Fibers Estimation from Arbitrarily Spaced Diffusion Weighted MRI

, where we define a transverse isotropic diffusion tensor model of the form:



$$\begin{aligned} \varvec{D}(\theta (\varvec{x}),\lambda _1,\lambda _2)= & {} (\lambda _1 - \lambda _2) \varvec{f}(\theta (\varvec{x}), \varvec{x}) \otimes \varvec{f}(\theta (\varvec{x}), \varvec{x}) + \lambda _2 \varvec{I} \end{aligned}$$

(1)



$$\begin{aligned} \varvec{f}(\theta (\varvec{x}), \varvec{x})= & {} \cos (\theta (\varvec{x})) \varvec{c} (\varvec{x}) + \sin (\theta (\varvec{x})) \varvec{\ell }(\varvec{x}) \end{aligned}$$

(2)
for all $$\varvec{x} \in \varOmega $$, with $$\theta (\varvec{x})$$ the helix angle of the fiber and $$\varvec{c}, \varvec{\ell }$$ the local circumferential and long-axial directions, respectively (see Fig. 1(a)–(b)). The values $$\lambda _1>\lambda _2>0$$” src=”/wp-content/uploads/2016/09/A339585_1_En_23_Chapter_IEq5.gif”></SPAN> are the diffusivities in fiber and cross-fiber direction, respectively.<br /> <DIV id=Fig1 class=Figure><br /> <DIV id=MO3 class=MediaObject><IMG alt=A339585_1_En_23_Fig1_HTML.gif src=


Fig. 1.
Circumferential and long-axis coordinate system.


We assume the helix angle distribution is given by a set of degrees-of-freedom $$\varvec{\varTheta }\in \mathbb {R}^\kappa $$ (DOF), such that the following linear relation holds: $$\theta (\varvec{x}_j)= \varvec{H}_j \varvec{\varTheta }$$, $$\varvec{x}_1, \dots \varvec{x}_p \in \varOmega $$, with $$\varvec{x}_1, \dots \varvec{x}_p$$ the set of discrete mesh points. Each element of $$\varvec{\varTheta }$$ is associated to one vertex of the surface “patchs” depicted in Fig. 2(a)–(b). Hence, the operator $$\varvec{H} \in \mathbb {R}^{p \times \kappa }$$ summarizes the linear interpolations from the patchs-corners to the rest of the surface, and then from the surfaces to the heart’s volume using Poisson interpolation. Two sets of degree of freedom were used in the numerical examples, a low resolution set with 82-DOFs (Fig. 2(a)) and a high-resolution set with 322 DOFs (Fig. 2(b)). We assume that we have a set of measured diffusion weighted signal values $$A_{i,j}$$, for $$j=0,\dots ,N_i$$ voxels for each diffusion encoding direction $$\varvec{b}_0 = \varvec{0}$$, $$\Vert \varvec{b}_i \Vert = 1$$, $$i=1,\dots , N_g$$. The fiber estimation method consists then of the following steps:

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Fig. 2.
Patches definition on the prolate spheroid geometry (white spheres at the corners represent the DOFs used for the diffusion tensor model).




  • Step1: Being $$\varvec{A}_0$$ the values of the zero-weighted images at the measured voxels, we first reconstruct these values in the whole heart domain as


    $$ \varvec{\alpha }_{0} = \varvec{H} (\varvec{H}_0{^\intercal } \varvec{H}_0)^{-1} \varvec{H}^\intercal _0 \varvec{A}_0. $$
    The linear operator $$\varvec{H}_0$$ consists of the rows of $$\varvec{H}$$, which maps the DOFs to the zero-weighted slices.


  • Step2: We then minimize the functional


    $$\begin{aligned} \text {argmin}_{\varvec{\varTheta }, \lambda _1,\lambda _2} J(\varvec{\varTheta },\lambda _1,\lambda _2) = \sum _{i=1}^{N_g} \sum _{j=1}^{N_{i}} \left( exp\left( - \varvec{b}_i^{\intercal } \varvec{D}( \theta _j(\varvec{\varTheta }), \lambda _1, \lambda _2) \varvec{b}_i\right) - \frac{A_{i,j}}{\alpha _{0,j}} \right) ^2 \end{aligned}$$

    (3)
    Note that since we directly compare the mismatch in the graduation itself, each voxel and diffusion gradient are equally weighted in the functional.


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Fig. 3.
Synthetic fibers organization for healthy (left) and infarcted (right) cases. The infarction is located in the area with more horizontal (yellow) fibers (Color figure online).


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Fig. 4.
Positions of the measured DWIs (Color figure online).


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Fig. 5.
Sample DWI slice for the healthy (a) and infarcted (b) case without (left) and with (right) noise. Infarcted region in (b) is indicated with a red circle. The voxels are coloured by the signal intensity ($$A_{i,j}$$-value) (Color figure online).

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