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

Abstract

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 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, which are termed as the activation and reaction terms (assimilated as nucleation and growth in chemical 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 and to provide a smooth solution at the crossing points in contrast to the classical PDF techniques.2

Mathematical modelling of T‐cells activation dynamics for CD3 molecules recycling process

“…when an extracellular protein like IL-2 binds to its receptor CD25. [36,37] DISCLOSURE The authors declare that there is no conflict of interests regarding the publication of this paper.…”

“…concentration, behaves nonmonotonically during the course of time while the solution curves cross each other at least once. [36] The TM formulation for crossing curves is described below in more detail.…”

“…This is a direct approach to find the population density function which is very useful when the internal variable, i.e. concentration, behaves non-monotonically during the course of time while the solution curves cross each other at least once [36]. The TM formulation for crossing curves is described below in more details.…”

“…n ( c , t ) |Δ c |. This implies that n ( c , t ) |Δ c | = B ( τ ( c , t ))Δ τ . Moreover, without any loss of generality, the activation time step can be chosen as Δ τ = Δ t .…”

“…For such cases, techniques like the transport method help to solve the problem. [36,37] In the case of monotonic characteristic curves c ¼ c(t,t) (without crossing), the conservation equation can be written as a scalar transport equation as defined in Equation (8), with the concentration c as an internal variable. However, for the crossing problem, a fixed grid is required for concentration c in order to sum up the population at the crossing points.…”