Chemical Engineering Science 58 (2003) 1545–1563 www.elsevier.com/locate/ces Maximumlikelihoodestimationofpseudo-stoichiometryinmacroscopic biologicalreactionschemes Ph.Bogaerts * ,J.-L.Delcoux,R.Hanus Control Engineering and System Analysis Department, Universit e Libre de Bruxelles, Av. F.-D. Roosevelt, 50, CP 165/55, B-1050 Brussels, Belgium Received 1 February 2002; received in revised form 27 August 2002; accepted 3 December 2002 Abstract Identicationofpseudo-stoichiometric(oryield)coecientsisofprimaryimportanceforbuildingabioprocessmodel.Inmostofthe applications, the estimation of these coecients has to be performed without any knowledge of the kinetics and on the basis of a few experimentsforwhichnoisydiscretemeasurementsofcomponentconcentrationsareavailable.Thispaperproposesmaximumlikelihood estimators which are able to deal with measurement errors on all the signals, at each sampling time (including the initial one) and with intrinsic sign constraints on the parameters. This kind of realistic hypotheses exclude the use of the usual (weighted) least-squares estimators. The maximum likelihood estimators are proved to be unbiased (provided a rst-order approximation) and their estimation error covariance matrix can be computed (at the same level of rst-order approximation). The solutions are proposed in a very general framework, dealing with cell cultures (of bacteria, yeasts or animal cells) performed in stirred tank (continuous, semi-batch or batch) reactors, and without any a priori knowledge on the kinetics. The use of the estimators and their statistical properties are illustrated in a simulationcasestudy(fed-batchbacterialcultures)andinarealcaseone(batchanimalcellcultures). ? 2003 Elsevier Science Ltd. All rights reserved. Keywords: Parameter identication; Parameter estimation; Maximum likelihood; Stoichiometric models; Reaction schemes; Bioprocesses 1. Introduction Mathematical models of cell cultures in bioreactors are necessary for their design, simulation, optimization, state estimation, control, etc. Usually, these modelling aims lead to the use of macroscopic models consisting of the mass balances for the main species involved in the cell culture (Bogaerts & Hanus, 2000). The main chemical or biochemical reactions are summarized in a reaction scheme. For bioprocesses, the concept of macro- scopic reaction scheme has been introduced by Bastin and Dochain (1990). This kind of reaction network is limited to the main phenomena occurring in the cell culture (e.g., growth, maintenance and death of a biomass), involving a limited number of macroscopic species (e.g., the biomass andthemainsubstrates,metabolitesandproductsofinter- est). On the contrary to a classical chemical reaction net- work,theelementarymassbalanceshavenottoberespected: the yield (or pseudo-stoichiometric) coecients only de- scribe the number of component moles or number of cells * Correspondingauthor.Tel.:+32-2-650-26-75;fax:+32-2-650-26-77. E-mail address: philippe.bogaerts@ulb.ac.be (Ph. Bogaerts). produced or consumed upon occurrence of a given reac- tion.Thesepseudo-stoichiometriccoecientshavetobees- timatedonthebasisofmeasurementsofthecomponentcon- centrationsinvolvedinthereactionscheme.Inmostcases,a fewexperimentsaremadeindierentoperationalconditions (e.g.,dierentinitialconditionsforbatchcellcultures)and, foreachone,thecomponentconcentrationsaremeasuredin discretetime.Thesesamplesarealwayscorruptedbynoise comingfromallthemeasurementandexperimentalerrors. Itisnecessarytotaketheseuncertaintiesintoaccountinor- dertoobtainanaccurate(unbiasedinthestatisticalsense) estimationoftheyieldcoecientsonthebasisofthenoisy measurement samples. This is also necessary for comput- ingagoodapproximationoftheestimationerrorcovariance matrix,whichallowstoquantifytheprecisionoftheparam- eter identication and to associate a condence interval to eachestimatedpseudo-stoichiometriccoecient. Althoughalotofworkhasbeendoneinordertoidentify the pseudo-stoichiometry in macroscopic reaction schemes independentlyofanyknowledgeaboutthekinetics(Bastin &Dochain,1990; Chen&Bastin,1996),littleattentionis paid to the eects of the measurement errors. As will be 0009-2509/03/$-see front matter ? 2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0009-2509(02)00680-2