Pharmacodynamics

Pharmacodynamics describes how a drug changes a biological system as a function of exposure. It connects target binding to a measured response, whereas pharmacokinetics describes the time course of exposure. A simple illustrative model iswhere $E_{max}$ is the maximum increment above baseline and $E{C}_{50}$ is the concentration producing half that increment. Doubling $C$ does not generally double the response. ^[ref]

Potency and maximal effect differ: two drugs can have different $E{C}_{50}$ values but the same $E_{max}$, or the same $E{C}_{50}$ but different maximal effects. Moreover, $E{C}_{50}$ need not equal a binding dissociation constant, because downstream amplification and measurement time also shape the response.

A dose-response relationship connects an administered amount or exposure level with a defined biological response under specified conditions. Dose, dosing schedule, route and observation time all matter. Pharmacokinetics links administration to the concentration history, while pharmacodynamics links exposure with the response; a dose-response curve combines those influences.

Graded dose response describes the magnitude of a response, such as a measured change in activity. Quantal dose response describes the fraction of a population meeting a specified response criterion. Half of a maximum effect and a response in half of a population are different quantities; a midpoint parameter must say which is meant.

For an original graded-response illustration, a saturating Hill equation model isHere $E_{\max }$ is an increment above baseline, $d_{50}$ gives half that increment and $n>0$ controls the shape in the stipulated model. With $n=1$, $E_0=0$, $E_{\max }=100$ response units and $d_{50}=3$ arbitrary dose units, doses $1$, $3$ and $9$ produce effects $25$, $50$ and $75$. Tripling dose from $3$ to $9$ does not triple the response. These arbitrary values are a mathematical example, not a drug regimen.

Estimation requires a design that separates dose from other changing conditions. If dose is increased on successive days while the condition also improves over time, a rising response cannot automatically be attributed to dose. The primary ICH-E4 guidance explicitly discusses such titration/time confounding and distinguishes population-average from individual response curves.^[ref]

A controlled experiment specifies its endpoint, comparator, exposure interval and uncertainty. If the measured range never reaches a plateau, the fitted maximum and midpoint may be poorly determined. Desirable and undesirable responses can have different curves, so one endpoint’s midpoint does not summarize all consequences of exposure. The scientific purpose is a reproducible relationship for a stated endpoint and setting, rather than an assumption that more exposure must always be better.

Quantal dose response describes the probability or proportion of subjects meeting a specified binary response criterion at each exposure level. The response is yes/no for that criterion, rather than the magnitude of the underlying change. A curve therefore requires both the exposure definition and the endpoint used to classify a responder.^[ref]

Let $B_i(d)$ be $1$ when subject $i$ meets the criterion at dose $d$ and $0$ otherwise. The population quantity isUnder independent subjects with the same probability, $k(d)$ has a binomial distribution. With $12$ responders among $20$ subjects, the estimate is $0.60$ and its plug-in standard error isThat is a sampling calculation, not a claim that each person has a $60\%$ effect magnitude or that the estimate is exact.

An original threshold model illustrates a possible origin of the curve. Suppose subjects have response thresholds $T$, with response occurring when $d\ge T$. Then $p(d)=\mathrm{Pr}(T\le d)$, a cumulative distribution function. Equal proportions with thresholds $1$, $2$, $4$ and $8$ arbitrary dose units give a responder fraction $1/2$ at dose $2$. Those subjects’ effect magnitudes are still unspecified. Real response need not follow a deterministic threshold for each person.

A dose associated with $50\%$ responders is consequently not the same parameter as a dose producing half a maximum graded dose response. Changing the responder criterion changes the quantal curve even if the underlying observations do not change.

The study design must handle observation time, population composition and repeated measurements. Several observations from one person are not automatically independent extra subjects. ICH-E4’s distinction between individual and population-average dose-response information is relevant to interpreting the resulting curve.^[ref] The curve summarizes a defined population and endpoint rather than supplying an individual dosing rule.

Graded dose response describes how the magnitude of a defined biological effect changes with dose or concentration under a specified protocol. The observable can be an activity, force, concentration or other quantitative endpoint. It differs from quantal dose response, which records whether a specified criterion is met. The EPA’s primary terminology explicitly separates degree of response from incidence or probability.^[ref]

For an original saturating concentration-response model, writeThe plateau $E_{\mathrm{top}}$ is an absolute response level, while $E_{\mathrm{top}}-E_0$ is the maximum change above baseline. At $c=K$, the response is halfway between baseline and plateau. If they are $20$ and $100$ response units, that midpoint is $60$, not $50$. Stating the baseline convention is essential when comparing midpoint parameters.

The graded mean also does not determine a binary responder fraction. Two original four-observation examples, $(20,20,20,20)$ and $(0,0,40,40)$, both have mean $20$. With a response criterion of at least $30$, their responder fractions are $0$ and $1/2$. Converting a continuous measurement into a binary outcome discards information about response magnitude.

A controlled assay measures matched concentrations at stated observation times with suitable baseline and reference conditions. Pharmacokinetics can make an administered dose produce different concentration histories, and biological time dependence can change the effect even at the same nominal concentration. ICH-E4 discusses concentration-response information and the interaction of response with time.^[ref]

A fitted curve is a model for the observed endpoint, not proof of a particular receptor mechanism. An apparent plateau may instead reflect an instrument limit or the tested range. Graded dose response is most useful when the measured quantity, normalization, exposure definition and uncertainties are all explicit.