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Building Generalized HVL and NLC Models
The HVL and NLC models represent two of the simplest pure theoretical treatments of chromatographic peak shapes.
The HVL Theoretical Treatment for Diffusion Models
The HVL represents the simplest diffusion-based model that accounts concentration-based distortion of peak shapes. In such a diffusion model, the a2 peak widths are retention time-independent, provided the a3 chromatographic distortion is scaled for the a1 retention time.
The first four HVL peaks in the plot below have a common a3 distortion value. The second four peaks in the plot use an a3 scaled for retention time.
The HVL model assumes the chromatographic distortion increases linearly with time. When the a3 is inversely scaled for the a1 retention time , exactly the same peak shape is reproduced at the different times.
The a2 widths in the HVL are Gaussian standard deviations, this being the Gaussian upon which the HVL model is based at infinite dilution. A Gaussian peak, like many statistical peaks, uses an (x-a1)/a2 positioning/scaling. The peaks above were combined into a single data set and fitted to the HVL model:
Measured Values
Peak Type Amplitude Center FWHM Asym50 FW Base Asym10
1 HVL 0.40750753 1.67399398 0.28966479 2.77626412 0.59551110 3.35218463
2 HVL 0.38406471 2.11593528 0.30849837 3.26778413 0.63495100 3.99698588
3 HVL 0.36371930 2.56353684 0.32692153 3.75581634 0.67282349 4.63495816
4 HVL 0.34603920 3.01554511 0.34473714 4.23826396 0.70903897 5.26500116
5 HVL 0.30482544 4.39034912 0.39434730 5.64843872 0.80875944 7.10965504
6 HVL 0.30482544 4.89034910 0.39434730 5.64844050 0.80875944 7.10965643
7 HVL 0.30482544 5.39034912 0.39434730 5.64843869 0.80875944 7.10965508
8 HVL 0.30482544 5.89034911 0.39434731 5.64843960 0.80875944 7.10965571
Peak Type Area % Area Mean StdDev Skewness Kurtosis
1 HVL 0.12500000 12.5000000 1.77533286 0.13352303 0.67484528 3.00087122
2 HVL 0.12500000 12.5000000 2.23659126 0.14378816 0.69436193 2.95967534
3 HVL 0.12500000 12.5000000 2.70175235 0.15351410 0.70177067 2.92082127
4 HVL 0.12500000 12.5000000 3.16990421 0.16271466 0.70334774 2.88612831
5 HVL 0.12500000 12.5000000 4.58693879 0.18766825 0.69593864 2.80549533
6 HVL 0.12500000 12.5000000 5.08693879 0.18766825 0.69593864 2.80549533
7 HVL 0.12500000 12.5000000 5.58693879 0.18766825 0.69593864 2.80549533
8 HVL 0.12500000 12.4999999 6.08693878 0.18766822 0.69593770 2.80549006
All Total 1.00000000 100.000000
Note that the peaks 5-8 with the a1 scaled a3 are identical except for the a1 location.
The NLC Theoretical Treatment for Kinetic Models
The NLC model represents the simplest kinetics-based model that accounts concentration-based distortion of peak shapes. In such a kinetics model, a constant a2 peak width also broadens with time, but the shapes are not identical when scaled for the a1 retention time. As in the HVL example above, the first four NLC peaks in the plot below have a common a3 distortion value. The second four peaks in the plot use an a3 scaled for retention time.
Measured Values
Peak Type Amplitude Center FWHM Asym50 FW Base Asym10
1 NLC 0.53080784 1.63105674 0.22134683 5.67010362 0.47843385 7.52428700
2 NLC 0.46995069 2.08559275 0.25063219 5.66884003 0.53873042 7.48045546
3 NLC 0.42578687 2.54448238 0.27713322 5.66775894 0.59325556 7.44810477
4 NLC 0.39190275 3.00667254 0.30151937 5.66683285 0.64340477 7.42296716
5 NLC 0.32400320 4.40714762 0.36575132 5.66469994 0.77541533 7.37173059
6 NLC 0.31804191 4.91259212 0.37243663 5.24429838 0.78745266 6.77945146
7 NLC 0.31258453 5.41816972 0.37872156 4.88975847 0.79895143 6.28395551
8 NLC 0.30752884 5.92381511 0.38470175 4.58668597 0.81002875 5.86315077
Peak Type Area % Area Mean StdDev Skewness Kurtosis
1 NLC 0.12500000 12.5000007 1.74826670 0.11239863 0.75559166 2.93940264
2 NLC 0.12500000 12.5000007 2.21746040 0.12638425 0.74921253 2.92458015
3 NLC 0.12500000 12.5000007 2.68960462 0.13903212 0.74451616 2.91374482
4 NLC 0.12500000 12.5000007 3.16398554 0.15066553 0.74087344 2.90538547
5 NLC 0.12500000 12.5000007 4.59655070 0.18129077 0.73346707 2.88851041
6 NLC 0.12500000 12.5000007 5.10001418 0.18351527 0.73386619 2.90403378
7 NLC 0.12500000 12.5000007 5.60353579 0.18561971 0.73382469 2.91886982
8 NLC 0.12499995 12.4999953 6.10708344 0.18763030 0.73327994 2.93276196
All Total 0.99999995 100.000000
The a2 widths in the NLC are Giddings time constants. An NLC is a Giddings model at infinite dilution. The Giddings zero-distortion density (ZDD), unlike the Gaussian ZDD of the HVL, is a kinetic peak where the a2 width is based on first order adsoprtion-desorption kinetics which have a nonlinear relationship with the a3 distortion. Although the Giddings model is based on the first order kinetics of adsorption and desorption, the model is equally applicable to any form of mass transfer which can be described by the differences in first order kinetic rates. In practice, the a2 is a 'lumped parameter' that includes all kinetics in a first order assumption.
Sufficiency of the HVL and NLC Theory
The discovery of a common chromatographic distortion in these two vastly different theoretical treatments of chromatographic peak broadening suggests the only difference in the NLC and HVL models rests in the ZDD (zero distortion density) assumption. The main limitation of the HVL and NLC models rest with the fact the Gaussian and Giddings ZDDs are very simple and inflexible densities with fixed third and fourth moments unlikely to fully model real-world chromatographic data.
The generalized HVL and generalized NLC models are based on the fact these restrictive Gaussian and Giddings density assumptions need not be imposed, that there exist generalized densities which can model both the Gaussian and Giddings densities, as well as a broad range of greater deviations from ideality, simply by allowing a well-designed variation in skew, as in the GenHVL and GenNLC models.
In building PFChrom, we deemed the HVL, the simplest of the diffusion-based models, and the Wade-Thomas NLC, the simplest of the kinetics models, sufficient as the base models upon which to build generalizations using statistical densities. Instead of seeking more elaborate and specialized diffusion or kinetics models, or hybrid models which address each form of broadening, something that has never been done successfully, we simply employ well-known and readily derived statistical densities for the ZDD of generalized models. In the HVL and NLC generalizations, any statistical density can be used, although the models that have been consistently the best performing in real-world fitting happen to be those that can model both the Gaussian of the HVL and the Giddings of the NLC.
By opening up the higher moments in the modeling, it is possible to separate the a3 non-linearity in the diffusion and generate a pure HVL and Gaussian deconvolution in the modeling. In so doing, it is also possible to quantify the intrinsic deviations from the HVL theory in an a4 skew that should be zero if a pure Gaussian ZDD is present in the distorted peak.
Similarly, it is possible to separate the non-first-order kinetics that would otherwise be lumped in the a2 kinetic parameter and realize a pure NLC and Giddings deconvolution in the fitting. Here the deviations from NLC theory are also mapped in this a4 skew parameter which is typically higher than that expected from the Giddings.
We thus derive pure deconvolved HVL and NLC peaks from these generalized chromatographic models, and pure Gaussians and Giddings peaks as representative of pure ideality.
In PFChrom one uses a GenHVL type model to deconvolve pure HVL and Gaussian peaks, and a GenNLC model to deconvolve pure NLC and Giddings shapes. In doing this, one reduces a chromatographic peak to what it would actually look like if these simplest of the diffusion or kinetics models were actually present, or further, to the zero distortion Gaussian or Giddings models, where the peak is deconvolved with no chromatographic distortion, as if the peak was generated at infinite dilution.
We deemed this approach sufficient from the simple pragmatism of realizing fits of single real world chromatographic peaks where the unaccounted variance approaches one part per million (r2=0.999999). We have built a modeling application where the actual errors in fitting analytic peaks are more associated with the inefficiencies of baseline subtraction and the variation in flow and temperature controllers, than with the peak shape. This approach of fitting the higher moments is essential for gradient HPLC and for fitting the unusual preparative shapes that arise from high overload of columns.
Instrumental Non-Idealities
Instruments are imperfect and add their own distortions. This is the well known science of system identification. In PFChrom, data is generally fitted to a model containing an Instrumental Response Function or IRF, or the instrumental distortions are removed in a Fourier deconvolution preprocessing.
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