An Approach to Quantify Anisotropic Multiaxial Failure of the Annulus Fibrosus

Abstract

Tears in the annulus fibrosus (AF) of the intervertebral disk (IVD) occur due to multiaxial loading on the spine. However, most existing AF failure studies measure uniaxial stress, not the multiaxial stress at failure. Delamination theory, which requires advanced structural knowledge and knowledge about the interactions between the AF fibers and matrix, has historically been used to understand and predict AF failure. Alternatively, a simple method, the Tsai-Hill yield criteria, could describe multiaxial failure of the AF. This yield criteria uses the known tissue fiber orientation and an equation to establish the multiaxial failure stresses that cause failure. This paper presents a method to test the multiaxial failure stress of the AF experimentally and evaluate the potential for the Tsai-Hill model to predict these failure stresses. Porcine AF was cut into a dogbone shape at three distinct angles relative to the primary lamella direction (parallel, transverse, and oblique). Then, each dogbone was pulled to complete rupture. The Cauchy stress in the material's fiber coordinates was calculated. These multiaxial stress parameters were used to optimize the coefficients of the Tsai-Hill yield. The coefficients obtained for the Tsai-Hill model vary by an order of magnitude between the fiber and transverse directions, and these coefficients provide a good description of the AF multiaxial failure stress. These results establish both an experimental approach and the use of the Tsai-Hill model to explain the anisotropic failure behavior of the tissue.

Introduction

Tears in the annulus fibrosus (AF) of the intervertebral disk (IVD) occur both radially and circumferentially [1,2] and lead to further degeneration of the IVD [3,4]. However, determining the multiaxial loading conditions that lead to AF tears is challenging, as is determining how the different loading conditions drive tissue damage. Previous experiments on AF failure have used ASTM standard pull-to-failure tests, peel testing, lap shear, and delamination experiments [5–21] (Table 1). While these experiments provide valuable information about the amount of uniaxial stress that cause AF failure, the multiaxial stress states of the AF are often ignored. Thus, the potential for simple failure theories that can predict the multiaxial failure strength of the AF should be established.

Table 1.

Summary of experimental studies of the annulus fibrosus (AF) that were pulled to failure

Loading modality	Tissue source	Loading direction	Yield stress	Ultimate stress	Ultimate strain	Citation
Uniaxial tension	Human	Circumferential	—	3.2 MPa (AO) 0.9 MPa (AI) 1.1 MPa (PO) 1.0 MPa (PI)	10% (AI) 14% (AO) 11% (PI) 18% (PO)	[5]
Uniaxial tension	Human	Vertical	—	3.9 MPa (AO) 8.6 MPa (PO)	65% (AO) 34% (PO)	[8]
Uniaxial tension	Ovine	Vertical	—	4.46 MPa	50.80%	[16]
Uniaxial tension	Human	Radial	0.23 MPa (Ave) 0.25 (Inner)0.27 (Mid) 0.21 (Outer0.24 (Ant) 0.24 (PL)	0.3 MPa (Ave)0.34 (Inner) 0.33 (Mid)0.28 (Outer) 0.32 (Ant) 0.31 (Post)	124% (Ave)139% (Inner) 102% (Mid) 132% (Outer)125% (Ant) 123% (Post)	[9]
Uniaxial tension	Bovine	Circumferential	—	2.94 MPa	21.30%	[17]
Uniaxial tension	Human	Longitudinal	—	2.37 MPa	19.00%	[21]
Uniaxial tension	Human	Circumferential	—	1.27 MPa	17.00%	[21]
Uniaxial tension	Human	Radial	—	0.195 MPa	91.00%	[21]
Uniaxial tension	Bovine	Circumferential	—	11.19 MPa	54%	[12]
Uniaxial tension	Bovine	Circumferential	—	2.25 MPa–4.67 MPa	47%–55%	[13]
Uniaxial tension	Bovine	Circumferential	—	2.25 MPa	36%	[11]
Uniaxial tension	Bovine	Oblique relative to lamella	0.5 MPa	0.77 MPa	94%	[19]
Uniaxial tension	Porcine	Oblique relative to lamella	0.48 MPa	0.7 MPa	112%	[19]
Uniaxial tension	Ovine	Oblique relative to lamella	0.28 MPa	0.39 MPa	154%	[19]
Uniaxial tension	Porcine	Oblique relative to lamella	1.09 MPa	1.44 MPa	117%	[7]
Uniaxial tension	Human	Oblique relative to lamella	—	10.3 MPa (AO) 3.6 MPa (AI)5.6 MPa (PO)5.8 MPa (PI)	9.2% (AO) 11.3% (AI) 12.7% (PO)15.4% (PI)	[18]
Uniaxial tension	Ovine	Oblique relative to lamella	—	0.2 MPa (PL) 0.25 MPa (Ant)	—	[10]
Peel	Bovine	Between lamella	—	0.37 N/mm	—	[6]
Peel	Human	Between lamella	—	0.57N/mm (Ant) 0.53 N/mm (Post)	—	[7]
Ring test	Rat	Circumferential	0.43 MPa	0.63 MPa	140%	[15]
Lap shear	Porcine	Between lamella	—	0.3N (Peak Load)	—	[14]

To understand AF failure, composite analysis techniques, including delamination theory [1,6,9,22], have been applied. However, delamination theory requires an accurate understanding of the fiber structure and the interactions between the fibers and the matrix. Meanwhile, a simpler failure model, the Tsai-Hill, only requires knowing the primary fiber direction. The Tsai-Hill model was originally created to describe multiaxial failure of anisotropic polymers and composites [23]. Recently, the model has provided adequate characterization of tear formation in biological tissues including the aorta [24]. Meanwhile, a similar failure theory (Tsai-Wu) has been used in studying fracture of bones [25,26] and has been used in finite element models of the AF [2,27]. However, coefficients for the Tsai-Hill (or Tsai-Wu) yield criteria have yet to be determined experimentally for AF failure.

Therefore, the goal of this study was to develop an experimental approach that could be used to identify the Tsai-Hill model coefficients for the AF. The approach considers the anisotropy of the tissue using the primary orientation of the lamellae [28].

Methods

Experimental Analysis.

Intervertebral disks were isolated from the lumbar spine (L1–L5) of healthy male Yorkshire swine (80–90 kg) sacrificed as part of an unrelated study. Since some previous work has shown no mechanical difference between porcine lumbar spinal levels [29,30], the level was not recorded and samples were taken from four lumbar spine levels (L1–L2, L2–L3, L3–L4, L4–L5). Then, the anterior region of the AF was isolated by making a vertical cut at the point where the AF curves into the right and left sides of the disk (Fig. 1(a)). Although the posterior side may be more clinically relevant due to the presence of posterior herniation, the posterior AF from our chosen porcine source was too small to notch and grip in these failure studies.

Fig. 1.

The process of experimentally testing the annulus fibrosus based on fiber direction: (a) isolate the annulus fibrosus, (b) create a dogbone shape relative to the primary lamella direction and load the tissue, and (c) calculate the stress tensor in the loading coordinates, then rotate into the fiber coordinates

To collect multiaxial anisotropic experimental failure properties, the primary fiber direction relative to the loading direction needs to be varied [24]. In the current study, the lamellar orientation in the transverse plane is the primary fiber direction and the ply structure of the collagen fibers within each lamellae was not considered. Therefore, the dogbone shape in each sample was cut such that the primary direction of the lamella was located parallel (N =2), transverse (N = 2), or at an oblique angle to the primary loading direction (N = 3, Fig. 1(b)). The width and thickness of the narrowest region of the dogbone was measured with a caliper and the cross-sectional area was calculated. The average notch thickness was 2.7 ± 1.3 mm and the average cross-sectional area was 9.7 ± 5.6 mm². The cross-sectional area for each sample was used to calculate tissue stress at the failure site.

Prior to mechanical testing, the AF was preconditioned for five cycles at a rate of 0.5% per second up to 10% grip stretch. This grip stretch was calculated by dividing the displacement of the loading arms by each sample's initial grip-to-grip distance (13.5 ± 2.9 mm). DIC analysis revealed the global tissue stretch for all samples was within 3% of the grip-to-grip stretch. Then, the dogbone shape was pulled at 0.5% per second until total tissue rupture occurred, as defined by the force on each loading arm reaching zero (Fig. 1(b)). Prior to rupture, the initial failure stress was determined at the first moment the force dropped by at least 5%. This initial drop in the force correlated with visible signs of tissue failure. These visible failure signs were used to identify the failure location and calculate the failure stress. Samples that did not fail at the narrowest region of the dogbone were not used for the analysis (∼40%). The ultimate stress state was also identified as the maximum stress experienced in the primary loading direction.

All stress analysis was completed using the Cauchy stress, which was calculated by assuming the tissue behaves as an incompressible material with little to no shear strain in the loading coordinate system. Therefore, the Cauchy stress was calculated as the force divided by the instantaneous cross-sectional area, where the instantaneous area is defined as the original area (measured with a caliper) divided by the tissue stretch. Based on previous work, the assumption about incompressibility may not be accurate for the AF [31–34]. Thus, a second analysis was performed where the Cauchy stress was calculated using an instantaneous cross-sectional area based on previously measured Poisson's ratios of the AF [31]. This approach and the results can be seen in Table and Figure available in the Supplemental Materials on the ASME Digital Collection.

Since the Cauchy stress analysis was calculated in the loading coordinate system, the stress tensor had to be rotated into material-based fiber coordinates. To transform the Cauchy stress from loading coordinates to fiber coordinates, a two-dimensional rotation matrix was created by measuring the primary lamella angle relative to the primary loading direction (Fig. 1(c)). This two-dimensional coordinate system assumes the tissue operates under plane stress conditions. Since the ratio of the initial grip-to-grip distance to the tissue thickness was always greater than 3.5, this assumption seemed valid. The primary direction was identified using images of the tissue mounted in the uniaxial loading system and taken prior to loading. The fiber stress, ${\sigma }_{\mathrm{fiber}}$, transverse stress, ${\sigma }_{\mathrm{trans}}$, and shear stress, $\tau ,$ at the initial failure point and the ultimate stress point were recorded for all AF samples, and then used to find coefficients for the Tsai-Hill model.

Tsai-Hill Yield Criteria.

The Tsai-Hill model is designed to describe anisotropic failure of composite materials undergoing tensile and shear stresses [23,35,36]. The model claims that failure occurs when $\Phi$ in Eq. (1) equals 1

$$\mathrm{\Phi }={\left(\frac{{\sigma }_{\mathrm{fiber}}}{A_{\mathrm{fiber}}}\right)}^2-\hspace{0.17em}\frac{{\sigma }_{\mathrm{fiber}}}{A_{\mathrm{fiber}}}\times \frac{{\sigma }_{\mathrm{trans}}}{A_{\mathrm{trans}}}+{\left(\frac{{\sigma }_{\mathrm{trans}}}{A_{\mathrm{trans}}}\right)}^2+{\left(\frac{\tau }{A_{\mathrm{shear}}}\right)}^2$$ (1)

In Eq. (1), ${\sigma }_{\mathrm{fiber}},\hspace{0.17em}{\sigma }_{\mathrm{trans}},\hspace{0.17em}\mathrm{and}\hspace{0.17em}\tau$ are the multiaxial failure stress states as determined by the experimental data. The constants $A_{\mathrm{fiber}},\hspace{0.17em}{A}_{\mathrm{trans}},\hspace{0.17em}\mathrm{and}\hspace{0.17em}{A}_{\mathrm{shear}}$ were found by optimizing the fit of the experimental data. A total of seven experimental datasets were used to optimize the Tsai-Hill parameters via a linear least square fitting algorithm such that $\Phi$ was close to 1. The goodness of fit for the Tsai-Hill model was visualized using the equivalent stress magnitude ( ${\sigma }_{\mathrm{eq}}$, Eq. (2)) and a root-mean-square (RMS) error analysis. The Von Mises yield criterion for this data was also optimized to verify that isotropic fits could not accurately explain experimental data

$${\sigma }_{\mathrm{eq}}^2\hspace{0.17em}={\sigma }_{\mathrm{fiber}}^2+\hspace{0.17em}{\sigma }_{\mathrm{trans}}^2+3\times {\tau }^2-{\sigma }_{\mathrm{fiber}}\times {\sigma }_{\mathrm{trans}}$$ (2)

Results

Experimental analysis shows that all AF samples can be cut into dogbone shapes and loaded parallel, transverse, and oblique angles relative to the primary lamella direction. Changing the primary lamellar angle relative to loading caused differences in the failure mechanism. Parallel loading often resulted in fibers breaking and pulling out of the surrounding matrix (Fig. 2(a)). Transverse failure appeared to occur via interlaminar failure (Fig. 2(b)). Finally, oblique loading resulted in initial sliding of lamina, typically followed by fiber sliding (Fig. 2(c)).

Fig. 2.

Failure of the annulus fibrosus (AF) based on loading direction. (a) Parallel loading appears to cause fiber pull out (arrows), (b) transverse loading appears to cause interlaminar splitting (arrows), (c)oblique loading shows fiber sliding (arrows), (d)–(f) Stress versus stretch curves for all tissue samples in the loading coordinate system revealed differences in failure magnitude, stretch at failure, and behavior between (d) parallel, (e) transverse, and (f) oblique loading cases. Stretch was measured between the grips and actual tissue stretch could vary. Each line type (solid or dashed) is associated with one sample. The plus sign indicates the initial yield point, while the star indicates the ultimate stress.

Stress-stretch curves also showed differences between the loading directions. Parallel loading resulted in catastrophic tissue failure, in which the stress—stretch curve showed a large, abrupt decrease in force very soon after the initial failure (Fig. 2(d)). Failure stresses in the loading direction were almost an order of magnitude larger than stresses calculated for all other directions (Table 2). Transverse loading resulted in a series of subcritical failures as identified by small but significant drops in the force during loading prior to the ultimate stress (Fig. 2(e)). Failure stresses for the transverse case were the smallest observed (Table 2). Oblique loading experiments showed both subcritical failures and large abrupt decreases in forces after the initial failure event (Fig. 2(f)). Finally, the loading arms were not always on the same axis, so some shear loading was present during all failure experiments (Fig. 2(d–f)).

Table 2.

Summary of the multiaxial stress associated with experimental multiaxial failure of the annulus fibrosus (AF)

		Sample angle (deg)	σ fiber	τ	σ trans	Equivalent stress (MPa)
Initial stress	Parallel	0	8.50	0.09	0.0	8.51
		0	10.99	0.31	0.0	11.0
	Oblique	25	2.48	0.84	0.24	2.78
		30	2.24	0.84	0.23	2.59
		47	0.60	0.51	0.40	1.03
	Transverse	90	0.0	0.04	0.60	0.60
		90	0.0	0.24	1.40	1.46
Ultimate stress	Parallel	0	9.26	0.21	0.0	9.27
		0	10.99	0.31	0.0	11.0
	Oblique	25	2.58	0.86	0.24	2.88
		30	2.24	0.84	0.23	2.59
		47	0.89	0.74	0.57	1.50
	Transverse	90	0.0	0.33	1.15	1.29
		90	0.0	0.29	1.80	1.87
$\left[\begin{array}{cc}{\sigma }_{\mathrm{fiber}}& \tau \\ \tau & {\sigma }_{\mathrm{trans}}\end{array}\right]$			Initial tear coefficients		Ultimate tensile coefficients
			Afiber = 10.66 MPa		Afiber = 10.91 MPa
			Atrans = 1.33 MPa		Atrans = 1.74 MPa
			Ashear = 0.82 MPa		Ashear = 0.84 MPa

Fitting the experimental data of Table 2 to the Tsai-Hill model revealed fit coefficients of $A_{\mathrm{fiber}}=10.66\hspace{0.17em}\mathrm{MPa},\hspace{0.17em}{A}_{\mathrm{trans}}=1.33\hspace{0.17em}\mathrm{MPa},\mathrm{and}\hspace{0.17em}{A}_{\mathrm{shear}}=0.82\hspace{0.17em}\mathrm{MPa}$ for the initial failure and $A_{\mathrm{fiber}}=10.91\hspace{0.17em}\mathrm{MPa},\hspace{0.17em}{A}_{\mathrm{trans}}=1.74\hspace{0.17em}\mathrm{MPa},\hspace{0.17em}\mathrm{and}\hspace{0.17em}{A}_{\mathrm{shear}}=0.84\hspace{0.17em}\mathrm{MPa}$ for the ultimate tensile stress fits (Table 2). These coefficients capture the observed failure stresses well. The calculated RMS error between the predicted $\Phi$ and theoretical $\Phi$ was 0.40 for initial yield stress and 0.21 for the ultimate tensile stress. The equivalent stress RMS errors of the Tsai-Hill model (1.8 MPa initial stress and 1.7 MPa for ultimate stress) are much lower than the RMS error of the Von Mises Fit (2.98 MPa, Fig. 3). Visualization of the fits using the equivalent stress shows that the Tsai-Hill anisotropic failure theory, applied to both the initial failure stress and the ultimate stress, can describe failure in the AF much better than the isotropic Von Mises Yield criteria. When the Poisson's ratios of the AF are used to calculate the Cauchy stress instead of an incompressibility assumption, similar fits of the Tsai Hill model with different coefficient are observed (See Table 1 and Figure 1 available in the Supplemental Materials on the ASME Digital Collection). The most drastic change in the coefficients can be seen in the $A_{\mathrm{fiber}}$ and $A_{\mathrm{shear}}$, while $A_{\mathrm{trans}}$ doesn't change between the two fits. These differences highlight the importance of establishing accurate volumetric changes and Poisson's ratios of the AF.

Fig. 3.

The Tsai Hill Yield criteria fit our experimental data better than the Von mises yield criteria for the initial yield stress (a) and the ultimate stress (b)

Discussion

In this work, we used a multidirectional experimental protocol to evaluate the potential of the Tsai-Hill model to predict the anisotropic failure behavior of the AF. Our results suggest that the Tsai-Hill yield model could be a good choice for predicting AF failure including failure predictions in finite element studies [2]. Previously, one of the most used failure theories for the AF was based on delamination theory [14,22]. However, using delamination theory to understand patient-specific tissue failure requires complex multiscale modeling [1]. In contrast, the Tsai-Hill model is a simple way to describe the contribution of multiaxial stress on tissue failure.

The Tsai-Hill model is useful in characterizing the role of anisotropy in the AF. However, the AF is also heterogeneous [10,31,37], with a stiffer outer AF than inner AF [10]. Additionally, the AF has been shown to have heterogeneous failure properties [16]. In the current approach, the effects of heterogeneity were considered to be mitigated by cutting the location of the narrowest region of the dogbone in approximately the middle of the AF for every sample. Additionally, the size of the narrowest region of the dogbone was small (1–4 mm), so mechanical variations across the region would be minimal. Differences between the stress in our study versus previous studies could be attributed to heterogeneity in the AF failure stress (Table 1). Future work could modify our technique to test failure in other tissue regions and predict the variation in the failure coefficients across the AF.

A few interesting failure mechanisms associated with the unique structure of the AF can be observed in this paper. First, the oblique loading case exhibited laminar sliding often followed by fiber sliding or pull out. The laminar sliding is expected since the oblique loading case should create high shear stresses between the lamina [22]. The area between the lamina is considered to be a weak region of the AF [22]. Additionally, the presence of fiber pullout during the oblique case may occur because the collagen fibers can reorient into the direction of pull [38,39]. The second interesting result is the presence of a stress minimum near 52 deg and 65 deg for the ultimate failure stress and initial failure stress respectively (Fig. 3). These minima are caused by the optimization of the Tsai Hill model where the shear stress coefficient $A_{\mathrm{shear}}$ in this study was smaller than the transverse stress coefficient $A_{\mathrm{trans}}$. Our observation of a minima may indicate that shear stresses contribute more to failure than normal stress. This observation is similar to a shear stress-induced delamination theory, where increased shear stresses between the lamina is believed to be a clinical cause of delamination and failure between AF laminar sheets [9,22]. However, the reader is cautioned to not overinterpret our results, the results from this study must be added to other failure studies to further understand mechanistic failure of the AF.

Additionally, this work is not without limitations. First, the failure analysis performed in this study all occurred on the circumferential radial plane. Thus, our analysis cannot predict how cranial-caudal loading will alter the three-dimensional failure stress state of the IVD. Nevertheless, our results show that the Tsai Hill model could be helpful in predicting AF failure. Additionally, this work was intended to show feasibility of the Tsai Hill model to explain failure of the AF. As such, this work used a relatively small sample size. More samples and further analysis should be completed to increase the reliability of the Tsai Hill theory to predict tissue failure. Finally, our porcine source included only male porcine samples because only male porcine samples were used in the other study that provided the tissue. As a result, the coefficients and trends found in this study may not hold true for female porcine samples. Adding female data to our male data may also increase the variability of our results and the goodness of fit. Since this paper shows the Tsai Hill model could be helpful to understand AF failure, future work could increase the sample size, expand testing into the cranial-caudal direction, and add female porcine samples to the analysis.

Transverse failure did not always span the entire width of the dogbone's narrow region. This behavior may be a result of the angle ply structure of the collagen fibers within each lamella. If the collagen fiber angle in each lamella were to change due to disease or aging the circumferential failure stress [13,18,19] and the $A_{\mathrm{fiber}}$ coefficient may also change. Similarly, the failure behavior of AF samples under oblique loading entailed sliding of the collagen fibers relative to each other. Thus, changes in collagen crosslinking due to disease and aging may affect the $A_{\mathrm{shear}}$ component in a similar manner to other mechanical properties of the AF [16,28,32,40]. Overall, the approach outlined in this study may be helpful in understanding failure in aged and diseased AF tissue.

Author Contribution Statement

All authors contributed to this work. JM—Data collection and analysis, interpretation of results, and paper writing. VB—Oversee experiments, data interpretation, and paper writing.

Supplementary Material

Supplementary Material

Supplementary Figures

Funding Data

• University of Minnesota (Grant No. U01 AT010326; Funder ID: 10.13039/100007249).