Comparison of the Trueness of Fits of the Biphasic Transverse Isotropic and Kelvin Models to the Tensile Behavior of Temporomandibular Joint Disc

Abstract

This technical brief explores the validity and trueness of fit for using the transverse isotropic biphasic and Kelvin models (first and second order generalized) for characterization of the viscoelastic tensile properties of the temporomandibular joint (TMJ) discs from pigs and goats at a strain rate of 10 mm/min. We performed incremental stress-relaxation tests from 0 to 12% strain, in 4% strain steps on pig TMJ disc samples. In addition, to compare the outcomes of these models between species, we also performed a single-step stress-relaxation test of 10% strain. The transverse isotropic biphasic model yielded reliable fits in reference to the least root mean squared error method only at low strain, while the Kelvin models yielded good fits at both low and high strain, with the second order generalized Kelvin model yielding the best fit. When comparing pig to goat TMJ disc in 10% strain stress-relaxation test, unlike the other two Kelvin models, the transverse isotropic model did not fit well for this larger step. In conclusion, the second order Kelvin model showed the best fits to the experimental data of both species. The transverse isotropic biphasic model did not fit well with the experimental data, although better at low strain, suggesting that the assumption of water flow only applies while uncrimping the collagen fibers. Thus, it is likely that the permeability from the biphasic model is not truly representative, and other biphasic models, such as the poroviscoelastic model, would likely yield more meaningful outputs and should be explored in future works.

1 Introduction

Tensile behaviors of many musculoskeletal tissues have been thoroughly characterized, however, data regarding the tensile properties of tissues of the temporomandibular joint (TMJ) lag behind in both quantity and validity of published research. With regards to the TMJ disc, there have not been consistent protocols to measure the viscoelastic properties, which have time and strain rate dependent behaviors. This variability of testing speed and strain step, along with different orientation of the samples, make it difficult to compare results. Moreover, there remains uncertainty around the biomechanical origins of disc failure or dysfunction, understandably, since the TMJ disc endures both tensile and compressive loads [1,2]. More recently, studies have been taking these important factors into consideration to develop protocols for tensile testing with the aim of finding a gold-standard for future research [3–13]. However, not all studies have used theoretical models to predict the tensile behaviors, and the studies that use models do not report the errors of the fits.

Of the studies that have used mathematical models, many have fit their data to combinations of standard linear models, such as variations of the Kelvin model and Maxwell model [14–17]. Beyond these classical models, Tanaka et al. used the Burger model due to its ability to represent residual strain after load removal, as well as relaxation and creep [18]. Other linear models, such as the power law model and the Prony series model, have also been implemented in attempts to accurately fit the viscoelastic behaviors of the TMJ disc in tension [19,20]. Aside from the more commonly used standard linear models, the quasi-linear viscoelastic theory and the nonlinear viscoelastic material finite element model (four-mode Maxwell model with second-order Mooney–Rivlin models as elastic elements) have also been employed in TMJ disc studies [21]. While each of these models represents great progress, they have not obtained exceptional agreement between their theoretical data and the corresponding experimental data. Therefore, exploration of other models is crucial to establish a standard for modeling viscoelastic behavior of the TMJ disc.

Our group has previously applied a transversely isotropic biphasic model to the unconfined compressive behavior of the TMJ disc [22], while others have applied it to the compressive data of articular cartilage [23,24]. However, to our knowledge, this model has never been used to describe the TMJ disc tensile properties. The linear transverse isotropic biphasic model, first described by Cohen et al. [25], models the resistance of fluid flow through solid matrix components in radial expansion. Moreover, they proposed that this resistance creates high tensile strain in the transverse axis. One of the objectives of this technical brief is to investigate for the first time the trueness of fit of the transversely isotropic biphasic model, as well as the formerly investigated Kelvin models, for predicting the tensile properties of the disc. In this study, we assumed the isotropy of the coronal plane of the strips since the fibers are bundled anterior-posteriorly at the center of the disc. Thus, the tensile properties in the sagittal and transverse planes of the center strips will be the same, and different from the coronal plane. By curve fitting our data to this model, we obtained the Young's moduli and Poisson's ratios in the transverse and axial planes, as well as the transverse permeability constant. This also assumes that water (fluid) will flow out of the collagen fibers (solid) as they are pulled closer together, squeezing water out as the space between fibers becomes tighter. Additionally, we fit our data to both the first and second order generalized Kelvin models as they have been shown to produce good fits for mandibular condyle cartilage [16,26]. Thus, we will compare different models using a 4% strain incremental stress-relaxation test for the pig TMJ disc, as well as a 10% strain stress-relaxation test for pig and goat discs to compare different species and explore the impact of the amplitude of strain on these models.

Methods

Sample Preparation.

In this study, six porcine heads and three goat heads were acquired from the local abattoir at slaughter age (6–9 months, breed and sex unknown). Only the left TMJ discs were extracted and used in this study, so a total of six porcine discs (n = 6 biological replicates) and three goat discs (n = 3 biological replicates) were harvested. The central portion of each disc was then cut anterior-posteriorly into a rectangular strip with a length of around 8 mm and width of around 5 mm. The samples were then placed into optical cutting temperature compound and frozen at −20 °C overnight. After, the superior surface of each strip was frozen to a cryotome platen using optical cutting temperature compound as previously described by Lowe et al. [27]. The strips were then sectioned in a cryotome machine at approximately 250 μm thickness with the blade cutting parallel to the strip surface. In this way, we collected multiple layers (n = 3–6 per disc) for the pig samples but only one layer for the goat samples (goat discs are thinner than pig). A 4 mm biopsy punch was then used to trim the sectioned strips into a dog-bone shape with the middle width being around 1–2 mm (Fig. 1). Following that, samples were stored in 1x phosphate buffered saline (PBS) at 4 °C until further procedure (usually within 3 days).

Fig. 1.

Dog-bone-shape strip placed between customized clamps

Sample Measurement.

The disc strips were allowed to equilibrate for 1 h at room temperature in 1× PBS. Then, the widths at the middle of each strip were measured with a caliper. The thickness was calculated as the averaged thickness measured 3 times while the sample was loaded to 0.05 N at a crosshead speed of 10 mm/min using a testing apparatus (MTS, Insight) with a 10 N load cell. The cross-sectional area at the middle of the strips was calculated as width times thickness.

Stress Relaxation Test.

The incremental stress relaxation test was conducted on an Instron mechanical testing apparatus (Instron 5566) in an attached container filled with 1× PBS at room temperature. Each end of the strip was sandwiched by 400 grit sandpapers. “Krazy Glue” (ethyl 2-cyanoacrylate) was applied between the strip and the sandpaper to prevent slipping. The samples were then gripped by customized clamps and preloaded to 0.05 N (Fig. 1). Twenty to thirty cycles of preconditioning were performed at a strain rate of 10 mm/min to 5% strain. For the 4%-strain-step stress relaxation test, three pig discs (n = 3 biological samples), each with three sectioned layers (three technical repeats), were loaded at a strain rate of 10 mm/min to 4%, 8%, 12%, and 16% strains successively or until failure with a 10-minute relaxation between strains. For the 10%-strain-step stress relaxation test, three pig discs (n = 3 biological sample) each with one sectioned layer (one technical repeat) and three goat discs (n = 3 biological samples) each with one sectioned layer (one technical repeat) were loaded at a strain rate of 10 mm/min to 10% followed by a 30-minute relaxation. Time, load and displacement were recorded at a 10 Hz frequency by the instron bluehill 2.0 software.

Curve Fitting.

The curve fitting procedure was run with matlab R2019a. The peak force was determined as the highest reading for load (N) at the end of each strain step recorded by the bluehill 2.0 software, and stress was calculated as the force (N) divided by the cross-sectional area (mm²). The viscoelastic parameters were computed using the transverse isotropic biphasic theory Eqs. (1)–(9) described in previous studies [22,28], which also detailed the steps on how the recorded data were incorporated into the equation to produce the estimated variables. As a comparison, the data were also fitted with the first and second order generalized Kelvin models [16,26].

With the assumption of transverse isotropy of the coronal plane of the strips, the transverse isotropic model determines the Young's moduli and Poisson's ratios in the transverse and axial planes (E₁, E₃ and ν₂₁, ν₃₁, respectively), and the transverse permeability coefficient (k) based on the initial values given. The constant-rate ramp condition was defined as an imposed constant strain rate $\left({\dot{\mathrm{\epsilon }}}_0\right)$, followed by a constant strain $\left({\mathrm{\epsilon }}_0\right)$. The time at which the peak load of the stress-relaxation curve occurred was defined as t₀, and only the data in the relaxation portion (t > t₀) were used for this analysis. The transverse isotropic biphasic theory equations for the relaxation portion can be written as

$$\mathrm{\epsilon }={\dot{\mathrm{\epsilon }}}_0{\mathrm{t}}_0={\mathrm{\epsilon }}_0,\hspace{1em}t\hspace{0.17em}>{\hspace{0.17em}t}_0$$ (1)

The stress–strain relationship is described by

$$\sigma \left(t\right)=E_3{\dot{\mathrm{\epsilon }}}_0{\mathrm{t}}_0-\hspace{0.17em}{E}_1\frac{{\dot{\mathrm{\epsilon }}}_0{\alpha }^2}{C_{11}k}{\mathrm{\Delta }}_3\hspace{0.17em}\times \sum _{n=1}^{\mathrm{\infty }}\frac{\exp \left(-{\alpha }_n^2{C}_{11}\left(kt/{\alpha }^2\right)\right)-\hspace{0.17em}\exp \left[-\left({\alpha }_n^2{C}_{11}k\left(t-t_0\right)\right)/{\alpha }^2\right]\hspace{0.17em}}{{\alpha }_n^2\left[{\mathrm{\Delta }}_2^2{\alpha }_n^2\hspace{0.17em}-\hspace{0.17em}\left({\mathrm{\Delta }}_1/\left(1+{\mathrm{\nu }}_{21}\right)\right)\right]},\hspace{0.17em}t\hspace{0.17em}>{\hspace{0.17em}t}_0$$ (2)

where

$${\mathrm{\Delta }}_1=\hspace{0.17em}1-\hspace{0.17em}{\nu }_{21}\hspace{0.17em}-\hspace{0.17em}2{\nu }_{31}\left(\frac{E_1}{E_3}\right)\hspace{0.17em}$$ (3)

$${\mathrm{\Delta }}_2=\hspace{0.17em}\frac{1-{\nu }_{31}^2\left(E_1/E_3\right)\hspace{0.17em}}{1+{\nu }_{21}}\hspace{0.17em}$$ (4)

$${\mathrm{\Delta }}_3=\hspace{0.17em}\left(1-{2\nu }_{31}^2\right)\hspace{0.17em}\frac{{\mathrm{\Delta }}_2}{{\mathrm{\Delta }}_1}\hspace{0.17em}$$ (5)

$$C_{11}=\hspace{0.17em}\frac{E_1\left(1-{\nu }_{31}^2\right)\left(E_1/E_3\right)\hspace{0.17em}}{\left(1+{\nu }_{21}\right){\mathrm{\Delta }}_1}\hspace{0.17em}$$ (6)

$${\mathtt{J}}_1\left(x\right)-\left(\hspace{0.17em}\frac{1-{\nu }_{31}^2\left(E_1/E_3\right)\hspace{0.17em}}{1-{\nu }_{21}-{2\nu }_{31}^2\left(E_1/E_3\right)}\right)x{\mathtt{J}}_0\left(x\right)=\hspace{0.17em}0$$ (7)

With the thermodynamic constraints

$$1-{\nu }_{31}^2\frac{E_1}{E_3},\hspace{1em}1-{\nu }_{21}^2>\hspace{0.17em}0$$ (8)

$$1-{\nu }_{21}^2-{\nu }_{31}^2,\hspace{1em}\frac{E_1}{E_3}-{2{\nu }_{21}\nu }_{31}^2\frac{E_1}{E_3}\hspace{0.17em}>\hspace{0.17em}0\hspace{0.17em}$$ (9)

In Eq. (2), α_n were the roots of Eq. (7), where J₀ and J₁ are Bessel functions of the first kind, and the number of summations (n) was the number of convergences to 0 for values of x in the range from 0 to 20 (in steps of 0.01). The root equaling zero was set to be between −0.02 and 0.04. The constants Δ₁, Δ₂, Δ₃, and C₁₁ in Eqs. (3)–(6) were calculated after Eq. (7) [25]. These constants were then used to determine the relaxation stress in Eq. (2). Equations (1)–(9) did not need to be adjusted in order to fit tensile behavior. Instead, the meanings of transverse and axial planes were changed from Cohen [25]. The axial plane runs anterior-posterior and aligns with the collagen fibers, while the transverse plane is parallel to the axial plane, and the collagen fibers are not as aligned in this direction. Therefore, the relaxation behavior still is described by E₃, but in this case, that plane should have a higher modulus than E₁, due to collagen fiber alignment. As such, we only allowed the code to look for solutions for E₁ that were between 0.3 to 0.999 of E₃. We further bounded solutions for k between 10⁻¹⁹ and 10⁻¹⁶.

From multiple fittings, we observed that values for ν₂₁ and ν₃₁ remained fairly constant and therefore were set to ν₂₁ = 0 and ν₃₁ = 0.3. Observations of the tissue behavior during the tensile test also confirmed that these Poisson's ratios make physical sense. As in Cohen et al. 1998 [25], E₃ is calculated as the average stress at equilibrium divided by the strain step

$$E_3={\sigma }_{\mathrm{eq}}/{\dot{\mathrm{\epsilon }}}_0{\mathrm{t}}_0$$ (10)

For this study, ${\sigma }_{\mathrm{eq}}$ is calculated as the average stress for the last 40% of the total test time. Then, the initial guess for E₁ was set to 0.5 times of E₃. The initial guess for k was estimated in a similar method as described by Yin and Elliott 2004 for tensile behavior of tendon fascicles [29], with

$$k\cong \frac{{\alpha }^2}{{\alpha }_1^2{C}_{11}\left(t_2-t_0\right)}\mathrm{l}\mathrm{n}\frac{\sigma \left(t_1\right)t_0/t_1-E_3{\epsilon }_0}{\sigma \left(t_2\right)-E_3{\epsilon }_0}$$ (11)

where $t_1<t\le t_2$ is the region early in the relaxation response where most of the stress decay occurs. In this study, t₁ was set to 0.2 s, and t₂ was set to 150 s.

After the initial guess, a procedure was performed in matlab R2019a to give optimized E₁ and k values that can best fit the experimentally obtained relaxation data using the least square method. Depending on the boundary set for the viscoelastic parameters, multiple answers might be generated due to the nature of the equation. The final parameters complied with defined thermodynamic restrictions [30].

As described in previous studies of the tensile properties of the mandibular condyle cartilage in pigs [16,26], the first order generalized Kelvin model was developed by assuming the tested samples to be Kelvin solid, which can be represented with a mechanical circuit composed of two components connected in parallel: (1) a Maxwell fluid (a spring of stiffness K₂ and dashpot of damping coefficient μ), and 2) an elastic solid (a spring of stiffness K₁)

$$\sigma \left(t\right)=\frac{E_{\mathrm{r}}u}{Z}\times \left\{1+(\frac{{\tau }_{\mathrm{\sigma }}}{{\tau }_{\mathrm{\epsilon }}}-1)e^{(-(t-t_0)/{\tau }_{\mathrm{\epsilon }})}\right\}$$ (12)

$$E_{\mathrm{r}}=K_1$$ (13)

$${\tau }_{\mathrm{\epsilon }}=\frac{\mu }{K_2}$$ (14)

$${\tau }_{\mathrm{\sigma }}=\frac{\mu }{K_1}\left(1+\frac{K_1}{K_2}\right)$$ (15)

$$E_{\mathrm{i}}=K_1+K_2\hspace{0.17em}$$ (16)

Equations (13)–(16) can be derived into

$$E_{\mathrm{i}}=\frac{{\tau }_{\mathrm{\sigma }}}{{\tau }_{\mathrm{\epsilon }}}{E}_{\mathrm{r}}$$ (17)

The second order generalized Kelvin model is developed by adding another Maxwell element in parallel with the first order Kelvin element as described in previous studies [16,26]

$$\sigma \left(t\right)=E_r\left[1\hspace{0.17em}+\hspace{0.17em}\left(\frac{{\tau }_{{\sigma }_1}}{{\tau }_{{\epsilon }_1}}\hspace{0.17em}-\hspace{0.17em}1\right)\mathrm{e}\mathrm{xp}\left(-\hspace{0.17em}t/{{\tau }_{\epsilon }}_1\right)+\left(\frac{{\tau }_{{\sigma }_2}}{{\tau }_{{\epsilon }_2}}\hspace{0.17em}-\hspace{0.17em}1\right)\mathrm{e}\mathrm{xp}\left(-\hspace{0.17em}t/{\tau }_{{\epsilon }_2}\right)\right]{\epsilon }_0\hspace{0.17em}$$ (18)

$$E_{\mathrm{i}}=\frac{\sigma \left(0\right)}{{\epsilon }_0}=E_{\mathrm{r}}\left(\frac{{\tau }_{{\mathrm{\sigma }}_1}}{{\tau }_{{\mathrm{\epsilon }}_1}}\hspace{0.17em}+\hspace{0.17em}\frac{{\tau }_{{\mathrm{\sigma }}_2}}{{\tau }_{{\mathrm{\epsilon }}_2}}\hspace{0.17em}-\hspace{0.17em}1\right)$$ (19)

For the first order generalized Kelvin model, the relaxation modulus (E_r) was set to 30 MPa, stress relaxation time constant (τ_ε) was set to 50 s, and creep time constant (τ_σ) was set to 100 s as the initial guesses in Eq. (12), where σ was stress, u was the specimen deformation, and Z was the specimen height. For the second order generalized Kelvin model, E_r, τ_ε1, τ_σ1, τ_ε2, and τ_σ2 were set to 3 MPa, 10 s, 200 s, 10 s, and 200 s, respectively. The optimization of E_r, τ_ε, and τ_σ for the first order and E_r, τ_ε1, τ_σ1, τ_ε2, and τ_σ2 values for the second order was performed using the matlab R2019a curve fitting package. Least square analysis was used to fit each viscoelastic model to the stress relaxation data. The instantaneous modulus (E_i) of the samples for each strain step can be calculated by applying the E_r, τ_ε, and τ_σ values from the first order curve-fitting, and E_r, τ_ε1, τ_σ1, τ_ε2, and τ_σ2 values from the second order curve-fitting into Eq. (17).

In order to evaluate the trueness of fit of the transverse isotropic model and the first and second order generalized kelvin models, we computed the mean absolute error, the root mean squared error (RMSD), and the coefficient of determination or R².

Results

Transverse Isotropic Biphasic Model Fit for 4% Strain Increments.

The transverse isotropic biphasic theory equations seemed to provide a moderate curve-fit to the relaxation behavior at low strain (0–4%) (Fig. 2). When looking at the trueness of the fit (Table S1 available in the Supplemental Materials on the ASME Digital Collection), the mean absolute error ranged from 0.17 to 661, the RMSD from 0.005 to 0.33, and the R² ranged from 0.42 to 0.94. We also observed that with increasing strain, all of the error calculations were higher. The calculated value for E₃ and the fitted values for E₁ and k were reported after the curve-fitting optimization for each sample was performed (Table S2 available in the Supplemental Materials on the ASME Digital Collection). The 12–16% strain-step data were not available for pig 2 layer 2 and pig 3 layer 3 due to the failure of the testing samples. The data were pooled by combining the replicates of each biological repeat, which were then calculated for the average (ave.) and standard deviation (std.) (Table 1). We found that E₁ and E₃ showed positive linear regression, while k showed a negative power regression in relation to strain (Fig. 3).

Fig. 2.

An example of the transverse isotropic biphasic model fit for 0–4% strain step in a load intensity-time (σ–t) curve. Thin line: optimized fitting curve. Thick line: experimental data.

Table 1.

Transverse isotropic biphasic model fit outcomes for 4% strain increments

	E1 (MPa)		E3 (MPa)		k (×10−18m4/N·s)
Strain step	Ave.	Std.	Ave.	Std.	Ave.	Std.
0–4%	7.2	1.3	13.5	1.8	32.4	22.9
4–8%	29.2	5.2	41.2	9.6	8.6	9.7
8–12%	40.1	6.8	66.3	13.4	12.6	13.2
12–16%	58.6	7.9	105.1	23.0	4.6	3.5

Fig. 3.

Plots of E₁ (a), E₃ (b), and k (c) outputted from the transverse isotropic biphasic model fit relative to strain (ε). (a) and (b) showed positive linear regression, while (c) showed negative power regression as shown by the dotted lines.

First and Second Order Generalized Kelvin Model Fits for 4% Strain Increments.

The first order Kelvin model seemed to provide a good fit to the tensile relaxation data (Fig. 4(a)). We also implemented the second order generalized Kelvin model reported in previous studies [16,26]. The curve generated from the second order generalized Kelvin model provided a better fit at the early stage of relaxation compared to the other two fittings (Figs. 2 and 4(b)). In terms of the trueness of the first order Kelvin fit (Table S1 available in the Supplemental Materials on the ASME Digital Collection), the mean absolute error ranged from 0.14 to 107, with RMSD ranging from 0.005 to 0.17, and R² ranging from 0.7 to 0.92. Also, the mean absolute error and RMSD were lower for the second order Kelvin model, ranging from 0.014 to 13.4 and 0.003 to 0.047, respectively, with the R² being between 0.9 and 0.996. For both the first and second order Kelvin models, the RMSD and mean absolute error also increased by each strain-step within each sample, but in a much smaller magnitude compared to that of the transverse isotropic biphasic model. The Kelvin model outputs, E_i and E_r (Table S3 available in the Supplemental Materials), were pooled and calculated for the average and standard deviation in the same manner as in the transverse isotropic biphasic model fit (Table 2). Both variables showed positive linear regression in relation to strain(ε) in both Kelvin models (Fig. 5).

Fig. 4.

Examples of the first (a) and second (b) order Kelvin model fit for 0–4% strain step in a load intensity-time (σ–t) curve using the same set of data as for the transverse isotropic biphasic model fit (Fig. 2). Thin line: optimized fitting curve. Thick line: experimental data.

Table 2.

First and second order Kelvin model fit outcomes for 4% strain increments

	First order Kelvin					Second order Kelvin
	Ei (MPa)			Er (MPa)		Ei (MPa)		Er (MPa)
Strain step	Ave.	Std.		Ave.	Std.	Ave.	Std.	Ave.	Std.
0–4%	16.7	1.9		13.5	1.8	11.7	1.5	6.6	0.8
4–8%	57.2	11.3		41.2	9.5	48.9	8.3	19.9	4.7
8–12%	88.5	16.5		66.3	13.4	79.3	18.0	32.3	6.6
12–16%	134.5	30.0		104.6	22.9	104.2	24.1	51.0	11.2

Fig. 5.

Plots of E_i, E_r outputted from the first (a–b) and second (c–d) order Kelvin model fit relative to strain (ε). All parameters showed positive linear regression as shown by the dotted lines.

Differences in the Curve-Fit Outcomes Between Pig and Goat for Single 10% Strain Step.

Single-step stress relaxation tensile tests were performed on pig and goat TMJ discs to investigate the variation in the curve-fit outcomes between two species and to explore a different strain amplitude (Table 3). The transverse isotropic biphasic model and both Kelvin models were used to curve-fit the data (Table S4 available in the Supplemental Materials). Mean absolute error, RMSD, and R² calculated from the pig disc tests ranged from 63 to 196, 0.06 to 0.10, and 0.59 to 0.84, respectively, when using the transverse isotropic biphasic model fit; versus 21–44, 0.03–0.05, and 0.93–0.94, respectively, when using the first order Kelvin model fit; and 3–5, 0.01–0.02, and 0.99, respectively, when using the second order Kelvin model fit (Table S5 available in the Supplemental Materials on the ASME Digital Collection). These numbers were slightly different for the goat discs, ranging from 103 to 893, 0.08 to 0.22, and 0.48 to 0.86, respectively, using the transverse isotropic biphasic model fit; 67–128, 0.06–0.08, and 0.91–0.95, respectively, using the first order Kelvin model fit; and 11–17, 0.025–0.03, and 0.98–0.99, respectively, using the second order Kelvin model fit (Table S5 available in the Supplemental Materials).

Table 3.

Single-step tensile test with goat and pig TMJ discs

Transverse isotropic biphasic fit
		E1 (MPa)		E3 (MPa)		k (×10−18m4/N·s)
Animal	Strain step	Ave.	Sth.	Ave.	Sth.	Ave.	Sth.
Pig	0–10%	10.4	4.2	13.4	2.5	8.7	2.2
Goat	0–10%	19.4	3.2	32.1	4.7	1.9	0.5

First and second order Kelvin fit
		First order Kelvin				Second order Kelvin
		Ei (MPa)		Er (MPa)		Ei (MPa)		Er (MPa)
Animal	Strain step	Ave.	Std.	Ave.	Std.	Ave.	Std.	Ave.	Std.
Pig	0–10%	19.9	3.6	13.3	2.5	18.2	3.4	6.4	1.2
Goat	0–10%	42.4	7.1	31.7	4.6	34.8	7.5	15.2	2.4

Discussion

Tensile tests on three layers per 3 pig TMJ discs were performed at 4% strain steps. After first fitting the data with the transverse isotropic model, we observed that the variability of results within the layers of the disc (technical repeats) was larger than the biological variability between individuals (biological replicates). For example, the technical variability between the layers of each disc was 11% (Pig 1), 37% (Pig 2), and 63% (Pig 3) for E₁ during the 8–12% strain steps. However, when the values of each layer were first averaged within the animal, the variability between animals was only 17%. We, therefore, reported this type of tensile data as the average of the biological replicates after the average of the technical replicates. Despite pooling and averaging the data from the technical replicates, we do believe that there might be differences between different layers of the pig TMJ disc, as shown in Tables S1–S3 available in the Supplemental Materials, but there are not enough samples to observe any trends. Thus, future studies will take note of the layers and block the statistical analysis based on tissue depth.

We noticed that the trueness of the transverse isotropic biphasic fit decreased as the strain increased, while that of the first and second order Kelvin fit remained almost the same (Table S1 available in the Supplemental Materials). With that said, we found that at the 4% strain step, the fits produced by the transverse isotropic biphasic model were comparable to the ones outputted from the first order Kelvin model. A possible reason for the transverse isotropic biphasic model to lack good fits for higher strains might be that this model assumes that the permeability comes from the flowing of fluid inside the sample, which is minimal when stretching the TMJ discs at higher strain. From 0 to 4% strain, the collagen fibers are uncrimping, which probably allows for the most water flow. This agrees with what we observed in Fig. 3 and Table 1 in which the average permeability was very high in the 0–4% strain step, but then quickly decreased and remained fairly consistent in the remaining strain steps. As such, the permeability output from the biphasic model beyond 4% strain likely has no physical meaning.

Elliot et al. [29] previously published a paper that reported similar results regarding the relationship of tensile strain levels in the tendon using the biphasic transverse isotropic model for tensile properties, such as the strain-dependent relationships between k and E₃. They reported the same negative exponential relation between the permeability and the strain as we report here. However, while we present a linear relationship of E₃ with strain, Elliot et al. presented an exponential relationship. Furthermore, they reported a fairly constant value of E₁ with increasing strain which may be due to the slower strain rate [29]. Differences in assumptions, animal models, and testing methods could also account for the difference in results.

While the Kelvin fits cannot be directly compared to the transverse isotropic biphasic model due to assumptions of differing properties and the limitation of excluding ramping contributions, both Kelvin models had good fits to our experimental data observationally. However, it is worth pointing out that both the transverse isotropic biphasic model and the first order Kelvin model did not seem to catch the early changes of the relaxation stage very well (Figs. 2 and 4(a)), whereas the second order Kelvin model provided a better fit at the early stage of the relaxation (Fig. 4(b)). The improved trueness of the second order Kelvin fit also verified that the second order Kelvin model provides an overall better fit to our data.

We also report the use of transverse isotropic biphasic model and the two Kelvin models for a 10% strain step tensile test data of both pig and goat TMJ disc samples (Table 3). In consensus with what we found in the 4% incremental test, the transverse isotropic biphasic model did not provide a good fit with the larger strain used in this test for both pig and goat samples (Table S5 available in the Supplemental Materials). In terms of the Kelvin models, we found that the E_i and E_r from the first and second order Kelvin model seem to be higher for the goat than the pig, which is different from what was reported by Kalpakci et al. in which the moduli of the pig were not significantly different from that of the goat [5]. This could be due to the fact that Kalpakci et al. were combining different zones in different directions when doing the interspecies comparison and/or the limitation that the sample size was small in this study.

In this study, the disc strips were trimmed into a dog-bone shape, resulting in a nonuniform width, which might cause inaccuracy in the outcome without using digital image correlation. Future studies should consider using uniform width strips in similar tests. Although the small sample size (n = 3) may seem small/inadequate, it is important to note that enough samples were obtained for both the strain step and the species investigations to ensure that the tensile behavior observed was repeatable, and thus the repeatability of the fits using each model. Increasing the sample size would not impact the findings that the second order Kelvin model fits better than the biphasic and first order Kelvin models. Nevertheless, future studies that compare regions or species statistically, and use the second order Kelvin model, may wish to consider using n = 5–9 samples to account for the 30–50% variance of the means of tensile parameters such as the moduli.

In conclusion, the transverse isotropic biphasic model only provided a good fit of the tensile stress-relaxation data from pig and goat TMJ discs at low strain (0–4%). Thus, this model is not the best choice for a similar test at high strain for two major reasons: (1) it did not capture the fast relaxing phase at the beginning of the relaxation; (2) it did not fit well in a higher strain. The former was probably due to the added contribution of the long term stress relaxation, which is weighted more heavily by the solver. This was also true in the first order Kelvin model, which can be addressed with the second order Kelvin by using two sets of time constants for the fast and slow relaxation phases, respectively. As mentioned above, we think the second issue is mainly due to the assumption of fluid flow through a solid matrix in the biphasic theory which may not be applicable to the tensile behavior of TMJ disc fibers beyond low strain. At the beginning of loading, fluid flow between the fibers is likely, but at higher strains, the main viscoelastic component is the solid matrix. Other biphasic models, such as the poroviscoelastic model [24], that describe the solid components as viscoelastic will likely yield more meaningful outputs and should be explored in future works.

Supplementary Material

Supplementary Material

Supplementary Material Tables PDF

Funding Data

• National Institutes of Health (Grant Nos. R21 DE027873 and 5.31228.xxxx.00000.132428.00000.00000; Funder ID: 10.13039/100000002).

• University of Pittsburgh School of Dental Medicine (Funder ID: 10.13039/100007921).