Capstone Project: Complete Tablet Formulation Optimization

📋 Project Overview

Duration: 45 minutes

🎯 Project Objective

Develop a robust immediate-release tablet formulation for a new API using systematic Design of Experiments approach.

Quality Target Product Profile (QTPP)

Critical Quality Attribute	Target Specification	Justification
Hardness	8-12 Kp	Sufficient mechanical strength for handling and packaging
Friability	< 1%	USP compliance; minimal tablet breakage during transport
Dissolution	> 85% at 30 min	Immediate release profile; bioavailability assurance

📚 Learning Integration

This capstone project integrates all concepts from Module 4:

Fractional factorial screening designs
Response Surface Methodology (RSM)
Statistical analysis and model fitting
Design space definition and optimization
Quality by Design (QbD) principles

Phase 1: Screening

Phase 2: Optimization

Phase 3: Design Space

🔍 Phase 1: Factor Screening

Fractional Factorial Design: 2^(7-3)

Step-by-Step Screening Analysis

1Understanding the Screening Design

Let's think about what this means...

We have 7 factors to evaluate but running a full 2^7 = 128 experiments would be expensive and time-consuming. Instead, we use a fractional factorial design 2^(7-3) = 2^4 = 16 runs.

⚠️ Trade-off: We sacrifice some information (confounding of interactions) to dramatically reduce experimental effort.

2Factors to be Screened

Factor	Low Level (-1)	High Level (+1)	Expected Impact
A: Binder (%)	2.0	5.0	Hardness, Friability
B: Lubricant (%)	0.5	1.5	Hardness, Dissolution
C: Disintegrant (%)	2.0	8.0	Dissolution, Friability
D: Filler Type	Lactose	MCC	All responses
E: Granulation Time (min)	5	15	Hardness, Dissolution
F: Drying Temp (°C)	40	60	Hardness, Friability
G: Compression Force (kN)	10	30	All responses

3Effect Calculation Method

Main Effect Calculation:
Effect = (Average response at +1 level) - (Average response at -1 level)

Excel Formula:
=AVERAGEIF(FactorColumn,1,ResponseColumn) - AVERAGEIF(FactorColumn,-1,ResponseColumn)

Example Calculation for Binder Effect on Hardness:

Step 1: Find average hardness when Binder = +1 (5%)

Average at high level = (9.2 + 8.8 + 10.1 + 9.5 + ... ) / 8 = 9.45 Kp

Step 2: Find average hardness when Binder = -1 (2%)

Average at low level = (7.1 + 6.8 + 7.9 + 7.3 + ... ) / 8 = 7.28 Kp

Step 3: Calculate effect

Binder Effect = 9.45 - 7.28 = 2.17 Kp

Interpretation: Increasing binder from 2% to 5% increases hardness by approximately 2.17 Kp.

📊 Data Analysis Tools

📁 Upload Screening Dataset

Click here to upload the provided screening data file
Expected format: tablet-optimization-screening.xlsx

📈 Pareto Chart Generator

Automatically generates a Pareto chart showing effect magnitudes in descending order.

📊 Half-Normal Plot

Creates a half-normal probability plot to identify significant effects.

📋 Effect Summary Table

Calculates and ranks all main effects for each response.

📋 Phase 1 Deliverable

Required Output: Pareto and Half-Normal plots identifying the 3 most significant factors affecting each response.

Excel Analysis Steps:
1. Calculate effects using AVERAGEIF functions
2. Create Pareto chart: Data → Insert → Column Chart
3. Add secondary axis for cumulative percentage
4. Apply Lenth's method for significance threshold

🎯 Phase 2: Response Surface Optimization

Central Composite Design (CCD)

🔗 Connection to Phase 1

Based on screening results, we've identified the 3 most critical factors. Now we'll optimize these factors using a more detailed experimental design that can model curvature and interactions.

Step-by-Step Optimization Analysis

1Central Composite Design Setup

Let's think about the design structure...

A CCD for 3 factors consists of:

Factorial points: 2³ = 8 corner points
Center points: 6 replicates for pure error estimation
Axial points: 6 star points (±α on each axis)
Total runs: 8 + 6 + 6 = 20 experiments

Rotatability condition:
α = (2^k)^(1/4) = (2³)^(1/4) = 1.682

Coded levels:
Factorial: ±1
Center: 0
Axial: ±1.682

2Critical Factors from Screening

Based on Phase 1 analysis, assume these are our critical factors:

Factor	-α (-1.682)	-1	0	+1	+α (+1.682)
X₁: Binder (%)	1.5	2.0	3.5	5.0	5.5
X₂: Lubricant (%)	0.3	0.5	1.0	1.5	1.7
X₃: Compression Force (kN)	6.6	10	20	30	33.4

3Quadratic Model Fitting

Understanding the quadratic model...

For each response, we fit a second-order polynomial model:

General Quadratic Model:
Y = β₀ + β₁X₁ + β₂X₂ + β₃X₃ + β₁₂X₁X₂ + β₁₃X₁X₃ + β₂₃X₂X₃ + β₁₁X₁² + β₂₂X₂² + β₃₃X₃² + ε

Where:
β₀ = Intercept (center point response)
βᵢ = Linear effects
βᵢⱼ = Two-factor interaction effects
βᵢᵢ = Pure quadratic effects

Example: Hardness Model Development

Step 1: Perform multiple regression analysis

Excel: Data → Data Analysis → Regression

Step 2: Evaluate model significance

ANOVA F-test: H₀: All βᵢ = 0 vs H₁: At least one βᵢ ≠ 0

Step 3: Check individual term significance (p < 0.05)

Step 4: Assess model adequacy (R² > 0.80, lack-of-fit test)

Example Final Model:

Hardness = 9.2 + 2.1X₁ - 0.8X₂ + 1.5X₃ + 0.3X₁X₂ - 0.6X₁² - 0.4X₂² - 0.5X₃²

🔬 RSM Analysis Tools

📁 Upload CCD Dataset

Upload the Central Composite Design results
Expected format: tablet-optimization-ccd.xlsx

📊 ANOVA Table Generator

Performs regression analysis and generates ANOVA tables for each response.

📈 Response Surface Plots

Creates 3D surface plots and 2D contour maps for visualization.

🔍 Model Diagnostics

Residual analysis and model adequacy checking.

📋 Phase 2 Deliverable

Required Output: ANOVA tables, final model equations, and contour plots for all three responses (Hardness, Friability, Dissolution).

Excel Analysis Workflow:
1. Set up regression with coded variables
2. Use Data Analysis Toolpak for regression
3. Create response surface plots using 3D Surface Chart
4. Generate contour plots for each response

🎨 Phase 3: Design Space Definition & Optimization

Desirability Function & QbD Integration

Step-by-Step Design Space Development

1Desirability Function Approach

Let's think about multi-response optimization...

We have three responses with different targets and importance. The desirability function converts each response to a 0-1 scale and combines them.

Individual Desirability Functions:

For Hardness (Target-is-best, 8-12 Kp):
d₁ = 0 if Y₁ < 8
d₁ = (Y₁ - 8)/(10 - 8) if 8 ≤ Y₁ ≤ 10
d₁ = 1 if Y₁ = 10
d₁ = (12 - Y₁)/(12 - 10) if 10 ≤ Y₁ ≤ 12
d₁ = 0 if Y₁ > 12

For Friability (Smaller-is-better, < 1%):
d₂ = 1 if Y₂ ≤ 0.5
d₂ = (1 - Y₂)/(1 - 0.5) if 0.5 < Y₂ ≤ 1
d₂ = 0 if Y₂ > 1

For Dissolution (Larger-is-better, > 85%):
d₃ = 0 if Y₃ < 85
d₃ = (Y₃ - 85)/(95 - 85) if 85 ≤ Y₃ ≤ 95
d₃ = 1 if Y₃ ≥ 95

Example Desirability Calculation:

For a formulation with Hardness = 9.5 Kp, Friability = 0.8%, Dissolution = 92%:

Step 1: Calculate individual desirabilities

d₁ = (12 - 9.5)/(12 - 10) = 2.5/2 = 1.25 → capped at 1.0

d₂ = (1 - 0.8)/(1 - 0.5) = 0.2/0.5 = 0.4

d₃ = (92 - 85)/(95 - 85) = 7/10 = 0.7

Step 2: Calculate overall desirability

D = (d₁ × d₂ × d₃)^(1/3) = (1.0 × 0.4 × 0.7)^(1/3) = (0.28)^(1/3) = 0.65

2Design Space Mapping

Understanding the design space concept...

The design space is the multidimensional combination of input variables that have been demonstrated to provide quality assurance.

🔑 Key QbD Concepts

Proven Acceptable Range (PAR): Factor ranges proven to meet specifications
Normal Operating Range (NOR): Narrower ranges for routine production
Edge of Failure: Boundaries where specifications start to fail
Design Space: Overlap region where all CQAs meet specifications

3Overlay Plot Construction

Step-by-step overlay analysis...

Creating Design Space Boundaries:

Step 1: Define constraint equations from fitted models

Hardness ≥ 8: 9.2 + 2.1X₁ - 0.8X₂ + 1.5X₃ - 0.6X₁² - 0.4X₂² - 0.5X₃² ≥ 8
Hardness ≤ 12: 9.2 + 2.1X₁ - 0.8X₂ + 1.5X₃ - 0.6X₁² - 0.4X₂² - 0.5X₃² ≤ 12
Friability ≤ 1: [Friability model] ≤ 1
Dissolution ≥ 85: [Dissolution model] ≥ 85

Step 2: Create contour plots for each constraint

Step 3: Overlay all acceptable regions

Step 4: Identify intersection (Design Space)

🎨 Design Space Tools

🎯 Desirability Optimizer

Find optimal factor settings using desirability function approach.

🗺️ Design Space Mapper

Create overlay plots showing the design space boundaries.

⚠️ Risk Assessment

Evaluate robustness and edge of failure analysis.

📊 Sensitivity Analysis

Evaluate how changes in factors affect responses within the design space.

Binder (%): 3.5

Lubricant (%): 1.0

Compression Force (kN): 20

Predicted Responses:

📋 Phase 3 Deliverable

Required Output: Complete QbD documentation package including:

Technical Report Contents:

Optimal operating conditions
Design space boundaries
Risk assessment summary
Control strategy recommendations
Robustness evaluation

Excel Workbook Contents:

Complete experimental data
Statistical analysis results
Desirability calculations
Design space mapping
Validation experiments

Final Excel Implementation:
1. Create desirability function using nested IF statements
2. Use Solver Add-in for optimization
3. Generate overlay plots using multiple data series
4. Apply conditional formatting for design space visualization

🎓 Project Completion Summary

🏆 Learning Outcomes Achieved

Technical Skills

Fractional factorial design
Response surface methodology
Statistical model fitting
Multi-response optimization

Analytical Skills

Effect significance testing
Model adequacy checking
Design space definition
Risk assessment

Industry Application

QbD implementation
Regulatory compliance
Process optimization
Control strategy development

📈 Next Steps in Your DoE Journey

This capstone project represents the foundation of your DoE knowledge. Consider exploring:

Advanced response surface designs (D-optimal, I-optimal)
Mixture experiments for formulation optimization
Split-plot designs for hard-to-change factors
Robust parameter design (Taguchi methods)
Process analytical technology (PAT) integration