# Dynamic Vascular Imaging Using Active Breast Thermography

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Method

#### 2.1. Goal

#### 2.2. Approach

#### 2.3. Mathematical Analysis

**I = T M**where

**I**is the original image, and the time matrix

**T**is

**W**of Equation (A8). The image matrix

**M**becomes a virtual wave with wave-like properties; each column represents an image at a different depth. The transformation is simple as it requires only matrix multiplication. The matrix

**W**has no information unless we want to reconstruct the original data. Performing VWT also rejects images that do not follow diffusion propagation in response to the cooling. As the transformation is ill-posed, time focusing is imperfect, and images in matrix

**M**still overlap. We remedy the overlap in two stages; in the first step, we use PCA. PCA factors the converted matrix based on similarities between images; it cannot separate similar-looking images that arrive at different times. Next, we separate the images based on their arrival time by applying ICA to the resulting time matrix. The process uses three alternate matrix factorizations; time-based and image-based, followed by a time-based. We give more details in the examples.

## 3. Results

#### 3.1. Data

#### 3.2. Data Preparation

#### 3.3. Analysis

^{−8}m

^{2}s

^{−1}. We applied PCA next using six components with 100 virtual time points at the sequence t

_{i}= 0.025 × 2

^{(i − 100)/20}. One hundred points are more than necessary but result in smoother tracings. We show the images in Figure 2, images 1 to 6. We present the amplitudes of the columns of the time matrix in Figure 3. The waveforms of different patients and contralateral breasts are very similar. On the log time scale, columns one and two appear similar but are shifted. In these traces, we recognized first a small negative peak, then a larger positive peak at twice the time of the first peak. The PCA algorithm separates the components based on a similar appearance but cannot distinguish between almost identical images at different depths. Tracings of Figure 2, column 4 peaks are collocated with the peaks of column 2, but in column 4, both are positive. We interpret it as image 2 being composed of two very similar images, while image 4 is the difference between them. The traces of columns 1 and 3 are very similar to those of 2 and 4 but less pronounced. The original application of PCA to the images separates similar images but mixes the time. We unmixed the components by applying ICA to the time matrix. At the same time, we apply the inverse to the corresponding image to maintain the product matrix. Block diagram of the processing is depicted in Figure 4. We show the first four traces of the unmixed time matrix in Figure 5 and the images in Figure 6. Assuming a fat layer, the estimated depth of trace 1 is 13 mm deep and 9 mm for trace 2.

## 4. Discussion

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Virtual Wave Transform (VWT)

#### Appendix A.1. Virtual Wave

_{w}is the amplitude. The thermal propagation is described by

_{w}is the amplitude of the diffusive component (temperature, k is conductivity, ρ is density, C is the specific heat, and g

_{d}is the heat source., Initially, we use Dirac’s delta function for g

_{d}to obtain Green’s function and then convolve it with g

_{w}to get the desired solution, g

_{w,}which is virtual and arbitrary; we limit it to differential or integrals of Dirac’s delta function. Applying the Laplace transform to the time variable, Equations (A1) and (A2) become:

**r**.

_{n}(x) is Hermite polynomials of order n. An important conclusion is that the transformation is linear and local.

_{d}, we get

**x**and

**W**.

#### Appendix A.2. Modified Truncated Singular Value Decomposition

_{w}from x

_{d}. To do this, one must invert the matrix

**W**. However, the determinant of

**W**is very small, making it an ill-posed problem. Therefore, truncated singular value decomposition (TSVD) or Tikhonov inversion must be used to invert the matrix

**W**and calculate x

_{w}from x

_{d}. In this work, we used a modified TSVD. We can write

**W**as

**W = UˑV**and

^{T}UˑU^{T}= I VˑV^{T}= I**W**

^{−1}=

**V D**

^{−1}

**ˑU**

**D**is diagonal; therefore, its inverse is the inverse of the individual element of

**D**. Truncate singular value decomposition replaces all inverse elements larger than specific values with zeros. We used a slightly modified version where we replaced 1/d

_{i}with d

_{i}/(d

_{i}

^{2}+λd

_{0}

^{2}), which resulted in less ringing in the inversion. We used Hermit polynomial order n = 2 and λ = 0.003, equivalent to keeping four eigenvalues out of 20. Reflections of waves arrive at precise timing; therefore, missing them means lost information, while reflections in diffusion propagation spread over time. Recording a significant portion of the signal in diffusion propagation is sufficient to perform the transform.

## Appendix B. Matrix Factorization

**I,**where each column represents a single frame of the thermal data. In matrix factorization, we separate the data into:

**T**contains time information, while matrix

**M**contains image information. VWT, PCA, and ICA are all matrix factorizations that operate on

**T**or

**M**. To maintain

**I**of Equation (A9), we apply the inverse to the other matrix when we operate on one matrix.

## Appendix C. Excitation Extraction

_{d}(t) to perform the inversion. Excitation was in the form of cooling by a fan; duration varied from patient to patient (but for less than three minutes) and was not recorded. Thermal images were recorded at the end of the cooling. Using a similar fan, we verified that ramps up and down are less than 15 s (sampling period); therefore, the excitation can be regarded as a square wave. To determine the start and the stop time of g

_{d}(t), we run the inversion algorithm followed by the PCA analysis over a grid of start and stop values. We inspect the earlier recovered reflection, which we assume is a superficial reflection from the skin’s top surface. The value of ‘latency’ (eigenvalues of the inversion) of the PCA of earliest reflection indicates the fit’s quality. We search over a grid of durations and excitations stop time for the most significant latent point, which we use for the inversion.

## Appendix D. Image Registration

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**Figure 2.**Thermal images of patient “T281”; right breast after VWT and PCA. Images A to F are individuzal PCA vectors converted to images (

**A**–

**F**).

**Figure 4.**A simplified block diagram of the processing. n is a count of all the pixels included in the individual breast.

**A**and

**C**are image matrixes, and

**B**and

**D**are time matrixes. In the example given, matrix

**I**is n × 100,

**A**dimensions are 100 × 6,

**B**dimensions are 6 × n,

**C**dimensions are n × 4, and

**D**dimensions are 4 × 100.

**Figure 5.**Traces 1:4 after application of ICA to the time matrix of Figure 3 on log time scale with the sequence t

_{i}= 0.025 × 2

^{(i − 100)/20}.

**Figure 6.**Healthy right breast of patient “T281.” Images 1A to 1D after application of the inverse ICA transform. (

**A**,

**B**) are images of thermal reflection, while (

**C**,

**D**) are images of vasoconstriction.

**Figure 7.**Sick left breast of patient “T281.” Images A to D after application of the inverse ICA transform. (

**A**,

**B**) are images of thermal reflection, while (

**C**,

**D**) are images of vasoconstriction.

**Figure 8.**“Healthy” left breast of patient #T285. Images 1 to 4 after application of the inverse ICA transform. (

**1**) and (

**2**) are images of thermal reflection, while (

**3**) and (

**4**) are images of vasoconstriction.

**Figure 9.**“Sick” left breast of patient #T285. Images 1 to 4 after application of the inverse ICA transform. (

**1**) and (

**2**) are images of thermal reflection, while (

**3**) and (

**4**) are images of vasoconstriction.

**Table 1.**Relevant medical information. QSL = Upper Side Quadrant, QIL = Lower Side Quadrant. QIM = Upper Middle Quadrant, None of the patients had mammography.

Patient | Age | Exam Details | Diagnosis | Quadrant | Biopsy | Complaint |
---|---|---|---|---|---|---|

T281 | 43 | Left | Sick | QSL | left | Pain in both breasts with burning |

T282 | 46 | Left | Sick | QIM | left | Pain in both breasts |

T285 | 57 | Right | Sick | QIL | right | Pain in the right breast |

T286 | 50 | Left | Infiltrating Ductal Carcinoma | QSL | left | Itching in both breasts, secretion in the left breast |

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**MDPI and ACS Style**

Gershenson, M.; Gershenson, J.
Dynamic Vascular Imaging Using Active Breast Thermography. *Sensors* **2023**, *23*, 3012.
https://doi.org/10.3390/s23063012

**AMA Style**

Gershenson M, Gershenson J.
Dynamic Vascular Imaging Using Active Breast Thermography. *Sensors*. 2023; 23(6):3012.
https://doi.org/10.3390/s23063012

**Chicago/Turabian Style**

Gershenson, Meir, and Jonathan Gershenson.
2023. "Dynamic Vascular Imaging Using Active Breast Thermography" *Sensors* 23, no. 6: 3012.
https://doi.org/10.3390/s23063012