title: A Tensor B-Spline Approach for Solving the Diffusion PDE With Application to Optical Diffusion Tomography.
abstract: Optical Diffusion Tomography (ODT) is a modern non-invasive medical imaging modality which requires mathematical modelling of near-infrared light propagation in tissue. Solving the ODT forward problem equation accurately and efficiently is crucial. Typically, the forward problem is represented by a Diffusion PDE and is solved using the Finite Element Method (FEM) on a mesh, which is often unstructured. Tensor B-spline signal processing has the attractive features of excellent interpolation and approximation properties, multiscale properties, fast algorithms and does not require meshing. This paper introduces Tensor B-spline methodology with arbitrary spline degree tailored to solve the ODT forward problem in an accurate and efficient manner. We show that our Tensor B-spline formulation induces efficient and highly parallelizable computational algorithms. Exploitation of B-spline properties for integration over irregular domains proved valuable. The Tensor B-spline solver was tested on standard problems and on synthetic medical data and compared to FEM, including state-of-the art ODT forward solvers. Results show that 1) a significantly higher accuracy can be achieved with the same number of nodes, 2) fewer nodes are required to achieve a prespecified accuracy, 3) the algorithm converges in significantly fewer iterations to a given error. These findings support the value of Tensor B-spline methodology for high-performance ODT implementations. This may translate into advances in ODT imaging for biomedical research and clinical application.
authors.forename: Dmytro
authors.initials: D
authors.lastname: Shulga
authors.forename: Oleksii
authors.initials: O
authors.lastname: Morozov
authors.forename: Patrick
authors.initials: P
authors.lastname: Hunziker
mesh_terms: D000465:Algorithms; D004058:Diffusion; D006801:Humans; D012815:Signal Processing, Computer-Assisted; D041622:Tomography, Optical