International Journal on Magnetic Particle Imaging IJMPI
Vol. 8 No. 1 (2022): Int J Mag Part Imag
https://doi.org/10.18416/IJMPI.2022.2208001
Improving Model-Based MPI Image Reconstructions: Baseline Recovery, Receive Coil Sensitivity, Relaxation and Uncertainty Estimation
Main Article Content
Copyright (c) 2022 Mark-Alexander Henn, Klaus Natorf Quelhas, Thinh Bui, Solomon Woods
This work is licensed under a Creative Commons Attribution 4.0 International License.
Abstract
Image reconstruction is an integral part of Magnetic Particle Imaging (MPI). Over the last years, several methods have been proposed for reconstructing MPI images more efficiently and accurately. One major challenge for model-based MPI image reconstruction methods is the realistic modeling of the measurement system; effects like non-linear gradient fields, non-uniform drive fields, space-dependent coil sensitivities, drive frequency filtering and particle relaxation, if not properly accounted for in the model, may yield inaccurate reconstructions. This work addresses these issues by means of an image reconstruction method that accounts for the coil sensitivity, baseline recovery and particle relaxation. We investigate the proposed approach for a 1D MPI setup, and provide an approach for the calculation of the uncertainties of the reconstructed images.
Int. J. Mag. Part. Imag. 8(1), 2022, Article ID: 2208001, DOI: 10.18416/IJMPI.2022.2208001
Article Details
References
[1] B. Gleich and J. Weizenecker. Tomographic imaging using the nonlinear response of magnetic particles. Nature, 435(7046):1214–1217, 2005, doi:10.1038/nature03808.
[2] T. Knopp and T. M. Buzug,Magnetic Particle Imaging: An Introduction to Imaging Principles and Scanner Instrumentation. Berlin, Heidelberg: Springer Berlin Heidelberg, 2012, doi:10.1007/978-3-642-04199-0.
[3] T. Knopp, N. Gdaniec, and M. Möddel. Magnetic particle imaging: from proof of principle to preclinical applications. Physics in Medicine & Biology, 62(14):R124–R178, 2017, doi:10.1088/1361-6560/aa6c99.
[4] 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(21):6363–6374, 2007, doi:10.1088/0031-9155/52/21/001.
[5] T. Knopp, S. Biederer, T. F. Sattel, J. Rahmer, J. Weizenecker, B. Gleich, J. Borgert, and T. M. Buzug. 2D model-based reconstruction for magnetic particle imaging.Medical Physics, 37(2):485–491, 2010, doi:10.1118/1.3271258.
[6] P. W. Goodwill and S. M. 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(11):1851–1859, 2010, doi:10.1109/TMI.2010.2052284.
[7] T. Hatsuda, T. Takagi, A.Matsuhisa, M. Arayama, H. Tsuchiya, S. Takahashi, and Y. Ishihara. Basic Study of Image Reconstruction Method Using Neural Networks with Additional Learning for Magnetic Particle Imaging. International Journal on Magnetic Particle Imaging, 2(2), 2016, doi:10.18416/IJMPI.2016.1611002.
[8] S. Dittmer, T. Kluth, M. T. R. Henriksen, and P. Maass. Deep image prior for 3D magnetic particle imaging: A quantitative comparison of regularization techniques on Open MPI dataset. International Journal on Magnetic Particle Imaging, 7(1), 2021, doi:10.18416/IJMPI.2021.2103001.
[9] J. Rahmer, J.Weizenecker, B. Gleich, and J. Borgert. Signal encoding in magnetic particle imaging: properties of the system function. BMC Medical Imaging, 9:4, 2009, doi:10.1186/1471-2342-9-4.
[10] T. Kluth, P. Szwargulski, and T. Knopp. Towards accurate modeling of the multidimensional magnetic particle imaging physics. New Journal of Physics, 21(10):103032, 2019, doi:10.1088/1367-2630/ab4938.
[11] M. Gruettner, M. Graeser, S. Biederer, T. F. Sattel, H. Wojtczyk, W. Tenner, T. Knopp, B. Gleich, J. Borgert, and T. M. Buzug, 1D-image reconstruction for magnetic particle imaging using a hybrid system function, in 2011 IEEE Nuclear Science Symposium Conference Record, 2545–2548, 2011. doi:10.1109/NSSMIC.2011.6152687.
[12] H. Schomberg, Magnetic particle imaging: Model and reconstruction, in IEEE International Symposium on Biomedical Imaging: From Nano to Macro, 992–995, IEEE, 2010. doi:10.1109/ISBI.2010.5490155.
[13] T.März and A. Weinmann. Model-based reconstruction for magnetic particle imaging in 2D and 3D. Inverse Problems and Imaging, 10(4):1087–1110, 2016, doi:10.3934/ipi.2016033.
[14] K. Lu, P. Goodwill, B. Zheng, and S. Conolly, The impact of filtering direct-feedthrough on the x-space theory of magnetic particle imaging, in SPIE Medical Imaging, J. B.Weaver and R. C.Molthen, Eds., 79652I, 2011. doi:10.1117/12.878446.
[15] K. Lu, P. W. Goodwill, E. U. Saritas, B. Zheng, and S. M. Conolly. Linearity and Shift Invariance for Quantitative Magnetic Particle Imaging. IEEE Transactions on Medical Imaging, 32(9):1565–1575, 2013, doi:10.1109/TMI.2013.2257177.
[16] J. J.Konkle, P.W.Goodwill, D.W.Hensley, R.D. Orendorff, M. Lustig, and S. M. Conolly. A Convex Formulation for Magnetic Particle Imaging X-Space Reconstruction. PLOS ONE, 10(10):e0140137J. Najbauer, Ed., 2015, doi:10.1371/journal.pone.0140137.
[17] T. Knopp, M. Grosser, M. Graeser, T. Gerkmann, and M. Moddel. Efficient Joint Estimation of Tracer Distribution and Background Signals in Magnetic Particle Imaging Using a Dictionary Approach. IEEE Transactions on Medical Imaging, 40(12):3568–3579, 2021, doi:10.1109/TMI.2021.3090928.
[18] T. Knopp, N. Gdaniec, R. Rehr, M. Graeser, and T. Gerkmann. Correction of linear system drifts in magnetic particle imaging. Physics in Medicine & Biology, 64(12):125013, 2019, doi:10.1088/1361-6560/ab2480.
[19] L. R. Croft, P.W.Goodwill, and S. M. Conolly. Relaxation in X-Space Magnetic Particle Imaging. IEEE Transactions on Medical Imaging, 31(12):2335–2342, 2012, doi:10.1109/TMI.2012.2217979.
[20] R. Dhavalikar, D. Hensley, L.Maldonado-Camargo, L. R. Croft, S. Ceron, P.W. Goodwill, S. M. Conolly, and C. Rinaldi. Finite magnetic relaxation in x -space magnetic particle imaging: comparison of measurements and ferrohydrodynamic models. Journal of Physics D: Applied Physics, 49(30):305002, 2016, doi:10.1088/0022-3727/49/30/305002.
[21] A. A. Ozaslan, M. T. Arslan, and E. U. Saritas, Image Reconstruction with Relaxation Estimation for Non-Cartesian Magnetic Particle Imaging, in 2020 28th Signal Processing and Communications Applications Conference (SIU), 1–4, IEEE, 2020. doi:10.1109/SIU49456.2020.9302276.
[22] T. Knopp, P. Szwargulski, F. Griese, and M. Gräser. OpenMPIData: An initiative for freely accessible magnetic particle imaging data. Data in Brief, 28:104971, 2020, doi:10.1016/j.dib.2019.104971.
[23] K. N.Quelhas, M.-A.Henn, T. Q.Bui, H. R.Wages,W. L.Tew, and S. I. Woods, Flexible Software for Rigorous Simulations of Magnetic Particle Imaging Systems, in InternationalWorkshop on Magnetic Particle Imaging, 2022. doi:10.18416/IJMPI.2022.2203081.
[24] T. Q. Bui, W. L. Tew, and S. I. Woods. AC magnetometry with active stabilization and harmonic suppression for magnetic nanoparticle spectroscopy and thermometry. Journal of Applied Physics, 128(22):224901, 2020, doi:10.1063/5.0031451.
[25] A. N. Tikhonov. On the solution of ill-posed problems and the method of regularization. Dokl. Akad. Nauk SSSR, 151(3):501–504, 1963.
[26] P. Virtanen, R. Gommers, T. E.Oliphant, et al. SciPy 1.0: fundamental algorithms for scientific computing in Python. NatureMethods, 17(3):261–272, 2020, doi:10.1038/s41592-019-0686-2.
[27] R. H. Byrd, P. Lu, J. Nocedal, and C. Zhu. A Limited Memory Algorithm for Bound Constrained Optimization. SIAM Journal on Scientific Computing, 16(5):1190–1208, 1995, doi:10.1137/0916069.
[28] C. R. Rao, Linear Statistical Inference and its Applications, C. R. Rao, Ed., ser. Wiley Series in Probability and Statistics. Hoboken, NJ, USA: John Wiley & Sons, Inc., 1973, doi:10.1002/9780470316436.
[29] C. R. Rao, Shalabh, H. Toutenburg, and C. Heumann, Linear Models and Generalizations, ser. Springer Series in Statistics. Berlin, Heidelberg: Springer Berlin Heidelberg, 2008, doi:10.1007/978-3-540-74227-2.
[30] P.W. Goodwill, E. U. Saritas, L. R. Croft, T. N. Kim, K. M. Krishnan, D. V. Schaffer, and S. M. Conolly. X-Space MPI: Magnetic Nanoparticles for Safe Medical Imaging. Advanced Materials, 24(28):3870–3877, 2012, doi:10.1002/adma.201200221.
[31] P.W. Goodwill and S. M. Conolly. Multidimensional X-Space Magnetic Particle Imaging. IEEE Transactions on Medical Imaging, 30(9):1581–1590, 2011, doi:10.1109/TMI.2011.2125982.
[32] R. M. Ferguson, K. R.Minard, and K. M. Krishnan. Optimization of nanoparticle core size for magnetic particle imaging. Journal of Magnetism and Magnetic Materials, 321(10):1548–1551, 2009, doi:10.1016/j.jmmm.2009.02.083.
[33] R. M. Ferguson, K. R.Minard, A. P. Khandhar, and K. M. Krishnan. Optimizing magnetite nanoparticles for mass sensitivity in magnetic particle imaging. Medical Physics, 38(3):1619–1626, 2011, doi:10.1118/1.3554646.
[34] L. R. Croft, P.W. Goodwill, J. J. Konkle, H. Arami, D. A. Price, A. X. Li, E. U. Saritas, and S. M. Conolly. Low drive field amplitude for improved image resolution in magnetic particle imaging. Medical Physics, 43(1):424–435, 2015, doi:10.1118/1.4938097.
[35] A.Wirgin. The inverse crime, 2004. arXiv: 0401050 [math-ph]. URL: http://arxiv.org/abs/math-ph/0401050.
[36] A. Tarantola, Inverse Problem Theory and Methods for Model Parameter Estimation. Society for Industrial and Applied Mathematics, 2005, doi:10.1137/1.9780898717921.
[37] M.-A. Henn, K. N. Quelhas, and S. I. Woods, Uncertainty estimation for 2D magnetic particle imaging, in International Workshop on Magnetic Particle Imaging, 2022. doi:10.18416/IJMPI.2022.2203082.
[2] T. Knopp and T. M. Buzug,Magnetic Particle Imaging: An Introduction to Imaging Principles and Scanner Instrumentation. Berlin, Heidelberg: Springer Berlin Heidelberg, 2012, doi:10.1007/978-3-642-04199-0.
[3] T. Knopp, N. Gdaniec, and M. Möddel. Magnetic particle imaging: from proof of principle to preclinical applications. Physics in Medicine & Biology, 62(14):R124–R178, 2017, doi:10.1088/1361-6560/aa6c99.
[4] 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(21):6363–6374, 2007, doi:10.1088/0031-9155/52/21/001.
[5] T. Knopp, S. Biederer, T. F. Sattel, J. Rahmer, J. Weizenecker, B. Gleich, J. Borgert, and T. M. Buzug. 2D model-based reconstruction for magnetic particle imaging.Medical Physics, 37(2):485–491, 2010, doi:10.1118/1.3271258.
[6] P. W. Goodwill and S. M. 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(11):1851–1859, 2010, doi:10.1109/TMI.2010.2052284.
[7] T. Hatsuda, T. Takagi, A.Matsuhisa, M. Arayama, H. Tsuchiya, S. Takahashi, and Y. Ishihara. Basic Study of Image Reconstruction Method Using Neural Networks with Additional Learning for Magnetic Particle Imaging. International Journal on Magnetic Particle Imaging, 2(2), 2016, doi:10.18416/IJMPI.2016.1611002.
[8] S. Dittmer, T. Kluth, M. T. R. Henriksen, and P. Maass. Deep image prior for 3D magnetic particle imaging: A quantitative comparison of regularization techniques on Open MPI dataset. International Journal on Magnetic Particle Imaging, 7(1), 2021, doi:10.18416/IJMPI.2021.2103001.
[9] J. Rahmer, J.Weizenecker, B. Gleich, and J. Borgert. Signal encoding in magnetic particle imaging: properties of the system function. BMC Medical Imaging, 9:4, 2009, doi:10.1186/1471-2342-9-4.
[10] T. Kluth, P. Szwargulski, and T. Knopp. Towards accurate modeling of the multidimensional magnetic particle imaging physics. New Journal of Physics, 21(10):103032, 2019, doi:10.1088/1367-2630/ab4938.
[11] M. Gruettner, M. Graeser, S. Biederer, T. F. Sattel, H. Wojtczyk, W. Tenner, T. Knopp, B. Gleich, J. Borgert, and T. M. Buzug, 1D-image reconstruction for magnetic particle imaging using a hybrid system function, in 2011 IEEE Nuclear Science Symposium Conference Record, 2545–2548, 2011. doi:10.1109/NSSMIC.2011.6152687.
[12] H. Schomberg, Magnetic particle imaging: Model and reconstruction, in IEEE International Symposium on Biomedical Imaging: From Nano to Macro, 992–995, IEEE, 2010. doi:10.1109/ISBI.2010.5490155.
[13] T.März and A. Weinmann. Model-based reconstruction for magnetic particle imaging in 2D and 3D. Inverse Problems and Imaging, 10(4):1087–1110, 2016, doi:10.3934/ipi.2016033.
[14] K. Lu, P. Goodwill, B. Zheng, and S. Conolly, The impact of filtering direct-feedthrough on the x-space theory of magnetic particle imaging, in SPIE Medical Imaging, J. B.Weaver and R. C.Molthen, Eds., 79652I, 2011. doi:10.1117/12.878446.
[15] K. Lu, P. W. Goodwill, E. U. Saritas, B. Zheng, and S. M. Conolly. Linearity and Shift Invariance for Quantitative Magnetic Particle Imaging. IEEE Transactions on Medical Imaging, 32(9):1565–1575, 2013, doi:10.1109/TMI.2013.2257177.
[16] J. J.Konkle, P.W.Goodwill, D.W.Hensley, R.D. Orendorff, M. Lustig, and S. M. Conolly. A Convex Formulation for Magnetic Particle Imaging X-Space Reconstruction. PLOS ONE, 10(10):e0140137J. Najbauer, Ed., 2015, doi:10.1371/journal.pone.0140137.
[17] T. Knopp, M. Grosser, M. Graeser, T. Gerkmann, and M. Moddel. Efficient Joint Estimation of Tracer Distribution and Background Signals in Magnetic Particle Imaging Using a Dictionary Approach. IEEE Transactions on Medical Imaging, 40(12):3568–3579, 2021, doi:10.1109/TMI.2021.3090928.
[18] T. Knopp, N. Gdaniec, R. Rehr, M. Graeser, and T. Gerkmann. Correction of linear system drifts in magnetic particle imaging. Physics in Medicine & Biology, 64(12):125013, 2019, doi:10.1088/1361-6560/ab2480.
[19] L. R. Croft, P.W.Goodwill, and S. M. Conolly. Relaxation in X-Space Magnetic Particle Imaging. IEEE Transactions on Medical Imaging, 31(12):2335–2342, 2012, doi:10.1109/TMI.2012.2217979.
[20] R. Dhavalikar, D. Hensley, L.Maldonado-Camargo, L. R. Croft, S. Ceron, P.W. Goodwill, S. M. Conolly, and C. Rinaldi. Finite magnetic relaxation in x -space magnetic particle imaging: comparison of measurements and ferrohydrodynamic models. Journal of Physics D: Applied Physics, 49(30):305002, 2016, doi:10.1088/0022-3727/49/30/305002.
[21] A. A. Ozaslan, M. T. Arslan, and E. U. Saritas, Image Reconstruction with Relaxation Estimation for Non-Cartesian Magnetic Particle Imaging, in 2020 28th Signal Processing and Communications Applications Conference (SIU), 1–4, IEEE, 2020. doi:10.1109/SIU49456.2020.9302276.
[22] T. Knopp, P. Szwargulski, F. Griese, and M. Gräser. OpenMPIData: An initiative for freely accessible magnetic particle imaging data. Data in Brief, 28:104971, 2020, doi:10.1016/j.dib.2019.104971.
[23] K. N.Quelhas, M.-A.Henn, T. Q.Bui, H. R.Wages,W. L.Tew, and S. I. Woods, Flexible Software for Rigorous Simulations of Magnetic Particle Imaging Systems, in InternationalWorkshop on Magnetic Particle Imaging, 2022. doi:10.18416/IJMPI.2022.2203081.
[24] T. Q. Bui, W. L. Tew, and S. I. Woods. AC magnetometry with active stabilization and harmonic suppression for magnetic nanoparticle spectroscopy and thermometry. Journal of Applied Physics, 128(22):224901, 2020, doi:10.1063/5.0031451.
[25] A. N. Tikhonov. On the solution of ill-posed problems and the method of regularization. Dokl. Akad. Nauk SSSR, 151(3):501–504, 1963.
[26] P. Virtanen, R. Gommers, T. E.Oliphant, et al. SciPy 1.0: fundamental algorithms for scientific computing in Python. NatureMethods, 17(3):261–272, 2020, doi:10.1038/s41592-019-0686-2.
[27] R. H. Byrd, P. Lu, J. Nocedal, and C. Zhu. A Limited Memory Algorithm for Bound Constrained Optimization. SIAM Journal on Scientific Computing, 16(5):1190–1208, 1995, doi:10.1137/0916069.
[28] C. R. Rao, Linear Statistical Inference and its Applications, C. R. Rao, Ed., ser. Wiley Series in Probability and Statistics. Hoboken, NJ, USA: John Wiley & Sons, Inc., 1973, doi:10.1002/9780470316436.
[29] C. R. Rao, Shalabh, H. Toutenburg, and C. Heumann, Linear Models and Generalizations, ser. Springer Series in Statistics. Berlin, Heidelberg: Springer Berlin Heidelberg, 2008, doi:10.1007/978-3-540-74227-2.
[30] P.W. Goodwill, E. U. Saritas, L. R. Croft, T. N. Kim, K. M. Krishnan, D. V. Schaffer, and S. M. Conolly. X-Space MPI: Magnetic Nanoparticles for Safe Medical Imaging. Advanced Materials, 24(28):3870–3877, 2012, doi:10.1002/adma.201200221.
[31] P.W. Goodwill and S. M. Conolly. Multidimensional X-Space Magnetic Particle Imaging. IEEE Transactions on Medical Imaging, 30(9):1581–1590, 2011, doi:10.1109/TMI.2011.2125982.
[32] R. M. Ferguson, K. R.Minard, and K. M. Krishnan. Optimization of nanoparticle core size for magnetic particle imaging. Journal of Magnetism and Magnetic Materials, 321(10):1548–1551, 2009, doi:10.1016/j.jmmm.2009.02.083.
[33] R. M. Ferguson, K. R.Minard, A. P. Khandhar, and K. M. Krishnan. Optimizing magnetite nanoparticles for mass sensitivity in magnetic particle imaging. Medical Physics, 38(3):1619–1626, 2011, doi:10.1118/1.3554646.
[34] L. R. Croft, P.W. Goodwill, J. J. Konkle, H. Arami, D. A. Price, A. X. Li, E. U. Saritas, and S. M. Conolly. Low drive field amplitude for improved image resolution in magnetic particle imaging. Medical Physics, 43(1):424–435, 2015, doi:10.1118/1.4938097.
[35] A.Wirgin. The inverse crime, 2004. arXiv: 0401050 [math-ph]. URL: http://arxiv.org/abs/math-ph/0401050.
[36] A. Tarantola, Inverse Problem Theory and Methods for Model Parameter Estimation. Society for Industrial and Applied Mathematics, 2005, doi:10.1137/1.9780898717921.
[37] M.-A. Henn, K. N. Quelhas, and S. I. Woods, Uncertainty estimation for 2D magnetic particle imaging, in International Workshop on Magnetic Particle Imaging, 2022. doi:10.18416/IJMPI.2022.2203082.