Two-step Anomalous Diffusion Tensor Imaging

Transcription

1 Two-step Anomalous Diffusion Tensor Imain Thomas R. Barrick 1, Matthew G. Hall 2 1 Centre for Stroke and Dementia, Division of Cardiac and Vascular Sciences, St. Geore s University of London, 2 Department of Computer Science, University Collee London

2 Introduction: Reular Diffusion Assumption: Mean squared-displacements of diffusin spins increases linearly with time Reular diffusion natural if: 1. In an unrestricted environment 2. In presence of restrictin structures in short-time limit i.e. no structure yet encountered by spins 3. In presence of structure in lon-time limit in which a tortuosity approximation applies Modified by eometric factor

3 Sinal Intensity Introduction: Sinal Attenuation Assumption of reular diffusion Stejskal-Tanner equation Prediction of monoexponential decay 300 White Matter Voxel Monoexponential Estimate Short time limit for free diffusion Lon time limit representin Tortuosity limit 100 Diffusion free, but slower b-value (s mm -2 )

4 Introduction: Anomalous Diffusion Intermediate reime: Anomalous diffusion Postulate bioloical complexity of diffusion environment well approximated by disordered or fractal environment Ozarslan et al., 2006 Non reular behaviour Hall and Barrick, 2008 Rane of timescales of intermediate reime dependent on sizes of structures restrictin diffusion Spins impeded by structure on many lenth scales arraned in disordered fashion Leads to mean-squared displacement varyin with non-inteer power of diffusion time Time-dependent diffusivity Questions for diffusion imain: Is the timescale of this transition important? What can be determined reardin the diffusion environment?

5 Introduction: Stretched Exponential Sinal intensity, S b-value, b Diffusion sensitisation (s mm -2 ) Distributed diffusivity, Overall amount of diffusion distributed across timescales (mm 2 s -1 ). S Anomalous exponent, b b S 0 e Complexity of diffusion environment. Dimensionless parameter (1) Formulations Bennett et al., 2006 Ozarslan et al., 2006 Hall and Barrick, 2008 Main et al., 2008

6 Objective To present an anomalous DTI technique that; Allows full analysis of anomalous diffusion in multiple radient directions Provides independent tensors representin Distributed diffusivity, Anomalous exponent, Allows analysis of tensors usin standard DTI techniques We refer to this technique as Two-step Anomalous Diffusion Tensor Imain

7 Two-step Anomalous DTI (adti) We propose an analoous technique to reular DTI First step Compute and alon radial directions in q-space Second step S b b S 0 e Fit tensor to the and values alon radient directions Obtain tensors and

8 Two-step adti Calculate anomalous diffusion in n diffusion radient directions, S n nb b S 0 e n Generalise stretched exponential to include directional dependence in both parameters, S n rˆ b S 0 e rˆ S 2 where and and are orientationally invariant spherical functions definin and. rˆ n n b (2)

9 We assume these spherical functions are well-described by Gaussian ellipsoids Two-step adti Anomalous tensors fitted usin eneral linear model x 2 x x 2 x A xx xx y y 2 y 2 y A z yy z yy A A A xx xy xz 2 z xx xy xz 2 z A zz A A A zz xy yy yz 2 xy yy yz A A A x 2 x xz yz zz y xz yz zz y A xy xy x y z 2 x y z x 2 x z (3) A z xz (4) xz 2 y 2 Distributed Diffusivity Tensor (A) y z A z yz Anomalous Exponent Tensor () yz

10 Distributed diffusivity tensor, Anomalous DTI A - overall amount of diffusion (mm 2 s -1 ). Anomalous exponent tensor, - complexity of diffusion environment. Both are 3 x 3 symmetric tensors. 3 positive real eienvalues, l 1, l 2 and l 3, Correspondin eienvectors, v 1, v 2 and v 3. Allows computation of standard tensor quantities (e.. Tr, FA). Averae of eienvalues (M)

11 Methods: Imae Acquisition Imae Acquisition St. Geore s, University of London, UK Philips 3T Achieva TX 32 channel head coil Enhanced radients Maximum of 80 mtm -1 at a slew rate of 100 mtm -1 ms -1 1 youn, healthy female volunteer Anomalous DTI acquisition (Spin Echo EPI) 12 axial slices, 3mm thick in ~35 minutes

12 # Data Acquisitions Methods: Imae Acquisition 12 diffusion radient directions b=0 and 11 b values 3mm isotropic voxels TE= 75 ms, TR = 4175 ms b value lo b lo b

13 Lo(Sinal) Methods: Data Quality 6 direction 1 direction 2 direction 3 direction 4 direction 5 direction 6 direction 7 direction 8 direction 9 direction 10 direction 11 direction b-value (s mm -2 )

14 Lo(Sinal) Methods: Data Quality 6 direction 1 direction 2 direction 3 direction 4 direction 5 direction 6 direction 7 direction 8 direction 9 direction 10 direction 11 direction b-value (s mm -2 )

15 Lo(Sinal) Methods: Data Quality 6 direction 1 direction 2 direction 3 direction 4 direction 5 direction 6 direction 7 direction 8 direction 9 direction 10 direction 11 direction b-value (s mm -2 )

16 Lo(Sinal) Methods: Data Quality 6 direction 1 direction 2 direction 3 direction 4 direction 5 direction 6 direction 7 direction 8 direction 9 direction 10 direction 11 direction b-value (s mm -2 )

17 Lo(Sinal) Methods: Data Quality 6 direction 1 direction 2 direction 3 direction 4 direction 5 direction 6 direction 7 direction 8 direction 9 direction 10 direction 11 direction b-value (s mm -2 )

18 Lo(Sinal) Methods: Data Quality 6 direction 1 direction 2 direction 3 direction 4 direction 5 direction 6 direction 7 direction 8 direction 9 direction 10 direction 11 direction b-value (s mm -2 )

19 Methods: Data Fittin Estimate and by fittin stretched exponential to DW measurements at b-values alon radial lines in q-space. Fit parameters usin a Levenber-Marquardt alorithm. If we let we have an alternate form for the equation, Partial derivatives with respect to fittin parameters are simplified. 1 a, S S b b S 0 e b S 0 e ab (5) Provides better converence of parameter estimates.

20 Tissue Maskin Semented mean of all DWIs at b = 5000 s mm -2 Statistical Parametric Mappin 8 ( Mean b=5000 s mm -2 Grey Matter White Matter Cerebrospinal Fluid

21 Lo(Sinal) Lo(Sinal) Sinal direction 1 direction Decay 2 direction 3 direction In 4 Brain Tissue 6 direction 5 direction 6 direction 7 direction 8 direction 9 direction 10 direction 11 direction 12 (a) Grey matter voxel direction 1 direction 2 direction 3 direction 4 2direction 5 direction 6 direction 7 direction 8 direction direction direction direction (b) White b-value matter (s mm -2 ) voxel Grey matter Similar sinal decay in diffusion directions White matter Anisotropy in sinal decay in different diffusion directions b-value (s mm -2 )

22 Tensor Summary xx yy zz xy xz yz A

23 Isotropy M D A

24 Isotropy M(D) M(A) M GM 0.81 GM 0.76 GM 0.82 WM 0.68 WM 0.56 WM 0.69 x 10-3 mm 2 s -1 x 10-3 mm 2 s -1 M M M l 1 D 1.04 l 2 D 0.60 l 3 D 0.36 l 1 A 0.94 l 2 A 0.46 l 3 A 0.26 l l l

25 Anisotropy D A

26 Anisotropy FA(D) FA(A) FA GM 0.14 GM 0.16 GM 0.08 WM 0.50 WM 0.58 WM 0.13 c l D 0.43 c p D 0.22 c s D 0.35 c l A 0.52 c p A 0.20 c s A 0.27 c l 0.12 c p 0.10 c s 0.79

27 Principal Directions

28 White Matter Reions Of Interest v 1 (A) v 3 (Γ)

29 Summary Of Results Tensor exhibits similar behaviour to D Hih FA in white matter Similar orientation of principal directions Tensor exhibits different behaviour to and D FA lower values l 3 ( estimates direction of reatest environment complexity?

30 Discussion Similarities between and D Represent analoous physical entities in stretched exponential reime D in reular diffusion reime Stretched exponential provides improved tissue contrast and reater imae fidelity Distributions not identical between D and A A refers to measures over continuously distributed diffusion compartments Overall estimate of diffusivity

31 Discussion Comparison of adwi studies (averaes) A mm 2 s Two-step approach Bennett et al., 2006 Gao et al., 2011

32 Discussion tensor very different shapes and orientations to D or Not a diffusion tensor and fractal dimension suested to be related Directional anisotropy in fractal dimension of spin/trajectories and/or sample tissue should result in concomitant anisotropy of the anomalous exponent Seem reasonable to consider as description of environmental complexity

33 Discussion Comparisons of directional dependence of Bennett et al., 2006, Hall and Barrick, 2008 Limited directional variation in No tensor model Hihlihts importance of usin full tensor form 0.35 Two-step approach Bennett et al., 2006 Hall Barrick & Barrick, et al., 2008

34 Discussion Comparison of invariant indices De Santis et al., 2011 Two-step approach De Santis et al., 2011 probe anisotropy alon principal directions obtained by reular tensor, D

35 Two-step process Discussion Applicable to other stretched exponential formulations e.. Main et al., 2008 One step formulation e.. fittin tensors directly to raw data Assumption of symmetric tensor too simplistic? Effects of crossin fibres? Greater environmental complexity in one direction than diametrically opposite direction?

36 Future Research Questions Better understandin of anomalous parameters Phantom experiments Tissue samples comparison with histoloy Optimisation of adti acquisition Reduce acquisition time Minimise number of diffusion directions Maximise sinal to noise ratio Maximise imae resolution and coverae

37 Future Research Questions New clinical biomarkers? Complement reular DTI Animal models of disease patholoy More sensitive to patholoical chane? Clinical application Disease dianosis Disease proression

38 Fundin Acknowledements St. Geore s Healthcare NHS Trust South West London United Kindom