Effects of Aging on the Mechanical Behavior of Human Arteries in Large Deformations

Transcription

1 International Academic Institute for Science and Technology International Academic Journal of Science and Engineering Vol. 3, No. 5, 2016, pp ISSN International Academic Journal of Science and Engineering Effects of Aging on the Mechanical Behavior of Human Arteries in Large Deformations Amin Safi Jahanshahi a,c, Alireza Saeedi b a PhD student, Shahid Bahonar Universiy, Kerman, Iran. b Professor, Shahid Bahonar University, Kerman, Iran. c Lecturer Sirjan University of Technology, Sirjan, Iran. Abstract The mechanical properties of living cells related to a lot of physical and Physiological factors. Aging is one important factor in changing physical and mechanical properties of living cells. The mechanical behavior of living cells and live tissues is nonlinear and their deformation is large. Finding a suitable mechanical model that can predict this behavior can have a lot of applications in prevention and treatment of various diseases. In this paper first based on the theory of nonlinear elasticity kinematic, equations has been extracted. Mooney-Rivlin nonlinear model considered on characteristic equations and using results of the report elastic stress test of biaxial Elastic constants have been calculated for the arteries. For simplicity, we considered artery into a long homogeneous cylinder. Elastic constants calculated for different age shows that arteries of people become coated by aging and their flexible capability decreases dramatically. Keywords: Nonlinear Elasticity, Large Deformations, Arteries, Biaxial Stress Test, Aging Introduction: Arteries and veins consisted of three layers (Figure 1) (Meyers et al.2008). Their Tunica intima is covered by a layer of endothelial cells. These cells are creating an inner membrane by mechanical and electrochemical connections for blood vessels. In fact, this membrane forms the boundary between the blood and the main lining of the vessels. Endothelial cells have consisted of collagen, fibronectin and laminin respectively (Meyers et al, 2008; Jafee 1987; Cox1978; Karsaj and Humphrey 2012). 57

2 The Tunica media or the main lining of the vessels includes the most vessels body and consisted of muscle cells and elastic by categories formation of collagen. Figure 1: Cross section of the vessel includes intima, media and adventitia layer. In arteries closer to the heart which have large-diameter, Elastic percentage is more than the muscle and in the arteries away from the heart that have the smaller diameter, this ratio is reversed. The outer layer of type 1 collagen, nerve cells of fibroblast (a type of cell that has the capability to become collagen fibers.) and formed elastic fibers (Meyers et al, 2008; Jafee 1987; Karsaj and Humphrey 2012). In macroscopic analysis, we can consider the entire arteries lining as isotropic or anisotropic, homogeneous and incompressible material (Carew et al. 1968; Vaishnav et al. 1972; Hudetz1980; Chuong and Fung1984; Papageorgiou and Jones1987). Deformation of large living tissues and their mechanical behavior is nonlinear. For mathematical modeling of arteries, we can consider them as long cylinder and double-layer or single-layer (Maltzahn 1981; Maltzahn 1982; Maltzahn et al. 1984; Hudetz 1980; Holzapfel and Weizsacker 1998). More modeling conducted, they have considered artery lining as homogenous with low thickness (Taber 1998; Rachev2000; Gleason et al. 2004; Tsamis et al. 2009; Valentin et al. 2009; Wan 2010) while to estimate actual behavior arteries, we need to provide a model with high thickness. In this paper, we consider a single-layer model with the actual thickness and calculate the Mooney-Rivlin constants based on biaxial stress test for presented model. Constants obtained from the biaxial stress test calculated and presented in different ages. Kinematic Suppose that we show the position of a point in the coordinate transformation to vector space X, after changing configuration, we illustrate this point with the position vector x screw. ( ) (1) Deformation gradient tensor expresses as follows: 58

3 In Cartesian coordinates, deformation gradient tensor components change as: that and place components of vectors x and X in the deformed and unreformed coordinate respectively. Polar decomposition theorem express as follows: That R is an arbitrary symmetric tensor and U and V are orthogonal and positive definite tensor, we can say them as right and left stretch tensor respectively and we write them as the shape of spectrum in terms of specific vectors as follow: (2) (3) ( ) ( ) ( ) ( ) (4) Where * + are the principal stretches, ( ) i.e.are the unit eigenvectors of, the so-called Lagrangian principal axes, ( ) the unit eigenvectors of, the Eulerian principal axes, and denotes the tensor product. We can write Jacobin of transformation based on the principal stretches as follows: (5) We can calculate Left Cauchy-Green deformation tensor by using the following formulas: and right Cauchy-Green deformation tensor (6) Green strain tensor is defined as follows: ( ) (7) That has equal invariables by the left and right Cauchy-Green strain tensors as follows: ( ), ( )- ( ) (8) That ( ) represents the trace of second order tensor. For an incompressible substance, equation (5) will be as follows: Stress and Equilibrium Equations According to Figure 2, consider an Element of the media in deformed and unreformed state. In the absence of body forces, we have: (9) (10) 59

4 Where is forming the outer surface of the object after deformation Figure 2: transfer an Element before and after deformation We can write the recent equation as follows: Now, that ( ) and also the relation between nominal and Cauchy stress, We can rewrite equation (10) as: (11) (12) That Forms the outer surface of the body before reshaping and is the first piola stress tensor. By applying the divergence theorem on, we have: The second Piola stress tensor is another stress tensor we show here with P: Now tensor showed another stress tensor called the Biot stress tensor with T and we express as following form: ( ) ( ) (15) That R is an arbitrary orthogonal tensor that we mentioned previously in the case of polar decomposition analysis. Characteristic Equations Strain energy function density W for a homogeneous material is only a function of the deformation gradient F. If the material was inhomogeneous, strain energy function in addition to deformation gradient (13) (14) 60

5 will also hinge to different parts to X. If there were no internal constraints such as incompressibility, nominal stress and deformation gradient will be co-producing that their multiplications produce work. So we have simply: We write recent equation in component form as. For an incompressible material equation (16) comes as following: That p is the Lagrange multiplier imported by adding an incompressible bond. We can write equation (16) according to Cauchy stress as follows: The mentioned equation for an incompressible material will be as follows: A logical consequence obtains from isotropy is that specific Cauchy stress tensor values, and also Baiot stress tensor and tensor U are equal. So we can write by similar process whatever we applied by writing the (4) equation: (16) (17) (18) (19) ( ) ( ) ( ) ( ) (20) That and are the principal stresses of Biot and Cauchy. Therefore: (21) And for an incompressible material, we have: (22) The equation (20) will be as following for Piola stress type two and nominal stress: ( ) ( ) ( ) ( ) (23) Which and. Biaxial Stress Test In this section, we review homogeneous deformation that in general we can consider it as follows: 61

6 (24) Deformation gradient tensor is: [ ] (25) And by considering the incompressibility conditions, we have: [ ] [ ] (26) We consider the following invariables based on principal stretches: (27) It will be as following by applying incompressibility condition: (28) Consider W potential energy subordinate as follows: ( ) ( ) (29) By eliminating p from the equation (22-b), we can write: (30) If we consider the biaxial test, we have and therefore the above equation will become: (31) If we consider: ( ) ( ) (32) Equations (31) conclude: 0 1 (33) And so it obtains a similar equation for that finally after simplification we have: ( )( ) ( )( ) (34) 62

7 By solving the mentioned equations, we have and : ( )( ) ( )( ) (35-a) ( )( ) ( )( ) In the above equations for simplicity in writing, we have used energy function for Mooney-Rivlin material is as follows: (35-b) equation. Helmholtz free ( ) ( ) (36) We can write Helmholtz free energy function for the Neo-Hockean by putting above equation as follows: equals zero in the ( ) (37) Comparing the equations (35) with (30) we have: This means that by doing a biaxial stress test, we can easily achieve Mooney-Rivlin constants. By using results of biaxial stress test that conducted by Mohan and Melvin (Mohan and Melvin 1983), we can easily calculate constants. Suppose we define two new parameters of μ and α: (38) (39) In this case we can write for the Mooney-Rivlin material: ( ) ( )( ) (40) In this case, it is clear that by placing equal amounts of α, we will have neo-hock in equation: ( ) (41) Mechanical analysis of arteries In this analysis, we consider whole lining arteries a homogeneous and isotropic substance. Deformation field for cylinder under tension and inside pressure is as follows (Batra &Bahrami 2009) ( ) (42) 63

8 That is stretch along the length of the cylinder. Given the right malformed deformation, we can write gradient tensor as follows: [ ] (43) That. By applying incompressibility condition, we have:. /. /. / (44) Integrating the last equation, we have: In equation (45), A is integration constant that we must identify it. By calculating left Cauchy deformation tensor and putting it in the equation (6), we have non-linear structure of Mooney-Rivlin equation: (45) (46) (47) In the recent equations If the constants p, D, A, distinguished, stress distribution is completely clear and solving the problem is accomplished. Balance equation in polar coordinates is expressed as follows: (48) ( ) (49) And we consider boundary conditions as follows: ( ) ( ) (50) (51) That and are internal and external artery pressure respectively and are axial force. By applying boundary conditions and a little math equations, we finally get the following equation. 64

9 ( ) 0 ( ) ( )1 (52) *. /. /. / 0 ( ) ( )1 +( ) (53) So equations (52) and (53) are two equation based on constants A, D, which we can calculate in this two unknown coefficients by solving simultaneously and finally stress distribution is fully characterized. Results The experimental results conducted by Mohan and Melvin using the biaxial stress test (Mohan and Melvin 1983). Mooney-Rivlin material constants calculated for artery in different ages and it displayed in Table 1 according to the equations (35). Table1. Mooney-Rivlin equation constants for artery in different ages ( ) ( ) Age(yaer) In order to show the results, in Table 2 hypertension environmental tensions and radius for 8 mmhg at different radii based on the theory of nonlinear elasticity Mooney-Rivlin modeling compared by resulting of numerical solution. Table 2 comparison between the calculated environmental stresses and radial on non-linear Mooney-Rivlin and Numerical solution Result r(mm) T θθ(mpa) -T rr(kpa) Mooney-Rivlin Numerical Mooney-Rivlin Numerical

10 T θθ (MPa) t t ( M P a ) International Academic Journal of Science and Engineering, In Figure 3 environmental stress distribution along the radius at different ages for internal pressure 16 mm Hg is drawn. This figure shows that with increasing age, increasing environmental stress in arteries. The reason for this is that with age, the flexibility of the arteries decreases r ( m m ) r (mm) Figure 3: Distribution of environmental tension in terms of the radius at different ages The graph shows that the inner layers in contrast to the outer layers of arteries play more role in stress tolerance.in This article arteries lining considered as single-layer that due to stress distribution obtained it is suggested that we should use a two-layer to investigate stress distribution References: A. G. Hudetz, (1980). Continuum mechanical methods and models in arterial biomechanics, Advances in Physiology, 8(6): A. Rachev, (2000). A model of arterial adaptation to alterations in blood flow, Journal of Elasticity, 61(1): A. Tsamis,N. Stergiopulos,A. Rachev, (2009). A structure-based model of arterial remodeling in response to sustained hypertension, Journal of Biomechanical Engineering, 131(10):

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