Narrow Results

This work introduces and analyzes new primal and dual-mixed finite element methods for deformable image registration, in which the regularizer has a nontrivial kernel, and constructed under minimal assumptions of the registration model: Lipschitz continuity of the similarity measure and ellipticity of the regularizer on the orthogonal complement of its kernel. The aforementioned singularity of the regularizer suggests to modify the original model by incorporating the additional degrees of freedom arising from its kernel, thus granting ellipticity of the former on the whole solution space. In this way, we are able to prove well-posedness of the resulting extended primal and dual-mixed continuous formulations, as well as of the associated Galerkin schemes. A priori error estimates and corresponding rates of convergence are also established for both discrete methods. Finally, we provide numerical examples confronting our formulations with the standard ones, which prove our finite element methods to be particularly more efficient on the registration of translations and rotations, in addition for the dual-mixed approach to be much more suitable for the quasi-incompressible case, and all the above without losing the flexibility to solve problems arising from more realistic scenarios such as the image registration of the human brain.

Approximately 60% of cancer patients are treated with external beam radiotherapy at some point during disease management. Despite the extended time frame of fractionated therapy (4–6 weeks), radiation therapy planning is carried out based on information that is currently limited to a single 3D anatomical computed tomography scan at the onset of treatment. This concept may result in severe treatment uncertainties, including the irradiation of risk organs and reduced tumor coverage. Repeat 3D single or multi-modality imaging acquired at various time intervals during and after a radiation course provides the opportunity to increase treatment accuracy and precision by optimizing treatment in response to anatomical changes; to improve target delineation through modality-specific complementary tumor representations, and to assess treatment response. Integration of multiple imaging sources into a single patient model requires compensation of geometric differences while maintaining modality-specific differences in information content. Deformable image registration aims to reduce such uncertainties by estimating the spatial relationship between the volume elements of corresponding structures across image data. This paper reviews the algorithmic components of deformation algorithms, and their application to treatment sites with evident geometric changes, including mono- and multi-modal image registration for cancer of the head and neck, lung, liver, and prostate.