Linking Syntactic Functions with Thematic Roles: Psych-Verbs and the Resolution of Subject-Object Ambiguity

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International Journal of Neural Systems, Vol. 15, Nos. 1 & 2 (2005) 1–11c© World Scientific Publishing Company

AUTOMATED DIAGNOSIS OF BRAIN TUMOURS ASTROCYTOMASUSING PROBABILISTIC NEURAL NETWORK CLUSTERING AND

SUPPORT VECTOR MACHINES

DIMITRIS GLOTSOSDepartment of Medical Physics, University of Patras,

Rio-Patras, 26500, [email protected]

JUSSI TOHKATampere University of Technology, Institute of Signal Processing, Tampere, Finland

andUniversity of California Los Angeles, Department of Neurology,

Laboratory of NeuroImaging, [email protected]

PANAGIOTA RAVAZOULAUniversity Hospital of Patras, Department of Pathology,

Rio-Patras, 26500, Greece

DIONISIS CAVOURASTechnological Institute of Athens, Department of Medical Instruments Technology,

Aigaleo, Athens, 12210, [email protected]

GEORGE NIKIFORIDISDepartment of Medical Physics, University of Patras,

Rio-Patras, 26500, [email protected]

A computer-aided diagnosis system was developed for assisting brain astrocytomas malignancy grad-ing. Microscopy images from 140 astrocytic biopsies were digitized and cell nuclei were automaticallysegmented using a Probabilistic Neural Network pixel-based clustering algorithm. A decision tree classi-fication scheme was constructed to discriminate low, intermediate and high-grade tumours by analyzingnuclear features extracted from segmented nuclei with a Support Vector Machine classifier. Nuclei weresegmented with an average accuracy of 86.5%. Low, intermediate, and high-grade tumours were identifiedwith 95%, 88.3%, and 91% accuracies respectively. The proposed algorithm could be used as a secondopinion tool for the histopathologists.

Keywords: Probabilistic neural network; support vector machines; microscopy; astrocytomas; grading.

1. Introduction

Astrocytomas are considered to be among the mostlethal and difficult-to-treat forms of cancer.1 In diag-nosing of astrocytomas, the most significant step is

the determination of the degree of tumour abnormal-ity (grading). Grading is performed by histopathol-ogists who visual inspect microscopic sections ofbiopsy material under the microscope.2 Accordingto the World Health Organization (WHO) grading

1

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2 D. Glotsos et al.

system,3 astrocytomas are classified into three grades(grade II–III and IV). Astrocytomas of grade II areconsidered as low-grade tumours, which generallyhave good prognosis. Astrocytomas of grade III andIV (high grade) are the most aggressive tumourswith survival time ranging on average from 6 to 12months.4

Although the WHO grading system is the mostpopular for grading astrocytomas, the WHO guide-lines are limited by the vagueness of the descriptionsused to define each grade. Some of these definitionsbecome clearer only by experience.5 The latter pro-motes relatively low inter and intra-observer repro-ducibility among histopathologists, especially in thecritical separation of grade II from grade III tumours,because malignancy from grade II to grade IIItumours develops along a biological continuum.6

From a clinical point of view it is important to dis-tinguish between low and high-grade tumours, sincemany high grade astrocytomas are often treated dif-ferently (frequently with radiation therapy) com-pared to low-grade astrocytomas.4 Hence, there is aneed for the development of reliable and reproduciblesystems for assessing tumor aggressiveness.

Although most studies have focused on valida-tion of visible features,7 a variety of automatic tech-niques have been introduced to improve diagnosticaccuracy.8–12 Reinhold et al.8 have evaluated withdiscriminant analysis the changes in topometry ofnuclei and developed a system that discriminatedgrade II from grade III tumours with an accuracyof 88%. Decaestecker et al.9 used a nearest-neighborclassifier that have discriminated less aggressivefrom highly aggressive low-grade astrocytomas with55% accuracy. Scarpelli et al.10 showed a signifi-cant change in quantitative nuclear features fromlow to high-grade tumours using linear discrimi-nant analysis, toluidine-blue-staining and the Burgesgrading system. Wangenheim et al.11 have devel-oped a new grading system (named HOM) that hasbeen shown to be more efficient in automatic grad-ing of gliomas compared to the WHO grading sys-tem. In Wangenheim’s study, different grade gliomaswere discriminated with 97% accuracy using theHOM system, with 49% accuracy using the Daumas-Duport system, and with 61% accuracy using theWHO grading system. Belacel et al.12 have devel-oped a fuzzy logic method for astrocytomas grading

that automatically classified different tumour gradeswith an accuracy of 66%.

However, many of these studies have intro-duced supervised computer-based systems8–12 thatwere developed based on modifications of theWHO grading scheme10–11 and specialized stain-ing protocols.8–11 Additionally, these studies havenot investigated the possibility of ‘suspicious’ cases,which are those critical cases that are character-ized by histopathologists as ranging form grade II(low grade) to grade III (high grade). It is mostsignificant to accurately specify these ‘intermedi-ate’ grade tumours (between grade II and gradeIII), because these tumours need re-examination.In this study we propose a novel method designedfor the automatic segmentation and classification ofastrocytomas microscopy images to be compatibleto the most widely accepted clinical protocols forastrocytomas grading, namely the WHO scheme andthe Hematoxylin-Eosin (H&E) staining protocol.3,13

In addition, the proposed method (i) is designedin an unsupervised manner to perform image seg-mentation by using a novel Probabilistic NeuralNetwork (PNN) pixel-clustering algorithm (ii) usesquantitative nuclear features and a Support VectorMachine (SVM) classifier for the automatic separa-tion of low from high grade tumours, and (iii) iden-tifies those intermediate-grade tumours that needre-examination using a decision tree classificationscheme.

2. Material and Methods

One hundred and forty (140) H&E stained biopsiesof astrocytomas were collected from the Departmentof Pathology of the University Hospital of Patras,Greece. Tumour grade was defined as low (61/140),high grade (67/140) or as ‘suspicious’ cases of inter-mediate grade (12/140) (see Table 1) according theWHO grading system by a histopathologist (P.R.).

Table 1. Astrocytomas biopsies dataset.

Low grade Suspicious cases High gradeof intermediate grade

Grade II Grade II–III Grade III Grade IV61 12 29 38

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Automated Diagnosis of Brain Tumours Astrocytomas 3

For each biopsy, the histopathologist specified themost representative region. From this region, images(Fig. 4) were acquired (768×576×8 bit) using a lightZeiss Axiostar-Plus microscope (ZEISS; Germany)connected to a Leica DC 300 F color video camera(LEICA; Germany).

2.1. Image segmentation

Nuclei were automatically segmented from surround-ing tissue in order to encode tumour malignancyby means of a set of features derived from the seg-mented nuclei. To design the segmentation algorithm5 × 5 windows around each pixel were considered.The center pixel of each window was assigned a3-dimensional textural feature vector based on whichpixels were classified as belonging to nuclei or sur-rounding tissue. The textural features were (1) thesum of autocorrelation function AF for all possibledisplacements m, n inside the window 5 × 5, (2) thecross relation S (1,1) and (3) the second degreespread S (2,2).14

S(1, 1) =5∑

m=0

5∑−5

(m − nm)(n − nn)AF (m, n)

(1)

S(2, 2) =5∑

m=0

5∑−5

(m − nm)2(n − nn)2AF (m, n)

(2)

AF (m, n) =∑

j

∑′k

F (i, j)F (i − n, k − m) (3)

where nm =∑5

m=0

∑5−5 mAF (m, n), nn =∑5

m=0

∑5−5 nAF (m, n), and F (i, j) is the image

intensity at the pixel (i, k).From Eq. (3) it is apparent that a window

region will exhibit higher sum of the autocorrela-tion function for a fixed shift (m, n) than a regionof fine texture. The sum of autocorrelation functionfor all displacements over the window 5 × 5 couldbe regarded as coarseness index; this feature takeshigher values for coarser regions. Since the sum ofautocorrelation function is estimated for all displace-ments, it would be possible that two different textu-ral fields will exhibit the same value. To improve thetexture discrimination, the textural features S(1,1)and S(2,2) were estimated additionally, which pro-vide the extra information of two-dimensional spread

measures of the autocorrelation function.14 In thisway formation of clusters (nuclei and surroundingtissue) was facilitated.

The feature vectors (sum of the autocorrela-tion function, spread and cross relation) generatedfor the window around each pixel were fed intoa Probabilistic Neural Network (PNN)-based clus-tering algorithm15 (see Sec. 2.1.1.). The algorithmassigned the central pixel of each window as belong-ing to either nuclei or surrounding tissue based onthe feature vector of that window. In the resultedbinary image (nucleus and surrounding tissue), pix-els that were classified as emanating from surround-ing tissue were marked with black colour and thosebelonging to a nucleus were marked with whitecolour (Fig. 5). To reduce noise the binary imagewas further processed by fill holes filters, morpholog-ical operations, elimination of nuclei less than 200pixels and elimination of nuclei intersecting imageboundaries (Fig. 6).13 Finally, nuclei segmentationwas achieved by superimposing the binary processedimage to the original image (Fig. 7).

To evaluate the performance of the segmentationprocedure, nuclei boundaries in 25 randomly selectedimages were manually delineated by an experiencedhistopathologist (P.R.). Subsequently, manually seg-mented nuclei were compared to the automaticallysegmented nuclei in terms of nuclei area, roundnessand concavity.

2.1.1. Probabilistic Neural Network-basedrobust clustering

In order to implement PNNs for robust cluster-ing, the network architecture is constructed with 5(input layer, pattern layer, summation layer, clus-tering layer and output layer) instead of 4 layers asoriginally proposed in Ref. 16 (Fig. 1).

The input layer comprises as many nodes asthe number of normalized to zero mean and unitdeviation17 input feature (or pattern) vectors. Thepattern layer consists of as many nodes as the inputfeature vectors; each feature vector corresponds toexactly one pattern node. At each pattern node,the Euclidean distances of the node’s correspond-ing feature vector from the rest of the input fea-ture vectors are calculated. The resulting Euclideandistances vector is then passed through a Gaussian

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4 D. Glotsos et al.

Fig. 1. PNN architecture for supervised (left) and unsupervised (right) classification.

activation function and summed to give an estima-tion of the PDF f(xi) at the location of each inputfeature vector. In other words, at each node i of thepattern layer we calculate

f(xi) =N∑

j=1

exp(−‖d‖

2σ2

)(4)

where xi = S(1,1) + S(2,2) + sum(AF ), d is theEuclidean distances vector and σ is the spread. Inthe summation layer the whole data PDF is cal-culated by combining the outputs from the patternlayer and interpolating. In symbols, the PDF is esti-mated based on the individual PDF estimates fi :

f(y; x1, . . . , xN ). (5)

The value of the f (yi) = fi and the values otherthan the given input feature vectors are obtainedby linear interpolation. The spread is an adjustableparameter, which has been shown18 to play animportant role in the estimation of the data PDFin Eq. (4). Too small σ causes very spiky approx-imation, whereas too large smoothes out details.Although the optimal value of σ is usually experi-mentally determined, in this work we estimated σ

according to:

σ =1

N − 1

N∑i=1

N∑j=1

‖xi − xj‖, i �= j. (6)

In this way sigma is independent of the typeof data and additionally sigma is automaticallyestimated.

In the clustering layer (Fig. 1b), the differentclusters are identified. Initially, peaks correspondingto class (cluster) centroids ci are identified on the1-dimensional representation of the PDF (see Fig. 2).

The most prominent peaks of the PDF are first deter-mined by locating the most prominent peak that cor-responds to the first cluster centroid. This peak andits surrounding points are then removed from consid-eration. The surrounding points are defined within aGaussian estimated spread calculated as:

spread =1M

max(∥∥x‖)2 (7)

where M is the number of data clusters. The algo-rithm then continues by specifying and removing thesecond peak, that corresponds to the second clus-ter centroid and this iterative procedure continuesby locating cluster centroids (c1, . . . , cM ) as manytimes as the a priori estimated number of data clus-ters M . This implies that we have a vague idea ofthe classes we are seeking for. A schematically viewof this procedure is illustrated in Fig. 2. Clusters arethen, formed by assigning each data point xi to clus-ter ci if the majority of its K nearest neighbors iscloser to ci than any other cluster centroid. K isdefined as K = 2 + dmax√

2M, where dmax is maximum

Euclidean distance between cluster centroids. Beforeassigning all points to clusters, a procedure to detectoutlying points is initiated. More distant probableoutliers are considered those feature vectors locatedat the two ends of the PDF satisfying max(‖x‖)2 (farright OR and far left OL edges in Fig. 2d). Featurevectors that are closer to OR or OLthan c1 of c2 inFig. 2d are considered outliers, provided that thesefeature vectors are outside the corresponding to c1 orc2 spreads (see Eq. (7)). These outliers are consideredas belonging to no class. Finally, in the output layereach data sample is assigned as belonging to clustercj or being an outlier.

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Automated Diagnosis of Brain Tumours Astrocytomas 5

(a) (b)

(c) (d)

Fig. 2. (a) Calculation of data PDF, (b) Identifying the first most prominent peak, (c) Removing this peak and itssurrounding points within a Gaussian estimated spread according to Eq. (7) and calculating the second most prominentpeak, (d) Formation of clusters and identification of outliers using a k-nearest neighbor procedure.

2.2. Image classification

After image segmentation, 33 morphological andtextural features (Appendix A) were extracted foreach case (patient) by selecting at least 50 non-overlapping nuclei. Textural features (15 features)described chromatin organization inside nuclei.19–20

Morphological features (18 features) encoded nucleisize and shape.21 A two level hierarchical decisiontree (Fig. 3) was constructed for the automatic iden-tification of initially low from high-grade tumoursand subsequently suspicious cases of intermediategrade in the low and high-grade groups respectively.At each level of the decision tree an SVM classifier22

was utilized alternatively constructed with an RBFand polynomial kernels. To reduce the dimensional-ity of the features an exhaustive search algorithm wasperformed in all possible combinations up to 6. Theperformance of the SVM classifier for each feature

combination was assessed by using a leave-one-outmethod.23

2.2.1. Support Vector Machine classification

By mapping input vectors into a higher dimensionfeature space and defining the hyperplane that hasthe maximum distance from the closest trainingdata, SVM can be utilized for binary classificationproblems24–25 with discriminant function as follows:

g(x) = sign

(N∑

i=1

αiyiK(x, xi) + b

)(8)

where xi training data belonging to either classyi ∈ {+1,−1}, N the number of training samples, αi,b weight coefficients and K the transformation or ker-nel function. Kernel functions utilized were the radialbasis function (RBF) with value of γ = 1/(2σ2) setequal to 0.5 after testing values from 0.005 to 6 and

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6 D. Glotsos et al.

Fig. 3. Decision tree classification scheme for discriminating low, high and intermediate cases of astrocytomas.

polynomial of degree d = 1 and 2. The adjustableparameter C that specifies the importance of mis-classifications was experimentally determined equalto 10.

KREF(x, xi) = exp(−‖x − xi‖2

2σ2

),

KPOLYNMIAL(x, xi) =((

xT xi

)+ 1)d

.

(9)

Using the routine quadprog provided with the MAT-LAB optimization toolbox, optimization problem ofcalculating parameters ai was solved.

3. Results

Figure 4 illustrates a digitized H&E stained image(768 × 576 × 8 bit) of a low-grade astrocytic tumor.The pixel classification procedure performed by thePNN algorithm resulted in the binary image ofFig. 5. In Fig. 6 nuclei were corrected by applying

Fig. 4. H&E stained image (768 × 576 × 8 bit) of anastrocytic tumor (×400).

Fig. 5. Resulting binary image after applying the PNNclustering segmentation algorithm that discriminatedpixels belonging to nuclei from pixels belonging to sur-rounding tissue.

Fig. 6. The binary image was further processed withmorphological filters, size filters (nuclei less than 200 pix-els were omitted) and fill holes operations.

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Automated Diagnosis of Brain Tumours Astrocytomas 7

Fig. 7. Final segmentation was achieved by superim-posing the original image to the morphological processedimage. Arrows indicate closely spatially located nucleithat were incorrectly identified by the algorithm as onestructure.

morphological-open filters, size filtering (nuclei lessthat 200 pixels were omitted) and fill-holes oper-ations. Final segmentation was obtained by super-imposing the original image to the morphologicalprocessed image (Fig. 7). According to the segmen-tation’s algorithm evaluation procedure describedin Sec. 2.2, the accuracy of correctly identifiednuclei in randomly selected images ranged between76.9–93.7%, with an average of 86.5% (seeTable 2).

Concerning image classification, high-gradetumours were classified with an accuracy of 91.0%,low-grade tumours with 95.0% and ‘suspicious’ casesof intermediate grade with 88.3%. Overall accu-racy was 92.1% (Table 3). From the 67 high-gradecases, 61 were correctly identified as high grade, 4incorrectly as low grade and 2 as of intermediategrade. From the 61 low-grade cases, 58 were cor-rectly classified as low grade, 1 incorrectly as highgrade and 2 as of intermediate grade. Finally, fromthe 12 ‘suspicious’ cases of intermediate grade, 10were correctly identified as of intermediate gradeand 2 incorrectly as high grade. Best feature vectorcombination for the discrimination of (i) low fromhigh-grade tumours was energy, roundness, correla-tion and short run emphasis; (ii) low from interme-diate grade tumours was concavity, angular secondmoment, roundness and energy, (iii) high from inter-mediate grade tumours was energy, area, gray levelnon-uniformity and concavity.

Table 2. Evaluation of the seg-mentation algorithm for 25-seg-mented H&E images of astrocytomas.

Diagnosis Accuracy

Image 1 Grade III 83.3Image 2 Grade III 88.9Image 3 Grade III 84.9Image 4 Grade II 93.7Image 5 Grade IV 88.6Image 6 Grade IV 76.9Image 7 Grade II 92.7Image 8 Grade II 88.7Image 9 Grade IV 79.1

Image 10 Grade III 81.5Image 11 Grade IV 81.2Image 12 Grade III 83.4Image 13 Grade II 92.0Image 14 Grade IV 81.3Image 15 Grade III 84.0Image 16 Grade III 84.6Image 17 Grade IV 83.3Image 18 Grade III 89.7Image 19 Grade II 92.2Image 20 Grade II 90.4Image 21 Grade II 94.1Image 22 Grade II 92.5Image 23 Grade II 92.2Image 24 Grade II 85.1Image 25 Grade IV 79.7

4. Discussion

In this study a computer-assisted diagnosis systemfor astrocytomas grading was introduced by devel-oping a novel PNN clustering algorithm for imagesegmentation and an SVM-based decision tree pro-cedure for image classification.

Regarding image segmentation results, nuclei seg-mentation accuracy reached approximately 86.5%.Accurate segmentation of nuclei is of crucial impor-tance to guarantee correct results in computer-assisted microscopy.26 Nuclei encode significantdiagnostic and prognostic information, that if quan-tified can potentially allow the prediction of the dis-ease course. Previous studies that have investigatedthe demanding task of nuclei segmentation27–30

have reported relatively high segmentation accu-racies, such as 85%,27 89%,28 and 99%.29 InRef. 27 a method has been presented for the auto-matic segmentation of nuclei in breast fine needleaspiration images using the Hough transform. InRefs. 28 and 29 images from Papanicolaou stained

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8 D. Glotsos et al.

Table 3. Truth table demonstrating classification results for the 140 cases of astrocytomas.

Diagnosis Low grade High grade Suspicious cases Classification results(grade I–II) (grade III–IV) (grade II–III) (accuracy)

Low grade 58 1 2 95.0%High grade 4 61 2 91.0%Suspicious cases 0 2 10 83.3%

Overall accuracy 92.1%

smears have been automatically segmented usingNeural Networks and active contours respectively. InRef. 30 a method for segmentation of immunohisto-chemically stained nuclei has been presented usinga supervised pixel based classification algorithm andwatershed operations. However, these studies havegreatly depended on a priori assumptions establishedfor nuclei morphology and size. The proposed PNNclustering algorithm requires no assumptions con-cerning either nuclei size or shape. Additionally, ithas to be stressed that in this study the routineH&E staining protocol was employed that is notas accurate in nuclei staining as other specializedstaining protocols such as those used in previousstudies (ki-67, Feulgen, Toluidine-blue and Papan-icolaou staining).10–11,27–30

The PNN-clustering algorithm is designed toidentify different data partitions based on data dis-tribution. In Ref. 31 a Gaussian mixture approachhas been presented based on the data distribution;however this algorithm has required a priori assump-tions on the data and noise distributions. In orderto let the data “speak for themselves” the pro-posed PNN clustering algorithm was designed tomake no a priori assumptions on data distribution,which was estimated by using a Gaussian Parzenkernel.32 Compared to Fuzzy C Means robust clus-tering methods,33–36 which have been designed todetect ellipsoid-shaped clusters and in some casescurves or surfaces, the proposed algorithm was moreflexible and was able to detect clusters of arbitrarysize and shape. Compared to NN-based robust clus-tering techniques,37–38 the PNN-based method wasrelatively faster since no feedback paths or itera-tive procedures were required. Finally, the algorithmneeded some a priori information regarding the num-ber of expected clusters, which in many cases doesnot present a problem.

Considering classification results, the methodgave high rates in correctly discriminating low from

high grade tumours and ‘suspicious’ cases of interme-diate grade with overall accuracy 92.1%. SVM withpolynomial kernel of degree 2 optimized classifica-tion performance in both levels of the decision tree.The decision tree structure was preferred to using asingle 3-class SVM classifier because it resembles thediagnostic procedure followed by the histopathologistand it is less complex and faster to implement. Thenumber of support vectors indicated the good gener-alization capacity24–25 of the SVM classifier: (a) 17for constructing the SVM classifier to discriminatelow from high-grade tumours (13.2% of classifiedcases), (b) 10 to identify low from intermediate-gradetumours (14.7% of all classified cases), and (c) 12to separate high from intermediate-grade tumours(15.6% of classified cases).

These results are promising compared to thosepresented in literature for automatic grading sys-tems that have utilized the WHO scheme but moredemanding staining procedures, such as 55% inRef. 9, 66% in Ref. 12, 88% in Ref. 8 and 89.7%in Ref. 21. In contrast to previous studies,9–12,21

the proposed method was designed to additionallyautomatically specify “suspicious” cases, i.e., thosecases that cannot be definitely characterized eitheras low or high grade (grade II to III cases). Thedifficulty for histopathologists to provide a definitediagnosis for cases between grade II and grade IIImight be explained due to that astrocytomas malig-nancy develops along a biological continuum.5 Itis, thus, very important for an automatic gradingalgorithm to specify “suspicious” cases of intermedi-ate grade that need re-examination because clinicalmanagement is strongly tied to grade assignment6:an incorrect assessment of a low grade tumour ashigh grade might lead to an excessively aggressivetherapy, whereas a false assignment of a high gradetumour as low grade might lead to a less aggres-sive therapy. In our dataset 12 cases were charac-terized by the histopathologist as “suspicious” cases

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Automated Diagnosis of Brain Tumours Astrocytomas 9

(between low and high grade). The PNN algorithmcorrectly identified 10 of the 12 “suspicious” cases,with accuracy 83.3%. The remaining two cases wereclassified by the algorithm as high grade. The firstof these 2 cases was a low-grade astrocytic tumourthat recurred after 2 years from first diagnosis andsurgical removal. The second case was an astrocytictumour with histological characteristics of grade II;however, the histopathologist assessed this tumour asbetween grade II and grade III due to the existenceof extended necrosis.

5. Conclusion

The method proposed in this study gave highaccuracy in automatically separating low fromhigh grade astrocytomas while specifying those‘suspicious’ cases of intermediate grade that needre-examination. The method was developed to becompatible with the WHO grading system and H&Estainingprotocol, which are the most common stan-dards in clinical routine.3,9 Therefore the proposedcomputer assisted microscopy system might be uti-lized in daily clinical practice without the needof performing specialized staining processes thatincrease the time and cost of diagnosis and withoutburdening histopathologists with the need to famil-iarize with new grading protocols.

Appendix A

In high-grade tumours the existence of giant nucleiwith irregular curvature and shape is frequently pro-nounced. On the contrary, in low-grade tumoursnuclei are smaller, round and look more or less alike.Thus, the quantification of morphological charac-teristics of nuclei may contain important diagnos-tic information concerning the grade of tumours.In this study 18 morphological features were com-puted. These features comprised measurements ofarea, roundness and concavity. Additionally, themean value, maximum value, minimum value, stan-dard deviation, skewness and kurtosis of each of the3 morphological features were calculated.

Textural features have been shown in manystudies as powerful descriptors of tumoursmalignancy.7–13 In this study three kinds of tex-tural features were calculated: First order statisticalfeatures that comprised measurements of the meanvalue, standard deviation, skewness and kurtosis

from nuclei histogram. Second order statistical fea-tures from the co-occurrence matrix.19 Although thefirst order statistical features describe the gray leveldistribution of nuclei, these features do not give anyinformation concerning the spatial distribution ofthe various gray levels inside nuclei. This type ofinformation can be extracted from the co-occurrencematrix that describes the frequency of appearance ofpairs of pixels at a distance d (inter-pixel distance)with gray values I1 and I2 inside each nucleus. Theco-occurrence features that were calculated were thefollowing:

1. Angular Second Moment ASM =∑Ng−1

i=0

∑Ng−1j=0(

p(i, j))2, where Ng is the number of gray lev-

els in the image, i, j = 1, . . . , Ng, and p(i, j) isthe co-occurrence matrix. ASM describes imagesmoothness and takes minimum values for smoothtextures nuclei.

2. Contrast CON =∑Ng−1

n=0 n2{∑Ng−1

i=0

∑Ng−1j=0(

p(i, j))2}

, |i − j| = n. CON increases for high

contrast nuclei. The factor n2 is enhances big dif-ferences.

3. Inverse Different Moment IDM =∑Ng−1

i=0

∑Ng−1j=0

p(i,j)1+(i−j)2 . IDM increases for low contrast nucleidue to the dependence on (i − j)2.

4. Entropy ENT = −∑Ng−1i=0

∑Ng−1j=0 p(i, j) log

(p(i, j)

).

ENT is a measure of randomness and takes lowvalues for smooth nuclei.

5. Correlation COR =PNg−1

i=0PNg−1

j=0 (ij)p(i,j)−mxmy

σxσy,

where mx, my, σx and σy the respective mean val-ues and standard deviations of px and py. CORencodes the gray tones dependencies in nuclei.

These 5 co-occurrence matrix based textural featureswere calculated with inter-pixel distance d = 1 andd = 3.

• Second order statistical features from the runlength matrix.20 The run length matrix describesthe frequency of appearance of a set of consecu-tive pixels (run) having the same gray value. Therun length features that were calculated were thefollowing:

1. Short run emphasis SRE =PNg−1

i=0PNr−1

j=0r(i,j)

j2PNg−1

i=0PNr−1

j=0 r(i,j),

where r(i, j) is the run length matrix, Ng is thenumber of gray values, Nr is the largest possiblerun, i = 1, . . . , Ng, j = 1, . . . , Nr. SRE tends to

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10 D. Glotsos et al.

emphasize short runs due to the division with j2

and takes high values for coarser nuclei.

2. Long run emphasis LRE =PNg−1

i=0PNr−1

j=0 j2r(i,j)PNg−1

i=0PNr−1

j=0 r(i,j).

LRE tends to emphasize long runs and is largefor smoother nuclei.

3. Gray level non-uniformity GLNU =PNg−1

i=0 (PNr−1

j=0 r(i,j))2

PNg−1i=0

PNr−1j=0 r(i,j)

. GLNU increases for nuclei

having many runs of the same gray level value.4. Run length non-uniformity RLNU =

PNg−1j=0 (

PNr−1i=0 r(i,j))2

PNg−1i=0

PNr−1j=0 r(i,j)

. RLNU takes low values for

nuclei with homogeneous distribution of runs.

5. Run Percentage RP =PNg−1

i=0PNr−1

j=0 r(i,j)

P whereP is the total possible number of runs in thenucleus image. This feature takes its lowest valuefor nuclei with linear structures.

Acknowledgments

We thank European Social Fund (ESF), OperationalProgram for Educational and Vocational Training II(EPEAEK II) and particularly the Program IRAK-LEITOS for funding the above work. J. Tohka’s workwas supported by the Academy of Finland under thegrants nos. 204782 and 104834 and the NIH/NCRRgrant P41 RR013642, additional support was pro-vided by the NIH Roadmap Initiative for Bioinfor-matics and Computational Biology U54 RR021813funded by the NCRR, NCBC, and NIGMS.

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