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 1 h and 1.3 mg/L at 9 h. Assume first-order elimination. Find the half-life t½.
C1 = 8 mg/L (1 h)C2 = 1.3 mg/L (9 h)
t½ = h
2.Clinical pharmacokineticsHard
Practice exercise (made-up values, not clinical guidance). A patient's tobramycin half-life is 4 h. The target is a peak of 6 mg/L and a trough of 1 mg/L. Assuming rapid IV doses and first-order elimination, what dosing interval τ lets the level fall from the peak to the trough? Round to one decimal place.
t½ = 4 hCpeak = 6 mg/LCtrough = 1 mg/L
τ = h
3.Pharmacy calculationsMedium
A solution has a ratio strength of 1:1000. You need 0.88 mg. Draw up the volume in the 1 mL syringe: what volume V is that?
1:10000.88 mg
V = mL
4.Pharmacy calculationsHard
Practice exercise (made-up values, not clinical guidance). You must prepare 1000 mL of 4% 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%4%1000 mL
V = mL
5.Pediatric & critical-care dosingMedium
Practice exercise (made-up values, not clinical guidance). An infusion contains 100 units of regular insulin in 250 mL. The order is 0.05 units/kg/h for a patient who weighs 64 kg. Find the pump rate R in mL/h. Round to one decimal place.
100 units/250 mL0.05 units/kg/h64 kg
R = mL/h
6.Pediatric & critical-care dosingHard
Practice exercise (made-up values, not clinical guidance). A bag contains dopamine 400 mg in 250 mL. The order is 10 mcg/kg/min for a patient who weighs 80 kg. At what rate R do you set the pump, in mL/h? Round to one decimal place.
400 mg/250 mL10 mcg/kg/min80 kg
R = mL/h
7.PharmacologyMedium
A man takes finasteride. What should be explained about handling the tablets?
AThe tablets must be crushed before swallowing
BThe tablets must be stored in the fridge
CPregnant women should not handle crushed or broken tablets
DAnyone can handle the tablets, as they contain no active drug
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?
AVery stable levels with almost no fluctuation
BLarge swings between peak and trough levels
CSteady state is never reached
DGradual accumulation to toxic levels
3Solutions
1.Clinical pharmacokinetics
Answer: t½ = 3.052 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(8/1.3)/8 = 1.817/8 = 0.2271 h⁻¹
Half-life
t½ = 0.693/0.2271 = 3.052 h
Answer
t½ = 3.052 h
2.Clinical pharmacokinetics
Answer: τ = 10.3 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 = 0.693/4 = 0.1733 h⁻¹
Dosing interval
τ = ln(6/1)/0.1733 = 1.792/0.1733 = 10.34 ≈ 10.3 h
Answer
τ = 10.3 h
In practice the interval is rounded to a convenient value such as 8, 12, 24 or 36 h, and the levels are checked again.
3.Pharmacy calculations
Answer: V = 0.88 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/1000 mL = 1 mg/mL
Volume
V = 0.88/1 = 0.88 mL
Answer
V = 0.88 mL
4.Pharmacy calculations
Answer: V = 800 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%: 4 − 0 = 4 parts
0%: 20 − 4 = 16 parts
total: 20 parts
Water
V = 1000·16/20 = 800 mL
Answer
V = 800 mL
Check: 200 mL of 20% contains 40 g of drug, which is 4% of 1000 mL.
5.Pediatric & critical-care dosing
Answer: R = 8 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
0.05·64 = 3.2 units/h
Concentration
100/250 = 0.4 units/mL
Pump rate
R = 3.2/0.4 = 8 mL/h
Answer
R = 8 mL/h
6.Pediatric & critical-care dosing
Answer: R = 30 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
400 mg = 400 000 mcg
400 000/250 = 1600 mcg/mL
Dose per minute and per hour
10·80 = 800 mcg/min
800·60 = 48000 mcg/h
Pump rate
R = 48000/1600 = 30 mL/h
Answer
R = 30 mL/h
7.Pharmacology
Answer: C · Pregnant women should not handle crushed or broken tablets
Why this is the answer
Finasteride can be absorbed through the skin and may affect genital development of a male fetus. Women who are or may become pregnant should avoid contact with crushed or broken tablets.
Answer
C · Pregnant women should not handle crushed or broken tablets
8.Pharmacology
Answer: B · 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.