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1 Int roduct ion
WHEN CURRENT is injected into or voltage is applied to a subject
in electrical impedance tomography (EIT), impe- dance cardiography,
impedance pneumography, cardiac defibrillation etc., the
current-voltage relationship is determined by Poisson's equation
with boundary condi- tions (Vp- IVV =f) . We need to solve the
equation to know how currents flow in the thorax, how a certain set
of
" boundary measurements is obtained etc. For a non-homogeneous,
anisotropic, and irregularly
shaped subject such as the thorax, an analytic solution of the
equation is impossible. Therefore numerical techniques are used,
such as the finite-element method (FEM) or finite-difference method
(FDM). As the FEM is advantag- eous in modelling an arbitrarily
shaped object, it has been used by many biomedical engineering
researchers in cardiology (KIM, 1982; KIM et aL, 1983; SIDEMAN,
1988), defibrillation (KOTHIYAL et al., 1988; RAMIREZ et al.,
1989); SEPULVEDA et aL, 1990; BLILIE et al., 1992), dispersive-
electrode study (KIM, 1982), electric-field distribution in the
human body (NATARAJAN and SESHADRI, 1976), impedance cardiography
(KIM et al., 1988), plethysmography (BHAT- TACHARYA and TANDON,
1988), and EIT (Hug et al., 1991; LIu et al., 1988; MURAl and
KAGAWA, 1985; WEBSTER, 1990; WEXLER, 1988; WOO et aL, 1993; YORKEY
et aL, 1987).
Even though the FEM is a powerful tool in studying the
current-voltage relationships of the human body, the design
First received 22nd October 1992 and in final form 20th May
1993
9 IFMBE: 1994
of the finite-element mesh (discretisation of the physical
medium) is very laborious and nontrivial, especiaUy in three
dimensions. Therefore there have been only a few three-dimensional
finite-element analysis studies in the biomedical engineering area,
mainly owing to the diffi- culties in generating three-dimensional
finite-element meshes and to the large number of computations.
Although several finite-element software packages are commercially
available, they are expensive and are not flexible when used for a
specific purpose such as EIT, where the FEM is used as a part of
the image-reconstruction algorithm.
To the best of our knowledge, the first three-dimensional
finite-element model of the thorax was by Kim et aL (KIM, 1982;
1983). Their torso model included 19 layers with an average of 232
elements between layers, giving 5340 nodes. They (KIM et al., 1988)
developed a three- dimensional finite-element model of the thorax
for the study of the origin of the impedance change in impedance
cardiology. Their model consisted of 22 eight-node trilinear
hexahedral elements in each of 29 layers, resulting in a total of
880 nodes and 658 elements. The most recently published
three-dimensional finite-element model of the thorax is (BL1LIE et
aL, 1992) for the study of defibrillation electrode systems. They
used 56 transverse CT images and averaged 4 voxels of CT density
data into one element. Their model consisted of 408 449 uniformly
sized linear brick elements with 429 829 nodes.
In static EIT imaging, where the absolute values of a
cross-sectional resistivity distribution are to be re- constructed,
it is necessary to construct a finite-element model of a subject
with an irregular boundary. Therefore
530 Medical & Biological Engineering & Computing
September 1994
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powerful and easy-to-use tools are required for generating
finite-element meshes, as well as very efficient numerical
techniques for solving a linear system of equations.
The main purpose of this paper is to describe two- and
three-dimensional finite-element analysis software packages with an
interactive graphical mesh generator, including automatic and
manual mesh generation and editing, mesh optimisation, automatic
node renumbering etc., for use in EIT and other biomedical
engineering research areas.
2 Fin i te-element method
When high-frequency (10 ~ 50 kHz) current is injected into a
subject, there is no internal source of such a high-frequency
component. Therefore the governing equa- tion of interest becomes
the following well known Laplace equation of voltage V for a
non-homogeneous domain with irregular boundary shape:
Vp-IVV = 0 (1)
In the FEM, we approximate the calculus problem (Vp-lVV = 0) by
the algebraic problem (Yv = c, a linear system of equations) by
discretising the domain into a finite number of small elements, and
then we solve the resulting linear system of equations. The
solution of the linear system of equations provides the piecewise
approximation of the voltage values within the domain.
2.1 Discretisation
First, we divide the domain into a finite number of elements. We
usually use triangular or quadrilateral elements in two-dimensional
problems, and tetrahedral or hexahedral elements in
three-dimensional problems. In EIT, we assume that the resistivity
in each element is homogeneous and isotropic. The discretisation
process converts the continuous problem into a problem with a
finite number of unknowns by expressing the unknown field variables
(voltages) in terms of certain interpolation functions within each
element. The interpolation functions are defined in terms of the
values of the field variables at nodes of the element. Therefore
the nodal values of the field variables become new unknowns, and
the values of the field variables inside the elements are
determined from the nodal values by the interpolation functions. We
know that the solution of the FEM converges to the true solution as
the element size decreases to zero (BURNETT, 1987; HUEBNER and
THORNTON, 1975).
2.2 Selection of interpolation functions
Usually, polynomials are used as interpolation functions as they
are easy to differentiate and integrate. They also provide
interelement continuity and completeness. The degree of the
polynomial depends on the number of nodes in the element, the
nature and number of unknowns at each node, and other boundary
conditions. Burnett describes various kinds of element and
interpolation functions.
We can increase the number of nodes of an element, the order of
the interpolation function and/or the number of elements until we
achieve the required accuracy. Sepulveda (SEPULVEDA, 1984)
discussed the pros and cons of elements with different numbers of
nodes and automatic determina- tion of their interpolation
functions. As an example, in EIT, sometimes it is better to
increase the number of elements rather than to increase the number
of nodes of one element, as the larger number of elements gives
better spatial resolution of resistivity, with the same total
number of nodes.
Medical & Biological Engineering & Computing
2.3 Characterisation of elements
Next, we evaluate the element matrix y, which formulates
solutions for individual elements. Huebner and Thornton (HUEBNER
and THORNTON, 1975) describe four different methods for evaluating
the element matrix. We use the variational approach, where we use
the constraint that the solution must have the minimum potential
energy within each element with the given boundary conditions.
Burner (BURNETT, 1987) describes more general methods, called the
methods of weighted residuals.
Element matrices for various kinds of two- and three-dimensional
element are derived in the literature (BURNETT, 1987; HUEBNER and
THORNTON, 1975; KIM, 1982; SEPULVEDA, 1984; TONG and ROSSETTOS,
1977). Kim described element matrices for hexahedral, tetrahedral
and prism elements, and Sepulveda described element matrices for
three-dimensional isoparametric elements.
2.4 Global node numbering and assembly of the master matrix
Y
All nodes are numbered globally in the mesh. Each node of an
element also has a local node number. Global node numbering could
be done in an arbitrary way. However, the way we globally number
the nodes greatly affects the amount of computation when we use the
methods described later.
After we have calculated element matrix y for each element and
assigned global node numbers to all nodes in the mesh, we assemble
the master matrix Y to obtain the solution over the entire region.
The basis of the assembly procedure is that the values of the field
variables are the same at the nodes where elements are
interconnected.
Now we have a linear system of equations Yc = c where Y is the
master matrix (or admittance matrix), v is the node voltage vector,
and c is the node current vector. The master matrix Y is N x N,
where N is the total number of nodes in the finite-element
mesh.
2.5 Application of appropriate constraints
We need to set one reference node and modify the master matrix
accordingly to have a rank of N. We also need to impose the
following boundary conditions when they are available.
2.5.1 Reference node: we select one node to be the reference
node and set the voltage at this node to 0. This is done by setting
the corresponding row and column of the master matrix to 0, with 1
at the diagonal, and setting the corresponding element of the
current vector to 0.
2.5.2 Dirichlet boundary condition (known surface voltages):
when we know some voltage values at the surface, we force the
solution of the corresponding node voltages to have those known
boundary values. For example, if the ith node is a boundary node
with a known voltage value of Vb, then we set the ith row and
column of the master matrix Y to be zero, with one at the diagonal
(Y, = t) and v~ = Vb.
2.5.3 Cauchy boundary condition (known surface currents): when
we know some current values at the surface (usually the injection
currents), we set the corresponding elements of the current vector
to those known values.
Now we have the following uniquely solvable linear system of
equations with all boundary data in it:
Yv = c (2)
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global decomposition. Heighway and Biddlecombe (HEIGH- WAY and
BIDDLECOMBE, 1982) described the two-dimen- sional automatic mesh
generator used in the finite- element electromagnetics package
PE2D. Cendes et al. (CENDES et aL, (1983) described the Delaunay
triangulation algorithm as an automatic mesh-generated method.
Some automatic mesh generators include a mesh- optimisation
method, which is based on the quality factor of an element. The
quality factor is determined by the size and shape of an element
(LINDHOLM, 1983). For a two-dimensional triangular element, any
angle near 180 ~ indicates a low quality factor. For a
two-dimensional quadrilateral element, if it is too flat or narrow,
its quality factor is low. Elements with low quality factor can
produce numerical problems when we solve the resulting linear
system of equations, as they may result in indefinite element
matrices. The Delaunay triangulation algorithm is widely used in
generating meshes with triangular elements as it generates elements
with good quality factors.
Fig. 1 Two-dimensional finite-element mesh generator developed
for EIT on a Macintosh I1; this mesh generator provides many
interactive graphical mesh-design tools; it can also use scanned
cross-sectional images to design realistic meshes of internal
organs
where Y is the modified master matrix, and v and e are the
modified voltage and current vectors, respectively, satisfying all
constraints.
2.6 Solution of the matrix equation
As the required computation time for the solution of the linear
system of equations is large, an efficient numerical technique is
essential in certain application areas. Usually, the master matrix
is symmetric, positive definite and very sparse. We can choose an
efficient numerical technique to solve eqn. 2 based on the
characteristics of the master matrix Y, which is determined by the
global node numbering method. In this paper, we used the
sparse-matrix and vector methods described later.
3 Mesh generation
As described in the preceding Section, the generation of
finite-element meshes is the most troublesome part of the use of
the FEM. When we design a finite-element mesh for impedance
imaging, for example, we can use any a priori information about the
structure of a subject. Therefore there is a need for interactive
graphical mesh-generation tools for adding, removing and modifying
elements and also for easy change of the boundary shape of the
mesh.
We have developed two- and three-dimensional finite- element
mesh generators on a Macintosh II personal computer to utilise
fully the graphical user interface features of the Macintosh
computer. The mesh generators provide many interactive graphical
tools for the ease of complicated mesh generation.
3.1 Automatic mesh generation and quality factor of element
Given the boundary definition of an object, the automatic mesh
generator will generate a mesh in a certain predefined way, and the
generated mesh can be modified later using the various mesh-design
tools. Sabonnadiere and Coulomb (SABONNADIERE and COULOMB, 1987)
described several mesh-generating techniques, including meshing by
block, grid superposition, frontal propagation, layer stacking,
and
3.2 Two-dimensional mesh generator
As shown in Fig. 1, our mesh generator provides many graphical
and interactive tools for creating and modifying two-dimensional
finite-element meshes. It also allows us to create a realistic mesh
of internal organs by overlapping a scanned cross-sectional image
of the human thorax as a background.
We also optimise the mesh to achieve better numerical stability
and accuracy. The current mesh generator uses the Delaunay
algorithm as an automatic mesh-generation and optimisation method.
We can also refine or optimise a manually generated mesh with the
same method. Table 1 summarises the developed mesh-generation tools
of the two-dimensional finite-element mesh generator.
3.3 Three-dimensional mesh generator
We expanded our two-dimensional finite-element mesh generator
into a flexible three-dimensional mesh generator, shown in Fig. 2,
by using the layer-stacking method. The three-dimensional mesh is
first created by stacking many layers of a two-dimensional mesh.
Then, one of the background images in the image database is
selected and
Table l Mesh generation tools of the two-dimensional finite-
element mesh generator
tool function tool function
new connected edges ~ selection arrow
9 new node A character tool
..~ inseff node r~.] range selection tool 1
.-~ delete node ~ range selection tool 2
"--" new edge 0~ zoom
"~ cut edge .."-. move node
~. triangular element ~ extend element
rectangular element 5 move element
I~ circular boundary A refine element
[ ] rectangular boundary ~ optimise element
voltage-sensing node (~)i current-injection node
J - reference node (~ non-electrode node
boundary element 1 ~ boundary element 2
boundary element 3 ~ new rectangular mesh
532 Medical & Biological Engineering & Computing
September 1994
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attached to each layer. Displaying the mesh for each layer on
top of the background image, we can modify the mesh using the
mesh-editing tools.
Fig. 3 shows a simple example of a three-dimensional mesh of the
human thorax. Each two-dimensional mesh in Fig. 3 was created using
cross-sectional images of the thorax (sections 20-27) from work by
Carter (CARTER et al., 1977). The entire three-dimensional mesh of
the thorax will require more than 30 layers,
Three-dimensional mesh generation is very difficult without
proper graphical feedback. We added a few three-dimensional
graphics capabilities into the mesh generator. We can view the
entire mesh in three dimensions by perspective transformation. We
can also view a single element or many elements belonging to a
certain organ. By having graphical feedback, we can confirm the
correct connections between nodes for the validity check of the
element. We can also determine how badly one element is distorted
in shape.
3.4 Boundary elements
We implemented three tools shown in Table 1 for generating
boundary elements that model electrodes (HUA et al., 1993). The
first tool is for point electrodes and generates a simple
one-dimensional element (a resistor). The other two tools are for
modelling large or compound electrodes. We can use either
three-node or four-node boundary elements to model large or
compound electrodes.
4 Solut ion of a l inear system of equations
To minimise the computation time to solve the linear system of
equations, we use sparse-matrix techniques as the master matrix is
very sparse. Our current mesh generator provides several different
node-renumbering algorithms that maximise the sparsity of the
master matrix, thereby minimising the amount of computation. It has
been shown that the sparse-matrix and vector method with node-
renumbering algorithms significantly reduces the computa- tion time
in reconstructing static images in EIT (Woo, 1990).
4.1 Sparse-matrix method
There are many different sparse-matrix techniques developed so
far in areas such as partial differential
Fig. 2 Three-dimensional mesh generator
equations, power system analysis, electric circuit analysis,
structural analysis etc. The basic idea of the sparse factorisation
and the sparse forward and back substitution routine is that we can
avoid any trivial computations using information about the matrix
structure and a carefully designed data structure.
4.1.1 Topology analysis and node renumberin,q: the analysis of a
matrix structure is the first step in the sparse-matrix method. The
analysis step includes ordering (node-renumbering) algorithms and
symbolic factorisation. Given a finite-element mesh, we find a
permutation vector using ordering algorithms that try to minimise
the number of fill-ins during LU factorisation. Then, the locations
of all fill-ins in the factorised matrix are found by symbolic
factorisation.
As the performance of most ordering algorithms depends on the
matrix, we need to compare the performance of different ordering
algorithms for a given sparse matrix (DUFF et al., 1986). Woo
compared six ordering algorithms: Tinney scheme 1 (SATO and TtNNEV,
1963), Tinney scheme 2 (TINNEY and WALKER, 1967), Tinney scheme 3
(TINNEY and WALKER, 1967), Cuthill-McKee (GEORGE and LIu, 1981),
reverse Cuthill-McKee (GEORCE and LIu, 1981) and
Gibbs-Poole-Stockmeyer (GxBBS et al.,
Fig. 3 Example of three-dimensional mesh generated using
cross-sectional images of the thorax from CARTER et al. (1977): (a)
sections 20-27. (b) two-dimensional mesh of section 23
Medical & Biological Engineering & Computing September
1994 533
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Fig. 4
(a) (b)
(a) Structure of the Y matrix (389 x 389, 2803 nonzero
(elements) for the finite-element mesh in Fig. 6; it has 2803
nonzero elements (1.85%); the global node numbers are assigned
radially from the inside; (b) structure of the L + U matrix after
LUfactorisation of Y: it has 44849 non-zero elements (29.64%); amon
9 them, 42 046 elements (27. 79%) are fill-ins
1976). The first three are general methods for all types of
sparse matrix, and the others are used only for brand matrices.
Based on the numerical experiments using various meshes, Woo
concluded that the Tinney scheme 2, which is also called 'the
minimum degree algorithm' or 'Markowitz ordering for symmetric
matrix', is the best in most cases.
Fig. 4a is the topology of the master matrix Y derived from the
two-dimensional circular mesh in Fig. 6. Fig. 4b is the matrix
after LU factorisation. We can see that many fill-ins are generated
by LU factorisation. Fig. 5a shows the structure of the Y matrix
shown in Fig. 4a after using the Tinney scheme 2 ordering, and Fig.
5b is the structure of the L + U matrix after factorisation. Fig. 5
shows that we could significantly reduce the number of fill-ins by
using the ordering algorithm.
The amount of computation time for the normal LU factorisation
increases with the cube of the number of nodes (~ O(NS)). However,
if we use the sparse-matrix method, the computation time for the
factorisation increases by less than the square of the number of
nodes (~O(Nl'Ss)). We also found that application of the ordering
algorithm could provide a maximum reduction to 1/32 of the
computation time, compared with the case when we do not use any
ordering algorithm.
4.1.2 Symbolic factorisation: in the sparse-matrix method, we
store only non-zeros of the matrix. In order not to create a new
storage area for a fill-in during the LU factorisation,
Fig. 5
we find all fill-ins before the actual numerical LU
factorisation. This procedure is called symbolic factorisa- tion,
and the sparse matrix with all fill-ins added is called the perfect
elimination matrix (ALVARADO, 1977).
4.1.3 Repeated solutions: one of the advantages of the direct
method in solving a linear system of equations is the ability to
perform many repeated solutions with different right-hand-side
(RHS) vectors without factorising the matrix again. This is
especially important in the EIT image reconstruction using the
modified Newton-Raphson method.
4.2 Sparse-vector method
When the RHS vector v in eqn. 2 is also sparse, we can further
reduce the computation time by analysing the forward and back
substitution path. We implemented the sparse-vector method of
Tinney et al. in solving eqn. 2 and found that the method reduces
the computation time by about 5-t0%.
4.3 Data structure
Many different data structures have been suggested for different
types of sparse matrix (DUFF et al., 1986; GEORGE and Liv, 1981).
The most used one in a general sparse-matrix code is the row
linked-list. As we set up a linked-list for each row of the matrix,
there are a total of N linked-lists, and each of them contains all
non-zero entries in the corresponding row of the perfect
elimination matrix.
4.4 Sparse factorisation and substitution
Using the information about the matrix structure and the row
linked-list data structure, we eliminated a sparse LU factorisation
and a forward and back substitution routine. Let z be the total
number of non-zeros in a matrix, then, for a sparse matrix,
z = O(N l+~) (3)
where N is the dimension of the matrix, and 7 < 1. Then, the
number of arithmetic operations involved in the sparse
factorisation and substitution step could be O(N 1 +2~) and
O(NI+~), respectively. Therefore we could reduce the computation
time significantly.
For a very large mesh, especially in three dimensions, we may
not be able to store the master matrix in memory even using
sparse-matrix techniques. Then the implementation. of the frontal
solver, where the equation is solved element by element (SEPULVEDA,
1984), will be necessary.
(a) (b)
(a) Structure of the Y matrix for the finite-element mesh in
Fig. 6 after Tinney scheme 2 ordering," (b) structure of the L + U
matrix after factorisation, it has 7869 non=ero elements (5.20%).
amony them, 5066 elements (3.35%) are fill-ins
Medical & Biological Engineering & Computing
5 Applications in EIT
The finite-element analysis software package is general, so that
we can use it in many different applications. To use it in static
EIT image reconstructions, we need the following special
considerations and customisations.
For a reconstruction algorithm such as the modified
Newton-Raphson method (YORKEY et at., 1987; HUA et al., 1991), the
number of elements limits the spatial resolution Of the resistivity
image. Therefore, in developing the finite-element model for EIT,
we must consider not only the accuracy of the amplitude of the
solution of the forward problem but also the spatial resolution it
provides. A fine mesh provides both good accuracy and good spatial
resolution. However, the design of the mesh is strictly limited by
the relationship between the number of electrodes and the number of
independent measurements.
534 September 1994
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5.1 Number of electrodes, number of independent measure- ments
and spatial resolution
Let the number of electrodes be E, then the number of
independent measurements is E(E - 1)/2 (WEBSTER, 1990). As this is
equivalent to the maximum number of equations, the maximum number
of resistivity values (unknowns) we can determine is limited by
this number. However, in practice, we can only determine a number
of resistivity values less than this number, owing to the
non-linear nature of the problem. Therefore the spatial resolution
of the resistivity image is primarily determined by the number of
electrodes used in the EIT system.
5.2 Groupings of elements
When a generated mesh contains many elements to provide a good
accuracy for the forward problem, we frequently need to constrain
some elements by forcing them to have the same resistivity value.
This is called element grouping and it decreases the spatial
resolution. If we already know the resistivity values of some
elements, we can fix the values during image reconstruction. This
feature can be used in the layer-stripping image reconstruction
(CHENEY et al., 1992).
5.3 Example of a finite-element mesh
Fig. 6 shows a mesh of a circular two-dimensional physical
phantom for a 32-electrode EIT system. This model consists of 389
nodes and 536 triangular or quadrilateral elements. The size of
each element was determined so that it could provide 5% spatial
resolution of resistivity images. We modelled the region around
each electrode with three elements (two under each-electrode and
one between pairs of electrodes), resulting in the total of 96
elements in the outermost layer. We used four-node boundary
elements to model the compound electrodes (HOA et al., 1993). This
permitted us to solve the forward problem accurately as voltage
changes are large in this region. Even though many elements are
needed in the outside layers for accuracy, high spatial resolution
is not usually needed there. Therefore we constrained the elements
in the two or three outermost layers by assigning the same
resistivity value to two or four elements as the resistance in this
region, including the electrode-skin resistance, does not change
rapidly from region to region. There are 376 resistivity values to
be determined by the reconstruction algorithm. As there are 496
independent measurements from the 32 electrodes, these 376
resistivity values can be determined from the measured voltage data
on the surface. Woo (Woo et al., 1993) used this mesh in the
reconstruction of static resistivity images.
6 Conclusions
We have developed a finite-element software package on a
Macintosh II computer for uses in EIT and other biomedical
engineering research areas, such as cardiology and impedance
pneumography, where finite-element analy- sis is required. It
includes two- and three-dimensional interactive graphical
finite-element mesh generators and efficient algorithms for solving
linear systems of equations using sparse-matrix and vector
techniques.
To improve the software package developed, more of the
three-dimensional graphics functions need to be developed. In
particular, hidden line removal will greatly improve the appearance
of the mesh. For graphical display of finite-element analysis
results, a contour plot of equi-
Medical & Biological Engineering & Computing
Fig. 6 Two-dimensional circular finite-element model with 389
nodes, 536 elements and 376 independent elements for a 32-electrode
EIT system; 32 pairs of annular current and voltage electrodes
cover 90% of the surface. (a) current electrode, (b) voltage
electrode
potential lines and display of current streamlines will be
useful. Acknowledgments--This work was supported by the National
Science Foundation under grant EET-8714618.
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Author 's biography Eung Je Woo received his BS and MS degrees
in Electronics Engineering from Seoul National University, Korea,
in 1983 and 1985, respec- tively. He received his PhD in Electrical
Engineering from the University of Wisconsin, Madison, in 1990. He
is currently Assistant Professor of Biomedical Engineering at Kon
Kuk University, Korea. His research interests include impedance
measurement, biomedical
computing and medical instrumentation. He is a member of IEEE,
KOSMBE and KITE.
536 Medical & Biological Engineering & Comouting
September 1994