Parameter estimation in the Montijano-Bergues-Bory-Gompertz stochastic model for unperturbed tumor growth

Different sources of noises endogenous and exogenous to the cancer are involved in its stochastic growth. The aim of this study is to propose the stochastic version of Montijano-Bergues-Bory-Gompertz equation for the unperturbed tumor growth kinetics. The maximum likelihood estimators for the intrinsic tumor growth rate and the growth decelerating factor, and their respective discrete time approximations were analytically calculated. Different simulations of the deterministic and stochastic of this equation were made for different values of their respective parameters. Limit conditions for the average diffusion coefficient and the growth decelerating factor were established. The tumor volume at the infinite was calculated for several values of parameters of the stochastic Montijano-Bergues-Bory-Gompertz equation. Furthermore, descriptive statistic for the maximum likelihood estimators of the intrinsic tumor growth rate was computed for several parameters of this equation. The results evidenced that solid tumors there are for values of the average diffusion coefficient and the growth decelerating factor less than their respective limit values. The transition between avascular and vascular phases of the unperturbed tumor growth kinetics was revealed in the plot of the discrete time approximation for the maximum likelihood estimator of the growth decelerating factor versus the discrete time approximation for the maximum likelihood estimator of the intrinsic tumor growth rate. These results were connected with different findings in the literature. In conclusion, the stochastic Montijano-Bergues-Bory-Gompertz equation may be applied in the experiment to describe the unperturbed tumor growth kinetics, as previously demonstrated for its deterministic version, in order to estimate the parameters of this equation and their connection with processes involved in the growth, progression and metastasis of unperturbed solid tumors. Author summaryIn order to comprehend the unperturbed tumor growth, we investigate a new mathematical model called the stochastic Montijano-Bergues-Bory-Gompertz equation. This study is made based on the ideas of Ferrante et al. and the deterministic version of the Montijano-Bergues-Bory-Gompertz equation. By applying this stochastic equation, we aim to provide valuable insights into how tumors grow and spread throughout the body. We focus on estimating key parameters that are essential for understanding the dynamic processes involved in the unperturbed tumor behavior. Our findings may help researchers to understand the stochastic nature of the unperturbed tumor growth; know the existence of transitions in the unperturbed tumor growth kinetics, probably between avascular and vascular phases; and reveal the values of the model parameters for which the solid tumor is functional, non-functional or does not exist. These aspects may be relevant to propose an individualized anticancer therapy aimed at minimizing the different noise sources that occur during the unperturbed tumor growth. Overall, this study contributes to our ongoing efforts to improve cancer treatment strategies and enhance patient outcomes by fostering a better understanding of tumor biology.