the university of north carolina at chapel hill adaptive weighted deconvolution model to estimate...
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The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL
Adaptive Weighted DeconvolutionModel to Estimate the Cerebral Blood Flow Function in Dynamic Susceptibility Contrast MRI
Jiaping Wang, Ph.D
Department of Mathematical Science
University of North Texas at Denton
Joint work with Drs. Hongtu Zhu and Hongyu An
from UNC-CH
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Outline
Background and Motivation
Adaptive Weighted De-convolution Model Simulation Studies
Real Data Analysis
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Part 1. Background and Motivation
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Dynamic Susceptibility Contrast (DSC) Perfusion MRI measures the passage of a bolus of a non-diffusible contrast through the brain. The signal decreases as the bolus passes through the imaging slices.
Background
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Convolution Relationship
= R(t) where Ca(t) is the given AIF, C(t) is the observed concentrationfunction, which is computed as S(t)/S0. We are interested in estimating the residue function R(t).
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Deconvolution Techniques
Fourier Transformation SVD TSVD at 0.01
TSVD at 0.05 TSVD at 0.1 TSVD at 0.2
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Part 2. Adaptive Weighted Deconvolution Model
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Notations
D : 3D volume
N : the number of points on D
d : a voxel in D
: : spatial-temporal process
: error process
: AIF function, constant along space : Residue function
}],,0[:),({ DdTtdtC
}],,0[:),({ DdTtdt )(tCa),( dtR
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Voxel-wise Approach
),()(),(),()(),(),( dtduuCdutRdttCdRdtC aa
Frequency-Domain
dtTiftdtCdfFT
C ))(/2exp(),(),(0
fdtFfFdfFdfFaCRC for ),()(),(),(
Temporal-Domain
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for ),()(),(),( fdfFfFdfFdfFaCRC
fdffdfdfaCRC for ),()(),(),(
))/(2exp(),(),(0
TiftdtCdfT
tC
( f ,d) ~ (0,1( f f ') ( f , f ';d,d'))
Discrete
Continuous
),(),(for smooth y piecewisel is ),( dfNdfdfH
Key Assumptions:
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Two Main Steps (Spatial Adaptive Approach): 1. Transform the time series into the Fourier or Wavelet domain.2. Smoothing the curves in the frequency domain by involving the local neighborhood information.
Voxel-wise vs. Spatio-Interdependence
Jumping Space Irregular Boundary
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),(),(for ),()(),(),( dfNdfdffdfdfaCRC
)','()'(),(
)','()'()','()','(
dffdf
dffdfdf
a
a
CR
CRC
UnknownApproximation
),(),(),);,(()','( hdBrfrfrhdfBdf
Spatial-Adaptive Approach
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),;',',,()]'(),()','([)),);,(();,(( 2
)','(
rhdfdfwfdfdfhrdfBdfLaCRC
dfR
Weighted LSE
),;',',,()'()'(/),;',',,()'()','(),(ˆ)','()','(
rhdfdfwffrhdfdfwfdfdfaaa CC
dfCC
dfR
)),(ˆ(Var dfR
)/2/()]/2cos(1)[2exp(),(ˆ),(ˆ 221
0
TtTttfidfdtRT
kkkR
Estimated HRF
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Being Hierarchical
Drawing nested spheres with increasing radiuses at each voxel and each frequency
S
S
rrr
hhh
10
10
…
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• Sequentially determine weights • Adaptively update
Being Adaptive
)(),(ˆ iR df
)(),;',',,( irhdfdfw
))','(ˆ,),(ˆ(|)'(||)'(|),;',',,( )1()1(2,1,
)( iR
iRhlochloc
i dfdfKddKffKrhdfdfw
)1(),;',',,( irhdfdfw
)1(),(ˆ iR df
)(),;',',,( irhdfdfw
)(),(ˆ iR df
)),(ˆ),(ˆ( )1()( iR
iR dfdfS
Stopping Statistics
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How to determine ?
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Part 3. Simulations
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(i) A temporal cut of the true images;
(ii) The true curves C(t)
(iii) The true curves R(t)
Simulation Set-up
The true residue curves
(iv) The AIF Curve
(i) (ii) (iii) (iv)
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Simulation Results
Result from SWADM Cluster Result Mean Curves of Clusters from SWADM
Comparison Statistics: Dd=
Where Xd is the estimated curve from the proposed method, Yd is from other methods including voxel-wise inverse Fourier Transformation (IFT), SVD, TSVD at thresholds 0.01, 0.05, 0.1 and 0.2, respectively. ||•|| is a norm operator.
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(1) Comparison with SVD;
Comparison Results
(2) Comparison with TSVD at 0.01; (3) Comparison with TSVD at 0.05;
(4) Comparison with TSVD at 0.1;(5) Comparison with TSVD at 0.2; (6) Comparison with Voxel-wise IFT;
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Comparison along SNRs
Average of Dd along different SNRs One sample t test for Dd
TSVD at 0.01
TSVDat 0.05
TSVDat 0.1
TSVDat 0.2
Voxel-wiseIFT
SVD
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Part 4. Real Data
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(e)
The DSC PWI data set obtained from an acute ischemic strokepatient at Washington University in St. Louis after receiving a signed consent form with Institutional Review Board approval.
MR images were acquired on a 3T Siemens whole body Trio system (Siemens Medical Systems, Erlangen, Germany). PWI imageswere acquired with a T2*-weighted gradient echo EPI sequence (TR/TE= 1500/43 ms,14 slices with a slice thickness of 5 mm, matrix= 128x128). This sequence was repeated 50 times and Gadolinium diethylenetriamine penta-acetic acid (Gd-DTPA, 0.1 mmol/kg) was injected at the completion of the 5th measure.
Data Description
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Slices from C(t) images
Sample of C(t) curves, the largest one can be considered as AIF.
Clustered pattern
Mean curves of clusters
Clustering Results
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Estimation Results from Different Methods
The curves from same voxel in Cluster I The curves from same voxel in Cluster II
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Thank You!