Hindawi Applied Bionics and Biomechanics Volume 2021, Article ID 5593037, 9 pages https://doi.org/10.1155/2021/5593037 Research Article Effect of Model Parameters on the Biomechanical Behavior of the Finite Element Cervical Spine Model 1 2 Suzan Cansel Dogru and Yunus Ziya Arslan Department of Mechanical Engineering, Faculty of Engineering, Istanbul University-Cerrahpasa, Turkey Department of Robotics and Intelligent Systems, Institute of Graduate Studies in Science and Engineering, Turkish-German University, Turkey Correspondence should be addressed to Yunus Ziya Arslan; yunus.arslan@tau.edu.tr Received 4 February 2021; Revised 11 May 2021; Accepted 14 June 2021; Published 28 June 2021 Academic Editor: Fahd Abd Algalil Copyright © 2021 Suzan Cansel Dogru and Yunus Ziya Arslan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Finite element (FE) models have frequently been used to analyze spine biomechanics. Material parameters assigned to FE spine models are generally uncertain, and their eﬀect on the characterization of the spinal components is not clear. In this study, we aimed to analyze the eﬀect of model parameters on the range of motion, stress, and strain responses of a FE cervical spine model. To do so, we created a computed tomography-based FE model that consisted of C2-C3 vertebrae, intervertebral disc, facet joints, and ligaments. A total of 32 FE analyses were carried out for two diﬀerent elastic modulus equations and four diﬀerent bone layer numbers under four diﬀerent loading conditions. We evaluated the eﬀects of elastic modulus equations and layer number on the biomechanical behavior of the FE spine model by taking the range of angular motion, stress, and strain responses into account. We found that the angular motions of the one- and two-layer models had a greater variation than those in the models with four and eight layers. The angular motions obtained for the four- and eight-layer models were almost the same, indicating that the use of a four-layer model would be suﬃcient to achieve a stress value converging to a certain level as the number of layers increases. We also observed that the equation proposed by Gupta and Dan (2004) agreed well with the experimental angular motion data. The outcomes of this study are expected to contribute to the determination of the model parameters used in FE spine models. 1. Introduction deﬁned through various empirical equations [6–9]. In many FE-based studies, the values of the elastic modulus were cal- culated by using these equations [10–12], while very few of Due to the ethical concerns and the requirement of invasive methods, determining the in vivo stress and strain values that them focused on the spinal region [9, 13, 14]. The reliability occur on the vertebrae under diﬀerent loading conditions is of these equations is still controversial, and a consensus on challenging [1]. The ﬁnite element-based computational the use of these equations has not yet been reached [15]. modeling and simulation approach provide a practical and In the literature, the eﬀect of assigned material properties eﬃcient solution to this problem. By using ﬁnite element of the ligament, intervertebral disc, and bone tissue on FE (FE) analysis, it is possible to simulate the biomechanical analysis results was investigated, but the eﬀect of the number behavior of the spinal components and calculate various bio- of layers of bone tissues with diﬀerent material parameters mechanical parameters such as stress, strain, and angular based on HU level was not investigated [16–18]. Kumaresan motion noninvasively [2–4]. Most of the FE models of the et al. [17] analyzed the sensitivity of the output of the FE spine are based on computed tomography (CT) data [5]. In analysis of cervical spinal components including the interver- the literature, the relationships between the Hounsﬁeld Unit tebral disc, posterior elements, endplates, ligaments, and cor- (HU), which is a dimensionless unit used in CT, density, and tical and cancellous bones to variations in the assigned modulus of elasticity of the anatomical structures were material properties. They considered the angular motion, 2 Applied Bionics and Biomechanics intervertebral disc stress, endplate stress, and vertebrae stress Table 1: Generated ﬁnite element vertebrae models consisted of diﬀerent layer numbers depending on HU values of the cortical as the output of the analysis. They concluded that the eﬀect of and cancellous bone. changes in material properties of the soft tissues was more determinant than that in the material properties of the bone Range of HU values deﬁned for the corresponding ﬁnite element [17]. However, bone tissue is the main load-bearing structure vertebrae models in the human body, and the mechanical composition of bone One-layer Two-layer Four-layer Eight-layer tissue should be accurately modeled in a FE analysis to obtain model model model model reliable stress and strain levels [19]. In the literature, models 148-300 were typically separated into three or four layers that are 301-500 representing vertebral body structures such as the endplates 148-300 501-661 and cortical and cancellous bones [17, 20]. In these studies, 148-661 301-661 662-900 148-1988 the distinction between these tissues was not made consider- 662-1988 662-1300 901-1100 ing the level of HU. Rayudu et al. [21] predicted the elastic 1301-1988 1101-1300 modulus by taking into account the level of HU, and a verte- 1301-1600 bra model was built up with nine layers. There is still little 1601-1988 knowledge about the required number of layers to be deﬁned for a FE vertebrae model to reﬂect the actual biomechanical behavior of the bone tissue. Moreover, the question of how elastic modulus in the literature were used to obtain the elas- sensitive the stress and strain results obtained from FE anal- tic modulus of the modeled bone tissues [25, 26]. In both yses are to variations in layer numbers and assigned material studies, CT data were taken from bone samples and compres- parameters of the vertebrae has not been answered yet. sive force was applied to them. The stress-strain curves were In this study, we aimed to analyze the eﬀect of model plotted to calculate the elastic modulus. In addition, tissue parameters on the range of motion, stress, and strain density measurements were measured. The empirical equa- responses of a FE cervical spine model. More speciﬁcally, tions were established between HU, density, and elastic we focused on (i) how many layers would be required to modulus. obtain an accurate model and (ii) which of the two widely Equation (1) was used for the relationships between HU used HU-elastic modulus relationships in the literature and density (ρ) expressed in kg/m [25, 26]: would provide a more accurate result in terms of joint range of motion. The FE spinal unit included C2 and C3 vertebrae, ρ =0:44 × HU + 527: ð1Þ intervertebral discs, ligaments, and facets at the same level. Gupta and Dan [25] proposed the following equation set (equation (2)) to establish the relationships between density 2. Materials and Methods (ρ) (kg/m ) and elastic modulus (E) (MPa): 2.1. Modeling of the Spinal Unit. CT data of a 55-year-old −6 3 male cadaver was processed to create the vertebrae model. E =3× 10 × ρ for 350 < ρ ≤ 1800, ð2Þ CT images of the nonpathological cervical spine were −6 2 E = 1049:25 × 10 × ρ for ρ ≤ 350: obtained from the archive of the Istanbul University-Cerrah- pasa, Turkey. The pixel size and slice thickness of the tomog- Morgan et al. [26] reported the following equation for the raphy data were 0.49 mm and 0.63 mm, respectively. The spinal functional unit was modeled between the C2 and C3 relationship between ρ (kg/m ) and E (MPa): vertebrae, including the intervertebral disc, ligaments, and 1:56 facets at the same level. Four diﬀerent FE vertebrae models E = 4730 × : ð3Þ were created separately, which were formed from one, two, four, and eight layers depending on HU values (Table 1) [22]. The vertebrae model comprising one layer was deﬁned Average HU values for all bone layers were speciﬁed from as a homogenized bone. In the two-layer model, one bone CT image data, and the related elastic modulus values were layer was deﬁned for each of the cancellous and cortical calculated from the abovementioned equations. Poisson’s bones. In the four-layer model, two bone layers were deﬁned ratio was 0.3 for all models. for each of the cancellous and cortical bones. In the eight- The intervertebral disc was modeled to ﬁll the space layer model, the ﬁrst three layers were represented as the can- between the endplates of the vertebrae. The intervertebral cellous bone and the remaining layers were represented as disc located between the C2 and C3 vertebrae consisted of the cortical bone [23]. Bonemat software (Bonemat, BO, the annulus ﬁbrosus and nucleus pulposus layers. The Italy) was used to build up the models depending on CT data. Mooney-Rivlin hyperelastic elements were used to model All vertebrae models were assumed to have linearly elas- the annulus ground substance [27]. For the nucleus pulposus, tic and isotropic behavior, and their mechanical properties the elastic modulus and Poisson’s ratio were 1 MPa and 0.49, were described by elastic modulus and Poisson’s ratio [24]. respectively, which were reported by Ruberte et al. [27] based The material properties for each layer were calculated by tak- on the experimental data available in the literature [28–30]. ing HU and related density values into account. To do so, the Anterior longitudinal, posterior longitudinal, facet cap- two most common empirical equations for relating HU- sular, supraspinous, interspinous ligaments, and ligamentum Applied Bionics and Biomechanics 3 Table 2: The force of ligaments relative to the displacement [31]. Interspinous Anterior longitudinal Posterior longitudinal Facet capsular Ligamentum ﬂavum ligaments/supraspinous Force Displacement Force Displacement Force Displacement Force Displacement Force Displacement (N) (mm) (N) (mm) (N) (mm) (N) (mm) (N) (mm) 32.5 1.24 26.8 1.02 59.5 2.02 8.6 1.38 29.2 1.71 60.8 2.46 49.5 2.12 122.8 4.00 16.9 2.74 54.9 3.37 82.4 3.63 65.0 3.13 170.2 5.92 22.7 4.12 71.9 5.10 100.3 4.78 79.8 4.23 206.5 7.99 28.8 5.55 94.5 6.68 C2 upper surface of the C2 vertebra was connected to the refer- ence point, which was created in line with the adjacent verte- bral body. The moment applied to the reference point was thus distributed over the upper surface of the C2 vertebra. The C3 inferior surface was ﬁxed in all directions. The con- tact between the intervertebral disc and the endplate was Intervertebral determined to be bounded (slip and clearance were not disc allowed). The facet joints were coupled between the C2 and C3 vertebrae as a continuum distributing type. For all cases, the loading conditions and analysis were assumed static. The eﬀects of layer number and elastic modulus on the angular motion, stress, and strain values obtained from the FE analysis of the vertebrae models were compared for four C3 diﬀerent loading conditions. As a result, a total of 32 analyses (two diﬀerent elastic modulus equations × four diﬀerent layer numbers × four diﬀerent loading conditions) were performed. All analyses were performed by using Ansys software (Ansys, Inc., Canonsburg, PA, USA). 3. Results Figure 1: Finite element model of the C2 and C3 vertebrae and intervertebral disc. Angular motion results at the C2/C3 segment were given for four diﬀerent layer numbers (one, two, four, and eight), two diﬀerent equations, and four diﬀerent loading conditions in ﬂavum were represented by tension-only spring-like connec- Figure 2. The model-predicted angular motions were also tors with nonlinear material properties (Table 2). The mate- compared with those obtained from the literature [35]. White rial property of the ligaments was deﬁned in terms of and Panjabi [35] analyzed various experimental results from stiﬀness. The experimentally obtained nonlinear stiﬀness the literature to describe the range of angular motion of the property was drawn from the literature [31]. Yoganandan C2-C3 vertebrae. The degree of motion of the C2-C3 verte- et al. [31] measured the tensile force-displacement of all liga- brae was experimentally obtained during the same loading ments at diﬀerent levels of the cervical region. conditions that we applied to the models [35]. It can be deduced from Figure 2 that the model-predicted angular 2.2. The Meshing of the Model. The hybrid quadratic tetrahe- motions under ﬂexion, extension, and axial rotation dral element was chosen for the vertebrae and intervertebral moments were consistent with the experimental data [35]. disc. The criteria for creating elements inﬂuence the number However, the results obtained for the lateral bending of elements [32]. The geometric criteria of the hybrid qua- moment were not in agreement with the experimental data. dratic tetrahedral element have been determined such that Angular motions obtained by using the equations by Gupta the lowest volume was 0.3 mm , the lowest internal angle ° ° and Dan [25] and Morgan et al. [26] were found similar to was 10 , and the highest internal angle was 130 [33]. Sizing each other, especially under the lateral bending moment. iterations were carried out up to the longest and shortest side The number of layers was found as an eﬀective parameter length ratio of 5 and the largest and lowest volume ratio of 2 in the calculation of the angular motion, while the major dif- [33]. All models had 120000 elements (Figure 1). ferences in terms of angular motion were found between one- 2.3. Loading and Boundary Conditions. Flexion, extension, and two-layer models. Maximum von Mises stress and strain values that lateral bending, and axial rotation moments, all of which were 1.5 Nm, were applied to the models [34]. The moment occurred on the C2/C3 intervertebral disc were given in value of 1.5 Nm was assumed to be suﬃcient to produce Figure 3. To determine whether the stress and strain values motion, but small enough to not injure the tissues [34]. The converge to a certain value as the number of layers increases, 4 Applied Bionics and Biomechanics Layer number 1248 1248 1248 1248 Lateral bending Axial rotation Flexion Extension Figure 2: Angular motion at the C2/C3 joint for four diﬀerent layer numbers (one-, two-, four-, and eight-layer models), two diﬀerent equations (equation (2) and equation (3)), and four diﬀerent loading conditions (ﬂexion, extension, lateral bending, and axial rotation moments). The results based on equation (2) were represented by a solid line with a circle [25] and those based on equation (3) by a solid line with a triangle [26]. Experimentally obtained angular motions were represented by grey zones [35]. the stress and strain values obtained from one-, two-, and solution. Finite element-based in silico analysis has become four-layer models were normalized to those obtained from widespread in the assessment of the spine over the last two decades [15]. The accuracy of such an analysis is critical to eight-layer models (Figures 3(a) and 3(b)). When the equa- tion proposed by Gupta and Dan [25] was taken into obtain clinically meaningful outcomes. In particular, model account, it was seen that the variations in the number of parameters play an important role in obtaining reliable bio- layers led to a 10% change in the stress and 30% in the strain mechanical results from spinal FE modeling and simulation values. As for the equation proposed by Morgan et al. [26], studies. The material properties of each element can be the variations in the number of layers caused a 62% change assigned depending on the HU value. On the other hand, in the stress and 30% in the strain values. It was also observed such a methodology would lead to a high computational cost that the stress and strain results obtained from four-layer and errors due to discontinuities in the internal structures of models converged to those of the eight-layer models for both the bone tissue. Therefore, there are many studies in the liter- equations. ature deﬁning the vertebral bone into diﬀerent layers, namely Table 3 and Table 4 indicate the maximum von Mises cortical and trabecular bone layers [36, 37]. Accordingly, we stress and strain values that occurred on the vertebrae, aimed to analyze the eﬀects of variations in elastic modulus respectively. The results were given for each layer and all and layer number on the model-predicted angular motion, loading conditions. In Table 3 and Table 4, as the number stress, and strain values that occurred at the C2/C3 level of of layers increases in the ﬁrst row, the level of HU value for the FE spine model under various loading modes. We also the corresponding layer increases. The ﬁrst and second compared the model-predicted angular motion with the values in each cell were based on equation (1) [25] and equa- experimentally obtained data. The strain and stress values tion (2) [26], respectively. It was observed from Table 3 that on the intervertebral disc and vertebrae were separately as the layer number in the vertebrae increased, the stress evaluated. values also increased. In terms of strain values, an opposite Angular motions under ﬂexion and extension moments trend was observed such that the strain values decreased as occurred in a similar range (Figure 2), and they agreed with the HU value of the layer increased (Table 4). The diﬀerence the experimental data [35], indicating that the mechanical in stress and strain values between layers decreased as the properties of the disc structure were well deﬁned. The number of layers increased, indicating that more homoge- model-predicted angular motion under the lateral bending nous stress and strain distributions occurred over the verte- moment was lower than the experimental data, which indi- brae as the number of layers increased. The use of equation cates that the stiﬀness value assigned to the FE model in the (2) and equation (3) did not result in signiﬁcant diﬀerences direction of the lateral bending movement was quite high. between the maximum stress values. The facet joint inﬂuences the lateral bending motion consid- erably [38], and hence the low level of model-predicted motion may be attributed to the uncertainty of the assigned 4. Discussion mechanical properties of the facet joint. Under the axial rota- To deepen our understanding of the eﬀect of various surgical tion moment, the calculated angular motion from the FE interventions on the spinal components, in silico analysis of analysis was within the range of the experimentally obtained the spine provides a practical and eﬃcient complimentary data. In the literature, it was reported that the level of angular Angular motion (degree) Applied Bionics and Biomechanics 5 1.7 1.6 1.5 1.4 1.3 1.2 1.1 Layer 0.9 number 1 24 1 24 1 24 1 24 Axial rotation Extension Lateral bending Flexion (a) 1.4 1.3 1.2 1.1 0.9 0.8 0.7 Layer 0.6 number 1 1 1 1 24 24 24 24 Axial rotation Lateral bending Flexion Extension (b) Figure 3: Maximum normalized stress (a) and strain (b) values on the C2/C3 intervertebral disc. The stress and strain values obtained from one-, two-, and four-layer models were normalized to that obtained from eight-layer models. The results based on equation (2) were represented by a solid line with a circle [25] and those based on equation (3) by a solid line with a triangle [26]. motion is highly associated with the material properties of Figure 3 illustrates how many numbers of layers would be the soft tissues [39]. The results of our study showed that adequate to deﬁne the vertebrae. It was revealed that the the soft tissues except facets were well deﬁned in the FE change in the material properties of the vertebrae plays a decisive role in the stress and strain values over the interver- model. Angular motions of the one- and two-layer models had a tebral disc. When the stress and strain on the intervertebral greater variation than those in the models with four and eight disc structure are examined in Figures 3(a) and 3(b), it was layers (Figure 2). This result indicates that deﬁning the verte- observed that the results of the model using both equations brae with the cancellous bone caused a big eﬀect on the angu- with four layers converged to the results of the model with lar motion results. Unlike the one-layer model, the two-layer eight layers. model characterized the cancellous bone. It was observed The diﬀerence in stress and strain distributions between that the angular motion level converged to a certain degree the layers decreased as the number of layers increased as the number of layers increased. The angular motions (Table 3 and Table 4). The stress and strain results obtained obtained for the four- and eight-layer models were almost by using equation (2) and equation (3) on the one-layer ver- the same, indicating that the use of a four-layer model would tebrae model were quite similar. However, the variation in be suﬃcient to achieve the stress value converging to a certain the results was greater between equation (2) and equation level as the number of layers increases. (3) when the number of layers increased. When compared Stress (rate) Stress (rate) 6 Applied Bionics and Biomechanics Table 3: The maximum von Mises stress values on the vertebrae. The ﬁrst and second values in each cell are based on equation (1) [25] and equation (2) [26], respectively. Maximum von Mises stress value (MPa) Number of layer 1 2345 678 One-layer model 17.8/19.0 Two-layer model 12.1/20.0 27.2/25.0 Flexion Four-layer model 4.9/7.0 15.5/23.8 19.4/20.2 26.7/26.0 Eight-layer model 5.8/9.0 7.0/9.7 7.9/14.1 13.3/25.8 16.3/18.6 24.0/23.6 23.7/23.9 17.8/18.7 One-layer model 17.8/20.0 Two-layer model 12.1/21.0 26.7/25.4 Extension Four-layer model 4.9/7.0 16.5/24.8 20.4/20.9 33.4/26.9 Eight-layer model 5.8/7.1 7.0/9.7 8.5/14.7 14.1/26.9 17.1/19.7 25.5/24.5 28.9/24.8 17.7/19.8 One-layer model 13.5/13.8 Two-layer model 10.3/5.8 24.4/0.4 Lateral bending Four-layer model 4.4/7.4 13.5/15.0 15.4/12.1 24.7/23.1 Eight-layer model 2.1/1.9 5.3/8.1 11.3/13.4 18.3/18.1 14.2/12.0 16.3/12.9 27.5/18.9 18.8/23.1 One-layer model 9.3/9.3 Two-layer model 5.4/10.1 12.4/11.1 Axial rotation Four-layer model 3.7/5.7 7.6/8.3 9.5/9.0 14.2/10.2 Eight-layer model 2.4/2.5 4.2/6.4 5.6/7.3 8.1/9.9 9.6/11.3 10.3/13.6 12.2/14.2 12.7/10.7 Table 4: The maximum strain (%) values on the vertebrae. The ﬁrst and second values in each cell are based on equation (1) [25] and equation (2) [26], respectively. Maximum strain value (%) Number of layer 1 2 3 45678 One-layer model 1.1/1.2 Two-layer model 1.9/1.1 1.0/0.8 Flexion Four-layer model 1.1/0.4 1.6/0.9 1.0/0.6 0.3/0.5 Eight-layer model 0.8/0.3 1.0/0.4 0.7/0.7 0.6/0.7 0.6/0.6 0.4/0.6 0.4/0.5 0.2/0.2 One-layer model 0.8/1.0 Two-layer model 1.2/0.7 0.7/0.6 Extension Four-layer model 1.6/0.6 1.0/0.6 0.6/0.5 0.4/0.5 Eight-layer model 1.1/0.7 0.6/0.4 0.9/0.6 0.9 0.6 0.7/0.6 0.6/0.5 0.4/0.5 0.4/0.2 One-layer model 0.8/0.8 Two-layer model 1.5/0.6 0.8/0.4 Lateral bending Four-layer model 1.0/0.4 1.2/0.4 0.7/0.3 0.4/0.2 Eight-layer model 0.4/0.1 0.9/0.4 1.2/0.5 1.4/0.6 0.8/0.3 0.4/0.2 0.3/0.2 0.3/0.2 One-layer model 0.5/0.5 Two-layer model 0.7/0.3 0.3/0.2 Axial rotation Four-layer model 0.7/0.3 0.6/0.3 0.3/0.2 0.3/0.2 Eight-layer model 0.5/0.2 0.6/0.3 0.6/0.2 0.5/0.3 0.4/0.2 0.3/0.2 0.3/0.1 0.2/0.1 more consistent with experimental data. When the stress to the cortical bone, the experimental mechanical testing on the vertebrae specimens showed that the strain level in the values of the vertebra are examined, it was observed that cancellous bone was higher and the stress level in the cancel- the stress levels in some layers decreased as the elastic mod- lous bone was lower [40, 41]. It was observed from the stress ulus of the layer increased, which is not compatible with results of our study that equation (2) agreed more with the the literature [41]. In 2001, Morgan and Keaveny reported in their experi- experimental data than equation (3). The fact that Gupta and Dan [25] created separate sets of equations for the can- mental study on cadavers that the yield stress of the cancel- cellous and cortical bone may have led to results that are lous bone during compression was 2.02 MPa, the yield Applied Bionics and Biomechanics 7 to only one sample of the vertebral body, caution should strain was 0.77%, the yield stress was 1.72 MPa, and the yield strain during tensile was 0.70% [40]. In addition, the cortical be taken when interpreting and generalizing the results bone with a modulus of elasticity of 18 GPa had a yield stress obtained from our study. of 70 MPa and a yield strain of 0.55% [41]. When these experimental values are taken into account, it was seen that 5. Conclusion the stress and strain values obtained from the models with We aimed to analyze the eﬀect of model parameters on the one and two layers exceeded the experimentally obtained range of motion, stress, and strain responses of the FE cervi- stress and strain values over the cancellous bone. On the cal spine model. We concluded that the angular motions of other hand, the eight-layer model provided more comparable the one- and two-layer models had a greater variation than results to the experimental data. When the model-predicted those in the models with four and eight layers. The angular stress results from the cortical bone were considered, it was motions obtained for the four- and eight-layer models were seen that all stress values remained within the speciﬁed range almost the same, indicating that the use of a four-layer model deﬁned by the experiments [41]. In terms of strain value, it would be suﬃcient to achieve the stress value converging to a was observed that the models with four and eight layers pro- certain level as the number of layers increases. We also vided more accurate results. observed that the equation proposed by Gupta and Dan in Studies on the investigation of the density and elastic 2004 agreed well with the experimental data because the cor- modulus relationship for diﬀerent bone structures are very tical and cancellous bones were modeled separately. In the few [25, 26]. The relationship between elastic modulus and next step, we plan to investigate the eﬀects of assigned apparent density does depend on the anatomic site, which mechanical parameters on the response of the entire spine was also experimentally proven by Morgan et al. [26]. Also, model under dynamic loading conditions. the empirical relations used for deﬁning elastic modulus are not only anatomical site speciﬁc, they can also vary between patients and computed tomography (CT) scan machine. Data Availability Although these limitations are present in creating the ﬁnite No data is available. element bone models, these models are most often derived from CT data in the literature [42]. Equation (2) was empir- Conflicts of Interest ically obtained based on the CT data of the scapula [25]. On the other hand, Morgan et al. [26] carried out their experi- The authors declare no conﬂict of interest. mental study on diﬀerent bone structures including verte- brae. The limitation in [26] is that the authors did not References provide separate equations for the cancellous and cortical bones depending on the individual bone density, but rather [1] M. Xu, J. Yang, I. H. Lieberman, and R. 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Applied Bionics and Biomechanics – Hindawi Publishing Corporation
Published: Jun 28, 2021
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