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Debye Relaxation in Model-Based Multi-Dimensional Magnetic Particle Imaging

Vladyslav Gapyak, Thomas März, Andreas Weinmann

Abstract

Model-based reconstruction approaches for the medical imaging modality Magnetic Particle Imaging (MPI) are typically based on the Langevin model, which assumes instantaneous alignment of the particles magnetic momenta with the applied field. Regarding the application to real data, Langevin model-based reconstruction methods require model transfer functions (MTF) obtained from calibrations to preprocess the data. There are also model-based reconstruction approaches that include relaxation effects and other particle-level dynamics. However, they are limited either to 1D or 1D-like scanning scenarios when considering real data, or are limited to simulated data in the case of multi-dimensional field-free point (FFP) MPI. Thus, fully model-based reconstructions from multi-dimensional FFP scanning data that incorporate relaxation effects without using an MTF have not yet been demonstrated. In this work, we incorporate relaxation effects by considering a multi-dimensional Debye model and provide reconstruction formulae. In particular, we show that the Debye model-based signal is the response of a linear time-invariant system with exponential memory applied to a Langevin model-based signal. We provide a reconstruction algorithm for the introduced multi-dimensional Debye model. To this end, we devise a relaxation adaption step. For the resulting relaxation-adapted Debye signal, we show that it can be expressed by the well-studied MPI core operator derived from the Langevin theory. This results in a three-stage algorithm with low additional cost over the Langevin model, as the relaxation adaption scales linearly in the input data. We provide numerical results for the proposed algorithmic approach. In particular, we obtain fully model-based reconstructions from real 2D MPI data without involving any specific MTF analogous to the Langevin model case.

Debye Relaxation in Model-Based Multi-Dimensional Magnetic Particle Imaging

Abstract

Model-based reconstruction approaches for the medical imaging modality Magnetic Particle Imaging (MPI) are typically based on the Langevin model, which assumes instantaneous alignment of the particles magnetic momenta with the applied field. Regarding the application to real data, Langevin model-based reconstruction methods require model transfer functions (MTF) obtained from calibrations to preprocess the data. There are also model-based reconstruction approaches that include relaxation effects and other particle-level dynamics. However, they are limited either to 1D or 1D-like scanning scenarios when considering real data, or are limited to simulated data in the case of multi-dimensional field-free point (FFP) MPI. Thus, fully model-based reconstructions from multi-dimensional FFP scanning data that incorporate relaxation effects without using an MTF have not yet been demonstrated. In this work, we incorporate relaxation effects by considering a multi-dimensional Debye model and provide reconstruction formulae. In particular, we show that the Debye model-based signal is the response of a linear time-invariant system with exponential memory applied to a Langevin model-based signal. We provide a reconstruction algorithm for the introduced multi-dimensional Debye model. To this end, we devise a relaxation adaption step. For the resulting relaxation-adapted Debye signal, we show that it can be expressed by the well-studied MPI core operator derived from the Langevin theory. This results in a three-stage algorithm with low additional cost over the Langevin model, as the relaxation adaption scales linearly in the input data. We provide numerical results for the proposed algorithmic approach. In particular, we obtain fully model-based reconstructions from real 2D MPI data without involving any specific MTF analogous to the Langevin model case.
Paper Structure (24 sections, 2 theorems, 33 equations, 6 figures, 1 table, 1 algorithm)

This paper contains 24 sections, 2 theorems, 33 equations, 6 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related work
  3. Time-domain model-based approaches (X-space family).
  4. Frequency-domain model-based approaches (Chebyshev family) and system-function family.
  5. X-space Debye relaxation models.
  6. Particle-level Debye-type and hysteresis models.
  7. Contribution
  8. Methods
  9. The Langevin model
  10. The Debye model and the relaxation adaption
  11. Reconstruction algorithm
  12. MPI Core Stage.
  13. Deconvolution Stage and multi-kernel deconvolution.
  14. Experiments
  15. Preprocessing of real data.
  16. ...and 9 more sections

Key Result

Lemma 2.1

With the assumptions (P1) and (P2) the Cauchy problem for any initial state $\bm{M}_0 \in L^1(\mathbb{R}^n; \mathbb{R}^n)$ is well defined for almost every $\bm{x} \in \mathbb{R}^n$. The magnetization $\bm{M}(\bm{x},t)$ is given by Duhamel's formula and $\bm{M}(\bm{x},t)$ is integrable over $\mathbb{R}^n$ for every fixed $t \geq 0$.

Figures (6)

  • Figure 1: Pictures of 3 of the phantoms that are scanned in the real dataset knopp2024equilibriumdata. Reproduced from knopp2024equilibriumdata, https://creativecommons.org/licenses/by/4.0/.
  • Figure 2: Overview of the average PSNR values over the simulated phantoms.
  • Figure 3: Reconstruction results after the MPI Core Stage (Trace) and the Deconvolution Stage (Deco.) of the 5 simulated phantoms for a variety of relaxation adaption parameters $\tau$ (rows). The ground truth delay $\tau_{\mathrm{GT}} = 5\cdot 10^{-6}$ is underlined. We have diplayed the traces obtained after the MPI Core Stage with the parameter $\gamma^*$. The final reconstructions are obtained using the deconvolution parameter $\nu_0^* = 10^{-7}$ using all entries of the MPI Core Response. We observe that the reconstructions obtained with $\tau = \tau_{\mathrm{GT}} = 5\cdot 10^{-6}$ are among the closest to the ground truths and exhibit the smallest amount of artifacts.
  • Figure 4: Reconstruction results on the dot phantom in figure \ref{['subfig:dot']} for a variety of relaxation adaption parameters $\tau_x$ and $\tau_y$. All other parameters are fixed. We observe in figure \ref{['subfig:dot:traces']}, that when $\tau_x$, $\tau_y < 10^{-6}$, the trace and the subsequent deconvolution are a blurry version of the dot phantom. Increasing the size of $\tau_x$, $\tau_y$ helps obtaining a trace and a reconstruction in which the dot-phantom is less blurred. Observing the reconstructions in figure \ref{['subfig:dot:recos']} row-wise and column-wise suggests that increased correction in one of the channels helps obtaining less blur in that direction. On the other hand, we observe that, when the relaxation parameters $\tau_x$, $\tau_y\geq 5\cdot 10^{-6}$, reconstruction artifacts appear more strongly, the bigger the relaxation parameter.
  • Figure 5: Reconstruction obtained with the relaxation adaption step of the three phantoms in \ref{['fig:real:phantoms']} from the real data in the dataset knopp2024equilibriumdata. Comparing the reconstructions with the pictures in figure \ref{['fig:real:phantoms']}, we observe that the shape and the main features of the phantoms are reconstructed.
  • ...and 1 more figures

Theorems & Definitions (4)

  • Lemma 2.1
  • proof
  • Theorem 2.2
  • proof