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). A patient receives an IV bolus of 160 mg of tobramycin. The level back-extrapolated to time zero is 9 mg/L and the half-life is 2.5 h. Find the clearance CL in L/h.
D = 160 mgC₀ = 9 mg/Lt½ = 2.5 h
CL = L/h
2.Clinical pharmacokineticsHard
Practice exercise (made-up values, not clinical guidance). A 57-year-old man weighs 57 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 gentamicin dose is 120 mg every 8 h for a creatinine clearance of 100 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 of penicillin G contains 5 000 000 units. Its practice label gives three strengths (see the table). You add 8.2 mL of diluent. The prescribed dose is 2 400 000 units. What volume V do you draw up? Round to one decimal place.
8.2 mL → 500 000 units/mL2 400 000 units
V = mL
4.Pharmacy calculationsHard
Practice exercise (made-up values, not clinical guidance). You must prepare 450 mL of 30% dextrose by mixing 50% and 10% dextrose. How many mL V of the 50% solution do you need? Round to the nearest whole mL.
50%10%30%450 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 20 units/kg/h for a patient who weighs 60 kg. At what rate R do you set the pump, in mL/h? Round to one decimal place.
25 000 units/500 mL20 units/kg/h60 kg
R = mL/h
6.Pediatric & critical-care dosingHard
Practice exercise (made-up values, not clinical guidance). A patient who weighs 75 kg has an order for norepinephrine at 2 mcg/min. The bag contains 4 mg in 250 mL. Find the pump rate R in mL/h. Round to one decimal place.
4 mg/250 mL2 mcg/min75 kg
R = mL/h
7.PharmacologyMedium
A drug inhibits the liver enzyme CYP3A4. What can happen to a second drug that is mainly metabolised by CYP3A4?
AIts plasma level falls, so it may stop working at the usual dose
BIts plasma level rises, increasing the risk of side effects
CIt is excreted faster by the kidneys
DNothing, because enzymes do not affect other drugs
8.PharmacologyHard
Which property makes a drug more likely to cross the blood–brain barrier?
AHigh water solubility and strong ionisation
BHigh lipid solubility and small size
CPositive charge at physiological pH
DLarge molecular weight and strong protein binding
3Solutions
1.Clinical pharmacokinetics
Answer: CL = 4.929 L/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.
Volume of distribution
Vd = 160/9 = 17.78 L
Rate constant
k = 0.693/2.5 = 0.2773 h⁻¹
Clearance
CL = 0.2773·17.78 = 4.929 L/h
Answer
CL = 4.929 L/h
2.Clinical pharmacokinetics
Answer: D = 49 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 − 57)·57/(72·1.6) = 41.07 mL/min
Adjusted dose
D = 120·41.07/100 = 49.28 ≈ 49 mg
Answer
D = 49 mg
In practice the dose is rounded to a convenient amount and checked against the product information.
3.Pharmacy calculations
Answer: V = 4.8 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
8.2 mL → 500 000 units/mL
Volume to give
V = 2 400 000/500 000 = 4.8 mL
Answer
V = 4.8 mL
4.Pharmacy calculations
Answer: V = 225 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
50%: 30 − 10 = 20 parts
10%: 50 − 30 = 20 parts
total: 40 parts
Share the volume
V = 450·20/40 = 225 mL
Answer
V = 225 mL
Check: 225 mL of 50% + 225 mL of 10% = 450 mL.
5.Pediatric & critical-care dosing
Answer: R = 24 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
20·60 = 1 200 units/h
Concentration
25 000/500 = 50 units/mL
Pump rate
R = 1200/50 = 24 mL/h
Answer
R = 24 mL/h
6.Pediatric & critical-care dosing
Answer: R = 7.5 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
4 000/250 = 16 mcg/mL
Dose per hour
The order is per minute for the whole patient, so the weight is not needed.
2·60 = 120 mcg/h
Pump rate
R = 120/16 = 7.5 mL/h
Answer
R = 7.5 mL/h
7.Pharmacology
Answer: B · Its plasma level rises, increasing the risk of side effects
Why this is the answer
Enzyme inhibitors (for example clarithromycin or grapefruit juice for CYP3A4) slow metabolism, so the second drug accumulates. Enzyme inducers have the opposite effect.
Answer
B · Its plasma level rises, increasing the risk of side effects
8.Pharmacology
Answer: B · High lipid solubility and small size
Why this is the answer
Small, lipophilic, non-ionised molecules diffuse across endothelial cell membranes. Charged or large, protein-bound drugs mostly stay out.