PK/PD Integration — Effect Compartment & Indirect Response Models

PK/PD Integration — Effect Compartment & Indirect Response Models

1. Effect-Compartment (Biophase) Model

The effect site does not contribute significantly to drug mass balance (negligible volume). Drug distributes from plasma to effect site with rate constant $k_{eo}$ (equilibration rate):

$$\frac{dC_e}{dt} = k_{eo}(C_p(t) - C_e(t))$$

where $C_e$ is the effect-site concentration and $C_p(t)$ is the plasma concentration (known from PK model). Effect is driven by $C_e$, not $C_p$:

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

The hysteresis loop (E vs $C_p$ plot) collapses to a monotonic relationship when $E$ is plotted against $C_e$. The peak effect-site concentration lags behind peak plasma concentration by $t_{lag} \approx \ln(k_{eo}/k_e)/(k_{eo}-k_e)$ for a one-compartment PK model.

2. Effect-Site Equilibration Half-Life

$t_{\frac{1}{2},k_{eo}} = \ln 2 / k_{eo}$. Small $k_{eo}$ (long $t_{\frac{1}{2},k_{eo}}$) = slow equilibration, late and blunted effect; large $k_{eo}$ (short $t_{\frac{1}{2},k_{eo}}$) = rapid equilibration, effect closely tracks plasma. Clinical examples:

• Fentanyl: $t_{\frac{1}{2},k_{eo}} \approx 4$ min (fast CNS equilibration)
• Morphine: $t_{\frac{1}{2},k_{eo}} \approx 17$ min (slower, less lipophilic)
• Digoxin: $t_{\frac{1}{2},k_{eo}} \approx 200$ min (very slow myocardial equilibration)

3. Indirect Response Models (Mechanism-Based)

Many drugs modulate a physiological response mediated by a turnover process — they stimulate or inhibit the production ($k_{in}$) or degradation ($k_{out}$) of a response variable $R$:

$$\frac{dR}{dt} = k_{in} \cdot [1 \pm H(C)] - k_{out} \cdot R \cdot [1 \pm H(C)]$$

where $H(C) = E_{max}C/(EC_{50}+C)$ is the Hill function. At baseline: $R_0 = k_{in}/k_{out}$. Model I (inhibit $k_{in}$): warfarin on prothrombin. Model II (inhibit $k_{out}$): corticosteroids on cortisol. Model III (stimulate $k_{in}$): erythropoietin on RBC. Model IV (stimulate $k_{out}$): β-agonists on heart rate. The indirect models inherently produce a temporal delay without an effect compartment.

4. Tolerance & Clock Models

Tolerance (reduced response to the same concentration over time) arises from receptor downregulation, counter-regulatory feedback, or precursor depletion. The counter-regulatory mechanism model:

$$E_{obs}(t) = E_{direct}(C(t)) - E_{tolerance}(C(t), R(t))$$

where $R(t)$ is a tolerance state variable driven by $C(t)$. Opioid analgesic tolerance, nitrate tolerance in cardiovascular disease, and methylphenidate tolerance in ADHD are classical examples studied with this framework.