PeakLab v1 Documentation Contents AIST Software Home AIST Software Support

GenHVL[E]

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:

a_{0} = Area

a_{1} = Center (as mean of underlying normal ZDD)

a_{2} = Width (SD of underlying Gaussian)

a_{3} = HVL Chromatographic distortion ( -1 > a_{3}
> 1 )

a_{4} = The EMG half-Gaussian convolution width

Built in model: GenHVL[E]

User-defined peaks and view functions: GenHVL[E](x,a_{0},a_{1},a_{2},a_{3},a_{4})

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 a_{4}
EMG convolution width from 0.01 to 0.25. Because the GenHVL[E] uses a convolution width, a right-sifted
(positive) a_{4} in the same direction as the a_{3}
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 a_{3},
the differences are vastly greater for the same magnitude of a_{4}.
Note that the GenHVL[E] model allows only positive a_{4}
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] a_{4} parameter is not
likely that which can be shared across a chromatogram with both tailed and fronted peaks.

GenHVL[E] Considerations

When a_{4 }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 a_{4}. There will be a built-in limiting point
which assures information in a minimum count of Fourier frequency channels.

This a_{4} 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 a_{4}
may be in the vicinity of 0.01-0.03 for analytic peaks . At these values, this positive a_{4}
will produce an increased tailing on tailed peaks and a positional shift on the fronted peaks.

This may be an instance there a_{4} cannot be assumed constant (shared) across all peaks in the
chromatogram.

The a_{4} 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 a_{4}, 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 a_{4} value, the greater this unwanted intrinsic distortion.

Note that the a_{4} 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 a_{2} will likely be varied (independently
fitted) for each peak.

Since peaks often evidence increased tailing with retention time, the a_{3}
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, a_{2} and/or a_{3} 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.