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Ultrasonic Nakagami Imaging: A Strategy to Visualize the Scatterer Properties of Benign and Malignant Breast Tumors
Ultrasound in Med. & Biol., Vol. 36, No. 2, pp. 209–217, 2010 Copyright Ó 2010 World Federation for Ultrasound in Medicine & Biology Printed in the USA. All rights reserved 0301-5629/09/$–see front matter
doi:10.1016/j.ultrasmedbio.2009.10.006
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Original Contribution ULTRASONIC NAKAGAMI IMAGING: A STRATEGY TO VISUALIZE THE SCATTERER PROPERTIES OF BENIGN AND MALIGNANT BREAST TUMORS PO-HSIANG TSUI,* CHIH-KUANG YEH,y YIN-YIN LIAO,y CHIEN-CHENG CHANG,*z WEN-HUNG KUO,x KING-JEN CHANG,x and CHIUNG-NIEN CHEN,x * Division of Mechanics, Research Center for Applied Sciences, Academia Sinica, Taipei, Taiwan, ROC; y Department of Biomedical Engineering and Environmental Sciences, National Tsing Hua University, Hsinchu, Taiwan, ROC; z Institute of Applied Mechanics, National Taiwan University, Taipei, Taiwan, ROC; and x Department of Surgery, National Taiwan University Hospital and College of Medicine, National Taiwan University, Taipei, Taiwan, ROC (Received 7 April 2009; revised 4 August 2009; in final form 3 October 2009)
Abstract—Previous studies have demonstrated the usefulness of the Nakagami parameter in characterizing breast tumors by ultrasound. However, physicians or radiologists may need imaging tools in a clinical setting to visually identify the properties of breast tumors. This study proposed the ultrasonic Nakagami image to visualize the scatterer properties of breast tumors and then explored its clinical performance in classifying benign and malignant tumors. Raw data of ultrasonic backscattered signals were collected from 100 patients (50 benign and 50 malignant cases) using a commercial ultrasound scanner with a 7.5 MHz linear array transducer. The backscattered signals were used to form the B-scan and the Nakagami images of breast tumors. For each tumor, the average Nakagami parameter was calculated from the pixel values in the region-of-interest in the Nakagami image. The receiver operating characteristic (ROC) curve was used to evaluate the clinical performance of the Nakagami image. The results showed that the Nakagami image shadings in benign tumors were different from those in malignant cases. The average Nakagami parameters for benign and malignant tumors were 0.69 ± 0.12 and 0.55 ± 0.12, respectively. This means that the backscattered signals received from malignant tumors tend to be more pre-Rayleigh distributed than those from benign tumors, corresponding to a more complex scatterer arrangement or composition. The ROC analysis showed that the area under the ROC curve was 0.81 ± 0.04 and the diagnostic accuracy was 82%, sensitivity was 92% and specificity was 72%. The results showed that the Nakagami image is useful to distinguishing between benign and malignant breast tumors. (E-mail: [email protected], or [email protected]. edu.tw (C.-C.C.)) Ó 2010 World Federation for Ultrasound in Medicine & Biology. Key Words: Ultrasound image, Nakagami distribution, Breast tumor.
tion, real-time display and comparatively low cost, making it convenient and suitable for routine and frequent breast screening. Ultrasound imaging can differentiate between cysts and solid masses (Zonderland 2000; Jackson 1990) and detect masses that are not visible in X-ray mammography of dense breasts (Jackson 1990). Until recently, the use of ultrasound to differentiate between benign and malignant tumors was not recommended because ultrasound image features are dependent on system settings and operators. Since the early 1990s, there has been significant progress in using ultrasound imaging to classify benign and malignant masses (Jackson 1995; Stavros et al. 1995; Jackson et al. 1996; Paulinelli et al. 2005). Several sonographic features based on margin, shape and image texture have been proposed for diagnosing tumors (Sehgal et al. 2006). Malignant tumors often have poorly defined margins and
INTRODUCTION Breast cancer is a public health problem worldwide. X-ray mammography is the most important imaging methodology for diagnosing breast cancer due to its high sensitivity and resolution, aiding early detection (Moore 2001). Considering that mammography does not reveal soft tissues and has a lower sensitivity for dense breasts (Kolb et al. 1998; Ma et al. 1992), the use of X-ray mammography in screening breasts is often complemented by ultrasound imaging. Ultrasound imaging has advantages including noninvasiveness, nonionizing radia-
Address correspondence to: Chien-Cheng Chang, Division of Mechanics, Research Center for Applied Sciences, Academia Sinica, Taipei, Taiwan, ROC. E-mail: [email protected], or mechang@ gate.sinica.edu.tw (C.-C.C.) 209
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irregular borders, with features such as spiculation, angular margins, microlobulations, marked hypoechogenicity, posterior shadowing, duct extension and tissue architectural distortion. Benign lesions have well-defined margins with rounder and smoother shapes (Skaane and Engedal 1998; Chou et al. 2001). To more quantitatively interpret the image features for discrimination of benign and malignant tumors, computer-aided diagnosis has been widely developed and used (Sehgal et al. 2006). In addition to the morphologic analysis of the tumor, the ultrasonic backscattered echoes contributed from scatterers inside the tumor may be another critical information resource associated with tumor properties. The reason is that the sizes, shapes, structures and numbers of cells (scatterers) in malignant breast tumors are different from those in benign masses (Rudland et al. 1993; Kashiwase et al. 2004). Different scatterer properties may cause different behaviors of ultrasonic backscattering. Therefore, the breast tumor may be characterized by backscattering analysis. Considering the randomness of ultrasonic backscattering, we can describe the probability density function (pdf) of backscattered echoes by using some statistical models (Shankar et al. 1993). Among all possibilities, the Nakagami statistical distribution has recently received considerable attention because its main parameter, the Nakagami parameter, can effectively quantify all backscattering conditions encountered in medical ultrasound, including pre-Rayleigh, Rayleigh and post-Rayleigh distributions (Shankar 2000). The Nakagami parameter estimated from the ultrasonic backscattered signals is only dependent on the statistical distribution of echo waveform and not affected by the echo amplitude and, thus, it is less operator-dependent. The Nakagami parameter has been demonstrated to have ability to assist conventional B-mode scanning for classifying breast tumors (Shankar et al. 2001, 2003a; Dumane et al. 2002a). Combination of the Nakagami parameter with additional techniques such as frequency diversity and compounding (Shankar et al. 2002; Dumane et al. 2002b), a multiparameter approach (Shankar et al. 2003b), and the Nakagami compounding distribution (Shankar et al. 2005) can also improve the effectiveness of using the Nakagami parameter to discriminate benign from malignant breast tumors. Although previous studies have demonstrated the usefulness of the Nakagami parameter in characterizing breast tumors by ultrasound, physicians or radiologists may need imaging tools in a clinical setting to visually identify the properties of breast tumors. Consequently, in our previous work, we further explored the feasibility of using the ultrasonic Nakagami parametric image to characterize breasts (Tsui et al. 2008a). Basically, the Nakagami image is a parametric map based on local
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Nakagami parameters. We found that the Nakagami image is less affected by different scanner settings and operators and not subject to the shadowing effect that frequently occurs beneath high-attenuation masses in B-scans (Tsui et al. 2008a). Moreover, the Nakagami image is able to classify cyst, tumor and fat because their backscattered statistics conform to extreme pre-Rayleigh, middle pre-Rayleigh and Rayleigh statistics, respectively (Tsui et al. 2008a). Note that the malignant tumor may have different levels of angiogenesis (Chaudhari et al. 2000), cellularity in the stroma (Goel et al. 2005) and fibrosis formation (Radisky et al. 2007; Radisky and Przybylo 2008) from those of benign tumor. These factors may cause different degrees of pre-Rayleigh distributions for backscattered envelopes of benign and malignant tumors. It implies that the degree of pre-Rayleigh statistics is a key point for classifying a tumor as benign or malignant by Nakagami imaging. This issue should be explored before the Nakagami image can be used as a reliable clinical imaging tool. In this study, clinical data obtained from 100 cases was used to investigate the difference of pre-Rayleigh backscattered statistics between benign and malignant breast tumors. The receiver operating characteristic (ROC) curve was used to compare both the clinical performances of the conventional B-scan and the Nakagami image in classifying benign and malignant tumors. Finally, we discussed possible advantages, limitations and contributions of the Nakagami imaging in breast ultrasound for evaluating the Nakagami image as a noninvasive tool to visualize the scatterer properties of breast tumors. MATERIALS AND METHODS Nakagami imaging methodology The pdf of the ultrasonic backscattered envelope R under the Nakagami statistical model is given by (Shankar 2000; Shankar et al. 2001) f ðrÞ5
2mm r 2m21 m 2 exp 2 r UðrÞ; GðmÞUm U
(1)
where G($) and U($) are the gamma function and the unit step function, respectively. Let E($) denote the statistical mean, then scaling parameter U and Nakagami parameter m associated with the Nakagami distribution can be respectively obtained from (2) U5E R2 and ½EðR2 Þ
2
m5 2: E½R2 2EðR2 Þ
(3)
Nakagami imaging of breast tumors d P.-H. TSUI et al.
Nakagami parameter m is a shape parameter determined by the pdf of the backscattered envelope. As m varies from 0 to 1, the envelope statistics changes from a preRayleigh to a Rayleigh distribution and the backscattered statistics conform to post-Rayleigh distributions if m is larger than 1, as shown in Figure 1. This property makes the Nakagami distribution a good general model for ultrasonic backscattering. We can further construct the Nakagami image based on the Nakagami parameter map. The Nakagami image belongs to the statistical parameter image, which is responsible for analyzing the backscattering statistics. The concept of Nakagami imaging originated from the suggestion of Shankar (2002), which prompted some preliminary studies (Kolar et al. 2004; Davignon et al. 2005). We have previously proposed a standard procedure for forming the Nakagami image (Tsui and Chang 2007) and confirmed its usefulness in visualizing local scatterer concentrations in biological tissues. In brief, the Nakagami image is constructed by using a square sliding window to process the envelope image (without logarithmic compression). This involves two main steps: (1) A square window within the envelope image is used to collect the local backscattered envelopes for estimating local Nakagami parameter mw, which is assigned as the new pixel located in the center of the window. (2) Step 1 is repeated with the window moving throughout the entire envelope image in steps of one pixel, which yields the Nakagami image as the map of mw values. Based on the conclusion in our previous study (Tsui and Chang 2007), the appropriate sliding window used to construct the Nakagami image is suggested as a square with a side length equal to three times the pulse length of the incident ultrasound.
Fig. 1. Nakagami distributions for different Nakagami parameters.
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Data acquisition The study was approved by the Institutional Review Board of National Taiwan University Hospital and the patients had signed informed consent forms. Breast ultrasound images were collected with a commercial portable ultrasound scanner (Model 2000; Terason, Burlington, MA, USA), with the raw radio-frequency (RF) data digitized at a sampling rate of 30 MHz. The applied probe is a wideband linear array with a central frequency of 7.5 MHz and 128 elements (Model 10L5; Terason). To estimate the pulse length of the incident wave, two-way pulse-echo testing of the ultrasound transducer was carried out. We placed both the transducer and a steel reflector in a water bath and adjusted the distance between the transducer and reflector to be the focal length of the transducer (focal length is adjustable for Terason system). Then, we acquired the echo signal from the reflector for estimating the pulse length. It showed that the pulse length was approximately 0.7 mm. We recruited 100 new volunteer female patients with breast tumors, aged 18 to 67, to participate in this clinical experiment after approval by the Institutional Review Board protocol. There were 50 benign (fibroadenoma) and 50 malignant (invasive carcinoma) cases, respectively. Breast tumors were identified as benign or malignant based on their biopsy reports. Although all of the patients had biopsy reports, not each one had suspicious features in the B-scan. The necessity of performing the biopsy examination was evaluated by physicians. A sonographer who was blind to the biopsy reports performed ultrasound scanning for the sonogram reports. Each image acquisition protocol involved obtaining a total of 128 scan lines of backscattered echoes. Each scan line was then demodulated using the Hilbert transform to obtain the envelope image and the B-mode image formed based on the logarithm-compressed envelope image at a dynamic range of 40 dB. The Nakagami image corresponding to each B-mode image was formed according to the algorithmic procedure described earlier. We constructed the Nakagami images of the breast tumor using a square window with a size of 2.1 3 2.1 mm2. A pseudocolor scale was applied to clearly reveal the information in the Nakagami image. Values of mw smaller than 1 were assigned blue shading, which changed from dark to light with increasing value, representing the backscattered envelope that conformed to various pre-Rayleigh statistics. Pixels with mw equal to 1 were shaded white to indicate a Rayleigh distribution and those larger than 1 were assigned red shading from dark to light with increasing value, indicating backscattered statistics with various degrees of a post-Rayleigh distribution. Finally, an investigator who did not know the tumor properties manually chose the region-of-interest (ROI) in the Nakagami image to calculate the average Nakagami image pixel for the
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tumor. The performances of using the B-scan and the Nakagami image to discriminate benign and malignant tumors were evaluated using the ROC curve and tests based on Bayesian theorem. RESULTS Figures 2(a) and (b) shows the B-mode image and the corresponding Nakagami image of a benign breast tumor, respectively. The tumor region of the Nakagami image was associated with less red and more blue shading, corresponding to an average Nakagami parameter of 0.78. It indicated that the global statistical distribution of the backscattered envelopes for the benign tumor conformed to pre-Rayleigh statistics. Figures 3(a) and (b) shows the B-mode and Nakagami images of a malignant tumor, respectively. The shading of the Nakagami image for the malignant tumor differed from that for the benign one. The malignant tumor region was associated with more blue shading corresponding to an average Nakagami parameter of 0.31, suggesting that the global backscattered statistics conformed to a higher degree of preRayleigh distribution than those for the benign tumor. The averages and standard deviations of the Nakagami parameter for benign (n 5 50) and malignant tumors (n 5 50) were 0.69 ± 0.12 and 0.55 ± 0.12, respectively, as shown in Figure 4. The experimental results agreed well with our hypothesis, indicating that on average, the ultrasonic backscattered signals returned from malignant tumors have a larger degree of pre-Rayleigh statistics than those from benign ones.
Fig. 2. The B-mode (a) and Nakagami (b) images of a benign breast tumor.
Fig. 3. The B-mode (a) and Nakagami (b) images of a malignant breast tumor.
Figure 5 shows the ROC curve for using the Nakagami image to classify benign and malignant tumors. The area under the ROC curve was 0.81 ± 0.04 in a 95% confidence interval from 0.72 to 0.89. The cut-off point for the best sensitivity and specificity was 0.64, which was determined by the closest point to (0, 1). The performance of the Nakagami image at the cut-off point of 0.64 for discriminating between benignancy and malignancy are listed and illustrated in Table 1 and Figure 6, respectively. The diagnostic accuracy was 82% (82 of 100), sensitivity was 92% (46 of 50) and specificity was 72% (36 of 50). The corresponding false positive proportion and false negative proportion were 8% and 28%, respectively. The misclassification rate was 18% and the
Fig. 4. The Nakagami parameters of benign and malignant tumors. The data is expressed by mean ± two standard errors.
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Fig. 5. The ROC curve for using the Nakagami image to classify benign and malignant breast tumors.
predictivities of positive and negative tests were 76.7% and 90%, respectively. On the other hand, according to the sonogram report of each case, we obtained 66% (66 of 100) in accuracy, 60% (30 of 50) in sensitivity and 72% (36 of 50) in specificity for the B-scan image. Evidently, the Nakagami image reduced the operator dependences to improve the classification of breast tumors. DISCUSSION AND CONCLUSIONS The results demonstrated that Nakagami parametric imaging has the ability to discriminate between benign and malignant tumors. Using Nakagami imaging to classify the breast tumors has some advantages. First, compared with using the Nakagami parameter based on one A-line of envelope signal, using the Nakagami image based on local envelope data provides a better spatial resolution to visualize the local properties of scatterers in the breast. In this study, the spatial resolution to characterize scatterers in a tumor is 2.1 3 2.1 mm2, which is determined by the window size used to construct the Nakagami image. Second, the estimation of the Nakagami parameter is mainly dependent on the shape of the backscattering statistical distribution and less affected by the echo amplitude (Shankar 2000; Shankar et al. 2001; Table 1. The performance of the Nakagami image at the threshold 5 0.64. Pathology Nakagami image mw #0:64 mw .0:64 Total
Malignant Benign 46 (TP) 4 (FN) 50
14 (FP) 36 (TN) 50
Total 60 40 100
Fig. 6. An illustration to show the normalized relationship between the parameter, subject proportion and diagnosis.
Tsui and Chang 2007); therefore, the breast Nakagami image is less affected by the shadowing effect and some system factors related to signal amplitude adjustment, such as amplifier gain or dynamic range of display (Tsui et al. 2008a). Third, the Nakagami image is constructed using the backscattered raw data. Therefore, the Nakagami image is relatively independent of signal and image processing built in the ultrasound system and may have ability to further complement B-scan texture analysis, which is a widely applied method to characterize breast tumor. Basically, both the Nakagami image and the B-scan texture analysis are able to analyze the speckle inside breast tumors. But, the difference is that texture analysis has several methods to show information of image gray-scale after logarithmic compression and image processing, rendering the performance of texture analysis to be system-dependent (Chang et al. 2005). For instance, a previous study showed that a sensitivity of 73.8% and a specificity of 54.2% were obtained from the texture analysis based on co-occurrence matrix (Bader et al. 2000). However, the other study using a different ultrasound system for the same texture analysis instead obtained 94.6% in sensitivity and 85.4% in specificity (Kuo et al. 2002). Different machines with different methods also have different performances in texture analysis. Some literatures showed that both the sensitivity and specificity larger than 80% can be obtained from texture analyses based on the wavelet transform (Chen et al. 2002), fractal dimension (Chen et al. 2005a) and three-dimensional (3-D) ultrasound image (Chen et al. 2005b). In addition, texture analysis combined with support vector machines and morphology analysis can have both the sensitivity
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and specificity higher than 90% (Huang et al. 2006; Huang 2009). The normal female breast is mainly composed of lobules (milk-producing glands), ducts (tiny tubes that carry milk from lobules to nipple) and stroma based on fatty and connective tissues surrounding the ducts, lobules, blood vessels and lymphatic vessels. Generally speaking, the speckle in the B-scan of the normal breast would be fully developed, corresponding to the speckle signal-to-noise ratio (SNR) 5 1.91 and the backscattered statistics of Rayleigh distribution (Shankar et al. 1993). Certainly, in this condition the Nakagami parameter is 1 (Shankar 2000). Compared with the normal breast, the benign tumor is a lump with some fibrocystic changes in the breast. The term ‘‘fibrocystic’’ refers to fibrosis and cysts. Fibrosis is the formation of fibrous or scar-like tissue and cysts are spaces filled with fluid. The benign tumors collected in this study were fibroadenomas, which are the most common benign breast condition in patients under 25 years of age (Smith and Burrows 2008). The fibroadenoma is composed of glandular tissues and some local fibrous tissues or calcification (Cotran et al. 1998) and fortunately it cannot spread outside the breast. Local fibrosis or calcification would increase the sound speed, density, hardness and echogenicities of scatterers and makes the scatterers in a tumor exhibit a higher degree of variability in scattering cross sections. This is why the fibroadenoma has the backscattered statistics of preRayleigh distribution corresponding to the Nakagami parameters smaller than 1 (Fig. 4). Unlike the benign tumor, the malignant tumor is a group of cancer cells that may invade the surrounding tissues or metastasize to distant areas of the body. Most malignant tumors begin in the cells that line the ducts (ductal cancers); some begin in the cells that line the lobules (lobular cancers) and the rest begin in other tissues (Cotran et al. 1998). In this study, the malignant cases are the most common malignant tumors, which are invasive (infiltrating) carcinomas, including ductal carcinomas and lobular carcinomas. For these two carcinomas, the cancer cells may spread to other parts of the body through the lymphatic system and bloodstream. In particular, the other difference between benign and malignant tumors is that malignant tumors may have different structure and calcification patterns (Busing et al. 1981; Olson et al. 1988; Shen et al. 1994). The malignant tumor has (1) developing density, asymmetry and isolated dilated ducts (Sickles 1990); (2) calcifications with larger hardness and density based on the fact that its intensity in the X-ray mammogram is stronger than that of the benign tumor (Olson et al. 1988); (3) calcifications with irregular sizes, shapes and nonuniform spatial distributions (e.g., branching, linear and clustered); and (4) a stronger vascular flow and angiogenesis effect (Chaudhari et al.
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2000; Kuo et al. 2008). These characteristics mean that the malignant tumor has more complex scatterer arrangement or composition, which may easily further enlarge the degree of variability in scattering cross section of scatterers, resulting in a much higher degree of pre-Rayleigh distribution for backscattered statistics (much smaller Nakagami parameters), as shown in Figure 4. It is critical to discuss some possible reasons to influence the accuracy of using the Nakagami image to classify benign and malignant breast tumors. Nakagami imaging may misclassify the benign tumors as malignancy mainly for three reasons. First, when the local sizes of the benign tumor are smaller than the resolution of the Nakagami image (i.e., the size of the sliding window), the tumor may easily be identified as malignant. The reason is that the subresolvable effect of Nakagami imaging makes the backscattered statistics of tumors smaller than the Nakagami image resolution shift toward an extremely preRayleigh distribution (Tsui et al. 2008a). In this condition, the Nakagami parameter becomes quite small, corresponding to the deep blue shading close to that of the malignant tumor, as shown in an example in Figure 7. To improve this weakness, higher frequency ultrasound may be used to enhance the resolution of Nakagami imaging. Second, when the benign tumor has a strong calcification effect, it may easily be classified as malignant. A calcification point with a high echogenicity would behave like a point reflector, corresponding to an
Fig. 7. The B-mode (a) and Nakagami (b) images of a benign breast tumor. The Nakagami image misclassified this benign tumor as malignancy because some local sizes of this tumor were smaller than the resolution of the Nakagami image (sliding window size).
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Fig. 8. The B-mode (a) and Nakagami (b) images of a benign breast tumor. The Nakagami image misclassified this benign tumor as malignancy because this benign tumor had a strong calcification point.
extremely pre-Rayleigh distribution (Tsui et al. 2008b) as shown in an example in Figure 8. Third, sometimes the tumor contour in the Nakagami image corresponds to very-dark-blue shading, which is not the part of the tumor but is also due to the subresolvable effect of Nakagami imaging at the interface (Tsui et al. 2007). The artifact of the Nakagami image at the interface may have an impact on the classification of benign tumors if the ROI is not well chosen as shown in Figure 9. There is another factor that may result in misclassification of malignant tumors as benignancy. In most cases, the tumor has a larger acoustic attenuation effect (Stavros et al. 1995; Harper et al. 1983; Sickles et al. 1984). A strong attenuation effect may cause a poor signal quality, making the signals received from the tumor just white noise, like an example in Figure 10. In this example, we can note that the signal intensity in the tumor was close to that of the background noise. Since white noise typically behaves as a random variable with Gaussian distribution of zero mean, its envelope will follow the Rayleigh statistics (Tsui et al. 2005). Therefore, the noise effect may tend to dominate the Nakagami parameter estimation, resulting in malignancy being classified as benignancy. Furthermore, the current study involves only two-dimensional (2-D) breast analysis. Different scanning angles and acquired frames may produce different Nakagami image features for the same tumor. We are now developing the 3-D Nakagami image to improve this shortcoming.
Fig. 9. The B-mode (a) and Nakagami (b) images of a benign breast tumor. The Nakagami image misclassified this benign tumor as malignancy because of the artifact of the Nakagami image at the interface due to subresolvable effect.
On the other hand, the Nakagami image may be combined with various imaging methods as a strategy to determine the final diagnosis of breast tumor by ultrasound. The reason is that different methods reveal different physiologic characteristics of tumors. For example, the gray-scale image based on the signal intensity reflects the echogenicities of scatterers and impedance mismatches between interfaces. Hence, the B-scan can be responsible for describing the tumor growth. In contrast, the Nakagami image is based on the backscattered statistics, supplying information about scatterer arrangements or concentrations inside a tumor. Note that the changes in the scatterer concentration may possibly affect the changes in the stiffness of tumor, implying that the Nakagami image may contain information of tumor stiffness. The tumor stiffness can be typically detected by the ultrasound elastography (Hiltawsky et al. 2001; Zhi et al. 2007). In such a condition, the results of the Nakagami image may correlate with those of the elastography to some degree. Combining the Nakagami image with the elastography is also possible to provide more detailed interpretations for the scatterer properties. In addition,
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Fig. 10. The B-mode (a) and Nakagami (b) images of a malignant breast tumor. The Nakagami image misclassified this malignant tumor as benignancy because this tumor has a serious acoustic attenuation effect, making the signals received from the tumor just white noise.
the vessel growth inside a tumor is also a critical scatterer property for classifying tumors (Chaudhari et al. 2000; Sehgal et al. 2006). Doppler images and the use of contrast agents can be involved to visualize the vascular topology and small vessels with low volume blood flow. In summary, the Nakagami image is useful to distinguishing between benign and malignant breast tumors. The Nakagami image may serve a clinical imaging tool to help physicians or radiologists better understand the scatterer properties of breast tumors in the future. Acknowledgments—This work was supported by Academia Sinica under Grant No. AS-98-TP-A02.
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Original Contribution ULTRASONIC NAKAGAMI IMAGING: A STRATEGY TO VISUALIZE THE SCATTERER PROPERTIES OF BENIGN AND MALIGNANT BREAST TUMORS PO-HSIANG TSUI,* CHIH-KUANG YEH,y YIN-YIN LIAO,y CHIEN-CHENG CHANG,*z WEN-HUNG KUO,x KING-JEN CHANG,x and CHIUNG-NIEN CHEN,x * Division of Mechanics, Research Center for Applied Sciences, Academia Sinica, Taipei, Taiwan, ROC; y Department of Biomedical Engineering and Environmental Sciences, National Tsing Hua University, Hsinchu, Taiwan, ROC; z Institute of Applied Mechanics, National Taiwan University, Taipei, Taiwan, ROC; and x Department of Surgery, National Taiwan University Hospital and College of Medicine, National Taiwan University, Taipei, Taiwan, ROC (Received 7 April 2009; revised 4 August 2009; in final form 3 October 2009)
Abstract—Previous studies have demonstrated the usefulness of the Nakagami parameter in characterizing breast tumors by ultrasound. However, physicians or radiologists may need imaging tools in a clinical setting to visually identify the properties of breast tumors. This study proposed the ultrasonic Nakagami image to visualize the scatterer properties of breast tumors and then explored its clinical performance in classifying benign and malignant tumors. Raw data of ultrasonic backscattered signals were collected from 100 patients (50 benign and 50 malignant cases) using a commercial ultrasound scanner with a 7.5 MHz linear array transducer. The backscattered signals were used to form the B-scan and the Nakagami images of breast tumors. For each tumor, the average Nakagami parameter was calculated from the pixel values in the region-of-interest in the Nakagami image. The receiver operating characteristic (ROC) curve was used to evaluate the clinical performance of the Nakagami image. The results showed that the Nakagami image shadings in benign tumors were different from those in malignant cases. The average Nakagami parameters for benign and malignant tumors were 0.69 ± 0.12 and 0.55 ± 0.12, respectively. This means that the backscattered signals received from malignant tumors tend to be more pre-Rayleigh distributed than those from benign tumors, corresponding to a more complex scatterer arrangement or composition. The ROC analysis showed that the area under the ROC curve was 0.81 ± 0.04 and the diagnostic accuracy was 82%, sensitivity was 92% and specificity was 72%. The results showed that the Nakagami image is useful to distinguishing between benign and malignant breast tumors. (E-mail: [email protected], or [email protected]. edu.tw (C.-C.C.)) Ó 2010 World Federation for Ultrasound in Medicine & Biology. Key Words: Ultrasound image, Nakagami distribution, Breast tumor.
tion, real-time display and comparatively low cost, making it convenient and suitable for routine and frequent breast screening. Ultrasound imaging can differentiate between cysts and solid masses (Zonderland 2000; Jackson 1990) and detect masses that are not visible in X-ray mammography of dense breasts (Jackson 1990). Until recently, the use of ultrasound to differentiate between benign and malignant tumors was not recommended because ultrasound image features are dependent on system settings and operators. Since the early 1990s, there has been significant progress in using ultrasound imaging to classify benign and malignant masses (Jackson 1995; Stavros et al. 1995; Jackson et al. 1996; Paulinelli et al. 2005). Several sonographic features based on margin, shape and image texture have been proposed for diagnosing tumors (Sehgal et al. 2006). Malignant tumors often have poorly defined margins and
INTRODUCTION Breast cancer is a public health problem worldwide. X-ray mammography is the most important imaging methodology for diagnosing breast cancer due to its high sensitivity and resolution, aiding early detection (Moore 2001). Considering that mammography does not reveal soft tissues and has a lower sensitivity for dense breasts (Kolb et al. 1998; Ma et al. 1992), the use of X-ray mammography in screening breasts is often complemented by ultrasound imaging. Ultrasound imaging has advantages including noninvasiveness, nonionizing radia-
Address correspondence to: Chien-Cheng Chang, Division of Mechanics, Research Center for Applied Sciences, Academia Sinica, Taipei, Taiwan, ROC. E-mail: [email protected], or mechang@ gate.sinica.edu.tw (C.-C.C.) 209
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irregular borders, with features such as spiculation, angular margins, microlobulations, marked hypoechogenicity, posterior shadowing, duct extension and tissue architectural distortion. Benign lesions have well-defined margins with rounder and smoother shapes (Skaane and Engedal 1998; Chou et al. 2001). To more quantitatively interpret the image features for discrimination of benign and malignant tumors, computer-aided diagnosis has been widely developed and used (Sehgal et al. 2006). In addition to the morphologic analysis of the tumor, the ultrasonic backscattered echoes contributed from scatterers inside the tumor may be another critical information resource associated with tumor properties. The reason is that the sizes, shapes, structures and numbers of cells (scatterers) in malignant breast tumors are different from those in benign masses (Rudland et al. 1993; Kashiwase et al. 2004). Different scatterer properties may cause different behaviors of ultrasonic backscattering. Therefore, the breast tumor may be characterized by backscattering analysis. Considering the randomness of ultrasonic backscattering, we can describe the probability density function (pdf) of backscattered echoes by using some statistical models (Shankar et al. 1993). Among all possibilities, the Nakagami statistical distribution has recently received considerable attention because its main parameter, the Nakagami parameter, can effectively quantify all backscattering conditions encountered in medical ultrasound, including pre-Rayleigh, Rayleigh and post-Rayleigh distributions (Shankar 2000). The Nakagami parameter estimated from the ultrasonic backscattered signals is only dependent on the statistical distribution of echo waveform and not affected by the echo amplitude and, thus, it is less operator-dependent. The Nakagami parameter has been demonstrated to have ability to assist conventional B-mode scanning for classifying breast tumors (Shankar et al. 2001, 2003a; Dumane et al. 2002a). Combination of the Nakagami parameter with additional techniques such as frequency diversity and compounding (Shankar et al. 2002; Dumane et al. 2002b), a multiparameter approach (Shankar et al. 2003b), and the Nakagami compounding distribution (Shankar et al. 2005) can also improve the effectiveness of using the Nakagami parameter to discriminate benign from malignant breast tumors. Although previous studies have demonstrated the usefulness of the Nakagami parameter in characterizing breast tumors by ultrasound, physicians or radiologists may need imaging tools in a clinical setting to visually identify the properties of breast tumors. Consequently, in our previous work, we further explored the feasibility of using the ultrasonic Nakagami parametric image to characterize breasts (Tsui et al. 2008a). Basically, the Nakagami image is a parametric map based on local
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Nakagami parameters. We found that the Nakagami image is less affected by different scanner settings and operators and not subject to the shadowing effect that frequently occurs beneath high-attenuation masses in B-scans (Tsui et al. 2008a). Moreover, the Nakagami image is able to classify cyst, tumor and fat because their backscattered statistics conform to extreme pre-Rayleigh, middle pre-Rayleigh and Rayleigh statistics, respectively (Tsui et al. 2008a). Note that the malignant tumor may have different levels of angiogenesis (Chaudhari et al. 2000), cellularity in the stroma (Goel et al. 2005) and fibrosis formation (Radisky et al. 2007; Radisky and Przybylo 2008) from those of benign tumor. These factors may cause different degrees of pre-Rayleigh distributions for backscattered envelopes of benign and malignant tumors. It implies that the degree of pre-Rayleigh statistics is a key point for classifying a tumor as benign or malignant by Nakagami imaging. This issue should be explored before the Nakagami image can be used as a reliable clinical imaging tool. In this study, clinical data obtained from 100 cases was used to investigate the difference of pre-Rayleigh backscattered statistics between benign and malignant breast tumors. The receiver operating characteristic (ROC) curve was used to compare both the clinical performances of the conventional B-scan and the Nakagami image in classifying benign and malignant tumors. Finally, we discussed possible advantages, limitations and contributions of the Nakagami imaging in breast ultrasound for evaluating the Nakagami image as a noninvasive tool to visualize the scatterer properties of breast tumors. MATERIALS AND METHODS Nakagami imaging methodology The pdf of the ultrasonic backscattered envelope R under the Nakagami statistical model is given by (Shankar 2000; Shankar et al. 2001) f ðrÞ5
2mm r 2m21 m 2 exp 2 r UðrÞ; GðmÞUm U
(1)
where G($) and U($) are the gamma function and the unit step function, respectively. Let E($) denote the statistical mean, then scaling parameter U and Nakagami parameter m associated with the Nakagami distribution can be respectively obtained from (2) U5E R2 and ½EðR2 Þ
2
m5 2: E½R2 2EðR2 Þ
(3)
Nakagami imaging of breast tumors d P.-H. TSUI et al.
Nakagami parameter m is a shape parameter determined by the pdf of the backscattered envelope. As m varies from 0 to 1, the envelope statistics changes from a preRayleigh to a Rayleigh distribution and the backscattered statistics conform to post-Rayleigh distributions if m is larger than 1, as shown in Figure 1. This property makes the Nakagami distribution a good general model for ultrasonic backscattering. We can further construct the Nakagami image based on the Nakagami parameter map. The Nakagami image belongs to the statistical parameter image, which is responsible for analyzing the backscattering statistics. The concept of Nakagami imaging originated from the suggestion of Shankar (2002), which prompted some preliminary studies (Kolar et al. 2004; Davignon et al. 2005). We have previously proposed a standard procedure for forming the Nakagami image (Tsui and Chang 2007) and confirmed its usefulness in visualizing local scatterer concentrations in biological tissues. In brief, the Nakagami image is constructed by using a square sliding window to process the envelope image (without logarithmic compression). This involves two main steps: (1) A square window within the envelope image is used to collect the local backscattered envelopes for estimating local Nakagami parameter mw, which is assigned as the new pixel located in the center of the window. (2) Step 1 is repeated with the window moving throughout the entire envelope image in steps of one pixel, which yields the Nakagami image as the map of mw values. Based on the conclusion in our previous study (Tsui and Chang 2007), the appropriate sliding window used to construct the Nakagami image is suggested as a square with a side length equal to three times the pulse length of the incident ultrasound.
Fig. 1. Nakagami distributions for different Nakagami parameters.
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Data acquisition The study was approved by the Institutional Review Board of National Taiwan University Hospital and the patients had signed informed consent forms. Breast ultrasound images were collected with a commercial portable ultrasound scanner (Model 2000; Terason, Burlington, MA, USA), with the raw radio-frequency (RF) data digitized at a sampling rate of 30 MHz. The applied probe is a wideband linear array with a central frequency of 7.5 MHz and 128 elements (Model 10L5; Terason). To estimate the pulse length of the incident wave, two-way pulse-echo testing of the ultrasound transducer was carried out. We placed both the transducer and a steel reflector in a water bath and adjusted the distance between the transducer and reflector to be the focal length of the transducer (focal length is adjustable for Terason system). Then, we acquired the echo signal from the reflector for estimating the pulse length. It showed that the pulse length was approximately 0.7 mm. We recruited 100 new volunteer female patients with breast tumors, aged 18 to 67, to participate in this clinical experiment after approval by the Institutional Review Board protocol. There were 50 benign (fibroadenoma) and 50 malignant (invasive carcinoma) cases, respectively. Breast tumors were identified as benign or malignant based on their biopsy reports. Although all of the patients had biopsy reports, not each one had suspicious features in the B-scan. The necessity of performing the biopsy examination was evaluated by physicians. A sonographer who was blind to the biopsy reports performed ultrasound scanning for the sonogram reports. Each image acquisition protocol involved obtaining a total of 128 scan lines of backscattered echoes. Each scan line was then demodulated using the Hilbert transform to obtain the envelope image and the B-mode image formed based on the logarithm-compressed envelope image at a dynamic range of 40 dB. The Nakagami image corresponding to each B-mode image was formed according to the algorithmic procedure described earlier. We constructed the Nakagami images of the breast tumor using a square window with a size of 2.1 3 2.1 mm2. A pseudocolor scale was applied to clearly reveal the information in the Nakagami image. Values of mw smaller than 1 were assigned blue shading, which changed from dark to light with increasing value, representing the backscattered envelope that conformed to various pre-Rayleigh statistics. Pixels with mw equal to 1 were shaded white to indicate a Rayleigh distribution and those larger than 1 were assigned red shading from dark to light with increasing value, indicating backscattered statistics with various degrees of a post-Rayleigh distribution. Finally, an investigator who did not know the tumor properties manually chose the region-of-interest (ROI) in the Nakagami image to calculate the average Nakagami image pixel for the
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tumor. The performances of using the B-scan and the Nakagami image to discriminate benign and malignant tumors were evaluated using the ROC curve and tests based on Bayesian theorem. RESULTS Figures 2(a) and (b) shows the B-mode image and the corresponding Nakagami image of a benign breast tumor, respectively. The tumor region of the Nakagami image was associated with less red and more blue shading, corresponding to an average Nakagami parameter of 0.78. It indicated that the global statistical distribution of the backscattered envelopes for the benign tumor conformed to pre-Rayleigh statistics. Figures 3(a) and (b) shows the B-mode and Nakagami images of a malignant tumor, respectively. The shading of the Nakagami image for the malignant tumor differed from that for the benign one. The malignant tumor region was associated with more blue shading corresponding to an average Nakagami parameter of 0.31, suggesting that the global backscattered statistics conformed to a higher degree of preRayleigh distribution than those for the benign tumor. The averages and standard deviations of the Nakagami parameter for benign (n 5 50) and malignant tumors (n 5 50) were 0.69 ± 0.12 and 0.55 ± 0.12, respectively, as shown in Figure 4. The experimental results agreed well with our hypothesis, indicating that on average, the ultrasonic backscattered signals returned from malignant tumors have a larger degree of pre-Rayleigh statistics than those from benign ones.
Fig. 2. The B-mode (a) and Nakagami (b) images of a benign breast tumor.
Fig. 3. The B-mode (a) and Nakagami (b) images of a malignant breast tumor.
Figure 5 shows the ROC curve for using the Nakagami image to classify benign and malignant tumors. The area under the ROC curve was 0.81 ± 0.04 in a 95% confidence interval from 0.72 to 0.89. The cut-off point for the best sensitivity and specificity was 0.64, which was determined by the closest point to (0, 1). The performance of the Nakagami image at the cut-off point of 0.64 for discriminating between benignancy and malignancy are listed and illustrated in Table 1 and Figure 6, respectively. The diagnostic accuracy was 82% (82 of 100), sensitivity was 92% (46 of 50) and specificity was 72% (36 of 50). The corresponding false positive proportion and false negative proportion were 8% and 28%, respectively. The misclassification rate was 18% and the
Fig. 4. The Nakagami parameters of benign and malignant tumors. The data is expressed by mean ± two standard errors.
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Fig. 5. The ROC curve for using the Nakagami image to classify benign and malignant breast tumors.
predictivities of positive and negative tests were 76.7% and 90%, respectively. On the other hand, according to the sonogram report of each case, we obtained 66% (66 of 100) in accuracy, 60% (30 of 50) in sensitivity and 72% (36 of 50) in specificity for the B-scan image. Evidently, the Nakagami image reduced the operator dependences to improve the classification of breast tumors. DISCUSSION AND CONCLUSIONS The results demonstrated that Nakagami parametric imaging has the ability to discriminate between benign and malignant tumors. Using Nakagami imaging to classify the breast tumors has some advantages. First, compared with using the Nakagami parameter based on one A-line of envelope signal, using the Nakagami image based on local envelope data provides a better spatial resolution to visualize the local properties of scatterers in the breast. In this study, the spatial resolution to characterize scatterers in a tumor is 2.1 3 2.1 mm2, which is determined by the window size used to construct the Nakagami image. Second, the estimation of the Nakagami parameter is mainly dependent on the shape of the backscattering statistical distribution and less affected by the echo amplitude (Shankar 2000; Shankar et al. 2001; Table 1. The performance of the Nakagami image at the threshold 5 0.64. Pathology Nakagami image mw #0:64 mw .0:64 Total
Malignant Benign 46 (TP) 4 (FN) 50
14 (FP) 36 (TN) 50
Total 60 40 100
Fig. 6. An illustration to show the normalized relationship between the parameter, subject proportion and diagnosis.
Tsui and Chang 2007); therefore, the breast Nakagami image is less affected by the shadowing effect and some system factors related to signal amplitude adjustment, such as amplifier gain or dynamic range of display (Tsui et al. 2008a). Third, the Nakagami image is constructed using the backscattered raw data. Therefore, the Nakagami image is relatively independent of signal and image processing built in the ultrasound system and may have ability to further complement B-scan texture analysis, which is a widely applied method to characterize breast tumor. Basically, both the Nakagami image and the B-scan texture analysis are able to analyze the speckle inside breast tumors. But, the difference is that texture analysis has several methods to show information of image gray-scale after logarithmic compression and image processing, rendering the performance of texture analysis to be system-dependent (Chang et al. 2005). For instance, a previous study showed that a sensitivity of 73.8% and a specificity of 54.2% were obtained from the texture analysis based on co-occurrence matrix (Bader et al. 2000). However, the other study using a different ultrasound system for the same texture analysis instead obtained 94.6% in sensitivity and 85.4% in specificity (Kuo et al. 2002). Different machines with different methods also have different performances in texture analysis. Some literatures showed that both the sensitivity and specificity larger than 80% can be obtained from texture analyses based on the wavelet transform (Chen et al. 2002), fractal dimension (Chen et al. 2005a) and three-dimensional (3-D) ultrasound image (Chen et al. 2005b). In addition, texture analysis combined with support vector machines and morphology analysis can have both the sensitivity
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and specificity higher than 90% (Huang et al. 2006; Huang 2009). The normal female breast is mainly composed of lobules (milk-producing glands), ducts (tiny tubes that carry milk from lobules to nipple) and stroma based on fatty and connective tissues surrounding the ducts, lobules, blood vessels and lymphatic vessels. Generally speaking, the speckle in the B-scan of the normal breast would be fully developed, corresponding to the speckle signal-to-noise ratio (SNR) 5 1.91 and the backscattered statistics of Rayleigh distribution (Shankar et al. 1993). Certainly, in this condition the Nakagami parameter is 1 (Shankar 2000). Compared with the normal breast, the benign tumor is a lump with some fibrocystic changes in the breast. The term ‘‘fibrocystic’’ refers to fibrosis and cysts. Fibrosis is the formation of fibrous or scar-like tissue and cysts are spaces filled with fluid. The benign tumors collected in this study were fibroadenomas, which are the most common benign breast condition in patients under 25 years of age (Smith and Burrows 2008). The fibroadenoma is composed of glandular tissues and some local fibrous tissues or calcification (Cotran et al. 1998) and fortunately it cannot spread outside the breast. Local fibrosis or calcification would increase the sound speed, density, hardness and echogenicities of scatterers and makes the scatterers in a tumor exhibit a higher degree of variability in scattering cross sections. This is why the fibroadenoma has the backscattered statistics of preRayleigh distribution corresponding to the Nakagami parameters smaller than 1 (Fig. 4). Unlike the benign tumor, the malignant tumor is a group of cancer cells that may invade the surrounding tissues or metastasize to distant areas of the body. Most malignant tumors begin in the cells that line the ducts (ductal cancers); some begin in the cells that line the lobules (lobular cancers) and the rest begin in other tissues (Cotran et al. 1998). In this study, the malignant cases are the most common malignant tumors, which are invasive (infiltrating) carcinomas, including ductal carcinomas and lobular carcinomas. For these two carcinomas, the cancer cells may spread to other parts of the body through the lymphatic system and bloodstream. In particular, the other difference between benign and malignant tumors is that malignant tumors may have different structure and calcification patterns (Busing et al. 1981; Olson et al. 1988; Shen et al. 1994). The malignant tumor has (1) developing density, asymmetry and isolated dilated ducts (Sickles 1990); (2) calcifications with larger hardness and density based on the fact that its intensity in the X-ray mammogram is stronger than that of the benign tumor (Olson et al. 1988); (3) calcifications with irregular sizes, shapes and nonuniform spatial distributions (e.g., branching, linear and clustered); and (4) a stronger vascular flow and angiogenesis effect (Chaudhari et al.
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2000; Kuo et al. 2008). These characteristics mean that the malignant tumor has more complex scatterer arrangement or composition, which may easily further enlarge the degree of variability in scattering cross section of scatterers, resulting in a much higher degree of pre-Rayleigh distribution for backscattered statistics (much smaller Nakagami parameters), as shown in Figure 4. It is critical to discuss some possible reasons to influence the accuracy of using the Nakagami image to classify benign and malignant breast tumors. Nakagami imaging may misclassify the benign tumors as malignancy mainly for three reasons. First, when the local sizes of the benign tumor are smaller than the resolution of the Nakagami image (i.e., the size of the sliding window), the tumor may easily be identified as malignant. The reason is that the subresolvable effect of Nakagami imaging makes the backscattered statistics of tumors smaller than the Nakagami image resolution shift toward an extremely preRayleigh distribution (Tsui et al. 2008a). In this condition, the Nakagami parameter becomes quite small, corresponding to the deep blue shading close to that of the malignant tumor, as shown in an example in Figure 7. To improve this weakness, higher frequency ultrasound may be used to enhance the resolution of Nakagami imaging. Second, when the benign tumor has a strong calcification effect, it may easily be classified as malignant. A calcification point with a high echogenicity would behave like a point reflector, corresponding to an
Fig. 7. The B-mode (a) and Nakagami (b) images of a benign breast tumor. The Nakagami image misclassified this benign tumor as malignancy because some local sizes of this tumor were smaller than the resolution of the Nakagami image (sliding window size).
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Fig. 8. The B-mode (a) and Nakagami (b) images of a benign breast tumor. The Nakagami image misclassified this benign tumor as malignancy because this benign tumor had a strong calcification point.
extremely pre-Rayleigh distribution (Tsui et al. 2008b) as shown in an example in Figure 8. Third, sometimes the tumor contour in the Nakagami image corresponds to very-dark-blue shading, which is not the part of the tumor but is also due to the subresolvable effect of Nakagami imaging at the interface (Tsui et al. 2007). The artifact of the Nakagami image at the interface may have an impact on the classification of benign tumors if the ROI is not well chosen as shown in Figure 9. There is another factor that may result in misclassification of malignant tumors as benignancy. In most cases, the tumor has a larger acoustic attenuation effect (Stavros et al. 1995; Harper et al. 1983; Sickles et al. 1984). A strong attenuation effect may cause a poor signal quality, making the signals received from the tumor just white noise, like an example in Figure 10. In this example, we can note that the signal intensity in the tumor was close to that of the background noise. Since white noise typically behaves as a random variable with Gaussian distribution of zero mean, its envelope will follow the Rayleigh statistics (Tsui et al. 2005). Therefore, the noise effect may tend to dominate the Nakagami parameter estimation, resulting in malignancy being classified as benignancy. Furthermore, the current study involves only two-dimensional (2-D) breast analysis. Different scanning angles and acquired frames may produce different Nakagami image features for the same tumor. We are now developing the 3-D Nakagami image to improve this shortcoming.
Fig. 9. The B-mode (a) and Nakagami (b) images of a benign breast tumor. The Nakagami image misclassified this benign tumor as malignancy because of the artifact of the Nakagami image at the interface due to subresolvable effect.
On the other hand, the Nakagami image may be combined with various imaging methods as a strategy to determine the final diagnosis of breast tumor by ultrasound. The reason is that different methods reveal different physiologic characteristics of tumors. For example, the gray-scale image based on the signal intensity reflects the echogenicities of scatterers and impedance mismatches between interfaces. Hence, the B-scan can be responsible for describing the tumor growth. In contrast, the Nakagami image is based on the backscattered statistics, supplying information about scatterer arrangements or concentrations inside a tumor. Note that the changes in the scatterer concentration may possibly affect the changes in the stiffness of tumor, implying that the Nakagami image may contain information of tumor stiffness. The tumor stiffness can be typically detected by the ultrasound elastography (Hiltawsky et al. 2001; Zhi et al. 2007). In such a condition, the results of the Nakagami image may correlate with those of the elastography to some degree. Combining the Nakagami image with the elastography is also possible to provide more detailed interpretations for the scatterer properties. In addition,
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Fig. 10. The B-mode (a) and Nakagami (b) images of a malignant breast tumor. The Nakagami image misclassified this malignant tumor as benignancy because this tumor has a serious acoustic attenuation effect, making the signals received from the tumor just white noise.
the vessel growth inside a tumor is also a critical scatterer property for classifying tumors (Chaudhari et al. 2000; Sehgal et al. 2006). Doppler images and the use of contrast agents can be involved to visualize the vascular topology and small vessels with low volume blood flow. In summary, the Nakagami image is useful to distinguishing between benign and malignant breast tumors. The Nakagami image may serve a clinical imaging tool to help physicians or radiologists better understand the scatterer properties of breast tumors in the future. Acknowledgments—This work was supported by Academia Sinica under Grant No. AS-98-TP-A02.
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