Overview of Modules on Statistical and Mathematical Modeling in the Pharmaceutical Sciences

Nov, 2007Dr. Gary Blau

Overview of Modules on Statistical and

Mathematical Modeling inthe Pharmaceutical Sciences

byGary Blau, Research Professor

E-enterprise CenterDiscovery Park

Purdue University


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.


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


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)


WHAT DOES IT MEAN TO “MODEL” SOMETHING?

Model (Verb)

• To produce a mathematical relationship representation or simulation of a problem


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 experimente.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 questionTherefore, 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)


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


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


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


MATHEMATICAL MODELS

Reverse Engineering

ProcessOptimization

PlantDesign

Process Debottlenecking

“What” Variables and “How” the work together. Questions

“Why” do processes work the way they do. Questions


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.


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


WHEN TO APPLY EXPERIMENTAL DESIGN

• When you know something about the process.

• When you can afford to make at least several runs


PHASE OF AN EXPERIMENTAL PROGRAM

A)EXPERIMENT:1) Statement of the Problem2) Choice of Response or

Dependent Variable3) Selection of Factors

(independent variables) that can be controlled or varied.

4) Determine feasible ranges and choice of levels of these factors.



B: DESIGN1) Number of Experiments2) Sequential Experimentation

3) Randomization/Blocking/Replication4) Postulated Mathematical Model

PROPER DESIGN AVOIDSExcessive data collectionFutile data analysisHigh GI/GO Ratio



ANALYSIS

1) Data Collection and processing

2) Computation of Test Statistics to Validate Model and Estimate

Model Parameters

3) Interpretation of Results


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 ExperimentsFactorial ExperimentationFractional Factorial ExperimentationResponse Surface ModelingProcess Optimization


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


HOW WILL THE MATERIAL BE COVERED

• Three Scenarios

• Lecture Examples– Software tutorials

• End of Section Problems


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


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!!!

Documents

Overview of Modules on Statistical and Mathematical Modeling in the Pharmaceutical Sciences