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The direct and the inverse magnetic encephalography problem

  • Ivan P. Pakhnenko , Arthur I. Sabirov and Tatyana V. Zakharova
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BIOKYBERNETIKA
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Abstract

Magnetic encephalography provides a unique opportunity for non-invasive study of neural processes occurring in the brain, but at the same time produces a large amount of data. Processing this data in order to reconstruct the signal sources with a given accuracy is an extremely difficult task. Our work is devoted to solving the direct and inverse problems of magnetoencephalography. The inverse problem o magnetoencephalography is ill-posed and difficult for both analytical and numerical solutions. Additional complications arise from the volume (passive) currents and the associated magnetic fields, which strongly depend on the brain geometry. In this paper, we find approximate analytical solutions for the forward and the inverse problems in the spherical and spheroid geometry. To localize the source of activity, it is first necessary to consider a direct problem, which in our case has an exact analytical solution. For this purpose, magnetic induction was calculated on a given surface of a complex system of dipoles, and a program was written to visualize magnetic induction on the surface of the head. In certain models of the human head (spherical and spheroid), the Biot-Savart law and IC analysis provide the basis for a detailed study in order to develop an algorithm for localization of the primary motor cortex. For the general case of the spheroid model, the inverse problem can be approximately solved by neglecting the volumetric magnetic field near the points of the maximum magnetic field and taking into account only the primary magnetic field. This paper presents a step-by-step algorithm for obtaining a solution to the inverse MEG problem under the assumptions of discreteness of signal sources originating from different functional areas of the brain and the surface location of signal sources. The new method is based on our obtained exact analytical solution of the inverse problem for the one-dipole model. Within the adopted constraints, the method is also applicable for a larger number of dipoles. The case of two dipoles has been investigated so far. Using the ICA to separate the magnetic field into its components and applying the analytical formula for the case of one dipole, we can obtain an approximate solution of the inverse MEG problem with high accuracy. The localization methods presented in this paper are undoubtedly important in real clinical practice. Thus, during neurosurgical interventions, various areas of the brain can be damaged, including irreparable ones. Since the location of functional zones in the human brain is individual, the doctor needs to be able to localize these areas in the preoperative period with high accuracy. The methods developed in this paper serve to solve such an important task.

Abstract

Magnetic encephalography provides a unique opportunity for non-invasive study of neural processes occurring in the brain, but at the same time produces a large amount of data. Processing this data in order to reconstruct the signal sources with a given accuracy is an extremely difficult task. Our work is devoted to solving the direct and inverse problems of magnetoencephalography. The inverse problem o magnetoencephalography is ill-posed and difficult for both analytical and numerical solutions. Additional complications arise from the volume (passive) currents and the associated magnetic fields, which strongly depend on the brain geometry. In this paper, we find approximate analytical solutions for the forward and the inverse problems in the spherical and spheroid geometry. To localize the source of activity, it is first necessary to consider a direct problem, which in our case has an exact analytical solution. For this purpose, magnetic induction was calculated on a given surface of a complex system of dipoles, and a program was written to visualize magnetic induction on the surface of the head. In certain models of the human head (spherical and spheroid), the Biot-Savart law and IC analysis provide the basis for a detailed study in order to develop an algorithm for localization of the primary motor cortex. For the general case of the spheroid model, the inverse problem can be approximately solved by neglecting the volumetric magnetic field near the points of the maximum magnetic field and taking into account only the primary magnetic field. This paper presents a step-by-step algorithm for obtaining a solution to the inverse MEG problem under the assumptions of discreteness of signal sources originating from different functional areas of the brain and the surface location of signal sources. The new method is based on our obtained exact analytical solution of the inverse problem for the one-dipole model. Within the adopted constraints, the method is also applicable for a larger number of dipoles. The case of two dipoles has been investigated so far. Using the ICA to separate the magnetic field into its components and applying the analytical formula for the case of one dipole, we can obtain an approximate solution of the inverse MEG problem with high accuracy. The localization methods presented in this paper are undoubtedly important in real clinical practice. Thus, during neurosurgical interventions, various areas of the brain can be damaged, including irreparable ones. Since the location of functional zones in the human brain is individual, the doctor needs to be able to localize these areas in the preoperative period with high accuracy. The methods developed in this paper serve to solve such an important task.

Chapters in this book

  1. Frontmatter I
  2. Prologue I VII
  3. Prologue II XI
  4. Prologue III XIII
  5. Preface XVII
  6. Overview XIX
  7. Contents XXXIII
  8. Part I: Theories
  9. Part I-A: Overarching theory
  10. Introduction 1
  11. Universal axioms in classical Chinese philosophy 5
  12. Category theory for structural characterization 15
  13. Axiomatic bipolar dynamics and their control 45
  14. Part I-B: Systems theories
  15. Introduction 75
  16. Stochastic formalization of agent-oriented systems 79
  17. Simplification of high-dimensional multitempo dynamic models 109
  18. Ideas of symmetry as a biophysical basis of system biomedicine 123
  19. Disorder of multiscale control 149
  20. Part II: Person’s life-sphere
  21. Part II-A: Person’s biosphere
  22. Introduction 185
  23. Mutations as activators of biological evolutionary processes at population levels 189
  24. Immunometabolism of T-cells in COVID-19 209
  25. Part II-A.2: Body’s vital functions
  26. Introduction 245
  27. Structural modeling of vascular networks 249
  28. Mathematical modeling of AI application for the diagnosis of blood flow disorders 283
  29. Modeling of glucose and insulin regulation within the framework of a self-consistent model of the cardiovascular system 303
  30. Hemodynamics in residual myocardial ischemia 319
  31. The quasi-one-dimensional model of the lymph flow in the human lymphatic system 335
  32. An integrate-and-fire mechanism for modeling rhythmicity in the neuroendocrine system 365
  33. Kinetic network modeling of the neuroendocrine hypothalamic-pituitary-adrenal axis dynamics with particular attention on the role of alcohol as a digestif 377
  34. Inflammation and immune response in atherosclerosis 393
  35. Part II-A.3: Body’s motor functions
  36. Introduction 423
  37. A magnetic resonance spectroscopy approach to quantitatively measure GABA and phosphorus level changes in the primary motor cortex elicited by transcranial direct current stimulation 427
  38. Part II-A.4: Body’s operational functions
  39. Introduction 441
  40. The fermionic mind hypothesis–a category-theoretic verification of consciousness 445
  41. Cross-task cognitive workload measurement based on the sample selection of the EEG data 459
  42. Part II-B: Person’s eco-sphere exposures
  43. Introduction 475
  44. The spread of SARS-CoV-2 in Russia and the evolution of the properties of the pathogen 479
  45. Agent-based modeling of epidemic spread via kinetic Monte Carlo method 491
  46. Control of SARS-nCoV outbreaks in China 2020 513
  47. Part II-B.2: Civilization
  48. Introduction 531
  49. Pesticide exposure: Toward holistic environmental modeling 535
  50. Part II-C: Person’s sociosphere exposures
  51. Introduction 559
  52. Evolution of the health system in Shanghai, China, 2016–2020 563
  53. Part III: Technologies
  54. Introduction 577
  55. Design-process automation using functional process blocks 581
  56. Slow/fast dynamic models with applications to engineering problems 601
  57. Part III-B: Information sciences
  58. Introduction 613
  59. Numerical modeling of medical ultrasound using the grid-characteristic method 617
  60. The direct and the inverse magnetic encephalography problem 635
  61. Part III-C: Data-analytic sciences
  62. Introduction 653
  63. Assessing the bioequivalence of two different drugs with the same active ingredient 655
  64. Estimation of adjusted relative risks in log-binomial regression using the Bekhit–Schöpe–Wagenpfeil algorithm 665
  65. Part IV: Clinical medicine
  66. Introduction 679
  67. Finding optimal two-stage combined treatment protocols for a blood cancer model 681
  68. Unraveling the mysteries: Mathematical perspectives on traditional Chinese medicine meridians 697
  69. Epilogue 721
  70. Index 723
https://doi.org/10.1515/9783111341996-040

Chapters in this book

  1. Frontmatter I
  2. Prologue I VII
  3. Prologue II XI
  4. Prologue III XIII
  5. Preface XVII
  6. Overview XIX
  7. Contents XXXIII
  8. Part I: Theories
  9. Part I-A: Overarching theory
  10. Introduction 1
  11. Universal axioms in classical Chinese philosophy 5
  12. Category theory for structural characterization 15
  13. Axiomatic bipolar dynamics and their control 45
  14. Part I-B: Systems theories
  15. Introduction 75
  16. Stochastic formalization of agent-oriented systems 79
  17. Simplification of high-dimensional multitempo dynamic models 109
  18. Ideas of symmetry as a biophysical basis of system biomedicine 123
  19. Disorder of multiscale control 149
  20. Part II: Person’s life-sphere
  21. Part II-A: Person’s biosphere
  22. Introduction 185
  23. Mutations as activators of biological evolutionary processes at population levels 189
  24. Immunometabolism of T-cells in COVID-19 209
  25. Part II-A.2: Body’s vital functions
  26. Introduction 245
  27. Structural modeling of vascular networks 249
  28. Mathematical modeling of AI application for the diagnosis of blood flow disorders 283
  29. Modeling of glucose and insulin regulation within the framework of a self-consistent model of the cardiovascular system 303
  30. Hemodynamics in residual myocardial ischemia 319
  31. The quasi-one-dimensional model of the lymph flow in the human lymphatic system 335
  32. An integrate-and-fire mechanism for modeling rhythmicity in the neuroendocrine system 365
  33. Kinetic network modeling of the neuroendocrine hypothalamic-pituitary-adrenal axis dynamics with particular attention on the role of alcohol as a digestif 377
  34. Inflammation and immune response in atherosclerosis 393
  35. Part II-A.3: Body’s motor functions
  36. Introduction 423
  37. A magnetic resonance spectroscopy approach to quantitatively measure GABA and phosphorus level changes in the primary motor cortex elicited by transcranial direct current stimulation 427
  38. Part II-A.4: Body’s operational functions
  39. Introduction 441
  40. The fermionic mind hypothesis–a category-theoretic verification of consciousness 445
  41. Cross-task cognitive workload measurement based on the sample selection of the EEG data 459
  42. Part II-B: Person’s eco-sphere exposures
  43. Introduction 475
  44. The spread of SARS-CoV-2 in Russia and the evolution of the properties of the pathogen 479
  45. Agent-based modeling of epidemic spread via kinetic Monte Carlo method 491
  46. Control of SARS-nCoV outbreaks in China 2020 513
  47. Part II-B.2: Civilization
  48. Introduction 531
  49. Pesticide exposure: Toward holistic environmental modeling 535
  50. Part II-C: Person’s sociosphere exposures
  51. Introduction 559
  52. Evolution of the health system in Shanghai, China, 2016–2020 563
  53. Part III: Technologies
  54. Introduction 577
  55. Design-process automation using functional process blocks 581
  56. Slow/fast dynamic models with applications to engineering problems 601
  57. Part III-B: Information sciences
  58. Introduction 613
  59. Numerical modeling of medical ultrasound using the grid-characteristic method 617
  60. The direct and the inverse magnetic encephalography problem 635
  61. Part III-C: Data-analytic sciences
  62. Introduction 653
  63. Assessing the bioequivalence of two different drugs with the same active ingredient 655
  64. Estimation of adjusted relative risks in log-binomial regression using the Bekhit–Schöpe–Wagenpfeil algorithm 665
  65. Part IV: Clinical medicine
  66. Introduction 679
  67. Finding optimal two-stage combined treatment protocols for a blood cancer model 681
  68. Unraveling the mysteries: Mathematical perspectives on traditional Chinese medicine meridians 697
  69. Epilogue 721
  70. Index 723
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