Jtam-A4.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 51, 4, pp. 949-958, Warsaw 2013 EXPERIMENTAL CHARACTERIZATION OF THE MECHANICAL PROPERTIES OF THE ABDOMINAL AORTIC ANEURYSM WALL UNDER UNIAXIAL TENSION Magdalena Kobielarz, Ludomir J. Jankowski Wroclaw University of Technology, Division of Biomedical Engineering and Experimental Mechanics, Wrocław, Poland e-mail: magdalena.kobielarz@pwr.wroc.pl; ludomir.jankowski@pwr.wroc.pl Althoughmany researchers havemade the assumption that the abdominal aortic aneurysm (AAA) wall behaves as an incompressible and isotropicmaterial, the experimental evidence for it is insufficient. Hence, the assumptions about the incompressibility and isotropy of the AAA wall were verified through analysis of stretch ratios of samples excised from the aneurysms walls. The stretch ratios were calculated on the basis of a real-time analysis of geometric dimensions of samples subjected to uniaxial tension. It was proved that the walls of abdominal aortic aneurysms can bemodelled as an incompressible and isotropicmaterial. Using histological techniques, the assumption concerning the negligence of shear stress in the analysis of AAA wall stresses was indirectly validated. The results were incorporated into a hyperelastic constitutive equation. Key words: abdominal aortic aneurysm, incompressibility, isotropy, shear stress 1. Introduction Theabdominal aortic aneurysm(AAA) is apermanentandprogressingdilation of theabdominal aorta by at least 50% as compared with its normal diameter (Sakalihasan et al., 2005; Li and Kleinstreuer, 2006). It results from the pathological multifactor remodelling the aortic wall connective tissue caused by enzymatic degradation of the main load-bearing components, i.e. elastin andcollagenfibres (Brady et al., 2004;Longo et al., 2005).The initiation anddevelopment of an AAA results in significant changes in the mechanical properties of the abdominal aorta wall (DiMartino et al., 2006; Geest et al., 2006a; Kobielarz et al., 2008). This means that a proper theoretical basis is essential for description of the mechanical properties of AAAwalls. Despite the intensive development of models and constitutive equations for pathologically unaffected blood vessels modelled as poroelastic materials (Simon et al., 1998; Johnson and Tarbell, 2001), viscoelastic materials (Veress et al., 2000; Holzapfel et al., 2002) or pseudoelastic materials (Fung, 1967; Chuong and Fung, 1986), for beehaviour of AAAwalls behaviour under mechanical loadsmodels based on the linear theory of elasticity (Mower et al., 1997; DiMartino et al., 1998; Vorp et al., 1998) or on the law of Laplace (Elger et al., 1996; Hall et al., 2000) are still commonly used. The application of the Laplace law to the assessment of the mechanical properties of AAAwalls undermechanical loads is incorrect for two reasons. Firstly, the AAA’s geometry does not correspond to a thin-walled cylinder or a sphere with a single curvature radius, forwhich theLaplace lawholds true.Each aneurysmhas adifferent shape, a complicated geometry with a different degree of eccentricity, and a variable wall thickness (Damme et al., 2005; Vorp and Geest, 2005). Secondly, the AAA diameter is not the only determinant of wall stresses (Vorp et al., 1998; Geest et al., 2006b). Neither is the application of the linear theory of elasticity to the assessment and analysis of the mechanical properties of AAA walls proper since this material stress-strain characteristic has been shown to be nonlinear (Raghavan et al., 1996; Kobielarz et al., 2004). Therefore, themechanical properties of aneurysmswalls should be 950 M. Kobielarz, L.J. Jankowski assessed on the basis of the nonlinear theory of elasticity. Moreover, the AAAwall is amaterial which is subjected to large strains (amounting to 20%-40%) prior to its failure (He ang Roach, 1994; Raghavan et al., 1996). Hence, it is necessary to use the theory of large strains in order to model the behaviour of AAA walls under mechanical loads. The few constitutive equations derived from the nonlinear theory of elasticity, i.e. hyperelastic models (Yamada et al., 1994; Raghavan and Vorp, 2000) are used assuming AAA wall incompressibility and isotropy and neglecting shear stresses without experimental evidence however. Therefore, themain objective of this research is to verify the a priori assumptions about abdominal aortic aneurysms walls for a large group of preparations and to evaluate the application of the results in a hyperelastic constitutive equation taking the theory of large strains into account. 2. Material and method 2.1. Test material The test material had the form of 96 AAA wall specimens intraoperatively taken from the anterior parts of the vessel. Thematerial was stored in a physiological salt solution at a tempe- rature of 4◦C until testing (no longer than 12h). The samples were obtained by permission of Bioethical Commission at theMedical University ofWroclaw, and the studieswere conducted in accordancewith the established procedures of preparation and storage of the biologicalmaterial. 2.2. Assumptions The assumption about the incompressibility and isotropy of AAA walls leads to many sim- plifications in the constitutive equation formulas and easier stress analysis. Thematerial incom- pressibility and isotropy assumptions are correct when proper conditions are satisfied, i.e. the product of the stretch ratios is constant and equal to 1 (λ1λ2λ3 =1) and the stretch ratios in di- rections perpendicular to the exciting force are equal to each other (λ2 =λ3). The assumptions were verified for uniaxial tension (at a constant rate of 2mm/min) of samples excised fromAAA walls in two directions orthogonal to the vessel long axis, i.e. in the circumferential direction (AAAc) and the longitudinal direction (AAAl). The excised quasi-planar samples, having an initial width s0 of 5.0mm, were mounted in a testing machine (Synergie 100, MTS, Fig. 1) by means of jaw chucks. The initial length l0 of each sample was 25mm (at a deviation less than 0.5mm). The Lagrangian stretch ratios in the three orthogonal directions (λ1,λ2,λ3) were cal- culated from the geometric dimensions of the samples recorded in real time in the course of the uniaxial tension test with a frequency of 5Hz by a videoextensometer (ME46-350,Messphysik). Fig. 1. Measurement set-up: testing machine (Synergie 100, MTS) and videoextensometer (ME 46-350, Messphysik) Experimental characterization of the mechanical properties ... 951 The uniaxial tensile test was preceded by pre-stretching: the sample was preloaded to 10% of its initial length and then unloaded to zero. The full cycle was repeated three times since the preliminary tests showed that the stress level stopped decreasing after three full stress cycles and thematerial behaved in a repeatable way during the further cyclic loading and unloading. The concept of neglecting shear stress in the analysis of AAA wall stresses is based on the commonly held view today that degeneration of the inner layer, containing endothelial cells sensitive to shear stress, takes place in the walls of abdominal aortic aneurysms (Holmes et al., 1995). The presence of degenerative changes in the inner layer of AAA walls has been proven through a histological analysis. Samples about 10mm2 of the full vascular wall thickness in size were excised from the test material, fixed in a 4% aqueous solution of formalin washed under running water for 24 hours and dehydrated through immersion in alcoholic baths with an ever higher concentration (from 70% alcohol to absolute alcohol). Then the samples were immersed in sodium benzoate for 24 hours. Thematerial prepared in this way was embedded in paraffin. The paraffin block containing thematerial was cut into 5µm thick slices bymeans of aMikron HM315 (Zeiss) microtome. The samples were dyed in two ways: with haematoxylin and eosin (H&E) and by Van Gieson’s method. The histological preparations were viewed under light microscopeAxioImager M1m (Zeiss). 2.3. Statistical analysis The results were presented as averages with standard deviations (X±SD). The statistical analysis was performed using Student’s t-test for dependent samples (Statistica 8.0, StatSoft). The tests were carried out assuming the limit significance level (p) of 0.05. 3. Results 3.1. Verification of assumptions The average stretch ratios in two directions perpendicular to the exciting force, and the product of the stretch ratios in three orthogonal directions were calculated (Table 1). Table 1.Product of the stretch ratios λ1λ2λ3 and values of the coefficients: λ2 and λ3 for sam- ples excised fromAAAwalls in the circumferential direction (AAAc) and longitudinal direction (AAAl) relative to the vessel long axis AAAc AAAl λ1λ2λ3 λ2 λ3 λ1λ2λ3 λ2 λ3 0.98±0.21 0.95±0.04 0.94±0.04 0.99±0.25 1.00±0.06 0.98±0.04 The obtained results show that the AAA wall incompressibility assumption (regardless of the sample excision direction) is correct since the product of stretch ratios is equal to approxi- mately 1 in each considered case. The results also indicate that aneurysms walls under uniaxial tension behave as an isotropic material since the statistical analysis did not show any statisti- cally significant differences in the results (λ2 vs. λ3) within the particular groups (for AAAc: p=0.43; for AAAl: p=0.19). Thehistological analysis revealeddisorders in the laminar structureof theAAAwalls (Fig. 2). Most of the analysed walls (63%) were found to be devoid of the inner layer. In the cases when the inner layer is not completely atrophied, there are series degenerative changeswhose principal feature is the lack of any visible layer of endothelium cells. 952 M. Kobielarz, L.J. Jankowski Fig. 2. Histological images of AAA walls: (a) with degenerated inner layer (TA – adventitia, TM –media, TI – intima), using H&E staining and (b) with atrophy of the inner layer, using Van Gieson’s staining 3.2. Model and constitutive equation Thebehaviour of AAAwalls under uniaxial loadingwas described using the generalized neo- Hookean model. For incompressible hyperelastic materials, the strain energy density function in the neo-Hookeanmodel depends on the first invariant (I1) of the Cauchy-Green deformation tensor as follows Ψ = c(λ21+λ 2 2+λ 2 3−3)= c(I1−3) (3.1) where c is an equation parameter; I1 – first invariant of the right Cauchy-Green transformation tensor. The constitutive equation for such a material assumes the form σ=−pI+2 ∂Ψ ∂I1 B (3.2) where σ is the Cauchy stress tensor; p – Lagrange multiplier; I – identity tensor; B – left Cauchy-Green deformation tensor. When the incompressibility (λ1λ2λ3 =1) and isotropy (λ2 =λ3) of the consideredmaterial is introduced, from equation (3.2) one can derive the following relation describing the Cauchy stress tensor component in the exciting force direction (σ1) dΨ dI1 = σ1 2(λ21−λ −1 1 ) (3.3) RaghavanandandVorp (2000) foundthat thedependencebetween dΨ/dI1 and I1−3hasa linear character. Ultimately, for uniaxial tension, the constitutive equation proposed byRaghavan and Vorp (2000) assumes the form σ1 = [2α+4β(λ 2 1+2λ −1 1 −3)](λ 2 1−λ −1 1 ) (3.4) Taking into account the relation for the normal component of the Green strain (E1) tensor in the exciting force direction, on the assumption that the shear components of the deformation gradient tensor are insignificant (Van Bavel et al., 2003; Geest et al., 2006b), one gets E1 = 1 2 (λ21−1) (3.5) where λ1 is the stretch ratio in the exciting force direction. Hence, the following form of the constitutive equation is σ1 = { 2α+4β [ (2E1+1)+2 √ 2E1+1−3 ]}[ (2E1+1)−2 √ 2E1+1 ] (3.6) Experimental characterization of the mechanical properties ... 953 The stress-stretch ratio curves obtained from the uniaxial tension test were described using constitutive equation (3.4), while the stress-strain curves were described using consti- tutive equation (3.6). Equations (3.4) and (3.6) fit the curves with a very good approximation: R2 min =0.962±0.011 and R2 min =0.969±0.027, respectively. The degree of fittingwas analyzed using theMicrocal Origin 7.0 software. The approximating function was determined for all the considered cases and the averaged data (Figs. 3 and 4). Fig. 3. Stress-stretch ratio curves described by constitutive equation (3.4) for samples excised fromwalls of tested blood vessels: (a) AAAc; (b) AAAl Fig. 4. Stress-strain curves described by new constitutive equation (3.6) for samples excised fromwalls of tested blood vessels: (a) AAAc; (b) AAAl Coefficients α and β are the best-fitmaterial parameters of constitutive equations (3.4) and (3.6), and they did not significantly differ statistically between the sample excision directions. The values of the material constants obtained by the authors are lower in comparison with the ones reported by Raghavan and Vorp (2000), although for the coefficient β, the order of magnitude is the same. Whereas the constant α obtained by the authors is at least two orders of magnitude lower for bothmodels (3.4) and (3.6). 4. Discussion In recent years, significantly increased interest in experimental studies of mechanical properties of biological tissues, including hard (Kot et al., 2011; Nikodem, 2012) and soft (Pezowicz, 2010; Żak et al., 2011) tissues.Now, to describe the behaviour of soft tissues under different conditions of mechanical loading, a nonlinear theory of elasticity (Holzapfel, 2000; Humprey, 2002) is com- monly used. For description of pathologically altered tissue, usually adjustedmodels previously developed for tissue without pathological changes are incorporated. Assumptions of adapted constitutive models require verification, however. Hence, the assumptions of incompressibility, 954 M. Kobielarz, L.J. Jankowski isotropy and shear stress in the abdominal aortic aneurysm wall were analysed, because many researchers have made the assumption without the experimental evidence. In the literature, it is commonlyassumed thatundermechanical loads, thewalls of abdominal aortic aneurysms, similarly as those of healthy vessels, are almost incompressible (Thubrikar et al., 2001; DiMartino et al., 2006; Raghavan et al., 2006). The assumption about the blood vessel wall incompressibilitywas introducedbyCarew et al. (1968)whoproved thatunderphysiological strains, the walls of blood vessels behave as an incompressible material. The incompressibility assumption is based on the principle of conservation of a structure volume during the deforma- tion of its material (Vito and Dixon, 2003). The incompressibility assumption makes sense in the case of biological tissues containing large amounts of water since water, is incompressible under physiological pressures (Vito and Dixon, 2003). This assumption is also valid for blood vessel walls which show negligible permeability to water (Chuong and Fung, 1986; Holzapfel and Ogden, 2003). Also in the present work, it was demonstrated that the walls of abdominal aortic aneurysms can be regarded as an incompressible material. The growth of an AAA does not result in a loss of the vascular wall ability to maintain its volume constant as the vessel structure is subjected to deformation (uniaxial loading). However, in the case of AAA walls showing signs of rupturing, permeability certainly increases, which may be the reason why the standard deviation was found to be quite high. The influence of the degree of advancement of the disease on the incompressibility of AAAs should be the subject of further research. The walls of healthy blood vessels are treated as anisotropic materials because of their complex and heterogeneous structure (Holzapfel andWeizsacker, 1998; Geest et al., 2004). It is known that themechanical properties of ahealthybloodvessel dependmainly on itsmiddle layer (Humphrey, 1995; Ogden and Schulze-Bauer, 2000). Histologically, the middle layer is a highly organized three-dimensional heterogeneous network built of three main structural components (elastin fibres, collagen fibres and smoothmuscle cells), but asOgden and Schulze-Bauer (2000) research shows, under mechanical loads, the middle layer behaves as a homogenous material. Moreover, Stergiopulos et al. (2001) showed that the middle layer in the pig aortic wall is characterized by a uniform distribution of matrix proteins and smoothmuscle cells, and similar mechanical properties along its entire thickness. Hence, in some papers, it is suggested that during mechanical tests the walls of blood vessels behave as isotropic structures (Weizsacker andPinto, 1988; Dobrin, 1999). For this reason, the walls of blood vessels are oftenmodelled as an isotropic material (Raghavan and Vorp, 2000; Geest et al., 2006a; Heng et al., 2008). It has been proved here that AAAwalls subjected to uniaxial tension can bemodelled as an isotropic material, which corroborates the hypotheses put forward by Raghavan et al. (1996), Kobielarz et al. (2004) andWitkiewicz et al. (2007). In the literature, it is generally believed that shear stress is not a significant factor in the analysis of AAAwall stresses, even though in vivoAAAwalls are subject tomultiaxial stresses, including normal and shear stresses produced by the blood flowing through the vessel. There are three reasons for the negligence of shear stresses. Firstly, the shear component is insignificant in comparison with the normal component of the stress vector (Truijers et al., 2007). Peattie et al. (2004) established that the shear stresses in AAA walls were below 2 ·10−6MPa, whereas the peak principal stress is at least 5 or even 6 orders of magnitude higher (Raghavan et al., 1996; Vorp, 2007;Kobielarz et al., 2008). Secondly, clinical observations and structural studies indicate that most of AAA walls have no distinguishable inner layer whereby the AAA wall is devoid of a functional layer of endothelial cells sensitive to shear stress (Holmes et al., 1995). Thirdly, the majority of AAAs contain mural thrombus which may shield the wall against the action of shear stresses generated by the flow of blood (Wang et al., 2002), and act as a damper (Vorp et al., 1996). The structural examinations carried out as part of the present research revealed atrophy or degeneration of the inner layer in most of the AAA wall preparations. Because of the lack of a properly developed inner layer, the walls of the abdominal aortic aneurysms were Experimental characterization of the mechanical properties ... 955 devoid of a functional layer of endothelial cells. Moreover, in 75% of the cases, mural thrombus sticking to the inner surface of the aortic wall was found to be present, although the significance ofmural thrombus is debatable (Hans et al., 2005). The shear stress neglect assumptionhas been proven through the indirect quality analysis. In the authors’ opinion, the insufficient number of studies evaluating shear stresses in the walls of AAAs is one of the limits to the development of constitutive models for the AAA. Therefore, more research is needed in this area. The lack of experimental verification of the assumptions concerning AAAwall incompressi- bility and isotropy is themain constraint of most constitutive models. The verified assumptions presented here have been incorporated to the constitutive equation derived from a hyperelastic model proposedbyRaghavanandVorp (2000) basedon thegeneralized neo-Hookeanmodel.Mo- reover, similarly as inYamada et al. (1994), large strains have been introduced into the equation. Thus, model (3.10) takes into account the theory of large strains, the experimentally verified assumption about the incompressibility and isotropy of AAAwalls and the structurally justified neglect of shear stresses. The proposed constitutive equationwell approximates the stress-strain characteristics. The analytically determinedmaterial constants (α and β) assume lower values than the ones calculated from the model by Raghavan and Vorp (2000), although in the case of coefficient β, the order of magnitude is the same. The constant α obtained in the present study is at least two orders of magnitude lower. The reduction is not due to the introduction of Green’s strains. The coefficient α assumes equally low values when the stress-stretch ratio curves are approximatedwith the constitutive equation proposedbyRaghavan andVorp (2000). It indicates that the best-fit material parameters depend on the results obtained for individual populations, particularlywhen the abdominal aortic aneurysm is a dynamicpathological process caused by structural changes of different intensity in the load-bearing elements. This indicates that it is necessary to take the degree of structural changes within thewalls of abdominal aortic aneurysms into account in the description of experimentally determined curves. 5. Conclusion On the basis of experimental verification, an evidence that the walls of abdominal aortic aneu- rysms behave as an incompressible and isotropic materials undermechanical loads is presented. Through indirect validation by using histological techniques, it was proved that AAA can be modelled on the assumption of negligence of shear stresses. 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