1. AME 60676 Biofluid & 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-Korteweg relationship (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. Damage modelingPennes equationComplete model 1. Hydrostatics Womersley modelMoens-KortewegWindkessel modelBernoulli applicationsRigid tube flow model

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 Damage modelingPennes equationComplete modelWomersley modelMoens-KortewegWindkessel modelBernoulli applicationsRigid tube flow model

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 Damage modelingPennes equationComplete modelWomersley modelMoens-KortewegWindkessel modelBernoulli applicationsRigid tube flow model

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 Damage modelingPennes equationComplete modelWomersley modelMoens-KortewegWindkessel modelBernoulli applicationsRigid tube flow model

8. 2. Rigid Tube Flow Model Damage modelingPennes equationComplete modelWomersley modelMoens-KortewegWindkessel modelBernoulli applicationsHydrostatics

9. Hagen Poiseuille Model Assumptions: – incompressible – steady – laminar – circular cross section From exact analysis: Damage modelingPennes equationComplete modelWomersley modelMoens-KortewegWindkessel modelBernoulli applicationsHydrostatics

10. Hagen Poiseuille Model Assumptions: – incompressible – steady – laminar – circular cross section From control volume analysis: Control volume Damage modelingPennes equationComplete modelWomersley modelMoens-KortewegWindkessel modelBernoulli applicationsHydrostatics

11. Hagen Poiseuille Model Validity considerations – Newtonian fluid: reasonable Casson model: linear at large shear rate – Laminar flow: reasonable Average flow: Re=1500 (< Re cr =2100) Peak systole: Re = 5100 – Blood vessel : compliance – Flow measurements: no evidence of sustained turbulence – No slip at vascular wall: reasonable Endothelial cell lining Damage modelingPennes equationComplete modelWomersley modelMoens-KortewegWindkessel modelBernoulli applicationsHydrostatics

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 Damage modelingPennes equationComplete modelWomersley modelMoens-KortewegWindkessel modelBernoulli applicationsHydrostatics

13. Blood Vessel Resistance On time-average basis: –  p : time-averaged pressure drop ( mmHg ) – Q : time-averaged flow rate ( cm 3 /s ) – R : resistance to blood flow in segment ( PRU, peripheral resistance unit) Damage modelingPennes equationComplete modelWomersley modelMoens-KortewegWindkessel modelBernoulli applicationsHydrostatics

14. Blood Vessel Resistance Series connection: Parallel connection: R3R3 R3R3 R2R2 R2R2 R1R1 R1R1 R1R1 R2R2 R3R3 Damage modelingPennes equationComplete modelWomersley modelMoens-KortewegWindkessel modelBernoulli applicationsHydrostatics

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 U region dominated by viscous effects region dominated by inertial effects parabolic velocity profile Entrance region Fully developed flow region Damage modelingPennes equationComplete modelWomersley modelMoens-KortewegWindkessel modelBernoulli applicationsHydrostatics

16. BL Thickness and Entrance Length Balance inertial force and viscous force: Entrance length definition: – Re > 50: – Re  0:  k = 0.06 Damage modelingPennes equationComplete modelWomersley modelMoens-KortewegWindkessel modelBernoulli applicationsHydrostatics

17. 3. Application of Bernoulli Equation Damage modelingPennes equationComplete modelWomersley modelMoens-KortewegWindkessel modelRigid tube flow modelHydrostatics

18. Bernoulli Equation Assumptions: – Steady – Inviscid – Incompressible – Along a streamline Damage modelingPennes equationComplete modelWomersley modelMoens-KortewegWindkessel modelRigid tube flow modelHydrostatics

19. Stenosis Narrowing of artery due to: – Fatty deposits – Atherosclerosis Effects of narrowing: – p 1, V 1, V 2 : known  a1a1 p 1, V 1 δ p 2, V 2 a2a2 Damage modelingPennes equationComplete modelWomersley modelMoens-KortewegWindkessel modelRigid tube flow modelHydrostatics

20. Stenosis Arterial flutter – Low pressure at contraction – Complete obstruction of vessel under external pressure a1a1 p 1, V 1 δ p 2, V 2 a2a2 – Decrease in flow velocity – Increase in pressure – Vessel reopening (cycle) Damage modelingPennes equationComplete modelWomersley modelMoens-KortewegWindkessel modelRigid tube flow modelHydrostatics

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 Damage modelingPennes equationComplete modelWomersley modelMoens-KortewegWindkessel modelRigid tube flow modelHydrostatics

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 C d : discharge coefficient (function of nozzle, tube, throat geometries) Damage modelingPennes equationComplete modelWomersley modelMoens-KortewegWindkessel modelRigid tube flow modelHydrostatics

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 (cm 2 ) MVA : mitral valve area (cm 2 ) MSF : mean systolic flow rate (cm 3 /s) MDF : mean diastolic flow rate (cm 3 /s)  p : mean pressure drop across valve (mmHg) Damage modelingPennes equationComplete modelWomersley modelMoens-KortewegWindkessel modelRigid tube flow modelHydrostatics

24. Heart Valve Stenoses Effects of flow unsteadiness and viscosity: Young, 1979  p = temporal acceleration convective acceleration viscous dissipation ++  based on mean valuesbased on peak-systolic values Damage modelingPennes equationComplete modelWomersley modelMoens-KortewegWindkessel modelRigid tube flow modelHydrostatics

25. 4. Windkessel Models for Human Circulation Damage modelingPennes equationComplete modelWomersley modelMoens-KortewegBernoulli applicationsRigid tube flow modelHydrostatics

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 Damage modelingPennes equationComplete modelWomersley modelMoens-KortewegBernoulli applicationsRigid tube flow modelHydrostatics

27. Windkessel Theory Definitions: Inflow: fluid pumped intermittently by ventricular ejection Outflow: calculated based on Poiseuille theory inflowoutflow Q(t) p(t), V(t), D i RSRS Windkessel chamber pVpV Damage modelingPennes equationComplete modelWomersley modelMoens-KortewegBernoulli applicationsRigid tube flow modelHydrostatics

28. Windkessel Solution Pressure pulse solution – Systole ( 0 < t < t s ): – Diastole ( t s < t < T ): Stroke volume p 0 : pressure at t=0 p T : pressure at t=T Windkessel (left) vs. actual (right) pressure pulse Damage modelingPennes equationComplete modelWomersley modelMoens-KortewegBernoulli applicationsRigid tube flow modelHydrostatics

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 Damage modelingPennes equationComplete modelWomersley modelMoens-KortewegBernoulli applicationsRigid tube flow modelHydrostatics

30. 5. Moens-Kortweg relationship Damage modelingPennes equationComplete modelWomersley modelWindkessel modelBernoulli applicationsRigid tube flow modelHydrostatics

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 Damage modelingPennes equationComplete modelWomersley modelWindkessel modelBernoulli applicationsRigid tube flow modelHydrostatics

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 Damage modelingPennes equationComplete modelWomersley modelWindkessel modelBernoulli applicationsRigid tube flow modelHydrostatics

33. Problem Statement z r R h V z (r, z, t) V r (r, z, t) flow Infinitely long, thin-walled elastic tube of circular cross-section Damage modelingPennes equationComplete modelWomersley modelWindkessel modelBernoulli applicationsRigid tube flow modelHydrostatics

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 (inertial force neglected on wall) Simplified Moens-Korteweg relationship Equations of vessel wall motion (with inertial force on wall) Equations of vessel wall motion (with inertial force on wall) Moens-Korteweg relationship Damage modelingPennes equationComplete modelWomersley modelWindkessel modelBernoulli applicationsRigid tube flow modelHydrostatics

35. Simplified Moens-Korteweg relationship Reduced Navier-Stokes equations: Inviscid flow approximation: Damage modelingPennes equationComplete modelWomersley modelWindkessel modelBernoulli applicationsRigid tube flow modelHydrostatics

36. Moens-Korteweg relationship Tube equation of motion: Coupling with fluid motion (without inertial effects): where Damage modelingPennes equationComplete modelWomersley modelWindkessel modelBernoulli applicationsRigid tube flow modelHydrostatics

37. Moens-Korteweg relationship Tube equation of motion: Coupling with fluid motion (with inertial effects): where Damage modelingPennes equationComplete modelWomersley modelWindkessel modelBernoulli applicationsRigid tube flow modelHydrostatics

38. Experimental vs. Theoretical c 0 Damage modelingPennes equationComplete modelWomersley modelWindkessel modelBernoulli applicationsRigid tube flow modelHydrostatics

39. 6. Womersley model for blood flow Damage modelingPennes equationComplete modelMoens-KortewegWindkessel modelBernoulli applicationsRigid tube flow modelHydrostatics

40. Problem Statement z r R h V z (r, z, t) V r (r, z, t) flow Infinitely long, thin-walled elastic tube of circular cross-section Damage modelingPennes equationComplete modelMoens-KortewegWindkessel modelBernoulli applicationsRigid tube flow modelHydrostatics

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) Damage modelingPennes equationComplete modelMoens-KortewegWindkessel modelBernoulli applicationsRigid tube flow modelHydrostatics

42. Equation of Motion Pressure gradient: Axial flow velocity: Axial flow velocity magnitude: where: Womersley number Damage modelingPennes equationComplete modelMoens-KortewegWindkessel modelBernoulli applicationsRigid tube flow modelHydrostatics

43. Examples of Womersley Number Damage modelingPennes equationComplete modelMoens-KortewegWindkessel modelBernoulli applicationsRigid tube flow modelHydrostatics

44. Flow Solution Flow solution: where: : Bessel function of order k Damage modelingPennes equationComplete modelMoens-KortewegWindkessel modelBernoulli applicationsRigid tube flow modelHydrostatics

45. Flow Solution Damage modelingPennes equationComplete modelMoens-KortewegWindkessel modelBernoulli applicationsRigid tube flow modelHydrostatics

46. Application Flow rate calculation for complex (non- sinusoidal) pulsatile pressure gradients Damage modelingPennes equationComplete modelMoens-KortewegWindkessel modelBernoulli applicationsRigid tube flow modelHydrostatics

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 Damage modelingPennes equationComplete modelMoens-KortewegWindkessel modelBernoulli applicationsRigid tube flow modelHydrostatics

48. 7. Complete model: Wave propagation in elastic tube with viscous flow Damage modelingPennes equationWomersley modelMoens-KortewegWindkessel modelBernoulli applicationsRigid tube flow modelHydrostatics

49. Elastic Tube Equations of Motion Stresses on a tube element Damage modelingPennes equationWomersley modelMoens-KortewegWindkessel modelBernoulli applicationsRigid tube flow modelHydrostatics

50. Elastic Tube and Fluid Stresses Tube stresses (from Hooke’s law) Fluid stresses (cylindrical coordinates) Damage modelingPennes equationWomersley modelMoens-KortewegWindkessel modelBernoulli applicationsRigid tube flow modelHydrostatics

51. Elastic Tube Equations of Motion Equations of motion for tube and flow must be solved simultaneously to obtain solutions for: Governing equations of motion for elastic tube Damage modelingPennes equationWomersley modelMoens-KortewegWindkessel modelBernoulli applicationsRigid tube flow modelHydrostatics

52. Flow Equations Non-linear inertial terms can be neglected (see order-of-magnitude study performed in Moens-Korteweg derivation) Along with the 2 tube equations, we obtain a set of 5 equations with 5 unknowns Governing flow equations Damage modelingPennes equationWomersley modelMoens-KortewegWindkessel modelBernoulli applicationsRigid tube flow modelHydrostatics

53. Flow Equations Boundary conditions at fluid-tube interface: No penetrationNo slip Governing flow equations Damage modelingPennes equationWomersley modelMoens-KortewegWindkessel modelBernoulli applicationsRigid tube flow modelHydrostatics

54. Solutions All variables are, at least, functions of z and t  Seek for solutions that vary as: where: k 1 : wave number (= 1/ ) k 2 : damping constant (decay along z ) Solutions: Damage modelingPennes equationWomersley modelMoens-KortewegWindkessel modelBernoulli applicationsRigid tube flow modelHydrostatics

55. Solutions After performing an order-of-magnitude study, the problem statement reduces to: BCs: Equation for  1 Equation for  1 NS / z continuity Damage modelingPennes equationWomersley modelMoens-KortewegWindkessel modelBernoulli applicationsRigid tube flow modelHydrostatics

56. Solutions By combining the 4 equations and applying the BCs, the problem can be expressed as a system of 2 equations with 2 unknowns: Damage modelingPennes equationWomersley modelMoens-KortewegWindkessel modelBernoulli applicationsRigid tube flow modelHydrostatics

57. Solutions Non trivial solutions if and only if determinant of the system = 0 If ( k / ω ) = φ is the root of the determinant: Damage modelingPennes equationWomersley modelMoens-KortewegWindkessel modelBernoulli applicationsRigid tube flow modelHydrostatics