3.1.1.

Average spectra of classification data

Average spectra of the first $\left(\mathbf{a}\right)$ and last $\left(\mathbf{e}\right)$ measurements of the training set aorta samples are shown in Fig. 1 . From these plots, it is clearly evident that the spectral differences between lesion and nonlesion tissue samples are very subtle. The experimental results obtained in this study convincingly demonstrate that sophisticated multivariate data analysis and classification techniques are essential to robust and reliable sample classification for diagnostic purposes.

Fig. 1

(a) Red—average $\mathbf{a}$ spectrum of lesion samples; black—average $\mathbf{a}$ spectrum of nonlesion samples. (b) Red—average $\mathbf{e}$ spectrum of lesion samples; black—average $\mathbf{e}$ spectrum of nonlesion samples (Color online only).

3.1.2.

Multivariate classification results using $\mathbf{a}$ data

In the following multivariate classification, lesion samples were assigned class 1, and nonlesion samples class 2. IR reflectance spectra were preprocessed by meancentering prior to further analysis.⁶⁵

PLS-DA is a discrimination method developed from PLS regression models.⁶⁶ Based on the root mean square error for cross validation (RMSECV) results for PLS-DA shown in Fig. 2a , four latent variables (LVs) are selected as optimal numbers to minimize error during classification and prediction. In general, four or six LVs were tested to build the statistical models. The corresponding classification and prediction results are shown in Figs. 2b and 2c. Ideally, lesion samples have a value of 0.5, and nonlesion samples have a value of $-0.5$ . However, the predicted values frequently deviate from the ideal hit values due to the variations of the samples within the same class.

Fig. 2

(a) RMSECV versus LV number using $\mathbf{a}$ data of training set samples. RMSECV—root mean square error for cross-validation. Class 1 (blue)—lesion samples; class 2 (green)—nonlesion samples. The minimum theoretically indicates the optimum number of LVs to build the model. (b) Classification and prediction results for PLS-DA model 6 LVs using $\mathbf{a}$ data. Red triangles—lesion training samples; green stars—nonlesion training samples; black dots—blind samples; red line—threshold $(-0.0507)$ . (c) Histograms for PLS-DA 6 LVs model using $\mathbf{a}$ data of training set samples. Threshold is $-0.0507$ (Color online only).

In all plots shown next, points 1 to 14 represent lesion training samples (class 1); 15 to 28 nonlesion training samples (class 2); and 29 to 40 samples from the first test set. The establishment of the model using the training samples (1 to 28) by Wang preceded the measurement of the unknown samples (by Chapman) by six months owing to tissue availability schedules. For the 12 samples from the first test set, only the raw single beam IR spectra were provided for evaluation without any indication of the number of lesion versus nonlesion cases among the 12 samples. The identity of the test samples was shared only after the classification had been made.

Threshold values were calculated using the observed distribution of the predicted values and the Bayesian theorem for discriminating the two different classes. As shown in Fig. 2c, blue bars are a histogram of the predicted values for class 1 samples; green bars are a histogram of the predicted values for class 2 samples. The threshold is the cross point of two normally fitted histograms.

The Bayesian statistics also provide the probability that a sample is a member of a certain class given the predicted value. The prediction probability results for both four LV and six LV PLS-DA models based on all investigated samples are shown in Table 1 . Given a sample, its probability belonging to class 1 is calculated using Eq. 1.

Table 1

Prediction probability results for PLS-DA models using a data. 1 to 28: training sample set; 1 to 14: lesion sample set; 15 to 28: nonlesion sample set; and 29 to 40: first set of test samples.

In the model using four LVs, sample 10 cannot be unambiguously classified, but its probability of belonging to class 1 is $>50\%$ (see Table 1). Using this model, only test sample 30 was incorrectly classified. If six LVs were applied to establish the model, all samples could be correctly classified or predicted.

Linear discrimination analysis (LDA) used before is a method to maximize the among-class difference relative to the within-class difference. The Mahalanobis distance^{67, 68} is a specific linear discriminant analysis method particularly suitable for classification, which was performed here by first compressing the spectral data to six latent variables and corresponding scores of a 6-D vector using PLS. Following this, the mean score vector ${\mathbf{S}}_{mn}$ and the mean-centered scores ${\mathbf{S}}_{mc}$ for each class (lesion or nonlesion) were calculated, and the covariance matrix $(6\times 6)$ $\mathbf{M}$ of ${\mathbf{S}}_{mc}$ for each class was computed. For the prediction of a blind sample, its score would be calculated from the measured spectrum and latent variables, and mean centered by ${\mathbf{S}}_{mn}$ of one class. The distance ${\mathbf{D}}_j^2$ of the mean-centered unknown score ${\mathbf{t}}_j$ from ${\mathbf{S}}_{mn}$ of this class was computed and normalized by $\mathbf{M}$ following Eq. 2.

The distance of an unknown sample to the classes determines which class the unknown belongs to. The class that has less distance to the unknown will incorporate the unknown sample. From Fig. 3 it is evident that the Mahalanobis distance method has provided 100% successful classification and prediction of all samples in the first test set, similar to PLS-DA.

Fig. 3

(a) Classification results of 28 training samples using the Mahalanobis distance method and $\mathbf{a}$ data of training set samples. Green stars—nonlesion training samples; red triangles—lesion training samples; diagonal line—discriminant line. (b) Prediction results of 12 blind samples using the Mahalanobis distance method and $\mathbf{a}$ data of blind samples. Diagonal line—discriminant line (Color online only).

3.1.3.

Multivariate classification results using $\mathbf{e}$ data

Based on the RMSECV results (not shown), six LVs have been determined as the optimal number for the PLS-DA classification model. The corresponding classification results are shown in Fig. 4a . All training samples could be clearly classified with this method, and only test sample 40 could not be classified with sufficient certainty. Most probably, it would be incorrectly classified as a nonlesion sample.

Fig. 4

(a) Classification and prediction results for PLS-DA 6 LVs model using $\mathbf{e}$ data. red triangles—lesion training samples (class 1); green stars—nonlesion training samples (class 2); black dots—blind samples; red line—threshold $(-0.1042)$ . (b) Histogram for PLS—DA 6 LVs model using $\mathbf{e}$ data. Threshold is $-0.1042$ (Color online only).

The corresponding histograms and the prediction probability results for the PLS-DA model using six LVs and $\mathbf{e}$ data are shown in Fig. 4b and Table 2 .

Table 2

Prediction probability results for PLS-DA models using e data. 1 to 28: training sample set; 1 to 14: lesion sample set; 15 to 28: nonlesion sample set; and 29 to 40: first set of test samples.

The Mahalanobis distance method was also applied to classify $\mathbf{e}$ data. The classification results are shown in Fig. 5 . Again, test sample 40 could not be correctly classified. Sample 30 could be classified more clearly using the Mahalanobis distance in contrast to using PLS-DA.

Fig. 5

(a) Classification results of 28 training samples using the Mahalanobis Distance method and $\mathbf{e}$ data. Green stars—nonlesion training samples; red triangles—lesion training samples; diagonal line—discriminant line. (b) Prediction results of 12 blind samples based on the PLS-DA 6 LVs model for $\mathbf{e}$ data (Color online only).

Using the PLS-DA models developed before, classification of the second set of test samples was attempted, however, with reduced hit quality using both $\mathbf{a}$ and $\mathbf{e}$ data. Yet 74% of the samples were classified correctly using $\mathbf{a}$ data, and 60% of the samples were classified correctly using $\mathbf{e}$ data. The sensitivity and specificity of the PLS-DA model for the test samples were calculated using the method introduced by Balchum, ⁶⁹ and are summarized in Table 3 . The lower classification rate is attributed to the limited diversity of the training sample set for developing the predictive models. Consequently, it is essential for introducing significantly more spectra for covering the variation among animals by using spectral imaging techniques, enabling the collection of large sets of model data in a reasonable period of time.

The possible reason that using $\mathbf{a}$ data provides (marginally) more accurate predictive results in contrast to using $\mathbf{e}$ data may result from the fact that the sample had significantly changed during ambient exposure and the experimental procedure. It has to be considered that the $\mathbf{e}$ dataset has been recorded as the fifth consecutive measurement starting after $11\min$ of an entire measurement series. Hence, due to water evaporation the sample was significantly drier compared to the beginning of the measurement series. In turn, this indicates that classification during hydrated conditions, which more closely resemble the in-situ environment, is more accurate.

Alternatively, the application of principal components regression (PCR) techniques was investigated for the $\mathbf{a}$ and $\mathbf{e}$ data series of the first test set to discriminate between lesion and nonlesion classes. 1 was the preset value for all lesion samples, and 0 was for all nonlesion samples.⁷⁰ All spectra were again mean centered prior to PCR. The predicted lesion value ideally centers at 0.5, and the nonlesion at $-0.5$ . However, PCR-based classification failed in accurately classifying $\mathbf{a}$ data. Figure 6 shows the PCR results using $\mathbf{e}$ data. A total of nine PCs were selected for the model, and all training samples could be accurately classified. Test sample 40 was incorrectly classified as nonlesion, similar to PLS-DA and the Mahalanobis distance method. In addition, test sample 35 could not be clearly predicted with the horizontal zero line as the discriminator, as the prediction value was only slightly above zero.

Fig. 6

Classification and prediction results for PCR model with 9 PCs using $\mathbf{e}$ data. Red triangles—lesion training samples (class 1); green stars—nonlesion training samples (class 2); black dots—blind samples (Color online only).

In contrast to PCR, the PLS-DA method not only considers the changes in the spectra, but instantaneously also considers the changes in concentration of the various constituents (or class difference in our case). Due to uncertainties introduced by the sample preparation process and ambient effects during the measurements, the among-group difference is not always larger than the within-group difference. Hence, PCR appeared to be the least able to provide satisfactory classification results.

The sensitivity and specificity of the investigated multivariate methods for test samples of the first test set without any a priori knowledge are summarized and compared in Table 3.