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
ceftriaxone 1 gAdd 9.6 mL sterile water.Powder volume: 0.4 mL.250 mg → ? mL

Practice exercise (made-up values, not clinical guidance). A vial contains ceftriaxone 1000 mg as a powder. The practice label says to add 9.6 mL of sterile water; the powder adds 0.4 mL to the volume. The prescribed dose is 250 mg. What volume V do you draw up? Round to one decimal place.

1000 mg9.6 mL+0.4 mL250 mg
V = mL
2.Pharmacy calculationsHard
30%hydrogen peroxide+0%water? mL10%900 mL

Practice exercise (made-up values, not clinical guidance). You must prepare 900 mL of 10% hydrogen peroxide from a 30% stock and purified water. How many mL V of water do you need? Round to the nearest whole mL.

30%0%10%900 mL
V = mL
3.Dosage calculationsMedium
63 kgBSA = 1.48 m²175 mg/m²D = ? mg

A patient weighing 63 kg has a body surface area of 1.48 m². The prescription is 175 mg/m². Find the dose D in mg.

63 kgBSA = 1.48 m²175 mg/m²
D = mg
4.Dosage calculationsHard
120 mg/5 mLV = ? mL

A child weighing 20 kg is prescribed 12.5 mg/kg per dose of an oral suspension that contains 120 mg in 5 mL. What volume V do you give? Round to one decimal place.

20 kg12.5 mg/kg120 mg/5 mL
V = mL
5.Clinical pharmacokineticsMedium
110100t (h)C (mg/L)C₀ = 12 mg/Lt½ = 3.5 hC₀/2

Practice exercise (made-up values, not clinical guidance). A patient receives an IV bolus of 270 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 = 270 mgC₀ = 12 mg/Lt½ = 3.5 h
CL = L/h
6.Clinical pharmacokineticsHard
69 kg88 · manSCr 1.9 mg/dLfe​ = 0.7D = ? mcg

Practice exercise (made-up values, not clinical guidance). An 88-year-old man weighs 69 kg and has a serum creatinine of 1.9 mg/dL (168 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 digoxin dose is 250 mcg for a creatinine clearance of 120 mL/min, and a fraction fe = 0.7 of the drug is excreted unchanged by the kidneys. Using the Dettli dose fraction Q = 1 − fe·(1 − CrCl/normal CrCl), find the adjusted dose D. Round to the nearest whole number.

88 years69 kgmanSCr = 1.9 mg/dL250 mcgfe = 0.7CrCl normal 120 mL/min
D = mcg

3Solutions

1.Pharmacy calculations

Answer: V = 2.5 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

9.6 + 0.4 = 10 mL

Concentration

1000/10 = 100 mg/mL

Volume to give

V = 250/100 = 2.5 mL

Answer

V = 2.5 mL
2.Pharmacy calculations

Answer: V = 600 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

30%: 10 − 0 = 10 parts
0%: 30 − 10 = 20 parts
total: 30 parts

Water

V = 900·20/30 = 600 mL

Answer

V = 600 mL

Check: 300 mL of 30% contains 90 g of drug, which is 10% of 900 mL.

3.Dosage calculations

Answer: D = 259 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 = 175·1.48 = 259 mg

Answer

D = 259 mg
4.Dosage calculations

Answer: V = 10.4 mL

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.

Dose from weight

12.5·20 = 250 mg

Volume

V = (250/120)·5 = 10.42 ≈ 10.4 mL

Answer

V = 10.4 mL
5.Clinical pharmacokinetics

Answer: CL = 4.456 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 = 270/12 = 22.5 L

Rate constant

k = 0.693/3.5 = 0.198 h⁻¹

Clearance

CL = 0.198·22.5 = 4.456 L/h

Answer

CL = 4.456 L/h
6.Clinical pharmacokinetics

Answer: D = 113 mcg

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 − 88)·69/(72·1.9) = 26.23 mL/min

Dose fraction (Dettli)

Q = 1 − 0.7·(1 − 26.23/120) = 0.453

Adjusted dose

D = 0.453·250 = 113.2 ≈ 113 mcg

Answer

D = 113 mcg

In practice the dose is rounded to a convenient amount and checked against the product information.

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