Optimization Practice

📈 Part 1: Response Surface Methodology Fundamentals

From Screening to Optimization

Response Surface Methodology (RSM) is the natural next step after factorial screening designs. While screening helps us identify the most important factors, RSM helps us find the optimal levels of those factors.

Step-by-Step Reasoning: Why Do We Need RSM?

Let's think about this step by step:

Factorial designs tell us direction: They show whether increasing a factor is good or bad
But they can't find the peak: The optimum might be between the tested levels
We need to model curvature: Real responses often show curved relationships
RSM provides the tools: Quadratic models can capture peaks, valleys, and saddle points

Pharmaceutical Example: Tablet Dissolution Optimization

Scenario: After screening, we found that polymer concentration (X₁) and mixing time (X₂) are the critical factors affecting dissolution rate. Now we need to find their optimal levels.

Why 2-level factorial isn't enough:

At low polymer (5%) and short mixing (5 min): Dissolution = 65%
At high polymer (15%) and long mixing (15 min): Dissolution = 45%
The question is: Is there a sweet spot in between that gives us >85% dissolution?

🎯 Design Space

The multidimensional region where we can operate while maintaining product quality. RSM helps us map this space visually.

📐 Response Surface

A mathematical model that describes how the response changes across the entire experimental region, including curved relationships.

🎪 Optimization

Finding the factor settings that give the best possible response, whether maximizing, minimizing, or hitting a target.

🏗️ Part 2: Central Composite Design (CCD)

Building a Response Surface Design

Central Composite Design is the most popular choice for response surface studies because it efficiently explores the experimental space while providing excellent model-fitting capabilities.

Step-by-Step Reasoning: CCD Structure

Let's understand how CCD works:

Start with a 2ᵏ factorial core: This gives us the corners of our experimental space
Add center points: These help us estimate pure error and detect curvature
Add star (axial) points: These extend beyond the factorial region to map curvature
Result: A design that can fit a full quadratic model

Pharmaceutical Example: 2-Factor CCD for Tablet Optimization

Factors:

X₁: Polymer concentration (5-15%)
X₂: Mixing time (5-15 minutes)

Run	Design Point	X₁ (Coded)	X₂ (Coded)	Polymer % (Actual)	Mixing Time (Actual)
1	Factorial	-1	-1	7.5	7.5
2	Factorial	+1	-1	12.5	7.5
3	Factorial	-1	+1	7.5	12.5
4	Factorial	+1	+1	12.5	12.5
5	Star	-1.41	0	6.5	10
6	Star	+1.41	0	13.5	10
7	Star	0	-1.41	10	6.5
8	Star	0	+1.41	10	13.5
9	Center	0	0	10	10
10	Center	0	0	10	10
11	Center	0	0	10	10

Why α = 1.41? This makes the design "rotatable" - the prediction variance is the same at all points equidistant from the center. Calculate as: α = (2ᵏ)^(1/4) = (2²)^(1/4) = 4^(1/4) = 1.414

Excel Implementation

Creating coded values:

Coded Value = (Actual Value - Center Value) / (Step Size)

Formula: =(B2-10)/2.5 where B2 contains actual value, 10 is center, 2.5 is step size

📐 Part 3: Quadratic Model Fitting and Analysis

Building the Response Surface Model

The power of RSM lies in fitting a quadratic model that can capture curved relationships between factors and responses.

Quadratic Model Equation:
Y = b₀ + b₁X₁ + b₂X₂ + b₁₁X₁² + b₂₂X₂² + b₁₂X₁X₂ + ε

Where:
• b₀ = Intercept (response at center point)
• b₁, b₂ = Linear effects
• b₁₁, b₂₂ = Quadratic effects (curvature)
• b₁₂ = Interaction effect
• ε = Random error

Step-by-Step Reasoning: Model Fitting Process

Let's work through the modeling process:

Collect response data: Run all experiments and measure dissolution rates
Set up the model matrix: Include X₁, X₂, X₁², X₂², and X₁X₂ columns
Use multiple regression: Fit the model using LINEST or Regression tools
Check model adequacy: Examine R², residual plots, and lack-of-fit tests
Interpret coefficients: Understand what each term means physically

Pharmaceutical Example: Dissolution Model Analysis

Sample Data Results:

Run	X₁	X₂	X₁²	X₂²	X₁X₂	Dissolution (%)
1	-1	-1	1	1	1	65
2	+1	-1	1	1	-1	72
3	-1	+1	1	1	-1	78
4	+1	+1	1	1	1	68
5	-1.41	0	2	0	0	70
6	+1.41	0	2	0	0	58
7	0	-1.41	0	2	0	75
8	0	+1.41	0	2	0	85
9-11	0	0	0	0	0	88, 86, 87

Fitted Model Equation:

Dissolution = 87 + 1.5X₁ + 5.2X₂ - 8.1X₁² - 2.3X₂² - 2.8X₁X₂

Interpretation:

87: Predicted dissolution at center point (10% polymer, 10 min mixing)
+5.2X₂: Increasing mixing time generally improves dissolution
-8.1X₁²: Strong negative curvature for polymer - there's an optimum level
-2.8X₁X₂: Negative interaction - effect of mixing depends on polymer level

Excel Implementation

Using LINEST function:

=LINEST(Y_range, X_matrix_range, TRUE, TRUE)

Where X_matrix includes columns for X₁, X₂, X₁², X₂², X₁X₂

Alternative: Data Analysis ToolPak

Data → Data Analysis → Regression → Select Y and X ranges

🎯 Part 4: Multi-Response Optimization

Finding the Sweet Spot

Real pharmaceutical optimization involves multiple responses that need to be balanced. The Desirability Function approach provides an elegant solution.

Step-by-Step Reasoning: Desirability Function Approach

Let's understand how multi-response optimization works:

Define goals for each response: Maximize, minimize, or target specific values
Transform each response to desirability (0-1): 0 = completely unacceptable, 1 = perfect
Combine individual desirabilities: Overall D = (d₁ × d₂ × ... × dₙ)^(1/n)
Optimize overall desirability: Find factor settings that maximize D

Pharmaceutical Example: Multi-Response Tablet Optimization

Three Critical Responses:

Dissolution: Maximize (target ≥85%)
Hardness: Target 8-12 kp (too soft breaks, too hard doesn't dissolve)
Friability: Minimize (target ≤1%)

Response Models:

Dissolution = 87 + 1.5X₁ + 5.2X₂ - 8.1X₁² - 2.3X₂² - 2.8X₁X₂
Hardness = 9.5 + 2.1X₁ + 0.8X₂ + 1.2X₁² + 0.5X₂² + 0.3X₁X₂
Friability = 0.8 - 0.3X₁ - 0.1X₂ + 0.2X₁² + 0.1X₂² + 0.15X₁X₂

Desirability Transformation Example:

For Dissolution (want ≥85%), if predicted value is 90%:

d₁ = (90-75)/(95-75) = 15/20 = 0.75

Where 75% is minimum acceptable and 95% is ideal target.

🧮 Interactive Optimization Calculator

Response Prediction

Polymer Concentration (X₁):

Mixing Time (X₂):

Enter values and click Calculate

Desirability Calculator

Dissolution Target (%):

Hardness Target (kp):

Overall Desirability: --

Response Surface Visualization

3D Response Surface Plot
(Dissolution vs Polymer Concentration vs Mixing Time)
Peak indicates optimal region

Interpretation: The surface shows a clear peak around X₁=-0.2, X₂=+0.8, indicating optimal conditions are slightly below center polymer concentration with higher mixing time.

✅ Part 5: Validation and Confirmation

Proving Your Optimization Works

No optimization is complete without validation. We must prove that our model predictions are accurate and that the optimal conditions actually work.

Step-by-Step Reasoning: Validation Strategy

How to validate optimization results:

Check model adequacy: R² > 0.80, residuals normally distributed
Run confirmation experiments: Test the predicted optimal conditions
Calculate prediction intervals: Account for model uncertainty
Verify robustness: Test nearby conditions to ensure stable performance

Pharmaceutical Example: Confirmation Experiment

Optimal Conditions from Desirability Function:

X₁ = -0.3 (Polymer: 9.25%)
X₂ = +0.8 (Mixing: 12 minutes)

Model Predictions:

Response	Predicted	95% PI Lower	95% PI Upper	Actual Result	Status
Dissolution (%)	91.2	88.1	94.3	90.8	✅ Confirmed
Hardness (kp)	10.1	9.3	10.9	9.9	✅ Confirmed
Friability (%)	0.65	0.52	0.78	0.71	✅ Confirmed

Conclusion: All responses fall within prediction intervals, confirming model validity. The optimization is successful!

Excel Implementation for Prediction Intervals

Calculating 95% Prediction Interval:

PI = Predicted Value ± t₀.₀₂₅,df × √(MSE × (1 + x'(X'X)⁻¹x))

Simplified approach in Excel:

=FORECAST(new_x, y_values, x_values) ± CONFIDENCE(0.05, STEYX(y,x), COUNT(y))

🏋️ Part 6: Hands-On Practice Exercise

Complete Optimization Study

Your Challenge: You're developing a sustained-release tablet formulation. Use the provided CCD data to optimize the formulation.

Exercise: Sustained-Release Tablet Optimization

Objective: Find optimal levels of hydroxypropyl methylcellulose (HPMC) concentration and tablet compression force to achieve target drug release profile.

Factors:

X₁: HPMC concentration (2-8%)
X₂: Compression force (5-15 kN)

Responses to Optimize:

Release at 2h: Target 25-35% (controlled release start)
Release at 8h: Target 65-75% (sustained release)
Release at 12h: Target >90% (complete release)

Run	X₁ (HPMC %)	X₂ (Force kN)	2h Release (%)	8h Release (%)	12h Release (%)
1	3.5	7.5	42	78	95
2	6.5	7.5	28	65	92
3	3.5	12.5	35	72	94
4	6.5	12.5	22	58	89
5	2.2	10	48	82	96
6	7.8	10	18	52	87
7	5	5.9	38	75	94
8	5	14.1	26	62	91
9	5	10	32	69	93
10	5	10	31	68	92
11	5	10	33	70	94

Your Assignment Tasks

Convert to coded values: Transform actual values to -1, +1 scale
Fit quadratic models: For each of the three responses
Check model adequacy: Calculate R² and examine residuals
Apply desirability function: Define targets and calculate overall desirability
Find optimal conditions: Identify the best factor settings
Validate results: Calculate prediction intervals and assess confidence

Excel Worksheet Setup

Suggested worksheet structure:

Sheet 1: Raw data and coded values
Sheet 2: Model fitting using LINEST
Sheet 3: Response surface predictions
Sheet 4: Desirability calculations
Sheet 5: Optimization summary and validation

🎯 Learning Objectives

From Screening to Optimization

Step-by-Step Reasoning: Why Do We Need RSM?

Pharmaceutical Example: Tablet Dissolution Optimization

🎯 Design Space

📐 Response Surface

🎪 Optimization

Building a Response Surface Design

Step-by-Step Reasoning: CCD Structure

Pharmaceutical Example: 2-Factor CCD for Tablet Optimization

Excel Implementation

Building the Response Surface Model

Step-by-Step Reasoning: Model Fitting Process

Pharmaceutical Example: Dissolution Model Analysis

Excel Implementation

Finding the Sweet Spot

Step-by-Step Reasoning: Desirability Function Approach

Pharmaceutical Example: Multi-Response Tablet Optimization

🧮 Interactive Optimization Calculator

Response Prediction

Desirability Calculator

Response Surface Visualization

Proving Your Optimization Works

Step-by-Step Reasoning: Validation Strategy

Pharmaceutical Example: Confirmation Experiment

Excel Implementation for Prediction Intervals

Complete Optimization Study

Exercise: Sustained-Release Tablet Optimization

Your Assignment Tasks

Excel Worksheet Setup