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 vancomycin, the level is 40 mg/L at 1 h and 20 mg/L at 7 h. Assume first-order elimination. Find the half-life t½.
C1 = 40 mg/L (1 h)C2 = 20 mg/L (7 h)
t½ = h
2.Clinical pharmacokineticsHard
Practice exercise (made-up values, not clinical guidance). A vancomycin infusion is started without a loading dose. The half-life in this patient is 6 h. How long t does it take to reach 95% of the steady-state level?
t½ = 6 h95% Css
t = h
3.Pharmacy calculationsMedium
Practice exercise (made-up values, not clinical guidance). A vial contains ampicillin 500 mg as a powder. The practice label says to add 1.8 mL of sterile water; the powder adds 0.2 mL to the volume. What is the concentration C in mg/mL after reconstitution?
500 mg1.8 mL+0.2 mL
C = mg/mL
4.Pharmacy calculationsHard
A vial holds 250 mg of powder. You add 9.6 mL of diluent, and the powder adds 0.4 mL of volume. The prescription is 112.5 mg. Draw up the dose: what volume V do you give?
250 mg9.6 mL + 0.4 mL112.5 mg
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 2 units/h. Find the pump rate R in mL/h. Round to one decimal place.
100 units/250 mL2 units/h
R = mL/h
6.Pediatric & critical-care dosingHard
Practice exercise (made-up values, not clinical guidance). A patient who weighs 96 kg has an order for norepinephrine at 4 mcg/min. The bag contains 8 mg in 250 mL. Find the pump rate R in mL/h. Round to one decimal place.
8 mg/250 mL4 mcg/min96 kg
R = mL/h
7.PharmacologyMedium
Why is propranolol usually avoided in patients with asthma?
AIt causes tolerance to inhaled corticosteroids
BIt blocks beta-2 receptors in the airways and can cause bronchospasm
CIt stimulates alpha-1 receptors in the airways and raises blood pressure
DIt increases mucus production by blocking histamine
8.PharmacologyHard
Beta-lactam antibiotics show time-dependent killing. What does this mean for administration?
ADoses can be given at any time as long as the daily total is right
BA single very large daily dose works best
CDoses should be given on time at evenly spaced intervals
DDoses should be given only when the patient has a fever
3Solutions
1.Clinical pharmacokinetics
Answer: t½ = 6 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(40/20)/6 = 0.6931/6 = 0.1155 h⁻¹
Half-life
t½ = 0.693/0.1155 = 6 h
Answer
t½ = 6 h
2.Clinical pharmacokinetics
Answer: t = 25.93 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 = 0.693/6 = 0.1155 h⁻¹
Time to the given fraction of steady state
t = −ln(1 − 0.95)/0.1155 = 2.996/0.1155 = 25.93 h
Answer
t = 25.93 h
That is 4.322 half-lives.
3.Pharmacy calculations
Answer: C = 250 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
1.8 + 0.2 = 2 mL
Concentration
C = 500/2 = 250 mg/mL
Answer
C = 250 mg/mL
4.Pharmacy calculations
Answer: V = 4.5 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
9.6 + 0.4 = 10 mL
C = 250/10 = 25 mg/mL
Volume
V = 112.5/25 = 4.5 mL
Answer
V = 4.5 mL
5.Pediatric & critical-care dosing
Answer: R = 5 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.
Concentration
100/250 = 0.4 units/mL
Pump rate
R = 2/0.4 = 5 mL/h
Answer
R = 5 mL/h
6.Pediatric & critical-care dosing
Answer: R = 7.5 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
8 000/250 = 32 mcg/mL
Dose per hour
The order is per minute for the whole patient, so the weight is not needed.
4·60 = 240 mcg/h
Pump rate
R = 240/32 = 7.5 mL/h
Answer
R = 7.5 mL/h
7.Pharmacology
Answer: B · It blocks beta-2 receptors in the airways and can cause bronchospasm
Why this is the answer
Non-selective beta blockers block both beta-1 and beta-2 receptors. Blocking beta-2 receptors in the bronchi removes their relaxing effect and can trigger severe bronchospasm.
Answer
B · It blocks beta-2 receptors in the airways and can cause bronchospasm
8.Pharmacology
Answer: C · Doses should be given on time at evenly spaced intervals
Why this is the answer
For beta-lactams, efficacy depends on the time the drug level stays above the minimum inhibitory concentration. Evenly spaced, on-time doses, or extended infusions, support this.
Answer
C · Doses should be given on time at evenly spaced intervals