Mathematical analysis of the 1D model and reconstruction schemes for magnetic particle imaging

Magnetic particle imaging (MPI) is a promising new in-vivo medical imaging modality in which distributions of super-paramagnetic nanoparticles are tracked based on their response in an applied magnetic field. In this paper we provide a mathematical analysis of the modeled MPI operator in the univariate situation. We provide a Hilbert space setup, in which the MPI operator is decomposed into simple building blocks and in which these building blocks are analyzed with respect to their mathematical properties. In turn, we obtain an analysis of the MPI forward operator and, in particular, of its ill-posedness properties. We further get that the singular values of the MPI core operator decrease exponentially. We complement our analytic results by some numerical studies which, in particular, suggest a rapid decay of the singular values of the MPI operator.

[1]  Peter Dencker,et al.  Multivariate polynomial interpolation on Lissajous-Chebyshev nodes , 2017, J. Approx. Theory.

[2]  Thorsten M. Buzug,et al.  Prediction of the Spatial Resolution of Magnetic Particle Imaging Using the Modulation Transfer Function of the Imaging Process , 2011, IEEE Transactions on Medical Imaging.

[3]  Thorsten M. Buzug,et al.  Single-sided device for magnetic particle imaging , 2009 .

[4]  Thorsten M. Buzug,et al.  Magnetic Particle Imaging: An Introduction to Imaging Principles and Scanner Instrumentation , 2012 .

[5]  R. Q. Edwards,et al.  Image Separation Radioisotope Scanning , 1963 .

[6]  H. Widom.  Asymptotic behavior of the eigenvalues of certain integral equations. II , 1964 .

[7]  Justin J. Konkle,et al.  Development of a field free line magnet for projection MPI , 2011, Medical Imaging.

[8]  Thorsten M. Buzug,et al.  Model-Based Reconstruction for Magnetic Particle Imaging , 2010, IEEE Transactions on Medical Imaging.

[9]  E. Hoffman,et al.  A positron-emission transaxial tomograph for nuclear imaging (PETT). , 1975, Radiology.

[10]  Bernhard Gleich,et al.  Signal encoding in magnetic particle imaging: properties of the system function , 2009, BMC Medical Imaging.

[11]  Jing-liang Cheng,et al.  Epidural spinal angiolipoma: a case series , 2017, BMC Research Notes.

[12]  Tobias Knopp,et al.  Edge Preserving and Noise Reducing Reconstruction for Magnetic Particle Imaging , 2017, IEEE Transactions on Medical Imaging.

[13]  Kevin R Minard,et al.  Optimization of nanoparticle core size for magnetic particle imaging. , 2009, Journal of magnetism and magnetic materials.

[14]  T Knopp,et al.  Hybrid system calibration for multidimensional magnetic particle imaging , 2017, Physics in medicine and biology.

[15]  F. Alberto Grünbaum.  Differential Operators Commuting with Convolution Integral Operators , 1983 .

[16]  B Gleich,et al.  Weighted iterative reconstruction for magnetic particle imaging , 2010, Physics in medicine and biology.

[17]  B Gleich,et al.  Fast reconstruction in magnetic particle imaging , 2012, Physics in medicine and biology.

[18]  Tobias Knopp,et al.  Magnetic particle imaging: from proof of principle to preclinical applications , 2017, Physics in medicine and biology.

[19]  Thorsten M. Buzug,et al.  Electronic Field Free Line Rotation and Relaxation Deconvolution in Magnetic Particle Imaging , 2015, IEEE Transactions on Medical Imaging.

[20]  F. Marchetti,et al.  Spectral filtering for the reduction of the Gibbs phenomenon for polynomial approximation methods on Lissajous curves with applications in MPI , 2017 .

[21]  D. Donev.  Prolate Spheroidal Wave Functions , 2017 .

[22]  Jochen Franke,et al.  On the formulation of the image reconstruction problem in magnetic particle imaging , 2013, Biomedizinische Technik. Biomedical engineering.

[23]  Patrick W. Goodwill,et al.  Multidimensional X-Space Magnetic Particle Imaging , 2011, IEEE Transactions on Medical Imaging.

[24]  Patrick W. Goodwill,et al.  Relaxation in X-Space Magnetic Particle Imaging , 2012, IEEE Transactions on Medical Imaging.

[25]  B Gleich,et al.  Three-dimensional real-time in vivo magnetic particle imaging , 2009, Physics in medicine and biology.

[26]  Hermann Schomberg,et al.  Magnetic particle imaging: Model and reconstruction , 2010, 2010 IEEE International Symposium on Biomedical Imaging: From Nano to Macro.

[27]  Bernhard Gleich,et al.  2D model-based reconstruction for magnetic particle imaging. , 2010, Medical physics.

[28]  T Knopp,et al.  Joint reconstruction of non-overlapping magnetic particle imaging focus-field data , 2015, Physics in medicine and biology.

[29]  Matthias Graeser,et al.  Magnetic particle imaging: current developments and future directions , 2015, International journal of nanomedicine.

[30]  Bo Zheng,et al.  Magnetic particle imaging (MPI) for NMR and MRI researchers. , 2013, Journal of magnetic resonance.

[31]  Thorsten M. Buzug,et al.  Bivariate Lagrange interpolation at the node points of non-degenerate Lissajous curves , 2014, Numerische Mathematik.

[32]  Tobias Knopp,et al.  Effiziente Rekonstruktion und alternative Spulentopologien für Magnetic-Particle-Imaging , 2010, Ausgezeichnete Informatikdissertationen.

[33]  A. Weinmann,et al.  Model-Based Reconstruction for Magnetic Particle Imaging in 2D and 3D , 2016, 1605.08095.

[34]  Wolfgang Erb,et al.  An orthogonal polynomial analogue of the Landau-Pollak-Slepian time-frequency analysis , 2011, J. Approx. Theory.

[35]  Thorsten M. Buzug,et al.  Non-Equispaced System Matrix Acquisition for Magnetic Particle Imaging Based on Lissajous Node Points , 2016, IEEE Transactions on Medical Imaging.

[36]  H. Engl,et al.  Regularization of Inverse Problems , 1996 .

[37]  Patrick W. Goodwill,et al.  Linearity and Shift Invariance for Quantitative Magnetic Particle Imaging , 2013, IEEE Transactions on Medical Imaging.

[38]  Bernhard Gleich,et al.  Analysis of a 3-D System Function Measured for Magnetic Particle Imaging , 2012, IEEE Transactions on Medical Imaging.

[39]  G. Burton.  Sobolev Spaces , 2013 .

[40]  A. Louis.  Inverse und schlecht gestellte Probleme , 1989 .

[41]  A. Kirsch.  An Introduction to the Mathematical Theory of Inverse Problems , 1996, Applied Mathematical Sciences.

[42]  E. Akl,et al.  Vitamin D supplementation in children with asthma: a systematic review and meta-analysis , 2015, BMC Research Notes.

[43]  Bo Zheng,et al.  Quantitative stem cell imaging with magnetic particle imaging , 2013, 2013 International Workshop on Magnetic Particle Imaging (IWMPI).

[44]  T. M. Buzug,et al.  1D-image reconstruction for magnetic particle imaging using a hybrid system function , 2011, 2011 IEEE Nuclear Science Symposium Conference Record.

[45]  B Gleich,et al.  A simulation study on the resolution and sensitivity of magnetic particle imaging , 2007, Physics in medicine and biology.

[46]  Bernhard Gleich,et al.  Tomographic imaging using the nonlinear response of magnetic particles , 2005, Nature.

[47]  Patrick W. Goodwill,et al.  The X-Space Formulation of the Magnetic Particle Imaging Process: 1-D Signal, Resolution, Bandwidth, SNR, SAR, and Magnetostimulation , 2010, IEEE Transactions on Medical Imaging.

[48]  Thorsten M. Buzug,et al.  A Fourier slice theorem for magnetic particle imaging using a field-free line , 2011 .

[49]  D. Slepian,et al.  Prolate spheroidal wave functions, fourier analysis and uncertainty — II , 1961 .

[50]  Emine Ulku Saritas,et al.  X‐Space MPI: Magnetic Nanoparticles for Safe Medical Imaging , 2012, Advanced materials.

[51]  B Gleich,et al.  Experimental results on fast 2D-encoded magnetic particle imaging , 2008, Physics in medicine and biology.

[52]  Christian Kaethner,et al.  A survey on bivariate Lagrange interpolation on Lissajous nodes , 2015 .

[53]  T. Buzug,et al.  Simultaneous patch reconstruction in Magnetic Particle Imaging , 2015, International Workshop on Magnetic Particle Imaging.

[54]  B Gleich,et al.  Trajectory analysis for magnetic particle imaging , 2008, Physics in medicine and biology.

[55]  B. Hofmann,et al.  Analytical and numerical studies on the influence of multiplication operators for the ill-posedness of inverse problems , 2005 .

[56]  Mario Bertero,et al.  Introduction to Inverse Problems in Imaging , 1998 .