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 vancomycin, the level is 40 mg/L at 2 h and 15.9 mg/L at 10 h. Assume first-order elimination. Find the half-life t½.
C1 = 40 mg/L (2 h)C2 = 15.9 mg/L (10 h)
t½ = h
2.Clinical pharmacokineticsHard
Practice exercise (made-up values, not clinical guidance). A patient's gentamicin half-life is 3 h. The target is a peak of 10 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 = 10 mg/LCtrough = 0.5 mg/L
τ = h
3.Pharmacy calculationsMedium
A solution has a ratio strength of 1:10 000. You need 0.7 mg. Draw up the volume in the 10 mL syringe: what volume V is that?
1:10 0000.7 mg
V = mL
4.Pharmacy calculationsHard
Practice exercise (made-up values, not clinical guidance). You must prepare 450 mL of 13% sodium chloride by mixing 23% and 3% sodium chloride. How many mL V of the 23% solution do you need? Round to the nearest whole mL.
23%3%13%450 mL
V = mL
5.Pediatric & critical-care dosingMedium
Practice exercise (made-up values, not clinical guidance). A practice protocol calls for a heparin bolus of 60 units/kg for a patient who weighs 76 kg. The vial contains 1 000 units/mL. What volume V do you draw up? Round to one decimal place.
60 units/kg76 kg1 000 units/mL
V = mL
6.Pediatric & critical-care dosingHard
Practice exercise (made-up values, not clinical guidance). A dopamine infusion of 800 mg in 500 mL runs at 31 mL/h. The patient weighs 59 kg. What dose D in mcg/kg/min is the patient receiving? Round to one decimal place.
800 mg/500 mL31 mL/h59 kg
D = mcg/kg/min
7.PharmacologyMedium
Which approach is usually recommended to prevent constipation in a patient on regular opioids?
AA stimulant laxative such as senna, often with a stool softener
BBulk-forming fibre alone, with fluid intake restricted
CAn antidiarrhoeal drug at night to slow the gut
DNo laxative until the patient has not opened the bowels for a week
8.PharmacologyHard
A man on long-term testosterone therapy has a rising haematocrit. Why is this important?
ATestosterone raises red cell production, increasing clot risk
BA high haematocrit shows that the testosterone is not being absorbed
CIt is a sign of liver failure caused by testosterone
DIt means the testosterone dose is too low
3Solutions
1.Clinical pharmacokinetics
Answer: t½ = 6.011 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(40/15.9)/8 = 0.9226/8 = 0.1153 h⁻¹
Half-life
t½ = 0.693/0.1153 = 6.011 h
Answer
t½ = 6.011 h
2.Clinical pharmacokinetics
Answer: τ = 13 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(10/0.5)/0.231 = 2.996/0.231 = 12.97 ≈ 13 h
Answer
τ = 13 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 = 7 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/10 000 mL = 0.1 mg/mL
Volume
V = 0.7/0.1 = 7 mL
Answer
V = 7 mL
4.Pharmacy calculations
Answer: V = 225 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
23%: 13 − 3 = 10 parts
3%: 23 − 13 = 10 parts
total: 20 parts
Share the volume
V = 450·10/20 = 225 mL
Answer
V = 225 mL
Check: 225 mL of 23% + 225 mL of 3% = 450 mL.
5.Pediatric & critical-care dosing
Answer: V = 4.6 mL
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 needed
60·76 = 4 560 units
Volume
V = 4560/1000 = 4.56 ≈ 4.6 mL
Answer
V = 4.6 mL
6.Pediatric & critical-care dosing
Answer: D = 14 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
800 000/500 = 1600 mcg/mL
Micrograms per minute
31·1600 = 49600 mcg/h
49600/60 = 826.7 mcg/min
Per kilogram
D = 826.7/59 = 14.01 ≈ 14 mcg/kg/min
Answer
D = 14 mcg/kg/min
7.Pharmacology
Answer: A · A stimulant laxative such as senna, often with a stool softener
Why this is the answer
Opioid-induced constipation does not usually resolve with time, so a laxative is started with the opioid. Stimulant laxatives counter reduced motility; bulk-forming laxatives alone can worsen symptoms if fluid intake is low.
Answer
A · A stimulant laxative such as senna, often with a stool softener
8.Pharmacology
Answer: A · Testosterone raises red cell production, increasing clot risk
Why this is the answer
Testosterone increases erythropoiesis and can cause polycythaemia, which increases blood viscosity and thrombosis risk. Haematocrit is monitored, and the dose may be reduced or paused.
Answer
A · Testosterone raises red cell production, increasing clot risk