# Inverse problems and magnetic particle imaging

Since the discovery of X-ray tomography in the 1970s, imaging techniques have continuously revolutionized medical diagnostics. Nowadays, there are various tomographic procedures in the clinical environment, which are applied differently due to their specific advantages and disadvantages. The most important procedures are computed tomography (CT), magnetic resonance imaging (MRI), and various functional technologies such as PET (positron emission tomography) and SPECT (single-photon emission computed tomography).

Since the early 2000s, Magnetic Particle Imaging (MPI) has made its way into research-based medicine. The basic idea is to inject patients with magnetic nanoparticles, iron oxide particles called tracers, and locate their concentration in the body with externally applied magnetic fields. The technique is radiation-free and has a very high temporal resolution. Currently, this technology has significant challenges that are being investigated in our team. These include detailed mathematical modeling of the measurement process. In addition, due to the current lack of data, many of the most common data-driven methods are not directly applicable.

## Modelling

Finding a sufficiently accurate model to reflect the behavior of large numbers of particles for MPI remains an open problem. As such, reconstruction is still computed using a measured forward operator obtained in a time-consuming calibration process. The model commonly used to illustrate the imaging methodology and obtain first model-based reconstructions relies on substantial model simplifications. One needs to take into account the magnetization dynamics of the nanoparticles' magnetic moment (red) such as Brownian (left) and Neel rotation (right) into the direction of the applied field (green). By neglecting particle-particle interactions, the forward operator can be expressed by a Fredholm integral operator of the first kind which yields the the inverse problem for image reconstruction.

## Modelling

Finding a sufficiently accurate model to reflect the behavior of large numbers of particles for MPI remains an open problem. As such, reconstruction is still computed using a measured forward operator obtained in a time-consuming calibration process. The model commonly used to illustrate the imaging methodology and obtain first model-based reconstructions relies on substantial model simplifications. One needs to take into account the magnetization dynamics of the nanoparticles' magnetic moment (red) such as Brownian (left) and Neel rotation (right) into the direction of the applied field (green). By neglecting particle-particle interactions, the forward operator can be expressed by a Fredholm integral operator of the first kind which yields the the inverse problem for image reconstruction.

## Projects

### D-MPI - Dynamic Inverse Problems in Magnetic Particle Imaging

DFG-Projekt**Duration**: 01.05.2020 - 30.04.2023**PI**: Tobias Kluth

Magnetic Particle Imaging (MPI) is an imaging technique with promising medical applications based on the behavior of superparamagnetic iron oxide nanoparticles. In D-MPI, the dynamic aspects of concentration dynamics, magnetic field dynamics, and particle magnetization dynamics are studied to facilitate modeling and reconstruction of the data.

### MPI² - Model-based parameter identification in Magnetic Particle Imaging

BMBF-Project**Duration**: 01.12.2016 - 31.05.2020**PIs:** Peter Maaß, Tobias Kluth

In MPI², model-based methods and their efficient algorithmic implementation are explored. Magnetic particle imaging (MPI), a tomographic method based on tracking iron oxide nanoparticles in the human body, serves here as an application example.

## Publications

T. Kluth, C. Bathke, M. Jiang, P. Maaß.**Joint super-resolution image reconstruction and parameter identification in imaging operator: Analysis of bilinear operator equations, numerical solution, and application to magnetic particle imaging.**

Erscheint in Inverse Problems

online unter: https://arxiv.org/abs/2004.13091

**L1 data fitting for robust reconstruction in magnetic particle imaging: quantitative evaluation on Open MPI dataset.**

Erscheint in International Journal on Magnetic Particle Imaging

online unter: https://arxiv.org/abs/2001.06083

S. Dittmer, T. Kluth, P. Maaß, D. Otero Baguer.

**Regularization by architecture: A deep prior approach for inverse problems.**

Journal of Mathematical Imaging and Vision, :456-470, Springer Verlag, 2020.

DOI: 10.1007/s10851-019-00923-x

online unter: http://link.springer.com/article/10.1007/s10851-019-00923-x

T. Kluth, B. Jin.**Enhanced Reconstruction in Magnetic Particle Imaging by Whitening and Randomized SVD Approximation.**

Physics in Medicine and Biology, Article ID 125026 64(12), 2019.

T. Kluth, P. Szwargulski, T. Knopp.**Towards Accurate Modeling of the Multidimensional Magnetic Particle Imaging Physics.**

New Journal of Physics, Article ID 10303 21, 10 pp., 2019.

online unter: https://iopscience.iop.org/article/10.1088/1367-2630/ab4938/pdf

T. Kluth.**Mathematical models for magnetic particle imaging.**

Inverse Problems, Article ID 083001 34(8), 2018.

T. Kluth, B. Jin, G. Li.**On the Degree of Ill-Posedness of Multi-Dimensional Magnetic Particle Imaging.**

Inverse Problems, Article ID 095006 34(9), 2018.

C. Bathke, T. Kluth, C. Brandt, P. Maaß.**Improved image reconstruction in magnetic particle imaging using structural a priori information.**

International Journal on Magnetic Particle Imaging, Article ID 1703015, 3(1), 10 pages, 2017.

DOI: 10.18416/ijmpi.2017.1703015

T. Kluth, P. Maaß.**Model uncertainty in magnetic particle imaging: Nonlinear problem formulation and model-based sparse reconstruction.**

International Journal on Magnetic Particle Imaging, Article ID 1707004 3(2), 10 pages, 2017.

DOI: 10.18416/ijmpi.2017.1707004

H. Albers, T. Kluth, T. Knopp.**A simulation framework for particle magnetization dynamics of large ensembles of single domain particles: Numerical treatment of Brown/Néel dynamics and parameter identification problems in magnetic particle imaging.**

Zur Veröffentlichung eingereicht.

online unter: https://arxiv.org/abs/2010.07772

S. Dittmer, T. Kluth, M. Henriksen, P. Maaß.**Deep image prior for 3D magnetic particle imaging: A quantitative comparison of regularization techniques on Open MPI dataset.**

Zur Veröffentlichung eingereicht.

online unter: https://arxiv.org/abs/2007.01593

T. Kluth, B. Jin.**Exploiting heuristic parameter choice rules for one-click image reconstruction in magnetic particle imaging.**

Zur Veröffentlichung eingereicht.

S. Dittmer, T. Kluth, D. Otero Baguer, P. Maass.**A Deep Prior Approach to Magnetic Particle Imaging.**

Machine Learning for Medical Image Reconstruction 2020.

Springer International Publishing, F. Deeba, P. Johnson, T. Würfl, J. C. Ye (Hrsg.), S. 113-122, 2020.

M. Möddel, F. Griese, T. Kluth, T. Knopp.**Estimating orientation using multi-contrast MPI.**

10th International Workshop on Magnetic Particle Imaging 2020, Würzburg, 07.09.-09.09.2020.

Erscheint in Infinite Science Publishing, T. Knopp, T. M. Buzug (Hrsg.), S. 2 pages.

H. Albers, T. Kluth, T. Knopp.**MNPDynamics: A computational toolbox for simulating magnetic moment behavior of ensembles of nanoparticles.**

10th International Workshop on Magnetic Particle Imaging 2020, Würzburg, 07.09.-09.09.2020.

Erscheint in Infinite Science Publishing, T. Knopp, T. M. Buzug (Hrsg.), S. 2 pages.

T. Kluth, P. Szwargulski, T. Knopp.**Towards accurate modeling of the multidimensional MPI physics.**

10th International Workshop on Magnetic Particle Imaging 2020, Würzburg, 07.09.-09.09.2020.

Erscheint in Infinite Science Publishing, T. Knopp, T. M. Buzug (Hrsg.), S. 2 pages.

T. Kluth, B. Hahn, C. Brandt.**Spatio-temporal concentration reconstruction using motion priors in magnetic particle imaging.**

International Workshop on Magnetic Particle Imaging 2019.

International Workshop on Magnetic Particle Imaging (IWMPI) Book of Abstracts 2019, S. 23-24, Infinite Science Publishing, 2019.

T. Kluth, B. Jin.**Exploiting Ill-Posedness in Magnetic Particle Imaging - System Matrix Approximation via Randomized SVD.**

International Workshop on Magnetic Particle Imaging 2018.

International Workshop on Magnetic Particle Imaging (IWMPI) Book of Abstracts 2018, S. 127-128, Infinite Science Publishing, 2018.

J. Flötotto, T. Kluth, M. Möddel, T. Knopp, P. Maaß.**Improving Generalization Properties of Measured System Matrices by Using Regularized Total Least Squares Reconstruction in MPI.**

International Workshop on Magnetic Particle Imaging 2018.

International Workshop on Magnetic Particle Imaging (IWMPI) Book of Abstracts 2018, S. 53-54, Infinite Science Publishing, 2018.

C. Bathke, T. Kluth, P. Maaß.**MPI Reconstruction Using Structural Prior Information and Sparsity.**

International Workshop on Magnetic Particle Imaging 2018.

International Workshop on Magnetic Particle Imaging (IWMPI) Book of Abstracts 2018, S. 129-130, Infinite Science Publishing, 2018.

C. Bathke, T. Kluth, C. Brandt, P. Maaß.**Improved image reconstruction in magnetic particle imaging using structural a priori information.**

International Workshop on Magnetic Particle Imaging 2017.

International Workshop on Magnetic Particle Imaging (IWMPI) Book of Abstracts 2017, S. 85, Infinite Science Publishing, 2017.

T. Kluth, P. Maaß.**Model uncertainty in magnetic particle iamging: Motivating nonlinear problems by model-based sparse reconstruction.**

International Workshop on Magnetic Particle Imaging 2017.

International Workshop on Magnetic Particle Imaging (IWMPI) Book of Abstracts 2017, S. 83, Infinite Science Publishing, 2017.