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). Vancomycin is given as a continuous infusion to a patient whose vancomycin clearance is 3.5 L/h. What infusion rate R in mg/h gives a steady-state level of 25 mg/L?
CL = 3.5 L/hCss = 25 mg/L
R = mg/h
2.Clinical pharmacokineticsHard
Practice exercise (made-up values, not clinical guidance). A 74 kg patient takes oral aminophylline (S = 0.8, F = 1) every 12 h. The theophylline clearance is 0.045 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 of penicillin G contains 5 000 000 units. Its practice label gives three strengths (see the table). You add 3.2 mL of diluent. The prescribed dose is 600 000 units. What volume V do you draw up? Round to one decimal place.
3.2 mL → 1 000 000 units/mL600 000 units
V = mL
4.Pharmacy calculationsHard
Practice exercise (made-up values, not clinical guidance). You must prepare 200 mL of 4% chlorhexidine from a 20% stock and purified water. How many mL V of water do you need? Round to the nearest whole mL.
20%0%4%200 mL
V = mL
5.Pediatric & critical-care dosingMedium
Practice exercise (made-up values, not clinical guidance). An infusion contains 250 units of regular insulin in 250 mL. The order is 0.05 units/kg/h for a patient who weighs 84 kg. Find the pump rate R in mL/h. Round to one decimal place.
250 units/250 mL0.05 units/kg/h84 kg
R = mL/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 74 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/min74 kg
R = mL/h
7.PharmacologyMedium
How does metformin mainly lower blood glucose?
AIt replaces missing insulin
BIt stimulates insulin release from the pancreas
CIt blocks glucose absorption in the kidney
DIt reduces glucose production by the liver
8.PharmacologyHard
An 80-year-old man with an enlarged prostate starts oxybutynin for an overactive bladder. He becomes confused and cannot pass urine. What is the most likely explanation?
AAnticholinergic adverse effects
BOpioid toxicity
CAn allergic reaction to the drug
DSerotonin syndrome
3Solutions
1.Clinical pharmacokinetics
Answer: R = 87.5 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.
Infusion rate
R = 3.5·25 = 87.5 mg/h
Answer
R = 87.5 mg/h
2.Clinical pharmacokinetics
Answer: D = 499.5 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 = 0.045·74 = 3.33 L/h
Dose
D = Css,avg·CL·τ/(S·F) = 10·3.33·12/(0.8·1) = 499.5 mg
Answer
D = 499.5 mg
3.Pharmacy calculations
Answer: V = 0.6 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.
Concentration from the label
3.2 mL → 1 000 000 units/mL
Volume to give
V = 600 000/1 000 000 = 0.6 mL
Answer
V = 0.6 mL
4.Pharmacy calculations
Answer: V = 160 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
20%: 4 − 0 = 4 parts
0%: 20 − 4 = 16 parts
total: 20 parts
Water
V = 200·16/20 = 160 mL
Answer
V = 160 mL
Check: 40 mL of 20% contains 8 g of drug, which is 4% of 200 mL.
5.Pediatric & critical-care dosing
Answer: R = 4.2 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
0.05·84 = 4.2 units/h
Concentration
250/250 = 1 units/mL
Pump rate
R = 4.2/1 = 4.2 mL/h
Answer
R = 4.2 mL/h
6.Pediatric & critical-care dosing
Answer: R = 16.7 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
7.5·74 = 555 mcg/min
555·60 = 33300 mcg/h
Pump rate
R = 33300/2000 = 16.65 ≈ 16.7 mL/h
Answer
R = 16.7 mL/h
7.Pharmacology
Answer: D · It reduces glucose production by the liver
Why this is the answer
Metformin mainly reduces hepatic gluconeogenesis and improves insulin sensitivity. Because it does not stimulate insulin release, hypoglycaemia is rare when used alone.
Answer
D · It reduces glucose production by the liver
8.Pharmacology
Answer: A · Anticholinergic adverse effects
Why this is the answer
Antimuscarinic drugs reduce bladder contraction, which can cause urinary retention in men with prostate enlargement. In older people they can also cause confusion, so their total anticholinergic burden should be kept low.