Original. Stresses and Strains Distributions in Three-Dimension Three-Layer Abdominal Aortic Wall Based on in vivo Ultrasound Imaging

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1 Original Stresses and Strains Distributions in Three-Dimension Three-Layer Abdominal Aortic Wall Based on in vivo Ultrasound Imaging P. Khamdaengyodtai 1, T. Khamdaeng 1, P. Sakulchangsatjatai 1, N. Kammuang-lue 1, P. Terdtoon 1 Abstract The aorta is an elastic conduit for blood flow and a portion of it in the abdomen called abdominal aorta is where many cardiovascular diseases caused by structural changes often occur. One of the implications of the structural changes is the change in mechanical properties of the blood vessel. This research therefore proposes to predict stresses and strains distributions in three dimensional coordinate and in three major layers of abdominal aortic wall based on in vivo ultrasound experimental imaging. In experiment with five healthy mice, luminal pressure acted to the inside wall is caused of wall movement and local aortic diameters could be obtained from crosscorrelation technique on the ultrasound signal. Continuum mechanics is used to approach the results. Geometry of the vascular wall is modeled as cylindrical circular and thick wall tube consisted of three layers corresponding to tunica intima, tunica media and tunica externa. Constitutive equation of fiber reinforcement is used for each layer to representing nonlinear elastic behavior of the aorta and Nelder-Mead simplex algorithm is used to estimate relevant parameters. The model is fitted to the experimental data on abdominal aorta. Stresses and strains distributions could be predicted for all of directions and also for series of time along cardiac cycle. The behaviors of stresses distributions from the model and previous study are also consistent. This model could be as preliminary model to extend to abnormal aortic wall in which because this model also provide measures of stresses and strains in tunica intima layer where many vascular diseases often occur. Keywords: three dimensions, three layers, aorta, stress, strain, ultrasound, fiber reinforce material Introduction Vascular system has the aorta which acts as both conduit and an elastic chamber. The elastic of aorta serves to convert pulsatile flow pumped by heart to steady flow in peripheral vessels. Atherosclerosis, the common disease of arterial wall, is a disease usually located within large arteries [6] and buildup of atherosclerosis plaque usually associates with pathological changes of intimal component of vascular layers [2, 3]. One of the implications of the structural changes is the change in mechanical properties of the blood vessel. Stress and strain are used to represent the mechanical properties. As mention, it could be observed that disease location associates with position on artery tree and inside wall. So, stress and strain distribution in multidimensional and multilayers should be predicted. In order to friendly for patient, advantage of imaging technology should be used as additional data, especially the ultrasound scanning. Ultrasound scanning is noninvasive medical test that has no needles or injections and is usually painless. Hence, this study proposes to predict stresses and strains 1 Department of Mechanical Engineering, Faculty of Engineering, Chiang Mai University, Thailand Tel ext.911, Fax pannathai@hotmail.com

2 586 P. Khamdaengyodtai et al. J Sci Technol MSU distributions in coordinate of three dimensions and in three major layers of abdominal aortic wall based on in vivo ultrasound imaging. Analysis Anatomy The blood vessel wall consists of three major layers which are intima (tunica intima), media (tunica media) and adventitia (tunica externa). The innermost layer is called intima where contains a single layer of endothelium cell. In healthy artery, thickness of intima is very thin and make insignificant to mechanical property of arterial wall [2]. However, buildup of atherosclerosis plaque or increasing of age, intima may become significant. The middle layer is called media where contains the alternating layers of smooth muscle cell and elastic connective tissue. Arrangement form of its components gives the media high strength and ability to resist load. The outermost layer of artery is called adventitia. In high level of pressure, the adventitia change to be like a stiff tube to prevent the artery from rupture [9, 10]. Experimental procedure Luminal pressure acted to the inside wall is cause of vascular wall movement. With five healthy mice (see Figure 1), an abdominal aorta of mouse is canulated using an ultraminiature pressure catheter through the mouse carotid artery and introduces into the abdominal aortic region to provide both of luminal pressure and ECG signals. The inside and outside aortic diameters of abdominal aortic wall are clearly obtained through manual tracing on the B-mode image of ultrasound imaging. Then, the radial incremental displacements of the inside and outside walls are determined which then accumulate to obtain the wall diameter variation over one cardiac cycle. The luminal pressure and aortic wall diameter variations are matched using the corresponding ECG by align the maximum and minimum peaks of the luminal pressure and diameter variations. All experimental data based on ultrasound then are used as importance inputs in mechanical model (see more details in [1] and [5]). (Experimental data had supported from UEIL, Biomedical Engineering and Radiology, Columbia University, NY, US). Figure 1 Experimental methodology with healthy mouse in abdomen. Continuum-mechanical framework The mathematical description of deformation, the body occupy in the reference configuration. When the body is deformed, every particle at point transforms to new position at point in deformed configuration. The transformation gradient could be determined by following equation. (1) Right and left Cauchy Green tensor associate with as following., (2) So, the Green-Lagrange strain tensor could be introduced as equation (3). (3) Where denotes the second order unit tensor. For stress response of the artery, Cauchy stress tensor relates to Green-Lagrange strain tensor via transformation the Piola

3 Vol 31, No 5, September-October 2012 Stresses and Strains Distributions in Three-Dimension Three-Layer Abdominal Aortic Wall Based on in vivo Ultrasound Imaging 587 Kirchhoff stress tensor which is the first derivative of strain energy function respected to Green- Lagrange strain tensor. Cauchy stress tensor could be expressed as the sum of two other stress tensors which are volumetric stress tensor and stress deviator tensor as equation (4). (4) Where and is Lagrange multiplier to descript the incompressibility of the artery wall [4, 8, 9, 10]. According to the artery structure composted of fiber and non-collagen matrix of material, fiber reinforced strain energy function suggested by G.A. Holzapfel (2000) had been suitable used to relate stress and strain. The strain energy function could be written in two terms of isotropic and anisotropic deformations as equation (8). (8) Where, are stress-like parameter and is dimensionless parameter, subscript refer to intima, media and adventitia layers (see Figure 2) and subscript refer to index number of invariants. In equation (8), is the first principal invariant of and the definitions of the invariants in equation (9) associate with the anisotropic deformation of arterial wall., (9) Figure 2 Geometry and boundary conditions. Mechanical model Geometry and boundary conditions shown in Figure 2 is in reference configuration, thickness ratio of media and adventitia is set as 2:1 [2] and thickness ratio of intima and media is set as 1:20 [6]. Kinematics of the artery in cylindrical coordinate, deformation equations [2, 3] are as following., (5), (6) (7) Where, is stretch ratio in longitudinal direction, and are opening angle and overall length of artery in reference configuration and subscript in equation (5) refers to inside. The collagen fiber is assumed that it do not support compressive stress. Thus, in case of and the response is similar to the response of rubber like material that descripted by Neo-Hookean functions. The tensor and characterizing the structure are given by equation (10)., (10) Component of the direction vector and in cylindrical coordinate system are in forms as equation (11)., (11) Where is the angle between the collagen fibers and circumferential direction. Hence, the stress in Eulerian description could be determined by the expression in equation (12).

4 588 P. Khamdaengyodtai et al. J Sci Technol MSU Where and (12) denotes as response function denotes as Eulerian counter part of. The equilibrium equation in cylindrical coordination in equation (13) is used with boundary equations. Inside pressure from the model and the equations to predict stresses distributions are provided. (13) There are three parameters in each arterial layers which must be estimated to apply in equations of stresses. Nelder-Mead simplex algorithm [7] is used to estimate these relevant parameters by minimizes function of mean square error (MSE) of pressures in form as equation (14). Where is number of longitudinal data points. Results and discussions Computational program Parameter estimation (14) Table 1 Material parameters and heart rate data of murine abdominal aortas Mouse #061 #132 #133 #140,[bpm] , [position] , [deg] , [kpa] , [kpa] , [-] , [deg] , [kpa] , [kpa] 1.63E E E E-13, [-] , [deg] , [kpa] , [kpa] , [-] Intima Media Adventitia Table 2 Material parameters and heart rate data of murine abdominal aortas (Continue) Mouse #301 Average STD Intima Media Adventitia,[bpm] , [position] , [deg] , [kpa] , [kpa] , [-] , [deg] , [kpa] , [kpa] 2.92E E E-14, [-] , [deg] , [kpa] , [kpa] , [-] Material parameter cannot be arbitrarily chosen. Figure 3 show contour plot of the potential in adventitia layer of mouse#061 with parameter set in Table 1, as an example. If the contour is non-convexity, the physical meanings of parameters are not clear. Thus, it is important to perform optimization process within the range which convexity exists. The material parameters and heart rate data of five healthy mice show in Table 1 where denotes heart rate, denotes number of data positions along longitudinal direction and denotes correlation coefficient. Figure 3 Contour of strain energy potential (Pa).

5 Vol 31, No 5, September-October 2012 Stresses and Strains Distributions in Three-Dimension Three-Layer Abdominal Aortic Wall Based on in vivo Ultrasound Imaging 589 Boundary conditions Luminal pressure (pressure inside lumen of the artery) and outside pressure are constrains of inside wall and outside wall. For instance, Cauchy radial stress distribution in reference configuration of longitudinal distance ( =89 points) across reference intima, media and adventitia layers is illustrated in Figure 4 at the physiological state with =11 kpa. This ensures the computational program for threedimension boundary value problem as following. stretch of 1.7, reference inside and outside radius of 0.71 and 1.1 mm, respectively, without torsion and residual strains. Luminal pressure in cardiac cycle of the experiment varies from 5 kpa at end contraction phase to 12 kpa at end dilated phase. Non uniform value in longitudinal direction occurs because non uniform of diameter variation which are obtained from ultrasound scanning. Strains distributions The strains distribution in radial and circumferential directions in configuration of longitudinal distance across the reference intima, media and adventitia layer could be illustrated in Figure 5 and Figure 6, respectively. Figure 4 Cauchy radial stress (kpa) distribution. At certain longitudinal position, Cauchy radial stress is continuous from inside wall which equal to negative value of luminal pressure and varies from this negative value at the inside wall and is continuous across interfaces of layers (intima-media interface and mediaadventitia interface) toward outside pressure at the outside wall. This trend of the variation of radial stress in certain longitudinal position could be also observed in many previous studies [2, 3, 10]. Stresses and strains distribution in coordinate of three dimensions and in three major layers of abdominal aortic wall based on in vivo ultrasound data could be predicted successfully. To discussions the results, stresses and strains distributions in reference configuration of longitudinal distance across the reference intima, media and adventitia layers from experimental data of mouse#061 and its parameter set is used to interpret as instances in the physiological state with luminal pressure of 11 kpa (dilated phase), longitudinal Figure 5 Green radial strain distribution. Green radial strain tensor represents as negative value because this state the vascular has been dilated by luminal pressure. Lumen of the vascular increases while conservation of volume, thickness of this state is reduced resulting to radial strain is negative.

6 590 P. Khamdaengyodtai et al. J Sci Technol MSU Figure 6 Green circumferential strain distribution. It could be observed that the circumferential stress in media layer has relatively high value compared with the value in adventitia. Extend from two layers of media and adventitia of G.A. Holzapfel (2000) by intima layer is in to accounted, the relatively highest value of circumferential stress occurs in intima layer. Overall of trend of the magnitude that the highest occurs at inside wall and then decreases toward to outside wall is also found by Fung (1990). Green longitudinal strain is equal to for all positions because of no torsion. The continuous of strains at both of interfaces, intima-media interface and media-adventitia interface, is found because continuous body of vascular is assumed. Stresses distributions The stresses distribution in radial, circumferential and longitudinal directions in configuration of longitudinal distance across the reference intima, media and adventitia layer could be illustrated in Figure 4, Figure 7 and Figure 8, respectively. Figure 7 Cauchy circumferential stress (kpa) distribution. In contrast to Cauchy radial stress distribution, Cauchy stress distributions in circumferential, longitudinal directions are discontinuous at both of interfaces, intima-media interface and media-adventitia interface. The behavior shown in Figure 7 is similar to that found by G.A. Holzapfel (2000) by following. Figure 8 Cauchy longitudinal stress (kpa) distribution. Moreover, the magnitude of radial stress is smaller than in others directions. Conclusion Stresses and strains distributions in coordinate of three dimensions and in three major layers of abdominal aortic wall based on in vivo ultrasound imaging along cardiac cycle could be predicted. Parameter has already ensured with convexity contour of strain energy density function. The ultrasound scanning is noninvasive method which gives a clear picture of soft tissues. Ultrasound scanning thus could provide important data of aortic wall movement. This model could be as preliminary model to extend to abnormal aortic wall in which because this model also provides measures of stresses and strains in intima layer where many vascular diseases often occur.

7 Vol 31, No 5, September-October 2012 Stresses and Strains Distributions in Three-Dimension Three-Layer Abdominal Aortic Wall Based on in vivo Ultrasound Imaging 591 Acknowledgments Financial support from the Thailand Research Fund through the Royal Golden Jubilee Ph.D.Program (PHD/0181/2549) to P.Khamdaengyodtai and P.Terdtoon is acknowledged. The author would like to great thank for supporting from Associate Professor Elisa E. Konofagou of UEIL, Biomedical Engineering and Radiology, Columbia University, NY, US. References [1] A.Danpinid, In vivo characterization of the aortic wall stress-strain relationship, Ultrasonics, doi: /j.ultras , [2] G.A. Holzapfel and T. C. Gasser, A new constitutive framework for arterial wall mechanics and a comparative study of material model, J. Elasticity 61, pp.1-48, [3] Holzapfel, Lemaitre handbook of materials behavior models, Academic Press, pp , [4] Joseph E. Shigley, Charles R. Mischke, Mechanical engineering design, 6 th ed., McGraw-Hill, pp , [5] K.Fujikura et al., A novel noninvasive technique for pulse-wave imaging and characterization of clinically-significant vascular mechanical properties in vivo, Ultrasonic imaging 29, pp , [6] N. Yang and K. Vafai, Modeling of low-density lipoprotein (LDL) transport in the artery-effects of hypertension, Int. J. Heat and Mass Transfer 49, pp , [7] Steven C. Chapra, Raymond P. Canale, Numerical method for Engineerings, 6 th ed., McGraw-Hill, pp , [8] Y.C. Fung, A first course in continuum mechanics, 3 rd ed., Prentice-Hall, [9] Y. C. Fung, Biomechanics mechanical properties of living tissue, 2 nd ed., Springer-Verlag, [10] Y.C. Fung, Biomechanics motion, flow, stress, and growth, Springer-Verlag, 1990.