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 10 mg/L at 2 h and 1.8 mg/L at 7 h. Assume first-order elimination. Find the half-life t½.
C1 = 10 mg/L (2 h)C2 = 1.8 mg/L (7 h)
t½ = h
2.Clinical pharmacokineticsHard
Practice exercise (made-up values, not clinical guidance). A 66 kg patient has a phenytoin level of 10 mg/L; the target is 20 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?
66 kgVd = 0.7 L/kgC = 10 → 20 mg/LS = 0.92
LD = mg
3.Pharmacy calculationsMedium
A solution has a ratio strength of 1:10 000. You need 0.86 mg. Draw up the volume in the 10 mL syringe: what volume V is that?
1:10 0000.86 mg
V = mL
4.Pharmacy calculationsHard
Practice exercise (made-up values, not clinical guidance). You must prepare 600 mL of 5% sodium chloride by mixing 23% and 3% sodium chloride. How many mL V of the 23% solution do you need? Round to the nearest whole mL.
23%3%5%600 mL
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 14 units/kg/h for a patient who weighs 61 kg. At what rate R do you set the pump, in mL/h? Round to one decimal place.
25 000 units/500 mL14 units/kg/h61 kg
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 27 mL/h. The patient weighs 110 kg. What dose D in mcg/kg/min is the patient receiving? Round to one decimal place.
250 mg/250 mL27 mL/h110 kg
D = mcg/kg/min
7.PharmacologyMedium
Which drug is used for torsades de pointes?
AIntravenous magnesium sulfate
BIntravenous adenosine (rapid push)
COral digoxin
DIntravenous verapamil
8.PharmacologyHard
Why should a dose never be written as "5.0 mg" on a medication order?
AThe pharmacy cannot supply doses that contain a decimal point
BIt means the dose must be given five times
CThe decimal point may be missed, and the dose read as 50 mg
DIt is too precise for most drugs
3Solutions
1.Clinical pharmacokinetics
Answer: t½ = 2.021 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(10/1.8)/5 = 1.715/5 = 0.343 h⁻¹
Half-life
t½ = 0.693/0.343 = 2.021 h
Answer
t½ = 2.021 h
2.Clinical pharmacokinetics
Answer: LD = 502.2 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 = 0.7·66 = 46.2 L
Rise needed
20 − 10 = 10 mg/L
Loading dose
LD = 46.2·10/(0.92·1) = 502.2 mg
Answer
LD = 502.2 mg
In practice the dose is rounded to a convenient amount.
3.Pharmacy calculations
Answer: V = 8.6 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.
Strength in mg/mL
1000 mg/10 000 mL = 0.1 mg/mL
Volume
V = 0.86/0.1 = 8.6 mL
Answer
V = 8.6 mL
4.Pharmacy calculations
Answer: V = 60 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
23%: 5 − 3 = 2 parts
3%: 23 − 5 = 18 parts
total: 20 parts
Share the volume
V = 600·2/20 = 60 mL
Answer
V = 60 mL
Check: 60 mL of 23% + 540 mL of 3% = 600 mL.
5.Pediatric & critical-care dosing
Answer: R = 17.1 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
14·61 = 854 units/h
Concentration
25 000/500 = 50 units/mL
Pump rate
R = 854/50 = 17.08 ≈ 17.1 mL/h
Answer
R = 17.1 mL/h
6.Pediatric & critical-care dosing
Answer: D = 4.1 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
250 000/250 = 1000 mcg/mL
Micrograms per minute
27·1000 = 27000 mcg/h
27000/60 = 450 mcg/min
Per kilogram
D = 450/110 = 4.091 ≈ 4.1 mcg/kg/min
Answer
D = 4.1 mcg/kg/min
7.Pharmacology
Answer: A · Intravenous magnesium sulfate
Why this is the answer
IV magnesium is first-line for torsades de pointes, even when serum magnesium is normal. QT-prolonging drugs are stopped and potassium is corrected.
Answer
A · Intravenous magnesium sulfate
8.Pharmacology
Answer: C · The decimal point may be missed, and the dose read as 50 mg
Why this is the answer
Trailing zeros (5.0) and missing leading zeros (.5) are well-known causes of tenfold errors. Safe practice is to write 5 mg and 0.5 mg.
Answer
C · The decimal point may be missed, and the dose read as 50 mg