Summer Workshop of Applied Mechanics. Influence of residual stress in coronary arteries

Transcription

1 Summer Workshop of Applied Mechanics June 2002 Department of Mechanics Faculty of Mechanical Engineering Czech Technical University in Prague Influence of residual stress in coronary arteries Ing.Lukáš Horný 1 Ing. Karel Vítek, CSc. 2,, Prof.Ing.Svatava Konvičková,CSc. 1 1 Laboratoř biomechaniky člověka FS ČVUT Praha Technická 4 Praha Odbor pružnosti a pevnosti FS ČVUT Praha Technická 4 Praha horny@biomed.fsid.cvut.cz Keywords: opening angle residual stress, aging, constitutive laws 1 Introduction An analytical model has been used to simulate the effects of tissue aging on residual strain, constitutive relations in the main right and left (ramus circumflexus) man s coronary arteries, based on experimental data. The experimental opening angle θ scatters considerably with age. The optimum angle θ op 70, which makes the circumferential stress uniform in the arterial wall at a normal blood pressure, is approximately constant throughout aging 2 Age related constitutive laws of the arterial wall Ozolanta and Purinya [1] examined in vitro the passive mechanical properties of the coronary arteries of 121 subjects. The samples were divided into six age groups from 1 year to 80 years. Specimens with the most severe stage of atherosclerosis were not 76

2 included. The range of the loading (Lagrangian stress) is (0 to 110) kpa. On the basis of nonlinear regression analysis, the theoretical estimation and extrapolation of the passive stress-strain relationships (Valenta et al. [2]) for the circumferential direction of the male (Fig. 1a,b) right and left main coronary arteries (ramus circumflex) were determined. The maximum loss in wall compliance with age is comparable for the two arteries, but the left coronary artery is stiffer than the right one for the group aged 35 to 45. This study shows that after more than 30 years of life the arterial wall circumferential stiffness rapidly increases owing to long-time and increased cyclic stretching, and also due to pathological conditions which play a role in the nonoptimum regulation of vascular structure and function. It follows from these results that the circumferential stiffness of the artery wall tissue and the slope of the stressstrain curves increase while the limit strain and the toughness of the artery wall tissue decrease. 3 Stress distribution A comprehensive study of the mechanical properties of human coronary arteries, which depend on their age, structure and biochemical processes, is of importance because of the prophylactics and therapy of atherosclerosis, which is a major disease of the arteries. For example, the biomechanical properties of a.radialis are practically the same at the left and right coronary stems, and may be used for a study of their mechanical response as well as their potential replacement (Ozolanta et al. [3]). For a description of artery tissue passive 3-D stress-strain relationships, the following forms of the strain energy density function have been applied: polynomials, exponential functions, logarithmic expressions and some empirical models (Hayashi [4], Humprey [5]). On the basis of hyperelasticity theory, an age-related stress distribution through wall thickness has been determined, as follows (Matsumoto, Hayashi [6] ): σ i (R, τ) σ 3 (R, τ) = λ 2 i The boundary condition at biological time τ (age) : e i i = 1, 2 (1) σ 3 (τ) = p i = R o R i λ 2 1 dr e 1 R and the condition of the wall material incompressibility (2) R o = (R 2 i + λ 1 2 (r 2 o r 2 i )) (3) Here W(τ) is the tissue strain energy density function at biological time τ (aging of the tissue), e i = 1 2 (λ2 i 1) are Green s strain in the circumferential (e 1 ) and axial 77

3 directions (e 2 ), and λ i are the stretch ratio in the i direction; σ 1 (R,τ), σ 2 (R,τ) and σ 3 (R,τ) are the circumferential, axial and radial Cauchy stress at radius R and biological time τ; R i, and R o, are the internal and external radii, respectively, at the loaded state (R i R R o ); p i is the intraluminal pressure; r i, and r o, are the internal and external radii, respectively, in the unloaded state; λ 2 is the axial stretch ratio, which is constant throughout wall thickness. The following assumptions have been incorporated: cylindrical shape, homogeneity, incompressibility and no shear strain. The e 1 response function was determined from the uniaxial, age-related, stress-strain relationships (Fig. 1a,b ) i.e. λ 1 = λ with λ 2 = 1/λ, λ 3 = 1, σ 3 = 0, Green strain e 1 = (1/2)(λ 2 1), e 2 = (1/2)(λ 2 1). The general form of the exponential fitting function is e 1 = A (τ) exp [B (τ) e r ] + C (τ) (4) Here e r = (e e 2 2) 1/2 is the reduced Green strain. Parameters A(τ), B(τ) and C(τ) characterize the age-related mechanical properties of the arterial wall structure and for the right male (m) the main proximal part of the coronary artery are given by the relations : A m (τ) = j a jm τ j B m (τ) = j C m (τ) = j b jm τ j j = 1, 2 c jm τ j (5) and for the right female (f) main coronary artery A f (τ) = j a jf τ j B f (τ) = j C f (τ) = j b jf τ j j = 1, 2 c jf τ j (6) The conditions A m (τ) + C m (τ) = 0A f (τ) + C f (τ) = 0 (7) for λ 1 = λ 2 = 1.0 and given τ = (1 to 80) years are approximately fulfilled. Table 1 reports the numerical values of coefficients a jm, a jf, b jm, b jf, and c jm, c jf. We 78

4 Figure 1: Fig 1a,b Age related constitutive laes circumferential direction identified the second response function in the form : = 1 ( ) γπr 2 i (τ) p i + F (τ) e 2 λ 2 2 π (Ro 2 (τ) Ri 2 (τ)) + p i R i (τ) R o (τ) + R i (τ) (8) where F(τ) is the axial force at biological time τ and γ = 1 or 0 if the vessel is closed or open, respectively (Humphrey [5]). The theoretical model satisfactorily represents the coronary artery age-related, circumferential mechanical response. The coronary artery stem is deposited in the collagenous bed and therefore at the systole there is very low axial pre-stress. The relations (2) have been used to determine the vascular loaded state, including the residual stress. 4 Conclusion At the given luminal pressure p i and the opening angle θ, the loaded state (R i and R o ) of the coronary vessel was determined by the numerical solution of Eq.(3) together with condition (4). The stress distribution through wall thickness is provided by Eq.(2), and for the right male coronary artery stems, at biological times τ = 1 year, 10 years, 30 years and 70 years, it is shown in Fig. 2. Figure 2 shows that the real vascular wall membrane (homogeneous) stresses correspond to the opening angle θ op 70. This angle is practically independent of age. From age 15 years the experimental, respectively the estimated, opening angle θ [7] is higher than θ op. Next page shows multiaxiaxial stress distribution dependece on age (residual stress is include). In the course of aging, the arterial wall structure changes because collagen cross-linking increases, and the fibroblasts produce elastic and collagen fibers especially in the external layers of the media (in the neighborhood of the external elastic lamina) and in the adventitia. The external vascular wall layer is hardened and 79

5 Figure 2: 80

6 overloaded and the residual strain increases. The arterial wall biocomposite forms non-homogeneous stresses and, when aging, decreases the medial internal layers and the intima loading. At 10 to 15 years of age, where the residual strains are low, the circumferential stress distribution in the arterial wall has the classic form. The stress maximum is in the neighborhood of the internal elastic lamina. Additional experiments and studies, especially in vivo, are needed to check these results and contribute to the determination of a mechano-sensory mechanism by which the vascular tissue senses and adapts its structure due to the recent history of mechanical loading. Acknowledgement The research project was supported by MSM grant No Bibliography [1] I. Ozolanta and B. Purinya, Age related changes of the biomechanical parameters and structure of coronary artery walls, Contemporary Problems of Biomechanics (in Russian) 2 (1985), [2] J. Valenta et al., Mechanical properties of some parts of the human cardiovascular system, in: Biomechanics of the Cardiovascular System, B.Bo. Sramek, J. Valenta and F. Klimes, eds. Czech Technical University Press, Prague, Irvine, CA, U.S.A., 1995, p. 95. [3] I. Ozolanta, B. Purinya, R. Lacis and V. Kasyanov, Investigation of biomechanical properties of man coronary arteries : a risk group of years old, 12 th Conference of the European Society of Biomechanics, Dublin, 2000,p. 29. [4] K. Hayashi, Experimental approaches on measuring the mechanical properties and constitutive laws of arterial walls, ASME Journal of Biomechanical Engineering 115 (1993), [5] J.D. Humphrey, An evaluation of pseudoelastic descriptors used in arterial mechanics, ASME Journal of Biomechanical Engineering 121 (1999), [6] T. Matsumoto and K. Hayashi, Stress and strain distribution in hypertensive and normotensive rat aorta concidering residual strain, ASME Journal of Biomechanical Engineering 118 (1996),

7 [7] Valenta, J., et al., Age related constitutive laws and stress distribution in human main coronary arteries with reference to residual strain, BioMedical and Materials, in press 82