Numerical Model of the Influence of Shear Stress on the Adaptation of a Blood Vessel BMT 03-35

Transcription

1 Numerical Model of the Influence of Shear Stress on the Adaptation of a Blood Vessel BMT Mirjam Yvonne van Leeuwen Supervisor: Dr. Ir. M.C.M. Rutten Ir. N.J.B. Driessen TUE Eindhoven, The Netherlands 2003

2 Abstract Abstract Atherosclerosis contributes to more mortality and serious morbidity in the Western world than any other disorder. Intervention may be necessary to restore the blood flow. After this intervention the flow and stress in the vessel wall changes and the vessel wall adapts under the influence of these changes. The aim of this study was to make a 3-dimensional numerical model that describes blood vessel growth under the influence of shear stress. The Piosson equation is used to let the shear stress diffuse into the vessel wall and this is related to the growth law. The Cauchy stress for incompressible material that arise in the vessel wall is solved in SEPRAN using the Newton-Raphson method. Two simulations are done. One with a steady flow rate and the other with an unsteady flow rate. In both simulations the radius adapts so with the given flow rate the shear stress is at its baseline level. In the case of the unsteady flow there is a slide delay in the growth. Because of growth in circumferential direction stresses in radial and circumferential direction occur. Using the Poisson equation the diffusion of substances into the vessel wall can be simulated. The program can be expanded by implementing growth under the influence of stresses in the vessel wall so the stresses in a normal vessel wall become homogeneous. Eventually the adaptation of a blood vessel can be predicted after intervention. Eindhoven, September 2003

3 List of contents List of contents ABSTRACT...2 LIST OF CONTENTS...3 CHAPTER 1: INTRODUCTION...4 CHAPTER 2: MATERIALS AND METHODS MATERIALS Finger stretch tensor Constitutive equation Kinematics of the growth Geometry Influence of shear stress on sub-endothelial layers Growth law METHOD...8 CHAPTER 3: RESULTS STEADY FLOW UNSTEADY FLOW...12 CHAPTER 4: DISCUSSION...14 CHAPTER 5: CONCLUSION...15 REFERENCES...16 Eindhoven, September

4 Introduction Chapter 1: Introduction Like all tissues whose physiological function is associated with exposure to mechanical forces, arteries are sensitive to changes in their mechanical environment [3]. The mechanical forces arteries are subjected to are axial forces due to surrounding tissues, periodic transmural pressure and flow-induced shear forces applied at the inner surface. The smooth muscle cells (SMC) are exposed to circumferential and axial strain due to periodic pressure. Changes in the flow-induced shear stress are sensed by the endothelial cells, which cover the inner surface of the vascular wall. When an increase in shear stress is sensed an acute increase in vascular lumen occurs, resulting in a temporary dilation of the artery. The process is mediated by the endothelium through the release of factors such as endothelium-derived relaxing factor (EDRF), which is Nitro Oxide (NO). If the high flow is maintained this response is followed by long term reconstruction of the media due to proliferation and migration of the SMC in such a way that the undeformed lumen of the vessel increases. Over time, the deformed vascular radius increases and tends to restore the normal baseline levels of the mean shear stress of about 1.5 Pa. This vascular enlargement also causes an increase in wall tension and thereby increases the average circumferential stress [3]. A compensatory thickening of the vascular wall was experimentally observed, which seems to restore the normal values of the wall stress [8,9]. Due to atherosclerosis intervention may be necessary. After intervention the flow through the blood vessel and stress in the vessel wall are changed. Under the influence of these changes the blood vessels will adapt at the site where the intervention took place. The main goal of this study is to make a numerical model that predicts how a blood vessel will adapt after intervention. Taber and Rachev both built a 2-dimensional model that describes vessel growth under the influence of stresses. They suggested that the derivative of the circumferential growth is dependent of the circumferential stress and shear stress. In their growth law they included the possibility that a biochemical signal released from the endothelium regulates the response to the shear stress. They described this process phenomenologically, with an exponential function.[1,2,3,4] The formulae they used can not be used for arbitrary geometries, e.g. branched tubes. The aim of this project was to make a 3-dimensional numerical model that describes blood vessel growth under the influence of shear stress. N. Driessen [7] has already built a 3-dimensional numerical model of a blood vessel in SEPRAN. In that program the vessel grows under the influences of circumferential stress induced by the hydrostatic pressure. Driessens program is adapted such that now the vessel grows under the influence of shear stress. Two simulations are done with different flow rates. First the blood vessel is subjected to a steady flow after wards an unsteady flow is imposed. The inner radius is followed over time to see how it changes. Eindhoven, September

5 Materials and Methods Chapter 2: Materials and Methods In this chapter the equations that describe the adaptation of the vascular wall are introduced. A 3-D numerical model of a blood vessel is coupled to the growth of the vascular wall. The numerical model is implemented in SEPRAN. 2.1 Materials In this section the way the flow rate stimulates the vessel growth is described. The trigger for growth is a change in flow rate, from a normal to a higher flow. The difference in shear stress is diffused into the vessel wall that induces the growth of the vessel Finger stretch tensor Each infinitesimal material vector dx 0 in the reference state Ω 0 will stay infinitesimal after deformation but will stretch and rotate to a new vector dx, see figure 2.1. The relation between dx 0 and dx is defined by the deformation gradient tensor F: d x c ( x) d x0 F d x0 = 0 (1) with 0 the gradient operator with respect to the initial reference frame Ω 0. The deformation tensor F is an asymmetric tensor. Multiplying this tensor with its conjugate gives the symmetric Finger stretch tensor B: B = F F c (2) Subtracting the unity tensor from the Finger stretch tensor, B I, gives the Finger strain tensor. This strain measure is chosen such that if no deformation is applied the strain B I = 0 [5,6] Constitutive equation The deformation induces stresses inside the vessel wall that may be described by the Cauchy stress tensor. For incompressible material the Cauchy stress is split into a hydrostatic and a deviatoric part: σ = pι + τ (3) where p is the hydrostatic pressure, and τ is called the extra stress tensor. In this study a vessel wall is assumed to be an incompressible linear elastic solid. Therefore it can be modeled as Neo-Hookean material. For Neo-Hookean materials the extra stress tensor is linearly related to the Finger tensor B according to: τ = G ( B I ) (4) where G denotes the shear modulus Kinematics of the growth The vessel gets stimulated by a change in flow-rate and starts growing until the baseline value of the shear stress is reached, i.e. the vessel deforms from a state Ω 0 to a state Ω described by the deformation tensor F, see figure 2.1. Consider the vessel in configuration Ω 0 that is loaded and stress-free at t=0. Ω 0 is now divided into infinitesimal elements, each of which then grows according to a growth deformation tensor F g. This growth deformation tensor is assumed to be a diagonal matrix with the radial, circumferential and axial stretch ratios on its diagonal, F g = diag[λ gr, λ gφ, λ gz ]. At this point the elements are still stress-free. Next, the Eindhoven, September

6 Materials and Methods dx r 0 x r 0 O x r dx r Ω 0 F=F e F g Ω F g F e Ω * Figure 2.1 Schematic representation of the deformation and its configurations. elements are reassembled into configuration Ω. This last step is described by the elastic deformation gradient F e. The total deformation F is thus defined as: F = F e F g (5) Because Neo-Hookean material behaviour is assumed the Finger tensor B can be replaced by the elastic part of the Finger tensor B e. The deformation tensor F and the growth tensor F g are both known, so the elastic part of the Finger tensor B e can be calculated with: B e = F c 1 c ( F F ) F g g (6) The Cauchy stress can calculated now the elastic Finger tensor B e is known Geometry A vessel in its natural environment is subjected to axial forces due to surrounding tissues. One of these forces is the flow-induced shear stress applied at the inner surface due to friction between the vascular wall and the blood.[1] To simulate this mechanical loading, the vessel is assumed to be an infinite long straight tube. Since the vessel (straight tube) is symmetric, is a quarter of the cross-section from the geometry enough, see figure 2.2. To make it possible that the vessel grows surface 1 and 2 (s1 and s2) can move freely in y and x direction, respectively. A flow is imposed through the tube. For simplicity, the details of the oscillatory flow pattern are ignored, and the blood is treated as a Newtonian fluid. The average fluid shear stress τ at the wall, surface 3 (s3), is approximated by Poiseuille s formula: 4ηQ τ = π 3 r i (7) where η is the blood viscosity, Q is the mean blood flow rate and r i is the inner radius. Q=Q 0 +Q 1 sin(ωt) is used to describe the flow, for the steady flow Q 1 is zero. Eindhoven, S eptember

7 Materials and Methods s1 s3 s2 Figure 2.2 Mesh of a tube that represents a blood vessel Influence of shear stress on sub-endothelial layers When the shear stress on the inner surface is higher than the baseline value of 1.5 Pa, the endothelium secretes substances that trigger the cells in the vascular wall to relax so the vessel dilates. In this report the difference in shear stress is diffused into the vascular wall, using the Poisson equation: ( τ ) = f 2 0 τ (8) with τ the shear stress on the wall, τ 0 the baseline value and f the source term. The source term represents the uptake of the substances by the vascular cells Growth law As noted in the introduction this vessel grows only under the influence of a change in shear stress, i.e. the lumen increases in response to an increasing flow rate. The smooth muscle tone and developmental changes in material properties are neglected in this model. Since smooth muscle fibers are oriented primarily in the circumferential direction, this observation suggests that the muscle fibers grow longer due to increased flow. The growth law is here taken as: λ gϕ t 1 = T τ ( τ τ ) 0 (9) with λ gφ the stretch ratio in circumferential direction, T τ a time constant and τ-τ 0 the solution of the Poisson equation (8). The change in stretch ratio is then put into the growth deformation tensor F g. The derivatives of the stretch ratios in radial and longitudinal direction are taken zero (λ gr = λ gz = 0). Eindhoven, September

8 Materials and Methods Finish Yes τ 0 τ ε No ( τ τ ) = f 2 0 Start τ = 4ηQ 3 πr λ ϕ t = T τ ( ) τ τ 0 t =0 t =1 day Q 0 =14 mm 3 /s Q 1 =0 or 1 mm 3 /s ω = rad/s τ 0 =1.5 Pa η =3.5 Pa s ε = 10-4 Pa T τ =20 days ξ =10-9 f = Q = Q 0 + Q1 t t = t + t r r v x = x + u sin( ω ) λ gϕ = λ gϕ σ r δu r u Yes + λ = 0 ξ gϕ No Figure 2.3 Schematic representation how the program in SEPRAN works. 2.2 Method Blood vessels try to keep the base value of the shear stress at 1.5 Pa. In this model for vessel adaptation the flow is set 50% higher at t=0 than the normal flow. For the simulation with the steady flow rate the flow is changed from 9 mm 3 /s to 14 mm 3 /s in the blood vessel with a diameter of 6 mm. In the second simulation an unsteady flow with a mean value of 14 mm 3 /s and amplitude of 1 mm 3 /s is imposed. The blood vessel also had a diameter of 6 mm. Figure 2.3 shows a schematic representation how the program works. With the imposed flow the shear stress on the inner wall can be calculated. When the difference in shear stress is higher than the criterion ε the difference is diffused into the vascular wall. This diffusion is solved with the Poisson equation (8). The next step taken is the determination of the change of the stretch ratio in circumferential direction ( λ gϕ ). This value depends on the time constant T τ and the solution of the Poisson equation (9). λ gϕ is thus dependent of the position in the vascular wall. The new stretch ration can be calculated and subsequently the new growth deformation tensor F g. Comment: Moet nog een uitleg komen! To determine the deformation the constitutive model (3) has to be solved. This is done by solving the momentum equation: σ = ( pi + G( B I) ) = 0 e (10) Eindhoven, September

9 Materials and Methods The momentum equation is solved using the Newton-Raphson method. With this method nonlinear equations may be solved iteratively. The method tries to find an approximated solution for the problem. It calculates: r r r u = u ˆ + δu where u r is the solution, u s ˆ is the approximated solution and δu r the error. This step is iterated until convergence is reached: r δu r ξ u (11) (12) with ξ the convergence criterion. Now the geometry is deformed to its new state. The next time step is taken where the convergence criterion for the shear stress is tested again. The program iterates until the difference in shear stress satisfies the criterion. For the unsteady flow rate this criterion is not used. In that simulation the program is stopped after 5 cycles. Eindhoven, September

10 Results Chapter 3: Results The results that are obtained from the simulations in Sepran are presented in this chapter. First the results from the situation with the steady flow will be shown and then the results of the unsteady flow will follow. 3.1 Steady flow In the simulation of the steady flow the flow rate is raised from 9 mm 3 /s to 14 mm 3 /s at t=0. This step in flow rate caused a difference in the shear stress that resulted in the growth of the vessel wall in circumferential direction. As a result of this the lumen diameter becomes larger. Figure 3.1 is a graph of the growth of the inner radius of the vessel. The imposed flow is also drawn in the graph Inner radius [mm] Time [days] Radius flow Flow [mm3/s] Figure 3.1 Graph of the growth of the inner radius versus time and the flow versus time. The end state of the vessel growth is reached after 194 days steps. The difference in shear stress is then smaller than 10-4 Pa. The distribution of the displacement in radial direction u r over the vessel wall is presented in figure 3.2. The figure shows that the inner and outer radii have the same displacement and the wall thickness is not changed. The displacement of the inner part of the vessel wall is smaller. As a result from the lumen growth stresses occur in circumferential and radial direction. The distribution of the two extra stresses is plotted in figure 3.3. The extra stress in axial direction is not plotted because there is no extra stress in that direction. The extra stress in radial direction (τ rr ) is positive at the inner surface of the vessel wall and becomes smaller towards the outside. For the extra stress in circumferential direction (τ cc ) is this the other way around smaller values at the inside than at the outer surface. This means that pre-stresses originate in the circumferential and radial direction. Eindhoven, September

11 Results Figure 3.2 Displacement u r (in mm) of the vessel wall after 194 days. Figure 3.3 Extra stress in radial and circumferential direction (in Pa) after 194 days. Eindhoven, September

12 Results Inner radius [mm] Radius Flow Time [days] Figure 3.4 Inner radius-time and flow-time curves. 3.2 Unsteady flow The second simulation is done with an unsteady flow and was stopped after 5 cycli (time step 1050). The growth of the inner radius of the vessel with this unsteady flow is drawn in figure 3.4. The growth follows the flow well with some phase difference. There is only a small delay between the flow and the displacement. This delay is the same for the last 4 cycles. The distribution of the displacement in radial direction after 1050 days is shown in figure 3.5. In this picture it can be seen that the displacement of the inner and outer surface of the vessel wall are larger than the displacement of the inside part of the vessel wall. This is the same as the results for the steady flow simulation. Just as the results with the steady flow the extra stress in circumferential direction is negative at the inner surface and becomes larger to the outer surface. For the extra stress in radial direction is this just the other way around. Figure 3.7 shows how the extra stress in Flow [mm3/s] Figure 3.5 Distribution of the displacement(in mm) in radial direction after 1050 days with a cyclic flow rate. Eindhoven, September

13 Results Figure 3.6 The extra stresses in radial τ rr (right) and circumferential τ cc (left) direction (in Pa) after 1050 days. extra stress cc [Pa] t=839 t=854 t=896 t=925 t=942 t=959 t=1000 t=1031 t= radius [mm] Figure 3.7 Radius versus extra stress in cicumferential direction over one time cycle. circumferential direction is distributed over the vessel wall, nine curves over one cycle are plotted. The delay that is seen in figure 3.4 also is found in this figure. Eindhoven, September

14 Discussion Chapter 4: Discussion With the 3-dimensional model two simulations are done, one with a steady flow and the other with an unsteady flow. In both cases the difference between the actual shear stress and the base value is used to let the vessel grow. The difference is diffused into the vessel wall and taken up by the wall. This can be done because the endothelial cells release growth factors. These growth factors diffuse into the vascular wall and are absorbed by the cells in the wall. To make the program more precise, more knowledge must be gathered about how the growth is influenced by the release of these growth factors. For the deformation as a result of the growth the assumption is made that the shear stress influences only the growth in circumferential direction. This does not have to be the case. Like Taber and Rachev [1,2,3,4] we prescribed the growth deformation tensor as a diagonal matrix with on the diagonal the stretch ratios in radial, circumferential and axial direction. The precise growth of a blood vessel is not yet understood and therefore the growth law is an approximation. Also the changes in material properties due to remodeling are not taken into account. The results of the two simulations both give the expected results. When the flow rate is increased from 9 mm 3 /s to 14 mm 3 /s and the radius is 3 mm. The new radius should become mm. Figure 3.1 shows that this is the case. The simulation with the unsteady flow rate exhibits that the growth of the vascular wall follows the flow rate. The growth in circumferential direction causes a displacement in radial direction. As a result stresses in the radial and circumferential direction are induced. To compensate these stresses the vascular wall thickens. The implementation of the phenomenon can be the next step to make a model of adaptation of a vessel wall. Comment: Ontstaan?? Eindhoven, September

15 Conclusion Chapter 5: Conclusion In this report a start is made to built a program that describes the adaptation of a blood vessel. As the results show the program works for a steady flow rate. The lumen diameter increases when the flow rate is increased. The program also works for an unsteady flow, the lumen diameter adapts to the changes in flow rate. These changes in lumen diameter induce stresses in radial and circumferential direction. By expanding the program growth under the influence of these stresses can be incorporated. The intention for the future is to expand the program such that it predicts the adaptation of a blood vessel after intervention. Eindhoven, September

16 References References [1] Taber LA, Humphrey JD, Stress-Modulated Growth, Residual Stress, and Vascular Hetrogeneity, Journal of Biomechnical Engineering, 2001;123: [2] Taber LA, A model for aortic growth based on fluid shear and fiber stresses, Journal of Biomechnical Engineering, 1998; 120: [3] Rachev A, A Model of Arterial Adaptation to Alterations in Blood Flow, Journal of Elasticity, 2000; 61: [4] Rachev A, Remodeling of arteries in respons to changes in their mechanical environment, Institute of Mechanics, Bulgarian Academy of Sciences, Sofia, Bulgaria. [5] Baaijens F, Numerical Analysis of Continua, Eindhoven University of Technology. [6] Vosse FN van de, Dongen MEH van, Cardiovascular Fluid Mechanics, Eindhoven University of Technology. [7] Driessen NJB, Personal correspondance. [8] Masuda H et al, Artery wall restructuring in response to increased flow, Surg. Forum, 1989; 40: [9] Tulis A et al, Flow-induced arterial remodeling in rat mesenteric vasculature, Am. J. Physiol, 1998; 180: Eindhoven, September