Algorithms

Quantitation by spectral analysis can be done using a variety of mathematical techniques. This technique addresses the simplest of these methods which is based on the direct measurement of band height or area from one or more spectra. At times the analyst is only interested in the raw numeric values of this measurement which is then compared to previous measured values as in a quality assurance procedure. The raw value may also be scaled by a single concentration factor which relates the raw value directly to the concentration values of the analysis. A more accurate and advanced technique of spectral quantitative analysis is to create a calibration equation or series of equations which, when applied to spectra of "unknown" mixture samples, will accurately predict the quantities of the components of interest. In order to calculate these equations, a set of "standard" mixtures are made which reflect the composition of the "unknowns" as closely as possible. These standards are also designed to span the expected range of concentrations and compositions in the unknowns and are measured under the same conditions (sampling method, pathlength, instrument, etc.) as the unknowns. The spectra of the standards (in absorbance or other concentration-proportional units) are then measured by a spectroscopic instrument and saved in digital format. This set of spectra and the known quantities of the components in each individual sample form what is known as a training set or calibration set from which the calibration equations are built. The unknown sample(s) are then measured in the same manner on the same instrument and the equations are used to "predict" the concentration of the calibrated components in each unknown.

One of the keys to quantitative analysis is the assumption that the concentrations of the components of interest are somehow related to the data from the measurement technique used for analyzing the samples. This relationship must be able to be accurately described by a calculated equation in order to be useful for predicting the compositions of any "unknown" samples measured. In some cases, the components of interest may have well-resolved bands. In these cases, either the peak height or the peak area can be related to concentration by a single simple equation. These equations can take the form of a straight line or even a quadratic curve. In this case, concentrations of the components are calculated from equations such as:

where C_a is the concentration of component A, and the Bs are the calibration coefficients. By measuring the height or area of a component band in the spectrum, it is possible to compute the concentration using the equation and these coefficients.

However, the coefficients are not necessarily known ahead of time and must be calculated. This is accomplished by first measuring the spectra of some samples of known concentration. As you can see from the equations, there are either two or three unknown coefficients. This means at least two or three known samples must be measured in order to solve the equation. However, more are usually measured to improve the accuracy of the calibration coefficients. In fact, sometimes multiple runs of the same concentration are used to get an averaging effect.

The peak areas or heights of the component band and the known concentrations of all the calibration spectra can then be used to calculate the coefficients. The best way to find the calibration coefficients from this set of data is by a Least Squares Regression. This is a mathematical technique that calculates the coefficients of a given equation such that the differences between the known responses (peak areas or heights) and the predicted responses are minimized. (The predicted measurements are those calculated by reading the measurements off of the line at the known concentrations.) If there is more than one component in the samples, a separate band must be used for each component of interest. This also means one equation is necessary for each component. Once the equations are calculated for each component, the concentrations of these components in "unknown" samples can be calculated by substituting the peak areas or peak heights into the proper equation for that component band and solving for concentration. 

While this method is conceptually easy to understand and the calculations are straightforward, it will not produce accurate results for mixtures with overlapping bands. Since Least Squares Regression assumes that the absorbance measurement of peak height or total peak area is the result of only one component, the predictions will have large errors if there are interfering components that have spectral bands that overlap those of the components of interest. In these cases, more sophisticated mathematical techniques are necessary such as Inverse Least Squares (ILS), Partial Least Squares (PLS), Principal Component Analysis (PCA) or Principal Component Regression (PCR).

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