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 220 mg of gentamicin. The level back-extrapolated to time zero is 12 mg/L and the half-life is 3 h. Find the volume of distribution Vd.
D = 220 mgC₀ = 12 mg/Lt½ = 3 h
Vd = L
2.Clinical pharmacokineticsHard
Practice exercise (made-up values, not clinical guidance). A 55-year-old woman weighs 64 kg and has a serum creatinine of 2.1 mg/dL (186 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 dose is 120 mg every 8 h for a creatinine clearance of 100 mL/min. Keeping the interval, the dose is reduced in proportion to the creatinine clearance. Find the adjusted dose D. Round to the nearest whole number.
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 1000 mg of powder. You add 9.2 mL of diluent, and the powder adds 0.8 mL of volume. The prescription is 240 mg. Draw up the dose: what volume V do you give?
1000 mg9.2 mL + 0.8 mL240 mg
V = mL
5.Pediatric & critical-care dosingMedium
Practice exercise (made-up values, not clinical guidance). An infusion contains 50 units of regular insulin in 50 mL. The order is 7 units/h. Find the pump rate R in mL/h. Round to one decimal place.
50 units/50 mL7 units/h
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 39 mL/h. The patient weighs 61 kg. What dose D in mcg/kg/min is the patient receiving? Round to one decimal place.
250 mg/250 mL39 mL/h61 kg
D = mcg/kg/min
7.PharmacologyMedium
How many micrograms are in 0.5 mg?
A500 micrograms
B5000 micrograms
C5 micrograms
D50 micrograms
8.PharmacologyHard
Codeine is converted to morphine by CYP2D6. What happens in a patient who is a CYP2D6 ultra-rapid metaboliser?
AMore morphine is formed, increasing the risk of opioid toxicity
BCodeine gives almost no pain relief because little morphine is formed
CCodeine is excreted unchanged in the urine
DCodeine is converted to an inactive compound only
3Solutions
1.Clinical pharmacokinetics
Answer: Vd = 18.33 L
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 = 220/12 = 18.33 L
Answer
Vd = 18.33 L
2.Clinical pharmacokinetics
Answer: D = 37 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.
Creatinine clearance (Cockcroft–Gault)
CrCl = (140 − 55)·64/(72·2.1)·0.85 = 30.58 mL/min
Adjusted dose
D = 120·30.58/100 = 36.7 ≈ 37 mg
Answer
D = 37 mg
In practice the dose is rounded to a convenient amount and checked against the product information.
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 = 2.4 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
9.2 + 0.8 = 10 mL
C = 1000/10 = 100 mg/mL
Volume
V = 240/100 = 2.4 mL
Answer
V = 2.4 mL
5.Pediatric & critical-care dosing
Answer: R = 7 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
50/50 = 1 units/mL
Pump rate
R = 7/1 = 7 mL/h
Answer
R = 7 mL/h
6.Pediatric & critical-care dosing
Answer: D = 10.7 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
39·1000 = 39000 mcg/h
39000/60 = 650 mcg/min
Per kilogram
D = 650/61 = 10.66 ≈ 10.7 mcg/kg/min
Answer
D = 10.7 mcg/kg/min
7.Pharmacology
Answer: A · 500 micrograms
Why this is the answer
0.5 mg × 1000 = 500 micrograms. Writing "micrograms" in full (or "mcg" in US practice), not "µg", helps avoid confusion with "mg".
Answer
A · 500 micrograms
8.Pharmacology
Answer: A · More morphine is formed, increasing the risk of opioid toxicity
Why this is the answer
Ultra-rapid metabolisers convert more codeine to morphine, which can cause respiratory depression. Poor metabolisers form little morphine and get poor pain relief. This is a classic example of pharmacogenetics.
Answer
A · More morphine is formed, increasing the risk of opioid toxicity