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 92 kg patient receives a theophylline infusion. The clearance is 0.04 L/kg/h. What infusion rate R in mg/h gives a steady-state level of 15 mg/L?
92 kgCL = 0.04 L/kg/hCss = 15 mg/L
R = mg/h
2.Clinical pharmacokineticsHard
Practice exercise (made-up values, not clinical guidance). A patient's tobramycin half-life is 4 h. The target is a peak of 6 mg/L and a trough of 1 mg/L. Assuming rapid IV doses and first-order elimination, what dosing interval τ lets the level fall from the peak to the trough? Round to one decimal place.
t½ = 4 hCpeak = 6 mg/LCtrough = 1 mg/L
τ = h
3.Pharmacy calculationsMedium
A solution has a ratio strength of 1:1000. You need 0.36 mg. Draw up the volume in the 1 mL syringe: what volume V is that?
1:10000.36 mg
V = mL
4.Pharmacy calculationsHard
A vial holds 2000 mg of powder. You add 9.4 mL of diluent, and the powder adds 0.6 mL of volume. The prescription is 400 mg. Draw up the dose: what volume V do you give?
2000 mg9.4 mL + 0.6 mL400 mg
V = mL
5.Pediatric & critical-care dosingMedium
Practice exercise (made-up values, not clinical guidance). A practice protocol calls for a heparin bolus of 60 units/kg for a patient who weighs 97 kg. The vial contains 5 000 units/mL. What volume V do you draw up? Round to one decimal place.
60 units/kg97 kg5 000 units/mL
V = mL
6.Pediatric & critical-care dosingHard
Practice exercise (made-up values, not clinical guidance). A patient who weighs 70 kg has an order for nitroglycerin at 10 mcg/min. The bag contains 25 mg in 250 mL. Find the pump rate R in mL/h. Round to one decimal place.
25 mg/250 mL10 mcg/min70 kg
R = mL/h
7.PharmacologyMedium
Benzathine benzylpenicillin (penicillin G benzathine) is ordered. Which route is correct?
AIntravenous push
BSubcutaneous in the abdomen
CDeep intramuscular only
DIntravenous infusion over 30 minutes
8.PharmacologyHard
A patient taking metoprolol is also started on verapamil. Which adverse effect is most likely?
ASevere bradycardia and heart block
BHyperkalaemia and muscle weakness
CRebound tachycardia and hypertension
DHypoglycaemia and weight gain
3Solutions
1.Clinical pharmacokinetics
Answer: R = 55.2 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 = 0.04·92 = 3.68 L/h
Infusion rate
R = 3.68·15 = 55.2 mg/h
Answer
R = 55.2 mg/h
2.Clinical pharmacokinetics
Answer: τ = 10.3 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.
Rate constant
k = 0.693/4 = 0.1733 h⁻¹
Dosing interval
τ = ln(6/1)/0.1733 = 1.792/0.1733 = 10.34 ≈ 10.3 h
Answer
τ = 10.3 h
In practice the interval is rounded to a convenient value such as 8, 12, 24 or 36 h, and the levels are checked again.
3.Pharmacy calculations
Answer: V = 0.36 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/1000 mL = 1 mg/mL
Volume
V = 0.36/1 = 0.36 mL
Answer
V = 0.36 mL
4.Pharmacy calculations
Answer: V = 2 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 = 2000/10 = 200 mg/mL
Volume
V = 400/200 = 2 mL
Answer
V = 2 mL
5.Pediatric & critical-care dosing
Answer: V = 1.2 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
60·97 = 5 820 units
Volume
V = 5820/5000 = 1.164 ≈ 1.2 mL
Answer
V = 1.2 mL
6.Pediatric & critical-care dosing
Answer: R = 6 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
25 000/250 = 100 mcg/mL
Dose per hour
The order is per minute for the whole patient, so the weight is not needed.
10·60 = 600 mcg/h
Pump rate
R = 600/100 = 6 mL/h
Answer
R = 6 mL/h
7.Pharmacology
Answer: C · Deep intramuscular only
Why this is the answer
Benzathine penicillin is a depot suspension given deep IM. Accidental intravenous injection can cause cardiorespiratory arrest and death.
Answer
C · Deep intramuscular only
8.Pharmacology
Answer: A · Severe bradycardia and heart block
Why this is the answer
Beta blockers and non-dihydropyridine calcium channel blockers (verapamil, diltiazem) both slow heart rate and AV conduction. Together they can cause severe bradycardia, heart block and heart failure.