Pharmacology
Half-life, volume of distribution and clearance
Half-life is the time for a drug's plasma concentration to fall by half, and it depends on how widely the drug spreads and how fast the body clears it.
Remember
Half-life goes up when clearance (the volume of plasma that is completely cleared of drug in each unit of time) goes down, so with the volume of distribution (how widely the drug spreads through the body) unchanged, halving clearance doubles the half-life.
Half-life (t½) is the time taken for the plasma concentration of a drug to fall by half. Under first-order kinetics (a constant fraction of the drug present is removed in each unit of time), the half-life is the same at any starting concentration and equals 0.693 × Vd ÷ Cl (volume of distribution divided by clearance), where 0.693 is the natural logarithm of 2.
Vd is the volume of distribution, the theoretical volume of fluid the drug would need to be dissolved in to give the measured plasma concentration. Cl is clearance, the volume of plasma completely cleared of the drug per unit time.
Clearance is the number you divide by. Half-life lengthens when clearance falls, and also when volume of distribution rises. Picture a bath: the volume is its size and clearance is the width of the plughole, so a narrower plughole means a longer wait for half the water to drain.
If Vd is 50 L and Cl is 5 L per hour, the half-life is 0.693 × 50 ÷ 5 = 6.93, so about 6.9 hours. For a second drug with a 4-hour half-life, halving clearance with Vd unchanged doubles the half-life to 8 hours. A 50% fall in clearance gives a 100% rise in half-life, not a 50% rise.
Acute kidney injury (a sudden loss of kidney function) lowers the clearance of a drug cleared mainly by the kidneys. Digoxin is a heart medicine cleared mainly by the kidneys, with a half-life normally around 40 hours. When the half-life lengthens in acute kidney injury, the plasma concentration rises at the same dose, which raises the risk of digoxin toxicity. A longer half-life delays steady state (the point where the amount eliminated equals the amount given, so the plasma concentration stops rising), reached after roughly five half-lives.
Memory aid
A bath makes a useful picture: the volume is its size and clearance is the width of the plughole, so a narrower plughole means a longer wait for half the water to drain.
Think of the volume of distribution as the size of the bath and clearance as the width of the plughole.
Key facts
- Equation
- for first-order kinetics, half-life = 0.693 × Vd ÷ Cl.
- Clearance
- with volume of distribution unchanged, a fall in clearance lengthens the half-life, and halving clearance doubles it.
- Volume of distribution
- the half-life also lengthens when volume of distribution rises.
- Worked sum
- Vd of 50 L and Cl of 5 L per hour give a half-life of about 6.9 hours.
- Digoxin
- cleared mainly by the kidneys, with a half-life normally around 40 hours that is prolonged in renal failure.
- Steady state
- reached after roughly five half-lives, so a longer half-life delays it.
Exam traps
- A 50% fall in clearance does not raise the half-life by 50%, because with the volume of distribution unchanged half-life is inversely proportional to clearance, so the half-life doubles, a rise of 100%.
- A first-order half-life is constant, but that is not the same as unchangeable: it is the same at any starting concentration, yet it still depends on clearance and volume of distribution.
- With Vd of 50 L and Cl of 5 L per hour, Vd ÷ Cl on its own gives 10 hours, which is not the half-life, because the ratio must be multiplied by 0.693.
- Wide spread into tissues contributes to a long baseline half-life but does not explain a rise here, because the volume of distribution is unchanged and the fall in clearance is the cause.
- A longer half-life does not bring steady state sooner, because steady state is reached after roughly five half-lives, so a longer half-life delays it.
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