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Common Chromatographic Distortion Model

Equivalence

The Wade-Thomas NLC model, while derived for affinity chromatography, and the HVL, designed primarily for GC peak shapes, but widely used in many forms of chromatography, employ common distortion mathematics which can result in identical distorted shapes in chromatographic peak fitting. This discovery of equivalence, that of a common distortion model for chromatographic peak models, is central to the modeling technology in PFChrom v5.

Nonlinear Chromatography and Fronted Distortions

In the HVL and Wade-Thomas NLC models, a peak dramatically changes shape with concentration. In
the above plot with a unit area normalization for all peaks, concentrations are shown for a range to two
orders of magnitude. A intrinsically fronted peak will be fronted (have a negative a_{3 }distortion
value) at all concentrations, and further, we have found a_{3} to be linear with concentration
when the instrumental effects are accounted. That is, the greater the magnitude of the -a_{3}
value, the greater the fronting. In the above plot, we show the approximate S/N thresholds. The higher
concentrations produce significantly better S/N, but the fronting also sharply increases. Note also that
the two samples at the lowest concentrations appear tailed, even though the peaks are intrinsically fronted
in the chromatographic separation. The extracolumn or instrumental effects produce the observed tailing
and are more discernible than the intrinsic fronting.

Nonlinear Chromatography and Tailed Distortions

The same concentration dependency occurs with intrinsically tailed peaks. Again, the plot
is made with a unit area normalization for all of the tailed peaks, and the concentrations are again shown
for a range to two orders of magnitude. A intrinsically tailed peak will be tailed (have a positive a_{3
}distortion value) at all concentrations. And here as well we found a_{3} to be linear with
concentration when all extracolumn effects are properly fitted. The greater the a_{3} value, the
greater the tailing. For this example, the S/N thresholds are somewhat lower. The higher concentrations
produce significantly better S/N, but the tailing also increases. Note also that the two samples at the
lowest concentrations appear Gaussian, even though the peaks are intrinsically tailed. The extracolumn
or instrumental tailing combines with the intrinsic tailing and both are little visible when the
broadening is high at long elution times.

It is the concentration dependency shown above that is treated identically in the distortion mathematics of the HVL and Wade-Thomas NLC models. The models use a different parameterization, but when the models are templated and each is given a common zero-concentration (infinite dilution) density assumption, the distortion mathematics produce identical shapes.

The Generalized HVL and NLC Model Templates

In the ZDD concepts topic, we derived the HVL generalized template:

The HVL model is reproduced by substituting the Gaussian or normal PDF and CDF for the Density and Cumulative in the formula.

To simplify matters, we will revise the NLC
Template to use the cumulative of the density rather than the reverse cumulative or complement of
the CDF. After simplification, the NLC template looks surprisingly close to the HVL template. The only
difference is that a_{3} is scaled by a_{1}/a_{2} in the HVL template.

The NLC model is reproduced by substituting the Giddings PDF and CDF for the Density and Cumulative in the formula.

For both templates the parameters have the same correspondence with moments:

a_{0} = area of the overall model

a_{1} = the center value used in the density and cumulatives

a_{2} = the width value used in the density and cumulatives

a_{3} = the HVL or NLC chromatographic distortion parameter (fronted for a_{3}<0,
tailed for a_{3}>0)

Density = PDF for the specified ZDD (zero distortion density), area=1

Cumulative = CDF for the specified ZDD, magnitude=1

RevCumulative = CDF complement for the ZDD, magnitude=1

Note that a_{1} and a_{2} derive their meaning in the Density and Cumulative
elements. The HVL Template's a_{1}/a_{2} adjustment of the distortion linearizes the a_{3}
distortion with concentration for a Gaussian-based zero-distortion density. The NLC Template, when using
a kinetic peak such as the Giddings, requires no adjustment for an a_{3} distortion that is linear
with concentration.

Generalized HVL and Generalized NLC Template Conversion

Conversion between the templates is straightforward. The only difference is this center/width scale factor:

A Common Distortion Operator for Chromatographic Data

PFChrom's generalized models use statistical generalizations which reduce to the normal density through
one or two levels of generalization. Since there exist a wealth of well-researched statistical generalizations
of densities, and even the simplest of the kinetic models using this a_{3} common distortion (the
NLC with the Giddings ZDD) is at the bounds of computational viability, we use the HVL
Template and statistical generalizations to manage the third and fourth moments of chromatographic
peaks. We then use an equivalence
relationship for the generalized NLC models.

When the GenHVL
model fits perfect NLC data, its a_{3} distortion values will be twice that of the NLC a_{3}
distortion. In order to have a truly common distortion operator, we adjusted the GenNLC
a_{3} distortion to exactly match the GenHVL
a_{3} distortion. You will see the same a_{3} distortion value whether you are looking
at a chromatographic peak as a scaled diffusion-type model (GenHVL),
or as a first order kinetics peak (GenNLC).

The GenNLC's a_{3} will thus be twice the value of the NLC's a_{3} and identical to the
GenHVL's a_{3} distortion value.

The different aspects of the equivalence between the diffusion-type GenHVL and kinetic-type GenNLC models will become clearer when using the Equivalent Parameters section of the Numeric Summary when fitting either a GenHVL or GenNLC model.

The NLC and HVL a_{3} Distortion

Even though the GenHVL and GenNLC, the generalized models using a generalized normal can each fit both
the HVL and NLC shapes, there are issues which must be considered when peaks eluting at different times
share this a_{3} common distortion parameter. This is not normally done, but may be necessary
when fitting a small band of highly overlapped peaks. It will definitely be necessary when there are hidden
peaks.

This choice to implement statistical ZDDs to model the kinetics impacts the behavior of a_{3}
across elution time. To fully understand this common chromatographic distortion, we will create a sequence
of HVL and NLC peaks.

The NLC and HVL models cannot replicate the same chromatographic shapes. The NLC assumes a Giddings ZDD,
a kinetic peak which broadens with location at constant a_{2} values, even with no a_{3}
chromatographic distortion. The HVL assumes a Gaussian ZDD, a scaled statistical peak where a given a_{2}
produces an identical width independent of the a_{1} location. The two models can, however, be
parameterized to be somewhat close in shape. That was done for the HVL and NLC peaks at x=2 in the plot
above. There are four HVL peaks sharing a common a2 and a3 and varying only by a1 location. Similarly,
there are four NLC peaks with their own common a2 and a3 which also vary only by location.

These are the four HVL peaks with alignment of contour unit area maxima. With the maxima aligned, the
left side of the peak diminishes and the right side increases with elution time for a constant a_{3}:

Fitted Parameters

r^{2} Coef Det
DF Adj r^{2}
Fit Std Err
F-value
ppm uVar

1.00000000 1.00000000 2.3702e-7 3.9922e+14 1.5631e-6

Peak Type a0 a1 a2 a3 a4

1 GenHVL 0.24999995 1.99999991 0.09999959 0.04999978 -1.037e-6

2 GenHVL 0.24999993 3.99999927 0.09999938 0.04999958 -1.037e-6

3 GenHVL 0.24999993 5.99999864 0.09999923 0.04999944 -1.037e-6

4 GenHVL 0.24999992 7.99999801 0.09999911 0.04999934 -1.037e-6

Equivalent Parameters

Peak Type a0 a1 a2 a3 a4

1 GenNLC 0.24999995 1.99999991 0.00249998 0.04999978 -2.074e-5

2 GenNLC 0.24999993 3.99999927 0.00124998 0.04999958 -4.148e-5

3 GenNLC 0.24999993 5.99999864 0.00083332 0.04999944 -6.222e-5

4 GenNLC 0.24999992 7.99999801 0.00062499 0.04999934 -8.296e-5

Measured Values

Peak Type Amplitude Center FWHM Asym50 FW Base Asym10

1 GenHVL 0.81501472 1.67399392 0.28966507 2.77626888 0.59551070 3.35218457

2 GenHVL 0.66114972 3.47107970 0.36190008 4.71449073 0.74367882 5.88715727

3 GenHVL 0.56839084 5.31793008 0.42457202 6.55985968 0.86908116 8.30659069

4 GenHVL 0.50593121 7.19050875 0.47964442 8.32609035 0.97861980 10.6382853

Peak Type Area % Area Mean StdDev Skewness Kurtosis

1 GenHVL 0.24999995 25.0000017 1.77533278 0.13352289 0.67484249 3.00086395

2 GenHVL 0.24999993 25.0000001 3.64042572 0.17143874 0.70204398 2.85565192

3 GenHVL 0.24999993 24.9999993 5.53896668 0.20256406 0.68855370 2.76628498

4 GenHVL 0.24999992 24.9999990 7.45453595 0.22935499 0.67473377 2.70919751

All Total 0.99999973 100.000000

If we fit these four HVL peaks to the GenHVL,
we see very close to the generated parameters (the model cannot fit the a_{4} statistical asymmetry
perfectly to 0 because of a singularity). Note that the a_{2} SD of the underlying Gaussian is
constant as is the a_{3} distortion, but the FWHM increases from .290 to .480 and the second moment,
as an SD, increases from .133 to .229. The impact on the higher moments is even more pronounced. The half-height
asymmetry varies from 2.78 to 8.33. The skewness, the third moment, is strongly tailed and approximately
constant, while the kurtosis, the fourth moment, successively drops below the 3.0 of the Gaussian. For
the HVL, constant a_{2} and a_{3} parameters do not equate to identical peaks shifted
only by location. The peaks are becoming much more sharply tailed, as given by the half-height asymmetry,
even though the a_{3} distortion is constant.

Note the Equivalent
Parameters section. Had we fit a GenNLC,
we would have the same a_{0}, a_{1}, and a_{3} and the same goodness of fit, but
the a_{2} NLC time constant decreases with elution, not something one would expect to see in practice.
Indeed, our experience with the adsorption-desorption kinetics in IC is such that the NLC a_{2},
the Giddings time constant, increases somewhat with elution time. These differences in shape with constant
a_{3} are a consequence of the HVL definition and this a_{1}/a_{2} scaling. It
is unrelated to the asymmetry in the generalized model; in this instance there is none.

This is a similar plot of the four NLC peaks where the maxima are aligned in the gradient rendering. A
constant a_{2} kinetic width and a_{3} distortion result in increased breadth on each
side of the apex of the peak:

Fitted Parameters

r^{2} Coef Det
DF Adj r^{2}
Fit Std Err
F-value
ppm uVar

1.00000000 1.00000000 5.062e-16 7.3065e+31 0.00000000

Peak Type a0 a1 a2 a3 a4

1 GenNLC 0.25000000 2.00000000 0.00307000 0.05560000 0.50000000

2 GenNLC 0.25000000 4.00000000 0.00307000 0.05560000 0.50000000

3 GenNLC 0.25000000 6.00000000 0.00307000 0.05560000 0.50000000

4 GenNLC 0.25000000 8.00000000 0.00307000 0.05560000 0.50000000

Equivalent Parameters

Peak Type a0 a1 a2 a3 a4

1 GenHVL 0.25000000 2.00000000 0.11081516 0.05560000 0.02770379

2 GenHVL 0.25000000 4.00000000 0.15671630 0.05560000 0.01958954

3 GenHVL 0.25000000 6.00000000 0.19193749 0.05560000 0.01599479

4 GenHVL 0.25000000 8.00000000 0.22163032 0.05560000 0.01385190

Measured Values

Peak Type Amplitude Center FWHM Asym50 FW Base Asym10

1 GenNLC 0.81906786 1.67446070 0.28274674 2.64603623 0.60523050 3.33555734

2 GenNLC 0.56503247 3.53573765 0.41206691 2.63078712 0.87138950 3.26829031

3 GenNLC 0.45633328 5.42928735 0.51141649 2.62377353 1.07568609 3.23860912

4 GenNLC 0.39263260 7.33954438 0.59521224 2.61951551 1.24793836 3.22095144

Peak Type Area % Area Mean StdDev Skewness Kurtosis

1 GenNLC 0.25000000 25.0000000 1.77615211 0.13719600 0.74820263 3.20420926

2 GenNLC 0.25000000 25.0000000 3.68040697 0.19657134 0.72261567 3.14585260

3 GenNLC 0.25000000 25.0000000 5.60691585 0.24215590 0.71136114 3.12086748

4 GenNLC 0.25000000 25.0000000 7.54495203 0.28059376 0.70467533 3.10622230

All Total 1.00000000 100.000000

If we fit these four NLC peaks to the GenNLC,
we recover the generated parameters to full precision. Note that the a_{2} kinetic time constant
of the underlying Giddings model is constant as is the a_{3} distortion, and the a_{4}
asymmetry fits to a constant 0.5, the NLC. Here the FWHM increases from .283 to .595 and the second moment,
as an SD, increases from .137 to .280. The widening with a constant a_{2} is appreciably higher.
The half-height asymmetry, however, is close to constant for all four of the peaks. The skewness, the
third moment is approximately constant, higher than the GenHVL, while the kurtosis, the fourth moment,
successively drops but remains above the 3.0 of the Gaussian. For the NLC, constant a_{2} and
a_{3} parameters equally do not equate to identical peaks shifted only by location.

Note the Equivalent
Parameters for the GenHVL. Had we fit the GenHVL
to this data, we would have the same a_{0}, a_{1}, and a_{3}. The a_{2}
Gaussian deconvolved SD increases by about a factor of 2 across the peaks, but the a_{4} asymmetry
changes as well, decreasing by a factor of 2. In designing the GenHVL model, we retained the a_{4}
definition as the standard statistical asymmetry of the generalized normal.

Sharing a_{2} Width, a_{3} Distortion, and a_{4} Asymmetry across GenHVL and GenNLC
Peaks

The GenHVL and
GenNLC models
will each fit the theoretical HVL
and NLC
shapes completely. With the a_{3} adjusted to be identical across the two models, the only differences
rest with the a_{2} term describing the width, and the a_{4} term describing the ZDD third
moment asymmetry.

The GenHVL and GenNLC are in effect the same model with two different parameterizations of a_{2}
and a_{4}. If fitting only a single peak you will always get the same identical goodness of fit,
whether you fit the GenHVL or the GenNLC. The differences arise only when sharing parameters across multiple
peaks in an elution.

The a_{2} and a_{3} parameters shared only when absolutely necessary, usually when fitting
poorly defined overlapping and hidden peaks. On the other hand, the a_{4} asymmetry is almost
always shared, a measure of the ZDD's deviation from theoretical ideality.

You should see the GenHVL
as a diffusion theory or statistical peak. The a_{2} is the deconvolved Gaussian SD and the a_{4}
is the statistical asymmetry of the generalized normal. When there is no asymmetry, a_{4}=0, the
peak becomes the pure HVL. When sharing a_{4}, there is a constant asymmetry in the ZDD across
all peaks.

You should see the GenNLC
as a kinetic theory peak. The a_{2} is the time constant of the first-order adsorption-desorption
of the Giddings model, and the a_{4} is an asymmetry normalized to the Giddings. When the ZDD
is a Giddings shape, the a_{4} will have a value of 0.5, representing the pure NLC. When there
is no no asymmetry, a_{4}=0, the peak becomes the pure HVL. In the GenNLC, sharing the a_{4}
asymmetry is sharing a constant measure of deviation from this a4=0.5 Giddings ideality associated with
the NLC.

Since the a_{4} is the parameter that is typically shared, you are generally choosing between
having a constant statistical asymmetry, a constant measure of deviation from the Gaussian in the infinite
dilution density used to construct the GenHVL peak, and having a constant measure of the deviation from
the Giddings asymmetry in the ZDD which is used to construct the GenNLC peak.

In practice, the differences in goodness of fit between sharing the a_{4} in a GenHVL
model and doing so in a GenNLC
model are quite small for most analytic peaks. For example, if the 16 IC cation data sets of the Chromatographic
Experiments tutorial are fitted to the GenHVL<ge> model with this shared a_{4}, the
average goodness of fit across the sets is 9.92 ppm unaccounted variance. For the GenNLC<ge> model,
the average of the 16 different fits is 10.26 ppm error. If instead, we fit the GenHVL<e2> model,
we average 10.54 ppm, and for the GenNLC<e2> we average 9.76 ppm. This sufficiently summarizes our
experience. The sharing of a_{4} does have a very small impact, but we cannot confirm either the
absolute statistical asymmetry or the Giddings-reference asymmetry as offering better fits. The differences
are too small.

Since you can display the parameters for each model in this Equivalent Parameters option in the Numeric Summary, you will always have the diffusion and kinetic widths and the statistical and kinetic asymmetries available in any fit to either model.