DTI versus higher order models DTI Fibers 2nd-order tensor higher orders Raw data [Basser94] approaches Numerous approaches exist: GDTI [Liu03], PASMRI [Jansons03], HODT / DOT [¨ Ozarslan03,06], Spherical Deconvolution [Tournier04], ODF [Tuch04, Descoteaux-etal05], etc. (GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 23 / 68
Spherical Harmonics: Definition Spherical harmonics are a basis for complex functions on the unit sphere. We use a modified basis constrained to be real and symmetric (imaginary and non-symmetric parts = noise) ∀ ( θ q , φ q ) ∈ Ω S = [0 , π ] × [0 , 2 π ) , S : Ω S → R S ( θ q , φ q ) = � N j =0 c j Y j ( θ q , φ q ) = ˜ BC j ( θ q , φ q ) , (1) Y 1 ( θ 1 , φ 1 ) . . . Y N ( θ 1 , φ 1 ) . . ... with ˜ . . B = . . Y 1 ( θ n s , φ n s ) . . . Y N ( θ n s , φ n s ) (GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 24 / 68
Angular Atoms √ 2 R ( Y m l ) if 0 < m ≤ l Y 0 y m l , if m = 0 = with l ∈ 2 N (2) l √ 2 ℑ ( Y | m | ) if − l ≤ m < 0 l Figure: Real and symmetric spherical harmonics: first orders l = 0 , 2 , 4 , 6. Blue indicates a negative value, whereas indicates red a positive value. (GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 25 / 68
Angular profile using Spherical Harmonics (a) (a) 6 coefficients Figure: Square sampled along a Figure: Angular reconstruction along 5-subdivided icosahedron. with increasing truncation order L . (GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 26 / 68
Angular profile using Spherical Harmonics (a) (a) 15 coefficients Figure: Square sampled along a Figure: Angular reconstruction along 5-subdivided icosahedron. with increasing truncation order L . (GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 26 / 68
Angular profile using Spherical Harmonics (a) (a) 28 coefficients Figure: Square sampled along a Figure: Angular reconstruction along 5-subdivided icosahedron. with increasing truncation order L . (GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 26 / 68
Angular profile using Spherical Harmonics (a) (a) 45 coefficients Figure: Square sampled along a Figure: Angular reconstruction along 5-subdivided icosahedron. with increasing truncation order L . (GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 26 / 68
Angular profile using Spherical Harmonics (a) (a) 66 coefficients Figure: Square sampled along a Figure: Angular reconstruction along 5-subdivided icosahedron. with increasing truncation order L . (GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 26 / 68
Angular profile using Spherical Harmonics (a) (a) 91 coefficients Figure: Square sampled along a Figure: Angular reconstruction along 5-subdivided icosahedron. with increasing truncation order L . (GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 26 / 68
Angular profile using Spherical Harmonics (a) (a) 120 coefficients Figure: Square sampled along a Figure: Angular reconstruction along 5-subdivided icosahedron. with increasing truncation order L . (GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 26 / 68
Angular profile using Spherical Harmonics (a) (a) 153 coefficients Figure: Square sampled along a Figure: Angular reconstruction along 5-subdivided icosahedron. with increasing truncation order L . (GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 26 / 68
Orientation Density Function (ODF) The ODF Ψ at direction u is defined as the radial projection of the diffusion PDF � ∞ Ψ( u ) = o P ( α u ) d α (3) � = P ( r , θ, z ) δ ( θ, z ) rdrd θ dz ODF Problem: How to get the PDF ? (GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 27 / 68
Funk-Radon Transform (FRT) The FRT (4) is a smoothed estimation of the true ODF (3) [Tuch04] � G q ′ ( u ) = 2 π q ′ P ( r , θ, z ) J 0 (2 π q ′ r ) rdrd θ dz (4) FRT Besides, the ODF can be directly expressed from diffusion signal in spherical harmonics by a Least Square minimization [Descoteaux06] ODF ≈ G q ′ = ˜ P ˜ � B C = 2 π P l j (0) c j Y j (5) j (GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 28 / 68
Least Square ODF estimation Least Square ODF estimation : Enables resolution of any specific local structure (crossing fibers) Model-free method: no assumption on macroscopic diffusion Light matrix computations But... Least Square method not adapted to Rice noise model [Sijbers98] No guaranty on spatial coherence of the ODF field (GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 29 / 68
Variational estimation of ODF � � Ω S [ � n s � min C :Ω C → R N E ( C ) = k ψ ( | D k | )] + αϕ ( ||∇ C || ) d Ω S , (6) j ˜ B k , j ˜ P − 1 with D k ( p ) = S k ( p ) − � C j ( p ) j The best fitting coefficients are computed with a gradient descent coming from the Euler-Lagrange derivation of the energy E . This leads to a set of multi-valued partial derivate equation. C t =0 = U 0 (7) ′ ( ||∇ C || ) ∂ C j B q , j + α div( ϕ q ψ ′ ( | D q | ) sign( D q ) P − 1 ˜ = � ∇ C ) ∂ t j ||∇ C || (GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 30 / 68
Advantages: regularity Ensure a global regularity of the ODF field: ϕ ( ||∇ C ( p ) || ) Spherical harmonics coefficients characterize anistropy [Frank02] l = 2 Single fibers l = 4 l = 0 orientation Several fibers Isotropic diffusion orientation Example of possible regularization function ϕ (GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 31 / 68
Advantages: adaptive to noise distribution ψ -likelihood function adapted to MRI noise law: The best ψ function is the one specific to MR scanners, ie. Rice distribution We seek S r which maximizes a posteriori (MAP) the log-posterior probability log p ( S r | S ) = log p ( S | S r ) + log p ( S r ) − log p ( S ) (8) Consequently the pointwise likelihood is σ 2 − ( S 2 + S 2 � S · S r � ψ ( S r ) = log p ( S | S r , σ ) = log S r ) + log I 0 (9) 2 σ 2 σ 2 [Basu,2006] (GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 32 / 68
Simulation: influence of Rice model (GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 33 / 68
Results on synthetical data ODF field DTI field Results are good on perfect datasets, what about MRI acquisition noise ? (GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 34 / 68
Results on human brain hardi data DTI field GFA (GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 35 / 68
Results on human brain hardi data ODF field GFA (GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 36 / 68
Simulation: energy minimization (GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 37 / 68
Simulation: regularization without with zoom GFA (GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 38 / 68
DTI GFA (GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 39 / 68
Consequences on fiber-tracking LS GFA (GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 40 / 68
Consequences on fiber-tracking LS GFA (GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 41 / 68
Consequences on fiber-tracking PDE GFA (GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 42 / 68
Fibertracking: regularization As for DTI models, ODF fibertracking is very sensitive to noise. without regularization with regularization (GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 43 / 68
Context 1 DTI model : Diffusion Tensor Imaging 2 QBI model : Q-Ball Imaging 3 Measuring the PDF : DSI and Multi Q-Ball 4 Conclusions & Perspectives 5 (GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 44 / 68
Signal acquisition Figure: Diffusion MRI acquisition steps. Why measuring the PDF ? The PDF brings new important radial information. (GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 45 / 68
Interest of Radial part of the PDF Information on cells micro-structure that composed the organic tissue. Ex: axon diameter, number of compartments. Spinal cord [Cohen02] May increase detection of anomalies such as demyelinization, a symptom of multiple sclerosis. Figure: Myelination of an axone [www.jdaross.cwc.net] (GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 46 / 68
Diffusion in a bi-homogeneous environment Figure: Experimental graph: human erythocytes rate for decreasing values of hematocrites. [Kuchel97] Empirical approximation of signal by a bi-exponential function (compartiments: intra/extra diffusion). (GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 47 / 68
Diffusion in a complex environment Figure: Simulation plot: fibers set of various diameters. [Cohen02] Observations Important information are found in the radial diffusion profile. (GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 48 / 68
High Angular Resolution Diffusion Imaging The Fourier Transform � q E ( q ) exp( − 2 π i q T p ) d q [Cory90,Callaghan91] PDF ( p ) = DSI: Fourier transform [Wedeen00] Very long acquisition time Needs high gradients ⇒ Magnetic field distortion (GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 49 / 68
High Angular Resolution Diffusion Imaging The Fourier Transform � q E ( q ) exp( − 2 π i q T p ) d q [Cory90,Callaghan91] PDF ( p ) = DSI: Fourier transform [Wedeen00] Very long acquisition time Needs high gradients ⇒ Magnetic field distortion Problem The DSI is not clinical-compliant. (GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 49 / 68
High Angular Resolution Diffusion Imaging The Funk-Radon Transform � ODF ( k ) = u ⊥ k E ( u ) d u HARDI: High Angular Diffusion Imaging [Tuch02] Reduced acquisition time Lack of radial information (GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 49 / 68
High Angular Resolution Diffusion Imaging The Funk-Radon Transform � ODF ( k ) = u ⊥ k E ( u ) d u HARDI: High Angular Diffusion Imaging [Tuch02] Reduced acquisition time Lack of radial information Problem The ODF does not give any radial information. (GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 49 / 68
HARDI Extension: multi-sphere imaging Figure: Example of HARDI extension [Assaf05, ¨ Ozarslan06, Wu07, Khachaturian07, Assemlal-et.al08, Assemlal-et.al09]. Better distribution of samples on the Q -Space. (GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 50 / 68
HARDI Extension: multi-sphere imaging Figure: Example of HARDI extension [Assaf05, ¨ Ozarslan06, Wu07, Khachaturian07, Assemlal-et.al08, Assemlal-et.al09]. Better distribution of samples on the Q -Space. Problem Still insufficient number of samples for a Fourier transform. Which mathematical tool for the signal estimation ? (GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 50 / 68
Continuous representation of the signal Continuous representation of the MR signal E in the following basis (Spherical Polar Fourier SPF): � q ∞ ∞ l � � � � a nlm R n ( || q || ) y m E ( q ) = (10) l || q || n =0 l =0 m = − l where a nlm expansion coefficients, R n and y m are respectively are radial l and angular atoms. The basis is orthonormal in spherical coordinates: � q � q � � �� � �� R n ′ ( || q || ) y m ′ R n ( || q || ) y m · d q = δ nn ′ ll ′ mm ′ l ′ l || q || || q || q ∈ R 3 (11) (GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 51 / 68
Radial Atoms � 2 � 1 / 2 −|| q || 2 � || q || 2 n ! � � � L 1 / 2 R n ( || q || ) = exp (12) n γ 3 / 2 Γ ( n + 3 / 2) 2 γ γ (GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 52 / 68
Radial Atoms � 2 � 1 / 2 −|| q || 2 � || q || 2 n ! � � � L 1 / 2 R n ( || q || ) = exp (12) n γ 3 / 2 Γ ( n + 3 / 2) 2 γ γ Figure: Some radial atoms R n , Figure: Experimental plot [Regan06] γ = 100 (GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 52 / 68
Radial Atoms (a) Signal de diffusion Figure: Radial reconstruction along with increasing truncation order N . (GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 53 / 68
Radial Atoms 1.1 1 Truth 0.9 Sample MR attenuation 0.8 Reconstruction 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 5 10 15 20 25 30 35 40 q (a) 1 Coefficient Figure: Radial reconstruction along with increasing truncation order N . (GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 53 / 68
Radial Atoms 1.1 1 Truth 0.9 Sample MR attenuation 0.8 Reconstruction 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 5 10 15 20 25 30 35 40 q (a) 2 Coefficients Figure: Radial reconstruction along with increasing truncation order N . (GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 53 / 68
Radial Atoms 1.1 1 Truth 0.9 Sample MR attenuation 0.8 Reconstruction 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 5 10 15 20 25 30 35 40 q (a) 3 Coefficients Figure: Radial reconstruction along with increasing truncation order N . (GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 53 / 68
Radial Atoms 1.1 1 Truth 0.9 Sample MR attenuation 0.8 Reconstruction 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 5 10 15 20 25 30 35 40 q (a) 4 Coefficients Figure: Radial reconstruction along with increasing truncation order N . (GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 53 / 68
Radial Atoms 1.1 1 Truth 0.9 Sample MR attenuation 0.8 Reconstruction 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 5 10 15 20 25 30 35 40 q (a) 5 Coefficients Figure: Radial reconstruction along with increasing truncation order N . (GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 53 / 68
Fitting the data How to fit the data From the diffusion samples, how do we retrieve the SPF coefficients ? (GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 54 / 68
Linear signal estimation The coefficient estimation is computed by the linear damped least square method: || E − MA || 2 + λ l || L || 2 + λ n || N || 2 A = arg min (13) A = ( M T M + λ l L T L + λ n N T N ) − 1 M T E (14) where M is the basis matrix, E is the MR signal vector and A is the coefficient vector: � � � � q 1 q 1 R 0 ( || q 1 || ) y 0 R N ( || q 1 || ) y L . . . 0 L || q 1 || || q 1 || . . ... . . M = , (15) . . � � � � q ns R 0 ( || q n s || ) y 0 q ns R N ( || q n s || ) y L . . . 0 || q ns || L || q ns || E = ( E ( q 1 ) , . . . , E ( q ns )) T (16) A = ( a 000 , . . . , a NLL ) T (17) (GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 55 / 68
Simulation: linear least square reconstruction Figure: N = 0, L = 4, γ = 100, 1 sphere – 42 directions, PSNR: 33.337902, 30 Coefficients (GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 56 / 68
Simulation: linear least square reconstruction Figure: N = 3, L = 4, γ = 70, 3 spheres – 42 directions, PSNR: 45.172752, 45 Coefficients (GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 56 / 68
Simulation: linear least square reconstruction Figure: N = 5, L = 6, γ = 50, 10 spheres – 162 directions, PSNR: 50.255381, 168 Coefficients (GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 56 / 68
Features of the PDF Now that we have a continuous reconstruction of the diffusion signal E , how do we compute the PDF ? (GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 57 / 68
Features of the PDF Now that we have a continuous reconstruction of the diffusion signal E , how do we compute the PDF ? We don’t. This would require a lot of computation. Besides, the PDF is cumbersome to display. (GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 57 / 68
Features of the PDF Now that we have a continuous reconstruction of the diffusion signal E , how do we compute the PDF ? We don’t. This would require a lot of computation. Besides, the PDF is cumbersome to display. We are interesting in a data reduction suitable to display: features of the PDF. � G ( k ) = p ∈ R 3 PDF ( p ) H k ( p ) d p (18) (GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 57 / 68
Features of the PDF: projection Figure: Example: ODF feature. � G ( k ) = p ∈ R 3 PDF ( p ) H k ( p ) d p (19) (GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 58 / 68
Features of the PDF: projection Figure: Example: ODF feature. � G ( k ) = p ∈ R 3 PDF ( p ) H k ( p ) d p (19) (GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 58 / 68
Features of the PDF: projection Figure: Example: ODF feature. � G ( k ) = p ∈ R 3 PDF ( p ) H k ( p ) d p (19) (GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 58 / 68
Features of the PDF: projection Figure: Example: ODF feature. � G ( k ) = p ∈ R 3 PDF ( p ) H k ( p ) d p (19) (GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 58 / 68
Features of the PDF: projection Figure: Example: ODF feature. � G ( k ) = p ∈ R 3 PDF ( p ) H k ( p ) d p (19) (GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 58 / 68
Features of the PDF: projection Figure: Example: ODF feature. � G ( k ) = p ∈ R 3 PDF ( p ) H k ( p ) d p (19) (GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 58 / 68
Features of the PDF: projection Figure: Example: ODF feature. � G ( k ) = p ∈ R 3 PDF ( p ) H k ( p ) d p (19) (GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 58 / 68
Features of the PDF: projection Figure: Example: ODF feature. � G ( k ) = p ∈ R 3 PDF ( p ) H k ( p ) d p (19) (GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 58 / 68
Features of the PDF: projection Figure: Example: ODF feature. � G ( k ) = p ∈ R 3 PDF ( p ) H k ( p ) d p (19) (GREYC-ENSICAEN) Diffusion MRI JIRNFI’2009, Septembre 2009 58 / 68
Recommend
More recommend
