The elastance model implemented in a left ventricle finite element model. Pim van Ooij TU Eindhoven BMTE Juni 2006

Transcription

1 The elastance model implemented in a left ventricle finite element model Pim van Ooij TU Eindhoven BMTE Juni

2 Summary Since heart diseases are the second largest cause of death in the western world, the function and physiology of the heart are studied in various ways. Marco Stijnen developed a model to study the influence of different geometries and orientations of an artificial heart valve. A limitation of this model was that a flow profile was prescribed as a function of time at the bottom of the left ventricle. This flow was independent of preload and afterload, in contrast to the situation in the human left ventricle. Therefore, a different model, the so-called elastance model was implemented in a similar model to simulate the behavior of pressure and accompanying flow in the ventricle. The elastance model is linked to a lumped parameter model, that describes the systemic circulation in a mathematical fashion. First this is done in 1D in Matlab, by the use of simple equations and conditions for flow. This produced physiological results. Next, this model was implemented in a 2D finite element model where complicated mathematical conditions for flow were necessary to use. Flow was calculated by the boundary integral on the in-, and outlet boundaries and to simulate the closing of a valve, the viscosity of blood was set to a high value. Unfortunately, the model developed was not able to produce results when the physiological viscosity, which is P a s, was used. Therefore, a simulation was performed with a viscosity of 0.4 and 0.04 P a s. The simulation where a viscosity of 0.4 P a s is used, was also performed by Seabra and da Silva, but they used an algorithm that estimates the values of the flows, whereas in this model, that was not necessary. This simulation, however, will not be presented in this report. Instead, the results of a simulation with a viscosity of 0.04 P a s will be presented. Sepran was not able to calculate the second and third period of this simulation, so the results contain transient effects, since steady state is not obtained. The pressure and velocity fields, however, contain useful information. The model requires some adjustments, such as better chosen model parameters and different conditions for simulating the opening and closing of the valves. The model indicated, though, that it is not necessary to use an estimation algorithm for the flows and that the model developed by Stijnen, can be complemented by more realistic pressure conditions.

3 Contents 1 Introduction 2 2 Materials & Methods Physiology The cardiovascular system The cardiac cycle Model Introduction The elastance model The lumped parameter model Matlab model Finite element model Simulations performed Results Matlab Sepran Ten times physiological viscosity Discussion Matlab Sepran Ten times physiological viscosity Conclusion 29 6 Recommendations 30 1

4 Chapter 1 Introduction Heart diseases are the second largest cause of death after all sorts of cancer. The heart contains valves, and often, complications in the heartcycle are caused by valve diseases. When natural valves are replaced by artificial valves in heart surgery, different geometries and orientations of the artificial valve can be chosen. This influences the ventricular flow field and it is believed that the flow field in the left ventricle influences the efficiency of the heart as a pump. Therefore, the influence of different orientations of the mitral valve on the flow field in the left ventricle was studied by a 2D finite element model, developed by Marco Stijnen [1]. A limitation of the model he developed was that a flow profile was prescribed as a function of time on the bottom of the left ventricle. This flow was independent of preload and afterload, in contrast to the situation in the real left ventricle. To replace this prescribed flow by ventricular pressure, in this project, the so-called elastance model is used. In this model, the pressure in the left ventricle is determined by the volume of blood in the left ventricle and the state of activation of the left ventricle muscle. Furthermore, in the model of Stijnen the pressures in the circulation were constant, whereas pressure variations are a result of the variation in bloodflow. To simulate pressures as a result of flow, a lumped parameter model is chosen. The purpose of this study is to simulate velocity profiles in the left ventricle as a result of more realistic pressure conditions. This was done before by Seabra and da Silva [2], but they used an estimation algorithm for the flows and were not able to perform the simulations with a realistic viscosity of the blood. They used a viscosity of 0.4 P a s. Furthermore, they used an algorithm to estimate the values of the flow. In this project, it is attempted to perform finite element simulations of the left ventricle, using a lower blood viscosity, namely 0.04 P a s and P a s, and to obtain the values of the flows without estimating them. It is expected that this model obtains more realistic results than results found by Seabra and da Silva. 2

5 Chapter 2 Materials & Methods 2.1 Physiology The cardiovascular system In figure 2.1, a cross section of the heart is displayed. Figure 2.1: Anatomy of the heart, showing right atrium (RA), right ventricle (RV), left atrium (LA) and left ventricle (LV). [3] The circulatory system can be divided into two parts, the pulmonary circulation and the systemic circulation [4]. The pulmonary circulation consists of the right atrium (RA), right ventricle (RV), the artery pulmonaris (a. pulmonaris), that leads the blood to the lungs and the vein pulmonaris (v. pulmonaris), that leads the blood from the lungs to the left atrium (LA). Here starts the systemic circulation which consists of the left atrium, the left ventricle (LV), the aorta, the arteries, capillaries and veins that lead the blood through the human body back to the right atrium via the vena cava (v. cava) as depicted in figure 3

6 CHAPTER 2. MATERIALS & METHODS 2.1. PHYSIOLOGY 2.1. The systemic circulation contains two valves, also displayed in figure 2.1: the aortic valve, between the left ventricle and the aorta, and the mitral valve, between the veins and the left ventricle The cardiac cycle In figure 2.2 a schematic overview of the cardiac cycle in the left ventricle is given. Figure 2.2: Pressure in the left ventricle as a function of volume [4]. In figure 2.3 flow and pressure curves of one heartcycle are shown, as well as the opening and closing of the valves. Figure 2.3: Pressure in the left ventricle and aorta as a function of time (A) and flow through the aorta valve and the mitral valve as a function of time (B) [3]. In this figure mc stands for the closing of the mitral valve, ao for the opening of the aortic valve, ac for the closing of the aorta valve and mo for the opening of the mitral valve. 4

7 CHAPTER 2. MATERIALS & METHODS 2.2. MODEL The cardiac cycle of the heart consists of two different periods: systole and diastole. These can be subdivided into four periods: the period of isovolumic contraction and the period of ejection during systole, the period of isovolumic relaxation and the period of the period of filling during diastole [4]. During the period of isovolumic contraction both the aortic valve and the mitral valve are closed. Since a valve opens as a result as a pressure difference over the valves, this means that the pressure in the left ventricle is larger than the pressure in the left atrium and lower than the pressure in the aorta. In spite of the blood in the ventricle, no flow can occur between the left ventricle and the aorta or between the left atrium and the left ventricle, since both valves are closed. However, there is an increase in the pressure in the left ventricle, due to the increase in muscle tension of the ventricular wall. When this pressure exceeds the pressure in the aorta, the period of ejection is initiated and the aorta valve will open, causing an accelerating blood flow from the left ventricle to the aorta. As a result of the flow in the aorta, the pressure in the aorta will increase, due to a resistance of the aorta itself. Next, the pressure in the ventricle will decrease, as a result of relaxation of the ventricular wall. At a certain moment the pressure in the aorta will exceed the pressure in the left ventricle causing the flow in the aorta to decrease and eventually, when the flow reaches zero, the aortic valve will close. A little backflow is needed to close the aorta valve fully. This initiates the isovolumic relaxation phase of the heart cycle. When the pressure in the left ventricle drops below the pressure in the left atrium, the mitral valve will open, and blood will flow from the left atrium into the left ventricle. Obviously, this is called the period of filling. Due to the flow from the left atrium, the pressure in the left atrium will decrease while the pressure in the left ventricle increases, causing the flow from the left atrium to decrease. As described above, when the flow in the left atrium drops to zero, the mitral valve will close. The cardiac cycle will now start again. 2.2 Model Introduction In this project the pulmonary system is neglected for which two reasons are mentioned. First, since the left ventricle has to built up a pressure difference that causes the blood to flow through the whole body, the pressure generated by the left ventricle is about three times the pressure that the right ventricle generates to cause the blood to flow through the lungs. Secondly, the flow phenomena in both circulations are alike. As a consequence, in this case only the systemic circulation is studied. Since the atria act as primer pumps for the ventricles [4], and have no major influence on the flow-pressure relationships, these are also neglected. This means that in this project the left ventricle, the aorta, the arteries and the veins, along with the aortic valve and the mitral valve, are studied. In this chapter, first, the model that describes the pressure curve in the left ventricle, the elastance model, is discussed. Then, to describe the pressure and flows in the aorta and veins, the lumped-parameter model is discussed. Next, these models will be combined in Matlab (1D), and furthermore, a finite element model will be added in Sepran (2D). 5

8 CHAPTER 2. MATERIALS & METHODS 2.2. MODEL The elastance model The elastance model was first proposed by Suga and co-workers in the 1970 s [5]. They considered the contraction of the ventricle in the pressure-volume plane to be an elastance, a measure of cardiac muscle stiffness, that varies over the cardiac cycle, as depicted in figure 2.4A. Elastance, E, is equal to the ratio of intraventricular pressure, p Lv, and volume, V Lv, minus a correctional volume, V 0 : E(t) = p Lv (t) V Lv (t) V 0 (2.1) Figure 2.4: In figure A different pressure-volume curves are shown. The slope of the lines drawn through isochronic points represents the elastance, which varies in a cyclic manner. The lines intercept the x-axis in the point V 0 that remains constant. When these lines are lined up a sinusoid (B) occurs [5]. In diastole, the muscle is relaxed thus the stiffness is low. The low stiffness is renamed to passive elastance (E pas ) and is nearly constant, as figure 2.4(B) displays. In systole, the muscle contracts and becomes stiffer. This stiffness is termed active elastance (E act ). The total elastance is then described by: E(t) = E pas + Act E act (2.2) where the activation (Act) is given by a sinusoid, used to model the elastance curve: { cos(4π(t 0.5)) if 0 < t < 0.5 Act = 0 if 0.5 < t < 1 This Activation curve is displayed in figure

9 CHAPTER 2. MATERIALS & METHODS 2.2. MODEL Figure 2.5: The Activation curve. The elastance relates to the pressure in the following manner: p Lv = (E pas + Act E act ) (V Lv V 0 ) (2.3) The computation of V Lv will be explained in the next subsection The lumped parameter model A lumped-parameter model is a model where distributed properties of a phenomenon (i.e. the arterial system or venous system) are lumped into a limited number of discrete components, neglecting their spatial distribution [5]. Lumped-parameter models of the arterial system can be used for understanding the function of the arterial system. They are also used to derive aortic flow from arterial pressure [6], as is also done in this project. The first of a class of models known as lumped-parameter models was the windkessel model, displayed in figure 2.6. It consists of a peripheral resistance (R p ), mainly located in the small arteries and arterioles, and the total arterial compliance (C art ), mainly determined by the visco-elastic properties of the aorta and the remaining large arteries. The two-element windkessel model thus gives insight into the contribution of the different arterial properties to the load on the heart. In this project, R ao is added, to represent the input impedance of the arterial tree, and C ven, representing the compliance of the veins. 7

10 CHAPTER 2. MATERIALS & METHODS 2.2. MODEL Figure 2.6: The two-element windkessel model [3] Matlab model In this section the previously mentioned models are combined and the equations are given which are used to perform the calculations in Matlab. The complete model is displayed in figure 2.7. Figure 2.7: Schematic overview of the systemic circulation in Matlab with the elastance model. 8

11 CHAPTER 2. MATERIALS & METHODS 2.2. MODEL The equations that describe flow are given by: { pven p Lv q ven = R mit if p ven > p Lv or q ven > 0 0 else q ao = { plv p art R aov+r ao p Lv > p ao or q ao > 0 0 else q p = p art p ven R p (2.4) where q ven, q ao and q p are the flows through, respectively, the veins, the aorta and the microcirculation. p ven, p Lv and p art are the pressures in, respectively, the veins, the left ventricle and the arteries. R mit is the resistance of the mitral valve, R aov the resistance of the aorta valve, R ao the characteristic resistance of the arterial tree and R p the resistance of the microcirculation. When the flows are known, the changes in volume in the arteries ( V art ), veins ( V ven ) and left ventricle ( V Lv ) can be calculated according to: V art = q ao q p (2.5) V ven = q p q ven (2.6) V Lv = q ven q ao (2.7) These changes in volume are used to update the volume, by: V art,new = V art old + ( V art t) (2.8) V ven,new = V ven old + ( V ven t) (2.9) V Lv,new = V Lv old + ( V Lv t) (2.10) At this point the new pressures can be calculated, according to: p art,new = V art new C art (2.11) p ven,new = V ven new C ven (2.12) p ao,new = p art + qao Rao (2.13) where C art and C ven are the compliances in, respectively, the arteries and the veins and where p ao is the pressure in the aorta. The pressure in the left ventricle will be calculated with the use of the elastance model as given by p Lv,new = (E pas + Act E act ) (V Lv,new V 0 ) (2.14) A problem that arises when this scheme is applied, is that on the first timestep where p Lv is larger than p ao, p ao is underestimated, since q ao is taken 9

12 CHAPTER 2. MATERIALS & METHODS 2.2. MODEL at the previous timestep, where it was zero. As a result, on the next timestep, q ao is overestimated, and the situation is reversed. p ao will be overestimated as well. On the next timestep, q ao will be underestimated and this will go on throughout the ejection phase. This mechanism is displayed in figure 2.8. To minimize this problem, a sufficiently small timestep should be chosen. Figure 2.8: A small delay and error in calculation of p ao and q ao. The parameters and the initial values of the variables are given in table 2.1. The elastance variables and resistances in the aorta and veins are obtained by attempting to reproduce the pressure curve in figure 2.3 by trial and error and the initial pressures are obtained from figure 2.3A. The resistance of the capillaries is estimated by dividing the mean aortic pressure ( p ao ) by the mean aortic flow ( q ao ). The volume variables of the arteries and venes are obtained using Guyton [4], where the volume in the arteries contains 7% and the venes 64% of the total blood volume. The blood volume in the left ventricle is the rest volume. The compliances are obtained by dividing the initial volumes by the initial pressures. The period for one heartcycle (T p) is set to 1 second. The number of heartcycles (np) is set to 3, and the number of timesteps nstep is set to

13 CHAPTER 2. MATERIALS & METHODS 2.2. MODEL Table 2.1: Parameters and initial values in the elastance model for one timestep in Matlab parameters value unity E pas 1.12e7 P a/m 3 E act 3.02e8 P a/m 3 C art 5.9e-8 m 3 /P a C ven 2.9e-6 m 3 /P a R ao 1e7 P a s/m 3 R aov 1e6 P a s/m 3 R mit 1.2e6 P a s/m 3 V 0-5e-6 m 3 initial values value unity R p 1.7e8 P a s/m 3 p ao P a p art P a p ven 1100 P a p Lv 1008 P a V Lv 85e-6 m 3 V art 6.49e-4 m 3 V ven 3.19e-3 m 3 q ven 0 m 3 /s q ao 0 m 3 /s 11

14 CHAPTER 2. MATERIALS & METHODS 2.3. FINITE ELEMENT MODEL 2.3 Finite element model To study the velocity and pressure fields in the ventricular cavity, a finite element model of the left ventricle was developed. This model is schematically displayed in figure 2.9. Figure 2.9: Schematic overview of the finite element model In this figure, in comparison with figure 2.7, the elastance model is substituted for a finite element model, as displayed by the mesh in the middle of the figure. Furthermore, the resistances R aov and R mit are omitted, since a viscosity model is applied at the valves, which will be explained later. Also, since q ven in this model is defined as flow in the opposite direction, the equation for V ven and V Lv are written as: V ven = q p + q ven (2.15) V Lv = q ven q ao (2.16) Finite element computations are performed using the mesh displayed in figure The mesh consists of two tubes, the left tube representing the aorta and the right tube representing the venae cavae, and a chamber that represents the left ventricle. The width of the tubes measures 1 centimeter and the length 4 centimeters. The ventricle has a height of 5 centimeters and counts 2.4 centimeters in width. The in- and outlet surfaces contain 12 by 12 elements, and the ventricular area contains two times 12 by 12 elements and in the middle 6 by 12 elements. This means a total of 648 elements. To model flow of a fluid and pressure by use of the finite element method, the Navier-Stokes equation (2.17) has to be solved. 12

15 CHAPTER 2. MATERIALS & METHODS 2.3. FINITE ELEMENT MODEL Figure 2.10: The mesh ρ( δ v δt + v v ) = p + ν 2 v (2.17) Where ρ represents the density of the fluid,ν the viscosity of the fluid, v δ the velocity of the fluid, δt a time derivative, t the time, p the pressure and ν the viscosity of the fluid. To simulate the opening and closing of the valves, the viscosity ν in the aorta and the veins varies. When the aorta valve is closed, a high viscosity ( P a s) for the aorta in equation 2.17 is chosen, such that no flow occurs between the left ventricle and the aorta. The same holds for the veins. When a valve opens, an other viscosity is chosen to allow flow. A physiological value for the viscosity would be P a s, but because of numerical instabilities, a viscosity of 0.04 P a s is applied in this project. The opening and closing of the valves are dependent on the same pressure conditions as explained in the Matlab model section. The density ρ is chosen as a constant with a value of 1000 kg/m 3. The elements used are Crouzeix-Raviart elements, in which the internal pressure and its gradient are defined in the centroid and available to the user. This pressure is discontinuous over the element boundaries. The averaged pressure produced by DERIV is available. There is a total of 2725 nodes. On the left outlet, represented by T out, the pressure p ao is prescribed, on the right outlet, represented by T in, the pressure p ven. p Lv is prescribed on the bottom curve, represented by T ven. On the remaining boundaries of the mesh, represented by T wall, a no slip boundary condition is applied. The Reynolds number is defined on T out, the aorta, and calculated by: Re = ρq max ηl (2.18) 13

16 CHAPTER 2. MATERIALS & METHODS 2.3. FINITE ELEMENT MODEL where Q max is the maximum flow through the aorta, η the viscosity and L the width of the aorta. The finite element model is connected to the lumped parameter model through pressure and flow. The 2D flows, q ao and q ven, are calculated using an algorithm in Sepran called bounin, that calculates surface integrals: q 2D = ( u n )ds (2.19) where u is the solution vector containing velocities, n is the normal vector and ds is the surface over which the integral is calculated. The outcome is in m 2 /s, so to obtain flows in m 3 /s, the outcome is multiplied by the diameter of the aorta, which is 1 cm. It is assumed that the diameter of the vein has the same value as the aorta. The flow through the micro-circulation q p is calculated according to equation 2.4. The changes in volume are then estimated by the use of equations and the volumes by equations The pressures are then estimated by equations The error mentioned in the Matlab section applies to this model as well. The solution is obtained by the Newton- Raphson iterative process where the solution of a timestep is subtracted by the solution of the previous timestep. If this difference is smaller than a certain criterion (ε = ), the solution is accepted. The parameters and initial values for the simulation with a viscosity set to 0.04 P a s are given in table 2.2. Table 2.2: Parameters and initial values at 0.04 P a s parameter value unity Epas 1.12e7 P a/m 3 Eact 1.90e8 P a/m 3 Cart 5.9e-8 m 3 /P a Cven 2.9e-6 m 3 /P a Rao 1e8 P a s/m 3 Rp 1.65e8 P a s/m 3 V0-5e-6 m 3 initial values value unity pao P a part P a pven 1100 P a plv 1008 P a VLv 85e-6 m 3 Vart 6.49e-4 m 3 Vven 3.19e-3 m 3 The period for one heartcycle (T p) is set to 1 second. The number of heartcycles (np) is set to 3, and the number of timesteps (nstep) is set to

17 CHAPTER 2. MATERIALS & METHODS 2.4. SIMULATIONS PERFORMED 2.4 Simulations performed Matlab To see if the elastance model, in combination with the lumped parameter model, is able to calculate the pressure, flow and volumes in the heartcycles, a simple 1D Matlab model is developed and simulated. Sepran, viscosity of 0.04 P a s To see if it was possible to use a more realistic viscosity in the elastance model, in combination with the lumped parameter, than Seabra and da Silva used, a simulation is performed where a viscosity of 0.04 P a s is used. The results are presented in the next chapter. 15

18 Chapter 3 Results 3.1 Matlab In figure 3.1, on the next page, the pressure, flow and volume curves of the third heartcycle of three heartcycles, simulated in Matlab, are displayed. In figure 3.1(a), it is clear to see that p Lv ranges from 1100 P a, when activation is zero, to almost P a, when a maximal activation is reached. During the activation period, it also found that p ao increases significantly, from P a, to P a. There is also a slight increase in p art, of approximately 1000 P a, while the variation in p ven is too small to be seen. In figure 3.1(b), it can be seen that q ao and q ven are almost equally large, as the maximal value of q ao is m 3 /s, and the maximal value of q ven slightly larger than m 3 /s. There is a very small increase in q p during the activation period. Furthermore, it can be seen that there is a small negative q ven on the first timestep, and a small negative q ao around t = 2.3. In figure 3.1(c) the accompanying volumes are displayed. V ven is scaled by subtracting a value of m 3. It can be seen that there is a decrease in V Lv during the activation period to m 3 (50 ml), and that V Lv increases after this to its initial value of m 3 (85 ml). During activation period there is an increase in V art, and a decrease when minimal value of V Lv is reached. V ven increases slowly during activation period, and decreases rapidly when this period has ended. 16

19 CHAPTER 3. RESULTS 3.1. MATLAB (a) Left ventricular pressure (p Lv ), aortic pressure (p ao), arterial pressure (p art) and venous pressure (p ven) against time (t) (b) Aortic flow (q ao), venous flow (q ven) and peripheral flow (q p) against time (t) (c) Left ventricular volume (V Lv ), arterial volume (V art) and venous volume (V ven = V ven ) against time (t) Figure 3.1: Pressure, flow and volume of the third heart cycle in Matlab 17

20 CHAPTER 3. RESULTS 3.2. SEPRAN 3.2 Sepran Ten times physiological viscosity hemodynamics In this section the hemodynamics of the finite element simulation will be presented. In this simulation, the Reynolds number is 150. In figure 3.3, on the next page, the pressure, flow and volume curves of the first heartcycle of the simulation, simulated in Sepran and plotted in Matlab, are displayed. The reason that the first period of the simulation is displayed, is that due to the initial conditions, causing transient effects, no steady state is obtained, because Sepran was not able to compute pressures and flows during the second period in the ejection phase. This is shown in figure 3.2, where pressures and flows are displayed. Figure 3.2: Pressure until Sepran crashed In figure 3.3(a), the maximal values of p Lv and p ao are slightly higher than in figure 3.1(a). The curves of these parameters are also more parabolic. The minimal value of p Lv in this figure is higher than in figure 3.1(a). Furthermore, there is no increase in p art during the activation period, although the rate of decrease is smaller than during the remaining part of the heartcycle. In figure 3.3(b), q ao and q ven are a factor 10 lower than in figure 3.1(b), while q p remains equal. There is, however, a decline in q p. In figure 3.3(c), it can be seen that there is nearly no decrease in V Lv during the activation period, and that at the end of the cycle, V Lv has increased with regard to the initial value. There is no increase in V art, although the rate of decrease is smaller during the activation period than during the remaining part of the heartcycle, and no decrease in V ven. This means that end values differ from the initial values. 18

21 CHAPTER 3. RESULTS 3.2. SEPRAN (a) Left ventricular pressure (p Lv ), aortic pressure (p ao), arterial pressure (p art) and venous pressure (p ven) against time (t) (b) Aortic flow (q ao), venous flow (q ven) and peripheral flow (q p) against time (t) (c) Left ventricular volume (V Lv ), arterial volume (V art) and venous volume (V ven = V ven ) against time (t) Figure 3.3: Pressure, flow and volume in the first heartcycle in Sepran 19

22 CHAPTER 3. RESULTS 3.2. SEPRAN Finite Element Model In this section the results of the finite element simulation will be presented. In figure 3.4, the pressure field of the four phases of the heartcycle are presented. (a) Pressure in isovolumic contraction phase, t=0.05 s (b) Pressure in ejection phase, t=0.148 s (c) Pressure in isovolumic relaxation phase, t=0.364 s (d) Pressure in filling phase, t=0.5 s Figure 3.4: Pressure in the four phases of the heart cycle In the pressure figures, every figure is newly scaled. This means that the darkest red is the highest pressure in every figure, or the darkest blue the lowest, but differs in number. Therefore, no pressure numbers are indicated in the colorbar. The values of p ao, p Lv and p ven can be found in figure 3.3(a), at the accompanying time. In figure 3.4(a), it can be seen that the pressure difference between p Lv and p ao is large, and that p ao is larger than p Lv. This means that the aortic valve is closed and no blood can flow from the ventricle to the aorta. Furthermore, it can be seen that the pressure difference between p ven and p Lv is small, but that p ven is lower than p Lv, which means that the mitral valve is closed as well. In figure 3.3(a), it is shown that on timestep 0.05, p Lv has a value of approximately 2500 P a, p ao P a and p ven 1000 P a. Since p Lv is increasing, the heart is isovolumic contraction phase. In figure 3.4(b), the pressure difference between p Lv and p ven is large, where p ven is smaller than p Lv, which means that the mitral valve is closed. The pressure difference 20

23 CHAPTER 3. RESULTS 3.2. SEPRAN between p Lv and p ao is small, where p Lv is larger than p ao. This indicates that the aorta valve is open and blood will flow from the left ventricle to the aorta. The heart is in ejection phase. On the central part of the curve where the mitral valve is positioned, the pressure is slightly higher (approximately 200 kpa) than the pressure in the left ventricle. At time t= 0.148, from figure 3.3(a), it can be seen that p Lv has a value of approximately 11500, and p ao slightly lower. p ven has value of 1000 Pa. To verify if the heart ejects blood, q ao must be positive at this timestep, which is the case, according to figure 3.3(b). In figure 3.4(c), the pressure difference between p Lv and p ao is small, but p ao is larger than p Lv, which indicates that the aortic valve is closed. The pressure difference between p Lv and p ven is large, where p ven is smaller than p Lv, which indicates that the mitral valve is closed as well. The heart is isovolumic relaxation phase. Furthermore, it can be seen that the pressure on the bottom of the left ventricle is higher than the pressure at the top of the left ventricle. Also, the pressure on the central part of the curve that represents the mitral valve, is slightly higher (approximately 200 kpa as well) than the pressure in the left ventricle. In figure 3.3(a), on timestep 0.364, p Lv has a value of approximately P a, p ao of P a and p ven of 1000 P a. In figure 3.4(d), the pressure difference between p Lv and p ao is large, where p ao is larger than p Lv, thus the aortic valve is closed. The pressure difference between p Lv and p ven is small, where p ven is larger than p Lv. This means that the mitral valve is open and blood flows from the veins into the left ventricle. On a section of the curve that represents the aortic valve, the pressure is lower than the pressure in the left ventricle. On timestep 0.5, as can be seen in figure 3.3(a), p ven has a value of 1000 P a, and p Lv slightly lower. p ao has a value of approximately P a. 21

24 CHAPTER 3. RESULTS 3.2. SEPRAN The next figures display the flow fields in the different phases of the heartcycle. These figures are created using the command quiver in Matlab. A scaling factor of 0.01 is applied. (a) Flow in isovolumic contraction phase, t=0.05 s (b) Flow in ejection phase, t=0.148 s (c) Flow in ejection phase, t=0.3 s (d) Flow in isovolumic relaxation phase, t=0.36 s (e) Flow in filling phase, t=0.54 s (f) Flow in filling phase, t=0.96 s Figure 3.5: Flow in the four phases of the heart cycle 22

25 CHAPTER 3. RESULTS 3.2. SEPRAN It can be seen in figure 3.5(a), that during isovolumic contraction phase, there is no flow through the valves and in the left ventricle. In figure 3.5(b), during ejection phase, just after opening of the aorta valve, there is a small flow from the left ventricle to the aorta. In figure 3.5(c), also during ejection phase, the flow field has become larger and more parabolic than in figure 3.5(b). In figure 3.5(d), during isovolumic relaxation phase, in the middle of the ventricle, flow is upward. On the sides of the ventricle, flow is downward. This results in two small whirls, that rotate in the opposite direction. There is no flow through the aorta of veins, thus both valves are closed. The inflow profile in figure 3.5(e) through the mitral valve disrupts the two whirls found in figure 3.5(d). In figure 3.5(f), during filling phase, the whirls, found in the previous two figures have disappeared, and a new whirl is created on the left side of the ventricle. The flow field in the veins is larger and more parabolic than the flow field found in the previous figure. 23

26 CHAPTER 3. RESULTS 3.2. SEPRAN In the next figure, the backflow near the end of the ejection phase is displayed. Figure 3.6: Backflow just after ejection phase In figure 3.6, backflow occurs on the side of the aorta. In the middle of the aorta the outflow remains. In the left ventricle, two small opposite whirls can be seen. 24

27 Chapter 4 Discussion 4.1 Matlab Figure 3.1(a) shows that by use of the elastance model and the 2-element Wind Kessel model as described in section 2.2, it is possible to reproduce the pressure curves, as displayed in figure 2.3(a). Also, the timesteps are sufficiently small, since there is no visible variation in flows, that is predicted by the model. The maximum and minimum p Lv in figure 3.1(a) coincide with the maximum and minimum p Lv in figure 2.3(a) within a range of 500 P a, and within a range of 1000 P a for the maximum value of p ao. The shapes of p Lv and p ao in ejection phase, however, are different. In figure 2.3(a), the maximum pressure is attained approximately halfway the ejection phase, whereas in figure 3.1(a), this pressure is already obtained at one third of the ejection phase. An explanation for this difference is that the sinusoid used in the model, is different from the curve as presented in figure 2.4(b). The latter has a peak, that lies at the right of the sinusoid that is used in the model. As a result, the maximum p Lv in figure 3.1(a) is attained on an earlier time than the physiological p Lv. This can also be a result of the choice for a two-element windkessel model. It is demonstrated by Stergiopulos et al. [6], that four-element windkessel models produce better estimations of pressures, but it is difficult to estimate the parameters of this windkessel model. It is therefore not applied in this model. The flows in figure 3.1(b) are very similar to the flows displayed in figure 2.3(b). The decline of q ao in figure 3.1(b) is a bit steeper than in 2.3(b). This is also a result of the assumption that the activation is a sinus, while it is displayed in figure 2.4 that this is not exactly the case. The sudden increase in q ven at the end of the heartcycle in figure 2.3(b) is a result of bloodflow as a result of the contraction of the left atrium, which was not modeled in this project and therefore this increase of q ven is not seen in figure 3.1(b). Small backflows of both q ao and q ven are a result of the condition that the valves close when pressures are equal and when flows are zero or negative. This means that at some point, when q ao is positive, a p Lv smaller than p ao is calculated. Then, on the next timestep, a negative q ao is calculated, and a p Lv that remains smaller than p ao. On the next timestep, since the condition in the if-loop now holds, the aorta valve will close and q ao will be zero. This condition also applies for q ven, but then for one timestep, a negative q ven is calculated, while p Lv is larger than p ven. 25

28 CHAPTER 4. DISCUSSION 4.2. SEPRAN Although the backflow, necessary for the closing of the valves is not modeled in this project, it is a result of the model, due to the time scheme that is applied and the condition for the closing of the valves. Figure 3.1(c) produces expected results. During ejection phase, V Lv decreases, while V art and V ven increase. During filling phase, V Lv increases, while V art and V ven decrease. 4.2 Sepran Ten times physiological viscosity hemodynamics First, it must be mentioned that, since Sepran was not able to compute pressures and flows during the second period in the ejection phase, no steady state was obtained, which means that transient effects are studied here. To obtain steady state, more care should have been taken in adjusting the initial values. Unfortunately, there was not enough time to determine good initial values by trial and error, as a simulation time of three heartcycles lies around four and a half hours. It is clear, when comparing figure 3.3(a) with 3.1(a), that for pressure, similar results are obtained in Sepran as in Matlab when a higher viscosity than the physiological viscosity is used. The maximum values of p Lv and p ao in figure 3.3(a) are similar, within a range of 1000 P a. The minimum value of p Lv is similar, within a range of 500 P a. The main difference between figures 3.3(a) and 3.1(a) can be found in the curve of p art, where no significant increase during the ejection phase is noticeable. This can be a result of the curve of q ao, as displayed in figure 3.3(b). In this figure, it can be seen that q ao and q ven are a factor 8 lower than q ao and q ven in figure 3.1(b). This is probably a result of the higher viscosity that is applied here, as can be seen in the Navier-Stokes equation (2.17). This equation shows that when viscosity increases, velocity of the fluid decreases, if the rest of the equation remains constant. It is expected, however, due to the fact that transient effects are discussed, q ao will increase in further periodes, since p art, and therefore p ao, will decrease and the pressure difference between p Lv and p ao will increase. It is also thought that q ven will increase in further periodes, since there is an increase in V ven, according to figure 3.3(c), and thus an increase in p ven. However, when q ao increases, p art will increase during ejection phase and q ao can therefore not increase without limit, equally so for q ven, and it is certain that the physiological values for the flows can not be reproduced in a simulation that uses a viscosity 10 times higher than the physiological viscosity. Due to the diminishing curve of p art and the rising curve of p ven, q p decreases in time. Due to a small q ao and a decreasing q p, V art decreases in time as well, according to equation 2.5 and due to a decreasing q p and a small q ven, V ven increases in time. Finite element model The results of the pressure fields in the four phases of the heart cycle are as expected, except for the slightly higher pressure on the central part of the curve that represents the mitral valve compared to the the ventricular pressure in ejection phase and isovolumic relaxation phase, and the slightly lower pressure 26

29 CHAPTER 4. DISCUSSION 4.2. SEPRAN on the central part of the curve that represents the aortic valve compared with the ventricular pressure in the filling phase. No possible explanation can be found, observing only the pressure fields. It is clear from figure 3.5(a), that there is no flow during isovolumic contraction phase, as expected. From figure 3.5(b), it is clear that blood is pumped out of the ventricle during the ejection phase. Furthermore, it is seen that the outflow curve is not parabolic. This is a result of the fact that it takes time for a flow of uniform velocity, or plugflow, to develop into a poiseuille flow, as displayed in figure 4.1. Thus, it is logic that figure 3.5(b) results in figure 3.5(c). Figure 4.1: A plug flow develops into a parabolic flow The same phenomenon is observed in the figures that represent the filling phase (3.5(e) and 3.5(f)). The higher pressure, on the mitral valve, in comparison with the pressure in the left ventricle, could be a result of the flat flow field on the mitral valve in figure 3.5(a). It is not known, however, how these are correlated. The two vortices in figure 3.5(d), are a result of the backflow of the blood, displayed in figure 3.6. It is expected that the velocity field would diminish during the period, and faster when a high viscosity is used, but this was not observed. The relaxation period must be too short to observe the effect of viscosity on the velocity pattern. The higher pressure on the aorta valve (figure 3.4(c)), in comparison with the pressure in the left ventricle, might be a result of the flat flow field on the aorta valve, but this is not certain or understood. In figure 3.5(e), it can be seen that the flow field in the left ventricle is not at once disrupted by the inflow pattern. It takes a short while for the flow field in the left ventricle, as a result of viscosity, to develop into a clear inflow pattern seconds later, which is not displayed, a clear inflow pattern in the left ventricle is obtained. It can also be seen that the inflow pattern at the veins is not parabolic. In figure 3.5(f), however, the pattern is developed into a parabolic shape, as explained earlier. This development occurs fast, as on t=300 ms, or 0.1 second later, which is not displayed, the curve is reasonably parabolic. The velocity patterns in figure 3.6 could be a result of the Womersley curves as displayed in figure 4.2. In general, it seems that this model describes the flow and pressure patterns well in the different periods of the heartcycle. More care could have been taken in choosing the initial values and parameters, so that a second an third period can be simulated and transient effects have no influence. A simulation of 0.4 P a s was also performed, which led to results of three heartcycles, where an estimation algorithm as used by Seabra and da Silva was not necessary, which was also the case for the first heartcycle of the simulation where a viscosity of 0.04 P a s was used. A simulation of P a s was performed as well, but the model was incomplete and therefore the results incorrect, in combination with 27

30 CHAPTER 4. DISCUSSION 4.2. SEPRAN Figure 4.2: Womersley profiles for backflow at the aorta [4] large variations in p ao and q ao, caused by the model as described in section The model, overall, showed that it is possible to simulate heartcycles, by using more realistic pressure conditions than used by M. Stijnen. To obtain more realistic results, alterations could be made. For instance, natural boundary conditions on the curves that represent the valves can be implemented that indicate no flow (q ao /q ven = 0) through the valves when they are closed. A more complicated method that implements the valves as a moving part of the mesh can be constructed. This has been done by M. Stijnen [1]. 28

31 Chapter 5 Conclusion The elastance model combined with a lumped parameter model produced physiological results in a 1D simulation, that was performed by Matlab. When this model was implemented in a finite element model in Sepran, viscosity of the blood had a large influence on the results. When a viscosity was used that was hundred times higher than the viscosity of blood, Sepran could easily calculate flows, related to pressure. These flows, however, were lower than the physiological flows. When a viscosity of ten times higher than the physiological viscosity was used, Sepran was not able to calculate pressures and flows after the first period of the simulation. Although the results of the first period contain transient effects, they contain some useful information as well. Pressure patterns showed that the model was able to calculate the pressure without problems, except for a small deflection that might be caused by accompanying flow patterns. The velocity fields of the periods showed properties that could be explained by known flow phenomena. With more carefully chosen model parameters, this model might be able to compute further heartcycles as well, to be able to discard transient effects and so that it is possible to perform simulations with a lower viscosity than Seabra and da Silva used. Also, this model showed that it was able to compute the values of flow, without using an estimation algorithm. Unfortunately, this model could not perform simulations with the physiological value of blood, and was thus not able to complement the model developed by Stijnen, but it showed that it is possible to simulate velocity fields by more realistic pressure conditions. This model can be expanded by various algorithms to obtain more realistic results. 29

32 Chapter 6 Recommendations A first adjustment to the model could be to avoid the viscosity condition for the valves and use different boundary conditions for flow on the valves. A boundary condition could be that when the valves are closed, flow should be 0 on the valves and in the tubes. Another and more complicated adjustment to the model could be to implement moving valves in the mesh. 30

33 References [1] J. M. A. Stijnen. Interaction between the mitral valve and aortic heart valve. Eindhoven, Technical University, [2] J. C. Seabra and J. M. Silva. Computational model of elastance-based control concept in left ventricle [3] A. C. Guyton and J. E. Hall. Textbook of medical physiology. W.B. Saunders Company, Philadelphia, 9th edition, [4] F. N. van de Vosse and M. E. H. Van Dongen. Cardiovascular Fluid Mechanics. Eindhoven, Technical University, [5] P. Segers. N. Stergiopulos. N. Westerhof. P. Wouters. P. Kohl and P. Verdonck. Systemic and pulmonary hemodynamics assessed with a lumpedparameter heart-arterial interaction model. Journal of Engineering Mathematica, 47: , [6] N. Stergiopulos. B. E. Westerhof. and N. Westerhof. Total arterial inertance as the fourth element of the windkessel model. the American Physiological Society, pages 81 88,