## Proposed Model and Problem Formulation

A seventh order nonlinear kinetic model for a fed-batch culture of hybridoma cells  is used in this work. The mass balance equations of a fed-batch fermentation for a single-feed case are:

^X. = (M — kd)Xv — FXv Gt° = (Glcin — Glc) -y — qglcXv = (Glnin — CMn)E — qginXv

dAmm = q x — E Amm dt — qammXv yAmm dMdAb = qMAbXv — E MAb = F

dV dt 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; Both glucose and glutamine concentrations are used to describe the specific growth rate, /. The cell death rate, kd, is governed by lactate, ammonia and glutamine concentrations. The specific MAb production rate, qMAb, is estimated using a variable yield coefficient model related to the physiological state of the culture through the specific growth rate. The parameter values and detailed kinetic expressions for the specific rates, qgic, qgin, qiac, qamm and qMAb are presented in Appendix A.

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

dX. dt dGl

dt V Glnin v Gln qgln

idAmm = q X — Ei+E2 Amm dt — Hamm^v v dMA = qMAbXv — ^ MAb dd = Fi+F2

The criterion to be maximized is the total amount of MAb concentration obtained at the end of the fed-batch culture:

The constraints on the control variable and the culture volume are:

The following initial culture conditions and feed concentrations are used:

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