6.2: Pharmaceutical Calculations: Infusions, Percentages, & Tonicity

Pharmaceutical Calculations: Infusions, Percentages, & Tonicity

Advanced calculations in intravenous therapy, pharmaceutical preparation, and compounding are frequently assessed in the OPRA exam. This section details infusion flow rates, percentage and ratio strengths, dilutions, and tonicity adjustment using the sodium chloride equivalent ($E$-value) method.

1. Infusion Rate Calculations

Intravenous (IV) therapies are administered via electronic pumps (set in mL/hour) or gravity-fed lines (set in drops/minute). To calculate gravity-fed rates, the drip factor of the administration set must be known. The drip factor (expressed in drops/mL) represents the number of drops required to deliver 1 mL of fluid.

Types of Administration Sets

• Macrobore Sets: Typically deliver 10, 15, or 20 drops/mL. Used for standard adult saline or glucose infusions.
• Microbore (Paediatric) Sets: Deliver 60 drops/mL. Used for slow-rate infusions or paediatric therapies.

Formulas

$\text{Flow Rate (mL/hour)} = \frac{\text{Volume (mL)}}{\text{Time (hours)}}$ $\text{Drip Rate (drops/minute)} = \frac{\text{Volume (mL)} \times \text{Drip Factor (drops/mL)}}{\text{Time (minutes)}}$

Clinical Worked Example: Gravity-Fed Saline Infusion

An intravenous infusion of 1 Litre of 0.9% sodium chloride is ordered to run over 8 hours. The IV set has a drip factor of 20 drops/mL.

Step 1: Convert units

• Volume = 1,000 mL
• Time = $8 \times 60 = 480\text{ minutes}$

Step 2: Calculate the drip rate $\text{Drip Rate} = \frac{1,000\text{ mL} \times 20\text{ drops/mL}}{480\text{ minutes}} = \frac{20,000}{480} = 41.67\text{ drops/minute} \approx 42\text{ drops/minute}$

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2. Percentage and Ratio Strengths

Medication strengths can be expressed as percentages or ratio strengths. Misinterpreting these values is a common source of clinical errors.

Percentages

• Weight in Volume (% w/v): Grams of solute in 100 mL of solution. (Common for liquids, e.g., 0.9% w/v NaCl = 0.9 g NaCl per 100 mL).
• Weight in Weight (% w/w): Grams of solute in 100 g of mixture. (Common for semi-solids like ointments or creams, e.g., 1% w/w hydrocortisone = 1 g hydrocortisone in 100 g ointment).
• Volume in Volume (% v/v): Millilitres of solute in 100 mL of solution. (Common for alcohol solutions, e.g., 70% v/v ethanol).

Ratio Strengths (Parts)

Ratio strength is expressed as "1 in X". For liquids, this represents 1 gram of solute in X mL of solution.

• 1 in 1,000 w/v: 1 g in 1,000 mL (equivalent to 1 mg/mL).
• 1 in 10,000 w/v: 1 g in 10,000 mL (equivalent to 0.1 mg/mL).

Safety Alert: Adrenaline (epinephrine) is used in two main ratio strengths in Australian practice. Intramuscular adrenaline for anaphylaxis is 1:1,000 (1 mg/mL). Intravenous adrenaline for cardiac arrest is 1:10,000 (0.1 mg/mL). Injecting the 1:1,000 strength intravenously can result in fatal arrhythmias and severe hypertensive crisis.

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3. Dilutions, Concentrations, and the Alligation Method

When mixing solutions, the amount of active solute remains constant. This is represented by the formula:

$C_1 \times V_1 = C_2 \times V_2$

Where $C$ represents concentration and $V$ represents volume (or mass).

The Alligation Method

The alligation alternate method is a mathematical tool used to calculate the proportions of two different strengths of preparations that must be combined to obtain a desired intermediate strength.

The Alligation Grid

1. Place the higher concentration in the top left corner.
2. Place the lower concentration in the bottom left corner.
3. Place the desired concentration in the centre.
4. Subtract diagonally (always larger value minus smaller value):
   • $\text{Top-Left} - \text{Centre} = \text{Parts of the lower strength}$ (written in the bottom right corner)
   • $\text{Centre} - \text{Bottom-Left} = \text{Parts of the higher strength}$ (written in the top right corner)

Clinical Worked Example: Compounding Coal Tar Ointment

A dermatologist prescribes 200 g of 10% coal tar ointment. The pharmacy stocks a 20% coal tar ointment and pure white soft paraffin (0% coal tar). How much of each ingredient is required?

Step 1: Set up the grid

• High Strength = 20%
• Low Strength = 0%
• Desired Strength = 10%

Step 2: Subtract diagonally

• Parts of 20% ointment (high strength) = $10 - 0 = 10\text{ parts}$
• Parts of white soft paraffin (low strength) = $20 - 10 = 10\text{ parts}$
• Total parts = $10 + 10 = 20\text{ parts}$

Step 3: Calculate the weight of each component $\text{Weight of 20\% ointment} = \frac{10}{20} \times 200\text{ g} = 100\text{ g}$ $\text{Weight of white soft paraffin} = \frac{10}{20} \times 200\text{ g} = 100\text{ g}$

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4. Tonicity and the Sodium Chloride Equivalent ($E$-value) Method

Injectable and ophthalmic solutions must be isotonic with physiological fluids to prevent tissue damage, pain, or red blood cell lysis. Isotonic solutions have the same osmotic pressure as 0.9% w/v sodium chloride (0.9 g of NaCl per 100 mL, or 0.009 g/mL).

The Sodium Chloride Equivalent ($E$-value)

The $E$-value of a drug represents the quantity of sodium chloride (in grams) that has the same osmotic effect as 1 gram of the drug.

Step-by-Step Tonicity Calculation Method

To determine how much sodium chloride must be added to a compounded solution to make it isotonic:

1. Calculate the equivalent mass of NaCl represented by the drug: $\text{Equivalent NaCl (g)} = \text{Weight of drug (g)} \times E\text{-value}$
2. Calculate the total mass of NaCl needed to make the target volume isotonic: $\text{Total NaCl needed (g)} = \text{Volume of solution (mL)} \times 0.009\text{ g/mL}$
3. Calculate the mass of NaCl to be added: $\text{NaCl to be added (g)} = \text{Total NaCl needed} - \text{Equivalent NaCl}$

Clinical Worked Example: Compounding Isotonic Eye Drops

Prepare 30 mL of a 1% w/v ophthalmic solution of drug X ($E$-value = 0.20) isotonic with tears using sodium chloride.

Step 1: Calculate the weight of the drug in the solution $\text{Weight of drug X} = 1\%\text{ of } 30\text{ mL} = 0.01 \times 30 = 0.3\text{ g}$

Step 2: Find the equivalent NaCl mass of the drug $\text{Equivalent NaCl} = 0.3\text{ g} \times 0.20 = 0.06\text{ g}$

Step 3: Find the total NaCl required to make 30 mL of water isotonic $\text{Total NaCl needed} = 30\text{ mL} \times 0.009\text{ g/mL} = 0.27\text{ g}$

Step 4: Determine the NaCl to add $\text{NaCl to add} = 0.27\text{ g} - 0.06\text{ g} = 0.21\text{ g} \text{ (or 210 mg)}$