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). After an IV dose of gentamicin, the level is 8 mg/L at 2 h and 3 mg/L at 7 h. Assume first-order elimination. Find the half-life t½.
C1 = 8 mg/L (2 h)C2 = 3 mg/L (7 h)
t½ = h
2.Clinical pharmacokineticsHard
Practice exercise (made-up values, not clinical guidance). A patient's tobramycin half-life is 2.5 h. The target is a peak of 10 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 = 2.5 hCpeak = 10 mg/LCtrough = 1 mg/L
τ = h
3.Pharmacy calculationsMedium
A solution has a ratio strength of 1:1000. You need 0.35 mg. Draw up the volume in the 1 mL syringe: what volume V is that?
1:10000.35 mg
V = mL
4.Pharmacy calculationsHard
A vial holds 1000 mg of powder. You add 3.2 mL of diluent, and the powder adds 0.8 mL of volume. The prescription is 875 mg. Draw up the dose: what volume V do you give?
1000 mg3.2 mL + 0.8 mL875 mg
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 dopamine infusion of 400 mg in 250 mL runs at 39 mL/h. The patient weighs 102 kg. What dose D in mcg/kg/min is the patient receiving? Round to one decimal place.
400 mg/250 mL39 mL/h102 kg
D = mcg/kg/min
7.PharmacologyMedium
Which serious but rare adverse effect of statins should patients report straight away?
AA metallic taste
BA dry cough
CUnexplained muscle pain or weakness
DAnkle swelling
8.PharmacologyHard
Why is pyridoxine (vitamin B6) often given together with isoniazid for tuberculosis?
ATo prevent peripheral neuropathy
BTo reduce bacterial resistance
CTo prevent liver damage from isoniazid
DTo increase the absorption of isoniazid
3Solutions
1.Clinical pharmacokinetics
Answer: t½ = 3.533 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.
Elimination rate constant
k = ln(8/3)/5 = 0.9808/5 = 0.1962 h⁻¹
Half-life
t½ = 0.693/0.1962 = 3.533 h
Answer
t½ = 3.533 h
2.Clinical pharmacokinetics
Answer: τ = 8.3 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 = 0.693/2.5 = 0.2773 h⁻¹
Dosing interval
τ = ln(10/1)/0.2773 = 2.303/0.2773 = 8.305 ≈ 8.3 h
Answer
τ = 8.3 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 = 0.35 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 = 0.35/1 = 0.35 mL
Answer
V = 0.35 mL
4.Pharmacy calculations
Answer: V = 3.5 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
3.2 + 0.8 = 4 mL
C = 1000/4 = 250 mg/mL
Volume
V = 875/250 = 3.5 mL
Answer
V = 3.5 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: D = 10.2 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
400 000/250 = 1600 mcg/mL
Micrograms per minute
39·1600 = 62400 mcg/h
62400/60 = 1040 mcg/min
Per kilogram
D = 1040/102 = 10.2 mcg/kg/min
Answer
D = 10.2 mcg/kg/min
7.Pharmacology
Answer: C · Unexplained muscle pain or weakness
Why this is the answer
Statins can cause myopathy, which rarely progresses to rhabdomyolysis with kidney damage. Unexplained muscle pain should be checked, for example with a creatine kinase level.
Answer
C · Unexplained muscle pain or weakness
8.Pharmacology
Answer: A · To prevent peripheral neuropathy
Why this is the answer
Isoniazid causes pyridoxine deficiency, which can lead to peripheral neuropathy. The risk is greater in slow acetylators, malnourished patients and people with diabetes or alcohol dependence.