Pharmacodynamics
Pharmacodynamics describes how drug concentration is related to biological or clinical effect. Pharmacokinetics tells us how concentration changes through time; pharmacodynamics tells us what that concentration does.
From concentration to effect
The pharmacodynamic model is the mathematical rule connecting concentration \(C\) to effect \(E\).
The Emax model
A standard saturating model is
\[\boxed{E(C)=E_0+\frac{E_{\max}C}{EC_{50}+C}.}\]| Symbol | Meaning |
|---|---|
| \(E(C)\) | effect at concentration \(C\) |
| \(E_0\) | baseline effect when drug concentration is zero |
| \(E_{\max}\) | maximum additional drug effect above baseline |
| \(EC_{50}\) | concentration producing half of \(E_{\max}\) |
Why the response saturates
At low concentration, many drug targets are still unoccupied or the downstream system has unused capacity, so increasing concentration can strongly increase effect.
At high concentration, most of the available effect has already been recruited. The response therefore approaches
\[\boxed{E_0+E_{\max}.}\]The Emax curve
Why EC50 means half-maximal effect
Set \(C=EC_{50}\):
\[E(EC_{50})=E_0+\frac{E_{\max}EC_{50}}{EC_{50}+EC_{50}}.\]Therefore
\[\boxed{E(EC_{50})=E_0+\frac{E_{\max}}2.}\]So \(EC_{50}\) controls where the concentration–effect curve reaches half of its drug-induced maximum.
Effect of changing EC50
A smaller \(EC_{50}\) shifts the curve to the left: the same effect is achieved at a lower concentration. A larger \(EC_{50}\) shifts it to the right.
This parameter is often used as a measure of potency within the model.
Potency is not the same as maximum effect
A drug can require a low concentration to produce an effect but still have a relatively small maximum effect. Another drug may require a higher concentration but ultimately produce a larger effect.
Thus \(EC_{50}\) and \(E_{\max}\) describe different properties.
Effect of changing Emax
Increasing \(E_{\max}\) raises the plateau of the concentration–effect curve. It changes how large the maximum additional response can become.
Changing \(E_{\max}\) does not, by itself, imply that \(EC_{50}\) changes.
Hill Emax model
Some concentration–effect relationships are steeper than the simple Emax curve. A Hill coefficient can be added:
\[\boxed{E(C)=E_0+\frac{E_{\max}C^n}{EC_{50}^n+C^n}.}\]The parameter \(n\) controls steepness.
What the Hill coefficient means
When \(n=1\), the model reduces to the ordinary Emax equation. Larger \(n\) gives a sharper, more switch-like response.
Inhibitory effects
Some drugs reduce a biological response rather than increasing it. One common inhibitory model is
\[\boxed{E(C)=E_0\left(1-\frac{I_{\max}C}{IC_{50}+C}\right).}\]Here \(I_{\max}\) is the maximum fractional inhibition and \(IC_{50}\) is the concentration giving half of that maximum inhibition.
Direct-effect models
The simplest pharmacodynamic models assume effect responds immediately to concentration:
\[E(t)=E(C(t)).\]If concentration rises or falls, effect changes at the same moment according to the concentration–effect function.
Why effect can lag behind concentration
In many systems, effect does not respond instantly. Drug may need to distribute to an effect site, bind slowly, trigger a signalling cascade or alter gene expression.
Then the concentration measured in plasma can peak before the biological effect peaks.
Effect-compartment model
A simple way to represent delay is to introduce an effect-site concentration \(C_e\):
\[\boxed{\frac{dC_e}{dt}=k_{e0}(C-C_e).}\]The pharmacodynamic effect is then calculated from \(C_e\), not directly from plasma concentration:
\[E(t)=E(C_e(t)).\]The parameter \(k_{e0}\) controls how quickly the effect site follows plasma concentration.
Hysteresis
If effect is delayed relative to concentration, the same plasma concentration can correspond to different effects depending on whether concentration is rising or falling.
A plot of effect against plasma concentration can therefore form a loop rather than a single curve. This is pharmacodynamic hysteresis.
Indirect-response models
Some drugs do not directly create the measured response. Instead they alter the production or loss of a physiological quantity.
Suppose a biomarker \(R\) is produced at rate \(k_{in}\) and removed at rate \(k_{out}R\):
\[\frac{dR}{dt}=k_{in}-k_{out}R.\]A drug might inhibit production:
\[\boxed{\frac{dR}{dt}=k_{in}\left(1-\frac{I_{\max}C}{IC_{50}+C}\right)-k_{out}R.}\]Because the drug acts on a turnover process, the response can remain delayed even if the concentration–effect interaction itself is immediate.
Baseline in an indirect-response model
Without drug, the steady state satisfies
\[0=k_{in}-k_{out}R_0.\]Therefore
\[\boxed{R_0=\frac{k_{in}}{k_{out}}.}\]The drug perturbs this physiological balance.
Tolerance
Repeated or sustained exposure can sometimes produce a smaller effect over time even when concentration remains similar. This can be represented through receptor adaptation, feedback or a separate tolerance compartment.
A simple conceptual model introduces a tolerance variable \(T(t)\) that increases with exposure and reduces effect.
Rebound
If a drug suppresses a physiological process and compensatory regulation develops, stopping the drug can temporarily push the system beyond its original baseline. This is called rebound.
Feedback-based pharmacodynamic models can reproduce such behaviour.
Therapeutic and adverse effects
The same concentration can influence several outcomes. One effect model may describe desired response while another describes toxicity.
This allows mathematical comparison of benefit and harm across concentrations, but clinical interpretation requires drug-specific evidence.
PK–PD modelling
A complete PK–PD model combines a pharmacokinetic model for concentration with a pharmacodynamic model for effect:
\[\boxed{\text{Dose}\rightarrow C(t)\rightarrow E(t).}\]For example, an IV one-compartment PK model may give
\[C(t)=C_0e^{-kt},\]and the Emax model then gives
\[E(t)=E_0+\frac{E_{\max}C_0e^{-kt}}{EC_{50}+C_0e^{-kt}}.\]Concentration versus exposure
Some effects depend mainly on instantaneous concentration. Others depend on cumulative exposure, peak concentration, duration above a threshold or repeated stimulation.
The pharmacodynamic model should therefore be chosen according to the biological mechanism and the measured endpoint.
Parameter estimation
Pharmacodynamic parameters are estimated by fitting concentration–effect models to observed data. Important parameters may include \(E_0\), \(E_{\max}\), \(EC_{50}\), Hill coefficient, effect-site equilibration rate and turnover parameters.
Because concentration and effect are measured with uncertainty, statistical modelling is usually combined with the underlying dynamical equations.
Population pharmacodynamics
Different individuals can have different sensitivities and maximum responses. Population models describe typical parameter values together with between-person variability.
PK variability and PD variability are distinct: two individuals can have the same concentration but different effects, or different concentrations but similar effects.
Mechanistic versus empirical PD models
An Emax model is often an effective input–output description. More mechanistic models may include receptor binding, signalling pathways, cell turnover or disease dynamics.
The appropriate level of detail depends on the biological question and available data.
A practical modelling workflow
Define the biological effect being measured. Decide whether effect is immediate or delayed. Choose a direct Emax, Hill, inhibitory, effect-compartment or indirect-response structure as appropriate. Link it to the PK concentration model. Estimate parameters from concentration and effect data. Examine baseline, maximum effect, potency, delay and variability. Finally, test whether the model reproduces both the magnitude and timing of the observed response.