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The Hill equation: a review of its capabilities in pharmacological modelling
Author(s) -
Goutelle Sylvain,
Maurin Michel,
Rougier Florent,
Barbaut Xavier,
Bourguig Laurent,
Ducher Michel,
Maire Pascal
Publication year - 2008
Publication title -
fundamental and clinical pharmacology
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.655
H-Index - 73
eISSN - 1472-8206
pISSN - 0767-3981
DOI - 10.1111/j.1472-8206.2008.00633.x
Subject(s) - structural equation modeling , hill differential equation , mathematics , probabilistic logic , entropy (arrow of time) , principle of maximum entropy , statistical physics , differential equation , statistics , first order partial differential equation , mathematical analysis , thermodynamics , physics , exact differential equation
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.