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 2 h and 3 mg/L at 4 h. Assume first-order elimination. Find the half-life t½.
C1 = 6 mg/L (2 h)C2 = 3 mg/L (4 h)
t½ = h
2.Clinical pharmacokineticsHard
Practice exercise (made-up values, not clinical guidance). A vancomycin infusion runs at 80 mg/h with no loading dose. The patient's clearance is 4 L/h and the half-life is 6 h. What is the level C after 9 h?
R = 80 mg/hCL = 4 L/ht½ = 6 ht = 9 h
C = mg/L
3.Pharmacy calculationsMedium
A solution has a ratio strength of 1:10 000. You need 0.18 mg. Draw up the volume in the 10 mL syringe: what volume V is that?
1:10 0000.18 mg
V = mL
4.Pharmacy calculationsHard
A vial holds 500 mg of powder. You add 4.6 mL of diluent, and the powder adds 0.4 mL of volume. The prescription is 70 mg. Draw up the dose: what volume V do you give?
500 mg4.6 mL + 0.4 mL70 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 84 kg. Find the pump rate R in mL/h. Round to one decimal place.
100 units/250 mL0.05 units/kg/h84 kg
R = mL/h
6.Pediatric & critical-care dosingHard
Practice exercise (made-up values, not clinical guidance). A patient who weighs 74 kg has an order for norepinephrine at 5 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 mL5 mcg/min74 kg
R = mL/h
7.PharmacologyMedium
Why should long-term oral corticosteroids not be stopped suddenly?
AThe immune system would immediately attack the liver and adrenal glands
BThe drug forms toxic metabolites when stopped
CStopping causes a sudden rise in blood glucose
DThe adrenal glands are suppressed and may not make enough cortisol
8.PharmacologyHard
A nurse calculates a child's dose and finds it is ten times higher than the usual range in the drug reference. What should the nurse do?
AGive half the dose as a compromise
BWithhold it, recheck the calculation and clarify with the prescriber
CGive the dose, because the prescriber is responsible for the order
DGive one tenth of the dose without telling anyone
3Solutions
1.Clinical pharmacokinetics
Answer: t½ = 2 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)/2 = 0.6931/2 = 0.3466 h⁻¹
Half-life
t½ = 0.693/0.3466 = 2 h
Answer
t½ = 2 h
2.Clinical pharmacokinetics
Answer: C = 12.93 mg/L
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.
Steady-state level
Css = 80/4 = 20 mg/L
Rate constant
k = 0.693/6 = 0.1155 h⁻¹
Level after the given time
C = 20·(1 − e−0.1155·9) = 12.93 mg/L
Answer
C = 12.93 mg/L
3.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.
Strength in mg/mL
1000 mg/10 000 mL = 0.1 mg/mL
Volume
V = 0.18/0.1 = 1.8 mL
Answer
V = 1.8 mL
4.Pharmacy calculations
Answer: V = 0.7 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
4.6 + 0.4 = 5 mL
C = 500/5 = 100 mg/mL
Volume
V = 70/100 = 0.7 mL
Answer
V = 0.7 mL
5.Pediatric & critical-care dosing
Answer: R = 10.5 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
0.05·84 = 4.2 units/h
Concentration
100/250 = 0.4 units/mL
Pump rate
R = 4.2/0.4 = 10.5 mL/h
Answer
R = 10.5 mL/h
6.Pediatric & critical-care dosing
Answer: R = 18.8 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.
5·60 = 300 mcg/h
Pump rate
R = 300/16 = 18.75 ≈ 18.8 mL/h
Answer
R = 18.8 mL/h
7.Pharmacology
Answer: D · The adrenal glands are suppressed and may not make enough cortisol
Why this is the answer
Steroid therapy suppresses the hypothalamic–pituitary–adrenal axis. Sudden withdrawal can cause an adrenal crisis, so doses are reduced gradually.
Answer
D · The adrenal glands are suppressed and may not make enough cortisol
8.Pharmacology
Answer: B · Withhold it, recheck the calculation and clarify with the prescriber
Why this is the answer
Doses outside the safe range must be questioned before administration. The nurse holds the dose, rechecks the weight and calculation and contacts the prescriber or pharmacist.
Answer
B · Withhold it, recheck the calculation and clarify with the prescriber