A. Daub, A. Loewe, and B. Frohnapfel. Haemodynamics in an elasto-mechanic model of the human heart. In Annual Scientific Conference of the International Association of Applied Mathematics and Mechanics, 2018
Numerical modelling enables a quantitative evaluation of physiological and patho-physiological relationships within the human heart and the circulatory system. Surgical planning and optimisation of medical equipment using a virtual heart become possible by merging of empirical studies with physical and mathematical knowl- edge. These goals motivate a multi-physical coupling between electro-physiology, elasto-mechanics, blood flow and the circulatory system. In a first step a one-way coupling of all four relevant physical domains is considered. Simulation of electro- physiological excitation spread in conjunction with excitation contraction coupling yields the spatio-temporal distribution of cardiac active tension. This, as well as a closed loop model of the circulatory system, drive the continuum mechanics simulation of cardiac deformation and pressure, which in turn serve as a boundary condition for blood flow simulation. Physiological blood flow dynamics are dominated by the formation of a ring vortex that washes out the ven- tricles and thereby reduces the risk of thrombogenesis and flow stasis. This process is strongly affected by the heart valves. However, including the three dimensional leaflets and their interaction with the blood flow is computationally expensive. Further, the effort for construction is not negligible. Therefore, a simpler model is implemented as a first step. It comprises of three layers of porous cells that move with the valve plane and time dependently block or open the plane respectively. First results illustrate a high potential of the model to reliably reproduce the physiological vortex formation in the ventricles.
In order to be used in a clinical context, numerical simulation tools have to strike a balance between accuracy and low computational effort. For re- producing the pumping function of the human heart numerically, the physical domains of cardiac continuum mechanics and fluid dynamics have a significant relevance. In this context, fluid-structure interaction between the heart muscle and the blood flow is particularly important: Myocardial tension development and wall deformation drive the blood flow. However, the degree to which the blood flow has a retrograde effect on the cardiac mechanics in this multi-physics problem remains unclear up to now. To address this question, we implemented a cycle-to-cycle coupling based on a finite element model of a patient-specific whole heart geometry. The deforma- tion of the cardiac wall over one heart cycle was computed using our mechanical simulation framework. A closed loop circulatory system model as part of the simulation delivered the chamber pressures. The displacement of the endo- cardial surfaces and the pressure courses of one cycle were used as boundary conditions for the fluid solver. After solving the Navier-Stokes equations, the relative pressure was extracted for all endocardial wall elements from the three dimensional pressure field. These local pressure deviations were subsequently returned to the next iteration of the continuum mechanical simulation, thus closing the loop of the iterative coupling procedure. Following this sequential coupling approach, we simulated three iterations of mechanic and fluid simulations. To characterize the convergence, we evaluated the time course of the normalized pressure field as well as the euclidean distance between nodes of the mechanic simulation in subsequent iterations. For the left ventricle (LV), the maximal euclidean distance of all endocardial wall nodes was smaller than 2mm between the first and second iteration. The maximal distance between the second and third iteration was 70μm, thus the limit of necessary cycles was already reached after two iterations. In future work, this iterative coupling approach will have to prove its abil- ity to deliver physiologically accurate results also for diseased heart models. Altogether, the sequential coupling approach with its low computational effort delivered promising results for modeling fluid-structure interaction in cardiac simulations.