Carlos Castillo-Passi^{1,2}, Gabriel Varela-Mattatall^{1,2}, Claudia Prieto^{1,2,3}, Carlos Sing-Long^{2,4,5}, and Pablo Irarrazaval^{1,2,5}

Inspired in the solution of the diffusion equation in the restricted case, we propose to express the diffusion $$$q$$$-space information in a restricted basis. This representation allows to obtain the distribution of barriers separations, thus providing useful information about the micro-structure. Previous methods used multiple Diffusion Spectrum Imaging (DSI) images with different diffusion times, which is impractical to characterize barriers in multiple directions. Our method proposes to obtain the barrier distribution with only a single DSI image. Furthermore, the model does not use a strong assumption for the geometry of the barriers (or axons) nor for the probability distribution of the barrier separation.

Diffusion Spectrum Imaging acquires the Fourier transform of the Ensemble Average Propagator (EAP)^{7,8}. This is obtained theoretically by first solving the diffusion equation. The problem of interest is to solve this equation for a non-permeable enclosure of separation $$$s$$$. The mathematical formulation of this phenomena for a particle positioned at $$$x_0$$$ is described by the following differential equation and boundary conditions

$$ \begin{cases} p_{xx}=\mathrm{D}p_{t}\\ p\left(x,0\right)=\delta\left(x-x_{0}\right)\\ p_{x}\left(x,t\right)=0 & x\in\left\{ 0,\text{s}\right\} . \end{cases}$$

Solving this equation we can obtain a probability distribution (propagator) of the position of a particle which is

$$ p\left(x|x_{0}\right)=\frac{1}{s}\sqcap\left(\frac{x-s/2}{s}\right)\left[1+2\sum_{n=1}^{\infty}\cos\left(\frac{n\pi}{s}x\right)\cos\left(\frac{n\pi}{s}x_{0}\right)e^{-\mathrm{D}\Delta\left(n\pi/s\right)^{2}}\right].$$

Applying the ensemble average and then a Fourier transform we obtain the Tanner signal equation^{9,10}

$$ P_{s,\tau}\left(q\right)=\mathrm{sinc}^{2}\left(sq\right)+2\sum_{n=0}^{\infty}\left(\frac{sq}{sq-n/2}\right)^{2}\mathrm{sinc}^{2}\left(sq+\frac{n}{2}\right)e^{-\mathrm{D}\Delta\left(n\pi/s\right)^{2}}\quad[1]\\=\sum_{n=0}^{\infty}a_{n}\left(\frac{sq}{sq-n/2}\right)^{2}\mathrm{sinc}^{2}\left(sq+\frac{n}{2}\right),\quad\quad\quad\quad\quad\quad\quad\quad$$

where we grouped the constants in $$$ a_{n}$$$. Then using $$$\left\{ P_{s,\tau}\right\} _{s\in\mathbb{R}}$$$ (as seen in Figure 2) as an atom of a dictionary we will reconstruct the barrier distribution for a given diffusion direction.

The obtained signal given a distribution of barriers is

$$ E\left(q\right)=\int_{0}^{\infty}\omega\left(s\right)P_{\tau}\left(sq\right)\mathrm{d}s\quad[2],$$

with $$$w( s )$$$ the probability distribution of finding a given barrier separation $$$s$$$. If we discretize $$$q$$$ and $$$s$$$ in [2], we obtain

$$ \vec{E}=\mathrm{P}\vec{\omega}.\quad[3]$$

Where $$$ \vec{E}$$$ is the signal in $$$q$$$-space, $$$ \mathrm{P}$$$ is a matrix representing the dictionary made from [1] and $$$\vec{\omega}$$$ is the barrier distribution. Since the linear system in [3] is not determined, we propose to solve an optimization problem using a Tikhonov regularization

$$ \hat{\omega}=\underset{\vec{\omega}}{\mathrm{argmin}}\left\Vert \mathrm{P}\vec{\omega}-\vec{E}\right\Vert _{2}+\lambda\left\Vert \vec{\omega}\right\Vert _{2}.$$

The advantage of this optimization problem is that it has a close solution $$$ \hat{\omega}=\left(\mathrm{P}^{\mathrm{T}}\mathrm{P}+\lambda^{2}\mathbb{I}\right)^{-1}\mathrm{P}^{\mathrm{T}}\vec{E}$$$. Which can be solved without the use of a complex optimization algorithm. The method was tested on a simulated signal of 11 samples with $$$ q_{\max}=0.3\mu\mathrm{m}^{-1}$$$ and a $$$ s\sim\Gamma\left(5\mu\mathrm{m},1\mu\text{m}\right)$$$ adding Gaussian noise of different standard deviations $$$\sigma$$$ . Figure 3 shows the barriers in a pixel for an instance the $$$\Gamma$$$ probability distribution.

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Figure 1:
For various diffusion times $$$
\tau=\mathrm{D}\Delta$$$ and barrier separations $$$s$$$ , the felt restriction level varies as $$$
\rho=\tau/s^{2}$$$. In this example, if we want to retrieve each weight $$$
w_{n}$$$ we need at least four different diffusion times so the system $$$
\mathrm{P}\vec{\omega}=\vec{E}$$$ has a solution.

Figure 2:
The behavior of each $$$
P_{s,\tau}\left(q\right)$$$ is different depending on the barrier separation.

Figure 3:
Example of a gamma distribution of barriers (representing axons in one direction). The square represents a pixel of the image in where the barriers are present.

Figure 4: The barrier distribution is obtained from the signal $$$E(q)$$$. In the top row without noise and in the bottom row after adding Gaussian noise with std. $$$\sigma=0.1$$$ (10%). Even with noise the fit between the theoretical and the estimated barrier distribution is very good.

Figure 5:
An error bar plot of the NMRSE is shown. The error increases with the noise of the data.