Training Pharmacodynamics — Receptor Theory, Dose-Response & PK/PD Integration Hill–Emax Model & Receptor Occupancy Theory
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Hill–Emax Model & Receptor Occupancy Theory

60 min Pharmacodynamics — Receptor Theory, Dose-Response & PK/PD Integration

Hill–Emax Model & Receptor Occupancy Theory

The Hill–Emax (or Hill–Langmuir) equation is the canonical pharmacodynamic model relating drug concentration to biological effect. Its derivation from receptor occupancy theory (Langmuir adsorption isotherm) provides a mechanistic foundation, while the Hill coefficient $n$ (cooperativity parameter) captures the steepness of the concentration-effect relationship and determines the width of the therapeutic window.

1. Receptor Occupancy Theory (Clark, 1926)

Drug $D$ binds reversibly to receptor $R$ to form the drug-receptor complex $DR$:

$$D + R \xrightleftharpoons[k_{off}]{k_{on}} DR$$

At equilibrium, the dissociation constant $K_D = k_{off}/k_{on}$. Fractional occupancy (fraction of receptors occupied):

$$\rho = \frac{[DR]}{[R]_{tot}} = \frac{[D]}{[D] + K_D}$$

If effect $E$ is proportional to occupancy: $E = E_{max} \cdot \rho = E_{max}[D]/([D]+EC_{50})$ where $EC_{50} = K_D$. The simpler case of no baseline effect, $E_0 = 0$.

2. The Hill–Emax Equation

The empirical Hill equation generalises the hyperbolic model to allow a sigmoidal curve:

$$E(C) = E_0 + \frac{(E_{max} - E_0) \cdot C^n}{EC_{50}^n + C^n}$$

Parameters:

  • $E_0$ — baseline effect (in absence of drug)
  • $E_{max}$ — maximum drug effect (at saturating concentration)
  • $EC_{50}$ — concentration producing 50% of maximum effect (potency)
  • $n$ — Hill coefficient (slope factor, cooperativity): $n=1$ is hyperbolic; $n>1$ sigmoidal; $n<1$ flat

Derived concentrations for fractional effect $f = (E-E_0)/(E_{max}-E_0)$:

$$EC_{f} = EC_{50} \cdot \left(\frac{f}{1-f}\right)^{1/n}$$

Therapeutic window: $EC_{10}$ to $EC_{90}$ (or $EC_{10}$ to $TC_{10}$ for toxicity). Wider window → larger $EC_{50,tox}/EC_{50,eff}$ ratio (selectivity) and/or steeper Hill slope (narrower window for same selectivity).

3. Competitive Antagonism — Schild Analysis

A competitive antagonist $I$ competes for the same binding site. Occupancy by agonist $D$ in presence of antagonist at concentration $[I]$:

$$\rho = \frac{[D]}{[D] + K_D(1+[I]/K_I)}$$

The agonist $EC_{50}$ is shifted rightward by factor $(1+[I]/K_I)$ — the "dose ratio" DR. Schild plot: $\log(DR-1)$ vs $\log[I]$ is linear with slope 1 (for purely competitive antagonism) and $x$-intercept $= pA_2 = -\log K_I$.

4. Graded vs Quantal Dose-Response

Graded responses (one subject, vary dose): the Hill-Emax curve. Quantal responses (population, binary endpoint — response or no response): the cumulative log-normal distribution of threshold concentrations. $ED_{50}$ (quantal) is the dose producing a response in 50% of subjects — related to but not identical to $EC_{50}$ (graded). $TD_{50}$ and $LD_{50}$ are the toxic and lethal doses in 50% of subjects; $TI = LD_{50}/ED_{50}$ is the therapeutic index.

Worked Example — Hill Equation Computations

Drug with $E_0 = 0$, $E_{max} = 100\%$, $EC_{50} = 5$ mg/L, $n = 2$. At $C = 5$: $E = 100 \times 25/(25+25) = 50\%$. ✓ At $C = 10$: $E = 100 \times 100/(25+100) = 80\%$. $EC_{90} = 5 \times (9)^{1/2} = 15$ mg/L. $EC_{10} = 5 \times (1/9)^{1/2} = 1.67$ mg/L. Concentration range $EC_{10}$→$EC_{90}$: 9-fold (much narrower than n=1 case: 81-fold). A steep Hill coefficient narrows the effective dosing window.

Hill–Emax Dose-Response Model
Effect at dose =?%
% of E_max =?%
EC₁₀ =?mg/L
EC₉₀ =?mg/L

Practice Problems

1. A drug has $E_{max} = 80\%$, $EC_{50} = 2$ mg/L, $n = 1$, $E_0 = 10\%$. Compute the effect at concentrations 0.5, 2, 5, and 20 mg/L. Plot the curve on a semi-log scale and identify the linear region (within $\pm 10\%$ of the tangent at $EC_{50}$).
2. An antagonist at [I] = 10 nM shifts the agonist $EC_{50}$ from 5 nM to 35 nM. Compute the dose ratio and $K_I$ of the antagonist. Is this result consistent with competitive antagonism? How would you distinguish it from non-competitive antagonism experimentally?
3. Compare the Hill equations with $n=1$ and $n=4$ for drugs with the same $EC_{50}$ and $E_{max}$. Compute the $EC_{10}$-to-$EC_{90}$ concentration ratio for each. Comment on the implications for therapeutic window management.
PK/PD Integration — Effect Compartment & Indirect Response Models