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Wartman WA, Weise K, Rachh M, Morales L, Deng ZD, Nummenmaa A, Makaroff SN. An adaptive h-refinement method for the boundary element fast multipole method for quasi-static electromagnetic modeling. Phys Med Biol 2024; 69:055030. [PMID: 38316038 PMCID: PMC10902857 DOI: 10.1088/1361-6560/ad2638] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/20/2023] [Accepted: 02/05/2024] [Indexed: 02/07/2024]
Abstract
Objective.In our recent work pertinent to modeling of brain stimulation and neurophysiological recordings, substantial modeling errors in the computed electric field and potential have sometimes been observed for standard multi-compartment head models. The goal of this study is to quantify those errors and, further, eliminate them through an adaptive mesh refinement (AMR) algorithm. The study concentrates on transcranial magnetic stimulation (TMS), transcranial electrical stimulation (TES), and electroencephalography (EEG) forward problems.Approach.We propose, describe, and systematically investigate an AMR method using the boundary element method with fast multipole acceleration (BEM-FMM) as the base numerical solver. The goal is to efficiently allocate additional unknowns to critical areas of the model, where they will best improve solution accuracy. The implemented AMR method's accuracy improvement is measured on head models constructed from 16 Human Connectome Project subjects under problem classes of TES, TMS, and EEG. Errors are computed between three solutions: an initial non-adaptive solution, a solution found after applying AMR with a conservative refinement rate, and a 'silver-standard' solution found by subsequent 4:1 global refinement of the adaptively-refined model.Main results.Excellent agreement is shown between the adaptively-refined and silver-standard solutions for standard head models. AMR is found to be vital for accurate modeling of TES and EEG forward problems for standard models: an increase of less than 25% (on average) in number of mesh elements for these problems, efficiently allocated by AMR, exposes electric field/potential errors exceeding 60% (on average) in the solution for the unrefined models.Significance.This error has especially important implications for TES dosing prediction-where the stimulation strength plays a central role-and for EEG lead fields. Though the specific form of the AMR method described here is implemented for the BEM-FMM, we expect that AMR is applicable and even required for accurate electromagnetic simulations by other numerical modeling packages as well.
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Affiliation(s)
- William A Wartman
- Electrical and Computer Engineering Department, Worcester Polytechnic Inst., Worcester, MA 01609 United States of America
| | - Konstantin Weise
- Max Planck Institute for Human Cognitive and Brain Sciences, Stephanstr. 1a, D-04103 Leipzig, Germany
- Department of Clinical Medicine, Aarhus University, DNK-8200, Aarhus, Denmark
| | - Manas Rachh
- Center for Computational Mathematics, Flatiron Institute, New York, NY 10012, United States of America
| | - Leah Morales
- Electrical and Computer Engineering Department, Worcester Polytechnic Inst., Worcester, MA 01609 United States of America
| | - Zhi-De Deng
- Computational Neurostimulation Research Program, Noninvasive Neuromodulation Unit, Experimental Therapeutics & Pathophysiology Branch, National Institute of Mental Health Intramural Research Program, National Institutes of Health, Bethesda, MD, United States of America
| | - Aapo Nummenmaa
- Athinoula A. Martinos Ctr. for Biomedical Imaging, Massachusetts General Hospital, Harvard Medical School, Charlestown, MA 02129 United States of America
| | - Sergey N Makaroff
- Electrical and Computer Engineering Department, Worcester Polytechnic Inst., Worcester, MA 01609 United States of America
- Athinoula A. Martinos Ctr. for Biomedical Imaging, Massachusetts General Hospital, Harvard Medical School, Charlestown, MA 02129 United States of America
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Makaroff SN, Qi Z, Rachh M, Wartman WA, Weise K, Noetscher GM, Daneshzand M, Deng ZD, Greengard L, Nummenmaa AR. A fast direct solver for surface-based whole-head modeling of transcranial magnetic stimulation. Sci Rep 2023; 13:18657. [PMID: 37907689 PMCID: PMC10618282 DOI: 10.1038/s41598-023-45602-5] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/18/2023] [Accepted: 10/21/2023] [Indexed: 11/02/2023] Open
Abstract
When modeling transcranial magnetic stimulation (TMS) in the brain, a fast and accurate electric field solver can support interactive neuronavigation tasks as well as comprehensive biophysical modeling. We formulate, test, and disseminate a direct (i.e., non-iterative) TMS solver that can accurately determine global TMS fields for any coil type everywhere in a high-resolution MRI-based surface model with ~ 200,000 or more arbitrarily selected observation points within approximately 5 s, with the solution time itself of 3 s. The solver is based on the boundary element fast multipole method (BEM-FMM), which incorporates the latest mathematical advancement in the theory of fast multipole methods-an FMM-based LU decomposition. This decomposition is specific to the head model and needs to be computed only once per subject. Moreover, the solver offers unlimited spatial numerical resolution. Despite the fast execution times, the present direct solution is numerically accurate for the default model resolution. In contrast, the widely used brain modeling software SimNIBS employs a first-order finite element method that necessitates additional mesh refinement, resulting in increased computational cost. However, excellent agreement between the two methods is observed for various practical test cases following mesh refinement, including a biophysical modeling task. The method can be readily applied to a wide range of TMS analyses involving multiple coil positions and orientations, including image-guided neuronavigation. It can even accommodate continuous variations in coil geometry, such as flexible H-type TMS coils. The FMM-LU direct solver is freely available to academic users.
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Affiliation(s)
- S N Makaroff
- Electrical and Computer Engineering Department, Worcester Polytechnic Institute, Worcester, MA, 01609, USA
- Athinoula A. Martinos Ctr. for Biomedical Imaging, Massachusetts General Hospital, Harvard Medical School, Charlestown, MA, 02129, USA
| | - Z Qi
- Electrical and Computer Engineering Department, Worcester Polytechnic Institute, Worcester, MA, 01609, USA.
| | - M Rachh
- Center for Computational Mathematics, Flatiron Institute, New York, NY, 10010, USA
| | - W A Wartman
- Electrical and Computer Engineering Department, Worcester Polytechnic Institute, Worcester, MA, 01609, USA
| | - K Weise
- Max Planck Institute for Human Cognitive and Brain Sciences, Stephanstr. 1a, 04103, Leipzig, Germany
- Advanced Electromagnetics Group, Technische Universität Ilmenau, Helmholtzplatz 2, 98693, Ilmenau, Germany
| | - G M Noetscher
- Electrical and Computer Engineering Department, Worcester Polytechnic Institute, Worcester, MA, 01609, USA
| | - M Daneshzand
- Athinoula A. Martinos Ctr. for Biomedical Imaging, Massachusetts General Hospital, Harvard Medical School, Charlestown, MA, 02129, USA
| | - Zhi-De Deng
- Computational Neurostimulation Research Program, Noninvasive Neuromodulation Unit, Experimental Therapeutics and Pathophysiology Branch, National Institute of Mental Health, NIH 10 Center Drive, Bethesda, MD, 20892, USA
| | - L Greengard
- Center for Computational Mathematics, Flatiron Institute, New York, NY, 10010, USA
- Courant Institute of Mathematical Sciences, 251 Mercer Street, New York, NY, 10012, USA
| | - A R Nummenmaa
- Athinoula A. Martinos Ctr. for Biomedical Imaging, Massachusetts General Hospital, Harvard Medical School, Charlestown, MA, 02129, USA
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Wartman WA, Weise K, Rachh M, Morales L, Deng ZD, Nummenmaa A, Makaroff SN. An Adaptive H-Refinement Method for the Boundary Element Fast Multipole Method for Quasi-static Electromagnetic Modeling. bioRxiv 2023:2023.08.11.552996. [PMID: 37645957 PMCID: PMC10461998 DOI: 10.1101/2023.08.11.552996] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 08/31/2023]
Abstract
Objective In our recent work pertinent to modeling of brain stimulation and neurophysiological recordings, substantial modeling errors in the computed electric field and potential have sometimes been observed for standard multi-compartment head models. The goal of this study is to quantify those errors and, further, eliminate them through an adaptive mesh refinement (AMR) algorithm. The study concentrates on transcranial magnetic stimulation (TMS), transcranial electrical stimulation (TES), and electroencephalography (EEG) forward problems. Approach We propose, describe, and systematically investigate an AMR method using the Boundary Element Method with Fast Multipole Acceleration (BEM-FMM) as the base numerical solver. The goal is to efficiently allocate additional unknowns to critical areas of the model, where they will best improve solution accuracy.The implemented AMR method's accuracy improvement is measured on head models constructed from 16 Human Connectome Project subjects under problem classes of TES, TMS, and EEG. Errors are computed between three solutions: an initial non-adaptive solution, a solution found after applying AMR with a conservative refinement rate, and a "silver-standard" solution found by subsequent 4:1 global refinement of the adaptively-refined model. Main Results Excellent agreement is shown between the adaptively-refined and silver-standard solutions for standard head models. AMR is found to be vital for accurate modeling of TES and EEG forward problems for standard models: an increase of less than 25% (on average) in number of mesh elements for these problems, efficiently allocated by AMR, exposes electric field/potential errors exceeding 60% (on average) in the solution for the unrefined models. Significance This error has especially important implications for TES dosing prediction - where the stimulation strength plays a central role - and for EEG lead fields. Though the specific form of the AMR method described here is implemented for the BEM-FMM, we expect that AMR is applicable and even required for accurate electromagnetic simulations by other numerical modeling packages as well.
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Affiliation(s)
- William A Wartman
- Electrical and Computer Engineering Department, Worcester Polytechnic Inst., Worcester, MA 01609 USA
| | - Konstantin Weise
- Max Planck Institute for Human Cognitive and Brain Sciences, Stephanstr. 1a, 04103 Leipzig, Germany
- Department of Clinical Medicine, Aarhus University, DNK-8200, Aarhus, Denmark
| | - Manas Rachh
- Center for Computational Mathematics, Flatiron Institute, New York, NY 10012, USA
| | - Leah Morales
- Electrical and Computer Engineering Department, Worcester Polytechnic Inst., Worcester, MA 01609 USA
| | - Zhi-De Deng
- Computational Neurostimulation Research Program, Noninvasive Neuromodulation Unit, Experimental Therapeutics & Pathophysiology Branch, National Institute of Mental Health Intramural Research Program, National Institutes of Health, Bethesda, MD, USA
| | - Aapo Nummenmaa
- Athinoula A. Martinos Ctr. for Biomedical Imaging, Massachusetts General Hospital, Harvard Medical School, Charlestown, MA 02129 USA
| | - Sergey N Makaroff
- Electrical and Computer Engineering Department, Worcester Polytechnic Inst., Worcester, MA 01609 USA
- Athinoula A. Martinos Ctr. for Biomedical Imaging, Massachusetts General Hospital, Harvard Medical School, Charlestown, MA 02129 USA
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Makaroff SN, Qi Z, Rachh M, Wartman WA, Weise K, Noetscher GM, Daneshzand M, Deng ZD, Greengard L, Nummenmaa AR. A fast direct solver for surface-based whole-head modeling of transcranial magnetic stimulation. Res Sq 2023:rs.3.rs-3079433. [PMID: 37503106 PMCID: PMC10371170 DOI: 10.21203/rs.3.rs-3079433/v1] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 07/29/2023]
Abstract
Background When modeling transcranial magnetic stimulation (TMS) in the brain, a fast and accurate electric field solver can support interactive neuronavigation tasks as well as comprehensive biophysical modeling. Objective We formulate, test, and disseminate a direct (i.e., non-iterative) TMS solver that can accurately determine global TMS fields for any coil type everywhere in a high-resolution MRI-based surface model with ~200,000 or more arbitrarily selected observation points within approximately 5 sec, with the solution time itself of 3 sec. Method The solver is based on the boundary element fast multipole method (BEM-FMM), which incorporates the latest mathematical advancement in the theory of fast multipole methods - an FMM-based LU decomposition. This decomposition is specific to the head model and needs to be computed only once per subject. Moreover, the solver offers unlimited spatial numerical resolution. Results Despite the fast execution times, the present direct solution is numerically accurate for the default model resolution. In contrast, the widely used brain modeling software SimNIBS employs a first-order finite element method that necessitates additional mesh refinement, resulting in increased computational cost. However, excellent agreement between the two methods is observed for various practical test cases following mesh refinement, including a biophysical modeling task. Conclusion The method can be readily applied to a wide range of TMS analyses involving multiple coil positions and orientations, including image-guided neuronavigation. It can even accommodate continuous variations in coil geometry, such as flexible H-type TMS coils. The FMM-LU direct solver is freely available to academic users.
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Affiliation(s)
- S N Makaroff
- Electrical and Computer Engineering Department, Worcester Polytechnic Institute, Worcester, MA 01609 USA
- Athinoula A. Martinos Ctr. for Biomedical Imaging, Massachusetts General Hospital, Harvard Medical School, Charlestown, MA 02129 USA
| | - Z Qi
- Electrical and Computer Engineering Department, Worcester Polytechnic Institute, Worcester, MA 01609 USA
| | - M Rachh
- Center for Computational Mathematics, Flatiron Institute, New York, NY 10010 USA
| | - W A Wartman
- Electrical and Computer Engineering Department, Worcester Polytechnic Institute, Worcester, MA 01609 USA
| | - K Weise
- Max Planck Institute for Human Cognitive and Brain Sciences, Stephanstr. 1a, 04103, Leipzig Germany
- Technische Universität Ilmenau, Advanced Electromagnetics Group, Helmholtzplatz 2, 98693 Ilmenau Germany
| | - G M Noetscher
- Electrical and Computer Engineering Department, Worcester Polytechnic Institute, Worcester, MA 01609 USA
| | - M Daneshzand
- Athinoula A. Martinos Ctr. for Biomedical Imaging, Massachusetts General Hospital, Harvard Medical School, Charlestown, MA 02129 USA
| | - Zhi-De Deng
- Computational Neurostimulation Research Program, Noninvasive Neuromodulation Unit, Experimental Therapeutics & Pathophysiology Branch, National Institute of Mental Health, NIH 10 Center Drive, Bethesda, MD 20892 USA
| | - L Greengard
- Center for Computational Mathematics, Flatiron Institute, New York, NY 10010 USA
- Courant Institute of Mathematical Sciences, 251 Mercer Street, New York, NY 10012 USA
| | - A R Nummenmaa
- Athinoula A. Martinos Ctr. for Biomedical Imaging, Massachusetts General Hospital, Harvard Medical School, Charlestown, MA 02129 USA
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Hoskins JG, Rachh M, Schotland JC. Quantum electrodynamics of chiral and antichiral waveguide arrays. Opt Lett 2023; 48:1232-1235. [PMID: 36857256 DOI: 10.1364/ol.477807] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/11/2022] [Accepted: 01/04/2023] [Indexed: 06/18/2023]
Abstract
We consider the quantum electrodynamics of single photons in arrays of one-way waveguides, each containing many atoms. We investigate both chiral and antichiral arrays, in which the group velocities of the waveguides are the same or alternate in sign, respectively. We find that in the continuum limit, the one-photon amplitude obeys a Dirac equation. In the chiral case, the Dirac equation is hyperbolic, while in the antichiral case it is elliptic. This distinction has implications for the nature of photon transport in waveguide arrays. Our results are illustrated by numerical simulations.
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Rachh M, Iyer V, Greengard L, Zibman S, Daneshzand M, Hämäläinen M, Ahveninen JP, Raji T, Nummenmaa A, Makaroff S. Potential of fast multipole method for mesoscale and multiscale brain modeling: applicability to EEG, MEG, TES, TMS, and DBS. Brain Stimul 2023. [DOI: 10.1016/j.brs.2023.01.068] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 02/17/2023] Open
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Linderman GC, Rachh M, Hoskins JG, Steinerberger S, Kluger Y. Fast interpolation-based t-SNE for improved visualization of single-cell RNA-seq data. Nat Methods 2019; 16:243-245. [PMID: 30742040 PMCID: PMC6402590 DOI: 10.1038/s41592-018-0308-4] [Citation(s) in RCA: 217] [Impact Index Per Article: 43.4] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/03/2018] [Accepted: 12/12/2018] [Indexed: 11/29/2022]
Abstract
t-distributed stochastic neighbor embedding (t-SNE) is widely used for visualizing single-cell RNA-sequencing (scRNA-seq) data, but it scales poorly to large datasets. We dramatically accelerate t-SNE, obviating the need for data downsampling, and hence allowing visualization of rare cell populations. Furthermore, we implement a heatmap-style visualization for scRNA-seq based on one-dimensional t-SNE for simultaneously visualizing the expression patterns of thousands of genes. Software is available at https://github.com/KlugerLab/FIt-SNE and https://github.com/KlugerLab/t-SNE-Heatmaps .
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Affiliation(s)
| | - Manas Rachh
- Applied Mathematics Program, Yale University, New Haven, CT, USA
| | - Jeremy G Hoskins
- Applied Mathematics Program, Yale University, New Haven, CT, USA
| | | | - Yuval Kluger
- Applied Mathematics Program, Yale University, New Haven, CT, USA.
- Department of Pathology, Yale University School of Medicine, New Haven, CT, USA.
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Pulupa J, Rachh M, Tomasini MD, Mincer JS, Simon SM. A coarse-grained computational model of the nuclear pore complex predicts Phe-Gly nucleoporin dynamics. J Gen Physiol 2017; 149:951-966. [PMID: 28887410 PMCID: PMC5694938 DOI: 10.1085/jgp.201711769] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/03/2017] [Revised: 06/27/2017] [Accepted: 08/10/2017] [Indexed: 11/25/2022] Open
Abstract
The phenylalanine-glycine–repeat nucleoporins are essential for transport through the nuclear pore complex. Pulupa et al. observe reptation of these nucleoporins on a physiological timescale in coarse-grained computational simulations. The phenylalanine-glycine–repeat nucleoporins (FG-Nups), which occupy the lumen of the nuclear pore complex (NPC), are critical for transport between the nucleus and cytosol. Although NPCs differ in composition across species, they are largely conserved in organization and function. Transport through the pore is on the millisecond timescale. Here, to explore the dynamics of nucleoporins on this timescale, we use coarse-grained computational simulations. These simulations generate predictions that can be experimentally tested to distinguish between proposed mechanisms of transport. Our model reflects the conserved structure of the NPC, in which FG-Nup filaments extend into the lumen and anchor along the interior of the channel. The lengths of the filaments in our model are based on the known characteristics of yeast FG-Nups. The FG-repeat sites also bind to each other, and we vary this association over several orders of magnitude and run 100-ms simulations for each value. The autocorrelation functions of the orientation of the simulated FG-Nups are compared with in vivo anisotropy data. We observe that FG-Nups reptate back and forth through the NPC at timescales commensurate with experimental measurements of the speed of cargo transport through the NPC. Our results are consistent with models of transport where FG-Nup filaments are free to move across the central channel of the NPC, possibly informing how cargo might transverse the NPC.
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Affiliation(s)
- Joan Pulupa
- Laboratory of Cellular Biophysics, The Rockefeller University, New York, NY
| | - Manas Rachh
- Courant Institute of Mathematical Sciences, New York, NY
| | - Michael D Tomasini
- Laboratory of Cellular Biophysics, The Rockefeller University, New York, NY
| | - Joshua S Mincer
- Laboratory of Cellular Biophysics, The Rockefeller University, New York, NY .,Department of Anesthesiology, Icahn School of Medicine at Mount Sinai, New York, NY
| | - Sanford M Simon
- Laboratory of Cellular Biophysics, The Rockefeller University, New York, NY
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