An Analytical Study on Mechanical Behavior of Human Arteries A Nonlinear Elastic Double Layer Model Amin Safi Jahanshahi a,b, Ali Reza Saidi a a Department of Mechanical Engineering, Shahid Bahonar University, Kerman, Iran b Department of Mechanical Engineering, Sirjan University of Technology, Sirjan, Iran Abstract: The focus of this article is on analytical solution for stress and deformation of human arteries. The artery is considered as a long homogeneous isotropic cylinder. Hyperelastic, incompressible stress-strain behavior is used by adopting a classical Mooney-Rivlin material model. The elastic constants of the arteries have been calculated by using the reported results of biaxial test. The analysis is based on both single and double layer arterial wall models and. Radial and circumferential stress distribution on the minimum and maximum blood pressure is calculated. Variation of radii due to internal pressure within the arteries is found which is in a good accordance with the experimental results. The results containing the changes in diameter and thickness together with the stress distribution for both single and double layer models have been plotted. It is shown that the major difference between the single and double layer models is in their stress distributions. The circumferential stress distribution for different human s ages is plotted which shows that the stress increases by increasing the age due to decreasing the flexibility of the artery. It is also shown that despite the artery s inner layer is softer than the outer layer, the maximum stresses occur at the inner layer. Keywords: Artery; Analytical Solution; Non-Linear Elasticity; Large Deformation; Biaxial Test.. Introduction The structure and function of arteries change throughout the lifetime of humans and animals []. Blood vessels can be considered as long cylinders that their internal pressure varies between a minimum and maximum. The internal pressure as systolic and diastolic blood pressure caused by contraction of the heart muscle and blood flow through the arteries occurs. Arteries and veins collected from different region of body shows different characteristics. Artery walls mainly composed of three layers, tunica intima, tunica media and tunica adventitia (Figure )[]. Corresponding author, Department of Mechanical Engineering, Shahid Bahonar University of Kerman, Jomhouri Blvd,P.O. Box 7688-68366, Tel:. +98 34 404, Fax:. +98 34 0964, Mob: +98 93 98 46, Kerman, Iran. E-mail address: saidi@uk.ac.ir (A. Saidi); ajahanshahi@sirjantech.ac.ir (A. Safi Jahanshahi)
(Figure ) The inner layer covered by a layer of endothelial cells and it is very thin. These cells, by the mechanical and electrochemical connections with each other, cause an internal membrane for the blood vessels. In fact, this membrane forms a boundary between the blood and the vessel wall. Endothelial cells formed of collagen, fibronectin and laminin [3-6]. The middle layer or the main wall of the arteries, includes muscle cells, elastin and collagen fibers [3, 5, 6]. It has the greatest volume and it is responsible for most of arterial properties consist of a three dimensional network of smooth muscle cells, elastin and bundles of collagen fibrils (type I about 30% and type III about 70%) [6-8]. Adventitia means the outermost connective tissue covering any organs, vessels or any other structures. The outermost layer of artery is called adventitia because it is considered extraneous to artery. It consists of type I collagen, nerves, fibroblast (a cell capable of forming collagen fiber) and some elastin fibers [9]. It is also believed that adventitia has the most contribution in elastic modulus of vessel wall [0]. Tests are performed on live tissue, divided in two categories: in vitro and in vivo. García-Herrera et al. characterized the human aorta under in-vitro and in-vivo conditions [].Carew et al. [], by using in vitro experiments, show that for most practical purposes arteries may be considered incompressible. Karimi et al. [3] investigated the linear and nonlinear mechanical properties of human artery using a series of uniaxial tensile tests. Akhtar [4], by using in vitro experiments and tensile testing, studied the biomechanical properties of arterial tissue. Vashnav et al. [5] by using in vitro experiments, showed that the wall tissue can been considered to be incompressible. Using the results of in vivo tests, Hudetz [7] developed different continuum models for anisotropic, incremental elastic, nonlinear viscoelastic, and time dependent active responses of arterial walls. In macroscopic analyses, the whole artery wall can considered as a long homogenous, isotropic and incompressible material [7,, 5-7]. The stability behavior of thoracic aorta was investigated with two boundary conditions by Rastgar et al. [8] which represented two extreme cases of in vivo constraints A suitable mechanical model that could predict the artery behavior can be useful in the prevention and treatment of some artery diseases. In addition, it is an important step to the production of some artificial tissues. Taghizadeh et al. [9] examined different strain energy function to express the mechanical behavior of soft tissue. Moltzan et al. [0-] proposed a twolayer cylindrical model to describe the nonlinear properties of carotid arteries. Ogden and
Holzapfel [3] and Holzapfel and Weizsacker [4] considered a two layered, anisotropic mechanical model for arteries and they used the Neo-Hookian model to express the matrix material. Numerous biomechanical studies idealize the three-dimensional wall as a membrane (or twodimensional surface) [5-8]. Constitutive equations for the arterial wall can further be categorized by the types of biological processes. Deformations of live tissues are very large and their mechanical behavior are nonlinear [9]. For mathematical modeling of arteries, they can be considered as long single or double layer cylinder [0-4]. Safi Jahanshahi and Saidi [9] examined the mechanical behavior of human arteries taking a single-layer and isotropic model by using biaxial stress test results. In most mechanical modeling, the artery wall is assumed as a thin cylinder [-4, 30-33] which is not an accurate assumption. In this paper, a nonlinear elastic, thick long cylindrical shell model has been used to predict the stress and deformation of human arteries under internal pressure. Since the intima layer is so thin and it does not have a considerable role in carrying out the stresses, the artery has been modeled as a double layer cylindrical shell. Using a biaxial stress test, the Mooney-Rivlin elastic constants for this model have been calculated. Investigating an analytical solution, the Radial and circumferential stress distribution on the minimum and maximum blood pressure have been found. Numerical results containing the changes in diameter and thickness together with the stress distribution for both single and double layer models have been plotted. The circumferential stress distribution for different human s age have been depicted which show that the stresses increase by increasing the human s age.. Constitutive Equations The strain energy function W, for a homogeneous material depends only on the deformation gradient tensor F. If there is no internal constraint such as incompressibility, the nominal stress is work conjugate to the deformation gradient and given simply by [7]: W S () F where S is the nominal stress. For an incompressible material, equation () will be written as: W p S F, J det F () F 3
where p is the Lagrange multiplier associated with the compressibility constraint and J stands for the Jacobian. From equation (), the Cauchy stress tensor, T, can be written as: W T JF F For an incompressible material, the above equation is modified to: (3) W T F p I, det F F An important consequence of isotropy is that the Cauchy stress T has the same eigenvectors as the left stretch tensor V. So we can write: 3 i i i i T t v v (5) where t i, are the principal values of the Cauchy stress tensor and the symbol refers to the (4) tensor product. Then: Jt i W i, i,,3 i (6) where i are the principal stretches. For an incompressible material equation (6) can be written as: W t i i p, 3, i,,3 i Using equation (7) and having the energy function W, the principal stresses can be found for incompressible materials. 3. Biaxial Stress Test Biaxial mechanical tests are required to quantify mechanical properties of hyperelastic materials. Zem anek et al. [34] designed and produced an experimental rig for biaxial testing of hyperelastic materials (elastomers and soft tissues).the testing rig consists of a bedplate carrying two orthogonal ball screws, equipped with force gauges, two servo motors and four carriages ensuring symmetric biaxial deformation of the specimen and a programmable CCD camera located on a support stand [34]. They presented the procedure of biaxial tension tests for aortic walls. A pure homogeneous deformation, in general form, can be written as [3]: x X, x X, x X (8) 3 3 3 (7) 4
The deformation gradient tensor then will be found as: 0 0 F 0 0 (9) 0 0 3 Imposing the incompressibility on equation (9), give us: 0 0 0 0 T F 0 0, B FF 0 0 (0) 0 0 0 0 where B stands for the left Cauchy-Green deformation tensor. The invariants of B, according to principal stretches are considered as: I, I, I () 3 3 3 3 3 Enforcing the compressibility condition, equations () can be rewritten as: I, I, I () 3 The strain energy function W is defined as: W, W,, (3) Omitting p from equations (7), we can write: W W t t, t t 3 3 (4) For biaxial test we have t 3 0, thus, in this case, the above equations reduce slightly to: t W, W t (5) W Let us define W and W W. So we can write: I I,, W I I W (6) Equation (5) then reduces to: W I W I t W W I I Similarly, t can be found as: (7) 5
t W W (8) Solving these equations for W and W yields: W t t t t, W 3 3 3 3 For a Mooney-Rivlin material, the Helmholtz free energy function is []: 3 3 0 (9) W c I c I (0) By comparing equations (6) and (0) we have: W c, W c () It means that by a biaxial test, the Mooney-Rivlin material constants can be specified. In this research, the material constants c and c have been found by using biaxial test results reported by Mohan and Melvin [35] and equation (9). 4. Mechanical Analyses of Artery The artery wall has been considered as a homogenous and isotropic cylinder under inflation and tension. For such a cylinder, the deformations field can be considered as [36]: r r R,, z Z / D () where r, and z are the cylindrical coordinate system in current configuration and / D is stretch along the cylinder. For deformation field (), the components of the deformation gradient tensor are given by: r 0 0 F 0 r / R 0 (3) 0 0 / D where r dr / dr. By satisfying the incompressibility condition, detf, we have: rr RD (4) Solving equation (4) yields: r DR A (5) where A is an unknown constant which has to be determined. For a nonlinear elastic material, the constitutive equation is [37]: T pi cb cb (6) z 6
where T is the Cauchy stress tensor, p is an unknown constant that appears for incompressibility condition, I is the identity tensor and the constants cand c are the Mooney-Rivlin material constants which can be determined using the biaxial test. For deformation field (), the normal components of Cauchy stress tensor can be written as: D R r T rr p c c r D R (7) r R T p c c R r (8) T zz p c c D D (9) In these last three equations if constants pdand, A are known, the stress distribution is completely known and the problem is completely solved. The equilibrium equation in r-direction is: dt dr rr Trr T 0 (30) r For a single layer cylinder, the boundary conditions can be written as: rr, T r P T r P (3) in in rr ou ou F a rou T rdr (3) rin zz where Pin and Pou are the inner and outer pressure of the cylinder and Fa is the axial force. Solving equation (30), using boundary conditions (3) and integration from internal radius to radius, we have: r r D R r r R T r P c ( ) dr c ( ) dr rr (33) in 3 3 r R D R r rin rin By integrating and doing some mathematical operations, equation (33) can be written as: c A T rr r Pin cd ln r ln r A D r r rin (34) By letting r r ou in equation (34) and assuming that the external pressure be equal to zero, the following relationship between two constants A and D can be found: 7
c A 0 Pin cd ln r ln r A D r ou r rin (35) In addition, by comparing equations (35) and (7), the unknown constant p can be found in terms of the unknown constants A and D as follows: rou A c r ou c A p Pin cd c D ln r ln r A rou D rou A D r Furthermore, using the boundary condition (3), we have: rou rin (36) rou Fa p c c D rdr D r in (37) Substituting p from equation (36) into (37) and integration yields: r A c r c P c D c D Fa r r ln ou ou in rou D rou A D r ou in ou A r ln r A c c D r r D in By solving equations (35) and (38), the constants A and D will be found and finally the stress distribution will be fully determined. As previously mentioned, the arterial wall consists of two main layers, media and adventitia. So in order to have a more accurate model for the artery, it is considered as a double layer cylinder. The boundary conditions for this model are: rr, T r P T r P (39) in in rr ou ou r r in Fa π T zz rdr T zz rdr rin r (40) where r is the inner radius of the adventitia (which is equal to the outer radius of the media). By satisfying the boundary conditions (39) on equation (7), we can write: (38) c A T rr r Pin cm Dm r r A m D m r m m ln ln c A 0 Pou T rr r cada ln r ln r A a D a r ou a a r rin r r (4) (4) 8
where the indices m and a stand for media and adventitia, respectively, and Am, Aa, Da and Dm are four unknown constants which should be determined from the boundary conditions. Eliminating T rr r from equation (4) and (4), we obtain: r c m Am Pou Pin cm Dm ln r ln r Am D m r r in ca Am cada ln r ln r Am D a r rou r (43) This is a relationship between the constants Am, Aa, Da and D m, By doing the same procedure as previous, for a double layer model, we can write: r A c r p P c D m m m in m m r Dm r Am c m Am cm Dm ln r ln r A m D m r r in r A c r p P c D ou a a ou a ou a a rou Da rou Aa c a Aa cada ln r ln r Aa D a r Now, by applying the boundary conditions, equation (40) will be: rou r r (44) (45) r r ou Fa π pm c m c mdm rdr pa c a c ada rdr D rin m D r a (46) Substituting equations (44) and (45) into equation (46) and integration yields: F a r A c r P c D π rou A a c a r ou Pou c ada rou Da rou Aa c a c ( rou ada rou r ca Aa D a c ada ln r ln r A a D a r r m m in m m r Dm r Am r m in c c m Dm r r m m D c A m c m Dm ln r ln r Am Dm r rin (47) ) 9
On the other hand, the non- slip condition requires: D D D * a m (48) A A A * a m (49) Finally, by solving equations (43), (47) simultaneously and using equation (48) and (49), the constants * A and 5. Results * D can be obtained and the problem will be solved completely. At first, using the experimental results of biaxial stress test carried out by Mohan and Melvin [35], the Mooney-Rivlin material constants for a single layer artery at different ages, according to the relations (9) are calculated and presented in table. All the numerical results are presented according to these material constants. (Table ) To show the accuracy of our analytical solution, In figure, the results for changes in the inner radius versus the internal pressure have been presented and compared with the experimental results published by Moltzan et al [0]. From this figure, it can be found that the results are in a good agreement and therefore, our analytical solution is accurate. (Figure ) Based on single layer model, the results for radial and circumferential stress distributions have been plotted in figure 3 and 4, respectively. (Figure 3) The non-dimensional variation of the radial stress has been plotted according to the nondimensional internal radius in figure 3. The maximum blood pressure (or diastolic pressure) and the minimum blood pressure (or systolic pressure) have been considered to be 75 and 50 mm Hg, respectively. The non-dimensional circumferential stress distribution versus the nondimensional internal radius has been also depicted in figure 4. (Figure 4) In order to find much more accurate results, a double layer model has been assumed. The Mooney-Rivlin material constants for media and adventitia layers of the artery at different ages are calculated and presented in table. (Table ) 0
The non-dimensional results for radial stress distribution based on the single and double layer model have been depicted in figures 5. (Figure 5) The results for non-dimensional circumferential stress distribution based on the double layer model have been depicted in figures 6. (Figure 6) The variation of dimensionless inner radius is plotted versus the blood pressure in figure 7. (Figure 7) Also, the changes in outer radius are considerable but smaller than those of the inner radius. This revealed that the thickness of the arteries decreases by increasing the internal pressure which is shown in figure 8. It is clear that the change of thickness occurs due to incompressibility behavior of the artery. (Figure 8) In table 3, the dimensionless inner radius changes verses the internal pressure based on single and double layer models have been compared. (Table 3) It is concluded that the results are very close to each other and therefore, the results of single layer for inner radius changes are accurate. In table 4, the radial and circumferential stresses based on single and double layer models have been compared. (Table 4) Based on the double layer model, the circumferential stress distribution along the radius at different ages for the internal pressure of 50 mm Hg is drawn in figure 9. (Figure 9) 6. Conclusions It can be seen that the radial stress distribution is highly non-linear and it varies between the blood pressure in the inner wall and zero at the outer wall. According to figures 3 and 4, it is found that the surfaces near the inner lining of artery have a greater role in the carrying out the stress than the outer surfaces. Prior to this, we have said that the inner lining of the arteries composed of the muscle cells, elastin fibers, and collagen and external layer composed of collagen type I, nerve cells and elastin fibers. Muscle cells have
much greater elastic properties to collagen [3] and it justifies the resulting stress distribution. Also, from figure 4, it is concluded that the circumferential stresses close to the inner surface are very large and taper off very rapidly. By comparing the magnitude of circumferential and radial stresses, it can be found that the major stresses in mechanical analysis of arteries are circumferential stresses, which are almost 00 times larger than the radial stresses. D According to figure 7, it can be found that the internal radius expands more than twice of initial value due to an internal pressure about 5 kpa ( mm Hg); which represents that the deformation is too large. It can be found from table 4 that the difference in stresses based on single and double layer models are noticeable. So, the single layer model cannot suitably predict the radial and circumferential stresses. Figure 9 shows that the circumferential stress in the arteries increases by increase of the age. For this reason, the flexibility of the artery reduces. 7. Conflicts of Interest Statement We declare that we have no conflicts of interest. 8. References [] Lakatta, E. G. "Central arterial aging and the epidemic of systolic hypertension and atherosclerosis", Journal of the American Society of Hypertension, (5), pp. 30-340, (007). [] Staff, B. c. "Medical gallery of Blausen Medical 04", WikiJournal of Medicine, (), pp. -79, (04). [3] Meyers, M. A. Chen, P.Y. Lin, A. Y. M. and Seki, Y. "Biological materials: structure and mechanical properties", Progress in Materials Science, 53 (), pp. -06, (008). [4] Jaffe, E. A. "Cell biology of endothelial cells", Human pathology, 8 (3), pp. 34-39, (987). [5] Cox, R. H. "Regional variation of series elasticity in canine arterial smooth muscles", American Journal of Physiology-Heart and Circulatory Physiology, 34 (5), pp. 54-55, (978). [6] Karšaj, I. and Humphrey, J. D. "A multilayered wall model of arterial growth and remodeling", Mechanics of materials, 44, pp. 0-9, (0).
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[9] Taghizadeh, D. Bagheri, A. and Darijani, H. "On the Hyperelastic Pressurized Thick-Walled Spherical Shells and Cylindrical Tubes Using the Analytical Closed-Form Solutions", International Journal of Applied Mechanics, 7 (), pp. 5-7, (05). [0] Von Maltzahn, W. W. Besdo, D. and Wiemer, W. "Elastic properties of arteries: a nonlinear two-layer cylindrical model", Journal of biomechanics, 4 (6), pp. 389-397, (98). [] Von Maltzahn, W. W. "Stresses and strains in the cone-shaped carotid sinus and their effects on baroreceptor functions", Journal of biomechanics, 5 (0), pp. 757-765, (98). [] Von Maltzahn, W. W. Warriyar, R. G. and Keitzer, W. F. "Experimental measurements of elastic properties of media and adventitia of bovine carotid arteries", Journal of biomechanics, 7 (), pp. 839-847, (984). [3] Holzapfel, G. A. and Ogden, R. W. "Biomechanical modelling at the molecular, cellular and tissue levels", vol. 508, Springer Science & Business Media, (009). [4] Holzapfel, G. A. and Weizsäcker, H. W. "Biomechanical behavior of the arterial wall and its numerical characterization", Computers in biology and medicine, 8 (4), pp. 377-39, (998). [5] Kroon, M. and Holzapfel, G. A. "A theoretical model for fibroblast-controlled growth of saccular cerebral aneurysms", Journal of Theoretical Biology, 57 (), pp. 73-83, (009). [6] Watton, P. Hill, N. and Heil, M. "A mathematical model for the growth of the abdominal aortic aneurysm", Biomechanics and modeling in mechanobiology, 3 (), pp. 98-3, (004). [7] Wulandana, R. and Robertson, A. "An inelastic multi-mechanism constitutive equation for cerebral arterial tissue", Biomechanics and Modeling in Mechanobiology, 4 (4), pp. 35-48, (005). [8] Zasadzinski, J. Wong, A. B. Forbes, N. Braun, G. and Wu, G. "Novel methods of enhanced retention in and rapid, targeted release from liposomes", Current opinion in colloid & interface science, 6 (3), pp. 03-4, (0). [9] Safi Jahanshahi, A. and Saidi, A. "Mechanical behavior of human arteries in large deformation using non-linear elasticity theory", Modares Mechanical Engineering, 5 (), pp. 53-58, (05). [30] Tsamis, A. Stergiopulos, N. and Rachev, A. "A structure-based model of arterial remodeling in response to sustained hypertension", Journal of biomechanical engineering, 3 (0), pp. 0004, (009). 4
[3] Valentín, A. Cardamone, L. Baek, S. and Humphrey, J. "Complementary vasoactivity and matrix remodelling in arterial adaptations to altered flow and pressure", Journal of The Royal Society Interface, 6 (3), pp. 93-306, (009). [3] Wan, W. Hansen, L. and Gleason Jr, R. L. "A 3-D constrained mixture model for mechanically mediated vascular growth and remodeling", Biomechanics and modeling in mechanobiology, 9 (4), pp. 403-49, (00). [33] Holzapfel, G. A. and Ogden, R. W. "Modelling the layer-specific three-dimensional residual stresses in arteries, with an application to the human aorta", Journal of the Royal Society Interface, pp. 90-357, (009). [34] Zemánek, M. Burša, J. and Děták, M. "Biaxial tension tests with soft tissues of arterial wall", Engineering Mechanics, 6 (), pp. 3-, (009). [35] Mohan, D. and Melvin, J. W. "Failure properties of passive human aortic tissue. II Biaxial tension tests", Journal of biomechanics, 6 (), pp. 3-44, (983). [36] Batra, R. and Bahrami, A. "Inflation and eversion of functionally graded non-linear elastic incompressible circular cylinders", International Journal of Non-Linear Mechanics, 44 (3), pp. 3-33, (009). [37] Rubin, D. Krempl, E. and Lai, W. M. "Introduction to continuum mechanics", Ed., 3th Edn., pp. 34-347, Butterworth Heinemann, Woburn, (993). Alireza Saidi was born in Rafsanjan, Iran on 970. He received his B.Sc. in Solid Mechanic from the Amir Kabir University of Technology, Iran, in 996 and his M.Sc. in Applied Mechanic from the Sharif University of Technology, Iran, 998. He has obtained his PhD degree in Applied Mechanic from the Sharif University of Technology, Iran, 003. He is currently a professor at the Shahid Bahonar University of Kerman. His research interests include Nonlinear Continuum Mechanics, Constitutive Modeling of Hyperelastic Materials, Non- Linear elasticity, thermoelasticity, Nano Mechanics, Finite Deformation and Plasticity. 5
Amin Safi Jahanshahi was born in Sirjan, Iran on Aug., 978. He received his B.Sc. in Solid Mechanic from the Shahid Bahonar University of Kerman, Iran, in 00 and his M.Sc. in Applied Mechanic from the Amir Kabir University of Technology, Iran, 005. He has obtained his PhD degree in Applied Mechanic from the Shahid Bahonar University of Kerman, Iran, 07. He is currently an assistant professor at Sirjan University of Technology. He has been involved with teaching and research activities in the stress analyses areas at the Sirjan University of Technology for the last ten years. His research interests include Nonlinear Continuum Mechanics, Constitutive Modeling of Hyperelastic Materials, Non- Linear elasticity, Finite Deformation and Plasticity. 6
9. Figure Caption Figure. Cross-section of an artery and vein composed of the endothelium, tunica intima, tunica media, and tunica adventitia [] Figure Comparison of changes in the inner radius, according to internal pressure between the model and the experimental results obtained from Moltzan et al tests [0] Figure 3 The non-dimensional radial stress distribution for systolic and diastolic blood pressure Figure 4 The non-dimensional circumferential stress distribution for systolic and diastolic blood pressure Figure 5 The non-dimensional radial stress distribution for systolic and diastolic blood pressure of both single and double layers model Figure 6 The non-dimensional circumferential stress distribution for systolic and diastolic blood pressure of two layers model Figure 7 Dimensionless inner radius verses the internal pressure Figure 8 The thickness changes verses the internal pressure Figure 9 The circumferential stress distribution according to radius for different ages 0. Table caption Table Mooney-Rivlin constants of the arteries in different ages Table Mooney-Rivlin constants for media and adventitia in different ages Table 3 comparison of dimensionless inner radius changes according to internal pressure of single layer and double layer Table 4 comparison of stresses of single layer and double layer 7
Figure. Cross-section of an artery and vein composed of the endothelium, tunica intima, tunica media, and tunica adventitia [] 8
P in (kpa) 8 5 The Mooney-Rivlin Material Model Experimental 9 6 3 0.8 3. 3.6 4 4.4 4.8 5. r(mm) Figure Comparison of changes in the inner radius, according to internal pressure between the model and the experimental results obtained from Moltzan et al tests [0] 9
0.8 0.6 Diastolic Pressure Systolic Pressure T rr /P in 0.4 0. 0.0.04.06.08.. r /r in Figure 3 The non-dimensional radial stress distribution for systolic and diastolic blood pressure 0
T (MPa).75.5.5 Systolic Pressure Systolic Pressure 0.75 0.5 0.5.0.04.06.08.. r / r in Figure 4 The non-dimensional circumferential stress distribution for systolic and diastolic blood pressure
0.9 T rr / P in 0.8 0.7 0.6 0.5 0.4 Double Layer-Systolic Pressure Double Layer-Diastolic Pressure Single Layer-Systolic Pressure Single Layer-Diastolic Pressure 0.3 0. 0. 0.0.04.06.08.. r /r in Figure 5 The non-dimensional radial stress distribution for systolic and diastolic blood pressure of both single and double layers model
T (MPa).4. 0.8 Systolic Pressure Diastolic Pressure 0.6 0.4 0..0.04.06.08.. r / r in Figure 6 The non-dimensional circumferential stress distribution for systolic and diastolic blood pressure of two layers model 3
.8 r /r in.6.4. 0 5 0 5 0 P in (kpa) Figure 7 Dimensionless inner radius verses the internal pressure 4
h(mm).6.5.4.3.. 0.9 0.8 0 5 0 5 P in (kpa) Figure 8 The thickness changes verses the internal pressure 5
T (MPa) 3.5 5 years old 49 years old.5 60 years old 78 years old 0.5 0 5.4 5.5 5.6 5.7 5.8 5.9 6 6. r(mm) Figure 9 The circumferential stress distribution according to radius for different ages 6
Table Mooney-Rivlin constants of the arteries in different ages Age(year) c (Pa) c (Pa) A(mm ) D(mm mm) 5-347 0396 30.78 0.79 49-34 8 7.090 0.736 60-508 9400 4.36 0.74 87-675 75.536 0.749 7
Table Mooney-Rivlin constants for media and adventitia in different ages Age(year) c m (Pa) c m (Pa) c a (Pa) c a (Pa) A (mm ) D (mm mm) 5 6030-483 850-467 30.604 0.745 49 953-5043 975-849 9.56 0.75 60 783-749 64-4446 6.98 0.77 87 300-8774 875-504 4.73 0.785 8
Table 3 comparison of dimensionless inner radius changes according to internal pressure of single layer and double layer r in / R in r ou / R ou P in (kpa) single layer double layer difference % single layer double layer difference % 0 0 0 5.6048.64 0.0.4763.49 0.0 0.848.8657 0.03.687.7056 0.0 5.04.043 0.04.836.864 0.03 0.379.699 0.05.9458.974 0.04 9
Table 4 comparison of stresses of single layer and double layer r(mm) -T rr (kpa) T θθ (MPa) single layer double layer single layer double layer 5.4 0.75 0.76.50.665 5.5 6.579 6.7383 0.575 0.4507 5.6 4.3555 4.840 0.3386 0.666 5.7 3.097 3.7360 0.44 0.9 5.8.30.9756 0.99 0.59 5.9.5476.3058 0.609 0.3709 6.0 0.986.4693 0.389 0.798 6. 0.5087 0.7578 0.30 0.086 30