A 3D‐1D coupled blood flow and oxygen transport model to generate microvascular networks

Abstract

In this work, we introduce an algorithmic approach to generate microvascular networks starting from larger vessels that can be reconstructed without noticeable segmentation errors. Contrary to larger vessels, the reconstruction of fine-scale components of microvascular networks shows significant segmentation errors, and an accurate mapping is time and cost intense. Thus there is a need for fast and reliable reconstruction algorithms yielding surrogate networks having similar stochastic properties as the original ones. The microvascular networks are constructed in a marching way by adding vessels to the outlets of the vascular tree from the previous step. To optimise the structure of the vascular trees, we use Murray's law to determine the radii of the vessels and bifurcation angles. In each step, we compute the local gradient of the partial pressure of oxygen and adapt the orientation of the new vessels to this gradient. At the same time, we use the partial pressure of oxygen to check whether the considered tissue block is supplied sufficiently with oxygen. Computing the partial pressure of oxygen, we use a 3D-1D coupled model for blood flow and oxygen transport. To decrease the complexity of a fully coupled 3D model, we reduce the blood vessel network to a 1D graph structure and use a bi-directional coupling with the tissue which is described by a 3D homogeneous porous medium. The resulting surrogate networks are analysed with respect to morphological and physiological aspects. K E Y W O R D S 3D-1D coupled flow models, blood flow simulations, dimensionally reduced models, flows in porous media, oxygen transport, vascular growth

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

“…In Fig 8(c), we show the iteration numbers required to make the relative difference smaller than 1 × 10 −3 . We can see that the iteration numbers are comparable for the simple and refined vessel network structures, while both of them increase slightly with the mesh size N. The increase in iteration numbers is mainly due to the choice of the temporal step size Δt in Eq (14). As we have discussed above, a large Δt is helpful for convergence while a too large Δt can induce instability in the iteration.…”

“…., i d ) is the index of the mesh point. Due to the nonlinearity in the consumption function M(�), we use the standard multigrid algorithm combined with the Newton's method to solve Eq (14). According to our numerical tests, only a few steps of Newton-iteration are sufficient to make the numerical error small enough.…”

“…Modeling studies of angiogenesis are important in understanding the progression of tumor [10] and its therapy [11][12][13]. Accurate evaluation of oxygen supply is also important in generating artificial vascular networks [14,15] and non-invasively investigating the impact of diseases and therapeutic procedures on the vascular bed [16].…”

Biologically-Based Mathematical Modeling of Tumor Vasculature and Angiogenesis via Time-Resolved Imaging Data

“…The first area Figure 4 of focus (panel a in Figure 4 ) bridges the cell to tissue scale by modeling the formation and evolution of tumor-induced angiogenic networks which are predominately modeled using a discrete (lattice-based and lattice-free), continuous, or hybrid strategy similar to those discussed in Section 3 [ 87 , 88 , 123 , 130 , 131 , 132 ]. Discrete approaches typically model individual TEC movement, while continuous approaches model the change in a spatially averaged, continuous variable (e.g., vasculature density or vascular volume fraction).…”

Numerical evaluation reveals the effect of branching morphology on vessel transport properties during angiogenesis

“…However, attempts to compare transport properties of vascular networks with different morphological features are still quite limited. This could be achieved by artificially removing/adding vessels from a reference structure based on statistical or empirical rules [14,15]. However, the resultant structures are no longer real, and experimental validation is prohibitive.…”