## Fedbatch Model and Problem Formulation

A mathematical model for a fed-batch culture of hybridoma cells [24] is employed in this study. The details are given in Appendix A. The mass balance equations for the system in fed-batch mode of a multi-feed case and the problem formulation are presented below.

The multi-feed case, which involves two separate feeds Fi and F2 for glucose and glutamine respectively, is reformulated as follows:

(Glnin — Gln) Fl+F2 — qginXv qiacXv — Fl+F Lac (3.1)

q X — Fl+F2 Amm Hamm^v y Jriitiiti qMAbXv — MAb

where, Xv, Glc, Gln, Lac, Amm and MAb are respectively the concentrations in viable cells, glucose, glutamine, lactate, ammonia and MAb; V is the fermentor volume and F the volumetric feed rate; Glcin and Glnin are the concentrations of glucose and glutamine in the feed stream, respectively. The parameter values and kinetic expressions are given in Appendix A. In this work, there are two problems that need to be solved:

(1) The first problem is to estimate all sixteen parameters, Mmax, kdmax,

Yxv/glc, Yxv/gln, mglc, kmglc, Kglc, Kgln, a0, ft, kdlac, kdamm, kdgln,

Ylac/glc, and Yamm/gln, from the measured values of Xv, Glc, Gln, Lac, Amm, MAb and V at the beginning of the fed-batch fermentation fed with a deliberate single-feed stream. The structure of the kinetic model used for the study is known (as shown in Appendix A). The identification problem is to minimize the error between actual values of these state variables and their estimated values predicted from the estimated parameters. The objective function is as follows:

J (to,tN) = min \\I (to),I(ti) ■■■,I(tN )ll (3.2)

dX. dt dGlc dt dGln dt dLac dt dAmm dt dMAb dt dV dt with

I(ti) = min \\Xest(ti) - X(t<)H, i = 1, 2, ■■■,N (3.3)

where, P, Xest(ti) and X(ti) are the estimated parameters, the estimated state variables and the actual (measured) state variables, respectively; \\ ■ \\ is the notation for L2 norm; N is the total number of intervals of the reaction time.

(2) The second problem is to determine how the glucose and glutamine should be fed to the fermentor in order to drive MAb to the maximum, for a set of initial conditions and constraints. The criterion used is the total amount of MAb obtained at the end of the fed-batch fermentation:

The constraints on the control variable and the culture volume are: 0 < F < 0.5L/d

The following initial culture conditions and feed concentrations have been used:

Xv (0) =2.0 x 108cells/L Glc(0) = 25mM Gln(0) = 4mM

The above mathematical models and initial conditions have been used to generate a 'reality' for testing the schemes proposed in the study.