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 is to be loaded intravenously with aminophylline to reach a theophylline level of 12 mg/L. The volume of distribution is 0.5 L/kg and the salt factor is S = 0.8. Find the loading dose LD of aminophylline in mg.
70 kgVd = 0.5 L/kgC = 12 mg/LS = 0.8
LD = mg
2.Clinical pharmacokineticsHard
Practice exercise (made-up values, not clinical guidance). An 86 kg patient on regular gentamicin doses is at steady state. The target is a peak of 10 mg/L and a trough of 0.5 mg/L, and the volume of distribution is 0.3 L/kg. Assuming rapid IV doses, what dose D raises the level from the trough to the peak?
Practice exercise (made-up values, not clinical guidance). A vial contains ceftriaxone 2000 mg as a powder. The practice label says to add 19.2 mL of sterile water; the powder adds 0.8 mL to the volume. The prescribed dose is 1200 mg. What volume V do you draw up? Round to one decimal place.
2000 mg19.2 mL+0.8 mL1200 mg
V = mL
4.Pharmacy calculationsHard
Practice exercise (made-up values, not clinical guidance). You must prepare 100 mL of 25% dextrose by mixing 70% and 10% dextrose. How many mL V of the 70% solution do you need? Round to the nearest whole mL.
70%10%25%100 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 50 mL runs at 2 mL/h. How many units per hour D does the patient receive?
100 units/50 mL2 mL/h
D = units/h
6.Pediatric & critical-care dosingHard
Practice exercise (made-up values, not clinical guidance). A bag contains dobutamine 500 mg in 250 mL. The order is 7.5 mcg/kg/min for a patient who weighs 73 kg. At what rate R do you set the pump, in mL/h? Round to one decimal place.
500 mg/250 mL7.5 mcg/kg/min73 kg
R = mL/h
7.PharmacologyMedium
A drug inhibits the liver enzyme CYP3A4. What can happen to a second drug that is mainly metabolised by CYP3A4?
AIts plasma level falls, so it may stop working at the usual dose
BIts plasma level rises, increasing the risk of side effects
CNothing, because enzymes do not affect other drugs
DIt is excreted faster by the kidneys
8.PharmacologyHard
Why is vincristine labelled "for intravenous use only – fatal if given by other routes"?
CIntramuscular injection causes severe and prolonged hypoglycaemia
DOral vincristine causes immediate anaphylaxis
3Solutions
1.Clinical pharmacokinetics
Answer: LD = 525 mg
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 = 0.5·70 = 35 L
Amount of active drug needed
Vd·C = 35·12 = 420 mg
Loading dose
LD = 420/(0.8·1) = 525 mg
Answer
LD = 525 mg
In practice the dose is rounded to a convenient amount.
2.Clinical pharmacokinetics
Answer: D = 245.1 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 = 0.3·86 = 25.8 L
Dose
D = 25.8·(10 − 0.5) = 245.1 mg
Answer
D = 245.1 mg
In practice the dose is rounded to a convenient amount.
3.Pharmacy calculations
Answer: V = 12 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
19.2 + 0.8 = 20 mL
Concentration
2000/20 = 100 mg/mL
Volume to give
V = 1200/100 = 12 mL
Answer
V = 12 mL
4.Pharmacy calculations
Answer: V = 25 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
70%: 25 − 10 = 15 parts
10%: 70 − 25 = 45 parts
total: 60 parts
Share the volume
V = 100·15/60 = 25 mL
Answer
V = 25 mL
Check: 25 mL of 70% + 75 mL of 10% = 100 mL.
5.Pediatric & critical-care dosing
Answer: D = 4 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
100/50 = 2 units/mL
Units per hour
D = 2·2 = 4 units/h
Answer
D = 4 units/h
6.Pediatric & critical-care dosing
Answer: R = 16.4 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
500 mg = 500 000 mcg
500 000/250 = 2000 mcg/mL
Dose per minute and per hour
7.5·73 = 547.5 mcg/min
547.5·60 = 32850 mcg/h
Pump rate
R = 32850/2000 = 16.43 ≈ 16.4 mL/h
Answer
R = 16.4 mL/h
7.Pharmacology
Answer: B · Its plasma level rises, increasing the risk of side effects
Why this is the answer
Enzyme inhibitors (for example clarithromycin or grapefruit juice for CYP3A4) slow metabolism, so the second drug accumulates. Enzyme inducers have the opposite effect.
Answer
B · Its plasma level rises, increasing the risk of side effects
8.Pharmacology
Answer: A · Accidental intrathecal injection causes fatal ascending neurotoxicity
Why this is the answer
Vincristine given intrathecally causes ascending paralysis and death. To prevent this, it is usually given in a small IV bag rather than a syringe and kept separate from intrathecal drugs.
Answer
A · Accidental intrathecal injection causes fatal ascending neurotoxicity