1 Introduction

Finite element (FE) modeling is widely used in spine biomechanics research because it evaluates stresses and strains in bony and soft tissue structures more realistically [1]. Single functional spinal units (FSUs) [2,3,4], as well as L3–L5 [5, 6], L2–L4 [7], L1–L3 [8], L2–L5 [9], and L1–L5 [10,11,12,13,14] spinal levels have been modeled to study the biomechanical effect of material properties, surgical procedures, and loading and boundary conditions on the lumbar spine, where small segments or FSUs are often used to evaluate the effect of different types of surgical interventions [11]. Xu et al. [7] analyzed stress concentrations of fixation rods with different fixation methods using a two-level model, while Ambati et al. [6] compared the stability of different fusion constructs using an L3–L5 lumbar FSU instrument with interbody fusion cages. In addition, the sacrum to lumbar spine region has been included in multi-segment spine models with consideration of global parameters such as pelvic incidence and lordosis, which are of great interest to clinicians [15,16,17,18,19,20,21,22,23]. Haddas et al. [16] first modeled scoliotic lumbosacral spines in relation to the Cobb angle based on in vivo computed tomography (CT) scans of patients. Furthermore, Park et al. [22] investigated the effects of single-level degeneration on whole lumbosacral spine biomechanics. In parallel with these studies, several subject-specific detailed models have been developed [17, 24,25,26], such as that of Haj-Ali et al. [17], who generated an L1–S1 model using patient-specific CT scans to simulate the spondylolysis effect on lumbar segments. Recently, FE models have been combined with musculoskeletal representations to conduct comprehensive and realistic biomechanical analyses [27,28,29,30,31,32,33].

Several studies have presented FE analyses of the lumbar spine in relation to spine biomechanics [34], soft tissue modeling [35], intervertebral discs [36], facet joints [37], load-sharing behaviors of lumbar motion segments [38], and FE modeling methods [39,40,41,42,43,44,45]. Schmidt et al. [36] reviewed an FE simulation of lumbar intervertebral discs (IVDs) with a focus on functional biomechanics, while Mengoni [37] provided a systematic review of FE modeling of the facet joints. In addition, Ghezelbash et al. [38] discussed the relevant findings of in vitro and FE modeling studies of lumbar motion segments, specifically with regard to the load-sharing contributions of spinal tissues in both normal and perturbed conditions. Recently, Knapik et al. [43] systematically reviewed computational modeling methodologies of the lumbar spine and identified that musculoskeletal models are capable of evaluating whole-body motion and deformations with kinematics-driven data, whereas the FE model enables a detailed investigation into individual tissue deformations and stresses [43]. Furthermore, Dreischarf et al. [46] compared eight well-established FE models of the lumbar spine to in vitro and in vivo measurements in terms of intervertebral rotations, intradiscal pressure (IDP), and facet joint force (FJF) under pure moments and combined loadings to show the validity of FE analysis. Despite the depth of these reports, detailed analyses of disc degeneration or muscle force prediction have been little considered. This study focuses on recent developments in FE modeling of the lumbar spine, including disc degeneration, muscle force prediction, and clinical applications. Articles were searched through the PubMed and Science Direct databases between 2013 and 2023 with the following keywords: ‘lumbar’, ‘spine’, ‘muscle’, ‘disc generation’, ‘finite element’, and ‘model’. The inclusion criteria were language in ‘English’ and a study population of ‘humans.’ Several older significant and relevant studies were also included.

2 FE Modeling Methodologies

The FE model including bony structures, IVDs, ligaments, and facet cartilages in flexion, extension, lateral bending, and axial rotation was depicted in Fig. 1. The details for FE modeling details including material properties were summarized in Table 1.

Fig. 1
figure 1

FE model of the lumbar spine including bony structures, IVDs, ligaments, and facet cartilages in flexion, extension, lateral bending, and axial rotation (IVD—intervertebral disc; ALL—anterior longitudinal ligaments; PLL—posterior longitudinal ligament; ISL—interspinal ligament; SSL—supraspinal ligament; CL—capsular ligaments; LF—ligament flavum)

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Table 1 FE modeling details including material properties
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2.1 Bony Structures

Cancellous and cortical bones, as well as post-bone material, are generally included in FE models. Computed tomography (CT) imaging is frequently used for generating spinal bony parts, although CT-based modeling requires substantial manual intervention for segmentation, threshold, and region-growing [1, 43]. Some studies have utilized radiographs [47] and geometric parameters [48]. Nikkhoo et al. [47] used lateral and anterior–posterior X-Ray images to develop an FE model of L1–S1 that automatically updated the spinal geometry of patients based on independent parameters. Bashkuev et al. [48] developed a parametric model of the L4–L5 motion segment considering natural variations in spinal geometry using 40 different parameters. In addition, several modeling techniques have been proposed, including an automated landmark identification method for subject-specific FE modeling of the lumbar spine that identifies all necessary landmarks for model creation using CT geometry of a spinal bone [49]. A parametric computer-aided design (CAD) model was used to generate a parametric FE model of the lumbar spine that integrated independent tuning of morphometrical parameters [50], and the mapping block technique was applied to model T12–S1, which allowed for a continuous mesh model from the superior to inferior vertebrae [51]. Lalonde et al. [52] developed a free-form deformation technique to deform a detailed FE mesh of the spine to subject-specific geometry. It should be noted that meshing elements affect the accuracy of the model, and hexahedral meshing is usually recommended for solid components because it is computationally efficient and has better numerical stability than tetrahedral meshing [50, 51]. Generally, bony components have been modeled with elastic material properties [46].

2.2 IVDs, Ligaments, and Facet Joints

The IVD is important for the regulation of flexion responses, whereas capsular ligaments (CL), anterior longitudinal ligaments (ALL), and discs are more dominant under extension [39]. The posterior longitudinal ligament (PLL), ligament flavum (LF), interspinal ligament (ISL), and supraspinal ligament (SSL) are generally resistant to flexion [53], while intertransverse ligaments (ITL) mainly contribute to the lateral bending stiffness of the motion segment [2], and facet joints allow greater flexion, extension, and lateral bending movements but resist rotation [54]. Naserkhaki et al. [3] reported that ligaments resist 45–75% of flexion, whereas facet joints and discs resist 20–33% and 48–60% of the extension movement, respectively.

The IVD consists of the annulus fibrous containing ground substances and fibers, as well as the nucleus pulposus and endplates. Recent studies have used hexahedral elements with Mooney–Rivlin [22], neo-Hookean [55], and Yeoh [21] hyperelastic materials as annulus ground substances, while the nucleus pulposus, which represents approximately 44% of the surface of the disc, was modeled with fluid elements [22] or Mooney–Rivlin hyperelastic material [56,57,58]. Truss elements were used to model the fibers of the annulus fibrosus with tension-only material properties [22], as well as shell elements with rebar properties [16]. The fibers which reinforce the ground substance in the radial direction are oriented approximately 30 degrees from the horizontal surface (the bottom portion of the IVD) [59]. The number of fiber layers ranges from 2 to 16 depending on the study [46]. The stiffness of each annular fiber layer is different since external layers have a greater stiffness than their internal counterparts [25]. The endplate was modeled to an approximate 0.5–0.6 mm thickness with solid elements by extruding the surface of the vertebral body [6, 14, 58].

The ligaments were modeled using truss [60], spring [61], shell [16], connector [49], and solid [54] elements. During the creation of the FE model, ligament attachment points were manually defined using anatomical landmarks. Subsequently, the automatic modeling technique was developed for determining the attachment points of ligaments [49]. The lack of experimental data mandated the use of major assumptions regarding the material properties of the ligaments which were simulated with linear, bilinear, and nonlinear force–displacement and stress–strain properties [3].

Facet joints are among the most difficult elements to model and are distinguished by a uniform gap between the articulating surfaces [43]. Facet cartilages have often been modeled on the superior and inferior plane of the facet joints with isotropic linear elastic wedge elements since the actual location and thickness of the cartilage cannot be determined from a CT scan [22, 49]. Frictionless surface-to-surface soft contact and an initial gap of 0.1–0.5 mm were assumed to exist between cartilages [22, 62], and a friction coefficient of 0.1 was used for the contact of facet joints [57].

2.3 Muscle Force Modeling

An individual muscle in the lumbar region was represented in a model as a straight line between its origin and insertion to define the muscle force direction [63]. The origins and insertions were obtained from the literature and adapted to the FE model based on anatomical landmarks of bony structures [64]. Physiological cross-sectional areas (PCSAs) of the muscles were obtained from the literature to calculate muscle stress [65]. Recent FE models have included a large number of muscles [63,64,65,66]. Kim et al. [64] considered 58 pairs of superficial muscles: longissimus pars lumborum (5), iliocostalis pars lumborum (4), longissimus pars thoracis (12), iliocostalis pars thoracis (8), psoas (11), quadratus lumborum (5), external oblique (6), internal oblique (6), and rectus abdominus (1); and 59 pairs of deep muscles: thoracic multifidus (12), lumbar multifidus (20), interspinales (6), intertransversarii (10), and rotatores (11). El Ouaaid et al. [66] used a kinematics-driven thoracolumbar FE model consisting of 46 local and 10 global muscle fascicles to estimate muscle forces, spinal loads, and stability during elevations with an optimization algorithm. Muscle forces were then predicted to satisfy the force and moment equilibrium using the calculated net intersegmental forces and moments, ligament forces, and facet joint forces by the conventional optimization technique due to their complexity and redundancy [64,65,66]. The objective function was normally the summation of cubic muscle stresses and the maximum isometric muscle force was determined as the upper boundary of the individual muscle force.

The concept of the follower load (FL) has been used to assume that resultant muscle forces follow a path through the vertebrae [25, 61, 67]. The FL concept was applied to the lumbar spine FE model to simulate standing and compared with other loading modes [68]. It was indicated that FL delivers the most probable intersegmental rotations and FL of 500 N was suggested for the simulation of a standing position with the lumbar spine model. In addition, Han et al. [69] demonstrated that spinal muscle can create a compressive FL in the lumbar spine during a standing posture, and recommended this approach for experimental and numerical studies. Kim et al. [64] proposed a modified concept of the FL, whereby the compressive load aimed for a half-radius of the body near the body center, and analyzed muscle coordination and trunk muscle activation using an optimization scenario. Shih et al. [70] compared the effects of concentrated, follower, and muscular loads on lumbar biomechanics during flexion. The results showed that the FL can aid in the avoidance of unreasonably high flexion and anterior shear at the disc level. Moreover, a realistic location for the FL path was studied for the lumbar in a neutral standing position.

3 Clinical Applications

3.1 Model Validation

Models are validated using comparisons with available experimental or computational data, which provides confidence to the model predictions [37, 46]. Most FE models are validated using in vitro range of motion (ROM), FJF, and IDP data with single or combined loadings [14, 53,54,55, 71] where experimentally observed moment-rotation data are commonly available [46, 67, 72, 73]. Rohlmann et al. [72] measured the ROM of the human lumbar spine under pure moments of 3.75, 7.5, and 7.5 Nm with a follower load of 280 N in vitro and found that at 7.5 Nm the experimentally observed L1–5 ROMs were 24°–37° in flexion–extension, 18°–42° in left–right lateral bending, and 7°–17° in left–right axial rotation [72]. Moreover, models can be validated using FJF [74,75,76] and IDP [77, 78]. Wilson et al. [75] measured FJFs in cadaver lumbar spines using flexible resistive sensors (Tekscan) under pure moments of 7.5 Nm and reported values of 55–110 N for axial rotation and 10–50 N for extension. In addition, extra-articular strains were applied to measure the FJF during axial rotation (71 ± 25 N), extension (27 ± 35 N), and lateral bending (25 ± 28 N). Wilke et al. [78] measured the IDP in the L4–5 disc in vivo and returned values in flexion, extension, lateral bending, and axial rotation of 1.08, 0.60, 0.59, and 0.70 MPa, respectively.

3.2 Spinal Degeneration Modeling

Spinal degeneration has been the subject of many lumbar spine FE models, whereby various parameters have been altered to understand the role of each structure in spinal biomechanics [79]. Disc degeneration can be predicted with the FE model by determining the locations of the greatest stresses in the endplates and annulus where failures can initiate [80], and numerous studies have modeled the disc degeneration process [9, 22, 48, 79]. Li et al. [9] simulated lumbar decompression surgery for moderated disc degeneration where the disc height was reduced by 40%. Park et al. [22] investigated intersegmental rotation, nucleus pulposus IDP, and FJF under various grades of disc degeneration. Those authors simulated disc degeneration by changing the geometry and material properties based on clinical classifications. Three parameters were used to describe disc degeneration: disc height, compressibility increase, and material property changes of the annulus fibrosis and ligaments. Rohlmann et al. [81] reported that a 20% decrease in disc height can generate mildly degenerated discs with increased nucleus compressibility. Bashkuev et al. [48] used a probabilistic FE model and found that stiffening of the motion segment led to an increase in the disc degeneration process. Disc height has been indicated as the most influential parameter on the mechanical behavior of discs, and reducing the height by only 10% has opposite results to those previously identified [36]. Other studies have investigated spinal degenerative disease models including those of osteoporosis [82] and osteophytes [58]. Kang et al. [82] modeled the osteoporosis bone model with lower bone density and elastic modulus, while the osteophyte formation model was created with disc height losses from 16 to 82% [58]. However, there are few studies on the effects of ligament properties or failures on the lumbar spine [3, 79]. Ellingson et al. [79] investigated the impacts of ligament degeneration on the functional mechanics of the lumbar spine by gradually removing ligaments from the motion segment and reported that incremental ligament failure produced an increased ROM and decreased stiffness in the lumbar spine FE model.

3.3 Surgical Interventions

FE analysis of spine biomechanics can be used to assess scenarios for a range of spinal disorders or associated surgical interventions through the evaluation of tissue deformations and stresses [37, 43]. The majority of studies have simulated the impacts of various surgical procedures as well as designed and assessed new surgical instrumentation on the lumbar spine based on validated intact models. such as screw fixation and fusion cages [6, 7, 13, 60, 83,84,85,86,87,88,89,90], artificial discs [14] cement discoplasty [91], facetectomy [54], laminectomy [9, 62, 92], and osteotomy [93]. Stress concentrations in rods and pedicle screws have been investigated since the material properties and geometry variations in the fixation devices affect spinal biomechanics, which is related to the possible failure of spinal instrumentation such as broken screws and rods. Guo et al. [13] used topology optimization to determine the optimal rod and fixation design to reduce stress in the rods while decreasing pressures and stresses in spinal tissues. In surgical simulation studies, the ROM or instability of the spine segment has been mainly compared among surgical procedures when experimental investigations are difficult or impossible [62]. Furthermore, the FE method has been expanded to the fields of scoliosis [94] and spinal disorder prevention and treatment [20, 95, 96]. In addition, the influence of loading rate and frequency on the lumbar spine has been reported [8, 97].

3.4 Muscle Force Prediction

Recently, muscle force modeling and prediction have been effective for assessments of spinal injury risk and the design of effective prevention and treatment programs because muscles act important role in spine stabilization by generating the substantial increase in internal loads to support external loads on the human spine [98]. Some studies have investigated the effects of muscle volume on the lumbar spinal column by including a passive muscle volume in the model [99, 100], while others have predicted muscle activation and spinal loads using the optimization method. Jamshidnejad and Arjmand [101] investigated the effects of paraspinal muscle intraoperative injuries on muscle activation and spinal loads and reported that trunk strength was reduced by 23% as a result of reductions in the cross-sectional area of the extensor muscles. El Ouaaid et al. [65] predicted trunk muscle forces using an FE model of the thoracolumbar spine with an optimization algorithm during lifting activities and noted that spinal force prediction was helpful to improve rehabilitation and stabilization exercise designs.

4 Conclusion

We reviewed the recent advances in FE modeling of the lumbar spine, including modeling and analysis from the bone to soft tissue and muscle forces, as well as the validation and application of these models. The discussion was associated with material properties, element types, loading and boundaries, and geometric parameters. In addition, muscle force modeling and the follower load concept were introduced. Furthermore, we summarized experimental ROM, FJF, and IDP data for validation of the lumbar spine FE model since all new models should be verified based on recognized intact models. Application studies were briefly reviewed, in which the majority of FE models focused on spinal degeneration diseases and surgical instrumentation techniques. Although muscle force prediction and optimization are challenging with FE modeling due to their complexity and redundancy, several studies have predicted muscle activation and spinal forces for injury prevention assessments and treatment strategies. The level of modeling prediction and representation can be improved with subject-specific data, and integration of FE and musculoskeletal models could generate a comprehensive analysis of the lumbar spine in clinical applications.