Diagnostic Biomarker for Breast Cancer Applying Rayleigh Low-Rank Embedding Thermography †

: Thermography has found extensive application as a supplementary diagnostic tool in breast cancer diagnosis, notably complementing the clinical breast exam (CBE). Within dynamic thermography, matrix factorization methods have demonstrated their utility in accentuating thermal heterogeneities by generating thermal basis vectors. A signiﬁcant challenge in such approaches is to identify the leading thermal basis vector that effectively captures predominant thermal patterns. Embedding methods are used to fuse multiple projected basis vectors onto a single basis for the extraction of the thermal features, known as thermomics . In this study, we introduce Rayleigh embedding to project thermal basis vectors obtained from factorization techniques into a lower-dimensional space, highlighting thermal patterns. This enhances the reliability of the thermal system, thereby assisting in CBE. The best results of the embedding method combining clinical information and demographics yield 82.9% (66.7%, 86.7%) using a random forest. The results demonstrated promising preliminary outcomes, leading to the early detection of breast abnormalities, and can serve as a non-invasive tool to aid CBE.


Introduction
Despite the availability of advanced screening and therapeutic methods, breast cancer continues to be the most commonly diagnosed cancer worldwide and ranks as the second most prevalent cancer [1].In this study, we propose Rayleigh embedding (a condition of Weibull) for thermographic imaging, which can serve as a valuable aid to clinical breast exams (CBE) preceding mammography.Our hypothesis posits that leveraging low-rank embedding techniques can effectively reduce the dimensionality of thermal sequences while boosting the heterogeneity of thermal patterns leading to better thermal imaging features.This heterogeneity may provide insights into vasodilation and angiogenesis resulting from cancer metabolism [2][3][4][5].
Many studies have corroborated the efficacy of infrared thermography in detecting hypervascularity and hyperthermia associated with non-palpable breast cancer [6][7][8].Consequently, infrared thermography exhibits promise as a potential biomarker for the early detection of breast cancer; however, it is crucial to emphasize its utilization in conjunction with CBE and mammography rather than as a standalone screening modality.
A new embedding is defined, using a condition of the Rayleigh distribution function, Rayleigh.Techniques such as Gaussian [20] and Bell [22] embedding approaches have been introduced previously, and this study demonstrates the application of Rayleigh embedding in factorization analysis for thermography.The study proposes using the most predominant basis combined with embedding to extract thermomics and train a classifier for the early diagnosis of breast abnormality.

Method
Input data X are a stack of vectorized thermal images, a heat matrix.A low-rank representation model is shown as follows: where X ∈ R MN×τ , i.e., X = [x 1 , x 2 , . . . ,x τ ] and can be shown by a linear combination of τ bases (basis vectors), B = {β 1 , β 2 , . . . ,β τ }, B ∈ R MN×τ and A, a coefficient matrix, A ∈ R τ×τ , A = {α 1 , α 2 , . . . ,α τ }. x 1 , x 2 , . . . ,x τ are vectorized thermal images and correspond to τ frames.X is a normalized stack of many thermal tensors obtained from input thermal images.Bases and coefficients can be modified depending on which eigen decomposition or matrix factorization method is used.

Embedding
This study proposes a novel approach using Rayleigh distribution that harnesses thermal base embedding to extract the most salient aspects of the thermal sequence, denoted as low-rank representation, to extract more effective thermomics, utilizing different eigen decomposition techniques.This distribution is an example of Weibull distribution (for k = 2 and coefficient matrix λ = √ 2σ) which exhibits the Rayleigh distribution [23], notably serving as an interpolation between the exponential distribution (k = 1) and Rayleigh distribution.
The use of bases embedding has been proposed in previous studies [20,22], which involves combining multiple decomposed bases to reduce the dimensionality of thermal images.We previously applied low-rank representation methods to transform higher temporal dimensionality into lower temporal representations, which can be considered as bases computed using matrix factorization approaches [19].We generate a set of low temporal dimensional represented basis vectors (B = {β 1 , β 2 , . . . ,β τ }, where B ∈ R s×τ , s = MN)) and integrate their overall representation using two embedding membership functions [20,22].This approach allows us to effectively reduce the dimensionality of presenting thermal images while preserving important information.
where w i is a function of basis vector β i and defines by for element of β e i ≥ 0: where the shape parameter k is the same as above, while the scale parameter is b = λ −k .
Thermomics extracted from the embedded bases subsequently lead to the automatic detection of breast cancerous leading abnormalities (CLAs) which can be used for a CBE and screening (see Figure 1).The findings of this study validate the reliability of thermomics in the early detection of breast cancer and effectively highlight CLAs in patients.

Results
The analysis of thermal patterns in breast cancer screening datasets was conducted with meticulous attention to monitoring vasodilation and the process of blood formation [24].To achieve an approximation of the data, we employed convex factorization embedding, a powerful technique that extracts meaningful insights.Furthermore, a comparative analysis was performed, contrasting the results obtained from convex factorization embedding with those derived from various low-rank matrix approximation algorithms.The purpose of this particular examination was to unravel the subtle nuances and inherent discrepancies among the different methods employed, thereby fostering a more comprehensive understanding of their relative effectiveness.
This embedding approach was employed to embed low-dimensional (LD) representations of images obtained using low-ranking representation methods.In order to assess the degree of thermal heterogeneity within the breast region, a reference label was affixed between the participants' breasts, serving as a reference point and facilitating the normalization of the image representations.By applying the embedding technique, the thermal heterogeneity was significantly enhanced, resulting in a distinct differentiation between symptomatic and cancerous patients as compared to healthy participants.The findings, as presented in Table 1, showcase the efficacy of embedding in detecting cancer-related abnormalities.

Results
The analysis of thermal patterns in breast cancer screening datasets was conducted with meticulous attention to monitoring vasodilation and the process of blood formation [24].To achieve an approximation of the data, we employed convex factorization embedding, a powerful technique that extracts meaningful insights.Furthermore, a comparative analysis was performed, contrasting the results obtained from convex factorization embedding with those derived from various low-rank matrix approximation algorithms.The purpose of this particular examination was to unravel the subtle nuances and inherent discrepancies among the different methods employed, thereby fostering a more comprehensive understanding of their relative effectiveness.
This embedding approach was employed to embed low-dimensional (LD) representations of images obtained using low-ranking representation methods.In order to assess the degree of thermal heterogeneity within the breast region, a reference label was affixed between the participants' breasts, serving as a reference point and facilitating the normalization of the image representations.By applying the embedding technique, the thermal heterogeneity was significantly enhanced, resulting in a distinct differentiation between symptomatic and cancerous patients as compared to healthy participants.The findings, as presented in Table 1, showcase the efficacy of embedding in detecting cancer-related abnormalities.
In this study, a total of 354 thermomics were extracted from the regions of interest (ROI) within the breast areas.These thermomics were obtained by applying spectral embedding to the embedded low-rank generated avatar, effectively reducing the dimensionality to seven features.Subsequently, a random forest model with 10-fold cross-validation was utilized to predict the diagnosis based on these reduced-dimensional thermomics.

Figure 1 .
Figure 1.Eight random cases, four cancer patients (a-d) and four healthy cases (e-h) using Rayleigh embedding.

Figure 1 .
Figure 1.Eight random cases, four cancer patients (a-d) and four healthy cases (e-h) using Rayleigh embedding.

Table 1 .
The results of cross-validated random forest classification model.

Table 1 .
The results of cross-validated random forest classification model.