BloodSim: Coupled Solid-Fluid Simulation of Cardiovascular Systems D.R.Hose 1, P.V.Lawford, A.J.Narracott, J.M.T.Penrose 2, C.J.Staples, I.P.Jones 1 University of Sheffield, England, 2 AEA Technology, England Abstract P. Nuesser 3, L.Choub 3,D.Quinn 4 3 Berlin Heart AG, Germany, 4 AVE/Medtronic, Ireland Cardiovascular systems typically feature strong interaction between solid and fluid phases. Software has been developed to couple ANSYS to CFX, via an external interface, to produce the capacity for simulation of systems of this type. The interface software passes pressure data from the fluid phase to the solid phase, which returns boundary position and velocity information. This process iterates until consistent results are obtained across the wetted boundary. There is facility for relaxation of pressure, displacement or both to assist convergence of the process. Automatic remeshing of the fluid domain is performed when the distorting mesh violates quality constraints. This feature is necessary for systems like heart valves that feature severe topological distortion. Results are presented for a series of applications including comparison with Womersley's analytical solutions for pulse transmission in an elastic-walled artery, analysis of the fluid and structural characteristics of a stented artery and simulations of the operation of an extracorporeal heart assist device. Introduction There are several approaches to the solution of coupled solid-fluid systems. It is likely that the most stable approach is the development of a fully coupled solver, with degrees of freedom for both solid and fluid phases present in the same analysis system. The primary drawback of this approach in the context of the cardiovascular system is that it is unlikely to be the most versatile in terms of its functionality. The advanced features required to describe many of the phenomena associated with cardiovascular systems are unlikely to be resident within a single suite of programmes. On the structures side the tissues are primarily nonlinear and anisotropic, and appropriate functions for the representation of their stress/strain characteristics are featured only in advanced structural solvers. On the fluids side, the rheology of blood is complex: it is certainly non-newtonian at some shear rates, and indeed two-phase plasma/erythrocyte representations might be most appropriate. There is also the likelihood of turbulence in some cardiovascular systems, including distal to aortic heart valve prostheses and at some arterial stenoses. For real application to benefit our understanding of haemodynamic processes including haemostasis and thrombosis, multiple species and chemical reaction representation are important. In terms of postprocessing, transient particle tracking is an important feature for the assessment of haemolysis and its contribution to thrombotic processes. There have been a number of attempts to write a generic interface for the external coupling of structural and fluids solvers, including one that featured an analysis of a mechanical heart valve as a target application 1. Many investigators have coupled specific codes for a particular application, including for arterial segments 2 and bifurcations 3. In these analyses the procedure has been to couple the solid and fluid domains at a wetted boundary, typically passing surface components of a stress tensor from the fluid domain to the solid and displacement or velocity back from solid to fluid. This process is repeated until the pressure and displacements are converged. The most common approach to the solution is some variant of a Jacobi iteration scheme, and indeed this is the basis of the approach adopted in the current programme. It is often the case, particularly for incompressible fluids, that a relatively small movement of the solid wall causes a large change in the local pressure and so inevitably some form of relaxation is applied to stabilize the system to try to achieve convergence. It has been demonstrated 4 that the relative stability of a Newton scheme might be more appropriate for some problems.
The selection of codes for coupling is based on the requirements from the analysis. Clearly the solids code must include the appropriate tissue materials modeling, together with any necessary contact facilities. The fluids code must feature the facility for moving the mesh as the boundary deforms, and must handle the mesh deformation terms appropriately. The implementation of the mesh movement terms within CFX is based on the methodology described by Hawkins 5. In the study of systems such as heart valve prostheses it is likely that topological changes of the mesh will be required, and so a facility for automatic remeshing and appropriate results interpolation is required. It is unlikely that the most efficient mesh for the problem will be of similar density in the solid and fluid domains (almost always the fluid mesh will be denser) and so a surface load and displacement interpolation facility should be provided by the coupling software. ANSYS and CFX were chosen for the solids and fluids phases respectively. Significant developments were required in the CFX software to produce the required functionality. Procedure A standalone coupling program has been produced to provide a software interface between ANSYS to CFX to enable the passing of results across a common wetted boundary between solid and fluid domains. Details of the coupling strategy and of the interpolation routines are presented by Penrose and Staples 6. The fundamental procedure is to compute the components of the stress tensor at the wetted boundary in the fluid phase, and to pass these to the solid phase a surface loads. In the current implementation only the static pressure component of the surface load is passed. The motion of the solid phase under the imposed fluid pressures is computed, and the displacements of the wetted boundary passed back to the fluid phase. The fluid solve is performed with the consequent boundary motion to produce a new pressure distribution. This procedure is repeated until the process converges, measured by the convergence of a pressure norm, before progression to the next timestep. A displacement norm is also monitored. If the fluid is incompressible, as blood is, then a small change in the boundary motion can lead to a large change in pressure. The coupling program offers the facility to relax the values passed across the boundary towards those generated at the previous solution. It is our experience that quite high values of relaxation (of the order of 90% or more of the previous solution) are required to achieve convergence, and in extreme conditions convergence might not occur at all. In the course of the BloodSim project satisfactory solutions were obtained for eight out of nine test cases. A mesh of the solid domain is set up in ANSYS. Appropriate boundary conditions are prescribed and the nodes on the wetted boundary are assigned to a component. The fluids mesh is set up in CFX and again appropriate boundary conditions prescribed and the wetted boundary is identified. The CFX Build programme features the facility to read an ANSYS cdb file, and so it is possible to define both meshes within ANSYS if this is preferred. Analysis The program of work was undertaken by a European consortium of industrial and academic users. The technology providers were led by AEA Technology (the developers of CFX), supported by IDAC (a European ANSYS distributor), ASD (expert on blood damage and haemodynamic modeling) and PFD (for dissemination and exploitation in chemical plant simulation). The end users were led by Medical Physics at the University of Sheffield, with industrial support and input from the Berlin Heart Institute (Cardiac Assist Device), Angiomed and Medtronic AVE (stent manufacturers), and Autogenics (Tissue Heart Valve Prostheses). Sheffield had a dual role as technology provider and end user. At the start of the program a series of eight test and validation cases were prescribed. The first was a development vehicle, representing pulsatile flow in an elastic-walled vessel, for which linearised analytical solutions are available. Two were simplified representations of coupled systems for which experimental results were available. Four were direct simulations of devices of direct interest to the industrial partners. Simulations were also performed of an in vivo treatment procedure involving the stenting of a coronary arterty. In this paper we present a brief overview of results from the elastic-walled tube, for a tube containing a stent and for the Cardiac Assist Device.
Analysis Results & Discussion A comprehensive description of the results of a series of eight validation tests and a case study is presented in the final technical report for the project 7. Elastic Tube The first test vehicle was the elastic tube. The Moens Korteweg solution for the speed of progression, c 0, of a wave in a fluid of density ρ in an elastic tube of Young's modulus E and wall thickness t is: Et c0 =. 2Rρ This equation is based on assumptions of an incompressible inviscid fluid and linear elastic displacement of the tube. A modification by Womersley 8 to introduce viscosity and longitudinal tethering results in a phase difference between pressure and flow waves. The modified wavespeed for a wave of frequency ω is: 1 F 3 / 2 10 2J ( ) c = c0, where 1 i α 1 [ ] 2 10 1 υ 3 / 2 2 0 ( 3 / = 1 F. i J i ) α α α is the dimensionless Womersley parameter, where α 2 =R 2 ωρ/µ (µ is the fluid viscosity), and J 0,J 1 are Bessel functions of order zero and one respectively. Analyses were performed for a vessel of dimensions similar to those of a small artery. One quarter of the vessel was modeled with appropriate symmetry conditions on the boundaries. A transient pressure in the form of a Fourier series was applied to the inlet, with zero pressure specified at the outlet. Results are illustrated in figure 1. The pressure and wall displacement wave travels along the vessel at a finite speed of magnitude consistent with that computed from the Womersley solution. There is significant axial dispersion of the wave as it travels along the vessel. Investigations continue to check the validity of this dispersion. The analytical solution does not recognize any radial variation of pressure (dp/dr =0), wheras in the computational solution there is significant radial variation, particularly close to the moving wall.
Figure 1 - Travelling wave in elastic walled tube: Pressure and radial displacement (Snapshot from Justin's latest avi) Earlier analyses had indicated that the computed wavespeed varied with the square root of Young's modulus over the range 0.5 to 8 MPa, typical of literature values for the tangent modulus of arterial wall at physiological strains. An analysis was also performed with a hyperelastic wall model more consistent with arterial material characteristics and, although no analytical solution was available for comparison, plausible results were obtained. The ready implementation of the wide range of material models available in ANSYS is one of the justifications for the multi-code approach to the solid-fluid coupling problem. Detailed analyses were performed of a segment of the tube with a prescribed sinusoidal pressure at inlet and outlet. The anticipated velocity solution, w, a complex function of radius and time, was computed from the Womersley solutions for the elastic tube. 3 / 2 ( α y i ) ( ) i e 3 / 2 0 α i A * J = 0 ω t ω 1 iωρ J The computed radial velocity profiles and temporal peak velocity profiles are compared with the analytical values in figure 2. There was good agreement during the positive pressure phase between the computed velocity profile at each instant in time and the analytical solution. It was interesting and encouraging to note that the tube buckled during the negative pressure phase, and that significant distortion of the mesh was possible before the solution process failed. This is illustrated in figure 3.
Figure 2 - Comparison between numerical and analytical results for elastic walled tube (Composite of Figures 1.29 through 1.32, Final Deliverable D4.1) Figure 3 - Buckled Tube during external pressure phase (Composite of figures 1.33 and 1.34 from final report)
Stented Artery Balloon angioplasty is a treatment for diseased arteries. An uninflated balloon on a catheter wire is guided along the artery to the site of a partial occlusion, or stenosis, that has been identified clinically as producing an excessive restriction to flow, or an excessive pressure drop. The balloon is inflated to break up the plaques in the wall of the vessel that form the stenosis. There is a significant failure rate in this operation, associated with the relatively short term re-occurrence of the stenosis. Stents are wire scaffold devices introduced into the diseased arteries to attempt to prevent restenosis. Two of the companies involved in the BloodSim program are stent manufacturers, and stenting is an important application of the analysis code. It was considered necessary that the analysis should explicitly model the wires of the stent, because the flow disturbance associated with their intrusion into the fluid domain might promote separation and other fluid dynamic processes associated with the short and long term patency of the device. The modeling of the deployment of a stent is a significant undertaking in its own right 9, and it was not within the remit of the BloodSim program to perform detail modeling of this process. To simplify the model construction process, a metal stent was expanded thermally to expand an artery locally. Contact was defined between stent and artery. The artery tends to 'quilt' through the gaps between the wires of the stent. An illustration of the quilting effect on an idealised stent geometry is presented in figure 4. Typical coupled analysis results are indicated in figure 5. The displacement results are shown relative to the initial state of the tubing after the deployment of the stent. This represents the displacements caused purely by the fluid loading conditions and not by the force of the stent on the tubing. The fluid results show the presence of a linear pressure gradient with undeveloped flow profiles at both inlet and outlet. The contact pressure of the stent wires on the wall of the artery holds the artery at a fixed radius at these locations, because the applied arterial fluid pressure is not sufficient to lift the vessel away from the stent. In between the stent wires the artery wall pulses in synchrony with the fluid pressure. This variation in fluid domain geometry has a significant effect on the local fluid dynamics at the wall of the vessel. Since local wall shear stress and 'fluid age', or residence time, have been implicated in thrombotic and neo-endothelialisation processes it is suggested that the detail assessment of these parameters might represent an important tool in the stent design process. At this point in time the primary value would be for device comparison and incremental evolution because, in common with many other physiological design processes, quantitative data relating absolute magnitudes of physical parameters to clinical consequence are unavailable. Figure 4 - Idealised stent deployed in elastic artery (Composite of figures 7.16 and 7.17
Figure 5 - Radial displacement and fluid pressure along a generator at an instant in time in coupled simulation of idealized stented artery (Composite of 7.20 and 7.21) A model of a real stent geometry was constructed. The applied boundary conditions were a downstream pressure and an upstream flow rate, both functions of time. Results at a snapshot in time are illustrated in figure 6. A test rig was constructed to measure the transient pressure drop across the stented vessel during the application of an appropriate flow waveform. Results for the measured pressure drop over time are compared with those from a coupled analysis of a real stent geometry in figure 7. Gross qualitative agreement was found, but the accurate measurement of transient pressure drop in the experimental system proved difficult. This is illustrated in figure 8, in which total system pressures and measurements of difference and differential pressures are presented. The pressure drop across the stented segment is only a small proportion of system pressure. Clearly it is important in the coupled simulation to apply the appropriate system pressure in order to expand and to stress the vessel in accordance with the cardiovascular pressure waveform, whilst at the same time achieving the right pressure drop to generate the right amount of flow. For an equivalent rigid walled analysis of flow of an incompressible fluid only the pressure differential need be specified. A consistent theme in the application of coupled analysis to real physiological problems is the great care required in the specification of appropriate boundary conditions. Integration with external system models, as described by Formaggia et al 2, is represents a productive area of research for coupled systems. The coupling software developed in the current program supports a simple Windkessel model at both inlet and outlet, so that boundary pressures can be adjusted as a function of flow and flow gradient. This feature has proved invaluable for the simulation of medical devices, including the Berlin Heart, discussed in the next section.
Figure 6 - Stented vessel and results at an instant in time (Composite of 7.27 and 7.29) Figure 7 - Test rig for stented vessel and comparison between measured and simulated pressure drops (Composite of 7.5, 7.28)
Figure 8 - System and differential pressure measurements in stented artery test rig (Composite of figure 7.10 and 7.11) Berlin Heart The elastic artery and the stented vessel feature relatively small displacements. Many medical devices, such as heart valve prostheses, undergo very large displacement in the normal course of operation. Indeed in these systems topological changes to the mesh are required during the course of the simulation. Automatic remeshing and results interpolation are required features for an analysis system. Berlin Heart AG, members of the BloodSim program, have produced an extracorporeal blood pump designed to assist the heart, sometimes as a bridge to transplant. This device features large displacements and coupling of solid and fluid phases. One of the Berlin designs features a blood chamber bounded by two opposed flexible roll membranes, with valves at inlet and outlet. The device fills from the left ventricle of the patient heart and expels the blood under pressure into the aorta. During the fill phase the membranes move apart under the pressure generated by the left ventricle, providing a relatively low resistance to flow and thus relatively low load on the ventricle. During the pump phase the membranes are driven by external pistons to pump blood into the aorta. The valves at inlet and outlet control the direction of flow. Coupled simulation of this device has been undertaken by Berlin Heart AG, supported by Sheffield University. The geometry for both the solid and the fluid mesh models was imported directly from generic CAD models (via the IGES format) into the respective Cfx and Ansys pre-processors. This ensures that consistent mesh models can be obtained directly from the Berlin Heart designers, and that the solid and fluid domains overlap correctly at the solid-fluid coupling interface. A time-dependent velocity function was prescribed at the inlet to simulate the flow from the ventricle, and a Windkessel opening was specified at the outlet. A simple resistance model was specified at the outlet, with pressure proportional to instantaneous flow. For the pump phase the resistance was assigned a value of 2x10 8 PaSecm -3, consistent with a pressure increase of about 120mmHg at the peak outflow rate. For the fill phase the value is essentially arbitrary: it must be large enough to prevent significant leakage at the outlet, but small enough not to cause coupling instability A value of 8.3 x10 7 PaSecm -3 was found to limit outflow during filling, whilst maintaining stability of the analysis. Once again this analysis highlighted the requirement for careful specification of bpundary conditions for the coupled analysis. The deformation of the membrane during the fill phase of the pump cycle is illustrated in figure 9, together with an indication of the massless particle pathlines. These show how the fluid enters the domain and fills the void vacated by the deforming membrane. The pressure in the domain is determined by the stiffness and displacement of the membrane, with a controlled leakage through the outlet. In;let and outlet flow rates during a simulation cycle are illustrated in figure 10.
Figure 9 - Simulation of the operation of a Berlin Heart - membrane displacement and particle paths (D4.1c, version3.1, figure 6.7) Figure 10 - Flowrate at inlet and outlet over one simulated pulse (D4.1c, version3.1, figure 6.8) Conclusion Coupling software has been developed to interface ANSYS to CFX, iterating across the wetted boundary until pressure and displacement are converged. The software has been applied to the simulation of several cardiovascular and biomedical systems with encouraging results. Although significant development tasks
remain, the software represents an important resource able to address leading edge design and research questions for the biomedical community. It is apparent that specification of appropriate boundary conditions for coupled analyses is an issue that requires careful consideration in the analysis planning phase, as well as appropriate functionality in the software. References 1. MpCCI - Mesh-based parallel Code Coupling Interface, www.mpcci.org 2. Formaggia L, Gerbau JF, Nobile F, Quarteroni A, On the Coupling of 3D and 1D Navier Stokes Equations for Flow Problems in Compliant Vessels, 2000 3. Zhao SZ, Xu XY, et al. The numerical analysis of fluid-solid interactions for blood flow in arterial structures - Part 2: development of coupled fluid-solid algorithms. Proceedings of the Institution of Mechanical Engineers Part H-Journal of Engineering in Medicine 212(H4): 241-252, 1998 4. Klein A, Gerlach G, Simulation of fluid-structure interaction in micropumps by coupling of two commercial finite-element programmes. 5. Hawkins I, Wilkes N. Moving Grids in Harwell-FLOW3D, AEA-InTech-0608, 1991 6. JMT Penrose, CJ Staples, Implicit Fluid Structure Coupling for Simulation of Cardiovascular Problems, Accepted for publication in the International Journal of Numerical Methods in Fluids, November 2001 7. Penrose JMT, Narracott A, Jones IP, Hose DR, Validation Report D4.1c, BloodSim Project EP28350, CEC IT Program, Framework 4, 2001 8. Womersley J, Oscillatory flow in arteries: the constrained elastic tube as a model of arterial flow and pulse transmission. Physics in Medicine and Biology 2: 178-87, 1957 9. Narracott AJ, Balloon Folding Influences Stent Deployment, PhD Thesis, University of Sheffield, 2001