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 9 mg/L at 2 h and 3.2 mg/L at 5 h. Assume first-order elimination. Find the elimination rate constant k.
C1 = 9 mg/L (2 h)C2 = 3.2 mg/L (5 h)
k = h⁻¹
2.Clinical pharmacokineticsHard
Practice exercise (made-up values, not clinical guidance). A 41-year-old man weighs 90 kg and has a serum creatinine of 3.6 mg/dL (318 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 120 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 contains ampicillin 1000 mg as a powder. The practice label says to add 3.5 mL of sterile water; the powder adds 0.5 mL to the volume. The prescribed dose is 300 mg. What volume V do you draw up? Round to one decimal place.
1000 mg3.5 mL+0.5 mL300 mg
V = mL
4.Pharmacy calculationsHard
A vial holds 2000 mg of powder. You add 3.8 mL of diluent, and the powder adds 0.2 mL of volume. The prescription is 1050 mg. Draw up the dose: what volume V do you give?
2000 mg3.8 mL + 0.2 mL1050 mg
V = mL
5.Pediatric & critical-care dosingMedium
Practice exercise (made-up values, not clinical guidance). An infusion contains 100 units of regular insulin in 250 mL. The order is 0.05 units/kg/h for a patient who weighs 52 kg. Find the pump rate R in mL/h. Round to one decimal place.
100 units/250 mL0.05 units/kg/h52 kg
R = mL/h
6.Pediatric & critical-care dosingHard
Practice exercise (made-up values, not clinical guidance). A bag contains dobutamine 500 mg in 250 mL. The order is 5 mcg/kg/min for a patient who weighs 77 kg. At what rate R do you set the pump, in mL/h? Round to one decimal place.
500 mg/250 mL5 mcg/kg/min77 kg
R = mL/h
7.PharmacologyMedium
Why are patients taking simvastatin often advised to avoid large amounts of grapefruit juice?
AGrapefruit speeds up simvastatin breakdown in the liver, so the drug stops working
BGrapefruit inhibits CYP3A4 in the gut, raising simvastatin levels and myopathy risk
CGrapefruit blocks cholesterol absorption and causes low cholesterol
DGrapefruit makes the tablet dissolve in the mouth
8.PharmacologyHard
A patient starting sertraline is taught about the drug. Which statement shows a need for further teaching?
A"It may take a few weeks to feel the full effect."
B"If I feel better after a week, I can stop taking it."
C"I may have some nausea at first."
D"I will tell someone if I have thoughts of harming myself."
3Solutions
1.Clinical pharmacokinetics
Answer: k = 0.3447 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(9/3.2)/3 = 1.034/3 = 0.3447 h⁻¹
Answer
k = 0.3447 h⁻¹
2.Clinical pharmacokinetics
Answer: D = 529 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 − 41)·90/(72·3.6) = 34.38 mL/min
Dose fraction (Dettli)
Q = 1 − 0.66·(1 − 34.38/120) = 0.5291
Adjusted dose
D = 0.5291·1000 = 529.1 ≈ 529 mg
Answer
D = 529 mg
In practice the dose is rounded to a convenient amount and checked against the product information.
3.Pharmacy calculations
Answer: V = 1.2 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
3.5 + 0.5 = 4 mL
Concentration
1000/4 = 250 mg/mL
Volume to give
V = 300/250 = 1.2 mL
Answer
V = 1.2 mL
4.Pharmacy calculations
Answer: V = 2.1 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
3.8 + 0.2 = 4 mL
C = 2000/4 = 500 mg/mL
Volume
V = 1050/500 = 2.1 mL
Answer
V = 2.1 mL
5.Pediatric & critical-care dosing
Answer: R = 6.5 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
0.05·52 = 2.6 units/h
Concentration
100/250 = 0.4 units/mL
Pump rate
R = 2.6/0.4 = 6.5 mL/h
Answer
R = 6.5 mL/h
6.Pediatric & critical-care dosing
Answer: R = 11.6 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
500 mg = 500 000 mcg
500 000/250 = 2000 mcg/mL
Dose per minute and per hour
5·77 = 385 mcg/min
385·60 = 23100 mcg/h
Pump rate
R = 23100/2000 = 11.55 ≈ 11.6 mL/h
Answer
R = 11.6 mL/h
7.Pharmacology
Answer: B · Grapefruit inhibits CYP3A4 in the gut, raising simvastatin levels and myopathy risk
Why this is the answer
Grapefruit compounds inhibit intestinal CYP3A4, so more simvastatin is absorbed. Higher levels increase the risk of muscle damage; drugs such as some calcium channel blockers and ciclosporin are also affected.
Answer
B · Grapefruit inhibits CYP3A4 in the gut, raising simvastatin levels and myopathy risk
8.Pharmacology
Answer: B · "If I feel better after a week, I can stop taking it."
Why this is the answer
Antidepressants are continued for months after improvement to prevent relapse, and are tapered when stopped. Early nausea, delayed benefit and monitoring for suicidal thoughts are correct points.
Answer
B · "If I feel better after a week, I can stop taking it."