A new formula for calculating normal tissue complication probability

To facilitate the use of quantitative modeling of biological effects in treatment planning by introducing a simpler function equivalent to the Lyman formula for calculating normal tissue complication probability (NTCP).

We first provide an approximation of the Lyman-Kutcher-Burman (LKB) formula using three parameters (n, m, TD₅₀) as a function of equivalent uniform dose (EUD). The parameters for the new formula are defined in terms of the Lyman model's m and TD₅₀. Conversely, m and TD₅₀ are expressed in terms of the parameters of the new equation. The role of the Lyman volume-effect parameter n remains unchanged from its role in the Lyman model.

The new formula, which exhibits a sigmoidal shape, demonstrates symmetry about TD₅₀, akin to the LKB model. The difference in NTCP between the two formulas is less than 0.1%. The parameters (n, m, TD₅₀) are preserved through rigorous mathematical deduction and have been recalibrated to the tolerance data of Emani et al. using the proposed formula. This new model provides a better fit to these data than the model by Burman et al., which was fitted "by eye" rather than using statistical methods.

We have developed a formula that represents NTCP as a function of EUD, which proves to be potentially useful. The parameters derived in this study are mathematically robust and offer a superior fit to the data compared to previous efforts. Additionally, the new model fits brain data as well as, if not better than, the LKB model.