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 170 mg of tobramycin. The level back-extrapolated to time zero is 6 mg/L and the half-life is 2 h. Find the clearance CL in L/h.
D = 170 mgC₀ = 6 mg/Lt½ = 2 h
CL = L/h
2.Clinical pharmacokineticsHard
Practice exercise (made-up values, not clinical guidance). A patient's tobramycin half-life is 3 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 hCpeak = 6 mg/LCtrough = 0.5 mg/L
τ = h
3.Pharmacy calculationsMedium
A solution has a ratio strength of 1:1000. You need 0.24 mg. Draw up the volume in the 1 mL syringe: what volume V is that?
1:10000.24 mg
V = mL
4.Pharmacy calculationsHard
A vial holds 1000 mg of powder. You add 9.4 mL of diluent, and the powder adds 0.6 mL of volume. The prescription is 130 mg. Draw up the dose: what volume V do you give?
1000 mg9.4 mL + 0.6 mL130 mg
V = mL
5.Pediatric & critical-care dosingMedium
Practice exercise (made-up values, not clinical guidance). A bag contains 20 000 units of heparin in 500 mL. The order is 16 units/kg/h for a patient who weighs 61 kg. At what rate R do you set the pump, in mL/h? Round to one decimal place.
20 000 units/500 mL16 units/kg/h61 kg
R = mL/h
6.Pediatric & critical-care dosingHard
Practice exercise (made-up values, not clinical guidance). A patient who weighs 69 kg has an order for norepinephrine at 8 mcg/min. The bag contains 8 mg in 250 mL. Find the pump rate R in mL/h. Round to one decimal place.
8 mg/250 mL8 mcg/min69 kg
R = mL/h
7.PharmacologyMedium
During a rapid infusion of vancomycin, a patient develops flushing and a red rash on the face, neck and upper body. What is the most likely cause and action?
AAcute kidney failure; stop all IV fluids and antibiotics at once
BA normal effect; increase the infusion rate to finish sooner
CAn infusion reaction; stop or slow the infusion and inform the prescriber
DA sign that the infection is spreading; give a second dose
8.PharmacologyHard
Before starting rituximab, patients are tested for hepatitis B. Why?
ARituximab is a vaccine against hepatitis B
BRituximab can reactivate hepatitis B and cause liver failure
CHepatitis B infection makes rituximab completely ineffective
DRituximab is made from the hepatitis B virus
3Solutions
1.Clinical pharmacokinetics
Answer: CL = 9.82 L/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.
Volume of distribution
Vd = 170/6 = 28.33 L
Rate constant
k = 0.693/2 = 0.3466 h⁻¹
Clearance
CL = 0.3466·28.33 = 9.82 L/h
Answer
CL = 9.82 L/h
2.Clinical pharmacokinetics
Answer: τ = 10.8 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 = 0.693/3 = 0.231 h⁻¹
Dosing interval
τ = ln(6/0.5)/0.231 = 2.485/0.231 = 10.75 ≈ 10.8 h
Answer
τ = 10.8 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 = 0.24 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 = 0.24/1 = 0.24 mL
Answer
V = 0.24 mL
4.Pharmacy calculations
Answer: V = 1.3 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
9.4 + 0.6 = 10 mL
C = 1000/10 = 100 mg/mL
Volume
V = 130/100 = 1.3 mL
Answer
V = 1.3 mL
5.Pediatric & critical-care dosing
Answer: R = 24.4 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
16·61 = 976 units/h
Concentration
20 000/500 = 40 units/mL
Pump rate
R = 976/40 = 24.4 mL/h
Answer
R = 24.4 mL/h
6.Pediatric & critical-care dosing
Answer: R = 15 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
8 000/250 = 32 mcg/mL
Dose per hour
The order is per minute for the whole patient, so the weight is not needed.
8·60 = 480 mcg/h
Pump rate
R = 480/32 = 15 mL/h
Answer
R = 15 mL/h
7.Pharmacology
Answer: C · An infusion reaction; stop or slow the infusion and inform the prescriber
Why this is the answer
Rapid vancomycin infusion releases histamine, causing flushing and itching of the upper body (formerly called "red man syndrome"). It is managed by stopping or slowing the infusion, and sometimes with an antihistamine.
Answer
C · An infusion reaction; stop or slow the infusion and inform the prescriber
8.Pharmacology
Answer: B · Rituximab can reactivate hepatitis B and cause liver failure
Why this is the answer
Rituximab destroys CD20-positive B cells, weakening control of latent hepatitis B. Reactivation can cause fulminant hepatitis, so carriers receive monitoring or antiviral prophylaxis.
Answer
B · Rituximab can reactivate hepatitis B and cause liver failure