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Viscoelastic Modeling of Biological Tissue in an Idealized Cerebral Aneurysm Kristin Kappmeyer H-B Woodlawn High School, Mathematics Department Arlington, Virginia Project Mentor Dr. Padmanabhan Seshaiyer George Mason University Abstract This REU research project presents mathematical modeling of the fluid structure interaction between the arterial wall, the blood within, and the cerebrospinal fluid surrounding an idealized cerebral aneurysm. A Voigt damped mass-spring model for the biological tissue is utilized. The pulsatile blood flow is modeled by a sinusoidal function and the cerebrospinal fluid flow is modeled by a simplified Navier-Stokes Equation. Our objective was to study the behavior of the exterior wall of the aneurysm as we vary the amount of damping in the viscoelastic tissue and as we vary the periodic blood force. An exact solution is presented in which Laplace Transforms are used to solve the coupled system of partial differential equations with boundary conditions. An approximation of the boundary condition at the aneurysm wall is also presented in which the pressure from the cerebrospinal fluid is assumed to vary linearly over space. The mathematical modeling and solving processes are examined from the perspective of a high school mathematics teacher for the purpose of incorporating into the classroom. This research report is submitted in partial fulfillment of the National Science Foundation REU and the Department of Defense ASSURE Programs George Mason University June 1, 2009 – July 31, 2009 1. Introduction The primary goal of this project is to apply sound research methods to three broad areas of study: mathematics, teaching, and a biological model. Once this has been accomplished, the author will work with her mentor, her teaching colleagues, and her school administration to prepare a lesson study for high school classes based upon the process and product of this research. The following is a general overview of each area of study. 1.1 THE MATHEMATICS The author had studied differential equations approximately thirty years ago as a civil engineering student at Virginia Tech. A twenty year career in architecture and a six year career in teaching high school Algebra and Pre-calculus intervened. Much of the college level mathematics had been forgotten and needed to be relearned in order to understand the biological model involving aneurysm and associated computations. Topics that required special attention were Elementary Calculus, First and Higher Order Ordinary Differential Equations, Partial Differential Equations and Boundary Value Problems, Approximating Techniques for use in computer programming such as Euler’s Method and the Finite Difference Method, Fourier Series, Laplace Transforms, Matrix Algebra, and the Physics of mechanical vibrations, force balance, and fluid dynamics. The biggest question was not how to refresh the skills, but which skills to refresh. The answer lay in reading previous research papers on aneurysm models [15, 16]. After assessing which skills might be needed, the researcher read sections in several math textbooks [9, 11, 13], but not in the conventional straight-through fashion. Most often one topic required review of a 2 tangentially related topic which required another textbook. The sequence was not as seamless as it might have been, but since it was review, it was not detrimental to the overall process. However, it would be necessary to think very carefully about guiding students in their discovery of the mathematical skills needed for such research if it was the first time they were presented with the content. More will be said on this in the Teaching Research section. 1.2 THE TEACHING In thinking about the methods used to become reacquainted with the math skills utilized in this project, the author questioned whether that same idea of motivating learning through a rich mathematical problem might work well in her own classroom. Having a real problem at hand puts a finer focus on the skills one needs to acquire to solve the problem, and it certainly makes them memorable. The National Council of Teachers of Mathematics (NCTM) has been arguing for problem-based learning for decades. The problem solving standard in Principles and Standards for School Mathematics claims that “Problem solving … can serve as a vehicle for learning new mathematical ideas and skills. A problem-centered approach to teaching mathematics uses interesting and well-selected problems to launch mathematical lessons and engage students. In this way, new ideas, techniques, and mathematical relationships emerge and become the focus of discussion [1].” NCTM’s Teaching through Problem Solving argues that “the role of problem solving in mathematics instruction should change from being an activity that children engage in after they have studied various concepts and skills to being a means for acquiring new mathematical knowledge [2].” If problem based learning is to be successful, students must have some very basic tools for approaching and solving a problem. George Pólya’s four problem solving steps as outlined in 3 his book How to Solve It: A New Aspect of Mathematical Method [17] were utilized throughout this research. The steps are quite simple and yet extremely effective: 1) Understand the problem 2) Devise a plan 3) Carry out the plan 4) Look back Every aspect of this research project was viewed through this problem-solving lens on a macro and micro level. The researcher demonstrates this four step method with her Pre-calculus students on a regular basis and then requires that they do a project of their choice in which they demonstrate fluency with the process. The first step alone could take pages to describe in that it includes reading, rereading, discussing, diagramming, finding information, looking up new words and ideas, making analogies, etc. This research project will be a wonderful vehicle for presenting Pólya’s 4-step process to the students. In addition to motivating high school Algebra 2 and Pre-calculus class discussions about problem solving, this research project provides an opportunity to apply many algebra and trigonometry skills. The author has made annotations by topic throughout the paper. Each foundational skill that is used in this research paper is indicated in italics with superscript number corresponding to its number in the Appendix. 1.3 THE BIOLOGICAL MODEL An aneurysm is a balloon-like bulge in an artery, or less frequently in a vein. Whether due to a medical condition, genetic predisposition, or trauma to an artery, the force of blood pushing against a weakened arterial wall can cause an aneurysm. Aneurysms occur most often in the 4 aorta (the main artery from the heart) and in the brain, although peripheral aneurysms can occur elsewhere in the body. Aneurysms in the aorta are classified as thoracic aortic aneurysms if located in the chest and are abdominal aortic aneurysms if located in the abdomen. Aortic aneurysms are usually “fusiform” or cylindrical in shape [3]. These aneurysms can grow large and rupture or cause dissection which is a split along the layers of the arterial wall. Most of the 14,000 US deaths from aortic aneurysms each year are the result of rupture or dissection [4]. Aneurysms that occur in the brain are called cerebral aneurysms. They are called “saccular” or berry because of their small spherical shape [3]. There are approximately 27,000 reported cases of ruptured cerebral aneurysms in the US each year. It is estimated that 40% or these patients die in the first 24 hours and another 25% die from complications within 6 months. Frequently there are no symptoms until the aneurysm ruptures. Complications from rupture include “bleed stroke,” permanent nerve damage, and death [5]. Many aneurysms are due to atherosclerosis (which is a form of arteriosclerosis) in which artery wall thickens as a result of a build-up of fatty materials like cholesterol [3]. Arteries consist of cells that contain two types of elastic material – elastin and collagen. Under small deformations, the elasticity of a vessel is determined by elastin. Under large deformations, the mechanical properties are determined by the higher tensile strength collagen fibers. As people age, the properties of collagen change and arteries become less rigid, even as the potential for hypertension increases [7]. The research herein focuses on modeling the arterial wall in an effort to better understand how its elasticity, or lack thereof, affects the tendency towards rupture. 5 The tension in the walls of arteries and veins is directly proportional to the blood pressure and the radius of the cylindrical vessel in accordance with LaPlaces’s Law: T PR . (T is the ratio of force to area of longitudinal section of the vessel wall times the wall thickness, P is the transmural pressure inside the vessel, and R is the radius of the vessel.) Therefore a larger vessel must have stronger walls to withstand the greater tension. A localized weak spot may gain “temporary relief” by taking on a spherical shape since for a sphere, Laplace’s Law states that T PR or half the tension of a cylindrical vessel. However, as the sphere continues to 2 expand, the tension continues to grow [6, 7]. One ramification of Laplace’s Law is that as the size of the aneurysm increases, so does the arterial wall tension and the risk of rupture. This is the reason that physicians generally consider size when deciding on treatment. A National Institute of Neurological Disorders and Stroke (NINDS) recent study showed that the risk of rupture for very small aneurysms (less than 7 mm) is low. If an aneurysm is small, the treatment might be blood pressure medication in conjunction with close monitoring. If an aneurysm is large, however, the treatment is likely to be surgical clipping or endovascular coiling. Coiling is “less invasive”, resulting in quicker recovery times, but early studies indicate that the recurrence rate for coiling is higher [5]. Another ramification of Laplace’s Law is that the “temporary relief” notion supports the idealized spherical aneurysm used in this research. By using a “saccular” or spherical shaped aneurysm, the model becomes one-dimensional, making it possible to find an exact, analytical solution. Our overall structure for this work is as follows. In Section 2, we will develop the mathematical model for the idealized cerebral aneurysm and present our solution methodology for an exact 6 solution and an approximation of the boundary condition at the aneurysm wall. In Section 3, we will investigate the influence of the aneurysm wall damping and the influence of forcing function frequency as we present numerical studies for analyzing our solutions. In Section 4, we will summarize our research and outline future work. 2. Background and Research Methods In this section, we will develop a mathematical model for an idealized cerebral aneurysm, and then we will present our solution methodology. 2.1 THE MATHEMATICAL MODEL We will consider the interaction between three general components of an idealized cerebral aneurysm in this study – the blood within the aneurysm, the wall of the aneurysm, and the cerebrospinal fluid that surrounds the aneurysm. The geometry of the aneurysm is idealized as a sphere as shown in Figure 1. In this way, the problem becomes one-dimensional due to radial symmetry. We will define u( x, t ) as the displacement of the cerebrospinal fluid with respect to space and time, with x measured from the exterior wall of the aneurysm and with positive x measured in the direction of the spinal fluid. u(0, t ) represents displacement at the exterior face of the aneurysm wall, and it is of greatest interest to us. Artery Aneurysm – see Figure 2 for enlarged detail at wall Blood x=0 at exterior wall Cerebrospinal fluid FIGURE 1: Aneurysm Section 7 The one dimensional Navier-Stokes equation derived from the Law of Conservation of Momentum is used to describe the flow of the cerebrospinal fluid (CSF). v v P 2v v 2 F t x x x (1) where is unit density, v is velocity, P is pressure, and is viscosity of the CSF, and F is the body force. The first term denotes the rate of change of momentum. The second term is the convection term which pertains to the time independent acceleration of a fluid with respect to space. We will neglect this term since the space derivative of the velocity may be assumed to be small in comparison to the time derivative. This assumption allows us to eliminate the only nonlinear term. The third term is the pressure gradient. The last term on the left side is the diffusion term which depends on the viscosity of the fluid. We will assume that the CSF is inviscid, and so this term may also be neglected. Finally, we will assume that the body force is zero. Thus equation (1) becomes: (2) v P 0 t x z Let u vdt represent the displacement of the CSF as defined above. Then u 2 u v v (t ) and 2 In addition, we will assume that the CSF is slightly compressible and t t t obeys the Constitutive Law: (3) P c 2 u x where c is the speed of sound through the CSF. Taking the appropriate partial derivative of the pressure and substituting into equation (2) gives the wave equation: (4) 2 2 2 u ( x , t ) c u( x , t ) t 2 x 2 8 Note that this is a partial differential equation1 (PDE) with two derivatives in time and two derivatives in space. Therefore we will need two initial conditions and two boundary conditions for the problem to be well posed. Assuming that the fluid is initially at rest, the initial conditions are: (5) u( x,0) 0 u( x ,0) 0 t and Now we will turn our attention to the boundary conditions. The boundary condition at x L , some position far from the arterial wall, is based on the Plane Wave Approximation which says that the waves will die down some long distance away. It can be represented: (6) u( L , t ) c u( L , t ) t x The boundary condition at x 0 will take into account, among other things, the arterial wall. Since an arterial wall exhibits properties of both an elastic solid and a viscous liquid, it is called viscoelastic. Viscoelastic materials have features of stress relaxation (when a body is strained and then after some time, the stresses relax even though the strain is constant), creep (when a body is stressed and then after some time, it continues to deform as the stress is maintained constant), and hysteresis (when a body is subjected to cyclical loading, its stress-strain unloading process differs from its loading process [14]). For ease in modeling, we will model the arterial tissue with a Voigt model which is a spring and dashpot arranged in parallel. (See Figure 2 which is adapted from Wikepedia ‘A mass-spring-damper system drawn by Ilmari Karonen’ [10].) There are many other models of viscoelastic material (Maxwell, St Venant, and Kelvin to name a few), but they all have limitations. According to McDonalds’s Blood Flow in Arteries, 9 “‘viscosity’ must be regarded as something of a mathematical abstraction, and no arrangement of springs and viscous dashpots has yet been found which mimics the dynamic elastic behavior of the arterial wall [8].” Wall F spring F blood = kX0cosωt u fluid F damping mutt(0,t) FIGURE 2: Voigt Model for Viscoelastic Aneurysm Wall The boundary condition at x 0 , the exterior face of the arterial wall, will be derived from a force balance equation. By Newton’s Second Law, the change in momentum must be equal to the total forces acting on the wall. Thus the total force contributions will be from the cerebrospinal fluid and from the spring with dashpot. The force from the spinal fluid is its pressure times its area or Ffluid Pa . Using the assumption that the fluid is slightly compressible, we can write the pressure in terms of displacement Ffluid c 2 a u(0, t ) where x is the density of the CSF, c is the speed of sound, and a is the cross sectional area. The damping force is Fdamping u(0, t ) where γ is the damping constant. The force from the spring will t be based on Hooke’s Law which states that the force is directly proportional to the spring constant times the displacement. Therefore Fspring k ( Xm Xb) where k is the spring constant, Xm is the displacement of the mass which corresponds to the motion of the exterior face 10 of the arterial wall u(0, t ) , and Xb is the displacement of the interior face of the arterial wall due to the blood pressure from within. The latter can be modeled as a sinusoidal function X 0 cost , where X 0 is the maximum displacement of the inner wall and is the frequency of the periodic force from the blood pressure. Thus, the boundary condition at x 0 becomes (7) m 2 u(0, t ) c2 a u(0, t ) u(0, t ) ku(0, t ) kX 0 cost 2 t x t We investigated two different approaches to solving this boundary condition – one exact and one approximate - both of which entailed viewing kX 0 cost as a forcing function and adjusting its frequency to simulate resonance. From our reading, we knew that the amplitude of the response to forced vibration increases without bound when the frequency of the forcing function approaches the natural frequency of the system 0 k and the damping approaches zero. This m is resonance and we questioned whether it could be a contributing factor in aneurysm rupture [9]. For the exact solution, we took the Laplace Transform of the CSF pressure term and evaluated it at the boundary condition x 0 which resulted in the CSF pressure term being grouped with the damping force. As a result, the combined CSF-damping term could never approach zero and resonance could not be produced. For the approximate solution to the boundary condition, we used a linear approximation of the change in displacement due to the CSF pressure term from x 0 to x L . This assumption resulted in grouping the CSF pressure effect with the spring force. The damping term could therefore approach zero, and interesting resonance phenomena could be investigated. 11 In summary, we will find an exact and an approximate solution to the following model for the coupled fluid structure interaction problem. (4) 2 2 2 u( x , t ) c u( x , t ) t 2 x 2 (5) u( x,0) 0 and (6) u( L , t ) c u( L , t ) t x (7) u( x ,0) 0 t 2 m 2 u(0, t ) c2 a u(0, t ) u(0, t ) ku(0, t ) kX 0 cost t x t Next, we will employ the Laplace Transform technique to find an exact solution and an approximate solution to the coupled fluid structure interaction problem. 2.2 SOLUTION METHODOLOGY We begin with the PDE for the wave equation: 2 2u 2 u c t 2 x 2 (4) t 0, 0 x L The Laplace Transforms of the displacement u( x, t ) and its first and second derivatives respectively are defined below: mb gr U ( x, s) ze u( x, t )dt st L u x, t 0 R U ub x, t g sU ( x , s) u( x ,0) S Tt V W R x, t gU L S ub s U ( x, s) su( x,0) u( x,0) V t Tt W L 2 2 2 12 We shall take Laplace Transforms of the wave and boundary equations in order to find an exact solution of U (0, s) . (The idea behind Laplace Transforms is that we can change a complex problem for an unknown function u into a simpler problem for U, solve U and then take the inverse transform of U to find the desired function u. It is analogous to using logarithms to perform computations, transforming the more “difficult” multiplication to “easy” addition.) Then we shall take the inverse Laplace Transform of U (0, s) to find an exact solution of u(0, t ) which models the displacement of the outer wall of the aneurysm. Taking the Laplace Transform of the wave equation yields (8) 2 2 s U ( x , s) su( x ,0) u( x ,0) c U ( x , s) t x 2 2 Substituting the initial conditions u( x,0) 0 and (9) u( x ,0) 0 , the above equation becomes t 2 s U ( x , s) c U ( x , s) x 2 2 2 The general solution to this equation is (10) U ( x , s) c1 cosh s I F Fs I xJ c sinhGxJ G Hc K Hc K 2 Recall that the boundary condition at x L is (6) u( L , t ) c ( L , t ) t x Taking the Laplace Transform of this boundary condition yields: (11) sU ( L, s) u( L,0) c 13 U ( L , s) x Substituting the general solution (10), the initial condition, and the derivative of U ( x, s) with respect to x into equation (11), and evaluating all at L yields: s I s F F Fs IIJ cF Fs I s Fs IIJ LJ c sinhGLJ c sinhGLJ c coshGLJ G G G H Hc K Hc KK Hc Hc K c Hc KK s c1 cosh 2 1 2 which results in: c1 c2 Thus the general solution can be expressed U ( x , s) c1 cosh (12) s I F Fs I xJ c sinhGxJ G Hc K Hc K 1 Evaluating U ( x, s) at the boundary condition x 0 gives U (0, s) c1 This is where the two approaches to solving the boundary condition at x 0 diverge. 2.2.1 Exact Solution Rearranging the terms from equation (7), we have c2 (13) 2 u(0, t )a m 2 u(0, t ) u(0, t ) ku(0, t ) kX 0 cost x t t Taking the Laplace Transform of this boundary condition yields L M N O P Q (14) c 2 a U (0, s) m s2U (0, s) su(0,0) u(0,0) sU (0, s) u(0,0) kU (0, s) kX 0 x t Simplifying for initial conditions and grouping like terms gives (15) c 2 a s U (0, s) (ms2 s k )U (0, s) kX 0 2 x s 2 14 s s 2 2 U ( x , s) of the general solution (12) and evaluating at x 0 gives x Taking the derivative c2 a s F Fs I s Fs IIJ (ms c sinhG 0J c coshG 0J G Hc Hc K c Hc KK 1 1 2 s I F F Fs IIJ kX 0J c sinhG 0J G G H Hc K Hc KK s k ) c1 cosh 1 which simplifies to casc1 (ms2 s k )c1 kX 0 (16) s s 2 2 Dividing both sides by m and rearranging terms yields ca I k I F F G G Ks mJ H Hm J K bk mgX c1 s2 (17) Let 0 ca and 0 2 s s 2 2 k . Then for x 0 m U (0, s) c1 (18) 0 bk mgX s F F I Ics h s GJs J G H Hm K K 0 0 2 0 2 2 2 Taking the inverse Laplace Transform of U (0, s) and using partial fraction decomposition L 1 A B bU (0, s)g L F G Hs r s r 1 1 Cs D 2 2 s s 2 2 2 IJ K yields a solution of form u(0, t ) Aer 1t Ber 2 t C cost D sin t (19) where: 0 0 2 4m2 0 2 r 1,2 2m bk mgX r br r gcr h bk mgX c h C cr hcr h 0 1 A 1 2 1 2 0 2 1 2 2 0 2 2 2 2 2 0 ca 0 2 k m b g br r gcr h ck m hX D cr hcr h k m X 0r 2 B 1 2 2 2 2 1 15 2 2 2 0 2 2 0 2 0 s s 2 2 2.2.2 Approximation of the Boundary Condition at the Aneurysm Wall Returning to the boundary condition at x 0 equation 2 m 2 u(0, t ) c2 a u(0, t ) u(0, t ) ku(0, t ) kX 0 cost t x t (7) We use the standard linear approximation [11] for u( x h, t ) : u( x h, t ) u( x , t ) h u( x , t ) x At h L , some distance far away from the arterial wall, u( L, t ) u(0, t ) L u( L, t ) 0 , we can solve for u( x , t ) . Since x u(0, t ) u( x , t ) , and substitute the expression u( x , t ) into x x L the CSF pressure term in the boundary condition (7), giving: F G H IJ K 2 c2 a m 2 u(0, t ) u(0, t ) k u(0, t ) kX 0 cost t t L (19) ~ Let k k c 2 a L ~ k and 0 m 2 Taking the Laplace Transform of this boundary condition (19) yields L M N O P Q ~ (20) m s2U (0, s) su(0,0) u(0,0) sU (0, s) u(0,0) k U (0, s) kX 0 t Simplifying for initial conditions and grouping like terms gives: ~ (21) (ms2 s k )U (0, s) kX 0 s s 2 2 Dividing both sides by m and solving: (22) U (0, s) bk mgX s F F I Ics h s GJs J G H HmK K 0 2 0 2 16 2 2 s s2 2 Taking the inverse Laplace Transform of U (0, s) and using partial fraction decomposition L 1 A B bU (0, s)g L F G Hs r s r 1 1 Cs D 2 2 s s 2 2 2 IJ K yields a solution of the same form as the exact solution: u(0, t ) Aer 1t Ber 2 t C cost D sin t (19) where: 2 4m2 0 2 r 1,2 2m bk mgX r br r gcr h bk mgX c h C cr hcr h 0 1 A 1 2 1 2 0 2 1 2 2 0 2 2 2 2 2 ~ kk ~ c 2 a k 0 m 2 L b g br r gcr h ck m hX D cr hcr h k m X 0r 2 B 1 2 2 2 2 2 2 1 2 0 2 2 2 Note that the exact and approximate solutions appear very similar in spite of the fact that the solution processes were quite different. The exact solution is a solution to the entire system while the approximate solution is merely a solution to one boundary condition. It has value in that it allows us to investigate phenomena that cannot be produced by the exact solution, but it cannot be confused with a complete solution. The exact solution has a 0 ca term and the ~ approximate solution has a k k a coefficient of the c 2 a L term. Despite the fact that the CSF pressure term is not u(0, t ) term in the original partial differential equation, the process of t taking the Laplace Transform in the exact solution resulted in it effectively joining the damping 17 of the arterial wall. In the approximate solution, the linear approximation of the CSF pressure term resulted in it effectively joining that of the spring. The quadratic roots2 r1 and r2 relate to the exponential terms which form the general solution of the corresponding homogeneous equation. Since the effect of these terms dies away quickly, together they are called the transient solution. The trigonometric terms form the particular solution of the nonhomogeneous equation and their effect lasts as long as the external force is applied. Together they are called the steady-state solution or forced response U (t ) . Since C cost and D sin t have the same period, their sum can be rewritten as a single sinusoid3 with amplitude R, and phase shift δ. If one does rewrite the sum, the steady-state solution is: U (t ) R cos(t ) where R kX 0 m ( 0 ) 2 2 2 2 2 and 2 tan m( 0 2 2 ) We confirmed that the results from our solutions (both exact and approximate) are equivalent to these equations from Boyce and DiPrima’s Elementary Differential Equations and Boundary Value Problems, section 3.9 “Forced Vibrations” [9] as long as one is careful to apply 0 in the ~ exact solution and that 0 2 takes into account k for the approximate solution. Boyce and DiPrima state that “It is possible to show, by straightforward but somewhat lengthy algebraic computations...” the amplitude R and phase shift δ as per above. We did just this in order to check that our roots and coefficients matched not only the textbook R and δ, but that our Laplace Transform solution matched one in which we used the “Method of Undetermined Coefficients” for a nonhomogeneous equation [9]. The lengthy computations were aided by the sum and product of quadratic roots4. 18 3. Results and Discussion In this section, we will present graphs of the exact solution and of the approximation of the boundary condition at the wall. We will investigate the influence of aneurysm wall damping and the influence of the frequency of forcing function due to blood pressure. The following parameters were used in our numerical studies for analyzing the solutions. Symbol Quantity with units ρ 1000 kg m3 1500 m s c a γ* k m X0 ω* L 2 0.0001 m mean 2 kg s * 8000 N m 0.0001 kg 0.002 m Mean 6 rad s * 0.1 m Parameter Density of cerebrospinal fluid Speed of sound in CSF Area of aneurysm wall Damping constant [12] Spring constant Mass of aneurysm wall Max. displacement of interior wall of artery Frequency of pulse Distance far from outer wall of aneurysm * These parameters will vary as shown on the graphs. All programs used to produce the graphs were developed in MATLAB. Influence of Wall Damping: As the damping constant of the aneurysm wall becomes large, it is expected that the maximum displacement will decrease (Fig.3). Influence of Frequency: As the frequency of the forcing function becomes larger, the period will decrease and it is expected that the maximum displacement will also decrease. The frequency made little impact on the maximum displacement in the exact solution (Fig. 4). We used 2 as computed from nominal diameter, thickness of intima-media of wall, and viscosity constant for human carotid artery [12]. 19 FIGURE 3: Exact Solution – Influence of Wall Damping ( 2 ) FIGURE 4: Exact Solution – Influence of Frequency ( 2 ) 20 FIGURE 5: Approximation of Boundary Condition – Influence of Wall Damping ( 2 ) We used a first step of 200 for in the exact solution since it was the smallest increment that resulted in a noticeable difference in displacement. The approximation required a much smaller step size of 0.00002 in order to show a comparable difference. The displacement showed greater variation however in the approximation of the boundary condition, especially for no damping, and this prompted us to investigate the effect of higher frequencies at small damping (Fig. 6) for the approximation. We will discuss the phenomena of resonance and beats below. Note that the transient solution dies away quickly in all of the graphs. 21 FIGURE 6: Approximate Solution – Influence of Wall Damping at High Frequency ( 0 ) Since the amplitude of the steady-state solution (the maximum displacement) is R kX 0 m ( 0 2 ) 2 2 2 2 2 , then as , R 0 . Also, as 0 , R X 0 (the amplitude of the forcing function.) At some frequency between these extremes, there may be a maximum R. If one differentiates the amplitude R with respect to ω and sets the result equal to zero, one finds that the maximum amplitude occurs when max , where max 0 2 The corresponding maximum displacement is R max kX 0 c 0 1 2 4mk 2 2 2m2 . . As the damping h constant approaches zero, the frequency of the maximum R approaches 0 and the amplitude R increases without bound. In other words, for lightly damped systems, the amplitude of the forced 22 response can be quite large even for small external forces. This phenomenon is called resonance [9]. For our exact solution 0 8,944 and 0 ca which using our parameters becomes 0 150 . The frequency of maximum R occurs at 0 when the damping constant is zero, but 0 cannot equal zero due to the CSF effect. Substituting the smallest value 150 for 0 results in a negative frequency, so there is no possibility of true resonance in the exact solution. This is why we were not able to produce graphs like Figure 6 for the exact solution. For the approximation, however, the damping can equal zero and resonance can occur. There are two ways that forced vibrations without damping ( 0 ) can behave. When 0 , the amplitude will vary slowly in a sinusoidal fashion, while the function oscillates rapidly. This type of motion in which the amplitude is periodic is called a beat. Figure 7 shows an example of a beat. It was produced using different numerical parameters than those used for this project. FIGURE 7: Beats 23 When 0 , the frequency of the forcing function equals the natural frequency of the system and resonance will occur. This is what is illustrated in Fig. 6 where 150,266 which was our ~ k approximation for 0 . The fact that the graph looks like it might be one large beat could m be due to the fact that the frequency has been rounded. This observation is inconsequential given that the displacement is one meter by the time two seconds have elapsed. “The mathematical model on which (the solution) is based is no longer valid since the assumption that the spring force depends linearly on the displacement requires that u be small [9].” Figure 8 shows the maximum displacement (or amplitude R) of our approximation as a proportion of the forcing function amplitude graphed against the forcing function frequency as a proportion of the natural frequency of the system. Several damping levels are shown and the asymptotic behavior for zero damping when 0 can be observed. 2 m2 0 2 0 0.00009 0.00036 0.00080 0.00142 0.00222 FIGURE 8: Normalized amplitude of steady-state response versus frequency of driving force 24 4. Conclusion and Future Work This research has involved looking at the fluid-structure interaction of blood, wall, and cerebrospinal fluid of an aneurysm. Several assumptions were made to simplify the wave equation and boundary conditions so that an analytical solution could be found. A Voigt model was chosen to model the aneurysm wall. The force from the blood within was modeled with a periodic function. The effect of the damping coefficient of the wall and the frequency of the forcing function were studied for a specific set of numerical parameters. Future work will include integration of the Voigt spring model into an implicit finite difference solution which has been updated to include the nonlinear convection term. In addition, a generalized Maxwell model for the aneurysm wall may be investigated. The researcher will continue to work with her mentor in preparation of a lesson study centered on the Appendix topic - Differential Equations for use in her Intensified Precalculus classes. She will invite two of her teacher colleagues and one administrator to participate. This research and the projects by fellow REU participants are examples of rich problems that can be used for problem-based learning. While the skill level was above that of most high school classes, students will benefit from being introduced to applications for the algebra and trigonometry which underpin the fields of science, engineering, and applied mathematics. This researcher will use this project as an example of Pólya’s four-step problem-solving process to help guide students in the creative process. All teachers would benefit from such research because it places the mathematics that they teach into a larger framework. Only when one is aware of what content came before, and what is yet to come, can one truly teach effectively. 25 Acknowledgements Many thanks to S. Minerva Venuti (George Mason University mentor) whose previous work on modeling the fluid structure interaction of a cerebral aneurysm was a springboard for this project. Variations on the wave equation and numerical solution techniques were concurrently explored by S. Minerva Venuti, Courtney Chancellor (REU program colleague), and Avis Foster (George Mason University URCM student). I would like to extend a thank you for their support and fellowship. Thank you to all REU participants for sharing their talents. The author would also like to acknowledge Elizabeth White McGinnis, former REU participant, whose previous work on the project proved to be invaluable. Thank you to my husband for his gentle proofreading, to my daughter for her encouragement, to my father for steering me towards engineering (sorry that I so stubbornly resisted!), and to my mother, who I wish could have lived to see this work. Thank you to the National Science Foundation and the Department of Defense for funding this research. Thank you to George Mason University faculty members Dr. Dan Anderson and Dr. Maria Emelianenko for their helpful input. And finally I would like to thank Dr. Padmanabhan Seshaiyer for his enthusiastic support, his insightful advice, his patience, and his friendship. I am grateful to him for introducing me to research, for bringing me into the REU program, and for guiding me through. 26 Bibliography [1] NCTM. Principles and Standards for School Mathematics; pp. 182, 2000 [2] NCTM Teaching Mathematics through Problem Solving, Grades 6-12; pp. x, 2003 [3] (2009, July 1). Aneurysm. Retrieved July 3, 2009, from Wikipedia Web site: http://en.wikipedia.org/wiki/Aneurysm [4] (2009, April). Aneurysm. Retrieved July 3, 2009, from National Heart Lung and Blood Institute Diseases and Conditions Index Web site: http://www.nhlbi.nih.gov/health/dci/Diseases/arm/arm_what.html [5] (2008,August 19). NINDS cerebral aneurysm information page. Retrieved 7/3/2009, from National Institute of Neurological Disorders and Stroke Web site: http://www.ninds.nih.gov/disorders/cerebral_aneurysm/cerebral_aneurysm.htm [6] (2006). Danger of Aneurysms. Retrieved July 23, 2009, from Hyperphysics Web site: http://hyperphysics.phy-astr.gsu.edu/HBASE/ptens3.html#aneu [7] Bogdanov, Konstantin. Biology in Physics - Is Life Matter? (Academic Press, USA, 2000), pp. 45-48 [8] Nichols, Wilmer W. and O’Rourke, Michael F. McDonald’s Blood Flow in Arteries, 5th Edition (Hodder Arnold, Great Britain, 2005) pp. 54-56 [9] Boyce, William E.; DiPrima, Richard C. Elementary Differential Equations and Boundary Value Problems, 8th Edition (John Wiley & Sons, Inc., USA, 2005) [10] Karonen, Ilmari (artist), (2005, November 1). Damping. Retrieved July 21, 2009, from Wikipedia Web site: http://en.wikipedia.org/wiki/Damping. [11] Weir, Maurice D.; Hass, Joel; Giordano, Frank R. Thomas’ Calculus Early Transcendentals, 11th Edition (Pearson Education, Inc., USA, 2008) [12] Ren´ee K Warriner, K Wayne Johnston and Richard S C Cobbold, A viscoelastic model of arterial wall motion in pulsatile flow: implications for Doppler ultrasound clutter assessment; IOP Publishing, Physiological Measurement 29 (2008) 157–179 [13] Powers, David L. Boundary Value Problems, 5th Edition (Elsevier Academic Press, USA, 2006) [14] Fung, Y. C. Biomechanics – Mechanical Properties of Living Tissues, 2nd Edition (Springer Science and Business Media, LLC, USA, 2004) 27 [15] Venuti, S. Minerva, Modeling, Analysis and Computation of Fluid Structure Interaction Models for Biological Systems, George Mason University 2009 [16] McGinnis, Elizabeth White, Mathematical Modeling and Simulation of Fluid Structure Interaction for a High School Classroom, REU Texas Tech University 2007 [17] Pólya, George, How to Solve It: A New Aspect of Mathematical Method, (Princeton University Press, USA, 1945 original printing 1st edition) 28 Appendix This appendix is a collection of applications of mathematical concepts which are learned in a typical Algebra 2 and/or Precalculus classroom. For the most part, they are skills that I used in this research project. Some, however, are topics that other REU participants discussed in their presentations. Some of the topics are a “stretch” in that they are advanced level topics, but I think that an introduction to the topic of differential equations, for example, can put functions into a larger context. By thinking about how things change and how rates can be related and expressed, students may transition more easily into calculus. 1) Partial Differential Equations: In algebra and precalculus courses, students are usually required to think of a dependant variable in relation to one independent variable. In real life, quantities usually change with respect to more than one variable at a time. Before we can talk about partial differential equations which relate rates of change of a quantity with respect to more than one other quantity, we should introduce the concept of ordinary differential equations. Ordinary Differential Equation Introduction: Think of a situation where the rate of change of a quantity (or quantities) may be known, even though the actual quantity (or quantities) might not be known. An equation that relates that rate to a number or variable does not explicitly establish a relationship between the variables. This is a differential equation. Differential equations are very common in engineering, science, economics, etc. (Examples: fluid flow, population changes and interactions, heat dissipation, current flow in electrical circuits, propagation of all kinds of waves, chemical dilution rates, investment models, climate changes) 29 Start with linear (as in function, not differential equation) example What does dy 2 dt dy dy mean? The fact that is a constant tells you what about the function? dt dt Does it tell you where that line would be in the plane? Can you imagine an infinite number of solutions? (The general solution is y 2 x b ) Introduce the concept of a slope field and graph one. Can you find a particular solution if I tell you a point that the line passes through – for example the point where time is zero? This is the concept of an initial condition which can result in a particular (versus a general) solution. Quadratic example dy x dx The same questions from above can be explored. The topic can be incorporated easily into Algebra 2 and Precalculus sections on quadratic and polynomial modeling. Precalculus classes can go further with exponential and logistic examples. 2) Quadratic Roots (and the Discriminant): Briefly discuss the idea of a second order differential equation. The most tangible example would be that of acceleration or the rate of change of velocity, which is itself a rate of change of displacement. In solving these types of equations, we end up with a quadratic equation in which roots can be real, complex, or repeated, as determined by the discriminant. The type of root greatly defines the characteristics of the solution. If both roots are real and positive, the solution will be increasing exponentially. If both roots are real and negative, the solution will be decreasing exponentially. If the roots are purely 30 imaginary, the solution will be sinusoidal without damping. (A simple harmonic oscillator without damping is a good example.) If the roots are complex conjugates, the real part of the solution will determine the exponential damping function which bounds the sinusoidal solution. Angela Dapolite, REU researcher, explored chemical reactions of reagents used to render pollutants safe through the process of oxidation. She investigated a non-linear system of partial differential equations in which she used the quadratic formula to find roots. The discriminant determined the behavior of the chemical reaction as follows. For a negative discriminant (complex roots), she got a spiral graph. For a positive discriminant (real roots) she found that the chemical reactions were stable. 3) Sum (of sinusoids with the same period) can be rewritten as a single sinusoid: Given two functions of the same period : f (t ) A cost and g (t ) B sin t , their sum b g can be written as a single sinusoid s(t ) R cos t as follows: b g A cost B sin t R cos t A cost B sin t R cost cos R sin t sin (Cosine of difference) Setting the coefficients equal, A R cos and B R sin If one divides arctan B B R sin , one gets which is tan . Consequently, the phase shift is A R cos A B . Also, if one squares A and B and adds the squares, one finds: A A2 B 2 R 2 cos2 R 2 sin 2 A2 B 2 R 2 (cos2 sin2 ) (Common factor) 31 A2 B 2 R2 Thus R (Pythagorean Identity) A2 B 2 . In this way, the amplitude and phase shift of the sum of two sinusoids of equal frequency can be found. 4) Sum and product of quadratic roots: In checking over my solutions, I wanted to confirm that R C 2 D 2 where bk mgX c C cr hcr 0 2 1 2 0 hand D ck m hX were the coefficients of the h cr hcr h 2 2 2 2 2 2 2 1 2 trigonometric terms of my solution and R 2 4m2 0 2 r 1,2 2m 0 2 2 2 kX 0 m ( 0 2 ) 2 2 2 2 2 . I used the roots ~ k and 0 2 , and generated some very challenging m algebraic equations. The sum and product of quadratic roots came to the rescue. For the characteristic equation r 2 r1 r 2 m r k k 0 ; a 1, b and c . Therefore, m m m b c k or and r 1 r 2 or . These two facts were used extensively in a m a m simplifying the work. 5) Matrices: There were several applications of matrices that were “discovered” over the course of this research. There is a proposed revision to the Algebra 2 Virginia Standard of Learning (SOL) to drop matrices altogether. At this time, however, matrix operations including addition, subtraction and multiplication are part of the curriculum. Finding determinants 32 of 2x2 and 3x3 matrices is optional. Solving matrix equations using inverse matrices, however, is a tested topic. Students need to understand that a matrix times its inverse produces the identity matrix and this fact allows us to solve matrix equations. Students do not need to know how to find an inverse matrix “by hand,” but they do need to be able to find an inverse with a calculator in order to solve matrix equations. Since row operations and Gaussian elimination are not required by the SOL, the method of finding a matrix inverse by performing the same set of row operations that are used to solve a system on the identity matrix is not something that is likely to be taught in an Algebra 2 classroom. It could be explained in general terms, however, to inform students as to how their calculators are programmed. The determinant of a 2x2 matrix is a Wronskian when applied to a second order differential equation. If the Wronskian is not zero, the solutions will be linearly independent. Algebra 2 students study linear systems and learn the vocabulary for solutions that are inconsistent, consistent & independent, and consistent & dependant. When the determinant of a linear system is zero, students are encouraged to use other solution techniques (i.e. graphing, substitution, elimination) to determine whether the system is inconsistent (parallel lines with no solution) or dependant (same lines with infinite many solutions on the line.) Using the two facts – that finding the determinant of a square matrix can give information about the types of solutions and that the term “linear independence” resurfaces later in students’ studies - to motivate students to learn these ideas would be of value. 33