CARDIAC MRI FOR TRACKING PURKINJE FIBERS THROUGH BRANCHINGS. H. Ertan ... We thereby introduce a novel algorithm to track the Purkinje fibers in.
ESTIMATION OF MULTIMODAL ORIENTATION DISTRIBUTION FUNCTIONS FROM CARDIAC MRI FOR TRACKING PURKINJE FIBERS THROUGH BRANCHINGS H. Ertan C¸eting¨ul† , Gernot Plank‡ , Natalia Trayanova† and Ren´e Vidal† †
Department of Biomedical Engineering, Johns Hopkins University, Baltimore MD, USA ‡ Institute of Biophysics, Medical University Graz, Graz, Austria ABSTRACT
The inclusion of the free-running Purkinje network in computational simulations provides a significant insight into understanding the mechanisms of cardiac pathophysiologies. However, its automatic extraction is challenging due to the presence of abundant local complexities. We thereby introduce a novel algorithm to track the Purkinje fibers in high resolution magnetic resonance (MR) images. Our formulation successively identifies local fiber orientations by using a nonlinear oriented filter. Specifically, the filter is used to compute several local profiles, from which one can estimate the orientation distribution function (ODF). The algorithm then determines the directions to be followed by detecting the modes of the local ODF using different spherical clustering methods. We quantitatively compare the accuracy of the tracked fibers with manually delineated anatomical structures. Index Terms— fiber tracking, nonlinear filters, spherical clustering, cardiovascular systems, magnetic resonance imaging. 1. INTRODUCTION The extraction of fibrous structures is a well-studied problem in pattern recognition, which finds a wide range of applications in medical image analysis. A fibrous structure comprises a spatially coherent appearance pattern, which can be quantified via different feature-based and/or model-based representations [1]. Furthermore, these representations have been used to develop computational tools that can improve current diagnostic and therapeutic approaches [1, 2, 3, 4]. Most of the existing fiber extraction methods can be categorized into three groups: i) region-based segmentation and curve thinning, ii) operator-based fiber detection and grouping, and iii) successive orientation estimation and fiber tracking. The first group uses region-based segmentation algorithms to find “fibrous” regions followed by curve thinning for centerline extraction. There exist a plethora of works which employ this top-down strategy (see [3] and references therein). The second group of methods employs a local operator for fiber detection and assigns a label to each pixel/voxel. These bottom-up approaches often employ the image gradient/Hessian, the structure tensor, or linear/nonlinear/model-driven operators such as steerable and multiscale oriented filters (see [5] and references therein). However, these techniques require local detection/labeling at each pixel/voxel, which can be computationally expensive. The third group of methods estimates local orientations via a nonlinear operator and uses this information to delineate the fiber by means of tracking. These methods are initiated at user-specified points and reduce the aforementioned computational load. Selected works employ intensity-based medial strength [6] and locally coherent appearance patterns [7]. This work has been funded by the grant NIH HL082729 and by startup funds from the JHU Whiting School of Engineering.
Other tracking methods use diffusion weighted MR images (DWMRI) to estimate the orientation distribution function (ODF), which is the radial projection of the probability density function (pdf) of particle displacements. Fiber directions are identified as the directions at which the corresponding ODF attains its peaks. However, the local displacement is often modeled with a Gaussian distribution. Consequently, the streamlining techniques that employ such unimodal representations do not solve partial volume effects [8]. Therefore, the state-of-the-art methods estimate local fiber orientations from multimodal representations (see [9, 10, 11] and references therein). In fact, any tractography technique employing ODFs could be applied on structural MRI if one could estimate the ODFs from intensity data. 1.1. Problem Statement and Algorithm Overview The Purkinje network (PN) comprises specialized fibers of the cardiac conduction system, which are responsible for the propagation of the electrical impulse initiating myofiber contraction. The PN is implicated in the generation and sustenance of arrhythmias, hence modeling this network in conduction simulations provides an understanding of this pathophysiological mechanism. Novel ex vivo MRI techniques offer sufficient resolution to identify the free-running PN, which activates endocardial structures such as the papillary muscle (Fig. 1(a)). However, the automatic extraction of these fibers is difficult due to local complexities such as branchings (Fig. 1(b)). We thereby address the problem of tracking the free-running Purkinje fibers by employing a novel nonlinear filter for local orientation estimation. We represent a fibrous structure as a sequence . of 3-D points (x0 , x1 , . . . , xl ) = X , or equivalently as a sequence of vectors (s1 , s2 , . . . , sl ) with si = xi − xi−1 . We initialize the algorithm by specifying {x0 , x1 }, which places the filter along the fiber of interest. We then compute the ODF at x1 and subsequently estimate the local orientations by detecting the modes of the ODF using two different spherical clustering techniques. We repeat the same procedure by shifting the filter to the newly identified locations and successively obtain the points x2 , x3 , . . . on the fiber. Our main contribution is the method to estimate the ODF from structural MRI.
(a)
(b)
Fig. 1. Illustration of the cardiac PN: (a) 3-D rendering of the MR data, (b) A selected MR slice with manually extracted fibers (red).
2. THE 3-D PIVOTING FILTER In [5], inspired from the filter introduced in [12], we proposed a novel nonlinear pivoting filter for tracking 2-D tubular structures through branchings. This filter takes a spatial support around each pixel of interest and computes a nonlinear function of the intensities of these pixels to obtain a coarse estimate of the local fiber orientations. In this section, we extend the pivoting filter for the analysis of 3-D fibers. The proposed 3-D pivoting filter, depicted in Fig. 2(a), is centered at a point of interest x. The filter has a fixed backward point b and a pivoting forward point f located at a distance r from x. The points {b, x, f } define the main axes of the filter. The purpose of the remaining points {bk , f k }2K k=1 is to fully encapsulate the 3-D fiber of interest. Specifically, the points {bk } are placed on a circle of radius w orthogonal to bx. The consecutive points {bk , bk+1 } are separated by an angular step α and the points {bk , bk+K } form an antipodal pair. The forward portion {f k } is constructed analogously. si−1 f
r f1 f2 f 2K x-axis
This estimate is further refined by an appearance-based measure, which quantifies the intensity coherence along the orientation of the forwardRportion of the filter. This new measure is given by 1 g(x, s) = 0 |I(x + λrs) − I(x)|2 dλ and is minimized when the search orientation s aligns with the true fiber orientation. Then the appearance profile is defined as exp(−g(x, s)) . sn ∈S exp(−g(x, sn ))
z
hs, x − bi . − bi
sn ∈S hsn , x
α
b2K
b1
b2
y-axis
(a)
(3)
x
PS (x, s) = P b
(2)
PA (x, s) gives an estimate of the probability of having a small appearance variation along s ∼ xf . Moreover, in order to enforce a certain level of (local) smoothness in the tracked fibers, we define a smoothness factor from the cosine of the angle between the previous orientation x − b and the candidate orientation s as
w z-axis
h(x, s) . sn ∈S h(x, sn )
PD (x, s) = P
PA (x, s) = P
s
x
ODF. First, in order to have a finite number of candidate orientations, we discretize the unit hemisphere at N = 1281 predefined vectors {sn } obtained by a fourfold tessellation of an icosahedron. This . give us the set S = {sn : hsn , x − bi > 0, sn ∈ S2 }. Next, the filter response at x is directly used to obtain a coarse estimate of the probability of having a structure oriented along any s ∈ S. The resulting measure, which we call the directional profile, is defined as
y
(b)
Fig. 2. Essentials of local orientation analysis: (a) 3-D pivoting filter with key parameters, (b) Estimated ODF at a bifurcation (top view). The points {bk } and {f k }, together with {b, x, f }, are used to compute the response of the filter at x. Specifically, let us denote the intensity value at p ∈ Υ ⊂ R3 by I(p)1 , where Υ is the image domain. Having fixed the backward portion bx, consider now a particular filter orientation s ∈ S2 such that f = x + rs. For each pair of antipodes {bk , bk+K } and {f k , f k+K }, we compute min {|I(b)−I(bj )|, 1 if |I(b)−I(f )| ≤ j∈{k,k+K} hk (x, s) = |I(f )−I(f j )|} 0 otherwise. (1) This can be considered as a partial filter response for a fixed k ∈ {1, 2, . . . , K}. The total filter response is then computed by sumP ming (1) over all pairs of antipodes as h(x, s) = K k=1 hk (x, s). This response should be high when s aligns with the true fiber orientation at x, and low otherwise.
(4)
Finally, we define the ODF by multiplying the directional, appearance, and smoothness factors as p(x, s) ∝ PD (x, s) × PA (x, s) × PS (x, s) . (5) The ODF p(x, s) provides an estimate of the probability of having a locally linear structure at x oriented along s ∈ S. Fig. 2(b) illustrates the discrete ODF estimated at a bifurcation point where the probabilities of the orientation vectors are color-coded (blue∼low, red∼high). Local fiber orientations are then identified by detecting the modes of the ODF, as described in the next section. 4. DETECTION OF LOCAL FIBER ORIENTATIONS VIA SPHERICAL CLUSTERING Since the set of candidate orientations S lies on the unit hemisphere, estimating local fiber orientations is equivalent to finding the modes of a spherical distribution. Notice also that the ODF can be considered as a function that assigns a weight ρn to each candidate orientation, . i.e., ρn = p(x, sn ), ∀sn ∈ S. Therefore, we can interpret the mode detection problem as the problem of clustering directional data on S2 in such a way that points with higher weights contribute more to the mode. We solve this problem using two spherical clustering methods: i) spherical k-means and ii) spherical mean shift.
3. ESTIMATION OF THE LOCAL ODF 4.1. Weighted Spherical k-Means (WSpkM) Similar to the 2-D case previously described in [5], we now utilize the 3-D pivoting filter to compute two different, but intertwined profiles: the directional profile and the appearance profile, which are combined with an additional smoothing factor to obtain an estimate of the local 1 When the point p lies outside the discrete grid, we compute the corresponding intensity value by trilinear interpolation.
In standard k-means clustering, one aims at finding the optimal clustering of the data by minimizing the within-cluster scatter, i.e., the average Euclidean distance between the data vectors and their corresponding cluster centers. In the case of the spherical k-means, the distance is replaced with a similarity metric, i.e., the cosine of the angle between unit-length data vectors and the centroids. Specifically,
given a set of weighted points {(sn , ρn )}N n=1 and the number of clusters C, the weighted spherical k-means initializes the centroids {µc }C c=1 and maximizes a cosine similarity objective function, i.e., N X C X
max
µ1 ,µ2 ,...,µC
ωnc ρn hsn , µc i,
(6)
n=1 c=1
by iteratively updating {µc } and the binary weights {ωnc }, where ωnc encodes the membership of sn to the cluster associated with µc . Note that the number of clusters, C, should be given beforehand to the k-means algorithm. For this purpose, we exploit the anatomical topology of the free-running Purkinje fibers. In particular, it is known that a vast majority of these fibers can only bifurcate. Therefore, we set C = 2 and merge the centroids into one if the angle between them is smaller than a user-specified threshold δ > 0. 4.2. Weighted Adaptive Spherical Mean Shift (WASpMS) The mean shift (MS) algorithm [13] is a nonparametric kernel density estimator, which uses gradient ascent to find the number and the locations of the modes of an unknown pdf, as well as the points that belong to the cluster associated with each mode. Given a set of points m {sn }N n=1 ⊂ R , the MS approximates the true pdf f (·) at µ as ˆf (µ) = ZΦ
N X
Φ(sn , µ; h)
Algorithm 1 ODF-guided 3-D Fiber Tractography on MRI 1. At the i-th iteration, consider {xi−1 , xi } or equivalently si−1 . 2. Place the pivoting filter such that b = xi−1 and x = xi . . 3. Obtain the set S i = {sn : hsn , si−1 i > 0, sn ∈ S2 }. 4. Find p(xi , sn ) = ρin from (5) using (2), (3), (4), ∀sn ∈ S i . 5. Estimate the local fiber orientations by detecting the modes {µ(c) xi } of the ODF p(xi , ·) via the WSpkM or the WASpMS. (c)
ˆf (µ) = ZΦ
N X n=1
ρ n κn exp(κn µ> sn ). 4π sinh κn
(c)
We subsequently estimate the ODF at xi , i.e., p(xi , sn ) = ρin , for n = 1, 2, . . . , N , from (5) and detect the modes using either the WSpkM or the WASpMS. Having estimated the number of branches (c) C C C and the modes {µ(c) xi }c=1 as the local orientations {si }c=1 at xi , (c)
Then the modes of ˆf are located by gradient ascent. Specifically, at the j-th iteration, given µj together with the sets {ρn } and {κn } as˜ j+1 /kµ ˜ j+1 k, sociated with {sn }, the mean is updated as µj+1 = µ where N X ρn κ2n ˜ j+1 = µ exp(κn µ> (9) j sn )sn . sinh κ n n=1 This scheme is guaranteed to converge to the modes of the underlying distribution [14]. These modes give the local fiber orientations and can be used for tracking the fibers, as described next. 5. TRACKING ALGORITHM Our algorithm can be considered as a streamlining technique, i.e., initialized at two user-specified points {x0 , x1 }, it iteratively tracks the fiber of interest by following the local orientation estimates. Specifically, at the i-th iteration, the backward portion bx of the pivoting filter is aligned with the previous orientation, i.e., b = xi−1 , x = xi and bx ∼ si−1 , whereas the forward portion xf ∼ sn is allowed to freely rotate so as to scan N candidate orientations.
(c)
the next point along the c-th branch is found as xi+1 = xi + si . In particular, our tracking method proceeds as outlined in Algorithm 1.
(7)
(8)
(c)
8. Obtain the tracked fiber as the sequence of points {xi }, ∀c.
6. METHOD VALIDATION
n=1
using a kernel function Φ of bandwidth h > 0 and a normalization term ZΦ . The estimation is further improved by introducing a different (adaptive) bandwidth hn ≡ h(sn ) for each sn . We are particularly interested in clustering directional data on the unit 2-sphere. For this purpose, we employ the von Mises-Fisher κ kernel [10] for m = 3, i.e., Φ(s, µ; κ) = 4π sinh exp(κµ> s), κ 2 with the mean vector µ ∈ S and the concentration parameter κ ≥ 0. After incorporating the weights {ρn } and the bandwidths {hn ∝ 1/κn }, where hn is chosen as the geodesic distance between sn and its 10-th nearest neighbor, the density estimate is written as
(c)
6. Set si = µ(c) xi and find xi+1 = xi + si , i.e., the next point on the c-th branch, ∀c. P 7. Set i = i + 1 and go to 1 until n ρin becomes small.
6.1. Experimental Details We conduct our experiments on a structural MR image of a healthy rabbit heart, which was acquired on an 11.7 T (500 MHz) MR system at a resolution of 26.5 × 26.5 × 24.5 µm3 [4]. Our method is tested on different subvolumes that contain 60 single and 35 branching fibers from the free-running PN. Single fibers are non-branching fibers running from one Purkinje-myocardial junction (PMJ) to another, whereas branching fibers contain at least one Purkinje-Purkinje junction (PPJ). For comparison, we manually delineate the PN by placing points on the fibers in every slice. To quantify the spatial error, we compute the symmetrized Chamfer distance (in voxels) between . . the true fiber X t = {xti } and the estimated fiber X e = {xej } as (X t , X e ) = 0.5 d(X t , X e ) + d(X e , X t ) , where d(X t , X e ) = |X t |−1
X
min{kxti − xej k : xej ∈ X e }. (10)
xti ∈X t
6.2. Results In our experiments, the parameters {r, w, α} of the filter are set to either {4, 4, 20◦ } or {3, 2, 20◦ } depending on the widths of the fibers and on the proximity of the surrounding structures. For single fibers, the WSpkM selects the mean vector (C = 1) as the local orientation, whereas for branching fibers, it locates two centroids (C = 2) and merges them using an angular threshold of δ = 60◦ . The WASpMS automatically finds the number and the directions of the centroids. Table 1(a) shows the tracking errors for 4 single fibers using the aforementioned clustering schemes, along with the minimum, maximum and mean errors over 60 single fibers. We observe that i) the majority of the fibers are extracted with errors less than 1.5 voxels, which demonstrates the accuracy of our filter-based orientation analysis, and ii) the WASpMS and the WSpkM achieve comparable performances with mean errors of about 1 voxel. Table 1(b) shows
(a)
(b)
(e)
(f)
(c)
(g)
(d)
(h)
Fig. 3. Visualization of selected fibers and their extraction: 3-D rendering of the volume (green) and the tracked fiber (red).
the tracking errors for 4 branching fibers along with the minimum, maximum and mean errors over 35 branching fibers. In terms of mean errors, the WASpMS outperforms the WSpkM by about 1 voxel. In addition, the WASpMS achieves lower tracking errors for the selected fibers and a lower maximum error of 7.90 voxels. Notice that in a few cases, we obtain tracking errors higher than 10 voxels. This is due to the accumulation of erroneous local estimates, i.e., the streamlining nature of our method. Furthermore, the performance of our method is visualized in Fig. 3, which shows the volume meshes of the fibers in Table 1 along with their extractions using the WASpMS. Table 1. Spatial tracking errors (X t , X e ) on selected fibers (a) 60 single fibers
Error () Volume WSpkM WASpMS (a) 1.06 0.96 (b) 2.39 1.57 (c) 1.07 0.92 (d) 1.30 1.24 min 0.10 0.07 max 4.94 4.00 mean 1.10 1.05
(b) 35 branching fibers
Volume (e) (f) (g) (h) min max mean
Error () WSpkM WASpMS 1.12 1.07 5.11 1.23 1.32 1.37 7.62 1.09 0.86 0.86 10.23 7.90 2.94 1.92
7. DISCUSSIONS AND FUTURE WORK This paper focused on extracting selected free-running Purkinje fibers from high resolution MRI. We presented a tracking formulation that successively identifies and follows local fiber orientations. These orientations are given by the modes of the ODFs that are estimated via a nonlinear oriented filter. Our experiments clearly demonstrated the success of the proposed method. Nevertheless, due to the (linear) geometry of the filter and the streamlining nature of the method, there exist a number of cases that need further attention. For instance, tracking individual fibers in dense fibrous regions, i.e., in the presence of nearby structures, or tracking fiber segments with high curvature under noisy conditions may not be successful. Therefore, as a future work, we will perform synthetic experiments to analyze the method’s sensitivity to noise and to the filter parameters, and subsequently use the resulting Purkinje fibers to advance cardiac conduction models.
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