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 gentamicin, the level is 9 mg/L at 1 h and 4.5 mg/L at 3 h. Assume first-order elimination. Find the half-life t½.
C1 = 9 mg/L (1 h)C2 = 4.5 mg/L (3 h)
t½ = h
2.Clinical pharmacokineticsHard
Practice exercise (made-up values, not clinical guidance). A patient's gentamicin half-life is 3.5 h. The target is a peak of 7 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 = 7 mg/LCtrough = 1 mg/L
τ = h
3.Pharmacy calculationsMedium
Practice exercise (made-up values, not clinical guidance). A vial contains cefazolin 1000 mg as a powder. The practice label says to add 4.5 mL of sterile water; the powder adds 0.5 mL to the volume. The prescribed dose is 300 mg. What volume V do you draw up? Round to one decimal place.
1000 mg4.5 mL+0.5 mL300 mg
V = mL
4.Pharmacy calculationsHard
A vial holds 2000 mg of powder. You add 4.8 mL of diluent, and the powder adds 0.2 mL of volume. The prescription is 1360 mg. Draw up the dose: what volume V do you give?
2000 mg4.8 mL + 0.2 mL1360 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 250 mL. The order is 14 units/kg/h for a patient who weighs 92 kg. At what rate R do you set the pump, in mL/h? Round to one decimal place.
25 000 units/250 mL14 units/kg/h92 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 18 mL/h. The patient weighs 76 kg. What dose D in mcg/kg/min is the patient receiving? Round to one decimal place.
250 mg/250 mL18 mL/h76 kg
D = mcg/kg/min
7.PharmacologyMedium
A patient with a peanut allergy is taught to use an adrenaline auto-injector. Which statement is correct?
AThere is no need to see a doctor if the swelling goes down
BCall emergency services after using it, even if symptoms improve
CKeep it at home so it is not lost
DUse it only if the antihistamine has not worked after an hour
8.PharmacologyHard
A patient on digoxin and furosemide develops nausea, visual disturbances and an irregular pulse. Which electrolyte disturbance most likely made digoxin toxicity more likely?
AHypokalaemia
BHyperkalaemia
CHypocalcaemia
DHypernatraemia
3Solutions
1.Clinical pharmacokinetics
Answer: t½ = 2 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(9/4.5)/2 = 0.6931/2 = 0.3466 h⁻¹
Half-life
t½ = 0.693/0.3466 = 2 h
Answer
t½ = 2 h
2.Clinical pharmacokinetics
Answer: τ = 9.8 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(7/1)/0.198 = 1.946/0.198 = 9.826 ≈ 9.8 h
Answer
τ = 9.8 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 = 1.5 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.
Final volume
4.5 + 0.5 = 5 mL
Concentration
1000/5 = 200 mg/mL
Volume to give
V = 300/200 = 1.5 mL
Answer
V = 1.5 mL
4.Pharmacy calculations
Answer: V = 3.4 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
4.8 + 0.2 = 5 mL
C = 2000/5 = 400 mg/mL
Volume
V = 1360/400 = 3.4 mL
Answer
V = 3.4 mL
5.Pediatric & critical-care dosing
Answer: R = 12.9 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
14·92 = 1 288 units/h
Concentration
25 000/250 = 100 units/mL
Pump rate
R = 1288/100 = 12.88 ≈ 12.9 mL/h
Answer
R = 12.9 mL/h
6.Pediatric & critical-care dosing
Answer: D = 3.9 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
18·1000 = 18000 mcg/h
18000/60 = 300 mcg/min
Per kilogram
D = 300/76 = 3.947 ≈ 3.9 mcg/kg/min
Answer
D = 3.9 mcg/kg/min
7.Pharmacology
Answer: B · Call emergency services after using it, even if symptoms improve
Why this is the answer
Anaphylaxis can recur after the adrenaline wears off (a biphasic reaction). The person always calls for help after using it, carries two auto-injectors and uses adrenaline early.
Answer
B · Call emergency services after using it, even if symptoms improve
8.Pharmacology
Answer: A · Hypokalaemia
Why this is the answer
Digoxin binds the Na⁺/K⁺-ATPase at the same site as potassium. Low potassium, often caused by loop or thiazide diuretics, increases digoxin binding and toxicity.