The Hill equation: a review of its capabilities in pharmacological modelling

Authors

  • Sylvain Goutelle,

    Corresponding author
    1. Université Lyon 1, Lyon, F-69003, France; UMR CNRS 5558 “Biométrie et Biologie Evolutive”, Bat G. Mendel, 43 boulevard du 11 Novembre 1918, F-69622 Villeurbanne Cedex, France
    2. Hospices Civils de Lyon, Hôpital Antoine Charial, ADCAPT-Service Pharmaceutique, Francheville F-69340, France
    Search for more papers by this author
  • Michel Maurin,

    1. Institut National de Recherche sur les Transports et leur Sécurité INRETS, F-69500 Bron, France
    Search for more papers by this author
  • Florent Rougier,

    1. Hospices Civils de Lyon, Hôpital Antoine Charial, ADCAPT-Service Pharmaceutique, Francheville F-69340, France
    Search for more papers by this author
  • Xavier Barbaut,

    1. Hospices Civils de Lyon, Hôpital Antoine Charial, ADCAPT-Service Pharmaceutique, Francheville F-69340, France
    Search for more papers by this author
  • Laurent Bourguignon,

    1. Université Lyon 1, Lyon, F-69003, France; UMR CNRS 5558 “Biométrie et Biologie Evolutive”, Bat G. Mendel, 43 boulevard du 11 Novembre 1918, F-69622 Villeurbanne Cedex, France
    2. Hospices Civils de Lyon, Hôpital Antoine Charial, ADCAPT-Service Pharmaceutique, Francheville F-69340, France
    Search for more papers by this author
  • Michel Ducher,

    1. Hospices Civils de Lyon, Hôpital Antoine Charial, ADCAPT-Service Pharmaceutique, Francheville F-69340, France
    Search for more papers by this author
  • Pascal Maire

    1. Université Lyon 1, Lyon, F-69003, France; UMR CNRS 5558 “Biométrie et Biologie Evolutive”, Bat G. Mendel, 43 boulevard du 11 Novembre 1918, F-69622 Villeurbanne Cedex, France
    2. Hospices Civils de Lyon, Hôpital Antoine Charial, ADCAPT-Service Pharmaceutique, Francheville F-69340, France
    Search for more papers by this author

*Correspondence and reprints: sylvain.goutelle@chu-lyon.fr

Abstract

The Hill equation was first introduced by A.V. Hill to describe the equilibrium relationship between oxygen tension and the saturation of haemoglobin. In pharmacology, the Hill equation has been extensively used to analyse quantitative drug–receptor relationships. Many pharmacokinetic–pharmacodynamic models have used the Hill equation to describe nonlinear drug dose–response relationships. Although the Hill equation is widely used, its many properties are not all well known. This article aims at reviewing the various properties of the Hill equation. The descriptive aspects of the Hill equation, in particular mathematical and graphical properties, are examined, and related to Hill’s original work. The mechanistic aspect of the Hill equation, involving a strong connection with the Guldberg and Waage law of mass action, is also described. Finally, a probabilistic view of the Hill equation is examined. Here, we provide some new calculation results, such as Fisher information and Shannon entropy, and we introduce multivariate probabilistic Hill equations. The main features and potential applications of this probabilistic approach are also discussed. Thus, within the same formalism, the Hill equation has many different properties which can be of great interest for those interested in mathematical modelling in pharmacology and biosciences.

Ancillary