ECG Reverse and Direct Problem

8/12/2019 ECG Reverse and Direct Problem

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Direct and Inverse bioelectric

problems

Presented by

• Chao Shen• George Landon

• Chengdong Li

University of Kentucky



Outline

• Introduction of bioelectricity and bioelectric problems.

• Direct and inverse problems

• Model construction and mesh generation

• example(EEG)



History of research on

bioelectric field problems

Luigi Galvani, in 1786,

stimulated muscle contractions

by mechanical and electrical

means, respectively.

Bioelectricity occurs in allliving tissue and has been the

subject of investigation since

Swammerdam, in 1658, and later



What is bioelectricity?

• The origins of bioelectricity lie within cellmembranes, 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, andcommunication to occur over the whole

body.



Bioelectric field problems

• Bioelectric field problems can be found in a widevariety of biomedical applications which rangefrom 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



Applications

• The solutions to bioelectric problems haveapplications 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).



Early ECG machine, circa, 1911



Poisson's and Laplace's

Equation

• Poisson's Equation

• Laplace's equation



Bioelectric Volume Conductors

• A general volume conductor can be defined as aregion 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, , fieldseverywhere within the volume, , and/or on one of

the bounded surfaces, .



continue

A typical bioelectric volume conductor can

be posed as the following boundary value

problem

(1)* The associated boundary conditions depend on what type

of problem one wishes to solve



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 zeroon the surface interfacing with air (here denoted

by ).



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:



To Determine the FIELD from the Known Source and Conductor is

CalledDIRECT PROBLEM

To Determine the SOURCE from the Known Field and Conductor is

CalledINVERSE PROBLEM

Figure(Direct and inverse problem )



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, boundaryelement, finite difference, or multigrid) of a partial

differential equation.

Mesh generation of a red blood cell.



Why do Mesh Generation?

• Once we have stated or derived the mathematicalequations which define the physics of the system,we must figure out how to solve these equations forthe particular domain we are interested in.

• Most numerical methods for solving partialdifferential equations break up the continuous

domain(volume conductor) into discrete elementsand approximate the PDE using the particularnumerical technique best suited to the particular

problem.



Mesh Generation

(continue)

Two main approaches for mesh construction• divide and conquer

a. advantage b. disadvantage

• Delaunay triangulation strategya. advantage

b. disadvantage



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 currentsources, 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), .




EEG (continue)

and Neumann boundary conditions on the scalp:

where is a conductivity tensor and are the volume currents

density due to current dipoles placed within the head. The

unknown is the electric potential created in the head by the

distribution of current from the dipole sources.




EEG (continue)An ideal current dipole source can be described as two point

sources of opposite polarity with infinitely large currentdensity and infinitely small separation d :

To solve Poisson's equation numerically, we began with the

construction of a computational model. The realistic head geometry

was obtained from MRI data, where the volume was segmented and

the segmented head volume was then tetrahedralized via a mesh

generator. For each tissue classification, we assigned a conductivity

tensor.




EEG (continue)

Cut-through of the tetrahedral mesh,with elements colored according to

conductivity classification.

Green: elements correspond to skin,

blue: skull

yellow : cerebro-spinal fluidPurple : gray matter

light blue : white matter.




EEG (continue)

Using the FEM, we obtain the linear systemof equations:

where is an stiffness matrix, is a source vector

and is the vector of unknown potentials at the nodes within the

head volume. The matrix is sparse (containing approximately two

million non-zeroes entries), symmetric, and positive definite.




EEG (continue)

Solution:

An equipotential surface is

displayed in wire frame, the dipole

source is indicated with the red and

blue spheres.

Time:

required approximately 12 seconds

of wall-clock time on a 14

processor SGI Power Onyx with

195 MHz MIPS R10000 processors



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/E

CG-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



Direct and Inverse Bioelectric Fields

• Motivation• Usage Example

• Numerical Methods• Comparison of Methods

• More Examples



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



Usage Example

Heart Models

Normal Electrical Propagation

Electrical Propagation During

Heart Failure



Mathematical Model

• Bioelectric Volume Conductors

• A reduction to Poisson's equation for electricalconduction: 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 asimplified mathematical model.

V I −=Φ∇⋅∇ σ Ω

Ω

V I

V I

Φσ



Numerical Methods

Three methods examined

– Finite Difference

– Finite Element

– Boundary Element



Finite Difference Method

• The solution domain is approximated by a grid ofuniformly spaced nodes

• At each node, the differential equation isapproximated by an algebraic expression whichreferences adjacent grid points

• A system of equations is obtained by evaluating

the previous algebraic approximations for eachnode in the domain.

• The system is solved for each value of the

dependent variable at each node.



Finite Difference Method• The finite difference representation of

in

is

• This can be reformulated into the matrix equation

V I −=Φ∇⋅∇ σ Ω

)(6 ,,

2

,,1,,1,,

,1,,1,,,1,,1

v I h nml nml nml nml

nml nml nml nml

−=Φ−Φ+Φ+

Φ+Φ+Φ+Φ

−+

−+−+

b A =Φ



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



Finite Element Methodin

can equivalently be expressed as a system of N

equations with N unknowns

Now use the linear combination generated from theFinite Element method

Now the matrix equation can be generated

V I −=Φ∇⋅∇ σ Ω

N i ξ ξ ,...,

∑= Ψ−=Ψ∇Ψ∇

N

i jV ji jii I 0

, ),(),(σ ξ N j ,...,0=

b A =ξ



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

b A =ξ

A

A



Boundary Element Method

• Only used for problems with isotropic domainsand few inhomogeneities

• Utilizes information only upon the boundaries ofinterest

• Transforms the differential operator defined in thedomain to integral operators defined on the

boundary• Makes mesh generation simpler, but potentials and

gradients can only be evaluated after boundary

solutions are obtained



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 domainmust be discretized

• Boundary Element method only requires the

bounding surfaces be discretized



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




Finite Element

• Preferred for – Irregular boundary problems

– complex domain problems

– complex boundary condition problems




Boundary Element

• Preferred for – set of highly irregularly spaced points in the

domain

– problems where only the boundary solutionsare of interest

• Boundary solutions are obtained directly bysolving the set of linear equations



More Examples• Defibrillator Simulation

Visualization of theVisualization of theelectric current flowelectric current flow

andand isovoltageisovoltage

surface from asurface from asimulation of asimulation of a

cardiac defibrillatorcardiac defibrillator

designdesign



More Examples• Electrocardiological Visualization

This visualizationThis visualizationshows the electricshows the electric

potential distributionpotential distribution

on the heart and bodyon the heart and bodysurfacessurfaces



More Examples• Seizure Visualization

This visualizationThis visualizationillustrates a simulationillustrates a simulation

of epilepsy occurringof epilepsy occurring

within the temporalwithin the temporallobe of the brainlobe of the brain



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



Multigrid methods for an

inverse Potential Problem--A Case study

Chengdong Li



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



Motivation

Background



Bioelectric field problems

• Goal:

– Modeling of relationship between current

density I v 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

Background



Bioelectric field problems (cont.)

• Together with – (1)

– (2)

• The bioelectrical field equation becomes

Background



Two types of problems

Background



Model problem

Model problem

• Inner points

– 5 point-stencil

• Boundary points

– 5 point-stencil

&

– Central difference



Structure of matrix

Model problem



Structure of matrix (cont.)

Model problem



Analytical solution

• Direct problem:



Analytical solution• Inverse problem:



Eigenvalue Decompositon

• The discrete operator:

• that maps the source boundary values

(epicard voltages) to the data (sourcevoltages) has the eigenvectors:

• with eigenvalues:



Eigenvalue and decomposition

• Condition number of discrete inverse problem >cosh(n) in the inverse case 1/vk

are amplification factors for the error.



Basic solution method

• Forward calculation from line to line.



Need for regularization

• Test procedure: – Choose values on right boundary

– Numerically solve direct problem

data for inverse problem

– Solve inverse problem by forward calculation



Test with 33x33 grid



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.



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.

CG for model problem



CG for model problem

• The problem is slightly modified:

– The system is now overdetermined. We applythe CG-method to the normal equation:

– And using an “appropriate” stopping criterion

to find a regularization solution.



Result of CG



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.



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)



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



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.



Kaczmarcz’s Projection method



Kaczmarcz’s Projection method

• This method is equivalent to solving

• with with the Gauss-seidel method



Test problem

• We assume that u=1 on the left boundary

•

u=1 on whole domain• Results for Kaczmarz for 33x33 grid:



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



Result of W-cycle



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.



Reference• http://www.mgnet.org/mgnet/Conferences/CopperMtn01/Talks/r

uede-biomef/img15.htm• http://www10.informatik.unirlangen.de/~mohr/Diverses

• http://www.epcc.ed.ac.uk/csep/bf/bf.html



Thanks

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