4.1 Pharmaceutical Calculations
Key Takeaways
- Pharmaceutical calculations reduce to ratio-proportion setups with rigorous unit tracking; master the setup, not just the arithmetic.
- Concentration expressions (% w/w, % w/v, % v/v, mg/mL, molarity, molality, normality) are interchangeable once you know the definition of each.
- Isotonicity is adjusted using E-values (NaCl equivalents) or the freezing-point depression method; isotonic solutions depress freezing point by 0.52 degrees C.
- Henderson-Hasselbalch predicts ionization: weak acids are absorbed in the stomach (unionized at low pH) and weak bases in the intestine (unionized at higher pH).
- IV flow rates use drops/min = (volume in mL x drop factor) / time in min; mL/hr = total volume / total hours for infusion pumps.
4.1 Pharmaceutical Calculations
Quick Answer: Pharmaceutical calculations are the quantitative backbone of pharmacy practice. Master ratio-proportion, concentration unit conversions, isotonicity (E-values and freezing-point depression), osmolarity, and the Henderson-Hasselbalch equation. These are the high-yield calculation types on the FPGEE.
Pharmaceutical calculations underpin every dose, infusion, and compounded preparation. The FPGEE tests your ability to manipulate concentrations, calculate doses across populations, set IV flow rates, adjust tonicity, and predict ionization. Each problem reduces to a ratio-proportion setup with careful unit tracking.
Concentration Units
Pharmacy uses several interchangeable concentration expressions. Knowing the definition of each prevents the most common errors.
| Expression | Definition | Example |
|---|---|---|
| % w/w | g of solute per 100 g of total product | 1% hydrocortisone cream = 1 g HC in 100 g cream |
| % w/v | g of solute per 100 mL of solution | 5% dextrose = 5 g D5W per 100 mL |
| % v/v | mL of solute per 100 mL of solution | 70% ethanol = 70 mL EtOH per 100 mL |
| mg/mL | mass per volume (often % w/v x 10) | 10 mg/mL = 1% w/v |
| Molarity (M) | moles solute per L solution | 0.9% NaCl is approximately 0.154 M |
| Molality (m) | moles solute per kg solvent | Used in colligative property work |
| Normality (N) | equivalents solute per L solution | N = M x n (n = number of reactive units) |
Worked example - converting % w/v to molarity: 0.9% NaCl (w/v) means 0.9 g per 100 mL, or 9 g/L. NaCl molar mass = 58.5 g/mol. Moles = 9 / 58.5 = 0.154 mol/L, so 0.9% NaCl is 0.154 M NaCl.
Ratio and Proportion
The fundamental setup is a / b = c / d. Solve for the unknown by cross-multiplication.
Worked example: A syrup contains 250 mg drug per 5 mL. How much drug is in 240 mL? 250 mg / 5 mL = X mg / 240 mL, so X = (250 x 240) / 5 = 12,000 mg (12 g).
Dose Calculations
Pediatric mg/kg dosing
Worked example: Amoxicillin 40 mg/kg/day divided BID for a 22 lb child. 22 lb / 2.2 = 10 kg. Daily dose = 40 x 10 = 400 mg. Divided BID = 200 mg every 12 hours.
BSA-based dosing
Most chemotherapeutics use BSA-based dosing. The Mosteller formula is the exam favorite: BSA (m squared) = square root of (height(cm) x weight(kg) / 3600). The Du Bois formula is more exact: BSA = 0.007184 x W^0.425 x H^0.725.
Worked example: Vincristine 1.4 mg/m squared for a child who is 110 cm and 22 kg. BSA = square root of (110 x 22 / 3600) = square root of (2420 / 3600) = square root of 0.672 = 0.82 m squared. Dose = 1.4 x 0.82 = 1.15 mg (cap at 2 mg total per protocol).
Loading vs maintenance doses
Loading dose (LD) = Vd x Cp_target (volume of distribution x target plasma concentration). Maintenance dose (MD) = Cp_target x CL x dosing interval / F (clearance x bioavailability).
Worked example: Vd = 50 L, target = 2 mg/L, CL = 0.5 L/hr, dosing interval = 12 hr, F = 1. LD = 50 x 2 = 100 mg. MD = 2 x 0.5 x 12 / 1 = 12 mg.
Geriatric adjustments
Renal function declines with age. Use Cockcroft-Gault for creatinine clearance: CrCl = [(140 - age) x weight(kg)] / (72 x SCr), multiplied by 0.85 if female.
IV Flow Rates
Drops/min = (Volume in mL x drop factor) / Time in min. Macrodrop sets are 10, 15, or 20 gtt/mL; microdrop sets are 60 gtt/mL.
Worked example: Infuse 1 L over 8 hr with a 15 gtt/mL set. Time = 8 x 60 = 480 min. Rate = (1000 x 15) / 480 = 31.25, rounded to 31 gtt/min.
For infusion pumps set in mL/hr: mL/hr = Total volume / Total hours. 1000 mL / 8 hr = 125 mL/hr.
Isotonicity
Tears and blood plasma have an osmotic pressure equivalent to 0.9% NaCl. Adjusting a compounded solution to that pressure prevents tissue irritation, pain on injection, and hemolysis.
Sodium chloride equivalent (E-value)
The E-value = grams of NaCl that produce the same osmotic effect as 1 g of the drug. Formula: E = (17 x L) / MW, where L is the dissociation factor (number of particles times the Liso value) and MW is molecular weight.
| Compound | MW | E-value |
|---|---|---|
| NaCl | 58.5 | 1.00 |
| Dextrose (anhydrous) | 180 | 0.18 |
| Ephedrine sulfate | 429 | 0.23 |
| Phenylephrine HCl | 204 | 0.32 |
Worked example - isotonicity via E-values: Prepare 30 mL of 1% phenylephrine HCl isotonic with NaCl. Drug mass = 30 mL x 1% = 0.3 g. E-value of phenylephrine HCl = 0.32. NaCl equivalent of the drug = 0.3 x 0.32 = 0.096 g. NaCl needed to make 30 mL isotonic: 0.009 g/mL x 30 = 0.27 g. Additional NaCl required = 0.27 - 0.096 = 0.174 g NaCl (q.s. to 30 mL with water).
Freezing point depression method
Blood and lacrimal fluid freeze at -0.52 degrees C (depression of 0.52 degrees C below pure water). A 1% NaCl solution depresses freezing point by approximately 0.576 degrees C. The formula: % NaCl needed for isotonicity = (0.52 - dTf from drug) / 0.576.
Osmolarity and Osmolality
Osmolarity (mOsm/L) = molarity x particles per formula unit x 1000. Osmolality (mOsm/kg) is temperature-independent. Plasma is approximately 290 mOsm/L.
Worked example: 0.9% NaCl = 9 g/L. Moles = 9 / 58.5 = 0.154 M. NaCl dissociates into 2 particles, so 0.154 x 2 x 1000 = 308 mOsm/L (slightly hypertonic to plasma).
Dextrose 5%: 50 g/L. Moles = 50 / 180 = 0.278 M. Dextrose is a non-electrolyte (1 particle), so 0.278 x 1 x 1000 = 278 mOsm/L (slightly hypotonic on paper, but becomes hypotonic in vivo as dextrose is metabolized).
Ionization: Henderson-Hasselbalch Equation
For weak acids: pH = pKa + log([A-]/[HA]), so the unionized-to-ionized ratio = 10^(pH - pKa). For weak bases: pH = pKa + log([B]/[BH+]), ratio = 10^(pH - pKa).
Fraction ionized for a weak acid: % ionized = 100 / (1 + 10^(pKa - pH)). Fraction ionized for a weak base: % ionized = 100 / (1 + 10^(pH - pKa)).
Worked example: Aspirin pKa = 3.5. At gastric pH 1.5: ratio = 10^(1.5 - 3.5) = 0.01. % ionized = 100 / (1 + 10^(3.5 - 1.5)) = 100 / 101 = approximately 0.99% ionized. Aspirin is 99% unionized in the stomach and therefore well absorbed there.
Worked example: Morphine pKa = 8.0 (weak base). At intestinal pH 6.0: % ionized = 100 / (1 + 10^(6 - 8)) = 100 / 1.01 = approximately 99% ionized. Morphine is mostly ionized even in the intestine, explaining its poor and variable oral bioavailability and significant first-pass metabolism.
Percentage Strength and Serial Dilution
Serial dilution uses C1V1 = C2V2. Repeated dilution multiplies the dilution factors.
Worked example: Prepare 100 mL of 0.1% solution from a 10% stock. C1V1 = C2V2: 10 x V1 = 0.1 x 100, so V1 = 1 mL of stock, q.s. to 100 mL with diluent.
Alligation Method
Used when mixing two known strengths to obtain a desired intermediate strength. Set up as a tic-tac-toe grid: the differences between the desired strength and each original strength give the parts of the other solution to use.
Worked example: Mix 20% and 5% to make 15%. Parts of 20% = 15 - 5 = 10. Parts of 5% = 20 - 15 = 5. Total = 15 parts. Ratio 10:5 = 2 parts of 20% to 1 part of 5%.
These calculations appear consistently on the FPGEE. Drill them until the setup is automatic, the units are tracked explicitly, and the answer is sanity-checked (a pediatric dose should never exceed the adult dose; an isotonic solution should never require negative NaCl).
A 25-kg child is prescribed amoxicillin 40 mg/kg/day divided BID. What is the single-dose amount?
Which statement about isotonicity adjustments is correct?
A weak acid drug with pKa 4.0 is dissolved in a solution at pH 5.0. Approximately what percentage is ionized?