# Baranyi and Roberts Model Formulation

Baranyi and Roberts proposed the following system that governs cell growth:

dq d/ ßmaxq subject to the initial conditions t — 0 : x (0) — xo and q(0) — qo

where x represents the cell concentration and q(t) = Q[t)/Qs, Qs being the Michaelis-Menten constant in the Michaelis-Menten kinetics or the half-saturation

Vadasz and Vadasz derived an autonomous version of the Baranyi and Roberts model defined by eqns - in the form d ix\ (6 + x) dt \xJ (6- x)

or alternatively expressed in terms of the logarithm of the cell concentration in the form

The solution to  or  requires two initial conditions, one for the initial cell concentration, xo, and the second for the initial growth rate, x_ o. Note that once these two initial values are established, their corresponding

max x max values in terms of y for the solution of eqn  can be evaluated in the form

Now it is obvious that somehow the initial condition for the variable q, that is, qo, in the original nonautonomous version of the same model was replaced with initial conditions of initial cell concentration and initial growth rate. A relationship between the two becomes appealing. This relationship was derived by Vadasz and Vadasz in the form qo

[(Xo/Xo) - Mmax (1 - Xo/6)] .o - Mmax ( 1 - ey° /5)]

Now the accurate meaning of the ''physiological state of the cells at inoculation'', qo, becomes clear. It is related to the initial cell concentration, initial growth rate, and the parameters ^max and 6 of the post-inoculation environment, as accurately presented in eqn . We should emphasize that qo is not a measurable quantity while xo (or yo) is. One can evaluate x from experimental data assuming a fairly high sampling frequency by using two subsequent readings of cell count xi and x2 taken at times t1 and t2 and using a central finite difference approximation for the derivative in the form X = (x2 — x1)/(t2 — t1) to evaluate the growth rate at time (t1 + t2)/2. An equivalent forward finite difference can be used to evaluate the initial growth rate in the form xo = (x1 — xo)/t1, by setting to = 0. It is obvious now also that 'the previous history of the cells' are relevant only as far as creating the inoculation conditions in terms of the initial growth rate, xo. The initial cell concentration is usually varied, typically a diluted concentration compared to the cell concentration in the pre-inoculation environment. The only memory that the post-inoculation growth conditions have that the cells were previously grown in a different environment is the initial growth rate, xo. However, the latter does not imply a dependence on the whole 'previous history of the cells', because the latter might have misleading connotations to 'delay-type' of models that do not capture correctly the lag phase. Nevertheless we do not imply that the whole history of the cells is lumped together into the initial growth rate but rather that it is the physiological state of the cells at inoculation that affects the subsequent growth and this growth is independent of the path by which this physiological state was reached (i.e., the pervious history of the cells). One may present the second-order eqn  as an autonomous system of two first-order equations dz dt 