Semiautomatic Mammographic Parenchymal Patterns Classification Using Multiple Statistical Features

Rationale and Objectives

Our project was to investigate a complete methodology for the semiautomatic assessment of digital mammograms according to their density, an indicator known to be correlated to breast cancer risk. The BI-RADS four-grade density scale is usually employed by radiologists for reporting breast density, but it allows for a certain degree of subjective input, and an objective qualification of density has therefore often been reported hard to assess. The goal of this study was to design an objective technique for determining breast BI-RADS density.

Materials and Methods

The proposed semiautomatic method makes use of complementary pattern recognition techniques to describe manually selected regions of interest (ROIs) in the breast with 36 statistical features. Three different classifiers based on a linear discriminant analysis or Bayesian theories were designed and tested on a database consisting of 1408 ROIs from 88 patients, using a leave-one-ROI-out technique. Classifications in optimal feature subspaces with lower dimensionality and reduction to a two-class problem were studied as well.

Results

Comparison with a reference established by the classifications of three radiologists shows excellent performance of the classifiers, even though extremely dense breasts continue to remain more difficult to classify accurately. For the two best classifiers, the exact agreement percentages are 76% and above, and weighted κ values are 0.78 and 0.83. Furthermore, classification in lower dimensional spaces and two-class problems give excellent results.

Conclusion

The proposed semiautomatic classifiers method provides an objective and reproducible method for characterizing breast density, especially for the two-class case. It represents a simple and valuable tool that could be used in screening programs, training, education, or for optimizing image processing in diagnostic tasks.

While the etiology of breast cancer remains unclear, many studies have demonstrated a correlation between cancer risk and factors such as age, breastfeeding and pregnancy history, family history of breast cancer, hormonal treatments, genetic factors, and breast density ( ). Breast density as a factor of risk was first investigated by Wolfe ( ), who defined a four-grade density scale on the basis of the patterns and textures observed on mammograms. Later, the BI-RADS (Breast Imaging Reporting Data System) density scale was developed by the American College of Radiology to standardize mammography reporting terminology and assessment and recommendation categories ( ). The BI-RADS density classification was created to inform referring physicians about the decline in sensitivity of mammography with increasing breast density. BI-RADS defines breast density 1 as almost entirely fatty, density 2 as scattered fibroglandular tissue, density 3 as heterogeneously dense tissue and density 4 as extremely dense tissues. It was not intended to serve as a method of measuring breast density percentage, although as per Wolfe’s scale ( ), correlations with this more objective factor do exist ( ). In clinical American and European conditions, the breast density of a given patient is typically evaluated and reported by a radiologist using BI-RADS on the basis of the simultaneous display of two mammograms per breast.

However, one of the difficulties for correctly assessing breast density is that the BI-RADS density scale definitions are rather subjective. A certain interpretational freedom prevents perfect interobserver and even intraobserver reproducibility ( ). On the other hand, numerous pattern recognition and classification techniques have been developed and can be directly applied to this task ( ). This is why different statistical approaches have been explored in the last few years in order to develop an objective classifier of mammograms according to Wolfe or the BI-RADS scale. These techniques have made use of various pattern recognition parameters to statistically describe the whole breast or part of it: fractal dimension ( ), gray level histogram properties ( ), moments ( ), gray level variations matrices ( ), or maximum response filters ( ). These descriptions have been combined with several general classification algorithms: Bayesian classification ( ), linear discriminant analysis (LDA) ( ), nearest neighbor rules ( ), neural networks, and textons ( ).

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Materials and methods

Mammogram Database

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Selection of Regions of Interest

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Statistical Description

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Table 1

Summary of the Texture Analysis Methods and the Corresponding Features

Analysis Method Statistical Features Gray level histogram standard deviation skewness kurtosis balance Gray level co-occurrence matrices energy entropy cmax contrast homogeneity Primitives matrices short primitive emphasis long primitive emphasis gray level uniformity primitive length uniformity Fractal analysis fractal dimension Neighbourhood gray-tone difference matrix

The 18 parameters in this table were computed for two scales as described in the text, making a total of 36 features.

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Definition of Gold Standard From Radiologists’ Ratings

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Classification Algorithms

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Classic Bayesian classifier based on Mahalanobis distance

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ψk(v)=1(2π)NdetKk√⋅exp[−12(v−μk)TK−1k(v−μk)], ψ

(

)

(

)

det

⁡

⋅

exp

⁡

[

−

(

−

)

−

(

−

)

]

where μ k represents the mean vector of class k and K k is the covariance matrix of vectors in class k :

μk=1nk∑vi∈Skvi μ

∑

∈

Kk=1nk−1∑vi∈Sk(vi−μk)T(vi−μk) K

−

∑

∈

(

−

)

(

−

)

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p(k∣∣v)=p(v|k)p(k)p(v)=ψk(v)pa(k)∑kψk(v)pa(k) p

(

)

(

)

(

)

(

)

(

)

(

)

∑

(

)

(

)

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cR=∑4k=1k⋅p(k|v), c

∑

⋅

(

)

with c R being rounded to the nearest integer value to obtain the class attributed to the tested sample vector v.

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pa(k)=14, p

(

)

which represents the most conservative a priori assumption.

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Naïve Bayesian classifier

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ψk(vn,k)=1(2π)N√exp[−12vTn,kvn,k], ψ

(

)

(

)

exp

⁡

[

−

]

where v has been normalized in the same way as training samples of class k to obtain the normalized vector v n,k . The four a posteriori probabilities p( k |v) were then computed with Equation 4 , and the attributed class with Equation 5 .

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12π√∫−∞pj,kne−t2/2dt=pj,kpj,kmax, 1

∫

−

∞

−

max

⁡

where p j,k max is the highest value in the original distribution of feature j in class k .

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Linear discriminant analysis

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ψk(v)=1(2π)NdetKo√⋅exp[−12(v−μk)TK−1o(v−μk)] ψ

(

)

(

)

det

⁡

⋅

exp

⁡

[

−

(

−

)

−

(

−

)

]

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Averaging the individual ROIs classifications

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Reduction of the features’ space size and number of classes

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Evaluation of the performance

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κ=po−pe1−pe, κ

−

where p o is the observed agreement proportion and p e the agreement expected by chance alone. Both are calculated from the confusion matrix and the quadratic weights matrix, and the values of κ stand between −1 and 1 (the minimum value actually depends on p e but is always between −1 and 0). Benchmarks by Landis and Koch ( ) (adjusted by Fleiss et al. [41] for taking the weighting process into account) are commonly used: κ values below 0.4 reflect poor agreement, between 0.4 and 0.6 moderate agreement, while it is substantial between 0.6 and 0.75 and excellent above 0.75. Weighted κ is particularly well adapted to multiclass tasks and when the classes are rather subjectively defined, which is the case for the BI-RADS density scale. The weighting process indeed differentiates between serious (more than one BI-RADS class difference) and slight disagreement (immediate neighbor class choice), and has been chosen as an evaluation parameter in numerous previous works on mammogram classification ( ). Although much more sensitive to differences in class prevalence, the exact agreement proportion was also computed to be able to compare the performance with results from other studies ( ).

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Results

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Table 2

Radiologist Classifications Compared to the Gold Standard Classification Defined in Text

Radiologist # 1 Radiologist # 2 Radiologist # 3 Kappa 0.81 ± 0.07 0.88 ± 0.07 0.91 ± 0.08 Exact agreement 77% 89% 89%

Standard error for weighted κ was computed according to the formula given by Fleiss et al. ( ).

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Table 3

(a) Confusion Matrix Obtained for the Bayesian Classifier Based on Mahalanobis Distance. Results are Averaged Over Mammogram Pairs from the Same View. (b) Same for LDA Classifier

(a) Gold Standard Bayesian Classifier Density 1 Density 2 Density 3 Density 4 Density 1 14 3 0 0 Density 2 5 30 6 1 Density 3 0 14 86 3 Density 4 0 0 10 4

(b) Gold Standard LDA Classifier Density 1 Density 2 Density 3 Density 4 Density 1 16 3 0 0 Density 2 3 31 4 1 Density 3 0 13 95 3 Density 4 0 0 3 4

Table 4

Weighted κ Values Obtained with the Different Averaging Processes and Classifiers. Exact Agreement is Given in Parenthesis

Individual ROI Classification Average per Mammogram (4 ROIs) Average per View Type (8 ROIs) Naïve Bayesian 0.50 ± 0.02 (39%) 0.65 ± 0.05 (55%) 0.68 ± 0.07 (60%) Mahalanobis Bayesian 0.58 ± 0.03 (53%) 0.73 ± 0.05 (69%) 0.78 ± 0.07 (76%) LDA 0.71 ± 0.03 (70%) 0.81 ± 0.05 (80%) 0.83 ± 0.08 (83%)

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Discussion

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Conclusion

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Acknowledgments

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Appendix

Definition of the statistical parameters

Parameters Computed From the Gray Level Histogram

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mean≡x¯=1N∑ixi mean

≡

∑

standarddev.≡σ=1N−1√(∑i(xi−x¯)2)1/2 standard

dev

≡

−

(

∑

(

−

)

skewness=1Nσ3∑i(xi−x¯)3 skewness

∑

(

−

)

kurtosis=1Nσ4∑i(xi−x¯)4−3 kurtosis

∑

(

−

)

−

balance=x70−x¯x¯−x30, balance

−

where the summations are performed over the N pixels of the ROI, and x p is the gray level yielding to p th percentile of the gray level distribution ( ).

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Gray Level Co-occurrence Matrices (GLCM)

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energy(C)=∑i,jC2i,j energy

(

)

∑

entropy(C)=−∑i,jCi,jlogCi,j entropy

(

)

−

∑

log

⁡

cmax(C)=maxi,jCi,j c

max

(

)

max

⁡

contrast(C)=∑i,j|i−j|2Ci,j contrast

(

)

∑

−

homogeneity(C)=∑i,jCi,j1+|i−j| homogeneity

(

)

∑

−

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Primitives Matrix (PM)

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spe=1Btot∑a∑rBa,rr2 spe

∑

lpe=1Btot∑a∑rBa,rr2 lpe

∑

glu=1Btot∑a(∑rBa,r)2 glu

∑

(

∑

)

plu=1Btot∑r(∑aBa,r)2, plu

∑

(

∑

)

where Btot is the sum of the elements of the primitives matrix B : Btot=∑a∑rBa,r. B

∑

. Note that B could be defined for several directions, but we limited our investigations to one ( ), corresponding to a scan of the image along direction [1,0].

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Fractal Dimension

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L(ε)=λε1−D L

(

)

−

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A(ε)=λε2−D A

(

)

−

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Neighborhood Gray-Tone Difference Matrix (NGTDM)

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A¯¯¯xk,l=1W−1[∑dm=−d∑dn=−dxk+m,l+n],(m,n)≠(0,0), A

−

[

∑

−

∑

−

]

(

)

≠

(

)

where d = 3 is the neighbouring size and W = (2d+1) 2 . Denoting { X i } the set of all pixels with value i in the ROI, the i th entry of the NGTDM is given by:

s(i)=∑x∈Xi∣∣i−A¯¯¯x∣∣ s

(

)

∑

∈

−

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coarseness=[ε+∑imaxi=0pis(i)]−1 c

[

∑

max

⁡

(

)

]

−

contrast′=[1Ng(Ng−1)∑imaxi=0∑jmaxj=0pipj(i−j)2]⋅[1n2∑imaxi=0s(i)] contrast

′

[

(

−

)

∑

max

⁡

∑

max

⁡

(

−

)

]

[

∑

max

⁡

(

)

]

complexity=∑imaxi=0∑jmaxj=0|i−j|[pis(i)+pjs(j)]n2(pi+pj),pi>0,pj>0 complexity

∑

max

⁡

∑

max

⁡

−

[

(

)

(

)

]

(

)

strength=∑imaxi=0∑jmaxj=0(pi+pj)(i−j)2ε+∑imaxi=0s(i),pi>0,pj>0, strength

∑

max

⁡

∑

max

⁡

(

)

(

−

)

∑

max

⁡

(

)

where pi=|Xi|/∑imaxi=0|Xi| p

∑

max

⁡

| is the probability of occurrence of gray level i in the ROI, i max the highest gray level and N g the number of different gray levels effectively present in the ROI and ε a small number (10 −12 in our case) to prevent coarseness and strength becoming infinite. The feature representing the contrast given by Equation 30 is called here contrast′ , to make a distinction with the contrast derived from the primitives matrices (see Equation 19 ).

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Semiautomatic Mammographic Parenchymal Patterns Classification Using Multiple Statistical Features

Rationale and Objectives

Materials and Methods

Results

Conclusion

Materials and methods

Mammogram Database

Selection of Regions of Interest

Statistical Description

Definition of Gold Standard From Radiologists’ Ratings

Classification Algorithms

Classic Bayesian classifier based on Mahalanobis distance

Naïve Bayesian classifier

Linear discriminant analysis

Averaging the individual ROIs classifications

Reduction of the features’ space size and number of classes

Evaluation of the performance

Results

Discussion

Conclusion

Acknowledgments

Appendix

Definition of the statistical parameters

Parameters Computed From the Gray Level Histogram

Gray Level Co-occurrence Matrices (GLCM)

Primitives Matrix (PM)

Fractal Dimension

Neighborhood Gray-Tone Difference Matrix (NGTDM)

References

Further Reading

Abstracts of Funded National Institutes of Health Grants (December)

Computer-Aided Diagnosis Scheme for Detection of Lacunar Infarcts on MR Images

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