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

Abstract

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.

1. 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.

Computational thermographic techniques, such as principal component analysis (PCA) [9], non-negative matrix factorization (NMF) [10], fixed eigenvector analysis [11,12], sparse factorization [13], t-distributed stochastic neighbor embedding (t-SNE) [14], candid covariance-free incremental principal component thermography (CCIPCT) [15], sparse PCA [16], semi-NMF [17,18], sparse NMF [19], convex NMF [20,21], deep semiNMF [20], and deep learning convex [20], have been utilized in thermography.

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.

2. Method

Input data $\mathrm{X}$ are a stack of vectorized thermal images, a heat matrix. A low-rank representation model is shown as follows:

$$\mathrm{X}\approx \mathrm{BA}$$

(1)

where $\mathrm{X}\in {ℝ}^{\mathrm{MN}\times \tau }$, i.e., $\mathrm{X}=\left[x_1,x_2,\dots ,x_{\tau }\right]$ and can be shown by a linear combination of $\tau$ bases (basis vectors), $\mathrm{B}=\left\{{\beta }_1,{\beta }_2,\dots ,{\beta }_{\tau }\right\}, \mathrm{B}\in {ℝ}^{\mathrm{MN}\times \tau }$ and $\mathrm{A}$, a coefficient matrix, A$\in {ℝ}^{\tau \times \tau }$,$\mathrm{A}=\left\{{\mathsf{\alpha }}_1,{\mathsf{\alpha }}_2,\dots ,{\mathsf{\alpha }}_{\tau }\right\}$. $x_1,x_2,\dots ,x_{\tau }$ are vectorized thermal images and correspond to $\tau$ frames.$\mathrm{X}$ 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$\lambda =\sqrt{2\sigma }$) 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 ($\mathrm{B}=\left\{{\beta }_1,{\beta }_2,\dots ,{\beta }_{\tau }\right\},$where $\mathrm{B}\in {ℝ}^{\mathrm{s}\times \tau },s=MN$)) 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.

Definition 1. 

The embedded low-dimensional representation, $\ddot{\Phi }$,  defined as follows:

$$\ddot{\Phi }={\displaystyle \sum }_{i=1}^p{w}_{\mathrm{i}}$$

(2)

where $w_i$ is a function of basis vector ${\beta }_{\mathrm{i}}$ and defines by for element of ${\beta }_{\mathrm{i}}^{\mathrm{e}}\ge 0$:

$$w_i=bk {\beta }_{\mathrm{i}}{}^{k-1}\odot e^{-b{\beta }_{\mathrm{i}}{}^k}$$

where the shape parameter $k$ is the same as above, while the scale parameter is $b={\lambda }^{-k}$.

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.

3. 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. The results of cross-validated random forest classification model.

Model	Method	Rayleigh/Weibull Embedding	Classification Accuracy 2 (%)
Random Forest	Clinical 1	73.6 (± 9.5)	73.6 (±9.5)
	CCIPCT	80.0 (66.7, 80.0)	76.9 (45.7, 85.8)
	PCT	80.0 (73.3, 85.7)	76.9 (45.7, 85.6)
	NMF	80.0 (73.3, 85.7)	76.9 (45.8, 85.7)
	Sparse PCT	80.0 (53.3, 85.7)	77.6 (45.7, 87.4)
	Deep SemiNMF	82.9 (66.7, 86.7)	76.4 (45.7, 84.8)
	Convex-NMF	78.6 (73.3, 85.7)	76.9 (73.7, 86.1)

¹ The covariates used for the clinical and demographics were family history, age and marital status. ² Classification accuracy reported by median (±IQR) (interquartile range-IQR) from Bell embedding [22].

, 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.

4. Conclusions

This study tackled a significant challenge associated with the low-dimensional (LD) representation of thermal sequences, specifically the selection of a prominent representative basis. To address this challenge, the study introduced embedding approaches.

To assess the performance of the proposed approach, a comparative analysis was conducted against state-of-the-art thermographic methods, including PCT, CCIPCT, NMF, Sparse PCT, Convex-NMF, Deep semi-NMF, as well as Bell embedding approaches. The results demonstrated that Rayleigh embedding exhibited notable capability in preserving thermal heterogeneity, enabling effective discrimination between abnormal and healthy participants. Rayleigh embedding achieved the highest accuracy of 82.9% (66.7%, 86.7%) compared to other factorization methods, notably outperforming Deep SemiNMF. Future work can focus on the low-rank approximation and more diverse cohort of patients to ensure the hypothesis of this study.