Davis J M, Pozrikidis C
Department of Chemical Engineering, University of Massachusetts, Amherst, MA, 01003, USA,
Bull Math Biol. 2011 Aug;73(8):1857-80. doi: 10.1007/s11538-010-9595-3.
A numerical method is implemented for computing unsteady blood flow through a branching capillary network. The evolution of the discharge hematocrit along each capillary segment is computed by integrating in time a one-dimensional convection equation using a finite-difference method. The convection velocity is determined by the local and instantaneous effective capillary blood viscosity, while the tube to discharge hematocrit ratio is deduced from available correlations. Boundary conditions for the discharge hematocrit at divergent bifurcations arise from the partitioning law proposed by Klitzman and Johnson involving a dimensionless exponent, q≥1. When q=1, the cells are partitioned in proportion to the flow rate; as q tends to infinity, the cells are channeled into the branch with the highest flow rate. Simulations are performed for a tree-like, perfectly symmetric or randomly perturbed capillary network with m generations. When the tree involves more than a few generations, a supercritical Hopf bifurcation occurs at a critical value of q, yielding spontaneous self-sustained oscillations in the absence of external forcing. A phase diagram in the m-q plane is presented to establish conditions for unsteady flow, and the effect of various geometrical and physical parameters is examined. For a given network tree order, m, oscillations can be induced for a sufficiently high value of q by increasing the apparent intrinsic viscosity, decreasing the ratio of the vessel diameter from one generation to the next, or by decreasing the diameter of the terminal vessels. With other parameters fixed, oscillations are inhibited by increasing m. The results of the continuum model are in excellent agreement with the predictions of a discrete model where the motion of individual cells is followed from inlet to outlet.
实现了一种数值方法来计算通过分支毛细血管网络的非定常血流。通过使用有限差分法对一维对流方程进行时间积分，计算每个毛细血管段上排出血细胞比容的演变。对流速度由局部和瞬时有效毛细血管血液粘度确定，而管与排出血细胞比容的比率则从可用的相关性中推导得出。在分叉处排出血细胞比容的边界条件源于Klitzman和Johnson提出的涉及无量纲指数q≥1的分配定律。当q = 1时，细胞按流量比例分配；当q趋于无穷大时，细胞被引导至流量最高的分支。对具有m代的树状、完全对称或随机扰动的毛细血管网络进行了模拟。当树包含超过几代时，在q的临界值处会发生超临界霍普夫分岔，在没有外部强迫的情况下产生自发的自持振荡。给出了m - q平面中的相图以确定非定常流动的条件，并研究了各种几何和物理参数的影响。对于给定的网络树阶数m，通过增加表观固有粘度、减小一代到下一代的血管直径比或减小终端血管的直径，可以在足够高的q值下诱导振荡。在其他参数固定的情况下，增加m会抑制振荡。连续介质模型的结果与离散模型的预测非常吻合，在离散模型中跟踪单个细胞从入口到出口的运动。
