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 95 kg patient is to be loaded with digoxin tablets to reach a level of 1 mcg/L. The volume of distribution is 7 L/kg and the tablets have a bioavailability F = 0.7. Find the total oral loading dose LD in mcg.
95 kgVd = 7 L/kgC = 1 mcg/LF = 0.7
LD = mcg
2.Clinical pharmacokineticsHard
Practice exercise (made-up values, not clinical guidance). An 81 kg patient has a theophylline level of 3.6 mg/L; the target is 12 mg/L. The drug is given intravenously as aminophylline (S = 0.8), and the volume of distribution is 0.5 L/kg. What extra loading dose LD raises the level to the target?
81 kgVd = 0.5 L/kgC = 3.6 → 12 mg/LS = 0.8
LD = mg
3.Pharmacy calculationsMedium
Practice exercise (made-up values, not clinical guidance). A vial contains ceftriaxone 1000 mg as a powder. The practice label says to add 9.6 mL of sterile water; the powder adds 0.4 mL to the volume. What is the concentration C in mg/mL after reconstitution?
1000 mg9.6 mL+0.4 mL
C = mg/mL
4.Pharmacy calculationsHard
Practice exercise (made-up values, not clinical guidance). You have 30 g of 10% salicylic acid ointment. How many grams m of 2% salicylic acid ointment must you add to make a 5% ointment? Round to the nearest whole gram.
30 g · 10%2%5%
m = g
5.Pediatric & critical-care dosingMedium
Practice exercise (made-up values, not clinical guidance). An infusion of 250 units of regular insulin in 250 mL runs at 10 mL/h. How many units per hour D does the patient receive?
250 units/250 mL10 mL/h
D = units/h
6.Pediatric & critical-care dosingHard
Practice exercise (made-up values, not clinical guidance). A patient who weighs 92 kg has an order for nitroglycerin at 15 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 mL15 mcg/min92 kg
R = mL/h
7.PharmacologyMedium
Why is propranolol usually avoided in patients with asthma?
AIt blocks beta-2 receptors in the airways and can cause bronchospasm
BIt causes tolerance to inhaled corticosteroids
CIt stimulates alpha-1 receptors in the airways and raises blood pressure
DIt increases mucus production by blocking histamine
8.PharmacologyHard
A patient with asthma starts timolol eye drops for glaucoma and soon becomes wheezy. What is the most likely explanation?
AThe eye drops contain an allergen that causes hay fever and asthma
BThe wheeze is unrelated to the eye drops
CTimolol activates histamine receptors in the lungs
DTimolol is absorbed and blocks beta-2 receptors in the airways
3Solutions
1.Clinical pharmacokinetics
Answer: LD = 950 mcg
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 = 7·95 = 665 L
Amount of active drug needed
Vd·C = 665·1 = 665 mcg
Loading dose
LD = 665/(1·0.7) = 950 mcg
Answer
LD = 950 mcg
In practice the dose is rounded to a convenient amount.
2.Clinical pharmacokinetics
Answer: LD = 425.2 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 = 0.5·81 = 40.5 L
Rise needed
12 − 3.6 = 8.4 mg/L
Loading dose
LD = 40.5·8.4/(0.8·1) = 425.2 mg
Answer
LD = 425.2 mg
In practice the dose is rounded to a convenient amount.
3.Pharmacy calculations
Answer: C = 100 mg/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
9.6 + 0.4 = 10 mL
Concentration
C = 1000/10 = 100 mg/mL
Answer
C = 100 mg/mL
4.Pharmacy calculations
Answer: m = 50 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%: 5 − 2 = 3 parts
2%: 10 − 5 = 5 parts
Scale to the amount you have
m = 30·5/3 = 50 g
Answer
m = 50 g
Check: 30 g + 50 g = 80 g of 5% ointment.
5.Pediatric & critical-care dosing
Answer: D = 10 units/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.
Concentration
250/250 = 1 units/mL
Units per hour
D = 10·1 = 10 units/h
Answer
D = 10 units/h
6.Pediatric & critical-care dosing
Answer: R = 4.5 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.
15·60 = 900 mcg/h
Pump rate
R = 900/200 = 4.5 mL/h
Answer
R = 4.5 mL/h
7.Pharmacology
Answer: A · It blocks beta-2 receptors in the airways and can cause bronchospasm
Why this is the answer
Non-selective beta blockers block both beta-1 and beta-2 receptors. Blocking beta-2 receptors in the bronchi removes their relaxing effect and can trigger severe bronchospasm.
Answer
A · It blocks beta-2 receptors in the airways and can cause bronchospasm
8.Pharmacology
Answer: D · 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
D · Timolol is absorbed and blocks beta-2 receptors in the airways