Computational problems of compartment models with Michaelis-Menten-type elimination

The Michaelis-Menten equation has been applied successfully in the study of enzyme kinetics. It usually is used to estimate vmax and km from observations of the initial rate of reaction, v, at various substrate concentrations, Cs. A variation of this expression recently was used in pharmacokinetics, where it was assumed that the elimination rate of drug from some compartment is VC(t)/[K + C(t)], where C(t) is the drug concentration. The meaning of V and K in this context is not clear. Attempts were made to estimate V, K, and other model parameters by fitting the model to observed drug concentrations at sampling times after dosing. This paper discusses the ill-conditioning of the estimation of parameters of a differential equation that includes the so-called Michaelis-Menten output. The solution of the equation is bound by the solutions to two first-order differential equations. Parameter values in an infinite region of the parameter space are shown to have solutions also lying within these two bounds. Simulations show that a minor change in the data (observations) or in the initial estimate of the parameters may cause a large change in the final estimates. In many cases, estimation and comparison of parameter values are meaningless.