Population modeling of tumor growth curves and the reduced Gompertz model improve prediction of the age of experimental tumors.

Cristina Vaghi,Clair Poignard,Sébastien Benzekry,Raphaëlle Fanciullino,Michalis Mastri,Joseph Ciccolini,John M L Ebos,Anne Rodallec,Jonathan P Mochel,Z Agur

doi:10.1371/journal.pcbi.1007178

Abstract

Tumor growth curves are classically modeled by means of ordinary differential equations. In analyzing the Gompertz model several studies have reported a striking correlation between the two parameters of the model, which could be used to reduce the dimensionality and improve predictive power. We analyzed tumor growth kinetics within the statistical framework of nonlinear mixed-effects (population approach). This allowed the simultaneous modeling of tumor dynamics and inter-animal variability. Experimental data comprised three animal models of breast and lung cancers, with 833 measurements in 94 animals. Candidate models of tumor growth included the exponential, logistic and Gompertz models. The exponential and-more notably-logistic models failed to describe the experimental data whereas the Gompertz model generated very good fits. The previously reported population-level correlation between the Gompertz parameters was further confirmed in our analysis (R2 > 0.92 in all groups). Combining this structural correlation with rigorous population parameter estimation, we propose a reduced Gompertz function consisting of a single individual parameter (and one population parameter). Leveraging the population approach using Bayesian inference, we estimated times of tumor initiation using three late measurement timepoints. The reduced Gompertz model was found to exhibit the best results, with drastic improvements when using Bayesian inference as compared to likelihood maximization alone, for both accuracy and precision. Specifically, mean accuracy (prediction error) was 12.2% versus 78% and mean precision (width of the 95% prediction interval) was 15.6 days versus 210 days, for the breast cancer cell line. These results demonstrate the superior predictive power of the reduced Gompertz model, especially when combined with Bayesian estimation. They offer possible clinical perspectives for personalized prediction of the age of a tumor from limited data at diagnosis. The code and data used in our analysis are publicly available at https://github.com/cristinavaghi/plumky.

Highlights

In the era of personalized oncology, mathematical modeling is a valuable tool for quantitative description of physiopathological phenomena [1, 2]
Mathematical models for tumor growth kinetics have been widely used since several decades but mostly fitted to individual or average growth curves
The exponential and the logistic models failed to fit the experimental data while the Gompertz model showed excellent descriptive power

Summary

Introduction

In the era of personalized oncology, mathematical modeling is a valuable tool for quantitative description of physiopathological phenomena [1, 2]. It allows for a better understanding of biological processes and generates useful individual clinical predictions, for instance for personalized dose adaptation in cancer therapeutic menagement [3]. While the etiology of the Gompertz model has been long debated [9], several independent studies have reported a strong and significant correlation between the parameters α and β in either experimental systems [6, 10, 11], or human data [11, 12, 13]. While some authors suga gested this would imply a constant maximal tumor size (given by Vinjeb in (1)) across tumor types within a given species [11], others argued that because of the presence of the exponential function, this so called ‘carrying capacity’ could vary over several orders of magnitude [14]

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