Mapping Tissue Microstructure using Spherical PolarFourier Diffusion MRI

Jian Cheng

National Institutes of Health

May 5, 2016

Outline

1 Problems and Objective

2 Compressed Sensing Reconstruction Using SPFI

3 DMRITOOL

4 Conclusion

Outline

1 Problems and Objective

2 Compressed Sensing Reconstruction Using SPFI

3 DMRITOOL

4 Conclusion

Diffusion MRI Data Processing Pipeline

Diffusion Data in 6D space

3D x-space (spatial space) and 3D R-space (diffusion displacement space)

Ensemble Average Propagator (EAP) P(R)

EAP field: P(x,R); ODF field: Φk(x,R)

[Descoteaux 2008, Hagmann 2006]

EAP profile P(R0r)Orientation DistributionFunction (ODF) Φk(r)

Φk(r) def= 1Z

∫ ∞0

P(Rr)RkdR

Diffusion Weighted Imaging (DWI)

The Pulse Gradient Spin-Echo (PGSE)sequence [Stejskal and Tanner 1965]

Narrow pulse assumption (δ � ∆)E(q) F3D⇐⇒ P(R)

[Callaghan, 1991]

q = qu F3D⇐⇒ R = Rr

E(q) = S(q)S(0) q = γδG/2π

Fourier relationP(R) =

∫E(q)e−2πiq·Rdq

E(q) =∫

P(R)e−2πiq·RdR

Diffusion Tensor Imaging (DTI)

Free diffusion, Gaussian diffusion assumption [Basser 1994]

P(R) = 1√(4πτ)3|D|

exp(− 1

4τ RTD−1R)

D =3∑

i=1λiv ivT

i

Diffusion tensor model [Basser 1994]

E(q) = S(q)/S(0) = exp(−4π2τqTDq) = exp(−buTDu)

b = 4π2τ‖q‖2

Tensor estimation (> 6 DWIs)

Modeling Beyond DTI: HARDI

Sampling in DTI, in Cartesian grid, single shell and multiple shells

sampling in DTI

Diffusion Tensor Imaging (DTI)[Basser1994]

Dense Cartesian grid

Diffusion Spectrum Imaging (DSI)[Callaghan 1991; Wedeen 2000, 2005]

Single shell sampling

sHARDI:Q-Ball Imaging (QBI), exact QBI[Tuch 2004; Descoteaux 2007; Aganj 2010]Diffusion Orientation Transform (DOT)[Ozarslan 2006]......

Multiple shell or arbitrarily sampling

mHARDI:Diffusion Propagator Imaging (DPI)[descoteaux 2010]Spherical Polar Fourier Imaging (SPFI)[Assemlal 2009; Cheng 2010]Simple Harmonic Oscillator basedreconstruction and estimation (SHORE,MAP-MRI)[Ozarslan 2009, 2013; Cheng 2011a]...

P(R) =∫

E(q)e−2πiq·Rdq, b = 4π2τ‖q‖2

Problems

Reconstruction:1 Reconstruction of continuous diffusion signal from its limited number of

samples with noise.2 Reconstruction of diffusion propagator which is the Fourier transform of

diffusion signal under some condition.3 Extract some other quantities: ODF, return-to-origin probability, anisotropy,

axon dimeter, etc.

Objective

1 Robust reconstruction from limited number of samples with noise.Continuous diffusion signal modelRobus reconstruction algorithmGood sampling scheme

2 Closed form solutions: solve reconstruction problems simultaneously.Continuous diffusion signal model with good Mathematical properties.

Outline

1 Problems and Objective

2 Compressed Sensing Reconstruction Using SPFI

3 DMRITOOL

4 Conclusion

Prior knowledge and assumption in reconstruction

Reconstruction signal in 1D space from its samples.

q

E(q)

1

No assumption on signal (E(q) ∈ C1(R))): cannot perform reconstructionMaximal frequency (DSI): Nyquist sampling rate.Gaussian assumption (DTI) : one sample, break Nyquist rate.

Prior knowledge V.S. Assumption

Diffusion signal model

A general signal model and its reconstruction:

mina

∑s

(f (qs|a)− Es)2 + R(a)

DTI:min

D

∑s

(exp(−4π2τq2

s Dqs)− Es)2

minD

∑s

(q2

s Dqs + 14π2τ

log(Es))2

linear representation model:

E(q) =∑

iBi(q)ai

mina

∑s

(∑i

Bi(qs)ai − Es

)2

+ R(a)

Model and regularization are designed based on prior knowledge orassumption

Sparsity prior

Assumption: the latent signal can be sparsely represented by a givendictionary.

E(q) =∑

iBi(q)ai , E = Ma

Sparsity:‖a‖0 =

∑iδ(ai = 0)

Sparsity assumption is always true because the dictionary can bedevised to have atoms which are similar with the signal itself.

Basic of Compressed Sensing

Basis pursuit deionising:

minx‖x‖1 s.t. ‖Ax − y‖2 ≤ δ (Pδ

1 )

Theorem (Stability of (Pδ1 ) [Donoho 2006])

For the problem (Pδ1 ) defined by (A, y, δ), if x0 ∈ Rn satisfies ‖x0‖0 < 1+1/µ(A)

4and ‖Ax0 − y‖2 ≤ δ, then the solution xδ1 of the problem (Pδ

1 ) obeys

‖xδ1 − x0‖22 ≤4δ2

1− µ(A)(4‖x0‖0 − 1)

We have to choice to make the estimation error ‖xδ1 − x0‖22 small, consideringA = MD

1 make µ(A) small =⇒ sampling matrix design for M2 make ‖x0‖0 small =⇒ dictionary learning for D

Compressed Sensing Reconstruction

Several variants:Prior knowledge on signal representation error:

minx‖x‖1, s.t.‖Ax − b‖2 ≤ δ

Prior knowledge on sparsity:

minx‖Ax − b‖2, s.t.‖x‖1 ≤ δ

No above prior knowledges:

minx‖Ax − b‖2 + λ‖x‖1

Reconstruction in dMRI

Reconstruction:1 Reconstruction of continuous diffusion signal from its limited number of

samples with noise.2 Reconstruction of diffusion propagator which is the Fourier transform of

diffusion signal under some condition.3 Extract some other quantities: ODF, return-to-origin probability, anisotropy,

etc.Ideas: compressed sensing reconstruction using a good basis:

1 Complete continuous basis (capture complexity)2 Sparse representation (capture prior knowledge).3 Analytical Fourier transform: simultaneously reconstruction of diffusion signal

and diffusion propagator.4 Closed form solutions for features (ODF, RTO, anisotropy, etc).

Spherical Polar Fourier Imaging (SPFI)

Spherical Polar Fourier expression of signals [Assemlal 2008, 2009]

E(q) =N∑

n=0

L∑l=0

l∑m=−l

anlmGn(q|ζ)Y ml (u) BSPF

nlm(q) = Gn(q|ζ)Y ml (u)

Gaussian Laguerre polynomialGn(q) = κn(ζ) exp

(− q2

2ζ

)L1/2

n ( q2

ζ ), κn(ζ) =[

2ζ3/2

n!Γ(n+3/2)

]1/2

Sparsity of SPF basis

Scale parameterSignal anisotropy

Analytical Transforms in SPFI

P(R) is analytically obtained from Fourier dual SPF (dSPF) basis [Cheng 2010a]

P(Rr) =N∑

n=0

L∑l=0

l∑m=−l

anlmFnl(R)Y ml (r) BdSPF

nlm (R) = Fnl(R)Y ml (r)

Fourier dual SPF (dSPF) basis, orthonormal basis in R-space

BdSPFnlm (R) =

∫R3 BSPF

nlm(q)e−2πiqT Rdq = Fnl(R)Y ml (r)

Fnl(R) = 4(−1)l/2 ζ0.5l+1.5πl+1.5Rl0

Γ(l+1.5) κn(ζ)∑n

i=0(−1)i(n+0.5n−i

) 1i!2

0.5l+i−0.5Γ(0.5l + i + 1.5)1F1( 2i+l+32 ; l + 3

2 ;−2π2R2ζ)

A linear transform from {anlm} to EAP profile represented by SH basisFor a given R0, EAP profile

P(R0r) =L∑

l=0

l∑m=−l

cPlm(R0)Y m

l (r), cPlm(R0) =

N∑n=0

anlmFnl(R0)

Analytical Transforms in SPFI

Linear transform for ODF by Tuch Φt(r) def= 1Z∫∞

0 P(Rr)dR [Cheng 2010b]

Φt(r) =L∑

l=0

l∑m=−l

cΦtlmY m

l (r)

Linear transform for ODF by Wedeen Φw(r) def=∫∞

0 P(Rr)R2dR [Cheng 2010b]

Φw(r) =L∑

l=0

l∑m=−l

cΦwlm Y m

l (r)

Analytical transform avoids the numerical error{E(qi)}→{anlm}→{cP

lm(R0)}, {cΦtlm}, {c

Φwlm }, RTO, MSD, PFA, NG, RTAP

RTO: return-to-origin probability, P(0).MSD: mean-squared displacement,

∫R3 P(R2)R2RdR

PFA: propogator fractional anisotropy.NG: non-gaussianity.RTAP: return-to-axis probability.

Tensorial SPFI: affinely transformed representation

Limitation of SPF basis: inefficient to represent highly anisotropic signal.Tensorial SPFI (inspired by MAP-MRI [Ozarslan 2013] ): D = QΛ2QT , QTQ = I

E(qu | D) =√

|Λ|N∑

n=0

L∑l=0

l∑m=−l

anlmGn

(q√

uT Du | ζ0

)Y m

l

(ΛQT u

‖ΛQT u‖

)

P(Rr | D) =1√|Λ|

∑nlm

anlmFnl

(R√

rT D−1r | ζ0

)Y m

l

(Λ−1QT r

‖Λ−1QT r‖

)Isotropic tensor ⇒ general tensor.Efficient to represent anisotropic signal.

Tensorial SPFI: affinely transformed representation

CS Reconstruction: [Cheng 2010, 2011]

mina′‖M′a′ − e′‖22 + ‖Ha′‖1

Constraint: E(0) = 1CS Reconstruction using learned basis: [Cheng 2013, 2015]

minc‖M′Wc − e′‖22 + ‖Vc‖1

Learned basis M′WOriginal basis + learned regularization HW, if we set a′ = Wc and V = HW.Full regularization matrix.

My journey:L2 SPFI (2010) ⇒ L1 SPFI (2011) ⇒ DL-SPFI (2013) ⇒ DL-TSPFI (2015)

Learning the Dictionary using Synthetic Signals

Real signal: limited samples with noise.Synthetic signal:

tensor model, cylinder modelmany samples without noise from given modelsMSPF is orthonormal matrixLearn a dictionary in model-free method using model-based priors.

Sparsity of Tensors

321 orientations, different FA, same MD D0 = 0.7× 10−3.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

20406080

100120140160180

FA (MD=0.6× 10−3 mm2/s)

Num

ber

ofN

onze

roCo

effici

ents SPF, single tensor

DL-SPF, single tensor

TSPF, DL-TSPF, single tensor

SPF, mixture of tensors

DL-SPF, mixture of tensors

TSPF, mixture of tensor

DL-TSPF, mixture of tensor

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

20406080

100120140160180

FA (MD=1.1× 10−3 mm2/s)N

umbe

rof

Non

zero

Coeffi

cien

ts SPF, single tensor

DL-SPF, single tensor

TSPF, DL-TSPF, single tensor

SPF, mixture of tensors

DL-SPF, mixture of tensors

TSPF, mixture of tensors

DL-TSPF, mixture of tensors

Figure: Synthetic Experiments. The average number of non-zero coefficients for SPF,DL-SPF, TSPF and DL-TSPF basis.

Note: MD range in learning process is [0.5, 0.9]× 10−3mm2/s

Synthetic Data Using Cylinder Model

RMSE in reconstruction

30 35 40 45 50 55 60 65 70 75 80 85 900

0.01

0.02

0.03

0.04

0.05

Crossing Angle (no noise)

RMSE

l1-SPFI

DL-SPFI

DL-TSPFI

30 35 40 45 50 55 60 65 70 75 80 85 900

0.05

0.1

0.15

0.2

0.25

Crossing Angle (SNR=20)

RMSE

l1-SPFI

DL-SPFI

DL-TSPFI

Real DSI Data

DSI scheme. Results from 515 samples V.S. result from 171 samples,RMSE = 2.82%

Real DSI Data

Results from 515 samples V.S. result from 171 samples, RMSE = 2.82%

TSPFI coefficients SPFI coefficients TSPFI DWI SPFI DWI

Real HCP Data

3 shells with b values 1000,2000,3000. 90 samples per shell.

FA eap profile 15µm

Real HCP Data

MSD RTO RTAP

NG PFA FA

Outline

1 Problems and Objective

2 Compressed Sensing Reconstruction Using SPFI

3 DMRITOOL

4 Conclusion

DMRITOOL

https://diffusionmritool.github.ioOpen source software in c++ and matlab.What?

Reconstruction in diffusion MRI: EAP, ODF, fiber ODF, scalar maps.Uniform sampling scheme design, sub-sampling.Data visualization: nifti image, spherical function field (ODF, EAP profile).Data simulation using mixture of tensor (or cylinder) model.

Why?Reproducible research, fair comparison.Useful for users.

Outline

1 Problems and Objective

2 Compressed Sensing Reconstruction Using SPFI

3 DMRITOOL

4 Conclusion

Conclusion

Compressed sensing reconstructionL2 SPFI (2010) ⇒ L1 SPFI (2011) ⇒ DL-SPFI (2013) ⇒ DL-TSPFI (2015)Analytic dictionary ⇒ dictionary learningClosed form solutions for variant features (ODF, anisotropy, RTO, MSD).

Software: https://diffusionmritool.github.io

Acknowledgement

NIH: Peter J. Basser, Richard Leapman, Ruiliang Bai, Alexandru Avram, DanBenjamini, Elizabeth Hutchinson, Okan Irfanoglu, Michal KomloshUNC: Pew-Thian Yap, Dinggang Shen, Hongtu ZhuCAS: Tianzi JiangINRIA: Rachid Deriche, Aurobrata Ghosh

Thank you!Questions?