Pharmacodynamics · Interactive teaching lab

Dose–Response Lab

Concentration–effect curves, agonist efficacy and receptor reserve, competitive antagonism and Schild analysis, quantal response and therapeutic index. Move a slider and the curves are recalculated instantly in your browser.

  • Instant calculation
  • Shareable view links
  • PNG and CSV export
  • English / Turkish

Educational model. The curves come from idealised pharmacological models; they are not patient data and must not be used for clinical dosing decisions. For concentration–time simulation with real drug data, use the Pharmacokinetics Simulator Pro.

Concentration–effect: potency vs efficacy

Sigmoid Emax (Hill) and inhibitory Imax models. Overlay up to four curves and compare the effect of EC₅₀ (potency), Emax (efficacy) and the Hill slope.

Concentration–effect curves

How to read
  • Sigmoid Emax (Hill) model: E = E₀ + Emax·Cⁿ/(EC₅₀ⁿ + Cⁿ). E₀ is the baseline effect; the upper asymptote of the curve is E₀ + Emax (not Emax itself unless E₀ = 0).
  • Potency is measured by EC₅₀, the concentration giving the half-maximal effect (E₀ + Emax/2). The lower the EC₅₀ (the further left the curve), the more potent the drug. Efficacy is the maximal attainable effect (Emax). The two are independent: in the default example Drug C is the most potent, while Drugs A and B are the most efficacious. Clinically, efficacy usually matters more than potency, which only sets the dose needed.
  • Hill slope (nH): the steepness of the curve. In functional (response) curves nH reflects the whole transduction chain from receptor binding to the measured response (amplification, threshold and saturation effects); it is not a direct measure of binding cooperativity. With n = 1 the concentration must rise 81-fold (≈1.9 log units) for the effect to go from 10% to 90%; in general EC₉₀/EC₁₀ = 811/n.
  • ECx: ECx = EC₅₀·(x/(100 − x))1/n. For n = 1, EC₉₀ = 9·EC₅₀ and EC₁₀ = EC₅₀/9.
  • Inhibitory (Imax) model: E = E₀ − Imax·Cⁿ/(IC₅₀ⁿ + Cⁿ), for endpoints that fall, such as blood pressure or heart rate. The lower asymptote is E₀ − Imax. For responses expressed as change from baseline, E₀ may be negative.
  • Axis: on a log-concentration axis the curve is a sigmoid symmetric about EC₅₀; on a linear axis (n = 1) it is a rectangular hyperbola. That is why dose–response curves are usually drawn on a log axis.
  • Units: mg/L = mol/L × MW (g/mol) × 1000; 1 mg/L = 1 µg/mL. Compare the potency of different drugs in molar units; comparisons in mg/L are distorted by differences in molecular weight.
  • Assumptions: an equilibrium relationship between effect and the concentration at the effect site (usually the free, unbound concentration). Time delays (hysteresis), tolerance, active metabolites and between-subject variability are not modelled.
Pharmacokinetics Simulator Pro Concentration–time: single and multiple dosing, steady state, accumulation and loading dose with real drug data. Receptor–Ligand Binding Pro Binding level: receptor occupancy, saturation and competition binding, kon/koff kinetics. DruglyLex Pharmacology dictionary: EC₅₀, pA₂, therapeutic index, intrinsic activity and more.
Model assumptions and references

All models assume equilibrium (steady-state) conditions, a single receptor population and a response that depends instantaneously on the concentration at the effect site. Calculations run in the browser (JavaScript); the normal distribution function Φ uses an erfc approximation with relative error < 1.2×10⁻⁷ and Φ⁻¹ uses Acklam’s algorithm. Results are for education and research.

  1. Holford NHG, Sheiner LB. Understanding the dose–effect relationship: clinical application of pharmacokinetic–pharmacodynamic models. Clin Pharmacokinet. 1981;6(6):429–453.
  2. Black JW, Leff P. Operational models of pharmacological agonism. Proc R Soc Lond B Biol Sci. 1983;220(1219):141–162.
  3. Furchgott RF. The use of β-haloalkylamines in the differentiation of receptors and in the determination of dissociation constants of receptor–agonist complexes. Adv Drug Res. 1966;3:21–55.
  4. Arunlakshana O, Schild HO. Some quantitative uses of drug antagonists. Br J Pharmacol Chemother. 1959;14(1):48–58.
  5. Neubig RR, Spedding M, Kenakin T, Christopoulos A. International Union of Pharmacology Committee on Receptor Nomenclature and Drug Classification. XXXVIII. Update on terms and symbols in quantitative pharmacology. Pharmacol Rev. 2003;55(4):597–606.
  6. Finney DJ. Probit Analysis. 3rd ed. Cambridge University Press; 1971.
  7. Motulsky H, Christopoulos A. Fitting Models to Biological Data Using Linear and Nonlinear Regression. Oxford University Press; 2004.
  8. Kenakin T. A Pharmacology Primer: Techniques for More Effective and Strategic Drug Discovery. Academic Press.