American Institute of Mathematical Sciences

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November  2016, 10(4): 1087-1110. doi: 10.3934/ipi.2016033

Model-based reconstruction for magnetic particle imaging in 2D and 3D

Thomas März 1, and  Andreas Weinmann 2,

1. 

Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quater, Woodstock Road, Oxford OX2 6GG, United Kingdom
2. 

Department of Mathematics and Natural Sciences, Hochschule Darmstadt, Schöfferstraße 3, 64295 Darmstadt, Germany

Received  September 2015 Revised  July 2016 Published  October 2016

We contribute to the mathematical modeling and analysis of magnetic particle imaging which is a promising new in-vivo imaging modality. Concerning modeling, we develop a structured decomposition of the imaging process and extract its core part which we reveal to be common to all previous contributions in this context. The central contribution of this paper is the development of reconstruction formulae for MPI in 2D and 3D. Until now, in the multivariate setup, only time consuming measurement-based approaches are available, whereas reconstruction formulae are only available in 1D. The 2D and the 3D (describing the real world) reconstruction formulae which we derive here are significantly different from the 1D situation -- in particular there is no Dirac property in dimensions greater than one in the high resolution limit. As a further result of our analysis, we conclude that the reconstruction problem in MPI is severely ill-posed. Finally, we obtain a model-based reconstruction algorithm.
Keywords: Magnetic particle imaging, model-based reconstruction, inverse problems., phase space, reconstruction formulae.
Mathematics Subject Classification: Primary: 92C55, 94A12, 94A08; Secondary: 44A35, 65R3.
Citation: Thomas März, Andreas Weinmann. Model-based reconstruction for magnetic particle imaging in 2D and 3D. Inverse Problems & Imaging, 2016, 10 (4) : 1087-1110. doi: 10.3934/ipi.2016033
References:
[1]

L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, vol. 254,, Clarendon Press, (2000).   Google Scholar

[2]

M. Bertero and P. Boccacci, Introduction to Inverse Problems in Imaging,, CRC press, (1998).  doi: 10.1887/0750304359.  Google Scholar

[3]

F. Bornemann and T. März, Fast image inpainting based on coherence transport,, Joural of Mathematical Imaging and Vision, 28 (2007), 259.  doi: 10.1007/s10851-007-0017-6.  Google Scholar

[4]

S. Chikazumi and S. Charap, Physics of Magnetism,, Krieger Publishing, (1978).   Google Scholar

[5]

H. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems,, Kluwer Academic Publishers, (1996).  doi: 10.1007/978-94-009-1740-8.  Google Scholar

[6]

R. Ferguson, K. Minard and K. Krishnan, Optimization of nanoparticle core size for magnetic particle imaging,, Journal of Magnetism and Magnetic Materials, 321 (2009), 1548.  doi: 10.1016/j.jmmm.2009.02.083.  Google Scholar

[7]

B. Gleich and J. Weizenecker, Tomographic imaging using the nonlinear response of magnetic particles,, Nature, 435 (2005), 1214.  doi: 10.1038/nature03808.  Google Scholar

[8]

B. Gleich, J. Weizenecker and J. Borgert, Experimental results on fast 2D-encoded magnetic particle imaging,, Physics in Medicine and Biology, 53 (2008).  doi: 10.1088/0031-9155/53/6/N01.  Google Scholar

[9]

G. Golub and C. Van Loan, Matrix Computations,, JHU Press, (2012).   Google Scholar

[10]

P. Goodwill and S. Conolly, The X-space formulation of the magnetic particle imaging process: 1-D signal, resolution, bandwidth, {SNR}, {SAR}, and magnetostimulation,, IEEE Transactions on Medical Imaging, 29 (2010), 1851.  doi: 10.1109/TMI.2010.2052284.  Google Scholar

[11]

P. Goodwill and S. Conolly, Multidimensional X-space magnetic particle imaging,, IEEE Transactions on Medical Imaging, 30 (2011), 1581.  doi: 10.1109/TMI.2011.2125982.  Google Scholar

[12]

P. Goodwill, E. Saritas, L. Croft, T. Kim, K. Krishnan, D. Schaffer and S. Conolly, X-space mpi: Magnetic nanoparticles for safe medical imaging,, Advanced Materials, 24 (2012), 3870.  doi: 10.1002/adma.201200221.  Google Scholar

[13]

M. Grüttner, T. Knopp, J. Franke, M. Heidenreich, J. Rahmer, A. Halkola, C. Kaethner, J. Borgert and T. Buzug, On the formulation of the image reconstruction problem in magnetic particle imaging,, Biomedical Engineering, 58 (2013), 583.   Google Scholar

[14]

S. Helgason, The Radon Transform,, Springer, (1999).   Google Scholar

[15]

D. Jiles, Introduction to Magnetism and Magnetic Materials,, CRC press, (1998).   Google Scholar

[16]

A. Kirsch, An Introduction to the Mathematical Theory of Inverse Problems, vol. 120,, Springer, (2011).  doi: 10.1007/978-1-4419-8474-6.  Google Scholar

[17]

T. Knopp and T. Buzug, Magnetic Particle Imaging: An Introduction to Imaging Principles and Scanner Instrumentation,, Springer, (2012).  doi: 10.1007/978-3-642-04199-0.  Google Scholar

[18]

T. Knopp, Effiziente Rekonstruktion und alternative Spulentopologien für Magnetic Particle Imaging,, Dissertation, (2011).  doi: 10.1007/978-3-8348-8129-8.  Google Scholar

[19]

T. Knopp, S. Biederer, T. Sattel, M. Erbe and T. Buzug, Prediction of the spatial resolution of magnetic particle imaging using the modulation transfer function of the imaging process,, IEEE Transactions on Medical Imaging, 30 (2011), 1284.  doi: 10.1109/TMI.2011.2113188.  Google Scholar

[20]

T. Knopp, S. Biederer, T. Sattel, J. Rahmer, J. Weizenecker, B. Gleich, J. Borgert and T. Buzug, 2D model-based reconstruction for magnetic particle imaging,, Medical Physics, 37 (2010), 485.  doi: 10.1118/1.3271258.  Google Scholar

[21]

T. Knopp, M. Erbe, T. Sattel, S. Biederer and T. Buzug, A Fourier slice theorem for magnetic particle imaging using a field-free line,, Inverse Problems, 27 (2011).  doi: 10.1088/0266-5611/27/9/095004.  Google Scholar

[22]

T. Knopp, J. Rahmer, T. Sattel, S. Biederer, J. Weizenecker, B. Gleich, J. Borgert and T. Buzug, Weighted iterative reconstruction for magnetic particle imaging,, Physics in Medicine and Biology, 55 (2010), 1577.   Google Scholar

[23]

T. Knopp, T. Sattel, S. Biederer, J. Rahmer, J. Weizenecker, B. Gleich, J. Borgert and T. Buzug, Model-Based Reconstruction for magnetic particle imaging,, IEEE Transactions on Medical Imaging, 29 (2010), 12.   Google Scholar

[24]

J. Konkle, P. Goodwill and S. Conolly, Development of a field free line magnet for projection MPI,, in SPIE Medical Imaging, 7965 (2011).  doi: 10.1117/12.878435.  Google Scholar

[25]

D. Kuhl and R. Edwards, Image separation radioisotope scanning,, Radiology, 80 (1963), 653.  doi: 10.1148/80.4.653.  Google Scholar

[26]

J. Lampe, C. Bassoy, J. Rahmer, J. Weizenecker, H. Voss, B. Gleich and J. Borgert, Fast reconstruction in magnetic particle imaging,, Physics in Medicine and Biology, 57 (2012), 1113.  doi: 10.1088/0031-9155/57/4/1113.  Google Scholar

[27]

J. Lawrence, A Catalog of Special Plane Curves,, Courier Corporation, (2013).   Google Scholar

[28]

A. Louis, Inverse Und Schlecht Gestellte Probleme,, Teubner, (1989).  doi: 10.1007/978-3-322-84808-6.  Google Scholar

[29]

T. März, Image inpainting based on coherence transport with adapted distance functions,, SIAM Journal on Imaging Sciences, 4 (2011), 981.  doi: 10.1137/100807296.  Google Scholar

[30]

T. März, A well-posedness framework for inpainting based on coherence transport,, Foundations of Computational Mathematics, 15 (2015), 973.  doi: 10.1007/s10208-014-9199-7.  Google Scholar

[31]

J. Rahmer, J. Weizenecker, B. Gleich and J. Borgert, Signal encoding in magnetic particle imaging: Properties of the system function,, BMC Medical Imaging, 9 (2009).  doi: 10.1186/1471-2342-9-4.  Google Scholar

[32]

J. Rahmer, J. Weizenecker, B. Gleich and J. Borgert, Analysis of a 3-D system function measured for magnetic particle imaging,, IEEE Transactions on Medical Imaging, 31 (2012), 1289.  doi: 10.1109/TMI.2012.2188639.  Google Scholar

[33]

L. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms,, Physica D, 60 (1992), 259.  doi: 10.1016/0167-2789(92)90242-F.  Google Scholar

[34]

E. Saritas, P. Goodwill, L. Croft, J. Konkle, K. Lu, B. Zheng and S. Conolly, Magnetic particle imaging (MPI) for NMR and MRI researchers,, Journal of Magnetic Resonance, 229 (2013), 116.  doi: 10.1016/j.jmr.2012.11.029.  Google Scholar

[35]

T. Sattel, T. Knopp, S. Biederer, B. Gleich, J. Weizenecker, J. Borgert and T. Buzug, Single-sided device for magnetic particle imaging,, Journal of Physics D: Applied Physics, 42 (2009).  doi: 10.1088/0022-3727/42/2/022001.  Google Scholar

[36]

O. Scherzer, M. Grasmair, H. Grossauer, M. Haltmeier and F. Lenzen, Variational Methods in Imaging,, Springer, (2009).   Google Scholar

[37]

H. Schomberg, Magnetic particle imaging: Model and reconstruction,, in 2010 IEEE International Symposium on Biomedical Imaging: From Nano to Macro, (2010), 992.  doi: 10.1109/ISBI.2010.5490155.  Google Scholar

[38]

M. Storath and A. Weinmann, Fast partitioning of vector-valued images,, SIAM Journal on Imaging Sciences, 7 (2014), 1826.  doi: 10.1137/130950367.  Google Scholar

[39]

M. Storath, A. Weinmann, J. Frikel and M. Unser, Joint image reconstruction and segmentation using the Potts model,, Inverse Problems, 31 (2015).  doi: 10.1088/0266-5611/31/2/025003.  Google Scholar

[40]

M. Ter-Pogossian, M. Phelps, E. Hoffman and N. Mullani, A positron-emission transaxial tomograph for nuclear imaging (PETT),, Radiology, 114 (1975), 89.  doi: 10.1148/114.1.89.  Google Scholar

[41]

A. Weinmann, M. Storath and L. Demaret, The $L^1$-Potts functional for robust jump-sparse reconstruction,, SIAM Journal on Numerical Analysis, 53 (2015), 644.  doi: 10.1137/120896256.  Google Scholar

[42]

A. Weinmann and M. Storath, Iterative Potts and Blake-Zisserman minimization for the recovery of functions with discontinuities from indirect measurements,, Proceedings of the Royal Society A: Mathematical, 471 (2015).  doi: 10.1098/rspa.2014.0638.  Google Scholar

[43]

J. Weizenecker, B. Gleich and J. Borgert, Magnetic particle imaging using a field free line,, Journal of Physics D: Applied Physics, 41 (2008), 105.  doi: 10.1088/0022-3727/41/10/105009.  Google Scholar

[44]

J. Weizenecker, J. Borgert and B. Gleich, A simulation study on the resolution and sensitivity of magnetic particle imaging,, Physics in Medicine and Biology, 52 (2007), 6363.  doi: 10.1088/0031-9155/52/21/001.  Google Scholar

[45]

J. Weizenecker, B. Gleich, J. Rahmer, H. Dahnke and J. Borgert, Three-dimensional real-time in vivo magnetic particle imaging,, Physics in Medicine and Biology, 54 (2009).   Google Scholar

[46]

B. Zheng, T. Vazin, W. Yang, P. Goodwill, E. Saritas, L. Croft, D. Schaffer and S. Conolly, Quantitative stem cell imaging with magnetic particle imaging,, in IEEE International Workshop on Magnetic Particle Imaging, (2013).  doi: 10.1109/IWMPI.2013.6528323.  Google Scholar

show all references

References:
[1]

L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, vol. 254,, Clarendon Press, (2000).   Google Scholar

[2]

M. Bertero and P. Boccacci, Introduction to Inverse Problems in Imaging,, CRC press, (1998).  doi: 10.1887/0750304359.  Google Scholar

[3]

F. Bornemann and T. März, Fast image inpainting based on coherence transport,, Joural of Mathematical Imaging and Vision, 28 (2007), 259.  doi: 10.1007/s10851-007-0017-6.  Google Scholar

[4]

S. Chikazumi and S. Charap, Physics of Magnetism,, Krieger Publishing, (1978).   Google Scholar

[5]

H. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems,, Kluwer Academic Publishers, (1996).  doi: 10.1007/978-94-009-1740-8.  Google Scholar

[6]

R. Ferguson, K. Minard and K. Krishnan, Optimization of nanoparticle core size for magnetic particle imaging,, Journal of Magnetism and Magnetic Materials, 321 (2009), 1548.  doi: 10.1016/j.jmmm.2009.02.083.  Google Scholar

[7]

B. Gleich and J. Weizenecker, Tomographic imaging using the nonlinear response of magnetic particles,, Nature, 435 (2005), 1214.  doi: 10.1038/nature03808.  Google Scholar

[8]

B. Gleich, J. Weizenecker and J. Borgert, Experimental results on fast 2D-encoded magnetic particle imaging,, Physics in Medicine and Biology, 53 (2008).  doi: 10.1088/0031-9155/53/6/N01.  Google Scholar

[9]

G. Golub and C. Van Loan, Matrix Computations,, JHU Press, (2012).   Google Scholar

[10]

P. Goodwill and S. Conolly, The X-space formulation of the magnetic particle imaging process: 1-D signal, resolution, bandwidth, {SNR}, {SAR}, and magnetostimulation,, IEEE Transactions on Medical Imaging, 29 (2010), 1851.  doi: 10.1109/TMI.2010.2052284.  Google Scholar

[11]

P. Goodwill and S. Conolly, Multidimensional X-space magnetic particle imaging,, IEEE Transactions on Medical Imaging, 30 (2011), 1581.  doi: 10.1109/TMI.2011.2125982.  Google Scholar

[12]

P. Goodwill, E. Saritas, L. Croft, T. Kim, K. Krishnan, D. Schaffer and S. Conolly, X-space mpi: Magnetic nanoparticles for safe medical imaging,, Advanced Materials, 24 (2012), 3870.  doi: 10.1002/adma.201200221.  Google Scholar

[13]

M. Grüttner, T. Knopp, J. Franke, M. Heidenreich, J. Rahmer, A. Halkola, C. Kaethner, J. Borgert and T. Buzug, On the formulation of the image reconstruction problem in magnetic particle imaging,, Biomedical Engineering, 58 (2013), 583.   Google Scholar

[14]

S. Helgason, The Radon Transform,, Springer, (1999).   Google Scholar

[15]

D. Jiles, Introduction to Magnetism and Magnetic Materials,, CRC press, (1998).   Google Scholar

[16]

A. Kirsch, An Introduction to the Mathematical Theory of Inverse Problems, vol. 120,, Springer, (2011).  doi: 10.1007/978-1-4419-8474-6.  Google Scholar

[17]

T. Knopp and T. Buzug, Magnetic Particle Imaging: An Introduction to Imaging Principles and Scanner Instrumentation,, Springer, (2012).  doi: 10.1007/978-3-642-04199-0.  Google Scholar

[18]

T. Knopp, Effiziente Rekonstruktion und alternative Spulentopologien für Magnetic Particle Imaging,, Dissertation, (2011).  doi: 10.1007/978-3-8348-8129-8.  Google Scholar

[19]

T. Knopp, S. Biederer, T. Sattel, M. Erbe and T. Buzug, Prediction of the spatial resolution of magnetic particle imaging using the modulation transfer function of the imaging process,, IEEE Transactions on Medical Imaging, 30 (2011), 1284.  doi: 10.1109/TMI.2011.2113188.  Google Scholar

[20]

T. Knopp, S. Biederer, T. Sattel, J. Rahmer, J. Weizenecker, B. Gleich, J. Borgert and T. Buzug, 2D model-based reconstruction for magnetic particle imaging,, Medical Physics, 37 (2010), 485.  doi: 10.1118/1.3271258.  Google Scholar

[21]

T. Knopp, M. Erbe, T. Sattel, S. Biederer and T. Buzug, A Fourier slice theorem for magnetic particle imaging using a field-free line,, Inverse Problems, 27 (2011).  doi: 10.1088/0266-5611/27/9/095004.  Google Scholar

[22]

T. Knopp, J. Rahmer, T. Sattel, S. Biederer, J. Weizenecker, B. Gleich, J. Borgert and T. Buzug, Weighted iterative reconstruction for magnetic particle imaging,, Physics in Medicine and Biology, 55 (2010), 1577.   Google Scholar

[23]

T. Knopp, T. Sattel, S. Biederer, J. Rahmer, J. Weizenecker, B. Gleich, J. Borgert and T. Buzug, Model-Based Reconstruction for magnetic particle imaging,, IEEE Transactions on Medical Imaging, 29 (2010), 12.   Google Scholar

[24]

J. Konkle, P. Goodwill and S. Conolly, Development of a field free line magnet for projection MPI,, in SPIE Medical Imaging, 7965 (2011).  doi: 10.1117/12.878435.  Google Scholar

[25]

D. Kuhl and R. Edwards, Image separation radioisotope scanning,, Radiology, 80 (1963), 653.  doi: 10.1148/80.4.653.  Google Scholar

[26]

J. Lampe, C. Bassoy, J. Rahmer, J. Weizenecker, H. Voss, B. Gleich and J. Borgert, Fast reconstruction in magnetic particle imaging,, Physics in Medicine and Biology, 57 (2012), 1113.  doi: 10.1088/0031-9155/57/4/1113.  Google Scholar

[27]

J. Lawrence, A Catalog of Special Plane Curves,, Courier Corporation, (2013).   Google Scholar

[28]

A. Louis, Inverse Und Schlecht Gestellte Probleme,, Teubner, (1989).  doi: 10.1007/978-3-322-84808-6.  Google Scholar

[29]

T. März, Image inpainting based on coherence transport with adapted distance functions,, SIAM Journal on Imaging Sciences, 4 (2011), 981.  doi: 10.1137/100807296.  Google Scholar

[30]

T. März, A well-posedness framework for inpainting based on coherence transport,, Foundations of Computational Mathematics, 15 (2015), 973.  doi: 10.1007/s10208-014-9199-7.  Google Scholar

[31]

J. Rahmer, J. Weizenecker, B. Gleich and J. Borgert, Signal encoding in magnetic particle imaging: Properties of the system function,, BMC Medical Imaging, 9 (2009).  doi: 10.1186/1471-2342-9-4.  Google Scholar

[32]

J. Rahmer, J. Weizenecker, B. Gleich and J. Borgert, Analysis of a 3-D system function measured for magnetic particle imaging,, IEEE Transactions on Medical Imaging, 31 (2012), 1289.  doi: 10.1109/TMI.2012.2188639.  Google Scholar

[33]

L. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms,, Physica D, 60 (1992), 259.  doi: 10.1016/0167-2789(92)90242-F.  Google Scholar

[34]

E. Saritas, P. Goodwill, L. Croft, J. Konkle, K. Lu, B. Zheng and S. Conolly, Magnetic particle imaging (MPI) for NMR and MRI researchers,, Journal of Magnetic Resonance, 229 (2013), 116.  doi: 10.1016/j.jmr.2012.11.029.  Google Scholar

[35]

T. Sattel, T. Knopp, S. Biederer, B. Gleich, J. Weizenecker, J. Borgert and T. Buzug, Single-sided device for magnetic particle imaging,, Journal of Physics D: Applied Physics, 42 (2009).  doi: 10.1088/0022-3727/42/2/022001.  Google Scholar

[36]

O. Scherzer, M. Grasmair, H. Grossauer, M. Haltmeier and F. Lenzen, Variational Methods in Imaging,, Springer, (2009).   Google Scholar

[37]

H. Schomberg, Magnetic particle imaging: Model and reconstruction,, in 2010 IEEE International Symposium on Biomedical Imaging: From Nano to Macro, (2010), 992.  doi: 10.1109/ISBI.2010.5490155.  Google Scholar

[38]

M. Storath and A. Weinmann, Fast partitioning of vector-valued images,, SIAM Journal on Imaging Sciences, 7 (2014), 1826.  doi: 10.1137/130950367.  Google Scholar

[39]

M. Storath, A. Weinmann, J. Frikel and M. Unser, Joint image reconstruction and segmentation using the Potts model,, Inverse Problems, 31 (2015).  doi: 10.1088/0266-5611/31/2/025003.  Google Scholar

[40]

M. Ter-Pogossian, M. Phelps, E. Hoffman and N. Mullani, A positron-emission transaxial tomograph for nuclear imaging (PETT),, Radiology, 114 (1975), 89.  doi: 10.1148/114.1.89.  Google Scholar

[41]

A. Weinmann, M. Storath and L. Demaret, The $L^1$-Potts functional for robust jump-sparse reconstruction,, SIAM Journal on Numerical Analysis, 53 (2015), 644.  doi: 10.1137/120896256.  Google Scholar

[42]

A. Weinmann and M. Storath, Iterative Potts and Blake-Zisserman minimization for the recovery of functions with discontinuities from indirect measurements,, Proceedings of the Royal Society A: Mathematical, 471 (2015).  doi: 10.1098/rspa.2014.0638.  Google Scholar

[43]

J. Weizenecker, B. Gleich and J. Borgert, Magnetic particle imaging using a field free line,, Journal of Physics D: Applied Physics, 41 (2008), 105.  doi: 10.1088/0022-3727/41/10/105009.  Google Scholar

[44]

J. Weizenecker, J. Borgert and B. Gleich, A simulation study on the resolution and sensitivity of magnetic particle imaging,, Physics in Medicine and Biology, 52 (2007), 6363.  doi: 10.1088/0031-9155/52/21/001.  Google Scholar

[45]

J. Weizenecker, B. Gleich, J. Rahmer, H. Dahnke and J. Borgert, Three-dimensional real-time in vivo magnetic particle imaging,, Physics in Medicine and Biology, 54 (2009).   Google Scholar

[46]

B. Zheng, T. Vazin, W. Yang, P. Goodwill, E. Saritas, L. Croft, D. Schaffer and S. Conolly, Quantitative stem cell imaging with magnetic particle imaging,, in IEEE International Workshop on Magnetic Particle Imaging, (2013).  doi: 10.1109/IWMPI.2013.6528323.  Google Scholar

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