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Materials 2017, 10(10), 1116; doi:10.3390/ma10101116

Article
Improving the Process of Adjusting the Parameters of Finite Element Models of Healthy Human Intervertebral Discs by the Multi-Response Surface Method
Fátima Somovilla Gómez 1, Rubén Lostado Lorza 1,*Orcid, Marina Corral Bobadilla 1Orcid and Rubén Escribano García 2Orcid
1
Department of Mechanical Engineering, University of La Rioja, 26004 Logroño, La Rioja, Spain
2
IK4-LORTEK, 20240 Ordizia, Guipuzcoa, Spain
*
Correspondence: Tel.: +34-941-299-527
Received: 29 June 2017 / Accepted: 19 September 2017 / Published: 21 September 2017

Abstract

:
The kinematic behavior of models that are based on the finite element method (FEM) for modeling the human body depends greatly on an accurate estimate of the parameters that define such models. This task is complex, and any small difference between the actual biomaterial model and the simulation model based on FEM can be amplified enormously in the presence of nonlinearities. The current paper attempts to demonstrate how a combination of the FEM and the MRS methods with desirability functions can be used to obtain the material parameters that are most appropriate for use in defining the behavior of Finite Element (FE) models of the healthy human lumbar intervertebral disc (IVD). The FE model parameters were adjusted on the basis of experimental data from selected standard tests (compression, flexion, extension, shear, lateral bending, and torsion) and were developed as follows: First, three-dimensional parameterized FE models were generated on the basis of the mentioned standard tests. Then, 11 parameters were selected to define the proposed parameterized FE models. For each of the standard tests, regression models were generated using MRS to model the six stiffness and nine bulges of the healthy IVD models that were created by changing the parameters of the FE models. The optimal combination of the 11 parameters was based on three different adjustment criteria. The latter, in turn, were based on the combination of stiffness and bulges that were obtained from the standard test FE simulations. The first adjustment criteria considered stiffness and bulges to be equally important in the adjustment of FE model parameters. The second adjustment criteria considered stiffness as most important, whereas the third considered the bulges to be most important. The proposed adjustment methods were applied to a medium-sized human IVD that corresponded to the L3–L4 lumbar level with standard dimensions of width = 50 mm, depth = 35 mm, and height = 10 mm. Agreement between the kinematic behavior that was obtained with the optimized parameters and that obtained from the literature demonstrated that the proposed method is a powerful tool with which to adjust healthy IVD FE models when there are many parameters, stiffnesses, and bulges to which the models must adjust.
Keywords:
Finite Elements Method; Multi Response Surface; optimization; biomechanics; human intervertebral lumbar disc

1. Introduction

The human intervertebral disc (IVD) is a fibrocartilage structure that is located between the vertebrae of the spine and absorbs the shock and pressure that are associated with daily movement. The healthy intervertebral disc provides mobility and spine flexibility during body movement and prevents excessive wear of the facet joints during daily movement of the spine. The behavior of the IVD is analogous to a ball full of water. The IVD is usually considered to be incompressible, similar to elastomers, due to its soft tissue and high water content. It permits limited motion while transmitting loads from one vertebra to another. Its complex structure permits significant mobility between adjacent vertebrae while transmitting considerable compressive loads. The IVD has three distinct regions [1]. These are the nucleus pulposus, the annulus fibrosus and the cartilage endplates. The endplate is composed of hyaline cartilage with a thin structure that covers the entire nucleus pulposus and about one third of the annulus fibrosus. The composition of the endplate is similar to a particular cartilage, but with a lower water content. The firm outer region, which is called annulus fibrosus, maintains the shape of the intervertebral disc. The annulus fibrosus is a complex ring structure that surrounds and protects the nucleus pulposus. It can be found between each pair of adjacent vertebrae. It is composed of concentric layers of fibrous tissue that have an approximate thickness of 1 mm. The fibers are an aggregate of collagen fibers and are alternatively oriented to the transverse plane at an angle of approximately ±30° [2]. The annulus fibrosus serves to resist the nucleus pressure in the radial and tangential directions [3]. The fibers are surrounded by a hydrated proteoglycam gel, or annulus ground substance [4]. The inner portion of the annulus is fibrocartilage, which gradually blends into the nucleus pulposus. The inner portion, which is called the nucleus pulposus, is the soft spongy tissue that enables the disc. It functions as a shock absorber, absorbing the impact of the body’s daily movements. The disc may be subjected to a complex combination of loads, but the nucleus pulposus helps to distribute the pressure evenly across the disc. This prevents the development of stress concentrations that could cause damage to the underlying vertebrae or their endplates. Many researchers have experimentally studied the behavior of the intervertebral disc of human cadavers by standardized tests (e.g., compression, flexion, extension, shear, torsion, and lateral bending) based on stiffness and disc bulges [5,6,7,8,9,10,11]. Today, due to medical ethics, the study of human cadavers is declining. Instead, many researchers carry out their studies with animals whose behavior is similar to that of humans [12]. Other researchers conduct their studies by modeling the behavior of the human body’s biomechanics by numerical methods (e.g., Finite Element Method, Finite Volume method, Finite-Difference method, etc.). FEM is known to be a powerful tool to design and optimize mechanical devices, as well as different human body parts or even human prostheses for foot, knee or hip. This is especially true when their behavior is nonlinear [13,14,15,16,17,18,19,20]. Over the years, several researchers have used the FEM to study the mechanical behavior of the human IVD. However, the parameters that define its complex structures differ entirely in each of these studies. Also, the process of achieving the best parameters to correctly define the behavior of the IVD FE models according to the standard test by trial-error method is difficult. This may result in an unacceptable cost. A large number of FE studies have proposed methodologies to adjust the deformations [21], the reaction forces [22] and the intradiscal pressure [23] of the proposed FE models were developed essentially by the trial-error method. However, an optimization algorithm was developed to determine the relationship among the components of the collagen fibers, annulus fibrosus and ground substance of an FE model using the biomaterial properties that were extracted from the literature [24]. In this case, the stiffness of the fibers was varied to approximate Young’s modulus of the ground substance in order to fulfil the required range of motions possible with the properties found in the literature. A method that was based on differential evolution was proposed by Ezquerro [25] to calibrate the FE model of a functional spinal unit that consisted of ligaments, nucleus pulposus, vertebral arch, and facet joints. In this case, the standard loads that were used were: flexion, extension, lateral bending and torsion. More recently, some researchers have used FEM and regression models based on support vector machine (SVM) combined with genetic algorithms (GA) to adjust the IVD FE model parameters [26]. Eleven material parameters were considered in this paper to define the behavior of the human IVD FE model. They were adjusted with nine different variations of stiffness and bulges from some of the previously mentioned standardized tests. These were two parameters for the nucleus pulposus (C10 and C0); two parameters for the cartilage endplate (E, μ); five parameters for the annulus fibers (Fiber12, Fiber34, Fiber56, Fiber78, and Fiber910), and two for the annulus ground substance (E, µ). The lowest value of error (around 6%) between the stiffness and bulges that were obtained from the proposed FE models and those obtained from the selected standardized test indicated that the proposed methodology was very effective. To date, no further references have been found in the literature for adjustment of parameters that define the non-linear behavior of human IVD FE models. However, some researchers have used the MRS method combined with FEM in order to study and optimize mechanical problems [27,28,29]. The current paper attempts to show how the combination of the FEM and the MRS methods with desirability functions can be used to obtain the most appropriate material parameters to use in defining the behavior of FE models of the human intervertebral lumbar disc. The proposed adjustment methods were applied to a medium-sized human IVD corresponding to the L3–L4 lumbar level with standard dimensions of width = 50 mm, depth = 35 mm, and height = 10 mm and developed as follows: First, a three-dimensional parameterized FE model that consisted of nucleus pulposus, cartilage endplate, annulus fibers, and annulus ground substance was developed and simulated with different standard tests. As in Somovilla Gómez et al. [26], and based on standard tests (e.g., compression, torsion, shear, flexion, extension, and lateral bending), three-dimensional parameterized FE models were generated that consisted of nucleus pulposus, cartilage endplate, annulus fibers, and annulus ground substance. Eleven parameters were used to define the parameterized FE models. Then, for each of the standard tests, regression models were generated to model six stiffness and nine bulges of the healthy IVD models as the parameters of the FE models were changed. Finally, an optimal combination of the eleven parameters was achieved by applying MRS based on desirability functions according to three different adjustment criteria. The first considered stiffness and bulges to be equally important for adjustment of FE model parameters. The second considered stiffness as most important, whereas the third considered bulges to be most important. An agreement between the kinematic behavior that was obtained with the optimized parameters and those obtained experimentally from the literature demonstrated that the proposed method is a powerful tool to use in adjusting healthy IVD-FE models when the latter have a high number of parameters, stiffness values, and bulges to which the models must adjust.

2. Use of FEM for Modeling the Lumbar Intervertebral Disc

A majority of the authors consider the nucleus pulposus to be an incompressible material. Its behavior has been modeled with FEM. It is considered to be a non-linear incompressible solid [30] as well as an incompressible fluid [11]. This non-linear behavior is due to its soft tissue condition and high water content [31,32]. In contrast, other authors have modeled the nucleus pulposus considering it to be a linear isotropic incompressible solid, with a Young’s modulus of 0.1 MPa [2,33,34,35], 1 MPa [36,37,38,39,40] and in a range of 0.5 to 1 MPa [41,42] or in a range of 1 to 4 MPa [43], whereas its Poisson’s module was 0.4999 or 0.5. Other authors have considered the incompressible elastomer formulation based on the Mooney-Rivlin model for modeling nucleus pulposus behavior. In this case, C10 and C0 are the empirical constants parameters that are most frequently used. For example, Smit et al. [41] considered the nucleus pulposus as incompressible and used the Mooney-Rivlin model with the parameters C10 = 0.12 and C0 = 0.09 and μ = 0.4999 to model it. Some authors also have considered the Mooney-Rivlin model with values of C10 = 0.12 and C0 = 0.03 [3,30,44], whereas others have used values of C10 = 0.10 and C0 = 0.09 [45,46]. Similarly, Ibarz et al. [47] considered parameters of C10 = 0.0343 MPa and C0 = 0.1369 MPa (E = 1.0 MPa and μ = 0.49).
On the other hand, Lu et al. [48], Martínez et al. [49], and González Gutiérrez et al. [42] considered the endplates’ cartilage behavior. They assumed an isotropic formulation with an elastic modulus E and a Poisson ratio μ of 20 MPa and 0.3, respectively. In a similar fashion, Kim et al. [36] and Dicko et al. [37], used the values for the endplates cartilage of E = 24 MPa and μ = 0.4, whereas Rohlmann et al. [46] and Schmidt et al. [44] used values of E = 23.8 MPa and μ = 0.8. However, Denozière [2] considered values of E = 12 MPa and μ = 0.3. The literature provides a Young’s modulus for the annulus ground substance of 2 MPa to 10 MPa. However, the most commonly used value is 4.2 MPa. Denozière [50], Dietrich et al. [51], and Baroud et al. [30] selected values of 4.2 and 10 MPa, respectively, for Young’s modulus. For Poisson’s ratio, they used values of 0.45 to 0.35. Other researchers considered the annulus ground substance as an incompressible, and used the Mooney-Rivlin model with the parameters of C10 = 0.348, C0 = 0.3, and μ = 0.45 [45,46,52]. Schmidt et al. [44] used values of C10 = 0.10, C0 = 0.05 and μ = 0.45. Dicko et al. [37] considered C10 = 0.18, C0 = 0.04, and μ = 0.45. Ayturk et al. [38] considered the annulus ground substance as a Yeoh hookean hyperelastic material that is a model for the deformation of nearly incompressible nonlinear realistic materials such a rubber. For the collagen fiber layers, most researchers have chosen linear properties in their studies, which provide an average Young’s modulus of approximately 360 to 550 MPa and a Poisson’ ratio of 0.3 to 0.45 [2,30,33,35,36,38]. However, Grauer et al. [39] considered values of 175 MPa for all fiber layers. Other authors assumed a nonlinear stress-strain behavior of the annulus collagen fiber layers based on a nonlinear function [37,44,45,46,52] that was obtained from previous reports that were based on the stress-strain curve of Shirazi-Adl et al. [11]. Ayturk et al. [38] considered the annulus fibers as Yeoh hyperelastic material with C10 = 0.0146, C20 = −0.0189, C30 = 0.04; a3 = 0.03, and b3 = 120. Table 1 shows the large number of parameters that some researchers have used during the last 30 years to develop intervertebral disc FE models. The latter adjusted the intervertebral disc FE models according to standard tests by trial and error, although this is difficult and very costly. To date, no author has made a methodical adjustment of these parameters. Only Somovilla Gómez et al. [26] have used FEM and Data Mining Techniques to adjust intervertebral disc FE models. In this paper, the authors propose the adjustment of the parameters that best define the behavior of the intervertebral disc. They propose the use of FEM and MRS with desirability functions and consider three different adjustment criteria. The first considers stiffness and bulges to be equally important, the second considers stiffness as most important, and the third considers bulges as most important.

2.1. Biomechanics of the Intervertebral Disc

The intervertebral disc (IVD) is one of the most important constitutive elements of a functional spine unit (FSU). An FSU consists of two adjacent vertebrae, the intervertebral disc, and all adjoining ligaments between them. The FSL excludes other connecting tissues such as muscles. Its complex structure allows significant mobility between two adjacent vertebrae while transmitting considerable compressive loads from one vertebra to another. Over time, the normal aging process causes the intervertebral discs to degenerate. This reduces the water contained in the intervertebral disc and the ability to absorb the impacts that are associated with spinal movements. Excessive pressure, strain, or injury to a rigid disc can cause the disc to tear or bulge. The degeneration of the disc’s size and the loss of its functionality will bring the adjacent vertebrae closer to one another. This causes impingement and compression of a spinal nerve root. Nerve impingement can result in intermittent low back pain or leg pain, depending on the level of impingement [3]. This problem can be relieved by a surgical procedure called artificial disc replacement. In this procedure, the damaged disc is replaced by an artificial prosthesis (arthroplasty). Figure 1a,b show, respectively, an FSU with a healthy disc and an FSU with a herniated disc.

2.2. Spine Movement

Spinal movement during daily activities is complex because it involves a combination of plane movements with several motions in axial, coronal, and sagittal planes. In the coronal plane, the spinal movement takes place when an individual bends forward or backward. This typically occurs when one is exercising or performing heavy tasks. Forward bending of the spine is defined as flexion, whereas backward bending is called extension. Movement in the sagittal plane occurs when bending laterally to the right or to the left. Bending the spine to the right is called right bending and to the left is called left bending. Movement of the spine in the axial plane occurs after applying torsion clockwise and counterclockwise. Torsion of the spine is defined as torsion. Figure 2 shows the plane movements and the movements [53]: (a) Movement Planes; (b) Compression; (c) Lateral Bending; (d) Torsion; (e) Flexion; (f) Extension.

2.3. Bulge

In this work, the radial bulge of the lower disc was examined in the healthy model. The bulge was calculated at only three characteristic locations of the disc (anterior, lateral, and posterior) as shown in Figure 3, when axial compression, flexion, extension and lateral bending were simulated. The bulges for the torsion and shear tests were not considered. The disc bulged anteriorly during flexion, posteriorly during extension, and toward the concavity of the spinal curve during lateral bending. One of the mechanisms of nerve root irritation and herniated disc is root impingement by disc bulge. Clinical interest has led to several in vitro studies that have carefully measured the disc bulge. In the early studies, only the compression load was applied [54], but in later studies, the effects of various other loads also were considered in several directions, such as flexion, extension, and lateral bending [55,56,57,58,59].

2.4. Experimental Behavior of the Intervertebral Disc

For decades, most authors have studied the behavior of the human intervertebral disc by standard tests using human cadavers based on stiffness and disc bulges. The standard tests examine compression, flexion, extension, shear, lateral bending, and torsion. One of the first authors to study the behavior of the intervertebral disc was Virgin [60]. He investigated the elastic properties of the intervertebral disc in order to ascertain their contribution to the function of the vertebral column.

2.4.1. Compression

In this work, Virgin [60] obtained a stiffness value of 2500 N/mm in a compression tests when he applied a load of 4500 N. Other authors used loads of 5500 N to 1000 N for the compression test and obtained stiffness values of 3000 to 700 N/mm [3,54,60,61,62,63,64]. In other works, for lower compression values with loads between 850 and 500 N, stiffness values between 2420 and 510 N/mm were obtained [5,7,65,66,67]. Others authors applied loads with values of less than 400 N and obtained stiffness values of 800 and 250 N/mm [8,68,69,70]. In regard to the bulges in the standard compression test, many authors analyzed the anterior, posterior, and lateral bulges of the intervertebral disc. In this regard, Shirazi-Adl et al. [11] used loads of 500, 720, and 1000 N and obtained values of 0.5–0.7–0.8 mm for the anterior bulge, 0.75–1.0–1.5 mm for the posterior bulge, and 0.35–0.4–0.6 mm for the lateral bulge. However, Denozière [2] considered a compression load of 2500 N and obtained results for the anterior, posterior and lateral bulges of 0.5, 0.7, and 0.4 mm. Table 2 summarizes the loads that were used in the compression test and the different stiffness and bulge values that were obtained by different authors.

2.4.2. Flexion and Extension

One of the first studies of the behavior of Flexion and Extension of the human intervertebral disc was [61]. In this work, a load of 20 Nm was considered for both flexion and extension. Stiffness values of 4.5 Nm/° were obtained in both tests. Similarly, Brown et al. [69] considered the same load (20 Nm) but obtained a stiffness value of 2 Nm/° in both tests. Another study, Miller et al. [71], used higher loads (70 Nm), and obtained stiffness values of 5.51 and 7.60 Nm/° for the Flexion and Extension tests respectively. Finally, Adams et al. [72] used a load of 80 Nm and obtained stiffness values of 7.3 Nm/° for both tests. However, most researchers have used loads of 5 Nm to 10 Nm for their studies. For example, the work by Schultz et al. [9] considered a load of 10.6 Nm for a study of the behavior of the intervertebral disc, and obtained stiffness values of 1.92 Nm/° and 3.53 Nm/° for the Flexion and Extension test respectively. Adams et al. [73] considered a load of 10.7 Nm, and the stiffness obtained was 1.34 Nm/°. Other authors used a load of 10 Nm for the Flexion and Extension test. They obtained stiffness values that ranged between 0.8 and 2.04 Nm/° for Flexion, and between 2 and 3.53 Nm/° for Extension [7,53,67]. Guan et al. [74], Busscher et al. [75] and [76], González Gutiérrez [3], and Patwardhan et al. [77] considered a load range of 4 to 8 Nm for both the Flexion and Extension tests. The stiffness values that they obtained from these tests varied from 0.8 to 1.18 Nm/° for the Flexion test and from 0.8 to 1.53 Nm/° for the Extension test. In regard to the study of anterior, posterior and lateral bulges by the Flexion and Extension tests, Reuber et al. [58] considered loads of 3.9 Nm to 7.9 Nm and obtained values of 0.73 and 1.11 mm for the posterior bulge, respectively. For the lateral bulge, they obtained values of 0.07 and 0.21 mm respectively. Finally, Denozière [2] used a load of 10 Nm and obtained values of 1.3, 1.9, and 2.6 mm for the anterior, posterior, and lateral bulges, respectively. Table 3 summarizes the stiffness and bulge values for the Flexion and Extension standard tests by the various authors.

2.4.3. Lateral Bending

As in the previous standard test, Schultz et al. [61] obtained a value of stiffness equal to 2.8 Nm/° for Lateral Bending tests using a load of 20 Nm. Subsequently, Miller et al. [71] obtained a stiffness value of 4.35 Nm/° for Lateral Bending using a load of 60 Nm. Other authors considered a lower range of loads (between 4 and 10 Nm) and obtained values that ranged between 0.75 and 2 Nm/° [3,7,9,53,74,75,76,78]. In addition, one of the few authors who were found in the extensive literature to have studied the anterior, posterior and lateral IVD bulges under a Lateral Bending load was Reuber, M. et al. [58]. In this work, two different loads of 9.8 Nm and 3.9 Nm were used. They resulted in bulge values of 0.49 and 1.13 mm for the posterior bulge respectively, whereas the lateral bulge values were 0.83 and 2.11 mm respectively (See Table 4).

2.4.4. Shear and Torsion

Only a few studies in the literature have studied the stiffness of the IVD in a shear test. One of these was conducted by Schultz et al. in [9,61], who used loads of 1000 N and 980 N and obtained, respectively, 685 N/m and 1000 N/m for stiffness values. Similarly, Weisse et al. [79] obtained a stiffness value of 830 N/m when they applied a load of 950 N. Another research team to conduct this type of test was Liu et al. [6]. In this work, the load applied was 450 N and the stiffness value that was obtained was 300 N/m. Other authors, such as Miller et al. [71] and Markolf [62], applied loads of 150 N for the same shear test but obtained different stiffness values (115 N/m and 260 N/m). Finally, Moroney et al. [68] used a much lower load than those used by the previously mentioned authors (20 N) and obtained a stiffness value of 60 N/m.
When the disc is subjected to a torsion load, the stress distribution in the IVD depends on the disc’s degree of degeneration and whether the posterior elements are intact. Miller et al. [71] was one of the groups of investigators in the literature, who considered the highest torsion load. It was 70 Nm and produced a stiffness value of 10.9 Nm/°. Farfan et al. [80] and Schultz et al. [61] considered loads of 31 Nm and 30 Nm, and obtained stiffness values of 2 and 4.5 Nm/° respectively. However, most researchers considered torsion test loads of 10 Nm to 8 Nm and obtained stiffness values that ranged from 8.48 to 1.44 Nm/° [7,53,78,81,82]. Today, loads of 5 Nm to 4 Nm are applied in this type of standard test, and the stiffness values that are obtained vary from 4.4 to 1.6 Nm/° [42,75,76]. Table 5 summarizes the stiffness values and the loads applied in shear and torsion tests.
As described in the previous paragraphs, the published studies of the behavior of IVD are extensive. In this regard, the current work selected only fifteen different stiffness values and bulges, as well as the corresponding loads, for a search for the optimal combination of the disc’s FE model parameters. Table 6 summarizes the 15 different stiffness and bulge values that were selected (six for stiffness and nine for bulges) and the load that each author, whose work was selected, applied in the corresponding standard test. These values were used to adjust the parameters that define the behavior of the FE model.

3. FE Model for Modeling the Intervertebral Disc Proposed

Like the authors’ disc that was modeled by FEM and described in Section 2 (See Table 1), this paper considered that intervertebral discs have four main parts. They are the annulus fibrosus, the nucleus pulposus, and the two endplates that link it to the vertebrae. Figure 4 shows a 3D view of the proposed human healthy IVD FE model. In the lumbar spine, the nucleus is located between the middle and the posterior third of the sagittal diameter. Its volume is 30–50% of the total disc volume [53]. In some finite element analyses of the lumbar spine, the authors concluded that the nucleus was 43% of the total disc volume, which is the volume that the current paper assumes.
Also, the nucleus pulposus was considered to be a non-linear incompressible and hyper-elastic Mooney Rivlin solid material. It was formulated according to empirical constants C10 and C0 [30,41]. It was assumed that the nucleus pulposus is surrounded by the annulus fibrosus. In order to prevent volumetric blockage, as the nucleus was considered to be practically incompressible, a full integration with Herrmann formulation was used for the eight-node solid FE elements that constituted the nucleus pulposus [83,84]. The annulus fibrosus was assumed to be a homogenous ground substance of hydrated proteoglycan gel that is reinforced by collagen fibers. The current FE model includes five layers of fibers that were oriented at an angle of ±30° relative to the transverse plane. The fibers were simulated by three-dimensional unidirectional line FE elements. The homogenous ground substance was simulated as eight-node isoparametric solid FE elements. For the five layers (Fiber12, Fiber34, Fiber56, Fiber78, and Fiber910), a range of E of 550 to 360 MPa, respectively, was assumed for their elastic modulus values. The endplates, which are composed of hyaline cartilage, link the disc to the vertebrae. This work considered cartilage endplates with an isotropic formulation (an elastic modulus E and a Poisson ratio of μ) for the FE model). A total of 11 material parameters were selected to define the behavior of the human disc FE model. They were adjusted after considering the above mentioned standard test. Table 7 summarizes these eleven different proposed material parameters.

3.1. Configuration of the Finite Element Analysis and Mesh Size

Due to the large displacements and deformations that the intervertebral disc suffered in this study, as well as the hyper-elastic behavior of the nucleus pulposus, a nonlinear analysis that involved large strain procedure with an updated Lagrangian formulation was used. Also, the mesh size that was established for each of the four parts of the intervertebral disc was based on the extensive literature available [85,86]. In this case, the largest mesh size for hexahedral elements of the endplate and annulus fibrosus was 2.2 mm, whereas the lowest was 0.5 mm, which corresponds to the thickness of the endplate itself (See Figure 5a). Also, the largest mesh size for the nucleus pulposus was 1.6 mm, and the smallest size was 0.3 mm (See Figure 5b). The openings in all of the mesh sizes that were established for the proposed FE models were smaller than each of the four parts of the intervertebral disc than those sizes of the FE models proposed in the literature. In other words, the discretization that is chosen can obtain satisfactory results without depending on further mesh refinement.

3.2. Intervertebral Disc Dimensions

For decades, many authors have studied the kinematic behavior of human intervertebral discs of patients of different ages, sex, and stature, as well as different states of degeneration of the intervertebral disc itself. Many of these studies have been based exclusively on experimental studies using intervertebral discs from cadavers. Other works have been based mainly on FE models of discs for the study of their kinematics. In the studies that were based on FE models, the proposed models have been experimentally validated using data obtained by the authors themselves, or have been experimentally validated using data obtained from the literature. For a very broad range of patients that were studied, all lumbar intervertebral discs for the L1–L5 segment analyzed have had very similar widths, depths, and heights (See Figure 6a). For example, Zhou et al. [87] studied 126 patients with differing degrees of disc degeneration according to the Friberg and Hirsch scale [88]. Fifty-five male patients with a mean age of 50 ± 13.6 and 71 female patients with a mean age of 49 ± 12.04 and an age range of 22–80 years were studied. The approximate dimensions of the intervertebral discs analyzed in this case were, according to Figure 6a, a width of 46.1 ± 3.1 mm, a depth of 34.1 ± 2.6 mm, and a height of 12.4 ± 1.7 mm. Rostedt et al. [5] experimentally studied six lumbar FSUs from four subjects with different degrees of disc degeneration. The less degenerate discs (with a degree of degeneration equal to 1) were those of individuals of 45 years of age and a disc height of 12 mm. Schultz et al. [9] studied 16 UVF from 10 females and 26 UVF from 13 males with an age range of 21 to 60 and a mean age of 43 years with different degrees of degeneration. In this case, the less damaged discs were those of men of 35, 40, and 53 years of age with a disc surface of 15.9, 16.8, and 15 cm2, respectively. Other authors, such as Panjabi [89], studied UVF with healthy intervertebral discs from a sample of 60 patients of age 19–59 years (a mean of 46.3) and disc dimensions of width of 48.1 mm and a depth of 34.7 mm. Eijkelkamp [90] also studied a group of 60 patients of age 18 to 65 years, and observed that the mean intervertebral disc height was 13.5 mm. With a group of 157 patients and ages between 20 and 38 years, Nissan and Gilad [91] determined an average depth and height of the disc of 34.6 mm and 10.8 mm, respectively. With a smaller group (11 individuals) of age between 25 and 36 years, Tibrewal and Pearcy [92] found a depth and height of the disc of 33 mm and 9.8 mm, respectively. However, Wolf et al. [93] determined a mean width and depth of 44.1 mm and 31.7 mm respectively for a disc in a group of 55 individuals of 20 to 90 years of age. A more extensive study was that of Amonoo-Kuofi [94]. In the latter case, a group of 305 males and 310 females of 10 to 64 years of age and 10 to 61 years respectively were studied. The mean dimensions of the intervertebral discs were a depth of 42.8 mm and a height of 13.5 mm for males and a depth of 39.9 mm and a height of 13 mm for females. Other authors proposed FE models and validated them later with their own experimental data. For example, Schmidt et al. [24] developed a method to determine the individual contribution of the fibers and the ground substance to bending moments with four different magnitudes. They proposed a 3D, non-linear FE-model of an intervertebral disc from the L4–L5 FSU. In order to determine the appropriate geometry of the FE-model, six specimens were tested. The median anteroposterior dimension of the inferior endplate of L4 was 37.4 mm, whereas the median left to right dimension was 58.7 mm. The geometry used in the FE-model was determined from one of the tested specimens. The latter had dimensions nearest to the median with an anteroposterior distance of 38.1 mm and a left to right distance of 59.0 mm. Kim et al. [36] studied with the FEM the influence of the disc degeneration. A 3D nonlinear FE model of the lumbar spine was developed that consisted of four lumbar vertebrae, three intervertebral discs, and the associated spinal ligaments. Geometrical details of the human lumbar spine (L2–L5) were obtained from the high-resolution computed tomography (CT) images of a 46-year-old male patient who had no spinal deformities. In this case, the average area of the intervertebral disc was 1119 mm2. González et al. [42] studied the biomechanics of the intervertebral disc with different degrees of degeneration in compression using a FE model that was validated by experimental data. The study was performed on 5 patients aged 65–75 years for L2–L3 and L4–L5 lumbar levels. The height of the healthy intervertebral disc that was studied was 9.9 mm with an area of 1845 mm2 for the L2–L3 level. For the L4–L5 level, it was 10 mm with an area of 1951 mm2. Shirazi-Adl et al. [11] proposed an FE model for analyzing the stress of the UVF para el nivel lumbar L2–L3 in compression for a 29 years old female patient. The geometry of the FE model that was analyzed was based in in vitro measurements. It had a width of 49.2 mm, a depth of 34 mm, a heigth of 11 mm and an area of 1371 mm2. Smit et al. [41] proposed a complete UVF FE model and studied the stresses on the cortical bone of the vertebrae. In that case, the intervertebral disc that was proposed in the FE model had a width of 42 mm and a depth of 35 mm. Finally, other authors proposed FE models and later validated them with experimental data from other authors. For example, Ibarz et al. [47] proposed a FE model for the Lumbar Spine to study the degeneration of the intervertebral discs. To verify the FE model, radiological images (X-rays) were taken of a group of 25 healthy male individuals who had an average age of 27.4 and average weight of 78.6 kg. The results obtained from the simulation and from radiological measurements were compared to the results obtained from White and Panjabi [95]. Good agreement was obtained for the different movements. However, Ayturk [38] proposed an FE model to study the changes in lumbar spine mechanics due to degenerative disc disease. The experimental study was conducted on a 49-year-old female patient. The results obtained from the FE model were also compared to the studies that were conducted by Heuer et al. [96]. Other authors such as Weisse et al. [79] proposed an FE model for the determination of the translational and rotational stiffness of an L4–L5 FSU using the FEM. In this case, the geometry of the disc was obtained by computed tomography, and the results were compared subsequently to the studies by several of the aforementioned authors [87,89,90,91,92,93,94]. Once the proposed FE model was simulated, the results were compared experimentally to results obtained from the literature [96,97,98]. Denozière and Ku [50] proposed a 3D FE model of a two-level ligamentous lumbar segment. The proposed FE model was validated with the clinical data obtained from the literature [98,99,100,101]. These authors developed an FE model of the intervertebral disc according to Figure 6a. In this case, the intervertebral disc had a width of 50 mm, a depth of 35 mm, a height of 10 mm and an area of 1440 mm2. Table 8 summarizes the anatomical dimensions of the L1–L5 lumbar segment and the differents intervertebral discs that were studied. As has been observed from the references in Table 8, the dimensions of the intervertebral discs are very similar for the range of patients studied. Also, it can be seen in this table that many of the mentioned authors validate their FE models with the experimental data obtained from the studies by other authors. For this reason, the FE model of the intervertebral disc that is studied in the current paper has been elaborated on the basis of the study by Denozière and Ku [50], which had the following dimensions according to Figure 6a (width = 50 mm, depth = 35 mm, height = 10 mm, and area = 1440 mm2).

3.3. Boundary Conditions

Numerous authors have studied the behavior of intervertebral discs with the FEM. In doing do, they applied the contour conditions to these models and thus reproduced, as rigorously as possible, the corresponding experimental analysis [2,3,50]. In the current paper, the boundary conditions applied to the proposed FE models were applied in a similar way to the work that was developed by Denozière [2] and Denozière and Ku [50]. The proposed intervertebral disc FE model consisted of the intervertebral disc itself (nucleus pulposus, endplate, annulus fibrosus, and the fiber layers), as well as two steel supports to which the disc was glued (See Figure 6a). The boundary conditions applied to the proposed FE model to obtain the compression stiffness were applied as follows (see Figure 6b). On the upper steel support the pressure Pc was applied. It was calculated by the following equation:
P c = F max S
where Fmax is the maximum compression test load, whose value was 500 N (See Table 6). S is the upper surface of the steel support, which has an area of 1440 mm2. The compression stiffness value was obtained from the “Z” displacement. The boundary conditions to obtain the stiffness value for the bending test were applied as follows (See Figure 6c): the corresponding maximum torque of the bending test (Bmax) was applied on the anterior area of the steel upper support, which, in this case, is 5 Nm (See Table 6). The pressure Pb was calculated as follows:
P b = B max S a · CG
where Sa is the anterior area of the intervertebral disc, namely 745 mm2, and CG is the distance from the geometric center of the disc surface or instantaneous axis of rotation (IAR) to the center of gravity of Sa. The bending stiffness value was calculated once the angle α had been obtained. The boundary conditions of the FE model that were required to obtain the lateral bending stiffness were applied in similar fashion to the process to obtain the bending stiffness (see Figure 6d). Over half area of the upper steel support was applied the pressure Plb, which was calculated according to the following equation:
P lb = LB max ( S / 2 ) · CG
where LBmax is the maximum torque of the bending test, which in this case was to 10.6 Nm (see Table 6). S/2 is the area of the intervertebral disc that is considered for the lateral bending, namely 720 mm2 and CG is the distance from the geometric center of the disc to the center of gravity of S/2 area. The lateral bending stiffness was obtained from angle β. The boundary conditions necessary to obtain the torsional stiffness (see Figure 6e) are: on a group of nodes that belong to the periphery of the steel upper plate and a force Ft (in this case in the Y direction) that is applied in a way that twists the disc. In order that the load Ft is always at an angle of 90° to a given R, a “following force” analysis was considered. The value of this force Ft was calculated with the following equation:
F t = T max n · R
where Tmax is the maximum torque value. In this case, it is 10 Nm (see Table 6), Also, n is the number of nodes at which the load Ft was applied and R is one the half of the width of the intervertebral disc (in this case is 25 mm). The value of the stiffness of lateral flexion is obtained from the angle η. Finally, the boundary conditions that are necessary to obtain shear stiffness (see Figure 6f) are as follows: on all of the nodes on the upper surface of the steel support, an Fs load was applied in the Y direction. The value of this force Fs was calculated with the following equation:
F s = F max n
where Fmax is the value of the maximum torque, which is 450 N in this case (see Table 6), and n is the number of nodes to which the load Fs is applied. The value of the stiffness of shear was obtained from the Y displacement of these nodes. In addition, for all previously proposed simulations, a boundary condition of embedding all nodes was imposed on the lower support of steel.

4. Design of the Experiments and Design Matrix

The response surface method (RSM) needs to establish a design of experiments (DoE) in order to obtain accurate models with the least amount of data to support the initial hypotheses [102]. Several methods have been proposed to develop DoE. However, they generally require the construction of a design matrix (inputs) to measure the outputs or responses of the experiments. One of the most widely used methods is the full factorial design method [103]. This method is based on the adoption all possible combinations of each of the values (or levels) and each of the factors that are considered in the DoE. Another DoE that is widely used is the 2k factorial design. This DoE is a full factorial design that has two levels and generates 2k experiments in which k is the number of factors. In using this method to adjust the parameters of the FE model, the number of factors is k = 11 (C0, C10, Fiber12, Fiber34, Fiber56, Fiber78, Fiber910, Annulus_E, Annulus_μ, Cartil_E, and Cartil_μ) and the number of experiments or FE simulations needed is 2048. Similarly, a 3k factorial design generates 3k experiments or 177147 FE simulations. For both methods, this amount of data may be sufficient to cover completely the entire range of possibilities. However, it has the disadvantage of generating a large number of experiments or FE simulations. Another of the methods that are used to develop a DoE is Central Composite Design (CCD). This method is considered to be a fractional three-level design that is useful in obtaining regression models, thereby reducing the number of experiments in comparison to a factorial design 3k. A CCD generates 2070 experiments. This is very similar to the number that a 2k factorial generates. However, a Box-Behnken design (BBD) generates 176 experiments. This is an advantage because it significantly reduces the number of FE simulations [104]. In the current paper, the DoE was employed using a two-level fractional factorial design. In this case, 128 FE simulations were needed. This reduced amount of data may be sufficient to completely cover the entire range of possibilities and subsequently to search for the optimal parameters that define the behavior of the intervertebral disc. Table 9 shows some of the 128 combinations of the material parameters (C0, C10, Fiber 12, AnnulusE, etc.) that are generated with a two-level fractional factorial design when the ranges that appear in Table 7 are considered. The design matrix and their corresponding combination of parameters that define de behavior of the human lumbar Intervertebral disc FE models are generated by using the “R” statistical software (R 2014).

4.1. Response Surface Method for Modeling and Optimizing Problems

RSM is a method that attempts to determine the relationships between input variables and one or more response variables. The method was introduced by Box and Wilson in [104] and was used first to obtain a regression model from experimental data and an optimal response. Originally, RSM was developed to model experimental responses. However, it has been used recently to optimize products and industrial processes [105,106]. RSM is a group of statistical techniques that utilize a regression model that is based on a polynomial function (Equation (6)).
Y = f ( x 1 , x 2 , x 3 , , x k + e )
where Y is the response variables for the experiments and ( x 1 , x 2 , x 3 , , x k ) are the input variables, e is an error, and f is a function that consists of cross-products of the terms that form the polynomial. To optimize the response Y, it is necessary to find an approximation functional relationship between the inputs and the response surface. A polynomial is the functional relationship used by RSM ((Equation (7)) in this kind of work.
Y = b 0 + i = 1 n b i · X i + i = 1 n b ii · X i 2 + i = 1 n 1 j = i + 1 n b ij · X i · X j + e
where the first summation is the linear part, the second is the quadratic part and the third is the product of pairs of all inputs of the polynomial. The values of the coefficients b0, bi, bii, and bij are calculated by use of a regression analysis to determine the relationship between inputs and outputs. Also, the terms that are selected to form the equation are chosen according to their levels of significance [107]. One computes this level by the analysis of variance (ANOVA) and selecting terms according to the p-value obtained. When a problem has more than one output, as does this work, the optimization problem can be solved by using MRS [108]. In this regard, Harrington developed the desirability functions to obtain a compromise between the different outputs (Equations (8) and (9)) and the overall desirability. The latter is defined as the geometric mean of the desirability of each output (Equation (10)), [109].
d r max = { 0 if   f r ( X ) < A ( f r ( X ) A B A ) S if   A f r ( X ) B 1 if   f r ( X ) > B
d r min = { 1 if   f r ( X ) < A ( f r ( X ) B A B ) S if   A f r ( X ) B 0 if   f r ( X ) > B
D = ( r = 1 R d r ) 1 / R
In these equations, parameters A and B correspond to the limits of the input ranges, S is an exponent that determines the importance of reaching the target value, X corresponds to the input vector, and fr is the polynomial function that is used to predict the response. A second polynomial degree should be used to optimize one or several responses [108]. The desirability approach involves transforming each estimated response into a unitless utility that is bounded by 0 < dr < 1, where a higher value of dr indicates that the response value is more desirable. The optimization, which was developed with R statistical package, searches for a combination of importance factors that simultaneously satisfy the optimization criteria of each response and input [110].

4.2. Combining FEM and MRS to Optimize Mechanical Problems

Over the years, many researchers have used the FEM as an alternative to reduce costs during the design and optimization phases of mechanical problems. The FEM is widely used in industry to analyze engineering problems that are too complicated to solve by classical analytical methods. The object of its application is to reduce the expense incurred by experimental tests. FEM has the disadvantage of requiring a high computational cost especially when the process of fitting the proposed FE models is based only on the experience of the designer through simulations and trial and error. This adjustment process is greatly amplified when the FE model tries to solve non-linear problems, such as mechanical contacts, large deformations and hyper-elastic material behavior. However, the computational cost is reduced with the RSM, which provides an approximation of the same problem that is modeled with FEM [111]. Building regression models based on RSM is a good strategy to reduce the number of simulations that are necessary to model and optimize FE models. These models learn from the most characteristic samples that are obtained from FEM simulations and use their outputs. In this regard, the combination of RSM and FEM has been used widely to optimize many industrial processes. For example, Lin et al. [112] proposed a functionally graded material (FGM) as a potential upgrade to some conventional implant materials like titanium in prosthetic dentistry. In that work, computational bone remodeling and design optimization are used. Based on the results of remodeling, RSM has been adopted to develop a multi-objective optimal design for FGM implantation. Similarly, Sadollah and Bahreininejad [113] investigated the development of an optimal design of an FGM dental implant to promote long-term success. Also, Rungsiyakull et al. [114] combined RSM and FEM to establish a relationship between the surface morphology induced micromechanics and bone remodeling responses to a solid bead coated porous implant. In this case, the RSM was used to relate the major implant coating parameters to the bone responses. More recently, Bahraminasab et al. [115] studied the use of a functionally graded material (FGM) for the femoral component of knee implants. This is attractive because the properties can be designed to vary in a certain pattern to meet the desired requirements at different regions of the knee joint system, thereby decreasing the loosening problem. Therefore, a multi-objective design optimization of a FGM femoral component is conducted in this study by use of the FEM and RSM. The results of using an optimized FGM are then compared to those of using a standard Co–Cr alloy in a femoral component knee implant to demonstrate relative performance. In the current paper, the RSM used the results of FE analysis to search for the optimal parameter that defines correctly the behavior of healthy human lumbar intervertebral disc (IVD) FE models. Eleven parameters with which to define the parameterized FE models were selected. They were C0, C10, Fiber12, Fiber34, Fiber56, Fiber78, Fiber910, Annulus_E, Annulus_μ, Cartil_E, and Cartil_μ. For each of the standard tests, regression models were generated to model the six stiffness and nine bulges of the healthy IVD models when the parameters of the FE models were changed. The optimal combination of the eleven parameters was achieved by applying MRS based on desirability functions according to three different adjustment criteria.

5. Results and Discussion

5.1. FE Models’ Results

After the parameterized healthy IVD FE models were created, an automatic procedure conducted the 128 simulations according to the design matrix that appears in Table 9. Table 10 shows the stiffness and bulges that were obtained from the FE simulations. They are named as follows and grouped according to the corresponding standard test. For the compression test, the anterior, posterior and lateral bulges values (Comp_bulgeA, Comp_bulgeL, Comp_bulgeP), as well as the stiffness (Comp_stiff), were obtained. The same applies to Shear, Extension, Lateral Bending, Flexion and Torsion tests: Shear_stiff, Exte_bulgeL, Exte_bulgeP, Exte_stiff, LBend_bulgeL, LBend_bulgeP, LBend_stiff, Flex_bulgeL, Flex_bulgeP, Flex_stiff, and Tors_stiff. These 128 values formed the training dataset and were used to generate the regression models. The following subsections show the process to generate and optimize these models.

5.2. Analysis of Variance

Equation (6) was fitted with the data that appear in Table 9 and Table 10 to obtain the regression equations for all responses by the use of the RMS “R” package [116]. Each of the responses or outputs from Equations (6) through (20) is shown. They were obtained by a combination of polynomials that are formed by input variables. The responses are: Comp_bulgeA, Comp_bulgeL, Comp_bulgeP, Comp_stiff, Shear_stiff, Exte_bulgeL, Exte_bulgeP, Exte_stiff, LBend_bulgeL, LBend_bulgeP, LBend_stiff, Flex_bulgeL, Flex_bulgeP, Flex_stiff, and Tors_stiff.
Comp _ bulgeA = 0.453125 0.150922 × C 10 0.154598 ×   C 0 4.4 × 10 0.5 × Fiber 12 2.8 × 10 0.5 × Fiber 56 2.3 × 10 0.5 × Fiber 78 2.5 × 10 0.5 × Fiber 910 0.034134 × Annulus _ E + 0.262148 × Annulus _ μ 0.001137 × Cartil _ E + 0.040998 × Cartil _ μ
Comp _ bulge   L = 0.169454 0.084948 × C 10 0.087169 × C 0 5 × 10 0.5 × Fiber 12 0.006264 × Annulus _ E + 0.088348 × Annulus _ μ 0.001015 × Cartil _ E + 0.031074 × Cartil _ μ
Comp _ bulge   P = 1.206599 0.91178 × C 10 0.915255 × C 0 0.113434 × Annulus _ E + 0.442496 × Annulus _ μ 0.003303 × Cartil _ E + 0.117189 × Cartil _ μ
Comp _ stiff = 635.243327 + 822.048324 × C 10 + 834.364625 × C 0 + 180.06106 × Annulus _ E + 2051.947822 × Annulus _ μ + 5.673156 × Cartil _ E
Cort _ stiff = 70.312808 + 199.672441 × C 10 + 199.736145 × C 0 + 0.02166 × Fiber 12 + 0.055155 × Fiber 34 + 0.031639 × Fiber 56 + 0.057449 × Fiber 78 + 0.044255 × Fiber 910 + 58.151336 × Annulus _ E 72.981833 × Annulus _ μ + 0.926751 × Cartil _ E 16.717294 × Cartil _ μ
Exte _ bulgeL = 0.083398 + 0.00016 × Fiber _ 910 + 0.028063 × Annulus _ μ 0.001135 × Cartil _ E
Exte _ bulgeP = 0.243941 + 0.001987 × Fiber 910 + 0.76198 × Annulus _ μ 0.003522 × Cartil _ E
Exte _ stiff = 6.205107 0.01339 × Fiber 910 + 2.595841 × Annulus _ μ + 0.008305 × Cartil _ E
LBend _ bulgeL = 1.574761 1.014308 × C 10 0.929876 × Annulus _ μ 0.017706 × Cartil _ E + 1.620387 × Cartil _ μ
LBend _ bulgeP = 2.995354 3.019289 × C 10 3.118902 × C 0 0.212027 × Annulus _ E + 1.351171 × Annulus _ μ 0.023247 × Cartil _ E + 0.84824 × Cartil _ μ
LBend _ stiff = 1.428516 + 1.708987 × C 10 + 0.297374 × Annulus _ E + 3.014858 × Annulus _ μ + 0.041594 × Cartil _ E 1.896308 × Cartil _ μ
Flex _ bulgeL = 0.125918 0.064104 × C 10 0.060597 × C 0 3.1 e 0.5 × Fiber 12 0.00418 × Annulus _ E + 0.04945 × Annulus _ μ 0.000311 × Cartil _ E 0.019064 × Cartil _ μ
Flex _ bulgeP = 0.681909 0.920479 × C 10 0.921496 × C 0 0.048626 × Annulus _ E 0.113105 × Annulus _ μ + 0.039252 × Cartil _ μ
Flex _ stiff = 0.697824 + 1.433372 × C 10 + 1.443693 × C 0 + 0.269714 × Annulus _ E + 2.099688 × Annulus _ μ + 0.013196 × Cartil _ E - 0.368862 × Cartil _ μ
Tors _ stiff = 2.079685 + 1.056596 × C 10 + 0.901329 × C 0 + 0.00263 × Fiber 34 0.000469 × Fiber 56 + 0.001202 × Fiber 78 + 0.908448 × Annulus _ E 0.905043 × Annulus _ μ + 0.013001 × Cartil _ E 0.521362 × Cartil _ μ
In order determine whether the variables that were used in the linear regression models are statistically significant, an ANOVA test was developed. Table 11, Table 12, Table 13, Table 14, Table 15, Table 16, Table 17, Table 18, Table 19, Table 20, Table 21, Table 22, Table 23, Table 24 and Table 25 show the ANOVA results. The tables show that most of the variables have a p-value of less than 0.01. This means that the models are statistically significant. In addition, the multiple correlation coefficient (R2) was calculated as a measure of the variation around the mean that the regression model produced. The results show that all values of R2 are close to one. This indicates that these models possess good predictive capacity.
Also, MAE and RMSE are calculated to determine the generalization capacity of the regression models that were obtained by using the results of the design matrix (inputs) from Table 9 and their corresponding outputs from Table 10 according to Equations (26) and (27).
M A E = 1 m · k = 1 m | Y k   F E M Y k   Model |
R M S E = 1 m · k = 1 m ( Y k   F E M Y k   Model ) 2
Table 26 shows the prediction errors, when the maximum error corresponds to Exte_stiff (MAE equal to 13.97% and RMSE equal to 21.81%). and the minimum error corresponds to Shear_stiff (MAE equal to 1.74% and RMSE equal to 2.19%).
Figure 7 shows the relationship between some of the actual values that were obtained experimentally with FEM (Table 10) and the predicted (regression models) values of LBend_stiff (Figure 7a), Comp_stiff (Figure 7b), Flex_bulgeP (Figure 7c), Flex_bulgeL (Figure 7d), Comp_bulgeP (Figure 7e), and Comp_bulgeA (Figure 7f). These figures indicate that the correlations between the real values and the predicted ones are high, and the residuals that were obtained were small.
Additionally, 30 new FE models were created to test the proposed regression models with parameters that had not been used previously to generate regression models. Table 27 shows the errors incurred during the testing stage, when the maximum error corresponds to Tor_stiff (MAE equal to 10.06% and RMSE equal to 19.50%) and the minimum error corresponds to Flex_bulgeP (MAE equal to 2.86% and RMSE equal to 3.19%). The errors show that the adjustment between the regression models and the results obtained from the FE models is almost accurate. This also demonstrates its good capacity for generalization.
Figure 8 shows the relationship between some of the 30 additional FE models that were implemented to test the regression models and their corresponding FE simulations. Figure 7 shows the values of LBend_stiff (Figure 8a), Comp_stiff (Figure 8b), Flex_bulgeP (Figure 8c), Flex_bulgeL (Figure 8d), Comp_bulgeP (Figure 8e), and Comp_bulgeA (Figure 8f). The figures indicate that the correlations are high, whereas the residuals that were obtained are small, which indicate that these regression models are adequate for the prediction of IVD behavior.

5.3. Multiple Response Optimization

Table 28, Table 29 and Table 30 show a combination of the eleven material parameter (inputs) that were studied in searching for the optimal behavior of human intervertebral lumbar disc FE models when using the desirability functions with the “R” package [110]. Three different adjustment criteria were considered. The first column of the tables shows the material parameters (inputs) and stiffness values and bulges (outputs) that were studied. The second column shows the goals that were established in the goal setting process for both inputs and outputs when different adjustment criteria are considered. The third column shows the optimal values to define the behavior of the FE model and the last column shows the desirability values. Table 28 shows the results when all material parameters, as well as stiffness and bulges, were considered to have the same level of importance (equal to “inRange”). In this case, the value of the overall desirability was 0.625. Also, the table shows that some of the values that were obtained are very close to the targets that were proposed. For example, the target proposed for the Exte_bulgeL was 0.1 and the optimal value that was obtained was 0.100 with a desirability value of 1. In contrast, the proposed target for the Comp_bulgeL was 0.35 and the optimal value that was obtained was 0.0970 with a desirability value of 0.244.
Table 29 shows the results when the stiffness was considered to have a higher level of importance than bulges. This table shows that the material parameters that were obtained for all different goals required in the first criteria are very similar, although the value of overall desirability was 0.817, which is higher than the first criteria. Also, the target for the parameters of Shear_stiff and Tors_stiff were 300 and 2.1, and the values obtained were 300.000 (desirability = 1) and 3.456 (desirability = 0.428), respectively.
Table 30 shows the results that were obtained with the third criterion when the bulges were considered to be the most important. Analogously to the other adjustment criteria that were previously studied, the material parameters that have been obtained are very similar for all different goals. In this case, the value of overall desirability was 0.554, which is the lowest result of the three criteria that are studied in this paper. Also, it is observed that the targets for the parameters of Exte_bulgeL and Comp_bulgeL were 0.1 and 0.35, and the values obtained were 0.103 (desirability = 0.947) and 0.101 (desirability = 0.257), respectively.
Finally, three new FE models were simulated with the eleven different optimal material parameters that considered the three different adjustment criteria. These FE models were simulated again by the same standard test (Compression, Flexion, etc.) in order to compare the methodology proposed for adjust the optimal parameters. Table 31 shows a comparison of the results with the FE models using the optimized parameters, the optimal results from the regression models using the RMS with desirability functions and the experimental standard test. In order to compare the different errors that were obtained using three different adjustment criteria that were studied in this case, different MAE were obtained from the normalized data. The data is commonly normalized in statistical processes to transform all variables to the same scale (from 0–1). The transformation was achieved in this case by subtracting the minimum value from each original value and dividing the result by the range of each variable according to Equation (28).
Y k , n o r m = Y k min ( Y ) max ( Y )
where Yk,norm were the normalized outputs that were obtained from the results of the FE models with the optimized parameters and the outputs that were obtained experimentally from the standard test. The first column of the table shows the stiffness and bulges (outputs) that were studied. The second, third, and fourth columns show, respectively, the results obtained from the FE models with the optimized parameters for each of the different criteria that were considered. The fifth column shows the values of the standard test, and the last column shows the normalized MAE that was obtained. It can be seen in the table that the normalized MAE of the three criteria considered are very similar (Criteria 1 = 0.2782, Criteria 2 = 0.2795 and Criteria 3 = 0.2788). In contrast, the normalized MAE obtained for each of the outputs is lower when predicting the Shear_stiff (MAE = 0.01) and greater when predicting Comp_bulgeL (MAE = 0.73). The reason for this difference may be that the proposed FE model is generally less accurate in predicting bulges than stiffness. In addition, all of the values of MAE that were obtained for each of the different outputs or stiffness and bulges are in acceptable agreement.

6. Conclusions

This paper sets out a fully automated method that combines the finite element method (FEM) and multi-response surface method (MRS) based on desirability functions. This work looks for the optimal parameters that will correctly define the behavior of a medium-sized healthy human lumbar intervertebral disc (IVD) models based on FEM. First, based on standard tests (compression, flexion, extension, shear, lateral bending, and torsion), three-dimensional parameterized finite element (FE) models were generated. Then, 11 parameters were selected to define the parameterized intervertebral lumbar disc FE models. For each of the standard tests, regression models were generated for modeling the six stiffness and nine bulges of the healthy IVD models when the parameters of the FE models were varied. The optimal combination of the 11 parameters was achieved by applying MRS based on desirability functions according to three different adjustment criteria. The first criterion considered all of the material parameters (inputs), as well as the stiffness and bulges (outputs), with the same level of importance (equal to “inRange”). The second criterion considered, with a higher level of importance the stiffness, then bulges (goal of stiffness = target), whereas the third criteria considered, the bulges with a higher level of importance, then stiffness (goal of bulges = target). The best fit of the FE model parameters was achieved with the proposed second criterion. This produced a value for the normalized MAE of 0.2795. However, the results were very similar for the first and the third criteria that were considered. These were, respectively, MAE = 0.2782 and MAE = 0.2788. In contrast, the normalized MAE obtained for each of the outputs was lower when predicting the Shear_stiff (MAE = 0.01) and greater when predicting Comp_bulgeL (MAE = 0.73). This reason for this difference may be that the proposed FE model is generally less accurate in predicting bulges than stiffness. The MAE obtained from each criterion that was studied demonstrated that the proposed method is a powerful tool for adjusting healthy IVD FE models when there are many parameter and the stiffness and number of bulges to which the models must be adjusted, are high.

Acknowledgments

The authors wish to thank the La Rioja Government through Project ADER 2014-I-IDD-00162.

Author Contributions

Finite Element Models: Fátima Somovilla Gómez, Rubén Lostado Lorza, and Marina Corral Bobadilla. Development of predictive models and optimization: Rubén Escribano García. Results analysis and manuscript preparation: all authors.

Conflicts of Interest

The authors declare no conflict of interest.

List of Symbols/Nomenclature

MAEMean Absolute Error
C10Mooney-Rivlin parameter for annulus ground substance
C0Mooney-Rivlin parameter for annulus ground substance
Annulus_EYoung’s modulus for the annulus ground substance(MPa)
Annulus_μPoisson’s modulus for the annulus ground substance
Fiber12Collagen fibers layer 1(mm2)
Fiber34Collagen fibers layer 2(mm2)
Fiber56Collagen fibers layer 3(mm2)
Fiber78Collagen fibers layer 4(mm2)
Fiber910Collagen fibers layer 5(mm2)
Cartil_EYoung’s modulus for the endplate cartilage(MPa)
Cartil_μPoisson’s modulus for the endplate cartilage-
Comp_bulgeABulge Anterior for the Compression Test(mm)
Comp_bulgeLBulge Lateral for the Compression Test(mm)
Comp_bulgePBulge Posterior for the Compression Test(mm)
Comp_stiffStiffness for the Compression Test(N/mm)
Shear_stiffStiffness for the Shear Test(N/mm)
Exte_bulgeLBulge Lateral for the Extension Test(mm)
Exte_bulgePBulge Posterior for the Extension Test(mm)
Exte_stiffStiffness for the Extension Test(Nm/°)
LBend_bulgeLBulge Lateral for the Lateral Bending Test (mm)
LBend_bulgePBulge Posterior for the Lateral Bending Test(mm)
LBend_stiffStiffness for the Lateral Bending Test(Nm/°)
Flex_bulgeLBulge Lateral for the Flexion Test(mm)
Flex_bulgePBulge Posterior for the Flexion Test(mm)
Flex_stiffStiffness for the Flexion Test(Nm/°)
Tors_stiff Stiffness for the Torsion Test(Nm/°)

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Figure 1. (a) Healthy Functional Spinal Unit (b) Detailed view of the nerve impingement and herniated intervertebral disc.
Figure 1. (a) Healthy Functional Spinal Unit (b) Detailed view of the nerve impingement and herniated intervertebral disc.
Materials 10 01116 g001
Figure 2. (a) Movement Planes; (b) Compression; (c) Lateral Bending; (d) Torsion; (e) Flexion; and (f) Extension.
Figure 2. (a) Movement Planes; (b) Compression; (c) Lateral Bending; (d) Torsion; (e) Flexion; and (f) Extension.
Materials 10 01116 g002
Figure 3. (a) Three-dimensional view of bulges and (b) actual dimension of the anterior, posterior, and lateral bulges.
Figure 3. (a) Three-dimensional view of bulges and (b) actual dimension of the anterior, posterior, and lateral bulges.
Materials 10 01116 g003
Figure 4. (a) Details of the FE model that is formed by the nucleus cartilage endplates, nucleus pulposus, and annulus fibrosus; and (b) details of the orientation of the five different fiber layers.
Figure 4. (a) Details of the FE model that is formed by the nucleus cartilage endplates, nucleus pulposus, and annulus fibrosus; and (b) details of the orientation of the five different fiber layers.
Materials 10 01116 g004
Figure 5. (a) Details of the mesh size for the endplate and annulus fibrosus; and (b) details of the mesh size for the nucleus pulposus.
Figure 5. (a) Details of the mesh size for the endplate and annulus fibrosus; and (b) details of the mesh size for the nucleus pulposus.
Materials 10 01116 g005
Figure 6. Intervertebral disc (IVD) dimensions and boundary conditions necessary for the standard test: (a) IVD dimensions; (b) compression load; (c) flexion load; (d) lateral bending load; (e) torsion load; and (f) shear load.
Figure 6. Intervertebral disc (IVD) dimensions and boundary conditions necessary for the standard test: (a) IVD dimensions; (b) compression load; (c) flexion load; (d) lateral bending load; (e) torsion load; and (f) shear load.
Materials 10 01116 g006
Figure 7. Scatter diagram of (a) lateral bending stiffness; (b) compression stiffness; (c) flexion bulge posterior; (d) flexion bulge lateral; (e) compression bulge posterior; and (f) compression bulge anterior.
Figure 7. Scatter diagram of (a) lateral bending stiffness; (b) compression stiffness; (c) flexion bulge posterior; (d) flexion bulge lateral; (e) compression bulge posterior; and (f) compression bulge anterior.
Materials 10 01116 g007
Figure 8. Test scatter diagram of (a) lateral bending stiffness; (b) compression stiffness; (c) flexion bulge posterior; (d) flexion bulge lateral; (e) compression bulge posterior, and (f) compression bulge anterior.
Figure 8. Test scatter diagram of (a) lateral bending stiffness; (b) compression stiffness; (c) flexion bulge posterior; (d) flexion bulge lateral; (e) compression bulge posterior, and (f) compression bulge anterior.
Materials 10 01116 g008
Table 1. Range of the material parameters proposed to define the behavior of human intervertebral lumbar disc models based on the finite element method (FEM).
Table 1. Range of the material parameters proposed to define the behavior of human intervertebral lumbar disc models based on the finite element method (FEM).
Nucleus PulposusCartilage EndplateAnnulus GroundAnnulus Fibers
AuthorsFE ParametersFE ParametersFE ParametersFE ParametersFE Parameters
Mooney-Rivlin Max.IsotropicIsotropicMooney-RivlinIsotropicF1F2F3F4F5
C10C0EΜEμC10C0EμEμ
Kim and Chun (2015) [36]--10.4999240.40--4.20.45550–3580.3
Dicko et al. (2015) [37]--10.4999240.400.180.045-0.45Non-linear stress-strain curve
González et al. (2015) [42]0.120.030.5 < E < 10.4 < μ < 0.5200.3--0.75–50.35–0.5--
Ibarz, Elena et al. (2014) [47]0.03430.1369------4.20.455505034554083600.3
Tsouknidas et al (2012) [35]--0.20.4999----4.20.455504854404203600.45
Ayturk, U.M. (2010) [38]--10.499923.80.8C10 = 0.0146; C20 = −0.0189; C30 = 0.041a3 = 0.03; b3 = 120
Schmidt, Kettler (2007) [44]0.120.03-0.499923.80.80.100.05 0.45* Stress-strain curve by Shirazi: σ = 23,000 × ε1.9
Rohlmann et al. (2006) [46]0.100.09-0.499923.80.80.3480.30.420.45* Stress-strain curve by Shirazi: σ = 23,000 × ε1.9
Rohlmann, Zander (2006) [45]0.100.09-0.4999--0.3480.30.420.45* Stress-strain curve by Shirazi: σ = 23,000 × ε1.9
Grauer et al. (2006) [39]--10.4999----4.20.45175175175175175-
Dietrich, M. et al. (2005) [51]--0.0120.4999----100.35------
Denoziére, G. et al (2004) [2]--0.10.4999120.3--4.20.455504854404203600.3
Baroud et al. (2003) [30]0.120.03------80.45500485420360--
Pitzen et al. (2002) [33]--0.10.4999----4.20.45500485420360--
Dooris et al. (2001) [40]--10.49------------
Eberlain et al. (2001) [52]Incompress. Fluid--23.80.40.3480.340.4* Stress-strain curve by Shirazi:σ = 23,000 × ε1.9
Martínez et al. (1997) [49]----200.3-0.3--------
Lu et al. (1996) [48]----200.3--4.20.45------
Smit et al. (1997) [41]0.120.090.5 < E < 10.4999------------
Sharma et al. (1995) [34] 0.10.4999----4.20.5
Lavaste et al. (1992) [43]--1 < E < 40.5------------
Shirazi-Adl et al. (1984) [11]Incompress. Fluid------4.20.45σ = 23,000 × ε1.9
Yeoh material. Material coefficients: C10 = 0.0146, C20 = −0.0189, C30 = 0.041; a3 = 0.03, b3 = 120.0 (b3 is unitless). C10 = 0.0343 MPa; C0 = 0.1369 MPa. An elastic analysis with a Young modulus of 1.0 MPa and Poisson ratio of 0.49 was conducted with similar results and a volume change of less than 0. * is Stress-strain curve by Shirazi et al. [11]: in this case, ε is the value for the deformation and σ is the stress.
Table 2. Summary of stiffness, bulges, and compression loads from various authors.
Table 2. Summary of stiffness, bulges, and compression loads from various authors.
Compression Test
AuthorsStiffness (N/mm)Load (N)Range of Load (N)
Moroney et al. (1988) [68]50074
Brown et al. (2002) [69]400200<400
Keller et al. (1987) [70]247253
Berkson et al. (1979) [8]800400
Nachemson et al. (1979) [67]571500
Rostedt et al. (1998) [5]810500
Stokes et al. (2002) [65]510500850–500
Panjabi et al. (1984) [66]750600
Gardner-Morse et al. (2004) [7]2420850
Hirsh and Nachemson (1954) [54]7001000
Schultz et al. (1973) [61]15001000
González Gutierrez (2012) [3]8331000
González Gutierrez (2012) [3]9331000
González Gutierrez (2012) [3]108910005500–1000
Markolf (1970) [62]18001800
Virgin (1951) [60]25004500
Rolander and Blair (1975) [63]30005000
Brown et al. (1957) [64]23005300
Bulges Values
AuthorsAnterior/Post/Lateral (mm)Load (N)
Reuber et al. (1982) [58]-/0.24/0.66400
Schmidt, Kettler (2007) [44]0.7 to 0.9500
Shirazi-Adl et al. (1984) [11]0.5/0.75/0.35500
Shirazi-Adl et al. (1984) [11]0.7/1/0.4720
Reuber et al. (1982) [58]-/0.34/0.8800
Brinckmann et al. (1991) [59]0.151000
González Gutierrez (2012) [3]0.691000
Shirazi-Adl et al. (1984) [11]0.8/1.5/0.61000
Nachemson, A. (1960) [71]-2000
Denozière (2004) [2]0.5/0.7/0.42500
Klein et al. (1983) [57]0.6-
Table 3. Stiffness, bulge values, and flexion-extension loads from various authors.
Table 3. Stiffness, bulge values, and flexion-extension loads from various authors.
Flexion/Extension Test
AuthorsStiffness Values (Nm/°)Load (Nm)
Guan et al. (2007) [75]0.82/1.534
Busscher et al. (2009) [76]0.84
Busscher et al. (2010) [77]0.85
González Gutierrez (2012) [3]1.18/1.385
Patwardhan et al. (2003) [78]1.338
White and Panjabi (1978) [79]0.8/210
Nachemson et al. (1979) [67]2.03/3.5310
Gardner-Morse et al. (2004) [7]2.0410
Schultz et al. (1979) [9]1.92/3.5510.6
Adams et al. (1980) [74]1.3410.7
Schultz et al. (1973) [61]4.520
Brown et al. (2002) [69]220
Miller et al. (1986) [72]5.51/7.6070
Adams et al. (1996) [73]7.380
Bulges Values
AuthorsAnterior/Post/Lateral (mm)Load (Nm)
Denoziére, G. et al. (2004) [2]1.3/1.9/2.610
Reuber et al. (1982) [58]-/0.73/0.073.9
Reuber et al. (1982) [58]-/1.11/0.217.9
Table 4. Stiffness values and lateral bending loads from various authors.
Table 4. Stiffness values and lateral bending loads from various authors.
Lateral Bending Test
AuthorsStiffness Values (Nm/°)Load (Nm)
Guan et al. (2007) [75]0.764
Busscher et al. (2009) [76]0.54
Busscher et al. (2010) [77]0.65
González Gutierrez (2012) [3]1.585
White and Panjabi (1978) [79]0.910
Nachemson et al. (1979) [67]1.110
Gardner-Morse et al. (2004) [7]1.2910
Schultz et al. (1979) [9]210.6
Schultz et al. (1973) [61]2.820
Miller et al. (1986) [72]4.3560
Bulges Values
AuthorsAnterior/Post/Lateral (mm)Load (Nm)
Reuber et al. (1982) [58]-/0.49/0.833.9
Reuber et al. (1982) [58]-/1.13/2.119.8
Table 5. Stiffness and loads for the shear and torsion tests by various authors.
Table 5. Stiffness and loads for the shear and torsion tests by various authors.
Shear Test
AuthorsStiffness (N/mm)Load (N)
Moroney et al. (1988) [68]6020
Markolf (1970) [62]260150
Miller et al. (1986) [72]115150
Liu et al. (1975) [6]300450
Weisse et al. (2012) [80]830950
Schultz et al. (1979) [9]1000980
Schultz et al. (1973) [61]6851000
Torsion Test
AuthorsStiffness (Nm/°)Load(Nm)
Busscher et al. (2009) [76]2.54
Busscher et al. (2010) [77]1.65
González Gutierrez (2012) [3]4.45
Haughton et al. (1999) [83]76.6
Adams et al. (1981) [82]1.447.4
White and Panjabi (1978) [79]2.2210
Nachemson et al. (1979) [67]8.4810
Gardner-Morse et al. (2004) [7]2.110
Schultz et al. (1979) [9]7.0710.6
Schultz et al. (1973) [61]4.530
Farfan et al. (1970) [81]231
Miller et al. (1986) [72]10.970
Table 6. Stiffness and bulge values selected from the standard tests that are used to adjust the parameters that define the behavior of the finite element (FE) model of the intervertebral lumbar disc.
Table 6. Stiffness and bulge values selected from the standard tests that are used to adjust the parameters that define the behavior of the finite element (FE) model of the intervertebral lumbar disc.
TestAuthorLoad UsedStiffness
CompressionRostedt et al. (1998) [5]500 N810 N/mm
FlexionGonzález Gutierrez (2012) [3]5 Nm1.18 Nm/°
ExtensionGuan et al. (2007) [75]4 Nm1.53 Nm/°
Lateral Bending BendingSchultz et al. (1979) [9]10.6 Nm2.0 Nm/°
ShearLiu et al. (1975) [6]450 N300 N/mm
TorsionGardner-Morse et al. (2004) [7]10 Nm2.1 Nm/°
TestAuthorsLoad UsedBulgeBulgBulge
Anterior (mm)Posterior (mm)Lateral (mm)
CompressionShirazi-Adl et al. (1984) [11]500 N0.50.750.35
FlexionReuber et al. (1982) [58]3.9 Nm-0.730.07
ExtensionReuber et al. (1982) [58]3.9 Nm-0.240.1
Lateral BendingReuber et al. (1982) [58]9.8 Nm-1.132.11
Table 7. Range of the proposed material parameters for defining the behavior of the human intervertebral lumbar disc models based on FEM.
Table 7. Range of the proposed material parameters for defining the behavior of the human intervertebral lumbar disc models based on FEM.
TissueFE ParametersTissueFE Parameters
Min.Max.Min.Max.
Nucleus Pulposus Annulus Fibrosus
C100.110.14Fiber12515.0550.0
C00.020.04Fiber34503.0515.0
Endplate Fiber56455.0503.0
E23.055.0Fiber78408.0455.0
μ0.30.4Fiber910360.0408.0
---E Annulus Fibrosus4.04.2
---μ Annulus Fibrosus0.250.45
Table 8. Range of the material parameters proposed to define the behavior of human intervertebral lumbar disc models based on FEM.
Table 8. Range of the material parameters proposed to define the behavior of human intervertebral lumbar disc models based on FEM.
Summary of Anatomical Dimensions of L1–L5
AuthorsGroup Size (n)Lumbar LevelSexMean Age
(From…to…)
Width
(mm)
Depth
(mm)
Height
(mm)
Area (cm2)
Rostedt et al. (1998) [5]4L3–L4-45--12-
Schultz et al. (1979) [9]1L1–L5male35---1590
Schultz et al. (1979) [9]1L1–L5male40---1680
Schultz et al. (1979) [9]1L1–L5male53---1500
Zhou et al. (2000) [88]55L3–L5male50 (22–80)5337.512.21492 ± 173.8
Zhou et al. (2000) [88]71L3–L5female49 (22–80)50.535.411.31492 ± 173.8
Panjabi (1992) [90]60L1–L5-46.3 (19–59)48.134.7--
Eijkelkamp (2002) [91]60L1–L5-(18–65)--13.5-
Nissan and Gilad (1986) [92]157L1–L5-26.8 (20–38)-34.610.8-
Tibrewal and Pearcy (1985) [93]11L1–L5-29.5 (25–36)-339.8-
Wolf et al. (2001) [94]55L1–L5-(20–90)44.131.7--
Amonoo-Kuofi (1991) [95]305L1–L5male(10–64)-42.813.5-
Amonoo-Kuofi (1991) [95]310L1–L5female(10–61)-39.913-
Schmidt et al. (2006) [24]-L4–L5--58.737.4--
Kim and Chun (2015) [36]1L4–L5male46---1119
González et al. (2015) [42]5L2–L3male/female(65–75)--9.91739
González et al. (2015) [42]5L4–L5male/female(65–75)--101951
Shirazi-Adl et al. (1984) [11]1L2–L3female2949.234111371
Smit et al. (1997) [41]-L4--4235--
Ibarz, Elena et al. (2014) [47]25L5–S1 27.4
Ayturk, U.M. (2010) [38]-L1–L5female49 -
Weisse et al. (2012) [80]-L4–L5male4350.333.712.8-
Denozière (2004) [2]-L3–L4--5035101440
Table 9. Design matrix for the simulation of FE models when considering combination of 128 material parameters (inputs).
Table 9. Design matrix for the simulation of FE models when considering combination of 128 material parameters (inputs).
RunC10C0Fiber
12
Fiber
34
Fiber
56
Fiber
78
Fiber
910
Annulus
E
Annulus
μ
Cartil
E
Cartil
μ
10.110.0251550345540836040.25550.4
20.140.025155034554083604.20.45230.3
30.110.045155034554083604.20.45230.4
40.140.0451550345540836040.25550.3
50.110.025505034554083604.20.45550.3
60.140.0255050345540836040.25230.4
70.110.0455050345540836040.25230.3
80.140.045505034554083604.20.45550.4
90.110.025155154554083604.20.25230.4
100.140.0251551545540836040.45550.3
1200.140.0455050350345540840.45230.4
1210.110.0251551550345540840.25550.4
1220.140.025155155034554084.20.45230.3
1230.110.045155155034554084.20.45230.4
1240.140.0451551550345540840.25550.3
1250.110.025505155034554084.20.45550.3
1260.140.0255051550345540840.25230.4
1270.110.0455051550345540840.25230.3
1280.140.045505155034554084.20.45550.4
Table 10. Results of the simulation of FE models when a combination of 128 material parameters are considered in Table 9.
Table 10. Results of the simulation of FE models when a combination of 128 material parameters are considered in Table 9.
RunComp
BulgeA
Comp
BulgeL
Comp
BulgeP
Comp
Stiff
Shear
Stiff
Exte
BulgeL
Exte
BulgeP
Exte
Stiff
LBend
BulgeL
LBend
BulgeP
LBend
Stiff
Flex
BulgeL
Flex
BulgeP
Flex
Stiff
Tors
Stiff
10.2650.0890.593984.510305.7520.0950.5741.7440.7731.0851.9980.0760.3511.6353.593
20.3370.1330.7301271.883281.3710.1370.8302.1210.9701.9291.5810.0950.2881.7623.264
30.3470.1400.7581253.827275.9270.1390.8612.0941.3862.1361.3200.0930.3011.6463.150
40.2560.0850.5531017.076315.4250.0960.5661.7790.7981.0512.2570.0730.3031.6863.640
50.2980.0940.6651500.886306.5970.0960.6962.4060.6171.2543.2680.0800.3212.2273.618
60.2900.1110.694897.694284.6110.1350.6671.6271.5201.8051.1600.0760.3311.3483.191
70.2860.1080.683891.309285.8010.1350.6441.6191.3491.6531.3220.0820.3341.3943.271
80.2950.0920.6341548.473316.7040.0950.6852.4610.6721.1883.2150.0760.2802.3093.682
90.2900.1130.711901.414289.5520.1320.6611.6611.5311.8921.1300.0780.3491.3383.320
100.3040.0950.6701474.443300.6170.0990.7282.3510.6251.2373.2290.0810.3012.1923.447
1200.3450.1360.7521256.486278.2780.1390.8752.0631.3222.0701.3820.0900.2841.6733.016
1210.2600.0880.596989.416313.2130.0940.5741.7490.7781.0861.9900.0760.3521.6353.665
1220.3330.1310.7311276.741287.8400.1350.8292.1270.9611.9281.5830.0940.2891.7663.318
1230.3420.1390.7601258.926282.3580.1370.8602.1001.3742.1371.3250.0920.3021.6523.201
1240.2520.0850.5551022.062322.9970.0950.5651.7830.8011.0522.2500.0730.3041.6863.723
1250.2940.0920.6671507.286314.0670.0940.6932.4140.6141.2513.2620.0790.3222.2303.670
1260.2860.1100.697901.330291.3070.1330.6671.6301.5241.8091.1600.0760.3321.3503.255
1270.2820.1070.685894.800292.5340.1340.6441.6211.3511.6541.3210.0820.3351.3953.333
1280.2910.0910.6351555.367324.3280.0930.6842.4690.6701.1863.2100.0760.2812.3103.740
Table 11. Analysis of variance (ANOVA) table for a compression bulge anterior linear model.
Table 11. Analysis of variance (ANOVA) table for a compression bulge anterior linear model.
Compression Bulge Anterior BulgeL
Var.DfSum of Sq.Mean SquareF Valuep-ValueSignificance Code
C1010.000660.0006635.82.41 × 10−8***
C010.000310.0003116.78.01 × 10−5***
Fiber1210.000080.000084.14.47 × 10−2*
Fiber5610.000060.000063.17.96 × 10−2.
Fiber7810.000040.000042.01.58 × 10−1
Fiber91010.000050.000052.61.09 × 10−1
Annulus_E10.001490.0014981.54.25 × 10−15***
Annulus_μ10.087960.087964805.3<2.2 × 10−16***
Cartil_E10.042370.042372314.5<2.2 × 10−16***
Cartil_μ10.000540.0005429.43.24 × 10−7***
Residuals1170.002140.00002
Significance codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1.
Table 12. ANOVA table for a compression bulge lateral linear model.
Table 12. ANOVA table for a compression bulge lateral linear model.
Compression Bulge Lateral
Var.DfSum of Sq.Mean SquareF Valuep-ValueSignificance Code
C1010.000210.000218.34.73 × 10−3**
C010.000100.000103.95.12 × 10−2.
Fiber1210.000100.000104.04.89 × 10−2*
Annulus_E10.000050.000052.01.60 × 10−1
Annulus_μ10.009990.00999398.4<2.2 × 10−16***
Cartil_E10.033780.033781347.0<2.2 × 10−16***
Cartil_μ10.000310.0003112.36.31 × 10−4***
Residuals12000.003010.00003
Significance codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1.
Table 13. ANOVA table for a compression bulge posterior linear model.
Table 13. ANOVA table for a compression bulge posterior linear model.
Compression Bulge Posterior
Var.DfSum of Sq.Mean SquareF Valuep-ValueSignificance Code
C1010.024440.02444323.4<2.2 × 10−16***
C010.010720.01072141.9<2.2 × 10−16***
Annulus_E10.016470.01647217.9<2.2 × 10−16***
Annulus_μ10.250630.250633316.3<2.2 × 10−16***
Cartil_E10.357530.357534730.9<2.2 × 10−16***
Cartil_μ10.004390.0043958.26.11 × 10−12***
Residuals1210.009140.00008
Significance codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1.
Table 14. ANOVA table for a compression stiffness linear model.
Table 14. ANOVA table for a compression stiffness linear model.
Compression Stiffness
Var.DfSum of Sq.Mean SquareF Valuep-ValueSignificance Code
C10119,46219,46214.72.00 × 10−4***
C01891189116.71.06 × 10−2*
Annulus_E141,50041,50031.41.34 × 10−7***
Annulus_μ15,389,4275,389,4274074.1<2.2 × 10−16***
Cartil_E11,054,6281,054,628797.2<2.2 × 10−16***
Residuals122161,3871323
Significance codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1.
Table 15. ANOVA table for a shear stiffness linear model.
Table 15. ANOVA table for a shear stiffness linear model.
Shear Stiffness
Var.DfSum of Sq.Mean SquareF Valuep-ValueSignificance Code
C1011148.21148.2438.8<2.2 × 10−16***
C01510.6510.6195.1<2.2 × 10−16***
Fiber12118.418.47.09.15 × 10−3**
Fiber34114.014.05.42.24 × 10−2*
Fiber56173.873.828.25.34 × 10−7***
Fiber781233.3233.389.24.82 × 10−16***
Fiber9101144.4144.455.22.02 × 10−11***
Annulus_E14328.44328.41654.0<2.2 × 10−16***
Annulus_μ16817.76817.72605.3<2.2 × 10−16***
Cartil_E128,143.428,143.410,754.5<2.2 × 10−16***
Cartil_μ189.489.434.24.73 × 10−8***
Residuals116303.62.6
Significance codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1.
Table 16. ANOVA table for an extension bulge lateral linear model.
Table 16. ANOVA table for an extension bulge lateral linear model.
Extension Bulge Lateral Stiffness
Var.DfSum of Sq.Mean SquareF Valuep-ValueSignificance Code
Fiber91010.001890.0018912.94.62 × 10−4***
Annulus_μ10.001010.001016.99.74 × 10−3**
Cartil_E10.042210.04221288.7<2.2 × 10−16***
Residuals12440.018130.00015
Significance codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1.
Table 17. ANOVA table for an extension bulge posterior linear model.
Table 17. ANOVA table for an extension bulge posterior linear model.
Extension Bulge Posterior
Var.DfSum of Sq.Mean SquareF Valuep-ValueSignificance Code
Fiber91010.290.2917.94.54 × 10−5***
Annulus_μ10.740.7445.64.90 × 10−10***
Cartil_E10.410.4125.01.94 × 10−6***
Residuals1242.020.02
Significance codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1.
Table 18. ANOVA table for an extension stiffness linear model.
Table 18. ANOVA table for an extension stiffness linear model.
Extension Stiffness
Var.DfSum of Sq.Mean SquareF Valuep-ValueSignificance Code
Fiber910113.213.220.41.45 × 10−5***
Annulus_μ18.68.613.33.88 × 10−4***
Cartil_E12.32.33.56.42 × 10−2.
Residuals12480.40.6
Significance codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1.
Table 19. ANOVA table for a lateral bending bulge lateral linear model.
Table 19. ANOVA table for a lateral bending bulge lateral linear model.
Lateral Bending Bulge Lateral
Var.DfSum of Sq.Mean SquareF Valuep-ValueSignificance Code
C1010.030.033.27.57 × 10−2.
Annulus_μ11.111.11119.9<2.0 × 10−16***
Cartil_E110.2710.271112.5<2.0 × 10−16***
Cartil_μ10.840.8491.0<2.0 × 10−16***
Residuals1231.140.01
Significance codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1.
Table 20. ANOVA table for a lateral bending bulge posterior linear model.
Table 20. ANOVA table for a lateral bending bulge posterior linear model.
Lateral Bending Bulge Posterior
Var.DfSum of Sq.Mean SquareF Valuep-ValueSignificance Code
C1010.260.2653.72.84 × 10−11***
C010.120.1225.51.59 × 10−6***
Annulus_E10.060.0611.88.21 × 10−4***
Annulus_μ12.342.34478.3<2.2 × 10−16***
Cartil_E117.7117.713624.2<2.2 × 10−16***
Cartil_μ10.230.2347.13.06 × 10−10***
Residuals1210.590.00
Significance codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1.
Table 21. ANOVA table for a lateral bending stiffness linear model.
Table 21. ANOVA table for a lateral bending stiffness linear model.
Lateral Bending Stiffness
Var.DfSum of Sq.Mean SquareF Valuep-ValueSignificance Code
C1010.080.081.91.68 × 10−1
Annulus_E10.110.112.61.10 × 10−1
Annulus_μ111.6311.63265.9<2.2 × 10−16***
Cartil_E156.6956.691295.7<2.2 × 10−16***
Cartil_μ11.151.1526.31.11 × 10−6***
Residuals1225.340.04
Significance codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1.
Table 22. ANOVA table for a flexion bulge lateral linear model.
Table 22. ANOVA table for a flexion bulge lateral linear model.
Flexion Bulge Lateral
Var.DfSum of Sq.Mean SquareF Valuep-ValueSignificance Code
C1010.000120.0001215.71.25 × 10−4***
C010.000050.000056.21.38 × 10−2*
Fiber1210.000040.000045.02.74 × 10−2*
Annulus_E10.000020.000023.08.73 × 10−2.
Annulus_μ10.003130.00313416.0<2.2 × 10−16***
Cartil_E10.003160.00316420.0<2.2 × 10−16***
Cartil_μ10.000120.0001215.51.42 × 10−4***
Residuals1200.000900.00001
Significance codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1.
Table 23. ANOVA table for a flexion bulge posterior linear model.
Table 23. ANOVA table for a flexion bulge posterior linear model.
Flexion Bulge Posterior
Var.DfSum of Sq.Mean SquareF Valuep-ValueSignificance Code
C1010.0240.0242130.8<2.2 × 10−16***
C010.0110.011949.1<2.2 × 10−16***
Annulus_E10.0030.003264.3<2.2 × 10−16***
Annulus_μ10.0160.0161429.9<2.2 × 10−16***
Cartil_μ10.0000.00043.11.36 × 10−9***
Residuals1220.0010.000
Significance codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1.
Table 24. ANOVA table for a flexion stiffness linear model.
Table 24. ANOVA table for a flexion stiffness linear model.
Flexion Stiffness
Var.DfSum of Sq.Mean SquareF Valuep-ValueSignificance Code
C1010.0590.05913.04.45 × 10−4***
C010.0270.0275.91.68 × 10−2*
Annulus_E10.0930.09320.51.39 × 10−5***
Annulus_μ15.6435.6431243.7<2.2 × 10−16***
Cartil_E15.7065.7061257.6<2.2 × 10−16***
Cartil_μ10.0440.0449.62.42 × 10−3**
Residuals1210.5490.005
Significance codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1.
Table 25. ANOVA table for a torsion stiffness linear model.
Table 25. ANOVA table for a torsion stiffness linear model.
Torsion Stiffness
Var.DfSum of Sq.Mean SquareF Valuep-ValueSignificance Code
C1010.0320.03221.01.13 × 10−5***
C010.0100.0106.81.03 × 10−2*
Fibra3410.0320.03220.91.22 × 10−5***
Fiber5610.0160.01610.61.47 × 10−3**
Fiber7810.1020.10266.83.84 × 10−13***
Annulus_E11.0561.056691.1<2.2 × 10−16***
Annulus_μ11.0491.049685.9<2.2×10−16***
Cartil_E15.5395.5393623.7<2.2 × 10−16***
Cartil_μ10.0870.08756.91.04 × 10−11***
Residuals1180.1800.002
Significance codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1.
Table 26. Results of the predicted error criteria using the regression models.
Table 26. Results of the predicted error criteria using the regression models.
Errors and CorrelationsComp_BulgeAComp_BulgeLComp_BulgePComp_StiffShear_Stiff
Correlation99.18996.86199.18098.78499.636
MAE3.3847.3532.8395.1661.740
RMSE3.7107.7863.4335.2282.190
Exte_BulgeLExte_BulgePExte_StiffLBend_BulgeLLBen_BulgeP
Correlation84.45764.53248.03595.66398.603
MAE11.14113.24113.9738.0234.176
RMSE17.39319.93621.8129.4835.135
LBend_StiffFlex_BulgeLFlex_BulgePFlex_StiffTors_Stiff
Correlation96.37693.81798.75797.70998.881
MAE8.9568.8643.2036.0363.257
RMSE9.0729.8403.6076.4413.997
Table 27. Results of the predicted error criteria using the regression models.
Table 27. Results of the predicted error criteria using the regression models.
Errors and CorrelationsComp_BulgeAComp_BulgeLComp_BulgePComp_StiffShear_Stiff
Correlation98.80195.64895.42997.60394.763
MAE5.99812.7818.0634.8356.320
RMSE6.79214.1069.1676.0317.802
Exte_BulgeLExte_BulgePExte_StiffLBend_BulgeLLBend_BulgeP
Correlation92.22882.88069.06888.82591.281
MAE6.5955.5379.8078.5758.315
RMSE7.5746.88411.20411.66310.359
LBend_StiffFlex_BulgeLFlex_BulgePFlex_StiffTors_Stiff
Correlation87.63792.93396.93094.03164.698
MAE9.1226.9812.86425.59410.067
RMSE11.1258.1523.1936.88319.505
Table 28. The first criterion considered inputs and outputs that were considered equally important.
Table 28. The first criterion considered inputs and outputs that were considered equally important.
Var.GoalValueDesirability
C10inRange → 0.1250.1021.000
C0inRange → 0.030.0151.000
Fiber12inRange → 532.5518.1331.000
Fiber34inRange → 509500.0831.000
Fiber56inRange → 479517.6921.000
Fiber78inRange → 431.5463.0541.000
Fiber910inRange → 384366.7941.000
Annulus_EinRange → 4.13.9511.000
Annulus_μinRange → 0.350.2011.000
Cartil_EinRange → 3942.1211.000
Cartil_μinRange → 0.350.4301.000
Comp_bulgeAtarget → 0.50.2650.262
Comp_bulgeLtarget → 0.350.0970.244
Comp_bulgePtarget → 0.750.6500.651
Comp_stifftarget → 810826.1430.983
Shear_stifftarget → 300298.1240.964
Exte_bulgeLtarget → 0.10.1001.000
Exte_bulgePtarget → 0.240.4900.711
Exte_stifftarget → 1.532.1670.867
LBend_bulgeLtarget → 2.111.2350.534
LBend_bulgePtarget → 1.131.4590.792
LBend_stifftarget → 21.4650.634
Flex_bulgeLtarget → 0.070.0740.873
Flex_bulgePtarget → 0.730.3750.381
Flex_stifftarget → 1.181.3570.880
Tors_stifftarget → 2.13.4010.451
Overall Desirability0.625
Table 29. The second criterion considered: setting the target of the FE model parameters based only on stiffness.
Table 29. The second criterion considered: setting the target of the FE model parameters based only on stiffness.
Var.GoalValueDesirability
C10inRange → 0.1250.1051.000
C0inRange → 0.030.0151.000
Fiber12inRange → 532.5541.8671.000
Fiber34inRange → 509500.1231.000
Fiber56inRange → 479458.6431.000
Fiber78inRange → 431.5396.2911.000
Fiber91inRange → 384421.3201.000
Annulus_EinRange → 4.13.9522051.000
Annulus_μinRange → 0.350.22691.000
Cartil_EinRange → 3945.5751.000
Cartil_μinRange → 0.350.27561.000
Comp_bulgeAinRange → 0.50.2621.000
Comp_bulgeLinRange → 0.350.0891.000
Comp_bulgePinRange → 0.750.6281.000
Comp_stifftarget → 810900.1470.907
Shear_stifftarget → 300300.0001.000
Exte_bulgeLinRange → 0.10.1051.000
Exte_bulgePinRange → 0.240.6051.000
Exte_stifftarget → 1.531.5300.999
LBend_bulgeLinRange → 2.110.8951.000
LBend_bulgePinRange → 1.131.2701.000
LBend_stifftarget → 21.9840.989
Flex_bulgeLinRange → 0.070.0761.000
Flex_bulgePinRange → 0.730.3631.000
Flex_stifftarget → 1.181.5180.772
Tors_stifftarget → 2.13.4560.428
Overall Desirability0.817
Table 30. The third criterion considered: setting the target of the FE model parameters based only on the bulges.
Table 30. The third criterion considered: setting the target of the FE model parameters based only on the bulges.
Var.GoalValueDesirability
C10inRange → 0.1250.1021.000
C0inRange → 0.030.0151.000
Fiber12inRange → 532.5559.3411.000
Fiber34inRange → 509512.7901.000
Fiber56inRange → 479443.2431.000
Fiber78inRange → 431.5396.3481.000
Fiber91inRange → 384348.2821.000
Annulus_EinRange → 4.13.9511.000
Annulus_μinRange → 0.350.2141.000
Cartil_EinRange → 3936.9331.000
Cartil_μinRange → 0.350.4291.000
Comp_bulgeAtarget → 0.50.2770.298
Comp_bulgeLtarget → 0.350.1010.257
Comp_bulgePtarget → 0.750.6730.730
Comp_stiffinRange → 810821.9811.000
Shear_stiffinRange → 300287.0111.000
Exte_bulgeLtarget → 0.10.1030.947
Exte_bulgePtarget → 0.240.4810.722
Exte_stiffinRange → 1.532.4031.000
LBend_bulgeLTarget → 2.111.3140.576
LBend_bulgePtarget → 1.131.5950.706
LBend_stiffinRange → 21.2881.000
Flex_bulgeLtarget → 0.070.0750.846
Flex_bulgePtarget → 0.730.3730.378
Flex_stiffinRange → 1.181.3151.000
Tors_stiffinRange → 2.13.3111.000
Overall Desirability0.554
Table 31. Comparison of the results of the regression models; FEM and the experimental values.
Table 31. Comparison of the results of the regression models; FEM and the experimental values.
ParametersCriteria 1Criteria 2Criteria 3ExperimentsError
FEMFEMFEMStandard TestNormalized MAE
Comp_bulgeA0.2660.2620.2690.500.469
Comp_bulgeL0.0960.0900.0950.350.732
Comp_bulgeP0.6240.6020.6250.750.177
Comp_stiff915.640944.360922.7408100.125
Shear_stiff302.925304.920299.0903000.010
Exte_bulgeL0.1060.0990.1060.100.041
Exte_bulgeP0.5270.5590.5380.240.539
Exte_stiff1.6341.6781.6451.530.073
LBend_bulgeL0.9970.8710.9772.110.551
LBend_bulgeP1.2821.1731.2631.130.085
LBend_stiff1.4892.1751.4932.000.183
Flex_bulgeL0.0770.0800.0760.070.096
Flex_bulgeP0.3800.3730.3730.730.486
Flex_stiff1.4881.5241.5001.180.213
Tors_stiff3.5503.5493.5062.100.404
Normalized MAE 0.27820.27950.2788
Materials EISSN 1996-1944 Published by MDPI AG, Basel, Switzerland RSS E-Mail Table of Contents Alert
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