We present a new method for modeling tissue perfusion on the capillary scale. The microvasculature is represented by a network of one-dimensional vessel segments embedded in the extra-vascular space. Vascular and extra-vascular space exchange fluid over the vessel walls. This exchange is modeled by distributed sources using smooth kernel functions for the extra-vascular domain. It is shown that the proposed method may significantly improve the approximation of the exchange flux, in comparison with existing methods for mixed-dimension embedded problems. Furthermore, the method exhibits better convergence rates of the relevant quantities due to the increased regularity of the extra-vascular pressure solution. Numerical experiments with a vascular network from the rat cortex show that the error in the approximation of the exchange flux for coarse grid resolutions may be decreased by a factor of 3. This may open the way for computing on larger network domains, where a fine grid resolution cannot be achieved in practical simulations due to constraints in computational resources, for example in the context of uncertainty quantification.

我们提出了一种在毛细血管规模上模拟组织灌注的新方法。微血管由嵌入血管外空间的一维血管段的网络表示。血管和血管外空间在血管壁上交换流体。这种交换是由分布式源使用平滑核函数为血管外域建模的。结果表明，与现有的混合维嵌入问题方法相比，该方法可以显着提高交换通量的近似性。此外，由于血管外压力解决方案的规律性增加，该方法表现出更好的相关量收敛速度。