Modelling of stress-strain states in arteries as a pre-requisite for damage prediction

Transcription

1 2002 WT Press, Ashurst Lodge, Southampton, SO40 7AA, UK. All rights reserved. Paper from: Damage and Fracture Mechanics V, CA Brebbia, & S Nishida (Editors). SBN Modelling of stress-strain states in arteries as a pre-requisite for damage prediction J. Bursa nstitute of Solid Mechanics, Brno University of Technology, Czech Republic. Abstract Damage to arteries can be caused by a number of different factors ranging fkom the influence of mechanical load of arterial walls on endothelial cells and, consequently, on the genesis of atheromatous plaques on the inner surface of the walls, to a global tissue rupture, To predict these states, it is necessary to know not only the limit conditions and limit values but also the most credible stress and strain states. These states were solved using FEA (ANSYS program system), namely in an intact artery and in various geometric shapes of joints (anastomoses) with a vascular graft. The computational model used the axisymmetrical geometry with up to two different homogeneous coaxial layers in the artery wall, The material was supposed to be non-linear elastic, isotropic, nearly incompressible, showing finite displacements and strains under physiologic load. The model considered three major load factors: blood pressure, axial prestretch and residual stresses, The influence of axial prestretch and residual stress on the stress (and strain) distribution in the arterial wall was quantified using the maximum to minimum stress (strain) ratio, These ratios tend to 1 at some physiological blood pressure values. Using the same type of model, the stress and strain distribution was computed for an anastomosis of the aorta with a vascular graft, Two basic types of anastomoses were modelled: a classical end-to-end anastomosis and a,japped anastomosis. Due to a very complex tissue structure that is considerably simplified in the model, the results do not filly meet real needs, however they enable us to compare various shapes and geometric parameters of anastomoses. n this way, risks of clinical complications related to specific limit states of tissues, such as ruptures or growth of aneurysms near the anastomosis, can be compared.

2 2002 WT Press, Ashurst Lodge, Southampton, SO40 7AA, UK. All rights reserved. Paper from: Damage and Fracture Mechanics V, CA Brebbia, & S Nishida (Editors). SBN Damage and Fracture Mechanics V 1 ntroduction Blood vessels walls oflen succumb pathological changes connected with their stress and strain state which can endanger the person s life, Reduction of the artery inner diameter, together with changes in mechanical properties due to the atherosclerotic process are the most frequent pathological changes influenced by mechanical factors, A growth of aneurysms and their rupture risk or application of arterial stents and vascular grafts are examples of clinical problems requiring some knowledge about stress and strain states in arteries, as the fust step to create computational supports. 2 Objectives A solution to stress and strain states in soft tissues is too complex for detailed computational modelling because of the complicated structure with various wavy fibres and their non-elastic properties, However, models respecting global tissue properties, as hyperelasticity, anisotropy, viscoelasticity etc. can be usefi.d e.g. in comparing various surgical approaches, Our FE model of aorta created in program system ANSYS supposed axisymmetry of the aortic segment, ts cylindric wall was created by one or two coaxial layers. The material was modelled as isotropic, incompressible, non-linear elastic, showing large displacements and strains under physiologic conditions, The model was loaded by the blood pressure, axial prestretch and residual stresses. This model was used to evaluate the stress and strain states in an intact aorta and to assess the influence of axial prestretch and residual stress on the stress and strain states in the wall. A similar model evaluated stress and strain states in two types of anastomoses with vascular grafts J -+-no residus lstrese, sxlslstraln 0% -+-with reeidual stress, axislstrsin O A with residus lstreee, exisletra in 20% - -with residuslstrees, sx ial strain residual streas, axisl strain 40 /0 -O-with residusletress, sxislstrsin 50 /0 o 0,5 i 1,5 Diatsnce from the inner surface [mm] Fig,l: nfluence of axial prestretch on the circumferential stress distribution in homogeneous aortic wall (residual stress given by the opening angle 60 deg) T i

3 2002 WT Press, Ashurst Lodge, Southampton, SO40 7AA, UK. All rights reserved. Paper from: Damage and Fracture Mechanics V, CA Brebbia, & S Nishida (Editors). SBN Results of intact aorta modelling Damage and Fracture Mechanics V 489 The created 3D computational model of stress and strain state in an aortic wall segment was loaded by residual stress (defined by the opening angle ~), axial prestretch (up to 50% axial strain) and inner pressure (up to 20 kpa). When computed separately, the residual stresses are relatively low (a few kpa according to the opening angle value and to the boundary conditions of the model used). However, the stress distribution in the aortic wall (see fig. 1) shows that the changes of extreme stresses in the wall induced by the additional load factors are much higher than the values of residual stress. The residual stress and axial prestretch do decrease the stress gradients in the wall or in the wall layer very significantly, This surprisingly high influence can be explained by the ma$erial non-linearity character typical of soil tissues. The stress non-uniformity across the aortic wall (or across a layer in the twolayer model) can be expressed by maximum to minimum circumferential stress ratio po. n the one-layer model, the additional load decreases this extreme stress ratio from 1.94 to 1.19 at blood pressure value of 20kPa. Consequently, a homogeneous stress state can be achieved at a certain value of the blood pressure (9 kpa for the presented model). This,,optimized pressure value may lie within the usual physiologic blood pressure range when either the opening :&-_&=J...f-,..._,,J uhuh --7,.,,. _,...,_~, o 0,2 0,4 0,6 0,8 1 1,2 1,4 1,6!,8 Distance from the inner surface [mm],? Fig,2: Circumferential stress distribution in the aortic wall for various levels of the model

4 2002 WT Press, Ashurst Lodge, Southampton, SO40 7AA, UK. All rights reserved. Paper from: Damage and Fracture Mechanics V, CA Brebbia, & S Nishida (Editors). SBN Damageand Fracture Mechanics V angle or the axial prestretch increases slightly. The importance of the influence of axial prestretch can be seen in fig. 1, presenting curves for the one-layer model loaded by the same pressure value, with the same residual stress (opening angle), but with different values of axial prestretch. With increasing axial prestretch, the maximum stress value and the stress gradient in the wall decrease significantly, For example, the stress ratio p. decreases from 1,29 (when there is no axial prestretch) to 1,08 (when axial strain equals 50Yo), A similar influence of residual stress and axial prestretch on the stress gradient can be found in the two-layer model but across each of the different layers, of course. The extreme stress ratio in the inner layer decreases from 1,84 to 1.17 under the same conditions (see fig. 2). The circumferential stress gradient in any layer is much lower at any pressure value, and at 8 kpa the stress state is nearly uniform in each of them. 3,1 Solution to inverse problems The inverse problem of solving the opening angle or blood pressure when the stress distribution is known (for example, searching of the opening angle and blood pressure corresponding to the homogeneous stress state), cannot be solved directly. To fmd these values, several various opening angles ~ were used as input values in the computations, The opening angle value which invokes the uniform stress distribution at a certain blood pressure value can be found by interpolation of the resulting extreme stress ratios, (The results do not correspond to the previous paragraph, because other material curves were used in the models), The opening angle determined in this way (~ = 510, gives a uniform stress distribution in each of the layer of the two-layer model under the blood pressure value of 12 kpa, This can be verified by the direct computation with this opening angle used as an input value, as presented in fig.3. Under these conditions, the stress distribution across the one-layer wall is uniform for the &l,l 1 1,2 i 1.- :1. to a) 3 0,9 -g ~ 0,8 d + extreme stress ratio - 1-layer model + etireme strain ratio - 1-layer model -k u-l (),7 extreme stress ratio - 2-layer model 0, Blood pre &re [kpa] Fig. 3: Extreme stress and strain ratio as a ti.mction of blood pressure (opening angle ~=51 deg, axial prestretch 30%)

5 2002 WT Press, Ashurst Lodge, Southampton, SO40 7AA, UK. All rights reserved. Paper from: Damage and Fracture Mechanics V, CA Brebbia, & S Nishida (Editors). SBN Damage and Fracture Mechanics V 491 blood pressure of 11 kpa, The curves in fig, 3 present extreme strain ratio as well. They enable us to find the blood pressure value at which the strain distribution in the aorta or in the aortic layer is uniform, in accordance with the hypothesis presented by Hayashi and Li [1]. This blood pressure value for the one-layer model equals to 14 kpa, the two-layer model gives the blood pressure value of 17kPa. 4 Stress state modelling in artery-vascular graft joints The actual level of computational model enables us to model stress and strain states in anastomoses (joints) of arteries with vascular grafts. The models presented here are conformable to the models of the intact aorta but they do not consider residual stresses, yet. Two variants of anastomosis geometry have been modelled, a lapped anastomosis and a classical,,end-to-end anastomosis. The lapped anastomoses are not commonly used but the commercial tissue adhesives enable us to test their feasibility and to compare their properties with the classical anastomoses shapes. Additionally, the lapped anastomoses enable us to overcome another substantial problem of modelling. f the knitted grafts are modelled as a continuum, their show a nearly zero stiffness in compression. n the lapped anastomosis the tubes undergo no substantial bending and the load invokes biaxial tension in the graft (i.e. the principal membrane stress components are positive) so that the stress-strain curve in compression does not have to be used in the model. t can be expected that the extreme stresses in the lapped anastomosis will be lower than in the classical one on account of lower additional bending of the 1, / ~ 0,8 - ~ E. (n ; 0,6 < > % al E 0,4 0,2- o~ o 0,1 0,2 0,3 0,4 0,5 0,6 Natural atrain [-] Fig.4: Stress-strain curve of the knitted graft used in the model. Value &ois the strain value corresponding to the nominal graft diameter,

6 2002 WT Press, Ashurst Lodge, Southampton, SO40 7AA, UK. All rights reserved. Paper from: Damage and Fracture Mechanics V, CA Brebbia, & S Nishida (Editors). SBN Damage and Fracture Mechanics V wall. Allthough we do not know any limit stress or strain values for arterial tissues, lower extreme stresses should ensure lower risk of clinical complications such as aneurysms or ruptures. And for such a comparison the actual level of computational model could be sufficient, 4.1 Lapped anastomosis of aorta with a knitted prosthesis There were two basic problems to solve in this computational model, The former is the nearly zero stiffhess of the knitted vascular grail in compression, the later is the diameter mismatch between aorta and vascular graft which is commonly used on behalf of the higher tension stiffness of the graft in comparison with the natural artery which can be one of the causes of neointimal thickening. The ariastomosis model is created by two lapped cylindrical tubes (see fig,5), The inner diameter of the vascular graft in the unloaded state is larger than the outer diameter of the aorta. Therefore, in the unloaded state, there is no contact between the aorta and graft models, the contact comes into existence in consequence of their deformation caused by the inner pressure and influenced by the stiffhess mismatch between the tubes. Therefore it is not possible to load the joint in axial direction if the inner pressure is not yet high enough to ensure the contact between the aorta and the graft models. n solving this problem, a nearly zero stiffness of the graft in compression was used. The initial shape of the model was in accordance with the expected NODAL SOLUTON STEP=l SUB =4 TME=l EXPANDED Sz (AVG) RSYS=O D!4X =2.865 SMN = SMX =, unloaded shape of - MAR :39:13 vascular ~ graft w gj _ aorta l-s3w~ Fig. 5: Circumferential stress (MPa) distribution in the lapped anastomosis of aorta and knitted vascular graft.

7 2002 WT Press, Ashurst Lodge, Southampton, SO40 7AA, UK. All rights reserved. Paper from: Damage and Fracture Mechanics V, CA Brebbia, & S Nishida (Editors). SBN Damage and Fracture Mechanics V 493 reality, i.e. the aortic tube was unloaded and undeformed but the graft was deformed to achieve its inner diameter equal to the outer diameter of the aorta. As this range of deformation is accompanied by no relevant stress in the graft, the stress-strain curve with nearly zero modulus of elasticity was prescribed for this strain range. After the nominal graft diameter has been achieved, the real experimental non-linear stress-strain characteristic of the grail was used in the multilineal elastic material model (see fig,4). This model approximates the natural strain-true stress curve with several straight lines. An example of stress distribution achieved using this model is presented in fig Classical,,end-to-end anastomosis of two identical parts of aorta This type of anastomoses brings much more problems with the computation convergence because of much higher bending deformations. Our model presents another type of anastomosis, namely an anastomosis of two identical parts of natural aorta. This type of anastomosis is used in arterial surgery in the case of coarctation of aorta. The level of the model is the same as in par, 4.1. The undeformed and deformed shapes of the model and the resulting stress distribution can be seen in fig. 6, NODAL EQLUT ON STEP=3 SUB -25 TME-75 SNT DFX SN i t( x Fig. 6: Von Mises stress distribution (kpa) in the joint of two identical parts of aorta (coarctation)

8 2002 WT Press, Ashurst Lodge, Southampton, SO40 7AA, UK. All rights reserved. Paper from: Damage and Fracture Mechanics V, CA Brebbia, & S Nishida (Editors). SBN A Damage and Fracture Mechanics V 5 Discussion The comparison of the models solved enables us to draw the following conclusions: The tissue remodeling in arterial wall gives a nearly uniform strain and stress distribution at a certain (mean) value of the blood pressure, Any of the investigated anastomoses causes some increase in the extreme stress values and is stiffer than the intact artery. A higher stress value brings not only a higher damage risk, but a higher wall stiflhess as the consequence of the material nonlinearity, as well. Finally, the higher wall stiffhess, together with the high graft stiflhess and with the stiffening effect of the joint shape, can contribute to the thickening of the intimal layer and, consequently, to the decrease of arterial lumen. The pulse wave propagation in the arteries can be also influenced by a partial pu~se wave reflection by the anastomosis or by the stiffer parts of the arterial tube, The comparison of the models shows that in both of these criteria, the lapped anastomosis is better than the classical shape of the joint. Fig. 6 shows the substantially limited deformation of classical anastomosis shape in comparison with the intact aorta. The length b of the collar in fig. 6 is one of the most important parameters influencing the above mentioned negative properties of these anastomoses. 6 Conclusion The created model enables us to consider all the substantial load factors, i.e. blood pressure, axial prestretch and residual stress in assessment of stress and strain states in arterial wall. Allthough the presented models of anastomoses of aorta with vascular graft are still more simple, they can be solved on the same level. The model is able to be easily enlarged for a 3D geomem. ts main disadvantage is the assumption of material isotropy. t could be overcome by using another program system (e.g. LS-DYNA) in which a special form of strain energy density fi.mction convenient for soft tissues is implemented. The level reached in this way could be sufficient for the surgeons to help them to make decisions, e.g. by comparing stress and strain states in anastomoses with various shapes or geometric parameters. Acknowledgments This work was supported by the MSMT of the Czech Republic, Research Project No, MSM References [1] Hayashi, K., Li, X,Y.: Histo-Dimensional Analysis of Strain Distribution in the Arterial Wall. Proceedings of the Third Conference on Biomechanics, JSME, Tokyo, pp , 1993.

9 2002 WT Press, Ashurst Lodge, Southampton, SO40 7AA, UK. All rights reserved. Paper from: Damage and Fracture Mechanics V, CA Brebbia, & S Nishida (Editors). SBN Damage and Fracture Mechanics V 495 [2]Bur3a, J,: Computational Simulation of Stress State in Arteries. Proceedings of the 3st nternational Conference Modelling and Simulation of Systems MOSS 97, Hradec nad Moravici, CZ, pp , 1997, [3] Bur3a, J.: Computational modelling of residual stresses in arteries using fictive temperature (in Czech), Proceedings of the Conference Engineering Mechanics 99,, Svratka, CZ, Volume 2, pp , [4] Bur5a, J., Janifiek, P.: Computational modelling of stress and strain states in aorta. Proceedings of the 12 h Conference of the European Sociep of Biomechanics, Dublin, reland, p. 441,2000. [5] Bur3a, J.: Computational Modelling of Stress and Strain States in Artery Walls Using FEM, Zeszyty naukowe katedry mechaniki stosowanej, Nr. 6, pp , [6] Bur5ai J: Possibilities of using strain energy density fimctions to solve stress state in arteries. Proceedings of the V. Conference on Biomechanics, Olomouc, CZ, pp ,2000. [7] Bur3a, J,, Vajddk, M.: Stress-strain analysis of an overlapped connection of artery and vascular graft. Proceedings of the Conference Engineering Mechanics 2001, Svratka, CZ, pp ,2001. [8] Bur3a, J., Vajdtik, M.: Mechanical Optimization of Geometry of the System Artery-Vascular Graft, Proceedings of the 3r~nternational Conference on A4echatronics, Robotics and Biomechanics, Trest, CZ, pp.45-50, 2001.