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Adaptive Weighted DeconvolutionModel to Estimate the Cerebral Blood Flow Function in Dynamic Susceptibility Contrast MRI

Jiaping Wang, Ph.D

Department of Mathematical ScienceUniversity 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 HierarchicalDrawing nested spheres with increasing radiuses at each voxel and each frequency

S

S

rrrhhh

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!