The effect of the LVAD connecting point on the hemodynamics in the aortic arch BMTE 09.16

Transcription

1 The effect of the LVAD connecting point on the hemodynamics in the aortic arch BMTE Author: F.L. Boogaard Supervisor: Dr. Ir. A.C.B. Bogaerds

2 Abstract Left Ventricular Assist Devices (LVADs) are used in patients with end-stage heart failure to unload the failing left ventricle and restore normal blood pressure and flow. In this study the effects of different connecting points of the LVAD in the aortic arch on the hemodynamics in the aortic arch are investigated using a 2D finite element model. To investigate this we have created two different 2D finite element models of the aortic arch with an LVAD connected. One where the LVAD is connected at the ascending aorta, the second where the LVAD is connected at the descending aorta. We assume that the LVAD has taken of the task of the left ventricle. Meaning there is no blood going through the aortic valve. From a model of the complete circulation, which is based on a model by Bovendeerd et al. [1], the inflow boundary conditions of the LVAD and the outflow boundary conditions at the coronary arteries are extracted. From the results it can be concluded that the position of the LVAD has an influence on the hemodynamics in the aortic arch. From the streak-lines it might be concluded that if the LVAD is connected at ascending aorta a circular flow pattern occurs. This could result in a long residence time of particles here (e.g. blood cells). However when the LVAD is connected on the descending aorta only the flow going to the coronary arteries flows through the ascending aorta. This not only causes a long residence time of particles, but also a low velocity at the aortic valve, which could result in the appearance of blood cloths. Also we see that if the LVAD is connected at the descending aorta a much more unstable flow occurs, however when the LVAD is connected at the ascending aorta the flow seems to stabilize in the aortic arch. From the shear-rates we can conclude that if the LVAD is connected at the descending aorta the shear-rates in the aortic arch are much lower, this could result in remodeling of the aortic vessel wall.

3 Contents 1 Introduction 2 2 Methods Finite Element Model Mesh The Navier-Stokes equations Boundary conditions Results Streak-lines Shear-rate Conclusion and discussion Conclusion Discussion Future research

4 Chapter 1 Introduction Heart failure is one of the main health problems in Western countries [3]. There are a lot of treatments for heart failure, varying from a change of life-style, drug treatment, surgical intervention and eventually heart transplantation. There are however some restrictions to heart transplantation. The number of available transplant hearts is limited, not every patient is a suitable candidate for transplantation and the life expectancy is no more then ten years [2]. As a temporary solution it is now possible to implant Left Ventricular Assist Devices (LVADs). These LVADs are used to unload the failing heart by bypassing the left ventricle and restoring normal blood flow and pressure [5]. Although LVADs are already available for quite some time [4] there are still dangers with using a LVAD. Tromboembolism is a big problem. Partially, this problem can be dealt with by improving the biocompatibility of the LVAD. However tromboembolism does not occur purely because of this. Tromboembolism can also occur in stagnant flow. When the LVAD is connected there is a possibility that stagnant flow occurs. When the LVAD has completely taken over the task of the heart, the aortic valve remains closed, which could result in stagnant flow. Also the heart could receive less nutrients because the coronary flow might be changed. Finally because of a change in hemodynamics the stress on the vessel wall may change. This could lead to a local adaptation of the endothelial cells in the vessel wall [8]. 2

5 An LVAD can be implanted in two ways, either through a left thoracotomy with the outflow-graft anastomosis to the descending thoracic aorta or through a midline sternotomy with the anastomosis to the ascending aorta [6]. Kar et al [6] showed that the connecting point of the LVAD has an influence on the hemodynamics in the aortic arch. However the boundary conditions were not specified in this study. Also they just looked at the streamlines and not at the shear-rates in the aortic arch. To look at the differences in hemodynamics we created two 2D Finite Element (FE) models. In Section 2.1 we discuss the FE model created for these simulations. The LVAD we simulate is the HeartMate II left ventricular assist system [5] (shown in Figure 1.1). For this LVAD we have a model based on a model by Bovendeerd et al. [1] that has proven to simulate the human circulation when an LVAD is connected in a study by Cox (Msc Thesis) [7]. This is discussed in Section 2.2. In Section 3 we look at the results of our simulation by visualizing the streak-lines and the shear stresses in the system. Figure 1.1: Battery-powered HeartMate II LVAD System. [5] 3

6 Chapter 2 Methods To investigate the difference in flow patterns we use an Finite Element (FE) model. To simplify the problem a 2D model is used. The boundary conditions for this model are extracted from a model of the complete circulation, which is based on a model by Bovendeerd et al. [1]. From this, the LVAD flow and the coronary flows are extracted and used as boundary conditions. 2.1 Finite Element Model Mesh We devoloped two two-dimensional meshes of the humane aortic arch. The aortic arch was modeled with a length (l a ) of 4.6 cm. The diameter of the aorta (d ao ) was set at 2 cm, the diameter of the coronary arteries (d c a ) at 3 mm. The aortic arch itself was modeled with an inner curve diameter (d c ) of 2 cm. In the first simulation the LVAD was connected at a height of 1 cm above the coronary arteries on the ascending aorta. In the other the LVAD was connected at the beginning of the descending aorta, just under the aortic arch. The LVAD was given a diameter (d LVAD ) of 1 cm. 4

7 Figure 2.1: The curves and the mesh made of the human aortic arch with Γ 1, Γ 2 and Γ 4 the outflow boundaries, Γ 3 the inflow boundary and Γ 5 the boundary where v = 0. The left figures show the LVAD connected at the ascending aorta, the figure on the right shows the LVAD connected at the descending aorta The Navier-Stokes equations For these meshes we used Taylor-Hood elements with 9 points per element for the velocity and 4 points per element for the pressure. Figure 2.2: Taylor-Hood elements with 9 points per element for the velocity and 4 points per element for the pressure In these Taylor-Hood elements the Navier-Stokes equations together with the continuity equation is solved according to ρ( v t + v v ) = p + η 2 v, (2.1) v = 0. (2.2) 5

8 Here v is the velocity of the fluid, p the pressure, ρ the density and η the dynamic viscosity. In dimensionless form these equations become: with Sr = L ρv L, Re = T V η Sr v t + (v )v = p + 1 Re 2 v, (2.3) v = 0, (2.4) We take ρ and η of blood at 1025 k g /m 3 and Pa.s and assume them to be constant. This gives a Reynolds number of about 1000 in the LVAD. We calculated the Reynolds number using a length scale of L = O(1c m ). The velocity was calculated from the flow calculated by the model in Section 2.2. However in our simulations the Reynolds number had to be reduced to approximately 500 to be succesfull, this was done by reducing the density of blood. At higher Reynolds numbers the flow became turbulent and this was not possible to simulate using our current FE model. 6

9 2.2 Boundary conditions The boundary conditions are extracted from a model which is based on a model by Bovendeerd et al. [1] (Figure 2.3) This is a complete model of the circulation. It has proven to calculate the parameters of a LVAD in the circulation in a study by Cox (Msc Thesis) [7]. Figure 2.3: Circulatory model. LV and RV are the left and right ventricle, and MV, AV, TV and PV are the mitral, aortic, tricuspid and pulmonic valve. L a r t,li and L a r t,ri are the inertia of the blood in the aortic and pulmonary artery elements; R a r t,li, R a r t,ri, C a r t,li and C a r t,ri are the resistance and compliance of the aortic an pulmonary artery elements. R p,l and R p,r are the peripheral resistances of the systemic and pulmonary circulations, and R v e n,l, R v e n,r, C v e n,l and C v e n,r are the resistance and compliance of the systemic and pulmonary veins. L v e n,l and L v e n,r are the inertia of the blood in the veins. R a r t,c, R m y o,1, R m y o,2 and R v e n,c represent the coronary arterial, intramyocardial and venous resistances, C a r t,c, C m y o,c and C v e n,c are the coronary arterial, intramyocardial and venous compliance, and p i m is the intramyocardial pressure. The HeartMate II LVAD [5] is simulated in this simulation. The pump speed of the LVAD is set at RPM. This generates an average flow through the LVAD of 7.1 l /m i n, which is enough flow to completely close the aortic valve throughout a heartbeat. Thus simulating the case where the LVAD has completely taken over the task of the left ventricle. Figure 2.4 shows the calculated flows, q Cor A is the flow through both coronary arteries (Γ 1 and Γ 2 in Figure 2.1). This flow has to be divided by two to get the flows through each separate coronary artery. q LVAD is the flow through the LVAD (Γ 3 ) and q Aov a l v e is 7

10 Figure 2.4: Boundary conditions calculated using the modified model of Bovendeerd et al. the flow through the aortic valve, as illustrated, this flow is zero. At the outflow (Γ 4 ) stress free boundaries are applied. Figure 2.5 shows the boundary conditions applied. Figure 2.5: Applied boundary conditions at the first geometry With the model of Bovendeerd et al. 100 heartcycles were simulated to acquire a stable result. The last four cycles were used as input for the FEM simulation. The time-step ( t ) used in the simulations was 0.001s. 8

11 Chapter 3 Results To view the results of our simulations streak-lines and the shear-rate have been calculated at set points in time. Figure 3.1 shows these points in time and the corresponding flow through the LVAD at that time. Figure 3.1: The points in time where snapshots are taken with the corresponding flows through the LVAD 9

12 3.1 Streak-lines Figure 3.2 shows the streak-lines for the LVAD connected at the ascending aorta. This figure shows a circular velocity profile developing which seems to be constant for the duration of a heartcycle. The circular motions in the aortic arch occur because of the deceleration of the fluid. Figure 3.2: The streak-lines of the first problem at t = 0.02s ; q LVAD = 6, 4l /m i n, t = 0.57s ; q LVAD = 12l /m i n and at t = 0.92s ; q LVAD = 5, 9l /m i n Figure 3.3 shows the streak-lines when the LVAD is connected at the descending aorta, we see a irregular flow developing at the descending aorta. Also only few streak-lines go into the ascending aorta, this is because the only flow going through here is the flow going out through the coronary arteries. 10

13 Figure 3.3: The streak-lines of the second problem at t = 0.02s with q LVAD = 6, 4l /m i n, t = 0.57s with q LVAD = 12l /m i n and at t = 0.92s with q LVAD = 5, 9l /m i n 3.2 Shear-rate Figure 3.4 and Figure 3.5 show the shear-rates calculated at respectively t = 0.57 and at t = 0.92s for the different geometries. These figures show that the shear-rates at the ascending aorta are higher when the LVAD is connected at the ascending aorta and that it is higher at the descending aorta when the LVAD is connected at the descending aorta. 11

14 Figure 3.4: The shear-rates at t = 0.57s with q LV AD = 12l /m i n in the two different geometries Figure 3.5: The shear-rates at t = 0.92s with q LV AD = 5, 9l /m i n in the two different geometries 12

15 Chapter 4 Conclusion and discussion 4.1 Conclusion When the LVAD is connected at the ascending aorta, the streak-lines show (Figure 3.2) a circular motion at the base of the aortic arch. This could imply that a stagnant flow is appearing here and thus the residence time of particles there is long and blood cloths could appear. This also could cause the coronary arteries not getting enough nutrients. The streak-lines in the geometry where the LVAD is connected at the descending aorta (Figure 3.3) does not show these circular motions, but there is not much flow going trough the ascending aorta. The only flow going through the aortic arch is the flow going to the coronary arteries. This could also result into a long residence time of particles, but because the picture of the shear-rates shows a low shear-rate in the base of the descending aorta (Figures 3.4 and 3.5), it may be concluded that the velocity of the blood here is low. This is more likely to cause blood cloths then the circular motion in the first geometry since here the blood is still moving, although in a vortex. Of course the low flow going through the ascending aorta also results in less nutrients getting to the coronary arteries. From figure showing the shear-rates (Figures 3.4 and 3.5), it can be concluded that when the LVAD is connected at the descending aorta the shear-rates get much lower then when connected at the ascending aorta. This could result in remodeling of the vessel wall of the aortic arch because less stress is applied then normal and thus making the vessel wall weaker. 4.2 Discussion The first point of discussion is that blood is assumed to be a Newtonian fluid in these simulations, this is of course not accurate. In blood at high shear-rates shear thinning 13

16 occurs, this will probably have an influence on the simulations done here, since we assumed the dynamic viscosity of blood at high shear-rates throughout the simulations. However, we see low shear-rates at the aortic arch, this would lead to a higher dynamic viscosity. Secondly the Reynolds number is decreased to about 500. This decreased Reynolds number will probably have an influence on the hemodynamics. At higher Reynolds number turbulence will probably occur, this could not be simulated using our current FE model. Also the boundary condition applied on the coronary arteries (Γ 1 and Γ 2 ) is the flow through the coronary arteries. This may be inaccurate because the flow through the coronary arteries will probably be influenced by the hemodynamics in the region of the aortic valve. This will probably be most of a problem when the LVAD is connected at the descending aorta because the blood then has to travel back trough the ascending aorta to the coronary arteries. This might be solved by using the pressure at the coronary arteries and at the end of the aortic arch as boundary conditions instead of the flow. Finally in these models the brachiocephalic artery, left common carotid artery and left subclavian artery are neglected. The main reason for this is that we have no boundary conditions for these arteries (eg. pressures or flows). This also will have an influence when the LVAD is connected at the descending aorta, because blood has to travel through the aortic arch to get to the coronary arteries. 4.3 Future research For future research it would be interesting to implement the residence time of particles in the aortic arch. This could give an insight of the probability that blood cloths will occur in areas of stagnant flow. This could be done by implementing the convectiondiffusion equation ( c t + u c = ε 2 c ) for each time step in the problem. It would also be interesting to change the boundary conditions at the coronary arteries and at the end of the aortic arch to the pressure instead of the flow. To get these boundary conditions, the arterial resistance in model by Bovendeerd et al. [1] would have to be altered to calculate the pressure at the beginning of the coronary arteries. This could be done by creating and extra part in the arterial resistance which simulates the pressure drop over just a small length of the coronary arteries. Finally it would of course be better to create a 3D model of the aortic arch instead of a 2D model. In this 2D model the secondary flows occurring in the aortic arch are not simulated. These will have an influence in the flow profile developing here. 14

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