PK-PD curve-fitting problems with the Hill equation? Try one of the 1-exp functions derived from Hodgkin, Douglas or Gompertz.
Non-linear phenomena are observed with enzyme kinetics, protein binding, pharmacokinetics or pharmacodynamics. The Hill equation, the Michaelis-Menten equation extended by a power coefficient, is traditionally used for sigmoid curve fitting. Sigmoid saturation phenomena can also be described by exponential functions (1-exp), extended by a power coefficient such as those derived by Hodgkin, Douglas or Gompertz. Comparing the 4 equations, the sigmoid 1-exp function in the form of Hodgkin and Huxley comes closest to the principle of simplicity and succinctness with regard to definition, slope and flexibility of the inflection point. To compare the applicability, a standardized sample of 250 curves was generated by each I of the 4 equations and mutually fitted with the remaining 3. The Hill equation gives the closest fit with the data generated by the other functions. The Douglas variant exhibits the highest rate of convergence. The Gompertz function provides the basic feature of a baseline effect.


CONCLUSIONS
The sigmoid functions investigated (Hill, Hodgkin, Douglas, Gompertz) have differing characteristics and can be used interchangeably for solving specific problems in non-linear modeling.
Keller
F
F
Nephrology, University Ulm, Germany. frieder.keller@medizin.uni-ulm.de
Giehl
M
M
Czock
D
D
Zellner
D
D
D008433:Mathematics; D017711:Nonlinear Dynamics; D010599:Pharmacokinetics