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

Abstract

Angiogenesis plays an essential role in many pathological processes such as tumor growth, wound healing, and keloid development. Low oxygen level is the main driving stimulus for angiogenesis. In an animal tissue, the oxygen level is mainly determined by three effects—the oxygen delivery through blood flow in a refined vessel network, the oxygen diffusion from blood to tissue, and the oxygen consumption in cells. Evaluation of the oxygen field is usually the bottleneck in large scale modeling and simulation of angiogenesis and related physiological processes. In this work, a fast numerical method is developed 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. With this new method, accurate evaluation of the oxygen field in a fully vascularized tissue on the scale of centimeter becomes possible.

The Robust Lamina Cribrosa Vasculature: Perfusion and Oxygenation Under Elevated Intraocular Pressure

“…The process we used to complete LC vessel network reconstruction and how these were turned into numerical models for hemodynamics have been described elsewhere. 41 For the actual simulations of blood flow within the ONH vasculature, we followed the approach described in Lu et al 42 The vasculature was simulated as a network system, which was represented as a set of interconnected capillary elements. Due to the low Reynolds number in capillaries, the blood flow within a capillary can be approximately by the Poiseuille flow: where Q is the volume flow rate, r is the vessel radius, L is the vessel length, µ is the blood viscosity, and ∆ P is the pressure drop along the vessel.…”

“…We applied a fast and efficient method to simulate the convective and diffusive oxygen transport. 42 The ONH oxygen field within the LC was used to determine the regions of the LC suffering hypoxia.…”

“…For efficiency, we implemented recently developed techniques that allow much faster modeling of flow and oxygen/nutrient distribution in networks. 42 The effects of the vascular collapse on the blood flow and oxygenation within the LC thus give an indication of the robustness of the LC vasculature.…”

Mathematical modeling of cerebral oxygen transport from capillaries to tissue

“…To simulate the oxygen transport, the brain is considered to consist of two parts: blood and tissue subdomains. The corresponding mathematical model is a system of two differential equations [5][6][7]: the diffusion equation describing the oxygen transport and its consumption in the tissue and the first order differential equation simulating the oxygen transport in the blood. Both differential equations are non-linear.…”

“…A similar numerical approach is utilized in [6] to find and analyze the oxygen levels in skeletal muscle, brain, and tumor tissues. A fast numerical method for the simulation of oxygen supply in tissue with a large-scale complex vessel network is developed and implemented in [7]. Note that the oxygen transport model can be applied to simulations in various parts of the body.…”