Immunotherapy has gained great momentum with chimeric antigen receptor T cell (CAR-T) therapy, in which patient’s T lymphocytes are genetically manipulated to recognize tumor-specific antigens, increasing tumor elimination efficiency. In recent years, CAR-T cell immunotherapy for hematological malignancies achieved a great response rate in patients and is a very promising therapy for several other malignancies. Each new CAR design requires a preclinical proof-of-concept experiment using immunodeficient mouse models. The absence of a functional immune system in these mice makes them simple and suitable for use as mathematical models. In this work, we develop a three-population mathematical model to describe tumor response to CAR-T cell immunotherapy in immunodeficient mouse models, encompassing interactions between a non-solid tumor and CAR-T cells (effector and long-term memory). We account for several phenomena, such as tumor-induced immunosuppression, memory pool formation, and conversion of memory into effector CAR-T cells in the presence of new tumor cells. Individual donor and tumor specificities are considered uncertainties in the model parameters. Our model is able to reproduce several CAR-T cell immunotherapy scenarios, with different CAR receptors and tumor targets reported in the literature. We found that therapy effectiveness mostly depends on specific parameters such as the differentiation of effector to memory CAR-T cells, CAR-T cytotoxic capacity, tumor growth rate, and tumor-induced immunosuppression. In summary, our model can contribute to reducing and optimizing the number of in vivo experiments with in silico tests to select specific scenarios that could be tested in experimental research. Such an in silico laboratory is an easy-to-run open-source simulator, built on a Shiny R-based platform called CARTmath. It contains the results of this manuscript as examples and documentation. The developed model together with the CARTmath platform have potential use in assessing different CAR-T cell immunotherapy protocols and its associated efficacy, becoming an accessory for in silico trials.

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Machine learning methods have revolutionized studies in several areas of knowledge, helping to understand and extract information from experimental data. Recently, these data-driven methods have also been used to discover structures of mathematical models. The sparse identification of nonlinear dynamics (SINDy) method has been proposed with the aim of identifying nonlinear dynamical systems, assuming that the equations have only a few important terms that govern the dynamics. By defining a library of possible terms, the SINDy approach solves a sparse regression problem by eliminating terms whose coefficients are smaller than a threshold. However, the choice of this threshold is decisive for the correct identification of the model structure. In this work, we build on the SINDy method by integrating it with a global sensitivity analysis (SA) technique that allows to hierarchize terms according to their importance in relation to the desired quantity of interest, thus circumventing the need to define the SINDy threshold. The proposed SINDy-SA framework also includes the formulation of different experimental settings, recalibration of each identified model, and the use of model selection techniques to select the best and most parsimonious model. We investigate the use of the proposed SINDy-SA framework in a variety of applications. We also compare the results against the original SINDy method. The results demonstrate that the SINDy-SA framework is a promising methodology to accurately identify interpretable data-driven models. Supplementary Information The online version contains supplementary material available at 10.1007/s11071-022-07755-2.

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Mathematical and computational modeling have been increasingly applied in many areas of cancer research, aiming to improve the understanding of tumorigenic mechanisms and to suggest more effective therapy protocols. The mathematical description of the tumor growth dynamics is often made using the exponential, logistic, and Gompertz models. However, recent literature has suggested that the Allee effect may play an important role in the early stages of tumor dynamics, including cancer relapse and metastasis. For a model to provide reliable predictions, it is necessary to have a rigorous evaluation of the uncertainty inherent in the modeling process. In this work, our main objective is to show how a model framework that integrates sensitivity analysis, model calibration, and model selection techniques can improve and systematically characterize model and data uncertainties. We investigate five distinct models with different complexities, which encompass the exponential, logistic, Gompertz, and weak and strong Allee effects dynamics. Using tumor growth data published in the literature, we perform a global sensitivity analysis, apply a Bayesian framework for parameter inference, evaluate the associated sensitivity matrices, and use different information criteria for model selection (First- and Second-Order Akaike Information Criteria and Bayesian Information Criterion). We show that such a wider methodology allows having a more detailed picture of each model assumption and uncertainty, calibration reliability, ultimately improving tumor mathematical description. The used in vivo data suggested the existence of both a competitive effect among tumor cells and a weak Allee effect in the growth dynamics. The proposed model framework highlights the need for more detailed experimental studies on the influence of the Allee effect on the analyzed cancer scenario.