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 4 mg/L at 5 h. Assume first-order elimination. Find the elimination rate constant k.
C1 = 10 mg/L (1 h)C2 = 4 mg/L (5 h)
k = h⁻¹
2.Clinical pharmacokineticsHard
Practice exercise (made-up values, not clinical guidance). A 66-year-old woman weighs 72 kg and has a serum creatinine of 2.6 mg/dL (230 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 gentamicin regimen is 80 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 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 must prepare 200 mL of 2% 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%2%200 mL
V = mL
5.Pediatric & critical-care dosingMedium
Practice exercise (made-up values, not clinical guidance). A practice protocol calls for a heparin bolus of 60 units/kg for a patient who weighs 61 kg. The vial contains 5 000 units/mL. What volume V do you draw up? Round to two decimal places.
60 units/kg61 kg5 000 units/mL
V = mL
6.Pediatric & critical-care dosingHard
Practice exercise (made-up values, not clinical guidance). A bag contains dobutamine 500 mg in 250 mL. The order is 2.5 mcg/kg/min for a patient who weighs 72 kg. At what rate R do you set the pump, in mL/h? Round to one decimal place.
500 mg/250 mL2.5 mcg/kg/min72 kg
R = mL/h
7.PharmacologyMedium
Which description fits tardive dyskinesia?
AHigh fever with lead-pipe rigidity and confusion
BInvoluntary movements of the lips, tongue and face after long-term antipsychotic use
CA sudden painful spasm of the neck muscles and upward rolling of the eyes within days of the first dose
DA low white cell count with sore throat and fever
8.PharmacologyHard
A patient with severe liver cirrhosis takes oral propranolol, a drug with a high first-pass metabolism. What effect is expected?
AThe drug is excreted faster by the kidney, so it works for a shorter time
BMore drug reaches the circulation, so a lower dose may be needed
CThe first-pass effect increases, so the drug becomes inactive
DLess drug is absorbed from the gut, so a higher dose is needed
3Solutions
1.Clinical pharmacokinetics
Answer: k = 0.2291 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(10/4)/4 = 0.9163/4 = 0.2291 h⁻¹
Answer
k = 0.2291 h⁻¹
2.Clinical pharmacokinetics
Answer: τ = 33.1 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 − 66)·72/(72·2.6)·0.85 = 24.19 mL/min
New interval
τ = 8·100/24.19 = 33.07 ≈ 33.1 h
Answer
τ = 33.1 h
In practice the interval is rounded to a convenient value such as 12, 24, 36 or 48 h.
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: V = 180 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%: 2 − 0 = 2 parts
0%: 20 − 2 = 18 parts
total: 20 parts
Water
V = 200·18/20 = 180 mL
Answer
V = 180 mL
Check: 20 mL of 20% contains 4 g of drug, which is 2% of 200 mL.
5.Pediatric & critical-care dosing
Answer: V = 0.73 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
60·61 = 3 660 units
Volume
V = 3660/5000 = 0.732 ≈ 0.73 mL
Answer
V = 0.73 mL
6.Pediatric & critical-care dosing
Answer: R = 5.4 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
500 mg = 500 000 mcg
500 000/250 = 2000 mcg/mL
Dose per minute and per hour
2.5·72 = 180 mcg/min
180·60 = 10800 mcg/h
Pump rate
R = 10800/2000 = 5.4 mL/h
Answer
R = 5.4 mL/h
7.Pharmacology
Answer: B · Involuntary movements of the lips, tongue and face after long-term antipsychotic use
Why this is the answer
Tardive dyskinesia develops after months or years of dopamine-blocking drugs and may be permanent. Patients are screened regularly so that treatment can be changed early.
Answer
B · Involuntary movements of the lips, tongue and face after long-term antipsychotic use
8.Pharmacology
Answer: B · More drug reaches the circulation, so a lower dose may be needed
Why this is the answer
In cirrhosis, less of the oral dose is removed by the liver on its first pass, partly because blood bypasses the liver through shunts. Bioavailability and plasma levels rise, so doses of such drugs often need to be reduced.
Answer
B · More drug reaches the circulation, so a lower dose may be needed