Multiphasic Mixture Modeling and Simulation of Coronary Circulation

本研究以多相混合物理论为基础，将心肌基质作为固相，血管树每一层中的血液作为特定的流体相，建立了冠状动脉系统的分层冠状动脉理论模型，其中血管具有从动、静脉到毛细血管床逐渐分叉的树状结构。此外，采用拉格朗日乘子法对固液两相施加了不可压缩条件。引入残差权重法进行有限元列式，并对其进行描述。推导了用于非线性问题迭代计算的切线刚度矩阵。由于层次化结构，流体变量在结点具有较大的自由度，为提高计算效率，采用压力作为结点变量，而不是流体相的相对速度，对有限元方程进行了重新表述。该公式被实施到一个自制的程序中，并内置在一个多尺度、多物理的心脏模拟器中。数值算例表明，该模型能够反映舒张期(或收缩期)动脉(静脉)血流的特征优势波形，具有一定的临床应用价值。应用该方法可望阐明心肌收缩对心肌内压的影响，进而对冠脉系统血流灌注的影响，为缺血性心脏病的病因阐明和诊断性治疗做出贡献。

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.