🧪 Module 4 Part 5: Mixture Design Laboratory

Duration: 30 minutes | Focus: Simplex Designs for Formulation Optimization

🎯 Learning Objectives

By the end of this session, you will be able to:

📚 Part 1: Mixture Design Fundamentals (15 minutes)

🔗 The Mixture Constraint

💡 Key Concept: What Makes Mixture Experiments Special?

Let's think step-by-step about mixture experiments:

  1. Understanding the constraint: In mixture experiments, the components are not independent. The sum of all proportions must equal 1 (or 100%).
  2. Mathematical representation: Σ xᵢ = 1, where xᵢ represents the proportion of component i
  3. Geometric interpretation: The experimental space forms a simplex - a triangle for 3 components, tetrahedron for 4 components
  4. Practical implication: If you increase one component, you must decrease others
Mixture Constraint: X₁ + X₂ + X₃ + ... + Xₖ = 1.0

💊 Pharmaceutical Example: Tablet Coating Solution

Scenario: You're optimizing a tablet coating solution with three solvents:

  • Water (X₁): 0-80%
  • Ethanol (X₂): 10-60%
  • Propylene Glycol (X₃): 10-40%
Step-by-Step Reasoning:
  1. Setup: If X₁ = 0.50 (50% water), then X₂ + X₃ = 0.50
  2. Constraint application: We cannot independently choose both X₂ and X₃
  3. Design implication: Traditional factorial designs don't work here
  4. Solution: Use mixture-specific designs that respect this constraint

📐 Simplex-Lattice Designs

Understanding Simplex-Lattice Design Construction:

  1. Definition: Evenly spaced points on the simplex
  2. Notation: {q, m} lattice where q = number of components, m+1 = number of levels
  3. Point generation: All combinations where each component takes values 0, 1/m, 2/m, ..., 1
  4. Design points: Total points = (q+m-1)! / (m!(q-1)!)

🔧 Interactive Tool: Simplex-Lattice Design Generator

💊 Example: {3,2} Simplex-Lattice Design for Excipient Blend

Scenario: Optimizing a blend of three excipients (A, B, C) for tablet compression

Run Component A Component B Component C Design Point Type
11.00.00.0Pure A
20.01.00.0Pure B
30.00.01.0Pure C
40.50.50.0Binary AB
50.50.00.5Binary AC
60.00.50.5Binary BC
70.330.330.33Centroid

🎯 Simplex-Centroid Designs

Understanding Simplex-Centroid Design Logic:

  1. Concept: Includes all possible centroids of sub-simplexes
  2. Points included: Pure components, binary centroids, ternary centroids, etc.
  3. Advantage: Fewer runs than equivalent lattice design
  4. Trade-off: Less information about intermediate blends
⚠️ Important Note: Simplex-centroid designs are a subset of simplex-lattice designs. The {3,2} lattice shown above is actually a simplex-centroid design!

🚧 Handling Constraints

💊 Real-World Constraint Example: Controlled Release Tablet

Problem: Designing a polymer blend with constraints:

  • HPMC (X₁): 20-60% (provides sustained release)
  • Ethyl Cellulose (X₂): 10-40% (controls permeability)
  • PVP (X₃): 10-50% (improves dissolution)
Step-by-Step Constraint Handling:
  1. Convert to constraint form:
    • 0.20 ≤ X₁ ≤ 0.60
    • 0.10 ≤ X₂ ≤ 0.40
    • 0.10 ≤ X₃ ≤ 0.50
    • X₁ + X₂ + X₃ = 1.00
  2. Check feasibility: Minimum sum = 0.20 + 0.10 + 0.10 = 0.40 (✓ feasible)
  3. Find extreme vertices: Use linear programming to find corner points
  4. Design selection: Choose points within the constrained region

📊 Part 2: Mixture Model Analysis (15 minutes)

🧮 Scheffé Models (Canonical Polynomials)

💡 Why Regular Regression Models Don't Work for Mixtures

Step-by-step reasoning:

  1. Regular model: Y = b₀ + b₁X₁ + b₂X₂ + b₃X₃ + error
  2. Problem: With X₁ + X₂ + X₃ = 1, the columns in the design matrix are perfectly correlated
  3. Mathematical issue: This creates a singular matrix that cannot be inverted
  4. Solution: Use Scheffé models that account for the mixture constraint
Linear Scheffé Model: Y = β₁X₁ + β₂X₂ + β₃X₃

Quadratic Scheffé Model: Y = β₁X₁ + β₂X₂ + β₃X₃ + β₁₂X₁X₂ + β₁₃X₁X₃ + β₂₃X₂X₃

Understanding Scheffé Model Coefficients:

  1. β₁, β₂, β₃: Expected response when component is pure (all others = 0)
  2. β₁₂: Synergistic/antagonistic effect of X₁ and X₂ binary blend
  3. Interpretation: βᵢⱼ > 0 indicates synergy, βᵢⱼ < 0 indicates antagonism

💊 Worked Example: Drug Solubility in Solvent Mixture

Scenario: Predicting drug solubility (mg/mL) in a three-solvent system

Model fitted: Y = 12.5X₁ + 8.2X₂ + 15.7X₃ + 6.3X₁X₂ - 4.1X₁X₃ + 2.8X₂X₃

Step-by-Step Model Interpretation:
  1. Pure solvent effects:
    • Pure X₁: 12.5 mg/mL solubility
    • Pure X₂: 8.2 mg/mL solubility
    • Pure X₃: 15.7 mg/mL solubility (best single solvent)
  2. Binary interaction effects:
    • X₁X₂: +6.3 (synergistic, blend better than individual)
    • X₁X₃: -4.1 (antagonistic, blend worse than expected)
    • X₂X₃: +2.8 (slight synergy)
  3. Prediction example: For X₁=0.3, X₂=0.2, X₃=0.5:
    Y = 12.5(0.3) + 8.2(0.2) + 15.7(0.5) + 6.3(0.3)(0.2) - 4.1(0.3)(0.5) + 2.8(0.2)(0.5)
    Y = 3.75 + 1.64 + 7.85 + 0.378 - 0.615 + 0.28
    Y = 13.29 mg/mL
💻 Excel Implementation: =MMULT(MMULT(TRANSPOSE(X),X)^(-1),MMULT(TRANSPOSE(X),Y))
Where X is the design matrix and Y is the response vector

📈 Graphical Interpretation Tools

🎨 Ternary Plot Creator

The key visualization tool for mixture data. Shows response as contours on the triangular simplex.

📊 Cox Response Trace Plot

Purpose: Shows the effect of changing one component's proportion while keeping the relative proportions of others constant.

Creating a Cox Trace Plot - Step by Step:
  1. Choose reference blend: Starting mixture proportions (e.g., equal proportions)
  2. Select component to vary: Which component will be increased/decreased
  3. Define trace direction: How other components change as the selected one varies
  4. Calculate response: Use Scheffé model to predict response along the trace
  5. Plot results: Component proportion vs. predicted response

💊 Pharmaceutical Applications

Case Study 1: Solvent System Optimization

Objective: Maximize drug solubility in a three-solvent extraction system

  • Components: Water, Ethanol, Propylene Glycol
  • Response: Drug extraction efficiency (%)
  • Constraints: Water ≥ 20% (safety), Ethanol ≤ 60% (regulatory)

Case Study 2: Excipient Blend Optimization

Objective: Optimize tablet excipient blend for flowability and compressibility

  • Components: Microcrystalline cellulose, Lactose, Starch
  • Responses: Flow rate (g/s), Tablet hardness (kp)
  • Approach: Multi-response optimization using desirability function

Case Study 3: Coating Polymer Blend

Objective: Design enteric coating with optimal acid resistance and flexibility

  • Components: HPMCP, CAP, Plasticizer
  • Responses: Acid resistance time (min), Film flexibility (% elongation)
  • Design: Simplex-centroid with replicated center points

🔬 Hands-On Exercise: Mixture Design for Tablet Coating

Exercise: Design and Analyze a Coating Solution Mixture

Scenario: You need to optimize a tablet coating solution with three solvents to achieve maximum coating efficiency while maintaining proper viscosity.

Step 1: Define Your Mixture Problem

Components:

  • Water (X₁): Cost-effective, but may cause instability
  • Ethanol (X₂): Good solvent, but expensive and regulatory constraints
  • Isopropanol (X₃): Moderate properties, good alternative

Constraints:

  • Water: 10-70% (stability and cost considerations)
  • Ethanol: 10-50% (regulatory and cost)
  • Isopropanol: 10-60% (performance requirements)

Response: Coating efficiency (%) - higher is better

Step 2: Generate Your Design Points

Step 3: Simulate Data Collection

Step 4: Fit Scheffé Model

Step 5: Find Optimal Formulation

📝 Summary and Key Takeaways

🎯 Essential Concepts Mastered:

  1. Mixture Constraint Understanding: Components sum to 1, creating dependent relationships
  2. Simplex Geometry: Experimental space forms triangles (3 components) or higher-dimensional simplexes
  3. Design Selection: Simplex-lattice vs simplex-centroid based on information needs
  4. Scheffé Models: Special regression models that handle the mixture constraint
  5. Ternary Plots: Essential visualization tool for mixture optimization

💡 When to Use Mixture Designs

  • Solvent system optimization: Drug extraction, dissolution enhancement
  • Excipient blending: Optimizing filler, binder, disintegrant ratios
  • Coating formulations: Polymer blends for enteric or sustained release
  • API co-crystal formation: Optimizing co-former ratios
  • Emulsion/suspension formulation: Phase ratio optimization

⚠️ Common Pitfalls to Avoid

  • Using regular factorial designs: Ignores the mixture constraint
  • Forgetting constraints: Real formulations often have component limits
  • Over-interpreting pure component effects: May not be realistic in practice
  • Ignoring process variables: Temperature, mixing time may also be important

🧪 Quick Assessment

Question 1: Mixture Constraint

A pharmaceutical scientist is optimizing a three-component buffer system where Component A = 0.3, Component B = 0.4. What must Component C equal?

Step-by-Step Solution:

  1. Apply mixture constraint: X₁ + X₂ + X₃ = 1.0
  2. Substitute known values: 0.3 + 0.4 + X₃ = 1.0
  3. Solve for X₃: X₃ = 1.0 - 0.3 - 0.4 = 0.3
  4. Answer: Component C = 0.3 (30%)

Question 2: Model Interpretation

Given the Scheffé model: Y = 15X₁ + 12X₂ + 18X₃ + 8X₁X₂ - 3X₁X₃ + 5X₂X₃

What is the predicted response for the centroid point (X₁=X₂=X₃=1/3)?

Step-by-Step Calculation:

  1. Substitute values: X₁ = X₂ = X₃ = 1/3
  2. Calculate linear terms: 15(1/3) + 12(1/3) + 18(1/3) = 5 + 4 + 6 = 15
  3. Calculate interaction terms: 8(1/3)(1/3) - 3(1/3)(1/3) + 5(1/3)(1/3) = 8/9 - 3/9 + 5/9 = 10/9 ≈ 1.11
  4. Total response: Y = 15 + 1.11 = 16.11

Question 3: Design Selection

You need to screen 4 components with limited budget. Which design would be most efficient?

Options:

  • A) {4,3} Simplex-lattice: 20 runs
  • B) {4,2} Simplex-lattice: 10 runs
  • C) Simplex-centroid: 15 runs

Answer: B) {4,2} Simplex-lattice provides quadratic model capability with minimum runs for 4 components.

💼 Practical Implementation Tips

🔧 Excel Implementation Guide

Setting up Mixture Design in Excel:

  1. Create design matrix: Ensure all rows sum to 1.0
  2. Check constraints: Use conditional formatting to highlight violations
  3. Calculate responses: Use MMULT for Scheffé model fitting
  4. Validate model: Calculate R² and residual plots

Key Excel Formulas:

  • Row sum check: =SUM(A2:C2) (should equal 1)
  • Constraint check: =IF(AND(A2>=0.1,A2<=0.7,B2>=0.1,B2<=0.5),"OK","Violation")
  • Scheffé prediction: =B1*A2+B2*B2+B3*C2+B4*A2*B2+B5*A2*C2+B6*B2*C2

📊 Visualization Best Practices

  • Always use ternary plots: Standard for mixture data presentation
  • Include contour lines: Show response levels clearly
  • Mark constraint boundaries: Show feasible region limits
  • Highlight optimal regions: Use color coding for desirable zones
  • Add experimental points: Show actual data locations

🎯 Optimization Strategy

Systematic Approach to Mixture Optimization:

  1. Define constraints clearly: Practical, regulatory, and cost limits
  2. Start with screening: Use simplex-centroid for initial assessment
  3. Identify promising regions: Based on initial results
  4. Add augmentation points: In regions of interest
  5. Fit appropriate model: Linear vs quadratic based on data
  6. Validate predictions: Run confirmation experiments
← Part 4: Response Surface Methodology Part 6: Formulation Optimization →