Explicit Method

Values ofucan be deduced from (4.5) at irregular points. For a given irregular point, the value

is computed as the weighted sum of neighbouring points. In a cuboid medium, this would equate to points at the end of an axis being assigned values from the penultimate points. For realistic

geometry, we can use the components ofn,{α, β, γ}, to interpolate the partial derivatives using

values from neighbouring points. So (4.5) can be discretised to

∂u ∂n =α[sgn(α)u···−sgn(α)u(−sgn(α))··]+ β[sgn(β)u···−sgn(β)u·(−sgn(β))·]+ γ[sgn(γ)u···−sgn(γ)u··(−sgn(γ))] (4.12) =|α| u···−u−sgn(α)··+ |β|u···−u·−sgn(β)·+ |γ| u···−u··−sgn(γ) (4.13) = 0 (4.14) as sgn(i)i=|i|. Therefore, from (4.12–4.14), ∴u···= |α|u−sgn(α)··+|β|u·−sgn(β)·+|γ|u··−sgn(γ) |α|+|β|+|γ| (4.15)

For example, u₋sign(α)··, given a negative value for α, is the value ofu at position x+ 1.

Intuitively, (4.15) gives that the boundary value will be computed as the weighted mean of neighbouring points in the opposite direction along each axis to that given by the boundary

normal, as shown in Figure 4.5. The application of this method inBeatboxrequires two additional

4.9. NEUMANN BOUNDARY CONDITIONS IN THE ISOTROPIC MEDIUM 125

(a) (b) (c)

!

(d)

Key

Void Non-void Centre Mass Computed Normal

Figure 4.4: Centre mass and resultant boundary normals.

Identifying Irregular Points

In order to avoid references to void points in Laplacian computations, we identify irregular points in the mesh. Irregular points are those with one or more void neighbour. The neighbourhood to be considered depends on the stencil of the Laplacian algorithm used, as it is the Laplacian

algorithm that will make references to neighbouring points. AsBeatbox’sdiff device employs

a six-point ‘cross’ stencil, we need only test the status of these points when identifying irregular points.

Computing Boundary Normals

Relatively few of the available models of cardiac geometry include boundary normals. For this reason, boundary normals must usually be estimated from the geometry data, particularly the

statuses of points. The technique used byBeatboxaims to find the ‘centre mass’ of the irregular

point’s neighbourhood — i.e. the average coordinates of the irregular point’s tissue neighbours. The boundary normal can then be drawn from this location, through the irregular point and out of the tissue. The neighbourhood used for normal generation need not match the Laplacian

stencil. To make the most of the available data, the implementation of this method in Beatbox

uses all 26 immediate neighbours.

One potential pitfall of the ‘centre mass’ method is that there can arise situations in the data where, despite the presence of void points, the arrangement of tissue points around the irregular point is such that their average location will coincide with the irregular point, as illustrated in Figure 4.4. This prevents a meaningful boundary normal from being generated. In these cases, the boundary normal can be made to point directly towards any of the void points.

The values of boundary normals give a finer-grained description of the curvature of the medium than can be represented by the rasterised mesh. These differing resolutions present some difficulties in the computation of boundary values at some locations, where one or two of the desired neighbouring points are void.

Selecting Neighbours

For each neighbouring point to be found, a test is made at the desired location of that neighbour to see if the point is tissue. If so, the point can be used as a neighbour. If the point is void, a hierarchy of alternatives are checked until a tissue point is found.

Foru−sign(α)··, whereα,β andγare positive, andα > β > γ, the hierarchy is detailed below.

Each point in the hierarchy is tested, in sequence, until a suitable neighbour is found.

(a) (b)

Figure 4.5: Relationship between boundary normals and selected neighbours.

• Select the diagonal, by adding a shift along the axis with the greater corresponding vector component. Test the neighbour at {x−1, y−1, z}.

• Select the diagonal, by adding a shift along the remaining axis with the lesser corresponding vector component. Test the neighbour at {x−1, y, z−1}.

• Shift along all three axes. Test the neighbour at{x−1, y−1, z−1}.

• Shift along the remaining axis with the greater corresponding vector component. Test the

neighbour at{x, y−1, z}.

• Shift along the remaining axis with the lesser corresponding vector component. Test the

neighbour at{x, y, z−1}.

• Change direction on the same axis. Test the neighbour at{x+ 1, y, z}.

In the last case, where a neighbour is selected in the same direction as the boundary normal,

its contribution to the boundary value equation is made negative. For example, if thex-neighbour

were selected in this manner, (4.15) would be altered as shown in (4.16).

u···=

−|α|u−sgn(α)·+|β|u·−sgn(β)·+|γ|u··−sgn(γ)

−|α|+|β|+|γ| (4.16)

Anisotropy

As described in Section 4.9, isotropy allows the simplification of the Neumann boundary condition to exclude the scalar diffusion coefficient. Since anisotropic diffusion employs a diffusion tensor, this must be considered when solving boundary conditions for anisotropic media, as shown in (4.17). n·D∇u _G = 0, (4.17)

4.9. NEUMANN BOUNDARY CONDITIONS IN THE ISOTROPIC MEDIUM 127 For the reasons discussed below, the explicit method is not used to satisfy boundary conditions in Beatbox. Consequently, its implementation for anisotropy is not considered here.

Limitations of the Explicit Method

Computing boundary values explicitly for irregular points from their intramural neighbours can lead to some artefacts in the solution near the boundary. When a wave of excitation reaches the penultimate point on an axis, it will immediately ‘leap’ to an associated boundary point, along the boundary normal. This can distort the wavefront, creating fronts that travel directly along the boundary normal between the last points in the tissue.

There is a tradeoff to be made when selecting neighbouring points. A greater proportion of ideal neighbours are likely to be found if other irregular points are accepted as neighbours. This can, however, introduce additional error, as any error caused by the above issue can be allowed to spread to neighbouring irregular points.

(a) QUIneumdevice (b) Explicit method

Figure 4.6: Black points depict an excitation wavefront moving towards the boundary. In the explicit method, the wavefront ‘leaps’ from the last internal tissue point towards the irregular point.