Mathematical explanations for knee osteotomies: "Dear engineer, how would you explain it in a simple way?".

Introduction: Knee osteotomies are effective procedures to treat different deformities and to redistribute the load at the joint level, reducing the risk of wear and, consequently, the need for invasive procedures. Particularly, knee osteotomies are effective in treating early arthritis related to knee deformities in young and active patients with high functional demands, with excellent long-term results. Precise mathematical calculations are imperative during the preoperative phase to achieve tailored and accurate corrections for each patient and avoid complications, but sometimes those formulas are challenging to comprehend and apply.

Methods: Four specific questions regarding controversial topics (planning methods, patellar height, tibial slope, and limb length variation) were formulated. An electronic search was performed on PubMed and Cochrane Library to find articles containing detailed mathematical or trigonometrical explanations. A team of orthopedic surgeons and an engineer summarized the available Literature and mathematical rules, with a final clear mathematical explanation given by the engineer. Wherever the explanation was not available in Literature, it was postulated by the same engineer.

Results: After the exclusion process, five studies were analyzed. For three questions, no studies were found that provided mathematical analyses or explanations. Through independent calculations, it was demonstrated why Dugdale's method underestimates the correction angle compared to Miniaci's method, and it was shown that the variation in patellar height after osteotomy can be predicted using simple formulas. The five included studies examine postoperative variations in limb length and tibial slope, providing formulas applicable in preoperative planning. New formulas were independently computed, using the planned correction angle and preoperatively obtained measurements to predict the studied variations.

Conclusions: There is a strict connection among surgery, planning, and mathematics formulas in knee osteotomies. The aim of this study was to analyze the current literature and provide mathematical and trigonometric explanations to important controversial topics in knee osteotomies. Simple and easy applicable formulas are provided to enhance the accuracy and outcomes of this surgical procedure.