1. Non-invasive relative pressure
estimation using 4D PC-MRI
Doctoral Thesis
Candidate Supervisors
Fabrizio Donati Pablo Lamata, PhD
David A. Nordsletten, PhD
Nicolas P. Smith, Prof
Division of Imaging Sciences and Biomedical Engineering
Faculty of Life Sciences and Medicine
King’s College London
22nd March 2016
2. Motivation (Chapter 1)
Aortic Stenosis (AS)
Healthy AS
• Inefficient opening of the aortic valve
• Constriction to the blood flow
• Increase of LV workload and heart failure
• 50% 2-years mortality with severe AS (>65yo)1
1Otto C M et al. (2000) Heart, 84(2):211-218
2Baumgartner H et al. (2009) Eur J Echo, 10:1-25 1/17
Relative pressure estimation
as an accepted biomarker
Severe AS for Δp≥20 mmHg2
• Aortic coarctation
• Aortic dissection
• Hypertrophic cardiomyopathy
• Fractional-flow reserve coronary stenosis
3. Motivation (Chapter 1)
Invasive techniques
Catheterization
Non-invasive techniques
• Direct pressure measurements
• Surgical procedure
• Angiography to know cath position (X-ray)
• Relative pressure estimated from acquired velocity
• Wider interrogation area
• No surgical procedure
Medical imaging
Clinical interest on non-invasive relative pressure estimation
2/17
www.radiology.northwestern.edu
4. Motivation (Chapter 1)
B-Mode Continuous-Wave
Doppler Echocardiography
M-Mode Color Doppler
Echocardiography
Cardiovascular
Magnetic Resonance
Simplified Bernoulli Euler equation4 CFD Navier-Stokes
Simulations5
Amount of data available
Amount of assumptions in the mathematical model
Computational time
3/17
3Hatle L et al. (1978) Brit Heart J, 40:131-140
4Greenberg N L et al. (2001) Am J Physiol Heart Circ Physiol 280(6), 2507-2515
5Figueroa C A et al. (2009), Annu Rev Biomed Eng, 11:109-134
Objective: accurate non-invasive pressure difference
estimation method feasible in the clinic
5. Relative Pressure Mapping from PC-MRI (Chapter 2)
Poisson Pressure Equation, Finite Element Method6
Data driven method: relative pressure map from velocity
• Easy generation of computational mesh, masking from velocity field
• Reduction of computational costs w.r.t. CFD Navier-Stokes simulations
4/17
6Krittian S B S et al. (2011) Med Image Anal, 16:1029-1037
7cheart.co.uk
State-of-the-art data driven
method accounting for the
complete Navier-Stokes
Sensitivity analysis to masking
and velocity interpolation
CHeart solver7: Test method
performance for clinical use
6. Relative Pressure Mapping from PC-MRI (Chapter 2)
Impact of masking operator
εΔp
Δx =
2mm
Δx =
1mm
Δx =
0.5mm
No
mask
84.15 86.60 84.51
Partial 44.65 34.33 47.07
Global 6.25 2.66 0.99
In silico test, 3D, Poiseuille, noise-free
No convergence
Impact of velocity interpolation
In silico test, 1D, Poiseuille
Nodal, L2 and H1 projections
High sensitivity to FE/vascular edges position
5/17
Viscous component is small, but clinical value (irreversible energy loss)8
Noise-free:
constant viscous gradients
Real cases:
removes near-walls
(high dissipation)
Cancellation effect on viscous gradients
Marginal improvement with projections
Computation of viscous pressure gradients with FE-PPE
hampered by the interpolation of velocity at the walls
8Barker A J et al. (2014) Magn Reson Med, 72(3): 620-628
7. Boundary reconstruction / Viscous gradients (Chapter 3)
Near-walls velocity reconstruction
Stokes-enhanced Poisson Pressure Equation, Finite Element Method9
Hybrid method: relative pressure map from velocity image (Model,boundary / Data,FE-PPE)
• Mesh fitting with ad hoc parameters
• Avoids noise contamination in low-SNR regions
• Circumvents FE/vessels edges misalignment issue
• Computational time: FE-SePPE ≅ FE-PPE
6/179Donati F et al. (2014) Annu Int Conf IEEE, 5097-5100
Plugged into FE-PPE solver
8. Consistent increased accuracy w.r.t. FE-PPE
Performance tested w.r.t. FE-PPE
Boundary reconstruction / Viscous gradients (Chapter 3)
In silico, 3D, Poiseuille
Simulated Gaussian noise, 100x
Sensitivity to wall detection
Need accurate definition of fluid boundaries
Velocity reconstruction recovers viscous
gradients but careful clinical adoption
7/17
SNR = [5 ÷ 15]
S = [5% Vmax ÷ 15% Vmax]
9. Work-Energy Relative Pressure method (Chapter 4)
8/17
Work-Energy principle, Finite Difference Method10
MATLAB standalone solver
Data driven method: pressure difference from velocity image
10Donati F et al. (2015) Med Image Anal 26:159-172
• No computational mesh needed ≠ FE-PPE, FE-SePPE
• Integration directly from velocity image
• Enables isolated components analysis
• Viscous: 1st vs. 2nd order derivatives (FE-PPE, FE-SePPE)
• Computational time: WERP < FE-PPE, FE-SePPE, CFD
10. Work-Energy Relative Pressure method (Chapter 4)
Validation, CFD simulation aortic coarctation
Simulated image acquisition:
• Δx = 2mm3, Δt = 43ms
• Gaussian noise, SNR=5, 100x
Performance vs. SB11, UB12, FE-PPE13
Increased accuracy
10/17
Increased robustness to noise
11Oshinski J N et al. (1996) J Am Coll Cardiol, 28(7):1818-1826
12Firstenberg M S et al. (2000) J Am Coll Cardiol, 36(6):1943-1949
13Krittian S B S et al. (2012) Med Image Anal, 16:1029–1037
11. Work-Energy Relative Pressure method (Chapter 4)
WERP performs
well on real data
11/1714Lamata et al. (2014) Magn Res Med, 72(4):1162–1169
Clinical application, 4D PC-MRI, 9 healthy patients
4 regions, ascending/descending aorta
Performance vs. FE-PPE14 (published results)
Good agreement
12. Clinical application, 4D PC-MRI, 32 patients
Aortic stenosis severity (guidelines: Bernoulli)
WERP clinical application / Beyond Bernoulli (Chapter 5)
12/1715Baumgartner H et al. (2009) Eur J Echo, 10:1-25
WERP Simplified
Bernoulli
Pressure constant on planes ✔ ✔
Negligible compliance ✔ ✔
Predominant advective ✔
vVC >> vLVOT ✔
Single velocity value ✔
Validity of Bernoulli assumptions
Group I (Δp<20mmHg), Group II (Δp≥20mmHg)15
WERP, relative contribution of pressure terms
1/3: Predominant advective
Predominant advective effect
sensible hypothesis
13. WERP clinical application / Beyond Bernoulli (Chapter 5)
13/17
Hp Advective predominant
vVC >> vLVOT
Bernoulli high overestimation
SAW correlates with AW
Validity of Bernoulli assumptions
2/3: vVC >> vLVOT
Pro Accounts for 3D hemodynamics
2D velocity field at one plane
No volume integrals
2D PC-MRI/3D Doppler echo
Advective
WERP
Simplified
Bernoulli
Simplified
Advective WERP
Pressure constant on planes ✔ ✔ ✔
Negligible compliance ✔ ✔ ✔
Predominant advective ✔ ✔ ✔
vVC >> vLVOT ✔ ✔
Single velocity value ✔
Advective WERP vs. SAW
SAW low overestimation
vVC >> vLVOT
sensible hypothesis
Advective WERP vs. SB
Simplified Advective WERP
3/3: Single velocity value
14. WERP clinical application / Beyond Bernoulli (Chapter 5)
In silico test, 3D, stenosis
Pressure drop with SB (single) / SAW (3D)
Discrepancy between methods:
due to profile shape only
14/17
Validity of Bernoulli assumptions
3/3: Single velocity value
Healthy cases: blunt, constant value
Disease cases, non-blunt, skewed, eccentric
Single velocity value not sensible
choice in highly impaired cases
Unveiled hidden limitation
of Bernoulli principle
15. WERP clinical application / Beyond Bernoulli (Chapter 5)
15/17
Clinical translation of SAW
It works nicely
Variable partial
correction
Application to simulated Doppler echo data
Adapted formula for 2D color Doppler echo, 1D line integral
Bernoulli correction feasible with echo data,
optimal acquisition protocol to be investigated
16. Contributions (Chapter 6)
• Verified FE-PPE sensitive to masking, viscous component cancellation effect at
boundaries: reported marginal improvement by changing projection
16/17
• Proposed a method to reconstruct boundary velocity to improve viscous gradients
computation: reported improvement, sensitive to vascular walls detection
• Proposed WERP (full drop), a method to account for laminar hemodynamics,
avoiding meshing, with reduced derivatives order: reported speed, accuracy and
robustness to noise w.r.t. existing methods
• Reported hidden limitation of Bernoulli: proposed a formulation to improve
estimation, with large potential clinical impact at a cost of further velocity data
17. • Account for flow through secondary branches
• Account for regions with low-flow improvement on compliance model
• Account for turbulence effects, additional energy term in WERP method
17/17
Future work (Chapter 6)
• Phantom experiments to corroborate findings
• Development of clinical-friendly software tool
• Adaptation of the method to different imaging techniques
• Used to characterize outcome after aortic arch reconstruction on HLHS patients
• Characterization of the pressure recovery phenomenon
Editor's Notes
Good morning, I will now give you a 20-30 mins overview of my phd project, focusing on the motivations and the main contributions we achieved during these years
To introduce our work, it’s useful talking about 2 of the most common diseases of the human cvs: AS where calcification of the leaflets of the aortic valve produces an inefficient opening, and HCM, where a portion of the myocardium is thickened, creating a functional impairment of the cardiac muscle.
What is common between these two disease conditions is the obstruction they cause to the blood flow, producing an increase of the LV workload, with worrying consequences if left untreated for the elderly (AS) or even for young people (HCM)
Under these and other disease conditions, relative pressure, meaning the difference in central blood pressure at two locations in the cvs, is a currently accepted biomarker for the assessment of the severity of the constriction to the blood flow, which is diagnosed when pressure drops are larger than 20 mmHg (AS) and 50 mmHg (hcm)
----- Meeting Notes (3/21/16 17:14) -----
fractional flow reserve
To estimate relative pressures in the central cvs, different techniques have been developed over the years.
On one hand, there are invasive techniques, as catheterization. This provides direct pressure measurements via fluid filled or wire catheters inserted from the brachial or femoral artery up to the heart with a surgical procedure. Apart from its invasiveness, this technique has the major drawbacks of requiring CT angiography (with exposure to X-ray) to know the position of the catheter when inserted, and resyncronization of the pressure signals at two points to estimate the instantaneous relative pressure.
On the other hand, there is non-invasive techniques, relying on medical imaging, which has been rapidly developing in the last decades. This is the case of CMR or echo, where there is no need for surgery, the interrogation area can be usually visualized in real time, with no need for X-ray exposure, and pressure gradients are estimated instantaneously from acquired velocity field, using mathematical models.
Moreover these techniques provide a larger amount of information on the hemodynamics of the cvs, instead of simple point-to-point measurements, that can be used to estimate other useful parameters such as WSS, turbulence indices and many others…
These are the main reasons why clinical practice is currently driven by imaging techniques.
----- Meeting Notes (3/21/16 17:14) -----
surgival or invasive ?
We have several techniques availble, which provide different amount of data
We have b-mode continuous wave doppler echo, m-mode color doppler echo up to cmr, that provides at the same time information on blood velocity and geometry over 2d planes or 3d volumes
To estimate pressure gradients from these images, different mathematical models are employed, from the most complex being the CFD simulations of 3D NS (applicable to cmr flow images) to more simplified models such as the Euler or Bernoulli equation.
Of course, the smaller is the amount of available data, the simpler will be the mathematical model used to evaluate pressure, meaning an increasing amount of assumptions made on the blood flow, but with benefits on the computational time, which is instead too large with CFD simulations.
In this context a compromise has to be found between the high accuracy of the pressure estimation method and the need for low computational times and costs.
This motivated the main objective of this phd project which was the development of a method able to accurately estimate non-invasive relative pressure within clinically feasible times
To this purpose, we have firstly focused on an interesting approach developed in 2011 by Krittian, who introduced a data driven method based on the PPE solved with the FEM.
With this method the full laminar hemodynamics described by Navier-Stokes equations are accounted for, and rel pressure maps are estimated directly from the velocity field from PC-MRI
This method reduces the computational costs wrt to ns cfd simulations,
And the generation of the computational domain is straightforward from the PC-MRI data,
Also, the use of a masking operator based on the velocity field constraints the computations to the fluid volume only
We have implemented the FE-PPE solver described in krittian’s paper in the home-built software CHeart and we have investigated the sensitivity of this method to some factors which are crucial for the pressure gradients estimation in vivo.
These factors are the masking operator and the velocity interpolation from the image data on the FE mesh
We have done this focusing on the viscous component of the pressure gradient, which although being usually small compared to the inertial flow fields effects, we hypothesized adding an important clinical value as it reflects the inefficiencies and the amount of energy that is lost due to friction during the heart cycle
Through an in silico test on a 3D pipe with Poiseuille flow in a noise free configuration, we computed the error on the pressure drop along the pipe (compared to the analytic sol), testing the impact of different masks, progressively choosing as part of the fluid domain elements made up by larger number of voxels with non-zero velocity.
We reported that as you see here, there is no convergence with image resolution using the same mask, with the less conservative criterions, where we also include part of the static tissue surrounding the fluid domain.
Now, we see that in the noise-free case, with the higher resolution and a global masking, there is convergence and the estimated error is really small, but this is not as good as it seems.
First, the global masking is working so well in this case because we have no noise, and the elements boundary match the vessel boundaries almost perfectly, which is unlikely in real cases
Second, in the real, noisy cases, the global masking is not an appropriate choice, as it removes the contributions of the near-walls regions, which is where velocity gradients are higher and viscous dissipation is largest.
So, we explored the reasons behind the lack of convergence of the less conservative masks and the impact of the relative position between the finite elements edges and vascular wall boundaries.
We have tested this in silico on a 1D Poiseuille flow, and compared the impact of different velocity projections (from nodal to more constraining H1) on the estimation of second derivatives, which drive the viscous component of pressure. We have this case here, where the alignment between E and vessel is perfect, and this case, where they don’t match.
It’s clear that in both cases the improvement with different interpolation approaches is marginal, while we have reported a high sensitivity to the relative position of the E/vessel boundaries, which translates into inaccurate computations of viscous gradients for the FE-PPE
----- Meeting Notes (3/21/16 17:14) -----
remove hp
We have tried to recover the computation of viscous gradients by performing a velocity reconstruction at the near-walls vascular region enforcing a Stokes solution at the vascular boundary, in the hypothesis that blood flow is dominated by viscous effects at the walls.
This is an hybrid method, as it enables the computation of relative pressure maps from the velocity image, but after a model-driven repopulation of the boundary velocities
We get the image (with noise) from CMR, than we perform a mesh fitting procedure to an geometrical template with arbitrarily defined mesh, and we define two regions based on the local SNR.
Where the SNR is locally higher, in the core lumen, we hp that the velocity field from the image is reliable, than we just interpolate the image data on the warped mesh nodes.
Where the SNR is instead lower, close to the vascular walls, we solve a Stokes problem by just imposing the velocity field from the image at the inlet boundary and imposing null velocity at the outer boundary, therefore reconstructing the velocity field.
Than, we combine the original field in the core to the reconstructed field at the boundary and we solve for rel p using the FE-PPE method
By using this approach, we can selectively define the boundary parameters (so for example increase the mesh resolution) and we avoid the contamination of noise to the gradients estimate.
This way, the method recovers the laminar viscous pg, by circumventing the issue of the misalignment of the edges of FE and vessel.
All this, with computational times comparable to FE-PPE
We have tested the method against the standard FE-PPE on the original mesh on an in silico 3D pipe with Poiseuille flow, adding simulated Gaussian noise and repeating the test 100 times, and we evaluated the error on the estimated pg wrt using the analytic solution.
Here, we have investigated the importance of a correct detection of the vascular walls, showing an overall improvement of the gradients computation when testing the sensitivity to noise and segmentation threshold.
However, as shown in this plot where the error is compare to the FE-PPE estimates as a function on the lumen detection error, the sensitivity to the accurate definition of the vascular walls is quite high, with the error increasing rapidly as the lumen is overestimated.
The main contribution with this new approach is therefore the improvement of viscous computations when compared to the standard FE-PPE, but its high sensitivity to the boundary detection limits its applicability in the clinic
In the second part of my phd project, we have developed a brand new approach to make the best use of medical image data in order to estimate pressure non-invasively
We called it the WERP method which is obtained from the work-energy principle.
Again, this is a data driven approach that follows a simple workflow:
From the acquired PCMRI data, after segmentation, the roi is defined by arbitrarily selecting an inlet and outlet plane, and through a fd approach, the pressure difference is computed.
The pressure difference is estimated by evaluating integrals from the image, without need to define a computational mesh:
surface integrals Q and A that account for the flow through the inlet (or outlet plane), and for the inlet to outlet advected energy rate and
volume integrals K and V that account for the kinetic energy rate and viscous dissipation rate
Other significant advantages with this method are the ability to isolate the pressure terms contributions, the reduced maximum order of spatial derivation needed to account for the viscous effects and the reduced computational costs wrt to the FE based approaches and CFD simulations
----- Meeting Notes (3/21/16 17:14) -----
more automated
We initially tested the method convergence mimicking the acquisition process by adding Gaussian noise on an in silico pipe with steady and unsteady flows, progressively adding noise, from a noisefree to a high-noise configuration
We have quantified the error on the pressure difference estimates wrt to the analytic values using two different stencils for the evaluation of the velocity field and velocity field derivatives
Depending on the level of added noise, we have reported a convergent behaviour with space or spatiotemporal refinemen (for these case for example), but we also reported another trend, for the low-SNR cases.
Here, that convergence is blocked and the error (after reaching a minimum) is amplified with further refinement, due the nature of noise, that is uncorrelated between adjacent voxels.
This behaviour varies depending on the amount of noise and case selected: we see the turning point here for the std stencil or here for the filtered
In these cases, we have shown the beneficial effect of the filtered approach to mitigate the contamination due to noise
----- Meeting Notes (3/21/16 17:14) -----
remove
We have validated the WERP method using a CFD simulation of an aortic coarctation where the solution for pressure was available as well as the distribution of velocity over the cardiac cycle
Again, we have simulated the PC-MRI acquisition process by adding Gaussian noise to the images
Then we have compared the transients of pressure difference across the coarctation computed with simplified Bernoulli, unsteady Bernoulli and FE-PPE on meshes obtained from 2 masks, and we have shown competitive accuracy and robustness to noise over all the systolic and diastolic events
We have verified our method on PC-MRI data from 9 healthy patients.
In this case, we have estimated pressure gradients at 4 different aortic locations (2 in the ascending and 2 in the descending) and compared against previously published results obtained with the FE-PPE method
We have reported a nice agreement with these 99% confidence intervals, confirming on real data the good performance highlighted with the in silico tests
----- Meeting Notes (3/21/16 17:14) -----
range of cvariability
Then, we have focused on a clinical application of the method, which we have used to assess the severity of the aortic stenosis of 32 patients from 4D PC-MRI data
We have defined the tvr from the lvot to the vc, plane where the jet out of the av during systole is narrowest, producing the peak velocity
Making advantage of the ability of WERP 3d I wanted to analyze the
First, we have used a threshold of 20mmHg for the tv mean systolic pd with Bernoulli to differentiate our cohort in two groups, as prescribed by the currently guidelines for echo assessment of as
And taking advantage of the features of the WERP method, we have evaluated the contribution of isolated components of the pd
Focusing on the tvr, we have reported the significant differences between groups for both the advective and the viscous mean systolic pressure drops
On the other hand, looking at pressure difference transients during systole, we have reported the clear predominance of the advective component in the tvr, especially in disease cases.
This suggests that the pressure gradient in this region is mainly driven by the spatial transport of momentum, due to the narrowing of the pipe.
The clinical translation of this result is the apparent evidence that using Bernoulli (which is only accounting for the adv forces) is a sensible choice to asses the as
So to further explore this result, we have introduced a modified formulation from the WERP method, that quantifies the advective contribution, focusing on the energy advected on the vc plane only (region of high velocity), following the hp that leads from the corrected to simplified Bernoulli
But wrt to simplfiied Bernoulli, it has the main advantage of accounting for the full 3d hemodynamics.
From a clinical perspective, this formulation does require only information on the velocity field at the vc plane, with no need for the kinetic and viscous volume integrals, being therefore feasible with 2d pcmri and 3d doppler echo
We evaluated the correlation between mean systolic advective pd and mean drops obtained with sb and saw, showing the high correlation with the simplified approach, the overestimation of the sb (which doubles the drop, with a linear regression slopes below 0.5 for both groups) and the higher variability for patients with more severe impairment, with both approaches.
We hypothesized the reason behind the sb overestimation being the assumptions of the math model, which approximates the flow field as a 1d streamline, as opposite to the WERP based formulations, where the full 3d flow field is considered.
Using the sb approach on a 3d field means assuming that the velocity profile is blunt and with constant value over the whole plane, while real cases have clearly non-axisymmetric, skewed and eccentric distribution.
We therefore performed an in silico analysis on a 3d pipe with imposed flow field, evaluated the drop with sb and saw, and the sensitivity of the discrepancy between methods to several factors, to further investigate the sb overestimation.
We have clearly reported here that the discrepancy is only due to the profile shape, with the minimal discrepancy between methods for the most blunt shape.
On the contrary, a parabolic profile yielding a doubled pd estimate
----- Meeting Notes (3/21/16 16:29) -----
blunt in bold
As an ultimate contribution from the werp method, we have preliminarily investigated the applicability of the approach to echocardiography, by simulating the acquisition of 3d doppler echo from the real pcmri data.
Also, we have adapted the saw formulation from 2d to 1d-based integrals, to test the potential translation of werp to more easily available 2d echo doppler images.
To do so, we have evaluated the correlation with pcmri-derived estimates of peak systolic pd with saw against sb estimates, and saw estimates from 3d doppler and 2d doppler, slicing the velocity profile at vc with 12 planes with equally spaced angles, to simulate the arbitrary insonation plane during a 2d echo doppler acquisition.
And, to confirm our hp that the sb overestimation is all due to the shape of the velocity field, we have reported the high variability of the results due to the non-axisymmetric velocity at vc, here in a single case, and here in the linear regression study.
Despite the large variability as compared to the saw applied to 3d doppler, we see how the pd estimated with the saw on 2d doppler images is consistently lower than the sb values and closer to the ground truth of the full 3d advective pressure drop, suggesting that even when the data available is not as rich and comprehensive as with pc-mri, the use of the werp method might improve the clinical assessment of the as severity
To summarise, we have verified the FE-PPE sensitivity to masking, and its poor performance at boundaries, that compromises the computation of viscous component: in this case we have reported no improvement by changing masking/projection
Then we have proposed a method to improve viscous gradients computation by reconstructing boundary velocity, reported an overall improvement, but an high sensitivity to the vascular walls detection
After that, we have proposed a method to account for laminar hemodynamics, avoiding meshing, with reduced derivatives order, reported speed, accuracy and robustness to noise w.r.t. existing methods
Finally, we have proposed a formulation to improve Bernoulli, that might have a large clinical impact at a cost of further velocity data
----- Meeting Notes (3/21/16 16:18) -----
add contributions
contributions
sb first time seen
highlight the mechanism
At the moment the method is used to characterize the clinical outcome after aortic arch reconstruction in baby patients with hypoplastic left heart syndrome, and to better investigate the phenomenon of pressure recovery downstream of the stenosis
As for the future work, on the methodological front we are going to test a modified version of werp that accounts for flow through secondary branches of the vasculature and for regions with low-flow, which will also improve the performance of the method when accounting for compliance effects.
Moreover, we will work to include turbulent effects in the formulation, which will appear as an additional term in the work-energy balance
Finally we will set up phantom experiments to corroborate our findings and
together with the clinical staff at kcl we will develop a clinical-friendly tool to estimate non-invasively relative pressures using werp combined to cmr and other imaging techniques
