Classical mathematical models for prediction of response to chemotherapy and immunotherapy

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.

Improving Mathematical Models of Cancer through Game-Theoretic Modelling: A Study in Non-Small Cell Lung Cancer

“…Here we compare the fits of six mathematical models (exponential, logistic, classic Bertalanffy, general Bertalanffy, Gompertz and general Gompertz) with that of a novel game-theoretic model. The data were from patients with Non-small Cell Lung Cancer from four clinical trials (NCT01846416 GO28625 FIR, NCT01903993 GO28753 POPLAR, NCT02031458 GO28754 BIRCH, NCT02008227 GO28915 OAK) [24]. For each patient there were measurements of the diameter of a patient's tumor taken over time.…”

“…For each patient there were measurements of the diameter of a patient's tumor taken over time. Recently, Ghaffari Laleh et al (2022) had fitted classic models of tumor growth to this dataset and performed a comparison between them, showing that they can fit the trajectory of tumor growth relatively well [24]. They can provide a good estimation of the future response based on early treatment data.…”

“…They can provide a good estimation of the future response based on early treatment data. However, the models' predictive capabilities fail when tumor growth is not monotonic, i.e., for tumors that are neither continuously growing nor shrinking [24]. Yet, many tumor growth dynamics show a U-shape where therapy is initially effective but loses efficacy as the cancer cell population evolves resistance.…”

“…The model we utilize here has been proposed in [40] and further developed in [47]. Here, we ask whether this model outperforms the classical models of tumor growth when fitted with the NSCLC patient data [24]. By fitting the model to the data, we can additionally ask whether different treatment protocols would possibly have been better for some of the patients.…”

“…This paper is organized as follows. In Section 2, we introduce the model and fit it with a subset of the anonymized data used in [24]. The subset of patients with metastatic NSCLC were treated with either immune checkpoint inhibition or chemotherapy.…”