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

A nonlinear regression technique for estimating the Monod parameters descri bing biodegradation kinetics is presented and analyzed. Two model data sets were taken from a study of aerobic biodegradation of the polycyclic aromat ic hydrocarbons (PAHs), naphthalene and 2-methylnaphthalene, as the growth- limiting substrates, where substrate and biomass concentrations were measur ed with time. For each PAH, the parameters estimated were: q(max), the maxi mum substrate utilization rate per unit biomass; K-S, the half-saturation c oefficient; and Y, the stoichiometric yield coefficient. Estimating paramet ers when measurements have been made for two variables with different error structures requires a technique more rigorous than least squares regressio n. An optimization function is derived from the maximum likelihood equation assuming an unknown, nondiagonal covariance matrix for the measured variab les. Because the derivation is based on an assumption of normally distribut ed errors in the observations, the error structures of the regression varia bles were examined. Through residual analysis, the errors in the substrate concentration data were found to be distributed lognormally, demonstrating a need for log transformation of this variable. The covariance between In C and X was found to be small but significantly nonzero at the 67% confidenc e level for NPH and at the 94% confidence level for 2MN. The nonlinear para meter estimation yielded unique values for q(max), K-S, and Y for naphthale ne. Thus, despite the low concentrations of this sparingly soluble compound , the data contained sufficient information for parameter estimation. For 2 -methylnaphthalene, the values of q(max) and K-S could not be estimated uni quely; however, q(max)/K-S was estimated. To assess the value of including the relatively imprecise biomass concentration data, the results from the b ivariate method were compared with a univariate method using only the subst rate concentration data. The results demonstrated that the bivariate data y ielded a better confidence in the estimates and provided additional informa tion about the model fit and model adequacy. The combination of the value o f the bivariate data set and their nonzero covariance justifies the need fo r maximum likelihood estimation over the simpler nonlinear least squares re gression. (C) 2000 John Wiley & Sons, Inc.