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 230 mg of gentamicin. The level back-extrapolated to time zero is 12 mg/L and the half-life is 2 h. Find the volume of distribution Vd.
D = 230 mgC₀ = 12 mg/Lt½ = 2 h
Vd = L
2.Clinical pharmacokineticsHard
Practice exercise (made-up values, not clinical guidance). A vancomycin infusion is started without a loading dose. The half-life in this patient is 6 h. How long t does it take to reach 50% of the steady-state level?
t½ = 6 h50% Css
t = h
3.Pharmacy calculationsMedium
Practice exercise (made-up values, not clinical guidance). A vial contains oxacillin 2000 mg as a powder. The practice label says to add 11.5 mL of sterile water; the powder adds 1 mL to the volume. What is the concentration C in mg/mL after reconstitution?
2000 mg11.5 mL+1 mL
C = mg/mL
4.Pharmacy calculationsHard
A vial holds 500 mg of powder. You add 1.8 mL of diluent, and the powder adds 0.2 mL of volume. The prescription is 450 mg. Draw up the dose: what volume V do you give?
500 mg1.8 mL + 0.2 mL450 mg
V = mL
5.Pediatric & critical-care dosingMedium
Practice exercise (made-up values, not clinical guidance). A practice protocol calls for a heparin bolus of 80 units/kg for a patient who weighs 70 kg. The vial contains 5 000 units/mL. What volume V do you draw up? Round to one decimal place.
80 units/kg70 kg5 000 units/mL
V = mL
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 2 mcg/kg/min for a patient who weighs 82 kg. At what rate R do you set the pump, in mL/h? Round to one decimal place.
400 mg/250 mL2 mcg/kg/min82 kg
R = mL/h
7.PharmacologyMedium
Why do NSAIDs such as ibuprofen increase the risk of stomach ulcers?
AThey raise the pH of gastric acid
BThey reduce protective prostaglandins in the stomach lining
CThey directly dissolve the mucus layer
DThey increase the number of Helicobacter pylori bacteria in the stomach
8.PharmacologyHard
For a drug with first-order elimination, doubling the dose (without changing the interval) will:
ADouble the steady-state level but not change the time to reach steady state
BDouble the half-life
CLeave the steady-state level unchanged but halve the time needed to reach it
DHalve the time to reach steady state
3Solutions
1.Clinical pharmacokinetics
Answer: Vd = 19.17 L
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 = 230/12 = 19.17 L
Answer
Vd = 19.17 L
2.Clinical pharmacokinetics
Answer: t = 6 h
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.
Rate constant
k = 0.693/6 = 0.1155 h⁻¹
Time to the given fraction of steady state
t = −ln(1 − 0.5)/0.1155 = 0.6931/0.1155 = 6 h
Answer
t = 6 h
That is 1 half-lives.
3.Pharmacy calculations
Answer: C = 160 mg/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
11.5 + 1 = 12.5 mL
Concentration
C = 2000/12.5 = 160 mg/mL
Answer
C = 160 mg/mL
4.Pharmacy calculations
Answer: V = 1.8 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
1.8 + 0.2 = 2 mL
C = 500/2 = 250 mg/mL
Volume
V = 450/250 = 1.8 mL
Answer
V = 1.8 mL
5.Pediatric & critical-care dosing
Answer: V = 1.1 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
80·70 = 5 600 units
Volume
V = 5600/5000 = 1.12 ≈ 1.1 mL
Answer
V = 1.1 mL
6.Pediatric & critical-care dosing
Answer: R = 6.2 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
2·82 = 164 mcg/min
164·60 = 9840 mcg/h
Pump rate
R = 9840/1600 = 6.15 ≈ 6.2 mL/h
Answer
R = 6.2 mL/h
7.Pharmacology
Answer: B · They reduce protective prostaglandins in the stomach lining
Why this is the answer
NSAIDs inhibit COX-1, lowering prostaglandins that maintain mucus, bicarbonate secretion and blood flow in the stomach lining.
Answer
B · They reduce protective prostaglandins in the stomach lining
8.Pharmacology
Answer: A · Double the steady-state level but not change the time to reach steady state
Why this is the answer
In first-order kinetics the steady-state concentration is proportional to the dosing rate, but it is still reached after about 4–5 half-lives.
Answer
A · Double the steady-state level but not change the time to reach steady state