Prediction Correction Tractography through Statistical Tracking

Davide Imperati, Iuri Frosio, Marc Tittgemeyer, and Nunzio Alberto Borghese

Abstract–In this work we describe a novel approach to diffusion tractography that is a notion common to a class of techniques based on diffusion MRI data aiming on tracking axonal pathways in the brain. Our approach, named Prediction-correction Diffusion-based Tractography (PDT), is based on Extended Kalman Filtering: at each step the local fibers orientation is estimated from their orientation in the previous step. This estimate is then corrected from an estimate of the local diffusivity through a principled model of fibers orientation. PDT has been implemented using a diffusion tensor (DTI) as local diffusion model, but higher order models can be used as well. Results on both synthetic and in-vivo data are reported and discussed. PDT produces tractograms comparable to those obtained with the widely distributed tractography method provided in the FSL package [18], also in the case where crossing fibers are of relevance. From preliminary data, PDT proved superior when one fiber of low fractional anisotropy crosses a fiber with a higher fractional anisotropy, that is a critical condition for other tractography methods.

I. INTRODUCTION

IFFUSION-weighted magnetic resonance imaging (dwMRI) [1] is a non-invasive technique to infer local white matter

architecture of the human brain. It is based on the detection of the diffusion of endogen hydrogen protons through magnetic resonance (MR), and it produces, for each voxel a signal, which represents the local orientation of the diffusion direction inside the white matter of the brain. Therefore, it is well suited to directly infer the brains fiber structure in-vivo, providing an intimating basis for diffusion tractography [2]. Here, tractography refers to a class of novel techniques that are aimed to track the axonal pathways in the brain [3]. Furthermore, such methods can provide a powerful tool to assess the connectivity between cortical areas [4] and mapping functional to anatomical connectivity information [5]. In general, diffusion tractography let to reconstruct three dimensional fiber pathways by putting together discrete (voxel-based) diffusion information to create continuous pathways. To achieve this, first, from an adequate set of

Manuscript received November 15, 2008. D. Imperati is with the AIS-Lab, Computer Science Dept., University of

Milan, 20133 Italy and with the Max-Plank-Institute for Neurological Research, D-50199 Cologne, Germany (telephone: +49-221-4726-252, e-mail: davide.imperati@nf.mpg.de).

I. Frosio is with the AIS-Lab, Computer Science Dept., University of Milan, 20133 Italy (telephone: +39-02-503-14010, e-mail: frosio@dsi.unimi.it)

M.Tittgemeyer is with the Max-Plank-Institute for Neurological Research, D-50199 Cologne, Germany (telephone: +49-221-4726-215, e-mail: marc.tittgemeyer@nf.mpg.de).

N. A. Borghese is with the AIS-Lab, Computer Science Dept., University of Milan, 20133 Italy (telephone: +39-02-503-14011, e-mail: borghese@dsi.unimi.it)

dwMRI images a local model is computed for each voxel (e.g. through Diffusion Tensor - DTI) representing the orientation of the fibers in the voxel center. Fiber tracking algorithms can be classified into two types: deterministic and probabilistic. Initial work in the field focused on deterministic tractography [2]. This is based on the assumption that the orientation of the largest component of the diagonalized diffusion tensor represents the orientation of the largest axon inside that voxel. To compute the fibers pathways, one segment is associated to each voxel: it goes through the voxel center and it is oriented as the principal direction of the diffusion tensor. The fiber pathway is reconstructed smoothly joining adjacent segments forming a streamline. Within this scope a voxel is selected as a seed and the pathway starting from this voxel is created moving from the seed. These methods, based only on local diffusion information, do produce tracking errors in those points where fiber bundles with different orientations cross. In this case the diffusion tensor becomes oblate and the principal direction is not representative anymore of fiber direction. For this reason probabilistic approaches to tractography have been proposed, e.g [6]. For instance, in [3,7] a set of iterations is performed, where, at each iteration, the streamline-like pathways obtained with deterministic methods are perturbed depending on the image noise associated to each voxel. The final tractogram presents the most probable pathway given the image. However, the problems introduced by fibers crossings are not completely eliminated. This is due to the fact that these methods use local diffusion information only. A method that goes beyond this is proposed in [8]. It uses the information on all the voxels as it finds pathways as the minimum of a variational Bayesian problem. In [9] the whole tensor shape is considered, that allows a better estimate of the fibers direction. A better result could be obtained by using higher order local diffusion models [10]. However, these methods require higher gradients and a higher number of diffusion orientations. A different approach has been proposed in [11] where a regularized tracking, using a Kalaman filter on the principal direction is proposed. The fiber direction estimated, at a previous iteration, is used to step over a critical crossing point. This is identified as the point where the diffusion tensor’s principal direction changes abruptly. We propose to extend this approach to non-linear mapping using an extended Kalman Filter, to facilitate tracking by considering both the direction and the shape of the diffusion tensor, and we estimate the current direction using information from the fiber bundle tracked to the current point. With this approach, named prediction correction diffusion tensor based

D

tractography (PDT), tracking is able to step over critical regions without the use of heuristics. The filter delivers also a confidence measure, which is useful to evaluate the quality of tracking and connectivity

II. BACKGROUND

A. Fibers Model Nerve fiber bundles can be locally approximated as a three-

dimensional cylinder with a radius of the order of some micrometers (cf. [12, 13]). A fiber bundle can be as long as several centimeters. Moreover, they are supposed to curve smoothly; in particular, we suppose that a given pathway has the minimum curvature among all possible pathways compatible with the data. Therefore we can model a fiber bundle as a curve, )(lr , with

3: ℜ⊆→ℜ Sr . We can also assume that it is continuous and at least derivable up to the second derivative,

( )32 RCr ∈ . It follows that the tangent to r, x, exists, is defined, and continuous at each location of the fiber bundle,

( )31 RCx ∈ , and constraint 1=x .

We introduce a first approximation of r(l):

dllxlrdllr )()()( +=+ (1) We suppose that the tangent curve to the fiber bundle can be approximated as:

dllwlxdllx x )()()( +=+ (2) where the representation of the versor wx(.) is a random variable with Bingham distribution and it describes the probability that the fibers in l, deviate from straightness. Here we model (.)xw as an isotropic Gaussian distribution [14]:

),0(~(l) xx Gw Σ (3)

with

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=Σ

σσ

σ

000000

x .

In the following the value of σ has been chosen empirically.

B. Diffusion tensor measurements The model that links the signal decay measured by the

Magnetic Resonance and the diffusion coefficient is given in the seminal proposal of Steijskal and Tanner [15]:

DddTbeSbS −= 0)( (4)

b is the magnitude of the diffusion sensitization vector, d its direction and D is the diffusion tensor that can be considered a second order three dimensional tensor. From a set of (4), the local tensor D can be derived as shown in [16]. Under some quite general assumptions the variance of the six independent components of D , 6ℜ∈D , can be modeled as a multivariate Gaussian distribution [17]. Given the cylindrical fiber model used here, when the tensor is oriented in the absolute reference frame, chosen with the x axis oriented as the fiber major axis, 1e , it is described as

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

ββ

λ

000000

absD (5)

where λ is the diffusivity along the fiber axis and β is the diffusivity across it. A generic D tensor, oriented in the direction x, can be computed as the rotation of absD obtained through the multiplication by a quadratic form:

RDRD absT= (6) Matrix multiplications inside (6) can be computed to give an explicit analytical formula for each of the 9 elements of D . As D is symmetrical only 6 components are independent. D can therefore be written as:

)absD |H(xD = (7)

where )absD |H(x projects a generic direction x in the 3D space onto the absolute reference frame.

III. MATH

The method proposed here is composed of two steps: i) Prediction and ii) Correction. In the prediction step, information on fibers orientation is propagated, taking into account the variability in the fibers orientation and noise. In the Correction step, measured data are corrected taking into account the propagated information. The algorithm starts from a given voxel with a distribution of orientations. This is provided by the user in form of an orientation px0ˆ . This orientation should be congruent with the

data. An initial estimate of direction variability, px0Σ , has also

to be provided in form of covariance matrix, based on the noise on the images. px0ˆ and p

x0Σ identify our starting multivariate Gaussian distribution.

Discretizing Eqs. (1) and (2) in space, we can write the curve that describes the fibers bundle as:

11 −− += nnn xrr (8) and the curve of the tangent to the bundle as:

),(),0( 11 xnxnn xGGxx Σ=Σ+= −− (9)

),(~)( 11 xnnn xGxxp Σ−− is therefore a multi-variate

Gaussian with covariance xΣ . Here n-1 and n are two following steps.

From Eqs. (6) and (7), the probability of having a certain diffusion tensor at location n, nD , can be expressed as:

( )( )nDnn

nDabsnnn

n

n

DxHG

GDxHxDp

,

,

),|(

,0)|()(

Σ

=Σ+= (10)

We can now define 3ˆ ℜ∈p

nx to be our a priori estimate of the fibers direction at step n, given the direction in the

previous n-1 steps. At each step n, pnx̂ can be computed as:

)(),...,(

),...,(

1111

11

−−−

− =

nnnn

npn

xxpDDxp

DDxp (11)

as the joint probability of the fibers having direction 1−nx at

step n-1, given the measured tensor, 1−nD , and that the fibers

at step n have direction nx if they have direction 1−nx at step n-1. Since these distributions are independent, the center of the new distribution is given by

qn

pn xx 1ˆˆ −= (12)

and its covariance is the sum of the covariance matrices of the two distribution and it is given by

xqn

pn Σ+Σ=Σ −1 (13)

where xΣ is obtained from (3). q

n 1−Σ is the covariance matrix that expresses the direction uncertainty at the previous step.

We now define 3ˆ ℜ∈qnx the a-posteriori estimate of the

direction at step n given pnx̂ , nD , and q

nΣ the a-posteriori covariance of the fibers orientation.

These quantities define the probability distribution of fiber directions, given the prior and the data and are the quantities that have to be determined. To this aim we define the a-posteriori estimate error as

qnn

qn xxe ˆ: −= (14)

and the posterior error covariance on xn as:

[ ]Tqn

qn

qxn eeE )(:=Σ (15)

Following the theory of Kalman filtering, q

nx̂ can be

computed as a linear combination of the a priori estimate pnx̂

and a weighted difference between the measured diffusion tensor, nD , and its prediction )|ˆ( absDp

nxH . In particular, as H(.) is a non-linear function, it will be replaced by its first

order approximation leading to the linear mapping pnxHˆ~

where H~ is the linearized form of (.)H around pnx̂ . q

nx̂ can therefore be computed as:

)ˆ~(ˆˆ pnn

pn

qn xHDKxx −+= (16)

The gain matrix K is the optimal Kalman gain (see [17])

and it is chosen to minimize the a-posteriori error covariance (Eq. 15):

( ) 1~~~ −Σ+ΣΣ= D

Tpxn

Tpxnn HHHK (17)

DΣ is the noise covariance in the diffusion tensor components.

The difference )ˆ~( pnn xHD − can be viewed as the

innovation introduced by the observation (measured tensor,

nD ) on the predicted value of the tensor itself pnxHˆ~

, and it is also called residual or prediction error. If the residual is zero the prediction coincides with the observation. If the residual is nonzero the linear map K projects the residual onto the directions space, and updates the estimate of p

nx̂ to obtain qnx̂ . The tracking uncertainty in the posterior estimate is

corrected to account for the new information added by the observation of nD as:

pn

qn HKI Σ−=Σ )~( (18)

The most likely fibers orientation is assumed as the a-priori

orientation for the next step according to (12). The direction of the fibers bundle at the starting voxel, is

the only missing information to complete the method. We chose a set of M random directions at the starting voxel and compute the M entire paths with the above method. Typical values of M are of the order of 1000-5000.

The final tractogram is represented as a 3D image, where for each voxel the probability that a fiber goes through that voxel is represented as a gray level. To obtain this a voting

mechanism is introduced. A vote is a scalar associated to each step of each tractogram, and it represents the product of the vote associated to the previous steps of the tractogram itself, up to the actual position. The vote at a certain step, vn, is a function of the posterior estimate of the fibers direction and of its posterior covariance (15) through:

{ }ββ q

nT

nn vv Σ−= − 21

1 exp (19)

where qnn

T xDH 1ˆ~−−=β is the difference between predicted

direction and the direction of the tensor. Equation (19) represents a measure of likelihood of the fibers orientation expressed by the local tensor, given the tractogram up to the previous step.

Fig. 1. Pseudo code of the algorithm.

We explicitly remark that tractograms do not need to go through the voxels center but can go through any position in 3D space. Under this assumption, nD has to be provided for any position in 3D space. This is obtained applying tri-linear interpolation to the value of nD , measured in correspondence of the voxels center.

nv is then normalized across all the computed paths to be able to compare different tractograms. Tractograms with a higher vote are associated to paths with a higher probability, while unlikely tractograms receive a low vote. The tractogram image, )( kVT , is obtained by assigning to each voxel a gray level that is proportional to the sum of the votes associated to all the tractograms through it:

ijk

M

i

N

jVk vrIVT

j

k)()(

1 1∑∑

= =

= (20)

)( kVT represents therefore the likelihood that the fibers

bundle goes through voxel kV .

(.)kVI is an indicator function on the voxel kV , 1(.) =

kVI if

the j-th position of the i-th tractogram is included into kV , otherwise its value is 0. The whole algorithm for identifying one tractogram is summarized in Figure 1.

IV. MATERIALS AND METHOD Preliminary results on tractograms obtained with the present

method from simulated and in-vivo dwMRI data are reported. Results are compared with the widely diffused probabilistic tractography algorithm described in [7], provided in the probtrackx module inside the FSL software package [18]. probtrackx works on the estimate of the local diffusion directions computed according to [10] and implemented in FLS inside the software module bedpostx; this is aimed on modeling multiple fibers orientations per voxel. The PDT approach described here, instead, was applied on simpler local diffusion tensors, that describe the local diffusivity [10].

A. Synthetic data To evaluate the performance of the algorithm we tested it on

the synthetic dataset reported in [19]. This is constituted of sets of simulated dwMRI images produced considering different fibers configurations that can be found inside the Human brain. For our test we used five configurations: straight fibers, fibers that cross at different angles, fiber that cross on a curved trajectory with a long overlapping between the two fibers (curved crossing) and fibers that depart one from the other after running tangentially on a curved trajectory a long overlapping between the two fibers (curved kissing fibers). For the datasets that contain two set of fibers, one is stronger than the other with fractional anisotropy, FA = 0.8 and FA = 0.6 respectively. For each data set, results with three different levels of noise are considered: i) without noise, ii) with signal to noise ratio (SNR)=15, iii) SNR=30.

The test data simulate a protocol in which 30 diffusion images were acquired with diffusion sensitization b = 1000 s/mm2 and gradient orientation equally distributed on a semi-spherical shell, while 4 diffusion images were acquired with diffusion sensitization, b=0 s/mm2 (cf. (4)) according to the acquisition scheme reported in [22]. This is at the limit of feasibility in clinical studies where the acquisition has to be carried out in the shortest possible time [20, 21]. Registration of the synthetic data was omitted since the data are already aligned. From the dwMRI images, we reconstructed the diffusion tensors to be use as input to PDT. From the same

∀ starting point and direction

qn

pn xx 1ˆˆ −=

xqn

pn Σ+Σ=Σ −1

HHnx∇=~

1)~~(~ −Σ+ΣΣ= DTp

nTp

n HHHK

)ˆ~(ˆˆ pnn

pn

qn xHDKxx −+=

pn

qn HKI Σ−=Σ )~(

qnnn xrr ˆ1 += −

qnn

T xDH 1ˆ~−−=β

{ }ββ qn

Tnn ww Σ−= − 2

11 exp

dwMRI data, we derive the local diffusion, using a two fibers population model, through bedpostx and feed this into probtrackx.

For both PDT and probtrackx tracking was started inside all the voxels of a region of interest (ROI) defined by the user. The width of the ROI can be adjusted to span the diameter of the fibers. For the simulated data in Fig. 2, this is constituted of four adjacent voxels at one end of the tract.

B. In-vivo data Diffusion-weighted MRI data and high-resolution 3D T1 and

T2 weighted images were acquired on a Siemens 3T Tim Trio scanner with a 12-channel array head coil and maximum gradient strength of 45 mT/m. The diffusion-weighted data was acquired using dual-echo spin-echo echo planar imaging (SE-EPI) (TR=12 s, TE=100 ms, 72 axial slices of 128 x 128 pixels and a resolution of 1.72×1.72×1.7 mm, no cardiac gating). As parallel imaging scheme, a generalized autocalibration partially parallel acquisition (GRAPPA) technique (PAT-Factor 2.0) was chosen. Diffusion weighting was isotropically distributed along 60 directions (b-value=1000 s/mm2). Additionally, seven data sets with no diffusion weighting were acquired initially and interleaved after each block of 10 diffusion weighted images as anatomical reference for motion correction. The high angular resolution of the diffusion weighting directions improves the robustness of probability density estimation by increasing the signal-to-noise ratio (SNR) and reducing directional bias. To further increase SNR, scanning was repeated three times for averaging, requiring a total scan time for the diffusion weighted imaging (DWI) protocol of approximately 45 minutes. DWI data was acquired after the T2 weighted images in the same scanner reference system.

As a first step in pre-processing the data, the 3D T1 weighted (MPRAGE; TR=1300 ms, TI=650 ms, TE=3.97 ms, resolution 1.0×1.0×1.0 mm, flip angle 10°, 2 acquisitions) images were reoriented to the plane of anterior and posterior commissure. Upon reorientation, the 3D T2 weighted images (RARE; TR=2 s, TE=355 ms, resolution 1.0×1.0×1.0 mm, flip angle 180°) were co-registered to the reoriented 3D T1-weighted images using rigid-body transformations [23], implemented in [18]. The images without diffusion weightings were used to estimate motion correction parameters with the same registration method. The motion correction for the DWI data was combined with the global registration to the T1-weighted image. The gradient direction for each volume was corrected using the rotation parameters. The registered images were interpolated to an isotropic voxel resolution of 1.72 mm and the three corresponding acquisitions were averaged. Tractography was performed in original image space and then mapped onto T1 using the combined map.

V. RESULTS

A. Results on Synthetic Data

(a) (b)

(c) (d)

(a) (b)

(c) (d)

Figure 2 shows that both Probetrackx tractography and PDT tractography perform similarly on all these fibers types, but

Fig. 2. Tractography on Synthetic Data. In panels (a) and (b) tractograms produced by PDT are shown when the seed voxel is at the beginning of the first fibers bundle, that have lower fractional anisotropy (a) or the second fibers bundle (b). The same results produced by Probtrackx are shown in panels (c) and (d). For each row, different fibers arrangements are analyzed: linear, orthogonal crossing, wide angle crossing, curved crossing and kissing. For each of them three different noise levels are considered: no noise, SNR = 30 and SNR = 15.

PDT might be more suited to cope with crossing fibers when a fiber with a lower fractional anisotropy crosses a fiber with a higher anisotropy. When crossing fibers share long parts of their path both methods fail to produce a correct tracking (see fourth and ninth row of Fig. 2), while they both succeed in tracking kissing fibers along a curved trajectory (fifth and tenth row of Fig. 2). When fibers cross with a large angle, both methods trace the high fractional anisotropy fiber (rows two, three, seven and eight of panels (b) and (d), but only PDT is able to fit the correct fibers direction of the low anisotropy fiber (same rows, panels (a) and (c).

A. Results on In-Vivo Data The tractogram computed using FSL (A1-3) is reported in

the left column and the same tractogram computed with PDT on the right column (B1-3).

To analyze the difference between the two algorithms in a realistic situation, they have been applied to in-vivo diffusion MRI data acquired on a Human brain. For both tractography algorithms, a seed point has been placed in middle of the Corpus Callosum (MNI coordinate :-1.03, 18.2, 1.25).

Fig. 3. Tractography in in-vivo dwMRI Data. Left: probtrackx tractography, Right: PDT tractography. For both methods a tractogram was seeded in the middle of the Corpus Callosum (MNI coordinate:-1.03, 18.2, 1.25).

Visual inspection of the tractograms reveals that overall both tractographic methods are able to track the principal upgoing callosal connection over the confluence with the frontal part of the Corona Radiata up to the superior frontal cortex, and lateral connections across the Corona Radiata to the inferior frontal cortex. PDT evidentiates also a set of connections towards the Medial Frontal Cortex and an

unraveled, less intense (lower vote from (20)), branch of connections towards the orbito-frontal cortex. This is grossly in accordance with the anatomical knowledge on fiber pathways that has been recently validated by histological tracing studies in monkeys [24,25].

VI. DISCUSSION AND CONCLUSION Tractography algorithms perform tracking using, in general,

only local information and in most cases they use only directional information, neglect shape information and use eventually some heuristic on the path.

Our approach uses shape information and a prior based on a plausible model of the fibers to couple the local step with the information coming from all previous steps in the tracking process. The tracking process and the weight of the prior is modified runtime in an adaptive framework. Although our method is implemented upon a less powerful local model, the tensor, which is unable to encode multiple fibers, in difficult crossing problems, it outperforms a method based on a local model able to encode multiple fiber orientations, in particular when the data are noisy or when using a moderate number of diffusion gradient orientations.

On synthetic data our method has a comparable performance in tracking straight fiber or crossing fiber, as long as the crossing angle is odd. When fibers cross at sharp angle and when we are tracking a fiber with low anisotropy crossing with a fiber with higher anisotropy the inclusion of an adaptive prior based on the fiber model dramatically increase the accuracy. On the other side the inclusion of an adaptive prior does not force an erroneous prosecution of the fiber in fiber kissing problems.

Another advantage of our approach is that the method produces a measure related to the quality of the tractography given the data that can be used together with any integrable function to define a connectivity measure. The method proposed here is quite general and thus it can be implemented upon a more powerful local model, like rank-k tensor or orientation distribution functions. This upgrade is reserved for future work.

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