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 25 mg/L at 1 h and 12.5 mg/L at 13 h. Assume first-order elimination. Find the half-life t½.
C1 = 25 mg/L (1 h)C2 = 12.5 mg/L (13 h)
t½ = h
2.Clinical pharmacokineticsHard
Practice exercise (made-up values, not clinical guidance). A 56 kg patient takes oral theophylline (F = 1) every 8 h. The clearance is 0.035 L/kg/h. What dose D in mg gives an average steady-state level of 15 mg/L?
Practice exercise (made-up values, not clinical guidance). A vial of penicillin G contains 5 000 000 units. Its practice label gives three strengths (see the table). You add 8.2 mL of diluent. The prescribed dose is 600 000 units. What volume V do you draw up? Round to one decimal place.
8.2 mL → 500 000 units/mL600 000 units
V = mL
4.Pharmacy calculationsHard
Practice exercise (made-up values, not clinical guidance). You must prepare 1000 mL of 13% sodium chloride by mixing 23% and 3% sodium chloride. How many mL V of the 3% solution do you need? Round to the nearest whole mL.
23%3%13%1000 mL
V = mL
5.Pediatric & critical-care dosingMedium
Practice exercise (made-up values, not clinical guidance). An infusion contains 250 units of regular insulin in 250 mL. The order is 10 units/h. Find the pump rate R in mL/h. Round to one decimal place.
250 units/250 mL10 units/h
R = mL/h
6.Pediatric & critical-care dosingHard
Practice exercise (made-up values, not clinical guidance). A bag contains dopamine 800 mg in 500 mL. The order is 5 mcg/kg/min for a patient who weighs 52 kg. At what rate R do you set the pump, in mL/h? Round to one decimal place.
800 mg/500 mL5 mcg/kg/min52 kg
R = mL/h
7.PharmacologyMedium
Which common side effect is typical of ACE inhibitors?
ADry cough
BBradycardia
CAnkle swelling caused by vasodilation
DHypoglycaemia
8.PharmacologyHard
Why are antipsychotics used with great caution in older people with dementia?
AThey are not absorbed in people over 65
BThey cause severe hypoglycaemia in older people
CThey are linked to a higher risk of stroke and death
DThey always cure the dementia and stop all behaviour problems
3Solutions
1.Clinical pharmacokinetics
Answer: t½ = 12 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(25/12.5)/12 = 0.6931/12 = 0.05776 h⁻¹
Half-life
t½ = 0.693/0.05776 = 12 h
Answer
t½ = 12 h
2.Clinical pharmacokinetics
Answer: D = 235.2 mg
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.
Clearance
CL = 0.035·56 = 1.96 L/h
Dose
D = Css,avg·CL·τ/(S·F) = 15·1.96·8/(1·1) = 235.2 mg
Answer
D = 235.2 mg
3.Pharmacy calculations
Answer: V = 1.2 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.
Concentration from the label
8.2 mL → 500 000 units/mL
Volume to give
V = 600 000/500 000 = 1.2 mL
Answer
V = 1.2 mL
4.Pharmacy calculations
Answer: V = 500 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 = 1000·10/20 = 500 mL
Answer
V = 500 mL
Check: 500 mL of 23% + 500 mL of 3% = 1000 mL.
5.Pediatric & critical-care dosing
Answer: R = 10 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.
Concentration
250/250 = 1 units/mL
Pump rate
R = 10/1 = 10 mL/h
Answer
R = 10 mL/h
6.Pediatric & critical-care dosing
Answer: R = 9.8 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
800 mg = 800 000 mcg
800 000/500 = 1600 mcg/mL
Dose per minute and per hour
5·52 = 260 mcg/min
260·60 = 15600 mcg/h
Pump rate
R = 15600/1600 = 9.75 ≈ 9.8 mL/h
Answer
R = 9.8 mL/h
7.Pharmacology
Answer: A · Dry cough
Why this is the answer
ACE also breaks down bradykinin. When it is inhibited, bradykinin accumulates in the airways and causes a persistent dry cough in some patients.
Answer
A · Dry cough
8.Pharmacology
Answer: C · They are linked to a higher risk of stroke and death
Why this is the answer
In older people with dementia, antipsychotics increase the risk of stroke and death. Non-drug approaches are tried first, and antipsychotics are used at low doses for short periods if needed.
Answer
C · They are linked to a higher risk of stroke and death