关于乳腺癌检测的IEEE文章

IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 48, NO. 9, SEPTEMBER 2001 1053 Communications______________________________________________________________________ Computational Modeling of Three-Dimensional Microwave Tomography of Breast Cancer Alexander E. Bulyshev*, Serguei Y. Semenov, Alexandre E. Souvorov, Robert H. Svenson, Alexei G. Nazarov, Yuri E. Sizov, and George P. Tatsis Abstract—Microwave tomographic approach is proposed to detect and image breast cancers. Taking into account the big difference in dielectrical properties between normal and malignant tissues, we have proposed using the microwave tomographic method to image a human breast. Because of the anatomical features of the objects, this case has to be referred to the tomography with a limited angle of observation. As a result of computer experiments we have established that multiview cylindrical configurations are able to provide microwave tomograms of the breast with a small size tumor inside. Using the gradient method, we have developed a computer code to create images of the three-dimensional objects in dielectrical prop- erties on microwave frequencies. Index Terms—Dielectric model of breast, gradient method, inverse problem, microwave tomography. I. INTRODUCTION Many sources [1]–[3], have demonstrated that malignant tissues are different from normal tissues in dielectrical properties on microwave frequencies. Based on these possibilities, the authors of [4] proposed a method for detecting breast cancer. They have shown that even a small spot with high values of complex dielectrical permittivities can affect a back-scattered microwave signal. They also underlined significant ad- vantages of the microwave diagnostic abilities over other existing tech- niques. In [5] microwave imaging of another kind of cancer has been presented. In this paper, we have applied a tomographic approach to create microwave images which allows us to obtain three-dimensional (3-D) images of a woman’s breast with high quality. The 3-D microwave tomography of the breast has significant differ- ence in comparison with standard approaches: in this case, we cannot provide all around illumination and observation of the object. More- over, the object (breast) borders the high-contrast region (chest), and this circumstance could be a serious obstacle to obtaining high-quality microwave images of the breast. However, our computational experi- ments show that we can produce images even under these conditions using cylindrical geometry of the tomograph described in [6]. The reconstruction algorithm is also an important issue in any at- tempts to construct a tomographic device. We have figured out that in our case the gradient method [7] could be an acceptable choice. We de- scribed our modifications of this method previously [6], and we have applied it to solve breast cancer imaging problems. Manuscript receivedJanuary 13, 2000; revised June 2, 2001. Asterisk indi- cates corresponding author. *A. E. Bulyshev is with the Laser and Applied Technologies Laboratory, Carolinas Medical Center, Charlotte, NC 28203 USA (e-mail: ssemenov@car- olinas.org). S. Y. Semenov, A. E. Souvorov, R. H. Svenson and G. P. Tatsis are with the Laser and Applied Technologies Laboratory, Carolinas Medical Center, Char- lotte, NC 28203 USA. A. G. Nazarov and Y. E. Sizov are with the Kurchatov Institute of Atomic Energy, Moscow, Russia. Publisher Item Identifier S 0018-9294(01)07449-3. Fig. 1. A vertical slice of the device and the model. 1: Working chamber; 2: a set of transmitters; 3: a set of receivers; 4: a semisphere; 5: a small inhomogeneous area; 6: a layer of fat; 7: a layer of muscles; 8: a layer of bone. Using the experimental data about dielectrical properties of living tissues, we have developed a dielectrical model of the breast. We have taken into account dielectrical properties of breast tissues, fat, muscles, bone, and malignant tissues. In Section II, we describe geometry of the device and models which are proposed. In Section III, we formulate microwave tomographical problems from a mathematical point of view. Section IV contains the results of our computational experiments, which is followed by our conclusions. II. OBSERVATION SCHEME AND MODEL OF THE OBJECT The cylindrical multiview illumination scheme has been recently ap- plied to solve full-body microwave tomographic problems [6]. Trying to use this scheme to image the breast, we face two significant differ- ences. First, the angle of observation is strictly limited in vertical (z) direction (see Fig. 1). Second, the object (breast) borders high-contrast region (chest). In the present investigation, we are not interested in the chest imaging. The aim is the breast imaging, but existence of a high contrast area could distort images of the object. However, the results of our computational experiments show that even under these conditions this method can provide high-quality breast images and can visualize small size cancer zones. In Fig. 1, one can see that the tomographical chamber is a cylinder with the transmitter and receiver antennas attached to the cylindrical surface. In our computational experiments, we have used one, three, or five rows of transmitters with 32 transmitters in each row and 32 rows of receivers with 32 receiver in each row. Transmitters work one after another, and receivers work all together simultaneously. This is the main reason why the numbers of transmitters and receivers are so different. Both computational time and real working time of the tomo- graph are proportional to the number of transmitters and do not depend on the number of receivers. Using the experimental data of tissues dielectrical properties, we developed the dielectrical model of a woman’s breast. This model consists of semi-sphere with radius R c and three cylindrical layers on the top of this semi-sphere. The first layer represents a fat layer, the second—muscles, the third—bones. We were modeling tumors as spheres with a small radius placed inside the semi-sphere. The 0018–9294/01$10.00 © 2001 IEEE Administrator 高亮 Administrator 高亮 Administrator 高亮 快速收敛方法，用于成像的算法 Administrator 线条 Administrator 高亮 Administrator 高亮 Administrator 高亮 Administrator 高亮 Administrator 高亮 Administrator 高亮 Administrator 高亮 Administrator 高亮 Administrator 高亮 Administrator 线条 Administrator 线条 Administrator 高亮 Administrator 高亮 Administrator 线条 1054 IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 48, NO. 9, SEPTEMBER 2001 schematic of this model is drawn in Fig. 1. In our computational exper- iments, we accept the following values of the dielectrical properties. Fat: " f = 10 + i; bone: " b = 11 + 2i; muscles: " m = 25 + 15i; breast tissue: " bt = 10 + 1:5i; malignant tissue: " c = 23 + 15i. III. MATHEMATICAL MODEL The mathematical model of the microwave tomographic system was described in our previous paper [6]. This model is based on the Helmholtz equation for solving the direct problems and on the gradient method for inverse problem solving. As we have showed, the multiview illumination scheme has to be used for good quality imaging. The significant difference between this case and the situation discussed in [6] is that we can provide only limited angle for illumi- nation in z-direction because of anatomical features of the object (see Fig. 1). A. Direct Problem As we mentioned, we use the scalar Helmholtz equation to describe electromagnetic waves propagation in free space and in the object r 2 E s + k 2 E s = (k 2 0 � k 2 )E 0 (1) where E 0 and E s z component of the incident and scattered electric field; k wave number; k 2 = (2�=�) 2 "; � wavelength of the radiation in vacuum. As a rule, the scalar approximation has been used in the works devoted to two-dimensional (2-D) microwave tomographic problems. We used this approximation as the first reasonable step to solve the 3-D vector problem. In our computational experiments, we used the point source field as the incident field E 0 (r) = exp(ik 0 r) 4�r (2) where r is the distance between the current point and the phase center of the transmitter antenna, k 0 is the wave number for homogeneous space. In order to solve (1), we need to formulate boundary conditions for E s . As a computational region we used a cylinder placed inside of the tomographical chamber. We used approximated absorbed boundary conditions [8] on the surface of the cylinder. One can find more details about how to solve this problem in [6]. B. Inverse Problem Being a nonlinear problem, the inverse problem for the Helmholtz equation requires iterative methods to solve it. Three different ap- proaches [7], [9], [10] have been applied to such kinds of problems. The first method [9], [11], [12] is an iterative scheme based on the Born–Rytov approximation and has a limited area of applications. This method is very effective to image weak scattering objects, but is unable to image strong scattering objects. The best results were achieved applying Newton iterative schemes [10], [13]–[15] to solve 2-D problems. The key points of the Newton method are calculation and inversion of a matrix of a large dimension, and in the 3-D case these procedures become extremely time consuming. We successfully employed the gradient method [7] to solve 3-D mi- crowave tomographic problems [6]. This method was also applied in the present work. A gradient method operates with functional J ["] J ["] = i; j kU teor i; j � U expr i; j j 2 (3) where U teor i; j is the theoretical prediction of the scattered field mea- sured by receiver number j when transmitter number i was employed, and U expr i; j is the experimentally measured value of the scattered field. An inverse problem can be reformulated as a minimization problem for the functional J ["]. It is very important to know the gradient of the functional J ["] to solve a minimization problem. In this case, the so- lution is given in [7], [16]. It could be shown that gradient J 0 can be represented as J 0 = i U � i V i (4) where U i is the electrical field produced by transmitter with number i. Function V i can be found from r 2 V i +k 2 V i = (k 2 �k 2 0 ) j (U expr i; j �U teor i; j )G 0 (k 0 jr j �rj)) (5) where G 0 Green’s function for the uniform space; r current point; r j receiver location. In the 3-D case, the Green function can be represented as G 0 (k 0 r) = exp(k 0 r) 4�r : (6) Knowing gradient of the functional J , we can construct the minimizing procedure " n+1 = " n � J 0 [" n ]s n (7) where step sn was chosen according to a simple algorithm [6]. C. Computational Experiments We used computational experiments as investigational tools. The standard scheme of the closed circle was employed [17]. First, we choose the model of the object. Then, we calculate “experimental” data. Then, we restore the image of the object. Finally, we are able to com- pare the results of restoration with the original. The main goal of the experiments was to establish the fact that a small but high contrast in- homogeneous feature could be made visible under these experimental conditions. In our computational experiments, we can vary geometric characteristics of the system, numbers of transmitters and receivers, working frequency, and parameters of computational schemes. IV. RESULTS AND DISCUSSION In our computational experiments, we used the model with the fol- lowing parameters: a radius of the semi-sphere R c = 4.5 cm, heights of the layers H 1 = H 2 = H 3 = 1 cm, a radius of the small sphere R t = 0.3 cm, dielectrical properties " bt = 10 + 1:5i, " b = 11 + 2i, " m = 25+ 15i, " f = 10+ i, dielectrical properties of the immersion liquid were chosen " 0 = 10 + i. The radius of the tomograph was 10 cm and the height of the tomograph was 20 cm. In our computational experiments, we used three rows of transmitters with 16 transmitters in each, and we used 32 receivers on one vertical line (see Fig. 1). In our computational experiments, we used the dual mesh approach [18]. The direct problems were solved applying the cylindrical mesh with the numbers of cells 64, 64, and 127 in the radial, azimuthal, and z directions, consequently. The inverse problems were solved on the cubic mesh with 64 cells in all directions. In our experiments, we used a DEC ALPHA 8200 server; total computational time to solve the inverse problem is about 30 min. Three different frequencies have been used in the computational ex- periments: 2, 3.5, and 5 GHz. The results of restoration on frequency Administrator 线条 Administrator 线条 Administrator 椭圆 Administrator 高亮 Administrator 线条 Administrator 线条 Administrator 线条 Administrator 线条 Administrator 高亮 Administrator 矩形 Administrator 高亮 Administrator 线条 Administrator 高亮 Administrator 矩形 Administrator 线条 Administrator 矩形 Administrator 线条 Administrator 线条 Administrator 矩形 Administrator 矩形 Administrator 线条 Administrator 线条 Administrator 线条 Administrator 矩形 Administrator 矩形 Administrator 线条 Administrator 线条 Administrator 线条 Administrator 线条 IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 48, NO. 9, SEPTEMBER 2001 1055 (a) (b) Fig. 2. The vertical slice of the image of the model of the breast. Frequency= 3.5 GHz. (a) Real part. (b) Imaginary part. The circle shows the place of the tumor. 3.5 GHz are represented in Fig. 2. Analyzing both the real and imagi- nary parts of the complex dielectrical permittivity one can see the shape of the object, location of the muscles, and location and dielectrical properties of the “cancer zone.” We have to conclude that this frequency provides better quality of the image, sensitivity, and space resolution. The results of the computer modeling at 2 and 5 GHz show poor quality of images. Using 2 GHz, cannot provide sufficient space res- olution because of the large wavelength. In the 5-GHz case, images are much more distorted. This circumstance could be connected with the significant absorption of microwaves in biological tissues at high frequencies. Even though our results are preliminary, we can conclude that the 3-GHz frequency is close to the optimal frequency to image a woman’s breast. However, the important questions about the optimal frequency, space resolution, sensitivity are subject to future investigations. They have to be solved taking into account detailed schematic of the tomographic system, and they would be solved completely in combined theoretical and experimental investigations. V. CONCLUSION It has been shown in computational experiments that 3-D microwave images of the woman’s breast can be obtained on the tomographical system in low GHz region with quality sufficient for the small size tumor area detection. The cylindrical working chamber with transmitter and receiver antennas attached to the cylindrical surface was consid- ered. The modified gradient method was shown to be capable to solve 3-D inverse problems for Helmholtz equation and provide 3-D images of the breast in complex dielectrical permittivity. It has been shown that 1056 IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 48, NO. 9, SEPTEMBER 2001 the tomographic system are able to detect small high-contrast inhomo- geneous features. REFERENCES [1] C. Gabriel, S. Gabriel, and E. 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Hartov, and R. K. Crane, “Microwave imaging for tissue assessment: Initial evaluation in multitarget tissue- equivalent phantoms,” IEEE Trans. Biomed. Eng., vol. 43, pp. 878–890, Sept. 1996. [16] R. Kleinman and P. van der Berg, “A modified gradient method for two- dimensional problems in tomography,” J. Comput. Appl. Math., vol. 42, p. 17, 1992. [17] A. N. Tikhonov and V. Y. Arsenin, Solutions of Ill-Posed Prob- lems. Washington, D.C.: Winston, 1977. [18] K. D. Paulsen, P. M. Meaney, M. J. Moskowitz, and J. Sullivan, “A dual mesh scheme for finite element based reconstruction algorithms,” IEEE Trans. Med. Imag., vol. 14, pp. 504–514, June 1995. Variation in the Dominant Period During Ventricular Fibrillation Michael Small*, Dejin Yu, and Robert G. Harrison Abstract—Time-varying periodicities are commonly observed in biolog- ical time series. In this paper, we discuss three different algorithms to detect and quantify change in periodicity. Each technique uses a sliding window to estimate periodic components in short subseries of a longer recording. The three techniques we utilize are based on: 1) standard Fourier spectral estimation; 2) an information theoretic adaption of linear (autoregressive) modeling; and 3) geometric properties of the embedded time series. We compare the results obtained from each of these methods using artificial data and experimental data from swine ventricular fibrillation (VF). Spectral estimates have previously been applied to VF time series to show a time-dependent trend in the dominant frequency. We confirm this result by showing that the dominant period o