One compartment model

Ke relates amount of drug in the compartment to rate of elimination:

$$\frac{dX}{dt} = k_e \cdot X$$

The half life is

$$t_{1/2} = \frac{\ln 2}{k_e}$$

Clearance relates concentration in the compartment to rate of excretion:

$$\frac{dX}{dt} = Cl \cdot C_X$$

They are related by the volume of distribution (a larger volume of distribution dilutes the drug, so for the same clearance, \(k_e\) is lower):

$$k_e = \frac{Cl}{V_d}$$

After a bolus,

$$X = \text{Dose} \cdot e^{-k_et}$$

and

$$C = \frac{\text{Dose}}{V_d} \cdot e^{-k_et}$$

After initiation of an infusion at rate \(k_0\), we have:

$$C = \frac{k_0}{Cl} \cdot (1 - e^{-k_et})$$

The effect site is a theoretical compartment with vanishingly small volume where the drug target resides (e.g the brain, the neuromuscular junction). We can't measure the effect site, but we can measure the effect of the drug (e.g. BIS, train-of-hour, spectral edge) and use experiments to model it.

Its transfer to and from the central compartment is characterized by two transfer constants \(K_{1e}\) and \(K_{e0}\), which are related by the intercompartmental clearance \(Cl_1e = K_{1e} \cdot V_1 = K_{e1} \cdot V_e\)