# Direct Effects and Effect Compartments

pharmacologists have long recognized the direct relationship between drug concentration and response (or effect). There has also been a long-running understanding that drug concentration as a function of time was related to measured response as a function of time, leading to numerous studies designed specifically to assure that steady-state concentrations were achieved before the measurement of the response measurement. These steady-state experiments led to a host of models (still used today) where the assumption of steady-state drug concentration was or is experimentally upheld.

If the lag between effect and concentration is very short, or if the experiment is performed at steady-state conditions, plasma concentrations can be used as a surrogate for concentrations at the active site. hence, [D] can be substituted with C, the plasma concentration, and the hill equation becomes

Sometimes there is a disconnect between the concentration at the active site and the concentration in plasma, in particular when equilibration between the two sites does not occur rapidly. In those cases, it is convenient to include an *effect compartment* (also called a *link compartment),* in which a negligible amount of the drug in plasma (proportional to the administered dose) distributes with a specific rate constant k_{e0}; negligible because the compartment should have no influence on the overall pharmacokinetics of the drug. Hence, the effect compartment is a hypothetical kinetic drug compartment, described by

where C_{e} is the concentration in the effect compartment. The hill equation then becomes

Notice that the effect is still direct with respect to C_{e}.

Since the maximum effect is observed at C_{max} in direct-effect models, the maximum effect always occurs at t_{max}, and a change in dose should not affect the time of maximum effect, assuming that the drug obeys linear pharmacokinetics.

The Michaelis-Menten and Hill equations provide physiological reality, in that both have asymptotic maximum effects (E_{max}): despite increasing the dose to very large sizes, these models return an effect that is capped at E_{max}, and very little gain is achieved by increasing the dose to large sizes. While these pharmacodynamics models are quite useful, the parameters (*E*_{max}, EC_{50}, and *y)* are only a function of pharmacological and physiological cascades; the parameters are not influenced by dose or time.