Pharmacy Calculations: cram pack

Formula sheets, the key method steps for every topic and 6 practice problems, with full solutions at the back.

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1Formulas and methods

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%

Dosage calculations

Dose from strength
volume = dose / strength
Weight-based dose
dose = mg/kg × body weight
Tablets
number = dose / tablet strength
Units
1 g = 1000 mg, 1 mg = 1000 mcg

Method

  1. Convert the ordered dose and the stock strength to the same unit.
  2. Use desired over have times quantity: (D/H) × Q.
  3. For weight-based doses, multiply by the weight in kg first, then divide the daily dose by the number of doses.
  4. Estimate first and check that the answer is reasonable.

Clinical pharmacokinetics

Rate constant and half-life
k = ln 2/t½ = 0.693/t½
First-order decay
C = C₀·e−kt = C₀·(1/2)t/t½
Two levels
k = ln(C₁/C₂)/Δt
Volume and clearance
Vd = dose/C₀, CL = k·Vd
Loading dose
LD = Vd·C/(S·F)
Maintenance rate
R = CL·Css/(S·F)
Average steady state
Css,avg = S·F·D/(CL·τ)
Steady state
1 − (1/2)ⁿ after n half-lives; ≈ 97% after 5
Infusion
C = (R/CL)(1 − e−kt)
Interval and dose from peak and trough
τ = ln(Cpeak/Ctrough)/k, D = Vd·(Cpeak − Ctrough)
Cockcroft–Gault
CrCl = (140 − age) × kg/(72 × SCr mg/dL), × 0.85 for women
Renal adjustment
dose × CrCl/normal; Dettli Q = 1 − fe·(1 − CrCl/normal)

Method

  1. Convert the half-life to a rate constant: k = 0.693/t½.
  2. Turn per-kg values into totals: Vd (L) = L/kg × weight, CL (L/h) = L/kg/h × weight.
  3. Pick the relation: C = C₀·e−kt, LD = Vd·C/(S·F), maintenance rate = CL·Css/(S·F), τ = ln(Cpeak/Ctrough)/k.
  4. For kidney function, estimate CrCl with Cockcroft–Gault first, then scale the dose or the interval.
  5. Check the result: levels fall by half every half-life, and steady state takes about five half-lives.

2Practice problems

1.Pharmacy calculationsMedium
024681010 mL

A solution has a ratio strength of 1:10 000. You need 0.8 mg. Draw up the volume in the 10 mL syringe: what volume V is that?

1:10 0000.8 mg
V = mL
2.Pharmacy calculationsHard
0123455 mL

A vial holds 2000 mg of powder. You add 9.2 mL of diluent, and the powder adds 0.8 mL of volume. The prescription is 780 mg. Draw up the dose: what volume V do you give?

2000 mg9.2 mL + 0.8 mL780 mg
V = mL
3.Dosage calculationsMedium
39 kgBSA = 1.14 m²250 mg/m²D = ? mg

A patient weighing 39 kg has a body surface area of 1.14 m². The prescription is 250 mg/m². Find the dose D in mg.

39 kgBSA = 1.14 m²250 mg/m²
D = mg
4.Dosage calculationsHard
40k units500 mL? mL/h500 units/h

An infusion contains 40000 units of a drug in 500 mL. The prescription is 500 units per hour. At what rate R do you set the pump, in mL/h?

40000 units/500 mL500 units/h
R = mL/h
5.Clinical pharmacokineticsMedium
110t (h)C (mg/L)C₀ = 8 mg/Lt½ = 2 hC₀/2

Practice exercise (made-up values, not clinical guidance). A patient receives an IV bolus of 150 mg of tobramycin. The level back-extrapolated to time zero is 8 mg/L and the half-life is 2 h. Find the volume of distribution Vd.

D = 150 mgC₀ = 8 mg/Lt½ = 2 h
Vd = L
6.Clinical pharmacokineticsHard
t (h)C (mg/L)Css​16 hC = ?

Practice exercise (made-up values, not clinical guidance). A vancomycin infusion runs at 125 mg/h with no loading dose. The patient's clearance is 5 L/h and the half-life is 10 h. What is the level C after 16 h?

R = 125 mg/hCL = 5 L/ht½ = 10 ht = 16 h
C = mg/L

3Solutions

1.Pharmacy calculations

Answer: V = 8 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/10 000 mL = 0.1 mg/mL

Volume

V = 0.8/0.1 = 8 mL

Answer

V = 8 mL
2.Pharmacy calculations

Answer: V = 3.9 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

9.2 + 0.8 = 10 mL
C = 2000/10 = 200 mg/mL

Volume

V = 780/200 = 3.9 mL

Answer

V = 3.9 mL
3.Dosage calculations

Answer: D = 285 mg

Body-surface-area dosing: many cytotoxic and some paediatric doses are prescribed per square metre of body surface area: dose (mg) = dose per m² × BSA (m²). The Mosteller formula estimates BSA = √(height (cm) × weight (kg)/3600).

Dose

D = 250·1.14 = 285 mg

Answer

D = 285 mg
4.Dosage calculations

Answer: R = 6.25 mL/h

Dose calculation: the amount to give equals what is prescribed divided by what you have, times the quantity it comes in: amount = (desired dose/stock strength) × stock volume. Always keep the units the same (convert g to mg, mg to mcg first) and check that the answer is sensible.

Concentration

40000/500 = 80 units/mL

Pump rate

R = 500/80 = 6.25 mL/h

Answer

R = 6.25 mL/h
5.Clinical pharmacokinetics

Answer: Vd = 18.75 L

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 = 150/8 = 18.75 L

Answer

Vd = 18.75 L
6.Clinical pharmacokinetics

Answer: C = 16.75 mg/L

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.

Steady-state level

Css = 125/5 = 25 mg/L

Rate constant

k = 0.693/10 = 0.06931 h⁻¹

Level after the given time

C = 25·(1 − e−0.06931·16) = 16.75 mg/L

Answer

C = 16.75 mg/L

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