1 / 87

醫用流體力學

醫用流體力學. Arterial Fluid Dynamics 邵耀華 台灣大學應用力學研究所. Physiological Fluid Dynamics. Evolution of Arterial Pressure Away from the heart. Systemic Arteries. Conduct blood flow from Left ventricle (LV) to peripheral organs Aortic valve  Aortic arch (180° turn) Geometry changes :Tapering

darren
Download Presentation

醫用流體力學

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. 醫用流體力學 Arterial Fluid Dynamics 邵耀華 台灣大學應用力學研究所 Physiological Fluid Dynamics

  2. Evolution of Arterial Pressure Away from the heart

  3. Systemic Arteries • Conduct blood flow from Left ventricle (LV) to peripheral organs • Aortic valve  Aortic arch (180° turn) • Geometry changes :Tapering • Geometry changes : Branching • Mechanical properties changes

  4. Radius and wall thickness of human arteries change with age

  5. Incremental Elastic Modulus of human arteries change with age

  6. Stress-Strain relations of rabbit’s thoracic aorta

  7. Fluid Mechanics of Elastic Conduct • Mass Conservation • Conservation of momentum • Conservation of energy

  8. Background • Fundamental VariablesPressure、 Flow • Geometrical VariablesSize、 Thickness 、 Length、 Curvature • Mechanical Properties Stiffness 、Visco-Elasticity

  9. Equations of Viscous Pipe Flow • Consider a conduct filled with incompressible fluid of density  and pressure p, let u be the only non-zero velocity component

  10. Poiseuille’s Law (1840) • Assume steady flow, u= u(r ) with no body forces, the equation of motion

  11. Laminar Poiseuillean flow • Rate of flow through the tube • Mean velocity of flow • Shear stress at the wall

  12. Laminar Poiseuillean flow • Skin friction • Shear stress in terms of skin friction

  13. Implication of Poiseuille’s Law • Q is proportional to the fourth power of the radius. • Q is directly proportional to the pressure difference. • Q is inversely proportional to the length of the tube. • If the arteries becomes constricted, the bloodpressure requires to supply the blood flow adequately will risesubstantially,leading to the state of hypertension.

  14. Optimum design of Blood Vessel Bifurcation (Poiseuille’s formula) For a given pressure drop, 1% change in vessel radius results in a 4% changes in flow Murray (1926) Rosen (1967) Work done Metabolism Energy loss

  15. Minimum cost function for optimum vessel configuration With respect to radiusa  the optimum radius The optimum vessel radius is proportional flow to the 1/3 power, and

  16. Optimum vessel bifurcation that with minimum cost function Minimize P at the bifurcation point B An optimum location B would be for arbitrary movements of B.

  17. Let B displaced along A-B direction first The optimum is obtained when

  18. Again, let B displaced in the C-B direction The optimum is obtained when Similarly, displaced B along D-B direction, we find

  19. Similarly, displaced B along D-B direction, we find The continuity equation gives We find which is often referred to as Murray’s Law 37.5°

  20. Let ao denotes the radius of the aorta, and assume equal bifurcation in all generation If the capillary blood vessel has a radius of 5 um and the radius of the aorta is 1.5 cm. We find n=30. The total number of blood vessel is about 230109. Note: in fact arteries rarely bifurcation symmetrically (a1=a2). For human, only one symmetric bifurcation. For dog, there are none.

  21. Pulsatile Blood Flow • Consider pulsatile flow in a circular vessel, p=p(x, t) and u = u(r, t) • For a sinusoidal flow

  22. Pulsatile Blood Flow(2) • The general solution of the ODE in the form involves Bessel functions of complex arguments U(r=a)=0 (non-slip) U(r=0)=finite

  23. Pulsatile Blood Flow(3) • Introducing Womersley number  • As 0, the velocity profile becomes parabolic. • As , viscosity is negligible U(r)=-i P/.

  24. Analysis of Blood Flow using Elastic Theory • From Poiseuille’s Law the flux is proportional to the pressure difference (p1-p2). However, the blood flow in veins are remarkably non-linear. • The flow in elastic conduct gradually attains a maximum value as the pressure difference increases and then on longer increases.

  25. Arterial Flow in Elastic Tube • Axial velocity, v • Lumen area, S

  26. Pressure-Diameter relationship • Let T denotes the tension of the blood vessel per unit thickness, wall thickness h, vessel radius a • Let ro be the radius of zero tension state, the Hooke’s Law gives elastic constant E as

  27. Poiseuille’s flow in elastic tube • Consider steady flow in elastic tube of length L, assume the tube is long and the pressure is function of axial coordinate z, let P1 and P2 denote the inlet and outlet pressure and the external pressure surrounding the tube is P0 • Assume the flow through the tube obey Poiseuille’s law, the flow becomes

  28. Transmission of Pulse wave (Velocity) in elastic tube • Consider inviscid and incompressible fluid flow in elastic tube of lumen area A, • By linearizing the equations

  29. Transmission of Pulse wave (Velocity) in elastic tube • Combining the continuity and momentum equations, • The wave equations Pulse Wave Velocity (PWV)

  30. Analysis of Aortic Diastolic and Systolic Pressure Waveforms • Constitutive relationship between aortic volume and pressure where K is the volume elasticity of the aorta, and V0 is the end-systolic volume. • If the aorta is very soft (K is very small), let I(t) and Q(t) denote the inflow and outflow rates, we have

  31. During diastole, the aortic valve is closed and there is no flow into the aorta. Hence I(t) =0. where is a non-invasive measure aortic volume elasticity. Let Td be the duration of diastolic phase, the aortic pressure (Pd) at the end of this phase or just prior to ejection is given by • The volume elasticity that depicts the exponential drop of aortic pressure is given by

  32. Reynolds #; Strouhal #; Womersley • Reynolds number • Strouhal number • Womersley number

  33. Flows under the action of Oscillating pressure gradient

  34. Wave propagation in Blood Vessel • Pulse wave propagation in arteries • A(x, t) depends on transmural pressure, Here c is the wave propagation velocity.

  35. For thin walled elastic tube: • Consider the elasticity of the tube, arterial diameter  blood pressure For a thick walled elastic tube:

  36. Balance of Force in Arterial Wall

  37. Resonant vibration of flow in a circular tube • When the tube length is equal to the half wave length • This is called the fundamental frequency of the natural vibration. • Hemodynamics : Effects of Frequency on the Pressure-flow relationship of Arterial tree

  38. Boundary conditions

  39. Pressure-Flow

  40. Mean Velocity Profile (Dog Aorta)

  41. Velocity waveform at the upper descending aorta of a dog

  42. Effect of Womersly number on the velocity distribution

  43. Blood PressureEvolution

  44. Effect of sinusoidal pressure wave speed of various frequencies on the instantaneous aortic pressure

  45. Distribution of Atherosclerotic Sites in Human

More Related