Emax and Hill models (EC50)
The Emax model describes an effect that rises with concentration to a maximum. EC50 is the concentration giving half of the maximum effect, and the Hill exponent sets the steepness.
Definition
The Emax model is E = E0 + Emax × C / (EC50 + C), where C is the concentration, E0 the baseline effect, Emax the maximum effect and EC50 the concentration that gives half of Emax. For an inhibitory effect the form is E = E0 × (1 − Imax × C / (IC50 + C)).
The Hill exponent
The sigmoid Emax (Hill) model adds an exponent γ: E = E0 + Emax × C^γ / (EC50^γ + C^γ). A larger γ makes the curve steeper around EC50; γ = 1 gives the simple Emax model.
Use
These functions give the shape of the concentration–effect relationship in indirect response models and in direct-effect models. EC50 and γ often cannot be estimated well from sparse data; see practical identifiability.
On this site
In the warfarin PK/PD analysis the maximum inhibition is fixed to one and the Hill exponent is a typical-value parameter without between-subject variability. The RAS inhibitor analysis links drug concentration to a target-coverage response.
Frequently asked questions
What is the difference between EC50 and IC50?
EC50 is the concentration giving half of the maximum effect when the effect is stimulatory; IC50 is the concentration giving half-maximal inhibition. The functional form is the same.
What does a Hill exponent above one mean?
The concentration–effect curve is steeper than the simple Emax model around the EC50, so the effect switches on over a narrower concentration range.