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 70 kg patient receives a theophylline infusion. The clearance is 0.045 L/kg/h. What infusion rate R in mg/h gives a steady-state level of 10 mg/L?
70 kgCL = 0.045 L/kg/hCss = 10 mg/L
R = mg/h
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 50% of the steady-state level?
t½ = 6 h50% Css
t = h
3.Pharmacy calculationsMedium
A solution has a ratio strength of 1:10 000. You need 0.86 mg. Draw up the volume in the 10 mL syringe: what volume V is that?
1:10 0000.86 mg
V = mL
4.Pharmacy calculationsHard
A vial holds 1000 mg of powder. You add 3.8 mL of diluent, and the powder adds 0.2 mL of volume. The prescription is 525 mg. Draw up the dose: what volume V do you give?
1000 mg3.8 mL + 0.2 mL525 mg
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 12 units/kg/h for a patient who weighs 90 kg. At what rate R do you set the pump, in mL/h? Round to one decimal place.
20 000 units/500 mL12 units/kg/h90 kg
R = mL/h
6.Pediatric & critical-care dosingHard
Practice exercise (made-up values, not clinical guidance). A dopamine infusion of 400 mg in 250 mL runs at 33 mL/h. The patient weighs 106 kg. What dose D in mcg/kg/min is the patient receiving? Round to one decimal place.
400 mg/250 mL33 mL/h106 kg
D = mcg/kg/min
7.PharmacologyMedium
A patient starting sulfasalazine should be told about which harmless effect?
ABlue colouring of the tongue and gums
BGreen colouring of the sweat
CWhite spots on the nails
DOrange-yellow colouring of the urine
8.PharmacologyHard
A patient with severe liver cirrhosis needs a benzodiazepine for alcohol withdrawal. Why is lorazepam often chosen instead of diazepam?
ALorazepam is cleared by simple conjugation and has no active metabolites
BLorazepam is a stronger anticonvulsant than all other benzodiazepines
CLorazepam does not cause sedation in liver disease
DLorazepam is excreted unchanged in the urine without any metabolism
3Solutions
1.Clinical pharmacokinetics
Answer: R = 31.5 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.
Clearance
CL = 0.045·70 = 3.15 L/h
Infusion rate
R = 3.15·10 = 31.5 mg/h
Answer
R = 31.5 mg/h
2.Clinical pharmacokinetics
Answer: t = 6 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 − 0.5)/0.1155 = 0.6931/0.1155 = 6 h
Answer
t = 6 h
That is 1 half-lives.
3.Pharmacy calculations
Answer: V = 8.6 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 = 0.86/0.1 = 8.6 mL
Answer
V = 8.6 mL
4.Pharmacy calculations
Answer: V = 2.1 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.
Final volume and concentration
3.8 + 0.2 = 4 mL
C = 1000/4 = 250 mg/mL
Volume
V = 525/250 = 2.1 mL
Answer
V = 2.1 mL
5.Pediatric & critical-care dosing
Answer: R = 27 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
12·90 = 1 080 units/h
Concentration
20 000/500 = 40 units/mL
Pump rate
R = 1080/40 = 27 mL/h
Answer
R = 27 mL/h
6.Pediatric & critical-care dosing
Answer: D = 8.3 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
33·1600 = 52800 mcg/h
52800/60 = 880 mcg/min
Per kilogram
D = 880/106 = 8.302 ≈ 8.3 mcg/kg/min
Answer
D = 8.3 mcg/kg/min
7.Pharmacology
Answer: D · Orange-yellow colouring of the urine
Why this is the answer
Sulfasalazine can colour urine and sometimes skin and contact lenses orange-yellow. More serious adverse effects include blood disorders and rash, so blood counts are monitored.
Answer
D · Orange-yellow colouring of the urine
8.Pharmacology
Answer: A · Lorazepam is cleared by simple conjugation and has no active metabolites
Why this is the answer
Diazepam is metabolised by CYP enzymes into long-acting active metabolites, which accumulate in liver failure. Lorazepam, oxazepam and temazepam are glucuronidated, a process better preserved in liver disease.
Answer
A · Lorazepam is cleared by simple conjugation and has no active metabolites