In this study, we describe a simple and rapid liquid chromatography–mass spectrometry (LC–MS) method for the evaluation of caffeine, taurine, and aspartame in teas, soft drinks, and energy drinks using high performance liquid chromatography (HPLC) coupled with electrospray ionization (ESI)–MS detection and multiple injections in a single experimental run (MISER) analysis. The misergrams obtained from the injection of one sample per minute allows the convenient visualization of the outcomes of multiple analyses, allowing rapid comparison of analyte levels among a variety of products and brands.

	The technique of liquid chromatography–mass spectrometry (LC–MS) is evolving into a workhorse technology used across many industries and disciplines for routine chemical analysis and problem solving (1–7). Recent trends in miniaturization and cost reduction suggest an opportunity for even more widespread adoption of LC–MS beyond the traditional laboratory setting (8–13). Consequently, studies that familiarize students and other newcomers with the practical aspects of LC–MS are becoming increasingly important. While mastery of LC–MS may require many years of training and experience, a working knowledge can be picked up in a short time, especially when simplified versions of the technique are used.


	Photo Credit: Patrick Llewelyn-Davies/Getty Images

	Multiple injections in a single experimental run (MISER) (14) is a simplified form of LC–MS that is well suited to fast analysis of the presence of a given component within multiple samples (14–19). Minimal chromatographic separation is used, allowing fast analysis while separating the component of interest from substances that could potentially interfere with MS detection. The injection of multiple samples in the same experimental run allows a straightforward interpretation of the MS detector response (or "misergram") providing ready interpretation without the need for difficult and time consuming peak integration, data processing, and graphing.

	We previously investigated the application of MISER LC–MS to a study of the level of capsaicin in chili peppers and hot sauces, finding the experimental approach to be easily grasped by students (19). We now report an investigation into the use of MISER LC–MS for the analysis of caffeine, aspartame, taurine, and other components within teas, soft drinks, and energy drinks, with the hope that this approach may be utilized by students and others interested in rapidly developing a familiarity with LC–MS and high-throughput analysis.

Reversed-phase high performance liquid chromatography (HPLC) experiments were performed on an Agilent 1100 system. The Agilent stack comprised a G1312A binary pump, a G1367A WPALS autosampler, a G1315B diode-array detector, and a 6120 quadrupole LC–MS detector with electrospray ionization in the positive mode. The system was controlled by Chemstation software, with the flow injection analysis (FIA) mode enabled.

Chemicals, Reagents, and Stationary Phases: 

Acetonitrile (HPLC grade) was purchased from Fisher Scientific. Formic acid (HCOOH), ammonium formate (NH

), caffeine, aspartame, taurine, theanine, vitamin C, vitamin B5, and sucralose were purchased from Sigma–Aldrich. Ultrapure water was obtained from a Milli-Q Gradient A10 water purification system from Millipore. The 50 mm × 4.6 mm, 2.5-μm XBridge Phenyl column was purchased from Waters Corporation.

Solutions containing 2 mM ammonium formate in water (pH 3.5) and 2 mM ammonium formate in acetonitrile (pH 3.5): 12.6 g ammonium formate and 7.9 mL formic acid were dissolved in 1 L of Millipore water. A 100-fold dilution of this stock solution was performed in either pure water or a 90:10 acetonitrile–water mixture to prepare the 2 mM solutions.

 A single Lipton All Natural tea bag was placed in 8 oz (approximately 240 mL) of boiling water, and the bag was submersed using a kitchen thermocouple. A 2-mL sample was taken at t = 0 s (before tea bag addition), 30 s, 60 s, and every minute thereafter up to 8 min of total steeping time. (Note: The tea bag was not intentionally moved during the course of the experiment.)

Brewed tea samples were achieved by placing the noted amount of tea in a Williams–Sonoma Open Kitchen Tea Ball, or a single tea bag where noted, and placing the ball or bag in 8 oz of boiling water for 5 min. The temperature was recorded at the start and end of the steeping, and a 2-mL sample was taken for analysis at the end of brewing. (Note: The tea ball [or bag] was not intentionally moved during the course of the experiment.) The observed average brewing temperatures were T

Samples of various beverages, including Fuze iced tea, were obtained from a local supermarket by high school interns and directly analyzed without treatment. Authentic standards of caffeine, aspartame, and taurine were prepared by serial dilution and analyzed along with the beverage samples.

HPLC separations were carried out on a 50 mm × 4.6 mm, 2.5-μm XBridge Phenyl column by isocratic elution at a flow rate of 1 mL/min. The LC eluents were 60% solvent A (2 mM ammonium formate in water, pH 3.5) and 40% solvent B (2 mM ammonium formate in 90:10 acetonitrile–water, pH 3.5). The column and samples were maintained at a temperature of 25 °C. The misergrams were obtained from continuous sample injections (1 μL) every 1 min. The positive ion electrospray ionization (ESI) parameters were as follows: skimmer, 45 V; desolvation gas, nitrogen; temperature, 350 °C; and flow rate, 12 L/min. The nebulizer was adjusted to 35 psig, and the fragmentor and the capillary voltages were adjusted to 150 and 2500 V, respectively. G1969-85000 ESI-L Low Concentration Tuning Mix (Agilent Technologies) was used for tuning and calibration of the mass spectrometer.

Ion Suppression and Enhancement Between Target Analytes: 

Six sets of samples were prepared at 0.25 mg/mL. Set 1 contains aspartame and caffeine; set 2, caffeine; set 3, caffeine and taurine; set 4, aspartame; set 5, aspartame and taurine; and set 6, taurine. All six samples were analyzed per duplicated using the standardized MISER LC–ESI(+)–MS method (as described above). In addition, a faster MISER LC–ESI(+)–MS method was evaluated for comparison: Column: Zorbax Stablebond SB-C18 (20 mm × 3.0 mm, 1.8-μm, Agilent); temperature: 40 °C; detection: selected ion monitoring (SIM) ESI–MS(+) at 

 = 195, 295, and 265 amu; sample: 0.6-μL injection of each component solution in acetonitrile–water; flow rate: 1 mL/min; isocratic mobile phase: 10% eluent A: 2 mM ammonim formate in water (pH 3.5) and 90% eluent B: 2 mM ammonium formate in 10:90 water–acetonitrile (pH 3.5). Time between injections was 22 s. The extent of ion suppression and ion enhancement effects are clearly visualized in the corresponding misergrams.


	Figure 1: Structures of caffeine, aspartame, and taurine. 

	Students learning elementary principles in chemistry can find ready subject matter for study in the modern world. In addition to the practical advantages of sample availability, convenient disposal, and reduced requirements for personal protective equipment, the study of materials that can be found in a local supermarket provides a quick connection with the everyday world, reinforcing the important message that chemistry is omnipresent in the modern world. The analysis of caffeine in beverages is a popular laboratory experiment with a proven ability to engage students (20,21). Bier and Grabowski (22) have even implemented virtual MS approaches. In this study, we apply the MISER LC–MS technique to the measurement of caffeine and other substances in teas, soft drinks, and energy drinks collected at local supermarkets.

	Samples of commercial teas were brewed before analysis (see the experimental section). A timecourse study of the appearance of caffeine in brewed tea upon the addition of water is shown in Figure 2(a). Sample aliquots were removed at designated timepoints and analyzed using MISER LC–MS with a sample injection period of 1 min and detection at 

of 195, corresponding to the caffeine molecular ion. The resulting misergram shows the kinetic profile of the appearance of caffeine in the brewed tea. Inclusion of several caffeine standards allows approximate quantitation. A maximum caffeine concentration of about 0.1 mg/mL is reached by about 5 min. It should be noted that the steeping tea samples were not stirred, and that more rapid extraction of caffeine may be possible when teabags are stirred or "dunked."


	Figure 2: MISER HPLC–MS analysis of caffeine in various teas. Analysis of samples and authentic standards by MISER HPLC–MS with selected ion monitoring of the caffeine molecular ion at 195 amu reveals relative caffeine levels of timecourse extraction study from (a) Lipton all natural tea and (b) nine different teas. Chromatographic and MS conditions were as described in the experimental section. Detection: ESI–MS(+) at m/z = 195 amu; injections: 0.6 μL every 1 min.

	A comparison of several different commercial teas, each prepared with 5 min steeping is shown in Figure 2(b). An iced tea beverage is also included for comparison. While most of the teas show a caffeine level of about 0.1 mg/mL, the level in the iced tea is significantly less, and perhaps not surprisingly, only a trace amount of caffeine is observed in the sample of decaffeinated tea included in the study. In addition, it should be noted that the loose teas studied differed in particle size from large (Frutto Blanco Pearl), to small (English Breakfast tea ground for the Flavia brewing system), to almost powder-like (Matcha green tea). These results are in agreement with previous studies where caffeine concentrations in white, green, and black teas ranged from 14 to 61 mg per serving (6 or 8 oz) (23). Taken together, the MISER LC–MS experiment provides an easily interpreted graphical depiction of the experimental results.

	An advantage of the MISER LC–MS analysis is the ability to simultaneously monitor the presence of several different analytes. We next set out to exploit this capability, focusing on the simultaneous evaluation of caffeine, aspartame, and taurine in soft drinks and energy drinks. Accurate quantitation with the MISER LC–MS technique requires analytes of interest to be chromatographically resolved from potentially interfering substances in the sample mixture. When several analytes are simultaneously analyzed, it is important to demonstrate that the short analysis times for MISER LC–MS do not lead to sample interference. Model experiments can help to rapidly assess potential interference issues, identifying any possible "crosstalk" between the principal analytes under investigation. Figure 3(a) shows the results of such a method development study, clearly illustrating that the shorter method leads to a significant ion suppression of caffeine and taurine MS signal in the presence of aspartame, while the longer 1-min injection period is free from such problems.


	Figure 3: Signal response comparisons (m/z 195, 295, and 126) for caffeine, aspartame, and taurine standards and the respective two component mixtures: (a) MISER HPLC–MS profiles obtained from a fast method: column: SB-C18 (20 mm × 3.0 mm, 1.8-μm); temperature: 40 °C; detection: SIM ESI–MS(+) at m/z = 195, 295, and 265 amu; sample: 0.6-μL injection of each component solution at 0.25 mg/mL in acetonitrile–water; flow rate: 1 mL/min; isocratic mobile phase: 10% eluent A: 2 mM ammonium formate in water (pH 3.5) and 90% eluent B: 2 mM ammonium formate in 10:90 water–acetonitrile (pH 3.5). Time between injections: 22 s. (b) MISER HPLC–MS profiles obtained from the standard method as describe in the experimental section. Time between injections: 1 min.

	The simultaneous evaluation of caffeine and aspartame in various soft drinks is shown in Figure 4. No caffeine signal was observed for the orange soda or Pepsi caffeine-free products, which was consistent with the labels on these products. The levels of caffeine observed in Coke, Diet Coke, and Pepsi were found to be slightly higher than 0.1 mg/mL. Pepsi Max was shown to have about 0.2 mg/mL, in keeping with reported values (24).


	Figure 4: MISER HPLC–MS analysis of caffeine and aspartame in various sodas. Analysis of samples and authentic standards by MISER HPLC–MS with selected ion monitoring of molecular ions at 195 and 295 amu. Experimental conditions as described above. Detection: ESI–MS(+) at m/z = 195 and 295 amu; injections: 0.6 μL every 1 min.

	The levels of aspartame in Diet Coke and Diet Pepsi were found to be about 0.5 mg/mL, in keeping with reported values (25). Very small peaks for aspartame (<< 0.05 mg/mL) were observed in both Coke and Pepsi, although neither of these products contain aspartame as a listed ingredient. This finding highlights one of the potential pitfalls of using MISER LC–MS for detecting trace amounts of analytes in very complex mixtures: It may well be the case that the complex caramel mixtures or other components in these colas may contain a component that gives rise to a small amount of a coeluted component with mass 295 amu. Such interference problems could potentially be reduced or eliminated by the use of high-resolution MS detection using exact mass or multiple reaction monitoring (MRM), or the application of selected reaction monitoring (26,27) to multiple product ions from one or more precursor ions (28,29).

	We next investigated the analysis of caffeine, aspartame, and taurine in a variety of different "sports energy" drinks, which have risen tremendously in popularity in recent years. In addition to caffeine, energy drinks often contain additives ranging from high-fructose corn syrup (for non-diet versions) to taurine, B vitamins, sucralose, aspartame, maltodextrin, glucuronolactone, carnitine, creatine, inositol, and various extracts of açaí, ginseng, and 

	The misergrams shown in Figure 5 show the relative levels of caffeine, taurine and aspartame in a variety of different energy drinks, with Diet Coke and Coke included for comparison. Initial evaluations by direct sample injection showed much higher levels for the three components, with resulting poor baselines. Consequently, all samples were diluted 10 times with water, with the exception of Red Line Xtreme and 5 Hour Energy, which were diluted 100×. The resulting misergrams show much higher caffeine content in the energy drinks than in Coke and Diet Coke, with 5 Hour Energy showing a considerably higher amount of caffeine (similar signal response as Rock Star drinks, but the sample is 10 times more diluted). Red Bull sugar free and Total Zero show aspartame concentrations lower, but comparable to Diet Coke. Rock Star shows almost no aspartame, but checking the ingredients we found sucralose, another artificial sweetener commonly used in energy drink preparations. On the other hand, the taurine signal response from Rock Star, Monster, and Red Bull drinks were very similar at about 0.4 mg/mL, but higher than AMP Boost Cherry. All of these results are consistent with the caffeine, aspartame, and taurine levels published in the different literature sources (24,25,31–35).


	Figure 5: MISER HPLC–MS analysis of caffeine, aspartame, and taurine in various energy drinks and sodas. Analysis of samples and authentic standards with selected ion monitoring of the caffeine, aspartame, and taurine molecular ions at 195, 295, and 126 amu, respectively. Experimental conditions were as described in the text. Detection: ESI–MS(+) at m/z = 195, 295, and 126 amu; injections: 0.6 μL every 1 min. The first two energy drink samples (Extreme Triple Berry and 5 Hour Energy Berry) were 100× diluted, while all of the other samples were 10× diluted.

	During the course of our investigations it became clear that in addition to caffeine, aspartame, and taurine, many other components are present in some of these beverages. Many products contain vitamins (such as vitamins C, B5, B6, and B12), artificial sweeteners such as sucralose or even added compounds such as açaí, choline, various forms of ginseng, inositol, carnitine, creatine, and 

 Figure 6 illustrates how the MISER LC–MS approach can potentially be used for rapid investigation of the presence of some of these additional components, although additional analysis and investigation of potential interference issues would be needed for definitive studies.


	Figure 6: MISER HPLC–MS analysis of multiple analytes in different samples: Theanine levels of timecourse extraction study from (a) Lipton all natural tea and (b) nine different teas. (c) Vitamin C and B5 levels of three different vitamin waters. (d) Sucralose levels in 24 different energy drinks (all samples were 10× diluted, except AMP and Rock Star drinks). Chromatographic and MS conditions as described in the experimental section. Detection: ESI–MS(+) at m/z = 175 (theanine) and 220 amu (vitamin B5); ESI–MS(-) at m/z = 175 (vitamin C) and 397 (sucralose) amu, 0.6-μL injection every 1 min. Fragmentor voltage was kept at 150 eV for all compounds, except for vitamin C (100 eV).

	Theanine, a nonprotein derived amino acid, is a component of tea that has received considerable attention recently (36–38). The timecourse analysis for extraction of this component in English Breakfast tea is shown in Figure 6(a), with the level reaching a maximum value at about 5 min, similar to what was observed with caffeine (Figure 2[a]). The level of theanine in nine different teas is shown in Figure 6(b). The level in the bottled iced tea drink is seen to be quite low, a result that is consistent with the report that theanine tends to be highest in freshly brewed teas, with degradation over time being observed in bottled teas.

	The levels of vitamins C and B5 in three different vitamin waters are shown in Figure 6(c). Similar amounts of vitamin B5 (about 0.2 mg/mL) are seen in all three products, whereas dramatically different amounts of vitamin C are observed. The latter result may be an artifact caused by interference, as reported values suggest that the vitamin C content of Berry Punch should be only about 30% lower than that of Orange Mango. Finally, the evaluation of sucralose in a variety of energy drinks (Figure 6[d]) shows the highest levels for AMP Sugar Free, Rock Star Sugar Free, and Rock Star Zero Carbs; all of them were reported to contain this artificial sweetener. This is also consistent with the observation that no aspartame was detected in Rock Star drinks (Figure 5).

	The advantage of the MISER LC–MS (high throughput, simple readout of results) for simultaneous semiquantitative evaluation of caffeine, taurine, and aspartame levels in teas, soft drinks, and energy drinks can provide an entertaining introduction to the important analytical technique of LC–MS. While this approach is fast and simple, and easily grasped by students, it is important to note that MISER LC–MS is typically used for the analysis of relatively clean samples containing only a few potentially interfering substances, and may be less well suited for reliable quantitation when dealing with samples with highly variable and complex matrices. When more exact quantitation is required, model experiments can help to rapidly assess potential interference issues (39–41).

	The approximate levels of caffeine, aspartame, and taurine in soft drinks and energy drinks can be easily measured by MISER LC–MS, providing a useful introduction to the important analytical techniques of liquid chromatography and mass spectrometry. Considerable variation in these analyte levels is clearly visualized among a variety of these beverages, with the energy drinks showing a higher concentration of caffeine and taurine, and the diet soft drinks containing a higher amount of artificial sweetener (aspartame).

	We are grateful to MRL Postdoctoral Research Fellows Program for financial support provided by a fellowship (E.L.R) and also to the MRL Process and Analytical Chemistry Laboratory for making available the use of the HPLC–MS system used in this study.

 work in process and analytical chemistry at the Merck Research Laboratories in Rahway, New Jersey, USA. 

Christina Kraml, E. Celeste Welch, Margaret J. Welch, Hannah Semmelhack, Danielle Almstead, Abigail S. Kress, 

 are with Lotus Separations with the Department of Chemistry at Princeton University in Princeton, New Jersey, USA. Direct correspondence to: 

	(1) S.J. Hird, B.P.Y. Lau, R. Schuhmacher, and R. Krska, 

	(4) J. Zheng, J. Mehl, Y. Zhu, B. Xin, and T. Olah, 

	(6) K. Wilczewska, A. Kot-Wasik, and J. Namiesnik, 

	(7) E.L. Regalado, R.K. Dermenjian, L.A. Joyce, and C.J. Welch, 

	(8) S.-L. Lin, T.-Y. Lin, and M.-R. Fuh,

	(9) M.R. Gama, C.H. Collins, and C.B.G. Bottoli, 

	(11) A. Rocco, A. Maruska, and S. Fanali, 

	(13) L. Rieux, E.-J. Sneekes, R. Swart, and M. Swartz, 

	(15) W. Schafer, X. Bu, X. Gong, L.A. Joyce, and C.J. Welch, in 

 C.J. Welch, Ed. (Elsevier, Oxford, UK, 2014) 9, pp. 28–53.

	(17) R. Papp, U. Andersson, and L.-D. Cantin, 

	(19) C.J. Welch, E.L. Regalado, E.C. Welch, I.M.K. Eckert, and C. Kraml, 

	(21) R.E. Leacock, J.J. Stankus, and J.M. Davis, 

	(22) E. Bier Mark and J. Grabowski Joseph, in 

(American Chemical Society, 2007) 970, pp. 171–187.

	(23) J.M. Chin, M.L. Merves, B.A. Goldberger, A. Sampson-Cone, and E.J. Cone, 

http://www.caffeineinformer.com/the-caffeine-database

	(28) R.W. Kondrat, G.A. McClusky, and R.G. Cooks, 

	(30) B. Meier (1 January 2013), "Energy Drinks Promise Edge, but Experts Say Proof Is Scant". 

	(31) J.P. Higgins, T.D. Tuttle, and C.L. Higgins, 

	(33) H.K. Khurana, I.K. Cho, J.Y. Shim, Q.X. Li, and S. Jun, 

	(34) L.C. Tamamoto, S.J. Schmidt, and S.-Y. Lee, 

	(38) Q.V. Vuong, M.C. Bowyer, and P.D. Roach, 

	(39) B.K. Matuszewski, M.L. Constanzer, and C.M. Chavez-Eng, 

	(40) S. Murugan, N. Pravallika, P. Sirisha, and K. Chandrakala, 

	(41) A. Furey, M. Moriarty, V. Bane, B. Kinsella, and M. Lehane, 

Articles

Cover Story

This article describes a simple and rapid LC–MS method for evaluation of caffeine, taurine, and aspartame in teas, soft drinks, and energy drinks using HPLC coupled with electrospray ionization mass spectrometry detection and MISER analysis.

MISER LC–MS Analysis of Teas, Soft Drinks, and Energy Drinks

Spreadsheet computer simulations can identify the influencing factors for the set-up of a calibration function such as the number of calibration points and their distribution or the position of the experimental points. A simple linear least-squares fit (linear regression) is not always allowed from theory but a weighted fit may be needed. By using a Monte Carlo approach, that is, by generating a large number of calibration functions and associated sample data points (for example 1000 for each set of simulations), the quality of the experimental results (bias and standard deviation) can be studied under different conditions. This article presents a spreadsheet for the simulation of unweighted and weighted linear least-squares fit. In practice, weighted fitting is only needed if all of the following conditions are fulfilled: a large calibration range; an equal distribution of the calibration points; the absolute standard deviations of the calibration points are not constant; and a sample result is at the lower end of the calibration range. 


	Photo Credit: Andrey Prokhorov/Getty Images

	Quantitative chromatographic methods require calibration by a pure standard that is identical to the analyte. Single-point calibration is quick and can be used if the analyst was able to show during validation that the relation between analyte concentration and signal (peak area or height) is linear and that a blank sample gives a signal value of zero. However, in many real laboratory situations a multipoint calibration is used. Several standard solutions of well-known analyte concentration 

 are investigated by the analytical instrument, yielding their corresponding signals 

 = slope is calculated by linear least-squares fit (also called linear regression) with three or more calibration points (1–5). The sample is subsequently investigated, yielding its signal 

, and the analysis function is used to calculate the analytical result; namely the concentration (c) of the analyte in the prepared sample solution (which in most cases is not the analyte concentration in the sample proper because the latter was prepared, diluted etc.):

 values usually differ in their "quality", which is expressed as standard deviation or uncertainty. The standard will be very pure and, at its best, come with a certificate of analysis; it will be weighed and diluted with great care. Therefore the analyte concentrations of the standard solutions are known with low uncertainty, for example in the 10

 (0.1%) range or even better (6). (The uncommon case of noteworthy uncertainty of the 

 data was discussed by Tellinghuisen [7].)

 values may have a higher uncertainty. This is because it is difficult to keep the injection precision of a liquid chromatograph at ≤1% in the long term (8); the analytical process may have its own variability; and the signal-to-noise ratio (

 of the signal adds significantly to the repeatability of the signal if 

 is smaller than 100 (9). Moreover, the classical model of linear least-squares fit works with exactly known 

 values and a constant absolute standard deviation of all 

	The analyst should try to set up a calibration function that yields analytical results with low standard deviation and bias while keeping in mind that the laboratory workload increases with the number of calibration points. Questions of interest are:

		How many calibration points are necessary?

		How should the calibration points be distributed?

		Is a narrow calibration interval better than a wide one?

		Where should the calibration function be placed with regard to the expected or possible analytical result?

	It is instructive to find the answers to these questions by simulation. This article demonstrates how weighted fits can be simulated with spreadsheet calculations (by using random numbers). Unweighted fits are also calculated by the spreadsheet, and can evaluate if the complicated weighting process is really required. By using a Monte Carlo approach, the 

-fold repetition of the simulation (here 

= 1000) is possible, therefore showing the mean of the 

 result, its bias from the theoretically true value, and the standard deviation that can be expected.

	Simulations of calibration functions and experimental data can be performed with Excel software. The order of magnitude of the 

 data shown in this paper is 10 times larger than that of the corresponding 

 data to obtain a clear distinction. The 

 values have no uncertainty, as explained above. The 

 values, which represent peak areas or heights, achieved a normally distributed (or Gauss) standard deviation, either with a constant absolute standard deviation (SD) (simple linear least-squares fit) or with a constant relative standard deviation (RSD) (weighted linear least-squares fit).

	Although it would be possible to use the random number generator of Excel for this purpose, this approach is inferior because it is based on an unknown algorithm and therefore the generated numbers are not really random in nature. Numbers with genuine randomness are to be preferred; they can be obtained from the free website 

, which generates the required numbers from atmospheric noise. Their number, type of distribution (Gauss for this study), mean, standard deviation, and number of significant digits can be selected. A possible set of random numbers shown here has a mean of 50,000, a standard deviation of 5000, five digits, and a size of 1000 numbers - all formatted into one column. These 1000 data points can be directly pasted in a single operation from the website into a column of the spreadsheet.

	The spreadsheet discussed here is based on five calibration points. A simulation procedure runs as follows:

		5 × 1000 calibration points with fixed 

, select FREE services Numbers, Gaussian Generator), 10 times larger than the corresponding 

 values, are put into columns A–E of an Excel spreadsheet.

		The calibration function is calculated by either simple or weighted linear least-squares fit. In the case described above, its slope is about 10 because 

 values and with a defined SD or RSD, are taken from 

, put into column F, and the corresponding 

 values calculated using the calibration function generated before, once in the unweighted (usual) mode and once by weighting.

	To summarize: The scatter of the calculated mean 

 value is the result of the scatter of the 1000 calibration functions and of the corresponding 1000 

 values. This kind of simulation is a Monte Carlo approach (10–15). The data of interest is not found with a closed mathematical function or by an iterative procedure but by the repeated calculation of the possible result under varying starting conditions. In this article, the variations have a normal (Gauss) distribution with a defined and selected SD or RSD. Depending on the problem, that is, the number and complexity of the involved calculations, a Monte Carlo process should include a great number of simulations. For the issues discussed here 1000 simulations give 

 RSDs that are stable at two figures, such as 4.3%, but not at three figures; from one 1000-simulations-set to another, the result may differ in the second decimal place, for example 4.28% and 4.34%. It can be assumed that 10

 simulations are necessary for two stable decimal places, but in chemical analyses such excessive precision is usually not needed. It is up to the user to check this problem and to expand the number of runs, if necessary, up to 10

. If weighted linear least-squares fit is needed (see below), the accuracy (bias) of the 

 value can be of greater importance than its precision.

 data, that is, ordinary least-squares (1–4), is based on two prerequisites. Firstly, the 

 data have, as explained and justified above. Secondly, the scatter of the 

 values have the same absolute SD irrespective of their value. This second prerequisite is known as homoscedasticity (16). It will occur if an autosampler works with a constant standard deviation, such as 1 μL, irrespective of the injected volume, and if the calibration points are generated with different volumes of the same solution (instead of the same volume of several solutions with different concentrations).

	Obviously, it is not difficult to set up a spreadsheet for the calculation of a linear least-squares fit if the data is homoscedastic. The Excel commands for SLOPE and INTERCEPT are available on demand. Such a spreadsheet can be of great didactic value because it allows many aspects to be studied:

 Simulations show that the precision of the analytical result increases (SD decreases) with increasing number of points (data not shown here).

Is a Narrow Calibration Range Advantageous? 

No, the range of calibration has no influence on the precision of the analytical result as long as its position remains unchanged, for example in the middle of the range (data not shown here). However, it is questionable if the absolute SD of the 

 values is constant in real situations. Can we expect an experimental SD of 100 if 

 = 10,000? It is an advantage that such set-ups can be studied in advance by Monte Carlo simulations.

Where is the Best Position of the Sample Signal?

 has two influencing factors: the scatter of the position of the calibration line and the scatter of the measured signal 

. Since the SD of all signals in the Simple Case is constant, this number affects a low 

 much more than a high one. Even without simulation, it is no surprise that under these circumstances, the best position of the sample signal is at the upper end of the calibration function.

Does an Irregular Distribution of the Calibration Points Make Sense?

 Researchers frequently want "to kill two birds with one stone" and place more calibration points at low concentrations than at high ones. However, simulations with constant SDs of the 

 points show that an irregular distribution has practically no influence on the result (data not shown here).

Does a Repeat Determination (Double Determination) Make Sense?

 Some analysts perform a second, repeat measurement in the hope of obtaining a more reliable result. This approach is maybe a good idea, maybe not. In routine quality control, it is best to observe the analytical performance by control charts (17). If the method is under statistical control, that is, if the analysis of a reference material with constant properties yields constant results of accepted standard deviation, then one can trust the result of a single determination. If these preconditions are not fulfilled (in non-routine analysis for example), more than one analysis should be performed. It is then necessary to do at least two analyses from scratch by taking two samples, making individual sample preparations (preferably not in parallel but on different days), leading to individual measurements and to individual results. However, it is not good practice to simply analyze the final sample solution two or three times. Such a procedure yields nothing more than the repeatability of the instrument, which has nothing to do with the repeatability of the sample preparation or the inhomogeneity of the sample. This kind of "repeat determination" cannot be recommended. In addition, it is a matter of course that the analytical instrument should work with higher precision (lower SD) than the observed, "true" SD of the samples (18). Despite these concerns, the influence of a second analysis can be studied with a spreadsheet.

		Several calibration points of the same 

 value that results in a calibration with 3 × 5 = 15 points, for example.

		The influence of wrong calibration points, for example, resulting from wrong weighing or wrong dilution.

 Data with Variable Standard Deviation (or Constant Relative Standard Deviation as a Special Case)

	A common situation in analytical chemistry is the constant (or nearly constant) RSD of the 

 values (the peak areas or heights). This happens if the autosampler works with a constant RSD of 1% or similar, irrespective of the absolute injected volume. In such a case, the SDs of the calibration points are heteroscedastic (19) and one prerequisite for simple linear least-squares fit is no longer fulfilled. It must be replaced by the markedly more complicated weighted linear least-squares fit (20). Contemporary software packages for chromatographs come with tools for this procedure, but without disclosure of the algorithms used. Programmes for weighted linear least-squares fit have been proposed (21–23) and Tellinghuisen studied the problem with Monte Carlo computations (24–25). Kuss presented his approach in detail, not only in 

(23) but also in a book with accompanying CD-ROM (26). This disk contains an Excel spreadsheet for weighted linear least-squares fit. The following discussion in this section is based on the Kuss spreadsheet; however, what is calculated by Kuss in rows is presented here in columns so that the 1000 simulations appear one below the other. As a consequence, the "weighted" Monte Carlo spreadsheet with five calibration points has approximately 50 columns and more than 1000 rows (Figure 1).


	Figure 1: Example of an Excel spreadsheet for a 5-point weighted linear least-squares fit. It is 53 columns and 1022 rows. The first 12 simulations of 1000 are shown. All y

 values are random numbers (coloured cells). All data in a row are linked together, and an x

 value is calculated from the data in its row. The mean of all 1000 weighted x

 is 4.99 in this example; its true value is 5.00. The relative standard deviation (RSD) of the calibration function's intercept is not calculated because this information is not useful: the mean of the 1000 intercepts is close to 0 (remember that the y axis runs from 0 to 50,000), which would result in an absurd RSD.

 RSDs are identical, for example 10%, the values at the upper end of the calibration get too much weight in a classical linear least-squares fit; the SD of 

 = 1000 is only 100. As a consequence, the calibration function can be at the incorrect position in the lower range of the line (Figure 2). Therefore these SDs must be normalized or weighted with the factor 

) can be between 1 and > 2 because the SDs are potentially not generated by the autosampler alone. The 

 can be determined by investigating the SDs of the highest and the lowest calibration points, which is a pragmatic approach to this problem (instead of studying the SDs of all calibration points).


	Figure 2: An example where a weighted fit is needed. The calibration range runs from x

 = 2- 5000. The unweighted (black) and weighted (red) functions were obtained with 1000 simulations with all y

 points having RSD = 10%, but the graph shows only six calibration points per concentration (partially covered by each other). (a): Full calibration range. (b): The lowermost part of the range, showing that the unweighted function would yield incorrect analytical results.

	In ordinary least-squares fitting, the calibration function runs through the centroid, which is the point 

 values. In weighted least-squares fitting, the centroid is at 

The weighting exponent can be calculated pragmatically as follows (23,26):

	This procedure can also be performed stepwise in a spreadsheet:

		Determine the (absolute) SDs of the lowest and the highest calibration points by multiple measurements of 

		Calculate the quotient of the squared SDs and its logarithm.

		Calculate the quotient of the values obtained in step 2 and 3.

 has the value of 2, irrespective of the percentage of the RSD. With constant SD at 

 = 1, this is the unweighted case as discussed above. The experimental 

 values of real analytical problems usually result in 0 < 

	With a limited database it is not possible to know if 

 has a small RSD) or if this result was obtained by chance. As an example, if all of the 

 points have an RSD of 10%, the measured data at 

 points may be 4.732, 4.877, and 5.369, giving a RSD

 = 2.3 instead of 2.0. The problem arises from the fact that 3 experiments are not sufficient to obtain a robust, "true" SD or RSD (27). However, the influence of 

on the analytical result must not be over-estimated. Usually it is small, as can be demonstrated by changing the exponent in the spreadsheet for weighted fit.

Weighted Linear Least-Squares Fit With a Spreadsheet:

 The set-up of the spreadsheet was explained in detail by Kuss (23). It concludes with the calculations of slope b and intercept a as illustrated in equations [3] and [4]. It handles not only weighting operations but also needs normalizations. The normalization factor (NF) is:

-fold if a certain concentration is investigated 

 times; for example six different concentrations 

, analyzed twice each, gives 12 calibration points. The 

 is calculated with a separate spreadsheet or is selected from experience (or from curiosity to study its influence). Table 1 presents the operations performed in the different columns of the spreadsheet discussed here; in rows 23 to 1022 the 1000 simulations are performed (not shown). Columns A–I calculate the unweighted case; the (long) part for the weighted case stretches over columns J–BA. With less than five calibration points, the spreadsheet would have less columns; with more points it would be larger (with six points nine more columns are needed, with seven points 18 more columns).


	Table 1: Operations in the columns of the spreadsheet for weighted linear least-squares fit with n = 5 calibration points.

	The spreadsheet for unweighted and weighted fit, investigated by Monte Carlo simulations, is available from the author on request for free (

Under What Conditions is Weighting Needed?

	The most interesting question in this context is of course: Under which circumstances is it necessary to perform the complicated process of a weighted fit?

 = 2 that weighting is not really needed if 

 is in the middle of the calibration range, but that it is necessary for data at the lower end of the range (24). Monte Carlo simulations, always with 1000 runs, confirm this result. The wider the calibration range (for example three orders of magnitude), the more important weighting for low 

 data is (Figure 2). In this example, a value at the lower end of 

 of 8.3 in the unweighted case; the correct result should be 2.0. The weighted function with 

 are constant at 10%) gives an astonishingly accurate 

 of 1.95. In the middle of the calibration range, at 

 = 30,000, it does not matter if the weighted or the unweighted function is used; 

 is 2980 or 3030, respectively (should be 3000). The same situation is found at the upper end, namely 

 = 4960 or 5050, respectively (should be 5000).

	If the calibration range is narrow, the question "to weight or not?" has no clear answer. The 5-point, 10% RSD case of 

 = 100, 200, 300, 400, and 500 (less than 1 order of magnitude) was studied with 1000 simulations each for 

 points at different positions. At the lower end of the function, with 

 was 10.0 unweighted and 10.2 weighted (true value = 10.0). However, the RSD of 

 was 19% unweighted vs. 12% weighted. Weighting gave a mean (of 1000 simulations) with some small bias but better precision. In the middle and at the upper end (

 = 250 and 500, respectively), the bias was also smaller in the unweighted case (

 = 25.1 and 50.6) than in the weighted one (

 = 25.4 and 51.2) and the precision was comparable in all four cases (11.3%, 11.7%, 12.5%, and 11.4%).

Unequal Distribution of the Calibration Points: 

When considering these results, there is one question still unanswered: Is weighting needed for a wide calibration range with four points at the lower end and one single at the upper end? This was studied with 

 = 10, 20, 30, 40, and 5000 and corresponding 

 values with RSD = 10%. The simulations show that weighting is not necessary, not even when the analytical result is at the lowest end (

 = 9.84 unweighted vs. 10.14 weighted; should be 10.0). Precision, however, is worse (higher RSD) with the simple fit (RSD 

 = 24% vs. 12% at low end, 14% vs. 11% in the middle, 15% vs. 11% at high end). Analogous results are found when 

Spreadsheet computer simulations can identify the influencing factors for the set-up of a calibration function such as the number of calibration points and their distribution or the position of the experimental points. By using a Monte Carlo approach, the quality of the experimental results (bias and standard deviation) can be studied under different conditions. This article presents a spreadsheet for the simulation of unweighted and weighted linear least-squares fit.

LCGC Europe-04-01-2015

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Event: 7th International Symposium on Recent Advances in Food Analysis (RAFA 2015)

Vol 28 No 4 LCGC Europe April 2015 Regular Issue PDF