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 91 kg patient is to be loaded intravenously with phenytoin sodium to reach a phenytoin level of 15 mg/L. The volume of distribution is 0.7 L/kg and the salt factor is S = 0.92. Find the loading dose LD of phenytoin sodium in mg.
91 kgVd = 0.7 L/kgC = 15 mg/LS = 0.92
LD = mg
2.Clinical pharmacokineticsHard
Practice exercise (made-up values, not clinical guidance). A patient's gentamicin half-life is 3.5 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 = 3.5 hCpeak = 6 mg/LCtrough = 0.5 mg/L
τ = h
3.Pharmacy calculationsMedium
A solution has a ratio strength of 1:1000. You need 0.87 mg. Draw up the volume in the 1 mL syringe: what volume V is that?
1:10000.87 mg
V = mL
4.Pharmacy calculationsHard
Practice exercise (made-up values, not clinical guidance). You have 150 g of 10% salicylic acid ointment. How many grams m of 2% salicylic acid ointment must you add to make a 6% ointment? Round to the nearest whole gram.
150 g · 10%2%6%
m = g
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 65 kg. The vial contains 1 000 units/mL. What volume V do you draw up? Round to one decimal place.
70 units/kg65 kg1 000 units/mL
V = mL
6.Pediatric & critical-care dosingHard
Practice exercise (made-up values, not clinical guidance). A patient who weighs 67 kg has an order for nitroglycerin at 20 mcg/min. The bag contains 50 mg in 250 mL. Find the pump rate R in mL/h. Round to one decimal place.
50 mg/250 mL20 mcg/min67 kg
R = mL/h
7.PharmacologyMedium
Which serious but rare adverse effect of statins should patients report straight away?
AA dry cough
BAnkle swelling
CUnexplained muscle pain or weakness
DA metallic taste
8.PharmacologyHard
Why can an ACE inhibitor cause acute kidney injury in a patient with bilateral renal artery stenosis?
AThe drug crystallises in the renal tubules
BACE inhibitors constrict the afferent arteriole
CGFR depends on angiotensin II constricting the efferent arteriole
DACE inhibitors are directly toxic to the proximal tubule cells in the kidney
3Solutions
1.Clinical pharmacokinetics
Answer: LD = 1039 mg
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 = 0.7·91 = 63.7 L
Amount of active drug needed
Vd·C = 63.7·15 = 955.5 mg
Loading dose
LD = 955.5/(0.92·1) = 1039 mg
Answer
LD = 1039 mg
In practice the dose is rounded to a convenient amount.
2.Clinical pharmacokinetics
Answer: τ = 12.5 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 = 0.693/3.5 = 0.198 h⁻¹
Dosing interval
τ = ln(6/0.5)/0.198 = 2.485/0.198 = 12.55 ≈ 12.5 h
Answer
τ = 12.5 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 = 0.87 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 = 0.87/1 = 0.87 mL
Answer
V = 0.87 mL
4.Pharmacy calculations
Answer: m = 150 g
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
10%: 6 − 2 = 4 parts
2%: 10 − 6 = 4 parts
Scale to the amount you have
m = 150·4/4 = 150 g
Answer
m = 150 g
Check: 150 g + 150 g = 300 g of 6% ointment.
5.Pediatric & critical-care dosing
Answer: V = 4.6 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
70·65 = 4 550 units
Volume
V = 4550/1000 = 4.55 ≈ 4.6 mL
Answer
V = 4.6 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
50 000/250 = 200 mcg/mL
Dose per hour
The order is per minute for the whole patient, so the weight is not needed.
20·60 = 1200 mcg/h
Pump rate
R = 1200/200 = 6 mL/h
Answer
R = 6 mL/h
7.Pharmacology
Answer: C · Unexplained muscle pain or weakness
Why this is the answer
Statins can cause myopathy, which rarely progresses to rhabdomyolysis with kidney damage. Unexplained muscle pain should be checked, for example with a creatine kinase level.
Answer
C · Unexplained muscle pain or weakness
8.Pharmacology
Answer: C · GFR depends on angiotensin II constricting the efferent arteriole
Why this is the answer
When renal perfusion is low, angiotensin II constricts the efferent arteriole to maintain glomerular pressure. Blocking it lets the pressure and GFR fall, so kidney function can drop sharply.
Answer
C · GFR depends on angiotensin II constricting the efferent arteriole