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HPLC Gradient Peaks - Fitting the Unwound Data

Approach 3: Unwinding the Gradient and Fitting a Once-Generalized Model with or without an IRF

If the gradient is perceived as a non-linear transform that compresses a peak, one can use the program's convolution/deconvolution engine to seek to unwind or undo the gradient, approximating the peak that would exist if the gradient had not occurred. Unlike the IRF deconvolution, where one seeks to remove a distortion to recover the true peak, here we seek to recover the true peak by adding back that which occurred within the gradient. In other words, we treat the gradient process as a deconvolution which narrows the peak, and we apply a convolution to restore the original width and shape of the peak. We treat the gradient process as a deconvolution which is undone by a subsequent convolution with an IRF which represents its system response function.

You may wish to refer to the tutorial: HPLC Gradient Peaks - Fitting Unwound Data (Tutorial)

In the spirit of allowing the data to reveal the nature of the system IRF altering it, we can take the simple approach of convolving different IRFs to see if a peak shape can be restored which can be more accurately fitted with an isocratic model. We can make the inference that the better an isocratic model is fitted, the closer we are to the IRF or system model that represents the gradient process.

If the gradient process is a purely linear one, we can use a zero order kinetic model to restore the peak shape. If the gradient process is first order, we can use an order 1 kinetic model. We can convolve any order kinetic model we wish. We can also fit a probabilisitic IRF. If we assume the gradient process is akin that of a one-sided Gaussian deconvolution, we can restore the original peak shape by convolving a simple half-Gaussian response function. In this example, we apply a simple one-parameter Gaussian convolution data to the first data set (in blue above), the one we directly fit with the Gen2NLC model and realized a 12.5 ppm unaccounted variance. If this fit of the convolved data (the yellow curve) is significantly improved, we can assume we have unwound at least a portion of the gradient without introducing more harm than benefit. In fairness, convolution does smooth data, so we will see a modest improvement from this noise reduction, but only so long as the new shape is at least as applicable as the original to the model used in the fitting. The half-Gaussian convolved data fits to an astonishing 0.28 ppm unaccounted variance, a 50x reduction in the squared error.

Fitted Parameters

r2 Coef Det        DF Adj r2          Fit Std Err        F-value            ppm uVar

0.99999972         0.99999969         0.03589583         58,284,780         0.28480840

Peak        Type                 a0                a1                a2                a3                a4                a5

1        Gen2NLC        10.9961570        8.33755300        2.4757e-5         -1.98e-5          2.00155019        30.1860459

Equivalent Parameters

Peak        Type                 a0                a1                a2                a3                a4                a5

1        Gen2HVL        10.9961570        8.33755300        0.02031824        -1.98e-5          2.00155019        0.07356203

Parameter Statistics

Peak 1 Gen2NLC

Parameter           Value             Std Error         t-value           95% Conf Lo        95% Conf Hi

Area        10.9961570        0.00072634        15139.1556        10.9947124         10.9976017

Center        8.33755300        4.0391e-5         2.0642e+5         8.33747266         8.33763333

Width        2.4757e-5         4.543e-8          544.959992        2.4667e-5          2.4848e-5

Distortn        -1.98e-5          3.7678e-7         -52.540847        -2.055e-5          -1.905e-5

Q-power        2.00155019        0.00052882        3784.91919        2.00049839         2.00260200

NLCasym        30.1860459        0.21616022        139.646626        29.7561119         30.6159799

The a4 of the Gen2NLC fit is 2.002, essentially a pure Gaussian decay. Unlike the prior fit where the gradient was not unwound, here all of the parameters have strong significance. If you can identify and quantify the gradient's response function, as we did here, you can fit a GenNLC or GenHVL to the convolved data:

Fitted Parameters

r2 Coef Det        DF Adj r2          Fit Std Err        F-value            ppm uVar

0.99999969         0.99999967         0.03749380         66,778,129         0.31447412

Peak        Type                 a0                a1                a2                a3                a4

1        GenNLC        10.9967995        8.33749378        2.4886e-5         -2.046e-5         30.4568072

Equivalent Parameters

Peak        Type                 a0                a1                a2                a3                a4

1        GenHVL        10.9967995        8.33749378        0.02037084        -2.046e-5         0.07441454

Parameter Statistics

Peak 1 GenNLC

Parameter           Value             Std Error         t-value           95% Conf Lo        95% Conf Hi

Area        10.9967995        0.00072330        15203.5459        10.9953611         10.9982379

Center        8.33749378        3.6714e-5         2.2709e+5         8.33742077         8.33756678

Width        2.4886e-5         1.2767e-8         1949.29874        2.486e-5           2.4911e-5

Distortn        -2.046e-5         3.1904e-7         -64.124470        -2.109e-5          -1.982e-5

NLCasym        30.4568072        0.20405363        149.258834        30.0510241         30.8625902

Here we have a higher F-statistic in the simper model. A F-statistic represents a model's capacity to represent the data properly accounting the number of parameters thrown at the fitting problem.

Fitting an GenHVL-based vs GenNLC-based Model for HPLC Gradient Peaks

The GenHVL model fits an a2 deconvolved Gaussian SD and an a4 generalized normal asymmetry at infinite dilution. The GenNLC fits an a2 kinetic width, the time constant of first order Giddings kinetics, and an a4 asymmetry indexed to both the Gaussian(HVL) at a4=0 and to the Giddings(NLC) at a4=0.5.

Since the Equivalent Parameters section of the Numeric Summary offers the parameters for both, you may wish to fit whichever model offers the confidence statistics you are most interested in. For fitting single peaks, there is no difference between the fit of these models. We mention elsewhere the value of selecting a kinetic or probabilistic generalized model based on an asymmetry parameter shared across peaks. Whereas this can make small differences in non-gradient analytic peaks, for gradient peaks the differences are negligible because of the low a3 chromatographic distortion.

Drawbacks of a Convolution Preprocessing

There is the obvious issue of loss of resolution. In the 11 peak example, all 11 peaks are clear, even if several are not fully baseline resolved. If we apply the half-Gaussian convolution to undo the gradient and fit the 11-peak data set, there are just 10 local maxima peaks which fit to a higher 87 ppm unaccounted variance. The original tenth and eleventh peaks fit as a single peak of an asymmetric shape.

Fitted Parameters

r2 Coef Det        DF Adj r2          Fit Std Err        F-value            ppm uVar

0.99991271         0.99990859         0.03684701         248,671            87.2855663

Peak        Type                 a0                a1                a2                a3                a4                a5

1        Gen2HVL        0.34204975        4.49269628        0.02086406        -1.792e-5         1.99181957        0.13034307

2        Gen2HVL        0.01166148        4.61046668        0.02123436        -3.016e-5         1.99181957        0.13034307

3        Gen2HVL        0.31537769        5.04350031        0.02072665        -9.335e-7         1.99181957        0.13034307

4        Gen2HVL        1.13003945        5.13668427        0.02049991        1.936e-6          1.99181957        0.13034307

5        Gen2HVL        0.75112499        5.20832806        0.02024335        -2.882e-6         1.99181957        0.13034307

6        Gen2HVL        0.19236483        5.57414105        0.02173971        -3.214e-6         1.99181957        0.13034307

7        Gen2HVL        0.81058549        5.82764475        0.02048346        -4.459e-6         1.99181957        0.13034307

8        Gen2HVL        0.30362755        5.95064910        0.02061090        -9.616e-6         1.99181957        0.13034307

9        Gen2HVL        0.25567534        6.13617861        0.02083078        -9.519e-6         1.99181957        0.13034307

10        Gen2HVL        0.18790095        6.28622363        0.02883085        0.00043404        1.99181957        0.13034307

When applying the convolution and subsequently fitting the unwound data, we are also giving up the improved resolution derived from the gradient. While the a4 parameter fits very close to 2, the loss of resolution means that peak 11 must be fitted as a hidden peak where it was a straightforward matter to fit the Gen2HVL or Gen2NLC directly to the original data.

Note that a Gen2HVL or Gen2NLC fit to gradient data can uncover hidden peaks as evident by both a2 and a3 in this example.   