Senior Capstone Project Computational Fluid Dynamics within the Left Ventricle of Mice Craig Smith December 11, 2020 Department – Mechanical Engineering Faculty Advisor – Dr. Maysam Mousaviraad Introduction In the United States, cardiovascular diseases account for the largest number of deaths, with an approximate cost of $555 billion each year [1]. The prevalence of these diseases shows that more research is needed to improve the medical field and reduce this health threat. Myocardium is the tissue that composes the heart muscle with distinct physiological features to efficiently perform its function; however, heart failure occurs as the myocardial tissue is altered over time. These tissues become too stiff to allow proper contraction, resulting in poor ejection rates required to deliver oxygen throughout the body [2]. The cardiovascular system is a complex, closed-loop system, thus making for a complex problem when researching the way fluid flows inside the heart’s ventricles. This flow within the heart can be simulated through the Navier-Stokes (N-S) equations which contains complex differential equations. The N-S equations are the governing equations that describe the flow of fluid. Since the exact solution to these physics-based equations does not exist, an iterative numerical method with boundary conditions must be implemented through a computational approach. This approach divides the time and space domains into many small subdomains which enables a numerical solution to be obtained. This can be achieved through the engineering software called ANSYS. ANSYS is a program that implements the discretization (division into subdomains) of the N-S equations for numerical solutions. Through computational fluid dynamics (CFD) modeling, and the use of the N-S equations, an accurate description of hemodynamics (the flow of blood) can be achieved. The computational methods, as non-invasive tools [3], have improved in the last few decades, and are used to optimize devices and strategies in the medical field [4]. Using CFD in the biomechanical industry could potentially increase safety and lower health concerns involved by providing detailed information about cardiovascular disease in human hearts. Background and Significance: Two problems exist within CFD research of cardiac hemodynamics. These need to be addressed before the CFD approach can be widely used in clinical applications [5]. 1) 3D geometries need to be extracted from medical imaging data on a patient-specific basis. So far, geometry extraction has been mostly carried out manually, which is not timely and requires significant user experience. Semi-automated methods could help significantly reduce this shortcoming [6]. 2) Significant computational power is required to obtain the high-resolution solutions based on the physics-based governing equations. This is since approximately 1,000,000 to 3,000,000 tetrahedral elements are required to completely mesh the 3D model of the heart [3] which can take a considerable amount of time for a computer to process. Once a model of the patient’s heart has been constructed, computationally intensive techniques are required to analyze the model. Reynolds numbers of blood flow in the heart can reach approximately 4,000; this number explains that the streamlines in a fluid are turbulent rather than laminar (smooth) flows that are easier to analyze. Large volume changes in the heart are needed in the systole- diastole cardiac cycle. During the systole period, blood is pushed from the left ventricle out through the aorta. During the diastole period, blood is moved through the mitral valve into the left ventricle. Most problems within engineering tend to small displacements with beams and other similar structures. The third parameter that requires computationally intensive techniques is the fact that the hearts walls are flexible structures [5]. While CFD is currently in use to provide helpful information towards cardiac diseases, the process has the ability for improvement to quickly construct and analyze a patient’s heart. Through improvements this will reduce the problems associated with CFD discussed above. Currently, the preferred method of non-invasive evaluations is a simpler 2D echocardiographic model. This is due to either a lower-cost, or faster actuation time compared to a 3D model [6]. This means that further improvements into the process of constructing a 3D model from echocardiographic images are possible. As computers continue to improve, limitations of CFD modeling will evolve. This includes the study into hemodynamics as well since CFD heavily relies on simulations. The assistance of graphical processing units (GPU) is of importance due to their large amount of processing cores [5]. Luckily, the GPU market will constantly improve. GPU development is driven by applications like gaming and video rendering [5], making for a good incentive to increase the power of GPU’s in the future. Likewise, CFD will increase in power instead which bodes well for the study into hemodynamics. Through the improvements that will surely take place within CFD, the continued use of this technique in the research of cardiac diseases should be signified. Goals The purpose of this study is to understand the research into modeling mice ventricles for CFD processing and compare results between different simulations. The goals of this study include: 1. Develop boundary tracing techniques for detecting the moving cardiac wall 2. Develop automated methods to reconstruct 3D geometries from 2D images 3. Develop user-defined functions to define the reconstructed geometries and input to ANSYS software 4. Extract fluid boundary conditions from non-invasive measurements of arterial velocities 5. Post-process, visualize, and quantify the hemodynamic characteristics of each simulation 6. Study the differences between simulations by defining variables that can quantify the hemodynamic alterations 7. Provide enhanced understanding of the hemodynamic alterations in different stages of hypertrophy progression for transverse aortic constriction surgery towards end-stage heart failure Methods This research used the geometry in the echocardiogram video, where a single frame of the echo imaging is shown in Figure 1a to create a simple 2D simulation of a healthy mouse heart. The engineering software MATLAB, a program that utilizes matrices and mathematical arrays for the implementation of iterative processes, then created a trace of the of left ventricle (LV) wall to construct the proper geometry for 2D CFD analysis and imported to ANSYS, shown in Figure 1b. Figure 1: Segmentation for echocardiographic image to 2-dimensional boundary of mouse heart [7]. It is necessary to import the geometry to ANSYS so that the identification numbers (ID) for each point within the model match in both MATLAB and ANSYS. Once the point ID’s are discovered, the model was moved to MATLAB to write proper, user-defined functions that correctly simulate the motion of the cardiac wall as well as the inlet/outlet conditions throughout a cardiac cycle. Through the constructed 2D model and user-defined functions, three simulated cardiac cycles are required. The first cycle in the calculation is not accurate and will improve with multiple cycles. The model’s results are then opened in the post-processing software Paraview. This creates a visualization of streamlines and vortices is the flow that occur from the cardiac cycle. The visualized simulations are then compared to results from a previous cardiac study conducted by Dr. Maysam Mousaviraad. The original study utilized the same LV geometry of a healthy mouse heart echocardiogram with a simplified one-valve inlet/outlet at the aorta. The specific numerical discretization required a division in the special domain (the LV cavity) into approximately 40,000 elements and a division in the time domain of 4260 timesteps. Additionally, the computational cost of this model with the determined grid elements and timesteps is approximately 24 hours to calculate, which is a considerable cost for only achieving a two-dimensional simulation. Analysis The echocardiogram provided by Dr. Nellie Bruns had 144 frames in a single cardiac cycle, which is not smooth enough for computational modeling. Large variations in the model’s point coordinates causes inaccurate calculations in the flow of the fluid, so extra frames must be created in between each provided frame. By using MATLAB to look in between each frame and interpolate 11 frames for each timestep, the model was improved to 1420 timesteps. While increasing the number of frames, even artificially, will increase the accuracy of the simulation results, other problems arose from noise. All transmitted electrical signals, such as music and videos, include irregular fluctuations. These fluctuations in the echocardiogram cause the LV to move in a way that creates problems with the continuity of the heart. To remove this problem from the computational model of the heart, a low pass filter was used on the data to remove the noise. A low pass filter passes a signal that has a lower frequency normally while dampening the force of signals with a frequency higher than the pre-selected cutoff frequency. To make sure that the filtration was reasonable, plots comparing the area (shown in Figure 2) and area differentiation (shown in Figure 3) of the filtered and unfiltered cases can be created. It is important to filter the data for a smoother cycle, but over-filtration can lead to an unnatural physiological behavior of the heart shown in Figure 4 and Figure 5 when a 5 Hz filter was used. Figure 2: Left ventricle area over one cardiac cycle for unfiltered (red) data and filtered (blue) data with a filter of 40 Hz. Figure 3: Left ventricle area differentiation plotted over one cardiac cycle for unfiltered (red) data and filtered (blue) data with a filter of 40 Hz. Figure 4: Left ventricle area over one cardiac cycle for unfiltered (red) data and filtered (blue) data with a filter of 5 Hz. Figure 5: Left ventricle area differentiation plotted over one cardiac cycle for unfiltered (red) data and filtered (blue) data with a filter of 5Hz. For the simulation case with two valves, vortices appear and dissipate at different time steps through the cardiac cycle. Figure 6 displays multiple timesteps that occur, starting at timestep 1430 (end of diastole and beginning of systole), having a transition from systole to diastole at timestep 2145, and ending at timestep 2476 (middle of systole). In the transition to systole, three vortices form but quickly dissipate. As the blood is pushed out through the aorta, a small vortex also appears at timestep 1534 along the cardiac wall and travels toward the aorta. In the early stages of systole, a singular large vortex appears yet stays stationary while another small, secondary vortex near the wall is created. The flow for the simplified model with one valve evolves differently though, shown in Figure 7. While three vortices are present during the transition from diastole to systole, they do not completely disappear as fast as the vortices from the two-valve case. Like the two-valve case, a small vortex forms near the boundary wall and travels toward the aorta. However, in the transition to diastole, a large vortex near the mitral valve is formed while unlike the case with two valves. As the cavity fills, this vortex dissipates, and another is shown near the aorta. This vortex is not found in the two-valve case. a b c d e f g Figure 6: Visualized streamlines and vortices for the two-valve simulation at timestep (a)1442 (b)1509 (c)1534 (d)1571 (e)2132 (f)2393 (g)2476. b c d e f g Figure 7: Visualized streamlines and vortices for the one-valve simulation at timestep (a)1442 (b)1509 (c)1534 (d)1574 (e)2132 (f)2393 (g)2477. There are differences between the two simulations since a LV model where both the inlet and outlet occurring at the aorta is unrealistic. Both cases had similar flow during systole since the blood was ejected at the aorta, but differing flow occurred during diastole. Vortices formed near the inlets; the mitral opening for the two-valve case and the aorta for the one- valve case. Understanding how flow is altered under different parameters and boundary conditions help provide vital information that could one day help improve the technology currently implemented in the medical field. Discussion The current study met most of its goals but was not able to accomplish all criteria. Geometries were extracted from echocardiographic imaging, modeled in CFD software with appropriate user-defined functions, and processed to visualize the differences between simulations. However, the study did not construct 3D models or compare to unhealthy model simulations. Clinical Implications Improving CFD methods will result in better assistance for patients experiencing cardiovascular diseases by providing doctors with better resources. For example, CFD modeling can simulate cardiac diseases such as hypertrophic obstructive cardiomyopathy (HOCM). This is one of the most common inherited cardiac diseases and is shown as thickened cardiac tissue that obstructs the outflow of the LV. CFD engineers and doctors can then artificially create excisions of the thickened tissue to determine which surgical process will correct the structure to allow for a blood flow that is typically observed in healthy hearts [8]. Future Work One key focus for future research into this topic includes converting the models into 3D simulations since the vortex structures may differ from its 2D counterpart. Data for mice hearts experiencing cardiac disease will be gathered, modeled, and simulated for comparison to this study’s results. Another key focus for future research is creating a more physiologically correct model. This includes extending the mitral valve and aorta past the LV wall, as well as the addition of valves that open and close [9], shown in Figures 8 and 9. Figure 8. Left ventricle model with extended mitral and aorta, with valve geometry at timestep 1. Figure 9. Left ventricle model with extended mitral and aorta, with valve geometry at timestep 450. References 1. Schade, D. S., Obenshain, S. S., Ramo, B., & Eaton, R. P. (2019). A Feasible, Simple, Cost-Saving Program to End Cardiovascular Disease in the United States. The American Journal of Medicine. doi: 10.1016/j.amjmed.2019.03.043 2. Kehat, I., & Molkentin, J. (2010). Molecular pathways underlying cardiac remodeling during pathophysiological stimulation. Circulation 122:2727-35 3. Chen, L.-J., Tong, Z.-R., Wang, Q., Zhang, Y.-Q., & Liu, J.-L. (2018). Feasibility of Computational Fluid Dynamics for Evaluating the Intraventricular Hemodynamics in Single Right Ventricle Based on Echocardiographic Images. BioMed Research International, 2018, 1–11. doi: 10.1155/2018/1042038 4. Randles, A., Frakes, D. H., & Leopold, J. A. (2017). Computational Fluid Dynamics and Additive Manufacturing to Diagnose and Treat Cardiovascular Disease. Trends in Biotechnology, 35(11), 1049–1061. https://doi.org/10.1016/j.tibtech.2017.08.008 5. Mittal et al., “Computational modeling of cardiac hemodynamics: Current status and future outlook”, Journal of Computational Physics 305 (2016) 1065–1082. https://doi.org/10.1016/j.jcp.2015.11.022 6. Rajan, N. K., Song, Z., Hoffmann, K. R., Belohlavek, M., McMahon, E. M., & Borazjani, I. (2016, January). Automated Three-Dimensional Reconstruction of the Left Ventricle From Multiple-Axis Echocardiography. Retrieved from https://www.ncbi.nlm.nih.gov/pubmed/26548948. 7. Mehri, M., Fig, M, & Mousaviraad, M. (2019). 3D Computational Fluid Dynamics Modeling of Mouse Left Ventricle Based On 2D Echocardiography Images [Powerpoint Slides] Retrieved from https://uwy- my.sharepoint.com/:p:/g/personal/mmousavi_uwyo_edu/EbYNeYLfOO5CvyAV2pZlwnkBPfm5V u_tT8pyceKkc5jdnw?e=H1ZvOb&CID=8768198a-1c48-81fb-e766-6cf6f63a4b44 8. Zheng, X., Seo, J.H., Vedula, V., Abraham, T., & Mittal, R. (2012). Computational modeling and analysis of intracardiac flows in simple models of the left ventricle. European Journal of Mechanics B/Fluids, 35 (2012), 31-39. 9. Doost, S. N., Zhong, L., Su, B., & Morsi, Y. S. (2016). Two-dimensional intraventricular flow pattern visualization using the image-based computational fluid dynamics. Computer Methods in Biomechanics and Biomedical Engineering, 20(5), 492-507. doi:10.1080/10255842.2016.1250891