title: The Hill equation: a review of its capabilities in pharmacological modelling.
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.
authors.affiliation: Université Lyon 1, Lyon, F-69003, France. sylvain.goutelle@chu-lyon.fr
authors.forename: Sylvain
authors.initials: S
authors.lastname: Goutelle
authors.forename: Michel
authors.initials: M
authors.lastname: Maurin
authors.forename: Florent
authors.initials: F
authors.lastname: Rougier
authors.forename: Xavier
authors.initials: X
authors.lastname: Barbaut
authors.forename: Laurent
authors.initials: L
authors.lastname: Bourguignon
authors.forename: Michel
authors.initials: M
authors.lastname: Ducher
authors.forename: Pascal
authors.initials: P
authors.lastname: Maire
mesh_terms: D000818:Animals; D001665:Binding Sites; D054796:Drug Dosage Calculations; D004347:Drug Interactions; D006801:Humans; D016015:Logistic Models; D008954:Models, Biological; D015233:Models, Statistical; D015999:Multivariate Analysis; D017711:Nonlinear Dynamics; D010100:Oxygen / Q000097:blood; D010108:Oxyhemoglobins / Q000378:metabolism; D010599:Pharmacokinetics; D010600:Pharmacology / Q000379:methods*; D011336:Probability; D011956:Receptors, Cell Surface / Q000378:metabolism; D018160:Receptors, Cytoplasmic and Nuclear / Q000378:metabolism; D018570:Risk Assessment
chemical_list: D010108:Oxyhemoglobins; D011956:Receptors, Cell Surface; D018160:Receptors, Cytoplasmic and Nuclear; D010100:Oxygen