Fig. 3.

Results for the analytic tumor / analytic velocity reaction-only (ATAV-REAC) test case; ground truth: (ρ _f = 15, ρ_w = 1, ρ_g = 0, k _f = 0, p = p^⋆, v = −v^⋆). The figure shows probability maps for the labels of the healthy atlas brain and the patient brain with tumor generated from a tumor grown in the atlas and known atlas to patient advection velocity (see text for details; axial-slice 64). We show the initial configuration for the problem (top row; iteration k = 1), the final configuration after joint registration and tumor inversion (middle row; iteration k = 7; the atlas image probability maps are transported to the patient space), and the target patient data (reference image; bottom row). Each row contains (from left to right) the probability maps for WM, GM, CSF, and TU, the residual differences (if available) between the probability maps, and a hard segmentation based on the given probabilities for the individual tissue classes. The table on the bottom provides quantitative results for the inversion. We report the average mismatch for the probability maps for the brain tissue labels ${\mu }_{B,L^2}$ and the tumor ${\mu }_{T,L^2}$ , the mean DICE coefficient for brain tissue DICE_B and tumor DICE_T, respectively. The reconstruction quality is given in terms of convergence of v^k and $c_A^k$ (0) towards the ground truth v^⋆ and $c_A^k$ (0), respectively ($e_{\mathit{v},L^2}$ and $e_{c0,L^2}$). We cannot expect this error to go to zero for several reasons. First, we lose information when we construct the test case (zero gradients in the intensity of the image), second our numerical solver introduces errors (in particular, the solver for the transport equations). We in addition to that report the change in update in the velocity v across successive iterations $\delta v=\parallel v^k-v^k{}^{-1}\parallel /\parallel v^k{}^{-1}\parallel$. Finally, we also list the relative norm of the gradient for the coupled problem in (3) $\left({\Vert g\Vert }_{\text{rel}}\right)$.