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Medical Image Registration using optimal control of a linear hyperbolic transport equation with a DG discretization

Bastian Zapf, Johannes Haubner, Lukas Baumgärtner, Stephan Schmidt

TL;DR

This work tackles automated generation of patient-specific brain meshes by registering a template MRI to a target MRI through a PDE-constrained optimal control problem governed by a linear hyperbolic transport equation. The authors adopt a velocity-based, Eulerian formulation discretized with a high-order discontinuous Galerkin scheme, and they address non-differentiability in the flux via a smoothed upwind flux, enabling gradient-based optimization in FEniCS/dolfin-adjoint. A key contribution is the introduction of an auxiliary velocity variable and a mass-matrix based control transformation to fit a workable function-space setting, coupled with a multi-step deformation strategy to capture large-scale then fine-scale deformations. The approach is demonstrated on two subjects, Abby and Ernie, showing that the registered template mesh can be transformed to match target anatomy while preserving mesh quality, with discussion of limitations and potential extensions toward higher performance and neural-network-inspired implementations.

Abstract

Patient specific brain mesh generation from MRI can be a time consuming task and require manual corrections, e.g., for meshing the ventricular system or defining subdomains. To address this issue, we consider an image registration approach. The idea is to use the registration of an input magnetic resonance image (MRI) to a respective target in order to obtain a new mesh from a template mesh. To obtain the transformation, we solve an optimization problem that is constrained by a linear hyperbolic transport equation. We use a higher-order discontinuous Galerkin finite element method for discretization and motivate the numerical upwind scheme and its limitations from the continuous weak space--time formulation of the transport equation. We present a numerical implementation that builds on the finite element packages FEniCS and dolfin-adjoint. To demonstrate the efficacy of the proposed approach, numerical results for the registration of an input to a target MRI of two distinct individuals are presented. Moreover, it is shown that the registration transforms a manually crafted input mesh into a new mesh for the target subject whilst preserving mesh quality. Challenges of the algorithm are discussed.

Medical Image Registration using optimal control of a linear hyperbolic transport equation with a DG discretization

TL;DR

This work tackles automated generation of patient-specific brain meshes by registering a template MRI to a target MRI through a PDE-constrained optimal control problem governed by a linear hyperbolic transport equation. The authors adopt a velocity-based, Eulerian formulation discretized with a high-order discontinuous Galerkin scheme, and they address non-differentiability in the flux via a smoothed upwind flux, enabling gradient-based optimization in FEniCS/dolfin-adjoint. A key contribution is the introduction of an auxiliary velocity variable and a mass-matrix based control transformation to fit a workable function-space setting, coupled with a multi-step deformation strategy to capture large-scale then fine-scale deformations. The approach is demonstrated on two subjects, Abby and Ernie, showing that the registered template mesh can be transformed to match target anatomy while preserving mesh quality, with discussion of limitations and potential extensions toward higher performance and neural-network-inspired implementations.

Abstract

Patient specific brain mesh generation from MRI can be a time consuming task and require manual corrections, e.g., for meshing the ventricular system or defining subdomains. To address this issue, we consider an image registration approach. The idea is to use the registration of an input magnetic resonance image (MRI) to a respective target in order to obtain a new mesh from a template mesh. To obtain the transformation, we solve an optimization problem that is constrained by a linear hyperbolic transport equation. We use a higher-order discontinuous Galerkin finite element method for discretization and motivate the numerical upwind scheme and its limitations from the continuous weak space--time formulation of the transport equation. We present a numerical implementation that builds on the finite element packages FEniCS and dolfin-adjoint. To demonstrate the efficacy of the proposed approach, numerical results for the registration of an input to a target MRI of two distinct individuals are presented. Moreover, it is shown that the registration transforms a manually crafted input mesh into a new mesh for the target subject whilst preserving mesh quality. Challenges of the algorithm are discussed.
Paper Structure (17 sections, 1 theorem, 55 equations, 10 figures, 1 table)

This paper contains 17 sections, 1 theorem, 55 equations, 10 figures, 1 table.

Table of Contents

  1. Introduction
  2. Pre--processing
  3. Image registration
  4. Relation to other approaches and motivation for our choice
  5. Choice of the objective function
  6. Choice of the transformation
  7. Modeling of the optimization problem
  8. Discretization
  9. Upwind DG scheme
  10. Formal motivation of upwind DG formulation as approximation of space--time variational formulation
  11. Smoothed upwinding
  12. Optimization
  13. Numerical results
  14. Mesh transformation
  15. Mesh transformation pipeline
  16. ...and 2 more sections

Key Result

Lemma 2

Let $\Omega \subset \mathbb R^d$, $d \geq 1$, be a bounded Lipschitz domain, $T > 0$ and $v \in L^1 ((0,T), W^{1, \infty}(\Omega)^d)$ with $v \vert_{\partial \Omega} = 0$. Moreover, for $t_0 \in (0,T)$, let $X_{t_0}$ be defined as the solution of Then, the transformation $\tau: \Omega \to \Omega$ defined by $\tau(x) = \tau_{{t_0}, s}(x) := X_{t_0}(x, s)$ is bi-Lipschitz continuous for all $s \in

Figures (10)

  • Figure 1: Sketch of the proposed mesh deformation pipeline to generate a mesh for target "Ernie" from an existing mesh for "Abby". After registering the MRI from "Abby" to "Ernie", the transformation is used to transform the mesh of "Abby" to obtain a mesh for "Ernie". \newlabelfig:idea0
  • Figure 2: Sketch of pre--processing pipeline. Top: Abby; Bottom: Ernie; From left to right: T1-weighted MRI, output of FreeSurfer recon--all (brain.mgz), affine pre--registration (only changes Abby image), Nyul-normalization and crop to bounding box.
  • Figure 3: Illustration of our multistep registration approach. Starting from the affine pre--registered image $\phi_a$ (blue box), we apply a sequence of velocity field transformations to register the image to the target $\phi_e$ (green box). Note that the vectors representing the velocity field are scaled equally in all illustrations, indicating that the first transformations learn the large distance deformations, while the last transformations learn to register small--scale details. Further note that changing the hyperparameters $\alpha, \beta$ for $\mathcal{S}_4$ yields a velocity field with increased local variations as visible by the two large velocity vectors in the lower row. \newlabelfig:multistep0
  • Figure 4: Reduction of the $L^2$-norm between deformed image and target during optimization of the velocity field deformations $\mathcal{S}_i$, $i=1,2,3,4$, with the L-BFGS-B algorithm. $\phi_1,\phi_2,\phi_3$ are obtained with $\alpha=0, \beta=1$ and $\phi_4$ is obtained with $\alpha=0.5, \beta=0.5$. \newlabelfig:optimization0
  • Figure 5: The figure compares the results of our registration algorithm with three and four steps as well as CLAIRE mang2019claire. Background MR images: Slices from target image "Ernie" $\tilde{\phi}_e$ (brain.mgz, cf. Fig. \ref{['fig::preprocessing']}). The heatmap shows the absolute difference between target $\tilde{\phi}_e$ and images registered with different procedures based on "Abby" $\tilde{\phi}_a$ (brain.mgz, cf. Fig. \ref{['fig::preprocessing']}).
  • ...and 5 more figures

Theorems & Definitions (10)

  • Remark 1
  • Lemma 2: see jarde2018analysis
  • Proof 1
  • Remark 3
  • Remark 4
  • Remark 5
  • Remark 6
  • Remark 7
  • Remark 8
  • Remark 9