IV Bolus & Oral Dosing — One-Compartment Model

IV Bolus & Oral Dosing — One-Compartment Model

1. IV Bolus: Differential Equation and Solution

After instantaneous IV bolus of dose $D$, the plasma concentration $C(t)$ satisfies:

$$\frac{dC}{dt} = -k_e C, \quad C(0) = C_0 = D/V_d$$

Solution: $C(t) = C_0 e^{-k_e t}$. Key derived parameters:

• Half-life: $t_{1/2} = \ln 2 / k_e = 0.693/k_e$
• Clearance: $CL = k_e V_d$ (volume cleared per unit time, L/h)
• AUC$_{0\to\infty}$: $\int_0^\infty C(t)\,dt = C_0/k_e = D/CL$

These three relationships — $C_0 = D/V_d$, $CL = k_e V_d$, $AUC = D/CL$ — are the PK holy trinity. Clearance is a physiological parameter (renal + hepatic clearance); $V_d$ reflects drug distribution in tissues.

2. Oral Dosing: First-Pass Effect & Bioavailability

Oral absorption follows first-order kinetics with rate constant $k_a$. The plasma concentration:

$$C(t) = \frac{F \cdot D \cdot k_a}{V_d(k_a - k_e)}\left(e^{-k_e t} - e^{-k_a t}\right)$$

where $F$ is the bioavailability (fraction of dose reaching systemic circulation). Peak concentration occurs at:

$$t_{max} = \frac{\ln(k_a/k_e)}{k_a - k_e}$$

$C_{max} = C(t_{max})$. AUC$_{oral} = FD/CL$ — the bioavailability $F$ scales the AUC; $F < 1$ due to incomplete absorption and first-pass hepatic metabolism. For drugs with high hepatic extraction (e.g., morphine, lidocaine, propranolol), $F = 1 - E_H$ where $E_H$ is the hepatic extraction ratio.

3. Multiple Dosing & Steady State

After $n$ doses at interval $\tau$, the trough concentration (just before the next dose) accumulates geometrically:

$$C_{trough,n} = C_{trough,1} \frac{1 - e^{-nk_e\tau}}{1-e^{-k_e\tau}}$$

Steady-state accumulation factor: $R_{acc} = 1/(1-e^{-k_e\tau})$. At steady state:

$$C_{ss,av} = \frac{FD}{CL \cdot \tau}, \quad C_{ss,max} = \frac{FD/V_d}{1-e^{-k_e\tau}} e^{-k_e t_{lag}}$$

Time to reach 99% of steady state $\approx 6.6 \times t_{1/2}$ — crucial for loading dose considerations in critical care (e.g., digoxin, aminoglycosides).

4. Nonlinear (Michaelis-Menten) Pharmacokinetics

Some drugs (phenytoin, ethanol, aspirin at high doses) saturate hepatic enzymes, exhibiting nonlinear (Michaelis-Menten) kinetics:

$$\frac{dC}{dt} = -\frac{V_{max}\cdot C}{K_m + C}$$

When $C \ll K_m$: pseudo first-order; when $C \gg K_m$: zero-order elimination (constant $V_{max}$ mg/h regardless of concentration). Small dose changes near $K_m$ produce disproportionately large changes in $C_{ss}$, explaining phenytoin's narrow therapeutic index.