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). A patient has a lidocaine clearance of 850 mL/min. What infusion rate R in mg/min keeps the steady-state level at 2.5 mg/L?
CL = 850 mL/minCss = 2.5 mg/L
R = mg/min
2.Clinical pharmacokineticsHard
Practice exercise (made-up values, not clinical guidance). A 78-year-old man weighs 66 kg and has a serum creatinine of 2.1 mg/dL (186 µmol/L). Estimate the creatinine clearance with Cockcroft–Gault: CrCl (mL/min) = (140 − age) × weight (kg)/(72 × SCr in mg/dL), × 0.85 for women. The usual tobramycin regimen is 120 mg every 8 h for a creatinine clearance of 100 mL/min. Keeping the dose, the interval is lengthened in proportion to the fall in clearance. Find the new dosing interval τ. Round to one decimal place.
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 18.2 mL of diluent. The prescribed dose is 800 000 units. What volume V do you draw up? Round to one decimal place.
18.2 mL → 250 000 units/mL800 000 units
V = mL
4.Pharmacy calculationsHard
A vial holds 500 mg of powder. You add 3.5 mL of diluent, and the powder adds 0.5 mL of volume. The prescription is 300 mg. Draw up the dose: what volume V do you give?
500 mg3.5 mL + 0.5 mL300 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 250 mL. The order is 14 units/kg/h for a patient who weighs 74 kg. At what rate R do you set the pump, in mL/h? Round to one decimal place.
25 000 units/250 mL14 units/kg/h74 kg
R = mL/h
6.Pediatric & critical-care dosingHard
Practice exercise (made-up values, not clinical guidance). A dobutamine infusion of 500 mg in 250 mL runs at 11 mL/h. The patient weighs 98 kg. What dose D in mcg/kg/min is the patient receiving? Round to one decimal place.
500 mg/250 mL11 mL/h98 kg
D = mcg/kg/min
7.PharmacologyMedium
Which drug is the antidote for an opioid overdose?
ANaloxone
BAcetylcysteine
CFlumazenil
DAtropine
8.PharmacologyHard
Which property makes a drug more likely to cross the blood–brain barrier?
ALarge molecular weight and strong protein binding
BHigh lipid solubility and small size
CHigh water solubility and strong ionisation
DPositive charge at physiological pH
3Solutions
1.Clinical pharmacokinetics
Answer: R = 2.125 mg/min
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.
Clearance in L/min
850 mL/min = 0.85 L/min
Infusion rate
R = 0.85·2.5 = 2.125 mg/min
Answer
R = 2.125 mg/min
2.Clinical pharmacokinetics
Answer: τ = 29.6 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.
Creatinine clearance (Cockcroft–Gault)
CrCl = (140 − 78)·66/(72·2.1) = 27.06 mL/min
New interval
τ = 8·100/27.06 = 29.56 ≈ 29.6 h
Answer
τ = 29.6 h
In practice the interval is rounded to a convenient value such as 12, 24, 36 or 48 h.
3.Pharmacy calculations
Answer: V = 3.2 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
18.2 mL → 250 000 units/mL
Volume to give
V = 800 000/250 000 = 3.2 mL
Answer
V = 3.2 mL
4.Pharmacy calculations
Answer: V = 2.4 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
3.5 + 0.5 = 4 mL
C = 500/4 = 125 mg/mL
Volume
V = 300/125 = 2.4 mL
Answer
V = 2.4 mL
5.Pediatric & critical-care dosing
Answer: R = 10.4 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
14·74 = 1 036 units/h
Concentration
25 000/250 = 100 units/mL
Pump rate
R = 1036/100 = 10.36 ≈ 10.4 mL/h
Answer
R = 10.4 mL/h
6.Pediatric & critical-care dosing
Answer: D = 3.7 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
11·2000 = 22000 mcg/h
22000/60 = 366.7 mcg/min
Per kilogram
D = 366.7/98 = 3.741 ≈ 3.7 mcg/kg/min
Answer
D = 3.7 mcg/kg/min
7.Pharmacology
Answer: A · Naloxone
Why this is the answer
Naloxone reverses opioid-induced respiratory depression. Flumazenil reverses benzodiazepines, acetylcysteine treats paracetamol overdose and atropine treats bradycardia or organophosphate poisoning.
Answer
A · Naloxone
8.Pharmacology
Answer: B · High lipid solubility and small size
Why this is the answer
Small, lipophilic, non-ionised molecules diffuse across endothelial cell membranes. Charged or large, protein-bound drugs mostly stay out.