Proper likelihood functions for parameter estimation in S-shaped models of unperturbed tumor growth

The analysis of unperturbed tumor growth kinetics, particularly the estimation of parameters for S-shaped equations used to describe growth, requires an appropriate likelihood function that accounts for the increasing error in solid tumor measurements as tumor size grows over time. This study aims to propose suitable likelihood functions for parameter estimation in S-shaped models of unperturbed tumor growth. Five different likelihood functions are evaluated and compared using three Bayesian criteria (the Bayesian Information Criterion, Deviance Information Criterion, and Bayes Factor) along with hypothesis tests on residuals. These functions are applied to fit data from unperturbed Ehrlich, fibrosarcoma Sa-37, and F3II tumors using the Gompertz equation, though they are generalizable to other S-shaped growth models for solid tumors or analogous systems (e.g., microorganisms, viruses). Results indicate that error models with tumor volume-dependent dispersion outperform standard constant-variance models in capturing the variability of tumor measurements, particularly the Thres model, which provides interpretable parameters for tumor growth. Additionally, constant-variance models, such as those assuming a normal error distribution, remain valuable as complementary benchmarks in analysis. It is concluded that models incorporating volume-dependent dispersion are preferred for accurate and clinically meaningful tumor growth modeling, whereas constant-dispersion models serve as useful complements for consistency and historical comparability.