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Fast optical tracking of diffusion in time-dependent environment of brain extracellular space. (English) Zbl 1340.92025

Brandts, J. (ed.) et al., Proceedings of the international conference ‘Applications of mathematics’, Prague, Czech Republic, May 15–17, 2013. In honor of the 70th birthday of Karel Segeth. Prague: Academy of Sciences of the Czech Republic, Institute of Mathematics (ISBN 978-80-85823-61-5). 117-123 (2013).
The author presents the theoretical background for the Fast Optical Tracking Of Diffusion (FOTOD) method, an improved version of the Integrative Optimal Imaging (IOI) method. In practice, the direct problem is to find the extracellular concentration of a marker in 3D space and time. The mathematical description consists in the initial value problem for the diffusion equation. The problem is solved with the help of the 3D Fourier transform in space.
The algorithm presented uses the measured concentration data and provides the solution of the inverse problem, i.e., the effective diffusion coefficient in homogeneous anisotropic media, as a function of time. The FOTOD method improves the time resolution significantly as compared with the IOI method.
For the entire collection see [Zbl 1277.00032].

MSC:

92C55 Biomedical imaging and signal processing
35K15 Initial value problems for second-order parabolic equations
42B10 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
35R30 Inverse problems for PDEs
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