Kevin GINSBURGER1,2, Jean-François MANGIN2,3, and Cyril POUPON1,2
1I2BM / Neurospin / UNIRS, CEA, Gif-sur-Yvette, France, 2Université Paris-Saclay, Orsay, France, 3I2BM / Neurospin / UNATI, CEA, Gif-sur-Yvette, France
Synopsis
In
this work, we used the extended Multiple Correlation Function (MCF)
method to derive analytical expressions of the NMR signal in
multilayered cylinder geometries for an arbitrary direction of the
magnetic field gradient. Each layer of the cylinder is characterized
by a diffusion coefficient and a relaxation time and each boundary
between adjacent layers is characterized by a value of permeability
in order to allow the modeling of the multilayered structure of axons
surrounded by its myelin sheat.
Purpose
Diffusion MR microscopy has become an alternative to histology to
probe in vivo quantitative features of the white matter
microstructure, such as the local mean axon diameter and density.
Identifying the appropriate model is critical to providing reliable and quantitative information. Many degrees of freedom are needed in brain white matter
multi-compartmental models in order to capture diffusion effects1.
However, in state-of-the-art models2,3, a simple cylinder geometry is used to represent the
restricted compartment of water trapped within axons. Multilayered cylinders would better represent the axon structure (Fig2). Analytical
solutions for restricted diffusion in multilayered cylinders have
been provided using the powerful MCF formalism4. However, those solutions are only valid for a
fixed direction of the magnetic gradient waveform, which does not
allow to incorporate the effects of imaging gradients in
diffusion-weighted NMR scans. To alleviate this limitation, the MCF technique was extended by allowing for
variations in the direction of the gradient5, but no analytical
solution was provided for permeable multilayered cylinders. The aim
of this theoretical work is thus to compute this analytical solution
using the extended MCF approach.Methods
Laplace
operator eigenbasis
The solution to the Bloch-Torrey equations in multilayered cylinders (Fig1)
in terms of eigenfunctions unk(r,ϕ) of the Laplace operator is calculated using polar coordinates, enabling the separation
between radial and angular components :
unk(r,Φ)=CnkRnk(r)Φnk(ϕ)
The computation of Laplace operator eigenfunctions and of their
corresponding eigenvalues does not depend on the gradient direction.
The computation of the eigenfunctions radial part Rnk(r) and of the corresponding eigenvalues was provided with specific diffusion coefficient, relaxation times T2 per layer and allowing different permeabilities at each interface4. However, the angular part is different and takes the form Φnk(ϕ)=einϕ as we need
to express the generalized case of an arbitrary gradient orientation.
The normalization constant Cnk is set to fulfill
C2nk∫Rr0∫2π0|unk(r,ϕ)|2rdrdϕ=1
We thus obtain
unk(r,ϕ)=1√2π(R2−r20)InkRnk(r)einϕ
where
the integral Ink=1(R2−r20)∫Rr0Rnk(r)2rdr
has already been computed4.
Magnetic
field matrix
The reconstruction of the macroscopic signal using MCF necessitates
the computation of the magnetic field matrix A corresponding
to a linear gradient r . In standard MCF
approach, the linear gradient is oriented along the x-axis.
Thus, only Ax is computed . Following Bar and Sochen 6, we computed the y-component Ay
which generalizes the multilayered cylinder solution of Bloch-Torrey
equation to any gradient direction.
Results & Discussion
The Ax component is given by
Axnk,n′k′=∫Rr0∫2π0unk(r)u∗n′k′(r)r2cosϕdrdϕ
and the Ay component is given by
Aynk,n′k′=∫Rr0∫2π0unk(r)u∗n′k′(r)r2sinϕdrdϕ
For both x and y components, the radial integration
writes
∫Rr0r2Rnk(r)Rn′k′(r)dr=R(R2−r20)Knk,n′k′
where the expression of Knk,n′k′ has already been derived 4.
For the x component, integration over ϕ yields
\int_0^{2\pi}{ e^{i (n – n' ) \phi} \cos{\phi} d\phi } =
\pi\delta_{n', n \pm 1}
while for the y component,
\int_0^{2\pi}{ e^{i (n – n' ) \phi} \sin{\phi} d\phi} =
i\pi[\delta_{n', n – 1} - \delta_{n', n + 1}]
We finally get
\mathcal{ A }^x_{nk, n'k'} = \pi \delta_{n', n \pm 1}R( R^2 - r_0^2
)K_{nk, n'k'}
and
\mathcal{ A }^y_{nk, n'k'} = i\pi[\delta_{n', n – 1} -
\delta_{n', n + 1}]R( R^2 - r_0^2 )K_{nk, n'k'}
We have thus computed the magnetic field matrix A = (\mathcal{ A
}^x, \mathcal{ A }^y, 0) .
The present work shows that it is possible to generalize solutions of
Bloch-Torrey equations for multilayered geometries to the case of
arbitrary gradient directions. The same framework could be used in
other multilayered geometries to obtain more accurate analytical
solution of the diffusion inside cellular structures in white matter.Conclusion
We established solutions of the Bloch-Torrey equations for
a multilayered cylinder structure with specific diffusion
coefficient, relaxation times T_2 per layer and allowing different
permeabilities at each interface. This model is particularly adequate
to represent the microstructure of white matter fibers made up of an
axon surrounded by its myelin sheat, and could be used to integrate
the two components at the same time and to allow different
permeabilities at their interfaces.Acknowledgements
This work was partially funded by the European
FET Flagship ‘Human Brain Project’ (SP2)
FP7-ICT-2013-FET-F/604102.References
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