Computer-aided Analysis of Continuous US Images

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Woo Kyung Moon, MD Ruey-Feng Chang, PhD Chii-Jen Chen, PhD Dar-Ren Chen, MD Wei-Liang Chen, MS

Published online

10.1148/radiol.2362041095 Radiology 2005; 236:458 – 464

Abbreviation:

A_z⫽ area under receiver operating characteristic curve

1From the Department of Radiology and Clinical Research Institute, Seoul National University Hospital and the Institute of Radiation Medicine, Seoul National University Medical Research Center, Seoul, Korea (W.K.M.); De- partment of Computer Science and In- formation Engineering, National Chung Cheng University, Chiayi, Taiwan (R.F.C., C.J.C., W.L.C.); and Department of Sur- gery, Changhua Christian Hospital, 135 Nanhsiao St, Changhua, Taiwan 500 (D.R.C.).Received June 21, 2004; revision requested August 30; revision received September 13; accepted Octo- ber 20. Address correspondence to D.R.C. (e-mail: [email protected] .net).

Authors stated no financial relation- ship to disclose.

Author contributions:

Guarantors of integrity of entire study, W.K.M., R.F.C., D.R.C.; study con- cepts, W.K.M., R.F.C., D.R.C.; study design, all authors; literature research, R.F.C., C.J.C., W.L.C.; clinical studies, W.K.M., D.R.C.; data acquisition, W.K.M., D.R.C.; data analysis/interpretation, all authors; statistical analysis, R.F.C., C.J.C., W.L.C.; manuscript preparation, definition of intellectual content, editing, and revision/review, all authors; manuscript final version approval, W.K.M., R.F.C., D.R.C.

©RSNA, 2005

Solid Breast Masses:

Classification with

Computer-aided Analysis of Continuous US Images

Obtained with Probe Compression ¹

PURPOSE: To prospectively evaluate the accuracy of continuous ultrasonographic (US) images obtained during probe compression and computer-aided analysis for classification of biopsy-proved (reference standard) benign and malignant breast tumors.

MATERIALS AND METHODS: This study was approved by the local ethics com- mittee, and informed consent was obtained from all included patients. Serial US images of 100 solid breast masses (60 benign and 40 malignant tumors) were obtained with US probe compression in 86 patients (mean age, 45 years; range, 20 – 67 years). After segmentation of tumor contours with the level-set method, three features of strain on tissue from probe compression— contour difference, shift distance, area difference—and one feature of shape—solidity—were computed. A maximum margin classifier was used to classify the tumors by using these four features. The Student t test and receiver operating characteristic curve analysis were used for statistical analysis.

RESULTS: The mean values of contour difference, shift distance, area difference, and solidity were 3.52%⫾ 2.12 (standard deviation), 2.62 ⫾ 1.31, 1.08% ⫾ 0.85, and 1.70⫾ 1.85 in malignant tumors and 9.72% ⫾ 4.54, 5.04 ⫾ 2.79, 3.17% ⫾ 2.86, and 0.53⫾ 0.63 in benign tumors, respectively. Differences with P ⬍ .001 were statistically significant for all four features. Area under the receiver operating characteristic curve (A_Z) values for contour difference, shift distance, area difference, and solidity were 0.88, 0.85, 0.86, and 0.79, respectively. The AZvalue of three features of strain was significantly higher than that of the feature of shape (P⬍ .01).

The accuracy, sensitivity, specificity, and positive and negative predictive values of US classifications that were based on values for these four features were 87.0% (87 of 100), 85% (34 of 40), 88% (53 of 60), 83% (34 of 41), and 90% (53 of 59), respectively, with an A_Zvalue of 0.91.

CONCLUSION: Continuous US images obtained with probe compression and com- puter-aided analysis can aid in classification of benign and malignant breast tumors.

©RSNA, 2005

Ultrasonography (US) is a valuable adjunct to mammography in breast imaging. In addition to its use for distinguishing cysts from solid breast tumors, US can be used to help differentiate benign from malignant solid masses (1). By using a strict classification scheme, Stavros et al (1) achieved a sensitivity of 98.4% (123 of 125) and a specificity of 67.8% (424 of 625) for the classification of 750 solid breast masses. US criteria for the classification of solid breast masses included lesion shape, orientation, margin, echoge- nicity, and acoustic transmission (1). Recently, similar criteria were applied to computer-

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aided US diagnosis, and promising results were obtained for the classification of breast lesions (2).

With elastography (3–7), a method of evaluating the strain on soft tissues from the US probe, elastograms are produced by means of comparison of US radiofrequency waveforms obtained before and after light tissue compression.

After compression with a probe, a soft benign tumor becomes flatter than does a comparable stiffer malignant tumor.

This may be so because benign tumors generally have smooth regular borders and are loosely bound to the surrounding tissues, whereas malignant tumors are usually characterized by firm desmoplastic reactions with surrounding tissue. Garra et al (8) and Hall et al (9) proposed several diagnostic criteria, such as lesion visualization, relative brightness, margin regularity, and lesion size, after they compared elastograms and B-mode US images. They found that the measured transverse diameters of benign tumors on elastograms were almost always the same as or smaller than the diameters of the tumors on US images, whereas the diameters of malignant tumors on elastograms were invariably larger than those on US images.

Instead of using radiofrequency data, B-mode images and computer-aided analysis could be used to extract information about the elasticity of benign and malignant solid breast tumors. Steinberg et al (10,11) proposed a disparity processing technique to obtain a correlation map of B-mode images. Speckle motion rather than the echo structure of the underlying tissue was used to compute the elasticity and to differentiate between benign and malignant breast tumors. In their study, however, they used only two images for the analysis of strain, they did not apply image segmentation, and their database was lim- ited to only 25 solid breast tumors. To our knowledge, analysis of strain and computer-aided diagnosis that is based on automatic segmentation of continuous US images obtained during tissue compression have not been used for the classification of benign and malignant breast tumors.

The purpose of this study was to prospectively evaluate the accuracy of continuous US images obtained during compression and computer-aided analysis for the classification of biopsy-proved (reference standard) benign and malignant breast tumors.

MATERIALS AND METHODS Patients and US Examination

This study was approved by the local ethics committee, and informed consent was obtained from all included patients.

The database used in our study contained the US images obtained with probe compression of 100 consecutive pathologi- cally proved (reference standard) solid breast tumors. Sixty benign tumors (31 fibroadenomas and 29 fibrocystic lesions) and 40 invasive carcinomas (39 invasive ductal carcinomas and one medullary carcinoma) were included. These tumors were from 86 patients (mean age, 45 years; range, 20 – 67 years). The images obtained with probe compression were collected from January 2002 to August 2002 by an author (D.R.C.) with 13 years of experience in the performance of breast US. All lesions were seen as solid breast masses at conventional US and were prospectively classified by the same author as Breast Imaging Reporting and Data System category 3 or probable benign lesions in 36 cases, category 4 or suspicious lesions in 33 cases, and category 5 or highly suspicious lesions in 31 cases. Fine-needle aspiration cytologic analysis was performed in 36 cases with lesions assessed as category 3, and core needle biopsy with a 14-gauge needle was performed in 64 cases with lesions assessed as either category 4 or category 5.

All patients with malignant lesions un- derwent surgery within 2 weeks of US examination. Mammographic and sono- graphic follow-up was performed in 60%

(36 of 60) of benign lesions, and the mean duration of follow-up was 9.2 months (range, 6 –18 months).

A commercial real-time B-mode scan- ner (Voluson 530; Kretz Technik, Zipf, Austria) with a 5–10-MHz transducer was used in the present study. US examina- tions were performed with the patient in the supine position and the arms raised above the head. When lateral lesions in large breasts were scanned, the patient was rolled slightly to her contralateral side so that gravity would flatten the breast tissue in the region to be scanned.

For precompression imaging, light compression was applied to the breast with the transducer to help position the breast tissue and to produce similar echo inten- sity over the entire image. Approximately 60 serial images were obtained per patient by using a probe compression depth of approximately 0.4 – 0.6 cm, depending on the total thickness of the breast. The US images were recorded at 15 frames per

second, and compression was completed in 4 seconds. Since neighboring images were similar and variations in elasticity were slight, every eighth image was used, and, in total, eight images from the 60 images were used in our analysis; hereaf- ter, these images will be referred to as continuous US images. For comparison, the first and the last frames of the 60 images were used for the analysis; here- after, these images will be referred to as noncontinuous US images. The image resolution was 59.7 pixels/cm.

Image Segmentation

Before image segmentation, anisotropic diffusion filtering (12,13) was adopted to reduce the speckles on US images. Aniso- tropic diffusion filtering is a method that relies on the use of the local image gradient to control anisotropic diffusion. This technique can be used to get rid of the major drawback of conventional spatial filters and improve image quality sub- stantially while preserving important boundary information. After anisotropic diffusion filtering, the stick, which is a line segment of variable orientation, was used to approximate boundaries, to reduce speckles, and to enhance edge information (14). For a given square area of size N⫻ N on an image, a number of lines equal to 2N⫺ 2 of a length of N pixels can be drawn through its center. The sums of the pixel values along these lines were calculated for each line. Then the maximum of these sums was selected.

After each pixel in an image was replaced by the maximum sum of the lines pass- ing through that pixel, edge contrast was enhanced and speckles were reduced.

To segment tumor contours from the continuous US images, the level-set technique (15,16) was used. A detailed de- scription of this method is provided in the Appendix. During the preprocessing procedure for the eight continuous images extracted from the data set, we applied two-dimensional anisotropic filtering and two-dimensional stick methods to each image. Because the two-dimensional method is inadequate for calculating the volume of sideslip during compression, the three-dimensional level-set method was used for continuous US images. Time for the preprocessing and segmentation was 20 –25 seconds per case.

To verify the performance of the developed automatic segmentation algorithm, an experienced radiologist (W.K.M., with 10 years of experience in breast US) re- viewed the automatically detected edges

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of all images. All cases were acceptable according to the radiologist’s review after adjustment of some parameters, such as the iteration number of the anisotropic diffusion filtering in the preprocessing procedure.

Strain and Shape Value Computation

After tumor contours were segmented from the continuous US images, three values for features of strain— contour difference, shift distance, and area difference—and one value for the shape feature—solidity—were computed by one author (W.L.C.) to evaluate the findings of benign and malignant tumors. The computation time was 3– 4 seconds per case.

Contour difference.—The values for con- tour difference were used to evaluate changes in lesion shape between two images. Because of compression, the contours between two neighboring images show movement. Hence, these contours should first be registered. In this study, the center of gravity of the tumor was

used as the reference point (Fig 1). For each pixel (x, y) inside the tumor, the center of gravity (Gx, Gy) was defined as follows:

Gx⫽共 x,y兲 in tumor

冘

x N,

(1) and

Gy⫽共 x,y兲 in tumor

冘

y N,

where N is the number of pixels in a tumor. After the two images were registered, the pixels in only one contour, not in both contours, were identified and de- scribed as the different pixels.

The contour difference between two neighboring images was then defined as the number of different pixels divided by the number of pixels in the first image.

That is,

contour difference⫽Ncon_diff

N ⫻ 100, (2) where N_{con_diff}is the pixel difference be- tween the two registered contours, and N

is the number of tumor pixels in the first image.

The average value for contour difference of the eight images was defined as the value for contour difference for that particular case. We chose the average value because this was better than the maximum value of contour difference parameters at the preliminary experi- ment.

Shift distance.—To determine contour motion, the contour of an image was divided equally into eight sections, thus defining eight cutting points. Each cutting point was used as the center of a block for which we needed to find the motion vector. Finally, the motion estimation technique (17) was used for eval- uation of these eight blocks, and the shift distance of the largest block in pixels was chosen. For a block at (m, n) in the cur- rent image, if the most similar block in the previous image was at (m⫹ i, n ⫹ j), then the motion vector for the block (m, n) was (i, j), as shown in Figure 2. The shift distance was defined as follows:

shift distance⫽ av

A

冋

^max^j僆G ^共ShD^j^兲

册

^, ⁽³⁾

where av represents average, max is max- imum of shift distance, j is block j, G represents eight blocks in every frame, A represents all frames, and ShDjis the shift distance of block j.

Area difference.—The area difference was used to compare tumor areas between two images, as tumor area changes according to the pressure exerted. The area difference was defined as the difference between areas of tumors in two neighboring frames divided by the number of the pixels in the tumor of the previous image. That is,

Figure 1. Images demonstrate computation of contour difference.

(a) First image. (b) Second image. (c) Shifted second image after regis- tration to a. (d) Image shows process involved in determining the contour difference. In a, the tumor area is 14 pixels, and the center of gravity (black pixel) is at position (5, 5). In b, the center of gravity (black pixel) is at position (5, 4). The center of the two images is registered in c.

The number of different pixels between a and c is 5 (black pixels in d), and therefore the contour difference is 5 of 14 (36%).

Figure 2. Diagrams demonstrate computation of shift distance.

Gray square signifies block with size (N⫻ N). (a) Current image. (b) Previous image. For the block with size of (N⫻ N) at (m, n) in a, if the most similar block in b is at (m⫹ i, n ⫹ j), then the motion vector for the block is (i, j). The blocks in a search window of the previous frame are (N⫹ 2w, N ⫹ 2w), where w is the size of the search window.

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area difference⫽ av

A

冉

^兩arⁱ^ar^{⫺ ar}i⫺1^i⫺1^兩

冊

^{䡠 100,}

(4) where A represents all frames, ari⫺1and ariare the sizes of the tumor in the previous and current frames, respectively.

Solidity.—Shape values can be used to distinguish between benign and malignant tumors. Benign tumors usually have smooth shapes, whereas malignant tumors are irregular. The normalized solidity value (18) was used in this study. It is defined as:

solidity⫽ cvx_ar⫺ tumor_ar

冉

^i⫽1^N

^冘

^tot^cvx_arⁱ^{⫺ tumor_ar}ⁱ

冊

^/N^tot^,

(5) where N_totis the total number of cases, cvx_ar is the area obtained from the convex hull of a tumor, cvx_ariis the cvx_ar for case i, as shown in Figure 3, tumor_ar is the area of tumor, and tumor_ar_iis the tumor_ar for case i. The convex hull (19) is the smallest convex set containing a tumor and resembles a rubber band wrapped around the tumor. This shape feature was computed initially without compression by using the first image of the 60 serial US images.

Classification

A support vector machine (20 –22), a maximum margin classifier, was used to classify solid breast tumors on the basis of the proposed feature and shape values:

contour difference, area difference, shift distance, and solidity. Given a set of points that belong to one of two classes, the support vector machine can be used to identify the hyperplane that leaves the largest possible fraction of points of the same class on one side, while maximizing

the distance of either class from the hyperplane. This hyperplane, which pro- vides optimal separation, can minimize the risk of misclassifying test set exam- ples. A support vector machine previ- ously used to classify solid breast tumors at gray-scale US was employed in this study (23,24). The US images of 100 solid breast tumors were classified into five groups randomly. One group was used as the test set, and the remaining four were used as the training set. The results obtained by using the support vector machine were compared with those obtained by using Bayesian classification, a classic linear classification method (18,19).

Statistical Analysis

The mean values and standard devia- tions of the three values for strain— contour difference, area difference, and shift distance—and of the one value for shape—solidity—were calculated for benign and malignant tumors on continuous and noncontinuous US images. Dif- ferences between the four values for features of strain and shape in benign and malignant tumors were evaluated by us- ing the Student t test. A scatter graph was obtained to analyze the distribution of the three values for features of strain. The x-, y-, and z-axes of scatter graphs were used to represent the three values for strain on the continuous US images. The performance of the values for the four features was also evaluated with receiver

operating characteristic curve analysis by using a computer program (LABROC1, 1993; Charles E. Metz, MD, University of Chicago, Chicago, Ill). The area under the receiver operating characteristic curve (AZ) was used as an indicator of performance.

For the four features, the difference be- tween the AZvalue for benign tumors and that for malignant tumors on continuous and noncontinuous US images was evalu- ated by using the Student t test.

The diagnostic performance of the support vector machine and that of the Bayesian classifier based on the values for the four features used for the classification of solid breast tumors on continuous and noncontinuous US images were evaluated by comparing the accuracy, sensitivity, specificity, and positive and negative predictive values. The diagnostic performance of the support vector machine based on the four values for features used for the classification of solid breast tumors on continuous and noncontinuous US images was also evaluated by comparing values of the receiver operating char- acteristic area index, A_z. The receiver operating characteristic analysis could not be applied for the Bayesian classifier because the output of the Bayesian classifier is either 1 or 0. The McNemar test was used to compare diagnostic accuracy, sensitivity, specificity, and positive and negative predictive values of the support vector machine with those of the Bayes- ian classifier, as well as those values for the continuous images with those for the Figure 3. Drawings depict (a) contour of a

tumor and (b) convex hull of a tumor. The outer polygon (arrow) is the convex hull of this tumor. The convex hull is the smallest convex set containing the tumor.

Figure 4. (a– h) Transverse continuous US images of a malignant breast lesion. The computer-delineated margin is the white outline.

In this case, the values for the three features of strain— contour difference, shift distance, and area difference—and the value for the one feature of shape—solidity—were 1.20%, 1.24, 0.85%, and 3.69, respectively.

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noncontinuous images. For each analy- sis, a P value of less than .05 was consid- ered to indicate a significant difference.

Statistical analyses other than receiver operating characteristic analysis were performed by using software (SPSS, version 10 for Windows; SPSS, Chicago, Ill).

RESULTS Strain and Shape

The mean values of the three features of strain— contour difference, shift distance, and area difference—in cases of malignant tumors were 3.52% ⫾ 2.12, 2.62⫾ 1.31, and 1.08% ⫾ 0.85 on continuous US images, whereas those in cases of benign tumors were 9.72%⫾ 4.54, 5.04 ⫾ 2.79, and 3.17% ⫾ 2.86, respectively (Figs 4, 5). On noncontinuous US images, these values were 22.13%⫾ 12.44, 2.86⫾ 2.10, and 6.18% ⫾ 6.10 in cases of malignant tumors, and in cases of benign tumors, they were 56.23%⫾ 27.53, 5.40 ⫾ 3.05, and 17.95% ⫾ 16.79, respectively.

Differences between benign and malignant breast tumors were statistically significant for values for all three features of strain (P⬍ .001) on continuous and noncontinuous US images. Values for features of strain obtained by using the noncontinuous images were larger than those of features of strain obtained by using the continuous images, because the values for strain with the noncontinuous images were computed only from the first and the

last frames. Mean solidity was 1.70⫾ 1.85 in cases of malignant tumors and 0.53⫾ 0.63 in cases of benign tumors, and these values were significantly different (P ⬍ .001).

In the scatter graph of the three values for strain on continuous US images, the feature values of malignant tumors were found to concentrate around the origin, with values lower than those of benign tumors. The values for features in benign tumors were spread in a disorderly man- ner, as shown in Figure 6.

The AZ values of the contour difference, shift distance, and area difference on continuous US images were 0.88, 0.85, and 0.86, respectively, and the values of these features on noncontinuous US images were 0.87, 0.80, and 0.81. The difference between values with continuous and noncontinuous US images was statistically significant only for area dif- ference (P⬍ .01). The AZvalue for solidity, the shape feature evaluated with images obtained without compression, was 0.79. The AZvalues for three features of strain were significantly higher than the A_Zvalue for the shape feature (P⬍ .01), as shown in Figure 7.

Diagnostic Performance

By using the support vector machine, the accuracy, sensitivity, specificity, and positive and negative predictive values of the US classification for continuous US

images were 87.0% (87 of 100), 85% (34 of 40), 88% (53 of 60), 83% (34 of 41), and 90% (53 of 59), whereas those for noncontinuous US images were 80.0%

(80 of 100), 75% (30 of 40), 83% (50 of 60), 75% (30 of 40), and 83% (50 of 60).

For the Bayesian classifier, accuracy, sensitivity, specificity, and positive and negative predictive values of the US classification for continuous US images were 77.0% (77 of 100), 88% (35 of 40), 70%

(42 of 60), 66% (35 of 53), and 89% (42 of 47), whereas those for noncontinuous US images were 71.0% (71 of 100), 78% (31 of 40), 67% (40 of 60), 61% (31 of 51), and 82% (40 of 49), respectively. The continuous images were better than the noncontinuous images with use of both the support vector machine and the Bayesian classifier. The difference, how- ever, was not statistically significant (P⬎ .05). When the performances of the support vector machine and the Bayesian classifier were compared, the support vector machine was found to be better than the Bayesian classifier for both the noncontinuous and continuous images.

The difference in specificity for the support vector machine and that for the Bayesian classifier with the continuous US images was statistically significant (P ⫽ .01). The AZ value of the support vector machine based on the values for the four features of strain used for the classification of solid breast tumors was Figure 5. (a– h) Transverse continuous US images of a benign breast

lesion. The computer-delineated margin is the white outline. In this case, the values for the three features of strain— contour difference, shift distance, and area difference—and the value for the one feature of shape—solidity—were 4.73%, 1.90, 1.78%, and 1.86, respectively.

Compared with malignant tumors, benign tumors had larger values for features of strain and a smaller value for feature of shape because benign tumors were softer and had regular borders.

Figure 6. Scatter graph shows contour difference (CD), area differ- ence (AD) and shift distance (SD) of all benign and malignant breast tumors. The values for features of strain of malignant tumors con- centrated around the origin and were smaller than those for features of strain of benign tumors.

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0.91 for both continuous and noncontinuous US images.

DISCUSSION

Elastography is an imaging technique that depicts tissue stiffness by means of imaging tissue strain from external compression, and the technique has been shown to have the potential to aid in the discrimination between benign and malignant breast tumors (9). A complex radiofrequency extraction subsystem, however, is required to apply this technique to current US scanners. Here, we propose a method for imaging elasticity and for computer-aided analysis that employs B- mode images rather than radiofrequency data. The results of our study show that the proposed method that is based on image segmentation and analysis of features of strain with continuous B-mode images can be used successfully to classify benign and malignant breast tumors.

The AZvalue for our computer-aided diagnosis method was 0.91 in equivocal cases that required an interventional procedure for the determination of status.

The total computation time, which included preprocessing, image segmentation, and feature analysis, was less than 30 seconds per case.

In this study, a level-set segmentation approach was used to extract tumor contours from continuous US images. Be- cause the two-dimensional method was inadequate for calculating the volume of the sideslip during compression, the three-dimensional level-set method was used. The results of the automated segmentation were found to be acceptable in all cases with a radiologist’s review.

Changes in lesion shape, lesion contour shift, and lesion area were evaluated on continuous US images. The proposed values for features of strain showed that malignant tumors tend to be more rigid and to have more desmoplastic reactions within the surrounding tissue, which means that those features of strain change to a lesser extent on continuous US images. From the experimental results obtained, values for all three features of strain on images obtained with compression were better than was the value for the feature of shape on images obtained without compression.

The computer-aided classification of benign and malignant breast tumors on continuous US images was improved by using the support vector machine. In our study, the support vector machine was

found to be better than the Bayesian classifier for both noncontinuous and continuous images. The differences in their specificity values were statistically signif- icant (88% and 70%, respectively; P ⫽ .01). According to a study by Chang et al (23), the classification ability of the support vector machine at gray-scale US was equal to that of a neural network model, whereas the support vector machine had a much shorter training time (1 vs 189 seconds). Because of its performance, the support vector machine has become an effective tool for pattern recognition, machine learning, and data mining (20 –22).

We compared the results obtained with continuous US images with those obtained with noncontinuous US images; with the latter method, only two images, the first and the last frames, were used for the analysis. The continuous images were better than the noncontinuous images for feature extraction and for the US classification of benign and malignant tumors. With the noncontinuous images, analysis of features of strain was somewhat inaccurate because the time involved was too long and the data were decorrelated. The computed features of strain are unreliable for diagnosis if the image obtained with compression is changed a lot.

There were limitations in our study. In clinical applications of US imaging of

strain on tissue from probe compression, the operator must use a constant pressure on the scanning probe to avoid an in- clined scanning plane. Otherwise, the tumor may slip out of the scanning plane and create inaccuracies. Gradual compression with the US probe, however, can be performed easily after some practice or by using a compression plate. Compared with the accuracy of elastography with radiofrequency data, the accuracy of our method depends heavily on the results of lesion segmentation. We adjusted some parameters to control the segmented contours and to obtain better results with an expert radiologist’s input. Currently, the computerized segmentation in our method is only semiautomated. For tumors with poorly demarcated borders, a good preprocessing technique would be helpful to enhance tumor boundaries.

For tumors without a definable boundary or for isoechoic masses, other methods should be developed. For example, some representative reference points can be used to evaluate tissue motion. More- over, the harmonic imaging technique may improve tumor boundary delinea- tion (25).

In conclusion, we propose a method for imaging elasticity that is based on image segmentation and analysis of features of strain with continuous US images. The results obtained show that the Figure 7. Receiver operating characteristic curves of the values for

four features and the support vector machine classifier (all features).

The A_Zvalues (Az) of three features of strain were significantly higher than the A_Zvalue of the single feature of shape (solidity) (P⬍ .01).

The value for the support vector machine with all four features—three features of strain and one of shape—produced the best performance, with an A_Z value of 0.91. Contour_Diff⫽ contour difference, Area- _Diff⫽ area difference, Shift_Dis ⫽ shift distance.

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method could be used successfully to classify benign and malignant breast tumors.

APPENDIX

The main idea of the level-set approach is to represent a motion contour␥(t) as the set {␺(x,t) ⫽ 0} in one higher dimension.

In other words, given a moving closed hypersurface ␥(0), that is, ␥(t ⫽ 0):

[0,⬁)3ℜ^N, we wish to produce a formulation for the motion of the hypersurface that is propagating along its normal di- rection with a speed F, where F is a func- tion of the surface characteristics (such as its curvature and normal direction) and the image characteristics (such as the gray level or gradient). Hence, the idea of the level-set method is to embed this propagating interface as the zero level set of a higher-dimension function ␺. Let

␺(x,t ⫽ 0), where x ⑀ℜ^N, be defined by

⌿(x,t ⫽ 0) ⫽ ⫾d , where d is the distance from x to␥(t ⫽ 0), and the sign indicates whether the point x is outside (plus) or inside (minus) the initial hypersurface

␥(t ⫽ 0). Since F supplies the speed in the outward normal direction, we obtain the evolution equation ␺, inside which the

propagating surface is embedded as the zero level set, namely:

␺t⫹ F兩ⵜ␺兩 ⫽ 0, (A1) with a given value of␺(x,t ⫽ 0).

To help illustrate these ideas, Figure A1 shows the outward propagation of an initial curve and the accompanying motion of the level-set function␺. Let the initial front␥ at t ⫽ 0 be a circle in the x-y plane (Fig A1a). Thus, the circle is the level set {␺ ⫽ 0} of an initial surface ␺(x,y, t ⫽ 0) in R³in Figure A1b. We can then match the moving curves␥(t) in such a way that the level set {␺ ⫽ 0} always yields the moving front (Fig A1c, A1d).

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Figure A1. Diagram shows level-set formulation of equation of mo- tion. (a, b) Diagrams show curve␥ and the corresponding surface

␺(x,y) at t ⫽ 0. (c, d) Diagrams show curve ␥ and the corresponding surface␺(x,y) at time t. Diagrams show the outward propagation of an initial curve and the accompanying motion of the level-set function

␺. The concept of the level-set approach is to represent the front ␥(t) as the level set {␺ ⫽ 0} of a function ␺.

Computer-aided Analysis of Continuous US Images

Solid Breast Masses:

Classification with

Computer-aided Analysis of Continuous US Images

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