Davis Jeffrey M
Department of Chemical Engineering, University of Massachusetts, Amherst, MA, 01003, USA,
Bull Math Biol. 2014 Dec;76(12):2985-3015. doi: 10.1007/s11538-014-0041-9. Epub 2014 Nov 20.
Under the approximation that blood behaves as a continuum, a numerical implementation is presented to analyze the linear stability of capillary blood flow through model tree and honeycomb networks that are based on the microvascular structures of biological tissues. The tree network is comprised of a cascade of diverging bifurcations, in which a parent vessel bifurcates into two descendent vessels, while the honeycomb network also contains converging bifurcations, in which two parent vessels merge into one descendent vessel. At diverging bifurcations, a cell partitioning law is required to account for the nonuniform distribution of red blood cells as a function of the flow rate of blood into each descendent vessel. A linearization of the governing equations produces a system of delay differential equations involving the discharge hematocrit entering each network vessel and leads to a nonlinear eigenvalue problem. All eigenvalues in a specified region of the complex plane are captured using a transformation based on contour integrals to construct a linear eigenvalue problem with identical eigenvalues, which are then determined using a standard QR algorithm. The predicted value of the dimensionless exponent in the cell partitioning law at the instability threshold corresponds to a supercritical Hopf bifurcation in numerical simulations of the equations governing unsteady blood flow. Excellent agreement is found between the predictions of the linear stability analysis and nonlinear simulations. The relaxation of the assumption of plug flow made in previous stability analyses typically has a small, quantitative effect on the stability results that depends on the specific network structure. This implementation of the stability analysis can be applied to large networks with arbitrary structure provided only that the connectivity among the network segments is known.
在血液被视为连续介质的近似条件下，本文提出了一种数值方法，用于分析通过基于生物组织微血管结构的模型树状网络和蜂窝状网络的毛细血管血流的线性稳定性。树状网络由一系列分支分叉组成，其中一个母血管分叉为两个子血管，而蜂窝状网络还包含汇合分叉，即两个母血管合并为一个子血管。在分支分叉处，需要一个细胞分配定律来解释红细胞的非均匀分布，它是进入每个子血管的血流速率的函数。控制方程的线性化产生了一个延迟微分方程组，该方程组涉及进入每个网络血管的排出血细胞比容，并导致一个非线性特征值问题。使用基于围道积分的变换来捕获复平面指定区域内的所有特征值，以构建具有相同特征值的线性特征值问题，然后使用标准QR算法确定这些特征值。在控制非定常血流方程的数值模拟中，细胞分配定律在不稳定性阈值处的无量纲指数预测值对应于超临界霍普夫分岔。线性稳定性分析的预测结果与非线性模拟结果之间具有很好的一致性。在先前稳定性分析中所做的栓塞流假设的放宽通常对稳定性结果有较小的定量影响，这取决于具体的网络结构。只要知道网络段之间的连通性，这种稳定性分析方法就可以应用于具有任意结构的大型网络。
