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 93 kg patient is to be loaded intravenously with aminophylline to reach a theophylline level of 10 mg/L. The volume of distribution is 0.5 L/kg and the salt factor is S = 0.8. Find the loading dose LD of aminophylline in mg.
93 kgVd = 0.5 L/kgC = 10 mg/LS = 0.8
LD = mg
2.Clinical pharmacokineticsHard
Practice exercise (made-up values, not clinical guidance). An 84 kg patient has a phenytoin level of 7.5 mg/L; the target is 15 mg/L. The drug is given intravenously as phenytoin sodium (S = 0.92), and the volume of distribution is 0.7 L/kg. What extra loading dose LD raises the level to the target?
84 kgVd = 0.7 L/kgC = 7.5 → 15 mg/LS = 0.92
LD = mg
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 2 000 000 units. What volume V do you draw up? Round to one decimal place.
18.2 mL → 250 000 units/mL2 000 000 units
V = mL
4.Pharmacy calculationsHard
A vial holds 250 mg of powder. You add 1.2 mL of diluent, and the powder adds 0.8 mL of volume. The prescription is 75 mg. Draw up the dose: what volume V do you give?
250 mg1.2 mL + 0.8 mL75 mg
V = mL
5.Pediatric & critical-care dosingMedium
Practice exercise (made-up values, not clinical guidance). A practice protocol calls for a heparin bolus of 70 units/kg for a patient who weighs 106 kg. The vial contains 5 000 units/mL. What volume V do you draw up? Round to one decimal place.
70 units/kg106 kg5 000 units/mL
V = mL
6.Pediatric & critical-care dosingHard
Practice exercise (made-up values, not clinical guidance). A bag contains dobutamine 500 mg in 250 mL. The order is 2.5 mcg/kg/min for a patient who weighs 107 kg. At what rate R do you set the pump, in mL/h? Round to one decimal place.
500 mg/250 mL2.5 mcg/kg/min107 kg
R = mL/h
7.PharmacologyMedium
After instilling eye drops, why should the patient press gently on the inner corner of the eye for a minute or two?
ATo spread the drop evenly over the eyelid
BTo make the pupil dilate faster
CTo stop the eye from producing tears for about an hour
DTo reduce drainage into the nose and absorption into blood
8.PharmacologyHard
A patient taking furosemide has a serum potassium of 2.9 mmol/L (mEq/L). Which ECG change is typical of this level?
AA short QT interval with high R waves
BWidened QRS with a sine-wave pattern
CFlattened T waves and prominent U waves
DTall, peaked T waves
3Solutions
1.Clinical pharmacokinetics
Answer: LD = 581.3 mg
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 = 0.5·93 = 46.5 L
Amount of active drug needed
Vd·C = 46.5·10 = 465 mg
Loading dose
LD = 465/(0.8·1) = 581.3 mg
Answer
LD = 581.3 mg
In practice the dose is rounded to a convenient amount.
2.Clinical pharmacokinetics
Answer: LD = 479.3 mg
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.
Volume of distribution
Vd = 0.7·84 = 58.8 L
Rise needed
15 − 7.5 = 7.5 mg/L
Loading dose
LD = 58.8·7.5/(0.92·1) = 479.3 mg
Answer
LD = 479.3 mg
In practice the dose is rounded to a convenient amount.
3.Pharmacy calculations
Answer: V = 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
18.2 mL → 250 000 units/mL
Volume to give
V = 2 000 000/250 000 = 8 mL
Answer
V = 8 mL
4.Pharmacy calculations
Answer: V = 0.6 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 = 250/2 = 125 mg/mL
Volume
V = 75/125 = 0.6 mL
Answer
V = 0.6 mL
5.Pediatric & critical-care dosing
Answer: V = 1.5 mL
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 needed
70·106 = 7 420 units
Volume
V = 7420/5000 = 1.484 ≈ 1.5 mL
Answer
V = 1.5 mL
6.Pediatric & critical-care dosing
Answer: R = 8 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
500 mg = 500 000 mcg
500 000/250 = 2000 mcg/mL
Dose per minute and per hour
2.5·107 = 267.5 mcg/min
267.5·60 = 16050 mcg/h
Pump rate
R = 16050/2000 = 8.025 ≈ 8 mL/h
Answer
R = 8 mL/h
7.Pharmacology
Answer: D · To reduce drainage into the nose and absorption into blood
Why this is the answer
Nasolacrimal occlusion keeps the drop in the eye longer and reduces systemic absorption through the nasal mucosa. This matters for drugs such as timolol, which can cause bradycardia or bronchospasm.
Answer
D · To reduce drainage into the nose and absorption into blood
8.Pharmacology
Answer: C · Flattened T waves and prominent U waves
Why this is the answer
Hypokalaemia causes flat T waves, ST depression and U waves, and it increases the risk of arrhythmias. Peaked T waves and wide QRS complexes are signs of hyperkalaemia.