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The progression of a cell population where each individual is characterized by the value of an internal variable varying with time (e.g. size, mass, and protein concentration) is typically modelled by a population balance equation, a first‐order linear hyperbolic partial differential equation. The characteristics described by internal variables usually vary monotonically with the passage of time. A particular difficulty appears when the characteristic curves exhibit different slopes from each other and therefore cross each other at certain times. In particular, such a crossing phenomenon occurs during T‐cell immune response when the concentrations of protein expressions depend upon each other and also when some global protein (e.g. Interleukin signals) that is shared by all T‐cells is involved. At these crossing points, the linear advection equation is not possible by using the hyperbolic conservation laws in a classical way. Therefore, a new transport method is introduced in this article that is able to find the population density function for such processes. A multi‐scale mathematical modelling framework is employed. At the first scale, two processes, the activation and reaction terms (assimilated as nucleation and growth in particulate processes), are studied as independent processes. At the second scale, the dynamic variation in the population density of activated T‐cells is investigated in the presence of activation and reaction terms. The newly‐developed transport method is shown to work in the case of crossing characteristics and to provide a smooth solution at the crossing points in contrast to the classical numerical techniques.

Population balances in case of crossing characteristic curves: Application to T‐cells immune response