Starting from full-dimensional models of solute transport, we derive and analyze multi-dimensional models of time-dependent convection, diffusion, and exchange in and around pulsating vascular and perivascular networks. These models are widely applicable for modelling transport in vascularized tissue, brain perivascular spaces, vascular plants and similar environments. We show the existence and uniqueness of solutions to both the full- and the multi-dimensional equations under suitable assumptions on the domain velocity. Moreover, we quantify the associated modelling errors by establishing a-priori estimates in evolving Bochner spaces. In particular, we show that the modelling error decreases with the characteristic vessel diameter and thus vanishes for infinitely slender vessels. Numerical tests in idealized geometries corroborate and extend upon our theoretical findings.
翻译:从溶质传输的全维模型出发,我们推导分析了脉动血管和周围环境中的多维时间依赖对流、扩散和交换模型。这些模型被广泛应用于模拟血管化组织、脑周围血管空间、植物血管和类似环境中的传输。我们在适当假设域速度下展示了完整和多维方程的解的存在性和唯一性。此外,我们通过建立变化的 Bochner 空间先验估计量来量化相关的建模误差。特别地,我们表明建模误差随着特征血管直径的减小而减小,因此对于无限细长的血管,误差会消失。理想几何形状下的数值测试证实并扩展了我们的理论发现。