A Workflow for Computational Fluid Dynamics Simulations using Patient-Specific Aortic Models

Transcription

1 2.8.8 A Workflow for Computational Fluid Dynamics Simulations using Patient-Specific Aortic Models D. Hazer 1,2, R. Unterhinninghofen 2, M. Kostrzewa 1, H.U. Kauczor 3, R. Dillmann 2, G.M. Richter 1 1) University Hospital, Heidelberg - Department of Radiodiagnostics 2) University of Karlsruhe - Institute of Computer Science and Engineering 3) German Cancer Research Centre, Heidelberg - Department of Radiology Summary Purpose: In this paper, we developed a workflow to simulate the blood flow within patient-specific models of the aorta. The workflow allows Computational Fluid Dynamics (CFD) simulations based on the Finite Volume Methods (FVM) to compute velocity profiles and pressure distributions in the aorta. Methods: The workflow includes the segmentation of patient-specific tomographs and the generation of 3D geometrical models. Then, high-quality meshes required for running and converging the numerical simulations are generated. Realistic boundary conditions are set based on MR flow measurements. Finally, the simulation and analysis of flow patterns and pressure distribution in the aorta is performed using a CFD software (Fluent). Results: We applied the workflow to an abdominal aortic model obtained from a CT scan. Careful processing of all steps was performed in order to generate accurate simulations. The computational results showed mathematically stable solutions characterized by a fast convergence, few numbers of iterations and small residuals. Conclusion: In conclusion, our workflow allows a CFD analysis based on patient-specific data to simulate, describe and analyse the hemodynamics of blood flow within aortic models. It represents a milestone toward an optimal workflow that can be implemented clinically. This may offer a method to aid the diagnosis of aortic aneurysms and their risk of rupture, as well as to plan and control the efficiency of endovascular treatments. Keywords CFD, FVM, simulations, blood flow, patient-specific aortic models 1

2 0. Introduction Aortic aneurysms are one of the most dangerous cardiovascular diseases. They are associated with a dilatation of the aortic wall and tend to occur at sites with pathological hemodynamic conditions, such as high blood pressure acting on a weak spot on the vessel. The aneurysm may at some stage rupture, causing internal bleeding and leading to death unless treated rapidly. Endovascular stentgrafts are mesh tubes designed to be inserted into the vessel to prevent an aneurysm from exploding. Several studies have demonstrated that minimal invasive surgery using endovascular procedures may be advantageous over conventional surgery. However, an efficient minimal invasive treatment requires the knowlege of physical parameters of the blood flow. Thus, it is of primary importance to be able to non-invasively identify parameters that individually contribute in the development, growth and risk of rupture of the aneurysm. The current criterion to predict the risk of rupture is associated with the diameter of the aneurysm. However, recent research shows that even small aneuryms are subject to rupture, whereas larger ones sometimes may not. This is an indication that aortic aneurysms are still not fully understood, and that further development of better patient-specific models is necessary. Computational Fluid Dynamics (CFD) provides a tool to describe the hemodynamics of the blood flow within the arteries. It helps to understand the role of flow and pressure distributions by characterizing regions of weakened vessels. Our aim is to develop a modelling process to facilitate a clinical implementation of such CFD methods. An important feature of the workflow is to be able to simulate and reproduce the patient-specific flow field in the aorta in an accurate way. This is the main focus of the present paper. Building the elements of the process-chain is described in the following sections. 1. Material and Methods The Computational Fluid Dynamics simulations consist of six stages and is represented in Fig. 1. CT/ MRI data Dicom Boundary Surface (.stl) Segmentation Geometrical Model 3D Surface Mesh Model High-Quality Boundary Mesh CFD Simulations Boundary and Initial Conditions 3D Volume Mesh Model Hemodynamics Physical Model/ Solver MR flow measurements Fig. 1 Workflow for hemodynamic computations starting with the segmentation of the patient data. 2

3 1.1 Segmentation The patient-specific geometry used in this study is based on morphological data derived from a computed tomography (CT) scan. A total of 210 slices were acquired to reconstruct the abdominal aortic model from about 50mm above the renal arteries to the aortic bifurcation. The 2D image size was 128mm x 128mm and the slice thickness was 1mm, resulting in a voxel size of 0.25 x 0.25 x 1 mm 3 at a resolution of 512 x 512 pixels. For patient-specific hemodynamic simulations, great care has to be taken when generating the 3D description of the geometry of interest. The CT data were therefore accurately segmented, extracting the region of interest represented by the inner wall of the aortic model. A first order approximation of the boundary surface was first generated from the 3D scan using the region growing approach. Then, a manual segmentation of the 2D slices in axial direction was required to improve the 3D segmentation by removing pixels outside, or filling in pixels inside the lumen. The segmentation was performed using a commercial software package (Aquarius, TeraRecon Inc.). 1.2 Geometrical Model The processing of the segmented patient data and the generation of the 3D geometrical model is performed using MediFrame [1], our in-house (*) developed framework for medical applications. MediFrame is a software platform for diagnostics and surgical planning, and offers a tool for modelling, simulating and visualizing medical data. Using the integrated DicomCenter component, the segmented Dicom slices are imported into MediFrame and a 3D image model is created by combining all 2D slices into one image file. Then, the boundary surface of the model is constructed, using a surface triangulation based on the marching cubes method [2]. The algorithm generates isosurfaces from the volume based on a minimum/maximum specified scalar range. Next, the obtained geometry is idealized using Laplacian smoothing to adjust the vertices distribution and the cells shape. Then a surface-cleaning filter is applied to merge duplicate points within a specific tolerance as well as to remove unused points. Further, since the blood must flow orthogonal to the inlet and outlets faces, a clip filter is applied using the software Paraview (Kitware Inc.), in order to cut and readjust the boundary surfaces such that their normal is parallel to the flow direction. Finally, using MediFrame, the surface is exported into an.stl file (stereolithography format) consisting of 3D triangles. 1.3 Surface Mesh Model The obtained.stl surface doesn t yet guarantee the stability of the CFD simulations. High-quality three dimensional surface mesh models have therefore to be created using the mesh generator Gambit. Gambit is the pre-processor of the FLUENT solver. It provides a variety of mesh control functions which allow the generation of high quality controlled meshes. The process of generating the 3D patient-specific boundary surface of the aortic model can be described by the following steps: First the.stl created in the previous step is imported and the old mesh is reset. In order to describe as many surface elements as possible, a very fine mesh that better represents the real surface on the wall is then created and exported as a.msh file. Within a new session, this fine mesh is read, removed and a size function accounting for the curvature at the wall surface is created. This allows, based on a curvature angle and a growth factor, to produce fine cells at sites with high curvatures and larger ones at sites were the wall is rather flat. Finally, based on the defined curvature size function, the wall is meshed using Tri-elements. 1.4 Volume Mesh Model Blood simulations based on the Finite Volume Methods require the generation of volume meshes to represent the blood elements. The computation of the parameters then occurs on those discrete elements before it is integrated over the whole volume domain. In order to create the 3D finite meshes, the volume model represented by the closed surface of the fluid domain must be created first. This is done within Gambit, by stitching the inlet, outlets and the wall faces together. The surfaces in turn are created from the edges defining their boundaries. In order to control the distribution of the fluid cell size near the wall, a Boundary Layers (BL) mesh control function must be created. The boundary layers are defined such that the fluid domain contains * Institute of Computer Science and Engineering, University of Karlsruhe 3

4 small cells at locations close to the wall, and which increase in direction toward the inside of the fluid. They also prevent tetrahedrons from standing with one tip on the wall, which badly affects the computation of wall fluxes. Further, in order to control the propagation of the mesh from the surfaces into the volume, a second control function, a meshed Size Function (msf), should be defined from the wall into the fluid domain. Finally, based on the control functions described above, the volume mesh model is generated. 1.5 Boundary and Initial Conditions In order to solve the system of partial differential equations governing the blood flow, a set of boundary and initial conditions needs to be defined. For realistic simulations and to keep the model patientspecific, physiological data based on MR flow measurements were used to set the boundary conditions at the inlet of the aortic model. An unsteady, homogeneous flow profile is therefore measured above the renal arteries. The profile is then idealized, by smoothing the curve in order to get rid of undesirable oscillations in the simulations. The boundary conditions at the outlets are determined in terms of outflow rates based on data taken from [3]. At the wall, the no-slip boundary condition is defined, resulting in a zero-velocity at the wall. For the initial conditions at the end of last diastole, the whole model is initialized from the velocity inlet profile at time t= 0s. 1.6 Simulations Three dimensional computations have been carried out to describe velocity fields and pressure distributions along the aortic wall and within the blood domain at any instant of time. The hemodynamics of the blood through the aortic model are simulated using the CFD software FLUENT. A numerical code integrated in FLUENT is based on a Finite Volume Method to discretize the Navier- Stokes (NS) equations. These include the mass, momentum and energy conservation equations. In the lumen region, they represent a mathematical relationship between the pressure p, the velocity v in the flow direction, the mass density ρ and the viscosity µ of the blood. An important aspect in modelling the blood flow is to accurately describe the physical nature of the blood as a fluid. In this paper, we assume that the shear rate within the blood is greater than 100s -1 [4], and therefore we consider the blood as a homogeneous Newtonian fluid with a constant dynamic viscosity of Nsm -2. With a Reynolds number of Re 2000, the flow is assumed to be laminar, as well as incompressible with a constant density of 1050 kgm -3. The blood vessel was modelled as rigid. For an incompressible and Newtonian fluid, the Navier-Stokes and continuity equations are: r v r r r r T ρ + v. v = p + ( ( v + ( v) ) + f t. μ ) Navier-Stokes Equations. v r = 0 Continuity Equation In our model, the segregated (implicit) solver was used and the field variables were interpolated to the faces of the control volumes using a second-order scheme. For the pressure-velocity coupling, the solver employed the Pressure Implicit Splitting of Operators (PISO) algorithm, useful for unsteady problems, to solve the 3D Navier-Stokes equations. 2. Results and Discussion 2.1 Segmentation The results of the segmentation are shown in Fig. 2. The output of the 3D segmentation (a) based on the classical region growing technique showed many artifacts, inaccurate boundary contours and inhomogeneous lumen domain. Therefore, a manual segmentation of the axial slices (b) was necessary for correction and improvement of the 3D segmentation. The final segmentation showed good results (c) but was time-consuming. 4

5 (a) (b) (c) Fig. 2 Results of the a) 3D segmentation, b) manual segmentation and c) final segmentation. In order to facilitate a clinical implementation of this step, further developments in the segmentation techniques and the usage of more sophisticated algorithms are needed to improve the results in terms of accuracy and speed. 2.2 Geometrical Model Fig. 3 shows the geometrical model obtained in MediFrame. The triangulation of the set of points representing the image file is shown in (a). The smoothing and cleaning filter effects (b) show a clear improvement of the surface quality. Finally the clipped.stl surface, which allows the blood to flow in normal direction to the inlet and outlets faces, is shown in (c) and is a necessary prerequisite for blood simulations free of backflows. (a) (b) (c) Fig. 3 Results of the (a) created geometrical model, (b) effect of cleaning and smoothing the surface, (c) clipping the boundaries normal to the flow direction. 5

6 2.3 Surface Mesh Model The results of the wall surface mesh based on the curvature Size Function (csf) are shown in Fig. 4. The csf used in this model was defined with a 10 curvature angle, a 20% growth factor, and a minimum and maximum cell sizes of 0.5 mm and 5 mm respectively. A high quality surface mesh must be free of high-skewness cells (skewness > 0.97). The obtained wall mesh includes a total of triangle cells with a good quality range varying between 4e-8 and Fig. 4 The wall surface mesh based on a Size Function to control the curvature at the wall. 2.4 Volume Mesh Model The results of the volume mesh based on the Boundary Layers (BL) and the meshed Size Function (msf) are shown in Fig. 5. Note that the Boundary Layers represent a topology that allows for an accurate computation of parameters near the wall, such as velocity vectors, from which the gradients can be derived to define the shear stresses along the wall. The parameters used to define the BL function were: number of layers= 3, first row size= 0.3 mm and growth rate= 1.2, resulting in a BL domain of mm thickness. As for the msf, it was defined by a maximum size of 5 mm and a growth factor of 10% toward the inside of the fluid domain. The volume mesh model described here consists of a total of fluid volume cells: tetrahedrons at the inside, wedges resulting from the BL and 1488 hexahedrons resulting from two extra volumes appended at two of the outlets to reduce backflow effects. The obtained volume mesh showed high quality cells required for the CFD based blood flow simulations. 6

7 Fig. 5 The volume mesh model consisting of tetrahedral, wedge shaped and hexahedral fine elements and a cut inside the volume show the high-quality of the cells (blue represents the highest quality range). 2.5 Boundary and Initial Conditions The MR based homogeneous and unsteady profile used at the inlet of the model is shown in Fig. 6. It was measured in a plane above the renal arteries, smoothed and then transformed to describe the mean velocity of the flow within one cardiac cycle. The input flow showed its peak at t= 0.18s and became zero at t= 0.44s. For the boundary conditions at the outlets, the calculated scaled outflow rates were: celiac artery 12%, superior mesenteric 24%, renal arteries 19% each and iliac arteries 13% each. The defined no-slip boundary condition resulted in a zero velocity at the wall. The initialization from the velocity inlet profile resulted in a zero velocity at t =0s within the whole model. Fig. 6 Mean Velocity Profile at the inlet boundary of the model along 1 cardiac cycle (T= 0.85s). 7

8 2.6 Simulations Simulation results were obtained using the FVM software FLUENT 6.2. The CFD computations were performed on an 6.4 GHz 8 GB Ram PC and required approximately 20 hours. A whole cardiac cycle of period T= 0.85s was modelled, using 2000 time steps, with 1 step size =0.425ms. The maximum number of iterations per time step was set to 20. Near the systole, 5 to 7 iterations were required to converge the solution, whereas near the diastole, due to backflows effects, 19 to 20 iterations were needed. Fig. 7 shows the blood velocity vectors along a sagittal cut inside the celiac artery, at t=0.13s (left) and t=0.5s (right). It is characterized by the formation of recirculation zones due to deformation sites on the wall. At early systole, few of the vortices start to develop, and during the diastole the recirculation zones grow to large 3D vortices with reversed flow at the outlet boundary. The direction of the vector represents the direction of the blood flow and the colour refers to the magnitude of the velocity in m/s. Fig. 7 Velocity vectors inside the celiac artery at early systole (left) and during the diaslole (right). Fig. 8 shows the distribution of the dynamic pressure (left) and the corresponding velocity magnitude (right) along a sagittal cut within the superior mesenteric artery at peak systole (t= 0.18s). It is characterized by a proportional variation, showing lower/larger values at sites of lower/larger flow velocities respectively. This agrees with the law forms known from the Bernouilli equations. Fig. 8 Dynamic pressure distribution (left) and the corresponding velocity magnitude (right) within the superior mesenteric artery at peak systole. The color scale represents values in Pascal (left) and in m/s (right), increasing from blue to red. 8

9 As for the shear stress distributions along the wall, they were also computed from the resultant velocity gradient and are presented in [5]. 3. Conclusion We developed a workfow for CFD simulations, to model the blood flow in the aorta. We built the process chain required for the computation of the 3D pulsatile hemodynamic parameters of the blood flow. Our process consists of a fine segmentation of the patient data, the creation of an image model, of an.stl surface (stereolithography format) as well as the processing and the filtering of the surface, the generation of controlled surface and volume meshes that allow high-quality and stable simulations, the setting of realistic boundary conditions and finally of the computations using suitable parameters and solvers that allow mathematically stable solutions. 4. Further Development The hemodynamic computed quantities still need to be validated. An experimental setting as well as clinical trials will be developed to allow the validation. Indeed, a physical verification of the computations, based on the generation of results that are mesh-independent, will also be performed. Furthermore, a validation study of the workflow with a large dataset and complex aneurysm geometries before and after endovascular treatment is planned. We will extend our blood model to a non-newtonian fluid, considering the stresses within the blood as nonlinearly dependent on the deformation rate. A structural model will be developed to allow strain and stress analysis, as well as mechanical computations of the deformation along the vessel wall. And finally a coupled system between the blood model and the structural model will allow a patient-specific Fluid-Structure Interaction in the aorta for realistic simulations. Acknowledgment Grateful and deepest thanks to R. Kröger from Fluent Germany for the extensive contribution and the precious advices regarding the simulations. Thanks also to H. von Tengg-Kobligk, K. Ruf and J. Ziko for the help with the clinical data. The present study was conducted within the setting of the Research training group 1126: Intelligent Surgery - Development of new computer-based methods for the future workplace in surgery founded by the German Research Foundation (DFG). References [1] Seifert, S., "MEDIFRAME an extendable software framework for medical applications, Surgetica, Grenoble, France, [2] Lorensen W., "Marching Cubes: A High Resolution 3D Surface Reconstruction Algorithm, Computer Graphics, Vol.21, Number 4, [3] W. Bleifeld, C.Kramer, K.Meyer-Hartwig, Klinische Physiologie, Verlag Gerhard Witzstrock, Baden-Baden, Köln, New York, Bd. II, [4] Amornsamankul, S., "Effect of Non-Newtonian Behaviour of Blood on Pulsatile Flows in Stenotic Arteries. International Journal of Biomedical Sciences, Vol.1, Number 1, [5] Hazer, D., Wall Shear Stress simulations in a CT based human abdominal aortic model, 5. Jahrestagung der Deutschen Gesellschaft für Computer- und Roboterassistierte Chirurgie. CURAC Okt