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 6 mg/L at 2 h and 3.4 mg/L at 4 h. Assume first-order elimination. Find the half-life t½.
C1 = 6 mg/L (2 h)C2 = 3.4 mg/L (4 h)
t½ = h
2.Clinical pharmacokineticsHard
Practice exercise (made-up values, not clinical guidance). A 50-year-old man weighs 72 kg and has a serum creatinine of 2.6 mg/dL (230 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 tobramycin regimen is 120 mg every 8 h for a creatinine clearance of 120 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 18.2 mL of diluent. The prescribed dose is 2 400 000 units. What volume V do you draw up? Round to one decimal place.
18.2 mL → 250 000 units/mL2 400 000 units
V = mL
4.Pharmacy calculationsHard
A vial holds 1000 mg of powder. You add 9.4 mL of diluent, and the powder adds 0.6 mL of volume. The prescription is 280 mg. Draw up the dose: what volume V do you give?
1000 mg9.4 mL + 0.6 mL280 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 500 mL. The order is 12 units/kg/h for a patient who weighs 55 kg. At what rate R do you set the pump, in mL/h? Round to one decimal place.
25 000 units/500 mL12 units/kg/h55 kg
R = mL/h
6.Pediatric & critical-care dosingHard
Practice exercise (made-up values, not clinical guidance). A patient who weighs 100 kg has an order for norepinephrine at 12 mcg/min. The bag contains 4 mg in 250 mL. Find the pump rate R in mL/h. Round to one decimal place.
4 mg/250 mL12 mcg/min100 kg
R = mL/h
7.PharmacologyMedium
How should a topical corticosteroid cream usually be applied?
AUnder a tight plastic dressing every night without advice
BAs a thick layer over the whole body
CAs a thin layer to the affected area only, as directed
DOn broken, infected skin to speed healing
8.PharmacologyHard
A patient with angina is taught to use sublingual glyceryl trinitrate (nitroglycerin). Which statement shows a need for further teaching?
A"I will replace my tablets before the expiry date."
B"I will sit down before I take the tablet, because it can make me feel faint or dizzy."
C"If the pain is no better after one tablet, I will wait an hour before calling for help."
D"I may get a headache after taking it."
3Solutions
1.Clinical pharmacokinetics
Answer: t½ = 2.441 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(6/3.4)/2 = 0.568/2 = 0.284 h⁻¹
Half-life
t½ = 0.693/0.284 = 2.441 h
Answer
t½ = 2.441 h
2.Clinical pharmacokinetics
Answer: τ = 27.7 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 − 50)·72/(72·2.6) = 34.62 mL/min
New interval
τ = 8·120/34.62 = 27.73 ≈ 27.7 h
Answer
τ = 27.7 h
In practice the interval is rounded to a convenient value such as 12, 24, 36 or 48 h.
3.Pharmacy calculations
Answer: V = 9.6 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 = 2 400 000/250 000 = 9.6 mL
Answer
V = 9.6 mL
4.Pharmacy calculations
Answer: V = 2.8 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
9.4 + 0.6 = 10 mL
C = 1000/10 = 100 mg/mL
Volume
V = 280/100 = 2.8 mL
Answer
V = 2.8 mL
5.Pediatric & critical-care dosing
Answer: R = 13.2 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
12·55 = 660 units/h
Concentration
25 000/500 = 50 units/mL
Pump rate
R = 660/50 = 13.2 mL/h
Answer
R = 13.2 mL/h
6.Pediatric & critical-care dosing
Answer: R = 45 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
4 000/250 = 16 mcg/mL
Dose per hour
The order is per minute for the whole patient, so the weight is not needed.
12·60 = 720 mcg/h
Pump rate
R = 720/16 = 45 mL/h
Answer
R = 45 mL/h
7.Pharmacology
Answer: C · As a thin layer to the affected area only, as directed
Why this is the answer
Thin application to affected skin limits systemic absorption and skin atrophy. Occlusive dressings increase absorption and are used only when prescribed.
Answer
C · As a thin layer to the affected area only, as directed
8.Pharmacology
Answer: C · "If the pain is no better after one tablet, I will wait an hour before calling for help."
Why this is the answer
Unrelieved chest pain may be a heart attack, so help must not be delayed. In US teaching, emergency services are called if pain is not better 5 minutes after the first dose; UK advice is to repeat the dose after 5 minutes and call 999 if pain persists 5 minutes after the second. Sitting down prevents falls from hypotension, and headache is a common effect.
Answer
C · "If the pain is no better after one tablet, I will wait an hour before calling for help."