Jonathan Trevathan^{1}, Jonathan Scott^{1}, Joshua Trzasko^{1}, Armando Manduca^{1}, John Huston^{1}, Richard Ehman^{1}, and Matthew Murphy^{1}

^{1}Mayo Clinic, Rochester, MN, United States

While most existing inversion algorithms used in MR elastography assume that the mechanical properties of tissue are isotropic, many tissues exhibit spatial anisotropy in structure that is not accommodated by these algorithms.^{1,2} In this work we present a framework for developing a learned inversion to address transverse isotropy, the simplest anisotropic case. A transversely isotropic stiffness matrix was used in a feed forward finite difference model to generate simulated displacements. The squared wave speeds anisotropic inclusions were calculated using direct inversion to validate the model against the theoretical wave speeds.^{3
}

The wave equation for harmonic motion in an elastic material is given by $$$\nabla\cdot\sigma=\frac{\partial^2 u}{\partial t^2}$$$, where $$$u$$$ is the displacement field and $$$\sigma$$$ is the stress field. For Hookean materials under small strain, this wave equation can be approximated as a linear system according to:

$$-D^TR^TCRED(u+ub)=-\rho\omega^2*I(u+ub)$$

where $$$u$$$ are the displacements to be estimated, $$$ub$$$ are the boundary conditions that induce motion, $$$\rho$$$ is density, $$$\omega$$$ is angular frequency, $$$D$$$ is the gradient operator, $$$E$$$ is a strain operator, $$$R$$$ rotates the strains into the material reference frame, and $$$C$$$ is the stiffness matrix. This system can be rearranged into an overdetermined linear system, $$$Au = f$$$, whose least squares solution is given by $$$u = (A*A)^{-1}A*f$$$, where $$$*$$$ denotes the adjoint of an operator. After parameterizing the material properties, material coordinates, and boundary conditions, we solved this system for each simulation using conjugate gradient method, stopping after 10,000 iterations.

For a transversely isotropic material, the stiffness matrix is governed by 5 independent parameters. In this study we parameterized those values according to previously defined formulas,

First, simulated examples were made to confirm wave propagation conformed to the expected anisotropic directions. A cubic region with shear anisotropy of 1 and tensile anisotropy of 1 was created using the stiffness formulation in equations 1. This results in anisotropy in the $$$x$$$ direction. A simple case was devised in which a rod placed in the center of the volume, along the $$$z$$$ direction induces motion in the $$$z$$$ direction. As expected, the resulting shape of the displacements appears as an ellipse (Fig 1). To confirm rotation was working as expected, the fiber direction was rotated at $$$\pi/4$$$ and $$$\pi/2$$$ radians about the $$$z$$$-axis. As expected, the long axis of the ellipse was rotated $$$\pi/4$$$ and $$$\pi/2$$$ radians, respectively.

To validate the implementation, simulated examples were made to confirm that calculated wave speeds match the known theoretical values for fast and slow waves (Equation 2).

In the case of the anisotropy rotating around the $$$z$$$-axis the expected result is for the DI wave speed in the anisotropic cube to match the theoretical slow wave speed described in equation 2. For this rotation, the shear anisotropy, $$$\phi$$$, was $$$1$$$ and tensile anisotropy, $$$\zeta$$$, was $$$0$$$. When the anisotropy is rotating around the $$$y$$$ axis, the expected result is for the wave speed in the anisotropic inclusion to match the fast wave speed described in equation 2. For this rotation, shear anisotropy, $$$\phi$$$, was $$$0$$$ and tensile anisotropy, $$$\zeta$$$, was $$$1$$$. For each axis of rotation, the angle was varied from $$$0$$$ to $$$\pi/2$$$ uniformly in 16 steps.

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Equations 1: Numerical
notation of strain, stress, stiffness matrix relation and components of the
stiffness matrix.

Equations 2:
Theoretical equations for fast and slow wave speeds.

Figure 1:
Ellipses produced by driving a rod in the $$$z$$$ direction.

Figure 2: Left:
True wave speeds for background and inclusion. Middle: FDM generated wave
displacements with inclusion present. Right: Direct inversion with anisotropic
inclusion.

Figure 3: Anisotropic wave speed calculated with direct inversion with
anisotropy rotations from $$$0$$$ to $$$\pi/2$$$. Left: Rotation about $$$z$$$-axis, shear
anisotropy of $$$1$$$, tensile anisotropy of $$$1$$$. Middle: Rotation about $$$y$$$-axis, shear
anisotropy of $$$1$$$, tensile anisotropy of $$$0$$$. Right: Rotation about $$$y$$$-axis, shear
anisotropy of $$$0$$$, tensile anisotropy of $$$1$$$.

DOI: https://doi.org/10.58530/2022/4076