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Plats för ev logotype Simulation of Blood Flow in the Left Ventricle Using Moving Geometries Based on Ultrasound Measurements INTRODUCTION Cardiac illness is a major cause of health problems and the number one cause of death in the world. Many of the cardiac diseases lead to deformations of the geometry of the heart and thereby also to changes in the qualitative and quantitative properties of the blood flow in the heart. Our goal is to apply numerical simulation to study such effects of geometry changes on the blood flow in the left ventricle (LV) of the heart. The purpose of such studies is to gain understanding of the effects of different pathologies and furthermore to possibly study the effects of treatment. Our approach [1] is based on a complete process including patient specific measurements, automatic geometry modeling based on the measurements, meshing and simulation of the Navier-Stokes equations. From a patient specific ultra sound measurement to the geometric model of the heart In our approach the blood in the LV is driven by the prescribed movement of the inner heart wall. The model geometry of the LV is based on ultrasound measurements of the position of the inner wall at three different levels at twelve specific time points during the cardiac cycle, as the movement of the heart wall during one heart cycle which lasts about one second is not uniform. From these measured points surface meshes of the chamber are constructed for each sample time. We then build a three dimensional mesh of tetrahedrons at the initial time, apply Hermite interpolation to allocate the position of all the boundary nodes at every time step and use a Laplace mesh smoothing algorithm to deform the mesh to fit the given surface meshes during the cardiac cycle. Results Quantities which are interesting for medical examination and can as well be used for the evaluation of the model are the mean velocity at the outflow and inflow and the pressure. Remember that the mean velocity at the inflow is mainly given by the prescribed inflow profile and that the calculation of the discrete pressure corresponds to the Poisson equation. Figure 3a: Mean velocity at the inflow calculated during the simulation Figure 3b: Velocity at the inflow derived with help of ultrasound measurements Figure 4a: Mean velocity at the outflow calculated during the simulation Figure 4b: Velocity at the outflow derived with help of ultrasound measurements Figure 1: Specific points of time when the samples are taken for one heart cycle Mathematical Model We simulate the blood flow with the incompressible Navier-Stokes equations. To take into account the movement of the mesh the Arbitrary Lagrangian-Eulerian description [3] is used where u denotes the velocity, p the pressure and w the mesh velocity: u̇u−w ∇ u− u∇ p= f METHODOLOGY We simulate the blood flow with the incompressible Navier-Stokes equations using a weighted standard Galerkin/streamline diffusion method. CONCLUSION With our approach we are able to calculate the blood flow in the left ventricle without the complex mechanical and electrical model of the heart wall. The results for the validation of the model are satisfying and encouraging to complete the model with properties as the aortic valves and a Windkesselmodel. Different collaborations have started to further derive interesting properties from the existing data and to make the model more applicable for various fields. ∇⋅u=0 By defining different boundary conditions we divide the cardiac cycle into the basic stages of diastole, systole, isovolumetric relaxation and isovolumetric contraction. A no-slip boundary condition on the wall and closed valves is applied and the pressure is prescribed by a standard curve to model the inflow through the mitral valve and the outflow through the aortic valve. A flat inflow profile for the velocity is given as well. −3 kg =2.7⋅10 Considering the size of the LV, the dynamic viscosity and density m⋅s kg =1.06⋅103 3 the Reynolds number is about 20'000. m Figure 5: Mean pressure in the LV Software The simulations are done in DOLFIN, a differential equation problem solving environment, and UNICORN, a unified continuum mechanics solver, which are developed as a part of the software project FEniCS and where we are an active contributor. See https://launchpad.net/fenics and https://launchpad.net/unicorn Current and future work Our present efforts as shown in Fig. 5 are focused on parallelizing our code for making it possible to run the calculation on multi-cores [4]. Further, as the next major step in our project we would like to extend and improve our model by including the aortic valves. To secure, develop and make our model applicable the collaboration, discussions, dialogue and interaction with other universities, institutes and research areas are carried out and striven for. Figure 2:Snapshot of the pressure and velocity during systole CONTACT INFORMATION Jeannette Spühler Johan Hoffman Johan Jansson Computational Technology Laboratory School of Computer Science and Communication Royal Institute of Technology KTH (spuhler, jhoffman, jjan)@csc.kth.se Ulf Gustafsson Department of Public Health and Clinical Medicine Umeå University [email protected] Mikael Brommé School of Technology and Health Royal Institute of Technology KTH and ECMO Center, Karolinska University Hospital Karolinska Institutet [email protected] Mats Larsson Per Vesterlund Computational Mathematics Laboratory Department of Mathematics and Mathematical Statistics Umeå University (mats.larson,per.vesterlund)@math.umu.se Jonas Forslund Eva-Lotta Sallnäs Pysander Human-Computer Interaction Royal Institute of Technology KTH (evalotta, [email protected]) Alex Olwal School of Computer Science and Communication Royal Institute of Technology KTH [email protected]) Galerkin finite element method with least-squares stabilization We apply the G2 cG(1)cG(1) Finite Element Method with piecewise continuous linear solution in time and space for solving the governing equation [2]. For the standard FEM formulation of the model we only have stability of the discrete solution U but not of its spatial derivatives. This means the solution can be oscillatory, causing inefficiency by introducing unnecessary error. We therefore choose a weighted least-squares stabilized Galerkin method. The weak solution to be solved on the computational mesh is thus as following: Find the discrete solution U =[U , P ] such that n U −U n−1 −1 n n k U −W n−1 n n ⋅∇ U , v 2 U , v n n n n n n − P , ∇ v ∇⋅U , qSD U , P ; v , q= f , v ∀ v , q∈V 0 ×Z and the stabilization term is defined as: n n n SD U , P ; v , q=1 U −W n−1 n n n ⋅∇ U ∇ P − f ,U −W n−1 n n n−1 where [v , q ] are test functions and U =1/ 2 U U . n ⋅∇ v∇ q2 ∇⋅U , ∇⋅v Figure 5: Simulation run on several cores Figure 6: The interactive HEART demonstrator in collaboration with CSC/HCI within SimVisInt [1] M. Aechtner, Arbitrary Lagrangian-Eulerian Finite Element Modelling of the Human Heart}, Master thesis, {TRITA-CSC-E} 2009:022, 2009. [2] J. Hoffman and C. Johnson, Computational Turbulent Incompressible Flow: Applied Mathematics Body and Soul Vol 4, Springer-Verlag Publishing, 2006. [3] J. Donea, A. Huerta, J.-Ph. Ponthot and A. Rodríguez-Ferran Arbitrary Lagrangian-Eulerian Methods, p. 413-437, Wiley, 2004. [4] N. Jansson, J. Hoffman and J. Jansson, Parallel Adaptive FEM CFD, KTH-CTL-4008, http://www.publ.kth.se/trita/ctl-4/008/, 2010.