Error measure

In Sect. 4.3 we will investigate the impact of the stationalization error on the observed tumor front. In this course we will compare the reconstructed tumor front of a fully instationary simulation with the reconstructed tumor front using the stationalization approach.

Given a reference density distribution ua(x,t):Ω×T↦R, and a stationary approximation ub(x):Ω↦R and a threshold value θ∈R we define two domains A and B as

The medically relevant information is the spatial discrepancy between two level-set surfaces (∂A,∂B) of these density profiles. An absolute measure for this error is the symmetric difference |A⊕B|, as depicted in Fig. 3. It describes those volumes, which are either included A but not in B, or vice versa. That way, both over- and underestimations of the approximation ub are represented. The symmetric difference is however not comparable between 1D, 2D and 3D simulations.

left: Symmetric difference region between the two level-sets |A⊕B|: gray regions. right: localized sketch of the symmetric difference region, the surface of the level-set volume |∂A| and the distance between the two level-sets LB

The most expressive information in the medical context is the average distance between the two level-sets. We therefore introduce the global characteristic level-set distance.

(Global characteristic level-set distance) For a given level-set value θ, we define the characteristic level-set distance between ∂A and ∂B as

It quantifies the average distance between the two level-sets. Assuming a spherical reference geometry for A we can approximate this expression in the following way. Given the radius rA, LB simplifies to