Vascular OnlineFirst, published on June 29, 2015 as doi:10.1177/1708538115594095
Original Article
A computational fluid–structure interaction model of the blood flow in the healthy and varicose saphenous vein
Vascular 0(0) 1–10 ! The Author(s) 2015 Reprints and permissions: sagepub.co.uk/journalsPermissions.nav DOI: 10.1177/1708538115594095 vas.sagepub.com
Reza Razaghi, Alireza Karimi, Shahrokh Rahmani and Mahdi Navidbakhsh
Abstract Objective: Varicose vein has become enlarged and twisted and, consequently, has lost its mechanical strength. As a result of the varicose saphenous vein (SV) mechanical alterations, the hemodynamic parameters of the blood flow, such as blood velocity as well as vein wall stress and strain, would change accordingly. However, little is known about stress and strain and there consequences under experimental conditions on blood flow and velocity within normal and varicose veins. In this study, a three-dimensional (3D) computational fluid–structure interaction (FSI) model of a human healthy and varicose SVs was established to determine the hemodynamic characterization of the blood flow as a function of vein wall mechanical properties, i.e. elastic and hyperelastic. Methods: The mechanical properties of the human healthy and varicose SVs were experimentally measured and implemented into the computational model. The fully coupled fluid and structure models were solved using the explicit dynamics finite element code LS-DYNA. Results: The results revealed that, regardless of healthy and varicose, the elastic walls reach to the ultimate strength of the vein wall, whereas the hyperelastic wall can tolerate more stress. The highest von Mises stress compared to the healthy ones was seen in the elastic and hyperelastic varicose SVs with 1.412 and 1.535 MPa, respectively. In addition, analysis of the resultant displacement in the vein wall indicated that the varicose SVs experienced a higher displacement compared to the healthy ones irrespective of elastic and hyperelastic material models. The highest blood velocity was also observed for the healthy hyperelastic SV wall. Conclusion: The findings of this study may have implications not only for determining the role of the vein wall mechanical properties in the hemodynamic alterations of the blood, but also for employing as a null information in balloonangioplasty and bypass surgeries.
Keywords Varicose, saphenous vein, fluid–structure interaction, elastic, hyperelastic
Introduction The saphenous vein (SV) is a large elastic vessel running near the inside surface of the leg from the ankle to the groin which enables to move freely with blood stream.1 If the SV becomes varicose, it twists and enlarges up to the skin surface.2 It is believed that the varicose is the result of weakened valves and veins in the leg.3 It has also been indicated that the varicose veins would lose their elastin and collagen contents.4 In addition, the elastin is responsible for the small deformation/strain while the collagen has a key asset in the large deformation/strain of the vein wall.5–7 Therefore, weak mechanical properties of the varicose veins is anticipated as
they contain a small amount of elastin and collagen.4 The mechanical properties of the vein wall play a key role in the hemodynamic characterization of the blood, such as Blood Velocity (BV) as well as vein wall stress and strain.8,9 However, little is known about stress and strain and their consequences under experimental Tissue Engineering and Biological Systems Research Laboratory, School of Mechanical Engineering, Iran University of Science and Technology, Tehran 16887, Iran Corresponding author: Alireza Karimi, School of Mechanical Engineering, Iran University of Science and Technology, Tehran 16846, Iran. Email:
[email protected]
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conditions on blood flow and velocity within normal and varicose veins. There is a little amount of literature in the field. Abraham et al. investigated the changes in the diameter and blood velocity of the SV during thermal stress.10,11 They employed a duplex ultrasound for studying saphenous cross-sectional area (CSA) and mean maximal venous BV in 10 healthy volunteers. Their results indicated that CSA increases during heat stress and decreases during cold stress. They also showed that BV increases during warming and decreases during cooling. Bandyk et al. performed an experimental measurement on the hemodynamics characteristics of in situ SV arterial bypass.12 They benefitted from a Doppler-derived blood flow velocity and limb blood pressure measurements for characterizing the hemodynamics of 128 in situ SV arterial bypass. Their results revealed that high blood flow velocity and pressure gradient associated with flow-restrictive venous conduits, but limb ischemic symptoms resolved, and graft patency was not decreased. Gusic et al. worked on the shear stress and pressure modulation of SV remodeling.13,14 They investigated the role of mechanical environment in vein remodeling in an ex vivo perfusion system. Porcine SVs were subjected to five different ex vivo hemodynamic environments for one week in order to independently assess the effects of shear stress and pressure on vein remodeling. The extent of intimal hyperplasia decreased with culture under increasing shear stress, with veins cultured under the lowest levels of shear stress exhibiting the greatest ratio of intimal/medial area, which was greater than that of the fresh veins. Their results suggested that pressure and shear stress act independently to regulate vein remodeling, influencing changes in vessel size as well as the nature of the remodeling. Leask et al. performed an experimental/numerical study on human SV coronary artery bypass graft morphology, geometry, and hemodynamics as coronary artery bypass graft failure has been linked to graft hemodynamics, in particular wall shear stress.15 Graft hemodynamics was evaluated in two flow models, fabricated from the casts, under steady and pulsatile flow conditions. They concluded that in both flow models, a large increase in wall shear rate occurred on the hood, just proximal to the toe. The local geometry of the hood created this large spatial gradient in wall shear stress which is a likely factor in hood intimal hyperplasia. To this end, to the best of the authors’ knowledge, there is no experimental or numerical study on the hemodynamic characterization of the human varicose SV, and most studies in the field benefited from animal samples. Therefore, this study aimed to perform a computational three-dimensional (3D) fluid–structure
interaction (FSI) study to evaluate the hemodynamic alterations for different SV wall mechanical properties. Studying FSI between blood flow and vascular tissue, the wave propagation that causes in the arterial walls, local hemodynamics, and Wall Shear Stress (WSS) is important in understanding the mechanisms leading to various complications in blood flow function.6,16 Computational methods for problems with FSI were pioneered by Peskin with immersed boundary method.17 Bathe developed a series of finite element procedures dealing with FSI where the fluid and solid models were either solved iteratively (for small strain/ small deformation) or solved fully coupled systems (for finite strain/deformation).18 However, computational simulations for blood flow in vessels are largely limited to idealized physical and geometries.19,20 The proposed model benefited from the experimental data of the elastic and hyperelastic mechanical properties of the healthy and varicose human SVs. These experimental data were implemented into the computational model to get a more precise outcome.
Materials and methods Saphenous vein preparation The healthy and varicose SV were removed during autopsy (seven healthy vein) and surgery (seven varicose vein), respectively, from 14 individuals. All material removal was excised with permission of donators under the ethical rules of Tehran University of Medical Sciences (TUMS) based on 2008 Declaration of Helsinki. In order to minimize tissue degradation, portions of the veins were immediately preserved in a solution of 0.90% w/v of NaCl at 4–5 C before the uniaxial tensile test. The testing material was obtained by cutting a 12 mm long segment of SVs.
Experimental tensile test The outer diameter, initial wall thickness, and length of the segments of SVs were measured precisely using digimatic ruler having a resolution of 0.005 mm0.05% (Insize, Vienna, Austria). The tensile test was performed using a well-approved uniaxial tensile test apparatus adapted for testing delicate soft biological tissues.21,22 The vein tissues were taken out from the physiological saline and mounted on the tensile test machine. The sample’s length was measured after the application of the pre-load for preconditioning. Force was then applied to each specimen before the failure occurred. A low strain rate of 5 mm/min which is typical for surgical procedures and gives more insight into tissue behavior was employed by the action of an axial servo motor.23,24
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Model geometry and mesh
ij nj ti ¼ 0 on
A comparative three-dimensional numerical simulation of an idealized human healthy and varicose SV was established to predict hemodynamic conditions within vein tissue during blood flow for various SV mechanical behaviors, including elastic and hyperelastic. Four models were developed, each containing a healthy and varicose SV of different mechanical behavior, including elastic and hyperelastic. The models were built, meshed, and solved using the explicit dynamics finite element code LS-DYNA 970 (LSTC, Livermore, CA, USA).25 The artery (length, inner diameter, and thickness of 15, 1.655, and 0.5 mm, respectively) dimensions were obtained from the experimental data. The vein was built of 640 hexahedral elements and 1089 nodes. The blood was also contained 1120 hexahedral elements and 1551 nodes, and *Mat_Null element type in LS-DYNA 970 material library.
The fluid model The flow in this study was assumed to be laminar, Newtonian, viscous, and incompressible. The blood flow was governed by the Navier–Stokes equations of incompressible flow stood on a moving domain. The moving fluid domain was handled using the Arbitrary Lagrangian–Eulerian (ALE) formulation, which is a popular choice for vascular blood flow applications.26,27 At the interface between the blood flow and the elastic wall, velocity and traction compatibility conditions were assumed to be on hold (no slip boundary). The motion of the fluid domain was governed by the equations of linear elasticity subject to Dirichlet boundary conditions coming from the time-dependent displacement of the fluid–solid interface.28,29 In the discrete setting, this procedure was referred to as mesh motion.30,31 The governing equations for the fluid domain (blood) are the Navier–Stokes and Continuity equations,32,33 i.e. conservation of momentum ~ @V~ ~ ~ ~ rp ~ 2 r:D þ V:rV þ ¼ 0~ @t
ð1Þ
r:V~ ¼ 0
ð2Þ
where V is the fluid velocity, P is fluid pressure, is ~ is the gradient operdensity, is dynamic viscosity, r ator and D is the fluid rate of deformation tensor.34 The governing equations for the solid (SV wall) are the momentum and equilibrium equations,35 i.e. Newton’s laws of mechanics i ij, j fi ¼ 0
in
s
ðtÞ
ð3Þ
s
ðtÞ
ð4Þ
where s ðtÞ is the structural domain at time t, ti is surface traction vector, ij is stress of the solid, and i is the acceleration of the material point along the ith direction.
The solid model The SVs wall was defined by both elastic36,37 and Mooney–Rivlin38,39 hyperelastic material models. The elastic and hyperelastic data were obtained from the linear Hookean part of the stress–strain diagram while the hyperelastic ones were excised from the experimental stress–strain diagram using nonlinear constraint optimization–trust region algorithm, respectively. Although the elastic model cannot adequately address the mechanical properties of the SV tissue, such hyperelastic material models of the vein wall can capture its nonlinear mechanical properties. The Mooney–Rivlin strain energy density function was used to address the nonlinear hyperelastic mechanical behavior of the human SV wall.40–42 W ¼ C10 ðI1 3Þ þ C01 ðI2 3Þ þ C20 ðI1 3Þ2 1 þ C11 ðI1 3ÞðI2 3Þ þ C02 ðI2 3Þ2 þ ðJ2 1Þ D ð5Þ where W stands for strain energy function per unit volume, which is a function of two principal strain invariants: W¼W(I1,I2), where I1 and I2 are defined as24,43–46 I1 ¼ 21 þ 22 þ 23
ð6Þ
I2 ¼ 21 22 þ 21 23 þ 22 23
ð7Þ
where 21 ; 22 , and 23 are the squares of the principal stretch ratios, associated with the connection 1 2 3 ¼ 1, due to an incompressibility assumption (det J ¼ 1).47,48 The coefficients of the Mooney–Rivlin hyperelastic constitutive equation, such as C10, C01, C20, C11, C02, and D, were determined by fitting to a centrally lying tensile test curve for each human SV experimental data.49,50 The fully integrated artery–plaque model was solved by a commercial explicit dynamics finite element code LS-DYNA 970 (LSTC, Livermore, CA, USA). The hyperelastic material coefficients of vein/blood tissue are listed in Table 1.
Arbitrary Lagrange Eulerian method Fluid–structure interfaces (the interface of vein wall and blood flow) were defined at the lumen surfaces of
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Table 1. The identified material coefficients of the human healthy and varicose saphenous veins along with blood. Hyperelastic healthy SV Hyperelastic varicose SV Elastic healthy SV Elastic varicose SV Blood
C10 ¼ 9.392 (MPa), C01 ¼ 8.828, C02 ¼ 0.014, C20 ¼ 5.627, C11 ¼ 5.523, r ¼ 1160 kg/m3 C10 ¼ 3.147 (MPa), C01 ¼ 4.009, C02 ¼ 956.843, C20 ¼ 1005.986, C11 ¼ 1982.319 E ¼ 10.63 (MPa), n ¼ 0.499 E ¼ 5.88 (MPa), n ¼ 0.499 r ¼ 1060 kg/m3, m ¼ 3.5 103 Pa.s, EOS_ Gruneisen: C ¼ 1780 m/s
E: elastic modulus; : Poisson’s ratio; : density; m: viscosity; EOS: equation of state.
the vein, and the wall boundaries of the fluid. The ALE method was used. The ALE is an effective way to treat moving mesh problems. Instead of using either a single Lagrangian approach or a single Eulerian approach, the ALE describes the motion of fluid in a moving reference frame whose velocity is almost arbitrary with the sole constraint that the velocity of the fluid–solid boundary must equal to that of the boundary.51,52 The velocity of the reference frame is usually neither the fluid particle velocity such as in a pure Lagrangian description nor zero in a pure Eulerian description. After introducing a reference frame which moves with some velocity, the ALE-modified Navier–Stokes equation for a viscous incompressible flow follows as53,54 @ui @ui @ @ui @uj þ uj b uj @xj @t @xj @xj @xi @P @uj þ ¼ 0, ¼ 0, i,j ¼ 1, 2, 3, @xi @xj
ð8Þ
where uj, j ¼ 1, 2, and 3, are the components of the flow velocity; b uj , j ¼ 1, 2, 3, are the components of the domain velocity; p is pressure; is fluid density; and is dynamic viscosity. Note that the reference frame velocity only affects the convection term. The boundary conditions for the inflow boundaries could be measured experimentally. The boundary condition for the surface traction can be prescribed at the outflow boundaries.
Fluid–structure coupling The Navier–Stokes equations for the fluid and the momentum and equilibrium equations for the solid are coupled with the fluid–solid interface via the kinematic and dynamic conditions.55 The solid and fluid models can be coupled by the fluid nodal positions on the FSI interfaces which are determined by the kinematic conditions.18 The displacements of the other fluid nodes are determined so as to preserve the initial mesh quality.56 The ALE-modified governing equations for fluid flow are then solved. For the dynamic case, the
fluid stresses are integrated along the fluid–solid interface and applied to the corresponding solid nodes.57
Results and discussion Simulations were executed using 3D FSI models to determine the hemodynamic behavior of the blood flow inside the elastic and hyperelastic healthy as well as elastic and hyperelastic varicose SVs. The stress and strain of the vein wall due to a normal blood flow were also computed. Experimental estimation of the normal and shear stresses inside a vein is not plausible, and numerical simulation provides an alternative way to obtain detailed flow patterns and vessel wall stress and strain fields. Eight healthy and varicose human SVs were excised from the human body during surgery and autopsy. A typical human SV employed for the computational FSI model is indicated in Figure 1. Due to symmetry, onefourth of the vein blood was modeled. The vein material was considered to be isotropic, incompressible, and homogenous. The mechanical behavior of the SV was modeled by two approaches, including elastic and hyperelastic material models. The blood flow was assumed to be laminar, Newtonian, viscous, and incompressible. The blood flow was governed by the Navier–Stokes equations of incompressible flow posed on a moving domain. The stress distribution in healthy and varicose SVs under two differential vein wall mechanical behavior was investigated. The distribution of von Mises (normal) stress in (a) elastic healthy, (b) hyperelastic healthy, (c) elastic varicose, and (d) hyperelastic varicose SVs are displayed in Figure 2. Regardless of healthy and varicose SVs, the highest stresses were observed in the veins with hyperelastic mechanical behavior. In terms of the elastic behavior of the vein wall, the stress on the varicose vein was 4.05% higher than that of the healthy one. Similarly, in the hyperelastic mechanical behavior of the vein wall, there was less than 0.15% differences in the stresses induced on the vein wall due to the blood flow. To this end, it can be concluded that the stresses in the varicose SVs are slightly higher than that of the healthy ones. This has
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Figure 1. The healthy and varicose human saphenous veins were excised during surgery and autopsy. The morphological characteristics of the samples were implemented into the further fluid–structure interaction simulations.
Figure 2. The contour of von Mises stress of the vein wall, including (a) elastic healthy, (b) hyperelastic healthy, (c) elastic varicose, and (d) hyperelastic varicose, human saphenous veins. The fluid–structure interaction (FSI) method has been utilized to compute the normal stress.
been anticipated since the varicose SV loses its mechanical strength and, as a result, the normal stress is higher than healthy SV. The contour of resultant displacement for the whole elements on the (a) elastic healthy, (b) hyperelastic healthy, (c) elastic varicose, and (d) hyperelastic varicose SVs, is exhibited in Figure 3. A node was selected on each model to figure out the displacement that would induce due to normal blood flow. The resultant displacement of the selected node versus the simulation time is demonstrated in Figure 4. The results showed that a higher displacement occurs for hyperelastic SVs regardless of healthy and varicose conditions.
According to these figures, it is observed that during the simulation time, a higher vein wall displacement is seen in the varicose veins. It agrees well with our anticipation as the healthy veins are significantly stiffer than the varicose one, and therefore, the displacements that occur on its wall should be lower than that of the varicose ones. A comparative representation of shear stress contour for different vein types, including (a) elastic healthy, (b) hyperelastic healthy, (c) elastic varicose, and (d) hyperelastic varicose SVs, are presented in Figure 5. A comparative representation of blood velocity contour for different vein types, including (a) elastic healthy, (b)
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Figure 3. The resultant displacement distribution of the vein wall, including (a) elastic healthy, (b) hyperelastic healthy, (c) elastic varicose, and (d) hyperelastic varicose, human saphenous veins. The FSI method has been utilized to compute the resultant displacement.
Figure 4. A comparative diagram of the resultant displacement versus simulation time for the elastic healthy, hyperelastic healthy, elastic varicose, and hyperelastic varicose, human saphenous veins.
hyperelastic healthy, (c) elastic varicose, and (d) hyperelastic varicose SVs, are presented in Figure 6. In addition, a node was selected on blood to determine the velocity it takes during the simulation time. Consequently, the blood velocity for the selected node was plotted and exhibited in Figure 7. The computational results revealed that regardless of healthy and
varicose SV types, a higher blood velocity was observed for the hyperelastic vein walls compared to the elastic ones. Furthermore, from the results it has been revealed that the highest blood velocity is occurred in the healthy SV with hyperelastic wall; however, the lowest one is located in the healthy SV with elastic wall. Although among the elastic
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Figure 5. The shear stress contour of the vein wall, including (a) elastic healthy, (b) hyperelastic healthy, (c) elastic varicose, and (d) hyperelastic varicose, human saphenous veins. The FSI method has been utilized to compute the effective shear stress.
Figure 6. The resultant velocity contour of the vein wall, including (a) elastic healthy, (b) hyperelastic healthy, (c) elastic varicose, and (d) hyperelastic varicose, human saphenous veins The FSI method has been utilized to compute the resultant velocity.
material models blood velocity is in the highest value for varicose SVs, in hyperelastic material models the highest blood velocity is seen in healthy SVs. The healthy SVs showed the blood velocity of 0.0221 and 0.0268 m/s for the elastic and hyperelastic material models which are in agreement with literature.10–12 However, in an elastic varicose vein despite the blood velocity decreases about 9.50%, the difference compared to the hyperelastic ones is less than 3%. To sum up, it can be concluded that the varicose SV would lose its mechanical strength and shows a higher stresses on its wall. As a result, the
displacement of the vein wall is higher for varicose vein while the blood velocity in it is slightly lower. Although many efforts have been made to reach more precise results, there are still some factors which need to be added into the current model, including anisotropic properties of the vein,58 viscoelastic properties of the vein,59 blood conditions and remodeling. Despite these simplifications, the present model demonstrates the alteration of the blood flow factors in healthy and varicose conditions by proposing a FSI computational model which may have implications for varicose and bypass surgeries.
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Figure 7. A comparative diagram of the resultant velocity versus simulation time for the elastic healthy, hyperelastic healthy, elastic varicose, and hyperelastic varicose, human saphenous veins.
Conflicts of interest
Conclusions
None declared.
This study was aimed to perform a comparative FSI simulation to compute the blood velocity as well as stress and strain of the vein wall as a function of vein wall mechanical behavior. To do this, the elastic and hyperelastic mechanical data of the healthy and varicose SV were implemented into the FSI model. The results revealed that the elastic walls reach to the ultimate strength of the vein wall, whereas the hyperelastic wall could tolerate more stress. The highest von Mises stress was seen in the elastic and hyperelastic varicose SVs. In addition, analysis of the resultant displacement in the vein wall indicated that the hyperelastic wall experiences a higher displacement compared to the elastic wall. The highest blood velocity was observed for the healthy hyperelastic SV wall. The findings of this study can be employed for diversity of disciplines, including engineering and clinical. These data can help biomechanical experts and clinicians to understand the precise value of stresses and deformations in the varicose vein wall and compare them to the healthy veins. This is considered as the main outcome of this research as the current experimental methods that have been used by clinicians are unable to determine the normal and shear stresses in the varicose veins. Funding This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.
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51.
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