Practical computation of the diffusion MRI signal of realistic neurons based on Laplace eigenfunctions
The complex transverse water proton magnetization subject to diffusion‐encoding magnetic field gradient pulses in a heterogeneous medium such as brain tissue can be modeled by the Bloch‐Torrey partial differential equation. The spatial integral of the solution of this equation in realistic geometry...
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| Published in | NMR in biomedicine Vol. 33; no. 10; pp. e4353 - n/a |
|---|---|
| Main Authors | , , |
| Format | Journal Article |
| Language | English |
| Published |
Oxford
Wiley Subscription Services, Inc
01.10.2020
Wiley |
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| Online Access | Get full text |
| ISSN | 0952-3480 1099-1492 1099-1492 |
| DOI | 10.1002/nbm.4353 |
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| Abstract | The complex transverse water proton magnetization subject to diffusion‐encoding magnetic field gradient pulses in a heterogeneous medium such as brain tissue can be modeled by the Bloch‐Torrey partial differential equation. The spatial integral of the solution of this equation in realistic geometry provides a gold‐standard reference model for the diffusion MRI signal arising from different tissue micro‐structures of interest. A closed form representation of this reference diffusion MRI signal called matrix formalism, which makes explicit the link between the Laplace eigenvalues and eigenfunctions of the biological cell and its diffusion MRI signal, was derived 20 years ago. In addition, once the Laplace eigendecomposition has been computed and saved, the diffusion MRI signal can be calculated for arbitrary diffusion‐encoding sequences and b‐values at negligible additional cost. Up to now, this representation, though mathematically elegant, has not been often used as a practical model of the diffusion MRI signal, due to the difficulties of calculating the Laplace eigendecomposition in complicated geometries. In this paper, we present a simulation framework that we have implemented inside the MATLAB‐based diffusion MRI simulator SpinDoctor that efficiently computes the matrix formalism representation for realistic neurons using the finite element method. We show that the matrix formalism representation requires a few hundred eigenmodes to match the reference signal computed by solving the Bloch‐Torrey equation when the cell geometry originates from realistic neurons. As expected, the number of eigenmodes required to match the reference signal increases with smaller diffusion time and higher b‐values. We also convert the eigenvalues to a length scale and illustrate the link between the length scale and the oscillation frequency of the eigenmode in the cell geometry. We give the transformation that links the Laplace eigenfunctions to the eigenfunctions of the Bloch‐Torrey operator and compute the Bloch‐Torrey eigenfunctions and eigenvalues. This work is another step in bringing advanced mathematical tools and numerical method development to the simulation and modeling of diffusion MRI.
We present a simulation framework that we have implemented inside the MATLAB‐based diffusion MRI simulator SpinDoctor that efficiently computes the matrix formalism representation for realistic neurons using the finite element method.
The matrix formalism representation requires around 100 eigenmodes to match the reference signal when the cell geometry originates from realistic neurons. We convert the eigenvalues to a length scale and illustrate the link between the length scale and the oscillation frequency of the eigenmode in the cell geometry. |
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| AbstractList | The complex transverse water proton magnetization subject to diffusion-encoding magnetic field gradient pulses in a heterogeneous medium such as brain tissue can be modeled by the Bloch-Torrey partial differential equation. The spatial integral of the solution of this equation in realistic geometry provides a gold-standard reference model for the diffusion MRI signal arising from different tissue micro-structures of interest. A closed form representation of this reference diffusion MRI signal called matrix formalism, which makes explicit the link between the Laplace eigenvalues and eigenfunctions of the biological cell and its diffusion MRI signal, was derived 20 years ago. In addition, once the Laplace eigendecomposition has been computed and saved, the diffusion MRI signal can be calculated for arbitrary diffusion-encoding sequences and b-values at negligible additional cost. Up to now, this representation, though mathematically elegant, has not been often used as a practical model of the diffusion MRI signal, due to the difficulties of calculating the Laplace eigendecomposition in complicated geometries. In this paper, we present a simulation framework that we have implemented inside the MATLAB-based diffusion MRI simulator SpinDoctor that efficiently computes the matrix formalism representation for realistic neurons using the finite element method. We show that the matrix formalism representation requires a few hundred eigenmodes to match the reference signal computed by solving the Bloch-Torrey equation when the cell geometry originates from realistic neurons. As expected, the number of eigenmodes required to match the reference signal increases with smaller diffusion time and higher b-values. We also convert the eigenvalues to a length scale and illustrate the link between the length scale and the oscillation frequency of the eigenmode in the cell geometry. We give the transformation that links the Laplace eigenfunctions to the eigenfunctions of the Bloch-Torrey operator and compute the Bloch-Torrey eigenfunctions and eigenvalues. This work is another step in bringing advanced mathematical tools and numerical method development to the simulation and modeling of diffusion MRI.The complex transverse water proton magnetization subject to diffusion-encoding magnetic field gradient pulses in a heterogeneous medium such as brain tissue can be modeled by the Bloch-Torrey partial differential equation. The spatial integral of the solution of this equation in realistic geometry provides a gold-standard reference model for the diffusion MRI signal arising from different tissue micro-structures of interest. A closed form representation of this reference diffusion MRI signal called matrix formalism, which makes explicit the link between the Laplace eigenvalues and eigenfunctions of the biological cell and its diffusion MRI signal, was derived 20 years ago. In addition, once the Laplace eigendecomposition has been computed and saved, the diffusion MRI signal can be calculated for arbitrary diffusion-encoding sequences and b-values at negligible additional cost. Up to now, this representation, though mathematically elegant, has not been often used as a practical model of the diffusion MRI signal, due to the difficulties of calculating the Laplace eigendecomposition in complicated geometries. In this paper, we present a simulation framework that we have implemented inside the MATLAB-based diffusion MRI simulator SpinDoctor that efficiently computes the matrix formalism representation for realistic neurons using the finite element method. We show that the matrix formalism representation requires a few hundred eigenmodes to match the reference signal computed by solving the Bloch-Torrey equation when the cell geometry originates from realistic neurons. As expected, the number of eigenmodes required to match the reference signal increases with smaller diffusion time and higher b-values. We also convert the eigenvalues to a length scale and illustrate the link between the length scale and the oscillation frequency of the eigenmode in the cell geometry. We give the transformation that links the Laplace eigenfunctions to the eigenfunctions of the Bloch-Torrey operator and compute the Bloch-Torrey eigenfunctions and eigenvalues. This work is another step in bringing advanced mathematical tools and numerical method development to the simulation and modeling of diffusion MRI. The complex transverse water proton magnetization subject to diffusion‐encoding magnetic field gradient pulses in a heterogeneous medium such as brain tissue can be modeled by the Bloch‐Torrey partial differential equation. The spatial integral of the solution of this equation in realistic geometry provides a gold‐standard reference model for the diffusion MRI signal arising from different tissue micro‐structures of interest. A closed form representation of this reference diffusion MRI signal called matrix formalism, which makes explicit the link between the Laplace eigenvalues and eigenfunctions of the biological cell and its diffusion MRI signal, was derived 20 years ago. In addition, once the Laplace eigendecomposition has been computed and saved, the diffusion MRI signal can be calculated for arbitrary diffusion‐encoding sequences and b ‐values at negligible additional cost. Up to now, this representation, though mathematically elegant, has not been often used as a practical model of the diffusion MRI signal, due to the difficulties of calculating the Laplace eigendecomposition in complicated geometries. In this paper, we present a simulation framework that we have implemented inside the MATLAB‐based diffusion MRI simulator SpinDoctor that efficiently computes the matrix formalism representation for realistic neurons using the finite element method. We show that the matrix formalism representation requires a few hundred eigenmodes to match the reference signal computed by solving the Bloch‐Torrey equation when the cell geometry originates from realistic neurons. As expected, the number of eigenmodes required to match the reference signal increases with smaller diffusion time and higher b ‐values. We also convert the eigenvalues to a length scale and illustrate the link between the length scale and the oscillation frequency of the eigenmode in the cell geometry. We give the transformation that links the Laplace eigenfunctions to the eigenfunctions of the Bloch‐Torrey operator and compute the Bloch‐Torrey eigenfunctions and eigenvalues. This work is another step in bringing advanced mathematical tools and numerical method development to the simulation and modeling of diffusion MRI. The complex transverse water proton magnetization subject to diffusion-encoding magnetic field gradient pulses in a heterogeneous medium such as brain tissue can be modeled by the Bloch-Torrey partial differential equation. The spatial integral of the solution of this equation in realistic geometry provides a gold-standard reference model for the diffusion MRI signal arising from different tissue micro-structures of interest. A closed form representation of this reference diffusion MRI signal has been derived twenty years ago, called Matrix Formalism that makes explicit the link between the Laplace eigenvalues and eigenfunctions of the biological cell and its diffusion MRI signal. In addition, once the Laplace eigendecomposition has been computed and saved, the diffusion MRI signal can be calculated for arbitrary diffusion-encoding sequences and b-values at negligible additional cost.Up to now, this representation, though mathematically elegant, has not been often used as a practical model of the diffusion MRI signal, due to the difficulties of calculating the Laplace eigendecomposition in complicated geometries. In this paper, we present a simulation framework that we have implemented inside the MATLAB-based diffusion MRI simulator SpinDoctor that efficiently computes the Matrix Formalism representation forrealistic neurons using the finite elements method. We show the Matrix Formalism representation requires around a few hundred eigenmodes to match the reference signal computed by solving the Bloch-Torrey equation when the cell geometry comes from realistic neurons. As expected, the number of required eigenmodes to match the reference signal increases with smaller diffusion time and higher b-values. We also converted the eigenvalues to alength scale and illustrated the link between the length scale and the oscillation frequency of the eigenmode in the cell geometry. We gave the transformation that links the Laplace eigenfunctions to the eigenfunctions of the Bloch-Torrey operator and computed the Bloch-Torrey eigenfunctions and eigenvalues. This work is another step in bringing advanced mathematical tools and numerical method development to the simulation and modeling ofdiffusion MRI. The complex transverse water proton magnetization subject to diffusion‐encoding magnetic field gradient pulses in a heterogeneous medium such as brain tissue can be modeled by the Bloch‐Torrey partial differential equation. The spatial integral of the solution of this equation in realistic geometry provides a gold‐standard reference model for the diffusion MRI signal arising from different tissue micro‐structures of interest. A closed form representation of this reference diffusion MRI signal called matrix formalism, which makes explicit the link between the Laplace eigenvalues and eigenfunctions of the biological cell and its diffusion MRI signal, was derived 20 years ago. In addition, once the Laplace eigendecomposition has been computed and saved, the diffusion MRI signal can be calculated for arbitrary diffusion‐encoding sequences and b‐values at negligible additional cost. Up to now, this representation, though mathematically elegant, has not been often used as a practical model of the diffusion MRI signal, due to the difficulties of calculating the Laplace eigendecomposition in complicated geometries. In this paper, we present a simulation framework that we have implemented inside the MATLAB‐based diffusion MRI simulator SpinDoctor that efficiently computes the matrix formalism representation for realistic neurons using the finite element method. We show that the matrix formalism representation requires a few hundred eigenmodes to match the reference signal computed by solving the Bloch‐Torrey equation when the cell geometry originates from realistic neurons. As expected, the number of eigenmodes required to match the reference signal increases with smaller diffusion time and higher b‐values. We also convert the eigenvalues to a length scale and illustrate the link between the length scale and the oscillation frequency of the eigenmode in the cell geometry. We give the transformation that links the Laplace eigenfunctions to the eigenfunctions of the Bloch‐Torrey operator and compute the Bloch‐Torrey eigenfunctions and eigenvalues. This work is another step in bringing advanced mathematical tools and numerical method development to the simulation and modeling of diffusion MRI. We present a simulation framework that we have implemented inside the MATLAB‐based diffusion MRI simulator SpinDoctor that efficiently computes the matrix formalism representation for realistic neurons using the finite element method. The matrix formalism representation requires around 100 eigenmodes to match the reference signal when the cell geometry originates from realistic neurons. We convert the eigenvalues to a length scale and illustrate the link between the length scale and the oscillation frequency of the eigenmode in the cell geometry. |
| Author | Nguyen, Van‐Dang Li, Jing‐Rebecca Tran, Try Nguyen |
| Author_xml | – sequence: 1 givenname: Jing‐Rebecca orcidid: 0000-0001-6075-5526 surname: Li fullname: Li, Jing‐Rebecca email: jingrebecca.li@inria.fr organization: INRIA Saclay‐Equipe DEFI, CMAP, Ecole Polytechnique – sequence: 2 givenname: Try Nguyen surname: Tran fullname: Tran, Try Nguyen organization: INRIA Saclay‐Equipe DEFI, CMAP, Ecole Polytechnique – sequence: 3 givenname: Van‐Dang surname: Nguyen fullname: Nguyen, Van‐Dang organization: KTH Royal Institute of Technology |
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| CitedBy_id | crossref_primary_10_1016_j_media_2023_102979 crossref_primary_10_1002_nbm_4646 crossref_primary_10_1016_j_neuroimage_2020_117198 crossref_primary_10_1137_21M1439572 |
| Cites_doi | 10.1523/JNEUROSCI.2055-07.2007 10.1016/j.jmr.2014.08.016 10.1063/1.3082078 10.1016/j.jcp.2014.01.009 10.1093/oso/9780198537885.001.0001 10.1016/j.neuroimage.2011.06.006 10.1002/nbm.3998 10.1002/mrm.26832 10.1137/16M1107474 10.1016/j.jmr.2019.106611 10.1016/j.neuroimage.2017.12.038 10.7712/100016.1796.8619 10.1002/mrm.26548 10.1109/TBME.2019.2893523 10.1103/PhysRevLett.68.3555 10.1016/S1090-7807(02)00039-3 10.1016/j.neuroimage.2011.09.081 10.1016/j.jmr.2015.01.008 10.1103/RevModPhys.79.1077 10.1371/journal.pone.0076626 10.1007/978-3-319-46630-9_4 10.1016/j.jcp.2018.08.039 10.1016/j.jmr.2011.04.004 10.1002/mrm.21577 10.1016/j.neuroimage.2016.01.047 10.1016/j.jmr.2018.09.013 10.1006/jmre.1997.1233 10.1016/j.neuroimage.2019.116120 10.1016/j.jmr.2019.01.007 10.1002/cpa.3160430802 10.1002/mrm.27101 10.1016/j.jmr.2015.08.008 10.1016/j.jmr.2013.06.019 10.1002/mrm.10078 10.1016/j.neuroimage.2010.05.043 10.1109/TMI.2019.2902957 10.1016/j.jmr.2019.01.002 10.1063/1.1695690 10.1016/j.neuroimage.2012.03.072 10.1148/radiology.161.2.3763909 10.1016/j.jmr.2011.02.022 10.1073/pnas.1418198112 10.1016/j.neuroimage.2011.01.084 10.1006/jmre.1999.1778 10.1016/j.jmr.2019.06.016 10.1103/PhysRevB.47.8565 10.1073/pnas.1316944111 10.1016/j.neuroimage.2006.10.037 10.1016/j.neuroimage.2018.09.076 10.1016/j.neuroimage.2016.09.057 10.1109/TMI.2009.2015756 10.1073/pnas.1504327113 10.1007/s10334-018-0680-1 10.1002/mrm.22033 10.1109/TMI.2018.2873736 10.1073/pnas.1320223111 10.1063/1.1830432 10.1016/j.neuroimage.2015.03.061 10.1103/PhysRev.80.580 10.1016/j.neuroimage.2018.09.075 10.1016/j.neuroimage.2018.06.046 |
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| Keywords | diffusion MRI Bloch-Torrey equation Laplace eigenfunctions simulation Matrix Formalism finite elements |
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| Title | Practical computation of the diffusion MRI signal of realistic neurons based on Laplace eigenfunctions |
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