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 1.8 mg/L at 10 h. Assume first-order elimination. Find the half-life t½.
C1 = 9 mg/L (2 h)C2 = 1.8 mg/L (10 h)
t½ = h
2.Clinical pharmacokineticsHard
Practice exercise (made-up values, not clinical guidance). A 90 kg patient has a theophylline level of 9 mg/L; the target is 15 mg/L. The drug is given intravenously as aminophylline (S = 0.8), and the volume of distribution is 0.5 L/kg. What extra loading dose LD raises the level to the target?
90 kgVd = 0.5 L/kgC = 9 → 15 mg/LS = 0.8
LD = mg
3.Pharmacy calculationsMedium
Practice exercise (made-up values, not clinical guidance). A vial contains cefazolin 1000 mg as a powder. The practice label says to add 4.5 mL of sterile water; the powder adds 0.5 mL to the volume. What is the concentration C in mg/mL after reconstitution?
1000 mg4.5 mL+0.5 mL
C = mg/mL
4.Pharmacy calculationsHard
A vial holds 2000 mg of powder. You add 4.2 mL of diluent, and the powder adds 0.8 mL of volume. The prescription is 1760 mg. Draw up the dose: what volume V do you give?
2000 mg4.2 mL + 0.8 mL1760 mg
V = mL
5.Pediatric & critical-care dosingMedium
Practice exercise (made-up values, not clinical guidance). A bag contains 25 000 units of heparin in 250 mL. The order is 12 units/kg/h for a patient who weighs 92 kg. At what rate R do you set the pump, in mL/h? Round to one decimal place.
25 000 units/250 mL12 units/kg/h92 kg
R = mL/h
6.Pediatric & critical-care dosingHard
Practice exercise (made-up values, not clinical guidance). A bag contains dobutamine 250 mg in 250 mL. The order is 10 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.
250 mg/250 mL10 mcg/kg/min82 kg
R = mL/h
7.PharmacologyMedium
A child weighing 20 kg is prescribed a drug at 15 mg/kg per dose. What is the dose?
A150 mg
B3000 mg
C35 mg
D300 mg
8.PharmacologyHard
An older woman started on an SSRI two weeks ago is now confused, and her plasma sodium is low. What is the most likely mechanism?
ASSRI-induced inappropriate ADH secretion causing water retention
BIncreased aldosterone secretion
CDirect damage to the adrenal glands
DLoss of sodium through severe diarrhoea caused by the SSRI
3Solutions
1.Clinical pharmacokinetics
Answer: t½ = 3.445 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(9/1.8)/8 = 1.609/8 = 0.2012 h⁻¹
Half-life
t½ = 0.693/0.2012 = 3.445 h
Answer
t½ = 3.445 h
2.Clinical pharmacokinetics
Answer: LD = 337.5 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.
Volume of distribution
Vd = 0.5·90 = 45 L
Rise needed
15 − 9 = 6 mg/L
Loading dose
LD = 45·6/(0.8·1) = 337.5 mg
Answer
LD = 337.5 mg
In practice the dose is rounded to a convenient amount.
3.Pharmacy calculations
Answer: C = 200 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
4.5 + 0.5 = 5 mL
Concentration
C = 1000/5 = 200 mg/mL
Answer
C = 200 mg/mL
4.Pharmacy calculations
Answer: V = 4.4 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
4.2 + 0.8 = 5 mL
C = 2000/5 = 400 mg/mL
Volume
V = 1760/400 = 4.4 mL
Answer
V = 4.4 mL
5.Pediatric & critical-care dosing
Answer: R = 11 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
12·92 = 1 104 units/h
Concentration
25 000/250 = 100 units/mL
Pump rate
R = 1104/100 = 11.04 ≈ 11 mL/h
Answer
R = 11 mL/h
6.Pediatric & critical-care dosing
Answer: R = 49.2 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
250 mg = 250 000 mcg
250 000/250 = 1000 mcg/mL
Dose per minute and per hour
10·82 = 820 mcg/min
820·60 = 49200 mcg/h
Pump rate
R = 49200/1000 = 49.2 mL/h
Answer
R = 49.2 mL/h
7.Pharmacology
Answer: D · 300 mg
Why this is the answer
20 kg × 15 mg/kg = 300 mg. The result is also checked against the safe dose range and the adult maximum.
Answer
D · 300 mg
8.Pharmacology
Answer: A · SSRI-induced inappropriate ADH secretion causing water retention
Why this is the answer
SSRIs can cause the syndrome of inappropriate antidiuretic hormone secretion (SIADH), especially in older people. Water retention dilutes plasma sodium, causing hyponatraemia and confusion.
Answer
A · SSRI-induced inappropriate ADH secretion causing water retention