Melanoma is a type of skin cancer characterized by its high aggressiveness and potential metastasis. These characteristics make melanoma a major clinical challenge and highlight the need for quantitative approaches capable of describing accurately its dynamic growth. In this respect, mathematical modeling has emerged as a valuable tool to help the interpretation of longitudinal tumor data, enabling a deeper understanding of tumor evolution under different biological conditions and supporting the evaluation of therapeutic strategies. During the last few years, tumor growth models have been widely applied to characterize tumor behavior and to assess treatment responses. Recently, mathematical approaches have been increasingly employed to describe tumor growth dynamics and to evaluate the effectiveness of different therapeutic strategies.

Mathematical models calibration and selection based on ordinary differential equations using tumor volume data, selected model predictive capability evaluation, and perform computational simulations of different therapeutic conditions.

In this study, mathematical models of exponential, logistic, and Gompertz growth were employed to describe melanoma progression using experimental data from untreated tumors. Model parameters were estimated through nonlinear adjustment techniques based on the Least Squares method, with the ordinary differential equations numerically solved using the Runge-Kutta method. The most appropriate model selection was based on two criteria: the Intraclass Correlation Coefficient (ICC), which assesses the reliability of the estimated data with the actual data, and the Bayesian Information Criterion (BIC), which balances adjustment quality and model complexity.

The ICC and BIC indices, with the parameters for each equation, were obtained from a specific data set showing the Gompertz model as the best result. Based on this result, the Gompertz model was selected as the basic model for the second stage of the study, and an additional term representing the effect of treatment on tumor growth was incorporated. This modification, combined with the parameter calibration process, enables the model to be used as a predictable tool able to predict tumor behavior under different therapeutic conditions.

Under untreated conditions, the Gompertz model provided the most adequate representation of melanoma growth among the evaluated models. Further work focuses on extending the selected basic model by incorporating a mechanistic treatment-effect component to capture tumor response to CAR-T cell immunotherapy and to enable quantitative assessment of therapy-driven deviations from natural growth.