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
penicillin G 5 000 000 unitsdiluent addedconcentration18.2 mL250 000 units/mL8.2 mL500 000 units/mL3.2 mL1 000 000 units/mL750 000 units → ? mL

Practice exercise (made-up values, not clinical guidance). A vial of penicillin G contains 5 000 000 units. Its practice label gives three strengths (see the table). You add 18.2 mL of diluent. The prescribed dose is 750 000 units. What volume V do you draw up? Round to one decimal place.

18.2 mL → 250 000 units/mL750 000 units
V = mL
2.Pharmacy calculationsHard
0123455 mL

A vial holds 250 mg of powder. You add 4.2 mL of diluent, and the powder adds 0.8 mL of volume. The prescription is 170 mg. Draw up the dose: what volume V do you give?

250 mg4.2 mL + 0.8 mL170 mg
V = mL
3.Dosage calculationsMedium
1:1000ampouleV = ? mL

An ampoule of epinephrine (adrenaline) is labelled 1:1000 (1 mg/mL). The prescription is 0.5 mg IM for anaphylaxis. What volume V do you draw up?

1:10000.5 mg
V = mL
4.Dosage calculationsHard
250 mg/5 mLV = ? mL

A child weighing 24 kg is prescribed 12.5 mg/kg per dose of an oral suspension that contains 250 mg in 5 mL. What volume V do you give?

24 kg12.5 mg/kg250 mg/5 mL
V = mL
5.Clinical pharmacokineticsMedium
VCss​ = 25 mg/LR = ? mg/hCL = 4.5 L/h

Practice exercise (made-up values, not clinical guidance). Vancomycin is given as a continuous infusion to a patient whose vancomycin clearance is 4.5 L/h. What infusion rate R in mg/h gives a steady-state level of 25 mg/L?

CL = 4.5 L/hCss = 25 mg/L
R = mg/h
6.Clinical pharmacokineticsHard
81 kg77 · womanSCr 2 mg/dLCrCl → D = ? mg

Practice exercise (made-up values, not clinical guidance). A 77-year-old woman weighs 81 kg and has a serum creatinine of 2 mg/dL (177 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 dose is 120 mg every 8 h for a creatinine clearance of 100 mL/min. Keeping the interval, the dose is reduced in proportion to the creatinine clearance. Find the adjusted dose D. Round to the nearest whole number.

77 years81 kgwomanSCr = 2 mg/dL120 mg q8hCrCl normal 100 mL/min
D = mg

3Solutions

1.Pharmacy calculations

Answer: V = 3 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.

Concentration from the label

18.2 mL → 250 000 units/mL

Volume to give

V = 750 000/250 000 = 3 mL

Answer

V = 3 mL
2.Pharmacy calculations

Answer: V = 3.4 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

4.2 + 0.8 = 5 mL
C = 250/5 = 50 mg/mL

Volume

V = 170/50 = 3.4 mL

Answer

V = 3.4 mL
3.Dosage calculations

Answer: V = 0.5 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

1:1000 = 1 g/1000 mL = 1 mg/mL

Volume

V = 0.5/1 = 0.5 mL

Answer

V = 0.5 mL

US labels now state mg/mL (1 mg/mL = 1:1000; 0.1 mg/mL = 1:10 000). Never give 1 mg/mL epinephrine IV push.

4.Dosage calculations

Answer: V = 6 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

12.5·24 = 300 mg

Volume

V = (300/250)·5 = 6 mL

Answer

V = 6 mL
5.Clinical pharmacokinetics

Answer: R = 112.5 mg/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.

Infusion rate

R = 4.5·25 = 112.5 mg/h

Answer

R = 112.5 mg/h
6.Clinical pharmacokinetics

Answer: D = 36 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.

Creatinine clearance (Cockcroft–Gault)

CrCl = (140 − 77)·81/(72·2)·0.85 = 30.12 mL/min

Adjusted dose

D = 120·30.12/100 = 36.15 ≈ 36 mg

Answer

D = 36 mg

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

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