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 gentamicin, the level is 6 mg/L at 1 h and 3 mg/L at 4 h. Assume first-order elimination. Find the half-life t½.
C1 = 6 mg/L (1 h)C2 = 3 mg/L (4 h)
t½ = h
2.Clinical pharmacokineticsHard
Practice exercise (made-up values, not clinical guidance). A 66-year-old man weighs 50 kg and has a serum creatinine of 1.6 mg/dL (141 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 levetiracetam dose is 1000 mg for a creatinine clearance of 100 mL/min, and a fraction fe = 0.66 of the drug is excreted unchanged by the kidneys. Using the Dettli dose fraction Q = 1 − fe·(1 − CrCl/normal CrCl), find the adjusted dose D. Round to the nearest whole number.
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 750 000 units. What volume V do you draw up? Round to one decimal place.
3.2 mL → 1 000 000 units/mL750 000 units
V = mL
4.Pharmacy calculationsHard
Practice exercise (made-up values, not clinical guidance). You must prepare 500 mL of 2% 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%2%500 mL
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 20 units/kg/h for a patient who weighs 86 kg. At what rate R do you set the pump, in mL/h? Round to one decimal place.
20 000 units/500 mL20 units/kg/h86 kg
R = mL/h
6.Pediatric & critical-care dosingHard
Practice exercise (made-up values, not clinical guidance). A patient who weighs 73 kg has an order for nitroglycerin at 10 mcg/min. The bag contains 25 mg in 250 mL. Find the pump rate R in mL/h. Round to one decimal place.
25 mg/250 mL10 mcg/min73 kg
R = mL/h
7.PharmacologyMedium
Which side effects are most common when starting gabapentin?
AHair loss and weight loss
BInsomnia and agitation
CDrowsiness and dizziness
DCough and runny nose
8.PharmacologyHard
A man on long-term testosterone therapy has a rising haematocrit. Why is this important?
AIt is a sign of liver failure caused by testosterone
BTestosterone raises red cell production, increasing clot risk
CA high haematocrit shows that the testosterone is not being absorbed
DIt means the testosterone dose is too low
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(6/3)/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: D = 552 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 − 66)·50/(72·1.6) = 32.12 mL/min
Dose fraction (Dettli)
Q = 1 − 0.66·(1 − 32.12/100) = 0.552
Adjusted dose
D = 0.552·1000 = 552 mg
Answer
D = 552 mg
In practice the dose is rounded to a convenient amount and checked against the product information.
3.Pharmacy calculations
Answer: V = 0.8 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 = 750 000/1 000 000 = 0.75 ≈ 0.8 mL
Answer
V = 0.8 mL
4.Pharmacy calculations
Answer: V = 450 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%: 2 − 0 = 2 parts
0%: 20 − 2 = 18 parts
total: 20 parts
Water
V = 500·18/20 = 450 mL
Answer
V = 450 mL
Check: 50 mL of 20% contains 10 g of drug, which is 2% of 500 mL.
5.Pediatric & critical-care dosing
Answer: R = 43 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
20·86 = 1 720 units/h
Concentration
20 000/500 = 40 units/mL
Pump rate
R = 1720/40 = 43 mL/h
Answer
R = 43 mL/h
6.Pediatric & critical-care dosing
Answer: R = 6 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
25 000/250 = 100 mcg/mL
Dose per hour
The order is per minute for the whole patient, so the weight is not needed.
10·60 = 600 mcg/h
Pump rate
R = 600/100 = 6 mL/h
Answer
R = 6 mL/h
7.Pharmacology
Answer: C · Drowsiness and dizziness
Why this is the answer
Gabapentin and pregabalin commonly cause sedation, dizziness and unsteadiness, which raise the risk of falls. They also increase sedation when combined with opioids.
Answer
C · Drowsiness and dizziness
8.Pharmacology
Answer: B · Testosterone raises red cell production, increasing clot risk
Why this is the answer
Testosterone increases erythropoiesis and can cause polycythaemia, which increases blood viscosity and thrombosis risk. Haematocrit is monitored, and the dose may be reduced or paused.
Answer
B · Testosterone raises red cell production, increasing clot risk