A fast numerical method for oxygen supply in tissue with complex blood vessel network

Abstract

<jats:title>Abstract</jats:title><jats:p>Oxygen field evaluation is important in modeling and simulation of many important physiological processes of animals, such as angiogenesis. However, numerical simulation of the oxygen field in animal tissue is usually limited by the unusual coupling of different mechanisms, the nonlinearity of the model, and the complex geometry of refined blood vessel networks. In this work, a fast numerical method is designed for the simulation of oxygen supply in tissue with a large-scale complex vessel network. This method employs an implicit finite-difference scheme to compute the oxygen field. By virtue of an oxygen source distribution technique from vessel center lines to mesh points and a corresponding post-processing technique that eliminate the local numerical error induced by source distribution, square mesh with relatively large mesh sizes can be applied while sufficient numerical accuracy is maintained. The new method has computational complexity which is slightly higher than linear with respect to the number of mesh points and has a convergence order which is slightly lower than second order with respect to the mesh size. As an example, the oxygen field of a tissue irrigated by a blood vessel network with more than four thousand blood vessels can be accurately computed within one minute with our new method. The new method will definitely promote further researches based on evaluation of oxygen field, such as modeling of angiogenesis and pathogenesis of many cardiovascular diseases.</jats:p>

Optimisation of Biological Transport Networks

“…Diffusion is used for transport only between the tissue and terminal branches of the transport networks, such as capillaries, hairy roots, and terminal leaf veins. A typical spacing between neighboring capillaries and between terminal leaf veins is of the order 100µm [44,66]. On this spacial scale, diffusion is sufficiently efficient for the time scale of living organisms.…”

“…Second, the adaptation of vessel segments plays an important role in angiogenesis and root development. However, it is still far from a clear picture on what stimuli help to maintain the stability of capillary networks while meeting the needs of the tissue, and efficient numerical methods are extremely important in simulations of such large scale systems [44]. Finally, new mathematical structures of biological transport networks may provide new insights and tools for applications in other fields, such as graph theory and matrix optimisation.…”