A NONLINEAR VISCOELASTIC MOONEY-RIVLIN THIN WALL MODEL FOR UNSTEADY FLOW IN STENOTIC ARTERIES. Xuewen Chen. A Thesis. Submitted to the Faculty.

Transcription

1 A NONLINEAR VISCOELASTIC MOONEY-RIVLIN THIN WALL MODEL FOR UNSTEADY FLOW IN STENOTIC ARTERIES by Xewen Chen A Thesis Sbmitted to the Faclty of the WORCESTER POLYTECHNIC INSTITUTE in partial flfillment of the reqirements for the Degree of Master of Science in Applied Mathematics by May 3 APPROVED: Dr. Dalin Tang Thesis Adisor Dr. Bogdan Vernesc Head of Department

2 Abstract Seere stenosis may case critical flow conditions related to artery collapse plaqe cap rptre which leads directly to stroke and heart attack. In this paper a nonlinear iscoelastic model and a nmerical method are introdced to stdy dynamic behaiors of the tbe wall and iscos flow throgh a iscoelastic tbe with a stenosis simlating blood flow in hman carotid arteries. The Mooney-Rilin material model is sed to derie a nonlinear iscoelastic thin-wall model for the stenotic iscoelastic tbe wall. The mechanical parameters in the Mooney-Rilin model are calclated from experimental measrements. Incompressible Naier-Stokes eqations in the Arbitrary Lagrangian- Elerian formlation are sed as the goerning eqation for the flid flow. Interactions between flid flow and the iscoelastic axisymmetric tbe wall are handled by an incremental bondary iteration method. A Generalized Finite Differences Method (GFD) is sed to sole the flid model. The Forth-Order Rnge-Ktta method is sed to deal with the iscoelastic wall model where the iscoelastic parameter is adjsted to match experimental measrements. Or reslt shows that iscoelasticity of tbe wall cases considerable phase lag between the tbe radis and inpt pressre. Seere stenosis cases cyclic pressre changes at the throat of the stenosis cyclic tbe compression and expansions and shear stress change directions in the region jst distal to stenosis nder nsteady conditions. Reslts from or nonlinear iscoelastic wall model are compared with reslts from preios elastic wall model and experimental data. Clear improements of or iscoelastic model oer preios elastic model were fond in simlating the phase lag between the pressre and wall motion as obsered in experiments. Nmerical i

3 soltions are compared with both stationary and dynamic experimental reslts. Mooney- Rilin model with proper parameters fits the non-linear experimental stress-strain relationship of wall ery well. The phase lags of tbe wall motion flow rate ariations with respect to the imposed plsating pressre are simlated well by choosing the iscoelastic parameter properly. Agreement between nmerical reslts and experimental reslts is improed oer the preios elastic model. ii

4 Acknowledgements I wold like to express firstly sincere thanks to my adisor Dr. Dalin Tang. His dynamic thoghts his broad and profond knowledge and his patient instrction hae gien me the greatest help in all the time of research for and writing of this thesis. Withot his gidance and encoragement this master thesis wold not hae been possible. I also wold like to express my deep gratitde and appreciation to the faclty in Mathematical Sciences Department. I am gratefl to Dr. Homer Walker Dr. Mayer Hmi Dr. Arthr Heinricher Dr. William Martin Dr Christopher Larsen and Dr. Jng- Han Kimn from whom I hae learned a lot. Thanks also go to many many people in Mathematical sciences department who hae gien me helps and cares. My special gratitde is for my brother Xeke Chen for giing me his nlimited spport alable criticisms and all the benefits of his experience. I dedicate this thesis to my parents Jianming Chen and Chnxi Zho. iii

5 Contents Chapter Introdction... Chapter Mathematical Models The Flid Model The Wall Model Introdction Nonlinear Viscoelastic Thin-wall Model... Chapter 3 Nmerical Method Introdction Otline of the Nmerical Method Generalized Finite Difference method GFD based finite-olme method with staggered grids and pwind techniqes Discretization of N-S Eqations oer Irreglar Geometry with Non-niform Mesh Discretization of continity eqation The SIMPLER Algorithm th Order Rnge-Ktta Method Incremental Bondary Iteration Method Geometry and mesh Accracy The parameters in Nonlinear MR iscoelastic wall model Chapter 4 Reslt and Discssion i

6 4. Comparison between experiment and nmerical reslt with Elastic Non-linear Viscoelastic wall model Comparison the effect of normal pressre condition and high pressre condition Comparison the effect of seerity of stenosis on the tbe wall Chapter 5 Conclsion... 6 Bibliography... 6

7 List of Figres. The stenotic collapsible tbe and starling resistor chamber The nonlinear iscoelastic MR wall model....3 Stress-Pressre relationship on the artery cross section The staggered grids and nmbering of neighboring points; a) -eqation; b) - eqation; c) c-eqation Finer mesh is sed near the tbe wall and stenosis to get better resoltion there Comparison of pressre and stretch-ratio( λ / λ ) relationship of artery wall between experimental data and MR model The experimental pressre conditions imposed at the inlet and otlet of the tbe Pin=7~3mmHg Pot=5~5mmHg Comparison of tbe radis between nmerical elastic wall model and experiment reslts at x=. cm S=8% Pin=7~3mmHg Pot=5~mmHg Comparison of tbe radis between nmerical MR iscoelastic wall model and experiment reslts at x=. cm. S=8% Pin=7~3mmHg Pot=5~mmHg Comparison of tbe nmerical radis and experiment reslts at x=. cm S=8% Pin=7~3mmHg Pot=5~mmHg.. 44 i

8 4..5 Comparison of nmerical flow rate from elastic wall model and experiment reslts at a period. S=8% Pin=7~3mmHg Pot=5~mmHg Comparison of nmerical flow rate from MR iscoelastic wall model and experiment reslts at a period; S=8% Pin=7~3mmHg Pot=5~mmHg Comparison of nmerical flow rate and experiment reslts at a period; S=8% Pin=7~3mmHg Pot=5~mmHg Comparison of transmral pressre nder prescribed inlet pressre conditions High pressre case more negatie pressre. a) Normal Pressre: Pin=7~3 mmhg; b) High pressre: Pin=9~5mmHg Tbe wall radis cres nder two prescribed inlet pressre conditions. S=8% Pot=mmHg; axial pre-stretch=36.5%. a) Normal pressre: Pin=7~3mmHg; b) High pressre: Pin=9~5 mmhg Comparison of shear stress on the wall ( dyn / cm ) nder maximm and minimm inlet pressre conditions S=8%; Pot= mmhg; axial pre-stretch=36.5%. a) Normal pressre: Pin=7~3mmHg; b) High pressre: Pin=9~5mmHg Comparison of maximm axial elocity nder nsteady inlet pressre conditions. S=8%; Pot= mmhg; axial pre-stretch=36.5% Comparison of minimm pressre nder nsteady inlet pressre conditions. S=8%; Pot= mmhg; axial pre-stretch=36.5%...53 ii

9 4.3.. Comparison of transmral pressre nder different stonosis seerity conditions; a) S=8% Pin=7~3mmHg Pot=mmHg; b) S=5% Pin=7~3mmHg Pot= 68.6~8.6mmHg Plots of tbe wall radis cre nder different stenosis seerity conditions; a) S=8% Pin=7~3mmHg Pot=mmHg; b) S=5% Pin=7~3mmHg Pot= 68.6~8.6mmHg Comparison of shear stress on the wall ( dyn / cm ) nder different stenosis seerity conditions imposed to maximm and minimm inlet pressre conditions. a) S=8% Pin=7~3mmHg Pot=mmHg; b) S=5% Pin=7~3mmHg Pot= 68.6~8.6mmHg Comparison of minimm pressre Pmin nder different stenosis seerity conditions. Inlet and otlet pressre imposed on S=8% tbe: Pin=7~3mmHg; Pot=mmHg; Inlet and otlet pressre imposed on S=8% tbe Pin=7~3mmHg; Pot=68.6~8.6mmHg Comparison of maximm elocity Umax nder different stenosis seerity conditions. Inlet and otlet pressre imposed on S=8% tbe: Pin=7~3mmHg; Pot=mmHg; Inlet and otlet pressre imposed on S=8% tbe Pin=7~3mmHg; Pot=68.6~8.6mmHg....6 iii

10 List of Tables Table 3. Comparison of nmerical soltions with exact soltion for a rigid straight tbe. p in = mmhg p = 99. 8mmHg -max(exact)=3.98 cm/s dt=.5 time step compted=6. Relatie errors are defined as e n ( f ) = f n f exact /( f n +.) n=time step...35 Table 3. Order of accracy of the nmerical method. p in = mmhg p = 3mmHg S = 8%. Step size redction ratios are.9 for r and.95 for x. Step sizes gien in the table are the max-min x-steps along the tbe length and the max-min r-steps at the inlet of the tbe. Relatie errors are defined as e n ( f ) f n f n / f n =..35 ix

11 Chapter Introdction Stroke and ischemic heart disease which reslt from high grade stenoses are the single most common cases of death in the United States. Approximately 35 percent of all deaths reslt from this case. High grade stenoses increases flow resistance in arteries which forces the body to raise the blood pressre to maintain the necessary blood spply. Both the high pressre and the narrowing of blood essel case high flow elocity high shear stress and low or negatie pressre at the throat of the stenoses low shear stress flow separation wall compression or een collapse at the distal side of the stenoses. These may be related to thrombs formation atherosclerosis growth and plaqe cap rptre which leads directly to stroke and heart attack. The exact mechanism of this complicated process is still not well nderstood. A better stdy in this physiological process is of great importance to early diagnosis preention and treatment stenoses related diseases. A considerable nmber of experimental and nmerical research works hae been condcted to stdy the flow dynamics and stresses in elastic collapsible tbe. Many interesting phenomena sch as flow limitation choking fltter and wall collapse hae been identified and analyzed [ ] in the last thirty years. Recently Tang [ ] sed axisymmetric models to inestigate steady/nsteady iscos flow in elastic stenotic tbes with arios stenosis stiffness and

12 pressre conditions. Caalcanti [8] did nmerical simlation to examine the hemodynamics in a mild stenosis with consideration of plsatile wall motion. Giddens [9] sed comptational methods to inestigate the interaction between flid mechanics and the artery wall. Bathe [3] introdced an axisymmetric thick-wall model with flidstrctre interactions for plsatile blood flow throgh a compliant stenotic artery. Also a different flid-strctre interactions method was deeloped by Yamagchi [3] and was applied to axisymmetric and symmetric plaqe models of coronary artery diseases. K [7] et al. condcted a series of experiments sing rigid tbes compliant tbes with rigid stenoses thin-wall silicone tbes with stream-lined compliant stenoses and thick-wall PVA hydrogel models whose mechanical properties are close to boine carotid arteries. Powell [3] measred the tbe law for boine carotid artery and stdied the effects of seerity of stenosis. Their reslts showed that the tbe wall collapsed nder physiological conditions. While mch work has been reported the mathematical models for flow in stenotic collapsible tbes were primarily limited to -D models becase of the difficlties in handling flid-strctre interactions with nonlinear large wall deformation large strain and the critical flow conditions indced by the stenosis. And most research focs on elastic tbes in which stress prodces its characteristic strain instantaneosly and strain anishes immediately on remoal of the stress. Bt real arteries contain a ariety of tisses. When it is sbjected to a constant strain artery tisse creeps in time and when it is sbjected to a constant strain the indced

13 stress gradally relaxes. This transient behaior of artery is known as iscoelasticity [5] [39] [4] [46]. Some mathematical models sch as Maxwell Model Voigt Model and St. Venant Model [37] etc. hae been deeloped to simlate the iscoelastic properties. Stdies from Wesseling at al [34] Goedhard and Knoop [38] showed that the iscoelastic model with more than one time constant reflects the inflence of iscoelastic properties better when representing a hman artery. To better nderstand blood flow behaior in real arteries a ariable time constant in iscoelstic artery model may be considered. In this thesis firstly the mechanical properties of artery wall are taken from Yamagchi s experiments [4]. In experiments the PVA hydrogel is sed to make thickwalled stenosis model whose mechanical properties are ery close to that of hman carotid arteries. Based on the experimental data an axisymmetric nonlinear comptational model is introdced to simlate blood flow in stenotic carotid arteries. The Naier-Stokes eqations are sed for the flid model. Arbitrary Lagrangian-Elerian formlation (ALE) [ ] is sed which is sitable for problems with free moing bondaries. The SIMPLER [8] [45] algorithm based on Generalized Finite Differences with staggered grids and the pwind techniqe is sed to sole the flid model. A modified iscoelastic thin-wall model with ariable time constant is introdced to model the dynamic nonlinear properties of the stenotic tbe wall with the mechanical parameters controlled by the Mooney-Rilin material model which is based on the experimental measrements (Tang ). An incremental bondary iteration method techniqe is sed to handle the flid-wall interactions. The ranges of physical parameters and geometries of the tbe and flid domain are chosen to match the experimental set-p. 3

14 Both experimental and comptational reslts show that the iscoelasticity of tbe wall cases considerable phase lag between the oscillations of downstream flow rate and imposed pressre small phase lag between the oscillations of radis and pressre. The freqency of the oscillations is identical to that of the imposed pressre. Seere stenoses case cyclic pressre changes between positie and negatie ales at the throat of the stenosis cyclic tbe compression and expansions for the axisymmetric nonlinear model and rapid shear stress changing directions in the region jst distal to the stenosis which may case excessie artery fatige and possible plaqe cap rptre. Nmerical soltions are compared with experimental reslts and a reasonable agreement is fond. 4

15 5 Chapter Mathematical Models. The Flid Model We consider nsteady iscos flow in a stenotic compliant tbe simlating blood flow in stenotic carotid arteries. The flow is assmed to be laminar Newtonian iscos and incompressible. The shape of the tbe is nder zero transmral pressre and the tbe wall is assmed to hae no axial motion that is no slipping takes place between the flid and the wall. And we also assme that there is no penetration of the flid throgh the tbe wall. A diagram of the experimental set-p is gien in the Figre.. Using Arbitrary Lagrangian-Elerian (ALE) Formlation the Naier-Stokes eqations are gien by: ) ( ) ( ) ( ( r r r x x p r t R x t X t = + + µ ρ (.) ) ( ) ( ) ( ( r r r r x r p r t R x t X t = + + µ ρ (.) = + + r r x (.3) Where U= ) ( and are the axial and radial components of the flid elocity ρ is density µ is iscosity. t X and t R are elocities of the moing mesh at the position of

16 moing mesh point considered. To specify the shape of the tbe wall (which means the inner wall same throghot this thesis) we se ( X ( t) R( X t)) to label the material points of the wall nder zero pressre condition and ( x ( X t) r( X t)) to denote the position ector of the moing tbe wall (to be determined). In this thesis the tbe radis nder zero pressre condition is gien by (see Figre.) R = R( X ) = R S( X ) (.4) S S X ) = π ( X X R[ cos( X X ) )] 4 X X X ( otherwise (.5) Where R is the radis of the niform part of the tbe S (X ) specifies the shape of the stenosis S is the stenosis seerity by diameter i.e. redction of the tbe diameter cased by stenosis X and X specify the beginning and ending of the stenosis. Stenosis seerity is commonly defined as: ( R Rmin ) S = %. (.6) R For bondary conditions we assme that no slipping and no penetration take place and the flid and wall moe together: x( X t) r( X t) ( ) Γ = ( ). (.7) t t 6

17 Here Γ stands for the bondary of the flid domain bonded by the free moing tbe wall x = x( X t) r = r( X t) = H ( x t) = H ( x) + H ( x t) (.8) c Where H ( x t) is the radis fnction x is sed as an independent ariable in H and H c to simplify presentation of reslts H ( x( X )) = R( ) gies the resting shape of the tbe X H c ( x t) gies the tbe wall radial ariations. At the inlet and otlet of the tbe we set: p = pin ( t) (.9) x = p = l = pot ( t) const (.) x = x x= l =. (.) Where p in (t) and p ot (t) are the pressre imposed at the inlet and otlet of the tbe respectiely in or experiments. We start the comptations from zero flow and zero pressre conditions with the tbe taking the resting shape. The pressre at the inlet and otlet will be raised gradally to the specified pressre conditions and the model is soled ntil a periodic soltion is obtained. 7

18 P e H (x) x P=P x=. (straight) P=P Upstream Reseroir Downstream Reseroir External Pressre P Proximal e Distal Pressre P Pressre P Flowrate Q Pressre Chamber Compliant Tbe Fig..: The stenotic collapsible tbe and starling resistor chamber 8

19 . The Wall Model.. Introdction Arteries contain a ariety of tisses. When it is sbjected to a constant strain artery tisse creeps in time and when it is sbjected to a constant strain the indced stress gradally relaxes. This transient behaior of artery is known as iscoelasticity. Some mathematical models sch as Maxwell Model Voigt Model and St. Venant Model etc. hae been deeloped to simlate the iscoelastic properties. Stdies from Wesseling at al Goedhard and Knoop showed that the iscoelastic model with more than one time constant reflects the inflence of iscoelastic properties better when representing a hman artery. A new idea is sing a nonlinear elastic model to represent the elastic part of iscoelastic property instead of combination of strings. So a better nderstand blood flow behaior in real arteries a nonlinear model which of corse has a ariable time constant in iscoelstic artery model may be considered. In this thesis a nonlinear iscoelastic thin-wall model with ariable time constant is introdced to model the dynamic nonlinear properties of the stenotic tbe wall with the mechanical parameters controlled by the Mooney-Rilin material model which is based on the experimental measrements. An incremental bondary iteration method techniqe is sed to handle the flid-wall interactions. The ranges of physical parameters and geometries of the tbe and flid domain are chosen to match the experimental set-p. 9

20 .. Nonlinear Viscoelastic Thin-wall Model The tbe wall radial displacement is determined by the Nonlinear Viscoelastic Thin-wall Model which is a parallel combination of a nonlinear Mooney-Rilin (MR) string and a iscos dashpot (Figre.). Emr σ σ η Fig..: The non-linear iscoelastic MR thin wall model. The MR String sed to describe the nonlinear material properties where the strain energy density fnction assmes the form: Where: D ( I 3) W = c ( I 3) + c ( I 3) + D [ e ] (.) I = λ + / λ I = λ λ (.) + are the inariants of the deformation tensor (Bathe Book 6.7) λ is the axial stretch ratio C i s and be obtained: D i s are material constants. The Lagrange stress [5] of this MR string can σ MR D ( 3) 3 λ + λ = (λ λ ) + C ( λ ) + DD (λ λ ) e C (.3)

21 where π ( r + δ ) r λ = =. (.4) r r For the dashpot stress σ d is a fnction of the rate of strain mltiplied by the iscos damping coefficient: η therefore we hae: dε σ d = η. (.5) dt The strain can also be related to radis and ths related to stretch radio. Assming small strains the change in circmferential engineering strain in the artery wall is gien by the change in circmference diided by the preios circmference: π ( r + dr) πr d ε = = πr dr r. (.6) Integrating eqation (.8) yields: r ε = ln( ) + ε. (.7) r Here ε and r are artery stain and radis respectiely and the sbscript of denotes a known condition at an initial time. Taking the deriatie of eqation (.7) with respect to time yields: dε dr =. (.8) dt r dt

22 If we define σ MR can also be written as σ MR = EMRε where E MR is the string constant of MR string ε is the radis strain of the tbe wall then the time constant [5] of this model: EMR ζ = (.9) η is obiosly ariable which satisfies the discoery of Goedhard and Knoop (973). Combining the MR string and Dashpot the whole stress in this model therefore is gien by: σ = σ MR + σ d. (.) Figre.3 shows the forces acting on an artery cross section. By smming the horizontal forces across the control olme we get the circmferential stress in a thin walled cylindrical tbe: pr σ = (.) h Where h is the thickness of the artery wall and p represents the internal artery pressre which is the same as blood pressre. Assming the artery wall is incompressible the olme of the wall is a constant and we can hae: π hr = πh r (.)

23 Fig..3: Stress-Pressre relationship on the artery cross section. Combining eqation (.) and (.) yields an expression for the circmferential stress in an artery of incompressible material: pr σ = (.3) h r Sbstitting eqation (.3) (.4) (.5) (.8) (.3) into (.) yields the Nonlinear Viscoelastic Constittie Eqation for the tbe: ( D ( λ + 3) 3 λ ) + C ( λ ) + DD (λ λ ) e r x) pλ = C(λ λ h ( x) λ + η (.4) λ t Where r( x t) 5 λ = is the stretch ratio C =. dyn / cm r ( x) C 5 =. dyn / cm D 3 = 3.8 dyn / cm and =. 4 experimental data for the boine carotid arteries [4]. D were chosen for the artery wall to match In the aboe eqation if we know pressre and isoelastic parameterη initial strain ε 3

24 wall geometric parameters of artery r h initial radis r (x) the aboe Nonlinear Viscoelastic Constittie Eqation for the arterial wall can be sed to determine the tbe radis r ( x t). Howeer when the tbe collapses the tbe is no longer adeqate to determine the tbe deformation becase the tbe is no longer axisymmetric. A fll 3-d model is needed to determine the wall deformation nder collapsed condition. In this work we assme that the tbe remains axisymmetric and compresses axisymmetriclly r when <. With this assmption tbe compression obsered sing this model will be a r clear indication of tbe collapse. Longitdinal tension is soled from the following eqilibrim eqation T L = τ (.5) s where s is the arc length τ is flid shear stress acting on the tbe wall. The initial axial stretch seres as the needed bondary condition for (.3). The inlet and otlet of the tbe are not allowed to moe in the axial direction to preent the entire tbe from being pshed away by the flow. Since the tbe wall is pre-stretched 36.5% and additional axial strain and displacement dring the simlation are small linear elasticity is sed to determine the wall axial displacement which gies excellent approximation. Assming the axial is linear elastic deformation axial stress tension and strain are related by: TL σ L = Eε L = (.6) h where E is the Yong s modls which is the material physical characteristic parameter 4

25 h is the wall thickness σ L and ε L are the axial stress and strain respectiely. Once the axial strain is determined axial displacement follows easily. 5

26 Chapter 3 Nmerical Method 3. Introdction This model is a free moing bondary problem which inoles flid-wall interactions. The seere stenosis and iscoelasticity of the tbe wall make the deformation pattern more complicated. All these reqire the application of mlti nmerical methods and techniqes in this problem. In this thesis the conentional Arbitrary Lagrangian-Elerian (ALE) based staggered generalized finite differences (GFD) [6] oer an irreglar grid with pwind differencing [7 8] is sed for the flid model an incremental bondary iteration techniqe is introdced for the flid-wall interaction. The SIMPLER algorithm is sed to sole the Naier-Stokes eqations. Using of ALE formlation enables s to choose the mesh properly to aoid large mesh distortion and eliminates the needs of interpolating the flow ariables for preios steps at the new grids. GFD makes it possible for s to se finer mesh near the tbe wall and in the stenotic region to handle the critical flow conditions inoled in the problem. The incremental bondary iteration method is essentially a relaxation techniqe which is sed 6

27 to handle pressre oer-shooting and bondary oer-shooting [9 ] and improe on the reglar bondary iteration method to get conergence for this model with large strain and large deformations. Details of the nmerical method are explained below. 3. Otline of the Nmerical Method The model is soled by sing a bondary iteration method whose main steps are: ). Start from an initial bondary elocity and pressre. ). In each time step a). Flid Part: Using GFD to discretize the ALE Formlation and Continity Eqation do the iterations sing Simpler method till the corrections are small enogh then moe to wall part. b). Wall Part: Use the pressre and shear stress fields from the flid part to adjst the shape of the tbe by soling the iscoelatic constittie eqation of the tbe wall and longitdinal eqilibrim eqation. c). Repeat the flid part and wall part ntil the correction: f i f i / f i TOL (3.) Where f denotes the soltion ector and TOL is a specified tolerance i.e. the comptation is considered conerged if the relatie corrections of the ariables being soled (flow elocity pressre and wall displacement) became less than the tolerance specified. 3). Moe to next time step. Repeat aboe steps ntil a periodic soltion is reached. 7

28 3.3 Generalized Finite Difference method Generalized Finite Difference method GFD method has been sed in many engineering applications where irreglar geometries and free-moing bondaries are inoled. The adantage of the GFD method is that the generalized finite difference schemes can be deried for arbitrary irreglar grids. With the GFD method we will be able to se finer mesh near the tbe wall and in the stenotic region to handle the critical pressre and flow conditions and se coarser mesh where flow and pressre changes are less drastic. This leads to considerable redction of grid points and CPU saings. With limited compting power this may een be essential when we sole the corresponding model with flidwall interactions. The GFD concept can be explained by the following example. To derie the second order GFD schemes for the deriaties f x f r f xx f rr and xr f at a gien point p let X i = ( xi ri ) ( i = n n 5) be n neighboring points of X and a i = xi x bi ri r ( i i = ρ = a + b ) f = f x r ) i i ( i i se the Taylor expansion of f at X and omitting higher order terms we hae for each X i xr f i = f + ai f x + bi f r + ( ai f xx + bi f rr + aibi f ) We define: 8

29 9 = n n n n n n b a b a b a b a b a b a A = f f f f f n = xr rr xx r x f f f f f df the Taylor expansions lead to f df A = (3.) The finite difference schemes for the 5 deriaties can be obtained all at once from these eqations sing proper least-sqares approximations. Other GFD schemes can be deried similarly. We can get second-order scheme: f C f A A A df T T = = ω ω ) ( (3.3) where

30 = ρ n ρ ρ ω Becase there are irtally no limitations on the selection of the points i X and the deriation can be done atomatically in the compter program each time the domain and mesh are adjsted the GFD method is a sitable tool to handle the irreglar geometry non-niform grids and the free moing bondary which reqires freqent atomatic remeshing of the domain.

31 3.4 GFD based finite-olme method with staggered grids and pwind techniqes The finite olme method with staggered grids and pwind techniqes is chosen to oercome difficlties cased by the large pressre gradient and large conection terms. GFD makes the implementation of the finite-olme method to the irreglar geometry and non-niform mesh possible. Rewriting the ALE formlation and Continity eqation into the form: X µ R T ρ + ( ρ ρ µ µ )( x r xx rr xr ) + px = t t r t (3.4) X µ R T ρ + ( ρ ρ µ µ )( x r xx rr xr ) + µ + p = r t t r t r (3.5) T T ( )( x r xx rr xr ) + ()( x r xx rr xr ) + = r (3.6) for each grid point X sing the backward difference for the t-deriatie the generalized finite difference schemes for the space deriaties with the neighboring points chosen by the finite-olme method with staggered grids (Figre 3.) Discretization of N-S Eqations oer Irreglar Geometry with Non-niform Mesh -eqation: The location of star nodes is show in Figre 3.; we se 8 nodes to generate the space

32 deriaties term at in eqation and 5 nodes to deal with conectie term. Using i (i=...8) arond the star following the procedre listed aboe we can get d = C. It is worth to notice that althogh there are conectie terms in d to gain more physical meaning for or scheme we adopt pwind scheme. Upwind Scheme: Discretization of the Conection term at To deal with conectie terms we do not se the r x in d. Upwind method is traditional method to discretize the conection terms which is stable and proide physical meaning for the formlas. For x if and if > are sed to discretize the conection terms. For r if > and if < sed to discretize it. By sing 5 points arond we can get another d = C combining with C we can get a new d = C. C Other terms For p at we se only for points arond throgh weight aerage to approximate for p θ the central difference is sed. x x

33 Discretizing the eqation sing the generalized finite difference schemes and pwind scheme deriing and rearranging terms we get -eqation: 8 + ki i + k9 + k ( p p ) = ( eqation) i= (3.7) The eqation is discretized in the same way: 8 + ki i + k9 + k ( p p ) = ( eqation). (3.8) i= 3.4. Discretization of continity eqation We discretize the continity eqation at p 6 points arond p is sed. Using the procedre of GFD we can get 6 i= k c i i 6 + k i= c 6+ i i + k c 3 = ( c eqation) (3.9) where the notations p i i at i are as marked in Figre 3.. First order schemes are sed for the pressre deriaties since experiences indicate that lower order schemes shold be sed for pressre to get better performance. 3

34 a) U6 U7 U8 U4 P P U U5 U U U3 b) V6 V7 V8 P V4 V V5 P V V V3 c) U3 U6 V4 V5 V6 U P U5 V V V3 U U4 Fig. 3.: The staggered grids and nmbering of neighboring points; a) -eqation; b) - eqation; c) c-eqation. 4

35 5 3.5 The SIMPLER Algorithm Becase the flid-wall interaction model is ery complex we start from a well-tested SIMPLER (Semi-Implicity Method for Pressre Linked Eqation) method. Instead of soling the discretized eqations (3.7)-(3.9) directly we se the SIMPLER algorithm to sole for δ and p δ the corrections to the elocity and pressre respectiely [8]. By doing so we improe on the soltions obtained from last iteration ntil the desired accracy is reached. Let ) ( m m m p be the mth iteration of the soltions of (3.7)-(3.9) and the residals of (3.7)-(3.8) by ) ( m m m p be (omitting the sperscript m): ) ( ) ( 9 8 i i i p p k k k p R = = ) ( ) ( 9 8 i i i p p k k k p R = =. Assming ) ( p p δ δ δ satisfy (3.7)-(3.8) we hae: ) ( ) ( ) ( ) ( 8 = = + + = p R p p k k p p R i i i δ δ δ δ δ δ (3.) ) ( ) ( ) ( ) ( 8 = = + + = p R p p k k p p R i i i δ δ δ δ δ δ (3.) Neglecting ) ( i i δ δ terms in (3.)-(3.) leads to: ) ( ) ( p p k p R δ δ δ = (3.) ) ( ) ( p p k p R δ δ δ =. (3.3) Let *) * ( = ) ( δ δ + + be the (m+)th iteration and sbstitte it back into the

36 6 eqation (3.9) and sing (3.)-(3.3) p δ can be determined. Then ) ( δ δ follows from (3.)-(3.3) and the (m+)th iteration is obtained. The aboe procedre is repeated ntil desired accracy is reached th Order Rnge-Ktta Method Nonlinear Viscoelastic Constittie Eqation for the tbe: η λ λ λ λ λ λ λ λ λ λ + + = + p x h x r e D D C C t D ) ( ) ( ) ( ) ( ) ( 3) ( 3 (3.4) is soled by 4 th order explicit Rnge-Ktta Method [47]. Gien a general ODE: ) ( ) ( y t y t t y t f dt dy = = (3.5) The obiosly approach is to integral from n t to h t t n n + = + by sing a qadratre formla: )) ( ( ) ( )) ( ( ) ( ) ( = + = + + τ τ τ τ τ τ d h t y h t f h t y d y f t y t y n n n t t n n n n and to replace the second integral by a qadratre. The otcome might hae been the method:

37 γ y + = yn + h b j f ( tn + c jh y( tn + c jh)) n n = j=... except that we don t know the ale of y at the nodes t n + c h + c h + c h. t n t n γ We mst resort to an approximation. We denote or approximant of y( tn + c jh) by ξ j j= γ. To start with we let c = since then the approximation is already proided by the former step of the nmerical method ξ = yn. The idea behind explicit Rnge-Ktta (ERK) methods is the express each ξ j j= 3 γ by pdating y n with linear combination of f t n ξ ) f ( tn + hc ξ ) f ( t n + hc j ξ j ). Specifically we let: ( ξ = y n ξ ξ = y + ha f ( tn ) n ξ 3 = yn + ha3 + ξ f ( tn ξ) + ha3 f ( tn ch ) (3.6) γ i= ξ γ = yn + h aγ i f ( tn + cih ξi ) y γ n+ = yn + h b j f ( tn + c jh ξ j ) j= The matrix A = ( a ) j i =.. γ where missing elements are defined to be zero is j i called the RK matrix while 7

38 b b b = M b γ and c c c = M c γ are the RK weights and RK nodes respectiely. We say that (3.6) has γ stages. To obtain RK matrix the most obiosly way consists of expanding eerything in sight into Taylor series abot t y ) then recollect terms and compare with the Taylor Expansion ( n n of the exact soltion abot the same point t y ). A great deal of persistence and care ( n n of comptations are reqired to obtain the family of 4 th order implicit Rnge-Ktta method. The best-known 4 th order for-stage ERK method is: A / = / / 6 / 3 b = / 3 / 6 / c = (3.7) / Using stretch ratio of preios time step and pressre from flid the Nonlinear Viscoelastic Constittie Eqation for the tbe can be soled with the 4 th order Rnge- Ktta method dring each time step. The iscoelastic coefficient in the iscoelastic constittie eqation is properly chosen to simlate well of the phase lag of tbe wall motion with respect to the imposed plsating pressre. 8

39 3.7 Incremental Bondary Iteration Method Bondary iteration methods hae become poplar for soling problems with flidstrctre interactions recently where the flid and solid models are soled iteratiely ntil conergence is obtained. Howeer it has been known that the bondary iteration method may fail to conerge if the tbe wall is considerably compliant []. We se an incremental iteratie method to redce the displacement oer-shooting and improe the conergence. Displacement oer-shooting cases elocity oer-adjstment at the bondary which affects the conergence of the flid model. For a gien wall adjstment r( θ z) if the flid model fails to conerge we redce r to half and try to sole the flid model again. This is repeated ntil conergence is reached. A similar relaxation techniqe can also be sed to redce pressre oer-shooting which is the case of tbe wall oer-adjstment. When δ and δ p are obtained we pdate and p with: p new new = + ω δ (3.8) old = p + ω δp (3.9) old p where ω and ω p can be chosen between and to achiee best conergence. 9

40 3.8 Geometry and mesh The ranges of parameters and the geometry of the tbe sed in the comptations are chosen to match the experimental set-p []: R =. 4cm l = 8cm (tbe length) X = 3. cm X = 4. 8cm ν =.4cm / s = g cm ρ / ν µ / ρ =. (3.) The external pressre was set to zero and inlet pressre P in (t) and otlet pressre (t) are set to be consistent with experiments [3]. Reynolds nmber Re is defined as D U /ν where D is the entrance tbe diameter ( R ) and U is the entrance aerage elocity. The Reynolds nmber for an 8% stenosis with inlet pressre mmhg and otlet pressre mmhg is abot 3. The nits gien aboe are sed throghot this thesis. P ot Nn-niform grids (Figre 3.) were sed in the comptation to handle the critical flow conditions inoled in the collapse process. The step size in r and x directions are redced by fixed ratios towards the wall and the middle of the tbe length to get better resoltion there. The ratios are: q dr / dr.9 q dx / dx. 95 (3.) r = i+ i = x = i+ i = For a ( 3 r x ) mesh we hae dr = dr 3 =. 3 at the inlet of the tbe where the step size at the wall dr 3 is less that eleenth of the starting step size dr. For x we hae dx =. 667 at the inlet dx 5 =. 755 at the middle of the tbe length 3

41 which is less that one twelfth of dx. By sing the non-niform mesh mch better resoltion is achieed near the wall and stenotic region with fewer grid points which lead to considerable saings of CPU time. 3

42 Fig. 3.: Finer mesh is sed near the tbe wall and stenosis to get better resoltion there. 3

43 3.9 Accracy Since analytic soltions for flow in a compliant stenotic tbe are not aailable comptations are performed for flow in a rigid tbe oer three meshes and nmerical soltions are compared with the exact soltion (Fng 997) = ( R r ) 4ν p x = p x = const. (3.) To check the conergence of the method Table 3. gies a smmary of the errors which show that the algorithm conerged well. The accracy of the nmerical soltions may be better than what Table shows becase the exact soltion is assmed to be x-independent while the actal flow always has entrance and end effects. To check the accracy and conergence of the nmerical scheme for the complaint model three meshes were tested and the reslts are gien in Table 3.. While the aerage dx i dr j is redced abot 5% from one mesh to next mesh (notice the mesh is nonniform) the errors are redced abot 5%. This indicates that the scheme is of first order accracy. One may be wondering whether the remeshing regriding and recalclation of the coefficients of the eqations might affect the efficiency of the algorithm. While it does increase the complexity of the program it actally takes ery little CPU time dring actal rn. For the meshes sed in Table the CPU time for 5 eqation iterations (the 33

44 SIMPLER algorithm soling (3.7)-(3.9)) were and.77 secons respectiely sing an ALPHA station (5 MHz) while the CPU time for wall displacement remeshing regrinding and recalclation the coefficients of the eqations took less than. seconds (the bondary is adjsted eery 5 eqation iterations). Mesh 3 ( 3 ) is sed for the general comptations in this thesis. The tolerance for the eqation iterations is set to. 6 i.e. soltions of (3.7)-(3.9) are considered obtained if the relatie errors (corrections) of the elocity and pressre are less that the specified tolerance. The tolerance for the bondary iteration is set to. 4 i.e. the soltion for the tbe wall flow elocity and pressre were considered conerged for a gien time step if the relatie errors became less that the tolerance specified. Periodic soltions were considered obtained when the soltions started to repeat itself within % tolerance. Or calclations indicate that three periods are needed for the soltions to become periodic. 34

45 Mesh m n e n () e n () ( p) x r e n Mesh Mesh 3. 3 Mesh Table 3.: Comparison of nmerical soltions with exact soltion for a rigid straight tbe. p in = mmhg p = 99. 8mmHg -max(exact)=3.98 cm/s dt=.5 time step compted=6. Relatie errors are defined as e f ) f f /( f.) n=time step. n ( = n exact + n Mesh m x nr dx dx m dr dr e n () e n () e n ( p) (H ) n e n Mesh e-5.e e-5 6.5e-6 Mesh e-6.3e-4 9.e e-6 Mesh e e e-6.95e-6 Table 3.: Order of accracy of the nmerical method. p in = mmhg p = 3mmHg S = 8%. Step size redction ratios are.9 for r and.95 for x. Step sizes gien in the table are the max-min x-steps along the tbe length and the max-min r-steps at the inlet of the tbe. Relatie errors are defined as en ( f ) = f n f n / f n. 35

46 3. The parameters in Nonlinear MR iscoelastic wall model Mechanical Parameters in MR model The Mooney-Rilin material model (3-) is sed to derie a nonlinear iscoelastic thinwall model (3-33) for the stenotic iscoelastic tbe wall. σ MR D ( 3) 3 λ + λ = (λ λ ) + C ( λ ) + DD (λ λ ) e C (3-) ( D ( λ + 3) λ r x) 3 pλ = C(λ λ ) + C ( λ ) + DD (λ λ ) e h ( x) λ + η λ t (3-33) The mechanical parameters C C D and D in the Mooney-Rilin model are decided from experimental measrements of stationary stress-strain relationship of artery wall: C 5 =. dyn / cm C 5 =. dyn / cm D 3 = 3.8 dyn / cm and =. 4 D. Figre 3.3 comparies of pressre and stretch-ratio( λ / λ ) relationship of artery wall between experimental data and MR model with aboe parameters. One can see the MR model fits the experimental data qite well. Viscoelastic Parameter: Once mechanical parameters in MR model are decided the iscoelastic parameter η is adjsted so that the nmerical radis nder experiment pressre matches the experimental radis. η =. 36

47 The phase lags of tbe wall motion flow rate ariations with respect to the imposed plsating pressre are simlated well by choosing aboe parameters in the nonlinear MR iscoelastic wall model. 37

48 Mooney Riilin Model when η= Mooney Riilin Model Experiment Data 8 Pressre mmhg 6 4 c = c = D =38 D = Stretch Ratio Fig. 3.3: Comparison of pressre and stretch-ratio( λ / λ ) relationship of artery wall between experimental data and MR model. 38

49 Chapter 4 Reslt and Discssion 4. Comparison between experiment and nmerical reslt with Elastic Nonlinear Viscoelastic wall model The iscoelasticity of artery wall prodced a phase lag between the pressre gradient and tbe radis and between the pressre gradient and otlet flx rate. Ths if an elastic wall model is sed dring nmerical simlation [3 4] one can see a clear phase lag between nmerical radis changing and experimental radis changing. To determine the inflence of the iscoelastic properties of the tbe flows in elastic tbe iscoelastic tbe are compted and compared with experiment reslts. Figre 4.. shows the pressre conditions imposed at the inlet and ot let of the tbe in the experimental. Dring the experiment the inlet pressre was set to 7-3mmHg otlet pressre was set to 5-5mmHg and the enironment pressre is set to mmhg. Maximm inlet pressre occrs at nmerical simlation. o 89. The same pressre condition is sed in the Figre 3.3 shows Mooney-Rilin material model with C = dyn / cm = C D = 38dyn / cm 4 D =. fits the mechanical property of tbe wall qite well. The stress-stretch ratio λ / λ relation of the tbe tested in a niaxial tensile loading condition. It is worth noting the Mooney-Rilin model can not fit the experimental data well when 39

50 the load condition is less than zero. This is becase we assmed the tbe is still axisymmetric when it collapses nder negatie pressre bt in reality we don t hae enogh information abot the changing of stress-stretch ratio nder negatie loading. Actally the experimental data is not totally reliable in this case becase of the measring difficlty. Figre 4.. plots the nmerical and experimental radis changing at x=.cm in a tbe with stenosis 8% dring one period. The maximm radis in experiment occrs at o o θ = 5.. Maximm nmerical radis from elastic wall model occrs at θ = 87.. Figre 4..3 shows by sing non-linear MR iscoelastic wall model with η = the maximm nmerical radis occrs at o θ =.5. The phase difference between maximm nmerical radis and experimental obseration is redced 79.44% by sing iscoelastic MR wall model instead of elastic wall model. The phases lag simlation between inpt pressre and wall motion is improed 88.7% by sing iscoelastic MR Model. Figre 4..4 shows the comparison of tbe nmerical radis with elastic model iscoelastic MR model and experiment reslts at x=. cm dring one period. A phase lag between elastic nmerical radis ariation and experimental radis ariation is canceled by sing iscoelastic MR wall model. Figre 4..5 gies the elastic comptational and experimental flow rates at the otlet dring one period. While there is a rogh agreement one can see that there again has a clear phase shift between comptational and experimental data. Figre 4..6 shows the comparison between the nmerical flow rate from the non-linear iscoelastic MR wall model and experimental reslt at a period. This indicates that the 4

51 phase lag in Fig. is cased by iscoelastic properties of the tbe wall and can be decreased by iscoelastic MR wall model. Figre 4..7 plots the nmerical flow rates and experimental reslt in the same figre to gie a better description of the effects of iscoelasity of tbe wall. 4

52 8 6 Experimental Inpt and Otpt Pressre Aerage P in = mmhg Aerage P ot = mmhg 4 Inpt Pressre Phase Fig. 4..: The experimental pressre conditions imposed at the inlet and otlet of the tbe p in = 7 3mmHg p ot = 5 5mmHg 4

53 Radis at cm in artery Elastic Model Experimental Radis Nmerical Radis.64.6 radis (cm) phase (Degree) Fig. 4.. Comparison of tbe radis between nmerical elastic wall model and experiment reslts at x=. cm. S = 8% p in = 7 3mmHg p ot = 5 mmhg Radis at cm in artery Nonlinear Viscoelastic MR Model Experimental Radis Nmerical Radis.64.6 radis (cm) phase (Degree) Fig. 4..3: Comparison of tbe radis between nmerical MR iscoelastic wall model and experiment reslts at x=. cm. S = 8% p in = 7 3mmHg p ot = 5 mmhg. 43

54 Radis at cm in artery Experimental Radis Nmerical Radis from Elastic Model Nmerial Radis from Viscoelastic Model.64.6 radis (cm) phase (Degree) Fig. 4..4: Comparison of tbe nmerical radis and experiment reslts at x=. cm. S = 8% p in = 7 3mmHg p ot = 5 mmhg. 44

55 6 Flow Rate from MR model Flow Rate from experiment Flow Rate Comparson 5 4 Flow rate (ml/s) Phase (Degree) Fig. 4..5: Comparison of nmerical flow rate from elastic wall model and experiment reslts at a period. S = 8% p in = 7 3mmHg p ot = 5 mmhg. 6 Flow Rate from MR Viscoelastic Thin wall Model 5 4 Flow Rate (ml/s) 3 Flow rate from MR model Flow rate from experiment Phase(degree) Fig. 4..6: Comparison of nmerical flow rate from MR iscoelastic wall model and experiment reslts at a period. S = 8% p in = 7 3mmHg p ot = 5 mmhg. 45

56 6 Flow Rate Compason 5 4 Flow Rate (ml/s) 3 Flow rate from elastic model Flow rate from experiment Flow rate from MR model Phase (Degree) Fig. 4..7: Comparison of nmerical flow rate and experiment reslts at a period. S = 8% p in = 7 3mmHg p ot = 5 mmhg. 46

57 4. Comparison the effect of normal pressre condition and high pressre condition To see the inflence of the imposed plsatile pressre on the flow and wall behaiors the pressre imposed in the inlet of the tbe ( P in ) was set to 7 ~ 3mmHg and9 ~ 5mmHg representing high and normal pressres; the pressre imposed in the otlet of the tbe ( P ot ) was set to mmhg. Stenosis is still 8% by diameter. Comparison of the two cases is also inclded in the following figres. Figre 4.. plots the transmral pressres along the tbe wall on the high pressre and normal pressre cases. It is obsered that in both cases minimm pressres occrred at the tbe at the throat of the stenosis and the axial location of the minimm pressre for the Pinmax s are shifted slightly becase the tbe is pshed more to the downstream side by the flow. a). Normal pressre: P in = 7 ~ 3mmHg the minimm pressres are 8.36 and -.79 mmhg; b). High pressre: P in = 9 ~ 5mmHg the minimm pressres are -.87 and mmhg. Figre 4.. plots the wall compression distal to the stenosis nder high pressre with that nder normal pressre. High pressre case more tbe wall compression cyclic wall compression cased by the high pressre is more noticeable that that from the normal case. This indicates that high pressre is more likely to cases cyclic wall compression which leads to accelerated artery fatige. 47

58 Figre 4..3 plots the stress distribtion on the wall for these two cases maximm shear stress for the high pressre case is mch more greater that that for the normal pressre case. High shear stress more likely case cap rptre which leads directly to stroke and heart attack. Figre 4..4 plots the behaior of axial maximm elocity nder these two prescribed inlet pressre conditions dring one period. High pressre cases high elocity which is more likely to case rptre. Figre 4..5 plots the behaior of the minimm pressre nder these two prescribed inlet pressre conditions dring one period. Minimm pressre nder high pressre condition is obiosly less than that nder normal pressre condition. 48

59 a). Normal Pressre 4 Pin max =3mmHg Pin min =7mmHg 8 Pressre (mmhg) Axial Position (cm) b). High Pressre 6 4 Pin max =5mmHg Pin min =9mmHg Pressre distribtion along wall (mmhg) Axial Position (cm) Fig. 4..: Comparison of transmral pressre nder prescribed inlet pressre conditions High pressre case more negatie pressre. a) Normal Pressre: P in =7~3 mmhg; b) High pressre: 9~5mmHg. 49

60 a). Normal Pressre.7 Wall Shape at Pin max =3mmHg Wall Shape at Pin min =7mmHg.6.5 Wall Shape (cm) Axial Position (cm) b). High Pressre.7 Pin max =5mmHg Pin min =9mmHg.6.5 Wall shape (cm) Axial Position (cm) Fig. 4..: Tbe wall radis cres nder two prescribed inlet pressre conditions. Higher pressre cased more tbe wall compresssion. S = 8% ; Pot= mmhg; axial prestretch=36.5%. a) Normal pressre: Pin = 9 ~ 5mmHg. Pin = 7 ~ 3mmHg ; b) High pressre: 5

61 a). Normal Pressre 35 3 Shear Stress Comparison Shear stress at Pinmax=3mmHg Shear stress at Pinmin=7mmHg 5 Shear Stress (dyn/cm ) b). High Pressre Axial Position (cm) 4 35 Pin max =5mmHg Pin min =9mmHg 3 5 Shear Stress (dyn/cm ) Axial Position (cm) Fig. 4..3: Comparison of shear stress on the wall ( dyn / cm ) nder maximm and minimm inlet pressre conditions S = 8% ; Pot= mmhg; axial pre-stretch=36.5%. a) Normal pressre: Pin = 7 ~ 3mmHg ; b) High pressre: Pin = 9 ~ 5mmHg. 5

62 65 Pin=9 5mmHgPot aerage =mmhg Pin=7 3mmHgPot aerage =mmhg 6 55 Umax (cm/s) Phase (Degree) Fig. 4..4: Comparison of maximm axial elocity nder nsteady inlet pressre conditions. S = 8% ; Pot= mmhg; axial pre-stretch=36.5%. 5

63 5 Pin=9 5mmHg Pin=7 3mmHg 5 Pmin (mmhg) Phase (Degree) Fig. 4..5: Comparison of minimm pressre nder nsteady inlet pressre conditions. S = 8% ; Pot= mmhg; axial pre-stretch=36.5%. 53

64 4.3 Comparison the effect of seerity of stenosis on the tbe wall Since pressre decreases considerably when passing a seere stenosis the effect of the stenosis seerity on flow and pressre fields becomes mch more noticeable when comparison is made with flow rate fixed. Two cases are designed to compare the effct of seerity of stenosis on the tbe wall: Seerity 5% inlet pressre: 7~3 mmhg otlet pressre: 68.6~8.6 mmhg. Seerity 8% inlet pressre: 7~3 mmhg otlet pressre: mmhg. Figre 4.3. plots the transmral pressres along the tbe wall on S=8% and S=5% cases. It is obsered that in both cases minimm pressres occrred at the tbe at the throat of the stenosis too. And the negatie pressre for the high seerity case S=8% is -.79 mmhg when the imposed pressre conditions are: Pin=3 mmhg and Pot= mmhg while no negatie pressre is obsered for S=5% case when imposed pressre conditions are: Pin=3 mmhg and Pot=mmHg. Figre 4.3. plots the wall compression nder high seerity of stenosis with that nder normal seerity. One can obsere that normal stenosis will not affect the shape of the tbe while high seerity stenosis case more tbe wall compression. Figre plots the shear stress distribtion on the wall for the two cases. We can conclde that high stenosis case mch greater shear stress which is more likely to case cap rptre and leads directly to stroke and heart break. S=8%: Max shear stress=373.9 S=5%: Max shear stress=76. dyn / cm ; dyn / cm. 54