miguel lourenço rodrigues master’s thesis in biomedical engineering december 2011 1
TRANSCRIPT
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A Bayesion perfusion estimation using spatio-temporal priors in
ASL-MRIMiguel Lourenço Rodrigues
Master’s thesis in Biomedical EngineeringDecember 2011
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Outline
1. Introduction and Objectives
2. Methods: Problem Formulation, Simulations and Real Data
3. Results and Discussion
4. Conclusions
Outline
1. Introduction
2. Literature Review
3. Problem Formulation
4. Experimental Results and Discussion
5. Conclusions
3
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Introduction
-Cerebral Blood Flow (CBF):
Volume of blood flowing per unit time[2]
-Perfusion:
CBF per unit volume of tissues
Arterial Spin Labeling (ASL):
-Non invasive technique for generating perfusion images of the brain [1]
Se [1] e [2] são refs, deviam aparecer antes com nome e ano
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Introduction
Labeled acquisiton
1. Labeling of inflowingarterial blood
2. Image acquisition
ASL: Este slide e o seguinte deviam ser 1 só
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Introduction
ASL
Control image Labeled image CBF
A number of control-label repetitions is required in order to achieve sufficient SNR to detect the magnetization difference signal, hence increasing scan duration.
[C1, L1, C2, L2,…, Cn/2, Ln/2] n length vectorCi – ith control imageLi – ith labeled imageP- perfusion
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Introduction
ASL signal processing methods
Pair-wise subtraction:
[P1, P2,…, Pn/2]=[C1- L1, C2- L2,…, Cn/2-Ln/2]
Surround subtraction:
[P1, P2,…, Pn/2]=[C1- L1, C2- (L1+L2),…, Cn/2-(L(n/2)-1-Ln/2)] 2 2
Sinc-interpolated subtraction:
[P1, P2,…, Pn/2]=[C1- L1/2, C2- L3/2,…, Cn/2-Ln/2-1/2]
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Objectives
Objectives
-Increase image Signal to Noise Ratio (SNR)
-Reduce acquisition time
Approach
- New signal processing model
- Bayesian approach
- spatio-temporal priors
No drastic signal variatons
(except in organ boundaries)
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Outline
1. Introduction
2. Literature Review
3. Problem Formulation
4. Experimental Results and Discussion
5. Conclusions
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Problem Formulation
Mathematical model
Y(t)=F+D(t)+v(t)ΔM+Γ(t)
Y (NxMxL) – Sequence of L PASL images
F (NxM) – Static magnetization of the tissues
D(NxM x L) – Slow variant image (baseline fluctuations of the signal – Drift)
v(L x 1) - Binary signal indicating labeling sequences ΔM(NxM ) - Magnetization difference caused by the inversion
Γ(NxM xL) – Additive White Gaussian Noise ~N (0,σy2)
(1)
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Problem Formulation
Algorithm implementation
Y(t)=F+D(t)+v(t)ΔM+Γ(t) (1)
Vectorization
Y=fuT+D+ΔmvT+Γ
Y(NM x L)
f(NM x1)
u(L x 1)
D(NM x L)
v(L x 1)
Δm(NM x 1)
Γ(NM x 1)
(2)
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Problem Formulation
Algorithm implementation
Since noise is AWGN,
p(Y)~N (μ, σy2), where μ=fuT+D+ΔmvT
Maximum likelihood (ML) estimation of unknown images, θ={f,D, Δm}
θ=arg min Ey(Y,v,θ)θ
Ill-posed problem
(3)
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Problem Formulation
Algorithm implementation
Using the Maximum a posteriori (MAP) criterion, regularization isintroduced by the prior distribution of the parameters
θ=arg min Ey(Y,v,θ)θ
(3)
θ=arg min E (Y,v,θ)θ
(4)
E (Y,v,θ)=Ey (Y,v, θ) + Eθ(θ) (5)
Data – fidelity term Prior term
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Problem Formulation
Algorithm implementation
E (Y,v,θ)=Ey (Y,v, θ) + Eθ(θ) (5)
½ Trace [(Y-fuT-D-ΔmvT) T (Y-fuT-D-ΔmvT)] E (Y,v,θ)=
+αTrace[(φhD)T(φhD)+(φvD)T(φvD)+(φtD)T(φtD)]
+β(φhf)T(φhf)+(φvf)T(φvf)
+γ(φhΔm)T(φhΔm)+(φvΔm)T(φvΔm)
(6)
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Problem Formulation
Algorithm implementation
-In equation (6), the matrices φh,v,t are used to compute the horizontal, Vertical and temporal first order differences, respectively
1 0 0 . -1
-1 1 0 . 0
0 -1 1 0
. . . . .
. . . . .
. . . . 0
0 0 . -1 1
Φ=
-α, β and γ are the priors.
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Problem Formulation
Algorithm implementation
-MAP solution as a global mininum
-Stationary points of the Energy Function – equation (6)
- Equations implemented in Matlab and calculated iteratively
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Outline
1. Introduction
2. Literature Review
3. Problem Formulation
4. Experimental Results and Discussion
5. Conclusions
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Experimental Results and Discussion
Synthetic data
-Brain mask (64x64)
-Axial slice
-White matter (WM) and Gray matter (GM)
ISNR=SNRf-SNRi
∑100
NxM
N,M
i=1,j=1
|xi,j-xi,j|
xi,j
^
Mean error(%)=
SNR=Asignal
Anoise
2
- ;
-
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Experimental Results and Discussion
Synthetic data
Control acquisition Labeled acquisition
Parameters:
σ=1Δm(GM)=1Δm(WM)=0.5D=[-1,1]F=10000α=0β=0γ=0
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Experimental Results and Discussion
Synthetic data
Proposed algorithm
Pair-wisesubtraction
SurroundSubtraction
Parameters:
σ=1Δm(GM)=1Δm(WM)=0.5D=[-1,1]F=10000α=0β=0γ=0
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Experimental Results and Discussion
Synthetic data
Method ISNR(dB)
Mean Error (%)
Proposed algorithm 13.906 24.658
Pair-wise subtraction 13.906 24.658
Surround Subtraction 13.999 24.393
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Experimental Results and Discussion
Synthetic data
Parameters:
σ=1Δm(GM)=1Δm(WM)=0.5D=[-1,1]F=10000α=1β=1γ=5
Proposed algorithm
Pair-wisesubtraction
SurroundSubtraction
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Experimental Results and Discussion
Synthetic data
Parameters:
σ=1Δm(GM)=1Δm(WM)=0.5D=[-1,1]F=10000α=1β=1γ=5
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Experimental Results and Discussion
Synthetic data
Method ISNR(dB)
Mean Error (%)
Proposed algorithm 16.990 17.807
Pair-wise subtraction 14.026 24.492
Surround Subtraction 14.103 24.269
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Experimental Results and Discussion
Synthetic data
Method ISNR(dB)
Mean Error (%)
Proposed algorithm 16.990 17.807
Pair-wise subtraction 14.026 24.492
Surround Subtraction 14.103 24.269
3dB
7%
23%
-30%
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Experimental Results and Discussion
Synthetic data
Monte Carlo Simulation for different noise levels
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Experimental Results and Discussion
Real data
-One healthy subject
-3T Siemens MRI system (Hospital da Luz, Lisboa)
-PICORE-Q2TIPS PASL sequence
-TI1/TI1s/TI2=750ms/900ms/1700ms
-GE-EPI
-TR/TE=2500ms/19ms
-201 repetitions
-spatial resolution: 3.5x3.5x7.0 mm3
-Matrix size: 64x64x9
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Experimental Results and Discussion
Real data
Proposed algorithm
Pair-wisesubtraction
SurroundSubtraction
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Proposed algorithm
Pair-wisesubtraction
SurroundSubtraction
Experimental Results and Discussion
Real data
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Outline
1. Introduction
2. Literature Review
3. Problem Formulation
4. Experimental Results and Discussion
5. Conclusions
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Conclusion
-The proposed bayesian algorithm showed improvement of SNR and ME
-SNR increased by 3db (23%)
-ME decreased by 7% (30%)
-Applied to real data
Future work:
-Automatic prior calculation
-Reducing the number of control acquisitions
-Validation tests on empirical data
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[1] T.T. Liu and G.G. Brown. Measurement of cerebral perfusion with arterial spin labeling: Part 1. Methods. Journal of the International neuropsychological Society, 13(03):517-525, 2007.
[2]A.C. Guyton and J.E. Hall. Textbook of medical physiology. WB Saunders (Philadelphia),1995.
[4]ET Petersen, I. Zimine, Y.C.L. Ho, and X. Golay. Non-invasive measurement of perfusion: a critical review of arterial spin labeling techniques. British journal of radiology, 79(944):688, 2006.
[3]D.S. Williams, J.A. Detre, J.S. Leigh, and A.P. Koretsky. Magnetic resonance imaging of perfusion using spin inversion of arterial water. Proceedings of the National Academy of Sciences, 89(1):212, 1992.
[5]R.R. Edelman, D.G. Darby, and S. Warach. Qualitative mapping of cerebral blood flow and functional localization with echo-planar mr imaging and signal targeting with alternating radio frequency. Radiology, 192:513-520, 1994.
Bibliography
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[8]W.C. Wu and E.C. Wong. Feasibility of velocity selective arterial spin labeling in functional mri. Journal of Cerebral Blood Flow & Metabolism, 27(4):831{838, 2006
[9]GK Aguirre, JA Detre, E. Zarahn, and DC Alsop. Experimental Design and the Relative Sensitivity of BOLD and Perfusion fMRI. NeuroImage, 15:488{500, 2002.
[10]E.C. Wong, R.B. Buxton, and L.R. Frank. Implementation of Quantitative Perfusion Imaging Techniques for Functional Brain Mapping using Pulsed Arterial Spin Labeling. NMR in Biomedicine, 10:237{249, 1997.
[11] J.M. Sanches, J.C. Nascimento, and J.S. Marques. Medical image noise reduction using the Sylvester-Lyapunov equation. IEEE transactions on image processing, 17(9), 2008.