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 55 kg patient is to be loaded intravenously with phenytoin sodium to reach a phenytoin level of 15 mg/L. The volume of distribution is 0.7 L/kg and the salt factor is S = 0.92. Find the loading dose LD of phenytoin sodium in mg.
55 kgVd = 0.7 L/kgC = 15 mg/LS = 0.92
LD = mg
2.Clinical pharmacokineticsHard
Practice exercise (made-up values, not clinical guidance). A patient's tobramycin half-life is 2 h. The target is a peak of 7 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 hCpeak = 7 mg/LCtrough = 1 mg/L
τ = h
3.Pharmacy calculationsMedium
Practice exercise (made-up values, not clinical guidance). A vial contains ceftriaxone 1000 mg as a powder. The practice label says to add 9.6 mL of sterile water; the powder adds 0.4 mL to the volume. The prescribed dose is 300 mg. What volume V do you draw up? Round to one decimal place.
1000 mg9.6 mL+0.4 mL300 mg
V = mL
4.Pharmacy calculationsHard
A vial holds 1000 mg of powder. You add 4.4 mL of diluent, and the powder adds 0.6 mL of volume. The prescription is 840 mg. Draw up the dose: what volume V do you give?
1000 mg4.4 mL + 0.6 mL840 mg
V = mL
5.Pediatric & critical-care dosingMedium
Practice exercise (made-up values, not clinical guidance). An infusion of 50 units of regular insulin in 50 mL runs at 2 mL/h. How many units per hour D does the patient receive?
50 units/50 mL2 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 2 mcg/kg/min for a patient who weighs 96 kg. At what rate R do you set the pump, in mL/h? Round to one decimal place.
400 mg/250 mL2 mcg/kg/min96 kg
R = mL/h
7.PharmacologyMedium
Which rule applies to intravenous potassium chloride?
AIt must be diluted and given as an infusion, never as an IV push
BIt may be given intramuscularly if there is no IV access
CIt is mixed into blood products to save time
DIt is given as a rapid IV push to correct low levels quickly
8.PharmacologyHard
Why is clavulanic acid combined with amoxicillin (co-amoxiclav)?
AIt reduces the risk of penicillin allergy
BIt slows the kidney excretion of amoxicillin
CIt inhibits beta-lactamase enzymes that would destroy amoxicillin
DIt makes amoxicillin cross the blood–brain barrier more easily
3Solutions
1.Clinical pharmacokinetics
Answer: LD = 627.7 mg
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 = 0.7·55 = 38.5 L
Amount of active drug needed
Vd·C = 38.5·15 = 577.5 mg
Loading dose
LD = 577.5/(0.92·1) = 627.7 mg
Answer
LD = 627.7 mg
In practice the dose is rounded to a convenient amount.
2.Clinical pharmacokinetics
Answer: τ = 5.6 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 = 0.693/2 = 0.3466 h⁻¹
Dosing interval
τ = ln(7/1)/0.3466 = 1.946/0.3466 = 5.615 ≈ 5.6 h
Answer
τ = 5.6 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 = 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
9.6 + 0.4 = 10 mL
Concentration
1000/10 = 100 mg/mL
Volume to give
V = 300/100 = 3 mL
Answer
V = 3 mL
4.Pharmacy calculations
Answer: V = 4.2 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
4.4 + 0.6 = 5 mL
C = 1000/5 = 200 mg/mL
Volume
V = 840/200 = 4.2 mL
Answer
V = 4.2 mL
5.Pediatric & critical-care dosing
Answer: D = 2 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
50/50 = 1 units/mL
Units per hour
D = 2·1 = 2 units/h
Answer
D = 2 units/h
6.Pediatric & critical-care dosing
Answer: R = 7.2 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
2·96 = 192 mcg/min
192·60 = 11520 mcg/h
Pump rate
R = 11520/1600 = 7.2 mL/h
Answer
R = 7.2 mL/h
7.Pharmacology
Answer: A · It must be diluted and given as an infusion, never as an IV push
Why this is the answer
Concentrated potassium given as a bolus can cause fatal cardiac arrhythmias. It is always diluted and given at a controlled rate, usually with an infusion pump.
Answer
A · It must be diluted and given as an infusion, never as an IV push
8.Pharmacology
Answer: C · It inhibits beta-lactamase enzymes that would destroy amoxicillin
Why this is the answer
Many bacteria produce beta-lactamases that inactivate penicillins. Clavulanic acid inhibits these enzymes, so amoxicillin can work against these organisms.
Answer
C · It inhibits beta-lactamase enzymes that would destroy amoxicillin