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Overview of Modules on
Statistical and
Mathematical Modeling in
the Pharmaceutical Sciences
by
Gary Blau, Research Professor
E-enterprise Center
Discovery Park
Purdue University
Dr. Gary Blau
Nov, 2007
COURSE BACKGROUND
• Initial ideas developed for
Pharmaceutical Scientists at the Dow
Chemical Plant in Brindisi, Italy (1975)
• Subsequently evolved into a global
course on Process Optimization
presented to Dow Scientists and
engineers in Europe and North America.
Dr. Gary Blau
Nov, 2007
COURSE BACKGROUND
• Morphed into two courses in the chemical
engineering department (Module 1:
Statistical Model Building and Design of
Experiments for undergraduates) and
(Module 2: Mathematical Model Building for
Process Optimization for Graduate students)
• Reformulated into a Short Course for
Practicing Professionals in the
Pharmaceutical Sciences
Dr. Gary Blau
Nov, 2007
WHAT IS A MODEL?
Model (Noun)
• A miniature representation of something
• A person who serves as a pattern for an artist.
• A type of design of a product (car, airplane)
• A description or analogy used to visualize
something that cannot be observed directly (atom)
• A system of postulates, data and inferences
presented as a mathematical description of an
entity or state of affairs (a system)
Dr. Gary Blau
Nov, 2007
WHAT DOES IT MEAN TO “MODEL”
SOMETHING?
Model (Verb)
• To produce a mathematical
relationship representation or
simulation of a problem
Dr. Gary Blau
Nov, 2007
WHY BUILD A MATHEMATICAL
MODEL?
To Answer Questions:
More specifically, to predict the behavior of the system under
various conditions without running a test or experiment
e.g. Process Operations
Process Design and Scale-Up
Process Optimization
Process Control
One-to-One Relationship
Math Model  Question
Math Models are “built” to answer specific question
Therefore, never use a math model to try to answer questions
not addressed in its construction.
“Remember”: ALL MODELS ARE WRONG BUT SOME ARE
USEFUL (George Box)
Dr. Gary Blau
Nov, 2007
TYPE OF MATHEMATICAL
MODELS
Empirical
Response= Linear Function of Operating Conditions
yield = bo + b1*Temp + b2*Pressure +
b3*Agitation+…..
Semi-Empirical/Mechanistic
lnp = A + B/(C+T) (Vapour Pressure)
Q=UA(LMΔT) (Heat Transfer)
k=koexp(-E/RT) (Arrhenius Temp)
Mechanistic/Fundamental/First Principles
PV=nRT (Gas Laws)
Navier Stokes
Ficks Law
Dr. Gary Blau
Nov, 2007
TYPE OF MATHEMATICAL
MODELS
Mass/Energy Balances across
“units”
Input –Output + Generation=Accumulation
Generation: Many models can frequently be
postulated for this term so that “model building”
is associated with the identification of the proper
form of the model to ANSWER questions
Dr. Gary Blau
Nov, 2007
TAXONOMY OF MATHEMATICAL
MODELS
•
•
•
•
•
•
•
Black versus White
Empirical(Statistical) versus Mechanistic
Linear(Statistical) versus Nonlinear
Small versus Large
Complex versus Simple
Integer/Discrete versus Continuous
Algebraic versus Differential Equations
Dr. Gary Blau
Nov, 2007
MATHEMATICAL MODELS
Process
Optimization
“What” Variables and “How” the
work together. Questions
Process
Debottlenecking
Plant
Design
Reverse Engineering “Why” do processes work the way
they do. Questions
Dr. Gary Blau
Nov, 2007
STEPS IN MODEL BUILDING
1)Define the problem (the question to be answered
by the model)
2)Postulate one or model models that could be used
to solve the problem.
3)Design/Analyse a set of experimental data to
choose between these models and generate
statistically meaningful model parameter
estimates.
4)If the resultant model selected is inadequate
return to step 2.
5) Use the model to solve the problem.
Dr. Gary Blau
Nov, 2007
WHAT IS EXPERIMENTAL DESIGN
• A methodological approach to planning
and conducting experiments which
ensures:
– Experiments will contain the
necessary information content to
choose between models, estimate
model parameters and test model
adequacy
Dr. Gary Blau
Nov, 2007
WHEN TO APPLY EXPERIMENTAL
DESIGN
• When you know something about
the process.
• When you can afford to make at
least several runs
Dr. Gary Blau
Nov, 2007
PHASE OF AN EXPERIMENTAL
PROGRAM
A)EXPERIMENT:
1) Statement of the Problem
2) Choice of Response or Dependent
Variable
3) Selection of Factors (independent
variables) that can be controlled or
varied.
4) Determine feasible ranges and
choice of levels of these factors.
Dr. Gary Blau
Nov, 2007
PHASE OF AN EXPERIMENTAL
PROGRAM
B: DESIGN
1) Number of Experiments
2) Sequential Experimentation
3) Randomization/Blocking/Replication
4) Postulated Mathematical Model
PROPER DESIGN AVOIDS
Excessive data collection
Futile data analysis
High GI/GO Ratio
Dr. Gary Blau
Nov, 2007
PHASE OF AN EXPERIMENTAL
PROGRAM
ANALYSIS
1) Data Collection and processing
2) Computation of Test Statistics to
Validate Model and Estimate
Model Parameters
3) Interpretation of Results
Dr. Gary Blau
Nov, 2007
TOPICS TO BE COVERED IN
THESE MODULES
Module 1:
1) Quantification of Uncertainty in Experimental data and
impact on model analysis using Probability Theory
2) Review of Statistics for building Statistical Models
(Multilinear Regression analysis)
3) Design of Experiments for Building Statistical models
Single factor Experiments
Multifactor Experiments
Factorial Experimentation
Fractional Factorial Experimentation
Response Surface Modeling
Process Optimization
Dr. Gary Blau
Nov, 2007
TOPICS TO BE COVERED IN
THESE MODULES
Module 2
1) When is it necessary to use nonlinear models.
2) Design and Analysis of Experiments with
Nonlinear Models
(1) Liklihood Estimation
-Nonlinear Regression Methods
(2) Bayesian Estimation
-Markov Chain/Monte Carlo Methods
(3) Discrimination of Rival Nonlinear Models
(4) Statistical Properties of Estimators
(5) Properties of Predicted Values
Dr. Gary Blau
Nov, 2007
HOW WILL THE MATERIAL BE
COVERED
• Three Scenarios
• Lecture Examples
– Software tutorials
• End of Section Problems
Dr. Gary Blau
Nov, 2007
HELPFUL HINTS
• Review Probability and Statistics or have a text
available during Module 1 (e.g. Runger and
Montgomery, Applied Statistics and Probability for
Engineers)
• Work all lecture examples using your own version
of the software.
• Work all problems at the end of the lectures.
• Complete Module 1 before starting Module 2.
GOOD LUCK
Dr. Gary Blau
Nov, 2007
WHEN SHOULD YOU NOT APPLY
EXPERIMENTAL DESIGN
• When you are not trying to predict behavior
– Just making a product
– Just a demonstration
• When only a “couple” of runs are to be made
– We will get answer with “just one more” run.
– Can’t afford any more
• When you are not even close to the right operating
region
– Most runs are infeasible
– Your product is just junk
• When you don’t know much about process
– Brand new process
BUT DON”T USE THESE EXCUSES TOO LONG!!!
Dr. Gary Blau
Nov, 2007