Hadi Taghvafard (1), J.G. Coen van Hasselt (1), Piet H. van der Graaf (1,2)
(1) Leiden Academic Centre for Drug Research, Leiden University, Leiden, The Netherlands, (2) Certara QSP, Canterbury, UK
Introduction: An important topic in quantitative pharmacology is modeling the shape of concentration- effect (E/[A]) curves, which relate the concentration of an agonist [A] to an observed functional effect E. Mathematical modeling of E/[A] data can allow us to classify agonists and receptors, design more selective and potent therapeutic agents, and better understand physiological functions [1]. Owing to the fact that all measures of system sensitivity or drug activity originate from E/[A] curves, it is of prime importance to find the mathematical functions which fit experimental data precisely. The shape of E/[A] curves can also provide insights into mechanisms of signal transduction.
Several mathematical equations have been postulated to describe E/[A] curves (see, e.g., [1, 2, 3, 4, 5]). One of the most important equations which has been extensively used in pharmacology to model E/[A] data is the Hill equation, which is symmetric, i.e., the inflection point is exactly the same as the mid-point of a concentration-effect curve. However, even standard models of receptor theory predict asymmetric E/[A] curves, which cannot be described by the Hill equation [3]. Therefore, there is a need for alternative equations to describe asymmetric E/[A] curves [1,3], which was the focus of the present study.
Methods: We propose two novel equations to describe E/[A] curves. The first equation is based on the exponential function and describes symmetric E/[A] curves. The second equation, being more general than the first one, can be applied to both symmetric and asymmetric curves.
The first equation has three parameters, while the second one has four parameters. In the second equation, there are two parameters which contribute to the asymmetry and slope of an E/[A] curve. We mathematically compare them to the Hill equation and the Richard equation [4] and explore how these new equations can be used for parameter fitting of E/[A] datasets.
Results: We propose the following equations:
E/[A] = a*(2/(1+e^-([A]^n/(K+[A]^n))) – 1), K,a,n>0 (1)
E/[A] = a*(2/(1+b^-([A]/(K+[A]))^n) – 1), K,a,n>0, b>=1 (2)
Comparing our two proposed equations with the conventional Hill equation, we have the following results:
- 1) Equation (1) is identical to the Hill equation for describing symmetric E/[A] curves. In this equation, parameters K, a and n correspond, respectively, to EC50, Emax and the slope parameter in the Hill equation.
- 2) Equation (2) is applicable to both symmetric and asymmetric E/[A] curves, while the Hill equation is not a proper equation for describing asymmetric curves. In the Richards equation [5], there is only one parameter playing a crucial rule in the slope and asymmetry of an E/[A] curve [1]. However, in equation (2) there are two parameters, namely b and n, which make such changes.
Conclusions: We have proposed two new equations for modeling symmetrical and asymmetrical E/[A] curves, which may provide alternatives for the commonly-used Hill equation in pharmacokinetic-pharmacodynamic and systems pharmacology models.
References:
[1] Giraldo J, Vivas NM, Vila E, Badia A. Assessing the (a) symmetry of concentration-effect curves: empirical versus mechanistic models. Pharmacology & Therapeutics. 2002;95(1):21-45.
[2] Black JW, Leff P. Operational models of pharmacological agonism. Proceedings of the Royal Society of London. Series B. Biological Sciences. 1983;220(1219):141-62.
[3] Roche D, van der Graaf PH, Giraldo J. Have many estimates of efficacy and affinity been misled? Revisiting the operational model of agonism. Drug discovery today. 2016;21(11):1735-9.
[4] Van der Graaf PH, Schoemaker RC. Analysis of asymmetry of agonist concentration-effect curves. Journal of pharmacological and toxicological methods. 1999;41(2-3):107-15.
[5] Richards FJ. A flexible growth function for empirical use. Journal of Experimental Botany. 1959;10(2):290-301.
Reference: PAGE 28 (2019) Abstr 9040 [www.page-meeting.org/?abstract=9040]
Poster: Methodology - New Modelling Approaches