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Great Parameter Estimation Monod

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<ul><li><p>Biochemical Engineering Journal 24 (2005) 95104</p><p>Practical identifiability of parameters if rectorty, 300</p><p>28 Janu</p><p>Abstract</p><p>A system el thatpresented. T sum ofchoice of w statisthe estimati knownprocedure. F l data,are presente 2005 Else</p><p>Keywords: B</p><p>1. Introduction</p><p>The kinetics of growth of microorganisms under substrate-limited conThe Monodment, bioreplications i</p><p>The origmaintenancsynthesis onance. Maiby the micMonod moto equationdsdt= k</p><p>ks</p><p>dxdt= yks</p><p>ks +</p><p> CorresponE-mail ad</p><p>1 Present ad</p><p>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-</p><p>1369-703X/$doi:10.1016/jditions was quantitatively defined by Monod [1].kinetic model is widely used in wastewater treat-</p><p>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</p><p>+ s , s(0) = s0 (1)</p><p>x</p><p>s bx, x(0) = x0 (2)</p><p>ding author. Tel.: +1 504 865 5773; fax: +1 504 865 6744.dress: law@tulane.edu (V.J. Law).dress: Sensitron Semiconductor, Deer Park, NY, USA.</p><p>s</p><p>stant for growth (M L3), y= yield coefficient (M M1), andb= decay coefficient (T1).</p><p>The yield coefficient is that portion of the substrate that isused for the synthesis of the biomass. McCarty [4] developeda method for the theoretical calculation of yield based onthermodynamics.</p><p>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 [717]. 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.</p><p>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</p><p> see front matter 2005 Elsevier B.V. All rights reserved..bej.2005.01.028statistical analysis oPadmanaban Kesavan 1, Vi</p><p>Department of Chemical and Biomolecular Engineering, Tulane UniversiReceived 22 January 2004; accepted</p><p>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</p><p>d.vier B.V. All rights reserved.</p><p>iokinetics; Dynamic modelling; Microbial growth; Bioreactionsn Monod kinetics andsidualsJ. Law </p><p>Boggs Building, New Orleans, LA 70118, USA</p><p>ary 2005</p><p>can be estimated uniquely by nonlinear regression issquares of residuals. The assumptions inherent in the</p><p>tical tests. The procedure is illustrated by consideringmeasurement noise are used initially to illustrate theand the validity and the uniqueness of the parameters</p></li><li><p>96 P. Kesavan, V.J. Law / Biochemical Engineering Journal 24 (2005) 95104</p><p>and the validity of assumptions in determining the weightsof the residuals.</p><p>2. Theory</p><p>2.1. Modilimitation i</p><p>The biosents the vitechniquessity measubiomass coviable andand (2) is nmental mea</p><p>If an assconstant frathen Eqs. (dsdt= k</p><p>ks</p><p>dxtdt</p><p>= yksks +</p><p>where xt =(where xv =</p><p>The moparametersconcentratimated, namand (4) candSdT</p><p>= KK</p><p>dXtdT</p><p>= SKs</p><p>where S=B = b/k0, S</p><p>s*, x* an</p><p>s, x and k, ras the expeconcentratis* and x*chosen as 1</p><p>2.2. Param</p><p>Assumicentrationtive is to es(4) such thaA suitablethen has to</p><p>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-</p><p>it is imeters</p><p>criterised oentalate thereforureme</p><p>xampni=1</p><p>(S</p><p>e q=n froimensel = tot. (6), aass coen, therrors</p><p>ach ofntratiriance</p><p>here iher wndent</p><p>he var</p><p>erefoto verican b[18].</p><p>nce thation</p><p>nonlinis wormeth</p><p>formaters wly [17auss</p><p>(requi(5) anh-ordeausstained2]. Thuantit</p><p>e Gause valed no</p><p>, to avcation of the Monod model to consider then experimental measurements</p><p>mass concentration (x) in Eqs. (1) and (2) repre-able (living) biomass concentration. Experimentalused (volatile solids measurement, optical den-</p><p>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,</p><p>1) and (2) can be modified as:sxt</p><p>+ s , s(0) = s0 (3)</p><p>xt</p><p>s bxt, xt(0) = x0t (4)</p><p>total biomass concentration (M L3) and = xv/xtviable biomass concentration (M L3)).</p><p>del Eqs. (3) and (4) can be used to estimate theusing the measured substrate and total biomass</p><p>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:</p><p>SXt</p><p>s + S , S(0) = S0 (5)</p><p>Xt</p><p>+ S BXt, Xt(0) = X0t (6)</p><p>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.</p><p>eter estimation</p><p>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.</p><p>choice of the objective function to be minimizedbe made. This is a nontrivial task since the param-</p><p>tion,paramThebe baperimestim(6) thmeas</p><p>For e</p><p>q =</p><p>whertrationondXmodby Eqbiom</p><p>Thment</p><p>1. Ece</p><p>va</p><p>2. Totpe</p><p>3. T</p><p>ThsaryThisBard</p><p>OestimThein thtonstransrame</p><p>viousthe GtersEqs.fourtthe Gbe ob[17,2fied qof th</p><p>Thstraineterspossible to infer anything about the validity of theand the procedure reduces to simple curve fitting.</p><p>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.</p><p>le, if the objective function chosen is:</p><p>model Sdata)2 +n</p><p>j=1(Xmodel Xdata)2</p><p>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-[19] are:the measurements of substrate and biomass con-</p><p>ons are normally distributed, each with its own.</p><p>s no correlation between any measurements. Inords, all the measurements are completely inde-.iance of all the measurements is equal.</p><p>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</p><p>e objective function is chosen, then the parameterproblem becomes a nonlinear regression problem.ear regression algorithm, CONREG [20] is usedk. This algorithm uses a combination of New-od and weighted steepest descent and is calledtional discrimination [21]. 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-</p><p>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</p><p>from the sensitivity equations and the residualse uniqueness of the parameter estimates is identi-atively by analyzing the orthonormal eigenvectorssNewton matrix [17].</p><p>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-</p></li><li><p>P. Kesavan, V.J. Law / Biochemical Engineering Journal 24 (2005) 95104 97</p><p>mation B = exp(B) is used. Eq. (6) can then be rewrittenas:</p><p>dXtdT</p><p>= SXKs</p><p>The par(7) are K,the numberneed to bex0 and foyk and k (to yield twfore, one pEqs. (3) ancoefficient</p><p>3. Results</p><p>An analters of Montions in difof substrategion (S0region (S0</p><p>Simulatfirst generathe analysitype of meshould be nwill not be pdepletion. H(and providon the parainfer the rethen be repcreasing thparameterswas, therefbe estimatetion from smeasureme</p><p>3.1. First-o</p><p>Experimlated in theand (7) usirameter valfor substrasimulated s22 h and he22 h were utal data waas 0.3 g/L,were used</p><p>Table 1The values of the parameters of Monod kinetics used to simulate experimen-tal data in the first-order region</p><p>Parameter Value</p><p>))</p><p>))</p><p>)L)</p><p>ion rotive funi=1</p><p>(S</p><p>e orthesavan</p><p>minedhe ra</p><p>Even tmelyher thminedaussiariment</p><p>1) onrent se</p><p>differdata. Testimnoiseureme</p><p>minimni=1</p><p>11</p><p>e 1 =data a</p><p>al bioe step:</p><p>1: Sit erroxperim-orderer RungeKutta method using the parameters listed inle 1.2: Simulation of experimental data with proportionalsurement error.aussian noise of15% was added to the simulated dataubstrate depletion and biomass growth (from step 1) us-a pseudorandom number generator. The algorithm usedhe random number generator is described by Knuth [23].t</p><p>+ S exp(B)Xt, Xt(0) = X0t (7)</p><p>ameters that can be estimated from Eqs. (5) andKs, , B, S0 and X0 for a total of six. However,of parameters of the original Monod model that</p><p>estimated from Eqs. (3) and (4) are k, ks, , b, s0,r a total of seven. The product of the parametersthree parameters) in Eqs. (3) and (4) are lumpedo parameters and K in Eqs. (5) and (7). There-arameter of the original Monod model defined byd (4) has to be estimated independently. The yield(y) can be estimated from thermodynamics [4].</p><p>and discussion</p><p>ysis was first performed to determine the parame-od kinetics from substrate and biomass concentra-ferent regions of substrate depletion. The regions</p><p>depletion can be characterized as first-order re-Ks), mixed-order region (S0=Ks) and zero-orderKs) [11].</p><p>ed data containing known measurement noise wasted using specified parameters and were used fors. The objective function was chosen based on theasurement error added to the simulated data. Itoted that in the case of experimental systems, itossible to ascertain a priori, the region of substrateowever, once the experiment has been carried out</p><p>ed the measurement errors are known), then basedmeters estimated uniquely, it may be possible togion of substrate depletion. An experiment mayeated, if need be, by altering (increasing or de-e initial substrate concentration) to estimate the. The purpose of the analysis using simulated dataore, to identify the number of parameters that cand uniquely in different regions of substrate deple-ubstrate and biomass concentrations with knownnt noise.</p><p>rder region</p><p>ental data with no measurement noise was simu-first-order region (S0Ks) by solving Eqs. (5)</p><p>ng a fourth-order RungeKutta method. The pa-ues used are listed in Table 1. The detection limitte concentration was assumed to be 10 ppm. Theubstrate concentration was less than 10 ppm afternce the substrate and biomass concentration forsed to estimate the parameters. The experimen-</p><p>s nondimensionalized by choosing s*, x* and k01 g/L and 1 h1, respectively. The simulated datato estimate the parameters using the nonlinear re-</p><p>k (h1ks (g/Ly (g/gb (h1</p><p>s0 (g/Lx0t (g/</p><p>gressobjec</p><p>q =</p><p>Thby KdeterKs. T105.extrewhetdeter</p><p>GexpeTablediffethreelatedwere</p><p>mentmeas</p><p>to be</p><p>q =</p><p>wherstratein tot</p><p>Thbelow</p><p>Stepmen</p><p>EfirstordTabStepmea</p><p>Gon s</p><p>ingby t0.8330.60.010.50.31</p><p>utine. Since there is no measurement noise, thenction to be minimized was chosen as:</p><p>model Sdata)2 +n</p><p>j=1(Xmodel Xdata)2</p><p>onormal eigenvectors were analyzed as discussedand Law [17] to identify the parameters uniquely</p><p>. The ill-determined parameter was either K ortio of the highest to the smallest eigenvalue washough this ratio is high, it may not be consideredpoor. Therefore, it is worth investigating furthere parameters of Monod kinetics can be uniquelyin the first-order region.</p><p>n measurement noise of 15% was added to theal data (simulated using the parameters listed in</p><p>substrate depletion and biomass growth. Threets of experimental data were created by addingent sequences of random numbers to the simu-he parameters for different sets of simulated data</p><p>ated by nonlinear regression. Since the measure-is proportional to the measurements and since thent errors are uncorrelated, the objective functionized is [18,19]:(Smodel Sdata</p><p>Sdata</p><p>)2i</p><p>+ni=1</p><p>12</p><p>(Xmodel Xdata</p><p>Xdata</p><p>)2i</p><p>standard deviation of measurement errors in sub-nd 2 = standard deviation of measurement errors</p><p>mass concentration.s followed for each set of data are summarized</p><p>mulation of experimental data with no measure-r.</p><p>ental data with no noise were simulated in theregion by solving Eqs. (5) and (7) by a fourth-</p></li><li><p>98 P. Kesavan, V.J. Law / Biochemical Engineering Journal 24 (2005) 95104</p><p>Table 2The values of the parameters of Monod kinetics used to simulate experimen-tal data in the mixed-order region</p><p>Parameter</p><p>k (h1)ks (g/L)y (g/g)b (h1)</p><p>s0 (g/L)x0t (g/L)</p><p>Gaussianstrate andsubstratesubstratedatasets 1Step 3: D</p><p>The vainitial subvalue of kStep 4: Nwith prop</p><p>The simnondimenStep 5: Iniregression</p><p>The guwere chos</p><p>Three diters. The gof s*, x* an</p><p>The detaand the eigcase is givposed by Kare made. Tvalue (ill-drameters ofin the first-the paramebiomass cothe smalleswas probab</p><p>3.2. Mixed</p><p>Experimlated in theand (7) usirameter valfor substrasimulated s15 h and he15 h were utal data we</p><p>Table 3The nondimensionalized parameter values of the mixed-order region</p><p>Parameter Value</p><p>/L, 1paramrthonoated thaussiariment</p><p>mixeated fding</p><p>lated dch setcan bre we</p><p>ass das. Howl guesnalysihich iss. (5)</p><p>t X0aramee para</p><p>rary (bates ounknoned b:</p><p>rformta usie meaess forformthe pessesr B. T) wereValue</p><p>0.8330.60.010.531</p><p>noise of15% were also added to the initial sub-biomass concentration. The detection limit for</p><p>concentration was assumed to be 10 ppm. Theconcentration was below 10 ppm after 21.5 h forand 2 and after 22 h for dataset 3.etermination of nondimensionalizing constants.lue of s* and x* were chosen to be equal to thestrate and biomass concentration from step 2. The0 was chosen to be 1.ondimensionalizing simulated experimental dataortional error.ulated experimental data with noise (step 2) weresionalized using the values of s*, x* and k0.tial guess of the parameters required for nonlinear.</p><p>ess for initial substrate and biomass concentrationen as those simulated in step 2.</p><p>fferent guesses were chosen f...</p></li></ul>