# A Diagnostic Procedure for Identifying Isotherm Models in Liquid Chromatography

## TL;DR

This paper introduces a diagnostic method to improve isotherm models in liquid chromatography, enhancing accuracy without sacrificing interpretability.

## Contribution

A novel diagnostic procedure using a Lagrange multiplier test to refine isotherm models in chromatography.

## Key findings

- The diagnostic procedure improves model accuracy for complex separations like peptides.
- The method avoids black-box models, preserving physical insight and interpretability.
- Three in-silico case studies validated the effectiveness of the approach.

## Abstract

Liquid chromatography
is a pivotal purification process
widely
used in pharmaceutical development and manufacturing. Efficient optimal
design and control of the process rely heavily on mechanistic models
such as the lumped pore diffusion model (POR) and the Equilibrium
Dispersion Model (EDM), both popular choices owing to their simplicity
and good accuracy for a wide range of applications. However, the choice
of the functional form of the isotherm models, which describe the
component adsorption equilibria, strongly affects the predictions
of the chromatography model. While traditional isotherms perform well
for simple compounds (e.g., small molecules), they often fall short
for more complex separations (e.g., peptides), thus resulting in process-model
mismatch, even following rigorous parameter estimation. As a remedy
to this, recent advances have introduced hybrid models that integrate
data-driven elements to improve the predictive accuracy, although
at the cost of loss of process insight, low interpretability, and
increased complexity. To address the process-model mismatch in chromatography,
we have proposed a model diagnostic procedure, adapted from a diagnostic
framework in kinetic models, based on a Lagrange multiplier test,
to refine isotherm models that initially underfit. The procedure is
demonstrated by three in-silico case studies, showing improved accuracy
against experimental data without having to resort to black-box models,
thus providing models that retain physical insight.