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 tobramycin, the level is 10 mg/L at 2 h and 2.5 mg/L at 6 h. Assume first-order elimination. Find the elimination rate constant k.
C1 = 10 mg/L (2 h)C2 = 2.5 mg/L (6 h)
k = h⁻¹
2.Clinical pharmacokineticsHard
Practice exercise (made-up values, not clinical guidance). A 54-year-old man weighs 84 kg and has a serum creatinine of 3.4 mg/dL (301 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 atenolol dose is 100 mg for a creatinine clearance of 120 mL/min, and a fraction fe = 0.9 of the drug is excreted unchanged by the kidneys. Using the Dettli dose fraction Q = 1 − fe·(1 − CrCl/normal CrCl), find the adjusted dose D. Round to the nearest whole number.
A solution has a ratio strength of 1:10 000. You need 0.84 mg. Draw up the volume in the 10 mL syringe: what volume V is that?
1:10 0000.84 mg
V = mL
4.Pharmacy calculationsHard
Practice exercise (made-up values, not clinical guidance). You must prepare 600 mL of 6% hydrogen peroxide from a 30% stock and purified water. How many mL V of water do you need? Round to the nearest whole mL.
30%0%6%600 mL
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 110 kg. The vial contains 5 000 units/mL. What volume V do you draw up? Round to one decimal place.
70 units/kg110 kg5 000 units/mL
V = mL
6.Pediatric & critical-care dosingHard
Practice exercise (made-up values, not clinical guidance). A dobutamine infusion of 500 mg in 250 mL runs at 35 mL/h. The patient weighs 106 kg. What dose D in mcg/kg/min is the patient receiving? Round to one decimal place.
500 mg/250 mL35 mL/h106 kg
D = mcg/kg/min
7.PharmacologyMedium
How many micrograms are in 0.5 mg?
A500 micrograms
B5 micrograms
C50 micrograms
D5000 micrograms
8.PharmacologyHard
A patient recently started on haloperidol develops high fever, "lead-pipe" muscle rigidity, confusion and a very high creatine kinase. What is the most likely diagnosis?
AAcute dystonia
BNeuroleptic malignant syndrome
CSerotonin syndrome
DMalignant hyperthermia from anaesthesia
3Solutions
1.Clinical pharmacokinetics
Answer: k = 0.3466 h⁻¹
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(10/2.5)/4 = 1.386/4 = 0.3466 h⁻¹
Answer
k = 0.3466 h⁻¹
2.Clinical pharmacokinetics
Answer: D = 32 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 − 54)·84/(72·3.4) = 29.51 mL/min
Dose fraction (Dettli)
Q = 1 − 0.9·(1 − 29.51/120) = 0.3213
Adjusted dose
D = 0.3213·100 = 32.13 ≈ 32 mg
Answer
D = 32 mg
In practice the dose is rounded to a convenient amount and checked against the product information.
3.Pharmacy calculations
Answer: V = 8.4 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.
Strength in mg/mL
1000 mg/10 000 mL = 0.1 mg/mL
Volume
V = 0.84/0.1 = 8.4 mL
Answer
V = 8.4 mL
4.Pharmacy calculations
Answer: V = 480 mL
Alligation: to mix a higher strength H and a lower strength L into a target strength T, take (T − L) parts of H and (H − T) parts of L. The parts are crossed: each stock gets the distance from the target to the other stock. Water or a plain base counts as 0%. The numbers are for practice only; at the bedside always follow local protocols and have high-risk doses double-checked.
Alligation parts
30%: 6 − 0 = 6 parts
0%: 30 − 6 = 24 parts
total: 30 parts
Water
V = 600·24/30 = 480 mL
Answer
V = 480 mL
Check: 120 mL of 30% contains 36 g of drug, which is 6% of 600 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
70·110 = 7 700 units
Volume
V = 7700/5000 = 1.54 ≈ 1.5 mL
Answer
V = 1.5 mL
6.Pediatric & critical-care dosing
Answer: D = 11 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
35·2000 = 70000 mcg/h
70000/60 = 1167 mcg/min
Per kilogram
D = 1167/106 = 11.01 ≈ 11 mcg/kg/min
Answer
D = 11 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: B · Neuroleptic malignant syndrome
Why this is the answer
Neuroleptic malignant syndrome follows dopamine antagonists and develops over days, with lead-pipe rigidity, fever and raised CK. Serotonin syndrome develops faster and features clonus and hyperreflexia.