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Turbulence Significantly Increases Pressure and Fluid Shear Stress in an Aortic Aneurysm Model under Resting and Exercise Flow Conditions
Turbulence Significantly Increases Pressure and Fluid Shear Stress in an Aortic Aneurysm Model under Resting and Exercise Flow Conditions Khalil M. Khanafer, PhD,1 Joseph L. Bull, PhD,1,2 Gilbert R. Upchurch Jr., MD,2 and Ramon Berguer, MD, PhD,1,2 Ann Arbor, Michigan
The numerical models of abdominal aortic aneurysm (AAA) in use do not take into account the non-Newtonian behavior of blood and the development of local turbulence. This study examines the influence of pulsatile, turbulent, non-Newtonian flow on fluid shear stresses and pressure changes under rest and exercise conditions. We numerically analyzed pulsatile turbulent flow, using simulated physiological rest and exercise waveforms, in axisymmetric-rigid aortic aneurysm models (AAMs). Discretization of governing equations was achieved using a finite element scheme. Maximum turbulence-induced shear stress was found at the distal end of an AAM. In large AAMs (dilated to undilated diameter ratio ¼ 3.33) at peak systolic flow velocity, fluid shear stress during exercise is 70.4% higher than at rest. Our study provides a numerical, noninvasive method for obtaining detailed data on the forces generated by pulsatile turbulent flow in AAAs that are difficult to study in humans and in physical models. Our data suggest that increased flow turbulence results in increased shear stress in aneurysms. While pressure readings are fairly uniform along the length of an aneurysm, the kinetic energy generated by turbulence impacting on the wall of the distal half of the aneurysm increases fluid and wall shear stress at this site. If the increased fluid shear stress results in further dilation and hence further turbulence, wall stress may be a mechanism for aneurysmal growth and eventual rupture.
INTRODUCTION Of the known causes of aortic aneurysmsdproteolytic degradation of aortic wall connective tissue, inflammation and immune responses, molecular genetics,1 and mechanical wall stressdthis study focuses on the latter. The epidemiology of infrarenal abdominal aortic aneurysms (AAAs) has been well described, but the mechanisms leading to accelerated aneurysm 1 Department of Biomedical Engineering, Vascular Mechanics Laboratory, University of Michigan, Ann Arbor, MI. 2 Section of Vascular Surgery, Vascular Mechanics Laboratory, University of Michigan, Ann Arbor, MI.
Correspondence to: E-mail: [email protected] Ann Vasc Surg 2007; 21: 67-74 DOI: 10.1016/j.avsg.2006.10.009 Ó Annals of Vascular Surgery Inc. Published online: January 12, 2007
growth and eventual rupture are poorly understood. We have a rudimentary ability to predictd with the exception of some relationship to aortic diameterdwhich aneurysms are most likely to expand further and rupture. A number of patientspecific factors also have been shown to be associated with rupture of small aneurysms. Cronenwett et al.2 documented that initial anteroposterior diameter, diastolic hypertension, and the presence of chronic obstructive pulmonary disease significantly enhance the rupture of small AAAs. Some studies have focused on the characteristics of aortic flow that may contribute to AAA rupture. Raghavan et al.,3 using finite element analysis from human computed tomographic (CT) scan data, demonstrated that increased tensile stress along the aortic wall at various points contributes to an increased risk of aneurysm rupture, independent of aortic diameter. Fillinger et al.4,5 conducted 67
68 Khanafer et al.
finite element analysis to calculate peak wall stresses for aneurysms in vivo and to analyze rupture risk over time in patients under observation. Their results showed that those AAAs that ruptured had higher peak wall stresses than those that went on to elective repair. Further, Fillinger et al.4,5 showed that peak wall stress is a better predictor of rupture than diameter. This was validated by Venkatasubramaniam et al.6 using finite element analysis and three-dimensional geometries of AAAs derived from CT scans. In the present study, we defined the effects of local turbulence in local fluid shear stress in AAA. From a flow mechanics standpoint, one can predict that changes in the geometry of an aneurysm will alter the patterns of blood flow and, hence, the hemodynamic stresses on the aortic wall. It is presumed that aneurysmal dilation and risk of rupture depend on the hemodynamic stresses associated with (or leading to) changes in the aortic wall that affect its mechanical integrity. Previous studies examining the effects of flow through AAAs assumed laminar flow,7-12 which typically occurs for Reynolds numbers 2,000, although instability leading to the onset of intermittent turbulence in AAA is seen with Reynolds numbers between 2,000 and 2,600.7,13-15 Other studies have noted maximum fluid shear stress at the distal end of aneurysms under laminar flow conditions.16-18 Additionally, large aneurysms exhibit more turbulence than smaller aneurysms at their distal end during diastole, under physiological pulsatile flow.13 The analysis of flow patterns and the mechanical stresses within aneurysms can lead to better understanding of the mechanical forces associated with dilation of the aortic wall and its eventual rupture. The utility of numerical simulation lies in its ability to study in detail conditions that are difficult or impossible to measure in humans or in animal models of AAA. The objective of the present study was to study the flow and mechanical changes that take place in our computational model of AAA under turbulent, pulsatile, non-Newtonian flow at rest and during exercise.
METHODS Turbulent, non-Newtonian flow was studied in a rigid-walled axisymmetric aneurysm as depicted in Figure 1. Throughout this study, we used an axisymmetric representation of aneurysms. The x axis represents centerline flow. The following nomenclature was used: L, length of the aneurysm; D,
Annals of Vascular Surgery
aneurysm maximum diameter; d, undilated aortic diameter. The diameter and length of aneurysms were designated by dimensionless parameters D/ d and L/d, respectively. Governing equations for this study imbedded in our model were (1) the Reynolds time-averaged equations of fluid motion, (2) the kinetic-energy dissipation model (k-3) to simulate the turbulence characteristics of the convective flow, and (3) the Boussinesq eddy-viscosity model. The details of our mathematical model and boundary conditions have been presented elsewhere.19 The exercise20,21 and resting22 arterial waveforms depicted in Figure 2 were used at the inlet to simulate in vivo flow conditions. The time-averaged Reynolds number (Remean ¼ rumNm d) in Figure 2 was based on the time-averaged inlet mean velocity (um ) as reported by Pedersen et al.20,21 for exercise flow condition and Long et al.22 for resting flow condition. Maximum Reynolds number is based on maximum inlet mean velocity (Remax ¼ rummax d). mN is the N viscosity at the infinite shear rate and d is the undilated diameter. A finite element formulation based on the Galerkin method23-25 to solve the governing equations subject to the boundary and initial conditions was employed. The highly coupled and nonlinear algebraic equations resulting from the discretization of the governing equations was solved using an iterative solution scheme. Extensive numerical refinement was performed to attain grid-independent results. When the relative change in variables between consecutive iterations was less than 104, we assumed convergence.
Validation of Our Model In order to benchmark our numerical method, we validated our model against experimental and numerical results published in the literature26,27 for non-Newtonian turbulent flow through a pipe (Table I). Our model used a modified Reynolds number (Rem) that takes into account the powerlaw variation of viscosity. Our results were in excellent agreement (Table I) with published numerical26 and experimental27 results for the power-law model. As an additional check on the accuracy of the present results, Table II shows a comparison of turbulent flow data in a pipe, comparing the results from our model and from other numerical studies28 for Re ¼ 4,910. Correlation is excellent with direct numerical simulation for both Cartesian and cylindrical formulations of the Navier-Stokes equations.
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Turbulence increases pressure and fluid shear stress 69
Fig. 1. AAM (D/d ¼ 3.33). r, radial dimension; X, axial position; S, a position along the aneurysm wall length.
0.8 Remax = 3300, Remean = 1770 0.6 Remax = 2904, Remean = 794
u(t)
0.4 Exercise 0.2 Rest 0.0
-0.2
0.0
0.2
0.4
0.6
Time (s)
RESULTS Comparison of Fluid Shear Stress and Relative Pressure between Laminar and Turbulent Flows at Exercise for Large Aneurysms (D/d [ 3.33) The significance of turbulent flow on the relative pressure and fluid shear stress on the large aneurysm (D/d ¼ 3.33) compared with the laminar flow condition using the exercise waveform is demonstrated in Figures 3 and 4. A normal aortic diameter (d) is 2-2.5 cm depending on age and gender, however, so the aneurysm diameter (D) would be
0.8
1.0
Fig. 2. Pulsatile inlet mean velocity for exercise and rest flow conditions.20-22
6.7-8.3 cm. A 6.7 cm AAA for a woman or an 8.3 cm AAA for a man would be large in the practice of any clinician. Pressure values presented here are relative and correspond to the difference between the instantaneous pressure on the AAA wall and the instantaneous pressure at the exit of the computational model. Figure 3 shows that pressure along the aneurysm wall is fairly uniform at any given phase of the cycle for both laminar and turbulent flow conditions. A significant pressure difference on the aneurysm wall between laminar and turbulent is depicted in Figure 3. Furthermore, turbulent flow induces higher fluid shear stress at
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Table I. Pipe flow simulations: non-Newtonian flow (Rem ¼ 1.07 105) Vmax =V
Table II. Pipe flow simulations: Newtonian flow (Re ¼ 4,910)
Error (%) Vmax =Vinlet
Present Numerical26 Experimental27 2n
Rem ¼
1.16 1.15 1.21
rV dn n n1 ; Kð0:75þ0:25 n Þ 8
0.86 4.3
Present 1.29 DNS (Cyl.)28 1.302 DNS (Cart.)28 1.297
Vmax =Vshear
Vinlet =Vshear
Vmax =Vshear error (%)
18.47 18.98 18.91
14.315 14.58 14.58
2.69 2.38
n ¼ 0:825; m ¼ K g_ n1 . DNS, direct numerical simulation; Cyl, cylindrical; Cart, Cartesian; Vinlet , inlet velocity; Vmax , maximum (centerline) qffiffiffiffi velocity; Vshear , shear velocity ( trw ) (tw , wall shear stress).
Relative Pressure along AAA (Pa)
1200
Peak Systole 900
Exercise Flow Condition 600 Peak Diastole
300
Remax = 3300
Laminar Turbulent 0 0.0
2.0
4.0
6.0
Aneurysm wall length (cm)
the distal end of an AAA during peak systole compared with laminar flow, as depicted in Figure 4. Thus, an increase in turbulence dramatically increases relative pressure by 32.5% and fluid shear stress by 23.6% at the distal end of the AAA (for D/d ¼ 3.33 during peak systole). This increase in peak fluid shear stress corresponds to 20 dyn/cm2 which likely will have a significant effect on the endothelial cells. As such, areas of low shear stress (2 dyn/cm2) experience upregulation of vascular adhesion molecule-1 (VCAM-1), while areas of high shear stress (20 dyn/cm2) experience downregulation of VCAM-1.29 An inverse relationship between pressure and fluid shear stress in the axisymmetric-rigid aortic aneurysm model (AAM) for turbulent flows at peak systole during exercise is demonstrated in Figure 5. This correlation is consistent with Bernoulli’s
8.0
10.0
Fig. 3. Comparison of the relative pressure between laminar and turbulent flow conditions on the AAM during exercise for a bulge size of D/d ¼ 3.33.
equation, where the pressure decreases as the velocity increases for incompressible flow. Comparison of Streamline and Pressure Contours at Rest and during Exercise for Large Aneurysms (D/d [ 3.33) In all plots (Fig. 6), we see an internal jet of fluid surrounded by a recirculating vortex moving about the centerline (x axis) and impinging on the distal half of the wall in the aortic model. At peak systole (time ¼ 0.345 sec [rest] and 0.143 sec [exercise]), a small recirculating vortex develops at the proximal end of the AAA at rest and during exercise, while a large recirculating flow region fills the aneurysm sac during peak diastole for both waveforms. The only difference in the flow patterns in AAA at rest and exercise is that the recirculation
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Turbulence increases pressure and fluid shear stress 71
12.0 Exercise Flow Condition 10.0
Fluid Shear Stress (Pa)
Relative deviation = 23.6% 8.0
6.0
4.0 Laminar Turbulent
Remax = 3300
2.0
0.0
-2.0 0
2
4
6
8
10
Aneurysm wall length (cm)
Fig. 4. Comparison of the fluid shear stress between laminar and turbulent flow conditions on the AAM during exercise for a bulge size of D/d ¼ 3.33.
1200
900 Remax = 3300 6 600
2
Fluid Shear Stress (Pa)
Relative Pressure along AAA (Pa)
10
300
0 0.0
2.0
4.0
6.0
Aneurysm wall length (cm)
vortices are faster in the latter case. Further, a large recirculating flow region results in large residence time, which could be a contributing factor for thrombosis in the aneurysm. Normal and tangential stresses may be involved in the local dilation and eventual rupture of aneurysms. Fluid shear stress is defined as a measure of the tangential forces per unit area generated by the flow stream on the walls of the AAM. Figure 7 illustrates fluid shear stress values in the AAM at different points of the cardiac
8.0
-2 10.0
Fig. 5. Inverse relationship between relative pressure and fluid shear stress for turbulent flow during exercise (D/d ¼ 3.33).
cycle. This figure highlights the significant differences found between rest and exercise. Peak values are found at the distal half of the AAM during rest and exercise conditions (Fig. 7). Exercise waveforms generate larger fluid shear stresses than resting waveforms. We have reported that the distal half of an AAM exhibits higher fluid shear stress than other locations30 under laminar flow. Thus, increased shear stress will decrease thrombus deposition.
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Rest
Exercise (D/d = 3.33)
(Peak Systole)
max
: 1.32×10–4
max
: 1.99×10–4
min
:0
min
:0
(Peak Diastole)
max
: 8.02×10–5
max
: 1.42×10–4
min
:0
min
:0
Cardiac cycle period = 1.0 sec
Cardiac cycle period = 0.5 sec
Fig. 6. Temporal variation of streamlines for AAM under rest and exercise conditions. This is an axisymmetric representation of an aneurysm. The centerline of flow is the x axis. This two-dimensional representation
is a longitudinal view of the aneurysm with flow from left to right. The isobars represent sites of equal pressure. The streamlines represent flow fields of equal velocity (J).
12.0 Exercise
Time = 0.143 s (peak systole)
Rest
Fluid Shear Stress (Pa)
9.0 Time = 0.342 s (peak systole) 6.0
3.0 Time = 0.794 s (peak diastole) 0.0
Time = 0.42 s (peak diastole) -3.0 0.0
2.0
4.0
6.0
Aneurysm wall length (cm)
DISCUSSION Turbulence induced by sudden expansion of the flow stream results in additional stresses acting on the aneurysm wall that may be responsible for further aortic dilation. Dilation results in further radial expansion of the flow stream and greater turbulence. The latter then becomes a self-perpetuating mechanism for aneurysm dilation. To date, the role
8.0
10.0
Fig. 7. Comparison of fluid shear stress on the AAM between exercise and rest flow conditions.
of turbulent flow in AAA growth and rupture has not been well studied. Turbulent flow simulation should be included in the study and modeling of aneurysms because it increases fluid and wall stresses. Turbulent flow results in kinetic forces being directed toward the vessel wall instead of parallel to the wall, as seen under laminar flow conditions. Because ours is a rigid model, this deformation of the wall secondary to kinetic forces is not displayed.
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This study documents that higher fluid shear stress also occurs in the distal half of the aneurysm at peak systolic velocity under turbulent nonNewtonian flow conditions. At this time and at this site, turbulence is greater and the impact of this kinetic energy on the wall is added to the wall tension generated by the intraluminal pressure. Consequently, turbulence, by increasing wall tension and inducing wall vibration, may induce dilation of an aneurysm. Cyclic turbulent stresses are known to alter the structure and integrity of the arterial wall. Large eddies induce vibrations at frequencies associated with the dilation of arteries. The smallest turbulent eddies31-33 (Kolmogorov microscale), which are a function of the kinematic viscosity of blood and turbulent dissipation rate, can mechanically damage the blood cells and the components of the arterial wall. As such, turbulent eddies in the near-wall region of the distal half of the aneurysm generate steep velocity gradients, which induce further fluid shear stress. Giddens et al.34 showed that the abdominal aorta regulates its diameter chronically to sustain the shear stress below the physiologically optimal value of 12 ± 2 dyn/cm2. Turbulent shear stress near the distal end of the AAM is significantly higher than this optimal value. Thus, the long-term effects of turbulent flow result in arterial wall damage and enlargement of the aortic diameter. There is previous experimental work in the literature on transitional and turbulent flow of Newtonian fluids in aneurysms. Yip and Yu35,36 studied oscillatory Newtonian flows in an axisymmetric aneurysm using laser-Doppler anemometry and documented a transition to turbulence in the deceleration phase of flow. Onset of turbulence was also observed in steady flows at high Reynolds numbers.35,36 Budwig et al.7 demonstrated numerically that turbulence in axisymmetric rigid aneurysms was intermittent for Reynolds numbers between 2,000 and 2,500, assuming steady flows. Bluestein et al.37 pointed out that under fully turbulent, steady-state Newtonian flow (Re ¼ 3,600) fluid shear stress values were one order of magnitude larger than those for laminar flow. Egelhoff et al.13 observed a transition to turbulent flow in moderately large aneurysms under exercise flow conditions during late systole. Fukushima et al.10 showed that flow becomes unstable during deceleration in an AAM model. Egelhoff et al.13 confirmed this phenomenon experimentally. Under exercise and resting waveforms, the vortex is observed during late systole and early diastole and moves toward the distal half of the AAM. We analyzed physiological pulsatile turbulent flow in large AAMs at rest and during exercise.
Turbulence increases pressure and fluid shear stress 73
Under the exercise flow condition, a large primary vortex is exhibited in the distal half of the aneurysm along with a small vortex at the proximal end of a large bulge during systolic peak flow. However, at rest, one primary vortex is developed at the proximal end of the aneurysm. The major assumptions used in the present study include the anatomic dimensions of the AAA model, rigid wall approximation, and a fusiform AAA model. These assumptions represent a reasonable first approximation to the actual flow and hemodynamic conditions in the abdominal aorta. The dimensions of the AAA model used in this investigation were based on those reported in clinical studies. The use of rigid wall assumption for flow studies in idealized AAA models should be viewed as a first approximation. As such, we anticipate reduction in fluid shear stress when assuming a compliant wall compared with a rigid wall model based on clinical observation of vessel wall deformation during the cardiac cycle.
CONCLUSION We studied pulsatile turbulent flow in large AAMs at rest and during exercise. The large increase in pressure recorded along the aneurysm wall during maximum systolic flow acceleration is likely a factor in determining the risk of aneurysm rupture. In addition, the turbulence induced by sudden expansion of the flow stream generates higher fluid shear stresses on the aneurysmal wall. These additional stresses may be responsible for further wall dilation that will eventually result in greater turbulence, possibly a self-perpetuating mechanism for aneurysmal growth. This work contributes to the understanding of the mechanics of aneurysm growth and rupture by elucidating the stresses that may be attributed to turbulence. Numerical models of AAA should include the mathematical representation of viscosity and turbulence to better model the dynamics involved in blood flow through aneurysms.
This work was supported by the Frankel Vascular Research Fund.
REFERENCES 1. Wassef M, Baxter BT, Chisholm RL, et al. Pathogenesis of abdominal aortic aneurysms: a multidisciplinary research program supported by the National Heart, Lung, and Blood Institute. J Vasc Surg 2001;34:730-738.
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2. Cronenwett JL, Murphy TF, Zelenock GB, et al. Actuarial analysis of variables associated with rupture of small abdominal aortic aneurysms. Surgery 1985;98:472-483. 3. Raghavan ML, Vorp DA, Federle MP, Makaroun MS, Webster MW. Wall stress distribution on three-dimensionally reconstructed models of human abdominal aortic aneurysm. J Vasc Surg 2000;31:760-769. 4. Fillinger MF, Marra SP, Raghavan ML, Kennedy FE. Prediction of rupture risk in abdominal aortic aneurysm during observation: wall stress versus diameter. J Vasc Surg 2003; 37:724-732. 5. Fillinger MF, Raghavan ML, Marra SP, Cronenwett JL, Kennedy FE. In vivo analysis of mechanical wall stress and abdominal aortic aneurysm rupture risk. J Vasc Surg 2002;36:589-597. 6. Venkatasubramaniam AK, Fegan MJ, Mehta T, et al. A comparative study of aortic wall stress using finite element analysis for ruptured and non-ruptured abdominal aortic aneurysms. Eur J Vasc Endovasc Surg 2004;28:168-176. 7. Budwig R, Elger D, Hooper H, Slippy J. Steady flow in abdominal aortic aneurysm models. ASME J Biomech Eng 1993;115:419-423. 8. Asbury CL, Rwberti JW, Bluth EI, Peattie RA. Experimental investigation of steady flow in rigid models of abdominal aortic aneurysm. Ann Biomed Eng 1995;23:29-39. 9. Perktold K. On the paths of fluid particles in an axisymmetric aneurysm. J Biomech 1987;20:311-317. 10. Fukushima T, Matsuzawa T, Homma T. Visualization and finite element analysis of pulsatile flow in models of the abdominal aortic aneurysm. Biorheology 1998;26:109-130. 11. Taylor TW, Yamaguchi T. Three-dimensional simulation of blood flow in an abdominal aortic aneurysm-steady and unsteady flow cases. J Biomech Eng 1993;116:89-98. 12. Elger DF, Slippy JB, Budwig RS, Kraishi TA, Johansen KH. A numerical study of the hemodynamics in a model AAA. Biomed Fluids Eng 1995;212:15-22. 13. Egelhoff CJ, Budwig RS, Elger DF, Khraishi TA, Johansen KH. Model studies of the flow in abdominal aortic aneurysms during resting and exercise conditions. J Biomech 1999;32:1319-1329. 14. Peattie RA, Riehle TJ, Bluth EI. Pulstaile flow in fusiform models of abdominal aortic aneurysms: flow fields, velocity patterns and flow-induced wall stresses. ASME J Biomech Eng 2004;126:438-446. 15. Schrader T, Peattie RA, Bluth EI, Comstock CE. A qualitative investigation of turbulence in the flow through models of abdominal aortic aneurysms. Invest Radiol 1992;27: 515-519. 16. Viswanath N, Rodliewicz CM, Zajac S. On the abdominal aortic aneurysms: pulsatile state considerations. Med Eng Phys 1997;19:343-351. 17. Khraishi TA, Elger DF, Budwig RS, Johansen KH. The effects of modeling parameters on the hemodynamics of an abdominal aortic aneurysm (AAA). Proceedings ASME Fluids Engineering Division Forum on Unsteady Flows, San Diego, Californnia, 1996;237:349-356. 18. Yu SCM, Chan WK, Ng BTH, Chua LP. A numerical investigation on the steady and pulsatile flow characteristics in axisymmetric abdominal aortic aneurysm models with some experimental evaluations. J Med Eng Technol 1999;23: 228-239.
Annals of Vascular Surgery
19. Jones WP, Launder BE. The prediction of Iaminarization with a two equation model of turbulence. Int J Heat Mass Transfer 1972;15:310-314. 20. Pedersen E, Yoganathan A, Lefebvre X. Pulsatile flow visualization in a model of the human abdominal aorta and aortic bifurcation. J Biomech 1992;25:935-944. 21. Pedersen E, Sung H, Burlson A, Yoganathan A. Two-dimensional velocity measurements in a pulsatile flow model of the normal abdominal aorta simulating different hemodynamic conditions. J Biomech 1993;26:1237-1247. 22. Long Q, Xu XY, Bourne M, Griffith TM. Numerical study of blood flow in an anatomically realistic aorto-iliac bifurcation generated from MRI data. Magn Reson Med 2000;43: 565-576. 23. Taylor C, Hood P. A numerical solution of the Navier-Stokes equations using finite-element technique. Comput Fluids 1973;1:73-89. 24. Gresho PM, Lee RL, Sani RL. On the time-dependent solution of the incompressible Navier-Stokes equations in two and three dimensions. In: Recent Advances in Numerical Methods in Fluids. Swansea, UK: Pineridge, 1980. pp 27-81. 25. Hood P. A finite element solution of the Navier-Stokes equations for incompressible contained flow. M.Sc Thesis, University of wales, Swansea, 1970. 26. Malin MR. Turbulent pipe flow of power-law fluids. Int Comm Heat Mass Transfer 1997;24:977-988. 27. Bouge DC, Metzner AB. Velocity profiles in turbulent pipe flow-Newtonian and non-Newtonian, Industrial & Engineering Chemistry Fundamentals 1963;2:143-152. 28. Schmidt S, McIver DM, Blackburn HM, Rudman M, Nathan GJ. Spectral element based simulation of turbulent pipe flow. Presented at the 14th Australasian Fluid Mechanics Conference 2001, Adelaide University, Adelaide, Australia, 10-14 December: pp 917-920. 29. Mohan S, Mohan N, Valente AJ, Sprague EA. Regulation of low shear flow-induced HAEC VCAM-1 expression and monocyte adhesion. Am J Physiol Cell Physiol 1999;276: 1100-1107. 30. Khanafer K, Gadhoke P, Berguer R, Bull JL. Modeling pulsatile flow in aortic aneurysms: effect of non-Newtonian properties of blood. Biorheology 2006;43(5):661-679. 31. Liu JS, Lu PC, Chu SH. Turbulence characteristics downstream of bileaflet aortic valve prostheses. ASME J Biomech Eng 2000;122:118-124. 32. Papoutsaki ET. Fluid mechanical damage of animal cells in bioreactors. Trends Biotechnol 1991;9:427-437. 33. Kawase Y, Moo-Young M. Mathematical models for design of bioreactors: applications of Kolmogoroff’s theory of isotropic turbulence. Chem Eng J 1990;43:B19-B42. 34. Giddens DP, Zarins CK, Glagov S. Response of arteries to near-wall fluid dynamic behavior. Appl Mech Rev 1990;43:S98-S102. 35. Yip TH, Yu SCM. Oscillatory flows in straight tubes with an axisymmetric bulge. Exp Thermal Fluid Sci 2002;26:947-961. 36. Yip TH, Yu SCM. Cyclic transition to turbulence in rigid abdominal aortic aneurysm models. Fluid Dyn Res 2001;29: 81-113. 37. Bluestein D, Niu L, Schoephoerster RT, Dewanjee MK. Steady flow in an aneurysm model: correlation between fluid dynamics and blood platelet deposition. ASME J Biomech Eng 1996;118:280-286.
The numerical models of abdominal aortic aneurysm (AAA) in use do not take into account the non-Newtonian behavior of blood and the development of local turbulence. This study examines the influence of pulsatile, turbulent, non-Newtonian flow on fluid shear stresses and pressure changes under rest and exercise conditions. We numerically analyzed pulsatile turbulent flow, using simulated physiological rest and exercise waveforms, in axisymmetric-rigid aortic aneurysm models (AAMs). Discretization of governing equations was achieved using a finite element scheme. Maximum turbulence-induced shear stress was found at the distal end of an AAM. In large AAMs (dilated to undilated diameter ratio ¼ 3.33) at peak systolic flow velocity, fluid shear stress during exercise is 70.4% higher than at rest. Our study provides a numerical, noninvasive method for obtaining detailed data on the forces generated by pulsatile turbulent flow in AAAs that are difficult to study in humans and in physical models. Our data suggest that increased flow turbulence results in increased shear stress in aneurysms. While pressure readings are fairly uniform along the length of an aneurysm, the kinetic energy generated by turbulence impacting on the wall of the distal half of the aneurysm increases fluid and wall shear stress at this site. If the increased fluid shear stress results in further dilation and hence further turbulence, wall stress may be a mechanism for aneurysmal growth and eventual rupture.
INTRODUCTION Of the known causes of aortic aneurysmsdproteolytic degradation of aortic wall connective tissue, inflammation and immune responses, molecular genetics,1 and mechanical wall stressdthis study focuses on the latter. The epidemiology of infrarenal abdominal aortic aneurysms (AAAs) has been well described, but the mechanisms leading to accelerated aneurysm 1 Department of Biomedical Engineering, Vascular Mechanics Laboratory, University of Michigan, Ann Arbor, MI. 2 Section of Vascular Surgery, Vascular Mechanics Laboratory, University of Michigan, Ann Arbor, MI.
Correspondence to: E-mail: [email protected] Ann Vasc Surg 2007; 21: 67-74 DOI: 10.1016/j.avsg.2006.10.009 Ó Annals of Vascular Surgery Inc. Published online: January 12, 2007
growth and eventual rupture are poorly understood. We have a rudimentary ability to predictd with the exception of some relationship to aortic diameterdwhich aneurysms are most likely to expand further and rupture. A number of patientspecific factors also have been shown to be associated with rupture of small aneurysms. Cronenwett et al.2 documented that initial anteroposterior diameter, diastolic hypertension, and the presence of chronic obstructive pulmonary disease significantly enhance the rupture of small AAAs. Some studies have focused on the characteristics of aortic flow that may contribute to AAA rupture. Raghavan et al.,3 using finite element analysis from human computed tomographic (CT) scan data, demonstrated that increased tensile stress along the aortic wall at various points contributes to an increased risk of aneurysm rupture, independent of aortic diameter. Fillinger et al.4,5 conducted 67
68 Khanafer et al.
finite element analysis to calculate peak wall stresses for aneurysms in vivo and to analyze rupture risk over time in patients under observation. Their results showed that those AAAs that ruptured had higher peak wall stresses than those that went on to elective repair. Further, Fillinger et al.4,5 showed that peak wall stress is a better predictor of rupture than diameter. This was validated by Venkatasubramaniam et al.6 using finite element analysis and three-dimensional geometries of AAAs derived from CT scans. In the present study, we defined the effects of local turbulence in local fluid shear stress in AAA. From a flow mechanics standpoint, one can predict that changes in the geometry of an aneurysm will alter the patterns of blood flow and, hence, the hemodynamic stresses on the aortic wall. It is presumed that aneurysmal dilation and risk of rupture depend on the hemodynamic stresses associated with (or leading to) changes in the aortic wall that affect its mechanical integrity. Previous studies examining the effects of flow through AAAs assumed laminar flow,7-12 which typically occurs for Reynolds numbers 2,000, although instability leading to the onset of intermittent turbulence in AAA is seen with Reynolds numbers between 2,000 and 2,600.7,13-15 Other studies have noted maximum fluid shear stress at the distal end of aneurysms under laminar flow conditions.16-18 Additionally, large aneurysms exhibit more turbulence than smaller aneurysms at their distal end during diastole, under physiological pulsatile flow.13 The analysis of flow patterns and the mechanical stresses within aneurysms can lead to better understanding of the mechanical forces associated with dilation of the aortic wall and its eventual rupture. The utility of numerical simulation lies in its ability to study in detail conditions that are difficult or impossible to measure in humans or in animal models of AAA. The objective of the present study was to study the flow and mechanical changes that take place in our computational model of AAA under turbulent, pulsatile, non-Newtonian flow at rest and during exercise.
METHODS Turbulent, non-Newtonian flow was studied in a rigid-walled axisymmetric aneurysm as depicted in Figure 1. Throughout this study, we used an axisymmetric representation of aneurysms. The x axis represents centerline flow. The following nomenclature was used: L, length of the aneurysm; D,
Annals of Vascular Surgery
aneurysm maximum diameter; d, undilated aortic diameter. The diameter and length of aneurysms were designated by dimensionless parameters D/ d and L/d, respectively. Governing equations for this study imbedded in our model were (1) the Reynolds time-averaged equations of fluid motion, (2) the kinetic-energy dissipation model (k-3) to simulate the turbulence characteristics of the convective flow, and (3) the Boussinesq eddy-viscosity model. The details of our mathematical model and boundary conditions have been presented elsewhere.19 The exercise20,21 and resting22 arterial waveforms depicted in Figure 2 were used at the inlet to simulate in vivo flow conditions. The time-averaged Reynolds number (Remean ¼ rumNm d) in Figure 2 was based on the time-averaged inlet mean velocity (um ) as reported by Pedersen et al.20,21 for exercise flow condition and Long et al.22 for resting flow condition. Maximum Reynolds number is based on maximum inlet mean velocity (Remax ¼ rummax d). mN is the N viscosity at the infinite shear rate and d is the undilated diameter. A finite element formulation based on the Galerkin method23-25 to solve the governing equations subject to the boundary and initial conditions was employed. The highly coupled and nonlinear algebraic equations resulting from the discretization of the governing equations was solved using an iterative solution scheme. Extensive numerical refinement was performed to attain grid-independent results. When the relative change in variables between consecutive iterations was less than 104, we assumed convergence.
Validation of Our Model In order to benchmark our numerical method, we validated our model against experimental and numerical results published in the literature26,27 for non-Newtonian turbulent flow through a pipe (Table I). Our model used a modified Reynolds number (Rem) that takes into account the powerlaw variation of viscosity. Our results were in excellent agreement (Table I) with published numerical26 and experimental27 results for the power-law model. As an additional check on the accuracy of the present results, Table II shows a comparison of turbulent flow data in a pipe, comparing the results from our model and from other numerical studies28 for Re ¼ 4,910. Correlation is excellent with direct numerical simulation for both Cartesian and cylindrical formulations of the Navier-Stokes equations.
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Turbulence increases pressure and fluid shear stress 69
Fig. 1. AAM (D/d ¼ 3.33). r, radial dimension; X, axial position; S, a position along the aneurysm wall length.
0.8 Remax = 3300, Remean = 1770 0.6 Remax = 2904, Remean = 794
u(t)
0.4 Exercise 0.2 Rest 0.0
-0.2
0.0
0.2
0.4
0.6
Time (s)
RESULTS Comparison of Fluid Shear Stress and Relative Pressure between Laminar and Turbulent Flows at Exercise for Large Aneurysms (D/d [ 3.33) The significance of turbulent flow on the relative pressure and fluid shear stress on the large aneurysm (D/d ¼ 3.33) compared with the laminar flow condition using the exercise waveform is demonstrated in Figures 3 and 4. A normal aortic diameter (d) is 2-2.5 cm depending on age and gender, however, so the aneurysm diameter (D) would be
0.8
1.0
Fig. 2. Pulsatile inlet mean velocity for exercise and rest flow conditions.20-22
6.7-8.3 cm. A 6.7 cm AAA for a woman or an 8.3 cm AAA for a man would be large in the practice of any clinician. Pressure values presented here are relative and correspond to the difference between the instantaneous pressure on the AAA wall and the instantaneous pressure at the exit of the computational model. Figure 3 shows that pressure along the aneurysm wall is fairly uniform at any given phase of the cycle for both laminar and turbulent flow conditions. A significant pressure difference on the aneurysm wall between laminar and turbulent is depicted in Figure 3. Furthermore, turbulent flow induces higher fluid shear stress at
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Table I. Pipe flow simulations: non-Newtonian flow (Rem ¼ 1.07 105) Vmax =V
Table II. Pipe flow simulations: Newtonian flow (Re ¼ 4,910)
Error (%) Vmax =Vinlet
Present Numerical26 Experimental27 2n
Rem ¼
1.16 1.15 1.21
rV dn n n1 ; Kð0:75þ0:25 n Þ 8
0.86 4.3
Present 1.29 DNS (Cyl.)28 1.302 DNS (Cart.)28 1.297
Vmax =Vshear
Vinlet =Vshear
Vmax =Vshear error (%)
18.47 18.98 18.91
14.315 14.58 14.58
2.69 2.38
n ¼ 0:825; m ¼ K g_ n1 . DNS, direct numerical simulation; Cyl, cylindrical; Cart, Cartesian; Vinlet , inlet velocity; Vmax , maximum (centerline) qffiffiffiffi velocity; Vshear , shear velocity ( trw ) (tw , wall shear stress).
Relative Pressure along AAA (Pa)
1200
Peak Systole 900
Exercise Flow Condition 600 Peak Diastole
300
Remax = 3300
Laminar Turbulent 0 0.0
2.0
4.0
6.0
Aneurysm wall length (cm)
the distal end of an AAA during peak systole compared with laminar flow, as depicted in Figure 4. Thus, an increase in turbulence dramatically increases relative pressure by 32.5% and fluid shear stress by 23.6% at the distal end of the AAA (for D/d ¼ 3.33 during peak systole). This increase in peak fluid shear stress corresponds to 20 dyn/cm2 which likely will have a significant effect on the endothelial cells. As such, areas of low shear stress (2 dyn/cm2) experience upregulation of vascular adhesion molecule-1 (VCAM-1), while areas of high shear stress (20 dyn/cm2) experience downregulation of VCAM-1.29 An inverse relationship between pressure and fluid shear stress in the axisymmetric-rigid aortic aneurysm model (AAM) for turbulent flows at peak systole during exercise is demonstrated in Figure 5. This correlation is consistent with Bernoulli’s
8.0
10.0
Fig. 3. Comparison of the relative pressure between laminar and turbulent flow conditions on the AAM during exercise for a bulge size of D/d ¼ 3.33.
equation, where the pressure decreases as the velocity increases for incompressible flow. Comparison of Streamline and Pressure Contours at Rest and during Exercise for Large Aneurysms (D/d [ 3.33) In all plots (Fig. 6), we see an internal jet of fluid surrounded by a recirculating vortex moving about the centerline (x axis) and impinging on the distal half of the wall in the aortic model. At peak systole (time ¼ 0.345 sec [rest] and 0.143 sec [exercise]), a small recirculating vortex develops at the proximal end of the AAA at rest and during exercise, while a large recirculating flow region fills the aneurysm sac during peak diastole for both waveforms. The only difference in the flow patterns in AAA at rest and exercise is that the recirculation
Vol. 21, No. 1, 2007
Turbulence increases pressure and fluid shear stress 71
12.0 Exercise Flow Condition 10.0
Fluid Shear Stress (Pa)
Relative deviation = 23.6% 8.0
6.0
4.0 Laminar Turbulent
Remax = 3300
2.0
0.0
-2.0 0
2
4
6
8
10
Aneurysm wall length (cm)
Fig. 4. Comparison of the fluid shear stress between laminar and turbulent flow conditions on the AAM during exercise for a bulge size of D/d ¼ 3.33.
1200
900 Remax = 3300 6 600
2
Fluid Shear Stress (Pa)
Relative Pressure along AAA (Pa)
10
300
0 0.0
2.0
4.0
6.0
Aneurysm wall length (cm)
vortices are faster in the latter case. Further, a large recirculating flow region results in large residence time, which could be a contributing factor for thrombosis in the aneurysm. Normal and tangential stresses may be involved in the local dilation and eventual rupture of aneurysms. Fluid shear stress is defined as a measure of the tangential forces per unit area generated by the flow stream on the walls of the AAM. Figure 7 illustrates fluid shear stress values in the AAM at different points of the cardiac
8.0
-2 10.0
Fig. 5. Inverse relationship between relative pressure and fluid shear stress for turbulent flow during exercise (D/d ¼ 3.33).
cycle. This figure highlights the significant differences found between rest and exercise. Peak values are found at the distal half of the AAM during rest and exercise conditions (Fig. 7). Exercise waveforms generate larger fluid shear stresses than resting waveforms. We have reported that the distal half of an AAM exhibits higher fluid shear stress than other locations30 under laminar flow. Thus, increased shear stress will decrease thrombus deposition.
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Rest
Exercise (D/d = 3.33)
(Peak Systole)
max
: 1.32×10–4
max
: 1.99×10–4
min
:0
min
:0
(Peak Diastole)
max
: 8.02×10–5
max
: 1.42×10–4
min
:0
min
:0
Cardiac cycle period = 1.0 sec
Cardiac cycle period = 0.5 sec
Fig. 6. Temporal variation of streamlines for AAM under rest and exercise conditions. This is an axisymmetric representation of an aneurysm. The centerline of flow is the x axis. This two-dimensional representation
is a longitudinal view of the aneurysm with flow from left to right. The isobars represent sites of equal pressure. The streamlines represent flow fields of equal velocity (J).
12.0 Exercise
Time = 0.143 s (peak systole)
Rest
Fluid Shear Stress (Pa)
9.0 Time = 0.342 s (peak systole) 6.0
3.0 Time = 0.794 s (peak diastole) 0.0
Time = 0.42 s (peak diastole) -3.0 0.0
2.0
4.0
6.0
Aneurysm wall length (cm)
DISCUSSION Turbulence induced by sudden expansion of the flow stream results in additional stresses acting on the aneurysm wall that may be responsible for further aortic dilation. Dilation results in further radial expansion of the flow stream and greater turbulence. The latter then becomes a self-perpetuating mechanism for aneurysm dilation. To date, the role
8.0
10.0
Fig. 7. Comparison of fluid shear stress on the AAM between exercise and rest flow conditions.
of turbulent flow in AAA growth and rupture has not been well studied. Turbulent flow simulation should be included in the study and modeling of aneurysms because it increases fluid and wall stresses. Turbulent flow results in kinetic forces being directed toward the vessel wall instead of parallel to the wall, as seen under laminar flow conditions. Because ours is a rigid model, this deformation of the wall secondary to kinetic forces is not displayed.
Vol. 21, No. 1, 2007
This study documents that higher fluid shear stress also occurs in the distal half of the aneurysm at peak systolic velocity under turbulent nonNewtonian flow conditions. At this time and at this site, turbulence is greater and the impact of this kinetic energy on the wall is added to the wall tension generated by the intraluminal pressure. Consequently, turbulence, by increasing wall tension and inducing wall vibration, may induce dilation of an aneurysm. Cyclic turbulent stresses are known to alter the structure and integrity of the arterial wall. Large eddies induce vibrations at frequencies associated with the dilation of arteries. The smallest turbulent eddies31-33 (Kolmogorov microscale), which are a function of the kinematic viscosity of blood and turbulent dissipation rate, can mechanically damage the blood cells and the components of the arterial wall. As such, turbulent eddies in the near-wall region of the distal half of the aneurysm generate steep velocity gradients, which induce further fluid shear stress. Giddens et al.34 showed that the abdominal aorta regulates its diameter chronically to sustain the shear stress below the physiologically optimal value of 12 ± 2 dyn/cm2. Turbulent shear stress near the distal end of the AAM is significantly higher than this optimal value. Thus, the long-term effects of turbulent flow result in arterial wall damage and enlargement of the aortic diameter. There is previous experimental work in the literature on transitional and turbulent flow of Newtonian fluids in aneurysms. Yip and Yu35,36 studied oscillatory Newtonian flows in an axisymmetric aneurysm using laser-Doppler anemometry and documented a transition to turbulence in the deceleration phase of flow. Onset of turbulence was also observed in steady flows at high Reynolds numbers.35,36 Budwig et al.7 demonstrated numerically that turbulence in axisymmetric rigid aneurysms was intermittent for Reynolds numbers between 2,000 and 2,500, assuming steady flows. Bluestein et al.37 pointed out that under fully turbulent, steady-state Newtonian flow (Re ¼ 3,600) fluid shear stress values were one order of magnitude larger than those for laminar flow. Egelhoff et al.13 observed a transition to turbulent flow in moderately large aneurysms under exercise flow conditions during late systole. Fukushima et al.10 showed that flow becomes unstable during deceleration in an AAM model. Egelhoff et al.13 confirmed this phenomenon experimentally. Under exercise and resting waveforms, the vortex is observed during late systole and early diastole and moves toward the distal half of the AAM. We analyzed physiological pulsatile turbulent flow in large AAMs at rest and during exercise.
Turbulence increases pressure and fluid shear stress 73
Under the exercise flow condition, a large primary vortex is exhibited in the distal half of the aneurysm along with a small vortex at the proximal end of a large bulge during systolic peak flow. However, at rest, one primary vortex is developed at the proximal end of the aneurysm. The major assumptions used in the present study include the anatomic dimensions of the AAA model, rigid wall approximation, and a fusiform AAA model. These assumptions represent a reasonable first approximation to the actual flow and hemodynamic conditions in the abdominal aorta. The dimensions of the AAA model used in this investigation were based on those reported in clinical studies. The use of rigid wall assumption for flow studies in idealized AAA models should be viewed as a first approximation. As such, we anticipate reduction in fluid shear stress when assuming a compliant wall compared with a rigid wall model based on clinical observation of vessel wall deformation during the cardiac cycle.
CONCLUSION We studied pulsatile turbulent flow in large AAMs at rest and during exercise. The large increase in pressure recorded along the aneurysm wall during maximum systolic flow acceleration is likely a factor in determining the risk of aneurysm rupture. In addition, the turbulence induced by sudden expansion of the flow stream generates higher fluid shear stresses on the aneurysmal wall. These additional stresses may be responsible for further wall dilation that will eventually result in greater turbulence, possibly a self-perpetuating mechanism for aneurysmal growth. This work contributes to the understanding of the mechanics of aneurysm growth and rupture by elucidating the stresses that may be attributed to turbulence. Numerical models of AAA should include the mathematical representation of viscosity and turbulence to better model the dynamics involved in blood flow through aneurysms.
This work was supported by the Frankel Vascular Research Fund.
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