5.2 Concentration, Alligation, Dilution, & Dosing Calculations

5.2 Concentration, Alligation, Dilution, & Dosing Calculations

Concentration calculations represent the core mathematical competency in nonsterile compounding. Technicians must fluidly convert between various strength expressions (% w/v, % v/v, % w/w, ratio strengths, parts per million), compute dilutions using proportionality equations, solve complex intermediate strength blending problems using alligation alternate and alligation medial methods, and apply specific gravity conversions to translate between mass and volume.

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Expressions of Pharmaceutical Concentration

Concentration measures the amount of solute present in a given quantity of solution or mixture. In nonsterile compounding, four primary concentration expressions are encountered:

1. Percent Weight-in-Volume (% w/v)

Defined as the number of grams of solute per 100 mL of total liquid solution. Used for liquid oral solutions, suspensions, lotions, and topical rinses.

% w/v = (Grams of Solute / Milliliters of Solution) * 100%

• Key Standard: 1% w/v = 1 g of solute in 100 mL of solution (or 10 mg/mL).

2. Percent Volume-in-Volume (% v/v)

Defined as the number of milliliters of liquid solute per 100 mL of total liquid solution. Used for liquid-in-liquid preparations (e.g., alcohol solutions, aromatic waters, volatile oils).

% v/v = (Milliliters of Solute / Milliliters of Solution) * 100%

• Key Standard: 1% v/v = 1 mL of liquid solute in 100 mL of solution.

3. Percent Weight-in-Weight (% w/w)

Defined as the number of grams of solute per 100 g of total mixture. Used for solid and semisolid preparations (creams, ointments, pastes, suppositories, bulk powders).

% w/w = (Grams of Solute / Grams of Total Mixture) * 100%

• Key Standard: 1% w/w = 1 g of solute in 100 g of final ointment/cream base.

4. Ratio Strength (1:X)

Ratio strength expresses concentration as 1 part of solute in X total parts of solution or mixture.

• For w/v solutions: 1 g solute in X mL solution.
• For v/v solutions: 1 mL solute in X mL solution.
• For w/w mixtures: 1 g solute in X g mixture.

Conversion Formulas between Ratio Strength (1:X) and Percentage Concentration:

Percentage Concentration (%) = (1 / Ratio Strength Denominator X) * 100%

Ratio Strength Denominator X = 100 / Percentage Concentration (%)

Worked Example 1: Ratio Strength Conversions

• Task A: Convert a 1:2500 w/v epinephrine solution to percentage concentration (% w/v). % w/v = (1 / 2500) * 100% = 0.04% w/v

• Task B: Express a 0.5% w/v chlorhexidine solution as a ratio strength. X = 100 / 0.5 = 200 Ratio Strength = 1:200 w/v

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Dilution and Concentration Equations (C1 * V1 = C2 * V2)

When a concentrated stock solution or preparation is diluted by adding pure solvent/vehicle, the total volume increases while the total mass (or volume) of active solute remains constant. Consequently, concentration is inversely proportional to volume.

The fundamental dilution equation is:

C1 * V1 = C2 * V2

Where:

• C1 = Initial (stock) concentration
• V1 = Initial volume (or mass) of stock preparation needed
• C2 = Desired (final) concentration
• V2 = Desired (final) volume (or mass) of diluted preparation

NOTE: Concentrations C1 and C2 must be in identical units (% w/v, % w/w, mg/mL, etc.), and volumes V1 and V2 must be in identical units (mL, L, g, etc.).

Worked Example 2: Preparing a Dilute Liquid Preparation

A physician orders 500 mL of a 0.9% w/v sodium chloride solution. The pharmacy only has a stock solution of 23.4% w/v sodium chloride available. How many milliliters of the stock solution and how many milliliters of Purified Water are required?

• Given: C1 = 23.4%, V1 = ?, C2 = 0.9%, V2 = 500 mL

• Step 1: Set up the equation 23.4% * V1 = 0.9% * 500 mL 23.4 * V1 = 450 V1 = 450 / 23.4 = 19.23 mL of 23.4% Stock Solution

• Step 2: Calculate Diluent (Purified Water) Volume Diluent Volume = V2 - V1 = 500 mL - 19.23 mL = 480.77 mL Purified Water

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The Alligation Alternate Method

The Alligation Alternate method is a matrix calculation technique used when a technician must mix two available stock preparations of different strengths (a Higher strength H and a Lower strength L) to produce a desired intermediate Target strength (T).

• Rule: Lower Strength (L) < Target Strength (T) < Higher Strength (H).
• If pure diluent (petrolatum, water) is used as the lower component, its concentration is 0%.
• If pure active drug powder is used as the higher component, its concentration is 100%.

Alligation Grid Matrix Layout

Higher % (H)                      (T - L) Parts of Higher Component
                 Target % (T)
Lower % (L)                       (H - T) Parts of Lower Component
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                                  Total Parts = (T - L) + (H - T) = H - L

Steps for Alligation Alternate Calculations:

1. Place Higher strength (H) at top left, Lower strength (L) at bottom left, and Target strength (T) in the center.
2. Subtract diagonally across the matrix (take absolute differences):
   • Top Right: Target (T) - Lower (L) = Parts of Higher Component needed.
   • Bottom Right: Higher (H) - Target (T) = Parts of Lower Component needed.
3. Sum the parts to get Total Parts.
4. Calculate the required quantity for each component using proportions: Quantity of High Component = (Parts of High / Total Parts) * Total Target Mass or Volume Quantity of Low Component = (Parts of Low / Total Parts) * Total Target Mass or Volume

Worked Example 3: Compounding an Intermediate Ointment Strength

A prescription requires 240 g of 5% Hydrocortisone Ointment. The pharmacy stock contains 10% Hydrocortisone Ointment and 1% Hydrocortisone Ointment. How many grams of each stock ointment are needed?

• Step 1: Set up the Alligation Grid

  • Higher (H) = 10%
  • Lower (L) = 1%
  • Target (T) = 5%

• Step 2: Diagonal Subtraction

  • Parts of 10% (High) = 5 - 1 = 4 parts
  • Parts of 1% (Low) = 10 - 5 = 5 parts
  • Total Parts = 4 + 5 = 9 parts

• Step 3: Calculate Mass of Each Component

  • Mass of 10% Ointment = (4 / 9) * 240 g = 106.67 g
  • Mass of 1% Ointment = (5 / 9) * 240 g = 133.33 g

• Step 4: Check Total Mass and Solute Balance

  • Total Mass: 106.67 g + 133.33 g = 240.00 g
  • Active Drug Check: (106.67 g * 0.10) + (133.33 g * 0.01) = 10.667 g + 1.333 g = 12.0 g
  • Expected Drug in 240 g of 5%: 240 g * 0.05 = 12.0 g (Perfect match).

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The Alligation Medial Method

The Alligation Medial method is used to determine the final average concentration achieved when mixing specified quantities of three or more preparations of known concentrations.

Alligation Medial Formula:

Final % = (Total Mass or Volume of Pure Solute / Total Mass or Volume of Mixture) * 100%

Worked Example 4: Blending Multiple Compounded Stock Bases

A technician combines three batches of triamcinolone acetonide cream:

• Batch A: 100 g of 0.025% cream
• Batch B: 200 g of 0.1% cream
• Batch C: 100 g of 0.5% cream

What is the final percentage concentration (% w/w) of the combined mixture?

• Step 1: Calculate Pure Solute Mass in Each Component

  • Solute in Batch A: 100 g * 0.00025 = 0.025 g
  • Solute in Batch B: 200 g * 0.001 = 0.200 g
  • Solute in Batch C: 100 g * 0.005 = 0.500 g
  • Total Pure Solute = 0.025 g + 0.200 g + 0.500 g = 0.725 g

• Step 2: Calculate Total Mixture Mass

  • Total Mixture Mass = 100 g + 200 g + 100 g = 400 g

• Step 3: Calculate Final Percentage Concentration Final % w/w = (0.725 g / 400 g) * 100% = 0.18125% w/w (or approximately 0.18% w/w).

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Specific Gravity (SG) and Density Calculations

Specific Gravity (SG) is the ratio of the mass of a substance to the mass of an equal volume of pure water at 4°C. Because 1 mL of pure water weighs exactly 1 g at 4°C, specific gravity is a dimensionless quantity that is numerically equal to the substance's density in g/mL.

Specific Gravity (SG) = Mass of Substance (g) / Volume of Substance (mL)

From this relationship, conversion formulas between mass (grams) and volume (milliliters) are derived:

Mass (g) = Volume (mL) * Specific Gravity (SG)

Volume (mL) = Mass (g) / Specific Gravity (SG)

• Key Concept:
  • If SG > 1.0, the liquid is denser than water (1 mL weighs more than 1 g), such as Glycerin (SG = 1.25) or Syrup USP (SG = 1.31).
  • If SG < 1.0, the liquid is less dense than water (1 mL weighs less than 1 g), such as Alcohol 95% (SG = 0.81) or Mineral Oil (SG = 0.88).

Worked Example 5: Converting Mass to Volume using Specific Gravity

A compounding formula calls for 180 g of Glycerin (SG = 1.25). How many milliliters of glycerin should the pharmacy technician measure using a graduated cylinder?

• Calculation: Volume (mL) = Mass (g) / SG Volume = 180 g / 1.25 = 144 mL
• Operational Note: The conversion is mathematically valid, but select gravimetric or volumetric measurement according to the master formulation and equipment accuracy. For a viscous liquid, weighing may reduce drainage and meniscus error.

Worked Example 6: Converting Volume to Mass using Specific Gravity

What is the mass in grams of 250 mL of Syrup USP, which has a specific gravity of 1.31?

• Calculation: Mass (g) = Volume (mL) * SG Mass = 250 mL * 1.31 = 327.5 g