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.

Background Beta-lactam antibiotics (βLA) are the most commonly used antibiotics in the intensive care unit (ICU). ICU patients present many pathophysiological features that cause pharmacokinetic (PK) and pharmacodynamic (PD) specificities, leading to the risk of underdosage. The French Society of Pharmacology and Therapeutics (SFPT) and the French Society of Anaesthesia and Intensive Care Medicine (SFAR) have joined forces to provide guidelines on the optimization of beta-lactam treatment in ICU patients. Methods A consensus committee of 18 experts from the two societies had the mission of producing these guidelines. The entire process was conducted independently of any industry funding. A list of questions formulated according to the PICO model (Population, Intervention, Comparison, and Outcomes) was drawn-up by the experts. Then, two bibliographic experts analysed the literature published since January 2000 using predefined keywords according to PRISMA recommendations. The quality of the data identified from the literature was assessed using the GRADE® methodology. Due to the lack of powerful studies having used mortality as main judgement criteria, it was decided, before drafting the recommendations, to formulate only “optional” recommendations. Results After two rounds of rating and one amendment, a strong agreement was reached by the SFPT-SFAR guideline panel for 21 optional recommendations and a recapitulative algorithm for care covering four areas: (i) pharmacokinetic variability, (ii) PK-PD relationship, (iii) administration modalities, and (iv) therapeutic drug monitoring (TDM). The most important recommendations regarding βLA administration in ICU patients concerned (i) the consideration of the many sources of PK variability in this population; (ii) the definition of free plasma concentration between four and eight times the Minimal Inhibitory Concentration (MIC) of the causative bacteria for 100% of the dosing interval as PK-PD target to maximize bacteriological and clinical responses; (iii) the use of continuous or prolonged administration of βLA in the most severe patients, in case of high MIC bacteria and in case of lower respiratory tract infection to improve clinical cure; and (iv) the use of TDM to improve PK-PD target achievement. Conclusions The experts strongly suggest the use of personalized dosing, continuous or prolonged infusion and therapeutic drug monitoring when administering βLA in critically ill patients. Electronic supplementary material The online version of this article (10.1186/s13054-019-2378-9) contains supplementary material, which is available to authorized users.