A Quantifiable Risk Factor for ACL Injury: Applied Mathematics to Model the Posterolateral Tibial Plateau Surface Geometry
Bibliographic record
Abstract
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.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.005 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.002 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".