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). Vancomycin is given as a continuous infusion to a patient whose vancomycin clearance is 4.5 L/h. What infusion rate R in mg/h gives a steady-state level of 20 mg/L?
CL = 4.5 L/hCss = 20 mg/L
R = mg/h
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 95% of the steady-state level?
t½ = 9 h95% Css
t = h
3.Pharmacy calculationsMedium
Practice exercise (made-up values, not clinical guidance). A vial of penicillin G contains 5 000 000 units. Its practice label gives three strengths (see the table). You add 8.2 mL of diluent. The prescribed dose is 400 000 units. What volume V do you draw up? Round to one decimal place.
8.2 mL → 500 000 units/mL400 000 units
V = mL
4.Pharmacy calculationsHard
Practice exercise (made-up values, not clinical guidance). You must prepare 750 mL of 13% sodium chloride by mixing 23% and 3% sodium chloride. How many mL V of the 23% solution do you need? Round to the nearest whole mL.
23%3%13%750 mL
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 63 kg. The vial contains 1 000 units/mL. What volume V do you draw up? Round to one decimal place.
60 units/kg63 kg1 000 units/mL
V = mL
6.Pediatric & critical-care dosingHard
Practice exercise (made-up values, not clinical guidance). A dopamine infusion of 400 mg in 250 mL runs at 32 mL/h. The patient weighs 90 kg. What dose D in mcg/kg/min is the patient receiving? Round to one decimal place.
400 mg/250 mL32 mL/h90 kg
D = mcg/kg/min
7.PharmacologyMedium
A patient with a peanut allergy is taught to use an adrenaline auto-injector. Which statement is correct?
AKeep it at home so it is not lost
BCall emergency services after using it, even if symptoms improve
CUse it only if the antihistamine has not worked after an hour
DThere is no need to see a doctor if the swelling goes down
8.PharmacologyHard
A patient receiving frequent salbutamol (albuterol) nebulisers develops a fine tremor and a heart rate of 120 per minute. What is the best explanation?
ASalbutamol blocks muscarinic receptors in the heart
BSalbutamol causes high potassium, which speeds the heart and causes muscle twitching
CThe patient is allergic to salbutamol and needs adrenaline
DAt high doses it also stimulates beta receptors in skeletal muscle and the heart
3Solutions
1.Clinical pharmacokinetics
Answer: R = 90 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.
Infusion rate
R = 4.5·20 = 90 mg/h
Answer
R = 90 mg/h
2.Clinical pharmacokinetics
Answer: t = 38.9 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 − 0.95)/0.07702 = 2.996/0.07702 = 38.9 h
Answer
t = 38.9 h
That is 4.322 half-lives.
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.
Concentration from the label
8.2 mL → 500 000 units/mL
Volume to give
V = 400 000/500 000 = 0.8 mL
Answer
V = 0.8 mL
4.Pharmacy calculations
Answer: V = 375 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
23%: 13 − 3 = 10 parts
3%: 23 − 13 = 10 parts
total: 20 parts
Share the volume
V = 750·10/20 = 375 mL
Answer
V = 375 mL
Check: 375 mL of 23% + 375 mL of 3% = 750 mL.
5.Pediatric & critical-care dosing
Answer: V = 3.8 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
60·63 = 3 780 units
Volume
V = 3780/1000 = 3.78 ≈ 3.8 mL
Answer
V = 3.8 mL
6.Pediatric & critical-care dosing
Answer: D = 9.5 mcg/kg/min
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
400 000/250 = 1600 mcg/mL
Micrograms per minute
32·1600 = 51200 mcg/h
51200/60 = 853.3 mcg/min
Per kilogram
D = 853.3/90 = 9.481 ≈ 9.5 mcg/kg/min
Answer
D = 9.5 mcg/kg/min
7.Pharmacology
Answer: B · Call emergency services after using it, even if symptoms improve
Why this is the answer
Anaphylaxis can recur after the adrenaline wears off (a biphasic reaction). The person always calls for help after using it, carries two auto-injectors and uses adrenaline early.
Answer
B · Call emergency services after using it, even if symptoms improve
8.Pharmacology
Answer: D · At high doses it also stimulates beta receptors in skeletal muscle and the heart
Why this is the answer
Tremor comes from beta-2 stimulation of skeletal muscle. Tachycardia comes from some beta-1 (and cardiac beta-2) stimulation and a reflex response to vasodilation. High doses also shift potassium into cells and can cause hypokalaemia.
Answer
D · At high doses it also stimulates beta receptors in skeletal muscle and the heart