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Hill Equation

Math

The Hill Equation block is the sigmoidal Emax model of pharmacodynamics: it turns a drug

concentration at the site of action into the effect that concentration produces. In

anaesthesia it converts the propofol effect-site concentration into the BIS index, a

depth-of-anaesthesia score that is about 93 awake and 40–60 during surgery.

The response is flat at low concentrations, steep around EC50 and saturates at high ones:

receptors fill up, and adding drug beyond that changes little. The Hill slope sets how

abruptly the patient goes from awake to deeply anaesthetised. The block is stateless — it

has no memory, so put the pharmacokinetics (how the concentration evolves) in front of it.

Mathematical Model

\text{occ} = \frac{C_e^{\gamma}}{EC_{50}^{\gamma} + C_e^{\gamma}}, \qquad E = E_0 - E_{max} \cdot \text{occ}

Where:

The occupancy is computed as 1 / (1 + (EC50/Ce)^γ) in logarithms, so a steep slope or a

very high concentration saturates cleanly instead of overflowing.

Direction	ID	Label	Type	Status
→ In	`in1`	Ce	number	Required
← Out	`effect`	Effect	number	Output
← Out	`occupancy`	Occ.	number	Output

Parameters	Label	Type	Default	Description
`e0`	Baseline (E0)	number	`93`	Effect at zero concentration, for example an awake BIS of 93
`emax`	Maximum Effect (Emax)	number	`93`	Largest possible drop from the baseline
`ec50`	EC50	number	`3.4`	Concentration giving half the maximum effect, in the unit of Ce
`gamma`	Hill Slope (γ)	number	`3`	Steepness of the curve: 1 is hyperbolic (Michaelis-Menten), above 1 sigmoidal

A plasma concentration step to 3.4 reaches the effect site through a first-order lag,

1/(10s+1), standing in for the equilibration rate ke0. The BIS starts at 93, lingers

while the concentration is still low, then falls steeply and settles at 46.5 — half the

maximum effect, because the concentration settles at EC50.

The defaults (E0 = 93, Emax = 93, EC50 = 3.4, γ = 3) are illustrative, not a validated

patient model. Published models fit these per population and often per patient.

Because the curve is steep near EC50, a closed-loop controller acting on the BIS sees a

gain that changes with depth: small near awake and near saturation, large in between.