Mathematical immunology: processes, models and data assimilation

The immune system is a complex multiscale multiphysical object. Understanding its functioning in the frame of systemic analysis implies the use of mathematical modelling, formulation of data consistency criterion, estimation of parameters, uncertainty analysis, and optimal model selection. In this work, we present some promising approaches to modelling the multi-physics immune processes, i.e., cell migration in lymph nodes (LN), lymph flow, homeostatic regulation of immune responses in chronic infections.

To describe the spatial-temporal dynamics of immune responses in lymph LN, we propose a model of lymphocyte migration, based on the second Newton’s law and considering three kinds of forces. The empirical distributions of three lymphocytes motility characteristics were used for model calibration using the Kolmogorov–Smirnov metric.

Prediction of lymph flow in a lymph node requires costly computations, due to diversity of sizes, forms, inner structure of LNs and boundary conditions. We proposed an approach to lymph flow modelling based on replacing the full-fledged computational physics-based model with an artificial neural network (ANN), trained on the set of pre-formed results computed using an initial mechanistic model. The ANN-based model reduces the computational time for some lymph flow characteristics by four orders of magnitude.

Calibration of Marchuk–Petrov model of antiviral immune response for SARS-CoV-2 infection was performed. To this end, we used previously published data on the viral load kinetics in nasopharynx of volunteers, and data on the observed ranges of interferon, antibodies and CTLs in the blood. The parameters, which have the most significant impact at different stages of infection process, were identified.

Inhibition of immune mechanisms, e.g., T cell exhaustion, is a distinctive feature of chronic viral infections and malignant diseases. We propose a mathematical model for the studies of regulation parameters of four exhausted T cell subsets in order to examine the balance of their proliferation and differentiation determined by interaction with SIRPa⁺ PD-L1⁺ and XCR⁺1 dendritic cells. The model parameters are evaluated, in order to study the reinvigoration effect of aPD-L1 therapy on the homeostasis of exhausted cells.