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IRF Estimation 3 - Standards Containing Only Tailed Peaks
In this final approach to estimating an IRF , we will estimate the IRF using only tailed peaks.
In the IRF Estimation 2 topic, the difficulty of estimating tailed peaks was discussed. Estimating the IRF from only intrinsically tailed peaks is a worst-case scenario. In estimating an IRF it is best if the a3 intrinsic chromatographic distortion is fronted, in the opposite direction of the IRF, which will be tailed. In other words, we would like one or more a negative a3 (intrinsically fronted) peaks to be used for the IRF determination.
If you must estimate the IRF from tailed peaks, such is possible, but it will require some additional effort. Because of the correlation between the instrumental tailing and the intrinsic chromatographic tailing, you will have to help the fitting process along. This can be straightforward. If you fit GenHVL or GenNLC peaks, you will generally have a good estimate of the deviation from non-ideality for the ZDD zero-distortion density. The program's default for the GenNLC in the ZDD dialog is 1.1858. The pure HVL is 0 and the pure NLC is 0.5. If your experience parallels ours, you will have a infinite dilution density with a skewness that is in the vicinity of 2-3x that of the Giddings density (which produces the NLC). Also, if you fit <ge> IRFs, you will rather quickly come to know the area fraction of your narrow and wide components. In the IRF dialog, our default for the <ge> is a g area fraction of 0.6113 (this is for the narrow component). We generally see about 5/8 of the overall IRF area attributable to the narrow width component, and about 3/8 area to the wider exponential. If we lock these two values in a fit, we can often stop much of this intercorrelation between the two different types of tailing.
Tailed Peak IRF Estimation
For this example, we will use this same six-peak cation standard, and we will isolate and fit the three tailed peaks in order to estimate the IRF. Here we will use a matrix of concentration (5-10-25-50 ppm) and temperature (30-35-40-45C) as the data we will average for the estimate. We use this data instead of the data in the fronted examples, since the additive impacted the IRF, and we know the tailed peak IRF estimation is going to be more challenging. Since this is different data, we will first do a full fit with the fronted peaks to establish an expectation for the tailed IRF estimation.
If we fit these 16 data sets to the GenHVL<ge> with the a4 ZDD asymmetry and a5-a6 IRF parameters shared across all peaks, and the IRF narrow width a7 fraction locked at 0.625, we see the following when these 16 sets are averaged using the Average Multiple Fits option in the Numeric Summary.
Average for 16 Fits
Fitted Parameters
r2 Coef Det DF Adj r 2 Fit Std Err F-value ppm uVar
0.99998921 0.99998918 0.01629336 50,476,703 10.7917053
Peak Type a0 a1 a2 a3 a4 a5 a6 a7
1 GenNLC<ge> 8.26155931 2.44289465 0.00024648 -0.0101086 1.20820068 0.00810942 0.04431287 0.62500000
2 GenNLC<ge> 2.32025938 3.85936993 0.00027942 -0.0018371 1.20820068 0.00810942 0.04431287 0.62500000
3 GenNLC<ge> 2.69783874 4.64491174 0.00027237 -0.0011021 1.20820068 0.00810942 0.04431287 0.62500000
4 GenNLC<ge> 1.21948046 7.11781529 0.00037808 0.00162510 1.20820068 0.00810942 0.04431287 0.62500000
5 GenNLC<ge> 2.07696389 12.6150205 0.00076793 0.01235987 1.20820068 0.00810942 0.04431287 0.62500000
6 GenNLC<ge> 4.21397004 14.1049378 0.00088484 0.02803543 1.20820068 0.00810942 0.04431287 0.62500000
With the 0.0081 narrow width half-Gaussian SD and the 0.0443 exponential time constant we have confirmed the two key IRF widths from the IRF Estimation 1 and IRF Estimation 2 examples. We also see the a4 ZDD asymmetry of 1.2082 very close to the program's default of 1.1858. We will need the a4 of 1.2082 and the a7 of 0.625 for this tailed peak estimation fitting.
It is sometimes instructive to use the above fit as a starting point to see if the a5 and a6 narrow and wide component of the IRF are changing in any discernible trend. We use the right click Set Common Parameters Across Peaks For All Data Sets... menu option to lock the a4 ZDD asymmetry at 1.2082 for all fits, to lock the a7 narrow component fraction of the IRF at .625 of the area, and we open up a5 and a6 to vary on a per peak basis:
Average for 16 Fits
Fitted Parameters
r2 Coef Det DF Adj r2 Fit Std Err F-value ppm uVar
0.99999123 0.99999120 0.01480487 45,916,924 8.77443472
Peak Type a0 a1 a2 a3 a4 a5 a6 a7
1 GenNLC<ge> 8.26338310 2.44283955 0.00024469 -0.0100655 1.20820000 0.00819214 0.04446035 0.62500000
2 GenNLC<ge> 2.31925724 3.86032380 0.00027733 -0.0018581 1.20820000 0.00547038 0.04391348 0.62500000
3 GenNLC<ge> 2.69220169 4.64808833 0.00027252 -0.0011007 1.20820000 0.00328424 0.04173921 0.62500000
4 GenNLC<ge> 1.22244426 7.11373675 0.00037659 0.00158623 1.20820000 0.01251984 0.04685827 0.62500000
5 GenNLC<ge> 2.07569340 12.6206726 0.00077245 0.01237467 1.20820000 0.01113443 0.02724279 0.62500000
6 GenNLC<ge> 4.21586319 14.1099190 0.00090909 0.02822314 1.20820000 0.00863092 0.02925939 0.62500000
With respect to the a6 exponential component, the one we expect to be close to constant, peak 4, potassium, which is only modestly tailed, is in reasonably good agreement with the three fronted peaks. The much later eluting calcium and magnesium, peaks 5 and 6, are extremely tailed, and their averages are not in agreement with the the first four peaks.
If we use the Explore option to look at the Conc vs Temp surfaces for the a6 exponential width in the IRF, only the slightly tailed fourth peak has a6 reasonably constant at most temperatures and concentrations. All is good at the 5,10,25 ppm concentration, but at the 30C and 35C temperatures and 50ppm, the a6 value falls off a cliff. For peaks 5 and 6, there is no constancy. We think it is due to the Ca+ and Mg+ interacting differently at different concentrations. If we average peak 5 or peak 6, we would have to limit the average to the 25 and 50 ppm concentrations. If we do this:
Peak 4: Potassium
Average for 14 Fits
Fitted Parameters
r2 Coef Det DF Adj r2 Fit Std Err F-value ppm uVar
0.99999232 0.99999230 0.01160998 49,648,269 7.67759619
Peak Type a0 a1 a2 a3 a4 a5 a6 a7
1 GenNLC<ge> 6.80469758 2.45061807 0.00023824 -0.0083379 1.20820000 0.00809840 0.04451166 0.62500000
2 GenNLC<ge> 1.91015205 3.85383405 0.00027029 -0.0015466 1.20820000 0.00474402 0.04398425 0.62500000
3 GenNLC<ge> 2.21744955 4.63842330 0.00026814 -0.0009203 1.20820000 0.00253383 0.04178367 0.62500000
4 GenNLC<ge> 1.00811714 7.06415436 0.00036583 0.00113263 1.20820000 0.01314113 0.05167537 0.62500000
5 GenNLC<ge> 1.71319123 12.6583478 0.00077102 0.01025733 1.20820000 0.00940577 0.02328284 0.62500000
6 GenNLC<ge> 3.47344205 14.1039501 0.00089252 0.02321374 1.20820000 0.00830972 0.02742569 0.62500000
Peak 5: Calcium, Peak 6: Magnesium
Average for 8 Fits
Fitted Parameters
r2 Coef Det DF Adj r2 Fit Std Err F-value ppm uVar
0.99998861 0.99998857 0.02513012 38,864,837 11.3910230
Peak Type a0 a1 a2 a3 a4 a5 a6 a7
1 GenNLC<ge> 13.7794738 2.41414124 0.00026231 -0.0166604 1.20820000 0.00917166 0.04430576 0.62500000
2 GenNLC<ge> 3.86755142 3.84323169 0.00029493 -0.0031048 1.20820000 0.00968468 0.04385163 0.62500000
3 GenNLC<ge> 4.49040808 4.63614358 0.00028285 -0.0018829 1.20820000 0.00576324 0.04120821 0.62500000
4 GenNLC<ge> 2.03869859 7.11493858 0.00038184 0.00278443 1.20820000 0.01568341 0.04447756 0.62500000
5 GenNLC<ge> 3.46263262 12.5467763 0.00072954 0.02020473 1.20820000 0.02115366 0.05390487 0.62500000
6 GenNLC<ge> 7.02979774 14.1154960 0.00093248 0.04690095 1.20820000 0.01267363 0.04340435 0.62500000
Although the overall six peaks in the shared parameter fit estimated to [0.00810942, 0.04431287] for the two IRF widths, with these independent estimates, we see both widths fitted to somewhat higher values.
Fitting the Tailed Peaks
If we fit the GenNLC<ge> to just the tailed peaks, with the a4 locked to the same 1.2082 ZDD asymmetry, and a7 locked to the .625 area fraction of the narrow IRF component, we see the following when all sixteen data sets are averaged:
Average for 16 Fits
Fitted Parameters
r2 Coef Det DF Adj r2 Fit Std Err F-value ppm uVar
0.99996608 0.99996602 0.00932128 20,179,722 33.9214551
Peak Type a0 a1 a2 a3 a4 a5 a6 a7
1 GenNLC<ge> 1.21997308 7.11478834 0.00037890 0.00169294 1.20820000 0.01751915 0.04459583 0.62500000
2 GenNLC<ge> 2.07707881 12.6113183 0.00076753 0.01243254 1.20820000 0.01751915 0.04459583 0.62500000
3 GenNLC<ge> 4.21419000 14.1015128 0.00088205 0.02821404 1.20820000 0.01751915 0.04459583 0.62500000
This match with the fronted peaks estimate for the exponential term may be due to the sixth peak with the largest area predominating in the fitting. The narrow width component fitted to almost twice the half-Gaussian SD observed when the fronted peaks were included in the fit.
If we refit with these values and allow a5 and a6 to vary, as we did when fitting all peaks, we can again average the 14 sets for the K+ peak and the 8 sets for Ca+ and Mg+ peaks:
Peak 1: Potassium
Average for 14 Fits
Fitted Parameters
r2 Coef Det DF Adj r2 Fit Std Err F-value ppm uVar
0.99997755 0.99997750 0.00664231 23,650,255 22.4514665
Peak Type a0 a1 a2 a3 a4 a5 a6 a7
1 GenNLC<ge> 1.00811700 7.06420650 0.00036583 0.00113221 1.20820000 0.01302649 0.05164986 0.62500000
2 GenNLC<ge> 1.71314332 12.6582763 0.00077087 0.01025646 1.20820000 0.00945635 0.02333329 0.62500000
3 GenNLC<ge> 3.47345967 14.1050224 0.00089319 0.02319855 1.20820000 0.00708538 0.02966004 0.62500000
Peak 2: Calcium, Peak 3: Magnesium
Average for 8 Fits
Fitted Parameters
r2 Coef Det DF Adj r2 Fit Std Err F-value ppm uVar
0.99997286 0.99997280 0.01309497 19,070,879 27.1416789
Peak Type a0 a1 a2 a3 a4 a5 a6 a7
1 GenNLC<ge> 2.03869919 7.11508092 0.00038183 0.00278430 1.20820000 0.01549019 0.04444166 0.62500000
2 GenNLC<ge> 3.46253250 12.5467267 0.00072935 0.02020250 1.20820000 0.02119142 0.05395209 0.62500000
3 GenNLC<ge> 7.02981958 14.1154983 0.00093258 0.04690051 1.20820000 0.01265206 0.04339503 0.62500000
Fitting Only Tailed Peaks
Fitting Both Tailed And Fronted Peaks
[0.01302649, 0.05164986]
[0.01314113, 0.05167537]
[0.02119142, 0.05395209]
[0.02115366, 0.05390487]
[0.01265206, 0.04339503]
[0.01267363, 0.04340435]
If we compare the independent values of the tailed fits with and without the fronted peaks in the data, the results are nearly identical. The values for three different +a3 tailed peaks, however, vary far more than was observed with the three -a3 fronted peaks.
We will be exploring the genetic algorithm in the IRF Deconvolution procedure with this first of these tailed peaks. With this separate analysis, we expect this K+ peak to fit to about 0.052 exponential width and 0.013 half-Gaussian width. The other two tailed peaks elute too closely together and most data sets lack a baseline between these two peaks, a requirement for the IRF Deconvolution genetic algorithm to be effective.
IRF Deconvolution using a Genetic Algorithm Optimization
If the correlation between the IRF and an intrinsic tailed shape causes the fitting a GenHVL<irf> or GenNLC<irf> model to be consistently overspecified even when locking the ZDD asymmetry and IRF component fractions, there may be an alternative. The Fourier IRF Deconvolution procedure offers a genetic algorithm optimization that seeks to maximize the amount of baseline after the deconvolution of the IRF.
If you look closely at an IRF deconvolution you will see that too weak of an IRF results in the deconvolved peak failing to decay fully to the baseline. If the IRF is too strong, the deconvolved peak will produce an oscillation that falls below the baseline, to negative values. By maximizing the amount of baseline, a good approximation to the IRF is realized.
The main difficulty with this approach on two component IRFs is that nearly all of the tailing rests with the higher width exponential component. If we isolate only this first of the three tailed peaks and use the genetic algorithm locking the .625 area fraction (vary 0%) and allowing the half-Gaussian SD to vary 0.01 ± 50% and the exponential width to vary 0.05 ± 50%, optimizing the BsLnZero procedure with a 0.5% tolerance for the baseline produces the following:
IRF Parm 1 Parm 2 Parm 3
<ge> 0.0050000 0.0476503 0.6250000 CationStd 5ppm 30C
<ge> 0.0050000 0.0542916 0.6250000 CationStd 5ppm 35C
<ge> 0.0050000 0.0525535 0.6250000 CationStd 5ppm 40C
<ge> 0.0050000 0.0469912 0.6250000 CationStd 5ppm 45C
<ge> 0.0050000 0.0562200 0.6250000 CationStd 10ppm 30C
<ge> 0.0050000 0.0517323 0.6250000 CationStd 10ppm 35C
<ge> 0.0050000 0.0508961 0.6250000 CationStd 10ppm 40C
<ge> 0.0050000 0.0476099 0.6250000 CationStd 10ppm 45C
<ge> 0.0050000 0.0560884 0.6250000 CationStd 25ppm 30C
<ge> 0.0050000 0.0536355 0.6250000 CationStd 25ppm 35C
<ge> 0.0050000 0.0504921 0.6250000 CationStd 25ppm 40C
<ge> 0.0050000 0.0498951 0.6250000 CationStd 25ppm 45C
<ge> 0.0050000 0.0588461 0.6250000 CationStd 50ppm 30C
<ge> 0.0050000 0.0573139 0.6250000 CationStd 50ppm 35C
<ge> 0.0050000 0.0547953 0.6250000 CationStd 50ppm 40C
<ge> 0.0050000 0.0511005 0.6250000 CationStd 50ppm 45C
The Half-Gaussian iterated to the lower bound in all of the optimizations. The average exponential width optimized to .0525 (we saw .0516 on the fits for this K+ peak when separate widths were fitted).
The above procedure works well for fronted peaks. For tailed peaks, there is a more sophisticated genetic optimization. We can look at the fits for the K+ peak and see that the IRF attenuates the peak by a certain measure. If you look at the amplitudes in the Numeric Summary for the K+ peak, we see about 1.10x higher amplitude in the deconvolution. The AmpAtten algorithm will isolate the parameters in the vicinity of this 1.10x and optimize those for this maximum baseline. If we again lock the .625 area fraction and allow the half-Gaussian SD to vary 0.01 ± 50% and the exponential width to vary 0.05 ± 50%, optimizing the AmpAtten procedure with a 1.10% amplitude factor, we see the following:
IRF Parm 1 Parm 2 Parm 3
<ge> 0.0069547 0.0565495 0.6250000 CationStd 5ppm 30C
<ge> 0.0064012 0.0591251 0.6250000 CationStd 5ppm 35C
<ge> 0.0056325 0.0566471 0.6250000 CationStd 5ppm 40C
<ge> 0.0055000 0.0522448 0.6250000 CationStd 5ppm 45C
<ge> 0.0061659 0.0614142 0.6250000 CationStd 10ppm 30C
<ge> 0.0115644 0.0575498 0.6250000 CationStd 10ppm 35C
<ge> 0.0140587 0.0571281 0.6250000 CationStd 10ppm 40C
<ge> 0.0055000 0.0534846 0.6250000 CationStd 10ppm 45C
<ge> 0.0124300 0.0575157 0.6250000 CationStd 25ppm 30C
<ge> 0.0127802 0.0554683 0.6250000 CationStd 25ppm 35C
<ge> 0.0115247 0.0539039 0.6250000 CationStd 25ppm 40C
<ge> 0.0141052 0.0528506 0.6250000 CationStd 25ppm 45C
<ge> 0.0085685 0.0517043 0.6250000 CationStd 50ppm 30C
<ge> 0.0122042 0.0510394 0.6250000 CationStd 50ppm 35C
<ge> 0.0107607 0.0495663 0.6250000 CationStd 50ppm 40C
<ge> 0.0133762 0.0491243 0.6250000 CationStd 50ppm 45C
The sixteen peaks average 0.00985 and 0.0547 for the two parameters.
The AmpAtten algorithm will strongly bind the principal tailing component, the exponential, but not the narrow width component. You can use this procedure as a second check on the constrained fits. Simply enter the amplitude factor for the deconvolution in the fit. You may or may not see this genetic optimization recover the parameters of the fits. Here there is no inter-correlation between the tailed intrinsic chromatographic shape and the tailing in the IRF. There is only the raw data, an expectation of the amplitude change arising from the deconvolution of the IRF, and a maximization of baseline within a narrow zone around that measure of sharpened amplitude.
The Wider Exponential Component
The above plots are of a series of zoomed-in deconvolutions with where the half-Gaussian width is set to the 0.009 and the narrow width fraction is set to .625. The exponential width varies .040, .044, .048, .052, .056, and .060. The exponential component's impact on the tailing is so prominent you can actually do a manual deconvolution optimization on this parameter (with the others locked). Here the optimum exponential would be seen as occurring between the .052 of the fourth graph and the .056 of the fifth.
The Narrow Width Component
The above plots are of a series of zoomed-in deconvolutions with where the exponential width is set to the 0.054 and the narrow width fraction is set to .625. The narrow width varies .002, .004, .006, .008, .012, and .020. The narrow width component's impact on the tailing is very subtle. Note also the oscillations around the baseline. This is why a genetic optimization is needed, and why the estimate of this narrow component is difficult. Both the data and the Fourier-filtered deconvolution are noisy and oscillating if you look closely at the baseline.
An IRF is an average of the instrumental and system distortions which alter the true shape of the chromatographic peak. The average parameters within an IRF may vary with the species and with the location of its elution. Further, it is likely that even a two-component IRF is an approximation of a series of complex processes which can perhaps be described sufficiently with a basic two-component model, but where those parameters may not be absolutely constant. This appears to be the case for the narrow component, but it may be that the narrow parameter is simply far harder to estimate since its tailing is mostly masked by the tailing of the much wider exponential. And as we noted in this example, be cautious of blind averaging. IRF parameters can iterate to zero and such fits should be excluded from any IRF averaging.
Estimating an IRF with only tailed peaks is certainly possible, and we appreciate there may well be instances where such is necessary. The procedures outlined in this topic should help get you a respectable estimate.
We will cover one more instance of estimating a tailed peak IRF. We will address the very different IRF shapes which occur with GC peaks.
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