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 intravenously with phenytoin sodium to reach a phenytoin level of 15 mg/L. The volume of distribution is 0.7 L/kg and the salt factor is S = 0.92. Find the loading dose LD of phenytoin sodium in mg.
63 kgVd = 0.7 L/kgC = 15 mg/LS = 0.92
LD = mg
2.Clinical pharmacokineticsHard
Practice exercise (made-up values, not clinical guidance). A 95 kg patient on regular tobramycin doses is at steady state. The target is a peak of 10 mg/L and a trough of 1 mg/L, and the volume of distribution is 0.25 L/kg. Assuming rapid IV doses, what dose D raises the level from the trough to the peak?
A solution has a ratio strength of 1:1000. You need 0.15 mg. Draw up the volume in the 1 mL syringe: what volume V is that?
1:10000.15 mg
V = mL
4.Pharmacy calculationsHard
Practice exercise (made-up values, not clinical guidance). You must prepare 300 mL of 8% sodium chloride by mixing 23% and 3% sodium chloride. How many mL V of the 3% solution do you need? Round to the nearest whole mL.
23%3%8%300 mL
V = mL
5.Pediatric & critical-care dosingMedium
Practice exercise (made-up values, not clinical guidance). An infusion of 100 units of regular insulin in 250 mL runs at 2 mL/h. How many units per hour D does the patient receive?
100 units/250 mL2 mL/h
D = units/h
6.Pediatric & critical-care dosingHard
Practice exercise (made-up values, not clinical guidance). A patient who weighs 67 kg has an order for norepinephrine at 6 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 mL6 mcg/min67 kg
R = mL/h
7.PharmacologyMedium
Which receptor do beta blockers such as metoprolol mainly block?
ABeta-3 receptors in fat tissue
BBeta-1 adrenergic receptors in the heart
CMuscarinic receptors in the heart
DAlpha-1 receptors in blood vessels
8.PharmacologyHard
A patient on long-term prednisolone has been vomiting for two days and could not take her tablets. She is now hypotensive and confused, with low sodium and high potassium. What is the most urgent treatment?
AIV insulin with glucose only
BIV hydrocortisone and IV fluids
CIV furosemide
DOral prednisolone with a glass of water
3Solutions
1.Clinical pharmacokinetics
Answer: LD = 719 mg
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 = 0.7·63 = 44.1 L
Amount of active drug needed
Vd·C = 44.1·15 = 661.5 mg
Loading dose
LD = 661.5/(0.92·1) = 719 mg
Answer
LD = 719 mg
In practice the dose is rounded to a convenient amount.
2.Clinical pharmacokinetics
Answer: D = 213.8 mg
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.
Volume of distribution
Vd = 0.25·95 = 23.75 L
Dose
D = 23.75·(10 − 1) = 213.8 mg
Answer
D = 213.8 mg
In practice the dose is rounded to a convenient amount.
3.Pharmacy calculations
Answer: V = 0.15 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 = 0.15/1 = 0.15 mL
Answer
V = 0.15 mL
4.Pharmacy calculations
Answer: V = 225 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%: 8 − 3 = 5 parts
3%: 23 − 8 = 15 parts
total: 20 parts
Share the volume
V = 300·15/20 = 225 mL
Answer
V = 225 mL
Check: 75 mL of 23% + 225 mL of 3% = 300 mL.
5.Pediatric & critical-care dosing
Answer: D = 0.8 units/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
Units per hour
D = 2·0.4 = 0.8 units/h
Answer
D = 0.8 units/h
6.Pediatric & critical-care dosing
Answer: R = 22.5 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.
6·60 = 360 mcg/h
Pump rate
R = 360/16 = 22.5 mL/h
Answer
R = 22.5 mL/h
7.Pharmacology
Answer: B · Beta-1 adrenergic receptors in the heart
This suggests an adrenal crisis triggered by missed steroids and illness. Treatment is immediate IV or IM hydrocortisone, IV fluids with glucose as needed, and treating the cause.