P a g e 1 To my Thesis Committee, This document seeks to clarify my research project. After describing what post-conditioning (PC) is, I will explain differences between my research and the recent peristaltic references that have been shared. Post-conditioning Post-conditioning (PC) is a therapeutic strategy aimed at limiting reperfusion injury to vital organs after a period of restricted blood flow due to a blocked coronary artery. PC uses an angioplasty balloon catheter (Fig. 1). The balloon catheter is guided to the blockage area where the distal balloon intermittently inflates and deflates, periodically stopping and allowing blood flow, causing stuttered reperfusion instead of a continuous rapid reperfusion. a. b. c. Fig. 1: Post conditioning inflation and deflation states. From top to bottom, the figures show the fully deflated (a), partially inflated (b), and fully inflated (c) state of the angioplasty balloon. The balloon will continue to inflate and deflate in this fashion based on a user-specified PC algorithm. Blood flows from left to right.
P a g e 2 Throughout PC, the balloon remains in the same position, only inflating and deflating in the radial direction of the blood vessel. Although the walls of a coronary vessel are compliant, when the balloon is fully inflated the balloon is pressing directly against endothelial tissue and the vessel is completely occluded, stopping blood flow (Fig. 1c). The balloon inflation starts from the centerline of flow and moves outward to the vessel wall. Modeling Objectives 1. Using progressively complex geometries, from simple straight tubes to actual arterial bifurcations based on human anatomy, we seek to predict the pressure and velocity profiles inside vessels during PC. 2. To build confidence in our models, our flow conditions will move from steady flow to pulsatile flow. 3. Based on these velocity and pressure profiles we seek to link wall shear stress findings to chemical transfer in the coronary artery bed. Model Geometries and Boundary Conditions Model Geometries Table 1 below shows the modeling geometries we investigate. In each model, the balloon catheter shaft will be placed concentrically in the vessel segment. The balloon is placed so that its distal edge is 1mm from the end of the catheter shaft. Boundary conditions for each model are described in the following section (Tables 2-4). Table 1: Modeling geometries Name Sketch 2D Axisymmetric Straight Tube 3D Symmetric Bifurcation
P a g e 3 3D Asymmetric Bifurcation 3D Human Anatomy-based Coronary Bifurcation Boundary Conditions For steady flow, a constant flow inlet condition with a zero pressure outlet condition will be implemented. For pulsatile flow, the inlet flow boundary condition is a function of time. A cardiac flow rate waveform will be used at the inlet for all pulsatile flow models. This waveform is a complex function of time, requiring MATLAB Fourier series analysis (Fig. 2).
P a g e 4 Fig. 2: MATLAB plot of the Fourier series representation of a cardiac flow rate waveform. To create this level of agreement between the waveform, 17 terms in the Fourier series were used. Tables 2-4 describe the boundary conditions for each the flow geometries given in Table 1. Table 2: Boundary conditions for straight tube geometry. Boundary Condition Inlet Steady Flow or Pulsatile Flow Waveform Outlet Zero Pressure Vessel Walls No Slip Catheter Shaft Walls No Slip Catheter End No Slip Centerline of Flow Symmetry Boundary (axisymmetric) Table 3: Boundary conditions for symmetric bifurcation geometry. Boundary Condition Inlet Steady Flow or Pulsatile Flow Waveform Branch Outlets Zero Pressure Vessel Walls No Slip Catheter Shaft Walls No Slip Catheter End No Slip
P a g e 5 Table 4: Boundary conditions for coronary artery model. Boundary Condition Inlet Branch Outlets Vessel Walls Catheter Shaft Walls Catheter End Steady Flow or Pulsatile Flow Waveform Zero Pressure No Slip No Slip No Slip Balloon Domain Modeling Our method to model the moving balloon domain is based off an infinite viscosity method. COMSOL example models have used this approach to simulate an oscillating valve pin. In this model fluid viscosity is changed from a normal value to a value several orders of magnitude greater, simulating valve movement. Unlike typical fluid-structure interaction models, a solid domain that represents the moving valve in the flow field is not needed. This approach simplifies the model significantly. As seen in Fig. 1, the time-dependent balloon domain is not a simple shape. For this reason, the balloon is broken down into three main sections to simply the infinite viscosity method. Two elliptical sections for the ends of the balloon and one cylindrical section for the middle of the balloon are created (Fig. 3). Fig. 3: Sectioning of balloon domain. Balloon is broken into two elliptical sections and one cylindrical section. To describe the motion of the balloon during inflation and deflation, an expression is written for each section of the balloon. The expression for each section is a step function with a specific argument such that the output is 1 in the balloon section and 0 elsewhere. Eqs. (1) through (3) show the expressions written for each balloon section:
P a g e 6 balloon xrange rrange 1 1 1 balloon xrange rrange 2 2 2 balloon xrange rrange 3 3 3 Eqs. 1-3 where xrange represents the length of each balloon section along the x-direction, and rrange is the radial distance of each balloon section from the catheter shaft. The subscripts 1, 2, and 3 are for the proximal balloon end, middle balloon section, and distal balloon end, respectively. For each simulation, xrange is a constant, and can be written as a product of step functions that will equal 1 if and only if an x value is between the specified limits on the balloon length (Fig. 4). Eqs. (4)-(6) are the functions of xrange for each balloon section. R X Fig. 4: Diagram showing x-ranges for each balloon section. x 1 is the lower limit on the proximal balloon end, x 2 is the upper limit on the proximal balloon end and the lower limit on the middle balloon section, x 3 is the upper limit on the middle balloon section and the lower limit on the distal balloon end, and x 4 is the upper limit on the distal balloon end. xrange flc2 hs( x x, scale) flc2 hs( x x, scale) 1 1 2 xrange flc2 hs( x x, scale) flc2 hs( x x, scale) 2 2 3 xrange flc2 hs( x x, scale) flc2 hs( x x, scale) 3 3 4 Eqs. 4-6 In Eqs. (4) through (6), flc2hs is COMSOL s smoothed Heaviside step function with a continuous second derivative without overshoot. In the function, the first argument ( x x or x x ) is a logical expression given by the user. It is evaluated as 1 when the expression is greater than 0 (true), and will be 0 otherwise (false). For example, when x 1 <x<x 2 in xrange1, both step functions in xrange 1 are true and will return a 1, stating that the certain x-value is within the proximal balloon section s x-range. The second argument scale is a user-defined value that tells flc2hs to smooth the step transition of the logical expression between (scale) and scale. This smoothing prevents COMSOL from diverging solutions. In our modeling cases, the smoothed transition needs to be extremely small so a clear distinction between the balloon wall and the surrounding fluid is made. For this reason, scale is set to 1*10-10. n n
P a g e 7 R Because the balloon will inflate and deflate in the radial direction, rrange for each section is timedependent (Fig. 5). Eqs. (7) through (9) below are the functions of rrange for each balloon section. r 1 (t) r 2 (t) r 3 (t) X r 0 Fig. 5: Diagram representing the r-ranges for each balloon section. Each section has the same lower radial limit r 0 that is equal to the diameter of the catheter shaft. The upper limits, which are a function of time, are r 1 (t), r 2 (t), and r 3 (t): the upper limit on the proximal balloon end, the middle balloon section, and the distal balloon end, respectively. rrange flc2 hs( r r, scale) flc2 hs( r ( t) r, scale) 1 0 1 rrange flc2 hs( r r, scale) flc2 hs( r ( t) r, scale) 2 0 2 rrange flc2 hs( r r, scale) flc2 hs( r ( t) r, scale) 3 0 3 Eqs. 7-9 In Eqs. (7) through (9), the step functions will return a 1 when r is greater than r 0 and less than r 1, r 2, or r 3. The time dependent functions r 1,r 2,r 3 are represented by Eqs. (10) through (12): ( x x ) 2 2 1( ) R( t) 1 ( 2 x2 x1 ) r t r ( t) R( t) 2 Eqs. 10-12 ( x x ) 2 3 3( ) R( t) 1 ( 2 x4 x3 ) r t Where r 1 and r 3 are derived from the ellipsoid equation, and R(t) is the varying distance of the balloon wall from the catheter shaft wall. R(t) is written as: R( t) d [ flc2 hs( t t, dt) flc2 hs( t t, dt) flc2 hs( t t, dt) flc2 hs( t t, dt)] Eq.13 on1 off 1 onk offk Where d is the distance from the catheter shaft wall to the vessel wall. The bracketed part of Eq. 13 represents the user-specified post-conditioning (PC) algorithm with k reperfusion/occlusion cycles. As time progresses in the model, the balloon will inflate when t becomes greater than t on, and will deflate when t is greater than t off. The inflation and deflation rates can be specified by the step function s smoothing parameter dt, where the total time for the balloon to fully inflate or deflate is 2*dt. Our models have dt = 0.5 seconds, allowing a realistic 1-second inflation/deflation rate (Fig. 6).
P a g e 8 1 2*dt 2*dt 0 t on Fig. 6: Diagram representing the step inflation and deflation rate for one cycle of a PC algorithm. Inflation will take place at t on, while deflation will occur at t off. By using the smoothing parameter of COMSOL s flc2hs function, the inflation and deflation step will be smoothed over an interval of 2*dt. This interval is centered over t on and t off. t off By substituting Eqs. (4) through (13) back into Eqs. (1) through (3), an expression for the viscosity becomes: balloon balloon balloon Eq. 14 0 inf 1 inf 2 inf 3 Where µ 0 is the fluid viscosity, 3e-3 Pa * s, and µ inf is a user-defined large viscosity, defined as 1e4 Pa*s. By applying Eq. (14) to the fluid subdomain, a pulsating balloon region can be represented with this infinite viscosity approach. Research Significance and Uniqueness Overview It is well-established that wall shear stress is directly linked to the formation and localization of atherosclerotic plaque in the coronary vessels. Areas of low wall shear are associated with low mass transfer rates between blood and the endothelial lining of the blood vessel (Fig. 7), promoting cholesterol buildup on the endothelial lining (Fig. 8). We have found no reference that investigates the relationship between wall shear stress and chemical transfer in the coronary bed during PC.
P a g e 9 Fig. 7: Cross-section view of artery, showing endothelial lining of inside wall of vessel. Low Wall Shear Stress Fig. 8: Representation of velocity streamlines in an arterial model. In regions of low flow, flow separation and recirculation occurs, leading to low wall shear stress and build up of atherosclerotic plaque (black). Related Literature Biofluids is a vast topic, however, and there are several areas of research that study fluid mechanics with fluid-structure interaction (FSI). Research areas range from modeling the fluid-structure interaction
P a g e 10 between compliant blood vessels and blood flow, to modeling the fluid interaction with medical devices such as cardiovascular stents and respiratory catheters. Below is a sample of these studies: 1. Zinovik, Igor N., and Federspiel, William J. Modeling of Blood Flow in a Balloon-Pulsed Intravascular Respiratory Catheter ASAIO Journal (2007) 464-468. Abstract: This paper looked at the interaction between blood flow through a pulsating balloon respiratory catheter with blood flow past the catheter. In the CFD model, flow was produced by the motion of the balloon, allowing flow to permeate through a blood-oxygenation fiber bundle. 2. Gay, M., and Zhang, L.T. Numerical studies on fluid-structure interactions of stent deployment and stented arteries Engineering with Computers 25.1 (2009) 61-72. Abstract: This study modeled pulsatile blood fluid dynamics before and after an angioplasty stent implantation with a finite element CFD method. By observing wall shear stress, the authors found platelet particles and blood clots form where wall shear is low, which occurs between the struts of the stent. This analysis will assist in the development of novel stent designs and stent deployment protocols to minimize vascular injury during stenting and reduce restenosis. 3. Scotti, Christine M., Shkolnik, Alexander D., Muluk, Satish C., and Finol, Ender A. Fluidstructure interaction in abdominal aortic aneurysm: effects of asymmetry and wall thickness Biomedical Engineering Online 64.4 (2005) Abstract: This paper used CFD to develop a fluid-structure interaction model that studied the effects of pulsatile blood flow on the walls of an abdominal aortic aneurysm (AAA). The models analyzed how wall shear stress, which is the ultimate cause of an AAA rupture, is dependent on the asymmetry of the vessel and the vessel s wall thickness. 4. Oscuii, H. Niroomand, Shadpour, M. Tafazzoli, and Ghalichi, Farzan. Flow Characteristics in Elastic Arteries Using a Fluid-Structure Interaction Model American Journal of Applied Sciences 4.8 (2007) 516-524. Abstract: This paper used FSI models to study the effects of pulsatile blood flow on the elastic arterial walls of an axisymmetric geometry. The authors used pulsatile pressure waveforms from the brachial artery as inlet and outlet conditions. Through their modeling, they found that wall displacements less than 1% of the vessel diameter had no significant affect on blood flow and wall shear stress.
P a g e 11 In addition to these biofluids areas, peristaltic flow is another fluid-structure interaction subject area. In peristalsis models, the tube walls are squeezed by an external mechanical force, creating a travelling wave along the walls that motivates flow. This type of flow is not only used in devices for blood pumping (Fig. 9), but it is also how our digestion process works (Fig 10). Outlet Rotating Rollers Inlet Pinched Tube Wall Fig. 9: Schematic of a peristaltic pump. Rollers squeeze the tube, causing fluid motion in the tube as they rotate. Fig. 10: Peristalsis in the digestive tract. Muscle contractions cause vessel walls to pinch inward, creating propagating waves that push food through. There is a wealth of existing studies related to peristaltic pump modeling. Figures 11-15 show the geometries and boundary conditions from several references:
P a g e 12 Fig. 11: Flow geometry modeled in A new model for study[ing] the effect of wall properties on peristaltic transport of a viscous liquid by Elnaby and Haroun (2008). In this model, an incompressible Newtonian fluid flows through a planar channel with a 2*h width. The walls of the channel are compliant and displace according to a sinusoidal function with amplitude a, wavelength λ, and wave speed c. The boundary conditions on the inlet and outlet of the geometry are specified to allow free inlet and free outlet flow, where normal stresses are zero. Any fluid motion is the result of the compliant wall s movement. Fig. 12: Flow geometry modeled in Interaction of peristaltic flow with pulsatile fluid through porous medium by Affifi and Gad (2003). The model geometry is a 2-dimensional channel with a uniform thickness of 2*d. Again the vessel walls are considered compliant and displace according to sinusoidal function with amplitude a, wavelength λ, and wave speed c. A periodically oscillating pressure gradient is imposed over the domain. The effect of this pressure gradient in addition to the wall motions on fluid flow is analyzed.
P a g e 13 Fig. 13: Flow geometry modeled in Peristaltic transport of non-newtonian fluid in a diverging tube with different waveforms by Harharan et al. (2008). The above article studied the peristaltic transport in an axisymmetric tube with a varying cross section. The fluid is considered a non-newtonian fluid, following the power law. At the walls, a no slip condition is imposed on the axial fluid velocity. A constant pressure gradient based on the power law parameters is specified, indicating a steady flow condition that is only influenced by wall motions. In this article, the authors look at different wall motion wave shapes: sinusoidal, multi-sinusoidal, triangular, square, and trapezoidal. Fig. 14: Flow geometry modeled in Peristaltic transport of a compressible viscous liquid through a tapered pore by Elshehawey et al. (2005). This model is used to observe the peristaltic motion of a viscous compressible fluid in an axisymmetric tube of varying cross section. The walls of the tube move as sinusoidal function with a constant amplitude and wavelength. The inlet and outlet conditions on the geometry are specified such that there is no flow through the tube without the imposed moving wall conditions.
P a g e 14 Fig. 15: Flow geometry modeled in Peristaltic flow of a couple stress fluid in an annulus: Application of an endoscope by Mekheimer and Abdelmaboud (2007). The peristaltic model above looks at axisymmetric flow of an incompressible non-newtonian fluid through an annulus. The inner tube is rigid with radius a 1, while the outer tube is flexible with radius a 2. The outer tube wall has a sinusoidal wave travelling down it with a constant amplitude b and wavelength λ. This model assumes that the fluid motion is dictated entirely by the wave motions of the walls, implying free inlet/outlet conditions. Table 5 compares the models from Figs. 11 through 15 with my research project: Author Elnaby and Haroun (2008) Table 5: Comparison of referenced models with my models Author Geometry 2-D planar channel Author Boundary Conditions Elastic, sinusoidal moving walls; free in/out flow La Barck Geometries (3 Types) 1) 3-D Straight tube with concentric shaft, pulsating region (not FSI); 2) 3-D symmetric and asymmetric bifurcation with inner catheter shaft, pulsating infinitely viscous balloon region; 3) 3-D coronary artery network with inner catheter shaft, pulsating region La Barck Boundary Conditions (2 Types) 1) Steady Flow constant inlet flow rate; zero-pressure outlet; no-slip on all walls 2) Pulsatile Flow inlet cardiac flow rate waveform; zeropressure outlet; no slip on all walls
P a g e 15 Affifi and Gad (2003) Harharan et al. (2008) Elshehawey et al. (2005) Mekheimer and Abdelmaboud (2007) 2-D planar channel 3-D Axisymmetric tube with varying cross-section 3-D Axisymmetric tube with varying cross-section 3-D Axisymmetric annulus Elastic, sinusoidal moving walls; periodic inlet pressure Elastic walls with varying travelling wave shapes; constant inlet pressure Elastic, sinusoidal moving walls; free in/out flow Elastic, sinusoidal moving outer walls; rigid inner walls; free in/out flow 1) 3-D Straight tube with concentric shaft, pulsating region; 2) 3-D symmetric and asymmetric bifurcation with inner catheter shaft, pulsating infinitely viscous balloon region; 3) 3-D coronary artery network with inner catheter shaft, pulsating region 1) 3-D Straight tube with concentric shaft, pulsating region; 2) 3-D symmetric and asymmetric bifurcation with inner catheter shaft, pulsating infinitely viscous balloon region; 3) 3-D coronary artery network with inner catheter shaft, pulsating region 1) 3-D Straight tube with concentric shaft, pulsating region; 2) 3-D symmetric and asymmetric bifurcation with inner catheter shaft, pulsating infinitely viscous balloon region; 3) 3-D coronary artery network with inner catheter shaft, pulsating region 1) 3-D Straight tube with concentric shaft, pulsating region; 2) 3-D symmetric and asymmetric bifurcation 1) Steady Flow constant inlet flow rate; zero-pressure outlet; no-slip on all walls 2) Pulsatile Flow inlet cardiac flow rate waveform; zeropressure outlet; no slip on all walls 1) Steady Flow constant inlet flow rate; zero-pressure outlet; no-slip on all walls 2) Pulsatile Flow inlet cardiac flow rate waveform; zeropressure outlet; no slip on all walls 1) Steady Flow constant inlet flow rate; zero-pressure outlet; no-slip on all walls 2) Pulsatile Flow inlet cardiac flow rate waveform; zeropressure outlet; no slip on all walls 1) Steady Flow constant inlet flow rate; zero-pressure outlet; no-slip on all walls 2) Pulsatile Flow inlet
P a g e 16 with inner catheter shaft, pulsating infinitely viscous balloon region; 3) 3-D coronary artery network with inner catheter shaft, pulsating region cardiac flow rate waveform; zeropressure outlet; no slip on all walls Conclusion This document described my research project and how it is unique amongst existing literature. This project seeks to simulate blood fluid dynamics in a realistic coronary structure during PC. The models will progress in complexity from straight tubes to bifurcations, and finally to a coronary artery model. Both steady and pulsatile flow will be analyzed, using a constant inlet flow rate and a zero pressure outlet condition for steady flow, and an inlet cardiac flow rate waveform (Fig. 2) and a zero pressure outlet condition for pulsatile flow. Many existing studies look at the interaction between medical devices and blood fluid dynamics, describing the effects of angioplasty stents and interventional catheters on blood flow. To the best of our knowledge, no one has studied the fluid characteristics of a pulsating PC balloon in coronary blood flow. What makes this research unique from other studies, specifically peristaltic pump modeling, is that peristaltic pump models have simple fluid geometries. Our research will initially model straight tube geometries, but will move into asymmetric bifurcations and realistic arterial networks. Furthermore, peristaltic flow is driven by a propagation of waves along the tube walls that push fluid through the tube. In contrast, our models assume rigid and static vessel walls. Flow is produced by the inlet steady or pulsatile flow boundary condition. The balloon pulsations will not be modeled as a FSI, but by the previously discussed infinite viscosity method. Unlike many computational studies where elastic walls contribute to fluid pumping, the moving boundaries of the balloon will not motivate fluid flow, but will only intermittently occlude and reperfuse the vessel. Based on these reasons and a literature review of CFD modeling in medical devices, arterial blood flow, and peristalsis models, we believe our research is unique.