1.

Introduction

Since the first report on the laser-induced fluorescence spectroscopy from biological tissues for cancer detection in the late 1980s,^1,2 fluorescence spectroscopy has been widely used for noninvasive diagnoses of cancer, developed into a field called optical biopsy. Fluorescence spectra of biochemicals play an important role in discrimination of cancerous tissue from normal tissue.^3,4 It is observed that a cancerous tissue contains more key molecules tryptophan and nicotinamide adenine dinucleotide (NADH), and less collagen than a normal tissue.^3,5,6 Quantification of the key biochemical components or biomarkers in a tissue provides a means of cancer detection. Multivariate curve resolution (MCR),^7–9 widely applied in chemometrics, can be employed to quantify the biochemical components by analyzing fluorescence spectrum. The fluorescence spectrum of a tissue is a superposition of spectra of various salient biochemicals and molecules. Thus component quantification is essentially an ill-posed problem. MCR, which is a model-free data fitting method, might result in invalid solutions; moreover, component quantification cannot guarantee to obtain sufficient statistics of the biomarkers, and the diagnostically useful information for cancer detection is inevitably reduced in terms of information theory. Spectral unmixing method (SUM),¹⁰ similar to MCR, is employed in remote sensing to determine the endmembers and quantify their fractions in a hyperspectral imaginary. Compared with MCR, SUM further requires the full additivity of component fractions that might further reduce the useful information for cancer detection. On the other hand, in the cancer detection point of view, without knowing biomarker components, a classifier can be trained directly using training samples of normal and cancerous fluorescence spectra. A fluorescence spectrum is sampled to become a high-dimensional vector. To obviate the effect of curse of dimensionality, principal component analysis (PCA)¹¹ can be applied to reduce the dimensionality by projecting fluorescence spectra of tissue samples onto a few of selected principal components and then a tissue is classified in the spectral subspace spanned by the selected principal components.¹² PCA can effectively employ most power of sample spectra with dimensionality reduction. The contribution of important biomarkers to the principal components is usually unknown, while many unknown biochemicals may contribute to the principal components. It is of interest to employ spectra of biomarkers in cancer detection and know the efficacy of each biomarker.

In detection of a signal coming from a set of known signals with observed data corrupted by noise and inference, a sufficient statistic can be obtained by projection of the data onto the subspace spanned by the set of signals. The sufficient statistic contains all information about the signal to be detected and removes the redundant dimension and reduces the power brought by the noise and inference. The sufficient statistic can be used in detection of the signal without loss of information. This principle is extensively applied in signal detection and telecommunications.¹³ Similar to but slightly different from the signal detection problem, cancer detection needs to determine whether an observed spectrum, corrupted by noise and the spectra of unknown biochemicals, comes from a cancerous or normal tissue based on a set of known spectra of biomarkers. Given a set of observed fluorescence spectra of tissues, a set of sufficient statistics can be obtained by projection of the observed spectra onto the subspace spanned by the spectra of biomarkers. The sufficient statistics contain all diagnostically useful information provided by the biomarkers. Hence cancer can be detected in this subspace without loss of diagnostically useful information while significantly reducing dimensionality.

In this paper, for the first time to our knowledge, a new theoretical approach of spectral analysis to cancer detection in the biomarkers spectral subspace (BSS) is proposed. Specifically, the bases of BSS are obtained by the Gram-Schmidt method using the spectra of biomarkers. The spectra of human cancerous and normal tissue samples are projected onto the BSS. The support vector machine (SVM)^14,15 in the BSS is trained by using training samples. To evaluate the efficacy of a classifier for cancer detection, the metrics of positive predictivity, Score1, maximum Score1, and accuracy (AC) are employed in addition to the commonly used sensitivity, specificity, and receiver operating characteristic (ROC). The BSS approach is applied in this paper to analyze fluorescence spectra for breast cancer detection with 340-nm excitation and using four biomarkers, collagen, NADH, flavin, and elastin.

2.

Methods and Materials

2.2.

Biomarkers Spectral Subspace

Since cancerous and normal tissues shall be discriminated by using the biomarkers, all information useful in cancer detection is embedded in the subspace spanned by the fluorescence spectra ${\mathbf{c}}_k$’s. The basis spectra of the subspace can be obtained by singular value decomposition (SVD) for $\mathbf{C}$. However, to clearly see how each biomarker contributes to the subspace and affects classification, the Gram-Schmidt method is employed to obtain the basis spectra as

Eq. (2)

$${\mathbf{b}}_1={\mathbf{c}}_1/\Vert {\mathbf{c}}_1\Vert$$

Eq. (3)

$${\mathbf{b}}_k=\frac{{\mathbf{c}}_k-\sum _{i=1}^{k-1}({\mathbf{c}}_k^T{\mathbf{b}}_i){\mathbf{b}}_i}{\Vert {\mathbf{c}}_k-\sum _{i=1}^{k-1}({\mathbf{c}}_k^T{\mathbf{b}}_i){\mathbf{b}}_i\Vert },k=2,\dots ,K.$$

The spectrum $\mathbf{y}$ of a tissue sample is projected onto the subspace and forms a $K$-dimensional vector

Eq. (4)

$$\mathbf{s}={\mathbf{B}}^T\mathbf{y},$$

where $\mathbf{B}=({\mathbf{b}}_1,{\mathbf{b}}_2,\dots ,{\mathbf{b}}_K)$ is the matrix of basis spectra. By the projection, $\mathbf{s}$ retains the component in the spectral subspace of biomarkers but gets rid of all other components orthogonal to the subspace. Statistically speaking, $\mathbf{s}$ is a sufficient statistic and contains all information useful in cancer detection when the biomarkers are used. Meanwhile, since $K$ is usually much smaller than $N$, the projection of $\mathbf{y}$ onto the subspace also significantly reduces the dimensionality and effectively obviates the curse of dimensionality when training a classifier. A tissue shall be classified in the spectral subspace of biomarkers using $\mathbf{s}$. The order of biomarker spectra presented in the Gram-Schmidt method affects $\mathbf{B}$ only by a unitary transform and therefore does not affect the topological relations of a set of tissue spectra in the subspace; therefore, the order does not affect classification of tissues for a given classifier.

The power of a sample spectrum $\mathbf{y}$ is defined as ${\Vert \mathbf{y}\Vert }^2$ and the total power of a set of spectra ${\mathbf{y}}_i$ for $i=1,2,\dots ,M$ is equal to $\sum _{i=1}^M{\Vert {\mathbf{y}}_i\Vert }^T$. By BSS, the spectral dimensionality is significantly reduced. Consequently, the spectral power of sample tissues in the subspace is usually less than the total spectral power of the original samples. The ratio of spectral power in the spectral subspace of biomarkers to the total spectral power measures the power usage of the biomarkers in the BSS. Denote by $\mathbf{Y}=({\mathbf{y}}_1,{\mathbf{y}}_2,\dots ,{\mathbf{y}}_M)$ the matrix of $M$ sample tissue spectra, the power usage of the biomarkers in the BSS is equal to

Eq. (5)

$$R=\frac{\mathrm{tr}[{({\mathbf{B}}^T\mathbf{Y})}^T({\mathbf{B}}^T\mathbf{Y})]}{\mathrm{tr}({\mathbf{Y}}^T\mathbf{Y})},$$

where $\mathrm{tr}$ means the trace of matrix. It is expected that a set of proper biomarkers can attain a high power usage while significantly reducing the dimensionality of spectral space.

SVM is well-known one of the most powerful classifiers.^14,15 By using training samples of cancerous and normal tissues, a classifier that is defined by a hyperplane ${\mathbf{w}}^T\mathbf{s}=b$ in the subspace is obtained by SVM with a linear kernel. In general, the SVM classifier is determined by and most effectively discriminates a number of samples, so called the support vectors, located at the boundary between cancerous and normal tissues in the spectral space. A schematic diagram of the proposed BSS approach to cancer detection in the spectral subspace of biomarkers is shown in Fig. 1.

Fig. 1

The BSS approach. Biomarkers spectra ${\mathbf{c}}_k$’s yield the basis spectra ${\mathbf{b}}_k$’s of the subspace by the Gram-Schmidt method. The spectrum of a sample tissue $\mathbf{y}$ is projected onto the subspace and yields $\mathbf{s}$, based on which the sample is determined as cancerous or normal tissue by the SVM classifier.

MCR and PCA are popularly used in tissue classification. The approach and performance of BSS will be compared with those of the MCR and PCA based classifiers using SVM, which for completeness are presented in the following subsections.

3.

Experimental Results

The objectives of experiments are to examine performance of the proposed BSS approach, to investigate the efficacy of biomarkers in cancer detection using the proposed BSS, and to compare the performance with those of MCR and PCA using SVM. BSS, MCR, and PCA are applied to breast cancer detection with four biomarkers—collagen, NADH, flavin, and elastin—with 340-nm wavelength excitation. Figure 2 shows the average spectra of the 74 tissues with standard deviations, which, for both cancerous and normal tissues, are large. The fluorescence spectra of the four biomarkers are shown in Fig. 3, and their crosscorrelations are given in Table 1. The crosscorrelation between biomarker spectra ${\mathbf{c}}_i$, ${\mathbf{c}}_j$ is defined as ${\mathbf{c}}_i^T{\mathbf{c}}_j/(\Vert {\mathbf{c}}_i\Vert \Vert {\mathbf{c}}_j\Vert )$. Flavin has low crosscorrelations with other biomarkers in the fluorescence spectrum. This implies that flavin can provide a significant component additional to the subspace spanned by {collagen, NADH, elastin}.

Fig. 2

Average spectra of normal and cancerous samples with standard deviations.

Fig. 3

Fluorescence spectra of four biomarkers.

Table 1

Crosscorrelations of biomarkers spectra.

In all the experiments, the linear kernel is applied in the SVM classifier. To classify the data in an identical space, the sample spectra in the subspace of biomarker spectra $\mathbf{s}$ are identically linearly transformed to locate in the cube ${[0,1]}^K$ (replacing $K$ by $L$ for the PCA approach). The linear transform retains the topological relationship among sample spectra and therefore does not affect the classification result. Then the parameter of $C$ in the SVM classifier can be identical in all experiments. By a number of tests on the data of fluorescence spectra in this paper and others for Raman, resonant Raman, and Stokes shifts spectra, it is found that $C=1000$ can achieve reasonably good performances and is then used in all experiments.

Figure 4 illustrates the basis spectra of the subspace spanned by the spectra of all four biomarkers, which are obtained by the Gram-Schmidt method in the order of collagen, NADH, flavin, and elastin. That is, ${\mathbf{b}}_1$ is obtained by normalizing spectrum of collagen to unit length, ${\mathbf{b}}_2$ is obtained by taking the component of NADH spectrum that is orthogonal to ${\mathbf{b}}_1$ and then normalized to unit length, etc. The correlations between the biomarker spectra and the basis spectra are illustrated in Table 2. If all the four biomarkers are used to classify the 74 samples using SVM, the power usage by BSS is 0.949, specificity is 0.973, sensitivity is 0.919, positive predictivity is 0.971, and therefore Score1 is 0.919, and AC is 0.946. By changing $b$ with the fixed $\mathbf{w}$, the ROC AUC is 0.993, close to one, the maximum Score1 is 0.973, and the number of support vectors (SV #) is 18 as listed in the last row of Table 3.

Fig. 4

Basis spectra of four biomarkers obtained by the Gram-Schmidt method in the order of collagen, NADH, flavin, and elastin.

Table 2

Correlations between the biomarker and basis spectra.

Table 3

Performance of BSS.

To investigate the efficacy of a biomarker in cancer detection, performance of all possible combinations of four biomarkers is calculated, and the results are presented in Table 3. In each row, a subset of biomarkers is given, the basis spectra of their spectral subspace are obtained by the Gram-Schmidt method, and then the SVM is trained. For example, Fig. 5 illustrates the data samples, support vectors, SVM classifier and NADH in the subspace spanned by the spectra of {collagen, NADH}. It is clear that collagen alone is incapable of properly classifying the normal and cancerous samples as shown in Fig. 5 and the first row of Table 3. However, collagen and NADH jointly can classify very well the samples with specificity of 0.973, sensitivity of 0.919, ${}^{+}P$ of 0.971, Score1 of 0.919, AC of 0.946, ROC AUC of 0.962, and maximum Score1 of 0.919 shown in Table 3. Figure 6 shows the sensitivity and positive predictivity (${}^{+}P$) versus 1−specificity. The ROC AUC, the area under the sensitivity curve, is equal to 0.962. As the specificity decreases, the sensitivity increases but ${}^{+}P$ decreases, and their intersection is the maximum $\text{Score}1=0.919$.

Fig. 5

Normal and cancerous samples, support vectors, SVM classifier and NADH component in the subspace spanned by the spectra of collagen and NADH. Specificity is 0.973, sensitivity is 0.919, ${}^{+}P$ is 0.971, Score1 is 0.919, and AC is 0.946.

Fig. 6

Sensitivity and positive predictivity (${}^{+}P$) versus 1—specificity when collagen and NADH are used, corresponding to the condition in Fig. 5. The ROC AUC is 0.962. As the specificity decreases, the sensitivity increases but ${}^{+}P$ decreases and their intersection is the maximum $\text{Score}1=0.919$.

From Table 3 the order of efficacy of biomarkers can be obtained. If one biomarker is used, collagen outperforms the other three biomarkers. If two biomarkers are used, {NADH, elastin} outperforms the other five combinations. If three biomarkers are used, {collagen, NADH, elastin} outperforms the other three combinations. If all four biomarkers are used, the performance is the best in terms of the maximum Score1 and ROC AUC. Hence the order of efficacy of biomarkers in breast cancer detection in this experiment is collagen, NADH, elastin, and flavin.

Table 3 also demonstrates that whenever collagen or elastin is used, about more than 85% power of data spectra is used in the cancer detection, which means a high power usage by the biomarkers. Table 3 also implies that when a combination of biomarkers makes samples easier to be classified in a subspace, the support vectors are fewer.

The performance of the MCR using SVM classifier is presented in Table 4. When one biomarker is employed, the performance of MCR is identical to that of BSS. However, when more biomarkers are employed, overall the BSS with ROC AUC of 0.993 and maximum Score1 of 0.973 outperforms the MCR with ROC AUC of 0.973 and maximum Score1 of 0.947 as some useful information might be lost by the MCR process.

Table 4

Performance of MCR.

The performance of the PCA using SVM classifier is presented in Table 5. Although taking into account 92.6% power of sample spectra, PC 1 is not the most effective PC in cancer detection; instead, taking only 3.6% power of sample spectra, PC 2 is the most effective PC in cancer detection. If two PCs are used, PCs 1, 2 outperform the other combinations. If three PCs are used, PCs 1, 2, 4 outperform the other combinations. Hence the order of efficacy of PCs in cancer detection is PC 2, PC 1, PC 4, and PC 3. In fact, PCs 1, 2, 4 outperform PCs 1, 2, 3, 4. In other words, PC 3 deteriorates the performance. These results imply that employing the PCs with more power of sample spectra may not result in a better performance, and using more PCs may not necessarily improve the performance in cancer detection. Overall, with properly chosen PCs, the PCA can achieve the same performance of the BSS using all biomarkers. However, it is unknown how each biomarker contributes to the PCs. Moreover, the PCs and their subspace are determined by the training samples, which may change as the number of tissue samples increases while the biomarkers spectral space is completely determined by the biomarkers.

Table 5

Performance of PCA.

In the above experiments, all 74 samples are used to train the SVM. To further compare the performance, cross validation with leave-one-out is also carried out for BSS, MCR, and PCA using the SVM classifier. The results are presented in Table 6. As expected, BSS outperforms MCR and achieves the highest accuracy of PCA among all combinations of PCs. All the analyses are performed in Matlab and the codes are available at http://www-ee.engr.ccny.cuny.edu/www/web/ysun/NanoscopeLab/Codes/BSS-JBO2012/BSS.htm.

Table 6

Performance in cross validation with leave-one-out.

4.

Conclusions

We propose a novel theoretical approach to cancer detection from the fluorescence spectral subspace of biomarkers. Projection of sample spectra onto the biomarkers spectra subspace (BSS) retains all useful information for cancer detection while significantly reducing data dimensionality. The efficacy of biomarkers in cancer detection can be determined in the subspace. In comparison, multivariate curve resolution (MCR) and spectral unmixing methods (SUM) may decrease the diagnostically useful information of sample spectra and make cancer detection less reliable. Principal component analysis (PCA) can achieve good performance if the best combination of PCs is employed, but PCA relies on the spectra of all sample tissues, and the contribution of biomarkers to PCs is unknown.

Applied to breast cancer detection with fluorescence spectrum of 340-nm wavelength excitation, the proposed BSS with support vector machine (SVM) outperforms the MCR-SVM and achieves the best performance of the PCA-SVM among all combinations of PCs. It is found that the efficacy of biomarkers in the breast cancer detection is in the order of collagen, NADH, elastin, and flavin. In principle, the BSS approach can be applied extensively to detection of other types of cancers using fluorescence and other spectra.

In addition to the commonly used sensitivity and specificity, the metrics of positive predictivity, Score1, maximum Score1, and accuracy (AC) can jointly provide a proper measurement of the sensitivity of a classifier to the true positive cancerous samples.

Finally, it needs to be pointed out that the efficacy of the BSS approach in cancer detection ultimately depends on the diagnostic power provided by the used biomarkers; moreover, the BSS approach does not attempt to quantify the fractions of biomarkers in a tissue. However, the BSS approach can retain all the diagnostic information of biomarkers, while significantly reducing the dimensionality, and provide a means to determine the efficacy of a biomarker in cancer detection.