Pick the relation: C = C₀·e−kt, LD = Vd·C/(S·F), maintenance rate = CL·Css/(S·F), τ = ln(Cpeak/Ctrough)/k.
For kidney function, estimate CrCl with Cockcroft–Gault first, then scale the dose or the interval.
Check the result: levels fall by half every half-life, and steady state takes about five half-lives.
Pharmacy calculations
Percentage strength
% w/v = g per 100 mL
Amount in a solution
g = % × mL / 100
Ratio strength
1:X = 1 g in X mL; % = 100 / X
Reconstitution
final volume = diluent + powder volume
Concentration
mg/mL = drug in vial / final volume
Alligation
parts of H = T − L, parts of L = H − T
Water or base
0%
Pediatric & critical-care dosing
Daily dose
mg/day = mg/kg/day × weight (kg)
Doses per day
number = 24 h / dosing interval (h)
Pounds to kilograms
kg = lb / 2.2
Infusion concentration
units/mL = units in bag / bag volume
Pump rate
mL/h = dose per hour / concentration
Titrated drip
mL/h = mcg/kg/min × kg × 60 / (mcg/mL)
Dose from pump rate
mcg/kg/min = mL/h × (mcg/mL) / (60 × kg)
Units
1 mg = 1000 mcg
2Practice problems
1.Clinical pharmacokineticsMedium
Practice exercise (made-up values, not clinical guidance). After an IV dose of vancomycin, the level is 35 mg/L at 2 h and 19.3 mg/L at 8 h. Assume first-order elimination. Find the elimination rate constant k.
C1 = 35 mg/L (2 h)C2 = 19.3 mg/L (8 h)
k = h⁻¹
2.Clinical pharmacokineticsHard
Practice exercise (made-up values, not clinical guidance). A patient's tobramycin half-life is 3.5 h. The target is a peak of 6 mg/L and a trough of 1 mg/L. Assuming rapid IV doses and first-order elimination, what dosing interval τ lets the level fall from the peak to the trough? Round to one decimal place.
t½ = 3.5 hCpeak = 6 mg/LCtrough = 1 mg/L
τ = h
3.Pharmacy calculationsMedium
Practice exercise (made-up values, not clinical guidance). A vial of penicillin G contains 5 000 000 units. Its practice label gives three strengths (see the table). You add 3.2 mL of diluent. The prescribed dose is 750 000 units. What volume V do you draw up? Round to one decimal place.
3.2 mL → 1 000 000 units/mL750 000 units
V = mL
4.Pharmacy calculationsHard
A vial holds 500 mg of powder. You add 1.2 mL of diluent, and the powder adds 0.8 mL of volume. The prescription is 175 mg. Draw up the dose: what volume V do you give?
500 mg1.2 mL + 0.8 mL175 mg
V = mL
5.Pediatric & critical-care dosingMedium
Practice exercise (made-up values, not clinical guidance). A bag contains 25 000 units of heparin in 500 mL. The order is 15 units/kg/h for a patient who weighs 75 kg. At what rate R do you set the pump, in mL/h? Round to one decimal place.
25 000 units/500 mL15 units/kg/h75 kg
R = mL/h
6.Pediatric & critical-care dosingHard
Practice exercise (made-up values, not clinical guidance). A bag contains dopamine 800 mg in 500 mL. The order is 2 mcg/kg/min for a patient who weighs 93 kg. At what rate R do you set the pump, in mL/h? Round to one decimal place.
800 mg/500 mL2 mcg/kg/min93 kg
R = mL/h
7.PharmacologyMedium
Which adverse effect of ciprofloxacin should a patient report at once?
AMild tiredness in the first few days
BA slightly bitter or metallic taste for a while after each dose
CDarker stools than usual
DPain or swelling in a tendon, such as the Achilles tendon
8.PharmacologyHard
A patient with myasthenia gravis took extra pyridostigmine doses. She now has increasing weakness, heavy salivation, sweating, small pupils and diarrhoea. What is the most likely cause?
AMyasthenic crisis from too little anticholinesterase drug
BCholinergic crisis from too much anticholinesterase drug
CA stroke unrelated to the medication
DAn allergic reaction to pyridostigmine
3Solutions
1.Clinical pharmacokinetics
Answer: k = 0.09921 h⁻¹
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.
Elimination rate constant
k = ln(35/19.3)/6 = 0.5952/6 = 0.09921 h⁻¹
Answer
k = 0.09921 h⁻¹
2.Clinical pharmacokinetics
Answer: τ = 9 h
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.
Rate constant
k = 0.693/3.5 = 0.198 h⁻¹
Dosing interval
τ = ln(6/1)/0.198 = 1.792/0.198 = 9.047 ≈ 9 h
Answer
τ = 9 h
In practice the interval is rounded to a convenient value such as 8, 12, 24 or 36 h, and the levels are checked again.
3.Pharmacy calculations
Answer: V = 0.8 mL
Reconstitution: the powder takes up space, so the final volume is the diluent plus the powder volume. Concentration = drug in the vial/final volume, and the volume to give = dose/concentration. On a multi-strength label, use the concentration that belongs to the amount of diluent you actually added. The numbers are for practice only; at the bedside always follow local protocols and have high-risk doses double-checked.
Concentration from the label
3.2 mL → 1 000 000 units/mL
Volume to give
V = 750 000/1 000 000 = 0.75 ≈ 0.8 mL
Answer
V = 0.8 mL
4.Pharmacy calculations
Answer: V = 0.7 mL
Strength to volume: turn the strength into mg per mL first. A w/v percentage is grams per 100 mL, so x% = 10x mg/mL. A ratio strength 1:n is 1 g in n mL. After reconstitution the concentration is the powder's dose over the final volume (diluent + powder volume). Then volume = dose/concentration.
Final volume and concentration
1.2 + 0.8 = 2 mL
C = 500/2 = 250 mg/mL
Volume
V = 175/250 = 0.7 mL
Answer
V = 0.7 mL
5.Pediatric & critical-care dosing
Answer: R = 22.5 mL/h
Heparin and insulin infusions: concentration = units in the bag/bag volume (units/mL). Pump rate (mL/h) = units per hour/concentration, and units per hour = mL/h × concentration. For a weight-based order, units/h = units/kg/h × weight first. A bolus volume = units needed/vial strength. The numbers are for practice only; at the bedside always follow local protocols and have high-risk doses double-checked.
Units per hour
15·75 = 1 125 units/h
Concentration
25 000/500 = 50 units/mL
Pump rate
R = 1125/50 = 22.5 mL/h
Answer
R = 22.5 mL/h
6.Pediatric & critical-care dosing
Answer: R = 7 mL/h
Titrated drips: first find the concentration in mcg/mL (mg × 1000/mL). For a weight-based order, mcg/min = mcg/kg/min × weight. Then mL/h = mcg/min × 60/concentration. Working backwards, mcg/kg/min = mL/h × concentration/(60 × weight). The numbers are for practice only; at the bedside always follow local protocols and have high-risk doses double-checked.
Concentration
800 mg = 800 000 mcg
800 000/500 = 1600 mcg/mL
Dose per minute and per hour
2·93 = 186 mcg/min
186·60 = 11160 mcg/h
Pump rate
R = 11160/1600 = 6.975 ≈ 7 mL/h
Answer
R = 7 mL/h
7.Pharmacology
Answer: D · Pain or swelling in a tendon, such as the Achilles tendon
Why this is the answer
Fluoroquinolones can cause tendinitis and tendon rupture. Patients should stop exercise and report tendon pain promptly.
Answer
D · Pain or swelling in a tendon, such as the Achilles tendon
8.Pharmacology
Answer: B · Cholinergic crisis from too much anticholinesterase drug
Why this is the answer
Excess acetylcholine causes muscarinic signs (salivation, sweating, miosis, diarrhoea) and depolarising weakness. In myasthenic crisis, weakness occurs without these cholinergic signs; both can cause respiratory failure.
Answer
B · Cholinergic crisis from too much anticholinesterase drug