Alternative parameterisations of saturable (Emax) models allowing for nesting of non-saturable models
Andreas V. Groth (1)
(1) Novo Nordisk A/S, Denmark
Objectives: It is often the case that non-saturable (linear) and saturable (Emax) models are fitted separately to experimental data, using parameters that are not directly comparable between the two types of model. Comparisons between the two may be facilitated if one type is nested within the other as a special case.
Methods: For the simple Emax model (Hill coefficient n=1), the parameterisation of Schoemaker et al.  is modified to introduce the parameter a = EC50-1, and the parameters Emax and EC50 are eliminated. It is then exploited that while no parameter can go to infinity, as in an Emax model would be required for both parameters Emax and EC50 in order to fit non-saturating data, a parameter can go to zero which, for the parameter a, creates a linear model nested within a simple Emax model. The concept is extended to Hill coefficients different from 1, in which case the above transforms into a power function model nested within a Hill Emax model. An example is given by simulating truncated data sets and re-estimate parameters from these, using Perl-speaks-NONMEM (PsN).
Results: The following parameterisation encompassing both Emax models with any positive Hill coefficient, linear models and power function models, is proposed:
E=ER * Cn / CRn * (1+(aCR)n) / (1+(aC)n)
CR and ER denote a reference concentration-effect pair within the range of the experimental data. Either one of these values are chosen, the other one is estimated. Thus, the model parameters to be estimated are the Hill coefficient n, the "curvature" parameter a (corresponding to EC50-1), and either CR or ER.
For the simpler special cases of n=1 and/or a=0, the expression is correspondingly simpler.
 Schoemaker RC et al., Journal of Pharmacokinetics and Biopharmaceutics, 26 (5), 1 (1998)
 Bachman WJ, Gillespie WR, Clin. Pharmacol. Ther. 63, 199 (1998)