Utilizing DeltaV to Perform PAT Calculations Real Time

Real-Time Non-Linear Regression

Chromatography Endpoint Detection

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Presenters

Michael Hausladen

Director of Manufacturing Technology, BMS

Paul Brodbeck

Engineering Manager,

Control Associates, LBP

Introduction

Chromatography Project Functional Requirements Chromatography Elution Modeling Prototyping Implementation Results Summary

Chromatography Project

BMS Syracuse Scientist: Mike Hausladen Pilot Plant Chromatography Skid Modification Purpose:

– Scale-up laboratory endpoint detection– Ensure process robustness

• Minimize incorrectly determined collection end-point

– Pilot scale model of full scale production system– Demonstrate capability for full scale production

Focus of this presentation: Robust chromatography elution end-point determination

Chromatography Basics

The basics of bind and elute chromatography:– Modify the conditions of the mobile phase to cause binding

or elution of the product (protein)– Aqueous systems: pH and conductivity

Stationary

Phase

Mobile phase:

1. Flush and prep column

2. Load protein on column

3. Wash

4. Elute protein

5. Clean and sanitize column

6. Store column

Monitor elution for product fraction

• UV adsorbance @ 280nm

• Collect the desired portion of the elution peak

Elution Curve

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mAU

Worse case chromatographygenerated in the lab. Data is not smooth., has multiple peaks.

z

w

h

The whole point – determine h (peak height) real time – to

calculate percent of peak maximum collection end-point

Percent of peak max (h) stop collecting

product

BMS Requirements

Robust endpoint of Elution Determination - real time Peak Maximum Calculation

– Endpoint = Percent of Peak Maximum• Model Predict Peak Max. of absolute optical density.• Percent of Peak Max lookup table of Sialic Acid vs. load material.

3 Models – in DeltaV Controller– Smoothing (1st Order Filter) Model– Linear Regression Algorithm - Polynomial Fit – Non-Linear Model

• Extreme Value Function fit with Real-time Data

– Column Volume (mobile phase volume) vs. UV absorbance Alarming/Auto-Switching

– Limits to ensure that the Peak Maximum is not determined early. – Ensure data is fitting the model with a sufficient level of accuracy. – Limits end Model if the algorithm is not converging.

Focus of this presentation is on the non-linear model real-time fit of the Extreme Value Function

Implemented

3 Models– Smoothing Model– Polynomial Fit – Non-Linear Model

1 Algorithm– Newton-Raphson

System Integrator Process

Select Model Equation Basic Curve Fitting

– Least Squares Error (LSE)

Select Numerical Method Algorithm– Gradient Descent– Newton Raphson– Levenberg-Marquardt

Machine Learning Parallel Grey Box Modeling Math

– Linear Algebra– Vector Based Programming Languages (MATLAB)– Solving of Partial Differential Equations

Research – Internet Literature– Example Programs– Textbook – Numerical Methods by Dahlquest &

Bjorck

Program Construction– Flowchart

DeltaV CALC Block Limitation

– 2000 lines per scan

– 64 If-Then loops per block

– 256 field arrays

– Dynamic Reference delays

Excel Solver

– Troubleshooting

– Math Checking

– Convergence Issues

Switch Gradient to Newton

– Bad Hardware

– Convergence Issues

Final Program

Model Statistics

IQ/OQ

Levenberg-Marquart Future

Numerical Integration

Non-linear Models

Controller vs. Application Station

Other Approaches– synTQ, MATLAB, SoftPhase

Summary

Non-Linear Model

Extreme Value Function: is an equation that approximates a chromatography peak

Using a dataset of x’s (volume of mobile phase) and y’s (elution response – UV adsorbance )– Determine h, w, and z that give the best approximation of the peak– From h, determine endpoint

There are many mathematical functions for the representation of chromatographic peaks, extreme value function chosen for simplicity (Journal of Chromatography)

Elution Curve Fitted

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mAU

fit

z

w

h

The whole point – determine h (peak height) real time – to

calculate percent of peak maximum collection end-point

Percent of peak max – stop collecting

product

Numerical Method – Basic Curve Fitting

Minimize the Squares of the Error– Least Squares Error

Curve fitting, minimizing error, finding best solution/ best fit

Analytical solutions for simple fit Iterative numerical solution required for complex

equations– Initial guess required– Convergence satisfied?

Solving in multidimensional space

Numerical Methods – Solving Least Squares

Gradient Descent– Make Initial Guess Xi-1

– New Guess: Xi = Xi-1 – e * F’(Xi-1)– e is tuning constant

• Too low = slow convergence, Too high = unstable

Newton - Raphson– Make Initial Guess Xi-1

– New Guess: Xi = Xi-1 - F’(xi-1)/2*(F’’(xi-1))– Can be unstable with a poor initial guess

Levenberg-Marquardt– Start with Gradient, end with Newton

Iteratively repeat, check for convergence

Sq

uare o

f Erro

r

Math

In order to mathematically determine the slopes of a multi-parameter system need to calculate the partial differentials.– Partials are slopes in n-dimensional space

Linear Algebra– When solving multivariate we need vectors, arrays or

matrices.– Solutions become complex when dividing matrices

Vector Based Programming– Matlab, Python, Octave, …

Partial Differentials – 3D Slopes

Solving the Extreme Value Function

Need to solve for h, z, w.– h = maximum peak height.– w = peak width.– z = retention time of column.

Vectorization Required Challenge here is dividing by a

Matrix Complex Linear Algebra Determinants Easy in Matlab Not so Easy Otherwise

Research

Internet Literature– Methods for Non-Linear Least Squares Problems1

• Nomenclature difficult• Theoretical

– Wikipedia– Dr. Math

• Least Squares Regression for Quadratic Curve Fitting2

– Example Programs• L-A Algorithm by Pradit Mittrapiyanuruck

– MATLAB• L-M Method for Non-Linear Least Squares by Henri Gavin

– MATLAB, extra code… weighting, CHI, R^2, lamda

Textbook – Numerical Methods by Dahlquist & Bjorck – 5.2.1 Numerical Linear Algebra– 5.6 Iterative Methods– 6.9.2 Newton-Raphson’s Method…– 10.5.1 Non-Linear Optimization Problems (Hessian)

Project Challenges

DeltaV Limitations Other Obstacles

No matrix math functions 2000 lines per scan 64 loops per scan Dynamic reference time

delay 256 cell arrays max Hardware problem

Literature for Matlab Solving Diff. Eq.’s Notation theory Matrix Algebra Numerical Methods Least Squared Error

Program Flowchart

Program – DeltaV CALC Block

Math – Solving PDE’s

Solving PDE’s

Prototyping in MS Excel

h = 15

z = 0.24

w = 270

Model Performance

Trust Zone 5 Second scan time 1000 pts each scan Model converges to solution each scan 40% MX Controller Usage @ 5 sec scan 90% MX Controller Usage @ 1 sec scan

IQ/OQ Validation

6 6.5 7 7.5 8 8.5 9 9.5 100

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Raw Data

Excel Prediction

DeltaV NonLinear

DeltaV Linear

Levenberg-Marqaudt Future Enhancement

Prototyped in Matlab Tested in DeltaV Ready to be upgraded Faster, more robust

Implemented Models

All 3 running simultaneously Auto Switch to best Model Non-Linear Model is primary Polynomial Model is secondary Smoothing Model last All 3 models tested within X% of each Monitor all 3 and Alarm if > X% Error

Other Options for Future

MATLAB– OPC communication– Runs in App station

synTQ– $$$– MATLAB or other math package

Soft Phase– Requires C, C++, C#, or VB

Controller vs. Application Station– Controller slightly more robust

• If App Station goes down Model still runs in controller

– Application Station• Run at higher speeds• Reduced controller memory & capacity

Summary DeltaV

Non-linear was successfully implemented to predict elution endpoint

Model updates 1000pt every 5 seconds Model is capable of handling model non-convergence Model passed IQ/OQ

Thank You for Attending!

Enjoy the rest of the conference.