The Radial Bulging and Axial Strains of Intervertebral Discs during Creep Obtained with the 3D-DIC System

Creep-associated changes in disc bulging and axial strains are essential for the research and development of mechano-bionic biomaterials and have been assessed in various ways in ex vivo creep studies. Nonetheless, the reported methods for measurement were limited by location inaccuracy, a lack of synchronousness, and destructiveness. To this end, this study focuses on the accurate, synchronous, and noninvasive assessment of bugling and strains using the 3D digital image correlation (3D-DIC) system and the impact of creep on them. After a preload of 30 min, the porcine cervical discs were loaded with different loads for 4 h of creep. Axial strains and lateral bulging of three locations on the discs were synchronously measured. The three-parameter solid model and the newly proposed horizontal asymptote model were used to fit the acquired data. The results showed that the load application reduced disc strains by 6.39% under 300 N, 11.28% under 400 N, and 12.59% under 500 N. Meanwhile, the largest protrusion occurred in the middle of discs with a bugling of 1.50 mm, 1.67 mm, and 1.87 mm. Comparison of the peer results showed that the 3D-DIC system could be used in ex vivo biomechanical studies with reliability and had potential in the assessment of the mechanical behavior of novel biomaterials. The phenomenon of the largest middle protrusion enlightened further the strength of spinal implants in this area. The mathematical characterizations of bulging and strains under different loads yielded various model parameters, which are prerequisites for developing implanted biomaterials.


Introduction
Creep is the progressive deformation of the disc that occurs under constant load and stress below the fracture threshold [1]. In the human body, the creep of intervertebral discs (IVDs) causes 1 to 2 cm [2] or a 1.1% difference [3] in normal heights, which is mainly attributed to the loss and gain of the disc height [4,5]. The disc height loss is often accompanied by lateral bulging, which is the major cause of disc protrusion and herniation [6]. It is also of importance as providing valuable data for further preclinical evaluations of spinal implants and biomechanical experiments of disc injuries.
Thus far, several studies have investigated disc bulging in in vitro creep tests in various ways. Heuer et al. [7][8][9] developed a laser-based three-dimensional contour scanner that was mounted in a spine tester to measure disc bulging during creep, providing a highprecision disc bulging assessment with complicated and customized mechanical structures. Another similar study reported by Fewster et al. [10] included applying the 3D laser scanner with several steel pins inserted into the disc to align the 3D surface profiles from each functional spinal unit and estimate the curvature of the endplate for acquiring bulges. Their 2 of 14 previous results [11] showed that the proposed system achieved errors of less than 0.2 mm. As this approach was destroyed, they may result in inaccurate data. Dupré et al. [12,13] adopted an optoelectrical tracking system to quantify a disc bulge after nucleus fortification for treating a degenerated nucleus. The results showed that a significant reduction in posterior, lateral, and posterolateral disc bulges was observed during lateral bending for the fortified treatment compared with disc-only. Pei et al. [14] applied five red oil points on the surface of specimens to mark the outer annular surface and two orthogonally placed cameras to capture images. Subsequently, photographic data were extracted using video editing software, and the markers were digitized using image processing software with a seven-pixel brush tool. The horizontal and vertical differential pixel numbers were computed as the radial and axial deformations. Though easily to be conducted, their method was limited in less automated processes.
As common imaging methods in clinics, computerized tomography (CT) and magnetic resonance imaging (MRI) can also be adopted in in vitro measurements. Mengoni et al. [15] used CT images to measure the disc bulge successfully, though with radiation and high costs. During specimen preparation, CT-compatible markers were embedded into the surfaces of the endcaps. Additionally, forty glass fiducial markers were attached to the surface of each disc using petroleum jelly. Coordinates for all glass markers on the CT images were automatically calculated to obtain the bulge data using image processing and recognition. MRI images were mainly used to acquire shape and texture features in in vivo measurements. Mayerhoefer et al. [16], Hung et al. [17], and Sundarsingh et al. [18] made efforts in the analysis of the parameter sensitivity. Their studies confirmed that texture features and geometric parameters were sensitive to posterior disc bulging and, thus, can be used as quantitative biomarkers that predict disease. To improve the accuracy, Beulah et al. [19] further proposed a computer-aided diagnostic system, which was based on a new segmentation method, to identify the disc bulge in axial lumbar spine T2-weighted MR images. On applying the proposed system, an accuracy of 92.78% was obtained. Lao et al. [20] also compared whether adding flexion and extension to the traditional neutral views would be beneficial in the diagnosis of cervical disc bulges in an MRI study. The results showed that a significant increase in the degree of cervical disc bulge was found by examining the extension views, thus confirming that the kinematic MRI views provided valuable added information, especially in situations where symptomatic radiculopathy was present without any abnormalities demonstrated on traditional neutral MRI.
To sum up, those methods mentioned above did have limitations in acquiring disc bulge data in in vitro practice. Fortunately, the digital image correlation (DIC) provided a new method to measure deformations with outcomes of high accuracy and noncontact, through which images of an object can be obtained in digital forms for further analysis [21]. In the early 1990s, the DIC technique was extended to stereovision systems, designating them 3D-DIC, which employed two or more cameras to record digital images of a tested object from multiple viewpoints. By using DIC to perform a cross-camera subset matching, a calibrated stereovision system obtains the true and 3D positions of each point on a nonplanar object. Over recent years, the 3D-DIC has seen remarkable growth, with applications in aerospace [22,23], micro-scale measurements [24], biomaterials [25], and fracture mechanics [26][27][28]. In this study, a commercial 3D-DIC system has been adopted to measure the axial strains and disc bulging of compressed discs under physiological creep successfully.
The purpose of this study was to investigate time-dependent disc bulging and axial strains as a cause of physiological creep using the 3D-DIC system. The relations between lateral bulging and axial strains were also revealed. Additionally, a new mathematical model for fitting disc bulging was also proposed. The results showed that the new model had high quality, and the relevant model parameters under different magnitudes of creep loads were calculated. The numerical model provided approximations of the physiological bulging of discs, thus facilitating further comparisons of mechanical properties, especially lateral deformation, between biomaterials and natural discs.

System Setup and Calibration
The physiological loads were axially applied with a universal testing machine (MTS-CMT 6104, MTS System (China) Co., Ltd., Shenzhen, China) and was registered at 1 Hz. The custom 3D-DIC system (VIC-3D LS-2M, Correlated Solutions Inc., Columbia, SC, USA) was adopted for measuring bulging and strains at 0.5 Hz, which included stereovision cameras (with a resolution of 1920 × 1200 for each pixel size of 5.86 µm) and postprocessing software (Vic3D). The camera system was set in front of the testing machine ( Figure 1a). The specimens were firstly exposed to 50 N axial compression for 30 min for reducing water. Subsequently, the formal creep tests were carried out under three different magnitudes of loads (300 N, 400 N, and 500 N) with a loading rate of 5 N/s for 4 h (Figure 1b), which was reported to be sufficient by Paul et al. [29]. The loading regime in this study was according to the earlier ex vivo study, conducted by O'Connell et al. [30] and Yang et al. [31,32], who adopted 4 h as the loading duration to quantify the time-response during creep. Five specimens were tested under the same conditions at room temperature; after which, the results were averaged. of creep loads were calculated. These numerical models provided approximations of the physiological bulging of discs, thus facilitating further comparisons of mechanical properties, especially lateral deformation, between biomaterials and natural discs.

System Setup and Calibration
The physiological loads were axially applied with a universal testing machine (MTS-CMT 6104, MTS System (China) Co., Ltd., Shenzhen, China) and was registered at 1 Hz. The custom 3D-DIC system (VIC-3D LS-2M, Correlated Solutions Inc., Columbia, SC, USA) was adopted for measuring bulging and strains at 0.5 Hz, which included stereovision cameras (with a resolution of 1920 × 1200 for each pixel size of 5.86 μm) and postprocessing software (Vic3D). The camera system was set in front of the testing machine ( Figure 1a). The specimens were firstly exposed to 50-N axial compression for 30 min for reducing water. Subsequently, the formal creep tests were carried out under three different magnitudes of loads (300 N, 400 N, and 500 N) with a loading rate of 5 N/s for 4 h (Figure 1b), which was reported to be sufficient by Paul et al. [29]. The loading regime in this study was according to the earlier ex vivo study, conducted by O'Connell et al. [30] and Yang et al. [31,32], who adopted 4 h as the loading duration to quantify the timeresponse during creep. Five specimens were tested under the same conditions at room temperature; after which, the results were averaged.  Before the tests, the calibration process was needed, taking images of the calibration grid in a range of orientations that included tilting the grid forward, backward, and rotating ( Figure 1c). Since the calibration grid was rigid, these images could be used to perform a shape measurement of the grid, thus determining the camera model and system parameters. The whole system should not be moved during or after calibration.
The error of the 30 calibration image pairs should be less than 0.100, indicating that the cameras were successfully calibrated. Through the comparison algorithm based on the calibration images and the random speckle patterns during the tests, subregions from the "deformed" and "undeformed" images were compared, thus generating a full field of sensor plane measurements.

Specimen Preparation
Ten porcine cervical spinal segments (C2-7) were used in this experiment. Soft tissue and facet capsules of the specimens were removed, and adjacent vertebral bodies were removed~3.5 mm from the disc with a hand saw. All specimens were stored at −20 • C in the refrigerator and used within a week. Before testing, the discs were thawed overnight in saline solution to promote rehydration.
The surfaces of the discs were prepared with matte speckles before being tested ( Figure 2a). In order to provide good tracking information, the speckles should ideally be 3-5 pixels (about 0.6 mm) in size to optimize the resolution. The speckles were recognized in the following data processing algorithms to compute the disc bulging and strains ( Figure 2b).
Before the tests, the calibration process was needed, taking images of the calibration grid in a range of orientations that included tilting the grid forward, backward, and rotating ( Figure 1c). Since the calibration grid was rigid, these images could be used to perform a shape measurement of the grid, thus determining the camera model and system parameters. The whole system should not be moved during or after calibration. The error of the 30 calibration image pairs should be less than 0.100, indicating that the cameras were successfully calibrated. Through the comparison algorithm based on the calibration images and the random speckle patterns during the tests, subregions from the "deformed" and "undeformed" images were compared, thus generating a full field of sensor plane measurements.

Specimen Preparation
Ten porcine cervical spinal segments (C2-7) were used in this experiment. Soft tissue and facet capsules of the specimens were removed, and adjacent vertebral bodies were removed ~3.5 mm from the disc with a hand saw. All specimens were stored at −20 °C in the refrigerator and used within a week. Before testing, the discs were thawed overnight in saline solution to promote rehydration.
The surfaces of the discs were prepared with matte speckles before being tested ( Figure 2a). In order to provide good tracking information, the speckles should ideally be 3-5 pixels (about 0.6 mm) in size to optimize the resolution. The speckles were recognized in the following data processing algorithms to compute the disc bulging and strains ( Figure 2b).

Disc Bulging and Strains Computation
During the creep tests, the speckles on the specimen surfaces were captured by the cameras and could be evaluated by means of image processing methods. Since the system parameters were determined during the calibration, it was possible to determine X, Y, and Z displacements and the corresponding strains by measuring the speckles distance and displacement. In principle, these deformations were determined by correlating surface image data from the calibration images and the deformed specimen images.
Disc strains were defined as the strain expressed along the virtual extensometer from the top of the disc to the bottom, which was an average value and the green Lagrange strain. Therefore, the extensometer should be located within the disc region and could be evaluated in both X and Y directions, named exx and eyy. Only eyy values were considered in this study. Disc bulging was computed by stereovision algorithms, and three points were equidistantly assigned from the upper part of the disc to the lower.
Since the specimens needed to be submerged in liquid during the tests, the refraction that occurred at the interfaces was inevitable, producing significant errors in the

Disc Bulging and Strains Computation
During the creep tests, the speckles on the specimen surfaces were captured by the cameras and could be evaluated by means of image processing methods. Since the system parameters were determined during the calibration, it was possible to determine X, Y, and Z displacements and the corresponding strains by measuring the speckles distance and displacement. In principle, these deformations were determined by correlating surface image data from the calibration images and the deformed specimen images.
Disc strains were defined as the strain expressed along the virtual extensometer from the top of the disc to the bottom, which was an average value and the Green-Lagrange strain. Therefore, the extensometer should be located within the disc region and could be evaluated in both X and Y directions, named exx and eyy. Only eyy values were considered in this study. Disc bulging was computed by stereovision algorithms, and three points were equidistantly assigned from the upper part of the disc to the lower.
Since the specimens needed to be submerged in liquid during the tests, the refraction that occurred at the interfaces was inevitable, producing significant errors in the measurements. To avoid the refractive error, the integrated algorithm of the Variable Ray Origin (VRO) camera model was used. This enabled the system to eliminate measurement bias when imaging through optical interfaces.

Fitting Models of Axial Strains and Lateral Bulging
During the creep period, the strain-time data were curve-fitted using the Kelvin function (Equation (1)). The model was typically used to characterize the creep behavior of natural IVDs [33][34][35].
The Kelvin model describes a linear elastic body connected with an ideal viscous body in parallel and then connected with a linear elastic body in a series (Figure 3). ε is the strain of the disc, E 1 is the viscous (equilibrium) modulus, E 2 is the elastic (instantaneous) modulus, δ 0 is the stress applied on the disc, t is the elapsed time, and τ is the time constant. η is the viscosity and can be derived from η = E 2 × τ. The curve-fitting procedure used the nonlinear least-squares fitting method to minimize the summed squares of residuals in MATLAB (MathWorks, Natick, MA, USA, version 19.2). measurements. To avoid the refractive error, the integrated algorithm of the Variable Ray Origin (VRO) camera model was used. This enabled the system to eliminate measurement bias when imaging through optical interfaces.

Fitting Models of Axial Strains and Lateral Bulging
During the creep period, the strain-time data were curve-fitted using the Kelvin function (Equation (1)). The model was typically used to characterize the creep behavior of natural IVDs [33][34][35].

1
(1) The Kelvin model describes a linear elastic body connected with an ideal viscous body in parallel and then connected with a linear elastic body in a series (Figure 3). is the strain of the disc, is the viscous (equilibrium) modulus, is the elastic (instantaneous) modulus, is the stress applied on the disc, and is the elapsed time. η is the viscosity and can be derived from η . The curve-fitting procedure used the nonlinear least-squares fitting method to minimize the summed squares of residuals in MATLAB (MathWorks, Natick, MA, USA, version 19.2). There was no suitable mathematical model used for describing disc bulging during the creep. The protrusion of the disc increases with prolonged time but is limited by structure, making bulging constant under the same load. Therefore, we proposed the parametric horizontal asymptote equations (Equations (2)-(4)) for fitting bulging. Their fitting precision R 2 were computed. The bulging data of three different points (upper, middle, and lower) were computed and fitted using the nonlinear least-squares method in MATLAB (MathWorks, Natick, MA, USA, version 19.2). * * , , and are the model parameters, and is the bulging value at the time, . There was no suitable mathematical model used for describing disc bulging during the creep. The protrusion of the disc increases with prolonged time but is limited by structure, making bulging constant under the same load. Therefore, we proposed the parametric horizontal asymptote equation (Equations (2)) for fitting bulging. Their fitting precision R 2 were computed. The bulging data of three different points (upper, middle, and lower) were computed and fitted using the nonlinear least-squares method in MATLAB (MathWorks, Natick, MA, USA, version 19.2).
a, b, and c are the model parameters, and x is the bulging value at the time t.

Statistical Analysis
Differences between the groups (loads) were determined using a one-way analysis of variance. The level of significance was set at p = 0.05. Both variance and correlation analyses were performed in SPSS v26.0 (SPSS, Chicago, IL, USA).

Effects of Loads on Disc Strains
An axial compression of 300 N resulted in a disc strain of 6.39% (±2.61) and increased with the applied load from 11.28% (±3.56) to 12.59% (±3.68) under 400 N and 500 N (Figure 4a). The typical strain results were fitted by Equation (1) with good quality (Figure 4b).

Effects of Loads on Disc Strains
An axial compression of 300 N resulted in a disc strain of 6.39% (±2.61) and increased with the applied load from 11.28% (±3.56%) to 12.59% (±3.68%) under 400 N and 500 N (Figure 4a). The typical strain results were fitted by Equation (1) with good quality ( Figure  4b). The associated parameters of Equation (1) are shown in Table 1. was derived from , where was the applied load (300 N, 400 N, and 500 N separately), and was the cross-sectional area of the porcine disc obtained from the literature (~800 mm 2 ) [36].

Effects of Loads on Disc Bulging
An  The associated parameters of Equation (1) are shown in Table 1. σ 0 was derived from σ 0 = F A , where F was the applied load (300 N, 400 N, and 500 N), and A was the cross-sectional area of the porcine disc obtained from the literature (~800 mm 2 ) [36].  (1)).

Effects of Loads on Disc Bulging
An axial compression of 300 N induced a bulging of 0.74 (±0.20) mm, 1.50 (±0.61) mm, and 0.76 (±0.24) mm in the upper, middle, and lower part of the discs. The maximum outward disc bulging occurred in the middle part under different magnitudes of loads and increased from 1.67 (±0.83) mm to 1.87 (±0.48) mm under 400 N and 500 N (Figure 5a). The typical bulging results were fitted by Equation (2) with good quality (Figure 5b).
The fitting precision (R 2 ) was determined by both the equation itself and the data quality. Based on the results, the new model had high fitting precision (Table 2). Additionally, the bulging of the discs in the present study was comparable to previous studies (Table 3).

Relations between Disc Strains and Bulging
The strains and bulging maps (Figure 6a) showed the distribution of deformation on the disc surface. With increasing the load, the disc strains and bulging grew larger. An axial load of 500 N produced the largest disc strains and outward bulging, while the maximum protrusion occurred in the middle of the disc under every load.
Although water outflowed during the creep, the value of the strain and bulging seemed to be positively correlated. Herein, we used the Pearson correlation coefficient to analyze the relations of the upper, middle, and lower bulging with the disc strains and the relations between every bulging. The results revealed a strong relationship between the bulging and strain and also within the bulging of each part (Figure 6b

Relations between Disc Strains and Bulging
The strains and bulging maps (Figure 6a) showed the distribution of deformation on the disc surface. With increasing the load, the disc strains and bulging grew larger. An axial load of 500 N produced the largest disc strains and outward bulging, while the maximum protrusion occurred in the middle of the disc under every load.
(a) Figure 6. (a) Disc strains and bulging maps of the frontal surfaces. The surface maps summarized the single-axial loading under compressions of 300 N, 400 N, and 500 N. The data could be extracted from every part or point of these maps. The blank areas were due to the poor imaging quality at this location and were not recorded. (b) The relation between disc strains and bulging (upper, middle, and lower parts of the disc). It was considered that two factors have a strong correlation when R (the absolute value of the correlation) > 0.8; R 0.3-0.8 was considered a weak relation; R < 0.3 was considered no correlation.

Discussion
Creep under axial compression is the main loading status of discs during physiological activities. The creep-associated axial strains and lateral bulging are affected by both the duration and magnitudes of the load. In a previous study, O'Connell et al. [40,41] reported that the measured disc height reduced 30% of the equilibrium displacement after creep for 4 h, thus suggesting that, by reducing the creep duration to 4 h in vitro, it would be enough to simulate the creep responses, as well as avoiding the decay of specimens. Thus, in this study, 4 h was adopted as our test duration and expected for assessing long-time physiological responses. According to the previous in vivo literature [42], 0.375-0.625 MPa, which was the range of compressive stress conducted by this study, was within the daily activity level in porcine discs. Axial deformation during the whole creep period was contributed by both the vertebral bodies and the cartilage endplates when using bone-discbone specimens; thus, they should be excluded to obtain an accurate assessment [7,33,43]. Accordingly, in this study, the 3D-DIC system was adopted to accurately measure the axial strains and lateral bulging of only the disc area for discarding interferences that soft tissues and bony structures brought. This method was proved to be effective in in vitro biomechanical tests by O'Connell et al. [40], who adopted Vic2D (Vic2D, Correlated Solutions Inc.) to evaluate the two-dimensional internal displacements and average strains of discs after 20 min under 1000 N compression. Gullbrand et al. [44] used a high-resolution digital camera (A3800; Basler, Exton, PA, USA) and a custom MATLAB program to track the axial displacements of two ink marks placed on each vertebral body adjacent to the disc in their study. Except for the failure in radial data acquisition and invasive preconditioning, the axial data from their study could be merely extracted from the two preprinted marks, and other areas of the disc could not be selected. The 3D-DIC system adopted in this study was superior in data acquisition from three dimensions, including the radial (X and Z axes) and axial deformation (Y axis), and from every point in the area of interest.
In Figure 4a, the disc height reduced 6.39% (±2.61) under 300 N, 11.28% (±3.56) under 400 N, and 12.59% (±3.68) under 500 N. It seemed that the reduction of the disc height was not proportional to the applied load, and this can be attributed to the fluid transfer during creep. From the perspective of the viscoelasticity of IVDs, the loss of disc height under prolonged creep was mainly governed by the process of fluid outflow, while the water content in natural IVDs was limited, within the range of 70-80%. Combined with the osmotic pressure in the body fluid environment, the disc height will ultimately reach an equilibrium, during which the reduction of the disc height may not be constant [45,46].
It can be seen from Figure 5a that the largest bulging occurred in the middle part of discs during a prolonged period. This can be attributed to the constant volume compression of the nucleus pulposus (NP) in a semi-confined configuration. During the axial compression, the gelatinous and spherical NP was compressed by the loaded upper vertebral body and, thus, was extruded along the transverse plane, forcing the outer annulus fibers (AF) outward, similar to the shape of a drum. Moreover, this phenomenon provided the biomechanical basis for the clinical observation that the middle part of the IVD was the place where disc herniation and subsequent nerve compression mostly occurred. Future implanted biomaterials of the total disc replacement (TDR) could enhance the strength of the middle region of the prosthesis so as to eliminate the secondary protrusion. For instance, in the recent design of the polyvinyl alcohol-bacterial cellulose (PVA-BC) hydrogel-based biomaterials [47], a higher amount of BC can be added at the middle part of the composite to enhance the toughness.
Bovine discs [48] and porcine discs [49,50] were common big animal models used in in vitro studies due to the similarities to human discs and the lack of facet joints and, thus, can be compared directly to human discs. These tissues were also usually used in the preclinical assessment of IVD repair [51][52][53]. It can be seen from Table 3 that the bulging data in this study were comparable to other published literature. Compressions of 300 N, 400 N, and 500 N were selected by this study and by Reuber et al. [37], Heuer et al. [7][8][9], and Fewster et al. [10]. Longer in the loading durations, the bulging data in this study were relatively higher than those previously reported in the literature while on the same order of magnitude, indicating the feasibility of this 3D-DIC system in in vitro biomechanics. It should be noted that different loading protocols were used in above aggregated studies, resulting in a wide range of bulging data in Table 3.
For mathematical characterization, the Kelvin model was adopted for fitting axial strains data. As studies on bulging under creep are lacking, there was no suitable function for fitting bulging data. The empirical equation of horizontal asymptotes was therefore proposed in this study to facilitate mathematical fitting. All R 2 of the model were higher than 0.9, indicating good fitting qualities of the equation. The proposed model could be used for further attempts of bulging fitting of biomaterials in mechanical tests. While it should be admitted that Equation (2) is purely empirical, further progress could be made towards developing functions with more physical significance. In the last few years, many researchers have studied the mechanism of disc responses after prolonged loading. The compression-induced herniation and the relation with the microstructure were studied by Sapiee et al. [54]. The results showed that shear differentially loaded the oblique counter-fibers in lateral annulus, thus increasing its vulnerability to disruption and herniation. From the perspective of chemo-mechanical coupling effects, both the osmotic pressure of the surrounding liquid [55], magnitudes of preloads [56], and loading rates [57] have coupling effects on the mechanical responses of IVDs. Feki et al. [55] also revealed that 18 h of unloading was still not enough to regain the original height of discs, either in hypo-osmotic, iso-osmotic, or in hyper-osmotic conditions. Recently, through scanning electron microscopy (SEM), Tavakoli et al. [58] firstly revealed the elastic fibers network that the straight and thick parallel fibers formed across the NP. Based on this, a novel structural model for the elastic fibers in the NP were proposed. With the rapid development of computational simulations, finite element modeling (FEM) with rational simplifications [59] can provide information on the underlying tissue that cannot be assessed from in vitro experiments [15,60] or original images [61]. Komeili et al. [62] and Castro et al. [63] systemically developed spinal models with linear elastic, hyper-elastic, and biphasic material constitutive models and applied various physiological conditions. Their studies confirmed the central role of the fluid pressure in spinal load sharing and highlighted the differences in energy distribution predicted by various models. As a major failure mode, the damage model in AF has been established to evaluate the fatigue responses [64] and interactions between the lamellar, interlamellar, and NP [65]. The outcome overturned the existing assumption of a uniform stress distribution in the AF and defined that the critical interlamellar spots had the highest delamination potential. Combined with the findings in this study, the area of spinal biomechanics will benefit from a more accurate and multiscale model, and the bio-inspired tissue-engineered scaffolds used for artificial IVDs could thus be developed.

Conclusions
In summary, the results from this study demonstrated that the 3D-DIC system could be used for measuring micro-deformations noninvasively and accurately. This provides a new method, combining stereo vision and image processing, in the area of biomechanical assessments. Additionally, this study helped us with understanding the disc responses under physiological loads, which have important clinical relevance. As the largest protrusion was observed at the middle part of the discs, it is necessary to reinforce IVD protheses at the middle part to avoid secondary herniation after implantation. Furthermore, the newly proposed model for lateral bulging fitting can facilitate future biomechanical studies and are prerequisites for developing implanted biomaterials and implants. Future studies of the mathematical characterization of bulging with more physical significance are warranted. The research and development of artificial disc prostheses with mechanic and structural mimicry are also urgently needed.

Conflicts of Interest:
There are no financial or personal relationships with other people or organizations that could inappropriately bias the contents of this paper.