Biomechanical Modeling of the Arterial Fluid-Structure interaction in hemodynamic Mostafa Salehi Fluid-Structure

Transcription

1 Biomechanical Modeling of the Arterial Fluid-Structure interaction in hemodynamic Mostafa Salehi Fluid-Structure Interaction ASEN 559 Instructor: Professor Felipa March 6 4

2 Outline zproblem Statement and Research Objectives ztechnical Background zmethodology/modeling zconclusions

3 Problem Statement and Research Objectives z Understanding of the biomechanical properties of pulmonary arteries under normal and hypertensive conditions. z Analyze theory of the soft-tissue z Formulation for fluid-structure interaction in hemodynamic functioning in the mechanics of blood circulation z Define pertinent function and relationships Mechanical Mathematical computation

4 Technical Background/Who really is at risk? zimportance:6.8 millions Americans suffer from Cardiovascular Disease CVD,.4 million procedures performed/year, aging population[ According to the American Heart Association] z 5 million adults approximately 8% of the adult population in the United states have been diagnosed as suffering from hypertension z 48, deaths related to Coronary Artery Disease CAD

5 Technical Background/ Problems to address Mechanism of disease, materials for treatment,assist physicians in positively identifying individuals ho exhibit the development of vascular disease and begin appropriate treatment-long before symptoms or clinical signs appear Using an advanced technology, Arterial aveform Analysis The matching of artificial vascular grafts to host vessels Accurate modeling of blood flo.

6 Technical Background ELASTIC ARTERY Intima Media Adventitia Fibroblasts 9% GS % Elastin % Collagen 78% Endothelium Orientation of collagen fibers Smooth muscle cells Elastic laminae SMCs 33% GS6%,EL4% Collagen 37% Collagen Vasa vasorum GS: Kind of a glue that keeps all components together

7 Technical Background /unexplored area z Properties of pulmonary arteries under normal and hypertensive conditions Pulmonary Hypertension: Raised pressure in pulmonary arterial system ¾Systolic Pressures > 3 mmhg ¾ Diastolic pressures > mmhg ¾Or Both ¾Systematic Hypertension :sys >6 mmhg or dias >9mmHg

8 Technical Background [Pressure mmhg] Data on age-related change in the aorta of 76 healthy human males Time s Normal variations in blood pressure P i as a function of time during the cardiac cycle, generated based on the Fourier Coefficients N P t P [ A cos nω t B sin nωt] i m n n n a.

9 Technical Background /unexplored zestimated Time Average mean arterial pressure P m [Berne and Levy,986] Pm Pdias Psys Pdias 3 Where: P dias :Local MinimaDiastolic P sys :Local maxima Systolic Classification of blood pressure in humans [Pickering,995] DiastolicmmHg SystolicmmHg Normotensive <85 <3 High blood pressure Hypertensive Stage Stage Stage Stage 4 > > area

10 Approach: Modeling Structure Arterial all tissue is extremely complex znonlinear stress-strain relationship zlarge deformations possible before yield zviscoelastic : Properties are function of time zinhomogeneous:properties vary through tissue thickness zanisotropic: Properties vary ith direction

11 Methodology/Modeling For all Assumptions : z The material is incompressible z We are consider to have a strain energy function for the hyperelastic material z Continuum medium is assumed to be isotropic in the form of Neo- Hookean z The energy density functions is exponential

12 Modeling / Structure Stress-strain relationship Mechanical properties of a hyperelastic material can be characterized by a strain energy function S ψ E ψ C We use the neo-hooken model to determine the Isotropic response c> :Stress-like material parameter ψ iso c I I 3 E: Green-Lagrange strain tensor C: Right Cauchy Green Deformation I :Second in variant of C

13 Strain-energy function proposed for arterial all ELASTIC ARTERY ψ C, A, A ψ ψ iso I aniso I 4, I c I 6 ψ 3 k k iso I i 4,6 ψ aniso {exp[ k k > :stress-like materials k > :Dimensionless parameter c: Materials parameter I I 4 i, I 6 ] } T intima T media T adventitia

14 Modeling Fluid TUBE CONSTRUCTION θ r r i r o Epithelium } Tunica Lamina mucosa propria l p i Tunica muscularis ρ nerve plexus vessel Tunica adventitia or serosa r, T z: Coordinates of the z Artery

15 Modeling :Fluid z Blood Exhibit non-netonian characteristic z It is reasonable to assume an incompressible, Netonian response ithin the large arteries under normal conditions z Blood flos through deformable vessels having complex geometrics, but it can be useful to consider flo through a straight, rigid, circular tube. z Consider as a steady flo

16 Methodology/Modeling Expression for steady flo Fluid-Structure Interaction in Hemodynamic z The Constitutive relation for an incompressible, Netonian response : t PI tcauchy stress µ D, D ν x ν x PLagrange multiplier that enforces the incompressibility constraint trd P viscosity of the fluid D stretching tensor X velocity Xcurrent position of a material particle T

17 Methodology/Modeling Expression for steady flo dt d a dt d p b ν ν ρ ν µ ρ Constitutive relation, the balance of linear momentum Navier-Stokes If e employ a cylindrical coordinate system r,t,z Eq. becomes: z z z z z r r r r r r a z r r r r r z p b a z r r r r r r r p r b a z r r r r r r r p b ρ ν θ ν ν µ ρ ρ ν θ ν θ ν ν µ ρ ρ ν θ ν θ ν ν µ ρ θ θ θ θ θ θ ] [ ] [ ] [ 3-a,b,c

18 Methodology/Modeling Expression for steady flo z Body force negligible b ν z A fully developed flo requires, z Axisymmetry, ν z θ ν z Steady flo, t z Axial flo, ν r, ν θ z D flo, ν z varies only ith r, PP z z So the Axial equation of motion becomes: p z ν z µ [ r ] 4 r r r

19 ν r z Methodology/Modeling Expression for steady flo The pressure is a linearly decreasing function of distance z and tice integration reveals that the axial velocity : ν r c z no slip boundary condition at the all so, c for this special case, z ν r r dp 4µ dz a a dp 4µ dz lnr ith determines c r a c First integral the final expression for the velocity Parabolic profile 5 This solution is knon as a Pineville flo Mercury manometer to measure the blood pressure

20 Modeling Expression for steady flo The spatial form of mass balance is satisfied identically by this velocity field,. v νν r e z z Given :Velocity, So it easy to compute the components of the Cauchy stress t in the fluid t rz ν ν µ r z z r r dp dz At ra The magnitude of the all shear stress is : 6 7 τ a dp dz 8

21 Modeling Expression for steady flo To find all shear stress in terms of the volumetric flo Q ν. n da π a 8 µ 4 dp dz 9 Where : n: Outard unit normal vector to the cross-sectional area of interest ne z 4 µ Q t π 3 a Equation ill be used ith regard to both in vivo and in vitro settings

22 Modeling Expression for unsteady flo Considered the Unsteady problem The governing Navier-Stoke equation is: p z ν z ν z µ r r r ν z ρ t blood pressure P i based on the Fourier Coefficients N P t P [ A cos nω t B sin nωt] i m n n n By assumed that blood pressure can be represented by a Furier series and suggested that the pressure-gradient be represented similarly, so assumed that m real solutions for the unsteady part

23 Modeling Expression for unsteady flo p ~ A z here : i ν r, t z ρνe d ν dr imωt Which - imωt ν ν ω imωt z z im Ae µ r r r can be ritten for each harmonic r m, ~ ia dv dr m : m e ν steady z λ ν A, µ A harmonic, m e ν imωt m λ rexp imωt imρω µ e imωt

24 Modeling Expression for unsteady flo The solution to equation is : ν h r cj λr cy λr, r [,a] To keep the solution finite, c and ith the boundary condition that v z a,t for all t, e have for each harmonic: v z r, t A J λr e µλ J λa Defining the so-called omersley parameter D auzx then ODDm / i 3/ im ω t

25 Methodology/Modeling Expression for unsteady flo The all shear stress for each harmonic a J a J e A t im m λ λ λ τ ω The Volumetric flo for each harmonic a aj a J e A a t Q t im m λ λ λ µλ π ω ] [ a J a aj a a J t Q t λ λ λ π λ µλ τ The all shear stress for each m

26 Part II z Fluid-all interaction z Obtain the frequencydependent transfer function of the fluidstructure interacting system

27 Biomechanical Modeling of the Arterial Fluid-Structure interaction in hemodynamic Mostafa Salehi Fluid-Structure Interaction ASEN 559 March 8 4 Part II

28 Fluid-Structure interaction in hemodynamic z Mathematical formulation for the study of three-dimensional flo ithin an elastic tube z The fluid flo equation is recast as a boundary Integral representation and coupled ith the all motion. z The Numerical solution response of the fluidall system to a pressure pulse is computed z Model dynamics of the elastic alls z Account for Fluid-all interaction

29 Consider a homogeneous incompressible inviscid irrotational fluid of density U,floing in a A compliant tube of finite length. The volume occupied by the fluid is V The boundary S is constituted by the elastic all of the arterial vessel surface Where: V: Volume Mathematical formulation V S: Arterial segment surface S : Elastic all surface S i : Inlet Arterial segment surface S o : outlet Arterial segment surfaces s s i s s o

30 Mathematical formulation z Kelvin s theorem: an initially irrotational flo of such remains irrotational at all times. So, v x, t φ x, t x, t: Velocity potential, v: Fluid velocity vector The continuity equation for incompressible flos. v φ

31 boundary conditions for Eq.: φ n Mathematical formulation On the Wall Surface S φ. n v. n W on S V :The all velocity, n: The inard normal V : The application considered is the time derivative of elastic displacement u So the conditions 3 becomes: 3 φ u. n 4 n

32 Mathematical formulation On the inlet section S i Assume the velocity to be uniform and sires as function time Typically periodic in time. φ n φ. n v t. n i i on S 5 V i : Incoming velocity N : inard normal

33 On the Outlet section S o The outflo rate is depends on the hole flo inside V. The continuity equation implies: by Gauss Theorem x ds n x ds n x ds n x dv or x dv o i s s s v v φ φ φ φ φ 6 The relationship beteen velocity potential and pressure is Bernoulli theorem ρ ρ φ p p 7 Mathematical formulation

34 The Boundary Integral Equation For the Laplace Equation e have G δ x y 8 Where: Boundary Integral Equation Gx - ydirac Gfunction, Representing an impulsive source located at y The solution of above equation 8 is: G x, y ith r x - y 4πr G: Free-Space Green Function for laplacian operator 9

35 v δ x Boundary Integral Equation Multiplying eqn by G and eqn.8 by integrating yields: y φ x dv x v φ, Subtracting and G G φ dv x Recalling the properties of the dirac Gfunction and applying the Gauss theorem finally obtain E φ G y φ y, t G φ ds x n n s Where E is a domain function defined as: E y if if y v y The value of the velocity potential at any point y The solution Iobtained on S v

36 Numerical Solution Approach Boundary Element Method BEM. The boundary S is partitioned into N surface Elements S j. NN i N N o 3. N i, N, N o : the number of the panels on S i, S, S o 4. A set of collection points is chosen on each S j to satisfy eq. 5. Discretization is able to describe complex geometries ith a good level of detail.

37 N j n S j S n j n n x ds y x n G x ds y x G E j j,, φ φ j N j nj j N j nj o j N j o nj i j N j i nj n C B B B n φ φ φ,, x ds y x n G C x ds y x G B n S nj S n nj j j The discretized form of eqn. Where Subscripts indicate evaluation at the corresponding collocation point 3 With Numerical Solution Approach

38 Numerical Solution Approach Equation represents a set of algebraic equations for the N unknons I n and its matrix form is: 4 φ i B i B o o B C φ Where: B and C being the N N matrices collecting in Eq.3 Bar indicating column vector

39 [ ] [ ] [ ] o B B B B B B o i i ˆ ˆ ˆ The column vector containing the boundary conditions Fis partitioned into the three N vectors x i, x o, and x as follos: ˆ ˆ ˆ i o i o i ˆ, ˆ, ˆ Where: are column vectors of length N i, N, N o 5 The coefficient matrices are defined as: Numerical Solution Approach

40 z Assuming a uniform velocity profile for the input and output sections, the discretized form of the eqn. 6 has the form:, A, A, A : o i o i N j j j o o i i s s s Where A A A or N j j j i i o o G A A A 6 7 From eq.6 e obtain: Eq.7:Gives the outflo as a function of inflo and allmotion induced normal velocity at the boundary S Numerical Solution Approach

41 Numerical Solution Approach z The outer flo condition in the case of rigid alls The G or matrix A A i o i G has... - A A j the form... Substituting eqn.7 into eqn.4 I C φ B i i B o or B o G B 8 The motion of the elastic boundary represents the coupling beteen the fluid and the mechanics of the structure.

42 Dynamic of the elastic alls z General form of the motion equation governing the structural dynamics is: Where: ρ U : The all density u Lu L:3D linear elastic operator F x,t: The forcing term due to the fluid flo u: Displacement ith time dependent The forcing term represents the Coupling beteen the all dynamics and the blood flo f

43 Dynamic of the elastic alls ds n p p x dv n p p f e t e t a t a l s l v l l l l l m m ϕ ϕ η δ ϕ ω.., M,..., l 5 4 Equation of the inner solid boundary

44 Fluid- all interaction z Blood flo in arteries is pulsatile z Fundamental frequency induced by the heart beat z We need to obtain the frequency-dependent transfer function of the fluid-structure interacting system, relating the inflo at S i to the velocity potential at arbitrary location inside the arterial district z To accomplish above statement, e need to rite the governing equation in the frequency domain I and S aˆ ˆ C Ω i i o or o φ 7 aˆ eˆ B B B G :Provides the coupling ith the all dynamics equ.8 B ˆ 8

45 Fluid- all interaction z The relationship beteen the velocity potential and the pressure perturbation is given by the Bernoulli theorem eqn.7hich, in the frequency domain becomes: P ˆ P ˆ o S ρφ ˆ 9 eˆ l s ρ S φˆ n. ϕ N N l j S j ds x eˆ Sρ ˆ φn. ϕ ds x sρ ˆ φ n. ϕds x l Substituting into eqn. 5 e obtain: We kno velocity potential along the boundary S e need to computing the forces at those location j j s j 3 3

46 Fluid- all interaction generalized force as a function of the value of the velocity potential on Jth panel eˆ ith E lj s ρ E S φ ˆ j n. ϕ ˆ a sρ S I Ω E ˆ φ ds x Substituting into eqn 8 and solving ith respect to a ~ One obtains as follo: H s ˆ φ H: M N Frequency-dependent matrix transfer function Relating the velocity potential at the collocation points ith the modal amplitude of the all elastic displacement We can also evaluated velocity potential on the sections S i and S o 3 33

47 Fluid- all interaction z To obtain the final form of complete coupled fluid-all dynamics z Relation beteen the all motion and the boundary conditions of the flo z The model representation of displacement as folloing equation In to eqn.9 yields: j u. n d m m xt, a t ϕ x. n a t ϕ x dt m m m m m m j j

48 Fluid- all interaction z The lap lace transform at J th Collocation point ˆ j m m In Vector notation: ˆ s aˆ ϕ x. n m sraˆ m j j Generic entry of the N M R jm n j. ϕ, m, x i S Otherise 34

49 Fluid- all interaction z The relationship beteen vectors φ and X φ ˆ ˆ s srh φ φ ˆ ˆ ˆ ˆ s RH B G B s B B C I o or o i i 35 The final fully coupled system of Equation is

50 Fluid- all interaction z Solution for velocity potential vector ˆ φ I C s BG B Where: T s RHs I C s B G B RH s [ ] Bx iˆi B o xˆ or T s qˆ Frequency-dependent matrix transfer function relating the Knon: Flux q Through S i and S o ith the velocity potential on s T includes the effects of the all dynamics through the matrix H 36

51 Conclusions zpa adventitial mechanics especially in pulm. hypertension still a relatively unexplored area. zmathematical models predicting the ave propagation characteristics of an arterial vessel are of interest for the clinicians

52 COMPUTER SIMULATION OF BLOOD FLOW WITH COMPLIANT WALLS

53 THE FLUID-STRUCTURE INTERACTION ALGORITHM LOOSE COUPLING Load transfer Loads CFD Solver INTERFACE CSD Solver Geometry of interface surfaces and velocity Surface tracking Information exchange for the coupled fluid-structure interaction problem

54 Finite element mesh for fluid domain of the carotid artery bifurcation Blood density U.5 g/cm 3 Kinematic viscosity Q.35 cm /s R e 3

55 Velocity distribution for t/t p.

56 Effective stress distribution at the carotid bifurcation alls, t/t p.35

57 Wall shear stres s field at t/t p. deformable alls

58 Blood velocity field in the carotid artery bifurcation for t/t p.

59 Biomechanical Modeling of the Arterial Fluid-Structure interaction in hemodynamic Mostafa Salehi Fluid-Structure Interaction ASEN 559 April 9 4 Part III

60 Overvie : / Problems to address Why do e need to analysis FSI In the Arterial? z Understanding many aspects of vascular physiology and clinical therapy z Mechanism of disease Hypertension, arteriosclerosis,.. z Materials for treatment z The matching of artificial vascular grafts to host vessels z Assist physicians in positively identifying individuals ho exhibit the development of vascular disease and begin appropriate treatment-long before symptoms or clinical signs appear z Accurate modeling of blood flo

61 Overvie Pulmonary: artery that supplies the lungs ith blood from the heart LPA,RPA

62 Modeling TUBE CONSTRUCTION θ r r i r o Epithelium } Tunica Lamina mucosa propria l p i Tunica muscularis ρ nerve plexus vessel Tunica adventitia or serosa r, T z: Coordinates of the z Artery

63 Fluid-Structure interaction in arterial z The main objective is to get to Relationship beteen :. arterial pressure. Wall deformation 3. Flo field in a large arterial vessel z Attention is focused on the harmonic response of the fluid-all interacting system to a pulsatile inflo predict the ave propagation velocity

64 Fluid-Structure interaction in arterial ƒfrom previous ork z The resulting integral-differential equation is solved numerically by a zero-order boundary element method z Pressure evaluated from the potential through the Bernoulli s theorem z The compliant all is modeled as an elastic shell z The problem is addressed ith a Galerkin approach and expressed in matrix form z Account for coupling beteen the structure Artery and fluid Blood

65 The final fully coupled system of Equation is: [ ] q s T B x B x RH s B B G s C I or o i i ˆ ˆ ˆ ˆ φ Where: s RH B B G s C I s T H: M N Frequency-dependent matrix transfer function relating the velocity Potential at the collocation point ith the model amplitude of the all elastic Displacement. o i ˆ, ˆ, ˆ q : Flux Through S i and So ith the velocity potential on s are column vectors of length N i, N, N o Function of the all velocity j j j n u. B, C: N N Matrices collecting coefficient

66 The hypotheses e are dealing ith are: Incompressible isotropic elastic material for the E, X Wall thickness h<<r << R o s u f R x u R Eh x h kg t h f x u x R Eh t u h ν ν ρ ν ν ρ l R o θ x Straight thin-alled cylindrical vessels X, T r, : cylindrical Coordinates of the artery h The membrane equations governing the structural dynamics are: G s : Shear modulus, k: Timoshenko s shear correction factor f u and f : Forcing terms due to action of the blood flo h: all thickness, E: Young modulus, Xpoisson s ratio U: Displacement in the longitudinal, : Displacement in the Radial direction 37 Elastic all relations

67 Wave propagation o s f R x u R Eh x kg h t h ν ν ρ ] [ R kg E L m kg s s m ν π ρ λ Inviscid fluid model implies f u, so eq.37 is satisfied, to yield: Evaluation of Wave velocity from resonance frequencies With using boundary condition for the elastic displacement, {,t l, t} and general solution of above equation e obtain:

68 Input impedance: To characterize the harmonic behavior of the system is represented by the input impedance of the arterial segment Z Wave propagation ~ p s p s i V s. n i P :Constant reference pressure v i :Incoming velocity The ave propagation velocity similarity ith an equivalent one-dimensional acoustic problem c kc f K γ l k,,... JWavelength of the traveling ave Relationship beteen the propagation velocity of a pressure perturbation and elastic all properties Eh c ρ R o ν { Artery: L. m, radius R o cm E6 ^5 Pa Kg/m^3 all 3 Kg/m^3 Corresponding estimated ave velocity6.976 m/s Agreement ith the physiological value

69 Conclusions zvery complex problem to coupling system Fluids, Structures, interaction for arterial z Models predicting the ave propagation characteristics of an arterial vessel are of interest for the clinicians zpa adventitial mechanics especially in pulmonary hypertension still a relatively unexplored area.

70 Future ork zexperimental RESEARCH ON THE ARTERIAL WALL zdevelopment OF COMPUTER METHODS AND SOFTWARE FOR BLOOD FLOW AND BLOOD VESSELS MODELING

71 THANKS Questions