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 98 kg patient receives a theophylline infusion. The clearance is 0.045 L/kg/h. What infusion rate R in mg/h gives a steady-state level of 10 mg/L?
98 kgCL = 0.045 L/kg/hCss = 10 mg/L
R = mg/h
2.Clinical pharmacokineticsHard
Practice exercise (made-up values, not clinical guidance). A 72-year-old woman weighs 78 kg and has a serum creatinine of 2.5 mg/dL (221 µmol/L). Estimate the creatinine clearance with Cockcroft–Gault: CrCl (mL/min) = (140 − age) × weight (kg)/(72 × SCr in mg/dL), × 0.85 for women. The usual gentamicin regimen is 80 mg every 8 h for a creatinine clearance of 100 mL/min. Keeping the dose, the interval is lengthened in proportion to the fall in clearance. Find the new dosing interval τ. Round to one decimal place.
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 1 200 000 units. What volume V do you draw up? Round to one decimal place.
8.2 mL → 500 000 units/mL1 200 000 units
V = mL
4.Pharmacy calculationsHard
A vial holds 500 mg of powder. You add 4.8 mL of diluent, and the powder adds 0.2 mL of volume. The prescription is 380 mg. Draw up the dose: what volume V do you give?
500 mg4.8 mL + 0.2 mL380 mg
V = mL
5.Pediatric & critical-care dosingMedium
Practice exercise (made-up values, not clinical guidance). An infusion contains 250 units of regular insulin in 250 mL. The order is 8 units/h. Find the pump rate R in mL/h. Round to one decimal place.
250 units/250 mL8 units/h
R = mL/h
6.Pediatric & critical-care dosingHard
Practice exercise (made-up values, not clinical guidance). A dobutamine infusion of 500 mg in 250 mL runs at 39 mL/h. The patient weighs 92 kg. What dose D in mcg/kg/min is the patient receiving? Round to one decimal place.
500 mg/250 mL39 mL/h92 kg
D = mcg/kg/min
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 reduce drainage into the nose and absorption into blood
CTo stop the eye from producing tears for about an hour
DTo make the pupil dilate faster
8.PharmacologyHard
A patient on disulfiram becomes flushed, nauseated and has palpitations after using a mouthwash. What is the most likely cause?
AAlcohol in the mouthwash caused a disulfiram–alcohol reaction
BDisulfiram toxicity from taking too many tablets by mistake
CLow blood glucose from skipping breakfast
DAn allergy to mint flavouring in the mouthwash
3Solutions
1.Clinical pharmacokinetics
Answer: R = 44.1 mg/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.
Clearance
CL = 0.045·98 = 4.41 L/h
Infusion rate
R = 4.41·10 = 44.1 mg/h
Answer
R = 44.1 mg/h
2.Clinical pharmacokinetics
Answer: τ = 31.9 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.
Creatinine clearance (Cockcroft–Gault)
CrCl = (140 − 72)·78/(72·2.5)·0.85 = 25.05 mL/min
New interval
τ = 8·100/25.05 = 31.94 ≈ 31.9 h
Answer
τ = 31.9 h
In practice the interval is rounded to a convenient value such as 12, 24, 36 or 48 h.
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
8.2 mL → 500 000 units/mL
Volume to give
V = 1 200 000/500 000 = 2.4 mL
Answer
V = 2.4 mL
4.Pharmacy calculations
Answer: V = 3.8 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 = 500/5 = 100 mg/mL
Volume
V = 380/100 = 3.8 mL
Answer
V = 3.8 mL
5.Pediatric & critical-care dosing
Answer: R = 8 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.
Concentration
250/250 = 1 units/mL
Pump rate
R = 8/1 = 8 mL/h
Answer
R = 8 mL/h
6.Pediatric & critical-care dosing
Answer: D = 14.1 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
500 000/250 = 2000 mcg/mL
Micrograms per minute
39·2000 = 78000 mcg/h
78000/60 = 1300 mcg/min
Per kilogram
D = 1300/92 = 14.13 ≈ 14.1 mcg/kg/min
Answer
D = 14.1 mcg/kg/min
7.Pharmacology
Answer: B · 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
B · To reduce drainage into the nose and absorption into blood
8.Pharmacology
Answer: A · Alcohol in the mouthwash caused a disulfiram–alcohol reaction
Why this is the answer
Disulfiram reactions can occur with small amounts of alcohol in mouthwash, cough syrups, sauces or aftershave. Patients are taught to check labels and avoid all alcohol-containing products.
Answer
A · Alcohol in the mouthwash caused a disulfiram–alcohol reaction