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). A patient receives an IV bolus of 150 mg of gentamicin. The level back-extrapolated to time zero is 9 mg/L and the half-life is 2 h. Find the clearance CL in L/h.
D = 150 mgC₀ = 9 mg/Lt½ = 2 h
CL = L/h
2.Clinical pharmacokineticsHard
Practice exercise (made-up values, not clinical guidance). A vancomycin infusion runs at 75 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 10 h?
R = 75 mg/hCL = 3 L/ht½ = 8 ht = 10 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 8.2 mL of diluent. The prescribed dose is 2 400 000 units. What volume V do you draw up? Round to one decimal place.
8.2 mL → 500 000 units/mL2 400 000 units
V = mL
4.Pharmacy calculationsHard
Practice exercise (made-up values, not clinical guidance). You have 120 g of 10% salicylic acid ointment. How many grams m of 2% salicylic acid ointment must you add to make a 5% ointment? Round to the nearest whole gram.
120 g · 10%2%5%
m = g
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 108 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/h108 kg
R = mL/h
6.Pediatric & critical-care dosingHard
Practice exercise (made-up values, not clinical guidance). A dobutamine infusion of 250 mg in 250 mL runs at 8 mL/h. The patient weighs 94 kg. What dose D in mcg/kg/min is the patient receiving? Round to one decimal place.
250 mg/250 mL8 mL/h94 kg
D = mcg/kg/min
7.PharmacologyMedium
A woman taking the combined pill should seek care urgently for which symptom?
ALight spotting between periods early on
BA painful, swollen calf or sudden chest pain
CMild breast tenderness in the first month
DSlight nausea in the first few weeks
8.PharmacologyHard
A patient has received a sodium nitroprusside infusion for several days. He becomes confused and develops a metabolic acidosis. What should be suspected?
AAn allergic reaction to the plastic of the infusion bag
BHyperkalaemia from the infusion
CHypoglycaemia from nitroprusside
DCyanide toxicity from nitroprusside breakdown
3Solutions
1.Clinical pharmacokinetics
Answer: CL = 5.776 L/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.
Volume of distribution
Vd = 150/9 = 16.67 L
Rate constant
k = 0.693/2 = 0.3466 h⁻¹
Clearance
CL = 0.3466·16.67 = 5.776 L/h
Answer
CL = 5.776 L/h
2.Clinical pharmacokinetics
Answer: C = 14.49 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 = 75/3 = 25 mg/L
Rate constant
k = 0.693/8 = 0.08664 h⁻¹
Level after the given time
C = 25·(1 − e−0.08664·10) = 14.49 mg/L
Answer
C = 14.49 mg/L
3.Pharmacy calculations
Answer: V = 4.8 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
8.2 mL → 500 000 units/mL
Volume to give
V = 2 400 000/500 000 = 4.8 mL
Answer
V = 4.8 mL
4.Pharmacy calculations
Answer: m = 200 g
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
10%: 5 − 2 = 3 parts
2%: 10 − 5 = 5 parts
Scale to the amount you have
m = 120·5/3 = 200 g
Answer
m = 200 g
Check: 120 g + 200 g = 320 g of 5% ointment.
5.Pediatric & critical-care dosing
Answer: R = 32.4 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
15·108 = 1 620 units/h
Concentration
25 000/500 = 50 units/mL
Pump rate
R = 1620/50 = 32.4 mL/h
Answer
R = 32.4 mL/h
6.Pediatric & critical-care dosing
Answer: D = 1.4 mcg/kg/min
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
250 000/250 = 1000 mcg/mL
Micrograms per minute
8·1000 = 8000 mcg/h
8000/60 = 133.3 mcg/min
Per kilogram
D = 133.3/94 = 1.418 ≈ 1.4 mcg/kg/min
Answer
D = 1.4 mcg/kg/min
7.Pharmacology
Answer: B · A painful, swollen calf or sudden chest pain
Why this is the answer
Combined hormonal contraception increases the risk of venous thromboembolism and, rarely, stroke. Leg pain or swelling, chest pain, breathlessness, severe headache or vision changes need urgent assessment.
Answer
B · A painful, swollen calf or sudden chest pain
8.Pharmacology
Answer: D · Cyanide toxicity from nitroprusside breakdown
Why this is the answer
Nitroprusside releases cyanide, which is normally converted to thiocyanate. With high doses, long infusions or poor organ function, cyanide and thiocyanate can accumulate; the infusion is also protected from light.