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 600 mL/min. What infusion rate R in mg/min keeps the steady-state level at 2.5 mg/L?
CL = 600 mL/minCss = 2.5 mg/L
R = mg/min
2.Clinical pharmacokineticsHard
Practice exercise (made-up values, not clinical guidance). A 63 kg patient has a theophylline level of 7.2 mg/L; the target is 12 mg/L. The drug is given intravenously as aminophylline (S = 0.8), and the volume of distribution is 0.5 L/kg. What extra loading dose LD raises the level to the target?
63 kgVd = 0.5 L/kgC = 7.2 → 12 mg/LS = 0.8
LD = mg
3.Pharmacy calculationsMedium
A solution has a ratio strength of 1:10 000. You need 0.56 mg. Draw up the volume in the 10 mL syringe: what volume V is that?
1:10 0000.56 mg
V = mL
4.Pharmacy calculationsHard
Practice exercise (made-up values, not clinical guidance). You have 200 g of 10% salicylic acid ointment. How many grams m of 2% salicylic acid ointment must you add to make a 6% ointment? Round to the nearest whole gram.
200 g · 10%2%6%
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 100 kg. The vial contains 5 000 units/mL. What volume V do you draw up? Round to one decimal place.
70 units/kg100 kg5 000 units/mL
V = mL
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 79 kg. At what rate R do you set the pump, in mL/h? Round to one decimal place.
800 mg/500 mL4 mcg/kg/min79 kg
R = mL/h
7.PharmacologyMedium
A patient uses a salbutamol (albuterol) inhaler and a corticosteroid inhaler at the same time of day. Which order is usually taught?
AUse the steroid first, then the salbutamol right after without any gap
BUse the salbutamol first, wait a few minutes, then use the steroid
CUse both in the same breath to save time
DThe order does not matter and no waiting is needed
8.PharmacologyHard
A drug with a short half-life is given at a dosing interval much longer than its half-life. What is the result at steady state?
ALarge swings between peak and trough levels
BGradual accumulation to toxic levels
CSteady state is never reached
DVery stable levels with almost no fluctuation
3Solutions
1.Clinical pharmacokinetics
Answer: R = 1.5 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
600 mL/min = 0.6 L/min
Infusion rate
R = 0.6·2.5 = 1.5 mg/min
Answer
R = 1.5 mg/min
2.Clinical pharmacokinetics
Answer: LD = 189 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.
Volume of distribution
Vd = 0.5·63 = 31.5 L
Rise needed
12 − 7.2 = 4.8 mg/L
Loading dose
LD = 31.5·4.8/(0.8·1) = 189 mg
Answer
LD = 189 mg
In practice the dose is rounded to a convenient amount.
3.Pharmacy calculations
Answer: V = 5.6 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 = 0.56/0.1 = 5.6 mL
Answer
V = 5.6 mL
4.Pharmacy calculations
Answer: m = 200 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
10%: 6 − 2 = 4 parts
2%: 10 − 6 = 4 parts
Scale to the amount you have
m = 200·4/4 = 200 g
Answer
m = 200 g
Check: 200 g + 200 g = 400 g of 6% 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
70·100 = 7 000 units
Volume
V = 7000/5000 = 1.4 mL
Answer
V = 1.4 mL
6.Pediatric & critical-care dosing
Answer: R = 11.9 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
4·79 = 316 mcg/min
316·60 = 18960 mcg/h
Pump rate
R = 18960/1600 = 11.85 ≈ 11.9 mL/h
Answer
R = 11.9 mL/h
7.Pharmacology
Answer: B · Use the salbutamol first, wait a few minutes, then use the steroid
Why this is the answer
The bronchodilator opens the airways so that the inhaled steroid can reach further into the lungs. A short wait between the inhalers is usually advised.
Answer
B · Use the salbutamol first, wait a few minutes, then use the steroid
8.Pharmacology
Answer: A · Large swings between peak and trough levels
Why this is the answer
When the interval is long compared with the half-life, most of each dose is eliminated before the next one. Levels rise high after each dose and fall low before the next, so there may be toxic peaks or ineffective troughs.