The progression of a cell population where each individual is characterized
by the value of an internal variable varying with time (e.g. size, weight, and
protein concentration) is typically modeled 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 crossing phenomenon occurs during T-cells
immune response when the concentrations of protein expressions depend upon each
other and also when some global protein (e.g. Interleukin signals) is also
involved which is shared by all T-cells. At these crossing points, the linear
advection equation is not possible by using the classical way of hyperbolic
conservation laws. Therefore, a new Transport Method is introduced in this
article which allowed us to find the population density function for such
processes. The newly developed Transport method (TM) 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.Comment: 18 pages, 10 figure
