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 text-book 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 chemo-therapy 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 re-sponse, 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 pro-vide 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

“…In general, we obtain a lower mean absolute error with the game-theoretic model than with the ones used in [12] (Figure 6). Of particular interest is the case of U-shaped tumors, that could not be well predicted by any of the classic ODE models explored in [12].…”

“…We have shown that metastatic non-small cell lung cancer data are more accurately fitted with game-theoretical models which allows for the evolution of therapy resistance than with classic ODE models. Especially tumors which exhibit a U-shape in their trend were fitted well by the game-theoretical models, while such trends could not be fitted at all with previous models, as stated in [12]. Furthermore, modeling resistance as an evolutionary trait has been shown to have a relevant impact on the results and thus, these types of models should be applied more often to further explore their relevance to clinical outcomes across many different cancer types.…”

“…In Section 3, we evaluate the results obtained, as well as the accuracy of the model in predicting the future dynamics of the tumor. We compare our results with those given by the classical population growth models explored in [12]. Moreover, we show how our model can be used to design alternative therapies for optimizing any pre-specified treatment goals.…”

“…We used data sets from four different clinical trials, namely the four trials from [12] which correspond to NSCLC. All the data correspond to patients with metastatic NSCLC treated with Atezolizumab (previously known as MPDL3280A), which is an immune checkpoint inhibitor directed against the Programmed Death Ligand 1 (PD-L1), or with Docetaxel, a chemotherapeutic agent.…”

“…The model we utilize here has been proposed in [16] and further studied in [19]. We will verify whether this mode outperforms the classical models of tumor growth when fitted with the NSCLC patient data t [12]. After fitting the model to the data, we analyze whether different treatment protocols would have been better for some of the patients, This application illustrates an application of Stackelberg evolutionary game theory.…”