Abstract
We present a multiscale model to combine reactions on thin blood vessel
capillaries and diffusion in bulk tissue domain in angiogenesis, and a
conservative multiresolution finite volume scheme. In angiogenesis, we study chemicals such as
growth factors that have two processes occurring at different spatial scales simultaneously: the
subcellular-level chemical reactions (such as ligand/receptor binding kinetics) on thin
capillaries, and the tissue-level diffusion (also including natural decay) in the three-dimensional
tissue domain. We first derive a new multiscale model where a line Dirac delta function is introduced
to integrate these two processes, and we compare this new model with existing models. Then we
develop a conservative finite volume method to solve the reaction and diffusion processes
where we use a finer mesh on capillary centerlines than the mesh in the tissue domain to accurately
capture faster and larger data changes along capillaries. In addition to the multiresolution
meshes, another challenge is the constantly-changing capillary shape and length. To overcome these
difficulties, we construct a data transferring algorithm between reaction and diffusion meshes,
which is proved to conserve the mass between two meshes and retain the variation in the reaction
domain. Numerical simulations of varying diffusion constants and two types of boundary
conditions are presented to demonstrate this multiscale model and multi-resolution algorithm.
Original language | English |
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State | Published - Jan 27 2012 |
Event | Applied Math Seminar - Lansing, MI Duration: Jan 27 2012 → Jan 27 2012 |
Seminar
Seminar | Applied Math Seminar |
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Period | 01/27/12 → 01/27/12 |