One of the key processes in living organisms is mass transport occurring from blood vessels to tissues for supplying tissues with oxygen, nutrients, drugs, immune cells, and - in the reverse direction - transport of waste products of cell metabolism to blood vessels. The mass exchange from blood vessels to tissue and vice versa occurs through blood vessel walls. This vital process has been investigated experimentally over centuries, and also in the last decades by the use of computational methods. Due to geometrical and functional complexity and heterogeneity of capillary systems, it is however not feasible to model in silico individual capillaries (including transport through the walls and coupling to tissue) within whole organ models. Hence, there is a need for simplified and robust computational models that address mass transport in capillary-tissue systems. We here introduce a smeared modeling concept for gradient-driven mass transport and formulate a new composite smeared finite element (CSFE). The transport from capillary system is first smeared to continuous mass sources within tissue, under the assumption of uniform concentration within capillaries. Here, the fundamental relation between capillary surface area and volumetric fraction is derived as the basis for modeling transport through capillary walls. Further, we formulate the CSFE which relies on the transformation of the one-dimensional (1D) constitutive relations (for transport within capillaries) into the continuum form expressed by Darcy's and diffusion tensors. The introduced CSFE is composed of two volumetric parts - capillary and tissue domains, and has four nodal degrees of freedom (DOF): pressure and concentration for each of the two domains. The domains are coupled by connectivity elements at each node. The fictitious connectivity elements take into account the surface area of capillary walls which belongs to each node, as well as the wall material properties (permeability and partitioning). The overall FE model contains geometrical and material characteristics of the entire capillary-tissue system, with physiologically measurable parameters assigned to each FE node within the model. The smeared concept is implemented into our implicit-iterative FE scheme and into FE package PAK. The first three examples illustrate accuracy of the CSFE element, while the liver and pancreas models demonstrate robustness of the introduced methodology and its applicability to real physiological conditions.

中文翻译：

---

用于毛细血管系统和生物组织中质量传输的复合涂抹有限元

生物体中的关键过程之一是从血管到组织的质量运输，为组织提供氧气、营养、药物、免疫细胞，以及 - 在相反的方向 - 将细胞代谢的废物运输到血管。血管与组织之间的质量交换通过血管壁发生，反之亦然。几个世纪以来，以及在过去几十年中，通过使用计算方法对这一重要过程进行了实验研究。然而，由于毛细血管系统的几何和功能复杂性和异质性，在整个器官模型中对单个毛细血管（包括穿过壁的运输和与组织的耦合）进行模拟是不可行的。因此，需要简化和稳健的计算模型来解决毛细血管组织系统中的质量传输问题。我们在这里引入了梯度驱动的质量传输的拖尾建模概念，并制定了新的复合拖尾有限元 (CSFE)。在毛细血管内浓度均匀的假设下，毛细血管系统的传输首先被涂抹到组织内的连续质量源。在这里，毛细管表面积和体积分数之间的基本关系被推导出来作为模拟通过毛细管壁的传输的基础。此外，我们制定了 CSFE，它依赖于将一维 (1D) 本构关系（用于毛细血管内的传输）转换为由达西张量和扩散张量表示的连续体形式。引入的 CSFE 由两个体积部分组成 - 毛细血管和组织域，并具有四个节点自由度 (DOF)：两个域中每个域的压力和浓度。这些域通过每个节点的连接元素耦合。虚拟连接元素考虑了属于每个节点的毛细管壁的表面积，以及壁材料属性（渗透性和分区）。整个 FE 模型包含整个毛细血管组织系统的几何和材料特性，生理上可测量的参数分配给模型内的每个 FE 节点。涂抹的概念被实现到我们的隐式迭代有限元方案和有限元包 PAK 中。前三个示例说明了 CSFE 元素的准确性，