Mathematical Modeling for Slide Tracheoplasty

The objective of this study is to create a mathematical model and calculate precise cut lengths for slide tracheoplasty (ST) to achieve better postoperative results. This study also is a pioneer in a novel field where mathematics and medicine intersect and is aimed to be used for achieving precise preoperative planning for many risky medical procedures.In ST, the trachea is divided into two segments with two longitudinal cuts and one oblique cut. These segments are then slid onto each other to widen the lumen of the stenotic segment. The postoperative stenotic segment diameter is directly determined by the longitudinal incisions and the subsequent oblique incision. We propose a mathematical model to estimate tracheal cut lengths preoperatively to achieve the desired tracheal lumen diameter.

The oblique cut creates an ellipse. After sliding, the suture line becomes the perimeter of another ellipse, equal to the sum of the initial ellipse perimeter and twice the longitudinal cut length. Longitudinal cuts determine the angle of the oblique cut and lengthen the perimeter of the final ellipse, the horizontal projection of which is a circle with the desired tracheal diameter. This mathematical model uses a novel perimeter equation, two geometric constraints, and a novel tissue area constraint. Exact solutions are found. Using this model, the longitudinal cut length to achieve the target diameter is determined preoperatively.

The mathematical model provides a closed-form solution to calculate the necessary longitudinal cut lengths to achieve a desired postoperative tracheal diameter. It integrates geometric and tissue area constraints to accurately predict the final lumen size. A practical implementation of the model was developed in Microsoft Excel, allowing for user-friendly, real-time calculation of cut lengths with adjustable inputs for preoperative tracheal dimensions and target diameters.

We present a novel mathematical model for preoperative planning in ST. By precisely linking incision geometry to final tracheal diameter, this tool supports more accurate and individualized surgical planning. The model offers a promising example of how mathematical approaches can enhance precision in complex surgical procedures and improve clinical outcomes.