Fourier Deconvolution and IRF Estimation

PFChrom’s Complete Solution

In order to model the chromatographic separation, instrument and measurement effects must be accounted in some manner.

A complete solution consists first of a comprehensive and effective fitting of the chromatographic model with IRF, the full convolution integral. When such fits are possible, this will maximize the accuracy, but at a computational cost where the fitting time for a large data set can require up to several minutes, depending on the power of your machine and the number of peaks being fitted within the data.

PFChrom also offers the option to independently estimate the IRF, the instrument response function, for a given instrument, column, and prep, using either nonlinear fitting or Fourier procedures, typically with well-behaved standards. Once the IRF has been accurately estimated, you can preprocess all data sets for which this IRF is applicable in the Fourier domain prior to fitting, even the messiest of data sets where peaks are not baseline-resolved. You can then fit up to 25 preprocessed data sets in a single step. By first removing all instrument effects, the fitting will involve only closed-form models and will be exceptionally fast, on the order of a second or so for each data set fitted.

Fourier Estimates of IRF Parameters

PFChrom uses its own specialized genetic algorithm for estimating the IRF parameters directly from a standard. The algorithm can be quite accurate, and further it is both simple and exceedingly fast.

In the above Fourier deconvolution, a two-component exponential IRF is estimated by PFChrom’s genetic algorithm (GA) and removed in a single step procedure. In this example, the GA beautifully manages four very different concentrations and shapes and only about 1 second was required to estimate and process all four of these data sets. The blue curves are the eluted peaks, the yellow the deconvolutions with the estimated IRF removed.

Fitted Estimates of IRF Parameters

A more rigorous and accurate IRF estimation involves fitting an appropriate standard to determine an average IRF for a given instrument, column, and prep. This will involve the fitting of the IRF-bearing chromatographic models, the convolution integrals.

In the above nonlinear fits, the three-parameter IRF consisting of the area-weighted sum of two exponentials, is estimated by directly fitting a GenHVL<e2> model to the standards. Here the fits ranged from 2 to 6 ppm error, meaning highly accurate estimates of the IRF parameters were achieved. The fitted IRF parameters from the four different peaks, varying one full order of magnitude in concentration, were nearly identical.

The IRF procedure in PFChrom offers left-sided, right-sided, and symmetric instrument response models, and both convolution and deconvolution with a dozen different IRFs built into the program. The GA algorithm is available for optionally estimating each IRF prior to fitting.

When an IRF has been determined by nonlinear fitting of standards, its parameters can be directly copied from one or more fits which can be averaged or individually selected. A sophisticated Fourier filter can then be optionally used to remove noise introduced by the deconvolution.

The threshold for the Fourier noise filter can be graphically set, or a second derivatve procedure can be used, as the one shown above, to automatically determine the optimum S/N separation.

Since Fourier deconvolution will add some noise to the data, even with a filter, and since in general, averaged IRF parameters will not precisely match a given data set, this preprocessing step of removing an IRF in the Fourier domain will not generate as low an error in the fitting. For those who want the swiftest possible processing, however, this is a methodology which will turn minutes into seconds since fits are nearly immediate when closed-form models can be used.

In the above fits of pre-subtracted IRF data, using estimates of the IRF parameters drawn from the four fits of the IRF-bearing models, the goodness of fit drops to 5-37 ppm error across the four fits. Note that in the worst case, the r² is still 0.99996.

PFChrom thus supports those who have a large amount of production data to process, where speed of processing is important, where an average IRF suffices, and where an absolute optimization of each data set’s specific IRF is not needed. There is also the instance of especially messy data sets with overlapping or hidden peaks where the IRF cannot be accurately fit in this single step using an IRF-bearing convolution model.

For the R&D scientists who want to model peaks to the absolute minimum of error, who have peaks amenable to this single step convolution model fitting, and who have no issue with a peak fit which may require a few minutes of computation, PFChrom offers the maximum of accuracy and the ability to characterize every part of the modeling.

PFChrom does more than accurately quantify the moments of the eluted solute peaks and their specific deconvolutions. You also model the measure of distortion in the instrument, flow path, and detector, and you quantify the non-ideality in the chromatographic model describing the separation.

Further details can be found in the IRF Deconvolution and Fitting tutorial.