15.1 Half-Life, Clearance & Kinetic Order

First-Order vs Zero-Order Kinetics

First-order (linear) kinetics means the rate of elimination is directly proportional to the plasma concentration: a constant fraction of drug is removed per unit time. Because the fraction is constant, the elimination half-life (t½) is also constant at any concentration. On a concentration–time graph with a linear y-axis the curve is a declining exponential that steeply falls then flattens; on a semi-logarithmic plot (log concentration versus time) it becomes a straight line with slope −k/2.303. The overwhelming majority of drugs used at therapeutic doses follow first-order kinetics, and every routine calculation in this chapter assumes it unless stated otherwise.

Zero-order (saturation) kinetics means the elimination pathways — enzymes or transporters — are fully saturated, so a constant amount of drug is removed per unit time (for example 10 mg per hour) regardless of concentration. The concentration–time graph is a straight downward line on linear axes, and there is no constant half-life: the apparent t½ gets longer as the concentration rises. The classic exam examples of drugs showing zero-order behaviour at therapeutic or commonly encountered doses are:

• Phenytoin — hepatic metabolism saturates within the therapeutic range
• Ethanol — alcohol dehydrogenase saturates even at social doses
• Salicylates (high-dose aspirin) — glycine conjugation saturates
• Theophylline — can switch to nonlinear elimination at higher levels, particularly in overdose

A small dose increase of such a drug can produce a disproportionately large rise in plasma concentration, which is exactly why phenytoin and theophylline are monitored so carefully.

The Elimination Rate Constant and Half-Life

The elimination rate constant (k) is the fraction of drug eliminated per unit time, with units of time⁻¹ (for example h⁻¹). Half-life is derived from it by:

t½ = 0.693 / k

where 0.693 is the natural logarithm of 2. Worked example: if k = 0.15 h⁻¹, then t½ = 0.693 / 0.15 = 4.62 hours. Conversely, if a level falls from 16 mg/L to 4 mg/L in 8 hours, that is two half-lives, so t½ = 4 hours and k = 0.693 / 4 = 0.173 h⁻¹.

Because half the drug is lost each half-life, the fraction remaining after n half-lives is (1/2)ⁿ:

This produces the most quoted rule in clinical pharmacokinetics: after 4–5 half-lives a drug is for practical purposes completely eliminated (about 94–97% gone), and — by exactly the same mathematics — a fixed maintenance regimen reaches steady state in 4–5 half-lives, regardless of dose size. Doubling the dose raises the eventual steady-state level but does not shorten the time to get there.

Clearance

Clearance (CL) is the volume of plasma (or blood) irreversibly cleared of drug per unit time, expressed in mL/min or L/h. It is the single most useful pharmacokinetic parameter because it directly determines the maintenance dose. Total (systemic) clearance is additive across organs:

CL_total = CL_renal + CL_hepatic + CL_other

For a drug eliminated mainly by glomerular filtration of unchanged drug (for example digoxin or gentamicin), renal clearance dominates and falls in parallel with creatinine clearance. For a drug eliminated mainly by hepatic metabolism (for example phenytoin or theophylline), hepatic clearance dominates and is sensitive to liver disease, enzyme induction and enzyme inhibition. Knowing which organ clears the drug tells you which organ failure will prolong the half-life.

Linking Vd, CL and Half-Life

Half-life is not an independent property; it is the visible consequence of volume and clearance:

t½ = 0.693 × Vd / CL

Worked example: a drug has Vd = 50 L and CL = 5 L/h. Then t½ = 0.693 × 50 / 5 = 6.93 hours. Now consider two clinical changes. If oedema or ascites doubles the apparent Vd to 100 L while clearance stays 5 L/h, t½ doubles to 13.9 hours — the drug is more spread out, so proportionally less is presented to the clearing organs each hour. If renal failure halves clearance to 2.5 L/h with Vd unchanged, t½ again doubles to 13.9 hours. The exam loves this logic: t½ rises when Vd rises or when CL falls, and falls in the opposite situations. This is why half-life alone is a poor guide to dosing — two patients can share a half-life for completely different reasons.

Michaelis-Menten (Saturation) Kinetics

Between pure first-order and pure zero-order behaviour lies capacity-limited (Michaelis-Menten) kinetics, described by:

Rate of elimination = Vmax × C / (Km + C)

where Vmax is the maximum rate of metabolism and Km is the concentration at half-maximal rate. When C is far below Km the denominator is approximately Km and elimination looks first-order; when C approaches or exceeds Km the rate approaches Vmax and elimination looks zero-order. Phenytoin sits in this transition zone at therapeutic doses, which makes its steady-state concentration rise nonlinearly with dose. Rearranged for maintenance dosing, the steady-state concentration is:

Css = Km × R / (Vmax − R)

Worked example: a patient has Vmax = 500 mg/day and Km = 4 mg/L. At a dose R = 300 mg/day, Css = 4 × 300 / (500 − 300) = 1200 / 200 = 6 mg/L. Increasing the dose by only a third, to 400 mg/day, gives Css = 4 × 400 / (500 − 400) = 1600 / 100 = 16 mg/L — a 33% dose increase produced a 167% concentration increase. The practical rule: adjust phenytoin in small increments (25–50 mg), allow extra time for the new steady state (nonlinear drugs can take weeks to stabilise), and never extrapolate dose and level proportionally.