A three-dimensional finite element model of lumbar sacral vertebrae was established
In this study, a three-dimensional finite element model of lumbar sacral vertebrae in adolescents was established based on the data of patients with lumbar sacral vertebrae, including 22 815 elements and 12 785 nodes (Table 3). The model is highly simulated, the geometric structure is good, the appearance of the model is close to the structure of sacral lumbar data, and has a high degree of reduction (Table 3 and Figure 6).








Stress analysis of lumbar vertebrae of sacral vertebrae under different working conditions in three-dimensional finite element model of adolescents
When the torque of 2 N·mm was applied to the lumbar vertebra finite element model of sacral vertebrae, the stresses of L5 and S1 under lumbar vertebrae of sacral vertebrae were as follows: during flexion, the stress was mainly concentrated in the facet joint of S1Unix 2, followed by the posterior structure of each vertebra. During the extension, the stress was mainly concentrated in the S1ax 2 facet joint, and decreased in turn from S1 to upward, which was basically consistent with the flexion movement. In the left and right side flexion, the stress is mainly concentrated in the S1hip 2 facet joint, followed by the superior articular process of S1, and decreases upward from S1. For the symmetry of human body development, the stress and strain of the model under the right bending condition is the same as that of the left bending. In the axial torsion of the left and right sides, the stress is greater in the anterior part of the C2-3 vertebral body, and the stress is also obvious in the facet joint and pedicle of the opposite side of the vertebral body. For L5 and S1 vertebrae, the maximum stress appeared in the inferior articular process of the vertebrae under flexion and extension, followed by the isthmus of the pedicle of the vertebrae, and in the condition of left and right lateral bending, the maximum stress appeared in the inferior articular process of the vertebrae, followed by the superior articular process of the vertebrae, and then in the isthmus of the pedicle of the vertebrae (Figure 7).




Strain and displacement analysis of lumbar vertebrae of sacral vertebrae under different working conditions in adolescents three-dimensional finite element model
In the stress analysis of the model, the maximum strain was located in the intervertebral disc tissue under different working conditions, and the strain law was that S1 decreased upward, and the maximum displacement was located at the anterior upper edge of L4 during flexion. Under the left bending condition, the maximum strain of the model was S1Scan2, and there was little difference in stress between L4-5 and L5/S1 (Figure 7). To analyze the displacement changes of the model under different working conditions, during flexion and extension, the maximum displacement is located at the upper edge of L4, and the displacement decreases successively from L4 to the sacrum. Under the condition of left and right side bending, the maximum displacement is located at the front and upper edge of L4, and the displacement decreases successively from L4 to downward (Figure 8).

[1] MITRA R, CARLISLE M. Bertolotti’s syndrome: a case report. Pain Pract. 2009;9(2): 152-154.  [2] LV YP, Lin W, Ma XM. The localization of iliolumbar ligaments in lumbar column segmentation. Fangshe Xue Shijian. 2016;31(11):1080-1083. [3] JANCUSKA JM, SPIVAK JM, BENDO JA. A Review of Symptomatic Lumbosacral Transitional Vertebrae: Bertolotti’s Syndrome. Int J Spine Surg. 2015;9:42.  [4] ZHAO S. Imaging study of percutaneous intervertebral foramen approach in lumbosacral 1. Suzhou: Suzhou University. 2015. [5] GONG H, SLUNSKY P, KLASS LG, et al. Brunnberg L. Prevalence of lumbosacral transitional vertebrae in dogs in Berlin. Pol J Vet Sci. 2020;23(2):261-265.  [6] SEKHARAPPA V, AMRITANAND R, KRISHNAN V, et al. Lumbosacral transition vertebra: prevalence and its significance. Asian Spine J. 2014;8(1):51-58.  [7] TANG M, YANG XF, YANG SW, et al. Lumbosacral transitional vertebra in a population-based study of 5860 individuals: prevalence and relationship to low back pain. 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Spine (Phila Pa 1976). 1986;11(9):914-927. [13] SCHMIDT H, HEUER F, DRUMM J, et al. Application of a calibration method provides more realistic results for a finite element model of a lumbar spinal segment. Clin Biomech (Bristol, Avon). 2007;22(4):377-384.  [14] LIU YK, RAY G, HIRSCH C. The resistance of the lumbar spine to direct shear. Orthop Clin North Am. 1975;6(1):33-49. [15] HU Y, XIE H, YANG SH. Utilization of three- dimensional finite element method in spinal biomechanics. Yiyong Shengwu Lixue. 2006;21(3):246-250. [16] TOKGOZ N, UCAR M, ERDOGAN AB, et al. Are spinal or paraspinal anatomic markers helpful for vertebral numbering and diagnosing lumbosacral transitional vertebrae?. Korean J Radiol. 2014;15(2):258-266.  [17] GIRDLER S, CHO B, MIKHAIL CM, et al. Emerging Techniques in Diagnostic Imaging for Idiopathic Scoliosis in Children and Adolescents: A Review of the Literature. World Neurosurg. 2020;136:128-135.  [18] JANCUSKA JM, SPIVAK JM, BENDO JA. A Review of Symptomatic Lumbosacral Transitional Vertebrae: Bertolotti’s Syndrome. Int J Spine Surg. 2015;9:42.  [19] MAHATO NK. Implications of structural variations in the human sacrum: why is an anatomical classification crucial?. Surg Radiol Anat. 2016;38(8):947-954.  [20] LUOMA K, VEHMAS T, KERTTULA L, et al. Chronic low back pain in relation to Modic changes, bony endplate lesions, and disc degeneration in a prospective MRI study. Eur Spine J. 2016;25(9):2873-2881.  [21] HOU LS, BAI XD, GE F, et al. Castellvi’s classification type-Ⅱa lumbarization combined with L5-S2 disc herniation: a case report. Dier Junyi Daxue Xuebao. 2018;39(10):1177-1179. [22] HANHIVAARA J, MÄÄTTÄ JH, NIINIMÄKI J, et al. Lumbosacral transitional vertebrae are associated with lumbar degeneration: retrospective evaluation of 3855 consecutive abdominal CT scans. Eur Radiol. 2020;30(6):3409-3416.  [23] DANG L, CHEN ZQ LIU XG, et al. Etiology of lumbar disc herniation in adolescents: excessive load on the lumbar spine and aberrant configurations of the lumbosacral region. Zhongguo Jizhu Jisui Zazhi. 2015;25(11):991-996. [24] FUJIWARA A, TAMAI K, KURIHASHI A, et al. Relationship between morphology of iliolumbar ligament and lower lumbar disc degeneration. J Spinal Disord. 1999; 12(4):348-352. [25] AHN SS, CHIN DK, KIM SH, et al. The Clinical Significance of Lumbosacral Transitional Vertebrae on the Surgical Outcomes of Lumbar Discectomy: A Retrospective Cohort Study of Young Adults. World Neurosurg. 2017;99:745-750.  [26] BERLEMANN U, JESZENSZKY DJ, BÜHLER DW, et al. Facet joint remodeling in degenerative spondylolisthesis: an investigation of joint orientation and tropism. Eur Spine J. 1998;7(5):376-380.  [27] HAMMERBERG KW. New concepts on the pathogenesis and classification of spondylolisthesis. Spine (Phila Pa 1976). 2005;30(6 Suppl):S4-S11.  [28] JIANG HC, WANG JX, YANG XD, et al. Analysis of the related factors of lumbar spondylolysis. Zhongguo Linchuang Jiepou Xue Zazhi. 2019;37(5):583-585,589. [29] KANEMATSU R, HANAKITA J, TAKAHASHI T, et al. Extraforaminal entrapment of the fifth lumbar spinal nerve by nearthrosis in patients with lumbosacral transitional vertebrae. Eur Spine J. 2020;29(9):2215-2221.  [30] ALBANO D, MESSINA C, GAMBINO A, et al. Segmented lordotic angles to assess lumbosacral transitional vertebra on EOS. Eur Spine J. 2020;29(10):2470-2476.

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Bertolotti first described the lumbosacral transitional vertebrae in 1917 and put forward a concept of Bertolotti syndrome[1]. It is believed that the unilateral or bilateral transverse process of the terminal lumbar vertebrae is enlarged and broadened due to abnormal development, and sometimes the enlarged transverse process forms pseudarthrosis or fusion with the sacroiliac. This deformity is related to chronic low back pain, lower limb pain and other symptoms. This is the earliest report about lumbago caused by lumbosacral transitional vertebrae. In recent years, different scholars[2-6] reported that the incidence of lumbosacral transitional vertebrae was different in different people, but in people with lumbar pain diseases such as lumbar disc herniation, the incidence of transitional vertebrae was significantly increased, and the overall incidence was higher[7-9]. Lumbar vertebrae as a whole system, the pathological changes and morphological abnormalities of any segment will lead to changes in the mechanical morphology of the whole system, and then lead to an increase in the probability of injury in a certain part, so the study of transitional vertebrae has a very important clinical significance.

For patients with lumbar sacral vertebrae, the analysis of the maximum load of each vertebral segment and the trend of mechanical changes is the basis of clinical treatment of low back pain caused by lumbar vertebrae, but at present, there are few mechanical simulation studies. Studies on the mechanics of lumbar pedicle in adolescent patients with transitional vertebrae are even less[10-11]. In this study, the CT images of adolescent patients with lumbar sacralization were selected to obtain three-dimensional finite element models of lower lumbar vertebrae of adolescent sacral lumbar vertebrae, and their stress and strain distribution and mechanical characteristics were analyzed, so as to provide carrier and basis for further study of mechanical changes and bone fatigue coefficient of lower lumbar vertebrae in transitional vertebrae patients. 
中国组织工程研究杂志出版内容重点：人工关节；骨植入物；脊柱；骨折；内固定；数字化骨科；组织工程

Design
A three-dimensional finite element analysis.

Time and settings
The experiment was conducted in 2019 at the Digital Medical Center of Inner Mongolia Medical University.

Subjects
One patient with lumbar sacral vertebrae was selected, a 16-year-old male of Han nationality, with a body mass of 60 kg, height of 1.75 m and body mass index (19.59 kg/m2). The volunteer had no previous history of lumbar trauma and related diseases. We obtain written informed consent from patient and his guardian. The research was approved by the Ethics Committee of Inner Mongolia Medical University (approval number YKD2018015) on March 5, 2018.

Methods
Acquisition of two-dimensional data
In this study, the Lightspeed dual-source 64-slice spiral CT (GE, USA) was used to scan the upper edge of the L1 vertebra to the lower edge of the L5 vertebra using the volume scan mode. Lie on your back with your head forward. Scanning parameters: slice thickness 5 mm, layer spacing 5 mm, scanning field 20 cm2, matrix 512 × 512, standard algorithm reconstruction. At the end of the scan, the data is divided into layer thickness 0.625 mm, layer spacing 0.625 mm and saved in DICOM format (Figure 1). 

3D reconstruction of 2D data
Import the original DICOM data into Mimics16.0 (Materialise Belgium), through the steps of threshold setting, image segmentation, region growth, image filling and so on, the L4-sacrum is segmented and smoothed, and then the Mask, of each vertebra is obtained through the Calculate3D module to get the 3D image of the vertebra. Then use the 3-matic software in Mimics16.0 software to optimize the model and export it in STL format (Figure 2).

Reconstruction of 3D solid model
After Mimics image extraction and 3-matic model repair, the complete vertebral models of L5 and sacrum were obtained. At present, the appearance of each model is close to the real vertebra, with regular vertebrae, transverse process, spinous process, upper and lower articular process and other main structures. After introducing the model into Geomagicstudio (Geomagic, USA) the model unit is first selected, which is millimeter by default, and the system automatically prompts for grid doctor examination. After this processing, the polygon model of feature and smoothness balance is obtained. Simplify the sacrum model, use the plane clipping function, retain only the part of the sacrum in contact with S1, and carefully close the clipping plane, using the grid doctor to check to avoid causing the model to be unclosed (Figure 3).

Making of lumbar intervertebral disc model
The real lumbar vertebra is a complex motion system composed of vertebrae, intervertebral discs, ligaments, muscles and so on. If you want to carry out finite element analysis, the most simplified model should at least include vertebrae and intervertebral discs. It is very easy to make intervertebral disc models with Geomagic Studio software. The shape of the intervertebral disc model made in the experiment is lifelike, showing a disk shape similar to the solid, the leading edge is thick, the trailing edge is flat, and the edge is smooth without edges and corners. The nucleus pulposus is located slightly behind the center of the intervertebral disc on the cross section and in the middle of the intervertebral disc on the coronal plane. After measurement, the relevant parameters of intervertebral disc and nucleus pulposus are consistent with the real human body. After assembling, the intervertebral disc and adjacent vertebrae can be closely attached, the relative position of intervertebral disc and vertebra is correct, and there are no pathological manifestations such as intervertebral disc herniation, which can meet the needs of further experiments (Table 1 and Figure 4).

Assembly and volume meshing of lumbosacral transitional vertebrae
Through the previously described work, a high-quality triangular model of the lumbar spine and intervertebral disc was obtained, but it is not a CAD model and it needs to be further transformed into a NURBS surface solid model and exported in IGES, sat, X_T, and other formats in order for the data to be smoothly imported into the WorkBench finite element analysis software (ANSYS, USA). The specific workflow is as follows, using the L4/5 disc as an example:
(1) After importing the model into Geomagic Studio, the Exact Surfaces Phase is clicked and Exact Surface is clicked.
(2) Click the Detect Curvature method and select Auto Estimate and Simplify Contour.
(3) Using the Construct Patch tool, the area was re-divided. There were occasional errors such as intersecting paths, different sizes of surface patches in the structure, or mismatch between the distribution pattern and the actual shape of the model, and further manual editing of the surface patches was required.
(4) Use the Move Panel function under the Move option to edit the patch.
(5) Construct the grid, click on the Construct Grid button, click on Repair Intersection Area, and check the geometry option, and use a resolution of 20 to get the model image that covers the grid.
(6) Click on the fitted surface to transform the model formed by the grid into a NURBS surface and optimize the smoothness option to further smooth the model. Then, save the NURBS surface in IGES or STEP format for the next step.

Import of finite element model into WorkBench and calculations
The assembled model was imported into WorkBench as described previously[11-13]. The corresponding material properties were assigned to the vertebrae, the annulus fibrosus, and the nucleus pulposus. Next, the assembled model and properties were assigned, the mesh was defined, and the contact surface, boundary conditions, and load were determined. For loading, a 2 N·mm force was applied for the analysis and calculation under the conditions of flexion, extension, lateral bending to the left, and lateral bending to the right. The stress and overall deformation of the model as a whole and the L5 and S1 finite element models were calculated when the flexion, extension, and bending moments were 2 N·mm (Table 2 and Figure 5).

Main outcome measures
To analyze the mechanical properties and stress-strain rules of lumbar vertebrae under simulated normal exercise conditions.

Advantages of finite element analysis in the study of spinal anatomy of dysplastic vertebrae
In 1975, LIU et al.[14] constructed the three-dimensional finite element model of vertebral body for the first time in the study of direct shear resistance of lumbar vertebrae, which marked the beginning of the application of three-dimensional finite element method in the study of spinal biomechanics. With the in-depth study of spinal diseases, three-dimensional finite element method has been used and developed to a great extent. Clinicians can collect data from practical problems, use three-dimensional finite element method to design and construct a simulation model, analyze the model, and determine the treatment plan and effect evaluation combined with the actual situation[15-16]. In lumbar diseases, lumbosacral transitional vertebra is a common spinal abnormality, including lumbar sacralization and sacral lumbar vertebrae, which are two distinct developmental abnormalities[17]. Estimates of the prevalence of lumbosacral transitional vertebrae in the general population vary widely throughout the literature, ranging from 4.0% to 31.9%, with an average of 12.3%. However, lumbosacral transitional vertebrae can affect the degeneration of intervertebral disc above lumbosacral segment and affect the function of innervating muscles or nerves[18-20]. The degenerative changes above the abnormal joints are caused by overactivity and abnormal torque above the lumbosacral transitional vertebrae and restricted movement between L5 and S1. Facet joint asymmetry (such as CastellviIIA) is mainly related to low back pain, leading to the possibility of discogenic pain[21-22]. From the point of view of human mechanics, the changes of lumbosacral transitional vertebrae on the structure of lumbar motion segments are completely different, one is the increase of lumbar motion segments, and the other is the decrease of lumbar motion segments. Theoretically, it should be two different mechanical models. In the related works, we can see that most of the studies on lumbosacral transitional vertebrae basically refer to lumbar sacralization, and the related research works on low back pain induced by this have been more mature. DANG et al.[23] reported that disc herniation in patients without transitional vertebrae or with lumbar sacralization mainly occurred in L4/L5. However, the disc herniation in patients with lumbar spondylolisthesis occurs in L5xS1, and the excessive limitation of L5/S1 lumbar joint will lead to stress conduction, which makes the intervertebral disc and vertebrae of L4cyan5 easily damaged, while insufficient restriction will lead to L5/S1 stress increase and damage[24]. According to the study of AHN et al.[25], patients with lumbar disc herniation with LSTV have a gap in postoperative pain symptoms and recovery of living ability compared with non-LSTV patients. And lumbar back pain increased in varying degrees 
12-24 months after lumbar disc-related surgery, so the study of transitional vertebrae has a very important clinical significance. Abnormal anatomical morphology of lumbosacral vertebrae also plays an important role in the occurrence of degenerative spondylolisthesis. BERLEMANN et al.[26] suggested that abnormal spinal arrangement and abnormal direction of lower lumbar endplate may be high risk factors for lumbar spondylolisthesis. HAMMERBERG [27] found that lumbar degenerative spondylolisthesis is closely related to the increase of pelvic incidence angle. In addition, the abnormality of joint paroxysm, the degeneration of articular process and the change of direction and asymmetry of articular process are all related to the occurrence of lumbosacral transitional vertebrae[28-30]. In comparison, there are few references for the mechanical causes of pain induced by lumbar sacral vertebrae. The finite element model of lumbar vertebrae of sacral vertebrae established in this study provides a mechanical reference for clinical low back pain caused by this cause by analyzing the stress, strain and displacement of L5/S1 vertebrae.

Characteristics and mechanical analysis of finite element model of lumbosacral transitional vertebrae
In this study, finite element analysis was used to simulate the mechanical analysis of a patient with lumbar vertebrae under four working conditions of flexion, extension, left bending and right bending. It was found that the maximum load on the lumbar spine of the patients with lumbar vertebra increased gradually to the lumbar pedicle isthmus of S1, so the stress of the pedicle isthmus of lumbar vertebra was significantly higher than that of lumbar 5 pedicle isthmus during lumbar movement. It is more likely to cause fatigue fracture and isthmus fissure, and its facet process and intervertebral disc bear greater stress, so it is more prone to degeneration. The stress of L5 and S1 pedicle isthmus and articular process on the same side of the movement direction was significantly higher than that of the contralateral pedicle isthmus during the left and right bending movement. Therefore, compared with the normal human body, the stress of the lumbar pedicle isthmus is greater than that of the normal human body, and the stress of each pedicle of the complete sacral lumbar vertebra model is also significantly higher than that of the lateral bending movement; and the stress change direction of S1 pedicle isthmus is the same as that of L5 pedicle isthmus in normal people. For the adolescent patients with thorough lumbar vertebrae, the stress in the isthmus of the pedicle of the transitional vertebra is significantly higher than that of the L5 pedicle isthmus, which is more likely to lead to fatigue fracture and isthmus, and the facet process bears greater stress and is more prone to degeneration. The corresponding abnormal S1 and S2 intervertebral disc becomes the most strained part of all lumbar and sacral intervertebral discs, in addition, it is generally poorly developed. Pathological changes such as disc degeneration and protrusion are more likely to occur in lumbosacral movement. Therefore, long-term engaged in weightlifting, volleyball, swimming and other sports activities or manual labor that requires repeated extension and flexion of the lumbar spine, due to frequent reciprocating one-way lateral bending of the lumbar spine, it is more likely to occur superior contralateral lumbar articular process degeneration and pedicle injury. It is also prone to transitional spondylolysis, and its pathogenic factors are consistent with lumbar spondylolysis in normal people. If the course of isthmus fracture is similar to that of normal L5 isthmus fissure, the same internal fixation can be used as L5 isthmus fissure, which is of guiding significance for clinical treatment.

Deficiency and prospect of the research
In this experiment, the finite element analysis can be used to make an effective mechanical analysis of the lumbar sacral vertebrae of a teenager, in order to find its stress characteristics and stress changes, but it is not representative. Only the use of this technique is only the mechanical analysis of adolescent sacral lumbar vertebrae, which provides a theoretical basis for clinical precision and personalized treatment in the later stage.
中国组织工程研究杂志出版内容重点：人工关节；骨植入物；脊柱；骨折；内固定；数字化骨科；组织工程

文题释义：#br# 腰骶移行椎：指腰椎数目变化及其伴随的L5 和S1 的形态结构变化。一般情况下整个脊柱的椎骨总数不变，只是各节段数目有所增减，可分为骶椎腰化和腰椎骶化。#br# 有限元模型分析法：是近年来研究人体生物力学的有效方法，其基本原理是把连续的物体离散为一组有限个，并按一定方式相互联结在一起的单元组合体，然后求其相互之间的关系及特征。

中国组织工程研究杂志出版内容重点：人工关节；骨植入物；脊柱；骨折；内固定；数字化骨科；组织工程

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