  In most biological studies and processes, cell proliferation and population
dynamics play an essential role. Due to this ubiquity, a multitude of
mathematical models has been developed to describe these processes. While the
simplest models only consider the size of the overall populations, others take
division numbers and labeling of the cells into account. In this work, we
present a modeling and computational framework for proliferating cell
population undergoing symmetric cell division. In contrast to existing models,
the proposed model incorporates both, the discrete age structure and continuous
label dynamics. Thus, it allows for the consideration of division number
dependent parameters as well as the direct comparison of the model prediction
with labeling experiments, e.g., performed with Carboxyfluorescein succinimidyl
ester (CFSE). We prove that under mild assumptions the resulting system of
coupled partial differential equations (PDEs) can be decomposed into a system
of ordinary differential equations (ODEs) and a set of decoupled PDEs, which
reduces the computational effort drastically. Furthermore, the PDEs are solved
analytically and the ODE system is truncated, which allows for the prediction
of the label distribution of complex systems using a low-dimensional system of
ODEs. In addition to modeling of labeling dynamics, we link the label-induced
fluorescence to the measure fluorescence which includes autofluorescence. For
the resulting numerically challenging convolution integral, we provide an
analytical approximation. This is illustrated by modeling and simulating a
proliferating population with division number dependent proliferation rate.
