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GenHVL[E] - [E] Exponentially-Modified Gaussian (EMG)

By inserting the [E] Exponentially-Modified Gaussian (EMG) ZDD for the PDF, CDF, and CDFc in GenHVL template, we produce the GenHVL[E] model:


a0 = Area

a1 = Center (as mean of underlying normal ZDD)

a2 = Width (SD of underlying Gaussian)

a3 = HVL Chromatographic distortion ( -1 > a3 > 1 )

a4 = The EMG half-Gaussian convolution width


Built in model: GenHVL[E]

User-defined peaks and view functions: GenHVL[E](x,a0,a1,a2,a3,a4)

The GenHVL[E] model uses an EMG (exponentially-modified Gaussian) ZDD to produce the asymmetry. Unlike the other single parameter ZDD models that vary the skew or third moment, the GenHVL[E] has a decidedly non-compact decay in the direction of the exponential.


The above plots vary the a4 EMG convolution width from 0.01 to 0.25. Because the GenHVL[E] uses a convolution width, a right-sifted (positive) a4 in the same direction as the a3 chromatographic distortion produces only small differences with tailed shapes. On the other hand, on fronted shapes where this convolution width is in the opposite direction of a3, the differences are vastly greater for the same magnitude of a4. Note that the GenHVL[E] model allows only positive a4 values.

While there is every indication the secondary exponential tailing in a chromatographic peak occurs externally, is modeled by a convolution IRF rather than as a part of the intrinsic ZDD, this model acknowledges the possibility of a theoretical underpinning for an exponential distortion in the ZDD itself, a measurable real-world kinetic delay impacting the ZDD. As suggested by the plots above, the GenHVL[E] a4 parameter is not likely that which can be shared across a chromatogram with both tailed and fronted peaks.

GenHVL[E] Considerations

When a4 approaches 0, the ZDD becomes a Gaussian and the model reduces to the HVL. Using discrete Fourier methods, it is not possible to have a zero a4. There will be a built-in limiting point which assures information in a minimum count of Fourier frequency channels.

This a4 EMG convolution width in the ZDD manages the deviations from the Gaussian ideality assumed in the theoretical infinite dilution HVL. Again, note that small widths in an opposite direction to the chromatographic distortion produce large deviations in shapes. For a retention scale, the a4 may be in the vicinity of 0.01-0.03 for analytic peaks . At these values, this positive a4 will produce an increased tailing on tailed peaks and a positional shift on the fronted peaks.

This may be an instance there a4 cannot be assumed constant (shared) across all peaks in the chromatogram.

The a4 can be an indicator of not only the deviation from this Gaussian ideality, but of the actual kinetic width of this additional distortion in the ZDD. Changes in the a4, in fitting a given standard, may be especially indicative of column health, assuming this model is able to accurately represent the peak. The greater the a4 value, the greater this unwanted intrinsic distortion.

Note that the a4 will be most effectively estimated and fitted when the peaks are skewed with some measure of fronting or tailing. Higher concentrations are very good for this model, assuming that one does not enter into a condition of overload that impacts the quality of the fit.

This model will be least effective in highly dilute samples with a poor S/N ratio since such peaks will generally have much less intrinsic skew.

Since peaks often increase in width with retention time, the a2 will likely be varied (independently fitted) for each peak.

Since peaks often evidence increased tailing with retention time, the a3 will probably be varied (independently fitted) for each peak.

If you are dealing with a small range of time, however, or of you are dealing with overlapping or hidden peaks in a narrow band, a2 and/or a3 can be held constant across the peaks in this band.

If you are addressing gradient peaks, or the overload shapes of preparative chromatography, you will need the GenHVL[Y] model where the fourth moment of the peak is also adjusted.

The GenHVL[E] model is part of the unique content in the product covered by its copyright.

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