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Viscoelastic modeling of aortic excessive enlargement of an artery
 

Viscoelastic modeling of aortic excessive enlargement of an artery

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    Viscoelastic modeling of aortic excessive enlargement of an artery Viscoelastic modeling of aortic excessive enlargement of an artery Document Transcript

    • International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 2, March - April (2013) © IAEME479VISCOELASTIC MODELING OF AORTIC EXCESSIVEENLARGEMENT OF AN ARTERYRajneesh Kakar*1, Kanwaljeet Kaur2, K. C. Gupta21Principal, DIPS Polytechnic College, Hoshiarpur, (Punjab), India2Research Scholar, Punjab Technical University, Jalandhar, 1446001, India2Supervisor, Punjab Technical University, Jalandhar, 1446001, IndiaABSTRACTIn this study we have considered a problem on application of viscoelasticity model. Amathematical model is purposed to study biological tissue. We have applied damped mass-spring Voigt model and the behavior of an excessive localized enlargement of an artery isstudied by varying the periodic blood force and amount of damping in the viscoelastic tissue.Also, the pressure inside the artery is assumed to vary linearly over space. We haveconsidered Voigt model by assuming blood flow as a sinusoidal function and viscoelastictissues as Navier-Stokes equation. The problem is solved with the help of LaplaceTransforms by using boundary conditions. The phenomena of damping are examined andevaluated using MATLAB.Keywords: Voigt model, Navier-Stokes Equation, Laplace Transforms, Resonance, Aortic.1. INTRODUCTIONAn excessive enlargement of an artery is a balloon-like bulge in an artery, or lessfrequently in a vein. Whether due to a medical condition, genetic predisposition, or trauma toan artery, the force of blood pushing against a weakened arterial wall can cause excessiveenlargement of an artery. Excessive enlargement of an artery occur most often in the aorta(the main artery from the heart) and in the brain, although peripheral excessive enlargementof an artery can occur elsewhere in the body. Excessive enlargement of an artery in the aortais classified as thoracic aortic excessive enlargement of an artery if located in the chest andare abdominal aortic excessive enlargements of an artery if located in the abdomen. AorticINTERNATIONAL JOURNAL OF MECHANICAL ENGINEERINGAND TECHNOLOGY (IJMET)ISSN 0976 – 6340 (Print)ISSN 0976 – 6359 (Online)Volume 4, Issue 2, March - April (2013), pp. 479-493© IAEME: www.iaeme.com/ijmet.aspJournal Impact Factor (2013): 5.7731 (Calculated by GISI)www.jifactor.comIJMET© I A E M E
    • International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 2, March - April (2013) © IAEME480excessive enlargement of an artery is usually cylindrical in shape. These excessiveenlargements of an artery can grow large and rupture or cause dissection which is a splitalong the layers of the arterial wall. Each year, most of the deaths are due to rapture of aorticexcessive enlargement of an artery.Many excessive enlargement of an artery are due to fatty materials like cholesterol.Arteries are made up of cells that contain elastin and collagen. Under small deformations,elastin demines the elasticity of the vessel and under large deformations; the mechanicalproperties are governed by the higher tensile strength collagen fibers. During the passage oftime, arteries become less rigid, which is a cause for hypertension. The tendency towardsrupture of an artery is explained with elasticity and viscoelasticity.Concerning viscoelasticity, some models are purposed by the researchers as Migliavacca andDubini (2005) discussed computational modeling of vascular anastomoses. Salcudean et al.(2006) presented viscoelasticity modeling of the prostate region using vibro-elastography;Wong et al. (2006) studied the theoretical modeling of micro-scale biological phenomena inhuman coronary; Keith et al. (2011) presented an article on aortic stiffness. Recently, Kakaret al. (2013) studied viscoelastic model for harmonic waves in non-homogeneous viscoelasticfilaments.In this study, we have developed a mathematical model for the aortic excessiveenlargement of an artery. The exact solution is obtained by using an approximation of theboundary condition at the excessive enlargement of an artery wall. We have also investigatedthe influence of the excessive enlargement of an artery wall damping. The influence offorcing function frequency is summarized numerically.2. THE MATHEMATICAL MODELThe purposed mathematical model under consideration for idealized aortic excessiveenlargement of an artery is composed of three sections:• The blood within the excessive enlargement of an artery.• The wall of the excessive enlargement of an artery.• The bodily fluid that surrounds the excessive enlargement of an artery.The geometry of the excessive enlargement of an artery is shown as a sphere in Fig. 1. Let( , )u x t represents the displacement of the bodily fluid w.r.t. space and time, with x measuredfrom the exterior wall of the excessive enlargement of an artery and it is in the direction of xi.e. in the direction of the spinal fluid. (0, )u t represents displacement at the exterior face ofthe excessive enlargement of an artery wall, and it is of greatest interest to us.Fig. 1: Excessive enlargement of an artery Sectionx=0 at exterior wallBodily fluidEnlargement of an arteryBlood
    • International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 2, March - April (2013) © IAEME481The flow of the bodily fluid can be described by one-dimensional Navier-Stokes equationwhich is derived from the Law of Conservation of Momentum22v v P vvt x x xρ ρ µ∂ ∂ ∂ ∂+ + − = Β∂ ∂ ∂ ∂(1)where ρ is unit density, v is velocity, P is pressure, and µ is viscosity of the bodily fluid,and Β is the body force. The first termvtρ∂  ∂ in Eq. (1) denotes the rate of change ofmomentum; the termvvxρ∂  ∂ is the time independent acceleration of a fluid with respect tospace and it is neglected because it is assumed that the space derivative of the velocity issmall in comparison to the time derivative. The third termPx∂  ∂ in Eq. (1) is the pressuregradient. The term22vxµ ∂ ∂ in Eq. (1) is known as the diffusion term it is dependent on theviscosity of the fluid. If we assume bodily fluid is in-viscid, then22vxµ ∂ ∂ term can also beneglected. Hence in the absence of body force, Eq. (1) reduces to0v Pt xρ∂ ∂+ =∂ ∂(2)The displacement of the bodily fluid is denoted by u vdt= ∫ as defined above. Then( )uv tt∂=∂and22u vt t∂ ∂=∂ ∂In addition, we will assume that the bodily fluid is slightlycompressible and obeys the Constitutive Law:2 uP cxρ∂= −∂(3)where c is the speed of sound through the bodily fluid. From Eq. (2) and Eq. (3), we getwave equation:2 222 2( , ) ( , )u x t c u x tt x∂ ∂=∂ ∂(4)Since Eq. (4) has two derivatives in time and two derivatives in space. Therefore, there aretwo initial conditions and two boundary conditions for the problem.
    • International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 2, March - April (2013) © IAEME4822.1 Initial conditionsAssuming that the fluid is initially at rest, the initial conditions are:( ,0) 0u x = and ( ,0) 0u xt∂=∂(5)2.2 Boundary conditionsThe problem has two boundary conditions1. The boundary condition at x L= , the plane wave will vanish at some long distance.( , ) ( , )u L t c u L tt x∂ ∂= −∂ ∂(6)Since an arterial wall is called viscoelastic because it has properties of both an elastic solidand a viscous liquid. For sake of convenience, in modeling, we have taken the arterial tissuein the form of model. In this model, spring and dashpot are arranged in parallel as shown inFig. 2.Fig. 2: Rheological Model for Viscoelastic Excessive enlargement of an artery Wall2. The boundary condition at 0x = , this condition is derived from a force balance equation.According to Newton’s Second Law, the rate of change in momentum must be equal to thetotal forces acting on the wall of the artery. Thus the total force will be the contributions ofbodily fluid and from the spring with dashpot. The force due to bodily fluid isfluidF Pa= − . (7a)If we assume the bodily fluid is slightly compressible, then we can write the pressure interms of displacement as2(0, )fluidF c a u txρ∂=∂(7b)where, ρ is the density of the bodily fluid, c is the speed of sound, and a is the crosssectional area. The damping force isuγF dampingmutt(0,t)F springWallF blood = kX0cosωt fluid
    • International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 2, March - April (2013) © IAEME483(0, )dampingF u ttγ∂= −∂(7c)where, γ is the damping constant.The force from the spring will be given by Hooke’s Law, Therefore( )spring m bF k X X= − − (7d)where k is the spring constant, mX is the displacement of the mass which arises due to themotion of the exterior face of the arterial wall (0, )u t , and bX represents the displacement ofthe internal wall of the artery because of blocking element which creates contraction of pathand increase in pressure, it can be given by a sinusoidal function 0cosX tω , where ω is thefrequency of the periodic force from the blood pressure and 0X is the maximum displacementof the inner wall and. Hence, the boundary condition at x = 0 becomesmtu t c axu ttu t ku t kX t∂∂=∂∂−∂∂− +22200 0 0 0( , ) ( , ) ( , ) ( , ) cosρ γ ω (7e)There are two solutions of Eq. (7)1 Exact solution2 Approximate solutionIn both the cases, 0 coskX tω is forcing function and its frequency is adjusted to simulateresonance. We will investigate whether this resonance is a contributing factor in excessiveenlargement of an artery rupture. In summary, we will find an exact and an approximatesolution of Eq. (4), Eq. (5), Eq. (6) and Eq. (7e). The exact solution is obtained by takingLaplace Transform of the bodily fluid pressure term and solved it at the boundary condition0x = which resulted in the bodily fluid pressure term being coupled with the damping force.As a result, the combined bodily fluid-damping term could never approach zero andresonance could not be produced. The approximate solution of the boundary condition canbe obtained by linear approximation of the change in displacement due to the bodily fluidpressure term from 0x = to x L= . This assumption leads to group the bodily fluid pressureeffect with the spring force. As a result of which the damping term could approach to zero,therefore an interesting resonance phenomenon is observed.3. SOLUTION OF THE MODELWe start with partial differential equation for the wave equation:2 222 2u uct x∂ ∂=∂ ∂0, 0t x L≥ < < (8a)The Laplace Transforms of the displacement ( , )u x t are
    • International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 2, March - April (2013) © IAEME484L ( ){ }0, ( , ) ( , )stu x t U x s e u x t dt∞−= = ∫ (8b)L ( ), ( , ) ( ,0)u x t sU x s u xt∂ = − ∂ (8c)L ( )222, ( , ) ( ,0) ( ,0)u x t s U x s su x u xt t ∂ ∂= − − ∂ ∂ (8d)Taking the Laplace Transform of the wave Eq. (8a), we get22 22( , ) ( ,0) ( ,0) ( , )s U x s su x u x c U x st x∂ ∂− − =∂ ∂(9)From Eq. (5) and Eq. (9), we have22 22( , ) ( , )s U x s c U x sx∂=∂(10)The general solution to this equation is1 2( , ) cosh sinhs sU x s c x c xc c   = +      (11)Taking the Laplace Transform of boundary condition Eq. (6), we get( , ) ( ,0) ( , )sU L s u L c U L sx∂− = −∂(12)Substituting the general solution (11), the initial condition, and the derivative of ( , )U x s withrespect to x into Eq. (12), and evaluating all at L, we get1 2 1 2cosh sinh sinh coshs s s s s ss c L c L c c L c Lc c c c c c          + = − +                    which gives1 2c c= −Thus the general solution can be expressed1 1( , ) cosh sinhs sU x s c x c xc c   = −      (13)Evaluating ( , )U x s at the boundary condition 0x = gives 1(0, )U s c=
    • International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 2, March - April (2013) © IAEME4853.1 Exact solutionRearranging the terms from Eq. (7.5), we have0222(0, ) (0, ) (0, ) (0, ) cosc u t a m u t u t ku t kX tx t tρ γ ω∂ ∂ ∂− + + + =∂ ∂ ∂(14)Taking the Laplace Transform of this boundary condition (Boyce et al (2005)), we get[ ] 02 22 2(0, ) (0, ) (0,0) (0,0) (0, ) (0,0) (0, )sc a U s m s U s su u sU s u kU s kXx t sρ γω∂ ∂ − + − − + − + = ∂ ∂ + (15)Simplifying for initial conditions and grouping like terms gives02 22 2(0, ) ( ) (0, )sc a U s ms s k U s kXx sρ γω∂− + + + =∂ +(16)Taking the derivative ( , )U x sx∂∂of the general solution (13) and evaluating at 0x = gives1 1 1 1 02 22 2sinh 0 cosh 0 ( ) cosh 0 sinh 0s s s s s s sc a c c ms s k c c kXc c c c c c sρ γω          − • − • + + + • − • =          +          which simplifies to1 1 022 2( )scasc ms s k c kXsρ γω+ + + =+(17)Dividing both sides by m and rearranging terms yields( )1 022 2ca k sc s s k m Xm m sρ γω + + + =  +  (18)Let 0 caγ ρ γ= + and 20kmω = . Then for 0x =( )( )01002 2 2 2(0, )k m X sU s cs s smγω ω= =  + + +    (19)Taking the inverse Laplace Transform of (0, )U s and using partial fraction decompositionL ( )1(0, )U s−= L1 212 2 2 2A B Cs Ds r s r s sωω ω−  + + + − − + + yields a solution of form
    • International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 2, March - April (2013) © IAEME4861 2(0, ) cos sinr t r tu t Ae Be C t D tω ω= + + + (20)where: 1,22 2 20 0 042mrmγ γ ω− ± −= 0 caγ ρ γ= + 20kmω =( )( )( )0 11 2 12 2k m X rAr r r ω=− +( )( )( )0 21 2 22 2k m X rBr r r ω−=− +( ) ( )( )( )0 01 22 22 2 2 2k m XCr rω ωω ω−=+ +( )( )( )01 2202 2 2 2k m XDr rγ ωω ω=+ +(21)3.2 Approximation of the Boundary Condition at the Excessive enlargement of anartery WallIn order to solve Eq. (7.5), we use the standard linear approximation for ( , )u x h t+ (Boyce etal. (2005)):( , ) ( , ) ( , )u x h t u x t h u x tx∂+ = +∂(22)At h L= , some distance far away from the arterial wall, ( , ) (0, ) ( , )u L t u t L u x tx∂= +∂.Since ( , ) 0u L t = , we can solve for ( , )u x tx∂∂, and substitute the expression(0, )( , )u tu x tx L∂= −∂into the bodily fluid pressure term in the boundary condition (7.5),giving:02 22(0, ) (0, ) (0, ) cosc am u t u t k u t kX tt t Lργ ω ∂ ∂+ + + = ∂ ∂  (23)Let2~ c ak kLρ= + and 20~kmω =Taking the Laplace Transform of this boundary condition (23) yields[ ] 0~22 2(0, ) (0,0) (0,0) (0, ) (0,0) (0, )sm s U s su u sU s u kU s kXt sγω∂ − − + − + = ∂ + (24)Simplifying for initial conditions and grouping like terms gives:0~22 2( ) (0, )sms s k U s kXsγω+ + =+(25)
    • International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 2, March - April (2013) © IAEME487Dividing both sides by m and solving:( )( )002 2 2 2(0, )k m X sU ss s smγω ω=  + + +    (26)Taking the inverse Laplace Transform of (0, )U s and using partial fraction decompositionL ( )1(0, )U s−= L1 212 2 2 2A B Cs Ds r s r s sωω ω−  + + + − − + + yields a solution of the same form as the exact solution Eq. (20)i.e. 1 2(0, ) cos sinr t r tu t Ae Be C t D tω ω= + + +where:01,22 2 242mrmγ γ ω− ± −=2~ c ak kLρ= + 20~kmω =( )( )( )0 11 2 12 2k m X rAr r r ω=− +( )( )( )0 21 2 22 2k m X rBr r r ω−=− +( ) ( )( )( )0 01 22 22 2 2 2k m XCr rω ωω ω−=+ +( )( )( )01 222 2 2 2k m XDr rγωω ω=+ +(27)DiscussionAlthough the two solution processes are quite different but exact and approximate solutionsappear very similar. The approximate solution is a solution corresponds to one boundarycondition whereas the exact solution is a solution to the whole system. In exact solution, weobtain the term 0 caγ ρ γ= + and in approximate solution, we get2~ c ak kLρ= + term (Boyceet al. 2005). 1r and 2r are the roots of the corresponding homogeneous equation. The effectof these terms vanishes quickly, it is the transient solution. The trigonometric termscontribute the particular solution of the non-homogeneous equation and their effect remainsas long as the external force is applied. The steady-state solution is: (appendix)( ) cos( )U t R tω δ= −where,002 2 2 2 2 2( )kXRm ω ω γ ω=− +02 2tan( )mγωδω ω=−
    • International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 2, March - April (2013) © IAEME4884. NUMERICAL ANALYSISVarious graphs are plotted for two solutions of the boundary condition at the wall. Inthese graphs, we have studied the effect of excessive enlargement of an artery wall dampingand the effect of the frequency of forcing function due to blood pressure by choosingfollowing parameters purposed by Salcudean et al. (2005).Table 1ρ( 3kg m ) c( m s) a( 2m ) γ(kg s ) k( N m ) m(kg) X0 (m) ω( rad s ) L(m)1000 1500 0.0001 2 8000 0.0001 0.002 6 0.1The Influence of Wall Damping and the influence of frequency are plotted for both exactand approximate solutions. It is observed that as the damping constant of the excessiveenlargement of an artery wall increases, the maximum displacement decreases (Fig.3). Thefrequency made small effect on the maximum displacement (Fig. 4). With the increase infrequency of the forcing function, the time period will decrease and the maximumdisplacement will also decrease. The value of γ = 200 is used in the exact solution but in forapproximation, it is required a much smaller γ i.e. equal to 0.00002. Fig. 5 and Fig. 6 areplotted to explain the approximate solutions. The concept of resonance and beats are alsoshown in Fig. 7 and Fig. 8. It is observed that the transient solution dies away immediately inall of the graphs.Phenomenon of resonanceResonance occurs when 0ω ω= . The maximum displacement is 002 2 2 2 2 2( )kXRm ω ω γ ω=− +,then as ω → ∞, 0R → . Also, as 0ω → , the forcing amplitude 0R X→ . when maxω ω= ,where max 022 222mγω ω= − . The corresponding maximum displacement is( )00max21 4kXRmkγω γ=−. As the damping constant 0γ → , the frequency of the maximum0R ω→ and the amplitude R increases without bound. i.e. in case of lightly damped systems,the amplitude of the forced vibrations becomes quite large even for small external forces.This phenomenon is known as resonance (Boyce et al. (2005)). (Fig. 8).Phenomenon of beatsBeats occur when 0ω ω≠ , in this case the amplitude varies slowly in a sinusoidal manner butthe function oscillates rapidly. This periodic motion is called a beat (Fig.7).
    • International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 2, March - April (2013) © IAEME489Fig. 3: Effect of Wall Damping ( 2ω π= ) for Exact SolutionFig. 4: Effect of Frequency ( 2γ = ) for Exact Solution
    • International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 2, March - April (2013) © IAEME490Fig. 5: Effect of Wall Damping ( 2ω π= ) for Approximation of Boundary ConditionFig. 6: Approximate Solution – Effect of Wall Damping at High Frequency ( 0ω ω= )
    • International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 2, March - April (2013) © IAEME491Fig. 7: BeatsFig. 8: Normalized amplitude of steady-state response v/s frequency of driving forceΓ = 0Γ = 0 00009.Γ = 0 00036.Γ = 0 00080.Γ = 0 00142.Γ = 0 00222.Γ =γω2202m
    • International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 2, March - April (2013) © IAEME4925. ConclusionWith the help of several assumptions, a complete solution of an excessiveenlargement of an artery is obtained. A Voigt model is taken to model the excessiveenlargement of an artery wall. It is observed that as the damping constant of the excessiveenlargement of an artery wall increases, the maximum displacement decreases. With theincrease in frequency of the forcing function, the time period decreases and the maximumdisplacement also decreases.Appendix1 The amplitude and phase shift of the sum of two sinusoids functions of equal frequencycan be determined as:Let ( ) cosf t A tω= and ( ) sing t B tω= be two sinusoidal functions with the same period andthey can be rewritten as a single sinusoid ( )( ) coss t R tω δ= − as follows:( )cos sin cosA t B t R tω ω ω δ+ = −cos sin cos cos sin sinA t B t R t R tω ω ω δ ω δ⇒ + = +Let cosA R δ= and sinB R δ=If we divideBA, we getsintancosB RA Rδδδ= =Consequently, the phase shift is1tanBAδ −  =   .Also, if we squares A and B and adding, we get2 2 2 2 2 2cos sinA B R Rδ δ+ = +2 2 2 2 2(cos sin )A B R δ δ+ = +2 2 2A B R⇒ + =Thus 2 2R A B= + .2 Sum and product of quadratic roots:In checking over our solutions, we wanted to confirm that 2 2R C D= + where( ) ( )( )( )0 01 22 22 2 2 2k m XCr rω ωω ω−=+ +and( )( )( )01 222 2 2 2k m XDr rγωω ω=+ +were the coefficients of the
    • International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 2, March - April (2013) © IAEME493trigonometric terms of the solution and 002 2 2 2 2 2( )kXRm ω ω γ ω=− +. we used the roots01,22 2 242mrmγ γ ω− ± −= and 20~kmω = , and generated some very typical algebraicequations. The sum and product of quadratic roots are found. For the characteristic equation20kr rm mγ+ + = ; 1a = , bmγ= andkcm= . Therefore, 1 2orbr ra mγ+ = − − and1 2ormc kr ra• = . These two facts are used to simplify the work.REFERENCES1. Boyce, William, E., DiPrima and Richard, C. (2005), “Elementary DifferentialEquations and Boundary Value Problems”, 8thEdition, John Wiley & Sons, Inc., USA.2. Kakar, R., Kaur, K., and Gupta, K.C. (2013), Study of viscoelastic model for harmonicwaves in non-homogeneous viscoelastic filaments, Interaction and MultiscaleMechanics, An International Journal, 6(1), 26-45.3. Keith, N., Cara, M., Hildreth, Jacqueline K. Phillips, and Alberto P. Avolio. (2011),“Aortic stiffness is associated with vascular calcification and remodeling in a chronickidney disease rat model” Am. J. Physiol. Renal Physiol.. 300, 1431-1436.4. Migliavacca, F. and Dubini, Æ G. (2005), “Computational modeling of vascularanastomoses Biomechan Model Mechanobiol” 3, 235–250. DOI 10.1007/s10237-005-0070-25. Powers and David L. (2006), “Boundary Value Problems”, 5thEdition, ElsevierAcademic Press, USA.6. Salcudean, S. E., French, D., Bachmann, S., Zahiri-Azar, R., Wen, X. and Morris, W.J. (2006), “Viscoelasticity modeling of the prostate region using vibro-elastography”,Medical Image Computing and Computer-Assisted Intervention – MICCAI 2006 ,Lecture Notes in Computer Science 4190, 389-396.7. Weir, Maurice, D., Hass, Joel, Giordano and Frank R. (2008), “Thomas’ Calculus EarlyTranscendentals”, 11thEdition, Pearson Education, Inc., USA.8. Wong, K., Mazumdar, M., Pincombe, B., Stephen, G., Sanders, P., and Abbott, D.(2006), “Theoretical modeling of micro-scale biological phenomena in human coronaryarteries Medical and Biological Engineering and Computing” 44(11), 971-982.