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GenNLC[T]

GenNLC[T] - [T] ZDD

The GenNLC[T] model is an asymptotic generalization of the GenNLC[Z] model where a₁ is the median of the asymmetric generalized normal ZDD.

By using the relationships of equivalence between the GenHVL and GenNLC models, we make the following simple substitution into the GenHVL[T] model equation to derive the GenNLC[T] model:

To convert the GenHVL[T] to the GenNLC[T], the a₂ is transformed to a Giddings kinetic time constant, the a₅ is transformed to a Gidding's indexed asymmetry.

a₀ = Area

a₁ = Center (as mean of generalized normal ZDD)

a₂ = Kinetic Width (Giddings time constant of ZDD)

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

a₄ = Student's t tailing, the nu or DOF ( 1 > a₄ > 1,000,000 ) (fourth moment)

a₅ = NLC indexed asymmetry ( -10 > a₅ > 10 ) a₅=0.5 NLC (Giddings), adjusts skew (third moment)

Built in model: GenNLC[T]

User-defined peaks and view functions: GenNLC[T](x,a₀,a₁,a₂,a₃,a₄,a₅)

In this plot, various Student's t a₄ nu or DOF values, which adjust the kurtosis or fourth moment, are shown for a series of fronted and tailed GenNLC[T] shapes at both the NLC and a higher a₅ ZDD asymmetry. The a₅ asymmetries are set at 0.5, the NLC, and at 1.5, a typical value for analytic peaks. An a₄ of infinity is Gaussian. PeakLab uses a practical limit of 1 million for the upper bound of a₄. This model cannot be used to fit the more compact gradient peaks in gradient HPLC. Only an increased fatness of tails can be modeled.

GenNLC[T] Considerations

When a₄ approaches infinity with a₅=0, the ZDD becomes a Gaussian and the model reduces to the HVL.

When a₄ approaches infinity with a₅=0.5, the ZDD becomes a Giddings and the model reduces to the NLC.

The GenNLC[T] model adjusts both the third and fourth moments, but is not likely to have value in fitting gradient HPLC peaks where a compression occurs from the gradient, and the decay of the peak will be more compact than that of a Gaussian (infinity in the a₄ parameter).

We have found this a₄ fourth moment adjustment in the GenNLC[T] model to often be overspecified (statistically insignificant) for analytic peaks. The nu is typically well below the upper limit, perhaps between 50 and 100. You should use the GenNLC[T] model cautiously for fitting analytic peaks. Use the F-statistic of the fit of the GenNLC[T] model against the F-statistic for the GenNLC or GenNLC[Z] models to ensure there is an actual improvement in the modeling. The GenNLC[T] F-statistic will increase in contrast with the GenNLC or GenNLC[Z] model when this adjustment to the fourth moment is statistically beneficial. A high S/N will definitely be needed to even see this benefit.

We have not found this model useful for fitting overload shapes.

The GenNLC[T]'s a₅ asymmetry parameter is indexed to the NLC and thus the absolute peak asymmetry is not independent of the peak's a₁ location. Use the GenHVL[T] if you wish to fit an absolute statistical asymmetry.

This a₅ skew adjustment in the ZDD manages the deviations from the Giddings ideality assumed in the theoretical infinite dilution NLC. This is an asymmetry parameter indexed to the NLC at a₅=0.5. For most IC and non-gradient HPLC peaks, you should expect an a₅ between 1.1 and 2.0 (the deviation from non-ideality is right skewed or further tailed from the Giddings).

In most instances, both a₄ and a₅ can be assumed constant (shared) across all peaks in the chromatogram. It is strongly recommended that a₄ and a₅ be shared across all peaks and only independently fitted with each peak if the parameter significance allows and you find such necessary. In our experience, across a wide range of concentrations, and across peaks ranging from highly fronted to highly tailed, the fitted a₄ and a₅ were very close to constant if the S/N was strong.

The addition of a shared a₄ and shared a₅ parameter to an overall fit can result in orders of magnitude improvement in the goodness of fit.

The a₄ and a₅ are indicators of deviation from this ideality. Changes in the a₄ and/or a₅, in fitting a given standard, may well be indicative of column health. The greater the a₅ value varies from 0.5, the greater the deviation from this Giddings ZDD assumption of the NLC. The lower a₄ happens to be, the more the ZDD tailing is increasing, possibly from a slow 'drizzling' off the column.

Note that the a₅ 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. Accurately fitting the tailing is even more demanding of data quality, especially with this [T] density.

The GenNLC[T]<irf> composite fits are available, the model with a convolution integral describing the instrumental distortions, in order to isolate the intrinsic chromatographic distortion from the IRF instrumental distortion, but this should be done with caution. The data must be of a sufficient S/N and quality to realize two independent deconvolutions within the fitting. For very dilute and noisy samples, you will probably have to remove the IRF prior using independent determinations of the IRF parameters.

The GenNLC[T]<ge> model uses the <ge>IRF, consistently the best of the convolution models as it fits both kinetic and probabilistic instrument distortions. Bear in mind, however, that this fit must extract the kinetic instrumental distortion, the probabilistic instrumental distortion, the a₅ intrinsic skew to the chromatographic distortion, the a₄ Student's t tailing, and the primary a₃ chromatographic distortion (very possibly for for each peak). It is recommended the IRF parameters be determined by fits of a clean standard, and the instrumental distortions removed by deconvolving the known IRF prior to fitting more complex peak data.

Since peaks often slow in kinetic rates with retention time, the a₂ will probably 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 segment, a₂ and/or a₃ can be held constant across the peaks in this band.

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