A Quantifiable Risk Factor for ACL Injury: Applied Mathematics to Model the Posterolateral Tibial Plateau Surface Geometry

BACKGROUND: The mechanism for traumatic ruptures of the native anterior cruciate ligament (ACL) is frequently a noncontact injury involving a valgus moment with internal rotation of the tibia. The abnormal rotation and translation of the lateral femoral condyle posteroinferiorly relative to the lateral tibial plateau is thought to be related to the geometry of the tibial plateau. PURPOSE/HYPOTHESIS: The purpose of the study was to mathematically model the posterior tibial plateau geometry in patients with ACL injuries and compare it with that of matched controls. The hypothesis was that increased convexity and steepness of the posterior aspect of the lateral plateau would subject knees to higher forces, leading to a potentially higher risk of ACL injury. STUDY DESIGN: Cross-sectional study; Level of evidence, 3. METHODS: We mathematically modeled the posterior curvature of the lateral tibial plateau in 64 patients with ACL injuries and 68 matched controls. Using sagittal magnetic resonance imaging scans of the knee, points on the articular cartilage of the posterolateral tibial plateau were selected and curve-fitted to a power function (y = a × x(n)). For coefficient a and coefficient n, both variables modulated the shape of the curve, where a larger magnitude represented an increase in slope steepness. Groups were compared using a Mann-Whitney test and α < .05. RESULTS: There was a significant difference in surface geometry between the patients with ACL injuries and matched controls. The equation coefficients were significantly larger in the patients with ACL injuries: coefficient a (ACL injury, 0.9 vs control, 0.68; P < .0001) and coefficient n (ACL injury, 0.34 vs control, 0.30; P = .07). For coefficient a, there was a 78.9% sensitivity, 77.5% specificity, and odds ratio of 12.6 (95% CI, 5.5-29.0) for ACL injury using a cutoff coefficient a = .78. CONCLUSION: Patients with ACL injuries had a significantly greater posterolateral plateau slope. The steeper drop off may play a role in higher anterior translation forces, coupled with internal rotation torques on the knee in noncontact injury, which could increase ACL strain and predispose to ACL injury.