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 tobramycin, the level is 8 mg/L at 2 h and 4 mg/L at 5 h. Assume first-order elimination. Find the elimination rate constant k.
C1 = 8 mg/L (2 h)C2 = 4 mg/L (5 h)
k = h⁻¹
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 10 h. How long t does it take to reach 95% of the steady-state level?
t½ = 10 h95% Css
t = h
3.Pharmacy calculationsMedium
Practice exercise (made-up values, not clinical guidance). A vial contains oxacillin 2000 mg as a powder. The practice label says to add 11.5 mL of sterile water; the powder adds 1 mL to the volume. What is the concentration C in mg/mL after reconstitution?
2000 mg11.5 mL+1 mL
C = mg/mL
4.Pharmacy calculationsHard
Practice exercise (made-up values, not clinical guidance). You have 30 g of 2.5% hydrocortisone ointment. How many grams m of 0.5% hydrocortisone ointment must you add to make a 1% ointment? Round to the nearest whole gram.
30 g · 2.5%0.5%1%
m = g
5.Pediatric & critical-care dosingMedium
Practice exercise (made-up values, not clinical guidance). A practice protocol calls for a heparin bolus of 70 units/kg for a patient who weighs 99 kg. The vial contains 5 000 units/mL. What volume V do you draw up? Round to one decimal place.
70 units/kg99 kg5 000 units/mL
V = mL
6.Pediatric & critical-care dosingHard
Practice exercise (made-up values, not clinical guidance). A dobutamine infusion of 500 mg in 250 mL runs at 39 mL/h. The patient weighs 103 kg. What dose D in mcg/kg/min is the patient receiving? Round to one decimal place.
500 mg/250 mL39 mL/h103 kg
D = mcg/kg/min
7.PharmacologyMedium
A patient receives lactulose for hepatic encephalopathy. Which finding shows that it is working?
ANo bowel movement for three days, with less confusion than before
BTen watery stools a day and a low serum potassium
CBetter alertness and orientation, with two to three soft stools a day
DHigher blood ammonia with improved appetite
8.PharmacologyHard
A patient recently started on haloperidol develops high fever, "lead-pipe" muscle rigidity, confusion and a very high creatine kinase. What is the most likely diagnosis?
ASerotonin syndrome
BMalignant hyperthermia from anaesthesia
CNeuroleptic malignant syndrome
DAcute dystonia
3Solutions
1.Clinical pharmacokinetics
Answer: k = 0.231 h⁻¹
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(8/4)/3 = 0.6931/3 = 0.231 h⁻¹
Answer
k = 0.231 h⁻¹
2.Clinical pharmacokinetics
Answer: t = 43.22 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 = 0.693/10 = 0.06931 h⁻¹
Time to the given fraction of steady state
t = −ln(1 − 0.95)/0.06931 = 2.996/0.06931 = 43.22 h
Answer
t = 43.22 h
That is 4.322 half-lives.
3.Pharmacy calculations
Answer: C = 160 mg/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
11.5 + 1 = 12.5 mL
Concentration
C = 2000/12.5 = 160 mg/mL
Answer
C = 160 mg/mL
4.Pharmacy calculations
Answer: m = 90 g
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
2.5%: 1 − 0.5 = 0.5 parts
0.5%: 2.5 − 1 = 1.5 parts
Scale to the amount you have
m = 30·1.5/0.5 = 90 g
Answer
m = 90 g
Check: 30 g + 90 g = 120 g of 1% ointment.
5.Pediatric & critical-care dosing
Answer: V = 1.4 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
70·99 = 6 930 units
Volume
V = 6930/5000 = 1.386 ≈ 1.4 mL
Answer
V = 1.4 mL
6.Pediatric & critical-care dosing
Answer: D = 12.6 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
500 000/250 = 2000 mcg/mL
Micrograms per minute
39·2000 = 78000 mcg/h
78000/60 = 1300 mcg/min
Per kilogram
D = 1300/103 = 12.62 ≈ 12.6 mcg/kg/min
Answer
D = 12.6 mcg/kg/min
7.Pharmacology
Answer: C · Better alertness and orientation, with two to three soft stools a day
Why this is the answer
Lactulose is adjusted to give about two to three soft stools a day and to lower ammonia, improving mental status. Excessive diarrhoea can cause dehydration and electrolyte loss.
Answer
C · Better alertness and orientation, with two to three soft stools a day
8.Pharmacology
Answer: C · Neuroleptic malignant syndrome
Why this is the answer
Neuroleptic malignant syndrome follows dopamine antagonists and develops over days, with lead-pipe rigidity, fever and raised CK. Serotonin syndrome develops faster and features clonus and hyperreflexia.