醫用流體力學

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

2. Hemodynamics is concerned with the forces generated by the heart and the resulting motion of blood through the cardiovascular system. Blood flow in living animal is far from the idealized motion of flow through smooth cylindrical tubes. *Non-homogeneous materials. *Viscous fluid. *Viscoelastic blood vessels. Biophysics of the circulation. Physicists : Mathematicians Physicians : Physiologists

3. Hemodynamics • Physical properties of the heart and blood vessels (Anatomy, Physiology) • The relationship between the above properties to the circulation of blood • Applications of the above results to physiological research or clinical science.

4. Background • The pressure and flow in blood vessels are pulsatile and with periodic waveforms. • Arterial hemodynamics describes the Pressure-Flow relationship in the frequency domain.

5. Laminar vs. turbulent flow:

6. Q Steady flow (Poiseuille’s Law) • Relation between volume flow (Q) and the pressure drop (P) along a tube of length (L) and inner diameter (D) Q P D4 /L • Poiseuille’s experiments (1846) showed that Q = P D4/128 L ; =1.3077 centi-poise Note: Girard (1813), Navier (1827): Q  D3; Hagen (1839): Q  D4.

7. Validity of Poiseuille’s Law in vivo • Newtonian fluid • Laminar flow • No slip at the vascular wall • Steady flow • Cylindrical shape • Rigid wall

8. R2 R1 R3 Re=R1+R2+R3 R1 R2 R3 Vascular resistance (R= P/Q) R= P/Q (electric analogy) R = 128 L/ D4 1/Re=1/R1+1/R2+1/R3 Poiseuille’s Law underestimates the ratio of pressure gradient to the flow in a blood vessel in vivo.

11. Hydraulic energies • Pressure energy (Wp= P ·Vol) • Kinetic energy (Wp= ½Vol · v2) • Gravitational energy (Wg= gh ·Vol) • Total hydraulic energies WT= (P +½v2+ gh ) ·Vol

12. A2 ; V2 A1 ; V1 Bernoulli’s Law (flow through orifices) Q= A1 V1 =A2 V2 • Continuity • Conservation of total hydraulic energy P1 +½ v12+ gh1 = P2 +½ v22+ gh2

13. Implications . if flow Q is constant: an  in radius (area) will result in a  in flow velocity • Resistance = 128L /D4 • Resistance to flow in a single vessel is: • increased with  viscosity and  length • decreased with  diameter to 4th power. • For elliptical cross section

14. Vascular Wall Properties • Law of Laplace (wall tension, T=P D/2) • Circumferential incremental Young’s modulus of a thick walled isotropic elastic cylindrical tube where  is the Poisson ratio, Ro the outer radius, Ri the inner radius, R the radial displacement and P is the pressure change,

15. Kelly (1994) Laurent & Safar (1994) CCA pressure-diameter

16. Arterial elasticity and Pulse Wave velocity • Moen-Korteweg equation • Modified Moen-Korteweg equation (thick wall) (Bergel) (Gow)

17. Pulse Wave Velocity (PWV) in Vena Cava canine’s vein

18. Pulsatile pressure and flow(Electrical Analogy) • Windkessel model • Volume Complicance C=dV/dP, • Resistance R=P/Q • The rate of outflow equals to the volume change, Q=dV/dt • Pressure of windkessel declines exponentially

19. Womersley number for fundamental harmonic in some mammals (Aorta) 2 =  ReSh =  (UD/) (f D/U)

20. Pulsatile pressure and flow(Electrical Analogy) • Noordergraaf (1963) Longitudinal Vascular Impedance :ZL = -(dP/dx)/Q Input Impedance : Zi = P/Q Transverse Impedance : Zw = P/(dQ/dx)

21. Pulsatile pressure and flow(Electrical Analogy) scaling Electrical Hemodynamics Longitudinal Vascular Impedance :ZL = R+jL Input Impedance : Zi = P/Q

22. Analogic Electric System

23. Governing Equations in integral form • Transport of Mass: • Transport of Momentum: • Transport of Energy

24. Human Circulatory System • Fundamental VariablesPressure、 Flow • Geometrical VariablesSize、 Thickness 、 Length、 Curvature • Mechanical PropertiesStiffness 、Visco-Elasticilty

25. Vascular Impedance as an index for arterial occlusion due to atherosclerosis

26. Harmonic Analysis of Pulsatile Flow Waves Harmonic Amplitude of Flow Wave

27. Harmonic Analysis of Pulsatile Pressure Waves

28. Vascular Impedance • Vascular impedance characterize the resultant of interactions of cardiac output with various organs and tissues • Input Impedance: Zi = P / Q • Longitudinal Impedance: Zl= (-dP/dx)/Q (correlation of pressure gradient to the flow) • Transverse Impedance:Zt = P / (-dQ/dx) • (correlation of pressure to flow gradient)

29. Quantification of Input Impedance (Z=P/Q)

30. Aortic Input Impedance • Resonance Frequency • Impedance Matching • Flow Distribution

31. Effects of exercise on arterial pressure-flow waveforms

32. Effects of Exercise on Doppler Spectra

33. Vascular impedance gives the changes in harmonic amplitudes thus provides more information than typical clinical indexes such as PI and RI ! Can it be accessed non-invasively ?  Color Duplex Sonography ! Vascular impedance

34. Non-invasive Impedance Measurements Ultrasound M-Mode

35. Non-invasive Impedance Measurements Ultrasound Doppler

36. M-Mode image processing D D = aD P (Elastic) a=D2/2Eh

37. Doppler Image Processing

38. Pressure-Flow waveforms

39. 頸動脈阻抗頻譜分析結果

40. Resonance

41. Measurement of Mechanical Properties for Blood Vessel and Soft-Tissues Correlation of Waves measured at Two Sites

42. PWV of Arterial Tree

43. Significance of Aortic Impedance clinical risk factors for developing foot ulceration Avolio, Aet al. (1994) Circulation

44. Comparative differences in changes in oscillatory and steady components of arterial hemodynamics in the early stages of cardiac failure in dogsThe importance of the pulsatile arterial function on the heart but also show that these occur before changes in peripheral resistance

45. Major change in the paced dogs is an increase in aortic impedance (i.e, characteristic impedance) average characteristic impedance is shown to increase from 121 to 186 dyne- s- cm-5 (an increase of 54%)

46. Pacing also produced a significant fall in mean arterial pressure a decrease in mean pressure from a baseline of 90 to 75 mm Hg after pacing (a reduction of 17%).

47.  implications in understanding the adaptive changes both in the heart and in the complex arterial load significance of passive effects on the aortic impedance in relation to the changes in arterial compliance

48. a reduction in mean arterial pressure should lead to a decrease in characteristic impedance due to passive effects of distending pressure • vascular compliance may have actually decreased to a much greater degree than that determined by characteristic impedance

49. Active effects of vascular tone or structural changes, with ensuing speculations involving the distribution of angio-tensin II receptors throughout the arterial tree. The increase in aortic characteristicimpedance can be almost totally explained by the passive effect of reduction in aortic diameter due to the decrease in mean pressure A. Avolio (1994)

50. Water hammer formula where Zc (characteristic impedance),  (blood density), c (wave velocity), A (lumen area), and R (radius)