## A Three-Decades Long Journey

#### PeakFit

When I began writing the software product PeakFit in 1990, I sought to create a new tool for researchers to extract additional information from chromatographic, spectroscopic, and statistical peak data. From 1990 through 2002, through four PeakFit releases, my company of that era, AISN Software, maintained the software and sought to add new innovations in modeling and nonlinear fitting.

It is possible that some of you know PeakFit from those times when Jandel Scientific, SPSS Science, or the current owners, Systat Software Inc., sold the software, first for AISN, and then independently, once Cranes Software acquired PeakFit in 2002.

Reserving certain rights to the original AISN technology, I began a next iteration PeakFit for chromatographers starting in 2015. I began with chromatography knowing its peaks were amongst the most challenging to effectively model. Five years of development later, PFChrom is finally ready. PFChrom is the first software product from AIST Software, AISN’s successor.

#### PFChrom’s Objectives

As the R&D work continued, the objectives became much more stringent, and the aims for PFChrom changed from those of PeakFit’s early updates. I began with the intention of finding better core models in the two decades’ time that had passed since I last created an update for PeakFit.

In that aspect of things, I was unsuccessful. Instead, I worked to extend the only two theoretical models that ever managed to accurately fit higher concentration chromatographic shapes, the Haarhoff-VanderLinde (HVL) diffusion model and the Wade-Thomas (NLC) kinetic model. In the course of this work, I came to see these as the two principal models for chromatographic peaks since they represent the simplest models born of well-crafted diffusion and kinetic theory.

#### A Common Chromatographic Distortion Model

In working extensively with these two core chromatographic models, it was discovered that the HVL, a model derived for GC, and the NLC, a model derived for LC, have common chromatographic distortion mathematics. The differences rest in this common distortion being applied to two different zero-distortion densities or ‘ZDDs’. This unexpected discovery meant it was now possible to not only create a fast, closed-form version of the NLC kinetic model, but to go much further, creating statistical generalizations of those densities which not only bridged these two core theoretical models, fitting each effortlessly, but which opened the way for complete third and fourth moment modeling of chromatographic peaks.

By treating the fitting problem as a statistical one, the HVL-NLC core theory has been expanded with a complete set of generalizations which manage any chromatographic shape, including the compression of gradient HPLC peaks, as well the dilation or overload of preparative shapes.

#### IRFs and the Real World

In authoring PFChrom v5, I had to acknowledge that the real-world will not conform even to a beautiful statistical generalization of these theoretical models. It was also necessary to implement convolutions of instrument response functions or ‘IRFs’. These are the different instrumental distortions which occur in real world instruments, flow systems, detectors, and where additional mass transfer resistances are not modeled by the statistical generalizations of the core models. Instead of fitting convolution integrals, however, PFChrom implements a much faster Fourier domain fitting. The new chromatography models you see in PFChrom weren’t even in the realm of possibility those decades ago when PeakFit was first authored.

#### A Next-Generation Modeling

These ZDD and IRF innovations made it possible to take peak fits of real world data from thousands of ppm unaccounted variance to goodness of fits as low as 1 ppm unaccounted variance, with full significance in all fitted parameters. In the tutorials, you will see real-world HPLC gradient peaks fitted to even lower error.

For analytic data, we realize this near-zero error of fitting with the addition of just three common IRF parameters, and one common higher moment ZDD parameter, irrespective of how many peaks are being fitted. PFChrom’s new fitting accuracy is at least three orders of magnitude improvement over the best fits PeakFit ever achieved. While we will not suggest that still better models are not possible, we can readily argue that a few ppm of statistical error isn’t likely to adversely impact analytic modeling, even of higher moments and instrument effects. You may wish to have a look at The Quest for a Universal Chromatographic Fit.

#### PFChrom is Significantly Different from the PeakFit You May Know

As a consequence of these innovations, you will have to start over in certain ways, much as we did, to get accustomed to entirely different approaches and methods for chromatographic peak modeling and analysis. As much as we may have felt PeakFit paved the way, there were certainly gaps in our original understanding. In fairness, modern data are of an exceptional quality. We can answer questions that were impossible those many years ago.

#### No Black-Boxes

I have always believed scientists had a profound disdain for ‘black box’ algorithms. The PFChrom documentation contains a large measure of information associated with the IRF and ZDD concepts you will need to understand in order to make maximum use of the new technology. In the topic Understanding PFChrom’s Models, there is an introductory explanation of the evolution of the software’s generalized chromatographic models.

#### Let the Data Reveal the Model

With the new technology in PFChrom, do we finally have it right? In the original PeakFit, we never assumed one model to be any more appropriate than another. We simply sought to present the existing state of the science as it could be rendered applicable to nonlinear modeling.

For PFChrom, we went a step further, seeking to produce fits that were as close to perfect as we could realize, with an open-mindedness as to how that would be achieved. A lovely principle of statistical modeling is to allow the data to inform as to the potential correctness or incorrectness of models and assumptions. In the course of developing PFChrom, over one hundred different IRFs, and as many ZDDs, were coded and evaluated.

#### The End Product

When all was finished, we had realized a simplicity and elegance we dared not believe possible. While we sought to be free of any form of bias with respect to models and the current state of the science, we did choose to dismiss all published models which failed to balance mass. If we harbored any bias at all, it was a very simple one. A model with 1 ppm error, and all parameters statistically significant, was deemed more ‘correct’ than a model with 100 ppm or 1000 ppm error, or with an over-specification resulting in parameter insignificance.

The once and twice generalized HVL and NLC models we have designated as the core models of PFChrom, and the three principal IRFs we routinely use, have consistently produced close to the expected minimum error based on the data S/N. You need not take our word for it. If we found an IRF or ZDD to be of promise in development, we retained these in the final program. You are welcome to quite swiftly confirm for yourself that which we ourselves discovered in the course of developing this technology.

#### Tutorials

There are more than ten tutorials, each of which illustrate aspects of chromatographic peak fitting which should significantly improve your analysis skills. If you take the time to explore these, you should be light years ahead of where we were in the PeakFit era of this technology. The higher moments, which were always in a nebulous state of estimation, if they were estimated at all, will now tell you a great deal about your chromatographic separations, columns, and instruments.

I sought to add much to the help system and to the tutorials to make this learning process as productive and fruitful as possible. The tutorials are intended to illustrate wholly new chromatographic technology, not simply how to navigate dialogs, menus, graphs, and reports. As you see the nuts and bolts of the software, you will also see a wholly new state of the science.

#### A Special Acknowledgement

PeakFit v1, completed in 1990, was the first product of its kind. Three decades have thus passed since that original work. Over that considerable expanse of time, I have had the honor of working with Dr. James Wade, a skilled scientist and chromatographer who derived the NLC chromatographic model in his thesis in the late 1980s. In the course of my work for PFChrom v5, my respect for the innovative work of Haarhoff, VanderLinde, Thomas, and Wade, grew as these generalized statistical models evolved. Some of the best of the data I was furnished for the R&D of PFChrom was generously given to me by Jim Wade and you will see his data in many of the tutorials. Jim’s assistance in those earliest times of the PeakFit era, and in this recent work with PFChrom, has been invaluable.

Ron Brown

Author of PFChrom and Founder of AIST Software