In recent years an increasing interest is registered in the direction of developing techniques to combine experimental data and mathematical models, in order to produce systems, i.e., in silico models, whose solutions could reproduce and predict experimental outcomes. Indeed, the success of informed models is mainly due to the consistent improvements in computational abilities of the machines and in imaging techniques that allow a wider access to high spatial and temporal resolution data. Here we present an interdisciplinary work in the framework of Organs-on-chip (OoC) technology, and, more precisely, in Canceron-Chip (CoC) technology.

The present work is motivated by the development of a mathematical model mimicking the mechanisms observed in lab-on-chip experiments, made to reproduce on microfluidic chips the in vivo reality. Here we consider the Cancer-on-Chip experiment where tumor cells are treated with chemotherapy drug and secrete chemical signals in the environment attracting multiple immune cell species. The in silico model here proposed goes towards the construction of a "digital twin" of the experimental immune cells in the chip environment to better understand the complex mechanisms of immunosurveillance. To this aim, we develop a tumor-immune microfluidic hybrid PDE-ODE model to describe the concentration of chemicals in the Cancer-on-Chip environment and immune cells migration. The development of a trustable simulation algorithm, able to reproduce the immunocompetent dynamics observed in the chip, requires an efficient tool for the calibration of the model parameters. In this respect, the present paper represents a first methodological work to test the feasibility and the soundness of the calibration technique here proposed, based on a multidimensional spline interpolation technique for the time-varying velocity field surfaces obtained from cell trajectories.