1. Pressure Mapping from Flow Imaging: Enhancing Computation of
the Viscous Term Through Velocity Reconstruction in Near-Wall Regions
Fabrizio Donati, Nicolas P. Smith, David A. Nordsletten, Pablo Lamata
fabrizio.donati@kcl.ac.uk
Department of Biomedical Engineering and Imaging Sciences
King's College London
Chicago, Illinois, 29
th
August 2014

2. Introduction and Motivation

3. Introduction and Motivation
What is pressure?

4. Introduction and Motivation
What is pressure?
• Pressure in a fluid system is a force over unit area that resists the changes in volume
• Pressure gradient is the variation of pressure over the vessels' length
• Described by Navier-Stokes' momentum conservation equation:

5. Introduction and Motivation
What is pressure?
• Pressure in a fluid system is a force over unit area that resists the changes in volume
• Pressure gradient is the variation of pressure over the vessels' length
• Described by Navier-Stokes' momentum conservation equation:
Time-dependent
Acceleration in time
Pulsatile flow
Function of heart pump

6. Introduction and Motivation
What is pressure?
• Pressure in a fluid system is a force over unit area that resists the changes in volume
• Pressure gradient is the variation of pressure over the vessels' length
• Described by Navier-Stokes' momentum conservation equation:
Time-dependent
Acceleration in time
Pulsatile flow
Function of heart pump
Advective
Acceleration in space
Function of vessel morphology
Function of vasculature geometry

7. Introduction and Motivation
What is pressure?
• Pressure in a fluid system is a force over unit area that resists the changes in volume
• Pressure gradient is the variation of pressure over the vessels' length
• Described by Navier-Stokes' momentum conservation equation:
Time-dependent
Acceleration in time
Pulsatile flow
Function of heart pump
Advective
Acceleration in space
Function of vessel morphology
Function of vasculature geometry
Viscous
Dissipation
Due to neighbouring laminae of
fluid moving at different velocity

8. Introduction and Motivation

9. Introduction and Motivation
Why do we care about pressure in a clinical arena?

10. Pressure gradient is accepted biomarker in clinical pre- and post- operative guidelines
Introduction and Motivation
Why do we care about pressure in a clinical arena?

11. Aortic coarctation
Kelly et al. (2005) EHJ Cardiov Imag, 6:288-290
Pressure gradient is accepted biomarker in clinical pre- and post- operative guidelines
Valvular stenosis
Baumgardtner et al. (2009) Eur J Echo, 10:1-25
Hypertrophic Cardiomyopathy
Allen et al. (2014) JCMR, 16
Introduction and Motivation
Why do we care about pressure in a clinical arena?

12. Aortic coarctation
Kelly et al. (2005) EHJ Cardiov Imag, 6:288-290
Pressure gradient is accepted biomarker in clinical pre- and post- operative guidelines
Valvular stenosis
Baumgardtner et al. (2009) Eur J Echo, 10:1-25
Hypertrophic Cardiomyopathy
Allen et al. (2014) JCMR, 16
Introduction and Motivation
Why do we care about pressure in a clinical arena?
Why viscous pressure gradient?

13. • Inefficiency of the flow
• Energy losses
• Related to changes in Wall Shear Stress
• Methodologically challenging: II order spatial derivatives
Aortic coarctation
Kelly et al. (2005) EHJ Cardiov Imag, 6:288-290
Pressure gradient is accepted biomarker in clinical pre- and post- operative guidelines
Valvular stenosis
Baumgardtner et al. (2009) Eur J Echo, 10:1-25
Hypertrophic Cardiomyopathy
Allen et al. (2014) JCMR, 16
Introduction and Motivation
Why do we care about pressure in a clinical arena?
Why viscous pressure gradient?

14. • Inefficiency of the flow
• Energy losses
• Related to changes in Wall Shear Stress
• Methodologically challenging: II order spatial derivatives
Aortic coarctation
Kelly et al. (2005) EHJ Cardiov Imag, 6:288-290
Pressure gradient is accepted biomarker in clinical pre- and post- operative guidelines
Valvular stenosis
Baumgardtner et al. (2009) Eur J Echo, 10:1-25
Hypertrophic Cardiomyopathy
Allen et al. (2014) JCMR, 16
Introduction and Motivation
Why do we care about pressure in a clinical arena?
Why viscous pressure gradient?
Why laminar pressure gradient?

15. • Inefficiency of the flow
• Energy losses
• Related to changes in Wall Shear Stress
• Methodologically challenging: II order spatial derivatives
Aortic coarctation
Kelly et al. (2005) EHJ Cardiov Imag, 6:288-290
Pressure gradient is accepted biomarker in clinical pre- and post- operative guidelines
Valvular stenosis
Baumgardtner et al. (2009) Eur J Echo, 10:1-25
Hypertrophic Cardiomyopathy
Allen et al. (2014) JCMR, 16
Introduction and Motivation
Why do we care about pressure in a clinical arena?
Why viscous pressure gradient?
Why laminar pressure gradient?
• Turbulence does need time to develop
• Pulsatile regime is not so prone for it

16. Introduction and Motivation

17. Introduction and Motivation
How can pressure gradient be measured?

18. Introduction and Motivation
Cardiac catheterization
Schlant et al. (1984), J Am Coll Cardiol, 3(4):1096-8
How can pressure gradient be measured?

19. Introduction and Motivation
Cardiac catheterization Echo Doppler Ultrasound
B-mode or M-mode
Image
acquisit
ion
Velocit
y trace
detecti
on
PG
estimat
ion
Bernoulli's principle
Euler equation
Yotti et al. (2005), J Am Coll Cardiol, 122:1771-1779Schlant et al. (1984), J Am Coll Cardiol, 3(4):1096-8
How can pressure gradient be measured?

20. Introduction and Motivation
Image
acquisition
Computer
model
Boundary
conditions
Pressure
solution
Cardiac catheterization Echo Doppler Ultrasound
B-mode or M-mode
Navier-Stokes simulations
Model driven
Figueroa et al. (2009), Annu Rev Biomed Eng, 11:109-134
Image
acquisit
ion
Velocit
y trace
detecti
on
PG
estimat
ion
Bernoulli's principle
Euler equation
Yotti et al. (2005), J Am Coll Cardiol, 122:1771-1779Schlant et al. (1984), J Am Coll Cardiol, 3(4):1096-8
How can pressure gradient be measured?

21. Introduction and Motivation
Image
acquisition
Computer
model
Boundary
conditions
Pressure
solution
Cardiac catheterization Echo Doppler Ultrasound
B-mode or M-mode
Navier-Stokes simulations
Model driven
Poisson Pressure Equation (PPE)
Data driven
Figueroa et al. (2009), Annu Rev Biomed Eng, 11:109-134
4D flow PC-MRI
acquisition
Quadratic
hexahedrals
Masking operator: nodal
intensity values Finite Elements
Method
Image
acquisition
FEM mesh
Masked field
+ velocity
field
PG mapping
Krittian et al. (2011), Med Image Anal, 16:1029-1037
Image
acquisit
ion
Velocit
y trace
detecti
on
PG
estimat
ion
Bernoulli's principle
Euler equation
Yotti et al. (2005), J Am Coll Cardiol, 122:1771-1779Schlant et al. (1984), J Am Coll Cardiol, 3(4):1096-8
How can pressure gradient be measured?

22. Introduction and Motivation
Which is the best technique?
Image
acquisition
Computer
model
Boundary
conditions
Pressure
solution
Cardiac catheterization Echo Doppler Ultrasound
B-mode or M-mode
Navier-Stokes simulations
Model driven
Poisson Pressure Equation (PPE)
Data driven
Figueroa et al. (2009), Annu Rev Biomed Eng, 11:109-134
4D flow PC-MRI
acquisition
Quadratic
hexahedrals
Masking operator: nodal
intensity values Finite Elements
Method
Image
acquisition
FEM mesh
Masked field
+ velocity
field
PG mapping
Krittian et al. (2011), Med Image Anal, 16:1029-1037
Image
acquisit
ion
Velocit
y trace
detecti
on
PG
estimat
ion
Bernoulli's principle
Euler equation
Yotti et al. (2005), J Am Coll Cardiol, 122:1771-1779Schlant et al. (1984), J Am Coll Cardiol, 3(4):1096-8

23. Introduction and Motivation
Which is the best technique?
Image
acquisition
Computer
model
Boundary
conditions
Pressure
solution
Cardiac catheterization Echo Doppler Ultrasound
B-mode or M-mode
Navier-Stokes simulations
Model driven
Poisson Pressure Equation (PPE)
Data driven
Figueroa et al. (2009), Annu Rev Biomed Eng, 11:109-134
4D flow PC-MRI
acquisition
Quadratic
hexahedrals
Masking operator: nodal
intensity values Finite Elements
Method
Image
acquisition
FEM mesh
Masked field
+ velocity
field
PG mapping
Krittian et al. (2011), Med Image Anal, 16:1029-1037
Image
acquisit
ion
Velocit
y trace
detecti
on
PG
estimat
ion
Bernoulli's principle
Euler equation
Yotti et al. (2005), J Am Coll Cardiol, 122:1771-1779Schlant et al. (1984), J Am Coll Cardiol, 3(4):1096-8
Noninvasive

24. Introduction and Motivation
Which is the best technique?
Image
acquisition
Computer
model
Boundary
conditions
Pressure
solution
Echo Doppler Ultrasound
B-mode or M-mode
Navier-Stokes simulations
Model driven
Poisson Pressure Equation (PPE)
Data driven
Figueroa et al. (2009), Annu Rev Biomed Eng, 11:109-134
4D flow PC-MRI
acquisition
Quadratic
hexahedrals
Masking operator: nodal
intensity values Finite Elements
Method
Image
acquisition
FEM mesh
Masked field
+ velocity
field
PG mapping
Krittian et al. (2011), Med Image Anal, 16:1029-1037
Image
acquisit
ion
Velocit
y trace
detecti
on
PG
estimat
ion
Bernoulli's principle
Euler equation
Yotti et al. (2005), J Am Coll Cardiol, 122:1771-1779
Noninvasive

25. Introduction and Motivation
Which is the best technique?
Image
acquisition
Computer
model
Boundary
conditions
Pressure
solution
Echo Doppler Ultrasound
B-mode or M-mode
Navier-Stokes simulations
Model driven
Poisson Pressure Equation (PPE)
Data driven
Figueroa et al. (2009), Annu Rev Biomed Eng, 11:109-134
4D flow PC-MRI
acquisition
Quadratic
hexahedrals
Masking operator: nodal
intensity values Finite Elements
Method
Image
acquisition
FEM mesh
Masked field
+ velocity
field
PG mapping
Krittian et al. (2011), Med Image Anal, 16:1029-1037
Image
acquisit
ion
Velocit
y trace
detecti
on
PG
estimat
ion
Bernoulli's principle
Euler equation
Yotti et al. (2005), J Am Coll Cardiol, 122:1771-1779
Noninvasive
Accurate
Automatic

26. Introduction and Motivation
Which is the best technique?
Image
acquisition
Computer
model
Boundary
conditions
Pressure
solution
Navier-Stokes simulations
Model driven
Poisson Pressure Equation (PPE)
Data driven
Figueroa et al. (2009), Annu Rev Biomed Eng, 11:109-134
4D flow PC-MRI
acquisition
Quadratic
hexahedrals
Masking operator: nodal
intensity values Finite Elements
Method
Image
acquisition
FEM mesh
Masked field
+ velocity
field
PG mapping
Krittian et al. (2011), Med Image Anal, 16:1029-1037
Noninvasive
Accurate
Automatic

27. Introduction and Motivation
Which is the best technique?
Image
acquisition
Computer
model
Boundary
conditions
Pressure
solution
Navier-Stokes simulations
Model driven
Poisson Pressure Equation (PPE)
Data driven
Figueroa et al. (2009), Annu Rev Biomed Eng, 11:109-134
4D flow PC-MRI
acquisition
Quadratic
hexahedrals
Masking operator: nodal
intensity values Finite Elements
Method
Image
acquisition
FEM mesh
Masked field
+ velocity
field
PG mapping
Krittian et al. (2011), Med Image Anal, 16:1029-1037
Noninvasive
Accurate
Automatic
Fast

28. Introduction and Motivation
Which is the best technique?
Poisson Pressure Equation (PPE)
Data driven
4D flow PC-MRI
acquisition
Quadratic
hexahedrals
Masking operator: nodal
intensity values Finite Elements
Method
Image
acquisition
FEM mesh
Masked field
+ velocity
field
PG mapping
Krittian et al. (2011), Med Image Anal, 16:1029-1037
Noninvasive
Accurate
Automatic
Fast

29. Introduction and Motivation
Relative pressure mapping using PPE FEM

30. Introduction and Motivation
Relative pressure mapping using PPE FEM
• Data-driven approach: pressure field is unknown but velocity field is provided
• Defined applying divergence to Navier-Stokes' momentum conservation equation:

31. Reduced computational time and costs
Formulation with Finite Elements Method is easy:
Introduction and Motivation
Relative pressure mapping using PPE FEM
• Data-driven approach: pressure field is unknown but velocity field is provided
• Defined applying divergence to Navier-Stokes' momentum conservation equation:

32. Reduced computational time and costs
Formulation with Finite Elements Method is easy:
Introduction and Motivation
Relative pressure mapping using PPE FEM
• Data-driven approach: pressure field is unknown but velocity field is provided
• Defined applying divergence to Navier-Stokes' momentum conservation equation:
MRI

33. Reduced computational time and costs
Formulation with Finite Elements Method is easy:
Introduction and Motivation
Relative pressure mapping using PPE FEM
• Data-driven approach: pressure field is unknown but velocity field is provided
• Defined applying divergence to Navier-Stokes' momentum conservation equation:
MESH

34. Reduced computational time and costs
Formulation with Finite Elements Method is easy:
Introduction and Motivation
Relative pressure mapping using PPE FEM
• Data-driven approach: pressure field is unknown but velocity field is provided
• Defined applying divergence to Navier-Stokes' momentum conservation equation:
MASK

35. Reduced computational time and costs
Formulation with Finite Elements Method is easy:
Introduction and Motivation
Relative pressure mapping using PPE FEM
• Data-driven approach: pressure field is unknown but velocity field is provided
• Defined applying divergence to Navier-Stokes' momentum conservation equation:
VELOCITY

36. Reduced computational time and costs
Formulation with Finite Elements Method is easy:
Introduction and Motivation
Relative pressure mapping using PPE FEM
• Data-driven approach: pressure field is unknown but velocity field is provided
• Defined applying divergence to Navier-Stokes' momentum conservation equation:
PRESSURE

37. Reduced computational time and costs
Formulation with Finite Elements Method is easy:
Introduction and Motivation
Relative pressure mapping using PPE FEM
• Data-driven approach: pressure field is unknown but velocity field is provided
• Defined applying divergence to Navier-Stokes' momentum conservation equation:
Finite elements edges do not match vessel's wall
Viscous term (II derivatives) compromises solution
Lower Signal-to-Noise Ratio at walls does not help!
PRESSURE

38. Reduced computational time and costs
Formulation with Finite Elements Method is easy:
Introduction and Motivation
Relative pressure mapping using PPE FEM
• Data-driven approach: pressure field is unknown but velocity field is provided
• Defined applying divergence to Navier-Stokes' momentum conservation equation:
Finite elements edges do not match vessel's wall
Viscous term (II derivatives) compromises solution
Lower Signal-to-Noise Ratio at walls does not help!
PRESSURE
Higher resolution is not the solution!

39. Reduced computational time and costs
Formulation with Finite Elements Method is easy:
Introduction and Motivation
Relative pressure mapping using PPE FEM
• Data-driven approach: pressure field is unknown but velocity field is provided
• Defined applying divergence to Navier-Stokes' momentum conservation equation:
Finite elements edges do not match vessel's wall
Viscous term (II derivatives) compromises solution
Lower Signal-to-Noise Ratio at walls does not help!
PRESSURE
Higher resolution is not the solution!

40. Introduction and Motivation
Relative pressure mapping using PPE FEM
• Data-driven approach: pressure field is unknown but velocity field is provided
• Defined applying divergence to Navier-Stokes' momentum conservation equation:
Integration of the boundary region by automatic fitting of a smooth tubular computational
mesh to the segmentation domain
Low-SNR and data issues are addressed by the reconstruction of the velocity profile via a
Stokes-driven degrees of freedom repopulation in the near-wall region
Stokes' enhanced PPE approach

41. Methods
Hybrid Stokes' enhanced PPE FEM approach

42. • Uses PPE FEM standard approach (data-driven) in the core region
• Uses Stokes' theory (model-driven) to reconstruct velocity in the boundary region
• Main hypothesis is that viscous term drives motion near the vessels' wall:
Methods
Hybrid Stokes' enhanced PPE FEM approach

43. • Uses PPE FEM standard approach (data-driven) in the core region
• Uses Stokes' theory (model-driven) to reconstruct velocity in the boundary region
• Main hypothesis is that viscous term drives motion near the vessels' wall:
Methods
Hybrid Stokes' enhanced PPE FEM approach

44. • Uses PPE FEM standard approach (data-driven) in the core region
• Uses Stokes' theory (model-driven) to reconstruct velocity in the boundary region
• Main hypothesis is that viscous term drives motion near the vessels' wall:
Methods
Hybrid Stokes' enhanced PPE FEM approach
Image
acquisit
ion
Segmen
tation +
Walls
detectio
n
Templat
e
definiti
on
Image registration +
Mesh personalization
Walls velocity
reconstruction (Stokes)
Relative pressure
mapping (PPE FEM)

45. • Uses PPE FEM standard approach (data-driven) in the core region
• Uses Stokes' theory (model-driven) to reconstruct velocity in the boundary region
• Main hypothesis is that viscous term drives motion near the vessels' wall:
Methods
Hybrid Stokes' enhanced PPE FEM approach
Image
acquisit
ion
Segmen
tation +
Walls
detectio
n
Templat
e
definiti
on
Image registration +
Mesh personalization
Walls velocity
reconstruction (Stokes)
Relative pressure
mapping (PPE FEM)

46. Materials
Validation test case

47. Materials
Validation test case
Image acquisition Template generationIn silico phantom
Cylindrical straight pipe
3D Poiseuille flow (viscous only)
Resolution: 2mm3
Simulated Gaussian noise,
8 longitudinal elms
8 circumferential elms
12 radial elms
8 boundary elms (25% lumen)
Original image
Noisy image, SNR = 5

48. Materials
Validation test case
Image acquisition Template generationIn silico phantom
Cylindrical straight pipe
3D Poiseuille flow (viscous only)
Resolution: 2mm3
Simulated Gaussian noise,
8 longitudinal elms
8 circumferential elms
12 radial elms
8 boundary elms (25% lumen)
Original image
Noisy image, SNR = 5

49. Materials
Validation test case
Image acquisition Template generationIn silico phantom
Cylindrical straight pipe
3D Poiseuille flow (viscous only)
Resolution: 2mm3
Simulated Gaussian noise,
8 longitudinal elms
8 circumferential elms
12 radial elms
8 boundary elms (25% lumen)
Original image
Noisy image, SNR = 5

50. Materials
Validation test case
Image acquisition Template generationIn silico phantom
Cylindrical straight pipe
3D Poiseuille flow (viscous only)
Resolution: 2mm3
Simulated Gaussian noise,
8 longitudinal elms
8 circumferential elms
12 radial elms
8 boundary elms (25% lumen)
Original image
Noisy image, SNR = 5

51. Materials
Validation test case
Image acquisition Template generationIn silico phantom
Cylindrical straight pipe
3D Poiseuille flow (viscous only)
Resolution: 2mm3
Simulated Gaussian noise,
8 longitudinal elms
8 circumferential elms
12 radial elms
8 boundary elms (25% lumen)
Original image
Noisy image, SNR = 5

52. Materials
Validation test case
Image acquisition Template generationIn silico phantom
Cylindrical straight pipe
3D Poiseuille flow (viscous only)
Resolution: 2mm3
Simulated Gaussian noise,
8 longitudinal elms
8 circumferential elms
12 radial elms
8 boundary elms (25% lumen)
Original image
Noisy image, SNR = 5
Stokes' enhanced PPE (SePPE) vs Direct PPE (DPPE)
pressure gradient percentage relative error:

53. Sensitivity analysis
• Lumen detection error
• SNR
• Segmentation threshold
Materials
Validation test case
Image acquisition Template generationIn silico phantom
Cylindrical straight pipe
3D Poiseuille flow (viscous only)
Resolution: 2mm3
Simulated Gaussian noise,
8 longitudinal elms
8 circumferential elms
12 radial elms
8 boundary elms (25% lumen)
Original image
Noisy image, SNR = 5
Stokes' enhanced PPE (SePPE) vs Direct PPE (DPPE)
pressure gradient percentage relative error:

54. Results

55. Sensitivity analysis to lumen detection error
Results

56. Sensitivity analysis to lumen detection error
Results
SePPE outperforms DPPE!
• With SePPE underestimation of lumen leads to pressure gradient overestimation and viceversa
• Pressure gradient is always underestimated with DPPE
• Performance is still acceptable with low-to-moderate εφ
• Lumen detection error computed as

57. Results

58. Sensitivity analysis to SNR and segmentation threshold
Results

59. SePPE performs better than DPPE!
• High sensitivity to segmentation threshold
• Both methods perform poorly for high level of noise
• 20 simulations run for each set of parameters to avoid variability of results
• SNR within the range [5÷15]
• Segmentation threshold as a percentage of the analytical peak velocity
• Physiologically relevant noise [7.5÷12.5] and segmentation threshold [7.5%÷12.5%] in red
Sensitivity analysis to SNR and segmentation threshold
Results

60. Discussion

61. Discussion
• Laminar viscous dissipation is higher at the vessels' boundary
• Viscous gradients are not captured correctly with PPE

62. ✓ SePPE method is fully automatic
✓ PPE enables computation of time-dependent, convective and viscous gradients separately
✓ Velocity reconstruction reduces error in viscous gradient irrespective of lumen detection error
x Segmentation is fundamental! (lumen detection error <5% to get 80% accuracy in gradients)
x Optimal segmentation threshold choice is function of SNR level
x 3D Poiseuille flow is not representative of physiological aortic flows (no inertial terms)
Discussion
• Performing validation tests on more realistic in silico cases (Womersley pulsatile flow field)
• Using Navier-Stokes' simulation to reconstruct velocity profile in boundary region
• Laminar viscous dissipation is higher at the vessels' boundary
• Viscous gradients are not captured correctly with PPE

63. ✓ SePPE method is fully automatic
✓ PPE enables computation of time-dependent, convective and viscous gradients separately
✓ Velocity reconstruction reduces error in viscous gradient irrespective of lumen detection error
x Segmentation is fundamental! (lumen detection error <5% to get 80% accuracy in gradients)
x Optimal segmentation threshold choice is function of SNR level
x 3D Poiseuille flow is not representative of physiological aortic flows (no inertial terms)
Hybrid method improves estimation of laminar viscous dissipation
Discussion
• Performing validation tests on more realistic in silico cases (Womersley pulsatile flow field)
• Using Navier-Stokes' simulation to reconstruct velocity profile in boundary region
• Laminar viscous dissipation is higher at the vessels' boundary
• Viscous gradients are not captured correctly with PPE

64. Thank you for your attention.
Questions?