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 6 mg/L at 2 h and 3 mg/L at 4 h. Assume first-order elimination. Find the elimination rate constant k.
C1 = 6 mg/L (2 h)C2 = 3 mg/L (4 h)
k = h⁻¹
2.Clinical pharmacokineticsHard
Practice exercise (made-up values, not clinical guidance). A 79-year-old woman weighs 92 kg and has a serum creatinine of 2.2 mg/dL (194 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 gentamicin dose is 120 mg every 8 h for a creatinine clearance of 120 mL/min. Keeping the interval, the dose is reduced in proportion to the creatinine clearance. Find the adjusted dose D. Round to the nearest whole number.
Practice exercise (made-up values, not clinical guidance). A vial contains ceftriaxone 1000 mg as a powder. The practice label says to add 9.6 mL of sterile water; the powder adds 0.4 mL to the volume. What is the concentration C in mg/mL after reconstitution?
1000 mg9.6 mL+0.4 mL
C = mg/mL
4.Pharmacy calculationsHard
Practice exercise (made-up values, not clinical guidance). You must prepare 200 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%200 mL
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 16 units/kg/h for a patient who weighs 96 kg. At what rate R do you set the pump, in mL/h? Round to one decimal place.
25 000 units/500 mL16 units/kg/h96 kg
R = mL/h
6.Pediatric & critical-care dosingHard
Practice exercise (made-up values, not clinical guidance). A bag contains dobutamine 250 mg in 250 mL. The order is 7.5 mcg/kg/min for a patient who weighs 51 kg. At what rate R do you set the pump, in mL/h? Round to one decimal place.
250 mg/250 mL7.5 mcg/kg/min51 kg
R = mL/h
7.PharmacologyMedium
Which adverse effect of ciprofloxacin should a patient report at once?
AMild tiredness in the first few days
BPain or swelling in a tendon, such as the Achilles tendon
CDarker stools than usual
DA slightly bitter or metallic taste for a while after each dose
8.PharmacologyHard
A man taking doxazosin for hypertension starts sildenafil for erectile dysfunction. What is the main risk?
AHyperkalaemia
BSymptomatic low blood pressure and fainting
CBronchospasm
DSevere hypertension and a pounding headache
3Solutions
1.Clinical pharmacokinetics
Answer: k = 0.3466 h⁻¹
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(6/3)/2 = 0.6931/2 = 0.3466 h⁻¹
Answer
k = 0.3466 h⁻¹
2.Clinical pharmacokinetics
Answer: D = 30 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.
Creatinine clearance (Cockcroft–Gault)
CrCl = (140 − 79)·92/(72·2.2)·0.85 = 30.11 mL/min
Adjusted dose
D = 120·30.11/120 = 30.11 ≈ 30 mg
Answer
D = 30 mg
In practice the dose is rounded to a convenient amount and checked against the product information.
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
9.6 + 0.4 = 10 mL
Concentration
C = 1000/10 = 100 mg/mL
Answer
C = 100 mg/mL
4.Pharmacy calculations
Answer: V = 50 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 = 200·15/60 = 50 mL
Answer
V = 50 mL
Check: 50 mL of 70% + 150 mL of 10% = 200 mL.
5.Pediatric & critical-care dosing
Answer: R = 30.7 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
16·96 = 1 536 units/h
Concentration
25 000/500 = 50 units/mL
Pump rate
R = 1536/50 = 30.72 ≈ 30.7 mL/h
Answer
R = 30.7 mL/h
6.Pediatric & critical-care dosing
Answer: R = 23 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
250 mg = 250 000 mcg
250 000/250 = 1000 mcg/mL
Dose per minute and per hour
7.5·51 = 382.5 mcg/min
382.5·60 = 22950 mcg/h
Pump rate
R = 22950/1000 = 22.95 ≈ 23 mL/h
Answer
R = 23 mL/h
7.Pharmacology
Answer: B · Pain or swelling in a tendon, such as the Achilles tendon
Why this is the answer
Fluoroquinolones can cause tendinitis and tendon rupture. Patients should stop exercise and report tendon pain promptly.
Answer
B · Pain or swelling in a tendon, such as the Achilles tendon
8.Pharmacology
Answer: B · Symptomatic low blood pressure and fainting
Why this is the answer
Alpha blockers and PDE5 inhibitors both cause vasodilation, so together they can cause marked hypotension. Patients should be stable on the alpha blocker, and the PDE5 inhibitor is started at a low dose.