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 130 mg of tobramycin. The level back-extrapolated to time zero is 6 mg/L and the half-life is 3 h. Find the clearance CL in L/h.
D = 130 mgC₀ = 6 mg/Lt½ = 3 h
CL = L/h
2.Clinical pharmacokineticsHard
Practice exercise (made-up values, not clinical guidance). A 48-year-old woman weighs 86 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 ciprofloxacin dose is 500 mg for a creatinine clearance of 120 mL/min, and a fraction fe = 0.6 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.
A solution has a ratio strength of 1:1000. You need 0.65 mg. Draw up the volume in the 1 mL syringe: what volume V is that?
1:10000.65 mg
V = mL
4.Pharmacy calculationsHard
Practice exercise (made-up values, not clinical guidance). You must prepare 600 mL of 6% hydrogen peroxide from a 30% stock and purified water. How many mL V of water do you need? Round to the nearest whole mL.
30%0%6%600 mL
V = mL
5.Pediatric & critical-care dosingMedium
Practice exercise (made-up values, not clinical guidance). An infusion contains 250 units of regular insulin in 250 mL. The order is 7 units/h. Find the pump rate R in mL/h. Round to one decimal place.
250 units/250 mL7 units/h
R = mL/h
6.Pediatric & critical-care dosingHard
Practice exercise (made-up values, not clinical guidance). A bag contains dopamine 400 mg in 250 mL. The order is 3 mcg/kg/min for a patient who weighs 83 kg. At what rate R do you set the pump, in mL/h? Round to one decimal place.
400 mg/250 mL3 mcg/kg/min83 kg
R = mL/h
7.PharmacologyMedium
A patient has severe kidney impairment. A drug is mainly excreted unchanged in the urine. What is the usual adjustment?
AIncrease the dose to make up for poor absorption
BReduce the dose or lengthen the dosing interval
CGive the drug orally instead of intravenously
DNo change is needed, because the liver takes over
8.PharmacologyHard
A 500 mg intravenous dose gives an initial plasma concentration of 10 mg/L. What is the volume of distribution?
A500 L
B50 L
C5000 L
D5 L
3Solutions
1.Clinical pharmacokinetics
Answer: CL = 5.006 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 = 130/6 = 21.67 L
Rate constant
k = 0.693/3 = 0.231 h⁻¹
Clearance
CL = 0.231·21.67 = 5.006 L/h
Answer
CL = 5.006 L/h
2.Clinical pharmacokinetics
Answer: D = 265 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 − 48)·86/(72·3.6)·0.85 = 25.95 mL/min
Dose fraction (Dettli)
Q = 1 − 0.6·(1 − 25.95/120) = 0.5297
Adjusted dose
D = 0.5297·500 = 264.9 ≈ 265 mg
Answer
D = 265 mg
In practice the dose is rounded to a convenient amount and checked against the product information.
3.Pharmacy calculations
Answer: V = 0.65 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 = 0.65/1 = 0.65 mL
Answer
V = 0.65 mL
4.Pharmacy calculations
Answer: V = 480 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
30%: 6 − 0 = 6 parts
0%: 30 − 6 = 24 parts
total: 30 parts
Water
V = 600·24/30 = 480 mL
Answer
V = 480 mL
Check: 120 mL of 30% contains 36 g of drug, which is 6% of 600 mL.
5.Pediatric & critical-care dosing
Answer: R = 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.
Concentration
250/250 = 1 units/mL
Pump rate
R = 7/1 = 7 mL/h
Answer
R = 7 mL/h
6.Pediatric & critical-care dosing
Answer: R = 9.3 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
400 mg = 400 000 mcg
400 000/250 = 1600 mcg/mL
Dose per minute and per hour
3·83 = 249 mcg/min
249·60 = 14940 mcg/h
Pump rate
R = 14940/1600 = 9.338 ≈ 9.3 mL/h
Answer
R = 9.3 mL/h
7.Pharmacology
Answer: B · Reduce the dose or lengthen the dosing interval
Why this is the answer
When renal clearance falls, drugs excreted by the kidney accumulate. The dose is lowered or given less often, guided by kidney function and, where possible, drug levels.
Answer
B · Reduce the dose or lengthen the dosing interval
8.Pharmacology
Answer: B · 50 L
Why this is the answer
Vd = 500 mg ÷ 10 mg/L = 50 L. This is close to total body water, so the drug spreads beyond the plasma but is not extensively stored in tissues.