Extending the hyper‐logistic model to the random setting: New theoretical results with real‐world applications

Abstract

We develop a full randomization of the classical hyper‐logistic growth model by obtaining closed‐form expressions for relevant quantities of interest, such as the first probability density function of its solution, the time until a given fixed population is reached, and the population at the inflection point. These results are obtained under very general hypotheses on the distributions of the random model parameters by taking extensive advantage of the so‐called random variable transformation method. To illustrate the practical implications of our findings, we apply them to model the growth of multicellular tumor spheroids using empirical data. In this context, we explore two methodologies—the Bayesian approach and the random least mean square method—aimed at effectively addressing the challenge of assigning appropriate distributions to model parameters. This ensures that probabilistic fits accurately capture the inherent uncertainties of tumor growth dynamics. Finally, we notably show that the results obtained using both approaches in the randomized hyper‐logistic model align closely with each other, surpassing those yielded by the randomized logistic model.