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 76 kg patient is to be loaded intravenously with aminophylline to reach a theophylline level of 15 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.
76 kgVd = 0.5 L/kgC = 15 mg/LS = 0.8
LD = mg
2.Clinical pharmacokineticsHard
Practice exercise (made-up values, not clinical guidance). A 47-year-old woman weighs 67 kg and has a serum creatinine of 2.1 mg/dL (186 µmol/L). Estimate the creatinine clearance with Cockcroft–Gault: CrCl (mL/min) = (140 − age) × weight (kg)/(72 × SCr in mg/dL), × 0.85 for women. The usual vancomycin regimen is 1000 mg every 12 h for a creatinine clearance of 120 mL/min. Keeping the dose, the interval is lengthened in proportion to the fall in clearance. Find the new dosing interval τ. Round to one decimal place.
Practice exercise (made-up values, not clinical guidance). A vial contains cefuroxime 750 mg as a powder. The practice label says to add 7.2 mL of sterile water; the powder adds 0.3 mL to the volume. What is the concentration C in mg/mL after reconstitution?
750 mg7.2 mL+0.3 mL
C = mg/mL
4.Pharmacy calculationsHard
Practice exercise (made-up values, not clinical guidance). You must prepare 100 mL of 8% sodium chloride by mixing 23% and 3% sodium chloride. How many mL V of the 3% solution do you need? Round to the nearest whole mL.
23%3%8%100 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 70 units/kg for a patient who weighs 110 kg. The vial contains 1 000 units/mL. What volume V do you draw up? Round to one decimal place.
70 units/kg110 kg1 000 units/mL
V = mL
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 12.5 mcg/kg/min for a patient who weighs 54 kg. At what rate R do you set the pump, in mL/h? Round to one decimal place.
500 mg/250 mL12.5 mcg/kg/min54 kg
R = mL/h
7.PharmacologyMedium
How does glyceryl trinitrate (GTN) relieve angina?
AIt releases nitric oxide, mainly dilating veins and reducing cardiac workload
BIt blocks beta-1 receptors and slows the heart
CIt dissolves the clot in the coronary artery
DIt increases the force of contraction so the heart pumps blood more efficiently
8.PharmacologyHard
Probenecid has been used to raise plasma penicillin levels. How does it do this?
AIt inhibits the liver metabolism of penicillin
BIt blocks the tubular secretion of penicillin in the kidney
CIt increases the absorption of penicillin from the small intestine
DIt displaces penicillin from plasma proteins
3Solutions
1.Clinical pharmacokinetics
Answer: LD = 712.5 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 = 0.5·76 = 38 L
Amount of active drug needed
Vd·C = 38·15 = 570 mg
Loading dose
LD = 570/(0.8·1) = 712.5 mg
Answer
LD = 712.5 mg
In practice the dose is rounded to a convenient amount.
2.Clinical pharmacokinetics
Answer: τ = 41.1 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.
Creatinine clearance (Cockcroft–Gault)
CrCl = (140 − 47)·67/(72·2.1)·0.85 = 35.03 mL/min
New interval
τ = 12·120/35.03 = 41.11 ≈ 41.1 h
Answer
τ = 41.1 h
In practice the interval is rounded to a convenient value such as 12, 24, 36 or 48 h.
3.Pharmacy calculations
Answer: C = 100 mg/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
7.2 + 0.3 = 7.5 mL
Concentration
C = 750/7.5 = 100 mg/mL
Answer
C = 100 mg/mL
4.Pharmacy calculations
Answer: V = 75 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%: 8 − 3 = 5 parts
3%: 23 − 8 = 15 parts
total: 20 parts
Share the volume
V = 100·15/20 = 75 mL
Answer
V = 75 mL
Check: 25 mL of 23% + 75 mL of 3% = 100 mL.
5.Pediatric & critical-care dosing
Answer: V = 7.7 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
70·110 = 7 700 units
Volume
V = 7700/1000 = 7.7 mL
Answer
V = 7.7 mL
6.Pediatric & critical-care dosing
Answer: R = 20.3 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
12.5·54 = 675 mcg/min
675·60 = 40500 mcg/h
Pump rate
R = 40500/2000 = 20.25 ≈ 20.3 mL/h
Answer
R = 20.3 mL/h
7.Pharmacology
Answer: A · It releases nitric oxide, mainly dilating veins and reducing cardiac workload
Why this is the answer
GTN is converted to nitric oxide, which relaxes vascular smooth muscle through cGMP. Venodilation lowers preload, so the heart works less and needs less oxygen.
Answer
A · It releases nitric oxide, mainly dilating veins and reducing cardiac workload
8.Pharmacology
Answer: B · It blocks the tubular secretion of penicillin in the kidney
Why this is the answer
Penicillins are excreted largely by organic anion transporters in the proximal tubule. Probenecid competes for these transporters, so penicillin is excreted more slowly and its level stays higher.
Answer
B · It blocks the tubular secretion of penicillin in the kidney