**3. FTIR microspectroscopy: Technical considerations**

192 Multivariate Analysis in Management, Engineering and the Sciences

**Figure 2.** FTIR absorption spectrum of a single intact murine oocyte. The measured absorption spectrum of a single intact murine oocyte (surrounded nucleolus, MI 10 H) is reported without any corrections. The oocyte - deposited on a BaF2 window - was measured in transmission by the IR microscope UMA 500, coupled to the FTIR spectrometer FTS 40A (both from Digilab), at a resolution of

We should add that to obtain reliable results on the studied process it is crucial to standardize firstly the sample preparation, since - for instance - metabolic changes due to cell aging could result in significant spectral changes that could, in turn, hide the IR response specifically due to the process of interest, as it has been recently reported in the literature [37]. For these reasons, it is fundamental to check accurately the stage of cell

We should also briefly mention that, before spectral analyses, the measured IR spectra could require some corrections due to artifacts that can interfere with the spectroscopic response. For instance, single cells, or subcellular compartments, or particles of the size of the same order of that of the incident infrared light (�3-10 microns) could give rise to Mie scattering, that significantly distorts the measured spectrum, causing misinterpretation of the results. For this reason, before further analyses, it is strongly recommended to correct the measured

Since the IR spectra of complex biological systems are due to the overlapping spectral features of multiple components, their analysis requires often the employment of resolution enhancement procedures to better resolve their absorption bands, an essential prerequisite for the identification of peak positions and their assignment to the vibrational modes of the different molecules. Among these, second derivative analysis is widely applied, as described in [39]. Since second derivative band intensity is inversely proportional to the square of the

2 cm-1. The absorption regions of the main biomolecules are indicated.

growth in culture before performing spectroscopic measurements.

spectra with opportune algorithms specifically developed to this aim [38].

FTIR microspectroscopy is realized coupling to a FTIR spectrometer an infrared microscope characterized by an all reflecting optics, since typical lenses and condensers of visible microscopy - being made of glass, not transparent to the IR radiation - cannot be employed.

The main advantage of FTIR microspectroscopy is that it offers the possibility to study selected areas of the sample under investigation, resulting particularly useful in the case of systems characterized by an intrinsic heterogeneity, such as biological systems.

Two main types of IR microscopy exist, depending on the detector employed, and both equipped with an IR thermal source (globar), whose spatial resolution is diffraction-limited.

The first, conventional, generally equipped with a nitrogen cooled mercury cadmium telluride (MCT) detector, makes it possible to measure IR absorption spectra from a microvolume within the sample, selected by a variable aperture of the microscope, whose side can be adjusted down to a few tens of microns.

The second type of IR microscope, more advanced, is equipped with a focal plane array (FPA), consisting of an array of infrared detector elements, that enables not only to collect the IR absorption spectrum of the sample, but also an IR chemical imaging, where the image contrast is given by the response of selected sample regions to particular IR wavenumbers. Depending mainly on the detection array, the spatial resolution in this kind of microscopy is approximately between 20 and 5 microns, making it possible to reach, therefore, a resolution near to the diffraction limit.

We should, however, add that the use of a synchrotron IR light source, with a brightness of at least two orders of magnitude higher than that of a conventional thermal source, makes it possible to achieve diffraction-limited spatial resolution with enhanced signal-to-noise ratio. In this way, synchrotron light could allow to explore the IR spectra at the subcellular level.

A final remark should be done concerning the spectral acquisition mode. Indeed, infrared measurements can be mainly performed in transmission, reflectance or attenuated total reflection (ATR) mode. Typically, measurements on complex biological systems are performed in transmission mode, using appropriate IR transparent supports for the deposition of the sample, such as BaF2, CaF2, ZnSe. In this case, the IR beam goes through the sample, that - depending mainly on its molar extinction coefficient - should have a uniform thickness, not exceeding 15-20 microns.

Moreover, in reflectance mode - where the sample is placed onto proper reflective slides the IR beam passes the sample, is reflected by the slide, and passes the sample again. In particular, the sample slides reflect mid-infrared radiation almost completely and usually are also transparent to visible light, allowing sample inspection by a conventional light microscope. This approach is, for instance, useful for tissue characterizations.

Multivariate Analysis for Fourier Transform Infrared Spectra of Complex Biological Systems and Processes 195

independent variables **Y** and another set of uni- or multivariate variables **Z**. Similarly to the **Y** matrix, the matrix **Z** has *n* rows, one for each observation and *m* columns, the dependent

11 12 1m 1 2 ... , ,...,

*T n*

(2)

...

**Z zz z**

The matrix **Y** (composed of the independent variables y) represents the only input for several multivariate techniques described here; in some other cases the matrices **Y** and **Z** 

In the following part, we will make a distinction between regression and classification techniques. However, it should be clear that the separation between these two domains is not always sharp and the same technique can be either used for regression or for

LMVR (or MLR) can be used to model linear relationships between one or more **z**  (dependent variable) and one or more **y** (independent variable). In the most general case, we have *n* independent multivariate variables **y** represented by the matrix **Y** and the

The LMVR is based, as many other statistical techniques, on the generalized linear model: **Z** *= +* **βY ε** where **β** is a matrix containing the parameters to be estimated, and *ε* is a matrix which models the errors or noise. The coefficients **β** are usually estimated using the ordinary least square, which consists of minimizing the sum of the square differences of the *n* observed *y*'s from their modeled values. Mathematically, the optimal values of **β** are

the number of observation must be larger than the number of variables, which is often not the case), otherwise the matrix *<sup>T</sup>* **Y Y** is singular and cannot be inverted. Another common problem is the correlation between variables; more specifically, none of the independent variables must be a linear combination of any other. This phenomenon is called

In some cases linear models cannot be used and one could try to apply non-linear models.

Common models which frequently apply to natural phenomena are the exponentials (which, indeed, is a transformed linear model. A linear model can be applied upon on the

"multicollinearity" [43, 44] and it will be explained in more details in section 4.2.3.

*4.2.2. Non-Linear Multivariate Regression (NLMVR)* 

**β YY YZ** . To apply the least square method we must have *n - 1 > m* (e.g.

corresponding response multivariate variable *z*, stored in the matrix **Z**.

*n1 n1 nm*

*zz z*

(composed of the dependent variables z) are both required.

**4.2. Multivariate regression techniques**

*4.2.1. Linear Multivariate Regression (LMVR)* 

*=z z z =*

variables.

classification purposes.

obtained by <sup>1</sup> *T T <sup>=</sup>*

Finally, in the ATR approach, where the sample is placed into contact with a higher refractive index and an IR transparent element (mainly germanium and diamond), samples with higher thickness than in transmission can be processed. In particular, the IR beam reaches the interface between the ATR support and the sample at an angle larger than that corresponding to the total reflection. In this way the beam is totally reflected by the interface and penetrates into the sample as an evanescent wave, where it can be absorbed. The beam penetration depth is of the order of the IR wavelength (a few micrometers) and depends on the wavelength, the incident angle, as well as on the refractive indices of the sample and of the ATR element. Furthermore, it should be noted that this kind of approach makes it possible to measure also samples not necessarily deposited onto an IR transparent support, as in ATR measurements it is only required that the sample be in close contact with the ATR element.

For a review of the technical aspects of FTIR microspectroscopy, see [40-42].
