Clearance and half-life: t½ = 0.693 × Vd / CL
How clearance and volume of distribution together set a drug's half-life, and why the half-life is the clock for reaching steady state and for washout.
Chapters (11)
Relate clearance and volume of distribution to half-life (t½ = 0.693 × Vd / CL) and predict the time to steady state (about four to five half-lives).
The short version
- Clearance is the volume of blood cleared of drug per unit time; the elimination rate equals clearance × plasma concentration.
- Half-life = 0.693 × Vd / CL, so it rises when the volume of distribution rises and when clearance falls.
- Half-life is not a measure of clearance: the same clearance can come with very different half-lives.
- Steady state, and washout, take about four to five half-lives, with about 97% of the change complete by five.
- For a first-order drug, a bigger dose changes the steady-state level, not the time it takes to arrive.
Same clearance, a 12-fold gap in half-life
Two drugs each have a clearance of 6 L/h. Drug A halves its concentration every 4 hours; drug B takes 2 days. The same clearance, yet a 12-fold gap in half-life.
This lesson explains clearance and half-life: how clearance and volume of distribution set the half-life, and how the half-life predicts the time to steady state. It builds on the previous lesson, where volume of distribution showed how much drug sits in tissue.[5][7]
Clearance: the volume of blood cleared per unit time
Clearance is the volume of blood (plasma) cleared of drug per unit time, in L/h or mL/min.[2][4]
Total clearance is the sum of the organ clearances, mainly kidney and liver.[2][6]
For first-order drugs clearance stays constant; it is the elimination rate that falls as the concentration falls.[1][4]
The half-life formula: 0.693 × Vd / CL
Half-life is the time for the plasma concentration to fall by half. It is not fixed for a drug: it depends on two other numbers, the volume of distribution and the clearance.[1]
0.693 is the natural log of 2, because we are halving.[1][4]
The formula comes from the elimination rate constant, k, which is clearance divided by volume of distribution. Half-life is 0.693 / k.[4]
| Change | Effect on half-life |
|---|---|
| Volume of distribution rises | Half-life rises |
| Clearance falls | Half-life rises |
| Volume doubled, or clearance halved | Half-life doubles |
| Volume doubled and clearance halved together | Half-life quadruples |
The two drugs solved: 4 hours versus 48 hours
Drug A has a volume of 35 L: 0.693 × 35 / 6 is about 4 hours. Drug B has a volume of 420 L, which gives about 48 hours.[1]
- Drug A: 0.693 × 35 / 6 ≈ 4 h
- Drug B: 0.693 × 420 / 6 ≈ 48 h
- Volumes differ 420 / 35 = 12-fold, and so do the half-lives
Clearance is the same, but most of drug B is hiding in tissue, out of reach of the clearing organs, so the plasma empties slowly. Half-life rises when the volume rises, and it rises when clearance falls.[1][5]
Worked example and time to steady state
A drug has a volume of 60 L and a clearance of 5 L/h. Given at a constant rate, the level climbs until the amount going in equals the amount cleared. That is steady state, and it takes about four to five half-lives to get there, whatever the dose.[3][8]
- t½ = 0.693 × 60 / 5 ≈ 8.3 h
- Steady state ≈ 5 × 8.3 ≈ 42 h
- Clearance falls to 2.5 L/h: t½ = 0.693 × 60 / 2.5 ≈ 16.6 h
- Steady state ≈ 5 × 16.6 ≈ 83 h, about 3.5 days
Each half-life closes half of the remaining gap: 50%, 75%, 87.5%, 93.75%, and about 97% after five.[1][8]
| Half-lives | Gap closed |
|---|---|
| 1 | 50% |
| 2 | 75% |
| 3 | 87.5% |
| 4 | 93.75% |
| 5 | about 97% |
Dose sets the level, not the time; washout
For a first-order drug, a bigger dose changes the level at steady state. It does not change how long it takes to arrive.[3]
The same five half-lives apply in reverse: five half-lives after stopping, about 97% of the drug has gone.[1][8]
Clearance and half-life: three things to keep
- Half-life formulaHalf-life equals 0.693 × volume of distribution / clearance.[1][4]
- Direction of changeA bigger volume or a lower clearance lengthens the half-life. Doubling the volume, or halving the clearance, doubles it.[1]
- Steady state and washoutBoth take about five half-lives, whatever the dose.[3][8]
Clearance itself is the efficiency of removal, while half-life also depends on how much of the drug sits in tissue. Keeping the two ideas apart is the point of the lesson.[1][2]
Practice question
Original · USMLE styleA 52-year-old man starts a constant intravenous infusion of a drug that follows first-order elimination. The drug's volume of distribution is 45 L and its clearance is 1.5 L/h.
Common questions
What is the half-life formula in pharmacokinetics?
Half-life = 0.693 × volume of distribution / clearance. The 0.693 is the natural log of 2, because the concentration is halving.[1]
How many half-lives does it take to reach steady state?
About four to five half-lives at a constant dosing rate. After five half-lives about 97% of the final change is complete.[3][8]
Does a bigger dose reach steady state sooner?
No. For a first-order drug, a bigger dose changes the level at steady state, not how long it takes to arrive.[3]
Transcript
Read the full narration, by chapter637 words
0:00Same clearance, a 12-fold gap in half-life
Two drugs, and each has a clearance of six litres per hour. Drug A halves its concentration every four hours. Drug B takes 2 days. Same clearance, a 12-fold gap in half-life. How? Today: relate clearance, volume of distribution and half-life, and predict the time to steady state. Last lesson, volume of distribution showed how much drug sits in tissue.
0:27Clearance: the volume of blood cleared per hour
Clearance is the volume of blood cleared of drug per unit time, in litres per hour or mL per minute. Total clearance is the sum of the organ clearances, mainly kidney and liver.
0:39Elimination rate = clearance x concentration
The elimination rate equals clearance times the plasma concentration. For first-order drugs, clearance stays constant. It is the elimination rate that falls as the concentration falls.
0:52Half-life: t1/2 = 0.693 x Vd / CL
Half-life is the time for the plasma concentration to fall by half. It is not fixed for a drug. It depends on two other numbers. Half-life equals 0.693, times the volume of distribution, divided by clearance. 0.693 is the natural log of two, because we are halving. It comes from the elimination rate constant, k, which is clearance divided by volume of distribution. Half-life is 0.693 over k.
1:25The hook solved: 4 h versus 48 h
Now our hook. Drug A has a volume of 35 litres, and 0.693 times 35 over six is about 4 hours. Drug B has a volume of 420 litres, which gives about 48 hours. Same clearance, but most of drug B is hiding in tissue, out of reach of the clearing organs, so the plasma empties slowly. Half-life rises when the volume rises, and it rises when clearance falls. Double the volume, or halve the clearance, and the half-life doubles. Do both and it quadruples.
2:04Worked example: Vd 60 L, CL 5 L/h
A worked example. A drug has a volume of 60 litres and a clearance of five litres per hour. The half-life is 0.693 times 60 over five, about 8.3 hours.
2:17Steady state takes four to five half-lives
Give it at a constant rate and the level climbs until the amount going in equals the amount cleared. That is steady state. It takes about four to five half-lives to get there, whatever the dose. Each half-life closes half of the remaining gap: 50%, 75, 87.5, 93.75, and 97% after five. Five times 8.3 is about 42 hours. Now clearance falls to 2.5 litres per hour. The half-life doubles to about 16 and a half hours, and steady state takes about three and a half days.
2:57Dose sets the level, not the time; washout
A bigger dose changes the level at steady state. It does not change how long it takes to arrive. The same five half-lives apply in reverse: after stopping, about 97% has gone.
3:11Practice question
Your turn. A 52-year-old man starts a constant intravenous infusion of a drug that follows first-order elimination. The drug's volume of distribution is 45 litres, and its clearance is 1.5 litres per hour. About how long will it take for his plasma concentration to reach steady state? A, about 1 day. B, about 2 days. C, about 4 days. D, about 6 days. E, about 9 days. Pause here and decide.
3:48Answer and explanation
A is out: 1 day is about one half-life, when only half of the steady-state level has been reached. B is out: 2 days is only a little over two half-lives, which is roughly four fifths of the way. D is out: that leaves out the 0.693. Forty-five over 1.5 is 30 hours, and five of those is about 6 days. E is out: 10 half-lives is far more than needed, because 97% is reached by five. So the answer is C. The half-life is 0.693 times 45 over 1.5, about 21 hours, and four to five of those is about 4 days.
4:33Recap
Three things to keep. One: half-life equals 0.693 times volume of distribution over clearance. Two: a bigger volume or a lower clearance lengthens the half-life. Three: steady state, and washout, take about five half-lives, whatever the dose. Next, zero-order versus first-order elimination.
Sources
Numbers in the text, e.g. [1], point here- Deranged Physiology. Half-life. derangedphysiology.com/main/cicm-primary-exam/pharmacokinetics/Chapter-322/half-life
- Deranged Physiology. Clearance. derangedphysiology.com/main/cicm-primary-exam/pharmacokinetics/Chapter-333/clearance
- Deranged Physiology. Maintenance dose and loading dose. derangedphysiology.com/main/cicm-primary-exam/pharmacokinetics/Chapter-401/maintenance-dose-and-loading-dose
- MSD Manual Professional Edition. Overview of Pharmacokinetics. msdmanuals.com/professional/clinical-pharmacology/pharmacokinetics/overview-of-pharmacokinetics
- MSD Manual Professional Edition. Drug Distribution to Tissues. msdmanuals.com/professional/clinical-pharmacology/pharmacokinetics/drug-distribution-to-tissues
- MSD Manual Professional Edition. Drug Metabolism. msdmanuals.com/professional/clinical-pharmacology/pharmacokinetics/drug-metabolism
- Deranged Physiology. Volume of distribution. derangedphysiology.com/main/cicm-primary-exam/pharmacokinetics/Chapter-202/volume-distribution
- Wikipedia (secondary). Biological half-life. en.wikipedia.org/wiki/Biological_half-life
- Benet LZ, Sodhi JK. AAPS J (Europe PMC PMC13171289). europepmc.org/article/PMC/PMC13171289
The two hook drugs, the worked example (Vd 60 L; CL 5 and 2.5 L/h) and the practice-question values are invented teaching numbers. Five half-lives give about 97 percent of the final change; four to five half-lives is the usual rule of thumb.
The practice question is original and its patient is synthetic. Every fact is checked against the sources above in independent accuracy reviews. Always follow your local guidelines and your own laboratory’s ranges.
Image credits. Illustrations: Servier Medical Art (smart.servier.com), CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/), adapted (recoloured, outlined): clock, kidney, liver. Intravenous bag icon: Health Icons (healthicons.org), CC0 1.0, adapted (recoloured, outlined). All tanks, graphs, formulas and handwriting are drawn in code (original).
Corrections. None so far. Spotted an error? Email info@medservatory.com with the lesson code MS-S1-PHARM-004.
For exam study only, not medical advice. Read the medical disclaimer.
In this series
Playlist on YouTube- EP 01Potency vs efficacy: EC50, Emax and agonist types5:17
- EP 02Competitive vs noncompetitive antagonists: effect on EC50 and Emax5:01
- EP 03Therapeutic index explained: TD50 / ED505:01
- EP 04First-pass metabolism and bioavailability: calculating oral F from AUC5:18
- EP 05Volume of distribution: why Vd can be bigger than the body5:17
- EP 06Clearance and half-life: t½ = 0.693 × Vd / CL5:01
- EP 07Zero-order vs first-order elimination explained5:12
- EP 08Loading dose and maintenance dose, explained5:02
- EP 09CYP450 inducers and inhibitors: how they change drug levels5:16
- EP 10Autonomic receptors: alpha, beta and muscarinic5:33
- EP 11Beta blocker selectivity: beta-1, ISA and alpha blockade5:24
- EP 12Vasopressors compared: receptors, pressure and heart rate5:46
- EP 13Organophosphate poisoning: muscarinic vs nicotinic signs5:36
- EP 14Receptor desensitization, downregulation and upregulation5:01
- EP 15Beta-lactams and vancomycin: enzyme versus substrate5:04
- EP 16Protein synthesis inhibitors: 30S vs 50S antibiotics5:07