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 25 mg/L at 1 h and 7.3 mg/L at 17 h. Assume first-order elimination. Find the half-life t½.
C1 = 25 mg/L (1 h)C2 = 7.3 mg/L (17 h)
t½ = h
2.Clinical pharmacokineticsHard
Practice exercise (made-up values, not clinical guidance). A theophylline infusion runs at 45 mg/h with no loading dose. The patient's clearance is 3 L/h and the half-life is 8 h. What is the level C after 5 h?
R = 45 mg/hCL = 3 L/ht½ = 8 ht = 5 h
C = mg/L
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 18.2 mL of diluent. The prescribed dose is 600 000 units. What volume V do you draw up? Round to one decimal place.
18.2 mL → 250 000 units/mL600 000 units
V = mL
4.Pharmacy calculationsHard
Practice exercise (made-up values, not clinical guidance). You must prepare 750 mL of 30% dextrose by mixing 50% and 10% dextrose. How many mL V of the 50% solution do you need? Round to the nearest whole mL.
50%10%30%750 mL
V = mL
5.Pediatric & critical-care dosingMedium
Practice exercise (made-up values, not clinical guidance). A bag contains 20 000 units of heparin in 500 mL. The order is 12 units/kg/h for a patient who weighs 81 kg. At what rate R do you set the pump, in mL/h? Round to one decimal place.
20 000 units/500 mL12 units/kg/h81 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 6 mcg/kg/min for a patient who weighs 81 kg. At what rate R do you set the pump, in mL/h? Round to one decimal place.
800 mg/500 mL6 mcg/kg/min81 kg
R = mL/h
7.PharmacologyMedium
How should a patient take oral alendronate?
ACrushed into yoghurt with calcium supplements
BWith breakfast and coffee to protect the stomach
CWith milk at bedtime, then lying down straight away to aid absorption
DWith plain water on an empty stomach, staying upright for 30 minutes
8.PharmacologyHard
A patient took a large paracetamol (acetaminophen) overdose 6 hours ago and feels well. Why is treatment still urgent?
AParacetamol overdose only causes harm if symptoms start in the first hour
BThe drug will cause kidney stones within a few hours
CThe patient is at risk of immediate respiratory arrest
DLiver damage appears 1–3 days later, and early treatment works best
3Solutions
1.Clinical pharmacokinetics
Answer: t½ = 9.009 h
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(25/7.3)/16 = 1.231/16 = 0.07694 h⁻¹
Half-life
t½ = 0.693/0.07694 = 9.009 h
Answer
t½ = 9.009 h
2.Clinical pharmacokinetics
Answer: C = 5.274 mg/L
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.
Steady-state level
Css = 45/3 = 15 mg/L
Rate constant
k = 0.693/8 = 0.08664 h⁻¹
Level after the given time
C = 15·(1 − e−0.08664·5) = 5.274 mg/L
Answer
C = 5.274 mg/L
3.Pharmacy calculations
Answer: V = 2.4 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
18.2 mL → 250 000 units/mL
Volume to give
V = 600 000/250 000 = 2.4 mL
Answer
V = 2.4 mL
4.Pharmacy calculations
Answer: V = 375 mL
Alligation: to mix a higher strength H and a lower strength L into a target strength T, take (T − L) parts of H and (H − T) parts of L. The parts are crossed: each stock gets the distance from the target to the other stock. Water or a plain base counts as 0%. The numbers are for practice only; at the bedside always follow local protocols and have high-risk doses double-checked.
Alligation parts
50%: 30 − 10 = 20 parts
10%: 50 − 30 = 20 parts
total: 40 parts
Share the volume
V = 750·20/40 = 375 mL
Answer
V = 375 mL
Check: 375 mL of 50% + 375 mL of 10% = 750 mL.
5.Pediatric & critical-care dosing
Answer: R = 24.3 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
12·81 = 972 units/h
Concentration
20 000/500 = 40 units/mL
Pump rate
R = 972/40 = 24.3 mL/h
Answer
R = 24.3 mL/h
6.Pediatric & critical-care dosing
Answer: R = 18.2 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
6·81 = 486 mcg/min
486·60 = 29160 mcg/h
Pump rate
R = 29160/1600 = 18.23 ≈ 18.2 mL/h
Answer
R = 18.2 mL/h
7.Pharmacology
Answer: D · With plain water on an empty stomach, staying upright for 30 minutes
Why this is the answer
Alendronate is taken first thing in the morning with water only, before food or other drugs. Staying upright reduces the risk of oesophageal ulceration, and food or calcium blocks its absorption.
Answer
D · With plain water on an empty stomach, staying upright for 30 minutes
8.Pharmacology
Answer: D · Liver damage appears 1–3 days later, and early treatment works best
Why this is the answer
Paracetamol overdose often causes few symptoms at first, while the toxic metabolite damages the liver. Liver injury peaks after about 3 days, so the decision to treat is based on the timed blood level, not on symptoms.
Answer
D · Liver damage appears 1–3 days later, and early treatment works best