A Coupled Cubic Hermite Finite Element/Boundary Element Procedure for Electrocardiographic Problems
A J Pullan, C P Bradley Department of Engineering Science University of Auckland Auckland, New Zealand March 25, 1996
Address for correspondence: A J Pullan Department of Engineering Science University of Auckland Private Bag 92019 Auckland, New Zealand email:
[email protected]
1
Abstract
The problem considered is that of trying to determine the potential distribution inside a human torso as a result of the heart' s electrical activity. We describe here a high order (cubic Hermite) coupled finite element/boundary element procedure for solving such electrocardiographic potential problems inside an anatomically accurate human torso. Details of the cubic Hermite boundary element procedure and its coupling to the finite element method are described. We then present two and three dimensional test results showing the success, efficiency and accuracy of this high order coupled technique. Some initial results on an anatomically accurate torso are also given.
2
1 Introduction
The problem considered is that of trying to determine the potential distribution inside a human torso as a result of the heart' s electrical activity. Due to the material properties of the biological tissues involved and the relatively low frequency components of the heart' s cycle, the governing equation to be solved for the potential in the torso outside the heart is the generalised Laplace equation
r (r ) = 0 where is the electric potential and
(1)
is the conductivity tensor. The most common numerical meth-
ods used to solve this equation inside a model of a human torso are the finite element method (FEM) (Johnson, MacLeod & Ershler 1992, Karlon, Eisenberg & Lehr 1993) and the boundary element method (BEM) (Gale, Kilpatrick, Johnson & Nicholls 1983, Gulrajani & Mailloux 1983, Huiskamp & van Oosterom 1988). Both techniques have their relative merits for this problem. In relation to solving for a potential distribution inside a torso, the major advantages and disadvantages of each technique are FEM:
1. Handles anisotropic regions well (e.g. skeletal muscle). 2. Requires a full domain mesh. 3. Generates entire domain solution. 4. Produces symmetric sparse matrices.
3 5. Element integrals are easily evaluated numerically.
BEM:
1. Only requires a surface mesh for electrically homogeneous regions (e.g. lungs).
2. Generates solutions only on the boundary (domain solutions can be calculated later if required).
3. For a given mesh discretisation, yields more accurate results for potential and gradients.
4. Produces smaller matrices which are non symmetric and dense.
5. Requires accurate evaluation of integrals with singular integrands.
We propose here to utilise the strengths of each method in solving potential problems inside a torso by applying the appropriate mathematical technique in the appropriate region of a torso model. A torso model has been developed and a cross section of this model is shown in figure 1. The model consists of 6 regions namely the torso cavity region (the inner torso excluding the heart and lungs), the right and left lung, the skeletal muscle layer, the subcutaneous fat layer, and the heart. We propose using the BEM exterior to the heart' s surface in every region up to the inner skeletal muscle layer. In this muscle region, due to its high degree of anisotropy (and for simplicity the fat layer as well) we use the FEM. At this stage we are thinking in terms of trying to relate potential distributions on the heart' s surface to those on the body surface (and vice versa).
[Figure 1 about here.]
4 To our knowledge only one such coupled procedure has been proposed before - the so-called combination method of Stanley & Pilkington (1989). Low order finite elements were coupled with low order boundary elements and results were presented for a spherical test problem. Follow up results on real canine cross sections were described in Stanley, Pilkington, Morrow & Ideker (1991), although errors in these results were rather large. While these two papers repres ent an intelligent approach to torso modelling, there were several shortcomings. The BEM procedure used, its implementation and its coupling to FEM were all rather simple. The matching of potential gradients across boundaries was barely mentioned, although special care is required to do this correctly since the BEM and the FEM treat these quantities differently. Also no results were given for potential gradients, the accurate determination of which can be important for forward and inverse calculations (where potential distributions on the heart are calculated from measured potential distributions on the body surface and vice versa).
Our aim is to ultimately include a sophisticated heart model inside the torso, in particular the model described by Nielsen, Le Grice, Smaill & Hunter (1991) and Hunter, Nielson, Smaill, Le Grice & Hunter (1993). This model employs a high order (cubic Hermite) FEM procedure. For consistency, the same interpolation is used in the coupled procedure described here.
Every torso model mentioned above has used some combination of linear elements. Nielsen et al. (1991) have found high order elements, in particular cubic Hermites, to be more efficient (that is a higher accuracy for a given number of mesh degrees-of-freedom) at representing the geometry than linear elements. We describe below cubic Hermite interpolation in contrast to standard linear interpolation. We then discuss a cubic Hermite boundary element procedure and show how this can be coupled to the finite element procedure. Two and three-dimensional test results showing the success,
5 efficiency and accuracy of this high order coupled technique are given. Some initial results on an anatomically accurate three-dimensional torso are also given.
2 Theory
2.1 Cubic Hermite Interpolation
The standard one-dimensional linear Lagrange basis functions are
'1( ) = 1 ?
;
'2( ) =
(2)
where is a local coordinate on the element (varying from 0 at local node 1 to 1 at local node 2). The linear interpolation formula for this case is
( ) = ' ( )
(; = 1; 2)
'1( ) 1 + '2( ) 2
(3)
where is the value of at node . Such an interpolation preserves continuity between elements, but fails to preserve slope continuity. Extension of this idea to more than one variable is afforded by using a local variable in each direction. In contrast to this, the one-dimensional cubic Hermite basis functions are
01 ( ) = 1 ? 3 2 + 2 3 ; 11 ( ) = ( ? 1)2 02 ( ) = 2 (3 ? 2 ) ; 12 ( ) = 2 ( ? 1)
(4)
6 The expression
( ) = i ( ) ;i (; i = 0; 1; = 1; 2)
@ @ 01 ( )1 + 11 ( ) + 02 ( ) 2 + 12 ( ) (5) @ 1 @ 2
is an interpolation of that preserves continuity of function and derivative across element boundaries. Here ;0 is the value of at node and ;1
= @ @ at node , = 1; 2.
To apply the above cubic Hermite interpolation in practice a further step is required. The derivative
@ defined at node is dependent upon the local element -coordinate and is therefore, in general, @ n different in the two adjacent elements. Therefore we carry a physical derivative @ at nodes and use @s
@ = @ @s : @ n @s ( ;e) @ to determine
@ . @ n
Here
@ @s
!
!
e
is a physical arc-length derivative,
(6)
n = ( ; e) is the global node
@s is an element `scale factor' which scales the arc in element e and, @ e length derivative to the -coordinate derivative. Thus @ @s is constrained to be continuous across element boundaries rather than @ @ . The extension to two dimensions is described in Nielsen et al. @ ; @ and @ 2 as the nodal parameters. (1991) and gives ; @s @s1 @s2 1 @s2 number of local node
Apart from the already documented increase in efficiency of using cubic Hermite elements over linear Lagrange elements (Nielsen et al. 1991), a perceived advantage in electrocardiographic problems is the ability to interpolate gradients directly. It is well recognised that accurate modelling of potential gradients is important in both forward and inverse problems, and these gradients can be quite large on or near the heart. In a finite element approach using linear basis functions means that gradients are only represented as constants on each element (since gradients are obtained by differentiating the
7 linear interpolating functions) and thus large numbers must be used to resolve gradient information. Here nodal gradient information is obtained directly as part of the solution, which should result in a substantial reduction in mesh size to produce a given accuracy of gradient.
2.2 Governing Equations
The basic equation that must be solved in any region outside the heart' s surface is the generalised Laplace equation as shown in (1). For both forward and inverse electrocardiographic problems the following boundary conditions must be imposed across an interface between regions i and j
i = j
(7)
that is continuity of potential across an interface at a common node, and
( r )i ni = ?( r )j nj
(8)
or continuity of normal component of current across an interface at a common node (the negative sign is due to the reverse in direction of the two normals), where outward normal on the surface of region
is the electric potential, nk is the unit
k and (r )k is the component of (r ) (which can be
considered as a current) on the surface of region k. It should be noted that summation is only implied in this paper whenever indices are repeated on scalar quantities (for example in equation (8) there is no implied sum over i or j ).
8 If cubic Hermite interpolation is used, we also have
( r )i sl = ( r )j sl i
(9)
j
that is continuity of the component of current in the tangential or arc length direction at an interface
s
at a common node (assuming the directions for l are the same in each region), and
(r [( r )i sl ]) ni = ? i
r (r )j sl nj
h
i
(10)
j
or continuity of the normal component of the tangential derivative of current across an interface at a
s
common node, where lk is a unit vector in the tangential direction l on the surface of region k . The other boundary conditions required to define the problem completely depend on the class of problem being solved. In the typical forward problem, the potential
is specified over the heart' s
surface, and the normal component of the current across the outer torso boundary is put to zero (the conductivity of the air is taken to be zero). In the typical inverse problem the potential
and zero
normal flux is specified on the outer torso boundary.
2.3 Finite Element Formulation
Using a weighted residuals approach, the weak formulation of (1) is Z
r (r )wd = 0
(11)
9 where
is the domain of the (finite element) region and w is some as-yet unspecified weighting
function.
The use of the divergence theorem gives Z
where ? = @ and
Z
r rwd = (r ) nwd? @
(12)
n is the unit outward normal vector to the surface.
2.3.1 Discretisation
The discretisation involves putting =
M
[
m . On each element we approximate ( ) by i ( ) ;i
m=1 where i ( ) are the cubic Hermite interpolation functions described above. It should be noted that from now on for cubic Hermite interpolation bicubic Hermite interpolation
ranges from 1 to 2 and i from 0 to 1, however for
ranges from 1 to 4 and i from 0 to 3.
We adopt the notation that
i () means the appropriate basis functions for the given dimension of the problem. For example if
f1g then i () are the cubic Hermite basis functions but for f1; 2g, i () are the
bicubic Hermite basis functions and ;0
2 = @ ; = @ . This notation ; = ; ;1 = @ ; 2 ; 3 @1 @2 @1 @2
concept can be similarly extended to tricubic Hermite elements.
Here the geometry is also approximated using cubic Hermite interpolation. For a Galerkin finite element formulation, one chooses the weight function
w to be one of the basis functions used to
approximate the dependent variable (and thus one equation is generated for each basis function). This
10 results in the following equation, written in terms of the local
M
X
m=1 0 where
1
Z
J
r i ( ) ;i r j ( ) j ( )j d =
coordinates Z
@
( r ) n j ()d?
(13)
= f(1); (1; 2); (1; 2; 3)g and J () is the Jacobian of the transformation from m to local
coordinates. Numerical evaluation of the above integrals, using low order Gaussian quadrature (typically 2 or 3 quadrature points in each direction) and assembly into a global system yields a matrix system of equations of the form
K = f
(14)
contains the nodal values of and @ @s , the global stiffness matrix K is sparse and symmetric, while f contains “integrated normal derivative” values at the surface nodes (resulting The vector
from the right hand side of (13)). The boundary element formulation presented below results in a system of equations whose right hand side involves nodal values of two procedures one can reduce the vector
( r ) n. To combine these
f to a vector of nodal normal derivatives in the method
described in Brebbia, Telles & Wrobel (1984). To illustrate this, consider one of the integrals on the right hand side of (13). This integral can be broken up into expressions of the form
1
Z
0 where
( r ) n j () jJ ()j d
(15)
= f(1); (1; 2)g is a local coordinate along ?m, the boundary of m.
To extract point values of the normal derivative, one must firstly introduce some interpolation approx-
r ) n e.g. @@ n = N @ @n
imation for (
)g is some set of interpolation functions (for
where fN (
11 the problems considered here it will be cubic Hermite, but regardless of the choice the same interpolation should be used across interfaces shared by the BEM and FEM). Use of this in the above equation (with constant conductivity) yields
1 @ @ n N () j () jJ ()j d 0 Z
(16)
Thus, by evaluating integral expression similar to that in (16) we can reduce
f from a vector of
r ) n to nodal values of (r ) n i.e. we have
integrated values of (
f = N @@n
(17)
This will be useful when the two procedures are coupled together (see below).
2.4 Boundary Element Formulation
The BEM will be used in any region of the torso in which the conductivity can be reasonably taken to be constant (e.g. the lungs). Thus the equation to be solved in such a region is simply Laplace' s equation. The conventional boundary integral equation for Laplace' s equation, r2 = 0, in a (closed) domain
