1. ME 7980 Cardiovascular Biofluid Mechanics
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: Unsteady flow 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)

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

20. Bernoulli applications
Stenosis
Arterial flutter
Low pressure at contraction
Complete obstruction of vessel under external pressure δ a1 a2
p1, V1
p2, V2
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. Bernoulli applications
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. Bernoulli applications
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. Bernoulli applications
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. Bernoulli applications
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. Bernoulli applications
Windkessel Theory
Variables
Definition p
Windkessel chamber pressure V
Windkessel chamber volume Di
Chamber distensibility RS
Peripheral resistance Q
Ventricular ejection flow rate pV
Venous pressure
inflow
outflow pV RS
Q(t)
p(t), V(t), Di
Windkessel chamber
Definitions:
Inflow: fluid pumped intermittently by ventricular ejection
Outflow: calculated based on Poiseuille theory
Hydrostatics
Rigid tube flow model
Bernoulli applications
Windkessel model
Moens-Korteweg
Womersley model
Complete model
Pennes equation
Damage modeling

28. Windkessel (left) vs. actual (right) pressure pulse
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. Bernoulli applications
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 39

40. Bernoulli applications
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. Bernoulli applications
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. Bernoulli applications
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 43

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

46. Bernoulli applications
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. Bernoulli applications
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

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

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

53. Bernoulli applications
Flow Equations
Boundary conditions at fluid-tube interface:
Governing flow equations
No penetration
No slip
Hydrostatics
Rigid tube flow model
Bernoulli applications
Windkessel model
Moens-Korteweg
Womersley model
Complete model
Pennes equation
Damage modeling

54. Bernoulli applications
Solutions
All variables are, at least, functions of z and t
 Seek for solutions that vary as:
where:
k1 : wave number (= 1/)
k2 : damping constant (decay along z)
Solutions:
Hydrostatics
Rigid tube flow model
Bernoulli applications
Windkessel model
Moens-Korteweg
Womersley model
Complete model
Pennes equation
Damage modeling

55. Bernoulli applications
Solutions
After performing an order-of-magnitude study, the problem statement reduces to:
Equation for 1
Equation for 1
NS / z
BCs:
continuity
Hydrostatics
Rigid tube flow model
Bernoulli applications
Windkessel model
Moens-Korteweg
Womersley model
Complete model
Pennes equation
Damage modeling

56. Bernoulli applications
Solutions
By combining the 4 equations and applying the BCs, the problem can be expressed as a system of 2 equations with 2 unknowns:
Hydrostatics
Rigid tube flow model
Bernoulli applications
Windkessel model
Moens-Korteweg
Womersley model
Complete model
Pennes equation
Damage modeling

57. Bernoulli applications
Solutions
Non trivial solutions if and only if determinant of the system = 0
If (k/ω) = φ is the root of the determinant:
Hydrostatics
Rigid tube flow model
Bernoulli applications
Windkessel model
Moens-Korteweg
Womersley model
Complete model
Pennes equation
Damage modeling