Mechanical properties of anterior malleolar ligament from experimental measurement and material modeling analysis

Abstract

In this paper, mechanical properties of the anterior malleolar ligament (AML) of human middle ear were studied through the uniaxial tensile, stress relaxation and failure tests. The digital image correlation (DIC) method was used to assess the boundary effect in experiments and calculate the strain on specimens. The constitutive behavior of the AML was described by a transversely isotropic hyperelastic model which consists of a first-order Ogden model augmented by a I₄-type reinforcing term. The material parameters of the model were estimated and the viscoelasticity of the AML was illustrated by hysteresis phenomena and stress relaxation function. The mechanical strength of the AML was obtained through the failure test and the mean ultimate stress and stretch ratio were measured as 1.05MPa and 1.51, respectively. Finally, a linear Young’s modulus–stress relationship of the AML was derived based on constitutive equation of the AML within a stress range of 0–0.5 MPa.

1. Introduction

The anterior malleolar ligament (AML) is one of suspensory middle ear ligaments located between the anterior process of the malleus and the bony wall of the middle ear cavity. One arm of the AML reaches the capsule of the temporomandibular joint and the other attaches the pin of the sphenoid bone (Cesarani et al. 1991). The ligament is observed with a smooth arrangement of fibers within the connective tissue (Alkofide et al. 1997).

The AML provides mechanical support to the malleus head. The structural change of the AML caused by middle ear disease, such as otitis media with effusion, may affect the transfer function of the middle ear for sound transmission. Studies from human temporal bone experiments and finite element modeling (Huber et al. 2003; Nakajima et al. 2005) have shown that fixation of the AML could result in decrease of ossicular velocity around 3 to 8dB at low frequency (below 1,000 Hz). It is also reported that AML had the potential to cause middle ear damage on the tympanic membrane and associated structures when tension was applied to the ligament during the surgical manipulation of the temporomandibular joint (Kim et al. 2004). Therefore, mechanical properties of the AML are one of research interests in ear tissue biomechanics.

Measurement of AML mechanical properties has not been reported in literature. In published finite element (FE) models of human ear, mechanical properties of the AML were assumed through the cross-calibration process and a constant elastic modulus varying from 2 to 21 MPa were used for different FE models (Ferris and Prendergast 2000; Gan et al. 2004; Prendergast et al. 1999; Sun et al. 2002; Wada et al. 1992). The accuracy of mechanical properties of the AML will improve the model-predicted results and extend the applications of the model.

This paper reports mechanical properties of human AML through both experimental measurement and modeling analysis. The uniaxial tensile, stress relaxation and failure tests were conducted on AML specimens. The experimental results were analyzed by digital image correlation (DIC) method. Using a transversely isotropic hyperelastic material model, the material parameters were estimated and the constitutive equation of the AML was derived. Finally, a linear Young’s modulus–stress relationship of the AML was presented based on the constitutive equation of the AML.

2. Methods

2.1. Specimen preparation

Nine AML specimens were harvested from fresh-frozen human temporal bones through the University of Oklahoma Health Sciences Center. The average age of donors was 71 (ranging from 51 to 92 year, five male and four female). Experiments were performed within 6 days after harvesting the temporal bone. To maintain soft tissue compliance, the bones were immersed in 1:10,000 merthiolate in 0.9% saline solution at 5°C until use (Gan et al. 2004). Before taking out the ligament from the bone, the tissue was checked under the microscope for not having any degradation or abnormality. The AML sample was prepared with two bony ends (the malleus and cavity wall) attached. By using the same mounting fixture and following the same procedure as we described in stapedial tendon study (Cheng and Gan 2007), two bony ends were attached to the mounting fixture under a microscope by a tiny drop of cyanoacrylate gel glue as shown in Fig. 1. Care was taken not to allow the glue reaching the ligament. A ruler was attached to the top of the fixture as reference for dimension measurement. The AML was then installed and lined up with grips in the material testing system (MTS, Model 100R, TestResources, MN) and a preload of 0.001 N was applied to the specimen through the load cell to adjust the zero load state.

Fig. 1.

The AML specimen harvested from the human temporal bone and installed into metal fixtures. A ruler was attached to the metal holder at the load cell side as a dimensional reference

The still images of the ligament in front and side views were captured first using a digital CCD camera and the Measure Tool in Adobe photoshop 7.0 was used to measure the length, width and thickness of the specimen. Table 1 lists the dimensions measured from nine AML specimens with the mean and standard deviation (SD). Considering the variation of width along the length direction, the width was measured at three locations: top, middle and bottom part of the specimen.

Table 1.

Dimensions of Anterior Malleoar Ligament Specimens (N = 9)

	AML1	AML2	AML3	AML4	AML5	AML6	AML7	AML8	AML9	Mean	SD (±)
Length (mm)	2.05	2.38	1.66	1.71	2.20	1.76	1.45	1.46	1.77	1.83	0.32
Width (top) (mm)	1.70	2.21	1.45	1.50	1.68	1.64	2.11	1.56	1.52	1.71	0.27
Width (middle) (mm)	1.20	1.90	1.11	1.10	1.08	1.25	1.77	1.21	1.22	1.32	0.30
Width (bottom) (mm)	1.50	2.10	1.31	1.35	1.35	1.43	1.80	1.40	1.01	1.47	0.31
Thickness(mm)	0.81	1.54	1.30	1.10	1.02	1.41	1.27	1.14	1.19	1.20	0.22

SD standard deviation

2.2. Mechanical testing

The MTS with SMT1 (10.0 Newtons capacity) load cell (Interface, Inc.) was used to measure the stress-strain relation, stress relaxation function, and ultimate stress and strain of the AML. The preconditioning was performed first to reach the steady state for each specimen (Fung 1993). In preconditioning, the MTS machine was programmed to perform five cycles of uniaxial elongation at the elongation rate of 0.1mm/s and length of 10% of the original length. During repeated loading–unloading processes, the difference between load–displacement curves from the first three cycles was decreasing and a steady state was generally observed after the third cycle (Fig. 2).

Fig. 2.

Preconditioning of one AML specimen. The first and second cycles are pointed with loading and unloading separately. The curve tends to stable starting from the third cycle. A hysteresis loop was observed for the AML with the unloading curve lagging the loading curve

After preconditioning, the specimen was subjected to three tests: uniaxial tensile, stress relaxation and failure tests. At the end of uniaxial tensile or stress relaxation test, the specimen was returned to the initial state and waited 2–3 min for recovery from previous deformation until the live load reading from the load cell was zero (±0.0001 N). The deformation process for each specimen was recorded simultaneously using a digital CCD camera. The MTS machine and CCD camera were electronically synchronized so that the load and deformation data on images could be correlated simultaneously. The MTS grip-to-grip displacement and the images of deformation were collected to compute strains and assess the boundary effect. The protocols for three tests in MTS are listed as follows:

1. Uniaxial tensile test. The displacement rate was set at 0.01 mm/s, and elongation length was 40% of the original length. Three parameters: load, deformation and time, were recorded with a resolution level of 10⁻⁶ N in force, and 10⁻⁶ mm in displacement. These data were further used to calculate the stress–strain relationship of the ligament.

2. Stress relaxation test . An approximate step function of elongation was applied to the specimen at beginning (t = 0) with a displacement rate of 1.8mm/s and elongation length of 40% of the original length. The corresponding stress including the initial stress response σ₀ and relaxed stress σ(t) were recorded over a period of time until the rate of loading change was less than 0.1%/s, or fully relaxed. Then, the MTS data recording program was stopped manually and the specimen was returned to the initial unstressed state for the failure test.

3. Failure test . The specimen was elongated till it was broken. The displacement rate was set at 0.02 mm/s. The entire failure process including the load and displacement was recorded and breaking site of the specimen was observed.

Note that all the specimens were maintained in physiological condition by spraying normal saline solution to the specimen on the side opposite to the camera during the test.

2.3. Digital image correlation method

To verify the boundary effect in mechanical experiments on specimens with a length in the order of 2mm, the digital image correlation (DIC) method was employed to calculate the transverse strain distribution on the specimen during the uniaxial loading process. The simultaneously recorded images from the tensile test were analyzed based on a grayscale pattern of pixels in the image. The detailed procedure of DIC can be referred to our recently published paper on the tympanic membrane viscoelastic properties (Cheng et al. 2007). In this study, 24 steps of specimen images at a constant time interval over the uniaxial loading process were output first from the CCD camera and installed into the DIC software (WinDIC_LS 2.0). A grid (3 × 8) of 24 nodes was generated around the center of the first image (t = 0), or the reference image (Fig. 3). Two horizontal lines along the top and bottom grids were identified with eight vertical lines connecting grid nodes on the horizontal lines. The length of each vertical line was used as the original length (L₀), and was traced in deformed images as the deformed length (L). The normal strain ϵ in the vertical direction was then calculated as ε = (L – L₀)/L₀. The strain distribution in transverse direction over eight locations at 24 steps (from the beginning to the end of a loading process) and the average strain at each step were compared with the strain measured in MTS. The specimen with relatively uniform distribution of transverse strain in DIC and the comparable average strain from DIC and MTS was used to derive mechanical properties of the AML.

Fig. 3.

Illustration of the digital image correlation (DIC) method for calculating the strain distribution of the AML specimen during the uniaxial loading process. A grid (3×8) of 24 nodes was generated at the middle portion of each image. Two horizontal lines were identified along the top and bottom grids, as well as eight vertical lines connecting corresponding grid nodes on the horizontal lines. The length of each vertical line in the reference image was used as the original length (L₀) of the specimen, and the length of the corresponding vertical lines traced in subsequently deformed images was measured as the deformed length (L) of the specimen. The normal strain ϵ in the vertical direction was calculated by ε = (L – L₀)/L₀

2.4. Stress and strain calculation

In this study, the stress was calculated based on the load measured from MTS and the cross-sectional area of the middle portion of the specimen. The strain obtained from the DIC was also calculated from the deformation of the middle portion of the specimen. The stretch ratio λ, or the ratio of the deformed length to the original length, was used to describe the strain, and λ=ε+1. The results of nine specimens were reported as the mean and standard deviation.

History of the stress response during the stress relaxation test is a function of time, or called stress relaxation function. The normalized stress relaxation function G(t) is defined as the ratio between the stress σ(t) at time t and the initial stress σ₀. The normalized stress relaxation functions G(t) of nine AML specimen were calculated from recorded experimental data with the mean and standard deviation. The stress relaxation function and hysteresis loop observed in tensile tests represent viscoelastic properties of the AML.

The load and displacement values recorded for each specimen in failure test were converted into the ultimate stress and stretch ratio by using dimensions of the ligament listed in Table 1. The average width from the top, middle and bottom portion of the specimen was used to calculate the cross-sectional area. The mean ultimate stress and stretch ratio were reported with standard deviation.

2.5. Material modeling

The scanning electron microscope (SEM) picture of the AML at 5,000× magnification prepared by Samuel Roberts Noble Electron Microscopy Lab at University of Oklahoma is shown in Fig. 4. A parallel-bundled collagen fibrous microstructure of the AML was observed. The mechanical behavior of the AML is expected to be described by an unidirectional fiber reinforced composite model. In this study, we used a transversely isotropic model which consists of a first-order Ogden model augmented by a I₄-type reinforcing term proposed by Ogden (2003). The strain energy potential of the model, U, is described as

$$U=\frac{2{\mu }_1}{{\alpha }_1^2}({\lambda }_1^{{\alpha }_1}+{\lambda }_2^{{\alpha }_1}+{\lambda }_3^{{\alpha }_1}-3)+\frac{2k{\mu }_1}{{\beta }^2}(I_4^{\beta ∕2}+2I_4^{-\beta ∕4}-3),$$ (1)

where λ₁, λ₂, λ₃ are principal stretches of the incompressible material and λ₁λ₂λ₃ = 1. I₄ coincides with the square of material stretch in the fiber direction. μ₁ is the infinitesimal shear modulus of the material in a natural configuration, and α₁ and β are temperature dependent material parameters. Parameter k is a coefficient (> 0) which counts for the increase of material stiffness in the fiber direction. The case with k = 0 represents the isotropic Ogden model which has been well used to model soft tissues such as the tympanic membrane, brain tissue, and skin (Cheng et al. 2007; Miller and Chinzei 2002; Wu et al. 2003).

Fig. 4.

A scanning electron microscope picture of the AML at 5,000× magnification

The Cauchy stress tensor σ is then derived from the strain energy function U (Eq. 1) as

$$\sigma =\sum _{i=1}^3\left({\lambda }_i\frac{\partial U}{\partial {\lambda }_i}-p\right){\mathit{v}}^{(i)}\otimes {\mathit{v}}^{(i)},$$ (2)

where v⁽ⁱ⁾ are eigenvalues of the left stretch tensor V, p is the Lagrange multiplier associated with the incompressibility constraint, and ⊗ denotes the tensor product.

Assuming that fibers of the AML are aligned along the X₂-axis of a given Cartesian coordinate system (X₁, X₂, X₃) and the uniaxial loading is applied along the fiber direction (X₂-axis), the stretch in the loading direction would be λ ≡ λ₂, and the nonzero stress component of σ is σ = σ₂₂. Thus, λ₁ = λ₃ = λ^−1/2, p = 2μ₁λ^−α₁/2/α₁, I₄ = λ₂, and

$$U=\frac{2{\mu }_1}{{\alpha }_1^2}({\lambda }^{{\alpha }_1}+2{\lambda }^{-{\alpha }_1∕2}-3)+\frac{2k{\mu }_1}{{\beta }^2}({\lambda }^{\beta }+2{\lambda }^{-\beta ∕2}-3)$$ (3)

$$\sigma =\frac{2{\mu }_1}{{\alpha }_1}({\lambda }^{{\alpha }_1-1}-{\lambda }^{-0.5{\alpha }_1-1})+\frac{2k{\mu }_1}{\beta }({\lambda }^{\beta -1}-{\lambda }^{-0.5\beta -1}).$$ (4)

Equation (4) represents the stress (σ)~stretch (λ) relationship or constitutive equation of the AML expressed with material parameters of μ₁, α₁, β and k.

Differentiating Eq. (4) with respect to λ, we have the tangent modulus or Young’s modulus $\left(\frac{\mathrm{d}\sigma }{\mathrm{d}\lambda }\right)$ of the material,

$$\begin{gathered} \frac{\mathrm{d}\sigma }{\mathrm{d}\lambda }=\frac{2{\mu }_1}{{\alpha }_1}\left[({\alpha }_1-1){\lambda }^{{\alpha }_1-2}+\left(\frac{{\alpha }_1}{2}+1\right){\lambda }^{-(0.5{\alpha }_1+2)}\right] \\ +\frac{2k{\mu }_1}{\beta }\left[(\beta -1){\lambda }^{\beta -2}+\left(\frac{\beta }{2}+1\right){\lambda }^{-(0.5\beta +2)}\right] \end{gathered}$$ (5)

Three material parameters (μ₁, α₁, β) in this study were determined as k value changed from 0 to 5, 10 and 20 through iterative regression processes in MATLAB (v.7.0). Briefly, given experimental stress-stretch data and a user-defined function (Eq. 4 in this study), we find the material parameters that best fit the function to the data in a least-squares sense. Constitutive equation of the AML was then obtained by substituting the material parameters into Eq. (4). The Young’s modulus of the specimen was obtained from Eq. (5) and plotted against the stress to derive the Young’s modulus–stress relationship of the AML.

3. Results

Figure 5a shows the transverse strain distribution of an AML specimen from the DIC analysis at four selected steps during the uniaxial loading test. Figure 5b shows the comparison of the strain measured directly from the MTS (broken line) and that from the DIC (solid line). The strain distribution (Fig. 5a) was relatively uniform across the specimen at each step. In Fig. 5b, the strain from MTS and DIC are in general agreement, although the strain from DIC is slightly lower than that from MTS. These differences maybe induced by geometric and structural non-uniformity of the AML tissue and sensitivity of the DIC method on calculating the local deformation of the tissue. In general, Fig. 5 indicates that the boundary effect of experiment is limited and the results are reliable.

Fig. 5.

a Normal strain distribution across the transverse surface of the AML specimen calculated from DIC analysis at four time steps. The no. of nodes represents eight locations (from left to right) across the specimen in the middle part of the grid shown in Fig. 3. b Comparison of the strain obtained from MTS experiment (broken line) and DIC analysis (solid line)

Figure 6a shows stress–stretch curves of nine AML specimens derived from the stress measured in MTS and the stretch calculated from DIC for the loading process of uniaxial tensile tests. Figure 6b displays the mean stress-stretch curve with standard deviation. A nonlinear stress-stretch relationship is clearly seen in Fig. 6. The standard deviation was increasing with the increasing stress, but the relative standard deviation remained the same at 0.50, which indicated that the variation between individual specimens was not changing with the stress level. It was also seen that the stress increased slowly at the beginning, and the stress-stretch curve was almost flat when the stretch ratio was less than 1.2. The curve became stiffer when the stretch ratio continued increasing.

Fig. 6.

a Stress–stretch curves of nine AML specimens under uniaxial loading processes. The maximum stretch ratio λ was around 1.5 and the displacement rate was 0.01 mm/s. b The mean curve of stress–stretch relationships obtained from nine AML specimens with standard deviation bars

Figure 7a shows the stress relaxation behavior of the AML obtained from nine specimens. The normalized stress relaxation function G (t) decreased with time and finally reached a stable state at 120 s, or the change rate of stress was less than 0.1%/s. The stress was considered fully relaxed after 120 s. The mean normalized stress relaxation function with standard deviation is shown in Fig. 7b. The mean initial stress was 1.13 MPa, and finally 33% of the initial stress, on average, was totally relaxed after 120 s.

Fig. 7.

a Normalized stress relaxation function G(t) obtained from nine AML specimens in stress relaxation tests. b The mean curve of G(t) of nine AML specimens with standard deviation bars

Table 2 lists the ultimate stress and stretch ratio of nine AML specimens obtained from failure tests. The breaking location of all specimens occurred around midsubstance in all the tests.

Table 2.

Ultimate Stress and Stretch Ratio of Anterior Malleolar Ligament Specimens in Failure

	AML1	AML2	AML3	AML4	AML5	AML6	AML7	AML8	AML9	Mean	SD (±)
Ultimate stress (MPa)	1.00	0.43	1.34	1.83	2.06	1.54	0.44	0.71	0.68	1.11	0.60
Ultimate stretch ratio λ	1.50	1.46	1.33	1.71	1.50	1.56	1.51	1.42	1.37	1.48	0.11

SD standard deviation

The mean experimental stress–stretch curve of nine AML specimens shown in Fig. 6b was used to determine material parameters, μ₁, α₁, β and k of Eq. (4). Table 3 lists the calculated values of μ₁, α₁ and β when k value was varied from 0 to 5, 10, and 20. When k = 0, or the material is isotropic, we had μ₁ = 0.078MPa, α₁ = 13.69, and β was not accounted for isotropic material. When k increased to 5, 10 and 20, the values of α₁ and β were almost equal (13.71~13.68), or the same as the α₁ value at k = 0. However, the μ₁ value was decreased as the k value, or the stiffness of material along the fiber direction, increased.

Table 3.

Material parameters of the hyperelastic model for anterior malleolar ligament

	μ1	α1	β
k = 0	0.078	13.69	–
k = 5	0.013	13.71	13.68
k = 10	0.0071	13.68	13.69
k = 20	0.0037	13.68	13.69

In Weiss and Gardiner (2001) review on ligament mechanics, the modulus of the ligament along the fiber direction was considered an order greater than the modulus in the transverse direction. The AML in this study was assumed with k = 10, and the constitutive equation of AML was then derived as:

$$\sigma =1.04\times {10}^{-3}({\lambda }^{12.68}-{\lambda }^{-7.84})+0.01({\lambda }^{12.69}-{\lambda }^{-7.85})$$ (6)

for stress level of 0~0.5MPa, and stretch range of 1~1.4. Figure 8 displays the comparison of stress–stretch curve obtained from Eq. (6) and that measured from experiments. It shows that the material model (Eq. 6) is generally able to describe the AML mechanical properties.

Fig. 8.

The stress–stretch curve of the AML obtained from modeling analysis (k = 10) and the mean stress–strain curve measured from nine AML specimens in uniaxial tensile tests

Young’s modulus of the AML was calculated from Eq. (5) and plotted against the stress in Fig. 9. It is clearly seen that the Young’s modulus of the AML is linearly increasing with the stress and mathematically expressed as

$$\frac{\mathrm{d}\sigma }{\mathrm{d}\lambda }=8.95\sigma +0.22\text{for}0\le \sigma \le 0.5\text{MPa},1\le \lambda \le 1.4.$$ (7)

Fig. 9.

The Young’s modulus–stress relationship of the AML derived from material modeling analysis. The Young’s modulus is linearly increasing with the stress. The values varied from 0.22 to 4.70 MPa when the stress increases from 0 to 0.5 MPa

The value rises from 0.22 to 4.70MPa when the stress increases from 0 to 0.5MPa.

4. Discussion

In this study, mechanical properties of the AML in human middle ear were first reported through experimental measurement and modeling analysis. The DIC method was used to assess the boundary effect in experiments (Fig. 5a) and calculate the strain in middle portion of the specimen. The strain from DIC was compared with the strain measured in MTS (Fig. 5b). The consistent results from both methods were further analyzed to derive mechanical properties of the AML.

A transversely isotropic hyperelastic material model was employed to derive the constitutive equation of the AML. Four material parameters of the model, μ₁, α₁, β, and k, were determined based on regression of experimental data and material modeling results. The physical meanings of parameters μ₁ and k have been stated in Sect. 2.5. The parameters α₁ and β are numbers without direct physical meaning. It is seen in Table 3 that the value of k only affects μ₁, which indicates that the increase of stiffness along the fiber direction (increase of k) results in the decrease of infinitesimal shear modulus (decrease of μ₁). Since α₁ and β are almost equal as k changes, Eqs. (4) and (5) can be simplified with three material parameters as

$$\sigma =\frac{2{\mu }_1(1+k){\lambda }^{-0.5{\alpha }_1-1}({\lambda }^{1.5{\alpha }_1}-1)}{{\alpha }_1}$$ (8)

$$\frac{\mathrm{d}\sigma }{\mathrm{d}\lambda }=\frac{2{\mu }_1(1+k)}{{\alpha }_1}\left[({\alpha }_1-1){\lambda }^{{\alpha }_1-2}+\left(\frac{{\alpha }_1}{2}+1\right){\lambda }^{-(0.5{\alpha }_1+2)}\right].$$ (9)

The stress–strain relationships of the AML specimens shown in Fig. 6 are based on the stress measured in the middle portion of each specimen from the MTS and the strain measured at the same location from the DIC. For fibrous tissues like the ligament, the mechanical response of the tissue is mainly due to the strength of fibers inside (Weiss et al. 2002), especially when a uniaxial load is applied to the tissue along the fiber direction. In this study, a local response of fibers in the middle portion of the AML was used to describe the mechanical properties of the ligament, which is often a good practical choice. However, due to the irregular geometry and inhomogenous microstructure of biological tissues, the stress or strain distribution in the whole tissue is expected to be more complicated. A more accurate model including geometry and microstructure of the tissue, such as finite element (FE) model, is needed to investigate mechanical properties of the AML precisely.

The ligament is usually considered as a transverse isotropic material with collagen fibers embedded in substance matrix. The material behavior of ligament depends on fiber properties (main factor), matrix properties, fiber-matrix interaction, and fiber-fiber interaction. Thus, a single uniaxial test may be not sufficient to characterize the three-dimensional material behavior of the tissue. Additional experiments on multiaxial quasi-static and viscoelastic material properties of the AML are necessary for an accurate representation of ligament mechanics.

The nonlinear stress-strain relationship of the AML shown in Fig. 8 resulted in the stress-dependent elastic modulus for the ligament (Fig. 9). The modulus increases as the stress increases, a typical mechanical behavior of soft tissues. A varying modulus of the AML at different stress levels may be used to simulate the middle ear response to the change of ear physiological condition such as the otitis media with effusion in the FE modeling of human middle ear.

In conclusions, mechanical properties and dimensions of the AML in human ear are reported in this paper from experimental measurement and modeling analysis. This is the first investigation of material properties of the middle ear ligament. The data reported here may provide valuable information for understanding the AML and its function. However, the future work is needed such as using the FE modeling approach to include geometric configuration, microstructural arrangement and interactions between fibers and ground substances on mechanical properties of the AML. It is also considered to develop new experimental methods on measurement of 2-dimensional or 3-dimensional mechanical properties of the AML.