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In biotechnology and biomedicine reliable models of cell proliferation kinetics need to capture the relevant phenomena taking place during the mitotic cycle. To this aim, a novel mathematical model helpful to investigate the intrinsic kinetics of in vitro culture of adherent cells up to confluence is proposed in this work. Specifically, the attention is focused on the simulation of proliferation (increase of cell number) and maturation (increase of cell size and DNA content) till contact inhibition eventually takes place inside a Petri dish. Accordingly, the proposed model is based on a population balance (PB) approach that allows one to quantitatively describe cell cycle progression through the different phases the cells of the entire population experienced during their own life. In particular, the proposed model has been developed as a 2D, multi-staged, and unstructured PB, by considering a different sub-population of cells for any single phase of the cell cycle. These sub-populations are discriminated through cellular volume and DNA content, that both increase during the mitotic cycle. The adopted mathematical expressions of the transition rates between two subsequent phases and the temporal increase of cell volume and DNA content are thoroughly analyzed and discussed with respect to those ones available in the literature. Specifically, the corresponding uncertainties and pitfalls are pointed out, by also taking into account the difficulties and the limitations involved in the quantitative measurements currently practicable for these biological systems. A novel mathematical expression for contact inhibition in line with the PB model developed is also formulated, along with a proper comparison between modeled and measurable DNA distributions. The strategy for a reliable, independent tuning of the adjustable parameters involved in the proposed model along with its numerical solution is outlined in Part II of this work, where it is also shown how it can be profitably used to gain a deeper insight into the phenomena involved during cell cultivation under microgravity conditions.

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