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 92 kg patient is to receive an aminophylline infusion (salt factor S = 0.8: 80% of aminophylline is theophylline). The theophylline clearance is 0.045 L/kg/h. What aminophylline infusion rate R in mg/h gives a steady-state theophylline level of 12 mg/L?
92 kgCL = 0.045 L/kg/hCss = 12 mg/LS = 0.8
R = mg/h
2.Clinical pharmacokineticsHard
Practice exercise (made-up values, not clinical guidance). A 97 kg patient takes oral theophylline (F = 1) every 8 h. The clearance is 0.03 L/kg/h. What dose D in mg gives an average steady-state level of 10 mg/L?
Practice exercise (made-up values, not clinical guidance). A vial contains ampicillin 500 mg as a powder. The practice label says to add 1.8 mL of sterile water; the powder adds 0.2 mL to the volume. What is the concentration C in mg/mL after reconstitution?
500 mg1.8 mL+0.2 mL
C = mg/mL
4.Pharmacy calculationsHard
A vial holds 1000 mg of powder. You add 9.6 mL of diluent, and the powder adds 0.4 mL of volume. The prescription is 310 mg. Draw up the dose: what volume V do you give?
1000 mg9.6 mL + 0.4 mL310 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 16 units/kg/h for a patient who weighs 74 kg. At what rate R do you set the pump, in mL/h? Round to one decimal place.
20 000 units/500 mL16 units/kg/h74 kg
R = mL/h
6.Pediatric & critical-care dosingHard
Practice exercise (made-up values, not clinical guidance). A bag contains dopamine 800 mg in 500 mL. The order is 6 mcg/kg/min for a patient who weighs 106 kg. At what rate R do you set the pump, in mL/h? Round to one decimal place.
800 mg/500 mL6 mcg/kg/min106 kg
R = mL/h
7.PharmacologyMedium
A patient receives lactulose for hepatic encephalopathy. Which finding shows that it is working?
ANo bowel movement for three days, with less confusion than before
BTen watery stools a day and a low serum potassium
CBetter alertness and orientation, with two to three soft stools a day
DHigher blood ammonia with improved appetite
8.PharmacologyHard
A patient on simvastatin is prescribed gemfibrozil. What is the main concern?
ALoss of all lipid-lowering effect
BA sharp rise in LDL cholesterol and triglycerides
CSevere hypoglycaemia
DA much higher risk of myopathy and rhabdomyolysis
3Solutions
1.Clinical pharmacokinetics
Answer: R = 62.1 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 = 0.045·92 = 4.14 L/h
Infusion rate
R = CL·Css/S = 4.14·12/0.8 = 62.1 mg/h
Answer
R = 62.1 mg/h
2.Clinical pharmacokinetics
Answer: D = 232.8 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.
Clearance
CL = 0.03·97 = 2.91 L/h
Dose
D = Css,avg·CL·τ/(S·F) = 10·2.91·8/(1·1) = 232.8 mg
Answer
D = 232.8 mg
3.Pharmacy calculations
Answer: C = 250 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
1.8 + 0.2 = 2 mL
Concentration
C = 500/2 = 250 mg/mL
Answer
C = 250 mg/mL
4.Pharmacy calculations
Answer: V = 3.1 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
9.6 + 0.4 = 10 mL
C = 1000/10 = 100 mg/mL
Volume
V = 310/100 = 3.1 mL
Answer
V = 3.1 mL
5.Pediatric & critical-care dosing
Answer: R = 29.6 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
16·74 = 1 184 units/h
Concentration
20 000/500 = 40 units/mL
Pump rate
R = 1184/40 = 29.6 mL/h
Answer
R = 29.6 mL/h
6.Pediatric & critical-care dosing
Answer: R = 23.9 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
800 mg = 800 000 mcg
800 000/500 = 1600 mcg/mL
Dose per minute and per hour
6·106 = 636 mcg/min
636·60 = 38160 mcg/h
Pump rate
R = 38160/1600 = 23.85 ≈ 23.9 mL/h
Answer
R = 23.9 mL/h
7.Pharmacology
Answer: C · Better alertness and orientation, with two to three soft stools a day
Why this is the answer
Lactulose is adjusted to give about two to three soft stools a day and to lower ammonia, improving mental status. Excessive diarrhoea can cause dehydration and electrolyte loss.
Answer
C · Better alertness and orientation, with two to three soft stools a day
8.Pharmacology
Answer: D · A much higher risk of myopathy and rhabdomyolysis
Why this is the answer
Gemfibrozil reduces statin clearance and adds its own muscle toxicity. The combination greatly increases the risk of rhabdomyolysis and is generally avoided.
Answer
D · A much higher risk of myopathy and rhabdomyolysis