Numerical simulation and macroscopic model formulation for diffusion magnetic resonance imaging in the brain

Transcription

1 Numerical simulation and macroscopic model formulation for diffusion magnetic resonance imaging in the brain Jing-Rebecca Li Equipe DEFI, CMAP, Ecole Polytechnique Institut national de recherche en informatique et en automatique (INRIA) Saclay, France L' HABILITATION À DIRIGL' HABILITATION À DIRIGER DES RECHEHERCHES Séminaire d Analyse Numérique, IRMAR, 24/9/2015 1

2 DeFI Houssem Haddar Simona Schiavi (current PhD) Gabrielle Fournet (current PhD) Dang Van Nguyen (former PhD) Julien Coatleven (former Post-doc) Fabien Caubet (former Post-doc) Denis Le Bihan Cyril Poupon Luisa Ciobanu Khieu Van Nguyen (current PhD) Hang Tuan Nguyen (former PhD) Séminaire d Analyse Numérique, IRMAR, 24/9/2015 2

3 Outline 1. Brain micro-structure is complex 2. MRI using diffusion encoding to see micro-structure 3. DMRI signal due to tissue (neurons+other cells) a) Full-scale numerical simulation b) Macroscopic model formulation via homogenization c) Experimental validation in simpler organism 4. DMRI signal due to micro-vessels, experiments, preliminary simulations 5. DMRI signal due to iron, planning stage. Séminaire d Analyse Numérique, IRMAR, 24/9/2015 3

4 Timeline of our work on brain diffusion MRI DMRI for tissue widely used 1990/2000-present, simple models Formulate the mathematical problem for tissue (neurons and other cells) 2010-present Full-scale simulation and reduced model of dmri signal due to tissue Intra-voxel incoherent motion (IVIM) DMRI for micro-vessels started to be used 2000/ present IVIM experiments to characterize brain microvessels Future Simulation and modeling of dmri signal due to microvessels and iron Séminaire d Analyse Numérique, IRMAR, 24/9/2015 4

5 Séminaire d Analyse Numérique, IRMAR, 24/9/2015 5

6 Bock et al. Nature 471, (2011) Large-scale Electron Micrograph Pink: blood vessels Yellow: nucleoli, oligodendrocyte nuclei, and myelin Aqua: cell bodies and dendrites. Scale bars: a, b, 100 mm; c e, 10mm; f, 1 mm. Séminaire d Analyse Numérique, IRMAR, 24/9/2015 6

7 Magnetic resonance imaging (MRI) Non-invasive, in-vivo MRI signal: water proton magnetization over a volume called a voxel. MRI To give image contrast, magnetization is weighted by some quantity of the local tissue environment. Contrast: (tissue structure) 1. Spin (water) density Spatial resolution: One voxel = O(1 mm) Much bigger than micro-structure 2. Relaxation (T1,T2,T2*) 3. Water displacement (diffusion) in each voxel Séminaire d Analyse Numérique, IRMAR, 24/9/2015 7

8 MRI contrasts Gray: cortical surface. Teal: fmri activations Red: arteries in red Bright green: tumor Yellow: white matter fiber Diffusion Tensor and Functional MRI Fusion with Anatomical MRI for Image-Guided Neurosurgery. Sixth International Conference on Medical Image Computing and Computer-Assisted Intervention - MICCAI'03. Séminaire d Analyse Numérique, IRMAR, 24/9/2015 8

9 o Standard MRI: T2 relaxation (T2 contrast) at different spatial positions of brain o In diffusion MRI (recently developed) magnetization is weighted by water displacement due to Brownian motion over 10s of ms (called measured diffusion time).??? o Water displacement depends on local cell environment, hindered by cell membranes. o Right: T2 contrast does not show dendrite beading hours after stroke, diffusion weighted image (DWI) does. Séminaire d Analyse Numérique, IRMAR, 24/9/2015 9

10 DMRI measures incoherent water motion during diffusion time between 10-40ms. Root mean squared displacement: 6-13 mm Voxel : 2mm x 2mm x 2 mm (human, clinical scanner) 100mm x 100mm x 100mm (small animal, high field scanner) Séminaire d Analyse Numérique, IRMAR, 24/9/

11 Goal: quantify dmri contrast in terms of tissue micro-structure This problem difficult because: 1. Dendrites (trees) and extra-cellular (EC) space (complement of densely packed dendrites) are anisotropic, numerically lower dimensional (dendrites 1 dim, EC 2 dim). 2. Multiple scales (5 orders of magnitude difference). Extra-cellular space thickness 10-30nm Dendrite radius mm Soma diameter 1-10mm DMRI voxel 2mm 3. Cell membranes are permeable to water. Cells must be coupled together. Séminaire d Analyse Numérique, IRMAR, 24/9/

12 Simple (original) model of dmri Brain: 70 percent water Brownian motion of water molecules u x t, x (, 0 ) e x x 0 4 Dt (4 Dt) d 2 2 Mean-squared displacement Can be obtained by dmri MSD u( x, t, x 0) 2 2 x x 0 dx ddt D = 2 x 10-3 mm²/s if no cell membranes Experimentally measured = 1x10-3 mm²/s Séminaire d Analyse Numérique, IRMAR, 24/9/

13 How diffusion MRI assigns contrast to displacement Water 1 H (hydrogen nuclei), spin ½ Precession Larmor frequency: γb x, t dt t Proton: g/2 = MHz / Tesla B x, t = f t g x TE D Diffusion time f(t) g d Gradient duration RF180 d g Echo Pulsed gradient spin echo (PGSE) sequence (Stejskal-Tanner-1965) Séminaire d Analyse Numérique, IRMAR, 24/9/

14 t = 0: M δ = M 0 e iγδg x 0 t = Δ + δ, M Δ+δ = M 0 e iγδg x Δ+δ x 0 Séminaire d Analyse Numérique, IRMAR, 24/9/

15 u x, t, x 0 x x0 2 = e 4Dt 3 2 4πDt Experimental parameters S b = u x, Δ + δ x 0 e iγδg x Δ+δ x 0 dxdx 0 x V x 0 V g D, d can be varied = e D γ2 δ 2 g 2 Δ δ 3 b g, Δ, δ γ 2 δ 2 g 2 Δ δ 3, ADC d log (S b ): db apparent diffusion coefficient Fitted at every voxel MSD = ADC*(2D) Brain gray matter: ADC around10-3 mm²/s Root MSD: 6-13 mm Séminaire d Analyse Numérique, IRMAR, 24/9/

16 5 Diffusion is not Gaussian in biological tissues (In each voxel) Human visual cortex (Le Bihan et al. PNAS 2006). S ( ADC ) b S 0 e Log plot not a straight line. Simple model is wrong ln(signal) Free diffusion: ln(s/s0) = -bd b value S S 0 Physicists try a different simple model f fast e D fast b f slow e f fast = 65.9%, f slow = 34.1% D fast = mm²/s, D slow = mm²/s D slow b. Séminaire d Analyse Numérique, IRMAR, 24/9/

17 DMRI signal due to tissue (neurons+other cells) A. Direct simulation of reference model (Bloch-Torrey PDE) o Finite volume and finite elements (Ph.D. Dang Van Nguyen) codes B. Asymptotic models using homogenization C. Experimental validation in simpler organism: Aplysia Séminaire d Analyse Numérique, IRMAR, 24/9/

18 Reference model: Bloch-Torrey PDE M j x, t g = i γf t g x M j x, t g + D j M j x, t g, x Ω j. t PDE with interface condition between cells and the extra-cellular space D j M j x, t g n j (x) = D k M k x, t g n k (x), x Γ jk, D j M j x, t g n j x = κ M j x, t, g M k x, t g, x Γ jk, M j g, t = M j x, t g dx. x Ω j S g, T end = M j (g, t) exp ( ADC b experi ) j M: magnetization g: magnetic field gradient T end : diffusion time From signal, want to quantify cell geometry and membrane permeability. Ω i, D i Séminaire d Analyse Numérique, IRMAR, 24/9/ κ ie Ω e, D e

19 1. Numerical simulation of diffusion MRI signals using an adaptive timestepping method, J.-R. Li, D. Calhoun, C. Poupon, D. Le Bihan. Physics in Medicine and Biology, A finite elements method to solve the Bloch-Torrey equation applied to diffusion magnetic resonance imaging, D.V. Nguyen, J.R. Li, D. Grebenkov, D. Le Bihan, Journal of Computational Physics, D i D e M x, t g Séminaire d Analyse Numérique, IRMAR, 24/9/

20 To do: Define/segment -- mesh -- solve Bloch-Torrey PDE on realistic brain tissue geometry Séminaire d Analyse Numérique, IRMAR, 24/9/

21 Asymptotic models of dmri using periodic homogenization 1. Coupled ODE model Works for large diffusion displacement and wide range of gradient amplitudes. (Post-doc Coatleven) 2. Simpler PDE model for wide range of diffusion displacement and low gradient amplitudes (ADC). (Ph.D. Schiavi) Both are limited to low membrane permeability Séminaire d Analyse Numérique, IRMAR, 24/9/

22 Choose space and time scales intrinsic diffusion coefficient Valid when Spatial scale membrane s permeability coefficient diffusion displacement long compared to cell features. low membrane permeability. We fix a non-dimensional parameter ε and match powers. Séminaire d Analyse Numérique, IRMAR, 24/9/

23 Formulated macroscopic model for the compartment magnetizations for arbitrary pulse shape: is an ODE model M ODE e (g, t) t M ODE i (g, t) t = c(t)γ 2 D e g 2 e 1 M ODE (g, t) τ e M e 1 ODE (g, t) + τ i M i ODE (g, t) = c t γ 2 D i g 2 i 1 M ODE (g, t) τ i M i 1 ODE (g, t) + τ e M e ODE (g, t) t c(t) = f s ds κ Ωi = τi Ω i, τ e = Ωi Ω e τi S ODE b = M ODE j g, t, j 1. A new macroscopic model including membrane exchange for diffusion MRI, J. Coatleven, H. Haddard, J-R. Li, SIAM Journal of Applied Math, Numerical study of a macroscopic finite pulse model of the diffusion MRI signal, J.-R. Li, H. T. Nguyen, D. V. Nguyen, H. Haddar, J. Coatleven, D. Le Bihan, Journal of Magnetic Resonance Séminaire d Analyse Numérique, IRMAR, 24/9/

24 Simulation box 5.8 mm x 5 mm x 10 mm Membrane permeability k= 1e-5 m/s δ = 40ms, Δ = 40ms Séminaire d Analyse Numérique, IRMAR, 24/9/

25 Simulation box 7.5 mm x 14 mm x 7.5 mm Membrane permeability k= 1e-5 m/s δ = 40ms, Δ = 40ms Séminaire d Analyse Numérique, IRMAR, 24/9/

26 Diffusion displacement not long w.r.t. cell features intrinsic diffusion coefficient Spatial scale membrane s permeability coefficient Time scale We fix a non-dimensional parameter ε and match powers. gradient strength Séminaire d Analyse Numérique, IRMAR, 24/9/

27 Resulting macroscopic model We obtain the dmri signal can be approximated as where the apparent diffusion tensor has the form and the effective diffusion tensors are where ω is the solution of Homogeneous diffusion equation with constant coefficient and Neumann boundary condition on the interface Séminaire d Analyse Numérique, IRMAR, 24/9/

28 Numerical results To validate the model we compute the following quantities: - Reference Bloch-Torrey model approximated through a polynomial fit, - Our new asymptotic model, - Short time model (Novikov et all 2011), - Long time model (Cheng and Torquato 1997) for some 2D configurations and we plot the results against a normalized diffusion displacement defined as Séminaire d Analyse Numérique, IRMAR, 24/9/

29 Numerical Results Domain configuration: Periodicity box is square of side 50μm with 32 circular cells of radii μm and 5 cylinders of thickness 0.7-2μm. Experimental parameters: gradient direction [1,1], k=1x10-5 m/s, D e =3x10-3 mm 2 /s, D c =2x10-3 mm 2 /s, various Δ and d to obtain diffusion times 0.3ms -160ms. ADC fitted using several b<=100s/mm 2. V e =40% Séminaire d Analyse Numérique, IRMAR, 24/9/

30 To do: Model is not so simple like ODE, must work to simplify it to make it numerically efficient for parameter estimation. Use mix of layer potentials and eigenfunctions Account for multiple scales and anisotropic shape of neurons. Séminaire d Analyse Numérique, IRMAR, 24/9/

31 Estimating cell size and membrane permeability using reference PDE model in neuron network of the Aplysia (sea slug). (Ph.D. Khieu Van Nguyen) Cells are really big. Séminaire d Analyse Numérique, IRMAR, 24/9/

32 o Segment high resolution T2 image of buccal ganglia. Modeling and simulation of brain diffusion MRI o Use literature values of feathers that cannot be seen in T2 image o Simulate Bloch-Torrey PDE o Obtain diffusion images Carlo Musio et al. Zoomorphology. 1990;110:17-26 Séminaire d Analyse Numérique, IRMAR, 24/9/

33 DMRI signal due to micro-vessels (Ph.D. Gabrielle Fournet) experiments preliminary simulations Séminaire d Analyse Numérique, IRMAR, 24/9/

34 Experimental data acquired at Neurospin MR System 7T (300 MHz) Maximum gradient strength : 740 mt/m RF Coils Four-element phased-array receiver coil Volume transmit coil, Ø coil = 7.2 cm Animal model Dark agouti rat MR acquisitions PGSE-EPI d = 3 ms, D = 14, 24 and 34 ms, TE/TR = 45/1000 ms, in-plane resolution 250 x 250 μm², FOV 2 x 2 mm 2, 1 slice, slice thickness 1.5 mm, 1 segment, 6 averages, 8 repetitions gradient direction [1, 1, 1] 30 b-values: [2 2500] s/mm² Anesthesia Isoflurane (1-2 % in a 1:2 O 2 :air mixture) Schematic representation of the pulse sequence used Séminaire d Analyse Numérique, IRMAR, 24/9/

35 IVIM (perfusion) Zoom Diffusion (tissue) The IVIM (perfusion) signal is what remains after removing the diffusion (tissue) component of the MRI signal. Séminaire d Analyse Numérique, IRMAR, 24/9/

36 Séminaire d Analyse Numérique, IRMAR, 24/9/

37 We have connectivity graph of vessels Jingpeng Wu, Yong He, Zhongqin Yang, Congdi Guo, Qingming Luo, Wei Zhou, Shangbin Chen, Anan Li, Benyi Xiong, Tao Jiang, Hui Gong NeuroImage 2014 Séminaire d Analyse Numérique, IRMAR, 24/9/

38 Thank you for your attention! Séminaire d Analyse Numérique, IRMAR, 24/9/