The basic characteristics of variable filter array (VFA) spectrometers, both near- and mid-infrared, available from several manufacturers, make them well suited for routine infrared analysis applications outside and inside the laboratory. Although the infrared data gathered by a VFA spectrometer is similar to that obtained from an FT-IR instrument, it differs in one key respect — resolution. In the mid-IR region, FT-IR spectra usually are recorded at resolutions between 0.5 cm

 . A VFA spectrometer with 64 pixels has a theoretical resolution of 16 cm

 . Because of crosstalk from pixel to pixel, both thermal and optical, the resolution is reduced further to the neighborhood of 32 cm

What effect does low resolution have upon the performance of a VFA spectrometer? To answer this question, we have used a VFA spectrometer in several different applications to determine the effect, if any, of low resolution. 

A company uses a solvent that comes in three isomers, each specific to a different process. Sending a sample to the laboratory can take over an hour for verification, delaying the offloading of the material. Figure 1 shows VFA spectra of the three isomers. Even though the spectral information in the VFA spectra is considerably less than that of FT-IR traces, there is enough data from the VFA spectrometer to verify the isomer in the shipment. Because a VFA spectrometer has no moving parts and no free air path, it is unaffected by atmospheric absorptions and can be situated at the receiving dock. In less than 5 min, the incoming raw material can be verified and off-loaded to the correct location. 

 Figure 1. (a) Infrared spectra of three isomers. (b) Using Isomer 1 as a reference, a spectrum of Isomer 1 produces a straight line while the spectra from the other two isomers show deviations from a straight line.			

In a process, a material that has an absorption band "A" between 8.5 and 9 µm (Figure 2) converts to a material that has band "B" at 7.0 µm. The reaction can take several hours. The figure shows spectra periodically taken by the VFA instrument situated close to the production line, which follow the switch from "A" to "B". Again, in this situation, the resolution is sufficient to tell when a reaction is complete. 

Figure 2. Following a reaction. As the reaction progresses, the band between 8.5 and 9 Âµm decreases to be replaced by the band at 7.0 Âµm.			

There are many areas in flexible film manufacturing in which an onsite test is invaluable. With multilayer films, rolls of film are laminated together with individual rolls, often having different top and bottom materials. Figure 3 shows spectra of common materials used in flexible film production. The spectra differ enough so that an operator in the production area can verify the film and orient it in the right direction for lamination. In the warehouse, rolls of finished films can be verified on-site before shipment to ensure that the customer is receiving the desired product. Another application involves measuring an inside layer thickness, such as nylon, in a multilayer film. Nylon has an absorbance different from the spectra of the outside polyethylene layers, which can be distinguished with a lower resolution VFA spectrometer. 

Figure 3. Spectra of common materials used in flexible film production.			

Partly as a result of 9/11, and partly because materials are being examined frequently for identification purposes, mid-IR FT-IR spectrometers have become widely used for this purpose to the extent that libraries containing thousands of spectra of known materials have been constructed. Is it feasible to use a VFA spectrometer to search the libraries for identification purposes? 

The spectra of common pharmaceutical products shown in Figure 4 are a good example of the differences between VFA spectra and FT-IR spectra. While each VFA spectrum is distinctly different from the others, each also is quite different from its FT-IR counterpart. Because of these substantial differences, it appears to be impractical to compare spectra from FT-IR libraries directly.  

Figure 4. VFA (upper) and FT-IR (lower) spectra of common pharmaceuticals (a) aspirin, (b) Bufferin, (c) Advil, and (d) Tums.			

On the other hand, if one examines the VFA spectra, it becomes apparent that each product has a characteristic VFA spectrum, sometimes more readily identifiable than its FT-IR spectrum, so libraries of VFA spectra could serve to identify various groups of materials. A pharmaceutical company might use one or two dozen materials routinely; it would be easy to build VFA spectral libraries with the compounds. Likewise, a law enforcement agency might be concerned with a fixed number of narcotics or poisons for which libraries could be created. 

While it does not appear that the VFA spectrometer will replace the FT-IR instrument as a general purpose unknown material identifier, it can be used for this purpose when a relatively small number of materials is involved. Because VFA spectrometers are small, rugged, and usually contain no moving parts or exposed optical paths, they can be practical alternatives to FT-IR instruments when applicable. 

It is possible to degrade FT-IR spectra from spectral libraries to lower resolution, as in Figure 5, so that they tend to match VFA spectra. However, in so doing, search programs might be less specific and could produce a number of possible hits rather than one or two. 

Figure 5. Changing resolution on spectral libraries to match VFA resolution.			

From the examples presented here, it is apparent that low-resolution mid-IR data is not necessarily a deterrent and often can have a positive effect. As the examples show, low-resolution spectra frequently contain gross differences between spectra of materials, so that visual verification of materials is feasible. Automatic identification from a VFA spectra library also is feasible. Although FT-IR spectra can contain much more detail, visual differences tend to be less apparent, and therefore, require a more detailed systematic evaluation for positive identification. Obviously, both high-resolution FT-IR spectra and low-resolution VFA spectra are useful, depending upon the application. 

The VFA spectra shown in this article were run on a Wilks InfraSpec VFA spectrometer. The FT-IR spectra were run on a Nicolet Model 5PCFTIR spectrometer. 

 is president of Wilks Enterprise, Inc. (South Norwalk, CT). 

Articles

The basic characteristics of variable filter array (VFA) spectrometers, both near- and mid-infrared, make them well suited for routine infrared analysis applications outside and inside the laboratory. The performance of this spectrometer is discussed here.

Tech Note — The Difference Between VFA and FT-IR Spectra

Mass spectrometry (MS) instrumentation has undergone several improvements over the years, including increased sensitivity, ease-of-use, and possibilities for more sophisticated and time-efficient experiments. This article illustrates the technical principles and typical applications of the most common mass analyzers used in bioanalytical laboratories today. There are four basic types of mass analyzers found in modern MS: time-of-flight (TOF), quadrupole and a recent derivative, quadrupole linear ion trap (LIT), which were covered in Part I of this series (1), and quadrupole ion trap (QIT), and Fourier transform ion cyclotron resonance (FT-ICR), which are discussed in this installment. Rather than simply explaining the underlying physical principles, we concentrate on each mass analyzer's characteristic features, their limitations, and their applications. In addition, several unique techniques and unusual or exotic mass analyzer designs are highlighted throughout the text. We have chosen examples from our own research in small molecule applications. Therefore, some interesting recent hybrid designs (with mostly proteomics applications) only are described briefly. 

The analyzers described in Part I of this article are ion beam-type instruments (except for the linear ion trap). In this installment, we will concentrate on ion-trapping devices, which allow the storage of ions using electric potentials (ion trap) or magnetic fields (ion cyclotron resonance, ICR) with subsequent mass analysis in the same space. This section explains the principles of quadrupole ion trap (QIT) MS. 

The ion trap can be envisioned as a regular linear quadrupole wrapped around itself. It is composed of two end-cap electrodes and one ring electrode. By applying appropriate potentials on these electrodes, a pseudo-potential well, to which ions are confined, is formed within the trap, and ions can be stored for long intervals in this instrument. The ion trap is kept at high pressures relative to other mass analyzers, because of the presence of a damping gas, usually helium, which causes collisional cooling of the analyte ions with subsequent confinement in the center of the trap. The usual pressure inside the trap is ~10

Mass analysis in the ion trap is achieved by means of the so-called 

. In the mass selective instability scan, ions of specific 

 values are ejected from the trap selectively by linearly increasing the rf amplitude on the ring electrode. Detection of the ions occurs following the ejection process. The resonance ejection technique extends the mass range of the ion trap by exciting the ions in the trap via a supplemental potential applied to the end caps. If the amplitude of this additional potential is large enough, ions are ejected from the ion trap at ring electrode rf voltages lower than those required for the mass selective instability scan. 

Furthermore, specific scans can be used selectively to eject all unwanted ions from the trap. A selected precursor, for instance, can be isolated by applying a broadband waveform consisting of all frequencies except that of the secular frequency of the ion of interest. It is important to realize that an experiment on an ion trap is composed of several steps occurring sequentially, in the same space. This chain of events is called a 

. The separated time periods include ion injection, isolation, excitation, and analysis. In addition, very often an initial pre-scan is performed to determine the ideal injection time, thereby reducing 

 effects. This is important because a fundamental limitation of the ion trap is the dependence of ion stabilities upon the number of ions present in the trap. As the amount of ions is increased, space charging becomes an important factor. The electrostatic field is distorted by having too many ions in the trap, thus causing broadening of peaks, mass scale shifts, and loss in sensitivity. Eventually, ions are no longer trapped when the ion density becomes too large. Therefore, the pre-scan assesses the amount of ions entering the trap, and certain parameters, such as the injection time, are adjusted consequently to correct for possible space charging. Interestingly, quadrupole linear ion traps (1) also exhibit space charge effects, but they have larger ion storage capacity (2). 

After the ions are injected into the trap from the source, a suitable rf voltage on the ring electrode confines them to stable trajectories. An isolation scan can be performed subsequently in order to accumulate a specific ion or a range of ions selectively. The next possible step is ion excitation, in the case of tandem MS. Finally, ions are analyzed by ejection from the ion trap to the detector through openings in the end caps. Figure 1 exhibits representative scan functions in an ion trap. The type of scan dictates the length of time spent on each step. The differences in the operation of ion traps and quadrupoles have a direct effect upon the duty cycle of the instruments, and we see that ion traps are very sensitive in a full-scan experiment, but do not gain very much sensitivity, as one would imagine, using the selected ion monitoring (SIM) mode. On the other hand, a SIM experiment in a quadrupole instrument has much higher sensitivity than it does for a full scan experiment (1). 

 Figure 1. The events during an QIT micro-scan are: 1. ion injection, 2. isolation, 3. excitation, 4. mass analysis. (a) Full-scan MS: a waveform is used to eject unwanted ions from the trap and to isolate a mass range of interest during ion injection; the rf voltage is ramped during mass analysis to eject the ions out of the trap sequentially. (b) SIM: a waveform is used to isolate a narrow m/z range of interest [the waveform has a notch present, corresponding to that of the secular frequency of the ion of interest] for selective accumulation. All unwanted ions are removed by resonant ejection; the rf ramping during mass analysis is very short for the limited m/z range. (c) MSn: a waveform is used to isolate only the precursor ion; the resonance excitation step dissociates the precursor ion using an auxiliary rf voltage at the end caps. Product ions are analyzed subsequently.			

Further important differences between ion trap and quadrupole MS are seen in tandem MS experiments. The QIT can be described as a 

 for a triple quadrupole) and allows multiple stages of tandem mass spectrometry (MS

 ), whereby precursor ions can be activated selectively and dissociated to form product ions. The product ions are detected by subsequent ejection from the trap. The ion activation of the isolated precursor usually is achieved via "on-resonance" excitation by applying a small (tickle) potential, corresponding to the secular frequency of the ion, across the end caps. As ion activation times in the order of tens of milliseconds can be used without significant ion losses, multiple collisions occur during the excitation period. Because of this relatively long time-scale, this excitation technique falls in the category of so-called "slow heating" processes (3). For a slow-heating process, excitation in an ion trap still is quite fast, due to the high pressure of helium present in the trap. There are other slow-heating methods, such as "sustained off-resonance irradiation" (SORI, see next paragraph) with longer excitation periods. MS

  experiments are quite advantageous for the elucidation of fragmentation pathways, which in turn aid in the analysis of CID spectra for identifying molecular structures for precursors as well as product ions. The results are, in principle, similar to those obtained in quadrupole linear ion traps. 

Figure 2. Extremely high resolution can be achieved using an FT-ICR mass spectrometer. This figure shows a small m/z range from a crude phytoplankton extract analyzed in full scan on a 9.4-T FT-ICR. Under lower resolution conditions, these three peaks would coalesce into one unresolved peak.			

Sometimes, mass spectrometrists point out that ion traps yield CID spectra with only a limited number of fragment ions, in comparison to triple quadrupole spectra. This is, in fact, quite true and often one needs to conduct several stages of MS

  to generate a sufficient number of product ions. The reason for this limitation is, of course, based upon the resonance excitation used for ion activation, which does not excite the product ions, because of their different secular frequencies (that is, they are no longer "in resonance" with the excitation frequency and therefore do not pick up further energy). In fact, molecules that tend to cleave off a large number of unspecific small molecules (mostly H

O) are sometimes difficult to analyze by MS

 , because structurally informative backbone cleavages are hard to reach. The selective activation procedure, however, offers a tremendous advantage with its ability to yield exact relationships between precursor and product ions, allowing the "genealogical" mapping (4) of the entire fragmentation process. Triple quadrupole CID experiments, on the other hand, are much less selective, as the product ions tend to undergo further collisions, to give additional second or even higher-generation product ions. The linked information between precursor and different generation product ions is difficult to determine in such cases. 

The analytical figures of merit of QIT mass spectrometers are somewhat comparable to quadrupole mass analyzers (that is, mass ranges of up to 

 4000 can be analyzed, at unit mass resolution). Higher mass resolution can be achieved with special slow scanning routines, for much smaller 

 ranges. The detection sensitivity, as outlined earlier, generally is very good in the full-scan mode because of the ion-trapping abilities. The duty cycle in the SIM or MRM modes of the QIT-MS is much lower than on the quadrupole MS, because of the significant overhead contributions to the microscan sequence (Figure 1). The sensitivity enhancements in the SIM mode in comparison to the full scan mode, therefore, are not very pronounced.  

Finally, the QIT also has been combined recently with a TOF analyzer (5). In QIT-TOF, the QIT is used as a precursor ion selection and/or MS

  stage, prior to high-resolution, accurate mass TOF analysis. Such an instrument allows accurate mass data to be obtained for product ions of each individual stage of MS

 . This instrument originally had been available only with a MALDI source; therefore, most current applications include large molecules such as proteins and polysaccharides. 

The Fourier Transform Ion Cyclotron Mass Analyzer

FT-ICR MS has experienced a remarkable increase in use by analytical chemists recently. This instrument was employed historically for fundamental physics and physical chemistry applications. Recently, however, new developments such as efficient ion-transport devices have allowed the use of external ionization sources (particularly ESI and MALDI), significantly increasing the versatility of FT-ICR mass spectrometers. Furthermore, quadrupole and linear ion-trap analyzers have been implemented for precursor ion selection and collisional activation, before FT-ICR mass analysis. Consequently, FT-ICR has become an invaluable tool for biological MS, primarily for its flexibility in ion-activation techniques available and its extremely high resolution and mass accuracy.  

While the principles of mass analysis by ICR date back to 1932, it was not until 1974 that Marshall and Comisarow (6) added the use of Fourier transform pulse techniques to ICR. Before we illustrate the unique properties of FT-ICR MS, a basic description of this analyzer allows one to understand how it achieves these important advantages over other instruments. For readers interested in a more thorough treatment of ICR fundamentals, we suggest reference 7. 

 entering the magnetic field, will bend the ion's path into a circular motion, as follows: 

 are vectors, perpendicular to each other, 

 is the balancing centripetal force. If the ion is under extremely low pressure conditions, it does not experience any collisions with residual gas molecules in the cell. Consequently, the speed of an ion in the FT-ICR cell is constant. With ω= 

, it follows that the angular velocity is given by 

This is the fundamental equation for FT-ICR, where the ion's 

 value can be directly measured by its ion cyclotron frequency ω

 and no dependence on the ion's initial velocity upon injection into the cell (that is, no energy focusing is required for the precise determination of 

). The ion's cyclotron frequency ranges from kHz to MHz values, depending upon its mass, for small organic molecules to biomolecules in the kDa range. It also is informative to look at the radius of the cyclotron motion, 

 (see Equation 1). As illustrated by Marshall and coworkers (7), cyclotron radii between 0.08 mm for 

 10,000 are assumed by the ions at room temperature and a magnetic field strength of 3 T (tesla). 

In order to measure the cyclotron frequency, and thus the mass, of an ion (via Equation 2), ions must be accelerated coherently to a new radius. This is required because initially, not all ions are at the same point on their circular orbits. Therefore, their motions are incoherent, and their frequencies thus cancel out and cannot be measured. For all ions of a specific 

 value to become coherent, an rf field, rotating with the ICR frequency of the specific ion, is applied. After this excitation, the radius of the ion no longer depends upon its 

 is the peak-to-peak magnitude of the rf electric field, 

 is the distance between the plates, and 

 is the excitation time. Note that by exciting all ions of different 

 values with a broadband rf field, they all end up at the same radius after the excitation process. The resulting ions all have become spatially coherent, but ions of different 

 values still have a characteristic cyclotron frequency. Each 

 value can be differentiated based upon the speed at which they orbit the common radius occupied by all ions. 

A unique and quite interesting feature of FT-ICR mass spectrometers is the way the frequency measurement is performed. FT-ICR analyzers do not have a detector, where ions are discharged physically, as is the case with most other mass spectrometers. Rather, detection is based upon measuring the 

 that individual ion packages induce when they pass the detector plates repeatedly, at their cyclotron frequencies. In a typical FT-ICR experiment, the ions in the cell are excited simultaneously, and a composite image current for all ions is measured. Next, this transient signal is converted from the time domain to the frequency domain using a fast Fourier transform algorithm. The individual frequencies then can be transformed to 

 values by means of Equation 2. Note that this method of detection is non-destructive. Ions can be measured for extended transient times, if required, thereby increasing the mass resolution. The transient signals diminish eventually, because of collisions in the ICR cell slowing down the orbiting ions. 

Hybrid Instruments for Biological Applications			

It is evident that the strength of the magnetic field is an important factor in several parameters. First, the initial radius of an ion is inversely proportional to 

, so that for a higher magnetic field, a higher mass range can be analyzed. Ions of larger 

 occupy larger radii and there is a mass range limit based upon the size of the ICR cell and the magnetic field. Furthermore, as 

 increases, so does the cyclotron frequency of an ion, indicating that higher magnetic fields yield higher mass resolution. However, there is a reduction in the lower limit of 

 values with a higher magnetic field, because their frequencies become correspondingly higher and thus more difficult to measure. The most striking feature of FT-ICR MS is its outstanding mass resolution, significantly larger than any other MS technique available currently. (Figure 2 shows an illustrative example for biotoxins; additional detail is given in the sidebar "Mass Resolution Versus Mass Accuracy.") As seen with quadrupole ion traps, FT-ICR also is affected by space charge effects, if too many ions are present in the cell. The interactions between the ions lead to dephasing of the ion cloud and peak coalescence.  

As mentioned earlier, one of the primary advantages of FT-ICR is the flexibility in ion activation techniques for MS-MS experiments available. As isolation and excitation take place in the same confined space, this instrument belongs to the tandem-in-time category. SORI often is applied for collisional activation in FT-ICR instruments. In SORI, a neutral gas pulse is introduced into the ICR cell and the precursor ion is excited at a frequency slightly higher than the natural cyclotron frequency. Ions undergo multiple acceleration/deceleration cycles as they increase and decrease their orbit radii repeatedly in the FT-ICR cell before dissociation takes place. The activation times often are in the hundreds of milliseconds, sometimes up to seconds, and a large number of collisions take place. This approach is somewhat similar to quadrupole ion trap MS

  experiments. An alternative to SORI is 

 (IRMPD), which is available for all commercial FT-ICR instruments. IRMPD activates ions in the ICR cell by irradiation with a low-power CO

 infrared laser (10.6 µm) for tens to hundreds of milliseconds per experiment, resulting in the stepwise absorption of photons, followed by subsequent dissociation of the ion. This process is rather non-selective, therefore, all trapped ions are excited and secondary product ions also can be observed. This feature of IRMPD can be convenient, because the resulting spectra often are similar to low-energy CID spectra measured on triple quadrupole instruments (8). Finally, a fundamentally different technique of activation also is available on all commercial FT-ICR instruments, 

 (ECD). ECD involves the capture of low-energy electrons, with charge-state reduction and subsequent fragmentation. It is therefore only applicable to multiply charged cationic species, usually peptides or proteins. ECD is assumed to occur much faster than dissociation in other activation techniques, permitting the occurrence of direct bond cleavages only. 

As with other hybrid mass spectrometers, MS-MS experiments are conducted increasingly outside the ICR cell, either with a linear ion trap or a Qq arrangement, coupled to the FT-ICR mass analyzer (see the sidebar "Hybrid Instruments for Biological Applications"). They offer increased compatibility with fast chromatography runs, as ion activation experiments directly in the ICR cell often take too long to be compatible with chromatography experiments. 

In this tutorial, we have presented a short overview of the operating principles and selected applications for the most common mass analyzers used in biological MS today, namely TOF, quadrupole, ion trap, and FT-ICR. As illustrated, the way these analyzers function and their characteristic features are quite different. The TOF analyzer is an ideal match for pulsed-ion sources such as MALDI. Because of its high speed, it also is very useful for extremely fast chromatography applications. It offers high transmission efficiency and a very high mass range. On the other hand, the TOF has limits with respect to duty cycle, dynamic range, and precursor ion selection in MS-MS. Quadrupoles are rugged and versatile. They can be used as mass filters, ion guides, and collision cells. On the other hand, hyphenation with pulsed-ionization sources is less than ideal and their resolving power is limited. Quadrupole ion traps offer high sensitivity in the full-scan mode as well as the ability to perform multiples stages of MS-MS. Similar to quadrupoles, the resolving power is limited. In addition, careful tuning of multiple operating parameters is required to avoid space charge effects, to obtain meaningful MS-MS spectra, and to use it in quantitative applications. Finally, the FT-ICR offers the highest resolving power and mass accuracy of any analyzer available today. Because the ion-detection method is non-destructive, remeasurements for extremely high sensitivity, resolution, and accuracy are possible readily. FT-ICR is very flexible with respect to ion-activation techniques as well. To obtain reliable results, however, it requires careful adjustment of many operational parameters. The hyphenation with chromatography is more complicated because the scan sequences for high-resolution spectra can be time-consuming, sometimes too slow for acquiring a sufficient number of data points across chromatographic peaks. For all the analyzers, we also have shown that there is an increasing trend toward hybrid designs to increase versatility by combining the intrinsic advantages of the individual analyzers. 

Finally, we cannot finish this article without mentioning an exciting new type of mass spectrometer that has recently emerged, promising resolving powers similar to FT-ICR without the need for superconducting magnets. This mass analyzer is called the orbitrap, and its operating principles are summarized in the sidebar "The Orbitrap Mass Analyzer." 

2. D.J. Douglas, A.J. Frank, and D. Mao, 

4. R.J. Strife, L.C. Robosky, G. Garret, M.M. Ketcha, J.D. Shaffer, and N. Zhang, 

The second installment of this two-part series illustrates further technical principles and applications of the most common mass analyzers used in bioanalytical laboratories today, as well as novel techniques and mass analyzer designs. Examples are based upon the authors' research in small molecule applications.

Tutorial — Mass Analyzers: An Overview of Several Designs and Their Applications, Part II

In our last few columns (1–4), we discussed shortcomings of current methods used to assess the presence of non-linearity in data, and presented a new method that addresses those shortcomings. This new method is statistically sound, provides an objective means to determine if non-linearity is present in the relationship between two sets of data, and is inherently suitable for implementation as a computer program. 

A shortcoming of the method presented is one that it has in common with virtually all statistical tests: while it provides a means of unambiguously and objectively determining the presence of non-linearity, if we find that non-linearity is present, it does not address the question of 

 non-linearity is present. This column, therefore, presents results from some computer experiments designed to assess a method of quantifying the amount of non-linearity present in a data set, assuming that the test for the presence of non-linearity already has been applied and found that indeed, a measurable, statistically significant degree of non-linearity exists. 

The spectroscopic community, and indeed, the chemical community at large, is not the only group of scientists concerned with these issues. Other scientific disciplines also are concerned with ways to evaluate methods of chemical analysis. Notable among them are the pharmaceutical communities and the clinical chemistry communities. In those communities, considerations of the sort we are addressing are even more important, for at least two reasons: 					

These disciplines are regulated by governmental agencies, especially the Food and Drug Administration. In fact, it was considerations of the requirements of a regulatory agency that created the impetus for this series of columns in the first place (1).

The second reason is what drives the whole effort of ensuring that everything that is done, is done "right." An error in an analytical result can conceivably, in literal fact, cause illness or even death.

Thus, the clinical chemistry community also has investigated issues such as the linearity of the relationship between test results and actual chemical composition, and an interesting article provides the impetus for creating a method of assessing the degree of non-linearity present in the relationship between two sets of data (5).

The basis for this calculation of the amount of non-linearity is illustrated in Figure 1. In Figure 1a, we see a set of data showing some non-linearity between the test results and the actual values. If a straight line and a quadratic polynomial both are fit to the data, then the difference between the predicted values from the two curves gives a measure of the amount of non-linearity. Figure 1a shows data subject to both random error and nonlinearity, and the different ways linear and quadratic polynomials fit the data. As shown in Figure 1a, at any given point, there is a difference between the two functions that represents the difference between the 

Figure 1b shows that irrespective of the random error of the data, the difference between the two functions depends only upon the nature of the functions and can be calculated from the difference between the 

-value. If there is no non-linearity at all, then the two functions will coincide, and all the differences will be zero. Increasing amounts of non-linearity will cause increasingly large differences between the values of the two functions corresponding to each 

-value, and these can be used to calculate the nonlinearity. 

The calculation used is the calculation of the sum of squares of the differences (5). This calculation normally is applied to situations where random variations are affecting the data, and indeed, is the basis for many of the statistical tests that are applied to random data. However, the formalism of partitioning the sums of squares, which we have discussed previously (6) (also in [7], page 81 in the first edition or page 83 in the second edition) can be applied to data in which the variations are due to systematic effects rather than random effects. The difference is that the usual statistical tests (

, etc.) do not apply to variations from systematic causes because they do not follow the required statistical distributions. Therefore, it is legitimate to perform the calculation, as long as we are careful how we interpret the results. 

Performing the calculations on function fitted to the raw data has another ramification: the differences, and therefore the sums of squares, will depend upon the units in which the 

-values are expressed. It is preferable that functions with similar appearances give the same computed value of non-linearity regardless of the scale. Therefore, the sum-of-squares of the differences between the linear and quadratic functions fitted to the data is divided by the sum-of-squares of the 

-values that fall on the straight line fitted to the data. This cancels the units, and therefore, the dependency of the calculation upon the scale. 

A further consideration is that the value of the calculated non-linearity will depend not only upon the function that fits the data, but we suspect that it also will depend upon the distribution of the data along the 

-axis. Therefore, for pedagogical purposes, here we will consider the situation for two common data distributions: the uniform distribution and the normal (Gaussian) distribution. 

Figure 2 presents some quadratic curves containing various amounts of non-linearity. These curves represent data that was, of course, created synthetically. The purpose of generating these curves was for us to be able to compare the visual appearance of curves containing known amounts of non-linearity with the numerical values for the various test parameters that describe the curves. Figure 2 represents data having a uniform distribution of 

-values, although, of course, data with a different distribution of 

-values would follow the same curves. The curves were generated as follows: 101 values of a uniformly distributed variable (used as the 

-variable) was generated by creating a set of numbers from 0 to 1 at steps of 0.01. The 

-values for each curve were generated by calculating the 

-value according to the following formula: 

 in Equation 1 induces a varying amount of non-linearity in the curve. For the curves in Figure 1, 

 varied from 0 to 2 in steps of 0.2. The subtraction of the quadratic term in Equation 1 gives the curves their characteristic of being convex upward, while adding the term 

 back in ensures that all the curves, and the straight line, meet at zero and at unity. 

Table I presents the results of computing the linearity evaluation results for the curves shown in Figure 1, for the case of a uniform distribution of data along the 

-axis. It presents the coefficients of the linear models (straight lines) fitted to the several curves of Figure 1, the coefficients of the quadratic model, the sum-of-squares of the differences between the fitted points from the two models, and the ratio of the sum-of-squares of the differences to the sum-of-squares of the 

-data itself, which, as stated earlier, is the measure of non-linearity. Table I also shows the value of the correlation coefficient between the linear fit and the quadratic fit to the data, and the square of the correlation coefficient. 

In Table I, we see an interesting result — the ratio of sums of squares we are using for the linearity measure is equal to 1 (unity) minus the square of the computed correlation coefficient value between the linear fit and quadratic fit to the data. This should not surprise us. As noted previously, the same formalisms that apply to random data also can be applied to data in which the differences are systematic. Therefore, the equality we see here corresponds to the well-known property of sums of squares from any regression analysis, that from the analysis of variance of the regression results, the correlation coefficient is related to sum-squared-error of the analysis in a similar way (8). It also is interesting to note that the coefficients of the models resulting from the calculations on the data (shown in Figure 1) are not the same as the original generating functions for the data. This is because the generating functions (from Equation 1) are not the best-fitting functions (nor, as we shall see, are they the orthogonalized functions), which is what is used to create the models, and the predicted values from the models. 

Figure 1. An illustration of the method of measuring the amount of non-linearity. (a) Hypothetical synthetic data to which each of the functions are fit. (b) The functions, without the data, show the differences between the functions at two values of X. The circles show the value of the straight line; the crosses show the value of the quadratic function at the given values of X.			

Because the correlation coefficient is an already-existing and known statistical function, why is there a need to create a new calculation for the purpose of assessing non-linearity? First, the correlation coefficient's roots in statistics directs the mind to the random aspects of the data for which it normally is used. In contrast, therefore, using the ratio of the sum of squares helps to remind us that we are dealing with a systematic effect whose magnitude we are trying to measure, rather than a random effect for which we want to ascertain statistical significance. 

Secondly, as a measure of non-linearity, the calculation conforms more closely to that concept than does the correlation coefficient. As a contrast, we can consider terms such as precision and accuracy, where "high precision" and "high accuracy" mean data with small values of <whatever measure is used, such as standard deviation> while "low precision" and "low accuracy" mean large values of the measure. Thus, for those two characteristics, the measured value changes in opposition to the concept. If we were to use the correlation coefficient calculation as the measure of non-linearity, we would have the same situation. However, by defining the "linearity" calculation the way we did, the calculation now runs parallel to the concept — a calculated value of zero means "no non-linearity," while increasing values of the calculation correspond to increasing non-linearity. 

Table II. Coefficients for Orthogonalized Functions			

Another interesting comparison is between the coefficients for the functions representing the best-fitting models for the data, and the coefficients for the functions that result from performing the linearity test as described in the previous column (4). We have not looked at these before because they are not involved directly in the linearity test. Now, however, we consider them for their pedagogic interest. These coefficients, for the case of testing a quadratic non-linearity of the data from Figure 1, are listed in Table II. We note that the coefficients for the quadratic terms are the same in both cases. However, the best-fitting functions have a constant intercept and varying slopes, while the functions based upon the orthogonalized quadratic term have a constant slope and varying intercept. 

Figure 2. Curves illustrate varying amounts of non-linearity.			

We now take a look at the linearity values obtained when the 

-data is normally distributed. The non-linearity used is the same as that used earlier in the case of uniformly distributed data, and the same diagram (Figure 2) applies, so we need not reproduce it. The difference is that the 

-data is normally distributed, so that there are more samples at 

 = 0.5 than at the extremes of the range of Figure 2, the falloff varying appropriately. The standard deviation of the 

 -values used was 0.2, so that the ends of the range corresponded to ±2.5 standard deviations. Again, synthetic data at the same 101 values of 

 were generated. In this case, however, multiple data at each 

-value were created, the number of data at each 

-value being proportional to the value of the normal distribution corresponding to that 

-value. The total number of data points generated, therefore, was 5468. 

We can compare the values in Table III with those in Table I; the coefficients of the models are almost the same. The coefficients for the quadratic model are, unsurprisingly, identical in all cases, because the data values are identical and there is no random error. The main difference in the linear model is the value of the intercept, reflecting the higher average value of the 

-data resulting from the center of the curves being weighted more heavily. The sums-of-squares are necessarily larger, simply because there are more data points contributing to this sum. 

The interesting (and important) difference is in the values for the ratio of sums-of-squares, which is the non-linearity measure. As we see, at small values of non-linearity (that is, 

 = 0, 1, 2), the values for the non-linearity values are almost the same. As 

 increases, however, the value of the non-linearity measure decreases for the case of normally distributed data, as compared to the uniformly distributed data, and the discrepancy between the two gets greater as 

 continues to increase. In retrospect, this also should not be surprising, because in the normally distributed case, more data is near the center of the plot, and therefore, in a region where the local non-linearity is smaller than the non-linearity over the full range. Therefore, the normally distributed data is less subject to the effects of the non-linearity at the wings, because less of the data is there. 

As a quantification of the amount of non-linearity, we see that when we compare the values of the non-linearity measure between Tables I and III, they differ. This indicates that the test is sensitive to the distribution of the data. Furthermore, the disparity increases as the amount of curvature increases. Thus, this test, as it stands, is not completely satisfactory because the test value does not solely depend upon the amount of non-linearity, but also upon the data distribution. 

In our next column, we will consider a modification of the test that will address this issue. 

serves on the Editorial Advisory Board of 

 and is director of research, technology, and applications development for the Molecular Spectroscopy & Microanalysis division of Thermo Electron Corp. He can be reached by e-mail at: 

 and runs a consulting service, Mark Electronics (Suffern, NY). He can be reached via e-mail at: 

, 1st ed. (Academic Press, New York, 1991). 

Fitting Equations to Data — Computer Analysis of Multifactor Data for Scientists and Engineers

, 1st ed. (John Wiley & Sons, Hoboken, NJ, 1971). 

Columns

This column presents results from some computer experiments designed to assess a method of quantifying the amount of non-linearity present in a dataset, assuming that the test for the presence of non-linearity already has been applied and found that a measurable, statistically significant degree of non-linearity exists.

Chemometrics in Spectroscopy Linearity in Calibration: Quantifying Non-linearity

The analysis of metals in clinical fluids such as whole blood, serum, and urine has been used for many years to provide information on toxicity, workplace exposure, nutrient availability, and as a diagnostic tool for a number of ailments. The fact that many trace metals are present at variable and often low concentrations (sub-nanogram-per-milliliter range) in different clinical sample types has presented clinical analysts with a variety of challenges. In addition, matrix components, such as organic compounds, proteins, or electrolyte salts that can interfere with the analysis of trace elements often are present at elevated levels (milligrams per milliliter or above). Depending upon the matrix to be analyzed, the amount of sample that can be taken and the means of sampling also can impose restrictions. Sufficient volumes of urine normally can be obtained from patients with non-invasive techniques, whereas the collection of whole blood or serum samples usually involves the use of needle and syringe and generally yields smaller sample volumes (often only microliters or milliliters) for analysis. The analysis technique employed should, therefore, provide the following capabilities — sufficiently low detection limits, the ability to overcome matrix-related interferences, sufficient linearity to measure a wide concentration range in unknown samples, simultaneous multi-elemental determinations, and the ability to cope with small sample volumes. 

Analysis of clinical matrices by inductively coupled plasma-mass spectrometry (ICP-MS) is becoming more widespread because ICP-MS meets a number of the above requirements, such as very low detection limits for many trace metals (sub nanogram per milliliter), relative freedom from interferences, simultaneous multielemental determination, and suitability for small sample volumes, as well as providing isotopic information and the possibility of employing isotope dilution mass spectrometry (IDMS) as a high-caliber reference calibration technique. Although some methodologies for the multi-elemental application of the ICP-MS technique to clinical sample matrices have been described in the literature (1–3), the majority of publications have focused on the analysis of single elements in these matrices. Part of the reason for this is that some of the elements of interest suffer from mass spectral and nonspectral interferences derived from the sample matrix when analyzed by ICP-MS (3). Recently, the routine determination of 23 elements in urine samples after fivefold dilution with 1% HNO

 using an octopole-based collision/reaction cell ICP-MS (CRC-ICP-MS) system, was described (4). Historically, matrix-based spectral interferences have been overcome by the use of sector field or high-resolution inductively coupled plasma mass spectrometry (HR-ICP-MS) (5) or by non-MS techniques such as atomic fluorescence (6) or atomic absorption (AA) spectroscopy (7). Another method of overcoming matrix effects has employed the use of sample digestion using concentrated acids or ashing techniques (8). These techniques can be expensive, time-consuming, and less suitable for high sample throughput. 

In our laboratory, magnetic sector HR-ICP-MS (Element 1, ThermoFinnigan) has been used (5, 9) for monitoring post- and preoperation samples from patients with metal-on-metal hip replacements. After 1:20 dilutions of blood and serum samples with ~0.7 mM ammonia, 0.01 mM EDTA, and 0.07% (v/v) Triton X-100, or 1:15 dilutions of urine samples with 1% HNO

, elements including Al, V, Co, Cr, Mo, Ni, and Ti were analyzed. The main drawbacks of this technique were cost, lack of practicality, and duration of instrument set-up, as well as instrument downtime and matrix tolerance during analytical runs containing more than ~30 blood or serum samples. 

The aim of this work was to develop a robust ICP-MS methodology based upon collision/reaction cell quadrupole ICP-MS (CRC-ICP-MS), capable of measuring a wide range of elements in a single analysis after only a simple dilution of the samples.  					

A simple dilution of the samples was selected as the preferred sample preparation method because acid digestion techniques can increase the sample turnaround time, cost, and the potential for contamination.

To achieve the required sample throughput of up to 100 samples per batch, the quantitation method had to be based upon external calibration. Minimal instrument drift, therefore, was paramount to reduce the need for frequent recalibration or drift correction.

The target elements included the trace metals Al, As, Cd, Co, Cr, Cu, Fe, Mn, Mo, Ni, Pb, Sb, Se, V, and Zn, as well as the electrolytes K, Mg, and Na.

The sensitivity achieved was required to match previously obtained detection limits using HR-ICP-MS in our laboratory of 0.2 ng/mL (for example, Co and Mo) and 1.0 ng/mL (Ni) in the undiluted samples.

All samples, standards, and QC materials were diluted 20-fold using a solution containing ~0.7 mM ammonia (BDH, Poole, UK), 0.01 mM EDTA (Aldrich, Poole, UK) and 0.07% (v/v) Triton X-100 (BDH, Poole, UK). Butan-1-ol (Fluka, Poole, UK) was added to the diluent as a carbon source at 1.5% (v/v) to improve matrix matching between standards and samples and thereby increase the accuracy for analytes such as As and Se whose ionization behaviors in the plasma are affected by the carbon content (10). 

To keep the chemistry of the sample introduction system stable throughout the run, the diluent also was used for pre- and postanalysis rinse functions. Use of common-rinse acids such as HNO

, even at dilute levels of 1%, can lead to coagulation or precipitation of clinical matrix components and result in blockages of tubing or nebulizers. 

The selection of internal standard elements and the internal standard concentration is very important. The choice of elements often is restricted in clinical analysis due to the presence of many of the elements that usually are used in environmental applications at nanogram-per-milliliter levels in clinical samples. Blood and serum samples were analyzed in semiquantitative mode to determine the most suitable internal standard elements — that is, those that were not present or present at the lowest levels in relative terms. For elements that were present in the samples, such as Sc, the concentration of the internal standard was added at such a level that the contribution from Sc in the sample to the total 

The internal standard elements (Sc, Ge, Rh, In, and Tl) were chosen according to the best possible match in terms of proximity in mass to charge ratio and ionization behavior and were added to the diluent at a concentration of 20 ng/mL. Addition of the internal standard in this way negated possible mixing problems if on-line addition of the internal standard via a T-piece was used. 

An Agilent (Palo Alto, CA) 7500ce collision/reaction cell ICP-MS was used in three different gas modes — hydrogen, helium, and standard, or no-gas, mode. The ICP-MS conditions and the isotopes, integration times, and gas modes for the multielemental determination are given in Tables I and II. Quantitation on all isotopes was performed using the three central points of the spectral peaks. 

A 100-µL/min PFA microflow nebulizer was used and sample uptake and washout times were reduced using the larger diameter peristaltic pumps of the Integrated Sample Introduction System (ISIS) (Agilent). The pump speed was set at 0.1 rps during the analysis and washout to minimize overloading of the sample introduction system and the plasma with matrix components. The torch was equipped with a 2.5-mm diameter injector and the Shield Torch System was used. Nickel (Ni) cones were used at all times. 

The total acquisition time per sample was 208 s. This included the sequential loading of the H

, He, and Std tune files and a 40-s equilibration and stabilization time between the different gas modes. Each sample–standard solution was analyzed sequentially in all gas modes before the autosampler probe moved to the next sample. After each sample, the autosampler probe was rinsed for 5 s using 5% HNO

 and the sample introduction system, then was rinsed using the diluent for 30 s. 

The stability of the proposed methodology was tested by running blood and serum samples in a sequence over a period of 10 h (a total of 90 samples including calibration standards and QC checks) and monitoring the behavior of internal standard elements, calibration slopes, and check standards. 

Instrument stability – signal variation for internal standard elements. 

Typical signal variation for the internal standard elements of choice (Sc, Ge, Rh, In, and Tl) was 4.8–9.3% in hydrogen mode, 5.5–8.2% in helium mode, and 6.7–10.0% in standard mode. This was assessed during a 90–sample sequence of blood and serum samples. Figure 1 shows the variation for the internal standard elements throughout the 10-h run. Sc is present in some clinical sample types at nanogram-per-milliliter levels, which can be seen here after sample 8. 

Figure 1. Variation of the internal standard signals in standard mode, throughout the 10-h run.			

Calibration repeatability and linearity. 

The robustness of the calibration technique was assessed by overlaying calibration curves from the beginning, middle, and end of the 10-h run. The correlation coefficients for the mean slope of three calibrations for V, Se, and Pb (Figure 2) during a 10-h sequence ranged from 0.9997 to 1.0000 and indicate the robustness of the method with these matrices. The calibration coefficients for all elements measured were better, generally, than 

Figure 2. Linearity of overlaid calibration curves for V, Se, and Pb, showing stability of the external calibration approach throughout a 90-sample sequence.			

Check standards at the 1-ng/mL level were analyzed throughout the run after every nine samples for the trace metals and were within 10% of the expected value for the elements tested. 

Effects of sample matrix on the sample introduction system. 

Using dilution factors of 20-fold or less for analysis of these matrices by HR-ICP-MS led to frequent problems with the sample introduction system, especially blocking of the torch injector, which had an internal diameter of 1 mm. When using quadrupole ICP-MS as described earlier, dilution factors of 15- and 10-fold can be used without detrimental effects on the sample introduction system (Figure 3) or instrument performance. The internal diameter of the torch injector used with the CRC-ICP-MS was 2.5 mm. Reagent blanks were monitored after the analytical run, and no significant deterioration in the detection limits or increase in the background levels was observed. 

Figure 3. Photos of the interface and sample introduction system after a 90-sample run. Both the sampler and skimmer cones show minor matrix deposits. The 2.5-mm injector torch used was relatively deposit-free. The blood deposits on the spray chamber and the nebulizer block were removed using a sodium hypochlorite solution.  			

Analysis of Certified Reference Materials (CRMs)

 = 4) of the certified reference material NIST SRM 1598 Bovine serum and the reference material Seronorm MR9067 (whole human blood, level 2) were diluted 20-fold as described above and analyzed using the conditions described in Tables I and II. These materials were chosen because they represented different clinical matrices and contained a wide range of analytes of interest ranging in concentration levels from sub-nanogram per milliliter to milligrams per milliliter. Levels for the same analytes often varied by more than an order of magnitude between the two materials. Certificate data for both materials, as well as method detection limits (calculated back to the undiluted sample and based upon 3σ of the blank concentration), for the method proposed here are shown in Table III. 

The analytical data for both materials were converted to percent recovery data relative to the certified or indicative values and are shown in Figures 4a and 4b. The combination of the reference materials chosen for this study provided certified values with uncertainty estimates for all of the elements determined except for Na, for which only an indicative value was available. The recovery for Na, compared to the indicative value, was 99.0%, and the data for the remaining elements measured fell within the uncertainty range for either one or both of the reference materials. Where the certificate values were not achieved for SRM 1598 (for example, Cr and Cd), the certified concentrations were below the detection limit for the method. Na, As, Ni, V, and Pb are quoted as indicative values only in SRM 1598 (see Table III). 

Figure 4. Data obtained expressed relative to the certified data for (a) human blood Seronorm MR9067 and (b) bovine serum SRM NIST 1598. Errors bars represent expanded uncertainties for data obtained and certified ranges for the reference materials. The numbers after the isotopes indicate the tune step used (1 = H2, 2 = He, 3 = Std).			

The data for As and Se in MR 9067 are slightly high compared to the certified mean value, and this could be due to a higher carbon content in this matrix. When increasing the level of butan-1-ol to in the diluent from 0% to 3% v/v, recoveries for these analytes decreased and approached 100% (Figure 5). When no butan-1-ol was added to the diluent, recoveries for As and Se were significantly higher than the mean certified values (by 94% and 72% respectively) in comparison to recoveries obtained with butan-1-ol addition at 1.5% (v/v). A complete matrix match was achieved for both samples by using the standard addition technique for As and Se in both reference materials. Recoveries for Se were 95.8% and 99.9% in NIST SRM 1598 and Seronorm MR9067, respectively, and 102.6% for As in Seronorm MR9067.  

Table I. ICP-MS parameters used in the different gas modes.			

Figure 5 also indicates that the effect of the carbon addition on both elements is slightly different. According to the data obtained here, an addition of 3% would be best for As (mean recovery of 95.4 ±2.3%), whereas the ideal volume of butan-1-ol addition for this sample and dilution level for Se is closer to 2%. 

Table II. Analysis parameters for the analytes of interest.			

For elements in which the ionization is affected by matrix components in the plasma, it is therefore imperative to obtain a good level of matrix matching for the greatest accuracy. If this is not possible, for example, if the carbon levels in different samples vary significantly, it might be better to use a different sample preparation procedure such as closed-vessel microwave digestion to destroy the organic carbon matrix. However, this can increase the sample turnaround time for large sample batches significantly. 

Table III. Certified concentrations for the analytes of interest in the SRM NIST 1598 and the reference material Seronorm MR9067. Method detection limits calculated back to the undiluted sample are given for comparative purposes.			

Spike recovery experiments were performed on both materials for the trace metal analytes at two to four different levels with concentrations ranging from two to five times of the original analyte concentrations. The only exception to this spiking regime was Fe in MR9067, for which no certificate or indicative value was available before the analysis and where the spike concentration added (20 ng/mL) was not sufficiently high above the determined sample concentration (400 ±5 µg/mL) to give meaningful recovery data. The mean data for all spike levels for the trace metal analytes are shown in Table IV. Spike recoveries for all elements fell within 100 ± 20%, and all except Fe, Se, and Mo were within 100 ±10%. The high Se recoveries are thought to be due to the fact that the matrix matching for carbon content consisted of only 1.5% butan-1-ol. High recoveries for Mo also were observed when the samples were analyzed by HR-ICP-MS, and this effect currently is under closer investigation. 

Table IV. Mean spike recovery data obtained for both reference materials.			

A robust CRC-ICP-MS method has been developed that is capable of high sample throughput (up to 100 samples per batch) for a large suite of elements in difficult clinical matrices after simple dilution. The method robustness has been demonstrated by minimal signal drift during analytical sequences of 10-h duration, negating the need for frequent recalibration. The method detection limits achieved matched those of a previously used HR-ICP-MS method. Further improvements in method detection limits can be achieved by reducing the dilution levels of the clinical matrices, which is possible due to the robustness of the sample introduction system. Good agreement within the uncertainty of certificate values was achieved for all of the target analytes in both reference materials, for which certified data were available across concentration levels ranging from nanograms per milliliter to milligrams per milliter. Spike recoveries for all elements fell within 100 ±20%, and all except Fe, Se, and Mo were within 100 ±10%. 

Figure 5.  Recovery data for As and Se in Seronorm whole blood (Level 2) with varying levels of butan-1-ol addition to the diluent.			

This study describes the development of a robust, high-throughput analytical method for the determination of 18 elements (15 trace elements and three electrolytes) in blood and serum samples using a collision/reaction cell quadrupole ICP-MS system.

Spectroscopy-12-01-2005

Tech Note — The Difference Between VFA and FT-IR Spectra

Tutorial — Mass Analyzers: An Overview of Several Designs and Their Applications, Part II

Chemometrics in Spectroscopy Linearity in Calibration: Quantifying Non-linearity

The Use of Collision/Reaction Cell ICP-MS for the Determination of Elements in Blood and Serum Samples