Clinical pharmacokinetics: practice problems with step-by-step solutions

Also called: pharmacokinetics · half-life calculations · loading dose · therapeutic drug monitoring

Free Clinical pharmacokinetics practice problems with diagrams, hints and full worked solutions. New values every time, at three levels: half-life, levels and loading doses, clearance, Vd, S and F, regimen design and renal adjustment.

Free practice problemEasy
110t (h)C (mg/L)6 mg/L2 hC = ?

Practice exercise (made-up values, not clinical guidance). Just after an IV dose, the tobramycin level is 6 mg/L. The half-life is 2 h, and the drug is eliminated by first-order kinetics. What is the level C 2 h later?

C₀ = 6 mg/Lt½ = 2 ht = 2 h
mg/L
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What you practise

  1. Easyhalf-life, levels and loading doses
  2. Mediumclearance, Vd, S and F
  3. Hardregimen design and renal adjustment

How to solve Clinical pharmacokinetics problems

  1. Convert the half-life to a rate constant: k = 0.693/t½.
  2. Turn per-kg values into totals: Vd (L) = L/kg × weight, CL (L/h) = L/kg/h × weight.
  3. Pick the relation: C = C₀·e−kt, LD = Vd·C/(S·F), maintenance rate = CL·Css/(S·F), τ = ln(Cpeak/Ctrough)/k.
  4. For kidney function, estimate CrCl with Cockcroft–Gault first, then scale the dose or the interval.
  5. Check the result: levels fall by half every half-life, and steady state takes about five half-lives.

Common mistakes

  • Using 1/t½ instead of 0.693/t½.
  • Using log₁₀ instead of ln.
  • Multiplying by S or F instead of dividing.
  • Forgetting the body weight for values per kg.
  • Giving the whole loading dose when part of the level is already there.

Key formulas

Rate constant and half-life
k = ln 2/t½ = 0.693/t½
First-order decay
C = C₀·e−kt = C₀·(1/2)t/t½
Two levels
k = ln(C₁/C₂)/Δt
Volume and clearance
Vd = dose/C₀, CL = k·Vd
Loading dose
LD = Vd·C/(S·F)
Maintenance rate
R = CL·Css/(S·F)
Average steady state
Css,avg = S·F·D/(CL·τ)
Steady state
1 − (1/2)ⁿ after n half-lives; ≈ 97% after 5
Infusion
C = (R/CL)(1 − e−kt)
Interval and dose from peak and trough
τ = ln(Cpeak/Ctrough)/k, D = Vd·(Cpeak − Ctrough)
Cockcroft–Gault
CrCl = (140 − age) × kg/(72 × SCr mg/dL), × 0.85 for women
Renal adjustment
dose × CrCl/normal; Dettli Q = 1 − fe·(1 − CrCl/normal)

Learn Clinical pharmacokinetics from scratch

Before this: Dosage calculations, Pharmacy calculations

1What happens to a dose

Pharmacokinetics describes how the level of a drug in the blood changes with time. Clinical pharmacokinetics uses a simple model to choose doses: the body is one well-mixed compartment with a volume of distribution (Vd) and a clearance (CL).

Most drugs are eliminated by first-order kinetics: in every hour the same fraction of the drug is removed, not the same amount. That is why the level halves after every half-life.

This model is used for drugs with a narrow therapeutic range, such as aminoglycosides, vancomycin, theophylline, phenytoin and digoxin. All values in this lesson are typical textbook numbers for practice.

2Half-life and the elimination rate constant

The elimination rate constant k (in h⁻¹) and the half-life t½ belong together:

k = ln 2 / t½ = 0.693 / t½ C = C₀ · e−kt = C₀ · (1/2)t/t½

On a graph with a logarithmic concentration axis (a semi-log plot), first-order decay is a straight line. From two levels C₁ and C₂ taken Δt apart:

k = ln(C₁/C₂) / Δt

With regular doses or an infusion, the level rises to steady state: after n half-lives it has reached 1 − (1/2)ⁿ of it, about 97% after 5 half-lives.

Self-check A level falls from 8 mg/L to 2 mg/L in 6 hours. What is the half-life?
  1. 1.5 h
  2. 3 h
  3. 4 h
  4. 6 h

Answer: 3 h. 8 → 4 → 2 is two halvings in 6 h, so t½ = 3 h. With the formula: k = ln(8/2)/6 = 0.231 h⁻¹ and t½ = 0.693/0.231 = 3 h.

3Loading dose and maintenance dose

A loading dose fills the volume of distribution to the target level at once. A maintenance dose replaces what is eliminated, so at steady state rate in = rate out.

LD = Vd × Ctarget / (S × F) maintenance rate = CL × Css / (S × F)
  • S is the salt factor: the fraction of the salt that is active drug, for example 0.8 for aminophylline (theophylline) and 0.92 for phenytoin sodium.
  • F is the bioavailability: the fraction of an oral dose that reaches the blood, 1 for IV doses.
  • Vd is often given per kg (L/kg): multiply by the weight first.

For doses every τ hours, the average steady-state level is Css,avg = S·F·dose/(CL·τ).

Self-check Vd = 0.5 L/kg, weight 80 kg, target 10 mg/L. The drug is given as a salt with S = 0.8 (IV). What is the loading dose?
  1. 320 mg
  2. 400 mg
  3. 500 mg
  4. 6.25 mg

Answer: 500 mg. Vd = 0.5 × 80 = 40 L. Active drug needed: 40 × 10 = 400 mg; divided by S = 0.8 gives 500 mg of the salt. 320 mg multiplies by S instead of dividing.

4Peaks, troughs and kidney function

With rapid IV doses, the level falls from a peak to a trough during the dosing interval τ. To reach a chosen peak and trough:

τ = ln(Cpeak/Ctrough) / k dose = Vd × (Cpeak − Ctrough)

Many drugs are cleared by the kidneys. The creatinine clearance is estimated with the Cockcroft–Gault equation, CrCl = (140 − age) × weight/(72 × SCr in mg/dL), × 0.85 for women.

For a drug cleared almost entirely by the kidneys, the dose is scaled in proportion to the clearance (dose × CrCl/normal), or the interval is lengthened (τ × normal/CrCl). When only a fraction fe is excreted by the kidneys, the Dettli dose fraction is Q = 1 − fe·(1 − CrCl/normal).

5A worked example

Follow every step and predict the next one before you open it. All values are made up for practice.

Worked example

A patient’s gentamicin half-life is 3 h. The target is a peak of 8 mg/L and a trough of 1 mg/L. Assuming rapid IV doses, what dosing interval is needed?

1

Rate constant

k = 0.693 / 3 = 0.231 h⁻¹
2

Interval

The level must fall from 8 to 1 mg/L, a factor of 8.

τ = ln(8/1) / 0.231 = 2.079 / 0.231 = 9.0 h
3

Check with half-lives

8 → 4 → 2 → 1 is three halvings: 3 × 3 h = 9 h. The answer agrees.

Answer

The dosing interval is 9 h; in practice it is rounded to a convenient value and the levels are checked.

6Common mistakes and exam questions

  • Using 1/t½ instead of 0.693/t½ for k.
  • Using log₁₀ instead of the natural logarithm ln.
  • Forgetting to multiply Vd or CL in “per kg” by the weight.
  • Multiplying by S or F instead of dividing.
  • Giving the full loading dose when the patient already has a level: only the difference is needed.

Easy: k and t½, the level after a time, steady state and a simple loading dose. Medium: maintenance rates, S and F, k from two levels, Vd and CL. Hard: interval and dose from a peak and trough, renal adjustment, average steady-state levels and infusions before steady state.

Self-check An infusion is started without a loading dose. The half-life is 6 h. About what fraction of steady state is reached after 12 h?
  1. 100%
  2. 25%
  3. 50%
  4. 75%

Answer: 75%. 12 h is two half-lives: 1 − (1/2)² = 0.75, so 75%. 25% is the part still missing. Steady state takes about 5 half-lives, 30 h here.

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Worked examples

Example 1Easy
110100t (h)C (mg/L)10 mg/L9 hC = ?

Practice exercise (made-up values, not clinical guidance). Just after an IV dose, the gentamicin level is 10 mg/L. The half-life is 3 h, and the drug is eliminated by first-order kinetics. What is the level C 9 h later?

C₀ = 10 mg/Lt½ = 3 ht = 9 h

Find C (mg/L)

Show hints
  1. Number of half-lives: n = t/t½.
  2. C = C₀·(1/2)ⁿ, or C = C₀·e^(−kt) with k = 0.693/t½.
  3. n = 9/3 = 3.
Show the answer and the step-by-step solution

First-order elimination: a constant fraction of the drug is removed per hour, so C = C₀·e−kt with k = ln 2/t½ = 0.693/t½. After n half-lives C = C₀·(1/2)ⁿ. With regular doses or an infusion, the level reaches 1 − (1/2)ⁿ of steady state after n half-lives: about 97% after 5. A loading dose fills the volume of distribution: LD = Vd × C. The numbers are for practice only; at the bedside always follow local protocols and have high-risk doses double-checked.

Number of half-lives

n = 9/3 = 3

Level

C = 10·(1/2)3 = 1.25 mg/L

Answer

C = 1.25 mg/L
Example 2Medium
Vd = 7 L/kgC = 1 mcg/LLD = ? mcg52 kg · S = 1 · F = 0.7

Practice exercise (made-up values, not clinical guidance). A 52 kg patient is to be loaded with digoxin tablets to reach a level of 1 mcg/L. The volume of distribution is 7 L/kg and the tablets have a bioavailability F = 0.7. Find the total oral loading dose LD in mcg.

52 kgVd = 7 L/kgC = 1 mcg/LF = 0.7

Find LD (mcg)

Show hints
  1. LD = Vd × C/(S × F).
  2. For the tablets S = 1 and F = 0.7.
  3. Vd = 7·52 = 364 L.
Show the answer and the step-by-step solution

Clearance and volume: at steady state the dosing rate equals the elimination rate, so the maintenance rate = CL × Css. A loading dose is LD = Vd × C/(S × F), with S the salt factor (fraction of active drug in the salt) and F the bioavailability (1 for IV). From two levels, k = ln(C₁/C₂)/Δt; for an IV bolus Vd = dose/C₀ and CL = k × Vd. The numbers are for practice only; at the bedside always follow local protocols and have high-risk doses double-checked.

Volume of distribution

Vd = 7·52 = 364 L

Amount of active drug needed

Vd·C = 364·1 = 364 mcg

Loading dose

LD = 364/(1·0.7) = 520 mcg

Answer

LD = 520 mcg

In practice the dose is rounded to a convenient amount.

Example 3Hard
VCss,avg​ = ?200 mg · q12hCL = 0.03 L/kg/h86 kg · S = 1 · F = 1

Practice exercise (made-up values, not clinical guidance). An 86 kg patient takes oral theophylline (F = 1) every 12 h. The clearance is 0.03 L/kg/h. Each dose is 200 mg of theophylline. Find the average steady-state level Css,avg.

CL = 0.03 L/kg/hτ = 12 hS = 1F = 186 kgD = 200 mg

Find Css,avg (mg/L)

Show hints
  1. At steady state the average rate in equals the rate out: S·F·D/τ = CL·Css,avg.
  2. CL (L/h) = CL (L/kg/h) × weight.
  3. CL = 0.03·86 = 2.58 L/h.
Show the answer and the step-by-step solution

Designing a regimen: with rapid IV doses the level falls from peak to trough during the interval, so τ = ln(Cpeak/Ctrough)/k and the dose that restores the peak is Vd × (Cpeak − Ctrough). The average steady-state level is Css,avg = S·F·dose/(CL·τ). An infusion rises as C = (R/CL)(1 − e−kt). For a drug cleared by the kidneys, the dose is scaled with the creatinine clearance. The numbers are for practice only; at the bedside always follow local protocols and have high-risk doses double-checked.

Clearance

CL = 0.03·86 = 2.58 L/h

Average steady-state level

Css,avg = S·F·D/(CL·τ) = 1·1·200/(2.58·12) = 6.46 mg/L

Answer

Css,avg = 6.46 mg/L

Questions and answers

What is Clinical pharmacokinetics?

Pharmacokinetics describes how the level of a drug in the blood changes with time. Clinical pharmacokinetics uses a simple model to choose doses: the body is one well-mixed compartment with a volume of distribution (Vd) and a clearance (CL).

Most drugs are eliminated by first-order kinetics: in every hour the same fraction of the drug is removed, not the same amount. That is why the level halves after every half-life.

How do I solve Clinical pharmacokinetics problems step by step?
  1. Convert the half-life to a rate constant: k = 0.693/t½.
  2. Turn per-kg values into totals: Vd (L) = L/kg × weight, CL (L/h) = L/kg/h × weight.
  3. Pick the relation: C = C₀·e−kt, LD = Vd·C/(S·F), maintenance rate = CL·Css/(S·F), τ = ln(Cpeak/Ctrough)/k.
  4. For kidney function, estimate CrCl with Cockcroft–Gault first, then scale the dose or the interval.
  5. Check the result: levels fall by half every half-life, and steady state takes about five half-lives.
What are common mistakes in Clinical pharmacokinetics?
  • Using 1/t½ instead of 0.693/t½.
  • Using log₁₀ instead of ln.
  • Multiplying by S or F instead of dividing.
  • Forgetting the body weight for values per kg.
  • Giving the whole loading dose when part of the level is already there.
What do the Clinical pharmacokinetics exercises cover?
  • Easy: half-life, levels and loading doses
  • Medium: clearance, Vd, S and F
  • Hard: regimen design and renal adjustment

Every problem has new values, so you learn the method instead of an answer. Stuck? Open a small hint first, and the full explanation only when you need it.

How practising works

  • Every problem has new values, so you learn the method instead of an answer.
  • Stuck? Open a small hint first, and the full explanation only when you need it.
  • Your progress, mistakes and saved problems are kept, so you can review before the exam.

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Printable worksheet

Clinical pharmacokinetics worksheet with answers
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