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 86 kg patient receives a theophylline infusion. The clearance is 0.03 L/kg/h. What infusion rate R in mg/h gives a steady-state level of 10 mg/L?
86 kgCL = 0.03 L/kg/hCss = 10 mg/L
R = mg/h
2.Clinical pharmacokineticsHard
Practice exercise (made-up values, not clinical guidance). A patient's tobramycin half-life is 3.5 h. The target is a peak of 6 mg/L and a trough of 0.5 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 = 3.5 hCpeak = 6 mg/LCtrough = 0.5 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 400 mg. What volume V do you draw up? Round to one decimal place.
1000 mg9.6 mL+0.4 mL400 mg
V = mL
4.Pharmacy calculationsHard
Practice exercise (made-up values, not clinical guidance). You have 20 g of 20% zinc oxide ointment. How many grams m of plain ointment base (0%) must you add to make a 10% ointment? Round to the nearest whole gram.
20 g · 20%0%10%
m = g
5.Pediatric & critical-care dosingMedium
Practice exercise (made-up values, not clinical guidance). An infusion contains 100 units of regular insulin in 100 mL. The order is 0.05 units/kg/h for a patient who weighs 90 kg. Find the pump rate R in mL/h. Round to one decimal place.
100 units/100 mL0.05 units/kg/h90 kg
R = mL/h
6.Pediatric & critical-care dosingHard
Practice exercise (made-up values, not clinical guidance). A dobutamine infusion of 500 mg in 250 mL runs at 13 mL/h. The patient weighs 83 kg. What dose D in mcg/kg/min is the patient receiving? Round to one decimal place.
500 mg/250 mL13 mL/h83 kg
D = mcg/kg/min
7.PharmacologyMedium
Why is IV promethazine given with great caution?
AIt can cause severe tissue injury if it leaks or enters an artery
BIt causes a severe rise in blood pressure when it is given intravenously
CIt causes immediate kidney failure
DIt is a strong opioid that causes addiction
8.PharmacologyHard
A malnourished patient with low serum albumin shows nystagmus and unsteady gait. Her total phenytoin level is within range. What is the most likely explanation?
ALow albumin speeds up phenytoin metabolism
BLow albumin makes phenytoin bind more tightly, so the drug is less active
CThe total level is high, but the laboratory is wrong
DMore phenytoin is unbound when albumin is low, so free levels rise
3Solutions
1.Clinical pharmacokinetics
Answer: R = 25.8 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 = 0.03·86 = 2.58 L/h
Infusion rate
R = 2.58·10 = 25.8 mg/h
Answer
R = 25.8 mg/h
2.Clinical pharmacokinetics
Answer: τ = 12.5 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 = 0.693/3.5 = 0.198 h⁻¹
Dosing interval
τ = ln(6/0.5)/0.198 = 2.485/0.198 = 12.55 ≈ 12.5 h
Answer
τ = 12.5 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 = 4 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 = 400/100 = 4 mL
Answer
V = 4 mL
4.Pharmacy calculations
Answer: m = 20 g
Alligation: to mix a higher strength H and a lower strength L into a target strength T, take (T − L) parts of H and (H − T) parts of L. The parts are crossed: each stock gets the distance from the target to the other stock. Water or a plain base counts as 0%. The numbers are for practice only; at the bedside always follow local protocols and have high-risk doses double-checked.
Alligation parts
20%: 10 − 0 = 10 parts
0%: 20 − 10 = 10 parts
Scale to the amount you have
m = 20·10/10 = 20 g
Answer
m = 20 g
Check: 20 g + 20 g = 40 g of 10% ointment.
5.Pediatric & critical-care dosing
Answer: R = 4.5 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
0.05·90 = 4.5 units/h
Concentration
100/100 = 1 units/mL
Pump rate
R = 4.5/1 = 4.5 mL/h
Answer
R = 4.5 mL/h
6.Pediatric & critical-care dosing
Answer: D = 5.2 mcg/kg/min
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
500 000/250 = 2000 mcg/mL
Micrograms per minute
13·2000 = 26000 mcg/h
26000/60 = 433.3 mcg/min
Per kilogram
D = 433.3/83 = 5.221 ≈ 5.2 mcg/kg/min
Answer
D = 5.2 mcg/kg/min
7.Pharmacology
Answer: A · It can cause severe tissue injury if it leaks or enters an artery
Why this is the answer
Promethazine is a vesicant. Deep intramuscular injection is the preferred parenteral route, and if given IV it is diluted, given slowly into a well-running large vein, and stopped at once if the patient reports burning or pain.
Answer
A · It can cause severe tissue injury if it leaks or enters an artery
8.Pharmacology
Answer: D · More phenytoin is unbound when albumin is low, so free levels rise
Why this is the answer
Phenytoin is highly protein bound. When albumin is low, the free fraction rises, so toxicity can occur with a "normal" total level; a free phenytoin level or corrected level is more useful.
Answer
D · More phenytoin is unbound when albumin is low, so free levels rise