In this paper, we present a systematic approach for obtaining qualitatively and
quantitatively correct mathematical models of some biological phenomena with
time-lags. Features of our approach are the development of a hierarchy of
related models and the estimation of parameter values, along with their
non-linear biases and standard deviations, for sets of experimental data.
We demonstrate our method of solving parameter estimation problems for neutral
delay differential equations by analyzing some models of cell growth that
incorporate a time-lag in the cell division phase. We show that these models are
more consistent with certain reported data than the classic exponential growth
model. Although the exponential growth model provides estimates of some of the
growth characteristics, such as the population-doubling time, the time-lag
growth models can additionally provide estimates of: (i) the fraction of cells
that are dividing, (ii) the rate of commitment of cells to cell division,
(iii) the initial distribution of cells in the cell cycle, and (iv) the
degree of synchronization of cells in the (initial) cell population.