  The complex transverse water proton magnetization subject to
diffusion-encoding magnetic field gradient pulses in a heterogeneous medium
such as brain tissue can be modeled by the Bloch-Torrey partial differential
equation. The spatial integral of the solution of this equation in realistic
geometry provides a gold-standard reference model for the diffusion MRI signal
arising from different tissue micro-structures of interest. A closed form
representation of this reference diffusion MRI signal has been derived twenty
years ago, called Matrix Formalism that makes explicit the link between the
Laplace eigenvalues and eigenfunctions of the biological cell and its diffusion
MRI signal. In addition, once the Laplace eigendecomposition has been computed
and saved, the diffusion MRI signal can be calculated for arbitrary
diffusion-encoding sequences and b-values at negligible additional cost.
  Up to now, this representation, though mathematically elegant, has not been
often used as a practical model of the diffusion MRI signal, due to the
difficulties of calculating the Laplace eigendecomposition in complicated
geometries. In this paper, we present a simulation framework that we have
implemented inside the MATLAB-based diffusion MRI simulator SpinDoctor that
efficiently computes the Matrix Formalism representation for realistic neurons
using the finite elements method. We show the Matrix Formalism representation
requires around a few hundred eigenmodes to match the reference signal computed
by solving the Bloch-Torrey equation when the cell geometry comes from
realistic neurons. As expected, the number of required eigenmodes to match the
reference signal increases with smaller diffusion time and higher b-values. We
gave the transformation that links the Laplace eigenfunctions to the
eigenfunctions of the Bloch-Torrey operator and computed the Bloch-Torrey
eigenfunctions and eigenvalues.
