2.1.

$q-r$ Curve of Thermal Tomography

Generally, the Pennes equation is considered to be the most suitable one in the bioheat transfer models so far.²⁶ It is written as

If the diseased area is not large, or the size of the diseased area can be neglected relative to the distance from the diseased area to the body surface, or the shape of the diseased area can be approximated as a sphere though its size is relatively large, then the diseased area can be regarded as a point heat source on these occasions. The mathematical function $\delta$ is introduced here to simplify the form of the bioheat transfer equation and make it more feasible to solve.

In order to simplify the model, $r$, $c$, and $k$ are set as constants, which means the heat from the point heat source in 3-D space has good spherical symmetry. Thus, in the spherical coordinate system, only the term $r$ is left in Eq. (3)

Equation (6) is used for a point source set in a free space with no boundary condition, and it is the solution of the heat transfer equation of the point heat source. Here, $r$ is the distance from the internal heat source to some point on the body surface and $T$ indicates the temperature function. In practical applications, the temperature value of various points on the body surface can be acquired with an infrared thermal detector.

On the body surface, the position of point O mapped from the internal heat source can be easily found, because it is the very point where the temperature is highest. Taking the point O as the origin, a polar coordinate OX can be established as in Fig. 1, where $d$ is the depth from the internal heat source to origin O and $x$ is the distance from origin O to an arbitrary point on the body surface. In consideration of the influence of environmental temperature $T(\infty )$, the temperature distribution function of the body surface can be written as

Fig. 1

Building of polar coordinate system on body surface.

Here, $q$ is the heat intensity and $d$ is the depth of the point heat source. $T(\infty )$ includes the influence of heat convective loss at the tissue–air interface in our model. This loss is proportional to the increase of the surface temperature. However, the temperature of a living human body is not possible to vary in a large range. In fact, the heat convective loss is relatively stable if the air speed on the body surface and the temperature around it can be controlled to a limited range. As can be seen from Eq. (6), the temperature distribution of the body surface can be acquired as long as the information for the internal heat source is known. However, in the practical application, the temperature of a body surface can be easily acquired with an infrared thermal detector, and the information of the internal heat source is just what we want to acquire. It is a typical inverse problem that is very difficult to be applied in clinical applications.^27,28 Based on Eq. (6), we propose a novel method of thermal tomography to acquire more information about the heat intensity.

In our model, the density, thermal conductivity, and thermal capacity of the biological tissue are all supposed to be constants, and the internal heat source is assumed to be an ideal point one. By setting a straight line through the point where the temperature is highest on the body surface, the temperature distribution along the straight line will fit the Lorenz curve very well.²⁸ In the clinical application, one can set a straight line through the center of the pathological high temperature area on the infrared image (Fig. 2). Twelve points are chosen equidistantly along the straight line, and one of the points should lie in the center of the high temperature area. The temperature of every point can be extracted (Table 1). In Fig. 3, $R$ denotes the distance from the center point of the pathological high temperature area to the point chosen on the straight line and $T$ denotes the temperature of the corresponding points. The sign before the abscissa values indicates the two sides of the center point. The fitting situation is shown in Fig. 3. We use the coefficient of determination ($R^2$) to quantify the fitting between the Lorentz curve and temperature distribution curve. After calculation, the value of $R^2$ is 0.8722, which means there is a good fitting between the Lorentz curve and temperature distribution curve.

Fig. 2

Straight line set on infrared image.

Table 1

Points on straight line and their corresponding temperatures.

Fig. 3

Fitting of temperature distribution curve and Lorentz curve.

Contrasting Eq. (6) with the standard Lorentz equation $y=A{\omega }^2/(x^2+{\omega }^2)+y_0$, it can be determined that $d=\omega$ and $q=4\pi kd\sqrt{A}$, which means that the heat intensity $q$ is relative to the peak height and peak width of the temperature distribution curve.²⁷ Taking the diseased tissue as the center of analysis, a $q–r$ curve can be drawn with the method of fitting the Lorentz curve. Figure 4(a) shows the $q–r$ curve of the diseased tissue where it can be seen that the temperature decreases gradually around the diseased tissue. Figure 4(b) shows the $q-r$ curve of normal tissue, from which it can be seen that there is no significant temperature difference between the chosen analysis point and the tissue around it. In Fig. 4, there are two numbers at the upper left corner, which express the coordinate values of the point chosen for analysis (The coordinate system is created in the original infrared image, the bottom left corner of the infrared image is taken as the origin, the bottom edge is taken as the abscissa axis, and the left edge is taken as the vertical axis. The coordinate values are expressed with the number of pixels). In Fig. 4(a), for instance, the analyzed point corresponds to the lateral 122nd pixel and the longitudinal 118th pixel from the bottom left corner of the original infrared image. It should be explained that the infrared images appearing in the following passage are cut out from the original ones, and are just for highlighting the diagnosed parts of the human body. Here, the coordinate values are very important for the positioning of the analyzed point.

Fig. 4

Plotting of $q-\mathit{r}$ curve: horizontal axis represents depth $r$, vertical axis represents quantity of heat $q$. (a) $q-\mathit{r}$ curve of diseased tissue and (b) $q-\mathit{r}$ curve of normal tissue.

2.2.

Establishment of Diagnostic Criteria for Breast Tumor

Tumors can be classified as malignant and benign. The benign tumor is composed of mature cells and grows slowly; the temperature difference between the body surface mapping the tumor tissue and the skin around is small, and in most instances, it is within 1°C; The malignant tumor is composed of immature cells and it grows rapidly; it has multiple blood vessels and vigorous metabolism, which leads to the obvious difference in temperature between the body surface mapping the tumor tissue and the skin around. However, in the middle and advanced stage, the liquefactive necrosis appears at the center of a malignant tumor, and it will affect the trend of the end part of the $q–r$ curve.

The shape of the $q–r$ curve reflects the status of heat distribution in the human body, i.e., the status of metabolism. Our study focuses on four kinds of cases; they are malignancy, benign, mastitis and hyperplasia or normal. The key problem is how to distinguish them according to the distribution and trend of the $q-r$ curve.

To resolve this problem, we collected 34,977 confirmed cases of breast disease and their corresponding $q–r$ curves from six large cooperative hospitals (People’s Hospital of Hubei province, Huaxi Hospital, General Hospital of Chinese PLA, Zhongnan Hospital, Hubei Tumor Hospital, and Hubei Hospital of Traditional Chinese Medicine). Among these confirmed cases, there were 10,566 cases of mastitis, 12,973 cases of malignant breast tumor, 9280 cases of benign breast tumor, and 2158 cases of hyperplasia of mammary glands or normal (Among the 34,977 confirmed cases of breast disease, we biopsied the tumor cases or suspected tumor cases diagnosed by imaging. If the cases can be confirmed by imaging not to be tumors, we did not do biopsy). We create a flat surface whose left border indicates the heat intensity and whose bottom border indicates the depth. All the $q–r$ curves must go through the origin set at the bottom left (including those of normal tissue and abnormal tissue), which means such a flat surface contains all the possibilities of the cases. The 34,977 $q–r$ curves were moved in this flat, and all the curves were classified with the type of disease. Different kinds of $q–r$ curves were indicated with different colors. Based on the distribution of $q–r$ curves and statistics, it was found the $q–r$ curves of most malignant tumors lay in section A, the $q–r$ curves of most benign tumors lay in section B, and the $q–r$ curves of most mastitides lay in section C. For most of the hyperplasia or normal cases, the $q–r$ curves lay in section D. The four sections were acquired with three straight lines whose included angles with the horizontal axis were 46, 29, and 15 deg (Fig. 5). (The criteria are only for female breasts at present.)

Fig. 5

Section division of thermal tomography curve for female breast.

The detailed statistics are shown as Table 2. The $P$-value is an important index of measuring the reliability of statistical data. After calculation, we found the $P$-value of our statistics is $<0.05$, which means our statistics are reliable. In order to get additional information, a further analysis, a posthoc test, is done to determine the difference between the two groups. In our study, multiple comparisons between the groups were performed using the Student-Newman-Keuls method. The result shows that there exists significant differences between the four groups (mastitis, malignancy, benign, and hyperplasia or normal).

Table 2

Distribution statistics of q−r curves.

2.3.

Device Scheme

The schematic diagram of the thermal tomography system is shown in Fig. 6. First, the infrared thermal detector is adopted to detect the infrared radiation of the human body and translate the thermal information into temperature information displayed with the infrared thermal image. Second, a point is chosen in the region of interest of the thermal image based on the data processing technology and the approach of thermal tomography presented in this paper, and the internal heat information $q–r$ curve can be acquired. Finally, according to the distribution and trend of the $q–r$ curve, the corresponding diagnosis can be suggested on the basis of diagnostic criteria. The highly sensitive uncooled focal plane infrared thermal imager, whose temperature resolution was 0.08°C, spatial resolution was 1.4 mrad, pixel was $320\times 240$ and focusing range was 0.5 m to $\infty$, is taken as the temperature measuring tool of the human body surface. In the examination, the angle of the patient toward the camera will not affect the measured temperature of the same location on the patient. But we still demand the abnormal temperature area of patients to be facing toward the camera to acquire enough analysis area.

Fig. 6

Schematic diagram of thermal tomography system for tumor diagnosis.

The flow diagram of thermal tomography analysis is shown in Fig. 7. In the original infrared thermal image, we set a reference object of scale whose size is constant, and it is the basis of ratiometric conversion between the infrared thermal image and its corresponding physical object.

Fig. 7

Flow diagram of thermal tomography analysis.

3.1.

Diagnosis of Malignant Breast Tumors

There were two confirmed cases of malignant breast tumor. One is diagnosed with typical malignant breast tumor that is very usual in the breast disease, such as invasive ductal carcinoma, and the other is diagnosed with atypical malignant breast tumor that is seldom seen in the breast disease such as medullary carcinoma.

Woman A, whose infrared thermal image of breast was shown in Fig. 8(a), is 32 years old. There was an abnormal high temperature area on the upper left side of the left breast. A point was chosen at the selected area to do a $q–r$ curve analysis of thermal tomography, and the result was shown in Fig. 8(b). As could be seen from Fig. 8(b), the vast majority of the $q–r$ curve lay in the section of the malignancy and the heat intensity $q$ increased conspicuously with the depth of the thermal tomography. Thus, it was diagnosed as a malignant tumor by the $q–r$ curve analysis. Figure 8(c) was the corresponding image of the MT x-ray, from which it could be seen that there was a high density mass behind and above the left papilla, the boundary was clear and appeared invasive, and a rare calcification could be seen. Figure 8(d) was the corresponding image of the B-ultrasonic, which showed that there was a hypoechoic mass at the upper side of the left breast, the profile was irregular, the boundary was obscure, and the Doppler signals were not strong inside and around the mass. The two examinations above gave a result of malignant tumor. By pathological examination, it was diagnosed as second stage invasive ductal carcinoma, which agreed with that of the $q–r$ curve, MT x-ray, and B-ultrasonic.

Fig. 8

$q-\mathit{r}$ curve analysis of woman A and the corresponding comparison. (a) The infrared thermal image of woman A. (b) The $q-\mathit{r}$ curve analysis of woman A. (c) The image of molybdenum target (MT) x-ray of woman A. (d) The image of B-ultrasonic of woman A.

Woman B, whose infrared thermal image of breast was shown in Fig. 9(a), was 41 years old. There was an abnormal high temperature area above the right breast. A point was chosen at the selected area to do the $q–r$ curve analysis of thermal tomography, and the result was shown in Fig. 9(b). As could be seen from Fig. 9(b), the vast majority of the $q–r$ curve lay in the section of malignancy and the heat intensity $q$ increased conspicuously with the depth of thermal tomography. Thus, it was diagnosed as malignant tumor by the $q–r$ curve analysis. Figure 9(c) was the corresponding image of the MT x-ray, from which it could be seen that there was a mass behind the right papilla, the boundary was clear, and a rare calcification could be seen. Figure 9(d) was the corresponding image of B-ultrasonic, which showed that there was a very hypoechoic mass behind the right papilla and the echo was uniform. The profile was irregular, the boundary was obscure, and the Doppler signals were strong. The two examinations above gave the result of a malignant tumor. By pathological examination, it was diagnosed as medullary carcinoma.

Fig. 9

$q-\mathit{r}$ curve analysis of woman B and the corresponding comparison. (a) The infrared thermal image of woman B. (b) The $q-\mathit{r}$ curve analysis of woman B. (c) The image of MT x-ray of woman B. (d) The image of B-ultrasonic of woman B.

3.2.

Diagnosis of Benign Breast Tumors

There were two confirmed cases of benign breast tumor. One is a cystic breast tumor, and the other is a solid breast tumor.

Woman C, whose infrared thermal image of breast was shown in Fig. 10(a), was 31 years old. There was an abnormal high temperature area on the upper left side of the left breast. A point was chosen at the selected area to do the $q–r$ curve analysis of thermal tomography, and the result was shown in Fig. 10(b). As could be seen from Fig. 10(b), the vast majority of the $q–r$ curve lay in the section of the benign area and the heat intensity $q$ increased with the depth of thermal tomography. Thus, it was diagnosed as benign tumor by the $q–r$ curve analysis. Figure 10(c) was the corresponding image of the MT x-ray, from which it could be seen that there was a low-density shadow at the posterolateral side of the left breast, the boundary was not clear, and clustered calcification could be seen. Figure 10(d) was the corresponding image of B-ultrasonic, which showed that there was a hypoechoic area at the posterolateral side of the left breast, the boundary was not clear, and at the posterior of the shadow, there was no significant change in the echo. The Doppler signals were weak. The two examinations above gave the result of benign tumor. By pathological examination, it was diagnosed as galactocele.

Fig. 10

$q-\mathit{r}$ curve analysis of woman C and the corresponding comparison. (a) The infrared thermal image of woman C. (b) The $q-\mathit{r}$ curve analysis of woman C. (c) The image of MT x-ray of woman C. (d) The image of B-ultrasonic of woman C.

Woman D, whose infrared thermal image of breast was shown in Fig. 11(a), was 36 years old. There was an abnormal high temperature area on the upper left side of the left breast. A point was chosen at the selected area to do the $q–r$ curve analysis of thermal tomography, and the result was shown in Fig. 11(b). As could be seen from Fig. 11(b), the vast majority of the $q–r$ curve lay in the section of the benign area and the heat intensity $q$ increased with the depth of thermal tomography. Thus, it was diagnosed as benign tumor by the $q–r$ curve analysis. Figure 11(c) was the corresponding image of the MT x-ray, from which it could be seen that there was a round high-density shadow at the posterolateral side of the left breast, the boundary was clear, and there were no obvious lobulated edge and calcification. Figure 11(d) was the corresponding image of B-ultrasonic, which showed that there was a hypoechoic mass at the posterolateral side of the left breast, the boundary was clear, and at the posterior of the shadow, there was no significant change in the echo. The Doppler signals were strong around the mass. The two examinations above gave the result of benign tumor. By pathological examination, it was diagnosed as breast fibroadenoma.

Fig. 11

$q-\mathit{r}$ curve analysis of woman D and the corresponding comparison. (a) The infrared thermal image of woman D. (b) The $q-\mathit{r}$ curve analysis of woman D. (c) The image of MT x-ray of woman D. (d) The image of B-ultrasonic of woman D.

3.3.

Diagnosis of Mastitis

Woman E, whose infrared thermal image of breast was shown in Fig. 12(a), was 28 years old. There was an abnormal high temperature area above the right breast. A point was chosen at the selected area to do the $q–r$ curve analysis of thermal tomography, and the result was shown in Fig. 12(b). As could be seen from Fig. 12(b), the vast majority of the $q–r$ curve lay in the section of mastitis and the heat intensity $q$ increased conspicuously with the depth of thermal tomography. Thus, it was diagnosed as mastitis by the $q–r$ curve analysis. Figure 12(c) was the corresponding image of the MT x-ray, from which it could be seen that there existed asymmetrical density and irregular structure in the right breast and there was no obvious mass and calcification. Figure 12(d) was the corresponding image of B-ultrasonic, which also showed that there was asymmetrical density and irregular structure in the right breast and there was no obvious mass and calcification. The Doppler signals were strong. The two examinations above gave the result of mastitis. By pathological examination, it was diagnosed as mastitis.

Fig. 12

$q-\mathit{r}$ curve analysis of woman E and the corresponding comparison. (a) The infrared thermal image of woman E. (b) The $q-\mathit{r}$ curve analysis of woman E. (c) The image of MT x-ray of woman E. (d) The image of B-ultrasonic of woman E.

3.4.

Diagnosis of Breast Hyperplasia

Woman F, whose infrared thermal image of breast was shown in Fig. 13(a), was 37 years old. There was an abnormal high temperature area on the upper left side of the left breast. A point was chosen at the selected area to do the $q-\mathit{r}$ curve analysis of thermal tomography, and the result was shown in Fig. 13(b). As could be seen from Fig. 13(b), the vast majority of the $q-\mathit{r}$ curve lay in the section of hyperplasia or normal breast and the heat intensity $q$ increased slowly with the depth of thermal tomography. Thus, it was diagnosed as hyperplasia by the $q-\mathit{r}$ curve analysis. Figure 13(c) was the corresponding image of the MT x-ray, from which it could be seen the hyperplasia was of the mammary gland and the distribution was dense. There existed nebulous masses at the inner side of the left breast. Figure 13(d) was the corresponding image of B-ultrasonic, which also showed that there was a hypoechoic area at the inner side of the left breast and the boundary was clear. The Doppler signals were weak. The two examinations above gave the result of hyperplasia. By pathological examination, it was diagnosed as hyperplasia.

Fig. 13

$q-\mathit{r}$ curve analysis of woman F and the corresponding comparison. (a) The infrared thermal image of woman F. (b) The $q-\mathit{r}$ curve analysis of woman F. (c) The image of MT x-ray of woman F. (d) The image of B-ultrasonic of woman F.

As can be seen from the six clinical practices and the comparisons above, the diagnosis results by $q–r$ curve are very consistent with those by MT x-ray and B-ultrasonic. This novel method of tumor diagnosis is not only noncontacting, noninvasive, and radiation free to patients, but also quick and convenient to implement.