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 10 mg/L at 1 h and 3.1 mg/L at 6 h. Assume first-order elimination. Find the half-life t½.
C1 = 10 mg/L (1 h)C2 = 3.1 mg/L (6 h)
t½ = h
2.Clinical pharmacokineticsHard
Practice exercise (made-up values, not clinical guidance). A patient receives vancomycin IV every 12 h; the vancomycin clearance is 4.5 L/h. What dose D in mg gives an average steady-state level of 15 mg/L?
CL = 4.5 L/hτ = 12 hS = 1F = 1Css,avg = 15 mg/L
D = mg
3.Pharmacy calculationsMedium
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 1 200 000 units. What volume V do you draw up? Round to one decimal place.
8.2 mL → 500 000 units/mL1 200 000 units
V = mL
4.Pharmacy calculationsHard
Practice exercise (made-up values, not clinical guidance). You must prepare 500 mL of 5% chlorhexidine from a 20% stock and purified water. How many mL V of water do you need? Round to the nearest whole mL.
20%0%5%500 mL
V = mL
5.Pediatric & critical-care dosingMedium
Practice exercise (made-up values, not clinical guidance). A bag contains 25 000 units of heparin in 250 mL. The order is 15 units/kg/h for a patient who weighs 62 kg. At what rate R do you set the pump, in mL/h? Round to one decimal place.
25 000 units/250 mL15 units/kg/h62 kg
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 4 mcg/kg/min for a patient who weighs 69 kg. At what rate R do you set the pump, in mL/h? Round to one decimal place.
800 mg/500 mL4 mcg/kg/min69 kg
R = mL/h
7.PharmacologyMedium
Besides lithium levels, which blood tests are monitored in long-term lithium treatment?
AINR and fibrinogen
BLiver enzymes, serum amylase and lipase
CTroponin and CK-MB
DKidney function and thyroid function
8.PharmacologyHard
A patient with early aspirin (salicylate) overdose has tinnitus and is breathing fast. Which acid–base pattern is classic?
ARespiratory alkalosis followed by metabolic acidosis
BPure metabolic alkalosis
CRespiratory acidosis with normal bicarbonate
DMetabolic alkalosis followed by respiratory acidosis
3Solutions
1.Clinical pharmacokinetics
Answer: t½ = 2.959 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(10/3.1)/5 = 1.171/5 = 0.2342 h⁻¹
Half-life
t½ = 0.693/0.2342 = 2.959 h
Answer
t½ = 2.959 h
2.Clinical pharmacokinetics
Answer: D = 810 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.
Dose
D = Css,avg·CL·τ/(S·F) = 15·4.5·12/(1·1) = 810 mg
Answer
D = 810 mg
3.Pharmacy calculations
Answer: V = 2.4 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 = 1 200 000/500 000 = 2.4 mL
Answer
V = 2.4 mL
4.Pharmacy calculations
Answer: V = 375 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
20%: 5 − 0 = 5 parts
0%: 20 − 5 = 15 parts
total: 20 parts
Water
V = 500·15/20 = 375 mL
Answer
V = 375 mL
Check: 125 mL of 20% contains 25 g of drug, which is 5% of 500 mL.
5.Pediatric & critical-care dosing
Answer: R = 9.3 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
15·62 = 930 units/h
Concentration
25 000/250 = 100 units/mL
Pump rate
R = 930/100 = 9.3 mL/h
Answer
R = 9.3 mL/h
6.Pediatric & critical-care dosing
Answer: R = 10.4 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
4·69 = 276 mcg/min
276·60 = 16560 mcg/h
Pump rate
R = 16560/1600 = 10.35 ≈ 10.4 mL/h
Answer
R = 10.4 mL/h
7.Pharmacology
Answer: D · Kidney function and thyroid function
Why this is the answer
Lithium can impair kidney function and cause hypothyroidism. Kidney function, thyroid function and calcium are checked regularly, along with lithium levels.
Answer
D · Kidney function and thyroid function
8.Pharmacology
Answer: A · Respiratory alkalosis followed by metabolic acidosis
Why this is the answer
Salicylates stimulate breathing, causing respiratory alkalosis early. They then disturb cell metabolism, causing a metabolic acidosis with a raised anion gap. Adults often show a mixed picture.
Answer
A · Respiratory alkalosis followed by metabolic acidosis