Application of Adomian Decomposition Method to Study Heart Valve Vibrations

Transcription

1 Applied Matheatical Sciences, Vol. 3, 009, no. 40, Application of Adoian Decoposition Method to Study Heart Valve Vibrations S. A. Al-Mezel, Moustafa El-Shahed and H. El-sely College of Education, P.O.Box 3771, Qasssi University - Unizah Kingdo of Saudi Arabia elshahed@yahoo.co Abstract The objective of this paper is to solve the equation of otion of seilunar heart valve vibrations. The vibrations of the closed seilunar valves were odeled with a Caputo fractional derivative of order α. With the help of Laplace transforation, closed-for solution is obtained for the equation of otion in ters of Mittag- Leffler function. We obtained the analytical solution for the nonlinear fractional differential equation using Adoian decoposition ethod. The siplicity of these solutions akes the ideal for testing the accuracy of nuerical ethods. These solutions can be of soe interest for a better fit of experiental data. Matheatics Subject Classifications: 51N0, 6J05, 70F99 Keywords: Fractional derivative; Seilunar heart valve vibrations; Adoian decoposition ethod 1 Introduction The heart valves are echanical devices that perit the flow of blood in one direction only. Four sets of valves are of iportance to the noral functioning of the heart. Two of these, the atrioventricular valves, guard the opening between the atria and the ventricles. The other two heart valves, the seilunar valves, are located where the pulonary artery and the aorta arise fro right and left ventricles, respectively [18]. The seilunar valves consist of half-oon shaped flaps growing out fro the lining of the pulonary artery and aorta. When these valves are closed, blood fills the spaces between the flaps and the vessel wall. Each flap then looks like a tiny, filled bucket. Inflowing blood soothes the flaps against

2 1958 S. A. Al-Mezel, M. El-Shahed and H. El-sely the blood vessel walls, collapsing the buckets and thereby opening the valves. Closure of the seilunar valves siultaneously prevents back flow and ensures forward flow of blood in places where there would otherwise be considerable back flow [18]. Wiggers (1915) suggested that there is silent approxiation of the seilunar valves and that after vibrations of the closed valve and the colun of blood cause the second heart sound. Stein (1981) supported the theory of silent valve closure and suggested that the valvular vibrations are a sufficient cause for production of the second heart sound [9]. Following coaption of the valve leaflets, a pressure difference is developed across the closed valve during the isovolic relaxation phase, and the leaflets distend slightly toward the ventricle. This causes a pressure reduction within the blood. Subsequent recoil of the valve copresses the blood in the aorta. Vibrations of the leaflets gradually diinishes due daping. The leaflet vibration produces transient pressure gradient in the surrounding blood ediu, which subsequently causes vibration of contiguous structures, which are transitted to the chest wall where they are recognized as audible heart sounds [9]. Blick [] odeled the aortic valve as a circular ebrane of radius a. In this odel the aortic valve is assued to be an elastic, hoogeneous ebrane secured around a circular edge and undergoing a parabolic displaceent. The valve close due to the intraventricular pressure falling below that of the aortic pressure. The force that drives the closed valve to vibrate is the pressure difference Δp that occurs across the valve. The differential equation of otion for forced and daped vibrations of a ebrane at any tie t, with onedegree-of-freedo equation was expressed as []: Dt x(t)+ c D tx(t)+ k x(t) =f(t), (1) where, c and k represent the effective ass of vibration, the daping force coefficient, and stiffness factor, respectively. f(t) = Δpπa is an external force. Gotolov [6] introduced the nonlinear odel of seilunar heart valve vibrations as Dt x(t)+ c D tx(t)+ k x(t)+ λ x3 (t) =f(t), () where λ represents the nonlinear paraeter. The fractional derivative approach provides a powerful tool for odeling systes with daping aterials. The odel based on fractional derivatives has been shown to be one of the ost effective approaches [16, 17]. If the fractional derivative odel is used to represent the daping characteristic, the equations of otion (1) and () assue the for: Dt x(t)+ c Dα t x(t)+ k x(t) =f(t), (3)

3 Heart valve vibrations 1959 D t x(t)+ c Dα t x(t)+ k x(t)+ λ x3 (t) =f(t), (4) The operator D t of the conventional odel was replaced by D α t. There are several definitions of derivative of fractional order. The fractional operator D α t based on Rieann-Liouville integral is defined as [7,13]: D α t x(t) = 1 d n Γ(n α) dt n t 0 x(u) du, n 1 <α<n (t u) α n+1 where n is an integer nuber and Γ is the gaa function. Also the Caputo s definition can be written as[7, 13]: Dt α x(t) = 1 t x (n) (u) du, n 1 <α<n. Γ(n α) 0 (t u) α n+1 The ain objective of the present paper is the atheatical study and the using of an easy ethod to obtain closed-for solution for the equation of otion for general value of α for linear case and also to obtain approxiate solution to the nonlinear fractional calculus odel of the seilunar heart valve vibrations. Adoian decoposition ethod The decoposition ethod does not change the proble into a convenient one for use of linear theory. It therefore provides ore realistic solutions. It provides series solutions which generally converge very rapidly in real physical probles. When the solutions are coputed nuerically, the rapid convergence is obvious. The advantage of the decoposition ethod relies on the fact that it provides an easily coputable schee and an efficient algorith. It is well known that the decoposition ethod decopose the linear ter u(x, t) into an infinite su of coponents u n (x, t) defined by [19] u(x, t) = u n (x, t). n=0 Moreover, the decoposition ethod identifies the nonlinear ter N(u(x, t)) by decoposition series N(u(x, t)) = A n, where A n are the so-called Adoian polynoials. These polynoials can be calculated for all fors of nonlinearity according to specific algoriths n=0

4 1960 S. A. Al-Mezel, M. El-Shahed and H. El-sely constructed and given in. In this specific nonlinearity, we use the general for of forula for A n Adoian polynoials as [19] A n = 1 d n [ ] N( λ n u n! dλ n n ),n 0. n=0 λ=0 This forula is easy to set coputer code to get as any polynoials as we need in the calculation of the nuerical as well as explicit solutions. The first few polynoials in our nonlinearity N(u) =u 3 are given by A 0 = u 3 0, A 1 =3u 0u 1, A =3u 0u +3u 1u 0, and so on, the rest of the polynoials can be constructed in a siilar anner. 3 Analysis One of the ost efficient and elegant ethods to solve Eq() is by eans of the Laplace transfor. The forula for Laplace Transfor of Rieann- Liouville fractional derivative is: 0 e st D α t x(t)dt = sα x(s) n 1 s j D α j 1 t x(0), (n 1 <α<n). where x(s) is the Laplace transfor of x(t). The Laplace transfor of the Rieann-Liouville fractional derivatives is well known. However, its practical applicability is liited by the absence of the physical interpretation of the liit values of fractional derivatives at t = 0. The forula for Laplace Transfor of Caputo fractional derivative is 0 e st D α t x(t)dt = sα x(s) n 1 s α j 1 D j t x(0), (n 1 <α<n). Since this forula for the Laplace transfor of the Caputo derivative involves the values of the function f(t) and its derivatives at t = 0, for which a certain physical interpretation exists, we can expect that it can be useful for solving applied probles leading to linear fractional differential equations with constant coefficients with accopanying initial conditions in traditional for [13, p. 106]. Applying the Laplace transfor to equation(3) we obtain: f(s) x(s) = s + c sα + k, (5) where x(s) and f(s) are the Laplace transfor of x(t) and f(t) respectively. Eq (5) can be written in the for: x(s) = ( 1) j ( k )j s α(j+1) f(s) (s α + c )j+1. (6)

5 Heart valve vibrations 1961 Using the convolution theore, the inversion of Eq(6) takes the for: where x(t) = t 0 G(t u)f(u)du, (7) ( 1) j G(t) = ( k j! )j t j+1 E α,+αj( j c t α ), (8) and E α,β is the two-paraeter function of the Mittag-Leffler type defined by the series expansion[13]: E α,β (z) = Equation(8) can be written in the for: G(t) = ( 1) j ( k )j i=0 (i + j)! j! i! z j Γ(αj + β). t j+i αi+1 ( 1) i ( c )i Γ(j +i αi +), (9) Measureents during catheterization of the instantaneous pressure gradient across the seilunar valve during diastole indicate that, to the first approxiation, the pressure gradient increases linearly with tie until a tie t 1 is reached following which the pressure gradient reains essentially constant[]. The deflection of the centerline for tie t<t 1 can be expressed as x(t) = πa A ( 1) j ( k )j i=0 (i + j)! j! i! t j+i αi+3 ( 1) i ( c )i Γ(j +i αi +4), (10) where A = dδp dt and the deflection of the centerline for t>t 1 is given by x(t) = πa B ( 1) j ( k )j i=0 (i + j)! j! i! t j+i αi+ ( 1) i ( c )i Γ(j +i αi +3). (11) where B = 150 is the constant value of Δp at t>t 1 and t 1 = see[9]. The velocity of the centerline deflection of the ebrane can be obtained by differentiation of equations (8) and (9). Instead of approxiating the pressure difference occurring across the valve by a rap function, Mazudard [8] used an exponential function Δp = C(1 e bt ) as a better approxiation to the physiological pressure gradient. The deflection of the centerline in this case can be expressed as x(t) = πa C ( 1) j ( k )j i=0 (i + j)! j! i! ( 1) i ( c F (t) )i Γ(j +i αi +), (1)

6 196 S. A. Al-Mezel, M. El-Shahed and H. El-sely where F (t) =t iα ( b) (i+j) t ( b t) (i+j) ( j +i αi + + e tb ( t b) iα (Γ(j +i αi + ) + Γ(j +i αi +, t b)) ). b The power series in equations (10), (11) and (1) ebodies the deflection of the centerline of the heart valve. The value α = 1 was adopted in this section because it has been shown that it describes that it describes the frequency dependence of the daping aterials quite satisfactorily [8]. The equation of otion (3) can be written in the for Dt x(t)+ηω 3 1 D t x(t)+ω x(t) =f(t), where ηω 3 = c and ω = k. The coefficient η is the daping ratio of an oscillator with fractional daping of order 1 and ω is the natural frequency. The exponent 3 was introduced for consistency of diensions. 4 Nonlinear fractional calculus odel Equation (4) can be written in the for: D t x(t) =f(t) c Dα t x(t) k x(t) λ x3 (t). (13) We adopt Adoian decoposition ethod for solving equation (13). Applying the inverse operator L 1 on both sides of (13), we obtain ( c x(t) =x(0) + tx (0) + L 1 f(t) L 1 Dα t x(t)+ k x(t)+ λ ) x3 (t). (14) We assue that x(t) = x 0 (t) +x 1 (t) +x (t) +..., to be the solution of Eq(14).When f(t) = A, using Adoian decoposition ethod we get x 0 (t) = At (15) ( c x n+1 (t) = L 1 Dα t x n (t)+ k x n(t)+ λ ) A n. (16) n=0 In view of (15), (16) we get x 1 (t) = cat4 α Γ(5 α) kat4 Γ(5) 90λA3 t 8, Γ(9)

7 Heart valve vibrations 1963 and x (t) = Ak t A3 kt 10 λ A5 t 14 λ Ac t 6 α Γ(7 α) + Ac t6 α Γ(7 α) + 90A3 ct 10 α λ Γ(11 α) + 3A 3 ct 10 α λ 4(90 19α + α )Γ(5 α), other coponents are deterined siilarly. The input pressure gradient was taken to be a rap function where Δp increased 8570 Hg/sec for tie less than t 1 = sec. The value of Δp was constant at 150 Hg per sec for tie equal to or greater than The effective stiffness factor for the ebrane was assued to be equal to the easured static value of k which was dyn/c,the daping force factor D = dyn sec c and the effective ass = 195 g[9]. 5 Results The theoretical curve for the ebrane vibration is plotted in [, 8, 9]. It can be seen that the calculated curve has a shape siilar to the curve calculated fro experiental observations [4]. The initial peak deflection were approxiately equal. The peak rise ties differed by about 0.01 sec. There was also a siilarity between the rates of deflection. Since the fractional derivatives odel approxiates the physical odels ore closely than other odels[16], it would be expected that the results using the fractional derivatives odel would be closer to the actual physiological values for deflection and rate of deflection of the centerline. The siplicity of the present solutions akes the ideal for testing the accuracy of nuerical ethods. The stiffness factor, k, in valve vibration and sound production explains the reason for an increased aplitude of the pulonary coponent of the second sound relative to the aortic coponent in pulonary hypertension. Another iportant factor is the effect of the viscosity factor, c, on the valve vibration. Since aneic patients have reduced viscosities, they are shown to have augented heart sounds[9]. Also, k/, it is clear that the higher the stiffness of the valve fro the relation ω = leaflet, the higher will be the natural frequency of vibration. Hence, patients with aortic stenosis can be expected to have higher frequency of the aortic coponent of the second sound. On the other hand, the increased ass of valve leaflet due to calcification will have only a little effect upon the frequency or aplitude of the second sound because the effective ass,, appearing in the expression of ω consists largely of the fluid surrounding he valve leaflets[9]. We hope that the above analysis helps to explain soe previously unexplained clinical observations by eans of factors that are identified to relate to valve vibration and heart sounds that result fro it.

8 1964 S. A. Al-Mezel, M. El-Shahed and H. El-sely References [1] O. P. Agrawal, Stochastic analysis of dynaic systes containing fractional derivatives, Journal of Sound and Vibrations, 47 (001) [] Blick. E. F, H. N. Sabbah and P. D. Stein, One-Diensional Model of Diastolic Seilunar Valve Vibrations Productive of Heart Sounds, Journal of Bioechanics,1 (1979) 3-7. [3] M. El-Shahed, Adoian decoposition ethod for solving burgers equation with fractional derivative, Journal of Fractional calculus, 4, (003) 3-8. [4] M. El-Shahed, Fractional calculus odel of the seilunar heart valve vibrations using Matheatica, International Matheatica Syposiu IMS 003, Iperial College- London, 7-11 July, (003) [5] M. El-Shahed, Application of He s hootopy perturbation ethod to Volterra s integro-differential equation, International Journal of Nonlinear Sciences and Nuerical Siulation, 6, (005) [6] V. Gotolv, R. Vadov and P. Kolobaev, Acoustic atheatical odel of the heart of the person-the theory and experient, XIII Session of the Russian Acoustical Society, Moscow, (003) [7] A. A. Kilbas, H. M. Srivastava and J. J. Trujillon, Theory and Applications of Fractional Differential Equations, Elseviesr, 006. [8] J. Mazuadar and D. Woodard, A Matheatical Study of Seilunar Valve Vibration Journal of Bioechanics, 17 (1984) [9] J. Mazuadar, An Introduction to Matheatical Physiology and Biology, Australian Matheatical Society, [10] K. Metzler and J. Klafter, The rando walk s guide to anoalous diffusion: A fractional dynaics approach Physics Reports, 339 (001) [11] K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, [1] E. Moonia, T. Selway and T. Jina, Analysis of Adoian decoposition applied to a third-order ordinary differential equation fro thin fil flow, Nonlinear Analysis 66 (007) [13] I. Podlubny, Fractional differential equations, Acadeic Press, New York, 1999.

9 Heart valve vibrations 1965 [14] R. Saha and K. Bera, Analytical solution of the Bagley-Torvik equation by Adoian decoposition ethod, Applied Matheatics and Coputation 168 (007) [15] S. Sakakbira, Properties of Vibration with Fractional Derivative Daping of Order 1, JSME International Journal 40 (1997) [16] L. Suarez and L. Shokooh, Response of Systes with Daping Materials Modeled using Fractional Calculus, Appl Mech Rev 48 (1995) S118- S16. [17] L. Suarez and L. Shokooh,An Eigenvector Expansion Method for the Solution of Motion Containing Fractional Derivatives, Journal of Applied Mechanics 64 (1997) [18] G. A. Thibodean and K. T. Patto, Anatoy and Physiology, Mosby, [19] A. Wazwaz, Adoian decoposition ethod for a reliable treatent of the Bratu-type equations, Applied Matheatics and Coputation, 166 (005) Received: October, 008