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Appendix AD. Spectroscopy and chromatography combined: time-resolved Classical Least Squares

The introduction of high-speed UV-Visible array detectors into high performance liquid chromatography (HPLC) instruments has significantly increased the power of that method. The speed of such detectors is such that they can acquire a complete spectrum multiple times per second over the entire chromatogram. An example of this is described in a technical report from Shimadazu Scientific Instruments (https://solutions.shimadzu.co.jp/an/n/en/hplc/jpl217011.pdf) which considers the separation of three positional isomers of methyl acetophenone: o-methyl (o-MAP), m-methyl a(m-MAP), and p-methyl (p-MAP). The ultraviolet absorption spectra of these three isomers at a concentration of 400 μg/mL each is shown below on the left, and the chromatographic separation, using the column and conditions specified in their report, are shown in the middle. The report goes on to describe their commercial software, which uses a complex iterative approach to extract the spectra and the chromatographic characteristics from the raw data.


CLSPercentErrors =        0.0021993   0.0020162    0.0015607
PerpDropPercentErrors =  -1.6315     -0.78697      3.272

Here I present a comparatively simple non-iterative technique based on the same chemical system, in which we consider each spectrum acquired by the detector as a separate sample mixture and apply the Classic Least-squares method previously introduced, in which the spectra of the components are known beforehand and where adherence to the Beer-Lambert Law is expected. The spectra and chromatographic peaks are simulated digitally in the Matlab/Octave script TimeResolvedCLS.m, shown in the figure below, by modeling the spectrum of each component as the sum of three Gaussian peaks and the chromatographic peaks as exponentially modified Gaussians. To make this simulation as realistic as possible, the parameters were carefully adjusted to match the graphics in the technical report as close as possible, and the other parameters, such as the spectral resolution, sampling rate, and detector noise (2 milliabsorbance units, mAU), were also directly based on that report. Note that the chromatographic peaks (middle figure) are nowhere near baseline resolved. Therefore, it is to be expected that quantitative calibration based on the measurement of peak areas in this chromatogram (for example by the perpendicular drop method might be inaccurate, especially if the peak heights are very different. In fact, in this case, even though the concentrations of the three components are much lower (0.05 μg/mL for each), the peak areas measured by perpendicular drop are only about 2% from the true values, mainly due to the slight asymmetry and nearly equal height of the three peaks. The spectra (left-hand figure) are even more highly overlapped than the chromatographic peaks, but they are distinct in shape, and that is the key. 

Basically, we treat this as a series of 3-component CLS calculations, one for each time slice of the detector. The actual calculations can be done in two ways, depending on whether the spectra are processed one by one or are collected for the entire chromatogram and then processed all at once, using either "Alternative calculation #1", lines 113-146, or "Alternative calculation #2", lines 150-170. The first method, shown on the left, looks like chromatography as it executes; it computes the chromatographic peaks of the three components point by point as they evolve in time and plots them in the first three quadrants of figure window 3 (on the right). The second method calculates the entire chromatogram in one step at the end and makes the same final plots. (The second method is faster computationally, but that's not significant because the chromatography takes much longer than the calculations). Either way, the result is the same; the chromatographic peaks of the three components are completely separated mathematically, so their areas are easily measured, no matter how much they overlap! Note that, although the three spectra must be known, no knowledge of the chromatography peaks is required; they emerge separate and intact from the data, purely computationally.

Stress test. In order to test the abilities and limitations of this method, I have prepared a series of increasingly challenging scenarios, starting with the one pictured above and becoming progressively more difficult by making the chromatographic peaks more closely spaced, making the peak more asymmetrical, making the spectra more similar, and making the concentrations unequal. These scenarios are listed in the table below, along with the typical percent errors in peak area measurement by the CLS method and links to the corresponding graphics and Matlab/Octave m-files. Each is a more challenging variation on the first one; #2 has much more chromatographic peak overlap; #3 has much more asymmetrical chromatographic peaks (higher tau); #4 has much more similar spectra - in fact, the peak wavelengths differ by only 0.1 nm, making them look identical; in #5, component 2 (the middle peak) has a concentration 100 times lower; and #6 is the same as #5 except that the peaks are highly asymmetrical. In all of these cases, the normal perpendicular drop area measurement technique is either impossible (because there are no distinct peaks for each component) or are very much in error, but the CLS techniques works well, giving very low errors except when the middle peak concentration is 0.0001, which approaches the random noise limit of the detector. (Another variation, TimeResolvedCLSbaseline.m, includes the correction for baseline shifts.)

Peak resolution
Spectral similarity
Peak asymmetry
Concentration ratios
% errors in area measurement
Links
1. Normal
Normal
Slight: tau=10
 .05   .05   .05
 0.0022%      0.002%    0.0016%
Graphic   m file
2. Unresolved
Normal Slight: tau=10  .01   .01   .01
  -0.06%       -0.053%    -0.041%
Graphic   m file
3. Partly resolved
Normal Great: tau=40
 .05   .05   .05
 -0.0004%    -0.013%    -0.066%
Graphic   m file
4. Unresolved Almost complete
Slight: tau=10  .01   .01   .01
 0.054%         0.049%     0.04%
Graphic   m file
5. Unresolved Almost complete Slight: tau=10
 .01  .0001  .01
 0.026%         2.4%       0.019%
Graphic   m file
6. Unresolved Almost complete Great: tau=40  .01  .0001  .01  -0.04%         -3.8%       -0.03%
Graphic   m file

Even when the peaks are resolved well enough for the perpendicular drop method to work, it can suffer from interaction between adjacent peak heights; that is, a change in the peak height of one peak can affect the measurement of the area of adjacent overlapped peaks, because of shifts in the valley point between them. This is illustrated by TimeResolvedCLScalibration.m, which simulates the measurement of 10 different three-component mixtures similar to the above (but modified so perpendicular drop measurement is possible), where the concentrations vary independently and randomly over a 1 x 10-4 to 9.5 x 10-4 microgram/mL range, and then plots measured peak area vs concentration for each component. (Each time you run this, you will get a different mix of concentrations). Linear least-squares fits of peak area vs concentration are calculated, as shown below. In this typical example, the average absolute percentage error in area measurement for the perpendicular drop method is about 5%, with an R2 of 0.995, and for the CLS measurement is less than 1%, with an R2 of 0.9995. (Even if the detector noise (line 22) is set to zero in this simulation, the errors in the perpendicular drop method remain, because they are caused by overlap between adjacent peaks, rather than by noise).


Though clearly the CLS method is very effective, all of this really only proves that the mathematics works well; the method still has the serious limitation that it requires that the spectra of all the components be known accurately. This requirement can be met in some applications, but in liquid chromatography there is a potential pitfall. If gradient elution and/or temperature programming are used, and if the spectra of those chemical compounds are sensitive to the solvent and/or to temperature, for example shifting their peaks slightly, then there will likely be additional errors in the CLS procedure. Obviously this depends on the particular chemical system and will have to be evaluated on a case-by-case basis.


But this suggests another interesting use for this method: speeding up a chromatographic method that normally would achieve complete baseline separation (from which accurate spectra of each component could be obtained in situ), and then adjusting the column and/or flow rate to achieve faster but incompletely resolved chromatograms to which the CLS method could be applied quickly and accurately to multiple samples.

In other applications, some or all the components may simply be unknown, and you may want to obtain their spectra. This can be done in situ if the peak separation is at least as good as that depicted in the first figure above, because you can see that at each peak maximum there is virtually no contribution from the adjacent peaks. Therefore, measuring the spectrum of each component at its chromatography peak maximum should give good spectra for each component in this case.

But what if the peaks are even more overlapped than this, so that pure component spectra are never achieved? In that case, more sophisticated methods must be used, such as the one described in the Shimadzu technical report. This involves making initial estimates of spectral and chromatographic peaks, followed by an iterative search for the best fit to the experimental data, subject to the imposition of some important known prior constraints, such as non-negativity of spectra and of the chromatography peaks (those peaks are always positive, except for random noise on the baseline), and the unimodality of the chromatography peaks (that is, each component gives one and only one chromatography peak). Methods of this type will be left to a future expansion of this book.

This page is part of "A Pragmatic Introduction to Signal Processing", created and maintained by Prof. Tom O'Haver , Department of Chemistry and Biochemistry, The University of Maryland at College Park. Comments, suggestions and questions should be directed to Prof. O'Haver at toh@umd.edu. Updated July, 2022.