Classical mathematical models for prediction of response to chemotherapy and immunotherapy.

Narmin Ghaffari Laleh,Julia Grajek,Heiko Enderling,Chiara Maria Lavinia Loeffler,Jan Poleszczuk,Christian Trautwein,Jakob Nikolas Kather,Kateřina Staňková,Alexander T Pearson,Hannah Sophie Muti,Douglas A Lauffenburger

doi:10.1371/journal.pcbi.1009822

Abstract

Classical mathematical models of tumor growth have shaped our understanding of cancer and have broad practical implications for treatment scheduling and dosage. However, even the simplest textbook models have been barely validated in real world-data of human patients. In this study, we fitted a range of differential equation models to tumor volume measurements of patients undergoing chemotherapy or cancer immunotherapy for solid tumors. We used a large dataset of 1472 patients with three or more measurements per target lesion, of which 652 patients had six or more data points. We show that the early treatment response shows only moderate correlation with the final treatment response, demonstrating the need for nuanced models. We then perform a head-to-head comparison of six classical models which are widely used in the field: the Exponential, Logistic, Classic Bertalanffy, General Bertalanffy, Classic Gompertz and General Gompertz model. Several models provide a good fit to tumor volume measurements, with the Gompertz model providing the best balance between goodness of fit and number of parameters. Similarly, when fitting to early treatment data, the general Bertalanffy and Gompertz models yield the lowest mean absolute error to forecasted data, indicating that these models could potentially be effective at predicting treatment outcome. In summary, we provide a quantitative benchmark for classical textbook models and state-of-the art models of human tumor growth. We publicly release an anonymized version of our original data, providing the first benchmark set of human tumor growth data for evaluation of mathematical models.

Highlights

The growth of solid tumors and their response to therapy is hard to predict on the level of individual patients
The theoretical foundation of mathematical oncology is provided by six classical mathematical models: the Exponential, Logistic, Classic Bertalanffy, General Bertalanffy, Classic Gompertz and General Gompertz model
We show that two of these models can predict tumor response to immunotherapy and chemotherapy at later time points when trained on early tumor growth dynamics

Summary

Introduction

The growth of solid tumors and their response to therapy is hard to predict on the level of individual patients. [6,7,8] In addition, they have resulted in a number of modeling approaches for cancer treatments including chemotherapy [9,10] and immunotherapy [11,12], improved drug dosage [13,14] and have yielded candidate biomarkers for treatment response [15]. The roots of tumor growth models go back to 1825, when Gompertz published a mathematical model to analyze the population growth [16]. He argued that the number of people alive as a function of their age L(x) declines faster than exponential functions which means that the death rate should be increasing with age. Tumor modeling provides information about the net tumor growth rate, facilitates their comparison among different tumor types [18] and makes it possible to predict the future growth of tumors [19]

Methods

Results

Discussion

Conclusion