Quantitative Analysis of Brain Asymmetry by Usingthe Divergence Measure:

Normal-Pathological Brain DiscriminationIhar Volkau, PhD,1 Bhanu Prakash, KN, PhD,1 Anand Ananthasubramaniam, ME,1 Varsha Gupta, PhD,1

Aamer Aziz, PhD, MD,2 Wieslaw L. Nowinski, PhD, DSc1

Rationale and Objectives. The human brain demonstrates approximate bilateral symmetry of anatomy, function, neuro-chemical activity, and electrophysiology. This symmetry reflected in radiological images may be affected by pathology.Hence quantitative analysis of brain symmetry may enable the normal and pathological brain discrimination. We proposea method based on the Jeffreys divergence measure (J-divergence), which attempts to quantify “approximate symmetry”and also aids to classify the brain as bilaterally symmetrical/asymmetrical (normal/abnormal).

Materials and Methods. The dataset included studies of 101 patients (59 without detectable pathologies and 42 with dif-ferent abnormalities). First, the midsagittal plane is computed for the volume data that divides the head into two hemi-spheres. The J-divergence is calculated from the density functions of intensities of both the hemispheres. Statistical analy-sis was conducted to find the best distribution for normal/abnormal datasets.

Results. Statistical tests showed that the lognormal distribution best characterizes the values of the J-divergence for bothnormal and abnormal cases, and the threshold value for the Jeffreys divergence measure to classify the brains with andwithout detectable pathologies is T � 0.007. The threshold value had a sensitivity of 88.1% and specificity of 90.9%.

Conclusion. The proposed method is fast and simple to compute. The high sensitivity and specificity indicate the resultsare encouraging. This method can be used for the initial analysis of data, detection of pathology, classification of datasetas presumably normal/abnormal, and localization of abnormality.

Key Words. Brain asymmetry; Jeffreys divergence measure; abnormality detection.© AUR, 2006

Although normal brains can be considered as approxi-mately bilaterally symmetrical, it possesses asymme-tries that are reflected in anatomy, function, neuro-chemical activity, and electrophysiology. An exhaustivereview on brain asymmetry can be found in Toga andThompson (1). Most pathology affects the brain sym-

Acad Radiol 2006; 13:752–758

1 From the Agency for Science, Technology and Research (A*STAR), Bio-medical Imaging Lab, 30 Blopolis Road, 07-01, Matrix, Singapore, 138671,Singapore; 2 From Regional Imaging, PO Box 5576, Wagga Wagga NSW,Australia, 2650. Received June 20, 2005; accepted January 17, 2006. Ad-dress correspondence to: I.V. e-mail: igor@sbic.a-star.edu.sg

©

AUR, 2006doi:10.1016/j.acra.2006.01.043

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metry/asymmetry and might change the radiologicalresponse of tissues. A lesion, for example, dependingon its nature and imaging conditions is typically hy-pointensive or hyperintensive, which affects the sym-metry of the image. The absence of normal hemi-spheric asymmetries suggests an anomaly in the devel-opment of laterally specialised cerebral systems inschizophrenia, and this may be associated with an ini-tial presentation of nonparanoid psychosis (2). Hence,capturing the brain asymmetry quantitatively might en-able us to distinguish between normal cases and neuro-anatomical abnormalities. But it is necessary first toquantify the term “approximate bilateral symmetry”

and find the corresponding range for normal cases.

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Thirion et al. (3) noted that “the quantification of ab-normal dissimilarity can also be a powerful tool to detectabnormalities.” Their method to analyze the brain asym-metries relies on the estimate of three-dimensional (3D)dissimilarity fields and uses nonrigid matching to obtain a3D significance map. Joshi et al. (4) explored the struc-tural and radiometric asymmetry in brain images by usinga large deformation image warping to register imageswith their reflections across the midsagittal plane. Hop-kins and Rilling’s (5) brain asymmetry measurement isbased on finding the widths from the midline to the lat-eral surface of the brain.

In our study we propose a quantitative global measure ofasymmetry and do not analyze qualitative asymmetries in-herent in the brain. The method evaluates an asymmetry as ascalar value using the Jeffreys divergence measure. Themethod does not require any prior segmentation and imagerealignment. The divergence value is calculated for 3D im-age and can be used for classifying datasets as presumablynormal or abnormal. If the asymmetry is a potential sign ofsome deviation or pathology, detection of the asymmetricalareas can be useful for initial analysis of data and for thefast preliminary localization of the region of interest.

MATERIALS AND METHODS

The dataset contained normal and pathological MRbrain scans. Sixty studies in 59 adults with normal brainmorphology were examined. In order to estimate thethreshold between normality and pathology, 108 studiesin 42 patients with detectable pathology were analyzed(the same patient could have studies of several modalities,e.g., T1WI, T1WI contrast enhanced, and FLAIR). Themajority of patients had brain tumors. Some cases of in-tracranial hemorrhages, multiple metastasis, arachnoidcyst, and hydrocephalous were also considered.

We use the Jeffreys divergence (J-divergence) measure(6) to analyze bilateral symmetry of both parts of the head(to the right and to the left from the midsagittal plane),which includes the cerebral hemispheres and skull. The mid-sagittal plane can be calculated rapidly, for example, by thealgorithm described by Hu and Nowinski (7). In our study,we analyze only the part of head above the base of thebrain, as the part below the level of the paranasal sinuses ishighly asymmetrical by nature. Detection of the starting sliceis done manually. If several studies are available for thesame patient, we use the maximum value of J-divergence

over all the studies as the measure of asymmetry.

Consider a 3D volume image as a union of two parts tothe left and right of the midsagittal plane (MSP). Let thevoxel intensity distribution of these two parts be p � {pi}and q � {qi}, where pi and qi are the probabilities of oc-currence of the voxel with intensity i in the left and rightparts, respectively. These probabilities are derived fromthe histogram as

pi � (No . of voxels of intensity i in the left part) ⁄

(total No . of voxels in the left part)

and

qi � (No . of voxels of intensity i in the right part) ⁄

(total No . of voxels in the right part)

The Jeffreys divergence measure is defined as

J(p, q) � �i

pi log(pi ⁄ qi) � �i

qi log(qi ⁄ pi) (1)

and gives the distance between two distributions p and q.This distance is non-negative, and it can be considered asa measure of the dissimilarity between the intensity distri-butions of each part.

To test the hypothesis of the underlying distribution ofthe J-divergence measure for normal and abnormal data-sets, a probability-probability (P-P) plot (8), Kolmogorov–Smirnov (KS) Lilliefors test (9), and Shapiro-Wilk test(10) were used. The distributions of J-divergence valuesfor both datasets were used to estimate the threshold forseparation of normal and abnormal cases by Bayes’ crite-rion (11). Also, to check the validity of our results, weperformed receiver operating characteristic (ROC) curveanalysis (12,13).

RESULTS

The analyzed dataset includes 59 cases without pathol-ogy (this dataset is denoted by [d�]) and 42 cases withdetectable pathologies [d�]. The J-divergences wereevaluated for each of the cases. The dataset [d�] showsan asymmetric distribution with range of J-divergencevalues equal to (0.00035, 0.0144). The (median, mean) ofthe distribution is equal to (0.0033, 0.004). The dataset[d�] also displays an asymmetric distribution of J-diver-

gences with extreme values equal to (0.003, 0.2256). The

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median and mean value of the sample of pathologicalcases is 0.0251 and 0.0385. The estimation of the crite-rion value of J-divergence was done by two independentmethods. For the analysis we used MATLAB, SYSTAT,and MEDCALC software packages.

Statistical Analysis

Lognormal FitsWe find lognormal function provides the best fit for

the two distributions, [d�] and [d�].The lognormal density is described as

f(x, �, �) �1

�x�2�e�

(ln(x) � �)2

2�2

where � and � represent the shape and the scale of theprobability density function, respectively.

The best-fit parameters (�, �) with 95% confidencebounds for [d�] cases and [d�] cases are (0.7701�0.12

�0.18,�5.7941�0.21

�0.21) and (1.0269�0.18�0.28, �3.7605�0.32

�0.32), respectively.Because the sample size is small, we do not use �2 as agoodness-of-fit measure. Instead, the goodness of the fit isdetermined by using the KS Lilliefors test and Shapiro-Wilktest for normality of data in the log-space using SYSTAT(see Appendix A for a brief summary). Table 1 summarizesthe goodness-of-fit test results. The fits are accepted becausetwo-tailed Lilliefors probability and P-value are �.05 (Table 1).

The P-P plot was used to define the distribution for thevalues of J-divergence of [d�] and [d�] cases. A P-Pplot compares the empirical cumulative distribution func-tion of a variable with the expected cumulative distribu-tion function (for our case, lognormal distribution). InFigure 1, top (bottom) box represents the P-P plot of[d�] ([d�]) cases. The points cluster close to the line ofunit slope indicates that lognormal is a good fit to thedistribution of J-divergences.

The best-fit probability density functions (pdfs) are

Table 1Statistics of Goodness-of-fit Tests

Test StatisticWithout

PathologyWith

Pathology

Kolmogorov-Smirnov (Lillieforsprobability, 2-tailed) .096 (.22) .085 (.61)

Shapiro-Wilk (P-value) .97 (.30) .99 (.94)

shown in Figure 2. In the figure, T is the point of inter-

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section of two pdfs, corresponding to [d�] and [d�] [sayP[d�](j) and P[d�](j)]. At any value j less than T, P[d�](j) isgreater than P[d�](j). In order to minimize the error onclassification, we use Bayes’ classification rule (11). Ac-cordingly, for any j � T, the patient is classified as with-out disease as P[d�](j) �j is always greater than P[d�](j)�j. The error on this decision is equal to P[d�](j) �j. Forj � T, vice versa is true. Thus, the point of intersectionof two pdfs (i.e., T � 0.008) corresponds to the total min-imum error on classification (which is the area of overlapof two pdfs) as the smaller of two probabilities is deter-mined as error.

The error region on threshold is estimated by incorpo-rating the errors because of fit on (�, �). The 95% confi-

Figure 1. Probability-probability plots for nonpathological cases(top) and pathological cases (bottom).

dence bounds on (�, �) are used to numerically estimate

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the point of intersection of both the distributions. Theextreme values of intersection are obtained as (0.006,0.011). These values represent the error region around thethreshold.

Receiver Operating Characteristic Curve AnalysisThe second method to study J-divergence as a tool to

classify brain symmetries is ROC curve analysis. We de-fine the few statistical terms we use in this analysis:

● Sensitivity: probability that the test will be positivewhen the disease is present (also called true-positiverate).

● False-positive rate (FPR): probability of a positive re-sult given the absence of a disease.

● Specificity: probability that a test result will be negativewhen the disease is not present (also called true-nega-tive rate).

● False-negative rate (FNR): probability of a negativetest given the presence of a disease.

● Positive likelihood ratio (LR�): ratio of true-positiverate to false-positive rate.

● Negative likelihood ratio (LR�): ratio of false-negativerate to true-negative rate.

An ROC curve is a graphical representation of the trade off

Figure 2. Threshold of classification T � .008 as obtained fromBayes’ criterion. The 95% lower and upper confidence bounds on(�, �) obtained from the fit are used to numerically estimate allthe possible points of intersection between the pdfs. The extremeintersection points obtained as (0.006, 0.0112), represent the up-per and lower limits of T (the dotted lines).

between the sensitivity and specificity for every possible

threshold of classification (Fig. 3). The ROC curve analysisis performed using the software package MEDCALC. Thearea under the ROC curve is 0.94 (95% confidencebounds equal to 0.88 and 0.981) with the standard errorequal to 0.026 (12). It is to be noted that the standarderror is dependent on the sample size. Since the area isclose to 1 (even with 95% confidence bounds beinggreater than 0.5), the J-divergence can be regarded as agood diagnostic test. The best criterion value for this testis determined such that the FNR and FPR are minimized.The optimum threshold is T � 0.007, as it corresponds tothe minimum false-positive (negative) cases. This valuecorresponds to LR� � 9.69 and LR� � 0.13. (Also seeTable 2.) The threshold is considered optimum if LR� �10 and LR� � 0.1). For T � 0.007 as a threshold, sensi-tivity of the test is 88.1% (95% confidence interval,74.4% to 96.0%) and specificity is 90.9% ( 95% confi-dence interval, 80.0% to 96.9%). The horizontal line at T� .007 as shown in Figure 4 indeed minimizes the false-positives and false-negatives.

DISCUSSION

In our study, we have analyzed the goodness of J-diver-gences as a method to classify symmetries in cases of patho-logical and nonpathological brains. The use of J-divergence

Figure 3. The ROC curve for the dataset.

measure is based on the fact that brains structures are “ap-

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proximately” symmetrical and a calculated distance betweentwo distributions can be considered as a measure of grossasymmetry of hemispheres. The J-divergence measure waschosen over other measures because it is a relative measure,and includes volume specific elements like noise, inhomoge-neity, etc. of both hemisphere in the calculation.

We use two independent methods to determine thecriterion of classification using J-divergences. In the firstmethod, we find that the lognormal functions are a goodfit to distributions of pathological and nonpathologicalcases. The KS Lilliefors test and Shapiro-Wilk test statis-tics are summarized in Table 1. Because the probabilitiesin each case are �.05, the normality of data in the log-space cannot be rejected. The P-P plot corresponding tobest fit lognormal functions with (�, �) for [d�] equal to(0.7701�0.12

�0.18, �5.7941�0.21�0.21) and for [d�] equal to

Table 2Sensitivity, Specificity, LR�, and LR� as a fu(J-divergence value)

Criterion(J-divergence value)

Sensitivity(95% CI)

0.0029 100.0 (91.5–100)0.0039 95.2 (83.8–99.3)0.0058 92.9 (90.5–98.4)0.0061 88.1 (74.4–96.0)

*0.007 88.1 (74.4–96.0)0.0071 85.7 (71.4–94.5)0.0081 85.7 (71.4–94.5)0.0091 81.0 (65.9–91.4)0.0127 66.7 (50.5–80.4)0.0134 64.3 (48.0–78.4)

The threshold of 0.007 is optimum in the neighbmum number of false-positive and false-negative

Figure 4. The horizontal line at T � 0.007 indicates the cutoff withbest separation (minimal false-negative and minimal false-positive).

(1.0269�0.18�0.28, �3.7605�0.32

�0.32) are shown in Figure 1. The

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clustering of points around line of slope 1 indicates good-ness of the fit.

The intersection point of the best lognormal fits isT � 0.008 (Fig. 2). This point has been considered as athreshold in accordance with the Bayes criterion as itminimizes the total error of classification. The error regionis found using errors on fitted parameters (�, �) whichplaces the lower bound on the criterion value to .0061.

By ROC curve analysis, we found T � .007 as a crite-rion of classification that minimizes the number of false-positives and false-negatives. It corresponds to sensitivityand specificity of 88.1 (95% confidence interval, 74.4 to96.0) and 90.9 (80.0 to 96.9), respectively. Figure 4 indi-cates the minimum number of false-positive and false-negative cases corresponding to criterion of T � 0.007.The area under the curve (Fig. 3) is 0.94 � 0.26 whichindicates that J-divergence is a good test (a value of 0.5would have indicated a tossing of a coin�like method).

It is to be noted from Table 2 that the threshold rangeof 0.007–0.0081 corresponds to LR� and LR� close to�10 and �0.1, respectively. This range also lies in theerror region indicated by the method of lognormal fits(Fig. 2). However, the minimum number of false-positivesand false-negatives correspond to T � 0.007 (Fig. 4). Basedon aforementioned information the threshold value is set toT � 0.007.

Eighty-six percent of brains without detectable abnor-malities had J-divergence value below 0.007, 99% of nor-mal cases corresponded to the values of J-divergencegreater than 0.0006, and approximately 85% of all the

n of Different Values of Criterion

Specificity(95% CI) LR� LR�

45.4 (32.0–59.4) 1.83 0.058.2 (44.1–71.3) 2.28 0.0880.0 (67.0–89.6) 4.64 0.0980.0 (67.0–89.6) 4.40 0.1590.9 (80.0–96.9) 9.69 0.1390.9 (90.0–96.9) 9.43 0.1692.7 (82.4–97.9) 11.79 0.1592.7 (82.4–97.9) 11.13 0.2196.4 (87.4–99.5) 18.33 0.3598.2 (90.2–99.7) 35.36 0.36

g values of criterion as it corresponds to mini-(indicated by LR� �10 and LR� �0.1).

nctio

ourin

normal cases were in the interval [0.0006, 0.007]. It gives

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a way to classify the brain under analysis as “approxi-mately” symmetrical if it is in the mentioned interval.

The accuracy of the result is influenced by many fac-tors. A smaller sample size will lead to a larger error dueto fitting the distribution curve as well as a larger error onthe area under the ROC curve. Another source of error isdata quality. Resolution of the image, artefacts, noise, andinhomogeneity affect the accuracy of classification be-cause they change the distribution of intensities and, inturn, the value of J-divergence. In some protocols, the

Figure 5. Plot of J-divergence: x is gray-scale intensity; y is itscontribution to the J-divergence. The filled area (rightmost peak)corresponds to the highlighted part of the 3D volume in Figure 6.

Figure 6. Three-dimensional view of asymmetries for the case oftwo tumors. The lines show the correspondence between pathol-ogy in two and three dimensions.

asymmetries caused by pathologies are not highlighted. In

such cases, the global symmetry will not be affected, andclassification could be erroneous. Also, in the case if apatient who did not have any detectable abnormality butfor whom the J-divergence measure is above the thresholdvalue (because of, for example, curved interhemisphericfissure), the classification may fail.

Some of the limitations of the proposed method are thefollowing:

● The manual detection of the slices for analysis abovethe base of brain

● The method considers not only asymmetries from pa-thologies and normal tissues substituted by pathologiesbut also from the skull, eyes, cerebral vasculature, andpart of the nose (Figs. 6 and 7).

J-divergence measure (1) is an additive one where we canseparate the contribution from different intensities. If weanalyze the plot of J-divergence (Fig. 5), we can highlightthe parts of the 3D volume that correspond to the intensi-ties making up every peak of the plot (Fig. 6) one by oneand can check for the localized asymmetry of interest. Acase of visualization of asymmetry for a smaller tumor isshown in Figure 7. The system based on the above-statedconcept was demonstrated at RSNA 2005, infoRAD (14).

CONCLUSIONS

We have proposed a method for quantifying the brain

Figure 7. Three-dimensional view of asymmetries for the case oftumor. The lines show the correspondence between pathology intwo and three dimensions.

asymmetry based on calculation of the J-divergence mea-

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sure. The datasets that have value of J-divergence lesserthan the threshold T � 0.007 can be classified as “ap-proximately” symmetrical. The results of the analysis canbe used for classifying the dataset as presumably normalor pathological, to visualize the asymmetries, and, hence,to aid in localization of abnormality.

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APPENDIX A

The KS Lilliefors test is a modified KS test. First themean and variance from given data are calculated. KSstatistic is the maximum discrepancy between the empiri-cal distribution function and cumulative distribution func-tion of the normal distribution with the estimated meanand variance. The Lilliefors test is used to test the nullhypothesis that data come from a normally distributedpopulation, when the null hypothesis does not specifywhich normal distribution, i.e., does not specify the ex-pected value and variance. The value of probability �.05implies that the data sample is normally distributed.

The Shapiro-Wilk test is a standard test for normalitywhen the sample size is between 3 and 5000. It tests thenull hypothesis that the sample is taken from a normaldistribution. The P-value corresponding to the Shapiro-Wilk statistics �0.5 implies that the data are normally

distributed.