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Direct and Inverse bioelectric problems

Outline. Introduction of bioelectricity and bioelectric problems.Direct and inverse problemsModel construction and mesh generationexample(EEG). History of research on bioelectric field problems. Luigi Galvani, in 1786, stimulated muscle contractions by mechanical and electrical means, respectively. .

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Direct and Inverse bioelectric problems

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    1. Direct and Inverse bioelectric problems Presented by Chao Shen George Landon Cheng dong Li University of Kentucky

    2. Outline Introduction of bioelectricity and bioelectric problems. Direct and inverse problems Model construction and mesh generation example(EEG)

    3. History of research on bioelectric field problems Luigi Galvani, in 1786, stimulated muscle contractions by mechanical and electrical means, respectively. Bioelectricity occurs in all living tissue and has been the subject of investigation since Swammerdam, in 1658, and later

    4. What is bioelectricity? The origins of bioelectricity lie within cell membranes, which maintain a small potential difference between the interior and exterior of each cell. Fluctuation of this potential acts as a signaling mechanism that permits nerves to interact, muscles to contract, and communication to occur over the whole body.

    5. Bioelectric field problems Bioelectric field problems can be found in a wide variety of biomedical applications which range from single cells, to organs, up to models which incorporate partial to full human structures To diagnose tissue as either healthy or diseased, bioelectric signals from the tissue have to be recorded. Measurement of Bioelectricity (1) Noninvasive Measurement (2) Invasive Measurement

    6. Applications The solutions to bioelectric problems have applications to defibrillation studies, detection and location of arrhythmias, and localization and analysis of spontaneous brain activity in epileptic patients, etc. Our focus EEG (electroencephalography) ECG (electrocardiography).

    7. Early ECG machine, circa, 1911

    8. Poisson's and Laplace's Equation Poisson's Equation Laplace's equation

    9. Bioelectric Volume Conductors A general volume conductor can be defined as a region of volume, , which has conductivity, , and permitivity, , in which resides a source current, , where the V signifies per-unit volume. Solving a volume conductor problem means finding expressions for the electric, , and potential, , fields everywhere within the volume, , and/or on one of the bounded surfaces, .

    10. continue

    11. Direct problems The direct problem would be to solve (1) for with a known description of and the Neumann boundary condition: The normal component of the electric field is zero on the surface interfacing with air (here denoted by ).

    12. Inverse problems The inverse problems associated with these direct problems involve estimating the current sources within the volume conductor from measurements of voltages on the surface of either the head or body. Thus one would solve (1) with the boundary conditions:

    13. To Determine the FIELD from the Known Source and Conductor is Called DIRECT PROBLEM To Determine the SOURCE from the Known Field and Conductor is Called INVERSE PROBLEM

    14. Mesh Generation Mesh generation can be defined as the process of breaking up a physical domain into smaller sub-domains (elements) in order to facilitate the numerical solution (finite element, boundary element, finite difference, or multigrid) of a partial differential equation.

    15. Why do Mesh Generation? Once we have stated or derived the mathematical equations which define the physics of the system, we must figure out how to solve these equations for the particular domain we are interested in. Most numerical methods for solving partial differential equations break up the continuous domain(volume conductor) into discrete elements and approximate the PDE using the particular numerical technique best suited to the particular problem.

    16. Mesh Generation   (continue) Two main approaches for mesh construction divide and conquer a. advantage b. disadvantage Delaunay triangulation strategy a. advantage b. disadvantage

    17. Example of direct problem in EEG (electroencephalography) The EEG forward problem(dipole current sources problem) can be stated as follows: Known: the positions, orientations and magnitudes of dipole current sources, as well as the geometry and electrical conductivity of the head volume,  . calculate: the distribution of the electric potential on the surface of the head (scalp),    .

    18. Example of direct problem in EEG (continue)

    19. Example of direct problem in EEG (continue)

    20. Example of direct problem in EEG (continue)

    21. Example of direct problem in EEG (continue) Using the FEM, we obtain the linear system of equations:

    22. Example of direct problem in EEG (continue)

    23. Links http://www.sci.utah.edu/ncrr/overview/background.html http://butler.cc.tut.fi/~malmivuo/bem/book/figures.htm  http://www.ruf.rice.edu/~gpotts/EEGlab/ERP.html http://www.cvrti.utah.edu/~macleod/bioen/be6900/notes/ECG-bw.pdf http://www.mgnet.org/mgnet/Conferences/CopperMtn01/Talks/ruede-biomef/slides.htm http://www.gg.caltech.edu/~zhukov/research/eeg_meg/ieee-emb/node2.html#conduct

    24. Direct and Inverse Bioelectric Fields -Numerical Methods George Landon

    25. Direct and Inverse Bioelectric Fields Motivation Usage Example Numerical Methods Comparison of Methods More Examples

    26. Motivation Defibrillation studies Changing the speed of a heartbeat Detection and location of arrhythmias Irregular heartbeat caused by the heart’s electrical system Impedance imaging techniques Measure the amount of blood being pumped by the heart Localization and analysis of spontaneous brain activity in epileptic patients

    27. Usage Example Heart Models Normal Electrical Propagation

    28. Mathematical Model Bioelectric Volume Conductors A reduction to Poisson's equation for electrical conduction: in = the electrical conductivity tensor = the electrostatic potential = the current source per unit volume = the solution domain In this form, one includes the source region and an understanding of the primary bioelectric sources, , usually in the form of a simplified mathematical model.

    29. Numerical Methods Three methods examined Finite Difference Finite Element Boundary Element

    30. Finite Difference Method The solution domain is approximated by a grid of uniformly spaced nodes At each node, the differential equation is approximated by an algebraic expression which references adjacent grid points A system of equations is obtained by evaluating the previous algebraic approximations for each node in the domain. The system is solved for each value of the dependent variable at each node.

    31. Finite Difference Method The finite difference representation of in is This can be reformulated into the matrix equation

    32. Finite Element Method The solution domain is discretized into a number of uniform or non-uniform finite elements that are connected via nodes. The change of the dependent variable with regard to location is approximated within each element by an interpolation function. The interpolation function is defined relative to the values of the variable at the nodes associated with each element. The interpolation functions are then substituted into the integral equation, integrated, and combined with the results from the solution domain

    33. Finite Element Method in can equivalently be expressed as a system of N equations with N unknowns Now use the linear combination generated from the Finite Element method Now the matrix equation can be generated

    34. Finite Element Method For volume conductor problems contains all of the geometry and conductivity information of the model The matrix is symmetric, has a unique solution, and is sparse

    35. Boundary Element Method Only used for problems with isotropic domains and few inhomogeneities Utilizes information only upon the boundaries of interest Transforms the differential operator defined in the domain to integral operators defined on the boundary Makes mesh generation simpler, but potentials and gradients can only be evaluated after boundary solutions are obtained

    36. Comparison of Methods The choice of a method depends on the nature of the problem Finite Element and Finite Difference methods are similar in that the entire solution domain must be discretized Boundary Element method only requires the bounding surfaces be discretized

    37. Comparison of Methods Finite Difference Easiest method to implement Special handling is needed for Irregular boundaries Abrupt changes in material properties Complex boundary conditions Better for Nonlinearity problems Highly heterogeneous problems True anisotropy problems

    38. Comparison of Methods Finite Element Preferred for Irregular boundary problems complex domain problems complex boundary condition problems

    39. Comparison of Methods Boundary Element Preferred for set of highly irregularly spaced points in the domain problems where only the boundary solutions are of interest Boundary solutions are obtained directly by solving the set of linear equations

    40. More Examples Defibrillator Simulation

    41. More Examples Electrocardiological Visualization

    42. More Examples Seizure Visualization

    43. References http://www.heartcenteronline.com http://www.bme.jhu.edu/ccmb/ccmbgallery.html http://www.ccs.uky.edu/csep/BF/BF.html http://www.sci.utah.edu/coe/images.html

    44. Multigrid methods for an inverse Potential Problem --A Case study Chengdong Li

    45. Overview Short introduction to problem background Model problem and discretization System of equations Analytical issues Application of CG-algorithm Hopes in Multigrid Some examples for mg

    46. Motivation

    47. Motivation

    48. Bioelectric field problems Goal: Modeling of relationship between current density Iv and electric/potential fields. Bioelectric characters: Temporal behavior of sources < 1KHZ Typically physiological conductivities. ?quasi-static behavior ?Displacement current can be neglected ?Ohm’s law takes the form

    49. Bioelectric field problems (cont.) Together with (1) (2) The bioelectrical field equation becomes

    50. Two types of problems

    51. Model problem Inner points 5 point-stencil Boundary points 5 point-stencil & Central difference

    52. Structure of matrix

    53. Structure of matrix (cont.)

    54. Analytical solution Direct problem:

    55. Analytical solution Inverse problem:

    56. Eigenvalue Decompositon The discrete operator: that maps the source boundary values (epicard voltages) to the data (source voltages) has the eigenvectors: with eigenvalues:

    57. Eigenvalue and decomposition ?Condition number of discrete inverse problem >cosh(n) in the inverse case 1/vk are amplification factors for the error.

    58. Basic solution method Forward calculation from line to line.

    59. Need for regularization Test procedure: Choose values on right boundary Numerically solve direct problem ?data for inverse problem Solve inverse problem by forward calculation

    60. Test with 33x33 grid

    61. Test with 33x33 grid

    62. Regularization The information of content of: Low frequencies is meaningful and must be represented in the solution High frequencies must be damped out to avoid pollution. Standard techniques (like Tikhonov regularization truncated Singular Value Decomposition,…) are based on adding a correction term to the operator that enforces a smooth solution.

    63. Iterative method Like Conjugate Gradient method can have implicit regularization property: So stop Late enough to reconstruct as much information from the smooth components as possible. Soon enough to not let the rough modes spoil the result.

    64. CG for model problem The problem is slightly modified: The system is now overdetermined. We apply the CG-method to the normal equation: And using an “appropriate” stopping criterion to find a regularization solution.

    65. Result of CG

    66. Result of CG

    67. Multigrid method Situation: The meaning of information one can extract from the data is connected to the lower frequency modes. For large problems direct methods and the commonly used iterative methods (CG, Landweber-iteration, …) are inefficient.

    68. Multigrid method Low frequency components: Can be represented on coarser grids Coarse grid correction (cgc) is efficient ?fast convergence. High frequency components: Must be resolved by relaxation. Relaxation is inefficient. ? are not well represented in the solution. ? desirable effect, since this is a way to regularize (like with CG)

    69. Multigrid method Intermediate frequency components Are partly resolved by relaxation and partly by cgc May by introduced by multigrid components, e.g. relaxation and interpolation. Cgc can over correct. ? need self regularizing effect of multigrid

    70. A first Multigrid Algorithm Components Standard grid coarsening H=2h Coarse grid correction scheme: Restriction by full weighting plus “trick” along boundaries. Prolongation by linear interpolation. Kaczmarz’s projection method for relaxation Forward calculation to solve on coarsest grid.

    71. A first Multigrid algorithm

    72. Kaczmarcz’s Projection method

    73. Kaczmarcz’s Projection method This method is equivalent to solving with with the Gauss-seidel method

    74. Test problem We assume that u=1 on the left boundary ? u=1 on whole domain Results for Kaczmarz for 33x33 grid:

    75. Results for W-cycle Using 10 pre- and post relaxation steps

    76. Result of W-cycle

    77. Future work Perform mode analysis for inverse model problem Get the grid transfer operators right Analyze regularization and convergence behavior Find the suitable stopping criterions Extend to more realistic situations Apply to inverse ECG/EEG-problem.

    78. Reference http://www.mgnet.org/mgnet/Conferences/CopperMtn01/Talks/ruede-biomef/img15.htm http://www10.informatik.unirlangen.de/~mohr/Diverses http://www.epcc.ed.ac.uk/csep/bf/bf.html

    79. Thanks

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