Abstract The value of a solid angle appears frequently when solving

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On the Approximation of the Auto Solid Angle for Solving
Integral Equations
DIANA RUBIO 1 and MARÍA INÉS TROPAREVSKY 2
1
Instituto de Ciencias, Universidad Nacional de General Sarmiento
1
Escuela de Ciencia y Tecnología, Universidad Nacional de General San Martín
2
Departamento de Matemática, Facultad de Ingeniería
Universidad de Buenos Aires
ARGENTINA
Abstract: - The value of a solid angle appears frequently when solving integral equations in potential theory.
The integral that represents it may become an improper one, known as the auto solid angle, and its value must be
evaluated carefully. A summary of different approaches to calculate the values of solid angles is included, and a
modified method for the calculation on any surface is proposed. Finally, this approach is used to calculate a
numerical solution of the Forward Problem (FP) in EEG. In addition, an error bound of the discretization of the
problem is stated.
Key-Words: - Solid Angle, Integral Equations, BEM, FP in EEG.
1 Introduction
The value of the solid angle appears in many fields as
potential theory, lighting design, astronomy,
luminosity,
cardiac
electrophysiology,
pharmaceutical chemistry and biophysics, among
others.
Here we focus on the case of a Poisson-type
equation numerically solved using Boundary Element
Methods (BEM). The differential equation is
transformed into an integral one and, in many cases,
one faces the problem of calculating an integral
whose value is the solid angle subtended by a surface
at a point. When the point does not belong to the
surface, the integral can be easily evaluated using
numerical methods with any desired accuracy. When
the point does belong to the surface, it is an improper
integral, known as the auto solid angle, and its value
must be approximated in a different way.
We review some formulas appearing in the
literature and propose an approach to approximate the
value of the auto solid angle. As an example, we
apply this approach to obtain a numerical solution of
the Forward Problem in EEG and we estimate the
error due to the discretization of the integral
equations that models the problem.
2 Weak Solutions of a Poisson equation
and the Solid Angle
to the weak solution u(x) and to the fundamental
solution of u  0 in G . Then, an equation that
contains
integrals
of
the
type
r  r'
dSr' , where r is a point on
| r  r' |3
S
the closed surface S  G , ( see [3], [4]) is obtained.
I ( r )   u( r' )
To calculate the value of I(r), we discretize the
surface S into m elements Ej and call E the collection
of the elements, i.e. E 
m
E
j
. The values of u on S
j 1
are approximated by linear combinations of chosen
basis functions defined on the elements Ej. In this
frame, the integral terms I(r) take the form
Cj
r  r'
 r  r'
3
dSr'
where C j  C j ( u , E j ) is a
Ej
constant that depends on the basis functions, on the
values of u at the nodal points and on the elements Ej.
Hence, we focus on the integral
r , j 
r  r'
 | r  r' |
3
dSr' ,
(1)
Ej
that is the value of the solid angle subtended by Ej at
the point r (see [5]).
Note that the integral (1) depends on the relative
position of r with respect to Ej. For r  E j , the value
of  r , j can be easily evaluated using numerical
Consider a Poisson-type equation .(  u )  F on
methods with any desired accuracy. For r  E j the
a bounded domain G  R 3 , G  C 1 , where  is a
given piecewise constant function, F is the source and
u is the unknown. The divergence theorem is applied
integral (1) becomes improper and it is usually
referred to as the auto solid angle. There exists
several approaches to calculate the auto solid angle
depending mostly on the shape of the elements Ej and
on the nodal points. Some of them are based on the
following geometric property: for any closed and
smooth surface S, the solid angle at any of its points
is  2 , (see [5]).
We will distinguish the elements that contain the
point r from those that does not. We will use j as the
index for the first ones, i.e. r  E j , and l for the
latter ones, i.e. r  El .
Let  r ,l be the solid angle subtended by El from
r, where r  El and let  r 

r,j
be the sum of
j
the solid angles subtended by the surface elements Ej
that share the point r, i.e. r  E j (auto solid angles

for r). Since E  



 E     E
l
l

 r    r ,l
 j
 2 .
j

 , we may write


some improvements in the accuracy of the
solution of the FP is obtained for a particular
problem.

where  r , j  lim  0 r , j and  r , j are integrals
on elements E j that differs from Ej in a small
surface of area less than ε that contains the point r
(see [12]).
 In [1] and [2] the authors suggest that the
property (2) can be used to update the value of
the approximated solid angles as follows. In [1]
the authors approximate all the integrals and then
use (2) to correct the auto solid angles. In [2], the
author calculates the integrals  r ,l and gives a
prescription to distribute the difference
 2   r ,l among the auto solid angles

l
(2)
r,j .
l

3 Some Approaches to Calculate The
Solid Angle
Some of the approaches to calculate the solid angle
proposed in the literature are based on the shape of
the elements chosen to discretized the surfaces. For
this reason we consider separately the case where the
meshed surface E is a polyhedral one.
The values of  r , j may be approached by  r , j ,
If the relative position of the point r with respect
to the surrounding elements are as shown in
Fig.1, the solid angle subtended by the union of
these elements from r can be approximated by

r 

Ej
r  r'
r  r'
3
dSr'   2 cos  0  , where
j
 0  arcsin( R' / R ) (see [1]).
r
3.1 Space Discretization using Planar
Elements
Many authors considered flat triangles to discretize a
surface. The solid angle is usually calculated by
means of the formulae given by van Oosterom and
Strackee (see [10]), which is an improvement of the
one developed by Barr et al.(see [6]). Other formulas
were developed by Barnard (see [11]) and de Munck
(see [12]).
Concerning the auto solid angle, we distinguish
the following situations:
 If r is an interior point of Ej, for instance the
center of mass, the value of  r , j is zero. Some
3.2 Space Discretization using No Planar
Elements
authors use this result to calculate the
approximated solution to a particular problem and
compare the results obtained (see [13]).
 If r is a vertex of Ej, the normal vector at the
point r does not exist, then  r , j is not defined. In
 The values of  r , j may be approached by  r , j ,
[7] the authors manage this situation choosing a
normal vector at the vertices by the method of the
weighted vertices. They showed examples where
Fig.1: The elements surrounding r.
As mentioned before, the solid angle can be evaluated
by any numerical integration method whenever the
integral  r , j is not an improper one.
Some approaches found in the literature can be
adapted to calculate the auto solid angle on curved
elements Ej:
as described above.
 The approaches proposed in [1] and [2] may be
generalized to not necessarily flat elements.
3.3 Another Approach to Calculate the Auto
Solid Angle
We present a slightly modification to the approaches
proposed in [1] and [2].
We numerically approximate the proper integrals
 r ,l with a desired precision. Then we calculate the
value of the sum of the auto solid angles from r based
on (2) as  r  2   r ,l . Finally, we apportion

l
the latter value among  r , j proportionally to the
areas of Ej. In consequence,  r , j is approximated by
Aj 

~
r , j 
  2    r ,l  ,
l

 Ai 
We apply integral theorems and the Boundary
Element Method (see [4]) to obtain integral equations
for points on the surfaces of conductivity transition
Sj
 k   k 1
2
u( r )   1v( r )
(4)
 j   j 1
r  r'
u
(
r
'
)
n
(
r
'
)
dS
'
3

4
j 1
r  r'
S
3

j
for r  S k , with v( r )  
.J i
1
41
 r  r'
dr' .
G
(3)
The solution of the integral equation (4) is the
potential distribution on the spherical surfaces S j .
where Ai is the area of Ei.
Note that this approach may be applied to either
planar or curved elements.
This approximation differs from the one presented
in [1] in that they approximate all the solid angles,
then use property (2) and distribute the error among
the auto solid angles.
The difference with [2] is just the way in which
we distribute the value of  r among the  r , j .
We discretized the surfaces S j by spherical
i
elements E j ,k such that S j 
E
where the
j ,k
k
nodal points are the vertices of elements E j ,k . Thus,
the surfaces S j are not being approximated, they are
just divided into simpler sets (see Fig. 2).
Next, we present a problem where the integrals
representing the solid angle appear. We show a
numerical solution to this problem using no planar
elements to discretize the surface. We apply the
approximation introduced above and we establish
error bounds for the discretization.
Fig. 2 The discretized surface.
4 Numerical Example
The Forward Problem in EEG consists in finding the
potential distribution on the scalp caused by current
sources within the brain (see [8]). The geometry of
the head is modeled by three concentric spherical
volumes: G1 the brain, G2 the skull and G3 the
scalp. The surfaces between them are denoted by S1 ,
S 2 and S 3 , respectively.
The problem may be described by the following
Poisson-type, second order elliptic equation on a
domain G representing the human head,
.(  ( x )u( x ))  .J i ( x ) ,
where  represents the conductivity (that is assumed
to be discontinuous across the surfaces S j and
constant on each S j ), u(x) is the electric potential
and J i (x) is the impressed current modeled as a
dipole (see [14]. [15]).
We approximate u over each E j ,k by the average
at the vertices, C j ,k
1

Nk
Nk
u
i 1
j ,k ,i
, where Nk is the
number of vertices of the element E j ,k . The surface
I S j ( r )   u( r' )
integral
Sj
approximated by
r  r'
r  r'
3
dS'
~
I S j ( r )   C j ,k  j ,k
is
(5)
E j ,k
~
where  j ,k is calculated as follows. If r  E j ,k ,
~
 j ,k 

Ê j ,k
r  r'
dS'  R j ,k where Rj,k is the error of
r  r'
the integration rule and if r  E j ,k we use the
approach (3).
We then obtain a linear system of equations of the
form
Mu=F
whose matrix M depends on the conductivity values
and on the approximated values of the solid angles
(see [14]). The solution vector u contains the values
of the potential u at the grid points and F depends
only on the dipole position and strength.
In Fig. 3 we show typical results of the scalp
potential distribution when the radii of Gi are .071m,
.078m and .085m and the conductivity values are
  (  1 , 2 , 3 )  ( 0.33,0.0042,0.33 ) Am/m for
the brain, skull and scalp, respectively (see [9]).
a)
5 Conclusion
In this communication we proposed a way to
calculate the solid angle subtended by a surface S at a
point r and we apply this approach to obtain an
approximated solution of the FP in EEG. We analyze
the error that is obtained and observed that the
approximation strongly depends on the discretization
of u on the surface.
b)
Fig. 3 Numerical Examples. a) Dipole Position: (0;
0.0065; 0), Dipole Moment: (0;0;1) 10-9 b) Dipole
Position:
(0;0.0062;0.04)
Dipole
Moment:
-10
(1.2;0.6;0.6) 10
We theoretically estimates the error in the
discretization process of the equations.
Approximating the proper integrals by the
Simpson Rule, an error of O( h 5 ) is obtained, where
h is the length of the subdivisions of the interval of
integration. In addition, since u is approximated by a
constant on E j ,k , the approximation of the integral
on
E j ,k becomes
r  r'
 u( r' ) r  r'
3
~
dS' C k , j  k , j
E j ,k

r  r'
 u( r' ) r  r'
3
dS' C j ,k
E j ,k
 C j ,k

E j ,k
 2 u  C j ,k
Note that since the way in which u is
approximated determined the error bound, choosing a
refined integration method may be worthless.
We want to point out that in order to estimate the
total error of the calculated solution, in addition to the
one produced by the discretization of the integral
equation, one must consider the error introduced by
the approximation of the domain where the equations
are solved and the error due to the estimated value of
the parameter  (see [14], [15]).

E j ,k
r  r'
r  r'
L ( E j ,k )
3
 u
r  r'
r  r'
3
dS'
~
dS' C j ,k  j ,k
L ( E j ,k )
O( h 5 ).
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