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). After an IV dose of gentamicin, the level is 6 mg/L at 1 h and 3.3 mg/L at 4 h. Assume first-order elimination. Find the elimination rate constant k.
C1 = 6 mg/L (1 h)C2 = 3.3 mg/L (4 h)
k = h⁻¹
2.Clinical pharmacokineticsHard
Practice exercise (made-up values, not clinical guidance). A vancomycin infusion is started without a loading dose. The half-life in this patient is 6 h. How long t does it take to reach 95% of the steady-state level?
t½ = 6 h95% Css
t = h
3.Pharmacy calculationsMedium
A solution has a ratio strength of 1:10 000. You need 0.84 mg. Draw up the volume in the 10 mL syringe: what volume V is that?
1:10 0000.84 mg
V = mL
4.Pharmacy calculationsHard
Practice exercise (made-up values, not clinical guidance). You must prepare 900 mL of 70% ethanol from a 90% stock and purified water. How many mL V of water do you need? Round to the nearest whole mL.
90%0%70%900 mL
V = mL
5.Pediatric & critical-care dosingMedium
Practice exercise (made-up values, not clinical guidance). An infusion of 100 units of regular insulin in 250 mL runs at 5 mL/h. How many units per hour D does the patient receive?
100 units/250 mL5 mL/h
D = units/h
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 15 mcg/kg/min for a patient who weighs 83 kg. At what rate R do you set the pump, in mL/h? Round to one decimal place.
400 mg/250 mL15 mcg/kg/min83 kg
R = mL/h
7.PharmacologyMedium
Patients on older MAO inhibitors are warned about foods rich in tyramine. What can happen?
AA hypertensive crisis
BAn allergic rash
CSudden kidney failure
DSevere hypoglycaemia
8.PharmacologyHard
Why is intravenous insulin given with glucose in severe hyperkalaemia?
AInsulin increases potassium excretion by the kidney
BGlucose binds potassium in the blood
CInsulin moves potassium from the blood into cells
DInsulin protects the heart muscle directly from potassium
3Solutions
1.Clinical pharmacokinetics
Answer: k = 0.1993 h⁻¹
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.
Elimination rate constant
k = ln(6/3.3)/3 = 0.5978/3 = 0.1993 h⁻¹
Answer
k = 0.1993 h⁻¹
2.Clinical pharmacokinetics
Answer: t = 25.93 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 = 0.693/6 = 0.1155 h⁻¹
Time to the given fraction of steady state
t = −ln(1 − 0.95)/0.1155 = 2.996/0.1155 = 25.93 h
Answer
t = 25.93 h
That is 4.322 half-lives.
3.Pharmacy calculations
Answer: V = 8.4 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/10 000 mL = 0.1 mg/mL
Volume
V = 0.84/0.1 = 8.4 mL
Answer
V = 8.4 mL
4.Pharmacy calculations
Answer: V = 200 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
90%: 70 − 0 = 70 parts
0%: 90 − 70 = 20 parts
total: 90 parts
Water
V = 900·20/90 = 200 mL
Answer
V = 200 mL
Check: 700 mL of 90% contains 630 mL of drug, which is 70% of 900 mL. This assumes the volumes simply add up; in practice ethanol is made up to volume.
5.Pediatric & critical-care dosing
Answer: D = 2 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
100/250 = 0.4 units/mL
Units per hour
D = 5·0.4 = 2 units/h
Answer
D = 2 units/h
6.Pediatric & critical-care dosing
Answer: R = 46.7 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
15·83 = 1245 mcg/min
1245·60 = 74700 mcg/h
Pump rate
R = 74700/1600 = 46.69 ≈ 46.7 mL/h
Answer
R = 46.7 mL/h
7.Pharmacology
Answer: A · A hypertensive crisis
Why this is the answer
MAO in the gut normally breaks down dietary tyramine. With MAO inhibited, tyramine from foods such as aged cheese reaches the circulation and releases noradrenaline, causing a dangerous rise in blood pressure.
Answer
A · A hypertensive crisis
8.Pharmacology
Answer: C · Insulin moves potassium from the blood into cells
Why this is the answer
Insulin activates the Na⁺/K⁺-ATPase and drives potassium into cells, lowering plasma levels within minutes. Glucose is given to prevent hypoglycaemia. The effect is temporary; potassium must still be removed from the body.
Answer
C · Insulin moves potassium from the blood into cells