Biochemical Engineering Journal 24 (2005) 95104
Practical identifiability of parameters if rectorty, 300
A system el thatpresented. T sum ofchoice of w statisthe estimati knownprocedure. F l data,are presente 2005 Else
The kinetics of growth of microorganisms under substrate-limited conThe Monodment, bioreplications i
The origmaintenancsynthesis onance. Maiby the micMonod moto equationdsdt= k
1 Present ad
where s= growth limiting substrate concentration (M L3),x= biomass concentration (M L3), k= maximum specificuptake rate of the substrate (T1), k = half saturation con-
1369-703X/$doi:10.1016/jditions was quantitatively defined by Monod .kinetic model is widely used in wastewater treat-
mediation and in various other environmental ap-nvolving growth of microorganisms.inal Monod model did not take into account thee metabolism. Microorganisms require energy forf new microorganisms as well as for their mainte-ntenance energy is the amount of energy requiredroorganisms even in the absence of growth. Thedel was modified by a number of researchers [2,3]s similar to the following:sx
+ s , s(0) = s0 (1)
s bx, x(0) = x0 (2)
ding author. Tel.: +1 504 865 5773; fax: +1 504 865 6744.dress: email@example.com (V.J. Law).dress: Sensitron Semiconductor, Deer Park, NY, USA.
stant for growth (M L3), y= yield coefficient (M M1), andb= decay coefficient (T1).
The yield coefficient is that portion of the substrate that isused for the synthesis of the biomass. McCarty  developeda method for the theoretical calculation of yield based onthermodynamics.
Various other models similar to Monod kinetics have beenproposed by a number of researchers [5,6]. However, theMonod model is the simplest and the one most widely appliedin practice. The parameters to be estimated are k, ks, y, b, s0and x0. The estimation of parameters in the Monod modelhas been carried out extensively . Different kinds ofobjective function, a variety of types of experimental data tobe measured and variations of the Monod model were con-sidered in these referenced works.
This paper discusses the statistical considerations inchoosing the objective function to be minimized to estimatethe parameters and the weights that should be assigned to theresiduals. This then provides a theoretical framework to an-alyze the validity of the parameters once they are estimated
see front matter 2005 Elsevier B.V. All rights reserved..bej.2005.01.028statistical analysis oPadmanaban Kesavan 1, Vi
Department of Chemical and Biomolecular Engineering, Tulane UniversiReceived 22 January 2004; accepted
atic procedure for identifying the number of parameters of a modhe objective function to be minimized is in the form of weightedeights and the validity of the parameters estimated are verified byon of parameters for Monod kinetics. Simulated data containinginally, parameters are estimated from different sets of experimenta
d.vier B.V. All rights reserved.
iokinetics; Dynamic modelling; Microbial growth; Bioreactionsn Monod kinetics andsidualsJ. Law
Boggs Building, New Orleans, LA 70118, USA
can be estimated uniquely by nonlinear regression issquares of residuals. The assumptions inherent in the
tical tests. The procedure is illustrated by consideringmeasurement noise are used initially to illustrate theand the validity and the uniqueness of the parameters
96 P. Kesavan, V.J. Law / Biochemical Engineering Journal 24 (2005) 95104
and the validity of assumptions in determining the weightsof the residuals.
2.1. Modilimitation i
The biosents the vitechniquessity measubiomass coviable andand (2) is nmental mea
If an assconstant frathen Eqs. (dsdt= k
= yksks +
where xt =(where xv =
The moparametersconcentratimated, namand (4) candSdT
where S=B = b/k0, S
s*, x* an
s, x and k, ras the expeconcentratis* and x*chosen as 1
Assumicentrationtive is to es(4) such thaA suitablethen has to
eter estimates will differ depending on the objective functionchosen. Therefore, even if the predicted profiles approximatethe measured profile for an arbitrary choice of objective func-
it is imeters
criterised oentalate thereforureme
e q=n froimensel = tot. (6), aass coen, therrors
here iher wndent
erefoto verican b.
formaters wly [17auss
(requi(5) anh-ordeausstained2]. Thuantit
e Gause valed no
, to avcation of the Monod model to consider then experimental measurements
mass concentration (x) in Eqs. (1) and (2) repre-able (living) biomass concentration. Experimentalused (volatile solids measurement, optical den-
rements, protein content, etc.) measure the totalncentration and do not distinguish between thedead biomass. Hence, a modification to Eqs. (1)ecessary to account for this limitation in experi-surements.umption that the viable biomass concentration is action of the total biomass concentration is made,
1) and (2) can be modified as:sxt
+ s , s(0) = s0 (3)
s bxt, xt(0) = x0t (4)
total biomass concentration (M L3) and = xv/xtviable biomass concentration (M L3)).
del Eqs. (3) and (4) can be used to estimate theusing the measured substrate and total biomass
ons. A total of seven parameters need to be esti-ely k, ks, , b, s0, x0t and (where = yk). Eqs. (3)be written in the following dimensionless form:
s + S , S(0) = S0 (5)
+ S BXt, Xt(0) = X0t (6)
s/s*, Xt = xt/x*, T= k0t, K= kx*/k0s*, Ks = ks/s*,0 = s0/s*, X0t = x0t/x*, and =/k0.d k0 are known constant values with dimensions ofespectively. The value of s* and x* can be chosenrimentally measured initial substrate and biomasson, respectively. However, the values chosen fordo not affect the result. The value of k0 can be.
ng that the substrate depletion and biomass con-are measured at suitable time intervals, the objec-timate the parameters of the model Eqs. (3) andt they approximate the measured profiles closely.
choice of the objective function to be minimizedbe made. This is a nontrivial task since the param-
tion,paramThebe baperimestim(6) thmeas
viousthe GtersEqs.fourtthe Gbe ob[17,2fied qof th
Thstraineterspossible to infer anything about the validity of theand the procedure reduces to simple curve fitting.
a for the choice of the objective function shouldn the measurement errors associated with the ex-data [18,19]. The choice of objective function toe parameters of the model defined by Eqs. (5) ande should be based on the errors associated with thents of the substrate and biomass concentrations.
le, if the objective function chosen is:
model Sdata)2 +n
objective function, Smodel = substrate concen-m the model defined by Eq. (5), Sdata =ionalized experimental substrate concentration,al biomass concentration from the model definedndXdata = nondimensionalized experimental totalncentration.e inherent assumptions made about the measure- are:the measurements of substrate and biomass con-
ons are normally distributed, each with its own.
s no correlation between any measurements. Inords, all the measurements are completely inde-.iance of all the measurements is equal.
re, once the parameters are estimated, it is neces-fy whether all the assumptions made are satisfied.e done based on statistical tests as discussed by
e objective function is chosen, then the parameterproblem becomes a nonlinear regression problem.ear regression algorithm, CONREG  is usedk. This algorithm uses a combination of New-od and weighted steepest descent and is calledtional discrimination . The scaling of the pa-as performed using the procedure developed pre-]. The function values, the gradient vector andNewton matrix for a given estimate of parame-
red by CONREG) are calculated by integratingd (6) along with the sensitivity equations by ar RungeKutta method. It should be noted thatNewton matrix as well as the gradient vector can
from the sensitivity equations and the residualse uniqueness of the parameter estimates is identi-atively by analyzing the orthonormal eigenvectorssNewton matrix .
ue of B is usually close to zero. Since uncon-nlinear regression is used to estimate the param-oid B being a small negative value the transfor-