Section 2.12: Pharmaceutical Calculations: Dosing and Strengths

Pharmaceutical Calculations: Dosing and Strengths

Accurate pharmaceutical calculations are critical to clinical efficacy and patient safety. Dosing errors are a major source of preventable adverse drug events. This section details the calculations required for determining individualized patient doses based on body weight and surface area, and converting between different units of concentration and strength.

Dosing Based on Body Weight

Many medications, particularly in pediatric, geriatric, and critical care medicine, are dosed based on body mass (e.g., $\text{mg/kg}$). In adults, body composition (fat versus muscle mass) affects drug distribution. This requires utilizing three weight definitions:

1. Actual Body Weight (ABW): The patient's physical measured weight.
2. Ideal Body Weight (IBW): The estimated lean body mass, calculated using the Devine formula:
   • Males: $IBW\text{ (kg)} = 50.0\text{ kg} + 2.3\text{ kg for every inch over 5 feet (60 inches)}$
   • Females: $IBW\text{ (kg)} = 45.5\text{ kg} + 2.3\text{ kg for every inch over 5 feet (60 inches)}$ Note: If the height is in centimeters, convert to inches ($1\text{ inch} = 2.54\text{ cm}$).
3. Adjusted Body Weight (AdjBW): Used for dosing hydrophilic drugs (e.g., aminoglycosides) in obese patients, where dosing based on ABW would cause toxicity, and dosing on IBW would lead to subtherapeutic levels. Obese is defined as $\text{ABW} > 120\%\text{ of IBW}$. $$AdjBW = IBW + 0.4 \cdot (ABW - IBW)$$

Body Surface Area (BSA) Dosing

BSA-based dosing is common in chemotherapy and pediatric protocols because it correlates closely with cardiac output and metabolic capacity (renal/hepatic function). BSA (in $\text{m}^2$) is calculated using the Mosteller formula:

$$\text{BSA (m}^2\text{)} = \sqrt{\frac{\text{Height (cm)} \cdot \text{Weight (kg)}}{3600}}$$

Dosing and Body Composition Formulas

Concentration and Strength Calculations

Pharmacists must convert fluid or solid compositions into different concentration formats:

1. Percentage Strengths

Percentage strength is defined as the amount of active solute per $100\text{ units}$ of total preparation:

• Weight-in-Volume (% w/v): Grams of solute in $100\text{ mL}$ of solution. (Common for IV fluids, liquid products).
• Weight-in-Weight (% w/w): Grams of solute in $100\text{ g}$ of mixture. (Common for ointments, creams).
• Volume-in-Volume (% v/v): Milliliters of solute in $100\text{ mL}$ of solution. (Common for liquid mixtures like alcohol solutions).

2. Ratio Strengths

Ratio strength is expressed as $1:X$, representing $1\text{ part}$ of active solute in $X\text{ parts}$ of total preparation:

• Liquid solutions ($1:X$): $1\text{ gram}$ of solute in $X\text{ mL}$ of solution.
• Solid mixtures ($1:X$): $1\text{ gram}$ of solute in $X\text{ grams}$ of mixture.
• Conversion to Percentage: $$\%\text{ Strength} = \frac{1}{X} \cdot 100\%$$
  • Example: Epinephrine $1:1,000$ = $1\text{ g / }1,000\text{ mL} = 0.1\%\text{ w/v}$ (or $1\text{ mg/mL}$).
  • Example: Epinephrine $1:10,000$ = $1\text{ g / }10,000\text{ mL} = 0.01\%\text{ w/v}$ (or $0.1\text{ mg/mL}$).

3. Parts-Per-Million (ppm)

Parts-per-million (ppm) represents parts of solute per $1,000,000\text{ parts}$ of total preparation. It is used for extremely dilute concentrations (e.g., fluoride levels in water, toxic impurities in pharmaceuticals):

• In aqueous solutions, $1\text{ ppm} = 1\text{ mg / }1,000,000\text{ mL} = 1\text{ mg / L} = 1\text{ mcg / mL}$.
• Conversion to Percentage: $$\%\text{ Strength} = \frac{\text{ppm}}{10,000}$$
  • Example: $250\text{ ppm} = 250 / 10,000 = 0.025\%\text{ w/v}$.

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Step-by-Step Worked Clinical Calculations

Example 1: Dosing Weight Calculation in an Obese Patient

A $55\text{-year-old}$ male patient is $5\text{ feet }10\text{ inches}$ tall ($70\text{ inches}$) and weighs $110\text{ kg}$ ($\text{ABW}$). He is prescribed amikacin for a serious pseudomonal infection. The clinical protocol requires dosing amikacin at $15\text{ mg/kg}$ based on the appropriate dosing weight.

1. Calculate Ideal Body Weight (IBW):

   $$IBW = 50.0\text{ kg} + 2.3\text{ kg/inch} \cdot (70 - 60) = 50.0 + 2.3 \cdot 10 = 73.0\text{ kg}$$

2. Compare ABW to IBW to check for obesity:

   $$\%\text{ Obesity} = \frac{ABW}{IBW} \cdot 100\% = \frac{110\text{ kg}}{73.0\text{ kg}} \cdot 100\% \approx 150.7\%$$

   Since $150.7\% > 120\%$, the patient is obese. We must use Adjusted Body Weight (AdjBW) for dosing.

3. Calculate Adjusted Body Weight (AdjBW):

   $$AdjBW = IBW + 0.4 \cdot (ABW - IBW) = 73.0 + 0.4 \cdot (110 - 73) = 73.0 + 0.4 \cdot 37 = 73.0 + 14.8 = 87.8\text{ kg}$$

4. Calculate the Amikacin dose:

   $$\text{Dose} = 15\text{ mg/kg} \cdot 87.8\text{ kg} = 1317\text{ mg}$$

   Clinical Recommendation: The dose is rounded to a standard $1300\text{ mg}$ dose.

Example 2: Ratio Strength to Milligrams

How many milligrams of epinephrine are contained in a $5\text{ mL}$ vial of a $1:10,000$ epinephrine solution?

1. Express the ratio strength as a concentration:

   $$1:10,000 = \frac{1\text{ g}}{10,000\text{ mL}} = 0.0001\text{ g/mL}$$

2. Calculate the mass of epinephrine in 5 mL:

   $$\text{Mass (g)} = 0.0001\text{ g/mL} \cdot 5\text{ mL} = 0.0005\text{ g}$$

3. Convert grams to milligrams:

   $$\text{Mass (mg)} = 0.0005\text{ g} \cdot 1000\text{ mg/g} = 0.5\text{ mg}$$