Multiphasic Mixture Modeling and Simulation of Coronary Circulation

本研究基于多相混合理论，提出了冠状动脉系统的分层冠状动脉理论模型，其中血管呈树形结构，从动、静脉向毛细血管床逐渐分叉，将心肌基质视为固相，将血管树各层中的血液视为特定的流体相。应用拉格朗日乘子法对固相和流相的不可压缩条件进行了求解。在有限元公式中引入残差加权法，并对其进行了描述。导出了切向刚度矩阵，用于非线性问题的迭代计算。由于分层结构导致流体变量在节点处具有较大的自由度，为了提高计算效率，将有限元方程的节点变量改为压力，而不是流体相的相对速度。该公式被实现到一个自制程序中，并构建在一个多尺度、多物理场心脏模拟器中。数值算例表明，应用该模型可以表示舒张期（或收缩期）动脉（或静脉）血流的特征主导波形，具有临床应用价值。期望应用该方法能够阐明心肌收缩对心内压及冠状动脉血流灌注的影响，为缺血性心脏病的病因阐明和诊断治疗做出贡献。

In this research, based on the multiphasic mixture theory, a hierarchical coronary theoretical model is proposed for the coronary system, in which the vessels have tree structure gradually bifurcating from artery and vein to capillary bed, by treating the myocardial matrix as solid phase and blood in each hierarchy of the vessel tree as specific fluid phase. Furthermore, the incompressible condition of the solid and fluid phases was imposed by applying Lagrange multiplier method. The residual weighted method was introduced for finite element formulation and followed by the descritization. The tangential stiffness matrix was derived for iterative computation of the nonlinear problems. Since the fluid variables have large degrees of freedom at a node due to the hierarchy structure, for the computational efficiency, the finite element equations were reformulated by adopting pressure as nodal variable instead of the relative velocity of fluid phase. The formulation was implemented into a home-made program and built in a multi-scale and multi-physics heart simulator. The numerical examples show that the characteristic predominate wave form of arteries (or veins) blood flow in diastolic (or systolic) can be represented by applying the proposed model and thus suggest the usability in the clinical application. It is expected that the effect of the myocardial contraction on the intramyocardial pressure and furthermore on the blood perfusion in coronary system can be elucidated by applying the proposed approach and to make contribution to etiology elucidation of the ischemic heart disease and diagnostic treatment.