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 94 kg patient is to receive an aminophylline infusion (salt factor S = 0.8: 80% of aminophylline is theophylline). The theophylline clearance is 0.035 L/kg/h. What aminophylline infusion rate R in mg/h gives a steady-state theophylline level of 10 mg/L?
94 kgCL = 0.035 L/kg/hCss = 10 mg/LS = 0.8
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 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 1:1000. You need 0.74 mg. Draw up the volume in the 1 mL syringe: what volume V is that?
1:10000.74 mg
V = mL
4.Pharmacy calculationsHard
A vial holds 2000 mg of powder. You add 4.2 mL of diluent, and the powder adds 0.8 mL of volume. The prescription is 280 mg. Draw up the dose: what volume V do you give?
2000 mg4.2 mL + 0.8 mL280 mg
V = mL
5.Pediatric & critical-care dosingMedium
Practice exercise (made-up values, not clinical guidance). An infusion contains 100 units of regular insulin in 100 mL. The order is 3 units/h. Find the pump rate R in mL/h. Round to one decimal place.
100 units/100 mL3 units/h
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 20 mL/h. The patient weighs 64 kg. What dose D in mcg/kg/min is the patient receiving? Round to one decimal place.
400 mg/250 mL20 mL/h64 kg
D = mcg/kg/min
7.PharmacologyMedium
Which assessment is most important during IV mannitol therapy?
APlatelet count and signs of bleeding
BUrine output, electrolytes and signs of fluid overload
CLiver enzymes and signs of jaundice
DBlood glucose before meals and signs of hypoglycaemia
8.PharmacologyHard
A woman with epilepsy who takes an antiepileptic drug plans to become pregnant. Which advice is standard?
AAvoid folic acid because it lowers drug levels
BNo change is needed, as antiepileptics carry no risk in pregnancy
CStop all antiepileptic drugs straight away as soon as she stops contraception
DTake folic acid before conception and seek specialist review of her medication
3Solutions
1.Clinical pharmacokinetics
Answer: R = 41.13 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 = 0.035·94 = 3.29 L/h
Infusion rate
R = CL·Css/S = 3.29·10/0.8 = 41.13 mg/h
Answer
R = 41.13 mg/h
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 = 0.74 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/1000 mL = 1 mg/mL
Volume
V = 0.74/1 = 0.74 mL
Answer
V = 0.74 mL
4.Pharmacy calculations
Answer: V = 0.7 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
4.2 + 0.8 = 5 mL
C = 2000/5 = 400 mg/mL
Volume
V = 280/400 = 0.7 mL
Answer
V = 0.7 mL
5.Pediatric & critical-care dosing
Answer: R = 3 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.
Concentration
100/100 = 1 units/mL
Pump rate
R = 3/1 = 3 mL/h
Answer
R = 3 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
20·1600 = 32000 mcg/h
32000/60 = 533.3 mcg/min
Per kilogram
D = 533.3/64 = 8.333 ≈ 8.3 mcg/kg/min
Answer
D = 8.3 mcg/kg/min
7.Pharmacology
Answer: B · Urine output, electrolytes and signs of fluid overload
Why this is the answer
Mannitol draws fluid into the circulation before it is excreted, which can cause heart failure and pulmonary oedema. The large diuresis can cause dehydration and electrolyte imbalance, so output and electrolytes are monitored.
Answer
B · Urine output, electrolytes and signs of fluid overload
8.Pharmacology
Answer: D · Take folic acid before conception and seek specialist review of her medication
Why this is the answer
Folic acid before and in early pregnancy reduces the risk of neural tube defects. Medication should be reviewed before conception, but drugs should not be stopped suddenly because uncontrolled seizures are also dangerous.
Answer
D · Take folic acid before conception and seek specialist review of her medication