failure. This parameter is nearly always assessed by measuring the peak of the stress-stretch curve during uniaxial tensile experiments (Robertson et al., 2015; Shah et al., 2014; Walsh et al., 2014; Monson et al., 2005; Costalat et al., 2011). However, imaging of the test specimens during uniaxial testing revealed little to no damage to the tissue when tissue stress reached the peak magnitude (Sang et al., 2018; Brunon et al., 2010; Wren and Carter, 1998). In addition, a variety of post-peak behaviors, ranging from a sudden drop to gradually diminishing tissue stress, were recently reported (Sang et al., 2018). This observation suggests that additional parameters, besides the peak stress, are needed to describe and quantify the rupture resistance of artery wall tissues. For example, fracture toughness, the energy expended by the material per unit area before complete failure (Anderson et al., 2005), has garnered recent attention for its importance in quantifying the failure process (Mantič et al., 2015; Taylor, 2018; Leng et al., 2015; Chen et al., 2017).

Uniaxial experiments, the primary method for evaluating tissue biomechanical parameters, are known to be sensitive to the geometry and boundary conditions of the test specimen (Sang et al., 2018; Walsh et al., 2014; Masouros et al., 2009). In addition, there is inherent variability in the structure and mechanical properties of arterial tissue across and even within subjects. As a consequence, how tissue material properties can be accurately assessed from ex vivo testing and how they are correlated with tissue failure response has yet to be established. This knowledge is essential for quantifying the effect of tissue biomechanical properties on its failure behavior, which will ultimately lead to the elucidation of the rupture mechanisms in aneurysms as well as other soft tissues.

The objective of the present work is to use in silico testing to analyze the impact of experimental protocol on mechanical response of the arterial tissue during uniaxial testing, and consequently assess how the intrinsic tissue failure properties control the uniaxial response of the tissue specimens. To accomplish these objectives, we constructed a finite element model of the uniaxial experiment protocol that reproduces initial clamping-induced tissue sample deformation followed by uniaxial extension of rectangular and dogbone shaped artery specimens, Fig. 1. Evolution of subfailure tissue damage that leads to the complete specimen failure was simulated using the cohesive volumetric finite element method (Nittur et al., 2008; Maiti et al., 2005). The stiffness, strength, and fracture toughness of the tissue were introduced as input material parameters to the model. Extensive parametric in silico experiments were performed to determine the impact of the specimen shape and insert material on the uniaxial experimental results, and then quantify the effect of these intrinsic material parameters on the uniaxial failure behavior of the tissue specimens.

Fig. 1. Schematic of computational domain and boundary conditions used for simulations of physical experiments.

In the physical experiments, the tissue specimen is often either (a) rectangular or (b) dogbone shaped and held by (c) metal clamps with a flexible insert material (black line) between the specimen and clamp. The computational model reproduced the (d) rectangular and (e) dogbone shaped specimens and the deformable insert material. In the simulations, the tissue is first clamped (f), modeled as a uniform pressure applied to the insert material (gripping BC), and is then loaded in the axial direction (g) by applying a tensile stretch on insert material (uniaxial extension BC) to recreate the physical uniaxial tensile testing conditions. Grips are schematically shown in (f-g) in grey color.

2. Methods

Following earlier works on material failure behavior (Dugdale, 1960; Barenblatt, 1962), we postulate that tissue failure is a localized effect operative within a process zone. In addition, we consider that the material failure in the process zone is a gradual and irreversible process. A finite amount of energy per unit area, termed as the fracture toughness, has to be expended to achieve complete failure of the process zone resulting in a failed region with two surfaces defining a tear within the tissue. In essence, mechanical response of the bulk (volumetric) domain is governed by a prescribed constitutive response (e.g. hyperelastic model) while a separate failure constitutive law, defined at the potential tear interface, governs the tissue failure response.

In this paper, we make a distinction between the peak stress and the intrinsic strength. The intrinsic strength is a local pointwise tissue material failure property that enters the model through the interfacial failure law (2.1.3). The peak stress is the maximum stress the specimen can globally carry during the uniaxial tensile testing, and is determined directly from the peak of the stress-stretch curve.

2.1. Governing Equations

2.1.1. Kinematics in the presence of a tear

We consider an arbitrary subset of the tissue sample, denoted as body $\mathcal{B}$ and occupying material region Ω₀ in the reference domain, chosen as the unloaded sample at time t = 0 (Fig.2). During the failure process, tears generated in the tissue sample introduce discontinuities in the displacement field, the displacement jump, with the creation of a new internal surface within the body where the displacement discontinuity is localized (Ortiz and Pandolfi, 1999). This internal surface of discontinuity can be mapped back to the reference configuration. In particular, an internal tear surface can be generated at some time t in the loaded domain Ω_t corresponding to a surface Γ₀ in Ω₀. This surface is then mapped to ${\Gamma }_t^{+}$ and ${\Gamma }_t^{-}$ in Ω_t, Fig.2. Internal surface Γ₀ divides the reference domain into two open regions (excluding surface Γ₀) ${\Omega }_0^{+}$ and ${\Omega }_0^{-}$ that are mapped to the regions ${\Omega }_t^{+}$ and ${\Omega }_t^{-}$ at time t. The function x = ϕ(X,t maps material points X from ${\Omega }_0^{+}$ and ${\Omega }_0^{-}$ to x in ${\Omega }_t^{+}$ and ${\Omega }_t^{-}$, respectively. We assume that ϕ(X,t) is continuous in these open regions for all times.

Fig. 2. Schematic of the tissue material domain with a tear surface.

(a) Closed region Ω₀ in the reference configuration is separated by the tear surface Γ₀ into open subregions ${\Omega }_0^{+}$ and ${\Omega }_0^{-}$. The two edges of the tear are co-located at position Γ₀. (b) In the deformed configuration at current time t, the subregions are mapped to subregions ${\Omega }_t^{+}$ and ${\Omega }_t^{-}$ along with the current locations of the edges of the tear surface at ${\Gamma }_t^{+}$ and ${\Gamma }_t^{-}$.

We define φ⁺ and φ⁻ as the restrictions¹ of ϕ(X, t) at ${\Gamma }_t^{+}$ and ${\Gamma }_t^{-}$, respectively, and assume these restrictions exist for all times t > 0. The jump in the displacement vector at the tear surface is then defined by

$$\mathbf{\Delta }:=〚\varphi 〛\text{where}〚\varphi 〛={\mathit{\phi }}^{+}-{\mathit{\phi }}^{-}.$$ (1)

Also, the mid-surface of the tear mediated discontinuity at time t can be tracked from the mean deformation map of the internal surface

$$\overline{\mathit{\phi }:}=\frac{1}{2}[{\mathit{\phi }}^{+}+{\mathit{\phi }}^{-}].$$ (2)

The deformation gradient F(x, t) := ∇x can then be defined to map points in the open regions ${\Omega }_0^{-}$ and ${\Omega }_0^{+}$ at t = 0 to ${\Omega }_t^{-}$ and ${\Omega }_t^{+}$, respectively, at time t. In the expression for the deformation gradient, ∇ is the gradient operator with respect to the Lagrangian coordinates X. The right Cauchy-Green tensor is then defined as C := F^T F. In what follows for the constitutive modeling, it is helpful to introduce the standard multiplicative volumetric-isochoric split of the deformation gradient tensor into volume changing and isochoric parts,

$$\begin{gathered} \mathbf{F}={\mathbf{F}}_{vol}{\mathbf{F}}_{iso} \\ \text{where}{\mathbf{F}}_{vol}:=J^{1/3}\mathbf{I}\text{and}{\mathbf{F}}_{\mathrm{iso}}:=J^{-1/3}\mathbf{F} \end{gathered}$$ (3)

interior to each subdomain, where J := detF and I is the second order identity tensor. Similarly, we define

$$\begin{gathered} \mathbf{C}={\mathbf{C}}_{vol}{\mathbf{C}}_{iso} \\ \text{where}{\mathbf{C}}_{vol}:=J^{2/3}\mathbf{I},{\mathbf{C}}_{iso}:=J^{-2/3}\mathbf{C} \\ \text{and}\text{det}({\mathbf{F}}_{iso})=\text{det}({\mathbf{C}}_{iso})=1. \end{gathered}$$ (4)

2.1.2. Material response away from the tear

Interior to the subdomains ${\Omega }_0^{+}$ and ${\Omega }_0^{-}$, the tissue is modeled as an anisotropic, incompressible, hyperelastic material with two families of fibers. Following standard approaches to soft tissue modeling, we employ a decoupled representation for the strain energy density function for both layers of the artery wall as an additive decomposition of volumetric and isochoric components,

$$\Psi (\mathbf{C})={\Psi }_{vol}(J)+{\Psi }_{iso}({\mathbf{C}}_{iso}).$$ (5)

For this work, we follow the formulation described in (Holzapfel et al., 2000). Accordingly, we choose following isochoric component of the strain energy function (Gasser et al., 2006) for both the medial and adventitial layers,

$$\begin{gathered} {\Psi }_{iso}({\mathbf{C}}_{iso})={\Psi }_g({\mathbf{C}}_{iso})+\underset{i=4,6}{\Sigma }{\Psi }_{fi}({\mathbf{C}}_{iso,}{\mathbf{H}}_i) \\ \text{where}{\Psi }_g=c_1({\overline{I}}_C-3) \\ \text{and}{\Psi }_{fi}=\frac{k_1}{2k_2}[\exp (k_2{\overline{E}}_i^2)-1],i=4,6. \end{gathered}$$ (6)

In the above equations,

$$\begin{gathered} {\overline{I}}_C:=\text{trace}({\mathbf{C}}_{iso}) \\ \text{and}{\overline{E}}_{\text{i}}={\mathbf{H}}_i:({\mathbf{C}}_{\mathit{iso}}-\mathbf{I}), \end{gathered}$$

where c₁ is the shear modulus of the non-fibrous ground matrix, and k₁ (with the dimension of stress) and k₂ (dimensionless) are two parameters associated with the fiber mechanical response. As the collagen fiber network was planar for all the arterial tissues we studied, we used the following form of the generalized structure tensor H_i,2D given by (Ogden, 2009),

$${\mathbf{H}}_{i,2D}=\kappa \mathbf{1}+(1-2\kappa ){\mathbf{M}}_i\otimes {\mathbf{M}}_i,$$ (7)

with 1 as a two dimensional identity tensor, κ as the dispersion parameter representing a transversely isotropic dispersion about M_i, the mean fiber direction of the fiber family i. It is important to note here, as emphasized by Holzapfel and Ogden (2015), that the fiber contribution of the strain energy function was considered only when the fibers are in tension, i.e., M_i; · (CM_i) > 1. Furthermore, we employ a commonly used form for the volumetric part of the strain energy of the nearly incompressible materials (Bonet and Wood, 2008)

$${\Psi }_{vol}(J)=\frac{1}{2}\mathbb{K}{(J-1)}^2$$ (8)

where $\mathbb{K}$ is a penalty parameter enforcing incompressibility condition. This parameter can be related to the material bulk modulus (Bonet and Wood, 2008) and was set at 25 MPa for all simulations.

2.1.3. Material response at the failure interface

With increasing subfailure damage, the displacement jump ⟦ϕ⟧ at the internal surfaces ${\Gamma }_t^{+}$ and ${\Gamma }_t^{-}$ increases. Note that these surfaces are two dimensional, but are oriented in the 3D space. Thus, it will be enough to define the failure response law locally to the internal surface in the normal and tangential directions in the current configuration. Accordingly, we will consider the decomposition of the displacement jump vector Δ into its normal and tangential components (Δ_n and Δ_t, respectively) relative to the current mid-surface ${\Gamma }_t^m$ defined by the mean deformation $\stackrel{‒}{\phi }$ at the discontinuity.

Gradual failure of the tissue material in the process zone is modeled using a scalar internal variable $\mathcal{I}$ that monotonically decreases as the magnitude of the displacement jump increases beyond a critical level, ranging from its initial value ${\mathcal{I}}_{\mathit{init}}$ (≈ 1) prior to the onset of damage to zero at complete failure. Following our prior work (Nittur et al., 2008; Maiti et al., 2005), we prescribe $\mathcal{I}$ to depend on $\tilde{\Delta }$, the magnitude of a normalized displacement jump vector,

$$\mathcal{I}:=\min [{\mathcal{I}}_{init,}\langle 1-\tilde{\Delta }\rangle ]$$ (9)

where $\langle 1-\tilde{\Delta }\rangle =\max (0,1-\tilde{\Delta })$ and

$$\tilde{\Delta }:=||\tilde{\mathbf{\Delta }}||={\left({\tilde{\Delta }}_n{}^2+{\tilde{\Delta }}_t{}^2\right)}^{1/2},{\tilde{\Delta }}_n:=\frac{{\Delta }_n}{{\Delta }_{nc}},{\tilde{\Delta }}_t:=\frac{{\Delta }_t}{{\Delta }_{tc}}.$$ (10)

Here, Δ_nc and Δ_tc are prescribed material properties denoting critical values of the normal and tangential displacement jumps at which the material at the internal surface separates completely resulting in tear propagation in the normal and shear modes, respectively. We note here that the coupling between the normal and shear modes of tear propagation is achieved through the incorporation of $\tilde{\Delta }$ in the expression for $\mathcal{I}$ (see Eq. 10). It follows from (Eq. 9) that, as $\tilde{\Delta }$ is increased from zero, that $\mathcal{I}$ remains at the initial value ${\mathcal{I}}_{\mathit{init}}$ until the onset of subfailure at $\tilde{\Delta }=1-{\mathcal{I}}_{\mathit{init}}$, after which $\mathcal{I}$ decays as a function (Eq. 10) of Δ_n and Δ_t to zero, Fig. 3e. For example, if the failure is simply in the normal direction (Δ_t = 0), then as the deformation progresses, Δ_n will increase from zero to Δ_nc with corresponding drop in $\mathcal{I}$ defined in Eq. 9 from ${\mathcal{I}}_{\mathit{init}}$ to zero (complete failure), Fig. 3c.

Fig. 3. Schematic of cohesive volumetric finite element and failure constitutive law.

Cohesive element in the (a) undeformed configuration where the two opposing surfaces of the cohesive element are collocated, and (b) the deformed configurations showing the displacement jump vector (Δ) that describes the separation of the two surfaces of the cohesive element. The displacement jump vector is decomposed into normal (Δ_n) and tangential (Δ_t) components. The components of the traction vector t in the normal (n) and tangental (t) directions at the interface are defined as functions of these components of the displacement jump vector, and were prescribed to have a (c,d) bilinear dependence on the respective normalized jump vector components (${\tilde{\Delta }}_n$,${\tilde{\Delta }}_t$), Eq. 11. The area under the traction-displacement jump curve is the fracture toughness in normal (G_I) and tangential (G_II) tissue tearing modes, respectively. The internal parameter $\mathcal{I}$ reduces as a function of ${\tilde{\Delta }}_n$,${\tilde{\Delta }}_t$, from ${\mathcal{I}}_{\mathit{init}}$ to 0 in (e)

Prior to defining a constitutive model governing the traction at the interface during the failure process, we recall that the Cauchy stress vector (traction) at an arbitrary surface ∂Ω_t in Ω_t with normal n can be written with respect to Cauchy stress tensor σ as, t = σ · n. The cohesive traction vector is assumed to be continuous across the tear surface, Γ_t, so that ⟦t⋧ = 0. We denote the normal and tangential components of t as t_n and t_t, respectively. Following our prior work, the failure constitutive law for the traction at the tear interface is chosen as, Fig. 3c–d,

$$\begin{gathered} t_n=\frac{\mathcal{I}}{1-\mathcal{I}}\frac{{\sigma }_{\max }}{{\mathcal{I}}_{init}}{\tilde{\Delta }}_n, \\ {t}_t=\frac{\mathcal{I}}{1-\mathcal{I}}\frac{{\tau }_{max}}{{\mathcal{I}}_{init}}{\tilde{\Delta }}_t\text{at}{\Gamma }_0 \end{gathered}$$ (11)

where it should be recalled $\mathcal{I}$ is a function of $\tilde{\Delta }$, Eq. 9. It follows from (11) that each of these components increases linearly with the respective component of $\tilde{\Delta }$ until the corresponding strength in normal or shear direction, σ_max and τ_max, respectively, is reached (Fig. 3cd). The material behaves elastically up to this instant, and will retrace the loading path upon unloading. Upon further increase in $\tilde{\Delta }$, progressive damage will occur to the material, causing a decrease in the respective component of interface stress until complete failure ($\mathcal{I}=0$, t_n = 0, t_t = 0 respectively), Fig. 3. The material unloads towards the origin in the descending regime of the failure law, and retraces the unloading path during subsequent loading events ensuring permanency of the previously experienced damage. Finally, this formulation employs a repulsive penalty force in the normal direction with a penalty parameter K to prevent interpenetration of the cohesive surfaces, Fig. 3b:

$$t_n=K{\tilde{\Delta }}_n{\tilde{\Delta }}_n<0.$$ (12)

2.1.4. Tissue fracture toughness

The shaded area under the failure envelope in the cohesive traction versus displacement jump curve, Fig. 3c–d corresponds to the intrinsic fracture toughness (the energy required to create a new unit surface area) of the tissue which we denote as G_I in the normal direction and G_II in the tangential direction. We note here that the material failure law for tangential direction law encompasses both mode II (in-plane shear) and mode III (out of plane shear) failures, and we have assumed fracture toughnesses in these two shearing modes to be equal due to the lack of any experimental fracture toughness data for these mode of failures in cardiovascular tissues.

It then follows that, for the bilinear failure law in (11), the cohesive parameters, (σ_max, Δ_nc) and (τ_max, Δ_tc) in the normal and tangential directions, respectively, are coupled to the corresponding fracture toughnesses GI and GII through the relations

$$G_I=\frac{1}{2}{\sigma }_{\mathit{max}}{\Delta }_{nc}\text{at}{G}_{\mathit{II}}=\frac{1}{2}{\tau }_{\mathit{max}}{\Delta }_{tc}$$ (13)

so that only two out of three of the material parameters need to be specified from each direction In this work, we chose to specify the intrinsic fracture toughness (G_I, G_II) and strength of the tissue (σ_max, τ_max). Further details of this material failure law can be found in Maiti et al. (2005).

2.2. In Silico Model of Uniaxial Testing

2.2.1. Finite element model of the tissue specimen and surface grip assembly

The solid model of the tissue specimen and mesh were created in Trelis Pro 14 (csimsoft, American Fork, UT) with specimen geometry and dimensions motivated from prior experimental work, Fig. 1a,b. The dimensional details of the solid model of the tissue specimen are listed in Table 1. The steel clamp, being of much higher stiffness than the tissue and the insert material layer, Fig. 1c, was idealized as rigid and not explicitly modeled. Rather its effect was modeled as a uniform gripping pressure on the contact surface of the insert material between the grip and tissue specimen. The undeformed insert layer was modeled as a cuboid with a thickness of 0.787 mm and a contact surface of 1.8 mm × 2 mm.

Table 1.

Solid Model Dimensions of Tissue Specimen

	Dogbone	Rectangle
Width at middle	0.6 mm	2.0 mm
Width at grip	2.0 mm
Grip-to-grip Length	3.5 mm
Total Length	7.1 mm
Media Thickness	0.1 mm
Adventitia Thickness	0.1 mm

The finite element mesh consisted of eight node hexahedral trilinear elements. Because our failure model only allows cracks to nucleate and propagate between the finite elements, the mesh design is an important feature of the simulation. We created non-rectilinear meshes for the specimen that enabled the cracks to nucleate and propagate in multiple non-orthogonal directions, which is not possible using a rectilinear mesh. The cohesive elements were inserted at the edges of each hexahedral element. The convergence criterion was taken as the change in peak von Mises stress anywhere within the domain to be within 2% between successive mesh refinements. For numerical convergence, 4,854 and 6,570 volumetric elements were required for modeling the rectangular and dogbone tissue specimens, respectively, while 4,032 elements were used for meshing the insert layer. The rectangular specimen required 10,108 cohesive elements resulting in 34,542 total nodes for the entire model. The dogbone shaped specimen model consisted of 12,898 cohesive elements and 42,534 total nodes.

With respect to the boundary conditions, the geometric symmetry of the finite element model and loading conditions was exploited so that only one-fourth of the entire model was analyzed. The two symmetry planes were provided with roller boundary conditions in appropriate directions to prevent out of plane motion. To mimic the experimental loading conditions, the load was applied to the model in two stages, as schematically shown in Fig. 1f,g, through a quasi-implicit load stepping scheme (Maiti and Geubelle, 2005). We first applied a uniform clamping pressure of 12.5 kPa onto the top surface of the insert in the transmural direction in 200 computational steps. The clamping pressure was calculated by matching clamping displacement typically used in our laboratory to fix the tissue specimen with the grips. The test specimen was then stretched uniaxially in 2000 computational steps by a grip stretch applied at the outer surfaces of the insert material until failure. The insert material and specimen could deform between the clamps. The unity stretch state (λ = 1.0) for the stress stretch curves was defined as the state under a tare load force of 0.005 N. Reaction forces at each load step, which corresponded to the load cell recordings in physical uniaxial experiments, were used to calculate the average Cauchy stress, defined as the reaction force divided by the current cross-sectional area computed mid-specimen.

2.2.2. Input material parameters for the finite element model

Image analysis of the cerebral arterial wall tissue, see Online Resource 1, revealed that the fibers were primarily oriented in the circumferential direction, the direction of uniaxial loading, with little dispersion. Accordingly, we took the mean fiber directions for the two families of fibers to be ±5°, and κ was fixed at 0.05 for all the simulations. The stiffness parameters c₁, k₁ and k₂ in Eq. 6 were determined by regressing the model predicted uniaxial stress-stretch curve against experimental pre-failure uniaxial constitutive behavior for human cerebral (n = 17) and sheep carotid (n = 21) arterial tissues reported in (Sang et al., 2018). The regression analysis was performed using nonlinear least squares fitting function lsqcurvefit in MATLAB 2016.b (Mathworks Inc, Natick, MA). The regressed parameters (c₁, k₁ and k₂) for all the categories are shown in Table 2. The magnitudes of the artery intrinsic strength was taken as 2 MPa and 6 MPa to represent two distinct groups of tissue peak stress covering the range of experimental values found in (Sang et al., 2018). Overall, the choice of input parameters resulted in a wide range of output peak stresses and high stretch stiffnesses that adequately represented experimentally observed values, as illustrated in Fig. 4.

Table 2.

Constitutive Model Parameters Used In Simulation

Material Parameters	Stiff-0	Stiff-1	Stiff-2
c1 (MPa)	0.2	0.2	0.2
k1 (MPa)	0.01	0.02	0.172
k2	0.1	0.1	0.1

Fig. 4. Healthy cerebral arterial tissues of human (open squares) and sheep (open circles), subjected to uniaxial tensile testing, exhibits a wide range of peak stress and high stretch stiffnesses.

For simulation purposes, five points (denoted by diamonds) were chosen to cover the range of experimental peak stress and high stretch stiffness. This data was gathered as part of an experimental study using the protocol defined in Sang et al. (2018).

As experimental data for tissue strength in shear are unavailable for human cerebral or sheep carotid arterial tissue, we modeled the strength of the media and adventitial layers as identical in the normal and shear directions, i.e., σ_max = τ_max. The magnitude of fracture toughness was considered in the range of 1kJ/m² to 8kJ/m² based on data for other cardiovascular tissue (Purslow, 1983), since no data was available specifically for arterial tissue used in this study. We modeled identical fracture toughness for both normal and tangential directions, i.e., G = G_I = G_II. Two types of fracture toughness based parametric studies were run, one with identical fracture toughness for the media and adventitia and one with the adventitia toughness twice that of media toughness. To distinguish these studies, the simulations were denoted with the letter G with subscripts indicating numerical values of the media and adventitia toughnesses, respectively, in (kJ/m²). For example, in the case of a media toughness of 2 kJ/m² and an adventitia toughness of 4 kJ/m², the simulation was denoted as G24.

For comparison with common uniaxial testing protocol in the literature, studies of rectangular tissue specimen were done with stiff insert material with properties relevant to sandpaper or cardboard (designated as Rect-CB). A newer experimental protocol designed to relieve stress concentrations at the clamp by using dog-bone tissue specimens and a PU foam layer introduced in Sang et al. (2018) was also modeled (designated as DB-FT). The cardboard material was modeled as a compressible neoHookean material with material properties λ =7.46 GPa and μ =2.35 GPa (Sato et al., 2007). The PU foam tape (McMaster-Carr, #7626A213) was modeled as an compressible neoHookean material with material properties λ = 166 MPa and μ = 1.67 MPa (Goods et al., 1997; Patel et al., 2008).

2.3. Model output parameters

Numerical data for the evolution of the Cauchy stress during uniaxial loading was visualized using ParaView (Kitware, Albuquerque, NM). Output data for each cohesive element were traction (t_n and t_t), displacement (Δ_n and Δ_t) in the normal and shear directions, and the internal parameter $\mathcal{I}$, as defined in Section 2.1.3. Three additional output parameters were extracted from the model simulations. First, the peak stress from the stress-stretch curve from each simulation was estimated. Second, we assessed the axial locations of the local complete failure of the tissue ($\mathcal{I}=0$).

2.4. Statistical Analysis

All statistical analyses were conducted using SPSS (IBM Corp., Armonk, NY). To determine if the data were normal, we ran a Kolmogorov-Smirnov test. For the comparison of two groups with normally distributed data, a two sample independent t-test was conducted. If the data were not normal, a Mann-Whitney test was conducted. Comparison between two or more groups was performed using a one-way independent ANOVA followed by a Bernoulli post hoc test. Data compared across multiple categorical groups were ranked and tested by Kruskal-Wallis test. Correlation analysis were performed using both Pearson’s (r) and Spearman’s (ρ) correlation coefficient. The alpha level of significance for all statistical tests was set at α < 0.05.

3. Results

3.1. CVFE modeling recapitulates the entire uniaxial response of soft tissue up to complete specimen failure

Simulated uniaxial stress-stretch curves for artery wall tissue specimens using both DB-FT and Rect-CB protocols and all the stiffness and strength cases considered in this work are presented in Fig. 5. Experimentally measured curves for the arterial wall tissue, as reported in (Sang et al., 2018), are also shown in this figure for the comparison purpose. To make appropriate comparison between experimentally obtained curves and simulation-derived ones, only experimental responses within ±1 MPa of the chosen intrinsic strength parameters (2 MPa and 6 MPa) and the high stretch stiffness within ±2 MPa of the simulated stiffness parameter values (see Fig. 4) were plotted. Different simulated curves in each panel were obtained by varying the fracture toughness of the two layers of the tissue systematically. This figure demonstrates that the simulated curves adequately recapitulated the pre-peak stiffness, peak stress, and the post-peak response exhibited by the experimentally measured curves. Notably, the transition of the post-peak response from an abrupt to a more gradual failure, as observed in experiments (Sang et al., 2018), was captured in the simulations by varying the fracture toughness.

Fig. 5. Simulated uniaxial stress versus stretch responses recapitulate the pre-failure as well as failure region of the experimentally obtained curves.

Experimental uniaxial response data from Sang et al. (2018) (grey) and in silico data (colored lines) were compared. For suitable comparison, experimental curves with similar stiffness and peak stress were included in each plot. The inclusion criteria being that the peak stress is within ±1 MPa from the intrinsic strength parameter (2 MPa or 6 MPa), and the high stretch stiffness is ±2 MPa from the high stretch stiffnesses used as input parameters. The multiple simulations curves represent varying toughness of the tissue to recapitulate the range of post-peak responses of the experimental data varying from abrupt to gradual failure.

To further demonstrate the points discussed above, Fig. 6 presents the stress-stretch curves along with the axial stress contours on the specimen surface for three representative uniaxial simulations with decreasing stiffness cases Stiff-2 (Fig. 6a), Stiff-1 (Fig. 6b), and Stiff-0 (Fig. 6c) and a constant fracture toughness combination (G24). Fig. 6a represent a high (6 MPa) intrinsic strength case while the low (2 MPa) strength cases are represented by Fig. 6b and Fig. 6c. The dogbone geometry is presented in Fig. 6a and Fig. 6c, while Fig. 6b depicts a rectangular sample. The stress distribution profiles were chosen at four preselected points (A, B, C, and D) on the evolution of the uniaxial response (top panels) to span the entire stress-stretch curve. In abrupt failure cases, the media and adventitia both failed together Fig. 6a, whereas in the gradual failure case the medial and adventitial layers fail sequentially Fig. 6b,c, corresponding to a delamination process as seen in the experiments. Black arrowheads in Fig. 6 and Fig. 7 designate the locations of complete failure ($\mathcal{I}=0$) in the tissue.

Fig. 6. Specimen stress distribution during loading to failure.

In row 1, Cauchy stress-stretch response and development of internal parameter $\mathcal{I}$ in each layer during uniaxial loading to failure for three representative specimens. Deformation of dogbone samples of high and low strength are seen in column (a) and (c), while deformation of a weak rectangular sample is shown in column (b), respectively. For these illustrative examples, all samples have the same fracture toughness (G24) while stiffness is decreasing from left to right thus Stiff-2 (a), Stiff-1 (b), and Stiff-0 (c). While $\mathcal{I}$ for each cohesive element is monitored, the minimum $\mathcal{I}$ for each layer as a function of stretch is shown in row 1. Below row 1, the corresponding undeformed samples are shown then the deformed samples with contours of axial Cauchy stress at four increasing levels of deformation denoted as (A-D). These same levels are also shown in the corresponding stress/stretch plots. Level O is the undeformed configuration, A is in the pre-peak region, B at peak stress, C closely following peak stress during softening and D at or slightly before failure of the media layer. Local failure can be seen in contour figures where failed elements are removed and failure regions are denoted by black arrowheads. In these three cases, the toughness of the media was half that of the adventitia (G24) so as expected the media failed first. At loading point D, elements in the middle (along the axis of symmetry) were removed for clarity as the cohesive elements have failed to the left and their presence is only a numerical artifact.

Fig. 7. Distribution of internal parameter $\mathcal{I}$, shows only local damage at peak stress.

For the same three cases as shown above in Fig. 6 the internal variable $\mathcal{I}$ is shown in the undeformed configuration (O) as well as four increasing levels of deformation denoted as (A-D). Level O is the undeformed configuration, A is in the pre-peak region, B at peak stress, C closely following peak stress during softening and D at or slightly before failure of the media layer. Local failure can be seen in contour figures where failed elements are removed and failure regions are denoted by black arrowheads. At loading point D, volumetric elements in the middle (along the axis of symmetry) were removed for clarity as the cohesive elements have failed to the left and right and their presence is only a numerical artifact.

To investigate the micromechanics of the failure process, we monitored the evolution of the internal parameter $\mathcal{I}$. In particular, we superimposed the evolution of the internal parameter $\mathcal{I}$ on the stress-stretch curves in the top panels of Fig. 6. In addition, we present the distribution of this parameter on the media surface for the entire specimen in Fig. 7 at the previously selected points A, B, C and D, for the three cases depicted in Fig. 6. Consistent with our experimental study of uniaxial failure in human cerebral and sheep carotid arteries (Sang et al., 2018), the peak stress (Point B) in Fig. 6 and Fig. 7 was not associated with complete tissue failure ($\mathcal{I}=0$) anywhere within the specimen or a large change in cross section for all these cases. Rather, various degrees of material softening due to distributed tissue damage ($0<\mathcal{I}<{\mathcal{I}}_{\mathit{init}}$) could be observed at the peak stress (Fig. 6). This finding held for the entire cohort of specimens tested in our study that covered a wide range of material properties, two specimen geometries, and boundary conditions.

3.2. Influence of uniaxial testing protocol on the tissue mechanical response

3.2.1. Dogbone samples focused stress at the middle

All our studies failed in the mid-region. The failure process was qualitatively similar to that shown in Fig. 6a with stress concentration primarily at the mid-region of the specimen with some samples also exhibiting localized stress concentrations near the clamp (e.g. Fig. 6a). The stress concentration spread throughout the specimen upon further stretching, and was relieved by local subfailure damage (point C), Fig. 6a, leading to eventual failure in the mid-region of the sample. For all dogbone specimens the stress concentration remains localized at the middle until failure occurs, Fig. 6a,c. In contrast, for all rectangular specimens, the stress concentration was distributed over a large region of the sample, Fig. 6b.

3.2.2. Peak stresses are relatively insensitive to testing protocol

Fig. 8 reveals that the testing protocol had little influence on peak stress. For example, peak stress of high intrinsic strength (6 MPa) samples were found to have no statistically significant difference between dogbone and rectangle protocols (p = 0.51). While we did find statistically significant differences in the peak stress between protocols for low strength specimens (p < 0.01), the magnitude of these differences is small enough to be of any practical significance for biomechanical testing of biological tissues. In particular, the mean simulated uniaxial peak stress of the tissue is 2.15 ± 0.02 MPa versus 2.20 ± 0.07 MPa for dogbone and rectangular samples, respectively, a difference in the mean value of 2.39%. The stretch associated with the peak of the stress-stretch curve was not statistically different between protocols for low (p = 0.24) or high strength tissue (p = 0.32).

Fig. 8. Simulated peak stress is insensitive to material stiffness, fracture toughness and testing protocol.

Stress stretch curves for simulated uniaxial loading to failure of 60 samples, divided into (a) low strength and (b) high strength groups. Within each strength group, materials with increasing high stretch stiffness from Stiff-0 (purple), Stiff-1 (red), and Stiff-2 (blue) are shown. For each of these sets, six different values of fracture toughness were considered for the media and adventitia ranging from 1-8 kJ/m². For each material parameter combination, both dogbone and rectangular samples were tested, shown as dashed and solid lines, respectively. Bars in (c) represent average peak stress across the simulations of varying toughness and stiffness (n=18 or 12 for low or high strength respectively). Error bars represent one standard deviation. Peak stress is correlated with the intrinsic strength of the material, but is nearly insensitive to the material stiffness, fracture toughness and even testing protocol. The fracture toughness has only minor influence on the pre-peak response and marked effect on the post-peak response. The influence of testing protocol is relatively unimportant with largest differences seen for the softest materials in the stronger group (purple curves in (b)).

3.2.3. Qualitative post-peak response is not altered by testing protocol

The post-peak response shows some modest visible differences between protocols, Fig. 8a–b, particularly for the high strength, low stiffness materials, Fig. 8b. Nonetheless, the qualitative nature of the response (gradual versus abrupt) did not change. This is consistent with the experimental findings in Sang et al. (2018). Fig. 9 presents the simulated uniaxial stress-stretch behavior of the tissue specimen for all DB-FT simulations with colors denoting each toughness case. Rect-CB simulations were found to show similar behavior and can be found in the Online Resource 2.

Fig. 9. Fracture toughness changes post-peak but not pre-peak response.

The stress versus stretch curves for different fracture toughness groups (G11-G48) for low (left column) and high (right column) material strength and different levels of stiffness, with increasing stiffness from Stiff-0 to Stiff-2. For all cases, fracture toughness has a negligible influence on the response prior to peak stress with curves being indiscernible pre-peak with the exception of the high strength specimens with the lowest stiffness for which the peak stress varies with fracture toughness. In contrast, the area under the post-peak curve increases with increasing fracture toughness, showing larger differences for low strength materials. Results are only shown for DB-FT samples.

3.3. Material parameters influencing mechanical response

3.3.1. Material parameters influencing pre-peak and peak response

Differences in tissue stiffness did not result in statistically significant differences in peak stress for high strengt tissues (p = 0.77). While differences in tissue stiffness for low strength tissues did result in statistical significance (p < 0.01) the peak stresses for each group of stiffness, Stiff-0, Stiff-1, Stiff-2, were 2.19 ± 0.07 MPa, 2.19 ± 0.04 MPa, and 2.13 ± 0.03 MPa respectively. The largest percent difference in the average values being < 2.8%. We did not find any statistically significant difference in peak stress due to fracture toughness for low (p = 0.05) or high (p = 0.11) strength tissue. Stretch associated with the peak stress was not statistically different between toughness groups for both low (p = 1.00) and high strength p = 0.99) tissues, Fig. 8. However, there were statistically significant differences in peak stretch between any of the three stiffness groups (p < 0.01) for both high and low strength tissues.

3.3.2. Material parameters influencing post-peak response

Fig. 9 illustrates how fracture toughness effects the post peak response. For instance, in Fig. 9b, cases of low toughness G11 − G22 demonstrate an abrupt failure while high toughness cases, G24 − G48, illustrate a gradual post-peak failure. In addition, the combination of the abrupt fall of the curve from the peak followed by a prolonged tail region, also observed in experiments (Sang et al., 2018), could be captured by choosing higher contrast of the fracture toughnesses between the layers. As an example, the abrupt fall from peak stress followed by prolonged tail region response can be seen for cases of contrasting fracture toughness such as a G24 case in Fig. 9b and a G12 case Fig. 9d. The post-peak responses of higher strength materials Fig. 8b are generally more abrupt than the response of lower strength materials Fig. 8a. Irrespective of material stiffness, for low material strength the post-peak behavior transitioned from abrupt to gradual failure with increasing fracture toughness in the media and adventitia layers. Interestingly, for high strength cases, the post-peak behavior was largely abrupt for Stiff-1 and Stiff-0 cases. The post-peak behavior exhibited a transition from abrupt to gradual for Stiff-2 cases.

The difference in post-peak response between toughness groups is illustrated in Fig. 9 showing the transition in post-peak response from abrupt to gradual with an increase in toughness.

4. Discussion

We studied the uniaxial failure behavior of the vascular tissue specimens with different testing geometry and boundary conditions using a cohesive-volumetric finite element method. Our computational methodology enabled the in silico realization of the entire uniaxial testing procedure including pre-failure and post-failure regimes. As a result, we could relate experimentally measurable uniaxial stress-stretch data to the intrinsic tissue material failure parameters. Our work revealed how intrinsic material failure properties, namely strength and fracture toughness, control the specimen scale tissue mechanical failure behavior. In addition, the onset and progression of the sub-tissue damage processes throughout the loading process was also revealed. We note that while the mechanical properties used in the current parametric studies were primarily motivated by experiments on human cerebral and sheep carotid tissue (Sang et al., 2018), the range of values considered here are also relevant to many other soft tissues such as kidney, liver, skin, and vaginal tissue (Brunon et al., 2010; Rubod et al., 2008; Holzapfel, 2001; Farshad et al., 1999).

4.1. Assessment of tissue uniaxial strength from the experimental data

Failure of vessel wall tissue leads to clinically significant events such as rupture of cerebral or abdominal aneurysms. To gain an in-depth knowledge of the failure process, the role of intrinsic material properties of the tissue failure mechanisms must be understood. It is important to emphasize that uniaxial tissue strength is generally estimated as the peak stress derived from the experimentally measured stress-stretch curve, while intrinsic tissue strength is a material property used in our cohesive model. Uniaxial testing of vascular tissue specimens has been extensively used to evaluate tissue failure response using many dilferent specimen shapes (Robertson et al., 2015; Shah et al., 2014; Walsh et al., 2014; Monson et al., 2005; Costalat et al., 2011). In some uniaxial experiments, the tissue was cut into rectangular shapes (Robertson et al., 2015; Pichamuthu et al., 2013; Raghavan et al., 2011; Di Martino et al., 2006), while in others dogbone shaped specimens were used (Shah et al., 2014; Ferrara et al., 2016; Smoljkic et al., 2017; Garcia-Herrera et al., 2012). Sandpaper, glue, rubber, or a combination of these materials have been used at the interface in an effort to avoid sample slippage at the grips during testing as well as to reduce stress concentrations in the specimen near the grips (Robertson et al., 2015; Pichamuthu et al., 2013; Ferrara et al., 2016; Smoljkic et al., 2017; Teng et al., 2009; Sichting et al., 2015). Further, some researchers explicitly discarded the samples that failed at the clamps (Holzapfel et al., 2005; Pichamuthu et al., 2013; Raghavan et al., 1996, 2011; Ferrara et al., 2016; Forsell et al., 2013; Shah et al., 2014). Interestingly, our group recently found experimentally that neither the clamping method nor the sample shape (dogbone versus rectangular) signilicantly impacted the strength and failure stretch for the uniaxial testing of small artery specimens (Sang et al., 2018). In this manuscript we recapitulated experimental sample configurations utilized in Sang et al (2018), in which a slightly curved contour specimen shape resulted in the desired consistent failure at the middle region. Also important in that study was the practical considerations of cutting such a small specimen size without creating jagged edges that could initiate failure. Our simulation results, in agreement with experiments, showed that prior to the onset of failure the middle region stretches enough to achieve a uniform cross section. As a consequence, the stress state became uniform at the middle, see Figure 6, which is the primary motivation behind using a dogbone sample.

We found that for all specimen shapes and insert materials considered here, the magnitude of peak stress is similar to the intrinsic tissue strength though the peak stress was consistently higher. For example, the average peak stress for the low and high strength cases were 8.60% and 4.37% greater than their respective input intrinsic strengths for all tissue stiffness cases. In addition, the percent difference in the peak stress between dogbone and rectangle protocols for the low and high strength cases were 2.39% and 1.81%. Therefore, we did not find any substantial effect from specimen shape and boundary condition, or tissue stiffness on the peak stress assessed from the specimen scale stretch-stretch curves typically recorded from uniaxial experiments. Taken together, our simulations revealed that the uniaxial peak stress is primarily governed by the tissue intrinsic strength, and the influence of specimen shape and boundary conditions on this parameter is negligible. Our results corroborate the conclusions by Taylor et al. (2012) that below a critical initial flaw size within the specimen, the failure of the tissue is strength-controlled, and as a result experimentally derived peak stress tends to the tissue intrinsic tissue strength.

Our stress analysis of the uniaxially loaded dogbone shaped specimens demonstrated the desired mid-region stress concentrations for all cases. As a result, the dog-bone shaped specimens always failed at the middle, an observation corroborated by the experimental observations on arterial tissue reported in Sang et al. (2018). All simulations reported here failed at the middle, but it is common in experiments for tissue to also fail at the clamp due to gripping damage (Sang et al., 2018). We observed an axial stress concentration at the clamp after clamping pressure was applied, Figs. 6 & 7. However, the stress concentration remained relatively stable and no cracks propagated through the thickness near the clamping location. Our result is in contrast to the findings in Peloquin et al. (2016), where the stress distribution in the dogbone sample of the bovine meniscus tissue was non-uniform even in the mid-region for dog-bone specimens when sandpaper inserts were used. This difference in stress distribution may stem from differences in biomechanical properties of the meniscus tissue and the arterial wall as well as different grip inserts being used. Further, the differences in specimen sizes (meniscus tissue grip to grip distance was four times higher than that considered in our work) may also play a role in the evolution of the stress distribution.

4.2. Relationship between tissue failure properties and failure progression

Prior to this work, there was very limited data on the relationship between tissue failure progression and the tissue failure properties. Our analysis demonstrates that tissue failure is a continuous process starting from localized softening followed by initiation, propagation and possible coalescence of multiple tears leading to complete separation of the specimen. It is of importance that uniaxial peak stress corresponds to localized tissue softening and partial damage in the highly stressed regions, rather than complete tissue failure. For example, as shown in Fig. 6 and Fig. 7, peak stress (Point B in the figure) was always associated with the reduction of the internal damage parameter, $\mathcal{I}$, from its initial value, indicating onset of material softening. This observation is consistent with the experimental results from Sang et al. (2018).

We consistently found that the progression of the damage process to complete tissue failure ($\mathcal{I}=0$) was influenced by the shape of the post-peak response. For example, as shown in Fig. 6 row 1, the dilference between the fast fracture in the sample shown in panel (a) is obvious in comparison to the more gradual failure of the sample in (b). To further quantify the dependence of post peak response on the intrinsic tissue material properties, we used the R Factor (Nittur et al., 2008; Sang et al., 2018), or the area under the post-peak curve divided by the area under the total response. We found that the R Factor is primary influenced by the tissue toughness, and a strong positive correlation between these two parameters was observed, see Online Resource 3.

We found that for low strength tissues, the higher tissue fracture toughness consistently resulted in a gradual post-peak response. However, for tissues with high strength, the fracture process tends to be abrupt for all combinations of media and adventitia fracture toughnesses studied. High strength tissues can absorb substantial amounts of strain energy prior to the onset of failure. As a result, there is a build up of mechanical energy released during fast tear propagation. In contrast, lower pre-failure strain energy enables the tissue to absorb additional post-peak energy before the complete failure. Additionally, we found that for samples where the fracture toughness of the media was at least twice that of the adventitia, the failure was consistently initiated within the media layer, consistent with uniaxial experiments (Sang et al., 2018; Iliopoulos et al., 2009).

Taken together, our results reveal that elastic and failure parameters influence different regimes of the entire stress-strain curve. To wit, elastic parameters control the mechanical behavior in the pre-peak regime while magnitude of peak stress is determined by the intrinsic tissue strength. Interestingly, we found that the shape of the post-peak regime is primarily governed by the tissue fracture toughness, while high intrinsic strength can further modulate it.

While tissue toughness has been identified as an important failure parameter, the evaluation of this parameter is difficult for soft tissues (Taylor et al., 2012). Hence, the post-peak behavior can be a surrogate to the direct evaluation of this failure parameter. In addition, our experiments reported in Sang et al. (2018) revealed that the tissue can bear substantial load even beyond the peak of the stress-stretch curve, and this additional load bearing capability is related to the post-peak biomechanical behavior. We anticipate that this mechanism of additional load bearing may have an important implication on the overall progression and eventual rupture of cerebral aneurysms as well as other soft tissues. Indeed, it has been suggested that causal progression of aneurysm failure may result from a “slow rupture” process (Steiger et al., 1989). Revealing the relationship between post-peak mechanical behavior and aneurysm rupture will, however, require additional work.

Our observations suggest that, at least for the uniaxial loading situations, there is a decoupling between the pre-peak and post-peak behavior. While the first part is governed by tissue stiffness, the second regime solely depends on the tissue failure properties. Further studies are required to ascertain whether this observation can be extended to general loading scenarios. Taken together, our results suggest that accurate quantification of tissue failure properties will require reliable measurements of tissue material properties in both the pre- and post-peak regimes.

4.3. Cohesive-volumetric finite element method as a modeling paradigm to assess tissue failure behavior

In silico studies of uniaxial testing offer several complementary features to physical experiments and provide a pathway for experimental design of failure testing. First, it is possible to systematically vary the experimental conditions, such as clamping method, while holding the mechanical properties and sample geometry (e.g. wall thickness) fixed. It is also possible to run numerous tests inexpensively that span the entire feasible space of the material properties in a controlled manner, enabling the exploration of the elfect of a wide range of material properties (for example, range of stiffness, strength, and fracture toughness). The material properties of the various wall components can be independently varied in such parametric studies. Furthermore, the stress distribution within the specimen, including local stress concentrations, can be assessed throughout the failure process making it possible to identify areas of high stress concentration. In addition, appropriate in silico models can reveal sub-tissue biomechanical failure phenomena such as regions of subfailure damage and stress release following damage. However, modeling post-peak softening response of the specimen remains numerically challenging.

We have introduced the cohesive-volumetric finite element method to predict the post-peak region of the stress-stretch curve. Fig. 6 demonstrates that our numerical method can accurately recapitulate the entire experimental stress-stretch curve until complete specimen failure. Cohesive zone methodology has recently been used to model biomechanics of delamination induced failure of vascular tissues. For example, delamination of atherosclerotic plaque from the abdominal aorta was modeled by placing cohesive elements at the interface of the plaque and the aortic media (Leng et al., 2015). Different research groups have proposed this technique to model arterial dissection, the delamination of arterial layers (Gasser and Holzapfel, 2006; Ferrara and Pandolfi, 2008; Noble et al., 2017; Wang et al., 2017). A single well-defined delamination propagation path was modeled in all these works, and accordingly the cohesive elements were assigned a priori on that path. In contrast, multiple micro-tears can initiate from arbitrary locations within the uniaxial testing specimen. These micro-tears can propagate and coalesce to eventually form a dominant tear that cleaves the specimen in separate pieces.

To model this scenario, we placed the cohesive elements along all the edges of the volumetric finite elements. Removal of the mesh bias on the potential tear propagation paths required randomization of the shape of these volumetric elements. Additionally, the cohesive model introduced in this work uses an internal damage parameter, $\mathcal{I}$, to indicate the status of the failure process locally. The evolution of this parameter enables the visualization of even regionally varying gradual failure processes anywhere within the tissue. This information is not accessible by typical uniaxial experiments. Thus, the CVFE method used here has the potential to complement our knowledge of tissue failure mechanisms by linking the biomechanical testing results at specimen scale with the localized evolution of the failure processes at the tissue scale. Quantification of the biomechanical integrity at the tissue microstructure scale as well as the organ scale is essential to elucidate the failure mechanics of the soft tissue.

4.4. Study limitations

Our computational model of the uniaxial tensile testing of arterial tissue has the following limitations. As discussed in Cortes et al. (2010), shortened fibers within a distribution cannot be excluded from contributing to the strain energy function using the GST formulation in uniaxial tension. However, given that the fiber dispersion in our model is very low (κ = 0.05) and the mean fiber direction is coincident with the loading direction, all the fibers within the fiber families are expected to be in extension. The internal parameter, $\mathcal{I}$, depends only on the elastic behavior of the tissue while future constitutive relations could include dynamic effects from the viscoelasticity of tissue. Finally, we have assumed a bilinear shape for the cohesive traction separation law in this study, because the actual shape of the failure law for arterial tissues is not known. However, it has been shown that the shape of the cohesive traction separation law does not significantly influence the overall mechanical behavior of the material (Alfano, 2006). Thus, our assumption of bilinearity of the failure law may not alter our findings reported in this article.

4.5. Conclusions

In summary, our parametric study shows the post peak response in failure testing deserves greater attention than it currently receives, as it provides information about the failure process that could have substantial consequences for living tissues. For example, a gradual failure process such as seen for cases with higher fracture toughness may allow for tissue remodeling, possibly avoiding rupture. Tissues that fail abruptly will likely not have adequate time to remodel and could be more prone to catastrophic failure or rupture. Also shown and expected, the uniaxial ultimate stress, a commonly used experimental parameter to characterize material strength, is controlled by intrinsic strength, a local material property. Finally, in agreement with Sang et al. (2018), the test protocol, specimen shape, & insert material does not greatly impact pre-peak, peak, or post-peak response. This provides important evidence that results of uniaxial failure testing are robust to common variants in experimental protocol, at least for the size and conditions used here.