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 has a lidocaine clearance of 600 mL/min. What infusion rate R in mg/min keeps the steady-state level at 2 mg/L?
CL = 600 mL/minCss = 2 mg/L
R = mg/min
2.Clinical pharmacokineticsHard
Practice exercise (made-up values, not clinical guidance). A vancomycin infusion is started without a loading dose. The half-life in this patient is 8 h. How long t does it take to reach 50% of the steady-state level?
t½ = 8 h50% Css
t = 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 600 mg. What volume V do you draw up? Round to one decimal place.
1000 mg9.6 mL+0.4 mL600 mg
V = mL
4.Pharmacy calculationsHard
Practice exercise (made-up values, not clinical guidance). You must prepare 600 mL of 15% dextrose by mixing 50% and 10% dextrose. How many mL V of the 10% solution do you need? Round to the nearest whole mL.
50%10%15%600 mL
V = mL
5.Pediatric & critical-care dosingMedium
Practice exercise (made-up values, not clinical guidance). A bag contains 25 000 units of heparin in 500 mL. The order is 12 units/kg/h for a patient who weighs 90 kg. At what rate R do you set the pump, in mL/h? Round to one decimal place.
25 000 units/500 mL12 units/kg/h90 kg
R = mL/h
6.Pediatric & critical-care dosingHard
Practice exercise (made-up values, not clinical guidance). A patient who weighs 62 kg has an order for nitroglycerin at 5 mcg/min. The bag contains 50 mg in 250 mL. Find the pump rate R in mL/h. Round to one decimal place.
50 mg/250 mL5 mcg/min62 kg
R = mL/h
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 immediate kidney failure
CIt is a strong opioid that causes addiction
DIt causes a severe rise in blood pressure when it is given intravenously
8.PharmacologyHard
A patient taking long-term hydroxychloroquine for lupus is advised to have regular eye examinations. Why?
AThe drug stains the cornea blue
BThe drug causes cataracts within a few weeks of starting
CThe drug raises pressure inside the eye
DThe drug can cause retinal toxicity that may be permanent
3Solutions
1.Clinical pharmacokinetics
Answer: R = 1.2 mg/min
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 in L/min
600 mL/min = 0.6 L/min
Infusion rate
R = 0.6·2 = 1.2 mg/min
Answer
R = 1.2 mg/min
2.Clinical pharmacokinetics
Answer: t = 8 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 = 0.693/8 = 0.08664 h⁻¹
Time to the given fraction of steady state
t = −ln(1 − 0.5)/0.08664 = 0.6931/0.08664 = 8 h
Answer
t = 8 h
That is 1 half-lives.
3.Pharmacy calculations
Answer: V = 6 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 = 600/100 = 6 mL
Answer
V = 6 mL
4.Pharmacy calculations
Answer: V = 525 mL
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
50%: 15 − 10 = 5 parts
10%: 50 − 15 = 35 parts
total: 40 parts
Share the volume
V = 600·35/40 = 525 mL
Answer
V = 525 mL
Check: 75 mL of 50% + 525 mL of 10% = 600 mL.
5.Pediatric & critical-care dosing
Answer: R = 21.6 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
12·90 = 1 080 units/h
Concentration
25 000/500 = 50 units/mL
Pump rate
R = 1080/50 = 21.6 mL/h
Answer
R = 21.6 mL/h
6.Pediatric & critical-care dosing
Answer: R = 1.5 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
50 000/250 = 200 mcg/mL
Dose per hour
The order is per minute for the whole patient, so the weight is not needed.
5·60 = 300 mcg/h
Pump rate
R = 300/200 = 1.5 mL/h
Answer
R = 1.5 mL/h
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 · The drug can cause retinal toxicity that may be permanent
Why this is the answer
Hydroxychloroquine can damage the retina, and the risk rises with dose and duration. Regular screening allows the drug to be stopped before vision is seriously affected.
Answer
D · The drug can cause retinal toxicity that may be permanent