Statistical Shape Modeling and Prediction of Lumbar Spine Morphology in Patients With Adolescent Idiopathic Scoliosis

Adolescent idiopathic scoliosis (AIS) is a three-dimensional (3D) spinal deformity affecting 2–4% of adolescents [1], predominantly females ages 10–18 [2]. Common curve patterns include thoracic, thoracolumbar, lumbar, and double major curves, with 40% involving lumbar curves [3,4]. Important clinical and statistical differences exist between two-dimensional and 3D measurements of scoliosis [5], with maximum differences of approximately 7 deg [6]. Additionally, there are notable differences in two-dimensional and 3D measurements of kyphosis between mild and severe cases of AIS [7], as well as in the sagittal plane, which is related to spinal curvature and pelvic rotation [5]. The true shape of AIS deformities is best described by 3D reconstructions from biplane X-ray images [8]. Thus, 3D parameters are crucial for evaluating the effects of AIS surgeries [9] and curvature progression [10].

Rehm et al. [11] modeled the 3D spinal rotation in patients with AIS using biplanar radiography data, which provided information on the deformity shape. Statistical shape models (SSMs) provide valuable information on bone morphology and anatomical variations and facilitate the generation of deformable models for multibody or finite element (FE) simulations. Moreover, they enable the analysis of pathology-related geometric differences, which is valuable for treatment plans and quantifying surgical outcomes. For example, Wai et al. [12] established a SSM for thoracic vertebrae T4–T6, which provided surgical guidelines for pedicle screws. SSMs also enable clinicians to quickly generate 3D vertebral models from computed tomography (CT) scans [13]. Assi et al. [14] used statistical modeling to predict postoperative 3D torso shapes in patients with AIS. Peters et al. [15] simulated the thoracic vertebral shape in children with AIS, accounting for age- and sex-related changes. González-Ruiz et al. [16] compared the 3D torsos of patients with and without AIS. Ali et al. [17] characterized the lumbar spine shape along the sagittal plane, while Campbell and Petrella [18] identified lumbar spine landmarks for FE models using statistical shape modeling. Sun et al. [19] established a statistical model of spinal shape and bone mineral density from CT images, providing age-related data for simulations. Hollenbeck et al. [20] developed SSMs for single vertebrae and lumbar arrangements (L1–S1) in adults and verified the relationships between the principal components and anatomical parameters. Tang et al. [21] created SSMs of adult lumbar vertebrae, which enabled the rapid generation of lumbar FE models based on characteristics like age, sex, height, and body mass index (BMI) through multivariate logistic regression. Clouthier et al. [22] developed deformable lumbar models to examine differences in shape with pathology, demographics (e.g., age and sex), and anatomical measurements. However, few studies have focused on lumbar spine SSMs for AIS.

Herein, we establish a SSM of the lumbar vertebrae (L1–L5) of a population with AIS and analyze the effects of four easily obtainable human characteristics, age, height, sex, and BMI, on the model. The SSMs of individual vertebrae characterize the morphological variations and quantify structural deformities, while the SSM of the whole lumbar spine describes the relative arrangement between bones, particularly focusing on lumbar scoliosis and lordosis. This model can be used to obtain anatomical parameters, generate specific geometric models, analyze the relationship between anatomical variations and demographics, and assist clinicians in diagnosing and treating AIS.

Forty-nine patients with AIS (14 males, 35 females; ages 10–18; height 1.3–1.8 m; and BMI 11–30 kg/m²) were examined using clinical supine CT (slice thickness: 0.625 mm, pixels: 512 × 512, and spatial resolution: 0.488 × 0.488 × 0.625 mm) as part of their usual clinical assessment. The subjects exhibited thoracolumbar, lumbar, or double major curve patterns. No significant correlations were found among their characteristics, except that height was related to sex and age. To maintain consistency, lumbar spines with right convexity were mirrored to left convexity.

For each patient, lumbar vertebrae L1–L5 were segmented from the CT images and reconstructed in 3D (Figs. 1 and 2). The SSMs were constructed through template model registration and remeshing of the 3D vertebrae models. The template mesh was deformed using landmark points to create vertebral meshes for specific samples. The geometric shapes and arrangements of vertebrae were analyzed statistically to establish a prediction model based on input variables such as age, sex, height, and BMI. An automatic measurement system based on a registration algorithm was established to measure anatomical parameters.

The CT data were imported into mimics for image segmentation. To generate bone masks using the CT Bone function in mimics, start by setting the threshold to the default value of 226–3071 Hounsfield Units. Next, select the Seed Point option to initiate the bone mask creation. Then, utilize the Smart Fill function to fill the medullary cavity at both the upper and lower sections of the bone. The middle section will automatically fill the gap, ensuring a seamless transition. The lumbar vertebrae of each subject were segmented using semi-automatic threshold segmentation and then reconstructed and imported into 3-matic. The template was prepared following a reported method [13]. The template model was prepared by an iterative process in which the new average model was compared with the previous one until convergence (root-mean-square error (RMSE) < 0.2 mm). All vertebrae models were remeshed in 3-matic with triangular meshes (∼2300 nodes per vertebra, 2 mm length per triangular surface element).

The iterative closest point (ICP) algorithm [23] was used for rigid body registration, with 15 landmarks on each vertebra [21]: seven at the superior and inferior articular, left and right transverse, and spinous processes of the vertebral tip; and eight at the anterior, posterior, left, and right points of the superior and inferior planes of the vertebral body. After identifying all 75 landmarks for each subject, the thin plate splines (TPS) method [24] was used to map the lumbar template to the lumbar vertebrae of each subject, establishing point correspondence.

where $X¯$ is the average shape model of the training set, v_s is the sth principal component, and ω_s is the weight coefficient varying in the interval [−3λ_s, +3λ_s]. The number of principal components m was defined based on the ratio of cumulative variance to total variance $(∑s=1mλs∕∑s=1n−1λs)$ reaching 0.95.

Following reported validation criteria [29,30], we used numerical indicators such as accuracy, compactness, generalization ability, and specificity to assess the SSMs. Accuracy evaluates the consistency between the statistical and real models. We selected the principal components explaining 95% of variance to calculate the accuracy and generalization ability. Compactness describes the number of principal components required to explain the variance of the training set. This variance correlates with the generalization ability of the SSM to anatomical variation (i.e., the ability to represent samples outside the training set). The generalization ability was assessed by leave-one-out cross-validation, removing each sample from the training set in turn to create new shape models and optimizing the models to fit the target sample. The results were expressed as the average and maximum RMSE between the optimized model and the corresponding nodes of the target sample. Specificity (the effectiveness of shapes generated by the SSM) was estimated by generating random parameter values from the normal distribution with zero mean and the corresponding standard deviation of PCA. The nearest matching distance between the generated shape and training set was averaged over many runs. The Levenberg–Marquardt (LM) nonlinear algorithm [28,31] was used to iteratively optimize the weight vector W and calculate the RMSE between shapes.

where SS_res is the sum of squares of all subjects’ weight vectors W_i and predicted weight vectors W′_i, and SS_tot is the sum of squares of the average weight vector and weight vectors W_i for all subjects.

To evaluate the accuracy of the multivariate logistic regression model, we used leave-one-out cross-validation to calculate the RMSE between the subject and predicted models for each node in the vertebral mesh. The P value was used to determine whether each feature had a statistically significant impact on each principal component.

An automatic measurement system was developed to measure the anatomical parameters of the sample model using ICP rigid registration and TPS elastic registration algorithms. The system involved the following steps (Fig. 3). First, the sample model underwent rigid transformation to create a Subject_ICP model, which was then used for TPS interpolation, resulting in a Temp_TPS model (a deformed mesh registered with the sample model). Next, anatomical marker points were defined on the template model. For each point, the nearest node (node 1) was identified, and its index (index 1) was recorded. Subsequently, using index 1, the corresponding node (node 2) was located in the deformed mesh. After the rigid transformation, the node (node 3) with the smallest distance from node 2 was identified, and its index (index 2) was recorded. Finally, in the sample model, node 4 was determined based on index 2. The required anatomical parameters were then calculated using the coordinate values of node 4. The advantage of this automatic measurement is that anatomical points need to be selected only once on the template model, after which corresponding points are automatically selected on the sample model based on the index and distance.

Figures 4(a) and 4(b) show the first five modes of variation of the lumbar SSMs after PCA, allowing for qualitative and intuitive evaluation. For the individual vertebral SSMs (Fig. 4(a)), mode 1 reflected the change in size; modes 2 and 3 represented changes in the anterior–posterior and anterior–downward directions, respectively (except for L2, where modes 2 and 3 were reversed); mode 4 described the relative changes between the vertebral body and articular processes; and mode 5 captured changes in the spinous process height or direction. Modes 4 and 5 together described changes in the right tilt of the upper endplate and the transverse processes. For the lumbar spine SSM (Fig. 4(b)), mode 1 indicated size changes; mode 2 related to apex position changes, reflecting the type of curvature from thoracolumbar bend (−3λ) to lumbar or double main bend (+3λ); mode 3 regulated the intervertebral angle and lordosis; mode 4 characterized the angular bend changes from convex (−3λ) to normal (+3λ), as well as the translatory shifts at the end of the coronal lumbar curve; and mode 5 represented left–right direction changes.

The shape changes associated with the first five principle components (PC1–PC5) were quantified using Pearson’s correlation coefficient after testing the data normality [20,32]. Quantification provides details that are not easily visualized. For the vertebral SSMs (Table 1), the vertebral body size closely correlated with PC1, with an average correlation coefficient of 0.73. The left and right articular process heights, defined as the distance from the lowest point on the inferior process to the highest point on the superior process, correlated not only with PC1 but also with principal components that represented changes in the upper and lower directions. Similarly, the spinous process length correlated with principal components representing anterior–posterior direction changes. The correlation coefficients between PC2 and the spinous process lengths of L1 and L3–L5 were 0.35, 0.50, 0.44, and 0.56, respectively, whereas that for L2(PC3) was 0.45. The left and right articular process widths correlated with PC1 for each vertebra. For L1, L2, and L4, the correlation coefficients between PC3 and pedicle diameter changes were 0.52, 0.59, and 0.53, respectively, whereas that for L3 was 0.45 with PC2. For L5, PC1–PC4 all correlated with pedicle width, possibly because of its relatively short transverse process and broad pedicle.

For the lumbar spine SSM, PC1–PC5 correlated with five anatomical parameters (Table 2), including the disk height and angle of the L4/L5 functional spinal unit. The height is the distance between the last point of the lower and upper endplates of the two vertebrae, and the angle is that between the lines connecting the most advanced points of these endplates. The mean correlation coefficients between these parameters and PC1 (size change) and PC3 (sagittal angle adjustment) were 0.3 and 0.35, respectively. The Cobb angle, which is widely used to quantify spinal curvature, was highly correlated with PC4 (left convexity to normal curvature), with a correlation coefficient of 0.81. Unlike the visual (qualitative) results, the quantitative results showed that PC2–PC5 all correlated with lumbar lordosis angle changes (Table 2).

Several evaluation indexes, including the compactness, accuracy, generalization ability, and specificity, were calculated for each SSM (Table 3). For the vertebral SSMs, 16–19 principal components were required to capture more than 95% of the shape change, with PC1 accounting for an average of 45%. The L1–L4 SSMs had similar accuracies, generalization abilities, and specificities, with RMSEs similar to that of the regression model, whereas the L5 SSM had worse accuracy (RMSE = 1.99±0.71 mm). For the lumbar spine SSM, ten principal components captured over 95% of the shape change, with PC1 representing 56% (Fig. 5). The accuracy, generalization ability, and specificity (RMSE = 1.52±0.35, 1.94±0.36, and 3.31±0.61 mm, respectively) were worse than those of the vertebral SSMs. The RMSE of the multiple logistic regression prediction model was 3.76±1.21 mm.

Figure 6 shows the RMSE distribution of the SSMs after leave-one-out cross-validation. The transverse and upper and lower articular processes had the largest errors, followed by the spinous process. By contrast, the pedicle, arch plate, and vertebral body errors were relatively small.

Figure 7 shows the lumbar geometry fitted by the prediction model, illustrating the effects of age, sex, height, and BMI on lumbar geometry, curvature, and vertebral size. The control variables were unique, and the values were input as the mean±standard deviation. With increasing age, the lateral curvature angle of the lumbar spine increased and the height increased slightly. With increasing height, the cross-sectional area and height of each vertebral body increased. Height also affected the size and curvature of the lumbar spine. Men had a slightly larger vertebral cross-sectional area than women, but smaller vertebral height. Increasing BMI resulted in enlargement of various processes and pedicles and a slight increase in the kyphotic angle. The effects of age, sex, and height were consistent across all vertebrae. Regression modeling predicted the variation of the vertebral SSMs with an overall R² of 0.62–0.75 and average R² of 0.69; thus, age, sex, body size, and BMI explained about 69% of the geometric variation.

Statistical shape models of the lumbar vertebrae and spine were constructed to describe the variability in vertebral shape and arrangement, capturing complex variations in the shape and size of individual bones and the multistructural anatomy of the lumbar spine.

As the vertebral level increased from L1 to L5, the vertebral height initially increased (L1 = 25.28 mm; L3 = 27.55 mm) and then decreased (L4 = 27.32 mm; L5 = 25.55 mm), while the width and length continually increased. This is consistent with the trend reported earlier [33,34]. After controlling for differences in age, height, and BMI, men had a larger vertebral cross-sectional area than women, but a lower vertebral height. This contrasts with findings that the cross-sectional area and vertebral height in male vertebrae are larger than in females within the normal adult group [21,35,36]. Age, height, and BMI were positively correlated with vertebral size, but age did not affect vertebral length.

Masharawi et al. [37] reported that, in a normal population (ages 20–80), the right lateral vertebral height was significantly greater than the left in 11 thoracic and lumbar vertebrae, while six (T4, T8, T9, T11, L3, and L4) had no significant differences. This may explain why adolescent scoliosis often involves right thoracic bending and left lumbar bending. Furthermore, this phenomenon was more common in women, which aligns with the higher prevalence of adolescent scoliosis in women. To investigate the structural deformity caused by scoliosis, we measured the left and right heights of L1–L5. The lateral height correlated strongly with mode 1, which represents the overall lumbar spine size. For L1–L3 and L5, the left side was taller than the right, while L4 showed no statistical differences.

The pedicle parameters in normal adult and adolescent groups have been measured previously [32,38]. In addition, Hong et al. [39] measured the pedicle parameters in patients with scoliosis, including AIS. The measurements in our study showed the same trend, with pedicle width increasing from L1 to L5. However, compared to previous data, the pedicle width of L1–L3 was approximately 2 mm greater than that in normal adolescent groups, and the pedicle width at each lumbar spine level exceeds the average pedicle width of the AIS group by 2 mm.

The articular process height increased from L1 to L2 and decreased from L3 to L5, while the upper articular surface width increased with increasing lumbar level (Table S1 available in the Supplemental Materials on the ASME Digital Collection). This is consistent with previous results [40]. PCA captured changes in facet height and width (Table 2). The articular facet joints are crucial to spinal biomechanics; they transmit spinal loads and provide translational, rotational, and axial stability [41]. Patient-specific predictions of spinal morphometrics (e.g., size and orientation the articular facet joints) are clinically valuable for evaluating spinal treatments. Our prediction model showed that age and height significantly affected the articular surface size, whereas sex and BMI had little effect.

The lumbar lordosis angle, associated with impaired lung function (a common complication of AIS) [42], was measured in the lumbar spine SSM. Visual analysis indicated that mode 4 correlated with a shift at the lower end of the lumbar curve. This deformity is associated with AIS-related back pain [3]. The Cobb angle, the gold standard for evaluating scoliosis severity and guiding surgical treatment [43], was measured by drawing line segments between the leftmost and rightmost points on the upper and lower planes at the top and bottom of the curve, respectively, and measuring the angle between them in 3D space. Quantitative analysis was consistent with the visual results; thus, mode 4 represented the lumbar scoliosis severity.

The generalization abilities of the individual vertebral SSMs had a mean RMSE of 1.33±0.52 mm, although that of L5 was significantly larger because L5 showed more variation, such as sacralization. The mean RMSE for the lumbar spine SSM was 1.94±0.36 mm, again due to L5 variation. The prediction models of the vertebrae had a mean RMSE of 2.10±0.66 mm, while that of the lumbar spine was 3.76±1.21 mm. The curvature variation of populations with AIS is far greater than that of normal populations, which is not sufficiently explained by demographics (e.g., age, sex, height, and BMI). Modeling the curvature and bone separately may improve accuracy [21]; however, the joint connections are crucial for FE models of the lumbar spine and its implants, and the facet surface and angle relate to the lumbar spine shape.

All models showed greater geometric error in the spinal processes than the vertebral bodies. This was attributed to the image resolution, segmentation process, method, and sample size (n = 49). The accuracy of the SSMs was slightly lower than that of other SSMs [20,21], primarily because of the young age of the population and deformity caused by AIS.

Three-dimensional geometric instances described by SSMs can inform implant design and sizing. For AIS populations, implants such as screws and vertebral space dilators are designed with limited sizes to reduce manufacturing costs while ensuring adequate endplate coverage and pedicle width for most people. PC1 of the vertebral SSMs (endplate size variation) can inform endplate implant sizing. In spinal fusion—a standard surgical procedure for severe scoliosis—the number and location of pedicle screws used largely depend on the surgeon’s preference and experience. The incidence of pedicle screw dislocation varies from 0% to 42% [44,45]. In addition, the incidence of complications related to misalignment is as high as 42% [46]. Thus, SSMs and prediction models can be used to improve spinal fusion devices and pedicle screw sizing.

Gonzalvo et al. [47] reported that surgeons must insert at least 60 thoracic pedicle screws under supervision before working independently. Simulation-based training uses either physical models (synthetic, animal, or cadaveric) or computer-based models. Computer-based models overcome some limitations of physical models, such as ethical and regulatory issues, disease transmission risk, and the need for supervision [48]. Accurate 3D models of patient anatomy, like the SSM in this study, are essential for immersive surgical simulations.

Statistical shape models and predictive models also provide a way to create age-, sex-, height-, and BMI-specific FE models for biomechanical studies. FE models are usually generated by manual segmentation of CT data or 3D reconstructions of X-rays. Predictive models are therefore useful for biomechanical modeling, parametric analysis, and FE simulations.

Statistical shape and intensity models can be established by combining bone density data with lumbar spine SSMs. Statistical shape and intensity models provide information on bone density and strength for more detailed FE models [49]. Further anatomical parameters can also be measured, such as the pedicle height, ridge length, and transverse pedicle angle [39], which would aid in determining the insertion angle of pedicle screws. The SSM of the lumbar spine enables fast segmentation of individual vertebrae from CT scans. Clogenson et al. [13] used a 3dslicer software to manually segment CT scans and generate surface mesh in an average of 1 h, while their study using Landmark’s SSM method took an average of 2 min.

The use of supine CT scans in this study may have introduced differences in spinal morphology compared to standing positions, as gravity and weight-bearing conditions influence spinal alignment, intervertebral angles, and vertebral spacing. Prior studies have highlighted these posture-induced changes and their impact on SSMs [50,51]. While our model effectively captures lumbar spine shape variability in AIS patients, its applicability is currently limited to the supine position. Incorporating weight-bearing imaging in future studies could improve model accuracy and generalizability to standing postures. Increasing the sample size, especially for subgroups like male participants, could enhance the precision and applicability of the SSM. While our sample size aligns with similar studies [29], a larger and more balanced dataset would improve the model’s ability to capture variability and better reflect anatomical differences within the AIS population.

We established a lumbar spine SSM of an AIS population and analyzed the effects of age, height, sex, and BMI. A prediction model was used to generate 3D models based on specific demographics. These models can be used to measure anatomical parameters; analyze relationships between anatomical variations, human characteristics, and pathology; and provide a theoretical basis for AIS examination and treatment.

Understanding the shape of the lumbar spine in AIS patients is clinically significant because it directly impacts spinal stability, balance, and function. Accurate lumbar spine modeling could improve the precision of treatment planning, including brace design, surgical interventions, and rehabilitation strategies, thereby enhancing patient outcomes. Furthermore, insights into lumbar spine morphology may help prevent the progression of deformities and optimize long-term management strategies.

All procedures performed for studies involving human participants were in accordance with the ethical standards stated in the 1964 Declaration of Helsinki and its later amendments or comparable ethical standards. Informed consent was obtained from all participants. The sponsors had no involvement in the study design; in the collection, analysis, or interpretation of data; in the writing of the report; or in the decision to submit the article for publication.

• China Jiangsu Province Science and Technology Project (Grant No. BE2023737; Funder ID: 10.13039/501100017547).

• China Zhejiang Province Medicine and Health Science and Technology Project (Grant No. 2019KY610; Funder ID: 10.13039/501100014996).

The authors have no competing interests to declare that are relevant to the content of this article.

This research was approved by the Kunshan Hospital of Jiangsu University, Jiangsu, China (License No. 2022-06-002-K01).

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.