2. Pharmacokinetics

After an intravenous bolus dose D, the drug distributes instantaneously into a single compartment of volume $V_d$. Elimination follows first-order kinetics:

Integrating with the initial condition $C(0) = D/V_d$:

The elimination rate constant relates to clearance and half-life:

The area under the concentration-time curve (AUC) from 0 to infinity:

This fundamental relationship $AUC = D/CL$ means clearance can be determined from any IV dosing study by measuring the AUC. The relationship is independent of the compartmental model and holds for all drugs.

For oral administration, drug must first be absorbed from the GI tract. With first-order absorption (rate constant $k_a$) and first-order elimination:

Solving these coupled differential equations yields the Bateman equation:

The time of peak concentration $t_{\max}$ is found by setting $dC/dt = 0$:

The peak concentration $C_{\max}$ is obtained by substituting $t_{\max}$back into the Bateman equation. Note that $t_{\max}$ depends only on rate constants, not on dose or bioavailability, while $C_{\max}$ is proportional to $F \cdot D$.

The apparent volume of distribution $V_d$ relates the total amount of drug in the body to the plasma concentration:

For a drug distributed between plasma (volume $V_p$) and tissue (volume $V_T$), with unbound fractions $f_u$ and$f_{uT}$ respectively:

This explains why $V_d$ can exceed total body water. If the drug binds extensively to tissues ($f_{uT}$ is small), the ratio $f_u/f_{uT}$is large, making $V_d$ very large. For the steady-state volume:

Clinically, $V_d$ determines the loading dose: $\text{Loading dose} = C_{\text{target}} \times V_d$. Drugs with large $V_d$ (chloroquine: 13,000 L) are extensively tissue-bound; those with small $V_d$ (warfarin: 8 L) are confined to plasma.

Clearance (CL) is the volume of blood completely cleared of drug per unit time. For an eliminating organ with blood flow Q and extraction ratio E:

For hepatic clearance, the well-stirred model gives:

Where $CL_{\text{int}}$ is the intrinsic clearance (enzyme capacity without flow limitations). Two limiting cases emerge:

Total systemic clearance is the sum of all organ clearances:$CL_{\text{total}} = CL_H + CL_R + CL_{\text{other}}$. Renal clearance can be further decomposed: $CL_R = f_u \cdot GFR + CL_{\text{secretion}} - CL_{\text{reabsorption}}$.

For repeated IV bolus doses D given every $\tau$ hours, the concentration after the $n$-th dose at time $t$ within the dosing interval:

As $n \to \infty$, the accumulation factor converges:

The steady-state peak and trough concentrations become:

The average steady-state concentration:

The fraction of steady state achieved after $n$ doses:$f_{ss} = 1 - e^{-nk_e\tau} = 1 - (1/2)^{n\tau/t_{1/2}}$. After 4-5 half-lives, $f_{ss} \geq 0.94$. This relationship is independent of dose, bioavailability, and clearance -- only $t_{1/2}$ determines the time to steady state.