Pharmacokinetics — ADME, Compartment Models and Drug Half-Life

When you swallow a pill, the drug doesn't instantly appear everywhere in your body at therapeutic concentration — it follows a precise mathematical trajectory governed by absorption, distribution, metabolism, and excretion. Understanding this trajectory is how physicians determine dose, frequency, and route of administration. It is also a beautiful example of exponential decay and ODE modeling in the real world.

1. ADME — The Four Processes

Absorption: the process by which drug moves from the administration site into the systemic bloodstream. Intravenous (IV) injection bypasses absorption entirely (100% bioavailability). Oral drugs must survive gastric acid, traverse the intestinal epithelium by passive diffusion or active transport, and pass through the liver before reaching the heart.
Distribution: drug spreads from plasma to tissues. The volume of distribution V_d = dose / plasma concentration expresses how extensively a drug distributes outside the bloodstream. Lipophilic drugs penetrate cell membranes and have very high V_d (hundreds of liters for chloroquine); hydrophilic drugs stay mostly in plasma (V_d ≈ 3–5 L).
Metabolism: primarily enzymatic transformation in the liver by cytochrome P450 enzymes (CYP3A4, CYP2D6, etc.) converting the active drug to typically more water-soluble, less active metabolites. Some prodrugs (codeine → morphine) require metabolism to become active.
Excretion: elimination of drug and metabolites from the body — primarily renal (glomerular filtration + tubular secretion − reabsorption) and biliary (elimination into bile → feces).

First-pass effect: orally administered drugs absorbed from the gut travel via the portal vein to the liver before reaching systemic circulation. High-clearance drugs (morphine, lidocaine, propranolol) are extensively metabolized in this first pass, dramatically reducing oral bioavailability (F). Morphine: F ≈ 30%; propranolol: F ≈ 25%.

2. One-Compartment IV Model

The simplest model treats the body as a single well-mixed compartment. After an IV bolus dose D, the drug is instantly present throughout the distribution volume V_d, then eliminated at a rate proportional to concentration — first-order kinetics:

dC/dt = −k_e · C Solution: C(t) = C₀ · e^{−k_e · t} C₀ = initial concentration = D / V_d k_e = elimination rate constant (1/h) Parameters interrelated by: k_e = CL / V_d CL = clearance (L/h) — volume of blood cleared of drug per unit time

The plasma concentration profile is a simple exponential decay. Plotting log(C) vs time gives a straight line with slope −k_e.

3. Oral Dosing — Absorption Phase

For oral administration, absorption is a first-order input (rate proportional to drug remaining at absorption site × bioavailability F):

dC/dt = (F · D · k_a / V_d) · e^{-k_a·t} − k_e · C k_a = absorption rate constant (typically k_a >> k_e for most drugs) Solution (one-compartment oral): C(t) = (F·D·k_a) / (V_d·(k_a−k_e)) · (e^{−k_e·t} − e^{−k_a·t}) Peak concentration (t_max): t_max = ln(k_a/k_e) / (k_a − k_e) Typical oral: t_max = 0.5–2 h (aspirin ~1h, ibuprofen ~1.5h, amoxicillin ~1h)

The concentration-time curve rises to a peak (C_max) then falls. The area under the curve (AUC) is proportional to total drug exposure and equal to F·D/CL.

4. Two-Compartment Model

Many drugs exhibit a biphasic decay: a rapid initial distribution phase (α, drug moving from plasma to peripheral tissues) followed by a slower elimination phase (β, drug returning from tissues and being eliminated):

Compartment 1 (central, plasma): dC₁/dt = −(k₁₂ + k₁₀)·C₁ + k₂₁·C₂ + I(t) Compartment 2 (peripheral, tissues): dC₂/dt = k₁₂·C₁ − k₂₁·C₂ k₁₂ = central-to-peripheral rate k₂₁ = peripheral-to-central rate k₁₀ = elimination from central compartment Biexponential solution: C₁(t) = A·e^{−αt} + B·e^{−βt} α ≈ fast distribution rate (t₁/₂α ≈ minutes to hours) β ≈ slow elimination rate (t₁/₂β ≈ hours to days)

Why it matters clinically: if you estimate half-life from only the elimination phase (β), you get a much longer value than the initial distribution phase suggests. For drugs with large peripheral volumes (digoxin, amiodarone), dosing based on the wrong half-life can cause toxicity or therapeutic failure.

5. Half-Life and Steady State

The elimination half-life t₁/₂ is the time for plasma concentration to fall by 50%:

t₁/₂ = ln(2) / k_e ≈ 0.693 / k_e = 0.693 · V_d / CL Drug accumulation (multiple dosing): After n doses, the trough concentration approaches steady state. C_ss_trough = F·D / CL · 1/(e^{k_e·τ} − 1) (oral, single compartment) ~97% of steady state reached after 5 half-lives. Common drug half-lives: Ibuprofen: t₁/₂ ≈ 2 h (dose every 6-8h) Warfarin: t₁/₂ ≈ 40 h (take days to reach therapeutic range) Amiodarone: t₁/₂ ≈ 58 days (accumulates for months!)

The loading dose concept: to instantly achieve therapeutic concentration without waiting 5 half-lives, administer a large loading dose (based on V_d) followed by smaller maintenance doses every τ hours (based on clearance):

6. Nonlinear Kinetics — Michaelis-Menten

Most drugs follow first-order kinetics because enzyme systems are not saturated at therapeutic concentrations. But some drugs (phenytoin, aspirin at high doses, ethanol) saturate their metabolic pathways — elimination switches to zero-order (constant rate regardless of concentration):

Michaelis-Menten elimination: dC/dt = -(V_max \cdot C) / (K_m + C) V_max = max metabolic rate (mg/L/h) K_m = Michaelis constant (concentration at ½ V_max) When C << K_m: dC/dt \approx -(V_max/K_m)\cdotC (first-order, k_e = V_max/K_m) When C >> K_m: dC/dt \approx -V_max (zero-order, constant elimination) Phenytoin (antiepileptic): K_m \approx 5 mg/L, therapeutic range 10-20 mg/L \to operates in the saturating zone: small dose increases \to large concentration jumps!

7. Clinical Applications

Therapeutic drug monitoring (TDM): measuring blood levels of narrow-therapeutic-index drugs (vancomycin, digoxin, lithium, cyclosporine) to ensure concentrations stay within the therapeutic window and avoid toxicity.
Renal dose adjustment: drugs cleared renally have reduced CL in patients with chronic kidney disease (CKD). Dose must be reduced proportionally to GFR decline (e.g., metformin, gabapentin).
Drug-drug interactions: CYP450 inhibitors (fluconazole, clarithromycin) reduce clearance of co-administered drugs → toxicity. Inducers (rifampin, phenobarbital) increase clearance → treatment failure.
Pediatric dosing: children are not small adults — V_d, protein binding, renal function, and enzyme expression all change dramatically from neonate to adolescent. Allometric scaling (dose ∝ weight^0.75) approximates metabolic clearance.

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