Edge-Based Partial Volume Averaging Estimation for ... - IEEE Xplore

32nd Annual International Conference of the IEEE EMBS Buenos Aires, Argentina, August 31 - September 4, 2010

Edge-Based Partial Volume Averaging Estimation for FLAIR MRI with White Matter Lesions April Khademi

Anastasios Venetsanopoulos

Alan R. Moody

Elec. and Comp. Eng. Dept. University of Toronto, Canada. Email: [email protected]

Elec. and Comp. Eng. Dept. Ryerson University, Canada. Email: [email protected]

Dept. of Medical Imaging. Sunnybrook Health Sciences Centre, Canada. Email: [email protected]

Abstract—Through the combination of intensity and fuzzy edge strength measures, a new partial volume averaging (PVA) quantification technique for FLAIR MRI with white matter lesions (WML) is developed. It is focused on an edge-based approach, which “probes” for PVA voxels via a global estimate for the change in the proportion of tissues α . This estimate is refined according to a probabilistic threshold, and the result is decoded to find the proportion of tissues fraction α - the percentage of one tissue found in a mixture voxel. The results from several images are shown illustrating how the technique may be used to segment PVA and pure tissue classes. The result is a non-model based approach to the detection and quantification of PVA.

I. I NTRODUCTION Magnetic Resonance Images (MRI) possess superior soft tissue contrast, which make them ideal for the non-invasive investigation of neurodegenerative diseases [1]. For example, white matter lesions (WML) were found to be both associated with ischemic brain disease and positively correlated with future stroke through the study of Fluid Attenuation Inversion Recovery (FLAIR) MRI [2]. Radiological detection and investigation of these WML were possible due the unique intensity characteristics of FLAIR MRI: ischemic damage such as WML appear as hyperintense regions, normal cerebral tissues (white/gray matter) are of lower and like intensities and the cerebrospinal fluid signal is nulled producing images that offer superior discrimination of ischemic pathology [1]. Despite the trend towards automating the analysis of WML, only few works solely focus on FLAIR MRIs (see [3] for a thresholdbased technique). Majority of the literature focuses on a multi-spectral approach (the use of T1, T2, PD and FLAIR MRI simultaneously [4] [5]). Since FLAIRs localize WML on their own [1] [6], development of processors for this single modality would be of great value since it reduces the need for protracted multiparametric scanning, eliminates the need for registration (no registration errors) and requires less processing time and memory requirements. One of the challenges faced for the automated analysis of FLAIR MRI with WML is that preprocessing techniques for neuro MRI are only very well developed for other sequences such as T1- and T2-weighted MRIs (see [7] [8]). Unfortunately, these model-based methods do not explicitly apply to FLAIR with WML since: 1) pathology modifies the intensity distribution in a manner that is difficult to model, 2) white matter (WM) and gray matter (GM) are often barely differentiable (single mode in the histogram [3]) and 3) neuro MRI are often reconstructed with multi-coil technologies which generate non-Gaussian noise properties [9], causing techniques that rely on the assumption of normality to be inaccurate [7]. To this end, the current work aims to further the research and development of preprocessors for FLAIR MRI with WML. Specifically, the focus is on the automated detection and quantification of partial volume averaging (PVA) artifacts, a type of image degradation present in all MRI which affects segmentation and classification performance.

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The method is tuned specifically to FLAIR MRI with WML and is derived based on inherent image properties, rather than specific statistical models. An edge-based approach is utilized to analyze PVA content and a global edge map derived from the combination of fuzzy edge and intensity information is used to robustly represent the PVA voxels. Based on probabilities, the edge map is thresholded to separate noise and prominent edges. The significant edges correspond to PVA voxels and these voxels are then used to decode the proportion of tissue parameter (quantifies the percentage of each tissue present in a mixture voxel). II. PARTIAL VOLUME AVERAGING In accordance to MR physics, PVA generates an image intensity which is linearly dependant on the proportion of each tissue in the voxel [10]. In neuro studies, there are only two tissue types present in mixture voxels [10]. Therefore, a PVA voxel’s intensity at a spatial coordinate x = (x1 , x2 ) ∈ Z 2 , is determined by the proportion of tissue j (Tj ) present at x: Yjk (x) = α(x) ∗ Yj (x) + (1 − α(x)) ∗ Yk (x),

(1)

where Yjk (x) is the resultant intensity of a PVA voxel, Yj (x) is the pixel value drawn from the distribution of Tj : Yj ∼ pj (y), Yk (x) is the intensity value selected from the intensity distribution of the second tissue (Tk ): Yk ∼ pk (y) and α is the proportion of Tj present in the PVA voxel where α ∈ [0, 1]. Traditional techniques to model PVA in T1- and T2-weighted MRI are focused around the construction of mixture models based on assumed distributions for p(y|α) and p(α) [7] [8] [10]. Bayes theorem is applied to find the statistical distribution of the proportion of tissues α, for a given voxel intensity y - p(α|y). A global estimate for the proportion of tissues α(y) is then computed from this PDF (see [10] for example). This global feature, α(y), quantifies the proportion of one tissue present in a voxel of intensity y and can be used to determine the volume of lesions with subvoxel accuracy. As stated, FLAIR images possess unique intensity characteristics which do not directly apply to traditional techniques. Consequently, the following work proposes a non-statistical-based method for PVA detection and quantification that is focused on an edge-based paradigm. The following sections discuss the proposed methods. III. M ETHODOLOGY In line with other PVA modeling techniques, the current technique estimates the global tissue mixing parameter α(y). However, instead of using models to estimate it directly, a data-driven estimate for the change of the proportion of tissues α (y) is first found via edge-based analysis. This edge map (α ), which is shown to be a function of edge content in boundaries that separates two tissues (PVA region), is then thresholded and integrated to decode α(y). This process results in the automatic detection and quantification of PVA in FLAIR with WML.

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A. Ideal Signal Model

the magnitude of the gradient, g, is estimated

Consider an ideal tissue model (no noise or other artifacts are present aside from PVA). Tissue intensities are simulated as constant quantities instead of values sampled from a random distribution and α can be approximated by deterministic functions (i.e. linear, nonlinear). With this no-noise model, the PVA governance equation of Equation 1 becomes Yjk (x) = α(x) ∗ Ij + (1 − α(x)) ∗ Ik ,

Y12 (x)

=

α12 (x) ∗ I1 + (1 − α12 (x)) ∗ I2 ,

(3)

Y23 (x)

=

α23 (x) ∗ I2 + (1 − α23 (x)) ∗ I3 ,

(4)

where I1 > I2 > I3 ≥ 0, Y12 (x) and Y23 (x) are the intensities of PVA voxels in the boundaries between T1 and T2 , and T2 and T3 tissue classes, respectively. The proportionality constant α is also class dependent. With the no-noise signal model, since the intensities of the two mixing tissues are constant, the final output intensity of the PVA voxel Yjk (x) is a function of the proportion of tissues αjk (x) (i.e. Yjk is bounded by Ik and Ij or Ik ≤ Yij (x) ≤ Ij ). These bounds indicate that PVA creates a lower intensity, fuzzy band/halo around the image objects such WML. This PVA band is created by a high contrast edge and consequently, this work takes an edge-based approach to model the edge content in PVA regions. The edge content in a PVA region is found by taking the gradient of Equations 3-4 to get  Y23

=

 α12 ∗ (I1 − I2 ) ,

(5)

=

 α23

(6)

∗ (I2 − I3 ) .

Solving for the change in the proportion of tissues (α ) yields  α12

=

 α23

=

 Y12 , I1 − I2  Y23 , I2 − I3

∂x1

(7) (8)

where each PVA parameter αjk is a normalized, class specific representation of edge information in PVA regions. It is a normalized  is Ij − Ik representation because the largest possible value of Yjk (step-edge or maximum intensity change in one pixel step) and the  ≤ 1. It is minimum is 0 in a constant region, resulting in 0 ≤ αjk class specific since the normalizing constant depends on the classes that are mixing and α12 ≤ α23 on average. As shown, the transition between the two tissue types occur in the PVA region, thus creating an ‘edge’ between the two objects. A datadriven edge-based estimate for α is used in this work to quantify this effect and is detailed in the following subsections.

ρk ρk

= =

Prob(g ≤ gk ),

(10)

pG (g),

(11)

gk  g=0

where ρk ∈ [0, 1]. It is a ‘fuzzyfication” of the edge information because it determines the membership of a pixel to the edge class. Majority of the edge information is comprised of a small gradient value created by low contrast edges, such as noise and white matter tracts. Since the edge map is cumulative, the final mapping ρk assigns large and similar values to high gradient values (significant edges), despite them occurring over a wide range of g and with few occurrences. This classifies both the WML and brain boundaries as having a significant edge presence. Moreover, this fuzzy edge measure retains unique features which are desired for the estimate of α : 1) it is a normalized measure 0 ≤ ρk ≤ 1 and 2) it is class specific since on average ρk12 ≤ ρk23 . Although such a nonlinear mapping function has the effect of separating relevant from irrelevant edges (noise), the measure is highly localized and is corrupt by noise. To combat noise and to robustly represent the change in proportion of tissues, the current work utilizes a global estimate of α (y). C. Global Edge Description To transform the localized fuzzy edge measure into a global one, several steps are performed. Initially, edge and intensity information are coupled through the conditional PDF of fuzzy edge strength ρk ≈ α, for a particular intensity y as in pP |Y (P = ρk |Y = y) =

# pixels with P = ρk |Y = y . # pixels with Y = y

(12)

This PDF quantifies the distribution of the edge information ρk ≈ α for a specific gray level value y. The interior of the brain and WML classes are defined by low and high intensity values, respectively. Since these regions are flat (no large edges), there is clustering in the PDF in low edge values over these intensity ranges (classes). Across anatomical boundaries (PVA), high edge values dominate the PDF for these class specific (PVA) intensity values. This conditional PDF, which couples intensity information with fuzzy edge information, is used to classify voxels into one of two categories: 1) PVA region (mixture tissue) or 2) non-PVA region (pure tissue). In order to perform this classification, p (ρk |y), the distribution of the edge information for a given intensity, is approximated using a kernel density estimator. Kernel density estimators use a series of kernels to smoothly approximate the shape of a histogram for some random data. In the current context, with a sample of fuzzy edge values ρk (x1 ), ρk (x2 ), · · · ρk (xn ), for a specific intensity y, the conditional PDF is approximated by n 1  pˆ(ρk |y) = K nhn

B. Fuzzy Edge Model

i=1

As an initial estimate of α , a fuzzy technique based on the CDF of the gradient [11] [12] is proposed. Consider an image with intensities y(x1 , x2 ), (x1 , x2 ) ∈ Z2 , which may be simply denoted as y. First,

(9)

∂x2

Based on the probability distribution function (PDF) of the gradient pG (g), the empirical CDF of the gradient magnitude ρk ≈ α may be found with [12]

(2)

where Yj = Ij and Yk = Ik are constant gray level values which uniquely depict the tissues Tj and Tk , respectively, and Ij > Ik . In brain extracted FLAIR MRI, there are typically three classes: 1) the background Y3 (includes CSF), 2) the brain Y2 (which is both the white and gray matter [3]) and 3) pathology Y1 (which in this case is WML). Therefore, a generalized PVA model may be generated for all three tissue classes:

 Y12

     ∂y 2  ∂y 2 g = ∇y =   +  .



ρk − ρk |y(xi ) hn



,

(13)

where a Gaussian kernel for K(·) is used and hn is the windowing function’s scale parameter. The optimal scaling parameter, h∗n , is determined through the optimization of the mean integrated squared

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error (MISE) in Matlab [13]. To classify PVA voxels, two bin locations at ρk = 0 and ρk = 1 are utilized. The result is a conditional PDF that describes edge presence in the following probabilistic manner: p (ρk = 0|y) - the probability that this pixel is located in a pure tissue class region (not located on an edge), or p (ρk = 1|y) - the probability that this voxel lies in a PVA region (significant edge). Therefore, estimation of the PDF in this fashion automatically determines which voxels are located in PVA (edgey) regions. To determine α (y) from the PDF, the conditional expectation operator is used since it offers the best prediction of the fuzzy edge measure α ≈ ρk given that the intensity is y in the Mean Square Error (MSE) sense. The result is an enhanced edge map, which provides a global representation of the edge information in the image. The estimate of α (y) is found by α (y)

= =

E{P = ρk |Y = y},



(14)

(b) PVA voxels

(c) Pure voxels

(d) Ex. 2

(e) PVA voxels

(f) Pure voxels

Fig. 1. Two example skull stripped FLAIR with WML and their respective PVA and pure tissue class segmentation.

ρk × pP |Y (ρk |y),

∀ρk

=

0 × p (ρk = 0|y) + 1 × p (ρk = 1|y) ,

=

p (ρk = 1|y)

indicating that the quantification of PVA content is directly proportional to the probability a voxel is located on an edge. D. Decoding the Proportion of Tissues α The global edge map estimate is directly proportional to the likelihood that a voxel is in an edgey PVA region. Since p (ρk = 1|y) ≤ 0.5 dictates which voxels y are not likely to be located on an edge in a probabilistic sense, this midpoint may be used as a thresholded to eliminate irreverent edge features, such as those caused by noise (low amplitude edges). This results in a final estimate for α (y): α (y)

(a) Ex. 1

→

p (ρk = 1|y) > 0.5

(15)

→ α (y) > 0.5.

After thresholding, two regions remain which depict the global edge   magnitudes for the two PVA regions, i.e. α12 (y) and α23 (y). The modified PVA map may be integrated over the corresponding intensity values to approximate the proportion of tissues

y

y αjk (y) =  ykj yk

 αjk (t)dt

αjk (t)dt

,

y k < y ≤ yj ,

measure is localized and corrupted by noise. To improve the estimate of PVA content, a globalized edge description is obtained with the kernel density estimator of Equation 13. The approximations of p (ρk = 0|y) and p (ρk = 1|y) for all y are shown in Figure 2. Note that the Savitzky-Golay smoothing filter (polynomial order 3, window size of 41) is used to robustly smoothen the probability estimates. As can be seen by these figures, specific intensity ranges have a high likelihood of being on an edge (PVA region). To retain only these significant PVA features and find the final PVA mapping function α (y), p(ρk = 1|y) is thresholded by 0.5, which corresponds to removing voxels that are not (in a probabilistic sense) located in a PVA region. The results for the final global mapping function α (y) are shown in Figure 2(b) and Figure 2(e), which are overlayed on the gray level PDF derived from the original image. As can be seen upon examination of the overlapping regions, the high PVA map values span a specific range of intensity values. These ranges correspond to voxels detected as PVA. Resultantly, the technique provides a set of thresholds that can be applied to the image to examine voxels from PVA and pure classes separately. To decode the proportion of tissues parameter α(y), Equation 16

(16)

where the denominator is a normalizing constant such that 0 ≤ α ≤ 1. The final PVA map, αjk (y), robustly detects and quantifies the way tissues are mixing due to PVA. IV. R ESULTS This research has been approved by the Research Ethics board at Sunnybrook Health Sciences Center, Toronto, Canada, in January 2008. Symptomatic patients are referred for FLAIR imaging (TR 8002 ms, TE 127.6 ms, TI 2000 ms, FOV 180×240 mm, 5 mm slice thickness, 6 mm gap). In total, 36 FLAIR MR images with WML were utilized but due to space constraints, results are shown for only a few patients. Brain extraction (skull stripping) is the only preprocessing applied [11]. The images used to illustrate the algorithm’s inner workings are shown in Figure 1. Although not shown, the localized fuzzy edge strength measures (Equation 11) for these images and the others show that the edge measure is a normalized, class specific feature since 0 ≤ ρk ≤ 1 and on average ρk12 ≤ ρk23 . However, as discussed this

(a) Ex1: p(ρk |y)

 (y) (b) Ex1: α

(c) Ex1: α (y)

(d) Ex2: p(ρk |y)

 (y) (e) Ex2: α

(f) Ex2: α (y)

Fig. 2.

Transformation functions for PVA quantification of Ex.1 and Ex.2

is utilized and the results are shown for the images of Figure 1 are shown in Figure 2(c) and Figure 2(f). The trends indicate that the PVA effect is nearly linear for both the brain-background and WMLbrain PVA. This map may be used in conjunction with a segmentation

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scheme to find how much each tissue contributes to the volume of an object (i.e. volume computation with subvoxel accuracy). Therefore, this map quantifies PVA. To examine the PVA voxels, any intensity value with a nonzero α(y) is retained. The result is an image that highlights the PVA regions (brain-background mixture tissues Y23 and WML-brain mixture tissues Y12 ), as shown in Figure 1. The remaining pixels belong to the pure tissue classes (brain Y2 and WML Y1 ) and are also displayed in this figure. The PVA segmentation shows that both the PVA from WML (fuzzy regions surrounding the lesion) and the brainbackground PVA (appears as dark brain in FLAIR) were detected with good results. The classification system automatically incorporated the edge information from all PVA voxels to determine the intensity ranges for PVA. In two of the images (Figure 1(e)) above the bottom right ventricle, Figure 3(h) the upper part of the interior of the brain), it appears that several pixels have been highlighted which may not be entirely comprised of PVA voxels. These voxels are comprised of the same intensity as PVA voxels and that is why they have been selected. From the original images, these voxels have an abnormal increase in intensity likely caused by the bias field affect. By applying a small threshold on the α(y) map (i.e. look at voxels only with α(y) > 0.1) these voxels were removed. Future works will investigate whether it is practical to include all PVA voxels in the final map (α(y) > 0), or whether α(y) > τ is more optimal for some image dependent τ . Moreover, the applicability of confidence intervals or error bounds on the threshold will also be investigated. We are in the process of developing synthetic models to validate these fine tunings. In the pure regions, it is easy to see that contiguous pixels from the interior of the brain and WML were also selected. Therefore, this technique may be used to objectively investigate the characteristics of specific tissue classes in FLAIR MRI. A few more sample images with brief results are shown in Figure 3. These images also exhibit the same observed properties. As a result, the current approach provides an objective and adaptive method for the detection and quantification of PVA since mixture voxels are robustly segmented from the pure tissue classes. Futures works will investigate the sensitivity of the proposed works to lesion loads, and we are currently compiling the new database. As the quality of the estimates for p(ρk = 0|y) and p(ρk = 1|y) depend on the number of voxels used in the approximation, small lesion loads may cause results to be skewed. The current database does not contain patients with small lesion loads, so this could not be verified and is the subject of future works.

V. C ONCLUSIONS The proposed work details a new method for the detection and quantification of PVA in FLAIR MRI with WML. Traditional works focus on the use of mixture models to define PVA and non-PVA regions, which do not directly apply to FLAIR with WML. To combat this, the current work details a non-statistical method of detecting PVA based on inherent image characteristics - namely intensity and fuzzy edge strength. A global edge description is generated by a conditional PDF that describes both intensity and edge information simultaneously. From this, the change in the proportion of tissues α (y) is generated and used to decode the PVA characterization map (the mixture parameter α(y), which may be used with a segmentation scheme to determine volume measurements with subvoxel accuracy). Such a map provides an objective mechanism for the segmentation of pure and mixed tissue classes.

(a) Ex.3

(b) PVA voxels

(c) Pure voxels

(d) Ex.4

(e) PVA voxels

(f) Pure voxels

(g) Ex.5

(h) PVA voxels

(i) Pure voxels

Fig. 3.

Examples of PVA and pure tissue segmentation

R EFERENCES [1] P. Malloy, S. Correia, G. Stebbins, and D. H. Laidlaw, “Neuroimaging of white matter in aging and dementia,” The Clinical Neuropsychologist, vol. 21, pp. 73–109, 2007. [2] N. Altaf, L. Daniels, P. Morgan, J. Lowe, J. Gladman, S. MacSweeney, A. Moody, and D. Auer, “Cerebral white matter hyperintense lesions are associated with unstable carotid plaques,” Eu. J. Vascular and Endovascular Surg., vol. 31, pp. 8–13, 2006. [3] C. Jack, P. O’Brien, D. Rettman, M. Shiung, Y. Xu, R. Muthupillai, A. Manduca, R. Avula, and B. Erickson, “Flair histogram segmentation for measurement of leukoaraiosis volume,” J. of Mag. Res. Img., vol. 14, no. 6, pp. 668–76, 2001. [4] P. Anbeek, K. Vincken, M. van Osch, R. Bisschops, and J. van der Grond, “Probabilistic segmentation of white matter lesions in MR imaging,” NeuroImage, vol. 21, no. 3, pp. 1037–44, 2004. [5] R. de Boer, F. van der Lijn, H. Vrooman, M. Vernooij, M. Ikram, M. Breteler, and W. Niessen, “Automatic segmentation of brain tissue and white matter lesions in MRI.” IEEE ISBI, 2007, pp. 652 – 55. [6] N. Altaf, A. Beech, S. D. Goode, J. R. Gladman, A. R. Moody, D. P. Auer, and S. T. MacSweeney, “Carotid intraplaque hemorrhage detected by MRI predicts embolization during carotid endarterectomy,” J. Vascular Surgery, vol. 46, no. 1, pp. 31–36, 2007. [7] M. Cuadra, L. Cammoun, T. Butz, O. Cuisenaire, and J.-P. Thiran, “Comparison and validation of tissue modelization and statistical classification methods in T1-weighted MR brain images,” IEEE Trans. on Med. Imag., vol. 24, no. 12, pp. 1548 – 1565, 2005. [8] P. Lin, Y. Yang, C.-X. Zheng, and J.-W. Gu, “An efficient automatic framework for segmentation of MRI brain image,” vol. 00. Int’l Conf. on Computer and IT, 2004, pp. 896–900. [9] A. Khademi, D. Hosseinzadeh, A. Venetsanopoulos, and A. Moody, “Nonparametric statistical tests for exploration of correlation and nonstationarity in images.” Int’l Conf on DSP, 2009. [10] M. G. Ballester, A. Zisserman, and M. Brady, “Estimation of the partial volume effect in MRI,” Med. Im. Anal., vol. 6, no. 4, pp. 389–405, 2002. [11] A.Khademi, A.Venetsanopoulos, and A. Moody, “Automatic contrast enhancement of white matter lesions in FLAIR MRI.” ISBI’09, 2009. [12] P. Meer and B. Georgescu, “Edge detection with embedded confidence,” IEEE Trans. on PAMI, vol. 23, no. 12, pp. 1351–1365, 2001. [13] P. H. Kvam and B. Vidakovic, Nonparametric Statistics with Applications to Science and Engineering. USA: Whiley-Interscience, 2007.

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