A 1D lumped-parameter/3d CFD approach for pressure drop in the Aortic Coarctation

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1 A 1D lumped-parameter/3d CFD approach for pressure drop in the Aortic Coarctation Eduardo Soudah, Maurizio Bordone, Pooyan Davdan and Riccardo Rossi CIMNE Biomedical Engineering Department. Centre Internacional de Mètodes Numèrics en Enginyeria. Edifici C1, Campus Nord-UPC, Gran Capità, s/n, Barcelona. Abstract. Aortic Coarctation is a congenital constriction of the aorta that increases blood pressure above the constriction and hinders the flow below it. Based on a 3D surface mesh of a moderate thoracic coarctation, a high quality volume mesh is created using an optimal tetrahedral aspect ratio for whole domain. In order to quantify the severity of this constriction, a coupled 1D lumped-parameter/3d CFD approach is used to calculate the pressure drop through the coarctation. The CFD computation is performed assuming that the arterial wall is rigid and the blood is considered a homogeneous Newtonian fluid with density r = gr/mm3 and a dynamic viscosity m = gr/mm/sec in laminar flow. The boundary conditions of the 3D model (inlet and outlet conditions) have been calculated using a 1D model. Parallelization procedures will be used in order to increase the performance of the CFD calculations. Keywords: Aortic coarctation, pressure drop, lumped model, 1D/3D coupled method. 1 Introduction Coarctation of the Aorta (CoA) is one of the most common congenital heart defects, and in the western world every year tens thousands of new cases are registered. In this pathology, the aorta narrows in the area where the ductus arteriosus inserts and depends where is this narrow the CoA is classified as: 1. Preductal coarctation if the narrowing is proximal to the ductus arteriosus. This condition could be life-threatening is the constriction is severe. This type of deformation is due to an intracardiac anomaly during fetal life. The blood flow is not regular on the left side of the heart producing a hypoplastic development of the aorta. 2. Ductal coarctation if the narrowing occurs at the insertion of the ductus arteriosus. 3. Postductal coarctation if the narrowing is distal to the insertion of the ductus arteriosus. This deformation could lead to an impaired blood flow in the lower part of the body. This condition is most common in adult beings. It can produce hypertension in the upper side of the body and insufficient peaks in the lower part. The deformation is generated during the fetal life as an extension of a muscular artery (ductus arteriosus) into an elastic artery (aorta). adfa, p. 1, Springer-Verlag Berlin Heidelberg 2011

2 Many clinical, pharmacological tests and analysis could be performed in order to study and treat of the CoA in the best way. The goal standard of the CoA treatment is the gradient pressure generated by the constriction in the ductus arteriosus. According to the pressure drop the clinicians determinate the best treatment for this patient. This pressure drop changes when the blood flow is modified by stress or exercises and is hard to be reproduced in a clinical way. Medical Imaging techniques and Mathematical Methods and Computational Fluid Dynamics (CFD) tools could reproduce a lot of different conditions if the problem is well known. 2 Methods 2.1 Geometry and mesh The geometry of study is obtained by an 8-year old female patient. It is possible to note a moderate coarctation of about 65% of area reduction. Image data come from Gadolinium-enhanced MR angiography (MRA) with the subject in the supine position. The scanner is a Signa 1.5-T GE that has a time of images acquisition of approximately 20 seconds, in which the patient holds her breath. The model includes the ascending aorta, aortic arch, descending aorta, right and left subclavia and finally right and left carotid in.stl format (surface mesh). To generate the 3D volume mesh GiD software [1] has been used. GiD is a CAD system that features the widely used for mesh generation. For generating a 3D volume mesh from the STL file, we have used an isosurface stuffing procedure. This technique is ideal to generate a volume mesh with quality warranty of the elements, this mean angle and volume bounds are guaranteed. Γ 0 Γ 2 Fig. 1. Representation of the 3D model rendering of stl model in the XZ plane. Detail of surface mesh. Γ 0 : inlet section (Aortic Root), : outlet sections (Supra-aortic arteries) and Γ 2 : outlet section (Descending aorta)

3 This technique increases the quality of the tetrahedral elements for computational simulations preserving the original shape and dimensions of original model. Using this method we have obtained a smooth element and an aspect radio for whole of the meshes upper than 0.9 (ideal ratio=1 for an equilateral triangle). This algorithm generates tetrahedral form a small set of precomputed stencils, and the boundary mesh is guaranteed to be a geometrically and topologically accurate approximation of the isosurface[1,2]. The result of the iso-stuffing procedure is a volume mesh of 6,423,037 tetrahedral elements and 356,835 triangular elements with 1,167,825 nodes. The original surface mesh has 114,514 triangle elements and 57,259 nodes. Figure 1 shows the rendering of.stl model and 3D volume mesh D model geometry. A 1D model replica of the 8-year old female patient has been created according to the model developed in [5]. For the following arteries: ascending aorta, aortic arch, innominate artery; left common carotid artery, left subclavia artery and descending aorta, the radius was measured directly from the 3D geometry. A 1D segment (thoracic coarctation artery) was created to define the aorta coarctation with a 65% of area reduction. For the rest of the arteries, the radius was scaled using the f R factor and the length of the arteries was scaled using f L factor. Arterial wall thickness and Young modulus values were taken directly from [5]. f R = R 8 year Aorta f [5] l = l Aorta R Aorta 8 year [5] (1) l Aorta 2.3 Boundary Conditions: inlet condition Realistic boundary conditions are essential for a good performance and a realistic result for the simulation. For this, blood flow data was acquired using 2D cardiacgated, respiratory compensated phase-contrast (PC) cine sequence with through-plane encoding. Figure 2 shows the inlet flow in mm 3 /sec. Through the flow profile measurement in the aortic root (inlet section Γ 0 ) the mean velocity describes in figure 2 is calculated. The inlet velocity profile is prescribed as uniform and flat in the inlet surface Γ 0. Fig. 2. Input data: aortic root flow profile.

4 During the acquisition procedure the patient breathed freely and each scan lasted approximately 3 min. The flow data is the results of a PC-MRI sequence encoding 20 phases over the cardiac cycle. Heartbeat has a frequency of 86 beats per minute (T=0.7 sec) and the cardiac output was 3,245 L/min. 2.4 Boundary Conditions: Pressure profile Through a measurement via sphygmometer was possible to register the upper-body systolic and diastolic pressure and values were of 115 and 65 mmhg. To determinate a realistic pressure profile in the outlet (, Γ 2 ) of the 3D model, a 1D lumped arterial model was used [4, 5]. The 1D outflow boundary conditions on (, Γ 2 ) were modeled using the relation: Q 1D = P 1D P out R p (2) where Q 1D and P 1D are the flow rate and pressure at the outlet of the 1-D terminal branch, and P out and R p are the outlet pressure and the peripheral resistance to the flow determined by [5]. The 1D model was fitted according the pressure values of the 8- year old female patient based on original model and calibration tests [6]. 2.5 CFD specifications The CFD computation is performed assuming that the arterial wall is rigid and the blood is considered an homogeneous Newtonian fluid with density ρ = 1000 k g/m and a dynamic viscosity µ = kg/(sec m) in laminar flow. A coupled 1D lumpedparameter/3d CFD approach was performed using KRATOS [7]. KRATOS is a fluid dynamics and multi-physics simulation environment based on the stabilized Finite Element Method that solved the 3D Navier Stokes equations. (3) that we assume to hold in the 3D domain where u is the velocity vector, f is the fluid forces and p is the pressure. For the 1D model, the model has been deduced using the same equations by making the following assumptions: axial symmetry, radial displacements, and constant pressure on each section, no body forces and dominance of axial velocity. Also, in the absence of branching, a short section of an artery may be considered a cylindrical compliant tube. In order to close the system equations (Eq.3) for the 1D, we provide a relation for the pressure completed with a pressure area (Eq.4) relation previously. It assumes a thin, homogeneous and elastic arterial wall and it takes the form:

5 (4) where A o and h o are the sectional area and wall thickness at the reference state (P o, U o ), with P o and U o assumed to be zero, E is the Young s modulus and ξ is the Poisson s ratio, typically taken to be ξ=0.5, since biological tissue is practically incompressible. The parameter β is related to the speed of pulse wave propagation, c (Eq.5), through Formaggia[5], 3 2 β c = A (5) 3ρA o 2.6 1D-3D Coupling A coupled 1D/3D CFD approach employed in this work is shown in figure 3. At every time step t n, we compute the velocity and pressure using the 1D approach over whole domain Ω 1D, and the variables over the interface sections (, Γ 2 ) of the Ω 1D - Ω 3D are determined. Then 3D problem is solved in Ω 3D using the boundary conditions obtained in, Γ 2 sections and the pressure drop is calculated. The variables computed in the interface sections (, Γ 2 ) are updated according to the 1D-3D problem solution to account the effect of the vessel elasticity. For the next time step t n+1, the process is repeated until the final time simulation is reached. Γ 0 1D -3D coupling Ω 3D Γ 2 Γ 0 Ω 1D Γ 0 : Aortic Inlet : supra-aortic outlets Γ 2 : descending aorta outlet Ω 1D: 1D domain Ω 3D: 3D domain Γ 2 Fig. 3. 1D-3D coupled approach schematics. Variables in the interface surfaces (*) are modified to account for vessel elasticity effects. On the left side, the 1D model of the 8-year old female patient. (551 elements and 586 nodes). To perform the simulation for the 1D and 3D approaches, we set the computational time step to sec for a total of 1400 computational step. The total time of the

6 simulation was of 1.4 sec in order to perform 2 complete heart cycles. We start the simulation with 50 initial steps in order to stabilize the initial condition solution. For the 1D approach second-order Taylor-Galerkin scheme and an explicit four-order Runge-Kutta time-integration scheme have been implemented. This scheme is appropriate for this kind of problem because of it can propagate waves of different frequencies without suffering from excessive dispersion and diffusion errors. For the 3D approach, time integration method chosen was a Backward Euler, using a Bi Conjugate Gradient Non-symmetric solver in order to accelerate the calculation time performance [8]. We used a pressure stabilization of 4º order and automatic velocity advection stabilization. Turbulence model was not included, but we used a laminar formulation. Parallelization procedures have been used in order to increase the performance of the CFD calculations [9]. Figure 4 shows the final pressure profiles imposed at the interface surfaces (*). Fig. 4. On the left pressure profile imposed at and on the right pressure profile imposed at Γ 2 3 Results 3.1 Branches flow distribution analysis To obtain the flow distributions in all the branches of the domain, we have calculated the flow values along the cycle time for the four outlets(, Γ 2 ) of the system related to the inlet flow. Table 1 shows the percentage of ascending aortic flow through the outlet sections of the aortic model as measured via PC-MRI and calculated via CFD, and the error between the CFD and the PC-MRI measurement. Section QIA QLCCA QLSA QDAo % CFD (Aortic Flow) % PC-MRI (Aortic Flow) % Error CFD-PC-MRI Table 1. % of ascending aortic flow through the outlet sections of the aortic model as measured via PC-MRI and calculated via CFD and the error. QIA: flow through innominate artery; QLCCA: flow through left common carotid artery, QLSA: flow through left subclavia artery; QDAo: flow through descending aorta.

7 Figure 5 shows the results of the volumetric flow at the four outlets of the system calculated via CFD. Fig. 5. Out flow: comparison calculated with CFD. QIA: flow through innominate artery; QLCCA: flow through left common carotid artery, QLSA: flow through left subclavia artery; QDAo: flow through descending aorta. 3.2 Pressure analysis To evaluate the pressure gradient, the average value of the pressure for all time steps between at the proximal (P1) and distal section (P2) is calculated: Plane 1 (P1) with a normal of (0.00, 0.27, 0.96) Plane 2 (P2) with a normal of (0.00, 0.14, 0.99) Figure 6 shows the pressure gradient obtained for P1 and P2. The maximum value of the pressure gradient (15,459 mmhg) is reached in the systolic phase. Fig. 6. Pressure drop (P1-P2)

8 4 Discussion The preliminary results show a good agreement with the pathological values; however an improvement of the methodology should be required to obtain a better approximation to the values measured using PC MRI at the outlets. The maximum error reached is 6% in the flow of the descending aorta; meanwhile the error in the supra-aortic arteries is less than 4%. Future work will be focused on the enhancement of the 1D lumped-3d coupling procedure, using physiological flow inlet profiles. These improvements, in conjunction with a fine adjustment of the lumped 1D model parameters, seem necessary to obtain a more accurate flow solution in the aortic arch and in the narrowing of the aorta. This will lead to a better computation of the pressure drop and the volumetric flow through artery outlets. 5 Bibliography 1. GiD. The personal pre and postprocessor, CIMNE (2011). 2. Franois Labelle and Jonathan Richard Shewchuk. Isosurface Stuffing: Fast Tetrahedral Meshes with Good Dihedral Angles. ACM Transactions on Graphics 26(3), August Special issue on Proceedings of SIGGRAPH E.Soudah, J. Pennecot, J.S. Pérez, M. Bordone and Eugenio Oñate. Medical-GiD: From Medical Images to Simualtion, 4D MRI Flow Analysis.Computational vision and Medical Image Processing, VIP Image 2009, pp 9-14, CRC Press Porto E.Soudah, E.Oñate. Validation of the one-dimensional numerical model in the ascendingdescending aorta with real flow profile. 8th. World Congress on Computational Mechanics (WCCM8). 5th. European Congress con Computational Methods in Applied Sciences and Engineering (ECCOMAS 2008) 30 June - 4 July 2008, Lido Island, Venice,Italy. 5. Luca Formaggia, Daniele Lamponi, Alfio Quarteroni. One dimensional models for blood flow in arteries. Journal of Engineering Mathematics, 2003, Volume 47, Numbers 3-4, Pages Koen S. Matthysa, Jordi Alastrueya,b, Joaquim Peiro, Ashraf W. Khirc, Patrick Segersd, Pascal R. Verdonckd, Kim H. Parkerb, Spencer J. Sherwina. Pulse wave propagation in a model human arterial network: Assessment of 1-D numerical simulations against in vitro measurements. J Biomech. 2007; 40(15): Dadvand, P.; Rossi, R.; Oñate, E.An object-oriented environment for developing finite element codes for multi-disciplinary applications. Archives of computational methods in engineering.17-3, pp /201. ISSN R. Barrett et al., Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods, 2nd Edition, Philadelphia, PA: SIAM, Dadvand, P.; Rossi, R.; Gil, Marisa; Martorell, X.; Cotela, J.; Juanpere, E.; Idelsohn, S.R.; Oñate,E. Migration of a generic multi-physics framework to HPC environments. Computers and fluids.pp. (to be published 2012)