2.3 Alligation Alternate & Concentration Adjustments

Alligation Alternate & Concentration Adjustments

Non-sterile compounding in Canadian community and hospital pharmacies regularly requires technicians to prepare customized drug concentrations that are not commercially available from pharmaceutical manufacturers. To achieve the exact prescribed strength, technicians must mix products of higher and lower strengths, fortify commercial bases with pure active drug powders, or dilute stock concentrates with inert bases.

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Mathematical Principles of Concentration Mixing

All concentration calculations are governed by the Law of Conservation of Mass: the total mass of active drug in the final compounded product must exactly equal the sum of the active drug masses contributed by each individual starting ingredient.

Alligation Medial vs. Alligation Alternate

• Alligation Medial: Used to find the resulting average concentration when known quantities of two or more preparations of differing strengths are blended together.
• Alligation Alternate: Used to determine the relative proportions (parts) of two starting preparations of differing strengths needed to compound a target intermediate strength.

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Setting Up the Alligation Alternate Grid (Tic-Tac-Toe Matrix)

The alligation grid simplifies proportional calculations without requiring complex algebraic systems of equations.

The Grid Layout

   Higher Strength (H)                  |Desired - Lower|  =  Parts of Higher Strength
                               \      /
                                Desired (D)
                               /      \
   Lower Strength (L)                   |Higher - Desired|  =  Parts of Lower Strength
                                       -------------------------------------------------
                                              TOTAL PARTS  =  Sum of Parts

Step-by-Step Rules for Constructing the Matrix

1. Verify Feasibility: The desired concentration ($D$) must lie between the higher concentration ($H$) and the lower concentration ($L$). You cannot prepare a 15% cream from 5% and 10% ingredients.
2. Place Strengths in Column 1:
   • Place the Higher Concentration ($H$) in the top left corner.
   • Place the Lower Concentration ($L$) in the bottom left corner.
3. Place the Desired Strength in the Center:
   • Place the Desired Concentration ($D$) in the center box.
4. Cross-Subtract Diagonally:
   • Subtract the lower concentration from the desired concentration: $|D - L|$. Write this number in the top right corner. This represents the Parts of Higher Strength needed.
   • Subtract the desired concentration from the higher concentration: $|H - D|$. Write this number in the bottom right corner. This represents the Parts of Lower Strength needed.
5. Sum the Total Parts:
   • Add the two values in the right-hand column: $\text{Total Parts} = \text{Parts of } H + \text{Parts of } L$.
6. Calculate Specific Amounts (Grams or Millilitres): $$\text{Amount of Higher Component} = \frac{\text{Parts of } H}{\text{Total Parts}} \times \text{Desired Final Quantity}$$ $$\text{Amount of Lower Component} = \frac{\text{Parts of } L}{\text{Total Parts}} \times \text{Desired Final Quantity}$$

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Working with 100% and 0% Ingredients

A frequent source of exam errors involves knowing what values to enter when an ingredient is pure active drug or an unmedicated base:

• Pure active drug substances (pure chemical powders, crystals, pure USP liquids) contain 100% active medication: enter 100% in the grid.
• Unmedicated bases, diluents, or vehicles (purified water, normal saline, white petrolatum, cold cream, simple syrup) contain no active medication: enter 0% in the grid.

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Single-Stock Dilution: $C_1 \times V_1 = C_2 \times V_2$

When diluting a single concentrated stock solution with an inert solvent (water or normal saline), the alligation grid can be used, or you can apply the standard dilution equation:

$$C_1 \times V_1 = C_2 \times V_2 \quad \text{or} \quad Q_1 \times C_1 = Q_2 \times C_2$$

Where:

• $C_1$ = Initial concentration of stock solution
• $V_1$ (or $Q_1$) = Initial volume (or quantity) of stock solution required
• $C_2$ = Target final concentration desired
• $V_2$ (or $Q_2$) = Target final volume (or quantity) of compounded solution

$$\text{Volume of Stock Needed } (V_1) = \frac{C_2 \times V_2}{C_1}$$ $$\text{Volume of Diluent to Add} = V_2 - V_1$$

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Practical Compounding Scenarios

Scenario 1: Compounding an Intermediate Alcohol Solution

Order: Prepare 500 mL of 70% v/v ethyl alcohol using 95% v/v ethyl alcohol and sterile water for irrigation (0%).

Method A: Dilution Equation

$$C_1 \times V_1 = C_2 \times V_2$$ $$95\% \times V_1 = 70\% \times 500\text{ mL}$$ $$V_1 = \frac{70 \times 500}{95} = 368.42\text{ mL}$$ $$\text{Volume of Water to Add} = 500\text{ mL} - 368.42\text{ mL} = 131.58\text{ mL}$$

Method B: Alligation Alternate Matrix

• $H = 95\%$, $D = 70\%$, $L = 0\%$
• Parts of 95%: $|70 - 0| = 70\text{ parts}$
• Parts of 0%: $|95 - 70| = 25\text{ parts}$
• Total Parts: $70 + 25 = 95\text{ parts}$
• Volume of 95%: $(70 / 95) \times 500\text{ mL} = 368.42\text{ mL}$
• Volume of Water: $(25 / 95) \times 500\text{ mL} = 131.58\text{ mL}$

Both methods yield identical results: 368.42 mL of 95% alcohol and 131.58 mL of sterile water.

Scenario 2: Blending Commercial Creams of Differing Strengths

Order: Prepare 120 g of 1.5% w/w hydrocortisone cream by blending commercial Hydrocortisone 2.5% cream and Hydrocortisone 0.5% cream.

Alligation Setup:

• Higher Strength ($H$): 2.5%
• Desired Strength ($D$): 1.5%
• Lower Strength ($L$): 0.5%

Calculations:

• Parts of 2.5%: $|1.5 - 0.5| = 1.0\text{ part}$
• Parts of 0.5%: $|2.5 - 1.5| = 1.0\text{ part}$
• Total Parts: $1.0 + 1.0 = 2.0\text{ parts}$
• Weight of 2.5% cream: $(1.0 / 2.0) \times 120\text{ g} = 60\text{ g}$
• Weight of 0.5% cream: $(1.0 / 2.0) \times 120\text{ g} = 60\text{ g}$

Result: Blend exactly 60 g of 2.5% cream with 60 g of 0.5% cream.

Scenario 3: Fortifying an Ointment with 100% Active Drug Powder

Order: Prepare 90 g of 5% w/w salicylic acid ointment using pure salicylic acid powder USP (100%) and an existing stock of 2% w/w salicylic acid ointment.

Alligation Setup:

• Higher Strength ($H$): 100% (pure powder)
• Desired Strength ($D$): 5%
• Lower Strength ($L$): 2% (stock ointment)

Cross-Subtraction:

• Parts of 100% powder: $|5 - 2| = 3\text{ parts}$
• Parts of 2% ointment: $|100 - 5| = 95\text{ parts}$
• Total Parts: $3 + 95 = 98\text{ parts}$

Component Quantities:

• Weight of Salicylic Acid Powder: $(3 / 98) \times 90\text{ g} = 2.76\text{ g}$
• Weight of 2% Ointment: $(95 / 98) \times 90\text{ g} = 87.24\text{ g}$

Quality Control Verification:

• Total product weight: $2.76\text{ g} + 87.24\text{ g} = 90.00\text{ g}$ (Matches target).
• Active drug mass in powder: $2.76\text{ g}$.
• Active drug mass in 2% ointment: $87.24\text{ g} \times 0.02 = 1.745\text{ g}$.
• Total active drug: $2.76\text{ g} + 1.745\text{ g} = 4.505\text{ g}$.
• Final concentration: $(4.505\text{ g} / 90\text{ g}) \times 100\% = 5.005\% \approx 5.0\%$. (Verified!).

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Compounding Verification & Quality Assurance

NAPRA's Model Standards for Pharmacy Compounding of Non-sterile Preparations require calculations to be documented on the compounding record and verified before compounding begins:

1. Check Component Sums: Ensure component weights or volumes sum to the desired final quantity.
2. Check Decimal Consistency: Always maintain consistent units (% w/w with grams, % w/v with millilitres).
3. Independent Double-Check: Follow the pharmacy's policy for independent verification of calculations. Best practice is for a second regulated technician or pharmacist to recalculate alligation values before compounding begins.