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). After an IV dose of tobramycin, the level is 10 mg/L at 1 h and 5 mg/L at 4 h. Assume first-order elimination. Find the half-life t½.
C1 = 10 mg/L (1 h)C2 = 5 mg/L (4 h)
t½ = h
2.Clinical pharmacokineticsHard
Practice exercise (made-up values, not clinical guidance). A 56-year-old man weighs 76 kg and has a serum creatinine of 3 mg/dL (265 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 tobramycin regimen is 120 mg every 8 h for a creatinine clearance of 100 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 oxacillin 2000 mg as a powder. The practice label says to add 11.5 mL of sterile water; the powder adds 1 mL to the volume. What is the concentration C in mg/mL after reconstitution?
2000 mg11.5 mL+1 mL
C = mg/mL
4.Pharmacy calculationsHard
A vial holds 500 mg of powder. You add 4.5 mL of diluent, and the powder adds 0.5 mL of volume. The prescription is 180 mg. Draw up the dose: what volume V do you give?
500 mg4.5 mL + 0.5 mL180 mg
V = mL
5.Pediatric & critical-care dosingMedium
Practice exercise (made-up values, not clinical guidance). A bag contains 25 000 units of heparin in 500 mL. The order is 12 units/kg/h for a patient who weighs 87 kg. At what rate R do you set the pump, in mL/h? Round to one decimal place.
25 000 units/500 mL12 units/kg/h87 kg
R = mL/h
6.Pediatric & critical-care dosingHard
Practice exercise (made-up values, not clinical guidance). A dobutamine infusion of 500 mg in 250 mL runs at 26 mL/h. The patient weighs 81 kg. What dose D in mcg/kg/min is the patient receiving? Round to one decimal place.
500 mg/250 mL26 mL/h81 kg
D = mcg/kg/min
7.PharmacologyMedium
Which statement about allopurinol is correct?
AIt lowers uric acid over time and does not relieve the pain of an acute attack
BIt works within an hour to relieve joint pain
CIt is taken only during an acute attack and stopped as soon as the pain settles
DIt increases uric acid production to flush out crystals
8.PharmacologyHard
Which heart failure drug improves symptoms and reduces hospital admissions but has not been shown to reduce mortality?
AA mineralocorticoid receptor antagonist such as spironolactone
BA beta blocker such as bisoprolol
CDigoxin
DAn ACE inhibitor
3Solutions
1.Clinical pharmacokinetics
Answer: t½ = 3 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.
Elimination rate constant
k = ln(10/5)/3 = 0.6931/3 = 0.231 h⁻¹
Half-life
t½ = 0.693/0.231 = 3 h
Answer
t½ = 3 h
2.Clinical pharmacokinetics
Answer: τ = 27.1 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 − 56)·76/(72·3) = 29.56 mL/min
New interval
τ = 8·100/29.56 = 27.07 ≈ 27.1 h
Answer
τ = 27.1 h
In practice the interval is rounded to a convenient value such as 12, 24, 36 or 48 h.
3.Pharmacy calculations
Answer: C = 160 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
11.5 + 1 = 12.5 mL
Concentration
C = 2000/12.5 = 160 mg/mL
Answer
C = 160 mg/mL
4.Pharmacy calculations
Answer: V = 1.8 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
4.5 + 0.5 = 5 mL
C = 500/5 = 100 mg/mL
Volume
V = 180/100 = 1.8 mL
Answer
V = 1.8 mL
5.Pediatric & critical-care dosing
Answer: R = 20.9 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
12·87 = 1 044 units/h
Concentration
25 000/500 = 50 units/mL
Pump rate
R = 1044/50 = 20.88 ≈ 20.9 mL/h
Answer
R = 20.9 mL/h
6.Pediatric & critical-care dosing
Answer: D = 10.7 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
500 000/250 = 2000 mcg/mL
Micrograms per minute
26·2000 = 52000 mcg/h
52000/60 = 866.7 mcg/min
Per kilogram
D = 866.7/81 = 10.7 mcg/kg/min
Answer
D = 10.7 mcg/kg/min
7.Pharmacology
Answer: A · It lowers uric acid over time and does not relieve the pain of an acute attack
Why this is the answer
Allopurinol reduces urate production and is taken daily to prevent attacks. Acute attacks are treated with NSAIDs, colchicine or corticosteroids, and a new rash on allopurinol must be reported.
Answer
A · It lowers uric acid over time and does not relieve the pain of an acute attack
8.Pharmacology
Answer: C · Digoxin
Why this is the answer
ACE inhibitors (or ARBs/ARNI), beta blockers, mineralocorticoid antagonists and SGLT2 inhibitors reduce mortality in heart failure with reduced ejection fraction. Digoxin improves symptoms and reduces admissions without reducing mortality.