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 vancomycin, the level is 25 mg/L at 1 h and 6.3 mg/L at 17 h. Assume first-order elimination. Find the half-life t½.
C1 = 25 mg/L (1 h)C2 = 6.3 mg/L (17 h)
t½ = h
2.Clinical pharmacokineticsHard
Practice exercise (made-up values, not clinical guidance). A 61 kg patient on regular gentamicin doses is at steady state. The target is a peak of 8 mg/L and a trough of 1 mg/L, and the volume of distribution is 0.3 L/kg. Assuming rapid IV doses, what dose D raises the level from the trough to the peak?
61 kgVd = 0.3 L/kgCpeak = 8 mg/LCtrough = 1 mg/L
D = mg
3.Pharmacy calculationsMedium
A solution has a ratio strength of 1:1000. You need 0.78 mg. Draw up the volume in the 1 mL syringe: what volume V is that?
1:10000.78 mg
V = mL
4.Pharmacy calculationsHard
Practice exercise (made-up values, not clinical guidance). You have 30 g of 2.5% hydrocortisone ointment. How many grams m of 0.5% hydrocortisone ointment must you add to make a 1% ointment? Round to the nearest whole gram.
30 g · 2.5%0.5%1%
m = g
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 83 kg. The vial contains 1 000 units/mL. What volume V do you draw up? Round to one decimal place.
70 units/kg83 kg1 000 units/mL
V = mL
6.Pediatric & critical-care dosingHard
Practice exercise (made-up values, not clinical guidance). A dopamine infusion of 400 mg in 250 mL runs at 32 mL/h. The patient weighs 69 kg. What dose D in mcg/kg/min is the patient receiving? Round to one decimal place.
400 mg/250 mL32 mL/h69 kg
D = mcg/kg/min
7.PharmacologyMedium
Which adverse effect of ciprofloxacin should a patient report at once?
ADarker stools than usual
BMild tiredness in the first few days
CA slightly bitter or metallic taste for a while after each dose
DPain or swelling in a tendon, such as the Achilles tendon
8.PharmacologyHard
Some receptors show activity even without an agonist. How does an inverse agonist differ from a neutral antagonist at such receptors?
AIt has no effect when given alone
BIt reduces activity below the baseline level
CIt destroys the receptor permanently
DIt increases activity above the baseline
3Solutions
1.Clinical pharmacokinetics
Answer: t½ = 8.046 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(25/6.3)/16 = 1.378/16 = 0.08615 h⁻¹
Half-life
t½ = 0.693/0.08615 = 8.046 h
Answer
t½ = 8.046 h
2.Clinical pharmacokinetics
Answer: D = 128.1 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.
Volume of distribution
Vd = 0.3·61 = 18.3 L
Dose
D = 18.3·(8 − 1) = 128.1 mg
Answer
D = 128.1 mg
In practice the dose is rounded to a convenient amount.
3.Pharmacy calculations
Answer: V = 0.78 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.
Strength in mg/mL
1000 mg/1000 mL = 1 mg/mL
Volume
V = 0.78/1 = 0.78 mL
Answer
V = 0.78 mL
4.Pharmacy calculations
Answer: m = 90 g
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
2.5%: 1 − 0.5 = 0.5 parts
0.5%: 2.5 − 1 = 1.5 parts
Scale to the amount you have
m = 30·1.5/0.5 = 90 g
Answer
m = 90 g
Check: 30 g + 90 g = 120 g of 1% ointment.
5.Pediatric & critical-care dosing
Answer: V = 5.8 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
70·83 = 5 810 units
Volume
V = 5810/1000 = 5.81 ≈ 5.8 mL
Answer
V = 5.8 mL
6.Pediatric & critical-care dosing
Answer: D = 12.4 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
400 000/250 = 1600 mcg/mL
Micrograms per minute
32·1600 = 51200 mcg/h
51200/60 = 853.3 mcg/min
Per kilogram
D = 853.3/69 = 12.37 ≈ 12.4 mcg/kg/min
Answer
D = 12.4 mcg/kg/min
7.Pharmacology
Answer: D · 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
D · Pain or swelling in a tendon, such as the Achilles tendon
8.Pharmacology
Answer: B · It reduces activity below the baseline level
Why this is the answer
A neutral antagonist simply blocks agonists and leaves baseline activity unchanged. An inverse agonist stabilises the inactive receptor state, so constitutive activity falls below baseline.