NORMALISED RADIAL LENGTH AND CONVEX POLYGONS TO CLASSIFY BREAST TUMOUR CONTOURS IN ULTRASOUND IMAGES

A. V. Alvarenga*, A. F. C. Infantosi*, W. C. A. Pereira* e C. M. Azevedo**

* Biomedical Eng. Program/COPPE, Federal University of Rio de Janeiro, Brazil
** Radiology Department/Brazilian National Cancer Institute (INCa), Rio de Janeiro, Brazil
victor@peb.ufrj.br

Abstract: This work aims to assess the potentiality of morphometric parameters in separating breast tumour contours, on ultrasonic images, as irregular or regular. A segmentation procedure, based on Mathematical Morphology, was applied to depict the tumour contours. Parameters calculated over normalised radial length and convex polygons were, then, calculated from 152 segmented tumour images. Linear Discriminant Analysis was applied and parameters performance assessed (accuracy, sensitivity and specificity). The best performances for individual parameters were the normalised residual mean square value and the circularity. Taking these last two and the roughness the best separation performance is obtained: high specificity (96.7%) and sensitivity (95.7%). These findings point out that these parameters are relevant to distinguish breast tumour contours.
Keywords: Convex polygon, normalised radial length, contour, breast ultrasound.

Introduction

Mammography has been considered as the only diagnostic technique that contributes, through a periodic accompaniment program, to early detection and mortality reduction by breast cancer [1]. However, the mammography accuracy depends on the composition of mammary parenchyma and tumour tissue characteristics, since a dense parenchyma can mask a tumour [2]. Thus the breast examination through ultrasound (US) images has been considered as the most important complementary exam for patients with palpable mass and inconclusive mammograms [1].

Morphologically, benign tumours generally present regular and well-defined contours on US images [3], while malign ones usually infiltrate adjacent tissues thus producing irregular and angled edges [4]. Therefore, the contour analysis from breast solid tumours, using US images, has the potential to aid in reducing biopsies carried through benign tumours [5].

In this work, the potentiality of morphometric parameters in distinguishing between irregular and regular breast tumour contours, on ultrasonic images, is investigated. Using a segmentation procedure [6], based on Mathematical Morphology, the tumour contours will be depicted and parameters from the Normalised Radial Length (NRL) [4] and from convex polygons [7] will be calculated. The contour discrimination parameters performance will then be assessed using Linear Discriminant Analysis.

Material and Methods

One hundred and fifty-two breast US images (with the respective diagnostics) were acquired in TIF format, using a 7.5 MHz US equipment (Sonoline . Sienna® Siemens) at the Brazilian National Cancer Institute (INCa). The tumour contours were determined using a segmentation method (the semi-automatic contour procedure - SAC) based on morphological operators [6]. An experienced radiologist classified all tumour contours as irregular (92) or regular (60).

For each SAC-defined contour with a perimeter P, the Normalized Radial Length (NRL) was calculated as [4]:

where (X0, Y0) and (x(i), y(i)) are the coordinates of the centroid and the i-th pixel on P, respectively. N is the number of pixels on P and max(d(i)) is the maximum value of the radial length (normalised factor).

From (1), three parameters were obtained: the standard deviation (D_NRL); the area ratio (RA) and the contour roughness (R). D_NRL as a measure of the contour

where d(i) is the mean value of d(i) and can be interpreted as the radius of a circular region. By taking into account the number of times d(i) is greater than d(i), RA measures the percentage of the tumour which is outside this circular region, that is [4]:

where . Therefore, RA increases with the number of pixels outside the circular region and so with the contour irregularity.

The contour roughness can be defined as the average distance between neighbouring pixels over the tumour contour [4]:

With such a definition, R increases with the contour irregularity.

Now, for each region (S) indicated by the SAC- defined contour, a convex polygon (S_o) that contains this contour was established. As illustrated by Fig. 1, the more irregular is the contour, the greater is the difference from its convex polygon. This feature can be quantified using two parameters: the overlap ratio (RS) and the normalised residual mean square value (NRV). The parameter RS is given by [9] as:

where the symbols ∩ and ∪ indicate the intersection and union, respectively. S, S_o and Area(.) are in number of pixels. If S and S_o have the same shape and size and also are in the same position, RS = 1.

Further, the residual region (S_r) can be calculated as [10]:

and if S and S_o are identical (shape and size) and also in the same position then S_r = 0. Based on (6), NRV is defined as:

where and are the mean squared values of S_r area and of the convex polygon perimeter (P_o), respectively. Instead of using Po, the So area could be used in the definition of (7), as pointed out by [10]. Nevertheless, considering NRV as a ratio between the residual area and convex polygon perimeter, the sensitivity of NRV is improved [7].

Two other parameters were also calculated: the circularity (C) and the morphological-closing ratio (Mshape). The first has been pointed out as an important parameter in the correct classification of breast tumours [4] and is defined as:

Mshape is defined as the ratio between the S area and its morphological-closing area. This morphological operator allows filling small holes and gaps (possible missing data) on SAC-defined contour [11]. By applying this operator, the morphological-closing area (white in Fig. 2) tends to be greater than the S area (grey in Fig. 2). Hence, the more irregular is the contour, the smaller is Mshape.

Linear discriminant analysis [12] was applied to all possible combinations of the seven parameters (normalised between 1 and .1). The performance of the parameters combination was assessed using the ROC curve, particularly the area A_z under this curve, sensitiv- ity (Se), specificity (Sp) and accuracy (Ac) [8], considering irregular tumours as positive cases and regular ones as negative.

Figure 1: Convex polygon (in white) of the segmented breast tumours using SAC: (a) regular and (b) irregular. The difference between the white and grey area is greater for the irregular tumour

Figure 2: Morphological-closing area (in white) of the defined segmented breast tumours using SAC: (a) regular and (b) irregular

Results

The performance of each one of the seven parameters in distinguishing irregular from regular tumours for all 152 US images is presented in Table 1. Based on A_z, the best performance was achieved by NRV (0.97) and the worst by Mshape (0.55). Also, the highest accuracy (93.4%) and specificity (91.7%) were obtained with NRV. On the other hand, although Mshape has presented the highest sensitivity (95.7%), it leads to a low accuracy (60.5%) and the lowest specificity (6.7%). Considering the Normalized Radial Length parameters, the highest Ac was reached by DNRL and RA, with similar values of Se and Sp, while R resulted in the second lowest accuracy among the seven parameters. Based on A_z, the parameter C was the second best in performance with very close Ac, Se and Sp values, around 88.0%. Further, it is worth to point out that RS is ranked as the third best based on A_z but with the lowest sensitivity.

Table 1: Individual performance of each of the 7 parameters in distinguishing tumour contours in irregular or regular

Figure 3: Scatter diagram for the pair (NRV, C) and the linear function (y=–2.26x+0.38) that best separates irregular from regular tumours (x refers to the normalised residual mean square value, NRV, and y to the circularity, C)

Table 2: Performance of the best combination of parameters in distinguishing tumour contours as irregular or regular

Figure 4: Scatter diagram for the parameters NRV x C x R and the best plane in separating irregular from regular tumours is z = –6.53x –3.03y +0.38 (x refers to the normalised residual mean square value, NRV, y to the circularity, C, and z to the contour’s roughness, R)

Using pairs of parameters, the highest performance was achieved with NRV and C (Table 2). Further, this pair of parameters allowed a performance improvement (Ac = 94.7%, Se = 95.7% and Sp = 93.3%) in compari- son to the results obtained when the individual parame- ters are undertaken. The scatter diagram of NRV x C (Fig. 3) indicates that NRV tends to concentrate irregular tumours (ð_NRV = 0.17, µ _NRV = 0.71) and spread out the regular ones (ðNRV = 0.44, µ NRV = 0.08). On the other hand, C has an opposite behaviour (irregular: ðC = 0.37, µC = .0.45; regular: ð C = 0.08, µ C = .0.87).

The best performance was reached by taking together NRV, C and R (Table 2). Compared to the results achieved with the pair (NRV, C), both Ac (96.1%) and Sp (96.7%) are increased whilst sensitivity remains the same (95.7%). Despite of R weak individual performance, the scatter diagram of NRV x C x R (Fig. 4) shows that R improves the performance of NRV and C in distinguishing the tumour contours. Moreover, increasing the number of parameters to four or more does not lead to a performance improvement.

The best performance was reached by taking together NRV, C and R (Table 2). Compared to the results achieved with the pair (NRV, C), both Ac (96.1%) and Sp (96.7%) are increased whilst sensitivity remains the same (95.7%). Despite of R weak individual performance, the scatter diagram of NRV x C x R (Fig. 4) shows that R improves the performance of NRV and C in distinguishing the tumour contours. Moreover, increasing the number of parameters to four or more does not lead to a performance improvement.

Discussion

Among all the individual parameters and based on Az, the normalised residual mean square value (NRV) leads to the best performance (0.97) in distinguishing irregular from regular tumours for the set of 152 US images used is this work. Another parameter also calcu- lated from the convex polygon, the overlap ratio (RS) has the third best individual performance (0.91). NRV and RS have already been used in a previous work [7] but with a better performance for the latter (0.93). The tumour circularity (C), ranked as the second parameter (0.93), has been considered by Chou et al [4] as an important parameter for classifying breast tumours in malign or benign, when it is associated with R and D_NRL.

The ROC curves of NRV, C and RS are depicted in Fig. 5. For the specificity of NRV ranging from 65.5% to 98.0%, the sensitivity of this parameter is higher than that of C (superior in ≈5.7%) and also that of RS (≈14.8%). Considering the pair of parameters (NRV, C), a decrease in the numbers of false-positive and false-negative has occurred. Hence an increasing in sensitivity and in specificity is noted in its ROC curve (Fig. 6). Joining the parameter R to this pair, leads to a further improvement in specificity but maintaining the sensitivity, as shown in the ROC curve of Fig. 6.

Figure 5: ROC curves of parameters NRV, RS and C. The arrows indicate the specificity range where the NRV sensitivity is higher than C and RS ones

Figure 6: ROC curves of the pair (NRV & C) and the combination (NRV, C & R)

For the set of 152 images, taking together the parameters NRV, C and R, the area under the ROC curve indicates the best global performance (A_z = 0.97) whilst D_NRL, C and R gives an A_z = 0.91. This finding is due to the lower sensitivity of the latter (Se = 81.5%) although with a very close specificity (96.7%). Using (D_NRL, C, R), Chou et al [4] have obtained the same value of A_z, and pointed out this set of parameters as the best for classifying breast tumour contours in malign or benign. This latter work reports a much lower specificity (80%) and a slightly superior sensitivity (97.2%).

Conclusion

Morphometric parameters calculated from the normalised radial length and convex polygons were used for distinguishing irregular from regular breast tumour contours in US images. With this aim, the normalised residual mean square value (based on the convex polygon) and the circularity can be considered the most relevant parameters. It is worth to emphasise that both parameters are calculated as a ratio between areas and perimeters. The best performance, sensitivity of 95.7% and specificity of 96.7%, was achieved by combining these two parameters with the contour roughness. Therefore, these three parameters should be used together for classifying breast tumour contours. Moreover, their ability in separating malign tumours from benign ones must be investigated.

Acknowledges
To FAPERJ and CNPq for the financial support.

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