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White Paper: Part IV - Fitting Preparative (Overload) Peaks

Part IV - Preparative (Overload) Peaks

In Part I of this series we described a generalization of the HVL and NLC models which accommodated higher moment differences in the zero distortion (infinite dilution) density underlying the peak shapes in chromatographic separations. In Part II, we added the complexity of fitting the mathematical convolution of this 'true' peak model and an IRF, or instrument response function, in the fitting of analytical isocratic peaks. In Part III, we addressed the further complication in the shape of chromatographic peaks when the strength of the mobile phase is changing in gradient HPLC separations. In Part IV, we will now address the sharply different chromatographic shapes which occur when the concentrations are high and major column overload is present.

The Challenge of Fitting Overload Shapes

Of all the types of chromatographic peak shapes, preparative or overload profiles have proven to be the most challenging to fit to the near zero error we see in the other types of chromatography. In an overload peak, the eluted shapes arise from the column and/or the detector function being saturated. In effect, the expected peaks are nowhere seen because of a breakdown in either the separation in the column, the signal response in the detector, or both. The high amount of eluting solute is processed very differently in this condition of saturation.

We have discovered overload models that are true theoretical chromatographic convolution models which address this saturation or breakdown and can reconstruct the peaks that would have been seen if the column and detector could fully manage infinite concentrations, as if having infinite capacity and response.

In fact, the same third and fourth moment twice-generalized models used in directly fitting the gradient shapes, as was done in Part III of this white paper series, are effective in fitting preparative shapes. There is one important difference. With HPLC gradient peaks, we treat the gradient as a compression in the fourth moment of the ZDD, or zero-distortion density. For overload shapes, it is the opposite, a dilation of the tails in the ZDD. Instead of fitting a higher power of fourth moment-related decay that represents a compression, we fit a lower power of fourth moment-related decay that represents a dilation of the tails. This dilation in the ZDD's tails manages this overload state and produces the overload shapes seen in preparative chromatography.

Just as with fitting constant mobile phase analytic peaks, IRF tailing must be addressed by fitting an IRF-bearing model, a convolution integral. Although we can modify the ZDD zero distortion density with this power of decay dilation to produce the overall core shape of the overload envelope, we cannot manage the high measure of instrumental/system distortion without fitting an <irf> convolution integral. With preparative shapes, a convolution integral is non-optional.

We also know that multiple non-idealities in fluid flow, mass transfer, and detection enter into the IRF components. We know that even the two component <ge>, <e2>, and <pe> IRFs we routinely use for analytic peaks represent simplifications that will not fully capture the error arising from all of these different distortions or non-idealities. If the tailing arising from the IRF in preparative shapes is much increased by this overload state, we would expect any error in fitting the IRF portion of the peak's tailing to likewise be amplified.

Overload Shapes

The following data consist of four TEA (triethanolamine) overload peaks with concentrations varying between 1 and 4 % (10000, 20000, 30000, and 40000 ppm in the titles). The dead time transform and baseline correction have already been applied.

Both the contour and non-normalized data plots clearly confirm the overload state. The additional concentration goes into the overload envelope. Note the differences in the strong tailing. If you were to place a tangent line on the decay of each of the envelopes, you see the tailing beginning at about 140, 110, 95, and 70 Y signal values across the four samples. If we treat this tailing as an IRF, the exposed portion of the IRF is greatest on the lowest concentration sample, and smallest on the highest concentration sample. If you look at the area-normalized contours, the tailing appears more drawn out for the higher concentration samples. The tailing is important because fitting the IRF is essential in accurately modeling overload peaks.

The GenHVL[Yp] Overload Model

The twice generalized Gen2HVL
(fitting both third moment and fourth moment effects), covered in Part III of this white paper series,
effectively manages these overload shapes. The Gen2HVL
model's ZDD
(zero distortion density) defines the a_{1} location parameter as the mean of the generalized
error ZDD. The [Y]
ZDD retains the a_{1} as the mean of the underlying Gaussian and we have chosen to use this
form of the [Y] ZDD for the overload models:

a_{0} = Area

a_{1} = Center

a_{2} = Width

a_{3}= Error model power (.25 > a_{3} > 4 , Laplace=1, Gaussian=2)

a_{4} = Asymmetry
( fronted -1 > a_{4} > 1 tailed)

When the power of decay (the fourth moment a_{3}
term above) in the [Y] generalized error class of ZDD drops from 2.0 toward 1.0, this dilation produces
a progression similar to those in the sequence above. In fact, by the time an 'envelope' has fully formed,
this fourth moment decay parameter fits to very close to 1.0, a ZDD which decays as a double-sided exponential
or Laplace density.

By inserting this [Y] Generalized Error Model ZDD for the PDF, CDF, and CDFc in the GenHVL template, we produce the GenHVL[Y] model. The GenHVL[Yp] (the appended 'p' for 'preparative') is identical to the GenHVL[Y] but offers a different starting estimate algorithm which assumes a high overload shape. In order to ensure the convergence of the non-linear fitting to an optimal solution, the starting estimates must be specific for this condition of overload. The GenHVL[Yp] model is as follows:

a_{0} = Area

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

a_{2} = Width (SD of underlying normal ZDD)

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

a_{4} = Power n in exp(-zn) decay ( .25 > a_{4} > 4 ) adjusts kurtosis (fourth moment)

a_{5} = ZDD asymmetry ( -1 > a_{5} > 1 ), adjusts skew (third moment)

When the a_{4} power of tail decay in the ZDD of the GenHVL[Yp] model is 2.0, the generalized
HVL peak is generated (the blue peak which has a traditional chromatographic peak shape). As the a_{4}
power decreases to the 1.0 of double-sided exponential or Laplace type of decay, the shape markedly changes
as an overload envelope forms. Strongly overload preparatory shapes will have an a_{4} value very
close to 1.0.

The a_{5} ZDD asymmetry primarily impacts the slope of the envelope when a high overload shape
(a_{4} approaching 1.0) is present. The most negative a_{5} value in the plot above produces
the sharpest upward slope for the envelope. The most positive of the a_{5} values produces the
sharpest downward slope in the envelope.

It is important to note that all of this overload model is a theoretical generalized chromatographic model which uses the common chromatographic distortion mathematics, just as was true of the models used in all of the other types of chromatographic peaks. In fact, with the proper starting estimates, this [Y] generalized error twice-generalized class of ZDD, can successfully fit virtually any chromatographic shape: analytic GC and LC peaks, gradient HPLC peaks, and overload preparative peaks. This two-higher moment class of chromatographic model could easily be deemed a universal model were it not for the fact the analytic peaks generally require only a third moment adjustment.

The GenHVL[Yp2] Overload Model

If one assumes there are two distinct behaviors in the formation of an overload shape, one prior to the
apex of the underlying zero distortion density, and one after, and one is constructing a model that will
be used near to this a_{5}=1.0 high overload envelope where there already exists a sharp reversal
in slope at the apex, it is a straightforward matter to create a two-width ZDD, one width used only for
the rise or left side of the ZDD, the other only for the decay or right side of the ZDD. This offers a
much greater control of the initial rise of the overload shape.

The GenHVL[Yp2] model is this modification of the GenHVL[Yp] where there are two widths fitted, a width for the ZDD peak consisting of the rise to the apex, and a separate width for the ZDD peak which decays from the apex. Although an additional parameter is required, this model can often be fitted successfully, and often with much better goodness of fits. With two components, one pre-apex, one post-apex, the ZDD is going to necessarily be more complicated. The [Yp2] ZDD is defined as follows:

a_{0} = Area

a_{1} = Center

a_{2} = Width

a_{3} = Width Asymmetry (right side width)/(left side width)

a_{4} = Error model power (.25 > a_{4}
> 4 , Laplace=1, Gaussian=2)

a_{5} =
Statistical Asymmetry ( fronted -1 > a_{5}
> 1 tailed)

By inserting this [Yp2] Generalized Error 2-width Model ZDD for the PDF, CDF, and CDFc in the GenHVL template, we produce the GenHVL[Yp2] model:

a_{0} = Area

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

a_{2} = Width (SD of underlying normal ZDD)

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

a_{4} = Width Asymmetry
(right side width)/(left side width)

a_{5} = Power
n in exp(-zn) decay ( .25 > a_{5}
> 4 ) adjusts kurtosis (fourth moment)

a_{6}
= ZDD asymmetry ( -1 > a_{6}
> 1 ), adjusts skew (third moment)

In this model, a_{4} is now the ZDD width ratio (right/Left), a_{5} is the ZDD power of
decay, and a_{6} is the ZDD asymmetry. The plot above illustrates the impact of separate rise
and decay widths. The overall a_{2} width was adjusted across the shapes to produce approximately
the same envelope width and inflection point. Here the right/left width ratio is varied from 1.0, the
curve with the softest rise and the most positive slope in the envelope, to 10.0, the curve with the sharpest
rise and the most negative slope in the envelope. The two-width model primarily adds the ability to adjust
the sharpness of the rise in the overload shape.

Fitting the Single-Width GenHVL[Yp]<e2> Model

If the four TEA data sets are fitted to the single width twice-generalized HVL with the two-component exponential IRF, the fits hare as follows:

While the fits are quite good, given the vastly distorted shapes, the errors of 243, 166, 367, 643 ppm are well removed from the sub-50 ppm typically seen on analytical shapes.

"TEA Std 10000ppm"

Fitted Parameters

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

0.99975686
0.99975643
0.99092398
2,624,370
__243.142042__

Peak Type a0 a1 a2 a3 a4 a5 a6 a7 a8

1
GenHVL[Yp]<e2>
165.712367
__7.62790616__
0.02915358
-0.0016850
__1.56108587__
0.16168679
__0.49264681__
0.00991622 __0.53729199__

Deconvolved Moments

Peak Type Area Mean StdDev Skewness Kurtosis Amplitude Center

1
HVL
165.712367
7.71770294
0.04489006
-0.7019512
2.91901600
__1649.72479__
7.75823777

Advanced Area Analysis

Peak Type Area % Area ApexAsym NonOverlap1 % PkArea NonOverlap2 % PkArea

1
GenHVL[Yp]<e2>
165.706546
100.000000
1.47867118
88.3333028
__53.3070690__
58.3726815
__35.2265393__

"TEA Std 20000ppm"

Fitted Parameters

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

0.99983345
0.99983316
1.32798346
3,831,649
__166.545420__

Peak Type a0 a1 a2 a3 a4 a5 a6 a7 a8

1
GenHVL[Yp]<e2>
323.814430
__7.62002945__
0.03298375
-0.0038178
__0.98442511__
-0.0025486
__0.45386384__
0.03646563 __0.33514355__

Deconvolved Moments

Peak Type Area Mean StdDev Skewness Kurtosis Amplitude Center

1
HVL
323.814430
7.76191624
0.06365485
-0.6934033
2.79079270
__2333.81036__
7.82965650

Advanced Area Analysis

Peak Type Area % Area ApexAsym NonOverlap1 % PkArea NonOverlap2 % PkArea

1
GenHVL[Yp]<e2>
323.807068
100.000000
0.46231684
229.137593
__70.7636168__
54.5986678
__16.8614812__

"TEA Std 30000ppm"

Fitted Parameters

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

0.99963293
0.99963229
2.52106273
1,738,143
__367.067242__

Peak Type a0 a1 a2 a3 a4 a5 a6 a7 a8

1
GenHVL[Yp]<e2>
501.196639
__7.62492405__
0.04796292
-0.0093378
__0.99198708__
-0.0067834
__0.45070353__
0.04086680 __0.24302761__

Deconvolved Moments

Peak Type Area Mean StdDev Skewness Kurtosis Amplitude Center

1
HVL
501.196639
7.85002070
0.09845306
-0.6871550
2.75980363
__2346.76645__
7.95814291

Advanced Area Analysis

Peak Type Area % Area ApexAsym NonOverlap1 % PkArea NonOverlap2 % PkArea

1
GenHVL[Yp]<e2>
501.184448
100.000000
0.29514841
362.272018
__72.2831722__
50.2737025
__10.0309782__

"TEA Std 40000ppm"

Fitted Parameters

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

0.99935605
0.99935491
3.93225491
990,502
__643.954884__

Peak Type a0 a1 a2 a3 a4 a5 a6 a7 a8

1
GenHVL[Yp]<e2>
698.602806
__7.63006608__
0.06314201
-0.0179053
__1.01030860__
-0.0080049
__0.45060925__
0.04512322 __0.19485141__

Deconvolved Moments

Peak Type Area Mean StdDev Skewness Kurtosis Amplitude Center

1
HVL
698.602806
7.94467842
0.13539268
-0.6823806
2.73911603
__2385.63939__
8.09634866

Advanced Area Analysis

Peak Type Area % Area ApexAsym NonOverlap1 % PkArea NonOverlap2 % PkArea

1
GenHVL[Yp]<e2>
698.568754
100.000000
0.22800778
509.408942
__72.9218046__
48.8806517
__6.99725709__

The a_{1} locations are nearly identical, despite the immensely different shapes. The a_{4
}power of decay value, is 1.56 for the least overloaded shape (1% concentration). The other three
(2,3,4% concentration) are all very close to the exponential 1.0 power of tailing. The a_{5} asymmetry
converges to very close to a symmetric zero-distortion density as the overload envelope fully forms. The
a_{6} slow exponential in the IRF is nearly constant across the four samples and far wider than
the highest distortions observed in analytic peaks. The a_{8} area fraction of this higher width
exponential component in the IRF dramatically decreases with increasing overload. There is a great deal
of information in a preparative peak fit, and we will now cover those in more detail.

The white peaks are those registered by the instrument. The red peaks are the peaks you would see if there existed no system/instrumental distortion. These are the fitted peaks, the GenHVL[Yp] with the <e2> IRF removed. Although the IRF attenuates the peak and significantly smears out the shape, especially with respect to a broad tailing, note that the overall shape of the overload envelope comes from the twice-generalized chromatographic model, not from the IRF.

If we use just the pure HVL parameters from the twice-generalized HVL model, we see the green peaks, the
pure HVL
without the influence of the ZDD's third and fourth moment parameters. This is the pure chromatographic
peak you would expect to see if the the column/detector were immune to overload and there existed no higher
moment nonidealities. As you will note, the green deconvolved HVL is sharply fronted, even to the point
of almost having a right-triangular shape. Even peaks with very little a_{3} chromatographic distortion
at analytical concentrations will have that small measure of fronting or tailing vastly amplified by concentration.
The deconvolved preparatory peaks, of very high concentration, as in the green HVL's above, should either
be strongly fronted or strongly tailed, depending on the sign of a_{3}. One of the key benefits
of preparative shape modeling is the recovery of the peak that would have been observed if the chromatographic
instrument had unlimited capacity.

There is a fair argument for seeking a refinement of this workhorse twice-generalized model for fitting
preparative shapes. Despite the IRF decreasing with overload (confirmed with a_{8}, the area fraction
of the IRF's very large width component), the error of fit actually increases, suggesting the lack of
fit rests with the core model, not the accounting of the IRF. While we would not expect such massively
distorted shapes to fit as well as analytic peak shapes, we would prefer to see sub-100 ppm fit errors,
if possible.

The dilation in the ZDD rests with parameter a_{4}, the power of decay. It fit to 1.56 for the
partial overload shape, and very close to 1.0 for all three of the full envelope (a plateau between the
rise and decay) data sets. A 2.0 power of decay is that of the Gaussian, 1.0 that of an exponential. Note
that a_{4 }maps a constant dilation of 1.0 once a full envelope has formed, and varies between
2 and 1 as the overload increases. The a_{4} dilation maps the overload shape, but only until
a full envelope has formed. The a_{4} parameter can thus be seen as mapping the extent to which
this envelope has formed.

The a_{7} high width exponential in the IRF fits to a constant .45 in the three full envelope
overload data sets. This is about ten times the highest width observed for this instrument with analytic
peaks. Since it was apparent just from visual inspection that the amount of this IRF tailing was itself
saturated (decreasing proportional to peak area), the a_{8} area fraction of this largest width
component also measures the overload, albeit in a very different way. Unlike analytic peaks where the
IRF is somewhat constant, it is far from such in overload shapes. With respect to a_{7}, the smaller
of the two exponential widths, we must refer to the analytic <e2> IRF of [tau 1=.007, tau 2=.044,
areafrac1=.62]. The a_{7} parameter certainly fit to the higher analytic width in the three full
envelope fits. In case you were wondering if the two exponentials are needed, fits with the GenHVL[Yp]<e>
model, a single exponential IRF, were terrible: 2376 ppm to 13140 ppm. As we have shared throughout these
white papers, fitting the IRF is essential in high accuracy chromatographic modeling, and overload shapes
have massive IRFs.

We will look at a_{0}-a_{3} parameter values at the four
concentrations:

In the first graph, the a_{0}
areas track concentration almost linearly.

One of the suggestions this model is theoretically sound for overload
peaks is that the a_{1}
locations are almost perfectly identical for the four vastly different overload shapes (the second graph
scales a_{1} from
7.62 to 7.63).

The a_{2}
widths increase with overload, and linearly once a full envelope is present. For this model, a_{2
}is the SD width of the underlying Gaussian.

The a_{3}
chromatographic shape, strongly fronted, becomes more negative (more fronted) with concentration, another
suggestion the model is tracking the overload well. Unlike the linearity of a_{3}
with concentration observed with analytic peaks, the distortion increases with a power greater than 1.
If a 0,0 point is inserted into the data (0 chromatographic distortion at 0 concentration), the data in
this plot fit to a power of 2.15. In a full accuracy analytic fit, it is our experience that a_{3
}is linear with concentration. Since these fits still have
a measure of error, however, we cannot automatically assume overload produces a disproportionate measure
of chromatographic distortion.

If the initial rise of the fits are zoomed-in, for the second through fourth data set, there is a significant error in the initial rise. It increases with concentration, and corresponds with the poorer goodness of fit:

This is the precisely the issue the two-width GenHVL[Yp2] model is designed to solve.

Fitting the Two-Width GenHVL[Yp2]<e2>

In the GenHVL[Yp2] peak the rise of the peak is independently fitted. The addition of an additional rise width parameter (expressed as a width asymmetry, a right-side width divided by a left-side width) changes the basic model from a six-parameter to a seven-parameter one. With a three parameter <e2> IRF added, we fit ten total parameters instead of nine.

We now better model the nuances in the rise of the overload shape. In the GenHVL[Yp]<e2> fits, we had an average F-statistic of 2.3 million and 355 ppm unaccounted variance across the four fits. In these GenHVL[Yp2]<e2> fits, this one additional parameter resulted in an average F-statistic of 4.3 million and 147.1 ppm unaccounted variance. This one parameter dramatically increased the power and efficacy of the fitting.

We see the about the same red IRF-removed overload shapes, and similar green deconvolved HVL peaks which now increase in amplitude as well as in width with concentration. Intuitively, we would expect at least some increase in amplitude of the deconvolved HVL with the increasing concentration. With the better fitting of the narrow rise of the overload shape, the ZDD is changed in shape, and this is translating differently in the common chromatographic distortion mathematics.

Analyzing the GenHVL[Yp2]<e2>

TEA Std 10000ppm"

Fitted Parameters

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

0.99975802
0.99975754
0.98866004
2,343,472
__241.984910__

Peak Type a0 a1 a2 a3 a4 a5 a6 a7 a8 a9

1
GenHVL[Yp2]<e2>
165.699591
__7.63050683__
0.02978956
-0.0017667
__0.95382969__
__1.59973390__
0.17562770
0.49175606 0.00753349 __0.53574798__

Deconvolved Moments

Peak Type Area Mean StdDev Skewness Kurtosis Amplitude Center

1
HVL
165.699591
7.72274368
0.04594607
__-0.7020460__
__2.91801540__
__1612.09703__
7.76429890

Advanced Area Analysis

Peak Type Area % Area ApexAsym NonOverlap1 % PkArea NonOverlap2 % PkArea

1
GenHVL[Yp2]<e2>
165.693877
100.000000
1.48240727
86.6831801
__52.3152586__
57.8075050
__34.8881360__

"TEA Std 20000ppm"

Fitted Parameters

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

0.99986153
0.99986126
1.21101447
4,095,737
__138.471680__

Peak Type a0 a1 a2 a3 a4 a5 a6 a7 a8 a9

1
GenHVL[Yp2]<e2>
323.663348
__7.61759861__
0.03119390
-0.0032068
__1.36941334__
__0.99724048__
__-0.0023158__
__0.46311223__ __0.03531645__ __0.32664782__

Deconvolved Moments

Peak Type Area Mean StdDev Skewness Kurtosis Amplitude Center

1
HVL
323.663348
7.74673995
0.05864179
__-0.6957895__
__2.80457939__
__2526.30873__
7.80823084

Advanced Area Analysis

Peak Type Area % Area ApexAsym NonOverlap1 % PkArea NonOverlap2 % PkArea

1
GenHVL[Yp2]<e2>
323.654534
100.000000
0.47529531
237.337848
__73.3306111__
53.8176966
__16.6281300__

"TEA Std 30000ppm"

Fitted Parameters

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

0.99988660
0.99988637
1.40142174
5,001,186
__113.404675__

Peak Type a0 a1 a2 a3 a4 a5 a6 a7 a8 a9

1
GenHVL[Yp2]<e2>
500.746011
__7.61641433__
0.04293464
-0.0062967
__2.47162746__
__1.02681407__
__-0.0073440__
__0.51548379__ __0.03705191__ __0.20461323__

Deconvolved Moments

Peak Type Area Mean StdDev Skewness Kurtosis Amplitude Center

1
HVL
500.746011
7.79795859
0.08191696
__-0.6944676__
__2.79675726__
__2801.67097__
7.88458338

Advanced Area Analysis

Peak Type Area % Area ApexAsym NonOverlap1 % PkArea NonOverlap2 % PkArea

1
GenHVL[Yp2]<e2>
500.709732
100.000000
0.32286923
391.254383
__78.1399597__
47.1495800
__9.41654954__

"TEA Std 40000ppm"

Fitted Parameters

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

0.99990538
0.99990519
1.50749340
5,993,965
__94.6232557__

Peak Type a0 a1 a2 a3 a4 a5 a6 a7 a8 a9

1
GenHVL[Yp2]<e2>
697.884878
__7.61608517__
0.05551610
-0.0110583
__3.66288634__
__1.06518094__
__-0.0085090__
__0.61958737__ __0.03773216__ __0.12932068__

Deconvolved Moments

Peak Type Area Mean StdDev Skewness Kurtosis Amplitude Center

1
HVL
697.884878
7.85788217
0.10811585
__-0.6925349__
__2.78610988__
__2963.63704__
7.97350246

Advanced Area Analysis

Peak Type Area % Area ApexAsym NonOverlap1 % PkArea NonOverlap2 % PkArea

1
GenHVL[Yp2]<e2>
697.677056
100.000000
0.26052019
559.392444
__80.1792806__
42.6207683
__6.10895369__

The most significant item is that the goodness of fit improves as the concentration and envelope increase,
and as the high width IRF component fraction diminishes in the overall IRF. We note that the GenHVL[Yp2]
did not substantially improve the fit of this first data set with its partial overload envelope. Note
that an a_{4} width asymmetry of 1.0 is an equal width to the left and right of the apex, reducing
to the GenHVL[Yp] model. On this first data set, a_{4} fit to 0.953 on this first peak, close
to equal widths. The

In the above plots, we see all ten parameters, from a0 through a9 lotted against the concentration of
overload. As with the single width overload model, a0 tracks concentration, and a1, the deconvolved Gaussian
mean, is close to a constant for all four overload shapes (the second graph scales a1 from 7.615-7.635).
The a_{2} deconvolved Gaussian SD in the third graph increases
with concentration and overload, again as was observed with the GenHVL[Yp] fits. The single width GenHVL[Yp]
and two-width GenHVL[Yp2] models differ little in a_{2} values until the overload becomes extreme.
The a_{3} chromatographic distortion in the fourth graph,
describing the fronting, increases with concentration, although not linearly as if often the case at analytic
concentrations. With this GenHVL[Yp2] model, the nonlinearity between a_{3} and concentration
is such that the data fit to a power of 1.7 as compared to 2.15, not the 1.0 linearity we perhaps wished
to see, but an improvement.

The new parameter a_{4} is a width-based asymmetry, the right side width divided by the left side
width. This parameter, shown in the fifth graph, also strongly tracks
the overload, linearly with a full envelope. For the last of the data sets, the a_{4}=3.66
width asymmetry means that the right side of the peak is computed using a width of a2*a4/(1+a4) or .0436
and the left side of peak is computed using a width of a2/(1+a4) or .0119. The right side width of the
[Yp2]
ZDD density is 3.66x larger than the left side.

The a_{5} ZDD power of decay in the sixth graph remains close to 1.0 for the full overload envelope
data sets, and at 1.60, we see close to the GenHVL[Yp] 1.56 power of decay value for the first data set
with the partial envelope. In both the GenHVL[Yp] and GenHVL[Yp2] fits, the power of decay increased slightly
with concentration for the full envelope fits. With the separate rise and decay widths, this deviation
from 1.0 is greater with the GenHVL[Yp2] model.

The a6 ZDD asymmetry in the seventh graph shows the first partial overload data set fitting to a strong right-skewed asymmetry. The three full overload sets fit to slightly negative values as did the GenHVL[Yp] fits.

The a_{7}, a_{8},
a_{9} parameters describing the <e2> IRF are in
the last three graphs. Unlike the GenHVL[Yp] fits, the two-width fits produce the more extended a_{7}
widths we saw in the original contours of the tails in the data. This a_{7}
exponential component which produces the huge tailing now increases with concentration. The amount of
the component a_{9} diminishes with concentration as did
the single-width fits.

The numeric results, shown above, furnish the moments of the deconvolved HVL peaks as well as an advanced area analysis which estimates the measure of non-overlap between the green HVL and the red curve absent the IRF, and this red curve relative to the peak as registered by the instrument.

We now add a last level of deconvolution. The blue curves above plot
the deconvolved Gaussian, the peak this two-width overload model predicts if there was no IRF, no higher
moment nonidealities, and no concentration-dependent a_{3}
chromatographic distortion.

The deconvolved HVLs are shown in the above plot with normalized areas.
As we observed in the fitted values, the a_{3} chromatographic
distortion, the fronting, continues to increase with concentration, but a_{2},
the Gaussian width also increases with concentration.

The deconvolved Gaussians remove the chromatographic distortion of the HVL, but the broadening with concentration is also evident. Note that the three full-envelope peaks have almost identical Gaussian locations and differ only in this broadening. Only the first shape with the partial overload, which did not fit quite as well, shows any locational variation.

Concentration and a_{2}

Since the overload model fits suggest that there is this broadening with very high levels of concentration in the deconvolved peaks, we conclude this white paper with a discussion of whether or not this premise has any support at analytical concentrations where no overload occurs.

In fitting analytic peaks, we have indeed observed an increase in a_{2}
with an increase in analytic concentrations. The above plot is for GenHVL<ge> fits of the cation
data that was used in the first of these white papers. The increase is a_{2}
appears to track the magnitude of the a_{3} distortion.
The a_{3} distortion is greatest in magnitude in the Ca+
(red) and Mg+ (magenta), the strongly tailed peaks, and those do show an increase in a_{2}
with concentration. The K+ peaks (cyan) have very little a_{3}
distortion, and they are close to constant in a_{2} across
concentration. The Li+ peaks (white) have the strongest fronting, and also have the highest increase in
a2 with concentration for the three different species that produce fronted peaks.

Higher Moment Chromatographic Models and IRFs

In these four white papers we have shown that higher moment models, based on the longstanding established science of the Haarhoff-VanderLinde and Wade-Thomas models, in conjunction with real-world IRF fitting, can result in close to zero error in fitting a wide variety of chromatographic shapes, including two of the most difficult challenges in chromatographic modeling, gradient HPLC and preparative shapes. The science of the core models has been in place for decades and in our experience it is complete with respect to the peak fitting of chromatographic shapes, sufficient for a rigorous implementation in both production and analytical environments. The additions we describe in these white papers include the addition of higher moment adjustments to these core chemical engineering models, this based on well-established advances in the statistical sciences, and the addition of IRF convolution and deconvolution, this drawn from the well-established digital signal processing sciences.