A Novel Mathematical Model of the Sagittal Spine

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Abstract
Development of a mathematical model with application to a cohort of healthy volunteers. To derive a smooth mathematical function representing the sagittal spinal curve from individual vertebral elements. To generate normative data using this model. Current concepts of spinal sagittal balance center on the C7 plumb line. While elegant in its ease of use, this method oversimplifies the true complexity of the spine. If the spinal curve could be expressed as a smooth mathematical function, the ability to analyze sagittal balance would be greatly enhanced. Lateral full-length radiographs of 18 normal volunteers were examined. The posterosuperior aspect of each vertebral body was chosen as a representative point for the spinal sagittal curve. A cubic spline function was derived from these points. From this function, the area under the curve (AUC) and average sagittal positions of the thoracic, lumbar, and thoracolumbar segments were calculated. Assuming an average vertebral column height of 60 cm with anterior being positive, the average position of the posterosuperior aspect of C7 relative to S1 was -2.8 cm (+/-3.0). The average thoracolumbar AUC was -161 cm2 (+/-83). The average lumbar and thoracic AUCs were 0 cm2 (+/-17) and -157 cm2 (+/-68), respectively. The average sagittal position over the thoracolumbar curve was -3.3 cm (+/-1.7). The average sagittal positions of the lumbar and thoracic subcurves were 0.1 cm (+/-1.1) and -5.1 cm (+/-2.2), respectively. Intraobserver and interobserver reliabilities were excellent. A mathematical model of the sagittal spine that retains the spine's segmental nuances was derived using cubic spline interpolation. The average sagittal position of the thoracolumbar spine, a calculation based on the AUC, is a less variable measure of sagittal balance than the C7 plumb line. The model and normative data generated from it will allow more insightful investigations of spinal deformity and more quantitative evaluations of corrective outcomes.