UNIVERSITY OF CINCINNATI

Transcription

1 UNIVERSITY OF CINCINNATI Date: 5-Nov-2010 I, Justin Niehaus, hereby submit this original work as part of the requirements for the degree of: Master of Science in Aerospace Engineering It is entitled: Use of Computational Fluid Dynamics to Evaluate Energy Loss in Three Palliative Strategies of Hypoplastic Left Heart Syndrome Student Signature: Justin Niehaus This work and its defense approved by: Committee Chair: Shaaban Abdallah, PhD Shaaban Abdallah, PhD Grant Schaffner, PhD Grant Schaffner, PhD Jeffrey Shuhaiber, MD Jeffrey Shuhaiber, MD 11/12/2010 1,156

2 Use of Computational Fluid Dynamics to Evaluate Energy Loss in Three Palliative Strategies of Hypoplastic Left Heart Syndrome In partial fulfillment of requirements for the degree of Master of Science In the Department of Aerospace Engineering Of the College of Engineering 2010 By Justin Niehaus Committee: Dr. Shaaban A. Abdallah(Chair) Dr. Jeffrey Shuhaiber Dr. Grant Schaffner Previous Degree: B.S. in Aerospace Engineering, University of Cincinnati

3 Abstract Hypoplastic Left Heart Syndrome is a congenital disease that affects neonates and leaves them with an under developed left ventricle. There are three palliative strategies for correcting this disease, which all involve redirecting the blood flow of the heart. In this study, computational fluid dynamics will be used to determine the energy loss in each of the palliative strategies. The losses of the three strategies will be compared to each other, along with the shear stress on the walls. The volumetric flow rate of the systemic and pulmonary circulatory systems of each palliative strategy will also be evaluated. In addition, the differences in systemic and diastolic losses will be compared.

4

5 Acknowledgement I would like to thank my committee chair, Dr. Abdallah for presenting this interesting project to me and guiding me through to the finish. Also, my adviser from the medical side, Dr. Shuhaiber was excellent in teaching me all about the heart and being available to help out in any way. I would like to thank my third committee member, Dr. Schaffner for all the teachings he has presented me in my undergraduate career, and guiding me to being a professional engineer. I would also like to thank my friends and family for being supportive throughout this whole process.

6

7 TABLE OF CONTENTS Nomenclature 1. Introduction 1.1 Hypoplastic Left Heart Syndrome 1.2 Palliated Strategies 1.3 Using CFD to model the heart 2. Background 2.1 The Healthy Heart 2.2 The Circulatory system 2.3 Hypoplastic Left Heart Syndrome 3. CFD Modeling 3.1 Fluid Mechanics 3.2 Steady Model 3.3 Unsteady and Turbulent Model 4. CFD Analysis 4.1 Steady Results 4.2 Unsteady Results 5 Conclusion 5.1 Summary of Results 5.2 Limitations of Model 5.3 Future Work 1

8 Nomenclature: HLHS = Hypoplastic Left Heart Syndrome BTS = Classic Norwood Procedure with Blalock-Taussig Shunt RVPAS = Norwood Procedure with right ventricle to pulmonary artery shunt CFD NSE = Computational Fluid Dynamics = Navier-Stokes Equations PF = Pulmonary Flow Q p Q s = Volume Flow Rate of Pulmonary Arteries = Volume Flow Rate of Systemic Circulation or Aorta 2

9 1. INTRODUCTION This section will outline the objectives of the project. Details of hypoplastic left heart syndrome will be discussed, along with the strategies to help reroute the blood flow throughout the heart. A short background on computational fluid dynamics and how it is used for fluid mechanics problems is given. 1.1 Hypoplastic Left Heart Syndrome Hypoplastic Left Heart Syndrome (HLHS) contributes the 2-3% of all congenital diseases [1]. The disease leaves neonates with little or no function of the left ventricle, which means the right side of the heart has to perform the systolic and pulmonary circulatory functions. This presents difficulty on the heart since only one ventricle is strong enough to pump blood throughout the body. HLHS also changes how blood and oxygen is distributed about the heart, the lungs, and the rest of the body. The disease requires that the infant be under constant care, and immediate surgery is needed to reroute the circulation of blood throughout the heart. Figure 1 shows a heart with HLHS [1]. 3

10 Figure 1 Hypoplastic Left Heart Syndrome 4

11 1.2 Palliated Strategies There are three different palliative strategies for correcting HLHS. The first is the Classic Norwood Procedure (BTS). It is shown in Figure 2 [1]. This procedure involves connecting a surgical conduit called a Blalock-Taussig Shunt that directs blood from the aorta to the pulmonary arteries to become oxygenized. The aorta is also sewed up with a homograft patch to allow blood to flow into the aorta while minimizing tissue damage. The second strategy for HLHS is also a Norwood procedure, except the shunt is moved to connect the right ventricle to the pulmonary artery (RVPAS). This is shown in Figure 3 [1]. The third strategy is called the hybrid procedure. As shown in figure 4, the pulmonary arteries and the aorta remain attached [1]. Also, the atrium septum is ballooned or stented to allow for better mixing of blood and oxygen and to prevent inflow and outflow obstruction. A stent is placed in the ductus arterious. There is little circulation modeling data available to compare these strategies for best patient outcome. The ideal strategy would have the least amount of energy loss across the heart, low shear stress on the heart wall, and distribute the blood evenly throughout the aorta and the pulmonary arteries, among other things. 1.3 Using CFD to model the heart Computational fluid dynamics (CFD) is frequently used to solve complex fluid problems that before could only be solved experimentally. When used correctly, it is a powerful tool that gives a great first answer that can cut down or even eliminate experimental runs, which helps the research become time and cost efficient. The CFD code uses numerical methods to solve the full 5

12 Navier-Stokes equations (NSE), which are highly nonlinear partial differential equations. Since no close form solution exists for the Navier-Stokes equations, numerical methods must be used. These equations give information about the fluid s mass, momentum and energy, and are ultimately solved to give pressure and velocity of the fluid. CFD has traditionally been used to solve aerospace and mechanical engineering problems, but it can be used to solve any problems that involve a fluid. The heart can be compared to a complex series of pipes, in which the fluid is blood. The models of the heart can be drawn using any computer aided drawing tool. From there, a grid will need to be applied to each model, which tells the CFD program where on the geometry the NSE need to be solved. In general, the more grid points or nodes that are added, the more accurate the solution becomes. However, as the number of nodes increases, the amount of time needed to solve the equations increases. There are many commercial programs for creating a grid. The next step is using the CFD program to solve for the pressure and velocity, among other things, at each of the grid points. This usually takes the longest as there can be millions of grid points and thousands of iterations. This process will be used to solve for the pressure difference across the heart for several different models. Three models for BTS procedure using a shunt size of 3, 3.5 and 4 mm in diameter will be solved for, as well as three models of the RVPAS strategy, using 4, 5 and 6 mm in diameter. The hybrid procedure will be solved for different sizes of pulmonary artery banding. The data used from the CFD models of each of the models will be used to determine which strategy produces the lowest pressure drop across the heart, as well as the lowest shear stress on the heart wall, and the most even distribution of blood to the pulmonary arteries and the aorta. 6

13 Figure 2 Norwood Procedure with Blalock-Taussig Shunt 7

14 Figure 3 Norwood Procedure with Right Ventricle to Pulmonary Artery Shunt 8

15 Figure 4 Hybrid Procedure with Pulmonary Artery Banding 2. Background This section details the necessary information about the heart. It will cover essential parts of the heart that will need to be modeled. It will then detail the circulatory system of both the healthy heart and the HLHS heart. From there, the necessary information about the three palliative strategies will be examined. 9

16 2.1 The Healthy Heart In order to complete the investigation, knowledge of fluid mechanics, computational mechanics, and the human heart is needed. The heart is a complex muscle that pumps blood through the circulatory system of the body. It is usually about the size of a human fist and grows accordingly. Blood travels through the circulatory system to deliver oxygen and nutrients to and from cells of the body to help fight disease and stabilize body temperature. The heart is divided into four chambers. The left and right ventricles are the lower half of the heart and the left and right atria are the upper half. In a normal heart, the right ventricle pumps the same amount of blood into the lungs as the left ventricle pumps out to the body. The heart beats at an average of 120 beats per minute for an infant, which is twice the amount as an adult heart. The average blood pressure is lower than the adult, which is 70 mmhg systolic over 40 mmhg diastolic. Systolic pressure signifies when the heart beats and the volume is the smallest, and diastolic signifies when the heart relaxes and is at its greatest volume. 2.2 The Circulatory System In a healthy heart, oxygen-depleted blood first enters through the superior vena cava from the top of the body and the inferior vena cava from the bottom, as shown in figure 5 [2]. The blood then flows into the right atrium where it is held by the tricuspid valve. When that valve is released, the blood enters the right ventricle, where it is pumped to the pulmonary semilunar valve into the pulmonary arteries. The pulmonary arteries lead to the lungs where the blood becomes oxygenized. The flow of blood from the vena cava to lungs is known as the pulmonary 10

17 circulation. Its purpose is to oxygenize the blood and is supported by the pumping of the right ventricle. After the blood is oxygenized from the lungs, it returns to the heart through the pulmonary veins where it enters the left atrium. The mitral valve prevents backflow from the left ventricle to the left atrium. The now oxygen rich blood is pumped through the aortic valve where it is then released through the aorta to the rest of the body. The descending aorta pumps blood to the lower half of the body, where it returns to the heart through the inferior vena cava. The three branches of the aortic arch pumps blood to the upper half of the body and then it returns to the heart through the superior vena cava. This part of the circulatory system is referred to as the systemic system. It pumps oxygenated blood from the lungs to the rest of the body. An anterior and posterior view of the heart is shown in figures 5 and 6 respectively [2]. Figure 5 Anterior View of Normal Heart 11

18 Figure 6 Posterior View of Normal Heart The systemic circulation has a higher fluid resistance than the pulmonary system. That is because the systemic system includes all the veins and capillaries of the entire body, which are very long and have a small diameter. The left ventricle which supports the systemic system by pumping blood throughout the body is stronger than the right ventricle because it pumps blood through a resistance of approximately ten times higher than the pulmonary system. Any deficiency of the left ventricle is a serious concern to the overall cardio vascular system. 12

19 2.3 Hypoplastic Left Heart Syndrome HLHS is classified as a congenital heart disease where the left ventricle is under developed and hence dysfunctional. The blood must now be pumped entirely throughout the body by the right ventricle. Much like in a healthy heart, the blood still flows into the heart through the superior and inferior vena cava where it is held in the right atrium by the tricuspid valve. The blood is then pumped throughout the body by the right ventricle, but the blood flow splits into the pulmonary arteries and the aorta. A mixture of oxygen rich and oxygen depleted blood enters the aorta to be pumped throughout the body. The same blood mixture also enters the pulmonary arteries, where the blood goes through the lungs and the pulmonary veins and returns to the vena cava through the patent foramen ovale. This opening only exists through the development of the heart, and eventually closes up. This makes it imperative that surgery is done immediately upon birth of the infant. HLHS is fatal without intervention, usually in the first few days of life. The first step to managing the disease is to stabilize the neonate s condition so that a diagnosis can be made, along with a treatment plan. There are a few ways to go about treating HLHS but the most popular is the Norwood procedure. The goal of the surgery is to make sure the remaining ventricle is unobstructed in systemic circulation. It is also important to make sure that the there is a secure and balanced flow into the pulmonary arteries. By removing the atrial septum and reconstructing the aortic arch to remove hypoplasia, unobstructed systemic circulation can be provided. The pulmonary artery homograft is then connected into the reconstructed arch so that the right ventricle will pump blood directly into the systemic circulation. To obtain balanced flow into the pulmonary and systemic circulations (Q p /Q s = 1), a correctly sized BT shunt or RVPA shunt is used. The Norwood procedure is referred to as a 13

20 palliative strategy for HPLH because it does not restore the function of the left ventricle. The Norwood is one of the most successful operations for this disease, but it remains one of the most challenging neonatal procedures that can be performed. The most common shunt size for the BTS procedure is 3.5 mm and the most common for the RVPAS procedure is 5 mm. When comparing the two Norwood procedures, retrograde diastolic flow in the aorta can be of a concern in the BTS model. This is due to flow occurring throughout the cardiac cycle, creating a flow run-off away from the aorta, which risks coronary steal syndrome. Patients with the BTS procedure tend to have low diastolic pressures that can lead to death. With the RVPA model, since most of the flow happens during systole and because there is not flow run-off during diastole, diastolic pressure is maintained. However, the RVPAS procedure requires a ventriculotomy, which might impair ventricle function. A recent study in the New England Journal of Medicine found that the mortality rates between infants with the BTS Norwood and RVPAS Norwood procedures are very similar [3]. However, there are also more instances of earlier than planned re-interventions with the RVPAS model. An alternative to the Norwood procedure has emerged in the last few years. This procedure came about in attempt to keep cardiopulmonary bypass to a minimum, considering the fragile state of the neonates. The hybrid procedure aims for the same balance of pulmonary to systemic circulation by placing constricting bands on the pulmonary arteries to limit the flow to the lungs. A stent is then placed in the arterial duct to hold it open. To then ensure atrial mixing, a ballooned septostomy is placed in the atrial septum. The procedure has had similar success in 14

21 mortality percentage as the Norwood. One critique of the procedure is the accurate placing of the pulmonary bands on the small pulmonary arteries is a very difficult challenge. No matter what method is used to treat the infant, energy loss across the heart becomes a major concern. The heart, which is now operating with only one ventricle or pump as opposed to two, has to work a lot harder to pump blood throughout the body. Minimizing local pressure losses and pressure losses due to shear stress becomes even more vital than in a healthy heart. Not much is known about the pressure loss generated from each procedure. Being able to generate data about the pressure loss in each case would help physicians determine which procedure would be better to perform. 3. CFD Modeling This section gives the necessary background of fluid mechanics needed to complete the investigation. Details of the CFD techniques will be discussed. The metrics of the models will be given. The time steady and unsteady model will be discussed in full detail. 3.1 Fluid Mechanics In order to complete the CFD process, knowledge of fluid mechanics is needed to set up the model and evaluate the results. The primary goal of this investigation is to evaluate the energy losses across the heart. In a fluid system, energy is related to pressure. The relation can be seen 15

22 by inspection of the units. Energy is in Joules, or Newton meters. Pressure is in Pascals, or Newton per square meter. So pressure is analogous to energy per unit volume. Energy losses in the heart are often evaluated using the pressure drop from one point to another point downstream. [4], [5]. Fluid moves from high pressure to low pressure, so a high pressure is needed to drive the fluid from start to finish. This high pressure is created by the beating of the heart. The more losses that occur in the heart, the harder the heart has to beat to create the same flow rate across the circulatory system. Much like in an electrical system, fluid will attempt to flow through the path of least resistance. Equation 1 is the resistance of, it can be seen that fluid will flow through the path that has the least amount of pressure [6]. The radius of the tube the fluid follows is also a very important factor in fluid resistance, as shown in equation 1. QL R r 4 P 1. Equation 1 is known as Poiseuille s Law. Length, volume flow, and viscosity are proportional to the fluid resistance, while pressure drop is inversely proportional. The radius of the tube to the fourth power is inversely proportional to resistance making it the biggest determining factor. As the fluid goes along its path, it will lose its energy, or pressure. These losses are due to shear stress losses at the walls, local losses from flow acceleration, decelerating, or turning, and turbulence. Shear stress is the friction the fluid particles encounter when reacting with each other and the wall of the heart [7]. It is more formally defined by equation 2. u 2. y 16

23 In equation 2, u is the axial velocity relative to the wall, and y is the distance from the wall. By examination of equation 2, an increase in the rate of axial velocity along the wall will increase the shear stress. Since stress is a force per unit area, an increase in the surface area, will increase the overall viscous force and hence losses. It has been found that larger arteries produce large shear [8]. Flow accelerates when going from a larger to smaller cross sectional area to comply with the conservation of mass, so having a smaller area may increase the shear. The viscosity of the fluid is, which for all models will be constant. Local losses and turbulence are similar in their nature. Turbulence is a function of Reynolds number, which is defined in equation 3. DV Re 3. A low Reynolds number, which for internal flow is around 2300, will put the flow in a laminar flow regime. Laminar flow will follow straight path lines. As the Reynolds number increases, it enters a transitional period and then eventually becomes turbulent. In the transitional and turbulent regimes, the fluid particles become erratic and time dependent in nature. Turbulence is usually modeled as fluid particles vibrating back and forth across a mean value as shown in equations 4 and 5. The following equations outline the k- model, a popular two equation model which gives the turbulent kinetic energy and the dissipation rate, respectively [9]. k t U j k x j ij U x j i x j T k ( ) k x j 4. 17

24 U t j x j C 1 ij k U x j i C 2 2 k x j T ( ) x j 5. The eddy viscosity if defined below: T C k 2 Table 1 gives the common constants used in this model, for simple internal flow. C μ 0.09 C C σ k 1 σ ε 1.3 Table 1 Commonly used k- model constants Similar to the damping of a mechanical spring system, there are losses associated by turbulence. In the fluid system, the losses are accounted for by a loss in pressure. Just like turbulence forces fluid particles off a straight path, changes in area or curvature of the conduit the fluid is following in will change the path of the fluid particle. The losses the fluid obtains from these changes are referred to as local losses. 18

25 19 Blood flow, like most fluids, can be quantified by the continuity equation and the Navier-Stokes equations, as shown in equations 6, 7, 8, and 9 [6]. These equations hold for Newtonian fluids, and while blood is not a Newtonian fluid, it can be modeled as such due to the large chambers it passes through, relative to the size of the oxygen molecules. 0 z w y v x u 6. g x z u y u x u x p z u w y u v x u u t u g y z v y v x v y p z v w y v v x v u t v g z z w y w x w z p z w w y w v x w u t w The first equation is the continuity equation, which is a statement of the conservation of mass. The following three equations are the momentum equations, which state momentum is conserved. These four equations, along with the k and ε equations if the flow is turbulent, is what the CFD solver will numerically solve to determine information about the flow at the various grid points designated. These equations are coupled partial differential equations that can not be solved by any known analytical means. This means numerical methods such as the finite difference methods, are needed to approximate the solution to the equations. CFD modeling is used to numerically solve the NSE.

26 The commercial software Fluent and Gambit were used to construct models of a neonatal heart with HLHS. Gambit was used to draw and grid the model and Fluent was used to solve the NSE for pressure and velocity, and also for some post processing. The program Tecplot was also used for post processing. Models of the BTS, RVPA, and Hybrid were constructed. Three models for the BTS were made, using 3 mm, 3.5 mm, and 4mm diameter for the conduit. Three models were also made for the RVPA, using 4mm, 5mm, and 6 mm for the conduit size. For the hybrid model, the bands on the pulmonary arteries shrunk the arteries to half of their diameter. A steady and unsteady version of each model was made. The steady version has a constant pressure at the outlets while the unsteady model has an oscillating pressure at the outlets, to account for the time dependant beating of the heart. Table 2 outlines metrics that are used on all models. They are the diameters of the major arteries of the heart, along with the properties of the blood used [10], [11]. Tricuspid Valve (inlet) 14 mm Pulmonary Artery 5 mm Left and Right pulmonary arteries (4 outlets) 2 mm Aorta (outlet) 12 mm Left subclavian artery (outlet) 6.5 mm Left common carotid Artery (outlet) 5.1 mm Brachiocephalic Artery (outlet) 4.8 mm coronary vessels (2outlets) 0.5 mm Density of Blood 1025 kg/m 3 Viscosity of Blood kg/(m-s) Table 2 Heart Metrics 20

27 3.2 Steady Model For the steady model, no turbulence was used. It is an open circuit model with a distinct inlet and several outlets. The inlet for the model is the inlet for the right ventricle, or the tricuspid valve. The pressure set at this inlet is variable for all the models. The outlets are the descending aorta, the three main branches of the aorta, which are the brachiocephalic artery, the left common carotid artery and the left subclavian artery, and the pulmonary arteries. These outlets are set to a pressure of 70 mmhg to simulate the end systolic state of the heart. The inlet is set to a pressure that will create.035 kg/s mass flow rate, or 2000 ml/minute of blood, which is average of infant cardiac output [12]. The pressure difference in each model will be determined by taking the variable inlet pressure and subtracting it with outlet pressure. Figures 7, 8 and 9 show the gridded model of the heart system for the BTS, RVPS, and Hybrid models respectively. Figure 7 shows the 3.5 mm shunt version of the model, and figure 8 shows the 4 mm shunt version of the model. Figure 9 is the hybrid model with no pulmonary artery banding. The average number of nodes and tetrahedral elements is approximately 500,000 and 1,200,000 respectively. The nodes were chosen to surpass the amount of nodes in literature that uses similar geometry [5], [13], [14]. In Bove et al, approximately 100,000 nodes were used with grid studies including fewer nodes. In surpassing this number, the model s results should be just as accurate. More nodes were placed close to the walls to capture boundary layer flow. 21

28 Figure 7 Grid of BTS 3.5 mm model 22

29 Figure 8 Grid of RVPAS 4 mm model 23

30 Figure 8 Grid of Hybrid Model 24

31 The gridded models get exported to fluent to be run as a three dimensional model. The model is first set as steady and laminar. The fluid properties of blood are used, which are 1025 kg/m 3 for the density and N-s/kg for the viscosity. Once the boundary conditions are set, the simulation can be run till a convergence criteria is met. In this case, the simulation runs until the average value of each of the 4 equations (mass, and three momentum equations) is within 10-4 error. Error is defined as the absolute value of the left hand side of the equations minus the right hand side of the equation. Once convergence is met, the data can be analyzed. The pressure difference between the inlet and the outlets is taken, along with the shear stress of the walls. The mass flow rate going to the pulmonary arteries and going through the aorta is also calculated. 3.3 Unsteady Model There are a couple major differences between the steady and unsteady model. First, as the name implies, the pressure at the outlets are unsteady, to give the heart a pulse. The other difference is now turbulence is modeled. A k-epsilon turbulence model is used due to its simplicity and accuracy in non-reacting flows. A user defined function must be made and imported into Fluent. A cosine model is used to oscillate between 40 and 70 mmhg pressure at 120 times a minute, or twice a second. Equation 10 shows the heart beat model, with the pressure being outputted in Pascals P out 2000Cos

32 In order to simulate the difference in resistance between the pulmonary circulation and the systemic circulation, the outlets of the left and right pulmonary arteries were set to the values shown in table 3 [13]. The pressures at the coronary arteries are also given. mmhg Pa Coronary Artery LPA RPA Table 3 Pressures in Unsteady Model The same geometry and grid is used in the steady model as in the unsteady model. Once exported to Fluent, the same set up is used, and the simulation is run to the same convergence criteria as the steady model. The only difference is a time step must be set. In order to oscillate between 80 and 120 mmhg fully, or the heart to beat once, a half of a second is needed. The simulation will run with time steps of.001 seconds, for 500 time steps, which will reach a simulated time of.5 seconds, or one complete heartbeat. The pressure difference is taken between the inlet and the outlets to evaluate the energy loss. The shear stress and mass flow rate of the systemic and diastolic systems were evaluated the same as in the steady model. 26

33 4. Results and Analysis The results for the BTS models are first discussed which are followed by the results for the RVPAS models, and then the hybrid models. Information on overall pressure loss and average shear stress is then given along with flow distribution between pulmonary and systemic circulation 4.1 Steady Results Results for percent pressure loss, shear stress and volume flow ratio between the systemic circulation and the pulmonary circulation (Q p /Q s ) is shown in table 4 for the seven steady simulations. Q p /Q s Shear (Pa) % Pressure Loss BTS BTS BTS RVPAS RVPAS RVPAS Hybrid.5 PF Table 4 Steady State Results As expected, the pulmonary to systemic flow ratio increases as the conduit size increase in the BTS and RVPAS models. Also, as predicted, the pulmonary to systemic flow rates are much smaller than one, since the model used the same pressure for the aorta as in the pulmonary 27

34 arteries. In a real heart, the systemic resistance is about ten times the amount of the pulmonary resistance. However, trends can be seen from the data. As the shunt increase, the pressure loss also increases. In the BTS model, shear also increase as shunt size increase. This makes sense because the area of the model increases. However, for the RVPAS models, the highest shear is at the 5 mm model. In order to discuss these results, the unsteady models will be evaluated. The unsteady results are discussed in much more detail because they are more realistic models compared with the steady models. 4.2 Unsteady Results The time unsteady results for the BTS models for the end systole phase are given below in table 5. It gives data on flow rates, shear stress and percent pressure loss for the BTS models. Q p /Q s Shear (Pa) % Pressure Loss BTS BTS BTS Table 5 End Systole BTS Results As the conduit size increases, the shear stress decreases. This is most likely due to less flow entering the descending aorta, which has an increased area opposed to the pulmonary arteries, and hence more losses can occur in that area. A look at the contours for the pressure loss and shear stress will help give a better idea as to the relationship with conduit size and pressure loss. 28

35 Figures 9 through 14 give the total pressure contours for the BTS systolic simulations with 3mm, 3.5mm and 4mm conduits respectively. An anterior view is always given first, followed by a lateral view. The figures always go in order from smallest to largest conduit size. Figure 9 Anterior View of BTS 3 mm Model Total Pressure at End Systole in Pa 29

36 Figure 10 Lateral View of BTS 3 mm Model Total Pressure at End Systole in Pa Figure 11 Anterior View of BTS 3.5 mm Model Total Pressure at End Systole in Pa 30

37 Figure 12 Lateral View of BTS 3.5 mm Model Total Pressure at End Systole in Pa Figure 13 Anterior View of BTS 4 mm Model Total Pressure at End Systole in Pa 31

38 Figure 14 Lateral View of BTS 4 mm Model Total Pressure at End Systole in Pa It can be seen through the pressure contours that a lot of pressure is lost in the conduit to the pulmonary arteries. This is due to the high velocity in that area, as flow rushes to the low resistance pulmonary arteries. As the conduit increases, the color in the aorta gets a darker shade of red, indicating that more pressure is in that area, and hence losses are decreased, as shown in table 5. A look at the shear stress in figures 15 through 20 gives a closer look at where some of the losses occur. 32

39 Figure 15 Anterior View of BTS 3 mm Model Shear Stress at End Systole in Pa Figure 16 Lateral View of BTS 3 mm Model Shear Stress at End Systole in Pa 33

40 Figure 17 Anterior View of BTS 3.5 mm Model Shear Stress at End Systole in Pa Figure 18 Lateral View of BTS 3.5 mm Model Shear Stress at End Systole in Pa 34

41 Figure 19 Anterior View of BTS 4 mm Model Shear Stress at End Systole in Pa Figure 20 Lateral View of BTS 4 mm Model Shear Stress at End Systole in Pa 35

42 It is clear that the pressure loss in the conduit, and hence entire heart, is due largly to the increased shear in the conduit. This can be explained by an increase in velocity in the conduit which is connected to the low resistance pulmonary ciculatory system. The BTS 3 mm model has the most average shear, because it has the most flow in the aorta, which has a high area. Looking at the shear contours, lighter colors, indicating higher shear are shown in the aorta for the BTS 3 mm model. Another reason that flow to the aorta creates more losses is because shows that most of the local pressure losses that are present due to turning into the descending aorta and the increased area which deceleates the flow. Similar results can be seen for the end diastole phase BTS models. Q p /s Shear (Pa) % Pressure Loss BTS BTS BTS Table 6 End Diastole BTS Results The diastolic losses in overall pressure are significantly higher. This is because as the pressure decreases in the ventricle, there will be some backflow in the ventricle. This creates higher losses, as the flow has to move more to get to its final destination. The shear stress follows the same pattern, with the most in the BTS 3 mm model and the least in the BTS 4 mm model. Now, the pressure loss increases from the BTS 3.5 mm to the BTS 4mm. 36

43 As mentioned earlier, one of the downfalls of the BTS model is the higher retrograde flow in the aorta compared to the RVPAS model. With more flow in the aorta for the BTS 3 mm model, the retrograde flow could have an amplified effect on the pressure loss as more flow enters the pulmonary arteries. Figures 21 through 26 give the total pressure of the BTS models in end diastole, while figures 27 through 32 gives the shear stress. Figure 21 Anterior View of BTS 3 mm Model Total Pressure at End Diastole in Pa 37

44 Figure 22 Lateral View of BTS 3 mm Model Total Pressure at End Diastole in Pa Figure 23 Anterior View of BTS 3.5 mm Model Total Pressure at End Diastole in Pa 38

45 Figure 24 Lateral View of BTS 3.5 mm Model Total Pressure at End Diastole in Pa Figure 25 Anterior View of BTS 4 mm Model Total Pressure at End Diastole in Pa 39

46 Figure 26 Lateral View of BTS 4 mm Model Total Pressure at End Diastole in Pa Figure 27 Anterior View of BTS 3 mm Model Shear Stress at End Diastole in Pa 40

47 Figure 28 Lateral View of BTS 3 mm Model Shear Stress at End Diastole in Pa Figure 29 Anterior View of BTS 3.5 mm Model Shear Stress at End Diastole in Pa 41

48 Figure 30 Lateral View of BTS 3.5 mm Model Shear Stress at End Diastole in Pa Figure 31 Anterior View of BTS 4 mm Model Shear Stress at End Diastole in Pa 42

49 Figure 32 Lateral View of BTS 4 mm Model Shear Stress at End Diastole in Pa Table 7 summarized the systolic losses for the RVPA models. Comparing them to the results for the BTS shows that the Q p /Q s values are higher for the RVPA model. This makes sense, referencing that fluid resistance is a fourth order function of tube radius. With larger conduit sizes for the RVPAS, it makes sense that more flow will enter the pulmonary arteries. Due to the decreased resistance to the pulmonary arteries, fewer losses are observed as Q p /Q s increases. For the RVPAS 4 and 5 mm models, the losses are about equal. That is most likely due to the Q p /Q s values being the same distance from 1. Figures 33 through 38 below show this decrease in pressure loss as pulmonary flow increases. 43

50 Q p /Q s Shear (Pa) % Pressure Loss RVPA RVPA RVPA Table 7 End Systole RVPAS Results Figure 33 Anterior View of RVPAS 4 mm Model Total Pressure at End Systole in Pa 44

51 Figure 34 Lateral View of RVPAS 4 mm Model Total Pressure at End Systole in Pa Figure 35 Anterior View of RVPAS 5 mm Model Total Pressure at End Systole in Pa 45

52 Figure 36 Lateral View of RVPAS 4 mm Model Total Pressure at End Systole in Pa Figure 37 Anterior View of RVPAS 6 mm Model Total Pressure at End Systole in Pa 46

53 Figure 38 Lateral View of RVPAS 6 mm Model Total Pressure at End Systole in Pa As obsvered in the BTS models, increased shear can be seen in the shunt area, for the same reason of the BTS model. The shear is greatest in the RVPAS 5 mm model. Shear stress contours can be seen in figures 39 through 44 below. In the 4 mm shunt, higher shear stress can be seen in the corner of the shunt. As the shunt diameter increases, the corner becomes less sharp, and hence less shear is observed in the corner. Significant decrease in shear happens in the RVPAS 6 mm model, due to less shear in that corner and significantly less flow in the aorta. 47

54 Figure 39 Anterior View of RVPAS 4 mm Model Shear Stress at End Systole in Pa Figure 40 Lateral View of RVPAS 4 mm Model Shear Stress at End Systole in Pa 48

55 Figure 41 Anterior View of RVPAS 5 mm Model Shear Stress at End Systole in Pa Figure 42 Lateral View of RVPAS 5 mm Model Shear Stress at End Systole in Pa 49

56 Figure 43 Anterior View of RVPAS 6 mm Model Shear Stress at End Systole in Pa Figure 44 Lateral View of RVPAS 6 mm Model Shear Stress at End Systole in Pa 50

57 Similar to the BTS model, the diastolic losses are higher for the RVPA models, as observed in Table 8. Q p /Q s Shear (Pa) % Pressure Loss RVPA RVPA RVPA Table 8 End Diastole RVPAS Results The RVPA diastolic models follow a similar pattern as the systolic. Both the shear and pressure loss decrease as the flow to the pulmonary arteries increases for both end systole and end diastole phases. Just like in the BTS model, large local losses are shown in the ascending aorta. It is logical that as less flow goes through the systemic system, there will be less overall losses. This is shown in figures 45 through 50 below. 51

58 Figure 45 Anterior View of RVPAS 4 mm Model Total Pressure at End Diastole in Pa Figure 46 Lateral View of RVPAS 4 mm Model Total Pressure at End Diastole in Pa 52

59 Figure 47 Anterior View of RVPAS 5 mm Model Total Pressure at End Diastole in Pa Figure 48 Lateral View of RVPAS 5 mm Model Total Pressure at End Diastole in Pa 53

60 Figure 49 Anterior View of RVPAS 6 mm Model Total Pressure at End Diastole in Pa Figure 50 Lateral View of RVPAS 6 mm Model Total Pressure at End Diastole in Pa 54

61 Also just like the BTS model, the conduits produce the most shear. However, this shear is concentrated in a small area, and in both the BTS and RVPAS models, shear stress decreases as more flow enters the conduit. This shows that the most overall shear is produced by the aorta. This is shown in figures 51 through 56. Figure 51 Anterior View of RVPAS 4 mm Model Shear Stress at End Diastole in Pa 55

62 Figure 52 Lateral View of RVPAS 4 mm Model Shear Stress at End Diastole in Pa Figure 53 Anterior View of RVPAS 5 mm Model Shear Stress at End Diastole in Pa 56

63 Figure 54 Lateral View of RVPAS 5 mm Model Shear Stress at End Diastole in Pa Figure 55 Anterior View of RVPAS 6 mm Model Shear Stress at End Diastole in Pa 57

64 Figure 56 Lateral View of RVPAS 6 mm Model Shear Stress at End Diastole in Pa It is interesting to compare the BTS and RVPAS models, as they both have high shear stress conduits. This is most likely due to the high velocity seen in this section. High velocity is generated because the fluid is introduced to the low pressure pulmonary arteries from the high pressure systemic circulation. Figure 57 shows velocity cross sectional areas at the entrance of the shunts. 58

65 Figure 57 Cross Sectional Area View of Velocity in Entrance of Shunt at End Systole in m/s With the average velocity in the models at approximately.3 m/s, very high velocity can be seen at the entrance of the shunts. Much higher velocity is seen in the BTS models, while more developed flow can be seen in the RVPAS models. Velocity decreases as shunt size increases to keep with the conservation of mass. It is also interesting to note the pressures at this point as well, which can be seen in figure

66 Figure 58 Cross Sectional Area View of Total Pressure in Entrance of Shunt at End Systole in Pa The pressure at the shunt entrance is much more developed and higher in the RVPAS models. This makes sense considering RVPA sees the higher pressure of the ventricle when it enters the shunt, as opposed to the BTS seeing the lower pressure down stream of the aorta. The more developed flow in the RVPAS models is explained by noticing that at the point of the shunt, the 60

67 flow only has two options to flow through, the shunt or aorta. On the other hand, in the BTS model, the flow can flow into the shunt, or any of the arteries in the aorta, as well as the coronary arteries. Figures 59 and 60 give the pressure cross section of the entire shunt for the BTS and RVPAS models respectively. Figure 59 Cross Sectional Area View of Total Pressure in Shunt at End Systole in Pa 61

68 Figure 60 Cross Sectional Area View of Total Pressure in Shunt at End Systole in Pa A study by Ohye et al in the New England Journal of Medicine showed little difference in mortality in the two different Norwood Procedures [3]. However, in the RVPAS Norwood procedure, there was a higher occurance of unscheduled interventions. This is most likely due to the higher pressure the RVPAS is exposed to, and perhaps the decreased velocity, which keeps flow in the conduit longer. The results for the hybrid procedure are shown below for different pulmonary artery banding in Table 9. Similar to the RVPA models, losses significantly decrease when pulmonary artery flow increases. 62

69 Systolic Q p /Q s Shear (Pa) % Pressure Loss.25 PF PF PF PF Table 9 End Systole Hybrid Results The hybrid procedure produced the least overall pressure loss and shear for the systolic phase. One of the major reasons for this is that a high shear stress conduit from the heart to the pulmonary arteries is not in the procedure. However, large amounts of shear still exist in near the pulmonary artery banding. This is because the reduced area creates a larger velocity, creating more shear as the flow accelerates. The total pressure contours, as well as the shear stress contours can be seen in figures 61 through

70 Figure 61 Anterior View of Hybrid.25 PF Model Total Pressure at End Systole in Pa Figure 62 Anterior View of Hybrid.25 PF Model Shear Stress at End Systole in Pa 64

71 Figure 63 Left Pulmonary Artery Close Up of Hybrid.25 PF Model Total Pressure at End Systole in Pa Figure 64 Left Pulmonary Artery Close Up of Hybrid.25 PF Model Shear at End Systole in Pa 65

72 Figure 65 Anterior View of Hybrid.50 PF Model Total Pressure at End Systole in Pa Figure 66 Anterior View of Hybrid.50 PF Model Shear Stress at End Systole in Pa 66

73 Figure 67 Left Pulmonary Artery Close Up of Hybrid.50 PF Model Total Pressure at End Systole in Pa Figure 68 Left Pulmonary Artery Close Up of Hybrid.50 PF Model Shear Stress at End Systole in Pa 67

74 Figure 69 Anterior View of Hybrid.75 PF Model Total Pressure at End Systole in Pa Figure 70 Anterior View of Hybrid.75 PF Model Shear Stress at End Systole in Pa 68

75 Figure 71 Left Pulmonary Artery Close Up of Hybrid.75 PF Model Total Pressure at End Systole in Pa Figure 72 Left Pulmonary Artery Close Up of Hybrid.75 PF Model Shear Stress at End Systole in Pa 69

76 Figure 73 Anterior View of Hybrid 1.00 PF Model Total Pressure at End Systole in Pa Figure 74 Anterior View of Hybrid 1.00 PF Model Shear Stress at End Systole in Pa 70

77 Figure 75 Pulmonary Artery Close Up of Hybrid 1.00 PF Model Total Pressure at End Systole in Pa Figure 76 Pulmonary Artery Close Up of Hybrid 1.00 PF Model Shear Stress at End Systole in Pa 71

78 The diastolic losses, like in all models are higher for the hybrid. They follow the same trend as the systolic losses in the hybrid, as seen in table 10. Diastolic Q p /Q s Shear (Pa) % Pressure Loss.25 PF PF PF PF Table 10 End Diastole Hybrid Results Figures 77 through 92 outline the hybrid end diastolic losses. Figure 77 Anterior View of Hybrid.25 PF Model Total Pressure at End Diastole in Pa 72

79 Figure 78 Anterior View of Hybrid.25 PF Model Shear Stress at End Diastole in Pa Figure 79 Left Pulmonary Artery Close Up of Hybrid.25 PF Model Total Pressure at End Diastole in Pa 73

80 Figure 80 Left Pulmonary Artery Close Up of Hybrid.25 PF Model Shear Stress at End Diastole in Pa Figure 81 Anterior View of Hybrid.50 PF Model Total Pressure at End Diastole in Pa 74

81 Figure 82 Anterior View of Hybrid.50 PF Model Shear Stress at End Diastole in Pa Figure 83 Left Pulmonary Artery Close Up of Hybrid.50 PF Model Total Pressure at End Diastole in Pa 75

82 Figure 84 Left Pulmonary Artery Close Up of Hybrid.50 PF Model Shear Stress at End Diastole in Pa Figure 85 Anterior View of Hybrid.75 PF Model Total Pressure at End Diastole in Pa 76

83 Figure 86 Anterior View of Hybrid.75 PF Model Shear Stress at End Diastole in Pa Figure 87 Left Pulmonary Artery Close Up of Hybrid.75 PF Model Total Pressure at End Diastole in Pa 77

84 Figure 88 Left Pulmonary Artery Close Up of Hybrid.75 PF Model Shear Stress at End Diastole in Pa Figure 89 Anterior View of Hybrid 1.00 PF Model Total Pressure at End Diastole in Pa 78

85 Figure 90 Anterior View of Hybrid 1.00 PF Model Shear Stress at End Diastole in Pa Figure 91 Pulmonary Artery Close Up of Hybrid 1.00 PF Model Total Pressure at End Diastole in Pa 79

86 Figure 92 Pulmonary Artery Close Up of Hybrid 1.00 PF Model Shear Stress at End Diastole in Pa It is difficult to find a correlation in the shear stress for the hybrid. The no banding case creates the most shear, most likely due to the increased area of the high shear stress pulmonary arteries. As the pulmonary banding contracts, several things ocure in the flow. First, the overall area becomes less, creating less shear. However this creates a higher velocity at the banding point, which creates more shear. Also, as the banding contracts, less flow goes into the high velocity pulmonary artery banding, which creates less shear, however, that sends more flow to the large surface area aorta, where a lot of shear is created. This creates an interesting optimization problem in the case of minimizing shear with in a certain Q p /Q s value. 80

87 With only a couple exceptions, as more flow entered the pulmonary arteries, overall losses decreased. This most likely has to do with the decrease shear the comes with less flow to the aorta, and less local losses from turning into the decending aorta. Another hypothesis can be that as flow heads towards all pulmonary flow, or all systolic flow, the losses decrease due to a simpler system. The RVPAS models demonstrate this. In the systolic models, losses are about the same at around.7, and 1.3 Q p /Q s values, but significantly decrease at a Q p /Q s of In the diastolic models, the Q p /Q s value was close to one for the RVPAS model, and created the most losses, while the RVPAS 4 and 6 mm created less. More evidence is shown in the steady state models, which start at very low Q p /Q s values but shear and losses increase as the conduit size increases. Directly comparing the steady and unsteady model is difficult due to the difference in turbulence modeling pressure boundary conditions, but it is interesting to point out the Q p /Q s values in the set of models. The steady models had small Q p /Q s values, and as those values increased, the pressure loss increased. However, the unsteady results had higher Qp/Qs values and the pressure loss generally decreased as Q p /Q s increased. This makes sense because as the Q p /Q s approaches 0, or all the flow in the systemic aorta, the system because much simpler and hence efficient. On the other end, as Q p /Q s heads to infinity, or all the flow into the pulmonary arteries, that also simplifies the system, making it more efficient. When flow is somewhere in the middle, and flow has to split between going to the pulmonary arteries and the aorta, the system is more complex and open to a lot of local losses from turning, accelerating, and decelerating. This correlation is probably the hardest to see in the BTS models. For the RVPAS and hybrid models, if the Q p /Q s values were to head towards infinity, no flow would have to enter the aorta. However, in the BTS 81

88 models, no matter what the Q p /Q s values are, the flow must enter the ascending aorta and the branches of the aortic arch in order to enter the conduit to get to the pulmonary arteries. 5. Conclusion and Future Work 5.1 Summary From the results of the unsteady simulations, the models produced less overall pressure loss when Q p /Q s values were higher in general. The RVPAS 5 mm model produced the most even flow between the pulmonary and systemic circulation for both systolic and diastolic models. The steady simulations, produced higher losses as Q p /Q s became higher, indicating that a maximum pressure loss can be obtained at some medium value of Q p /Q s which is most likely different for each model. The hybrid procedure produces the least amount of shear over all for all cases, and the least amount of pressure loss in the most common clinically used cases. Shear is definitely a function of pulmonary flow, but it is a different correlation with each model. 5.2 Limitations While much can be learned from this investigation, there are improvements that could be made to the model to make it represent a heart better. First is the rigid wall assumption. In a real heart, the ventricle stretches, and losses occur due to elastic losses in the wall. In all the models, the heart wall is assumed ridged. The next addition to the model could be the addition of valves. No valve was used in the model to stop blood flow from returning to the ventricle as the pressure in the ventricle fell at the 82

89 diastole phase. Even if valves were to be added, diastolic losses would be higher, but would probably be limited. The heart model used in this investigation is also an open loop model. The real circulatory system is a closed loop, which means some energy will return to the ventricle with every beat of the heart. While this energy is very small by the time the flow cycles through the entire body, it could yield more accurate results. Addition of the valve is a necessity to make sure the flow in the closed loop case will be flow the correct direction. Also, in an infant, the heart does grow rapidly, and grows different with each palliative strategy. When models of the growth of the heart for each strategy become more sophisticated, they could be used to model energy losses over period of months as the heart grows. 5.3 Future Work In the immediate future, optimization studies could be used to find out what pulmonary to systemic flow ratio produces the least amount of shear or energy loss within a certain boundary. Also, optimization studies can be done on the hybrid to learn about the best pulmonary artery banding size to use to minimize shear. Since a Q p /Q s value of 1 happens between the pulmonary artery banding of.25 PF and.5 PF, more data should be found between those two points. As more data is generated off the current model, more sophisticated models can be generated. Turning the model into a closed loop will help determine if energy dissipated back to the right 83