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 63 kg patient is to be loaded with digoxin tablets to reach a level of 0.8 mcg/L. The volume of distribution is 7 L/kg and the tablets have a bioavailability F = 0.7. Find the total oral loading dose LD in mcg.
63 kgVd = 7 L/kgC = 0.8 mcg/LF = 0.7
LD = mcg
2.Clinical pharmacokineticsHard
Practice exercise (made-up values, not clinical guidance). A 64-year-old woman weighs 58 kg and has a serum creatinine of 2.2 mg/dL (194 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 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. The prescribed dose is 800 mg. What volume V do you draw up? Round to one decimal place.
1000 mg3.5 mL+0.5 mL800 mg
V = mL
4.Pharmacy calculationsHard
Practice exercise (made-up values, not clinical guidance). You must prepare 300 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%300 mL
V = mL
5.Pediatric & critical-care dosingMedium
Practice exercise (made-up values, not clinical guidance). An infusion contains 50 units of regular insulin in 50 mL. The order is 8 units/h. Find the pump rate R in mL/h. Round to one decimal place.
50 units/50 mL8 units/h
R = mL/h
6.Pediatric & critical-care dosingHard
Practice exercise (made-up values, not clinical guidance). A patient who weighs 71 kg has an order for norepinephrine at 10 mcg/min. The bag contains 4 mg in 250 mL. Find the pump rate R in mL/h. Round to one decimal place.
4 mg/250 mL10 mcg/min71 kg
R = mL/h
7.PharmacologyMedium
A patient with a fentanyl patch is given home-care advice. Which instruction is correct?
AAvoid heating pads, hot baths and saunas over or near the patch
BCut the patch in half if the pain is mild
CApply each new patch over the same skin area
DThrow used patches loose into the household bin with the packaging
8.PharmacologyHard
A patient taking a GLP-1 receptor agonist such as semaglutide has severe, constant upper abdominal pain spreading to the back, with vomiting. What should the nurse suspect?
AAcute pancreatitis; the drug is stopped and urgent assessment arranged
BHypoglycaemia; the patient should eat sweets and continue as usual
CConstipation; a laxative will solve the problem
DExpected nausea; the patient should take the next dose with food
3Solutions
1.Clinical pharmacokinetics
Answer: LD = 504 mcg
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 = 7·63 = 441 L
Amount of active drug needed
Vd·C = 441·0.8 = 352.8 mcg
Loading dose
LD = 352.8/(1·0.7) = 504 mcg
Answer
LD = 504 mcg
In practice the dose is rounded to a convenient amount.
2.Clinical pharmacokinetics
Answer: τ = 33.8 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 − 64)·58/(72·2.2)·0.85 = 23.65 mL/min
New interval
τ = 8·100/23.65 = 33.82 ≈ 33.8 h
Answer
τ = 33.8 h
In practice the interval is rounded to a convenient value such as 12, 24, 36 or 48 h.
3.Pharmacy calculations
Answer: V = 3.2 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
1000/4 = 250 mg/mL
Volume to give
V = 800/250 = 3.2 mL
Answer
V = 3.2 mL
4.Pharmacy calculations
Answer: V = 270 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 = 300·18/20 = 270 mL
Answer
V = 270 mL
Check: 30 mL of 20% contains 6 g of drug, which is 2% of 300 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.
Concentration
50/50 = 1 units/mL
Pump rate
R = 8/1 = 8 mL/h
Answer
R = 8 mL/h
6.Pediatric & critical-care dosing
Answer: R = 37.5 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
4 000/250 = 16 mcg/mL
Dose per hour
The order is per minute for the whole patient, so the weight is not needed.
10·60 = 600 mcg/h
Pump rate
R = 600/16 = 37.5 mL/h
Answer
R = 37.5 mL/h
7.Pharmacology
Answer: A · Avoid heating pads, hot baths and saunas over or near the patch
Why this is the answer
Heat increases fentanyl absorption and can cause overdose. Sites are rotated, patches are never cut, and used patches are folded sticky sides together and disposed of safely, away from children and pets.
Answer
A · Avoid heating pads, hot baths and saunas over or near the patch
8.Pharmacology
Answer: A · Acute pancreatitis; the drug is stopped and urgent assessment arranged
Why this is the answer
Mild nausea is common with GLP-1 agonists, but severe persistent abdominal pain radiating to the back suggests pancreatitis. The drug is stopped and the patient needs urgent assessment, including serum lipase or amylase.
Answer
A · Acute pancreatitis; the drug is stopped and urgent assessment arranged