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 receives an IV bolus of 190 mg of gentamicin. The level back-extrapolated to time zero is 12 mg/L and the half-life is 3.5 h. Find the clearance CL in L/h.
D = 190 mgC₀ = 12 mg/Lt½ = 3.5 h
CL = L/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 10 h. How long t does it take to reach 75% of the steady-state level?
t½ = 10 h75% Css
t = h
3.Pharmacy calculationsMedium
Practice exercise (made-up values, not clinical guidance). A vial contains ampicillin 1000 mg as a powder. The practice label says to add 3.5 mL of sterile water; the powder adds 0.5 mL to the volume. What is the concentration C in mg/mL after reconstitution?
1000 mg3.5 mL+0.5 mL
C = mg/mL
4.Pharmacy calculationsHard
Practice exercise (made-up values, not clinical guidance). You must prepare 100 mL of 5% sodium chloride by mixing 23% and 3% sodium chloride. How many mL V of the 23% solution do you need? Round to the nearest whole mL.
23%3%5%100 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 80 units/kg for a patient who weighs 68 kg. The vial contains 1 000 units/mL. What volume V do you draw up? Round to one decimal place.
80 units/kg68 kg1 000 units/mL
V = mL
6.Pediatric & critical-care dosingHard
Practice exercise (made-up values, not clinical guidance). A patient who weighs 77 kg has an order for nitroglycerin at 5 mcg/min. The bag contains 25 mg in 250 mL. Find the pump rate R in mL/h. Round to one decimal place.
25 mg/250 mL5 mcg/min77 kg
R = mL/h
7.PharmacologyMedium
Which adverse effect is a well-known risk of zidovudine?
ABone marrow suppression, especially anaemia
BGum overgrowth
CRed–green colour blindness
DOrange-red colouring of the urine and tears
8.PharmacologyHard
Why is clavulanic acid combined with amoxicillin (co-amoxiclav)?
AIt reduces the risk of penicillin allergy
BIt slows the kidney excretion of amoxicillin
CIt makes amoxicillin cross the blood–brain barrier more easily
DIt inhibits beta-lactamase enzymes that would destroy amoxicillin
3Solutions
1.Clinical pharmacokinetics
Answer: CL = 3.136 L/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.
Volume of distribution
Vd = 190/12 = 15.83 L
Rate constant
k = 0.693/3.5 = 0.198 h⁻¹
Clearance
CL = 0.198·15.83 = 3.136 L/h
Answer
CL = 3.136 L/h
2.Clinical pharmacokinetics
Answer: t = 20 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 = 0.693/10 = 0.06931 h⁻¹
Time to the given fraction of steady state
t = −ln(1 − 0.75)/0.06931 = 1.386/0.06931 = 20 h
Answer
t = 20 h
That is 2 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
3.5 + 0.5 = 4 mL
Concentration
C = 1000/4 = 250 mg/mL
Answer
C = 250 mg/mL
4.Pharmacy calculations
Answer: V = 10 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
23%: 5 − 3 = 2 parts
3%: 23 − 5 = 18 parts
total: 20 parts
Share the volume
V = 100·2/20 = 10 mL
Answer
V = 10 mL
Check: 10 mL of 23% + 90 mL of 3% = 100 mL.
5.Pediatric & critical-care dosing
Answer: V = 5.4 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
80·68 = 5 440 units
Volume
V = 5440/1000 = 5.44 ≈ 5.4 mL
Answer
V = 5.4 mL
6.Pediatric & critical-care dosing
Answer: R = 3 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
25 000/250 = 100 mcg/mL
Dose per hour
The order is per minute for the whole patient, so the weight is not needed.
5·60 = 300 mcg/h
Pump rate
R = 300/100 = 3 mL/h
Answer
R = 3 mL/h
7.Pharmacology
Answer: A · Bone marrow suppression, especially anaemia
Why this is the answer
Zidovudine, a nucleoside reverse transcriptase inhibitor, can cause anaemia and neutropenia. Blood counts are checked regularly during treatment.
Answer
A · Bone marrow suppression, especially anaemia
8.Pharmacology
Answer: D · It inhibits beta-lactamase enzymes that would destroy amoxicillin
Why this is the answer
Many bacteria produce beta-lactamases that inactivate penicillins. Clavulanic acid inhibits these enzymes, so amoxicillin can work against these organisms.
Answer
D · It inhibits beta-lactamase enzymes that would destroy amoxicillin