A Fractional-Order Machine Learning Framework for Modeling Vertebral Column Pathology and Biomechanical Dynamics

Spinal disorders, such as disk hernia and spondylolisthesis, affect millions worldwide, leading to chronic pain and reduced quality of life due to disruptions in biomechanical alignment. Traditional diagnostic methods often overlook the viscoelastic memory effects in spinal tissues, necessitating advanced models that integrate machine learning with fractional calculus for improved accuracy and interpretability. The research introduces a new fractional-order machine learning system that analyzes vertebral column abnormalities through biomechanical motion analysis by using the University of California, Irvine (UCI) vertebral column dataset. The system selects the best machine learning model from Random Forest (RF), Gradient Boost (GB), XGBoost, Deep Neural Network (DNN), and Voting Ensemble (VE) models to work with Caputo fractional-order differential equations, which simulate spinal tissue viscoelasticity memory effects through pseudo-time analysis of pelvic incidence data. The study demonstrates that GB achieves its highest accuracy at 0.937, and RF attains its top Area Under the Curve (AUC) at 0.949, and the fractional model achieves a weighted Mean Squared Error (MSE) of 0.0017. The optimized parameters showed that the growth rates and coupling coefficients worked in an inhibitory manner. The biological findings demonstrated that patients with higher pelvic incidence and lumbar lordosis variability experienced greater spinal stress, which led to compensatory curvature patterns that: disk hernia and spondylolisthesis development. The results showed that pelvic incidence and sacral slope had the strongest correlation at 0.87, and pelvic incidence and lumbar lordosis angle showed a correlation of 0.74, which informs the fractional-order growth rates where higher-correlated features evolve faster under pathological stress due to their mechanical interdependence. Theoretical proofs establish solution existence and uniqueness and boundedness, and stability, and numerical efficiency emerges from the Adams-Bashforth-Moulton method, while a diagnostic Graphic User Interface (GUI) enables clinical application. The framework enables better spinal disease detection through fractional derivatives, which replicate real biomechanical operations.