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 140 mg of tobramycin. The level back-extrapolated to time zero is 7 mg/L and the half-life is 2.5 h. Find the volume of distribution Vd.
D = 140 mgC₀ = 7 mg/Lt½ = 2.5 h
Vd = L
2.Clinical pharmacokineticsHard
Practice exercise (made-up values, not clinical guidance). A patient's gentamicin half-life is 4 h. The target is a peak of 6 mg/L and a trough of 0.5 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 = 0.5 mg/L
τ = h
3.Pharmacy calculationsMedium
Practice exercise (made-up values, not clinical guidance). A vial contains ampicillin 500 mg as a powder. The practice label says to add 1.8 mL of sterile water; the powder adds 0.2 mL to the volume. The prescribed dose is 200 mg. What volume V do you draw up? Round to one decimal place.
500 mg1.8 mL+0.2 mL200 mg
V = mL
4.Pharmacy calculationsHard
A vial holds 500 mg of powder. You add 1.8 mL of diluent, and the powder adds 0.2 mL of volume. The prescription is 300 mg. Draw up the dose: what volume V do you give?
500 mg1.8 mL + 0.2 mL300 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 68 kg. The vial contains 5 000 units/mL. What volume V do you draw up? Round to two decimal places.
60 units/kg68 kg5 000 units/mL
V = mL
6.Pediatric & critical-care dosingHard
Practice exercise (made-up values, not clinical guidance). A bag contains dopamine 400 mg in 250 mL. The order is 3 mcg/kg/min for a patient who weighs 63 kg. At what rate R do you set the pump, in mL/h? Round to one decimal place.
400 mg/250 mL3 mcg/kg/min63 kg
R = mL/h
7.PharmacologyMedium
How should the nurse straighten the ear canal to give ear drops to a 2-year-old child?
APull the pinna up and back
BPull the pinna down and back
CLeave the ear untouched and drop into the canal
DPush the pinna forward and up
8.PharmacologyHard
A patient with asthma starts timolol eye drops for glaucoma and soon becomes wheezy. What is the most likely explanation?
ATimolol activates histamine receptors in the lungs
BThe wheeze is unrelated to the eye drops
CTimolol is absorbed and blocks beta-2 receptors in the airways
DThe eye drops contain an allergen that causes hay fever and asthma
3Solutions
1.Clinical pharmacokinetics
Answer: Vd = 20 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 = 140/7 = 20 L
Answer
Vd = 20 L
2.Clinical pharmacokinetics
Answer: τ = 14.3 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(6/0.5)/0.1733 = 2.485/0.1733 = 14.34 ≈ 14.3 h
Answer
τ = 14.3 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 = 0.8 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.
Final volume
1.8 + 0.2 = 2 mL
Concentration
500/2 = 250 mg/mL
Volume to give
V = 200/250 = 0.8 mL
Answer
V = 0.8 mL
4.Pharmacy calculations
Answer: V = 1.2 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
1.8 + 0.2 = 2 mL
C = 500/2 = 250 mg/mL
Volume
V = 300/250 = 1.2 mL
Answer
V = 1.2 mL
5.Pediatric & critical-care dosing
Answer: V = 0.82 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
60·68 = 4 080 units
Volume
V = 4080/5000 = 0.816 ≈ 0.82 mL
Answer
V = 0.82 mL
6.Pediatric & critical-care dosing
Answer: R = 7.1 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
400 mg = 400 000 mcg
400 000/250 = 1600 mcg/mL
Dose per minute and per hour
3·63 = 189 mcg/min
189·60 = 11340 mcg/h
Pump rate
R = 11340/1600 = 7.088 ≈ 7.1 mL/h
Answer
R = 7.1 mL/h
7.Pharmacology
Answer: B · Pull the pinna down and back
Why this is the answer
In children under about 3 years, the pinna is pulled down and back; in adults it is pulled up and back. The child then lies with the treated ear up for a few minutes.
Answer
B · Pull the pinna down and back
8.Pharmacology
Answer: C · Timolol is absorbed and blocks beta-2 receptors in the airways
Why this is the answer
Ophthalmic timolol is a non-selective beta blocker and can reach the circulation in amounts that cause bronchospasm or bradycardia. It is generally avoided in asthma.
Answer
C · Timolol is absorbed and blocks beta-2 receptors in the airways