Evaluation of Fourier Transform Coefficients for The Diagnosis of Rheumatoid Arthritis From Diffuse Optical Tomography Images

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1 Evaluation of Fourier Transform Coefficients for The Diagnosis of Rheumatoid Arthritis From Diffuse Optical Tomography Images Ludguier D. Montejo *a, Jingfei Jia a, Hyun K. Kim b, Andreas H. Hielscher *a,b,c a Department of Biomedical Engineering, Columbia University, New York, NY 10027, USA b Department of Radiology, Columbia University Medical Center, New York, NY 10032, USA c Department of Electrical Engineering, Columbia University, New York, NY 10027, USA ABSTRACT We apply the Fourier Transform to absorption and scattering coefficient images of proximal interphalangeal (PIP) joints and evaluate the performance of these coefficients as classifiers using receiver operator characteristic (ROC) curve analysis. We find 25 features that yield a Youden index over 0.7, 3 features that yield a Youden index over 0.8, and 1 feature that yields a Youden index over 0.9 (90.0% sensitivity and 100% specificity). In general, scattering coefficient images yield better one-dimensional classifiers compared to absorption coefficient images. Using features derived from scattering coefficient images we obtain an average Youden index of 0.58 ± 0.16, and an average Youden index of 0.45 ± 0.15 when using features from absorption coefficient images. Keywords: Diffuse Optical Tomography, Computer-Aided Diagnosis, Rheumatoid Arthritis 1. INTRODUCTION Rheumatoid arthritis (RA) is an autoimmune disorder that affects % of adults in industrialized countries, with up to 50 per 100,000 new cases each year [1]. Uncontrolled RA can lead to significant joint damage, characterized by the presence of synovitis, effusion, and erosion of the affected joint. These symptoms can lead to lower quality of life due to decreased joint mobility. Left untreated, RA can also lead to physical disabilities and cardiovascular disease, among other complications. Diffuse optical tomography (DOT) is an increasingly promising imaging modality for the diagnosis of RA [2, 3, 4, 5, 6]. A recent publication by Hielscher et al. showed that retrospective analysis of clinical data (99 joints affected with RA and 120 healthy control joints) resulted in clinical sensitivities and specificities up to 85.0% for the diagnosis of RA [2]. In that study, a frequency domain DOT (FD-DOT) imaging system was used to image each joint included in the study. Each joint was imaged at 0, 300, and 600 MHz. It was shown that the 600 MHz data resulted in optimal diagnostic results when compared to 0 and 300 MHz data. The process for diagnosing each joint as affected or not affected by RA was a two step process as follows: (1) Extract heuristic image features from each of the absorption and scattering maps corresponding to each imaged joint. (2) Use each of the heuristic image features to classify each imaged joint as affected or not affected with RA. In step 1, the extracted image features included the maximum and minimum absorption values. A third feature was the variation among the absorption values inside the imaged joint. A fourth feature was the ratio of maximum to minimum absorption coefficient values. The same procedure was applied to the scattering coefficient images, resulting in 4 additional features. Overall, 8 features were computed. Linear discriminant analysis (LDA) was used to classify each joint as affected or not affected by RA using only the extracted features. In this work we explore the ability to diagnose RA using only single features obtained from the leading coefficients of the Fourier Transform applied to each of the three-dimensional absorption and scattering coefficient images. We use a modified version of the leave-one-out cross validation (LOOCV) procedure and receiver operator characteristic (ROC) curves to validate our results. In Section 2 we present details on the feature extraction and classification method. In Section 3 we present classification results. Section 4 completes this paper with a conclusion and discussion. * Corresponding authors: ldm2106@columbia.edu (L.D. Montejo) and ahh2004@columbia.edu (A.H. Hielscher). Optical Tomography and Spectroscopy of Tissue X, edited by Bruce J. Tromberg, Arjun G. Yodh, Eva Marie Sevick-Muraca, Proc. of SPIE Vol. 8578, 85781N 2013 SPIE CCC code: /13/$18 doi: / Proc. of SPIE Vol N-1

2 2. METHODS 2.1 Clinical Data The clinical data used in this study consists of 33 subjects with RA and 20 healthy control subjects. PIP joints II-IV were imaged on the dominant hand of each of the subjects with RA (resulting in 99 imaged joints), while PIP joints II-IV were imaged in both hands of the control subjects (resulting in 120 imaged joints) [2]. Imaging data at a modulation frequency of 600 MHz is used in this work. For each joint we reconstruct the absorption and scattering coefficient distribution inside the finger joint using the equation of radiative transfer (ERT) as the light propagation model [7]. (a) Absorption coefficient Absorption Coefficient [cm 1 ] (b) Scattering coefficient Scattering Coefficient [cm 1 ] Figure 1 : Cross sections of the absorption (a) and scattering (b) coefficient maps inside a typical PIP joint of a healthy subject. 2.2 Feature Extraction The absorption and scattering coefficient maps of each joint are processed to extract features that capture the inherent spatial properties of the distributions. Applying the Fast Fourier Transform (FFT) to each image parameterizes the spatial distribution of the underlying optical property map. This is equivalent to reducing the dimensionality of the reconstruction data. Each FFT coefficient can be used as a heuristic image feature for classification purposes, thus replacing the role that the maximum, minimum, variance, and ratio features played in our previous analysis. The level of detail contained by the first 5 FFT coefficients leads to an accurate representation of the original image (Figure 2). Information contained by the first 5 leading FFT coefficients requires the storing of 63 real values obtained by keeping only the absolute value of the complex-valued FFT coefficients, and exploiting frequency space symmetries to reduce the number of coefficients from 125. Compared to these 63 features, the original image is defined by over 5,000 values (i.e. the optical property μ defined at each node of the unstructured tetrahedral mesh used during the reconstruction procedure). Furthermore, each PIP joint is defined on distinct meshes rendering it impossible to directly compare them. Here, however, the FFT coefficients from each finger are defined on the same coordinate system, thus allowing for direct comparison between different PIP joints. Proc. of SPIE Vol N-2

3 Absorption Coefficient [cm 1 ] 0.21 (a) Absorption coefficient Scattering Coefficient [cm 1 ] 7.0 (b) Scattering coefficient Figure 2 : Cross sections through the absorption (a) and scattering (b) coefficient maps inside a PIP joint. The original data in (a) and (b) is presented in the top rows, while the reconstructions of the same data using only the first 5 leading FFT coefficients obtained from the application of the FFT to the original image are presented for comparison on the bottom rows of (a) and (b). The original data and model data are highly correlated, with correlation factors of 0.85 and 0.86, respectively. As an additional feature we define the error between the original image and the model image (obtained from using only the 63 extracted FFT coefficients). For simplicity we number the features from 1 to 64 and specify whether they are derived from absorption or scattering images. Feature 1 corresponds to the error between the original image and the model image. Features 2 through 64 are the 63 FFT features arranged based on distance from the frequency space origin. Features with equal distances from the origin are sorted based on axis specific distances, where preference is given to the z-axis, then the y-axis, and then the x-axis. The extracted features are arranged into a single vector (with 64 elements) for each image, resulting in a matrix, where each column corresponds to an individual image feature and each row corresponds to an individual sample (or PIP joint image), as shown in Figure 3. There is one such matrix for absorption coefficient features and a separate matrix of equal dimensions for features derived from scattering coefficient images Figure 3 : Feature matrix with 64 features (columns) and 219 samples (rows) Proc. of SPIE Vol N-3

4 2.3 Classification With ROC Curves The resulting matrix is an input to the receiver operator characteristic (ROC) curve algorithm, which performs onedimensional classification on the data set. Using ROC curve analysis we identify features that are individually the best classifiers for diagnosing RA by determining the threshold value x!" that best separates the two groups (i.e. one group with RA and one without RA). The best threshold is the feature value x!" that maximizes the Youden index (Y), which is defined as Y = Se + Sp 1. Diagnostic sensitivity and specificity are defined below [8]. Se = Sp = TP TP + FN TN TN + FP In the above equations TP, FP, TN, and FN denote the assigned diagnosis to each PIP joint and are defined as true positives (TP), false positives (FP), true negatives (TN), or false negatives (FN). A feature that perfectly separates the affected from the healthy joints yields Y = 1.0, while a feature that completely fails to separate the two classes yields Y = 0.0. To avoid over-fitting our data we use a modified version of the leave-one-out cross-validation (LOOCV) procedure. The standard LOOCV procedure requires using N 1 (where N = 219, the total number of PIP joints) to find x!" from the ROC curve analysis, while the remaining data point is used to test the optimal threshold x!". The test sample is classified as TP, FP, TN, or FN. This process is repeated for each sample and an overall sensitivity and specificity is computed. In contrast to the standard LOOCV procedure, where one sample (finger) is used for testing while the remaining samples are used for training, we leave out all samples (fingers) belonging to one single subject (3 for subjects with RA and 6 for subjects without RA). The remaining samples are used for training the ROC curve algorithm and for finding x!". In the testing phase, each of the testing samples (3 for subjects with RA and 6 for subjects without RA) are classified as TP, FP, TN, or FN. This process is repeated for each of the 53 distinct subjects and called an iteration of the LOOCV procedure. The sensitivity and specific are computed for each feature by summing all TP, FP, TN, and FN from each of the 53 individual LOOCV iterations. In Section 3 we report the sensitivity, specificity, and Youden index computed for each of the FFT coefficients extracted from the absorption and scattering images. 3. RESULTS Results from ROC curve analysis of the 3D FFT coefficients extracted from the absorption and scattering coefficient reconstructions are presented in Figure 4. It is clear that features from the scattering coefficient images generally yield better classification results. The average sensitivity and specificity obtained using scattering coefficient features are 76.4 ± 12.3% and 81.1 ± 12.4%, respectively. Meanwhile, features from absorption coefficient images yield averages of 67.9 ± 12.8% and 78.0 ± 15.0% for sensitivity and specificity, respectively. Together, we can compare the Youden index of each set of data. The scattering coefficient yields an average Youden index of 0.58 ± 0.16, while the absorption coefficient yields an average Youden index of 0.45 ± There are 25 features with Youden indices over 0.7, with 5 features arising from absorption images and 20 features from scattering images. There are 3 features with Youden indices over 0.8, including 1 feature from absorption images (feature 47) and 2 features from scattering images (features 16 and 56). There is 1 scattering coefficient feature (feature 16) with a Youden index above 0.9. To highlight some specific results, there are 13 features with sensitivity above 90.0%, with 4 features coming from absorption images and the other 9 from scattering images. Meanwhile, there are 28 features with specificity above 90.0%, with 13 features coming from absorption images and the other 15 from scattering images. There are no Proc. of SPIE Vol N-4

5 absorption coefficient features with both sensitivity and specificity above 90.0%. There is, however, 1 scattering coefficient feature with both sensitivity and specificity above 90%. Feature 16 from the scattering coefficient images has 90.1% sensitivity and 100.0% specificity, resulting in a Youden index of Youden Index Se Sp Y Feature number from the absorption coefficient 3D FFT Youden Index Se Sp Y Feature number from the scattering coefficient 3D FFT Figure 4 : Sensitivity, specificity, and Youden index from ROC curve analysis of 3D FFT coefficients from absorption (top) and scattering (bottom) PIP joint images. 4. CONCLUSION We find that using coefficients obtained from the application of the Fourier Transform to the absorption and scattering images is a good method for identifying features that are effective at differentiating between PIP joints of subjects with and without RA. Using ROC analysis and LOOCV we determine that there are multiple FFT coefficients that perform well when used as one-dimensional classifiers. Furthermore, this strategy proves useful in its ability to reduce the dimensionality of the reconstruction data and allows us to directly compare PIP joints with one other, as the FFT coefficients are all defined on the same frequency space. This is important, as it is impossible to directly compare PIP joints using only the reconstruction data defined on unstructured tetrahedral meshes unique to each individual PIP joint. Using each of the FFT coefficients as image features in ROC curve analysis shows that there are 25 features that yield a Youden index over 0.7, 3 features that yield a Youden index over 0.8, and 1 feature that yields a Youden index over 0.9. Scattering coefficient images yield better one-dimensional classifiers compared to absorption coefficient images. The scattering coefficient image features yield an average Youden index of 0.58 ± 0.16, while the absorption coefficient features result in an average Youden index of 0.45 ± FFT coefficients are strong one-dimensional classifiers as the FFT is efficient at capturing the variations of spatially distributed parameters. As we have previously shown [2], the internal variations of optical properties within a finger joint can be strong indicators on the involvement of said joint. It is therefore not surprising that high sensitivities and specificities can be achieved with FFT coefficients. Proc. of SPIE Vol N-5

6 In our future work we will explore the utility of these FFT coefficients in a multi-dimensional setting, where more advanced image classification algorithms will be used to classify each joint as affected or not affected with RA. These algorithms include linear discriminate analysis, self-organizing maps, and support vector machines. Our previous work indicates that multidimensional classification improves diagnostic sensitivity and specificity, and that support-vector machines is a generally superior classification algorithm for the diagnosis of RA with DOT. ACKNOWLEDGMENTS This work was supported in part by a research grant from the National Institute of Arthritis and Musculoskeletal and Skin Diseases (NIAMS - 2R01 AR46255), which is part of the National Institutes of Health (NIH). Furthermore, L.D. Montejo was partially supported by a NIAMS training grant on Multidisciplinary Engineering Training in Musculoskeletal Research (5 T32 AR ). REFERENCES 1. Scott, D.L., Wolfe, F., and Huizinga, T.W.J., Rheumatoid arthritis, Lancet 376, (2010). 2. Hieslcher, A.H., Kim, H.K., Montejo, L.D., Blaschke, S., Netz, U.J., Zwaka, P.A., Illing, G., Müller, G.A. and Beuthan, J., Frequency-domain optical tomographic imaging of arthritic finger joints, IEEE Transactions on Medical Imaging 30(10), (2011). 3. Klose, C.D., Klose, A.D., Netz, U., Beuthan, J. and Hielscher, A.H., Multi-parameter classifications of optical tomographic images, J. Bio. Optics 13(5), (2008). 4. Montejo, L.D., Montejo, J.D., Kim, H.K., Netz, U.J., Klose, C.D., Blaschke, S., Zwaka, P.A., Müller, G.A., Beuthan, J. and Hielscher, A.H., Comparison of Classification Methods for Detection of Rheumatoid Arthritis with Optical Tomography, in Biomedical Optics, OSA Technical Digest, paper BWF2 (2010). 5. Klose, C.D., Klose, A.D., Netz, U.J., Beuthan, J. and Hielscher, A.H., Optical tomographic detection of rheumatoid arthritis with computer-aided classification schemes, Proc. SPIE 7171, 71710C (2009). 6. Klose, C.D., Klose, A.D., Netz, U.J., Scheel, A., Beuthan, J. and Hielscher, A.H., Computer-aided interpretation approach for opticaltomographic images, J. Bio. Opt. 15(6), (2011). 7. Kim, H.K and Hielscher, A.H., A PDE-constrained SQP algorithm for optical tomography based on the frequencydomain equation of radiative transfer, Inverse Problems 25, (2009). 8. Zweig, M.H. and Campbell, G., Receiver-operating characteristic (ROC) plots: a fundamental evaluation tool in clinical medicine, Clin. Chem. 39, (1993). Proc. of SPIE Vol N-6