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Chromatographic Peaks Overview

Overview of Chromatographic Models in PeakLab

The main delta-input/density
chromatographic models in PeakLab are a convolution of a chromatographic peak model and an instrument
response function:
** PeakFn** Ä
**IRF
**where

Sys = System-Dependent Delay

Instr = Fixed Instrument Delay

For the GenHVL
and GenNLC, **PeakFn**
is the common
chromatographic distortion applied to the GenNorm[m]
**ZDD **zero distortion (infinite dilution) density, a generalized
normal with its mean as a_{1}.

For the Gen2HVL
and Gen2NLC,
**PeakFn** is the common
chromatographic distortion applied to the GenError[m]
**ZDD **zero distortion (infinite dilution) density, a generalized
error model with its mean as a_{1}.

In a generalized model, a ZDD must be implemented as the PDF, the probability density function, and at least one of the CDF and CDFc, the cumulative distribution function and its complement.

For the <ge> two-component **IRF**, **Sys**
is the half-Gaussian probabilistic delay 'g' and **Instr** is
'e' first order exponential component

For the <e2> two-component **IRF**, **Sys**
is a first order kinetic or exponential delay and **Instr**
is the other of the two 'e' components

For the <pe> two-component **IRF**, **Sys**
is a 1.5 order dC/dt=-kC*sqrt(C) delay 'p' and **Instr** is
'e' first order exponential component

There are fixed delays in the solute transport and detection which distort (convolve) the pure chromatographic peak shape. We refer to these system invariant distortions as the "instrumental" component of a two-component instrument response function (IRF). There are also variable system-dependent delays in the solute transport which distort (convolve) the pure chromatographic peak shape. These are the "system" delays which produce distortions which will vary with system variables such as concentration, temperature, additives, and other variants in the prep. The two component IRFs consist of a fixed exponential component and a system-dependent variable component, which distort the "true" peak with a two-component convolution of delays.

Fitting the Full Chromatographic Model

Especially for the purpose of discovering the IRF parameters by fitting, or for achieving near perfect fits, one can fit the full convolution or IRF-bearing model to the data. In PeakLab this will be a fit to a set of convolution integrals, one for each peak. This is done in the Fourier domain to enable the fast convolutions needed. In such fitting, each peak is individually convolved with its own IRF. Although this allows the IRF parameters to be independently fitted for each peak, you will likely want to share these parameters across all peaks in the data unless you specifically wish to study the system-dependent component of the IRF as a function of retention time.

For a two component IRFs where the parameters are shared across all peaks, this full model fit will add three additional parameters. The considerably longer fitting times of the full IRF convolution models derives from the Fourier domain fitting of integrals, not from the small overhead of adding a handful of additional parameters to the fitting. Even with the Fourier domain processing, and a number of innovations to hasten fitting, the time to fit full convolution integral models will be appreciably greater. Fitting a dozen data sets to a closed form peak model might require a few seconds. In contrast, those same dozen data sets fitted to a full convolution integral model might require a few minutes.

In a higher moment fit, as in the GenHVL or GenNLC models, the additional ZDD zero distortion density parameter additionally adjusting the zero-distortion (infinite dilution) skew is usually also treated as having a common value across all peaks, adding one more parameter to the fit.

Fitting the full chromatographic models can be challenging. There is the intrinsic a_{3} chromatographic
distortion which can be tailed. Almost universally, the real-world ZDD asymmetry will increase the right
skew. A two-component IRF adds two distinct delays which influence tailing. Special fitting strategies
which hold these IRF parameters and ZDD parameters constant while the principal peak parameters are first
estimated, and where each peak is cyclically given its own slice of enhanced focus in the fitting, makes
the fitting of these full convolution models possible. The data will have to be sufficiently sampled and
with a good S/N to make such fits effective.

The IRF and ZDD parameters will dramatically impact the peak shapes, even though most of the fitted parameters will usually be assigned to the primary chromatographic model, four parameters per peak (area, retention, width, distortion). Note that the primary peak parameters are generally not shared in common across peaks unless the fit that is being done is in a narrow retention band where the widths and distortions can be treated as approximately constant.

Deconvolving an IRF and then Fitting the Closed-Form Chromatographic Peak Model

Fitting the Convolution Integral

For a given analysis, where the column and system variables are constant, you can fit standards to derive the known IRF values. It is often possible to determine the IRF from direct convolution integral fits of current standards, or by adding a component to the standard which enables the IRF to be more accurately estimated. Once the IRF has been estimated, an IRF Deconvolution step applies this singular set of IRF parameters to the entirety of the data set(s), irrespective of the number of peaks that may be present in the data. The impact of the IRF is removed by deconvolution prior to fitting the data.

Optimizing the Deconvolution

As an alternative to fitting to determine the IRF, it can be estimated using a genetic algorithm in a deconvolution optimization of the data. This procedure is based on a merit function that is designed to closely capture the optimal deconvolution parameters for the IRF. Using a Fourier domain filter, the output of this step will be a revised data set where the IRF has been deconvolved, ideally with little to no additional noise, and no bias from the Fourier processing. For this genetic optimization to work effectively, there must be baseline resolved peaks.

Fitting the Closed-Form Peak Model

Once an IRF has been determined in either of these approaches, these parameters can be used to deconvolve any data set where this known IRF can be deemed applicable. Once the IRF is removed, the resultant data is fitted in a subsequent non-linear fit to just the closed-form primary chromatographic model absent any IRF. The GenHVL or GenNLC closed-form models are fitted in the time-domain very quickly, and in most cases, there will only be the one or two additional shared parameters, basically the overall adjustments in the skew and possibly the kurtosis in the infinite dilution zero-distortion density (ZDD) to account the actual higher moments which may not precisely track the fixed theoretical ZDD shapes.

When the IRF is known and deconvolved in a preprocessing step, the fits should be effective ones. If a known IRF changes, you will see this readily. A fit that might be expected to produce a 5-25 ppm unaccounted variance will have a much higher error, in the hundreds or perhaps thousands of ppm. If the predetermined IRF is incorrect, the error in the fit will dramatically increase. In that sense, a fit using a pre-determined IRF will be self-validating. You should be able to detect small changes in your separations in this way. You can then fit a standard with the IRF in a direct model, or prefit the data to the IRF, to see what has changed.

Fitting a Full Higher Moment or Gradient Deconvolution Model

For gradient HPLC where the IRF is largely offset by the gradient, the isocratic convolution models described above cannot be fitted to a high accuracy.

Two Higher Moment Closed-Form Models

A gradient HPLC peak will not, however, be without specific higher moment changes as a consequence of
this gradient existing over the course of that peak's elution. However cleverly the gradients are engineered,
there will likely be a third moment skew to the peaks, and this may be in either direction. More importantly,
the natural decay of the elution will be compressed by the gradient, resulting in a more compact rise
and decay which will be strongly reflected in the fourth moment or kurtosis of the peak. The twice generalized
Gen2HVL and Gen2NLC models offer an additional parameter which will be higher than the 2.0 Gaussian decay.
In actual practice, the typical LC peak, if fully fit for this fourth moment, will typically come in at
a decay power of 1.96-1.99, a decay slightly less than that of the exp(-z^{2}) Gaussian. For a
gradient HPLC peak, this value will typically be much higher, from 2.1-2.2. This additional fourth moment
or kurtosis-related parameter will be a stable and accurate measure of the rate of change within the gradient
that was occurring during the peak's elution.

The Gen2HVL and Gen2NLC models are closed form, easily and accurately fitted, and require very little fitting time. Because the two additional parameters in these generalized models are directly related to the skew and kurtosis, this orthogonality of moments makes these twice generalized models as stable as the basic GenHVL and GenNLC models which only address the third moment of the ZDD. Admitted, one is fitting a compressed shape as opposed to truly fitting the peak with a pure chromatographic model which directly accounts the gradient, but the loss of fit is quite small. In these models the gradient is lumped-in with all other non-idealities and fitted in the higher moment generalizations in the ZDD.

Deconvolution Model Fits

One can directly model the gradient in the opposite of a Fourier convolution fit. Instead of adding an IRF convolution component to the model, one can add a gradient deconvolution component. This is more complex since deconvolution adds noise, and a special automated filter is needed since each iteration can look very different in noise. It is also a process comparable in fitting speed to the isocratic convolution integral models. The benefit is often superior goodness of fits and the recovery of the pure isocratic peak parameters. The limitations rest mostly with the complexity of Fourier domain discrete processing and the need for strong baseline-resolved peaks.

Unwound Gradient Fits

If one can identify a system response function for the gradient, one can convolve that IRF, and in so doing "unwind" the gradient as if the mobile phase had been constant for the duration of a peak's elution. The unwinding decreases noise and allows isocratic models to produce exceptional goodness of fits. The full recovery of the pure chromatographic peak parameters can be realized. The downside is that a measure of the resolution is lost in this decompression.