MULTI-EXPONENTIAL FIT OF NMRD DATA FOR OLIVE OIL ANALYSIS

1. INTRODUCTION NMR spectroscopy has been applied to edible oils and fats (2-9), but little is known about relaxivity properties of olive oil. NMRD could become an interesting tool to characterize olive oil quality from a novel point of view, and to acquire information on its composition in a rapid and non destructive way. 2. METHOD AND MATERIAL NMRD profiles of Tuscan olive oil samples were obtained by means of a Fast Field Cycling Relaxometer, in the 10kHz-35MHz range and processed (numerically and graphically) by means of programs written in the SAS (Statistical Analysis System) software environment. 3. RESULTS AND DISCUSSION 3.1 A mono-exponential model was tentatively fitted to the experimental data: The line linking the experimental points regularly crosses the graph of the corresponding monoexponential model three times. Thus a systematic error is apparent, either in the Pre-Polarized (PP, 0.001- 15MHz) or in the Not-Polarized (NP, 15-35MHz) experiments and the multi-exponential fitting is mandatory. 3.2 The bi-exponential model reduces the residuals of two-three orders of magnitude. No further reduction was achieved incrementing the number of exponential components. On the beginning, the bi-exponential models were fitted to experimental data allowing all the parameters to change freely, even the between-component ratio (q1 in the graphs, kk1bi in the tables). Then a unique value was estimated. 3.3 The between-component ratio is a very critical parameter, not only for its physical meaning, but also because it is not easy to estimate, as shown by the table and by the graph below. An ascending trend can be seen and the ratio seems to be field-dependent because it increases together with the frequency of the relaxing field (BRLX). The between-component ratio was estimated here on the whole BRLX range (FREE estimate in tables and graphs). Then the model has been re-fitted forcing the ratio to be equal to the estimated value (REFIT estimate in tables and graphs). In the same way a tri-exponential model (not reported) has been tested, but without further decrease of the error, neither systematic nor random, and it has been discarded. 3.4 The mono- and multi-exponential models have been fitted to NMRD data by means of Non-Linear Regression Analysis. In the following graphs the Sums of Squares of Errors (SSEs) are plotted and compared, for each relaxation field (BRLX) and for the three cases: 1) SSEmono (mono-exponential model): MAGNITUDES = a1mono+ b1mono*(exp(-TAU1/t1mono)) 2) SSEbiFREE (bi-exponential model; all parameters allowed to change) MAGNITUDES = a1bi+ b1bi*(exp(-TAU1/t1bi) + kk1bi*(exp(-TAU1/t2bi))) 3) SSEbiREFIT (bi-exponential model; all parameters allowed to change except the between-component ratio) MAGNITUDES = a1bi+ b1bi*(exp(-TAU1/t1bi) + kk1bi*(exp(-TAU1/t2bi))) with the bound kk1bi = 0.7380 The SSEs of the bi- and tri-exponential (not reported) FREE models are equivalent and better than the mono-exp.. The SSE of the bi-exp. model, REFITTED including the kk1bi value estimated on the whole BRLX range, is less than the corresponding one of the mono-exp, for all BRLX values. The FREE and the REFIT SSE of the bi-exp. model are alike. 3.5 R1 (t1) pre-analysis The following graphs show the trend of R1. R1 is derived from the mono-exp. model (diamonds) and from the first (circles) and the second (stars) component of the bi-exp. one. The first graph is the result of the FREE kk1bi fitting (the between-component ratio of the bi-exponential model was allowed to change with BRLX). In the second graph R1 has been calculated considering the bound: kk1bi = 0.7380 as explained above. No outlier is detectable and the trend of R1 is more smoothed if the kk1bi unique estimated value is included in the model and the between components ratio is not allowed to change freely. 3.6 Fitting lorentzian models to R1 data Multi-Lorentzian, “model-free” equations (see the table below) were fitted to the two components, by means of Non-Linear Regression, testing n values from 1 to 5. 4. CONCLUSIONS NMRD profiles of Tuscan olive oil samples have been acquired for their relaxometric characterization. The magnetization decays detected by the relaxometer could be well fitted by means of a bi-exponential function. Inclusion of a third exponential term did not reduce the sum of the differences between experimental and back-calculated values. Two relaxation rate profiles for each set of data were determined as a function of the magnetic field. The profiles were then analysed as a sum of Lorentzian functions. Comparative analysis of the NMRD profiles of different types of oils is in progress, to investigate whether the quality of the different oils may be correlated to the parameters related to relaxometric properties. The fitted mono- and multi-Lorentzian models were compared evaluated and selected within the Non- Linear Regression Analysis framework, as shown by the following tables and graphs. In the first table the selected models are summarized and the parameters are all normalized (norm). In the tables below, the parameters are not normalized and differently labeled: qq replaces x and the constant value 1 replaces (1-x). The models were evaluated taking in account their significance, the analysis of the residuals, and the confidence intervals of the parameters. Each selected model is illustrated by two graphs. In the former the fitting is plotted, together with the residuals. In the latter the selected Lorentzian is drawn with its components.