AME 60676 Biofluid & Bioheat Transfer

1. AME 60676Biofluid & Bioheat Transfer 4. Mathematical Modeling

2. Objectives • Applications of hydrostatics and steady flow models to describe blood flow in arteries • Unsteady effects: • pressure pulse propagation through arterial wall • Effects of inertial forces due to blood acceleration/deceleration • Effects of artery distensibility on blood flow

3. Outline • Steady flow considerations and models: • Hydrostatics in circulation • Rigid tube flow model • Application of Bernoulli equation • Unsteady flow models: • Windkessel model for human circulation • Moens-Kortewegrelationship (wave propagation, no viscous effects) • Womersley model for blood flow (wave propagation, viscous effects) • Wave propagation in elastic tube with viscous flow (wave propagation, viscous effects) • Bioheat transfer models: • Pennes equation • Damage modeling

4. 1. Hydrostatics Hydrostatics Rigid tube flow model Bernoulli applications Windkessel model Moens-Korteweg Womersley model Complete model Pennes equation Damage modeling

5. Hydrostatics in the Circulation • Blood pressure in the “lying down” position • Arterial: 100 mmHg • Venous: 2 mmHg • Distal pressure is lower Hydrostatic pressure differences in the circulation “lying down” position Hydrostatics Rigid tube flow model Bernoulli applications Windkessel model Moens-Korteweg Womersley model Complete model Pennes equation Damage modeling

6. Hydrostatics in the Circulation • Blood pressure in the “standing up” position • Head artery: 50 mmHg • Leg artery: 180 mmHg • Head vein: -40 mmHg • Leg vein: 90 mmHg • Pressure differences due to gravitational effects Hydrostatic pressure differences in the circulation “standing up” position Hydrostatics Rigid tube flow model Bernoulli applications Windkessel model Moens-Korteweg Womersley model Complete model Pennes equation Damage modeling

7. Hydrostatics in the Circulation • Bernoulli equation: • Tube of constant cross section: • Effects of pressure on vessels: • Arteries are stiff: pressure does not affect volume • Veins are distensible: pressure causes expansion Hydrostatic pressure differences in the circulation “standing up” position Hydrostatics Rigid tube flow model Bernoulli applications Windkessel model Moens-Korteweg Womersley model Complete model Pennes equation Damage modeling

8. 2. Rigid Tube Flow Model Hydrostatics Rigid tube flow model Bernoulli applications Windkessel model Moens-Korteweg Womersley model Complete model Pennes equation Damage modeling

9. Hagen Poiseuille Model • Assumptions: • incompressible • steady • laminar • circular cross section • From exact analysis: Hydrostatics Rigid tube flow model Bernoulli applications Windkessel model Moens-Korteweg Womersley model Complete model Pennes equation Damage modeling

10. Hagen Poiseuille Model • Assumptions: • incompressible • steady • laminar • circular cross section • From control volume analysis: Control volume Hydrostatics Rigid tube flow model Bernoulli applications Windkessel model Moens-Korteweg Womersley model Complete model Pennes equation Damage modeling

11. Hagen Poiseuille Model • Validity considerations • Newtonian fluid: reasonable • Casson model: linear at large shear rate • Laminar flow: reasonable • Average flow: Re=1500 (< Recr=2100) • Peak systole: Re = 5100 • Blood vessel : compliance • Flow measurements: no evidence of sustained turbulence • No slip at vascular wall: reasonable • Endothelial cell lining Hydrostatics Rigid tube flow model Bernoulli applications Windkessel model Moens-Korteweg Womersley model Complete model Pennes equation Damage modeling

12. Hagen Poiseuille Model • Validity considerations • Steady flow: not valid for most of circulatory system • Pulsatile in arteries • Cylindrical shape: not valid • Elliptical shape (veins, pulmonary arteries) • Taper (most arteries) • Rigid wall: not valid • Arterial wall distends with pulse pressure • Fully developed flow: not valid • Finite length needed to attain fully developed flow • Branching, curved walls Hydrostatics Rigid tube flow model Bernoulli applications Windkessel model Moens-Korteweg Womersley model Complete model Pennes equation Damage modeling

13. Blood Vessel Resistance • On time-average basis: • p: time-averaged pressure drop (mmHg) • Q: time-averaged flow rate (cm3/s) • R: resistance to blood flow in segment (PRU, peripheral resistance unit) Hydrostatics Rigid tube flow model Bernoulli applications Windkessel model Moens-Korteweg Womersley model Complete model Pennes equation Damage modeling

14. Blood Vessel Resistance R1 R2 R3 • Series connection: • Parallel connection: R1 R2 R3 Hydrostatics Rigid tube flow model Bernoulli applications Windkessel model Moens-Korteweg Womersley model Complete model Pennes equation Damage modeling

15. Transition Flow in Pipes • Entrance • V=0 at wall • Velocity gradient in radial direction • Downstream • Fluid adjacent to wall is retarded • Core fluid accelerates • Viscous effects diffuse further into center region dominated by inertial effects region dominated by viscous effects U parabolic velocity profile Entrance region Fully developed flow region Hydrostatics Rigid tube flow model Bernoulli applications Windkessel model Moens-Korteweg Womersley model Complete model Pennes equation Damage modeling

16. BL Thickness and Entrance Length • Balance inertial force and viscous force: • Entrance length definition: • Re > 50: • Re  0:  k = 0.06 Hydrostatics Rigid tube flow model Bernoulli applications Windkessel model Moens-Korteweg Womersley model Complete model Pennes equation Damage modeling

17. 3. Application of Bernoulli Equation Hydrostatics Rigid tube flow model Bernoulli applications Windkessel model Moens-Korteweg Womersley model Complete model Pennes equation Damage modeling

18. Bernoulli Equation • Assumptions: • Steady • Inviscid • Incompressible • Along a streamline Hydrostatics Rigid tube flow model Bernoulli applications Windkessel model Moens-Korteweg Womersley model Complete model Pennes equation Damage modeling

19. Stenosis δ • Narrowing of artery due to: • Fatty deposits • Atherosclerosis • Effects of narrowing: • p1, V1, V2: known  a2 a1 p2, V2 p1, V1 Hydrostatics Rigid tube flow model Bernoulli applications Windkessel model Moens-Korteweg Womersley model Complete model Pennes equation Damage modeling

20. Stenosis δ • Arterial flutter • Low pressure at contraction • Complete obstruction of vessel under external pressure a2 a1 p2, V2 p1, V1 • Decrease in flow velocity • Increase in pressure • Vessel reopening (cycle) Hydrostatics Rigid tube flow model Bernoulli applications Windkessel model Moens-Korteweg Womersley model Complete model Pennes equation Damage modeling

21. Aneurysm • Definition: Arterial wall bulge at weakening site, resulting in considerable increase in lumen cross-section • Characteristics: • Elastase excess in blood • Decrease in flow velocity • Limited increase in pressure (<5 mmHg) • Significant increase in pressure under exercise • Increase in wall shear stress • Bursting of vessel Hydrostatics Rigid tube flow model Bernoulli applications Windkessel model Moens-Korteweg Womersley model Complete model Pennes equation Damage modeling

22. Heart Valve Stenoses • Flow through a nozzle • Flow separation  recirculation region • Fluid in core region accelerates • Formation of a contracted cross section: vena contracta Cd: discharge coefficient (function of nozzle, tube, throat geometries) Hydrostatics Rigid tube flow model Bernoulli applications Windkessel model Moens-Korteweg Womersley model Complete model Pennes equation Damage modeling

23. Heart Valve Stenoses • Effective orifice area: • Gorlin equations (clinical criteria for surgery): Q: mean flow rate (CO) For aortic valve: Q: mean systolic flow rate AVA: aortic valve area (cm2) MVA: mitral valve area (cm2) MSF: mean systolic flow rate (cm3/s) MDF: mean diastolic flow rate (cm3/s) p: mean pressure drop across valve (mmHg) Hydrostatics Rigid tube flow model Bernoulli applications Windkessel model Moens-Korteweg Womersley model Complete model Pennes equation Damage modeling

24. Heart Valve Stenoses • Effects of flow unsteadiness and viscosity: temporal acceleration convective acceleration viscous dissipation + + p = Young, 1979  based on mean values based on peak-systolic values Hydrostatics Rigid tube flow model Bernoulli applications Windkessel model Moens-Korteweg Womersley model Complete model Pennes equation Damage modeling

25. 4. Windkessel Models for Human Circulation Hydrostatics Rigid tube flow model Bernoulli applications Windkessel model Moens-Korteweg Womersley model Complete model Pennes equation Damage modeling

26. Windkessel Theory • Simplified model • Arterial system modeled as elastic storage vessels • Arteries = interconnected tubes with storage capacity Unsteady flow due to pumping of heart Steady flow in peripheral organs Attenuation of unsteady effects due to vessel elasticity Hydrostatics Rigid tube flow model Bernoulli applications Windkessel model Moens-Korteweg Womersley model Complete model Pennes equation Damage modeling

27. Windkessel Theory Definitions: • Inflow: fluid pumped intermittently by ventricular ejection • Outflow: calculated based on Poiseuille theory inflow outflow pV RS Q(t) p(t), V(t), Di Windkessel chamber Hydrostatics Rigid tube flow model Bernoulli applications Windkessel model Moens-Korteweg Womersley model Complete model Pennes equation Damage modeling

28. Windkessel Solution • Pressure pulse solution • Systole (0 < t < ts): • Diastole (ts < t < T): • Stroke volume p0: pressure at t=0 pT: pressure at t=T Windkessel (left) vs. actual (right) pressure pulse Hydrostatics Rigid tube flow model Bernoulli applications Windkessel model Moens-Korteweg Womersley model Complete model Pennes equation Damage modeling

29. Windkessel Theory Summary • Advantages: • Simple model • Prediction of p(t) in arterial system • Limitations: • Model assumes an instantaneous pressure pulse propagation (time for wave transmission is neglected) • Global model does not provide details on structures of flow field Hydrostatics Rigid tube flow model Bernoulli applications Windkessel model Moens-Korteweg Womersley model Complete model Pennes equation Damage modeling

30. 5. Moens-Kortweg relationship Hydrostatics Rigid tube flow model Bernoulli applications Windkessel model Moens-Korteweg Womersley model Complete model Pennes equation Damage modeling

31. Wave Propagation Characteristics • Speed of transmission depends on wall elastic properties • Pressure pulse: • depends on wall/blood interactions • Changes shape as it travels downstream due to interactions between forward moving wave and waves reflected at discontinuities (branching, curvature sites)  Need for model of wave propagation speed Hydrostatics Rigid tube flow model Bernoulli applications Windkessel model Moens-Korteweg Womersley model Complete model Pennes equation Damage modeling

32. Moens-Korteweg Relationship • Speed of pressure wave propagation through thin-walled elastic tube containing an incompressible, inviscid fluid • Relationship accounts for: • Fluid motion • Vessel wall motion Hydrostatics Rigid tube flow model Bernoulli applications Windkessel model Moens-Korteweg Womersley model Complete model Pennes equation Damage modeling

33. Problem Statement h r Vr(r, z, t) R flow z Vz(r, z, t) Infinitely long, thin-walled elastic tube of circular cross-section Hydrostatics Rigid tube flow model Bernoulli applications Windkessel model Moens-Korteweg Womersley model Complete model Pennes equation Damage modeling

34. Derivation Outline Equations of fluid motion in infinitely long, thin-walled elastic tube of circular cross section Equations of vessel wall motion (inertial force neglected on wall) Equations of vessel wall motion (with inertial force on wall) Simplified Moens-Korteweg relationship Moens-Korteweg relationship Hydrostatics Rigid tube flow model Bernoulli applications Windkessel model Moens-Korteweg Womersley model Complete model Pennes equation Damage modeling

35. Simplified Moens-Korteweg relationship • Reduced Navier-Stokes equations: • Inviscid flow approximation: Hydrostatics Rigid tube flow model Bernoulli applications Windkessel model Moens-Korteweg Womersley model Complete model Pennes equation Damage modeling

36. Moens-Korteweg relationship • Tube equation of motion: • Coupling with fluid motion (without inertial effects): where Hydrostatics Rigid tube flow model Bernoulli applications Windkessel model Moens-Korteweg Womersley model Complete model Pennes equation Damage modeling

37. Moens-Korteweg relationship • Tube equation of motion: • Coupling with fluid motion (with inertial effects): where Hydrostatics Rigid tube flow model Bernoulli applications Windkessel model Moens-Korteweg Womersley model Complete model Pennes equation Damage modeling

38. Experimental vs. Theoretical c0 Hydrostatics Rigid tube flow model Bernoulli applications Windkessel model Moens-Korteweg Womersley model Complete model Pennes equation Damage modeling

39. 6. Womersley model for blood flow Hydrostatics Rigid tube flow model Bernoulli applications Windkessel model Moens-Korteweg Womersley model Complete model Pennes equation Damage modeling

40. Problem Statement h r Vr(r, z, t) R flow z Vz(r, z, t) Infinitely long, thin-walled elastic tube of circular cross-section Hydrostatics Rigid tube flow model Bernoulli applications Windkessel model Moens-Korteweg Womersley model Complete model Pennes equation Damage modeling

41. Problem Assumptions • Flow assumptions: • 2D • Axisymmetric • No body force • Local acceleration >> convective acceleration • Tube assumptions: • Rigid tube • No radial wall motion ( no radial fluid velocity) Hydrostatics Rigid tube flow model Bernoulli applications Windkessel model Moens-Korteweg Womersley model Complete model Pennes equation Damage modeling

42. Equation of Motion • Pressure gradient: • Axial flow velocity: • Axial flow velocity magnitude: where: Womersley number Hydrostatics Rigid tube flow model Bernoulli applications Windkessel model Moens-Korteweg Womersley model Complete model Pennes equation Damage modeling

43. Examples of Womersley Number Hydrostatics Rigid tube flow model Bernoulli applications Windkessel model Moens-Korteweg Womersley model Complete model Pennes equation Damage modeling

44. Flow Solution • Flow solution: where: : Bessel function of order k Hydrostatics Rigid tube flow model Bernoulli applications Windkessel model Moens-Korteweg Womersley model Complete model Pennes equation Damage modeling

45. Flow Solution Hydrostatics Rigid tube flow model Bernoulli applications Windkessel model Moens-Korteweg Womersley model Complete model Pennes equation Damage modeling

46. Application • Flow rate calculation for complex (non-sinusoidal) pulsatile pressure gradients Hydrostatics Rigid tube flow model Bernoulli applications Windkessel model Moens-Korteweg Womersley model Complete model Pennes equation Damage modeling

47. Flow Solution • Time history of the axial velocity profile (first 4 harmonics) • Characteristics: • Hagen-Poiseuille parabolic profile never obtained during the cardiac cycle • Presence of viscous effects near the wall makes the flow reverse more easily than in the core region • Main velocity variations along the tube cross section are produced by the low-frequency harmonics • High-frequency harmonics produce a nearly flat profile due to absence of viscous diffusion Hydrostatics Rigid tube flow model Bernoulli applications Windkessel model Moens-Korteweg Womersley model Complete model Pennes equation Damage modeling

48. 7. Complete model:Wave propagation in elastic tube with viscous flow Hydrostatics Rigid tube flow model Bernoulli applications Windkessel model Moens-Korteweg Womersley model Complete model Pennes equation Damage modeling

49. Elastic Tube Equations of Motion Stresses on a tube element Hydrostatics Rigid tube flow model Bernoulli applications Windkessel model Moens-Korteweg Womersley model Complete model Pennes equation Damage modeling

50. Elastic Tube and Fluid Stresses • Tube stresses (from Hooke’s law) • Fluid stresses (cylindrical coordinates) Hydrostatics Rigid tube flow model Bernoulli applications Windkessel model Moens-Korteweg Womersley model Complete model Pennes equation Damage modeling