Page 126 - YORAM RUDY BOOK FINAL

P. 126
BEM is derived from Green’s second theorem 260 , which relates the potential in a volume to the

potential on its bounding surfaces via integration:

2
2
∫ [Φ▼ (1/r) – (1/r)▼ Φ]dv = ∫ [Φ{∂(1/r)/∂n} – 1/r{∂ Φ/∂n}]ds (5.4)
v s

Where the surface S includes the epicardial and torso surfaces and v is the torso volume enclosed
by these surfaces; r is the Euclidian distance in three-dimensional space. Manipulation of equation
2
5.4 considering that ▼ Φ = 0 in the volume v, and discretization of the epicardial and torso
surfaces into triangular elements, leads to the following matrix equation relating Φ and Φ :
T
E
Φ = AΦ (5.5)
T E

Where Φ is a vector of torso potentials and Φ is a vector of epicardial potentials; A is a transfer
T
E
matrix that contains the geometric relationship between the heart surface and torso surface.
The potential over each triangular element can be assumed to be constant or to vary as a linear

or higher order function of the intrinsic element coordinates. Equation (5.5) computes Φ from a
T
given Φ in a specific heart-torso geometry (contained in the transfer matrix A). This formulation
E
constitutes the forward problem of electrocardiography. The objective of ECGI is to invert this
relationship to compute Φ from a known Φ , a formulation that constitutes the inverse
T
E
problem of electrocardiography. The inverse problem is mathematically ill-posed in the sense
that even low-level noise or error in the data (the measured torso potential, Φ ) can cause large,
T
unbounded errors in the solution (the epicardial potentials, Φ ). Because of this property, one
E
cannot simply invert the (ill-conditioned) matrix A to obtain Φ . To overcome this difficulty, we
E
have applied different computational schemes in the ECGI application: Tikhonov regularization
(TR) 260 and the generalized minimal residual (GMRes) 262 method.

TR stabilizes the computation by imposing constraints on the solution through the

following objective function:

2
2
Min [║Φ - AΦ ║ + t║RΦ ║ ] (5.6)
Φ E T E E
By minimizing this function with respect to Φ , one finds the Φ that best approximates (in the
E
E
least-square sense) a solution to equation (5.5) subject to a given constraint. The first term in
equation (5.6) represents the solution and the second term is the constraint (regularization) term
that imposes bounds on the magnitude of Φ (for R=I, the unit matrix), its steepness (for R=G, the
E
surface gradient operator), or its curvature (for R=L, the surface Laplacian operator); t is a regular-
ization parameter that determines the weight of the imposed constraint.