Comparison of different implementations of lymphocyte proliferation in QSP models of immune response

Objectives: To summarize conventional approaches and propose a new one of lymphocyte proliferation description in mathematical models. To compare the different implementations of cell proliferation to characterize pros and cons of their use in large scale QSP models. Results: Three different approaches to describe rate law of lymphocyte proliferation were considered: (1) linear proliferation, (2) saturable proliferation, (3) generation dependent proliferation (GDP). Two types of models imitating in vitro and in vivo conditions and describing lymphocyte proliferation, death and influx (for in vivo only) were constructed. Analytical expressions of model variables at steady states and their stability were studied. Concept of lymphocyte generation was introduced and rate laws GDP were derived via convolution of infinite ODE system describing dynamics of lymphocyte of different generations. It was found that conventional proliferation rate laws (1) and (2) do not allow to describe bell-shaped dependence of cell number on time which might be observed in vitro. Implementation of rate law (1) allows to observe stable steady state only if rate constant of degradation is larger than that of proliferation. Rate law (3) enables us to describe both bell-shaped dependence in vitro and stable positive steady state in vivo at any parameter values.