Concluding Remarks

Concluding Remarks Part I chapters give the necessary knowledge associated with cell behavior, especially when the cell reacts to various stimuli, particularly mechanical stresses. The magnitude and direction of these mechanical stresses applied by the flowing blood on the wetted surface of the endothelium as well as within the vessel wall varies during the cardiac cycle. The heart generates unsteady flows in a network of blood vessels characterized by complicated architecture and variable geometry both in space and time. Vessel geometry varies over short distances. The vascular network of curved blood vessels is composed of successive geometrical singularities, mainly branchings. Three-dimensional blood flow is thereby developing, being conveyed in en- trance length (App. C.7). Moreover, blood vessels are deformable. Changes in transmural pressure (the pressure difference between the pressure at the wetted surface of the lumen applied by the moving blood on the vessel wall and the pressure at the external wall side, which depends on the activity on the neighbor organs) can also affect the shape of the vessel cross-section, especially when it becomes negative. More generally, the change in cross-section shape is due to taper, possible prints of adjacent organs with more or less progressive constriction and enlargment, and adaptation to branching (transition zone). These changes can induce three-dimensional blood motions displayed by vir- tual transverse currents, even if the vessel is considered straight (Part II). Local changes in the direction of stress components are caused by flow separation and flow reversal during the cardiac cycle. Flow separation is set by an adverse pressure gradient when inertia forces and blood vorticity are high enough, especially in branching vascular segments. Due to the time-dependent feature of blood flow, the flow separation region spreads over a variable length during the cardiac cycle and can move. The location and variable size of the flow separation region depends on the flow distribution between the branches, which can vary during the cardiac cycle. Flow reversal occurs during the dias- tole of the left ventricle in elastic arteries, such as the aorta, and most of the muscular arteries, such as brachial and femoral arteries (but not in the carotid arteries). Flow reversal can be observed either in a region near the wall, more 490 Concluding Remarks or less wide with respect to the position of the local center of vessel curvature, or in the entire lumen. This book also provides a survey of flow modeling and simulations in the cardiovascular system. The basic knowledge of fluid mechanics necessary to understand blood flow behavior is given in part II. However, the Navier-Stokes equations, the set of partial differential equations (PDE) developed in the framework of continuum mechanics that governs the fluid flow is introduced in the following. Like any mathematical model, Navier-Stokes equations link the physical quantities that are involved in the studied process (both independent and dependent variables, time, space coordinates, temperature, pressure, velocity, energy, cross-section area, etc.) using equations which describe the system behavior. The mathematical model has a value only when realizable measurements validate the model, i.e., the model generates verifiable predictions. The Navier-Stokes equations correspond to the mathematical formulation of basic physical principles, especially the conservation of mass and momentum. Any fluid motion is indeed characterized by three principles: (1) the matter is neither lost nor created; (2) the variation rate of momentum of any fluid part is equal to the resulting forces (sum of all applied forces) on this fluid part; and (3) the energy is neither lost nor created. Conservation Equations Mass and momentum conservation equations are obtained from the analysis of the evolution of involved quantities in an infinitesimal control volume, the so-called fluid particle. The following differential operators are used: (1) the ∇ gradient operator =(∂/∂x1,∂/∂x2,∂/∂x 3), (2) the divergence operator ∇· ∇2 3 2 2 ∇ , and (3) the Laplace operator = i=1 ∂ /∂xi . The gradient p of scalar p is a vector of component (∇p)i = ∂ip. The gradient ∇v of vector v is a second order tensor with component (∇v)ij = ∂ivj. The first basic postulate states that matter is neither destroyed nor created, meaning that any matter finite volume neither disappers nor becomes infinite. The second basic postulate states that matter is impenetrable; any matter element does not share the same location with another element (continuity axiom). ∂ The elements in presence are: (1) the mass change rate ρdV,and ∂t Ve (2) the mass flux across the surface Γe of the infinitesimal control volume 1 Ve, of unit normal n outward oriented − ρv · n dA. Using the divergence Γe theorem (Gauss formula),2 the equation of mass conservation, or the so-called equation of continuity, is obtained: 1 · = 3 The scalar product of two vectors provides a scalar: v n i=1 vini. 2 ∇·(ρv) dV = (ρv) · n dA. V Γ Concluding Remarks 491 ∂tρ + ∇·(ρv)=0. (10.2) The equation of momentum conservation of a moving fluid of mass density ρ, of dynamic viscosity µ, of kinematic viscosity ν = µ/ρ, which is conveyed with a velocity v(x,t) (x: Eulerian position, t: time) is obtained from the balance of surface and remote forces applied to the fluid particle. The elements in presence are: (1) the body forces f dV , (2) the pressure Ve forces − pn dA, (3) the viscous forces Cn dA, and (4) the inertia forces Γe Γe Dv Dv 3 4 ρ dV ,where = ∂tv +(v ·∇)v. Using the Gauss formulas, the Ve Dt Dt momentum conservation is obtained: ρ(∂tv +(v ·∇)v)=f −∇p + ∇·C. (10.3) The Navier–Stokes equations governing an unsteady flow of an incompressible (ρ constant) fluid in the absence of body forces are then given by: ∇· 3 The divergence ρv of the vector ρv is the scalar i=1 ∂iρvi. 3 The convective inertia term (v ·∇)v is a vector of component i magnitude given by: 3 (v ·∇)v =( vj ∂j )vi. i j=1 The convective inertia term can also be expressed either in Cartesian coordinates by the contraction product of the velocity vector and of the velocity gradient tensor v∇v, or by the divergence of the tensorial product of the velocity vector by itself ∇·(v ⊗ v) when the fluid is incompressible (∇·v =0; v ⊗ w being ( ⊗ ) = ∇·( ⊗ ) a second order tensor of component v w ij viwj , and v v a vector 3 = 3 + 3 of component j i=1 ∂ivivj i=1 vi∂ivj vj i=1 ∂ivi). The convective inertia term is also equal to: ∇(v · v)/2 − v × (∇×v). The velocity curl has components (∇×v)eî = ∂j vk − ∂kvj , i = j = k using circular permutation. 4 ∇pdV = pn dA. V Γ (∇·T) dV = Tn dA. V Γ The divergence ∇·T of tensor T is the vector of component: 3 (∇·T)eî = ∂iTij eî. i=1 492 Concluding Remarks ∇ · v =0, 2 ρ(vt +(v · ∇)v)=−∇pi + µ∇ v. (10.4) The Navier–Stokes equations are associated with initial and boundary conditions be solved. Dv 2 When the inertia term is neglected, i.e., the diffusive term µ∇ v Dt is predominant, the momentum conservation equation is called the Stokes equation: 2 ν∇ v − ∇pi/ρ =0. (10.5) The matrix A of the coefficients {aij } of the second order derivatives ∂i∂jv (i, j =1, 2, 3) is diagonal. det(λI − A)=(λ − ν)3 =0. The eigenvalue λ = ν is positive and the Stokes equation belongs to the elliptic equation class of the set of partial derivative equations. Any point of the fluid domain receives information from all neighboring points (all directions) in the context of diffusive process. Downstream events thus intervene. 2 Dv When the viscous term µ∇ v is neglected, i.e., becomes the major Dt equation term, the momentum conservation equation is called the Euler equation: Dv + ∇pi/ρ =0. (10.6) Dt The Euler equation belongs to hyperbolic equation class. Informations travel in one direction given by the velocity field. The downstream region does not influence flow. A predominant convective inertia term (v · ∇)v) brings a hyperbolic feature, even in the presence of temporal inertia. The time inertia ∂tv brings a parabolic feature. Flow Boundary Conditions Boundary conditions, as well as initial conditions, are required to solve the set of partial derivative equations. The boundary conditions are applied at the computational domain surface, i.e., when carrying out flow computations at the wall, entry and exits of the vasculature model (Tables 10.8, 10.9, and 10.9). The boundary of the fluid domain Ω is partioned into three surfaces: the 5 entry cross-section Γ1, the exit cross-sections Γ2 and the vessel wall Γ3. The classical no-slip condition is applied to the rigid vessel wall. A time-dependent uniform injection velocity vΓ(t) can be prescribed, at least, at the inlet, which is obtained from the Fourier transform of in vivo MR flow signals.6 At the 5 Γ3 is the fluid-structure interface, i.e., the moving boundary when the deformable wall is taken into account. 6 Most often the spatial resolution of in vivo velocity measurements is not high enough to provide velocity distribution at vessel ends, especially at the domain Concluding Remarks 493 outlet cross-sections,7 either the pressure is set to zero or, better, a stress-free condition is commonly applied. Table 10.8. A first family of boundary conditions for the Navier-Stokes equation. The boundary Γ of the fluid domain Ω is partioned into three parts Γi.

Load more