Statistical analysis of nonlinear parameter estimation for monod biodegradation kinetics using bivariate data

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Abstract
A nonlinear regression technique for estimating the Monod parameters describing biodegradation kinetics is presented and analyzed. Two model data sets were taken from a study of aerobic biodegradation of the polycyclic aromatic hydrocarbons (PAHs), naphthalene and 2‐methylnaphthalene, as the growth‐limiting substrates, where substrate and biomass concentrations were measured with time. For each PAH, the parameters estimated were: qmax, the maximum substrate utilization rate per unit biomass; KS, the half‐saturation coefficient; and Y, the stoichiometric yield coefficient. Estimating parameters when measurements have been made for two variables with different error structures requires a technique more rigorous than least squares regression. An optimization function is derived from the maximumlikelihood equation assuming an unknown, nondiagonal covariance matrix for the measured variables. Because the derivation is based on an assumption of normally distributed errors in the observations, the error structures of the regression variables were examined. Through residual analysis, the errors in the substrate concentration data were found to be distributed log‐normally, demonstrating a need for log transformation of this variable. The covariance between ln C and X was found to be small but significantly nonzero at the 67% confidence level for NPH and at the 94% confidence level for 2MN. The nonlinear parameter estimation yielded unique values for qmax, KS, and Y for naphthalene. Thus, despite the low concentrations of this sparingly soluble compound, the data contained sufficient information for parameter estimation. For 2‐methylnaphthalene, the values of qmax and KS could not be estimated uniquely; however, qmax/KS was estimated. To assess the value of including the relatively imprecise biomass concentration data, the results from the bivariate method were compared with a univariate method using only the substrate concentration data. The results demonstrated that the bivariate data yielded a better confidence in the estimates and provided additional information about the model fit and model adequacy. The combination of the value of the bivariate data set and their nonzero covariance justifies the need for maximum likelihood estimation over the simpler nonlinear least squares regression. © 2000 John Wiley & Sons, Inc. Biotechnol Bioeng 69: 160–170, 2000.