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 330 mg of gentamicin. The level back-extrapolated to time zero is 12 mg/L and the half-life is 3.5 h. Find the volume of distribution Vd.
D = 330 mgC₀ = 12 mg/Lt½ = 3.5 h
Vd = L
2.Clinical pharmacokineticsHard
Practice exercise (made-up values, not clinical guidance). A patient's tobramycin half-life is 2.5 h. The target is a peak of 10 mg/L and a trough of 1 mg/L. Assuming rapid IV doses and first-order elimination, what dosing interval τ lets the level fall from the peak to the trough? Round to one decimal place.
t½ = 2.5 hCpeak = 10 mg/LCtrough = 1 mg/L
τ = h
3.Pharmacy calculationsMedium
A solution has a ratio strength of 1:1000. You need 0.42 mg. Draw up the volume in the 1 mL syringe: what volume V is that?
1:10000.42 mg
V = mL
4.Pharmacy calculationsHard
A vial holds 1000 mg of powder. You add 9.8 mL of diluent, and the powder adds 0.2 mL of volume. The prescription is 400 mg. Draw up the dose: what volume V do you give?
1000 mg9.8 mL + 0.2 mL400 mg
V = mL
5.Pediatric & critical-care dosingMedium
Practice exercise (made-up values, not clinical guidance). An infusion of 100 units of regular insulin in 250 mL runs at 15 mL/h. How many units per hour D does the patient receive?
100 units/250 mL15 mL/h
D = units/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 10 mcg/kg/min for a patient who weighs 73 kg. At what rate R do you set the pump, in mL/h? Round to one decimal place.
400 mg/250 mL10 mcg/kg/min73 kg
R = mL/h
7.PharmacologyMedium
How do SGLT2 inhibitors such as empagliflozin lower blood glucose?
AThey make the liver store more glycogen
BThey stimulate insulin release from beta cells
CThey slow the emptying of the stomach
DThey increase the loss of glucose in the urine
8.PharmacologyHard
Probenecid has been used to raise plasma penicillin levels. How does it do this?
AIt blocks the tubular secretion of penicillin in the kidney
BIt displaces penicillin from plasma proteins
CIt inhibits the liver metabolism of penicillin
DIt increases the absorption of penicillin from the small intestine
3Solutions
1.Clinical pharmacokinetics
Answer: Vd = 27.5 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 = 330/12 = 27.5 L
Answer
Vd = 27.5 L
2.Clinical pharmacokinetics
Answer: τ = 8.3 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 = 0.693/2.5 = 0.2773 h⁻¹
Dosing interval
τ = ln(10/1)/0.2773 = 2.303/0.2773 = 8.305 ≈ 8.3 h
Answer
τ = 8.3 h
In practice the interval is rounded to a convenient value such as 8, 12, 24 or 36 h, and the levels are checked again.
3.Pharmacy calculations
Answer: V = 0.42 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 = 0.42/1 = 0.42 mL
Answer
V = 0.42 mL
4.Pharmacy calculations
Answer: V = 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
9.8 + 0.2 = 10 mL
C = 1000/10 = 100 mg/mL
Volume
V = 400/100 = 4 mL
Answer
V = 4 mL
5.Pediatric & critical-care dosing
Answer: D = 6 units/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
100/250 = 0.4 units/mL
Units per hour
D = 15·0.4 = 6 units/h
Answer
D = 6 units/h
6.Pediatric & critical-care dosing
Answer: R = 27.4 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
10·73 = 730 mcg/min
730·60 = 43800 mcg/h
Pump rate
R = 43800/1600 = 27.38 ≈ 27.4 mL/h
Answer
R = 27.4 mL/h
7.Pharmacology
Answer: D · They increase the loss of glucose in the urine
Why this is the answer
SGLT2 reabsorbs most filtered glucose in the proximal tubule. Blocking it lets glucose leave in the urine, which also explains the higher risk of genital fungal infections.
Answer
D · They increase the loss of glucose in the urine
8.Pharmacology
Answer: A · It blocks the tubular secretion of penicillin in the kidney
Why this is the answer
Penicillins are excreted largely by organic anion transporters in the proximal tubule. Probenecid competes for these transporters, so penicillin is excreted more slowly and its level stays higher.
Answer
A · It blocks the tubular secretion of penicillin in the kidney