Efficient design of experiments in the Monod model

In this paper the estimation problem and the problem of designing experiments in a nonlinear regression model, used in microbiology, are studied. The model is called Monod model, defined imlicitly by a differential equation for the regression function and has numerous applications in microbial growth kinetics, water research, pharmacokinetics and plant physiology. It is proved that least squares parameter estimates are asymptotically unbiased and normally distributed. The asymptotic covariance matrix of the least squares estimator is the basis for construction of efficient designs of experiments. In particular locally D-, E-and c-optimal designs are determined and their properties are studied. Moreover the performance of the designs (determined by the asymptotic theory) is confirmed in simulation experiments for realistic sample sizes. If certain intervals for the nonlinear parameters can be specified based on microbiological background, locally optimal designs can be constructed, which are robust with respect to misspecification of the initial parameters and which allow efficient estimation of the parameters in the Monod model. The results indicate that parameter variances can be decreased by a factor two by simply sampling at optimal times during the experiment.