Chaos theory modeling improves olecranodiaphyseal angle prediction from proximal ulnar dorsal angulation in healthy elbows

To evaluate the relationship between Proximal Ulnar Dorsal Angulation (PUDA) and Olecranodiaphyseal Angle (ODA) in healthy elbows using both linear regression and advanced chaos theory approaches, analyzing the effects of age, sex, and side parameters. In this cross-sectional study, 295 healthy elbows (178 male, 117 female; 130 right, 165 left) were evaluated with standard radiographs. PUDA, Varus Angle (VA), and ODA measurements were performed by two independent observers. Linear regression analysis and chaos theory-based nonlinear modeling were used to establish mathematical relationships between PUDA and ODA. Phase space reconstruction, fractal dimension analysis, Lyapunov exponent calculation, and strange attractor identification were performed to characterize the underlying dynamical system. Linear analysis revealed an inverse relationship between PUDA and ODA (regression coefficient β = -0.340, significance level p < 0.001). However, chaos theory analysis uncovered complex nonlinear dynamics with a fractal attractor structure (correlation dimension D₂ = 2.34 ± 0.12, indicating non-integer dimensional geometry) and positive Lyapunov exponent (λ₁ = 0.127 ± 0.043, confirming sensitive dependence on initial conditions characteristic of chaotic behavior). The chaos-based local linear model achieved superior prediction accuracy (R² = 0.758, RMSE = 2.53) compared to linear regression (R² = 0.210, RMSE = 4.63), representing a 3.6-fold improvement. Bifurcation analysis identified critical PUDA threshold values where system behavior changed dramatically. Males exhibited higher dimensional complexity (correlation dimension D₂ = 2.51 ± 0.18) compared to females (D₂ = 2.14 ± 0.21, significance level p = 0.032), indicating that male proximal ulnar geometry is governed by more complex dynamical interactions. The chaotic dynamics underlying PUDA-ODA relationships provide superior predictive capability compared to traditional linear models. This chaos theory-based approach offers clinicians dramatically improved accuracy for estimating normal ODA values in complex elbow injuries, with precise predictions (± 2°) improving from 34.2 to 67.8% of cases. The identification of strange attractors and bifurcation points reveals that small variations in PUDA measurements can lead to dramatically different ODA predictions, emphasizing the critical importance of measurement precision in surgical planning.