Matrix formalism

The Matrix Formalism [17], [18] representation of the solution to the Bloch-Torrey PDE is based on the generalized Laplace eigenfunctions of the given geometrical configuration. These are the eigenfunctions of the generalized Laplace operator   subject to the same boundary and interface conditions as the BTPDE. It which does not depend on the gradient sequence. The Matrix Formalism truncates the number of Laplace eigenmodes at an acceptable level, by setting a minimum length scale for the eigenfunctions. The gradient sequence dependent Bloch-Torrey operator can then be expressed in the truncated Laplace eigenfunction basis. This allows for a simple analytical expression which approximates the solution to the BTPDE down to an arbitrary length scale.

Eigenvalue decomposition of the generalized Laplace operator

Let be the -normalized eigenfunctions and eigenvalues of the generalized Laplace operator  , defined almost everywhere on the domain  . Denoting the restriction of to compartment , the eigenfunctions respect the following set of equations, for  :

where the boundary conditions are the same as for the BTPDE:

Let the solutions to the above equations be denoted by . We assume the non-negative real-valued eigenvalues are ordered in non-decreasing order:

If the domain consists of only one contiguous group of compartments connected through a chain of permeable membranes, only the first eigenvalue will be zero, and the corresponding eigenfunction will be the only constant function. If there are   groups of connected compartments completely separated by interior hard wall membranes, the first eigenvalues will be zero:

and there will be corresponding groupwise constant eigenfunctions. In the latter case, the equations may be rewritten separately for each connected subdomain to obtain a set of eigenvalues with a multiplicity of one, but the formulation is also valid in the global form with multiple zero eigenvalues. These two formulations will lead to an identical eigenfunction basis (up to a linear combination for the eigenvalues with multiplicity higher than one), where the basis of each subdomain is a subset of the global eigenfunction basis.

Let be the diagonal matrix containing the first Laplace eigenvalues:

Then the matrix represents the generalized Laplace operator   in the truncated Laplace eigenfunction basis.

Bloch-Torrey operator in the Laplace eigenfunction basis

Let be the   matrix defined by:

where   is the gradient vector and , , and are three symmetric   matrices whose entries are the first order moments in the coordinate directions of the product of pairs of eigenfunctions:

Similarly, let be the   Laplace relaxation matrix defined by

Then the general time dependent Bloch-Torrey operator

in the truncated Laplace eigenfunction basis is given by the complex-valued matrix

Matrix Formalism approximation

Denoting   the vector of the first Laplace eigenfunctions, we may project the solution to to the Bloch-Torrey PDE onto the truncated Laplace eigenfunction basis:

where the vector of coefficients     is the solution to

The initial coefficients are given by

By using a piece-wise constant approximation of the gradient [18], we obtain

where are intervals such that  ,   for  ,  , and  . The constants may be computed through quadrature:

For the PGSE and double-PGSE sequences  , where    for all , the coefficients of the final magnetization are given by

and

respectively. These are the exact solutions, although there may still be truncation errors from .

The signal is given by

Apparent diffusion coefficient

From the Matrix Formalism signal, the analytical expression of its ADC for a gradient direction given by a unit vector and a sequence is the following:

where the coefficients of the three-row matrix    are given by

To clarify the relationship between the ADC and the diffusion encoding direction , we rewrite the Matrix Formalism ADC as

where the Matrix Formalism effective diffusion tensor is given by

with    depending on :

In the case of a constant initial spin density , we also allow for computing the Matrix Formalism Gaussian Approximation (MFGA) signal, given as

Eigenfunction length scale and orientation

On a line segment of length and diffusivity , the eigenvalues of the Laplace operator with Neumann boundary conditions are

To make the link between the computed eigenvalue and the spatial scale of the eigenmode, we will convert the computed into a length scale:

and characterize the computed eigenmode by instead of . The reference diffusivity is taken as a volume weighted mean of the trace average (spherical part) of :

To characterize the directional contribution of the eigenmode we use the fact that its contribution to the ADC in the direction is  . We thus call   the "diffusion direction" of the th eigenmode. We remind that the three components of are the first moments in the 3 canonical basis directions of the associated eigenfunction.