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). An 81 kg patient is to receive an aminophylline infusion (salt factor S = 0.8: 80% of aminophylline is theophylline). The theophylline clearance is 0.035 L/kg/h. What aminophylline infusion rate R in mg/h gives a steady-state theophylline level of 12 mg/L?
81 kgCL = 0.035 L/kg/hCss = 12 mg/LS = 0.8
R = mg/h
2.Clinical pharmacokineticsHard
Practice exercise (made-up values, not clinical guidance). An 82 kg patient on regular gentamicin doses is at steady state. The target is a peak of 6 mg/L and a trough of 1 mg/L, and the volume of distribution is 0.3 L/kg. Assuming rapid IV doses, what dose D raises the level from the trough to the peak?
82 kgVd = 0.3 L/kgCpeak = 6 mg/LCtrough = 1 mg/L
D = mg
3.Pharmacy calculationsMedium
Practice exercise (made-up values, not clinical guidance). A vial contains cefuroxime 750 mg as a powder. The practice label says to add 7.2 mL of sterile water; the powder adds 0.3 mL to the volume. The prescribed dose is 300 mg. What volume V do you draw up? Round to one decimal place.
750 mg7.2 mL+0.3 mL300 mg
V = mL
4.Pharmacy calculationsHard
A vial holds 250 mg of powder. You add 2 mL of diluent, and the powder adds 0.5 mL of volume. The prescription is 100 mg. Draw up the dose: what volume V do you give?
250 mg2 mL + 0.5 mL100 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 20 mL/h. How many units per hour D does the patient receive?
100 units/250 mL20 mL/h
D = units/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 71 kg. At what rate R do you set the pump, in mL/h? Round to one decimal place.
250 mg/250 mL10 mcg/kg/min71 kg
R = mL/h
7.PharmacologyMedium
After a vaccination, why is a person asked to wait for a short time before leaving?
ATo check that the vaccine has already created protective antibodies
BTo measure the person's temperature for fever
CTo watch for anaphylaxis, which usually starts within minutes
DTo make sure the injection site does not bruise
8.PharmacologyHard
Which heart failure drug improves symptoms and reduces hospital admissions but has not been shown to reduce mortality?
ADigoxin
BAn ACE inhibitor
CA mineralocorticoid receptor antagonist such as spironolactone
DA beta blocker such as bisoprolol
3Solutions
1.Clinical pharmacokinetics
Answer: R = 42.52 mg/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.
Clearance
CL = 0.035·81 = 2.835 L/h
Infusion rate
R = CL·Css/S = 2.835·12/0.8 = 42.52 mg/h
Answer
R = 42.52 mg/h
2.Clinical pharmacokinetics
Answer: D = 123 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 = 0.3·82 = 24.6 L
Dose
D = 24.6·(6 − 1) = 123 mg
Answer
D = 123 mg
In practice the dose is rounded to a convenient amount.
3.Pharmacy calculations
Answer: V = 3 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
7.2 + 0.3 = 7.5 mL
Concentration
750/7.5 = 100 mg/mL
Volume to give
V = 300/100 = 3 mL
Answer
V = 3 mL
4.Pharmacy calculations
Answer: V = 1 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
2 + 0.5 = 2.5 mL
C = 250/2.5 = 100 mg/mL
Volume
V = 100/100 = 1 mL
Answer
V = 1 mL
5.Pediatric & critical-care dosing
Answer: D = 8 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 = 20·0.4 = 8 units/h
Answer
D = 8 units/h
6.Pediatric & critical-care dosing
Answer: R = 42.6 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
10·71 = 710 mcg/min
710·60 = 42600 mcg/h
Pump rate
R = 42600/1000 = 42.6 mL/h
Answer
R = 42.6 mL/h
7.Pharmacology
Answer: C · To watch for anaphylaxis, which usually starts within minutes
Why this is the answer
Anaphylaxis after vaccination is rare but usually begins soon after the injection. Staff and adrenaline must be available during the observation period.
Answer
C · To watch for anaphylaxis, which usually starts within minutes
8.Pharmacology
Answer: A · Digoxin
Why this is the answer
ACE inhibitors (or ARBs/ARNI), beta blockers, mineralocorticoid antagonists and SGLT2 inhibitors reduce mortality in heart failure with reduced ejection fraction. Digoxin improves symptoms and reduces admissions without reducing mortality.