biomedical electromagnetic field modeling and source localization using the finite element method...
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Biomedical Electromagnetic Field Modeling and Source
Localization Using the Finite Element Method
Paul Schimpf, PhD
Copyright 2006
Eastern Washington UniversityComputer Science [email protected]
estimated completion time: 4.0-5.0 hrs
© 2006-07 P. Schimpf 2
Table of Contents
Tutorial Objectives Background Information
– Biomedical Applications of Electromagnetic Fields
– Volume Conduction and the Poisson Equation
– The Finite Element Method
Forward Modeling– Defining the Domain
– Defining Boundary Conditions
– Specifying Current Sources
– Solving and Visualizing Output
– Exercises 2 and 3
Inverse Modeling– Introduction to Inverse Modeling
– A Simple Inverse Algorithm
– Exercises 4 and 5
© 2006-07 P. Schimpf 3
Biomedical Tutorial Objectives
1. Understand the basic steps in developing a forward model from medical imagery
Define a head model from classified medical images (provided)
Simulate and visualize the electroencephalogram (EEG) for different source configurations:
2. Understand the principle of superposition Verify that the field due to two sources is the sum of the fields
due to the individual sources
3. Understand the basic issues of inverse problemsnonlinear dependence on source position requires
characterization of each possible source, resulting in a highly underdetermined problem
assumptions or a-priori information may be applied to make the problem tractable
© 2006-07 P. Schimpf 4
Biomedical Applications
Cardiology– optimal electrode design to minimize burning from edge
effects
– implanted defibrillation: where should the electrodes be placed to optimize efficacy?
– diagnosis: are there regions on the heart that are not generating a neural signal (infarct)?
Neurology– what neural regions are involved in performing task X?
– locating the foci of seizure activity
– detecting silent seizures
– brain to computer interfacing
Impedance Imaging– visualizing changes in electrical impedance, which
correlates to tissue structure
© 2006-07 P. Schimpf 5
Volume Conduction
Both artificial and biologic sources introduce current into the tissue– in most biomedical applications, inductive and capacitive effects are negligible, and the conduction is purely resistive
– voltage and current are thus governed by the Poisson equation:
– in two dimensions, with isotropic conductivity, this expands to:
– V, σ, and are all functions of position (x,y)
– V is voltage (V), σ is conductivity (Siemens/m), and ρ represents current source density (A/m2 in two dimensions)
V
2
2
2
2
y
V
x
V
)1(
)2(
© 2006-07 P. Schimpf 6
The Finite Element Method
Given the conductivity and the sources, our goal is an approximate solution for the voltage
We create a piecewise approximation to V using a collection of local basis functions, Ni
– the Ni are called local basis functions because each approximates the solution over a subregion called an element
– the simplest element shape in two dimensions is a triangle
– polynomial functions of (x,y) are usually used for the Ni– for this tutorial, we will use linear functions:
n
iii yxNayxUyxV
1
,,,
yaxaayxN i 321,
)3(
)4(
© 2006-07 P. Schimpf 7
The Finite Element Method
The end result of using linear basis functions is a faceted approximation to the true solution
– the approx. improves as the elements are made smaller
– we should use small elements where the voltage is rapidly changing
– an adaptive mesh solver shrinks the elements in such regions automatically
– this example illustrates a possible piecewise linear solution over a two-dimensional domain using triangular elements, each having 3 degrees-of-freedom (a1, a2, a3)
© 2006-07 P. Schimpf 8
Moving to 3D
In 3 Dimensions, the problem definition requires only an extra term for the added dimension (z):
– V, σ, and are now functions of (x,y,z), and represents volume current source density (A/m3)
– note that equation (1) remains unchanged, which is one of the advantages of the Divergence () and Gradient () operators
We also need 3D elements, and corresponding basis functions (polynomials in x, y, z)
V
2
2
2
2
2
2
z
V
y
V
x
V
)1(
)2(
© 2006-07 P. Schimpf 9
Moving to 3D
Note that our linear basis in 2D had 3 degrees of freedom (DOFs)
– DOFs are the parameters that fit the linear basis functions to the voltage (a1, a2, a3)
– the number of DOFs for a linear basis is the same as the number of vertices (or nodes) of the simplest element shape (a triangle in 2D)
The simplest element shape in 3D is a tetrahedra
– which would then have 4 DOFs:
zayaxaazyxN i 4321,,
n
iii zyxNazyxUzyxV
1
,,,,,, )3(
)4(
© 2006-07 P. Schimpf 10
Image Based Elements
When modeling a biological domain, it is convenient to define elements as collections of medical image pixels
– pixels are classified according to tissue type, and each tissue type is assumed to have a particular conductivity
– in 3D, we use a set of images, and pixels become voxels
– the pixels are usually square, but the space between images is often not the same as the pixel size, thus our elements will be brick-shaped (rectangular parallelipiped)
– such elements have 8 nodes, and thus 8 DOFs:
– this is more DOFs than a linear basis, but not enough to provide a full quadratic basis (Can you determine what the missing quadratic terms are? Can you see the cubic term?)
– it is thus called a super-linear element
xyzaxzayzaxyazayaxaazyxN i 87654321,, )4(
© 2006-07 P. Schimpf 11Classified Domain An element "mesh"
Image Based Elements
pixels/voxels in the medical images are classified according to tissue type, and each tissue type is assumed to have a particular conductivity
element boundaries are defined as collections of voxels
Typical FE programs draw the mesh as a wire-frame of the elements. In this case it is convenient to just color the groups of pixels that comprise an element, and ensure that neighboring elements use different colors.
© 2006-07 P. Schimpf 12
The Galerkin Method
Transforms the problem into a linear system of equations in terms of the ai by:
– substituting the approximation U (3) into the Poisson equation (2)
– formulating the residual, which is the difference between the right and left hand sides of the Poisson equation
– and requiring that the average weighted residual be zero
– averaging here means integrating the weighted residual over the entire domain:
Residual R(x,y,z;a) =
– the details of this transformation are beyond our scope
U
0,,;,,
dzyxNazyxR jj n1 2, , ,
Ka f
)5(
)6(
)7(
© 2006-07 P. Schimpf 13
Forward Modeling
– Define a head model from classified medical images (provided)
– Simulate the electroencephalogram (EEG) for 3 different source configurations:
1) a single dipolar source in the prefrontal cortex
2) a single dipolar source in the cingulate cortex
3) both of the above sources simultaneously
– Verify the 3rd solution is the sum of the first two
– We'll cover the simulation of the first source configuration here - the other two are similar
– Suggested directory structure:
(you may also find it convenient to put a copy of plotsamples.m in all 3 directories)
© 2006-07 P. Schimpf 14
Define the Domain
Use a text editor to view the file image.map– this is a list of the correspondence between image files and z coordinates
– note that it specifies model images in a subdirectory called HeadModel
Run GalerWin and select Edit / Domain Definition
– press the Import button, browse to your tutorial directory, and select the file image.map
– notice that the image dimensions (128 x 128) are deduced from the file size
You must define the dimensions of the pixels in whatever units you prefer to work with
– let's work with meters
– the pixel dimensions in these images are 2 mm x 2 mm and the spacing between images is 3.2 mm
– enter the appropriate values (in meters) and press the OK button
© 2006-07 P. Schimpf 15
Define the Domain
You should now see a grayscale rendering of the first slice
– each pixel is a byte code representing a particular tissue type, and the solver defaults to simply interpreting that byte value as a gray scale – we'll change that shortly
– these images are pretty small, so select View / Zoom 400%
– select View / Current Z Slice, enter 25, and press OK
– move the mouse around in the window and notice the updates of the voxel coordinates and tissue class number in the status bar
select File / Save– and save this project as Exercise1to3 \ Exercise1.gal
– you can now load that file to continue your work at a later time
– the .gal project files save the definitions of the problem, but not solutions
© 2006-07 P. Schimpf 16
Editing Material Properties
Select Edit / Material Properties– this is where we assign a conductivity and
a color to each tissue class
– try selecting some material numbers – you'll see that the assigned material color defaults to grayscale
– the conductivity is in the units you choose, but should be consistent with the other units – we'll use Siemens / m
– use a text editor to view the file tutorial.rst
– for historical reasons, the import and export of conductivities uses the inverse, which is resistivity, with units (in this case) of -m
– press Import and select the file tutorial.rst
– you should now see some conductivities and tissue descriptions in the window, but the colors are still grayscale
– to edit a tissue description, double-click the tissue in the scroll window
– we'll assign some colors to these tissues next
© 2006-07 P. Schimpf 17
Editing Material Properties Select tissue #90, Scalp, and move the color scrollbars to give it the color black. Color the EEG electrodes (#91) red, the Reference electrode (#94) green, and cortex
voxels (#93) magenta. feel free to color other
tissues as well, but this is purely for visual asthetics and does not effect the model solution(see next slide for moretissue colors used here)
press the Close button you should now see
a colored image ofthe domain
select File / Save
© 2006-07 P. Schimpf 18
Suggested Tissue Colors
tissue # name Color Red Green Blue
40 blood red 255 0 0
45 hard bone gray 128 128 128
50 soft bone light gray 196 196 196
55 gray matter lavender 204 128 204
60 white matter olive 196 228 128
65 cerebellum brown 196 128 64
70 CSF cyan 0 204 204
75 eye lt lavender 228 196 196
80 fat yellow 255 228 112
85 muscle blue 32 32 180
93 cortex magenta 255 0 255
95 soft tissue lt green 128 204 128
209 air white 255 255 255
© 2006-07 P. Schimpf 19
Material Properties
Some things to note:– "eeg electrodes" have been given their own tissue number
(91) so that they can be viewed on the image
• this also makes it convenient to sample the field solution at those points, but it is not necessary to define a tissue class at sample points
• note that their conductivity is the same as "scalp" (90)
– one particular electrode has class 94 "reference electrode"
• we'll define a 0 voltage boundary condition here later
• again, this is not necessary, but it makes the reference electrode easy to see in the image set
– a subset of the gray matter voxels (#55) have been classified as "cortex" (#93), which you've colored magenta
• note that the conductivities are the same
• these "cortex" voxels mark possible source sites when we discuss inverse problems later on
© 2006-07 P. Schimpf 20
Defining Boundary Conditions
Boundary Conditions (BCs) come in two types:– Fixed or Dirichlet: represent fixed voltages at any site in the domain
• used to simulate voltage sources or reference points
• in this case, we need only specify a 0 voltage at the location of the reference electrode
• note that these can be internal, even though they are called "boundary conditions" – fixing the voltage at internal points in the domain essentially removes that region from the model, creating an internal boundary
– Natural, or Neumann: specify values for the normal component of current density at boundary points (A/m2 in 3D)
• they are satisfied only approximately in the eventual solution
• Neumann BCs are rare in this application (we'll have none), except that any boundary point that has NO boundary condition explicitly specified automatically gets a 0 normal current boundary condition. This says that current does not escape from the modeled domain.
• with some pre-calculation, non-zero Natural BCs take the same form as Fixed BCs – with this S/W that is the only way to enter them
© 2006-07 P. Schimpf 21
Defining Boundary Conditions On slice 25, right-click the green pixel and select Set Boundary Conditions
– Fixed BCs are actually applied at nodes of the mesh, and nodes occur at the vertices of elements
– Because you've selected a voxel, the pop-up allows you to enter BCs at any of the vertices of that voxel. The terminology asks that you think of your screen view as looking down onto the top of the modeled object (this may be disorienting, but calling the first image "top" instead of "left" makes sense with axial medical images, which are more common for modeling than the saggital images used here)
– One BC would be sufficient toestablish a voltage reference,but since the entire voxel represents our knowledge of the location of the referencepoint, check all the vertices
© 2006-07 P. Schimpf 22
Defining Boundary Conditions
Press OK and select Edit / Boundary Conditions– this screen lists all fixed boundary conditions
– it also allows you to specify fixed boundary conditions in two other ways:
• Over a range of coordinates, in which case the boundary condition is applied to all nodes falling within the box defined by those two coordinates
• Over a material class, in which case the boundary conditions is applied to all nodes falling on a voxel of the specified class
– note that we could have specified these 8 reference BCs on this screen using the coordinate range:
(116, 60, 25) to (117, 61, 26)
– or using Material class 94
© 2006-07 P. Schimpf 23
Specifying Current Sources
The right-hand side of the Poisson equation represents current sources
– neuronal source activity is generally modeled as a dipolar source: a current source close to a current sink
– we will simulate this by placing identical, but opposite, current sources on adjacent nodes
right-click on a cortex voxel (class 93) in the frontal cortex and select Set Sources
© 2006-07 P. Schimpf 24
Specifying Current Sources This pop-up is similar to the pop-up for boundary conditions
– you can specify a total current for each node
– specify 625 uA at each of the "top" nodes, and -625 uA at each of the "bottom" nodes
– this simulates a dipolar source centered in the voxel, oriented in the z direction (ear to ear), with a moment of 8 uA-m
– this is 2.5 mA total for eachside, times a 3.2 mm separation(the voxel size in the z dimension)
Press OK and select Edit / Source
– the resulting dialog box listsall defined sources
Press OK and select File / Save
© 2006-07 P. Schimpf 25
Solving
Select Edit / Adaptation Controls– this screen allows you to specify parameters that effect the way
the mesh is (automatically) adapted– we'll take a look at a coarsely adapted mesh and at
a finely adapted mesh
– set the initial mesh settings to specify element sizes of 32 down to 4 voxels
– turn "an exact match ... to external border" off
– next to "Solve Equations using", press Params and select Both solution methods (discussed next)
– set the "Repeat, at most" loop counter to 5
– set the error estimate condition for termination to 5%
© 2006-07 P. Schimpf 26
Solving
The program will first build an initial mesh– It starts by populating the domain with the larger element size
– It will then automatically refine elements in an attempt to make each as homogeneous (containing a single conductivity region) as possible, but will not refine elements beyond the minimum specified size
It then applies the specified BCs, generates a linear system of equations from the mesh, and solves using the specified iterative solvers It then iterates on refining the mesh in areas that
have high flux, re-solving each new mesh– in this case, the "flux" is current density, which is proportional
to voltage gradient
– this terminates when the average estimated error in flux falls below the specified threshold
– many FE solvers do not have an adaptive meshing ability – it is generally up to the user to refine the mesh
© 2006-07 P. Schimpf 27
Iterative Solvers
Recall that the finite element method produces a linear system of equations from the mesh:
The matrix K generated by FE meshes are large and sparse
– the vector a that we are solving for contains the DOFs for the basis functions associated with the elements of the mesh
– there as thus as many rows (and columns) as there are nodes in the mesh
– there will be a non-zero entry in matrix K only where the two nodes associated with that row and column are connected through the edge of an element
– for example, if the mesh is uniform (each element is 1 voxel), each node will connect to at most 26 other nodes, and there will thus be at most 27 non-zero entries on any given row of K (including the diagonal entry)
Ka f )7(
© 2006-07 P. Schimpf 28
Iterative Solvers
You may be familiar with deterministic solution methods such as Gaussian elimination
When the matrices are sparse, it is more efficient to use an iterative approach
– example of a simple iterative approach called relaxation:
– start with an arbitrary initial guess a1=1 and a2=1
– refine this guess iteratively with using the following equations (derived from the system of equations above)
– where * indicates the next estimate
– repeat until the estimates don't change much or the difference between the right and left-hand sides is below some threshold
1931
2513
21
21
aa
aa
3/19
3/25
1*2
2*1
aa
aa
© 2006-07 P. Schimpf 29
Iterative Solvers
Evolution of a1 and a2
Optional Exercise– try this method for this rearrangement of the same problem:
a1 a21 1
8 6
6.3 3.7
7.1 4.2
6.9 4.0
7.0 4.0
7.0 4.0
3/19
3/25
1*2
2*1
aa
aa
2513
1931
21
21
aa
aa
1*2
2*1
325
319
aa
aa
© 2006-07 P. Schimpf 30
Iterative Solvers
This control allows the user tochoose from two iterativesolver algorithms
– Many FE programs wouldn't allow such choices. They were included here because this code was developed for research work.
– Select "Both" and leave the other parameters at defaults
SOR = Successive Over-relaxation– this algorithm is similar to the relaxation approach in the
preceding example, with some refinements
– updates are used as soon as available (i.e. a2* in the preceding
example would use a1* instead of a1)
– the RHS calculations are multiplied by a “relaxation factor” to speed convergence
JCG = Jacobi Conjugate Gradient– this is an advanced iterative algorithm that is beyond the scope
of this tutorial
© 2006-07 P. Schimpf 31
Solving
Select Action / Solve– a pop-up window shows status of the solution in progress,
including the number of elements & nodes in the mesh
Press Dismiss and select View / Field Solution– look at the solution on z slices 15-35
– notice how the energy of the sources "piles up" at the skull boundary
© 2006-07 P. Schimpf 32
Looking at the Mesh
Select View / Element Mesh– You'll get a warning that viewing the element mesh will overwrite the solution. This is
because the voltage solution is overwritten with color dataas the mesh is colored, which is done to conserve memory.
– press OK
– take a look at the mesh onthe same range of slices
– note how the elements aresmaller near the currentsource
– most of the other elementsare 4x4x4 pixels from theinitial attempt to representinternal conductivity boundaries
© 2006-07 P. Schimpf 33
Forcing a Uniform Mesh
Select Edit / Adaptation Controls– In order to test the principle of superposition, we'll now solve two more
problems: a different dipole source followed by both dipole sources
– we'll then compare the solution for both to the sum of the individual solutions
– the comparison will come out best if we ensure the mesh for all three problems is the same
– the easiest way to do this is to force a uniform mesh of single voxel elements
– modify the adaptation controls as shown here:
– Elements from size 1 downto 1
– Exact match of border on
– Repeat at most 0 times
© 2006-07 P. Schimpf 34
Solve Again
Select Action / Solve Press Dismiss and select View / Field Solution
– look at the solution on z slices 15-35
– notice the sharper resolution of energy piling up against both the skull and other localtissues (looking at slice 34 here)
© 2006-07 P. Schimpf 35
Extracting EEG Samples
Select Edit / Solution Points– press Import and select the file eegpts.sln
– we could also enter the class for EEG electrodes (91) in the box titled "Solutions for a specific material class"
– eegpts.sln defines the same samples in the order we'll need for inverse work later on
Press OK and select Action / Show Solution at Listed Points
– press Save to File and save in a file named frontal.txt
– use a text editor to view that file
© 2006-07 P. Schimpf 36
Plotting EEG Samples
One common way to get a quick idea of the voltage distribution on the scalp is to view a contour plot of the EEG samples as seen from above
– we'll do that by ignoring the y dimension of the sample locations
Run matlab and change (cd) to your tutorial directory
– execute the command: plotsamples('frontal.txt') ;
– note that the left / right orientation of the source is visible in the scalp EEG
– take a look at the file plotsamples.m in the matlab editor to see how it works
© 2006-07 P. Schimpf 37
Exercises 2 and 3
The next 2 exercises are similar to the simulation just completed
Exercise 2: simulate an 8uA-m dipolar source located in the posterior cingulate cortex
– oriented in the anterior / posterior (x) direction
– see next slide, and use the same slice as before (z=25)
– don't forget to remove the existing sources first
– don't forget that the voxel dimension in x is different than z
– plot the EEG in Matlab
Exercise 3: simulate the prefrontal and posterior cingulate sources at the same time
– plot the EEG in Matlab
– use Matlab to verify that the resulting EEG is the sum of the two EEGs taken separately
© 2006-07 P. Schimpf 38
Exercise 2
Posterior Cingulate Cortex
Resultant EEG:
© 2006-07 P. Schimpf 39
Exercise 3
both frontal (z) cingulate (x)
Put your matlab code that compares the simulation of both to the sum of the two simulated separately into a file called compare.m
Some of the values returned by the call to plotsamples() can help you verify that this is the case (to within some level of error)
=?
+
© 2006-07 P. Schimpf 40
Intro to Inverse Modeling
Given incomplete knowledge of the field (EEG)– determine the location, orientation & strength of the sources
First, model the EEG due to each possible unit source (location and orientation) as a column in a matrix
– Principle of Superposition: the EEG obtained from multiple sources is the same as the sum of the EEGs obtained from each source separately
mnnmmm
n
n
v
v
v
s
s
s
lll
lll
lll
2
1
3
2
1
32,1,
,22,21,2
,12,11,1
nm 3 13 n 1m
"Lead-field" matrix
source strengths
EEG
© 2006-07 P. Schimpf 41
Some things to note
m = number of EEG measurements n = number of possible source sites in the brain
– this can be a restricted number of sites using a-priori information– here it is restricted to sites near the cortical surface
each site has a dipolar source, with an orientation– an arbitrary dipole orientation is represented by 3 dipoles, each
oriented in one of the 3 ordinal directions of the model (summing the effects of the dipoles is just like summing vectors)
– thus the columns of the matrix come in triplets for each brain site
mnnmmmmm
n
n
v
v
v
s
s
s
lllll
lllll
lllll
2
1
3
2
1
3,4,3,2,1,
3,24,23,22,21,2
3,14,13,12,11,1
site 1x dipole
site 1y dipole
site 1z dipole
site 2x dipole
site nz dipole
© 2006-07 P. Schimpf 42
Inverse Modeling Challenges
There are a large number of possible sources– at a ~2.5 mm resolution roughly tens of thousands of possible
sources, and each of 3 ordinal orientations must be simulated
Problem 1: yikes, that's a lot of forward solutions– fortunately, there is a way around this using the Principle of
Reciprocity
– the theory is beyond our scope, but it allows us to build the lead-field matrix one row at a time, instead of one column at a time, using one forward simulation for each EEG measurement site, instead of 3 times each source site
Problem 2: there are many more columns (unknown sources) than rows (EEG measurements)
– so this problem is extremely underdetermined
– which means there are, in principle, an infinite number of source arrangements that can produce a given EEG measurement
© 2006-07 P. Schimpf 43
A Simple Inverse Algorithm
One solution to Problem 2 is to assume that there are fewer sources than measurements
– i.e., many of the entries in the s vector will be zero
– this leads to a useful solution in many applications
The simplest extreme – assume 1 source– iterate through each triplet of columns in the lead-field, find the
best fit of those 3 columns to the EEG measurement, which gives the best fitting orientation of that particular source site, and calculate the residual
– the column triplet giving the smallest residual represents the best fitting single source
– at each iteration, we now have an overdetermined problem to solve, because there are 3 columns (the 3 ordinal orientations of a particular source) and m rows (the number of EEG measurements)
– we shouldn't expect to find an exact solution (residual=0), unless there truly is only a single source, on our grid spacing, the model is perfect, and there is no noise present
© 2006-07 P. Schimpf 44
Overdetermined Pseudoinverse
The least squares strategy is to minimize the norm of the residual:
si is then the best fitting set of amplitudes for subset i of the set of possible sources– best in terms of minimizing the residual norm
– in this case each subset i is the three ordinal source orientations at a particular location
– Ri can be calculated from si and the smallest Ri represents our best fit for a single source
vLLLs
vLsLL
vLsLLRs
vvvLssLLsvsLvsLR
Tii
Tii
Tiii
Ti
Tiii
Tii
TTi
Tiii
Ti
Ti
iiT
iii
1
2
2
2
2
022
2
© 2006-07 P. Schimpf 45
Single Source Search
Algorithm [i, o, rr] = SingleSourceSearch(L, v) Inputs: matrix L: the lead-field for the model vector v: the measurement vectorOutputs: integer i: index of the 1st of 3 consecutive columns
that represent the best fitting source vector o: 3x1 source vector representing the best linear combination of those columns scalar rr: relative residual of the fit
remove the average value from v % IMPORTANTsmallestR2 = infinityfor each set of 3 consecutive columns in L miniL = L(those 3 columns) s = inv(miniL'*miniL)*miniL'*v R2 = (miniL*s-v)'*(miniL*s-v) if R2 is less than smallestR2 then smallestR2 = R2 i = those 3 columns o = s endifendforrr = sqrt(smallestR2) / sqrt(v'*v)
© 2006-07 P. Schimpf 46
Exercise 4
Implement the single source search algorithm in matlab
– store it in a file called singlesourcesearch.m
write a matlab script (solution.m) that inverts the simulated EEG of the frontal cortex source
– use your code above along with the lead-field in lf.mat
– the matrix in roi.mat gives the (x,y,z) voxel coordinates of each column triplet in lf.mat
– IOW, roi(:,1) contains the coordinates of lf(:,1:3) and roi(:,2) contains the coordinates of lf(:,4:6), etc.
– use the roi matrix to determine the location of the best fitting source
– your script should display the location, orientation, and relative residual of the best fitting source
– how well does the inverted location and orientation compare to the actual location and orientation (you can use GalerWin to look at the inverted coordinate)?
© 2006-07 P. Schimpf 47
Exercise 5
Now add an inversion of the simulated EEG of both sources
In this case your assumption is wrong– you're finding the best fitting single source
– but there are actually two
You should get a location somewhere between the two
The residual of the fit should be higher than the residual obtained in Exercise 4
– because there is no single source that fits well when there are actually two sources
For discussion: can you think of a modification to the algorithm that would give you a better inverse?
© 2006-07 P. Schimpf 48
Project Educational Objectives
The educational goal of this tutorial is to provide undergraduate engineering students with understanding of bioelectric fields and the use of finite element (FE) methods, along with an ability to apply FE to bioelectric problems. The educational goal is accomplished through four educational objectives based upon Bloom’s Taxonomy and ABET Criteria 3 as follows:
1. Engineering Topics (Comprehension: 3a, 3k). Understand the fundamental basis of engineering topics through the use of finite element models.
2. FE Theory (Comprehension; 3a). Understand the fundamental basis of FE Theory.
3. FE Modeling Practice (Application; 3a, 3e, 3k). Be able to implement a suitable finite element model using FE software.
4. FE Solution Interpretation and Verification (Comprehension and Evaluation; 3a, 3e) Be able to interpret and evaluate finite element solution quality.
© 2006-07 P. Schimpf 49
Acknowledgement
This tutorial was developed under National Science Foundation Division of Undergraduate Education Grant Number 0536197