CHAPTER 1
Measurements and Models
1 The Basic Measurements
The basic measurements made by an analyst on a chromatogram for purposes of quantitation are shown in Figure 1.1; they comprise all the 'Y's and 'X's of the trace, the voltages and times, their combinations, repetitions and trends. There are no other measurements to be made, but much analytical information can be derived from these measurements.
These quantities are obtained by integrators, lab computers or data processors – the names are used synonymously throughout this book. The integrator is the only window into the chromatograph to show what is happening. It is both measuring device and diagnostic tool.
Regulatory pressures from Good Laboratory Practice (GLP), Good Manufacturing Practice (GMP), IS0 9000 and bodies such as the US Food and Drug Administration (FDA) put increasing demands on the quality and credibility of analytical results. It is the aim of this book to show analysts how to use integrators to provide good quality results – provided that the chromatography is up to it.
Measurements and Their Use
Integrator measurements serve three analytical purposes:
(1) solute identification and quantitation;
(2) diagnostics, trouble-shooting and system measurement;
(3) results assessment and trend analysis;
and, increasingly, one corporate finance purpose for monitoring:
(4) performance measurement and lab resource utilization.
2 Quantitation
Peak Area and Peak Height
Solute quantity is measured from peak areas and/or heights. In a controlled analysis, peak area is the true measure of solute quantity if the solute elutes intact and is detected linearly. It can be shown to be so theoretically for peaks with or without a known shape, and experimentally by plots of area against solute quantity.
Peak height is an alternative measure of solute quantity. There is no theoretical proof of this unless a specific peak shape such as Gaussian is assumed, but experimentally it is easy to show that plots based on height and quantity are also linear over a usable though smaller range (Figure 1.2), even for peaks that are not particularly symmetrical as long as their shapes do not change. Peak height is an easier measurement to make and has advantages for the measurement of small overlapping peaks, but it ignores most of the data contained in the detector signal.
The choice of which is best to use in practice is discussed at the end of Chapter 2. Area is the normal choice when signal-to-noise ratio (SIN) is large because height is susceptible to peak asymmetry while area is not, and the linear dynamic range for area measurement is greater than for height. When signal-to -noise ratio is small, height is preferred because errors of baseline placement affect height less than area.
Peak 'Volume'
The ability of the photo diode array detector (DAD) to measure peaks over a range of wavelengths allows a third measure of solute quantity: peak 'volume'. Here, all the peak areas measured at the wavelengths to which the analyte species responds are added together to create a three-dimensional measure (time, amplitude, wavelength). It is proposed that this measure is better because it contains more data and more information. For this to be true, background noise must be proportional to analyte signal otherwise the signal-to-noise ratio degrades away from λ max and the benefits of the extra information become compromised by its lower quality. The usual case is that noise has origins independent of the solute signal and does degrade the signal.
Peak volume would be better than area or height if the signal-to-noise ratio is good at all measurements but this may be hard to achieve. With current diode array detectors and current integration software, peak area (measured at λmax) remains the safest measure of solute quantity, especially for trace peaks.
Retention Time and Solute Identity
Once solutes in a mixture have been identified, they are subsequently recognized on a daily basis by their retention times. Integrators assign peak names and response factors to peaks which elute inside a specified time window. If another peak elutes at that time it also will be recognized as the expected peak; if two peaks co-elute they will not be uniquely identified. If retention time varies with sample size due to increasing peak asymmetry, it is quite possible for incorrect identification to be made when a peak crosses from one window into the next. When this happens, the integrator will assign the wrong name, response factor and standard concentration to that peak, and these errors will continue into the final report.
Integrators can measure relative retention time, i.e. the peak retention time compared with the retention time of a standard peak. Experimental variations cancel but the errors of both measurements add together so that relative retention times are more accurate but less precise than absolute retention time. There is also the usual problem of finding a suitable standard.
To an integrator, retention time is the elapsed time from the moment of injection until the peak maximum emerges, which includes the gas or solvent hold-up time. The retention time of asymmetric peaks does not coincide with the centre of gravity of the peak. Separation of the observed retention time tR (mode), from the peak centre of gravity (mean) is one measure of asymmetry (see also: First Moment and Equation 40).
The mean retention has not achieved any common use in the analytical laboratory. It is difficult to measure manually but integrators which sample the peak signal at a fixed frequency could, with a simple addition to their software, measure it very easily. Its theoretical value lies in the separation of tR and tmean being equal to the Exponentially Modified Gaussian (EMG) time constant τ (Figure 1.3).
Column Hold-up Time
The journey time of a molecule or atom of mobile phase from the beginning to the end of the column is called the column hold-up time. Since the mobile phase is the propellant of a chromatograph, no solute can emerge before this time has elapsed. The hold-up time is the shortest retention time possible, it is also equal to the total residence time of a solute in the mobile phase as it traverses the column.
Integrators measure the column hold-up time as the retention time of an unretained solute. Knowing this time allows the analyst to optimize mobile phase flow rate to achieve maximum column performance.
3 Diagnostics and System Suitability Tests
Most of the instrument checks that an analyst makes before injection to assess instrument readiness can be made automatically by standard integrator routines, or by a software program running in the integrator. Absence of noise and a stable baseline in the correct place (near zero millivolts) are accepted indications that a chromatograph is ready for sample injection. Tests on peak shape, size and resolution indicate whether the autosampler, column and detector are working to specification. Injection of standard or known samples calibrates the detector response and allows comparisons with working samples ('unknowns') in order to confirm that the correct sample has been injected and is within specification or not. Finally the sheer number of analyses per day can be counted to show whether the chromatograph is fully utilized, whether the work load is building up, and other trends.
The drift away from stand-alone integrators towards computer systems linked to laboratory information management (LIM) systems has accelerated the creation of multiple reports and database storage of results. When properly introduced, this additional scrutiny of results for trends brings good improvements in quality control.
Baseline Noise
Baseline noise makes peak areas hard to measure and so reduces confidence in analysis results. Integrators assess baseline noise by means of 'Noise' or 'Slope Sensitivity' tests, and report its average value. On a daily basis this value should be recorded and the trend shown to be reasonably constant; analysts should know what noise levels are acceptable. Any worsening of noise must be addressed; it might indicate that the column needs reconditioning, the detector needs cleaning, or that some part of the electronics is about to fail.
Baseline Signal Level
Unexpected high signal levels warn the analyst that a peak which ought not to be there is eluting, or that the detector has become contaminated; it may be that temperature or flow control is drifting. Whatever the reason, some delay or corrective action is necessary before the next injection. Integrators indicate a value for the baseline level and some incorporate a 'Not Ready' warning if the level is outside a prescribed range.
Peak Boundaries
The start and end times of peaks are measured by noting the times when integration of a peak is commenced and terminated. These times are used to measure peak width and, when combined with retention time, to measure peak asymmetry (see below).
Knowing these times allows the analyst to assess column performance and judge whether the column needs conditioning or replacement.
Peak Width and Column Efficiency
The measurement of column performance in terms of efficiency, peak skewness or asymmetry, and peak resolution are key System Suitability tests.
A column is 'efficient' if the lateral diffusion of solute bands is restricted during their column residence time. The narrower the peaks are, the better the separation from neighbouring peaks and the more accurately they can be measured. Efficiency is measured as numbers of theoretical and/or effective plates per column. It is calculated from peak widths (actually, from peak variance σ2, see Figure 1.6) and retention times.
In repeated analyses, the retention time and width of a fully-resolved peak at a fixed height above the baseline should be constant. Variations in either quantity indicate that column performance is varying or that some experiment-controlling parameter such as temperature or flow is drifting.
Peak width is related to variance and contains components of spreading added by the injection port, system dead volume, and by the detector and electronics as well as the column.
If σ is the standard deviation of a peak (see Figure 1.6), σ 2 is the peak variance and:
σ2total = σ2inj + σ2col + σ2det + σ2etc. (1)
Of these, the contributions to variance from injection port and column are the most important as they can become worse with instrument use. Detector broadening is important until the cause is found and removed; once removed it tends not to recur. Extra-column broadening effects are observed in isolation by connecting the injection port directly to the detector with, for example, uncoated capillary tubing. If the injection technique is not repeatable, or if the column is deteriorating, there will be a variation in peak width and apparent variation in column efficiency.
Most integrators can measure peak area and height simultaneously and this allows empirical measurement of peak width. For symmetrical peaks the ratio of area to height gives the peak width at 45.6% of the height [= [square root of ((2π)σ), see Equation 13]; for asymmetric peaks, the ratio is equal to the width at some other height. Alternatively, peak width can be measured as base width, the time interval between the start and end of peak integration (= 6σ for a symmetrical peak). Once peak width is known, column efficiency can be calculated.
Column efficiency is generally defined as the number of theoretical plates N, calculated from:
N = TR2/σ2 (2)
However, the value of σ2 can vary with the peak model chosen and tR may be taken at the centre of gravity of the peak rather than the conventional peak top. For a Gaussian peak:
σ = wh/2 [[square root of (-2 ln(h))]] from (16) below
thus
N = 8tR2 ln(h)/wh2 (3)
where N = number of column plates
tR = retention time
wh = width of peak at fractional peak height h, usually h = 0.5 H
All are quantities the integrator can measure.
There are limitations in measuring column efficiency this way for real peaks – the theory is based on the Gaussian model and few peaks are exactly Gaussian, but monitoring the width of a well-resolved peak in the chromatogram over successive analyses will empirically indicate any column degradation or highlight imprecise control of instrument parameters. It will not necessarily identify what is causing the change or what is to be corrected. Alternative calculations of efficiency based on asymmetry or the second peak moment (q.v.) are not always available in integrators.
Peak Asymmetry
Asymmetry broadens the base of a peak, increases peak overlap with neighbours and makes peak areas harder to measure; an aim of method development is therefore to produce symmetrical peaks.
A peak will be symmetrical if
(i) injection technique is good;
(ii) dead volume in the solute flow path is absent;
(iii) the solute quantity is not large enough to overload the column or detector;
(iv) the residence time of the solute in the stationary phase is long enough to achieve dynamic equilibrium;
(v) the solute adsorption isotherm is linear.
It follows that where peak asymmetry is evident, some improvements to the analysis may be desirable: change to a different column, improve the injection technique, reduce the injection volume, etc.
Asymmetry can be described in terms of the separation of peak mean and mode times, but a more common measure of asymmetry compares the peak half widths on either side of the peak mode (see Figure 1.4). This is called the Asymmetry Ratio, or sometimes just 'Tailing', and it varies with peak height.
Such measurements can in principle be made at any height but are usually made near the peak base where asymmetry is greatest, typically at 10% of peak height for which much theory has been developed or at 5% for US Pharmacopoeia and FDA requirements. When peaks overlap, the widths at 10% peak height remain measurable for longer, but obtaining the '5% width' demands a better standard of chromatography.
Asymmetry can also be expressed as the Tailing Factor or Tailing Coefficient,
Tailing Factor = w0.052/2A = (B + A)/2A (4)
which is the average of B/A and 1, and therefore half as variable as B/A alone.
At the top of the peak, measurements of B and A would be relatively small, more sensitive to measurement errors, and B/A is closer to unity. Nevertheless, it has been suggested" that it is probably more accurate, although less precise, because there is less measurement interference from neighbours and noise at the top of a peak.
Measurement of the Asymmetry Ratio is one of the System Suitability measurements and a standard routine on most integrators. Experimentally, measurements can be taken from any peak in the chromatogram, but they should only be taken from fully-resolved peaks.
The asymmetry measurement, B/A, as shown in Figure 1.4 is used in the evaluation of the mean retention time, the number of column plates and variance derived from the Exponentially Modified Gaussian peak model.
Peak Resolution
Peak resolution is the degree of separation of two adjacent peaks measured in units of peak width rather than time. It compares the actual peak top separation to the average constructed peak base width and is used to judge whether the separation between two peaks is good enough for accurate quantitation and fast analysis. It is a dimensionless quantity. In Figure 1.5:
Resolution, RS = tR2 - tR1/ (w2 + w1)/2 (5)
Allowing that the constructed base width of a symmetrical peak is approximately twice the width at half height w0.5:
RS ≈ tR2 - tR1/(w0.5,1 + w0.5,2)/2 (6)
In theory, a resolution of 1 means that the peaks are just resolved down to baseline and areas can be measured accurately. In practice, if peaks are unequal in size and asymmetric, a resolution nearer 2 is needed to obtain near baseline separation, and even that is sometimes not enough.
Equation 6 is only an approximation based on Gaussian theory and can be as much as 100% in error for asymmetric peaks, but resolution of peaks calculated from the difference in their retention times and widths (derived from Area/Height) can be monitored as an empirical test to show when the column needs reconditioning or replacement.
Most analysts seek to develop analyses which have near symmetrical peak shapes and so avoid problems of non-Gaussian theory.
