3. Pharmacodynamics

The simplest pharmacodynamic model assumes a hyperbolic relationship between drug concentration and effect, directly analogous to Michaelis-Menten enzyme kinetics:

Where $E_{\max}$ is the maximum achievable effect and $EC_{50}$is the concentration producing 50% of maximum effect. The sigmoid Emax model introduces the Hill coefficient $\gamma$:

The slope of the concentration-effect curve at $EC_{50}$ is:

When $\gamma = 1$, the effect goes from 20% to 80% of maximum over an 81-fold concentration range. When $\gamma = 3$, this range narrows to only 4.3-fold, creating a steep "switch-like" response that can be clinically dangerous if the dose varies near $EC_{50}$.

The therapeutic index (TI) quantifies the margin between therapeutic and toxic doses:

For clinical safety, the certain safety factor (CSF) is more conservative:

Using the sigmoid Emax model, $ED_x$ (the dose producing x% of max effect) is:

Therefore:

Drugs with TI less than 2 (warfarin, lithium, digoxin, theophylline, phenytoin) require therapeutic drug monitoring. The narrower the TI, the more critical are factors affecting drug levels: renal/hepatic impairment, drug interactions, genetic polymorphisms, and adherence.

Selectivity is the ratio of potencies at different receptor subtypes. For a drug acting at receptor A (desired) and receptor B (undesired):

At a given concentration C, the differential activation is:

The concentration that maximizes the selectivity window (maximum $\Delta E$) is:

At this optimal concentration, the maximum selectivity window is:$\Delta E_{\max} = \left(\frac{\sqrt{S} - 1}{\sqrt{S} + 1}\right)^2$ where$S = EC_{50,B}/EC_{50,A}$. A 100-fold selectivity ratio gives only 67% differential activation, explaining why even "selective" drugs have off-target effects at high doses.

The effect compartment resolves PK/PD hysteresis by introducing a hypothetical compartment representing the biophase (site of action):

Where $k_{e0}$ is the equilibration rate constant. The effect compartment has negligible volume (does not affect PK). After an IV bolus:

The time to peak effect site concentration:

The $t_{1/2,k_{e0}}$ characterizes the delay between plasma concentration changes and effect. For propofol, $t_{1/2,k_{e0}} \approx 2.6$ min (rapid equilibration with brain). For warfarin, $t_{1/2,k_{e0}} \approx 24$ hours (slow clotting factor turnover).

Many drugs act by modifying the production or elimination of an endogenous mediator. The indirect response model describes the response variable R:

At baseline, $R_0 = k_{\text{in}}/k_{\text{out}}$. Four types exist based on whether the drug stimulates or inhibits production or elimination:

The maximum drug-induced change in the response at steady state (for Type I):

Type I: inhibition of production (warfarin inhibiting clotting factor synthesis, corticosteroids suppressing cortisol). Type II: stimulation of elimination (diuretics enhancing sodium excretion). Type III: stimulation of production (erythropoietin). Type IV: inhibition of elimination (probenecid blocking uric acid excretion).