3.1 Concentrations, Dilutions, Alligations & Reconstitution

3.1 Concentrations, Dilutions, Alligations & Reconstitution

Pharmaceutical calculations form the foundation of safe pharmacy practice and product compounding in Canada. Accuracy in posology, concentration expression, and volumetric compounding protects patients from subtherapeutic treatment failures and toxic overdoses. In community, hospital, and compounding practice environments, pharmacists must fluidly translate between percentage strengths, ratio concentrations, parts per million, and volumetric dilutions.

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Expression of Drug Concentrations in Canadian Pharmacy

Drug concentration defines the quantity of an active pharmaceutical ingredient (API) relative to the total quantity of the pharmaceutical preparation. In Canadian pharmacy practice, concentrations are conventionally categorized into three distinct percentage types based on the physical state of the solute and the solvent:

Ratio Strength

Ratio strength expresses concentration as parts of active drug relative to parts of the total formulation. By international convention:

• For solids in liquids: $1:X = 1\text{ g of solute in } X\text{ mL of solution}$.
• For liquids in liquids: $1:X = 1\text{ mL of solute in } X\text{ mL of solution}$.
• For solids in solids: $1:X = 1\text{ g of solute in } X\text{ g of preparation}$.

A critical patient-safety application in Canadian acute care is the posology of epinephrine (adrenaline), where decimal or ratio misinterpretations can cause fatal cardiac arrhythmias or cerebral hemorrhage:

• Epinephrine $1:1,000\text{ w/v}$: $\frac{1\text{ g}}{1,000\text{ mL}} = \frac{1,000\text{ mg}}{1,000\text{ mL}} = 1\text{ mg/mL} = 0.1\%\text{ w/v}$ Clinical Indication: Intramuscular injection into the anterolateral thigh for acute anaphylaxis.
• Epinephrine $1:10,000\text{ w/v}$: $\frac{1\text{ g}}{10,000\text{ mL}} = \frac{1,000\text{ mg}}{10,000\text{ mL}} = 0.1\text{ mg/mL} = 0.01\%\text{ w/v}$ Clinical Indication: Intravenous administration during Advanced Cardiac Life Support (ACLS) resuscitation protocols.

Parts Per Million (PPM) & Parts Per Billion (PPB)

For extremely dilute solutions, such as municipal water fluoridation, residual solvent limits, and trace heavy metal impurities, concentration is expressed as parts per million (ppm):

$\text{ppm} = \frac{\text{Parts of Solute}}{1,000,000\text{ Parts of Total Preparation}}$

In dilute aqueous systems where the specific gravity approximates $1.0\text{ g/mL}$ ($1\text{ L} \approx 1,000,000\text{ mg}$):

• $1\text{ ppm} = 1\text{ mg of solute per } 1\text{ L of water} = 1\ \mu\text{g/mL}$.
• In percentage terms: $1\text{ ppm} = \frac{1}{1,000,000} \times 100\% = 0.0001\%\text{ w/v}$.

For example, if Health Canada specifies that drinking water contains $0.7\text{ ppm}$ of fluoride ion, each litre of water contains $0.7\text{ mg}$ of fluoride, which equals $0.00007\%\text{ w/v}$.

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Dilution and Concentration Principles: The $C_1 V_1 = C_2 V_2$ Relationship

When diluting a concentrated stock solution with an inactive vehicle (such as sterile water for injection or petrolatum) or concentrating a preparation by vehicle evaporation, the absolute mass of the active pharmaceutical ingredient remains constant. This conservation of mass yields the universal inverse proportionality equation:

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

Where $C_1$ and $C_2$ represent initial and final concentrations, and $V_1$ and $V_2$ (or $Q_1$ and $Q_2$) represent initial and final volumes or weights.

Step-by-Step Worked Dilution Example

Clinical Scenario: A hospital pharmacist receives a physician order to compound $500\text{ mL}$ of a $0.05\%\text{ w/v}$ chlorhexidine gluconate topical wound rinse. The pharmacy stock inventory contains only concentrated $4.0\%\text{ w/v}$ chlorhexidine gluconate surgical scrub.

1. Identify the Knowns and Unknowns:

   • Stock concentration ($C_1$) = $4.0\%\text{ w/v}$
   • Stock volume needed ($V_1$) = $?\text{ mL}$
   • Desired target concentration ($C_2$) = $0.05\%\text{ w/v}$
   • Desired target volume ($V_2$) = $500\text{ mL}$

2. Set up the Equation: $(4.0\%) \times V_1 = (0.05\%) \times 500\text{ mL}$

3. Solve for $V_1$: $V_1 = \frac{0.05 \times 500\text{ mL}}{4.0} = \frac{25}{4.0} = 6.25\text{ mL}$

4. Determine the Required Diluent Volume: $V_{\text{diluent}} = V_{\text{final}} - V_1 = 500\text{ mL} - 6.25\text{ mL} = 493.75\text{ mL}$

5. Compounding Procedure: Accurately measure $6.25\text{ mL}$ of the $4.0\%$ stock solution using a calibrated syringe, transfer to a volumetric cylinder, and bring to a total volume of $500\text{ mL}$ with sterile water for irrigation.

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Alligation Methods: Alternate and Medial Compounding Math

When combining two or more preparations of differing strengths to obtain an intermediate strength, simple linear dilution formulas cannot be applied directly if both components contain active ingredient. Pharmacists employ alligation alternate to calculate the relative proportions of ingredients, and alligation medial to determine the resulting strength of a mixture.

Alligation Alternate (Grid Method)

Alligation alternate establishes the relative parts of a higher-strength component and a lower-strength component needed to achieve an intermediate target strength:

1. Place the desired target strength in the center of the grid.
2. Place the higher available strength at the top-left and the lower available strength (or pure vehicle at $0\%$) at the bottom-left.
3. Subtract diagonally across the grid (always taking the absolute difference):
   • Parts of Higher Strength $= |\text{Desired Strength} - \text{Lower Strength}|$
   • Parts of Lower Strength $= |\text{Higher Strength} - \text{Desired Strength}|$
4. Sum the parts to obtain the total parts of the final formulation.
5. Convert the proportional parts to actual weights or volumes.

 Higher Strength (H)                  Parts of Higher = |Desired - Lower|
                        Desired (D)
 Lower Strength (L)                   Parts of Lower  = |Higher - Desired|
 ------------------------------------------------------------------------
                                      Total Parts     = Sum of Parts

Step-by-Step Worked Alligation Alternate Example

Clinical Scenario: A dermatologist prescribes $120\text{ g}$ of a $1.5\%\text{ w/w}$ hydrocortisone cream. The pharmacy has in stock $2.5\%\text{ w/w}$ hydrocortisone cream and $0.5\%\text{ w/w}$ hydrocortisone cream. How many grams of each cream must be blended?

1. Set Up the Alligation Grid:

   • Higher strength: $2.5\%$
   • Lower strength: $0.5\%$
   • Desired strength: $1.5\%$

2. Diagonal Subtraction:

   • Parts of $2.5\%$ cream $= |1.5 - 0.5| = 1.0\text{ part}$
   • Parts of $0.5\%$ cream $= |2.5 - 1.5| = 1.0\text{ part}$
   • Total parts $= 1.0 + 1.0 = 2.0\text{ parts}$

3. Calculate Component Proportions and Weights: $\text{Weight of } 2.5\%\text{ cream} = \frac{1.0\text{ part}}{2.0\text{ parts}} \times 120\text{ g} = 60\text{ g}$ $\text{Weight of } 0.5\%\text{ cream} = \frac{1.0\text{ part}}{2.0\text{ parts}} \times 120\text{ g} = 60\text{ g}$

4. Proof / Verification:

   • Drug from $2.5\%$ cream: $60\text{ g} \times 0.025 = 1.5\text{ g}$
   • Drug from $0.5\%$ cream: $60\text{ g} \times 0.005 = 0.3\text{ g}$
   • Total active drug: $1.5\text{ g} + 0.3\text{ g} = 1.8\text{ g}$
   • Final concentration: $\frac{1.8\text{ g}}{120\text{ g}} \times 100\% = 1.5\%\text{ w/w}$. The calculation is confirmed.

Alligation Medial (Weighted Average)

Alligation medial calculates the final concentration obtained when blending known quantities of preparations with known strengths. It represents a weighted arithmetic mean:

$\text{Final Concentration (\%)} = \frac{\sum (\text{Quantity}_i \times \text{Concentration}_i)}{\sum \text{Quantity}_i}$

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Dry Powder Reconstitution and Volume Displacement

Many pharmaceutical anti-infective suspensions (e.g., amoxicillin, amoxicillin/clavulanate, cephalexin, azithromycin) are manufactured as dry powder mixtures because of aqueous instability. When reconstituted with purified water, the dry powder dissolves and occupies physical space, contributing to the total final volume. This is termed the powder volume or displacement volume ($V_{\text{powder}}$):

$V_{\text{final}} = V_{\text{diluent}} + V_{\text{powder}}$ $V_{\text{powder}} = V_{\text{final}} - V_{\text{diluent}}$

Failure to account for powder volume displacement when modifying commercial suspension concentrations leads to severe dosing errors.

Step-by-Step Worked Reconstitution Problem

Clinical Scenario: A community pharmacist receives a prescription for amoxicillin oral suspension for an infant requiring fluid restriction. The physician orders a customized concentration of $500\text{ mg}/5\text{ mL}$ ($100\text{ mg/mL}$) using a standard commercial bottle labeled: Amoxicillin Powder for Oral Suspension $250\text{ mg}/5\text{ mL}$, $100\text{ mL}$ when reconstituted. To reconstitute, add $68\text{ mL}$ of water in two portions and shake vigorously.

1. Determine the Total Active Drug in the Bottle: $\text{Total Drug} = 100\text{ mL} \times \frac{250\text{ mg}}{5\text{ mL}} = 5,000\text{ mg}$

2. Calculate the Dry Powder Volume ($V_{\text{powder}}$): $V_{\text{powder}} = V_{\text{standard final}} - V_{\text{labeled diluent}} = 100\text{ mL} - 68\text{ mL} = 32\text{ mL}$

3. Calculate the Desired Final Volume for the Custom Concentration: $\text{Target Final Volume} = \frac{\text{Total Drug (mg)}}{\text{Target Concentration (mg/mL)}} = \frac{5,000\text{ mg}}{100\text{ mg/mL}} = 50\text{ mL}$

4. Calculate the Diluent Volume to Add: $V_{\text{water to add}} = V_{\text{target final}} - V_{\text{powder}} = 50\text{ mL} - 32\text{ mL} = 18\text{ mL}$

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Compounding Pitfalls & Verification Best Practices

To ensure compounding accuracy and score maximally on calculations sections of entry-to-practice exams, candidates must recognize standard traps:

• Unit Discordance: Never mix milligrams and grams, or millilitres and litres, without explicit conversion factors.
• Specific Gravity Omission in % w/w to % w/v Conversions: For non-aqueous liquids or dense syrups (e.g., glycerin with specific gravity $1.26\text{ g/mL}$), $100\text{ g}$ does not equal $100\text{ mL}$. Use $\text{Volume} = \frac{\text{Weight}}{\text{Specific Gravity}}$.
• Non-Additive Volumes: Mixing concentrated alcohol with water causes volume contraction due to intermolecular hydrogen bonding. In compounding $70\%\text{ v/v}$ ethanol, pharmacists add water quantum sufficiat (q.s.) to the calibration mark in a volumetric flask rather than measuring separate additive volumes.