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 54 kg patient is to be loaded intravenously with phenytoin sodium to reach a phenytoin level of 20 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.
54 kgVd = 0.7 L/kgC = 20 mg/LS = 0.92
LD = mg
2.Clinical pharmacokineticsHard
Practice exercise (made-up values, not clinical guidance). A theophylline infusion is started without a loading dose. The half-life in this patient is 9 h. How long t does it take to reach 90% of the steady-state level?
t½ = 9 h90% Css
t = h
3.Pharmacy calculationsMedium
Practice exercise (made-up values, not clinical guidance). A vial contains cefazolin 1000 mg as a powder. The practice label says to add 4.5 mL of sterile water; the powder adds 0.5 mL to the volume. The prescribed dose is 750 mg. What volume V do you draw up? Round to one decimal place.
1000 mg4.5 mL+0.5 mL750 mg
V = mL
4.Pharmacy calculationsHard
Practice exercise (made-up values, not clinical guidance). You must prepare 200 mL of 5% chlorhexidine from a 20% stock and purified water. How many mL V of water do you need? Round to the nearest whole mL.
20%0%5%200 mL
V = mL
5.Pediatric & critical-care dosingMedium
Practice exercise (made-up values, not clinical guidance). A bag contains 20 000 units of heparin in 500 mL. The order is 16 units/kg/h for a patient who weighs 109 kg. At what rate R do you set the pump, in mL/h? Round to one decimal place.
20 000 units/500 mL16 units/kg/h109 kg
R = mL/h
6.Pediatric & critical-care dosingHard
Practice exercise (made-up values, not clinical guidance). A patient who weighs 71 kg has an order for nitroglycerin at 30 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 mL30 mcg/min71 kg
R = mL/h
7.PharmacologyMedium
Which group of signs suggests opioid withdrawal?
AConstipation, itching and a slow heart rate
BYawning, runny nose, sweating, dilated pupils and diarrhoea
CDry mouth, urinary retention, blurred vision and confusion
DPinpoint pupils, slow breathing and drowsiness
8.PharmacologyHard
Which heart failure drug improves symptoms and reduces hospital admissions but has not been shown to reduce mortality?
AA mineralocorticoid receptor antagonist such as spironolactone
BA beta blocker such as bisoprolol
CDigoxin
DAn ACE inhibitor
3Solutions
1.Clinical pharmacokinetics
Answer: LD = 821.7 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 = 0.7·54 = 37.8 L
Amount of active drug needed
Vd·C = 37.8·20 = 756 mg
Loading dose
LD = 756/(0.92·1) = 821.7 mg
Answer
LD = 821.7 mg
In practice the dose is rounded to a convenient amount.
2.Clinical pharmacokinetics
Answer: t = 29.9 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 = 0.693/9 = 0.07702 h⁻¹
Time to the given fraction of steady state
t = −ln(1 − 0.9)/0.07702 = 2.303/0.07702 = 29.9 h
Answer
t = 29.9 h
That is 3.322 half-lives.
3.Pharmacy calculations
Answer: V = 3.8 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
4.5 + 0.5 = 5 mL
Concentration
1000/5 = 200 mg/mL
Volume to give
V = 750/200 = 3.75 ≈ 3.8 mL
Answer
V = 3.8 mL
4.Pharmacy calculations
Answer: V = 150 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
20%: 5 − 0 = 5 parts
0%: 20 − 5 = 15 parts
total: 20 parts
Water
V = 200·15/20 = 150 mL
Answer
V = 150 mL
Check: 50 mL of 20% contains 10 g of drug, which is 5% of 200 mL.
5.Pediatric & critical-care dosing
Answer: R = 43.6 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
16·109 = 1 744 units/h
Concentration
20 000/500 = 40 units/mL
Pump rate
R = 1744/40 = 43.6 mL/h
Answer
R = 43.6 mL/h
6.Pediatric & critical-care dosing
Answer: R = 18 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.
30·60 = 1800 mcg/h
Pump rate
R = 1800/100 = 18 mL/h
Answer
R = 18 mL/h
7.Pharmacology
Answer: B · Yawning, runny nose, sweating, dilated pupils and diarrhoea
Why this is the answer
Opioid withdrawal causes rhinorrhoea, lacrimation, yawning, sweating, mydriasis, cramps, diarrhoea and anxiety. It is very unpleasant but rarely life-threatening in healthy adults.
Answer
B · Yawning, runny nose, sweating, dilated pupils and diarrhoea
8.Pharmacology
Answer: C · Digoxin
Why this is the answer
ACE inhibitors (or ARBs/ARNI), beta blockers, mineralocorticoid antagonists and SGLT2 inhibitors reduce mortality in heart failure with reduced ejection fraction. Digoxin improves symptoms and reduces admissions without reducing mortality.