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

Convert the ordered dose and the stock strength to the same unit.
Use desired over have times quantity: (D/H) × Q.
For weight-based doses, multiply by the weight in kg first, then divide the daily dose by the number of doses.
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·C_ss/(S·F)
Average steady state: C_ss,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(C_peak/C_trough)/k, D = Vd·(C_peak − C_trough)
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

Convert the half-life to a rate constant: k = 0.693/t½.
Turn per-kg values into totals: Vd (L) = L/kg × weight, CL (L/h) = L/kg/h × weight.
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.

2Practice problems

1.Pharmacy calculationsMedium

A solution has a ratio strength of 1:10 000. You need 0.16 mg. Draw up the volume in the 10 mL syringe: what volume V is that?

1:10 0000.16 mg

V = mL

2.Pharmacy calculationsHard

A vial holds 1000 mg of powder. You add 9.5 mL of diluent, and the powder adds 0.5 mL of volume. The prescription is 60 mg. Draw up the dose: what volume V do you give?

1000 mg9.5 mL + 0.5 mL60 mg

V = mL

3.Dosage calculationsMedium

A dose of 250 mg is needed from a 5% (w/v) solution. What volume V contains this dose?

250 mg5% w/v

V = mL

4.Dosage calculationsHard

The prescription is 0.88 mg. The ampoule contains 4000 micrograms per mL. Draw up the dose in the 1 mL syringe: what volume V do you give?

0.88 mg4000 mcg/mL

V = mL

5.Clinical pharmacokineticsMedium

Practice exercise (made-up values, not clinical guidance). After an IV dose of gentamicin, the level is 6 mg/L at 1 h and 1.2 mg/L at 9 h. Assume first-order elimination. Find the elimination rate constant k.

C₁ = 6 mg/L (1 h)C₂ = 1.2 mg/L (9 h)

k = h⁻¹

6.Clinical pharmacokineticsHard

Practice exercise (made-up values, not clinical guidance). A vancomycin infusion runs at 80 mg/h with no loading dose. The patient's clearance is 4 L/h and the half-life is 10 h. What is the level C after 29 h?

R = 80 mg/hCL = 4 L/ht½ = 10 ht = 29 h

C = mg/L

3Solutions

1.Pharmacy calculations

Answer: V = 1.6 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 = 0.16/0.1 = 1.6 mL

Answer

V = 1.6 mL

2.Pharmacy calculations

Answer: V = 0.6 mL

Final volume and concentration

9.5 + 0.5 = 10 mL

C = 1000/10 = 100 mg/mL

Volume

V = 60/100 = 0.6 mL

Answer

V = 0.6 mL

3.Dosage calculations

Answer: V = 5 mL

Solution strengths: a percentage strength is grams per 100 mL, so 1% = 1 g/100 mL = 10 mg/mL. A ratio strength 1:1000 means 1 g in 1000 mL = 1 mg/mL; 1:10 000 is 0.1 mg/mL. For a dilution the amount of drug stays the same: C₁·V₁ = C₂·V₂.

Concentration

5% = 50 mg/mL

Volume

V = 250/50 = 5 mL

Answer

V = 5 mL

4.Dosage calculations

Answer: V = 0.22 mL

Drawing up a dose: volume = (prescribed dose/stock strength) × stock volume. Convert to the same units first, then check that the volume fits the syringe and lands on one of its graduations.

Same units

0.88 mg = 880 mcg

Volume

V = 880/4000 = 0.22 mL

Answer

V = 0.22 mL

5.Clinical pharmacokinetics

Answer: k = 0.2012 h⁻¹

Clearance and volume: at steady state the dosing rate equals the elimination rate, so the maintenance rate = CL × C_ss. 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.

Elimination rate constant

k = ln(6/1.2)/8 = 1.609/8 = 0.2012 h⁻¹

Answer

k = 0.2012 h⁻¹

6.Clinical pharmacokinetics

Answer: C = 17.32 mg/L

Designing a regimen: with rapid IV doses the level falls from peak to trough during the interval, so τ = ln(C_peak/C_trough)/k and the dose that restores the peak is Vd × (C_peak − C_trough). The average steady-state level is C_ss,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

C_ss = 80/4 = 20 mg/L

Rate constant

k = 0.693/10 = 0.06931 h⁻¹

Level after the given time

C = 20·(1 − e^{−0.06931·29}) = 17.32 mg/L

Answer

C = 17.32 mg/L

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