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 French-Vietnam Master in Applied Mathematics, Sept 23,

2 DeFI Houssem Haddar Simona Schiavi (PhD) Gabrielle Fournet (PhD) Dang Van Nguyen (PhD) Julien Coatleven (Post-doc) Fabien Caubet (Post-doc) Denis Le Bihan Cyril Poupon Luisa Ciobanu Khieu Van Nguyen (PhD) Hang Tuan Nguyen (PhD) Master 2 interns Vu Duc Thach Son (April-August 2017), Hoang An Tran (April-August 2016), Hoang Trong An Tran (April-August 2015), Thi Minh Phuong Nguyen (April-August 2014), Khieu Van Nguyen (April Aug 2013), Tri Tinh Nguyen (March July 2011), Thong Nguyen (March July 2011), Cesar Retamal Bravo (March July 2007), Sonia Fliss (Sept Mar. 2005). French-Vietnam Master in Applied Mathematics, Sept 23,

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 French-Vietnam Master in Applied Mathematics, Sept 23,

4 MRI useful for studying 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. French-Vietnam Master in Applied Mathematics, Sept 23,

5 Large-scale electron microscopy of serial thin sections, mouse primary visual cortex. Pink: blood vessels Yellow: nucleoli, oligodendrocyte nuclei, and myelin Aqua: cell bodies and dendrites. Bock et al. Nature 471, (2011) Scale bars: a, b, 100 mm; c e, 10mm; f, 1 mm. French-Vietnam Master in Applied Mathematics, Sept 23,

6 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 French-Vietnam Master in Applied Mathematics, Sept 23,

7 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. French-Vietnam Master in Applied Mathematics, Sept 23,

8 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) French-Vietnam Master in Applied Mathematics, Sept 23,

9 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. French-Vietnam Master in Applied Mathematics, Sept 23,

10 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 French-Vietnam Master in Applied Mathematics, Sept 23,

11 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) French-Vietnam Master in Applied Mathematics, Sept 23,

12 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 French-Vietnam Master in Applied Mathematics, Sept 23,

13 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 French-Vietnam Master in Applied Mathematics, Sept 23,

14 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. French-Vietnam Master in Applied Mathematics, Sept 23,

15 DMRI signal due to tissue (neurons+other cells) A. Direct simulation of reference model (Bloch-Torrey PDE) o Finite elements + RKC time stepping (PhD Dang Van Nguyen) B. Asymptotic models using homogenization C. Experimental validation in simpler organism: Aplysia French-Vietnam Master in Applied Mathematics, Sept 23,

16 Reference model: Bloch-Torrey PDE M j x, t g = i γf t g x M j x, t g + D M j x, t g, x Ω j. t PDE with interface condition between cells and the extra-cellular space D M j x, t g n j x = D M k x, t g n k (x), x Γ jk, D 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 κ ie Ω e, D French-Vietnam Master in Applied Mathematics, Sept 23,

17 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, M x, t g French-Vietnam Master in Applied Mathematics, Sept 23,

18 To do: Define/segment -- mesh solve Bloch-Torrey PDE on realistic brain tissue geometry French-Vietnam Master in Applied Mathematics, Sept 23,

19 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, PhD Hang Tuan Nguyen) 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 French-Vietnam Master in Applied Mathematics, Sept 23,

20 Choose space and time scales σ = σ e in Y e σ c in Y c intrinsic diffusion coefficient L = εl 0 κ = εκ 0 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 which goes to 0 French-Vietnam Master in Applied Mathematics, Sept 23,

21 Define two space scales: coarse and fine scales And expand magnetization in terms of y = x ε M = i i=1 ε i ε i M ie x, y, t in Ω e M ic (x, y, t) in Ω c Match powers (0,1,2) of using the PDE, get relationships between coefficient functions French-Vietnam Master in Applied Mathematics, Sept 23,

22 M ODE e (g, t) t M ODE i (g, t) t Formulated macroscopic model for the compartment magnetizations for arbitrary pulse shape: is an ODE model = 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 f(t) S ODE b = M ODE j g, t, j D Diffusion time d Gradient duration g d RF180 TE g Echo French-Vietnam Master in Applied Mathematics, Sept 23,

23 Simulation box 7.5 mm x 14 mm x 7.5 mm Membrane permeability k= 1e-5 m/s δ = 40ms, Δ = 40ms French-Vietnam Master in Applied Mathematics, Sept 23,

24 Diffusion displacement not long w.r.t. cell features σ = σ e in Y e σ c in Y c L = εl 0 intrinsic diffusion coefficient Spatial scale κ = εκ 0 t = ε 2 t 0 q = q 0 ε 2 membrane s permeability coefficient Time scale gradient strength French-Vietnam Master in Applied Mathematics, Sept 23,

25 Homogenized model for Deff D eff σ il 1 e i e l δ 2 Δ δ 3 Δ+δ 0 ADC new D eff u g u g F t p il (t) p il t 1 Y Y x i ω l (x, t) t F t = f s ds 0 TE D Diffusion time d Gradient duration d F(t) d RF180 Echo French-Vietnam Master in Applied Mathematics, Sept 23,

26 Homogenized model for Deff t ω l div σ ω l = 0 in Y 0, T σ ω l (, t) ν = F(t)σe l ν on Γ 0, T ω l, 0 = 0 in Y ω l is Y periodic Homogeneous diffusion equation with constant coefficient and Neumann boundary condition on the interface Need to analyze 1 p il t Y x i Y ω l (, t) French-Vietnam Master in Applied Mathematics, Sept 23,

27 t ω 1 div σ ω 1 = 0 in Y 0, δ σ ω 1 ν = tσe 1 ν g, t on Γ 0, δ ω 1, 0 = 0 in Y t ω 1, t = SL, t = G x y, t τ μ y, τ ds y dτ σ 2 μ x0, t + σk μ x 0, t = g(x 0, t) K μ (x 0, t) First Pulse: t 0, δ, single layer potential 0 t Γ 0 Γ G x 0 y, t τ μ y, τ ds n y dτ x 0 Fundamental solution G x, t = (4πσt) d 2e x 2 4σt p 11 t 1 Y Y x ω l (, t) 4 3 π σ SA x VOL t3 2 SA x VOL 1 Y n x 2 Γ French-Vietnam Master in Applied Mathematics, Sept 23,

28 t ω 1 div σ c ω 1 = 0 in Y c δ, Δ σ ω 1 ν = δσe 1 ν on Γ δ, Δ ω 1, δ = SL, δ in Y Between pulses: t [δ, Δ], eigenfunction expansion ω 1, t ω 1 +δx t ω 1 div σ ω 1 = 0 in Y δ, Δ σ ω 1 ν = 0 on Γ δ, Δ ω 1, δ = SL, δ δx in Y ω 1, t = δx + c n φ n e λ nσ(t δ) n=1 French-Vietnam Master in Applied Mathematics, Sept 23,

29 Between pulses: t [δ, Δ] eigenfunction expansion ω 1, t = δx + (b n + δa n )φ n e λ nσ(t δ) n=1 b n SL, δ φ n Y a n Y x φ n ( ) h in 1 Y φ n x a n are the first moment of the eigenfunctions. p 11 t = 1 Y Y x ω 1 (, t) δ + n=1 δa n + b n h in e λ nσ(t δ) French-Vietnam Master in Applied Mathematics, Sept 23,

30 Second Pulse: t [Δ, Δ + δ] single layer potential t ω 1 div σ ω 1 = 0 in Y Δ, Δ + δ σ ω 1 ν = Δ + δ t σe 1 ν on Γ Δ, Δ + δ ω 1, Δ = ω 1, Δ in Y t ω 1 div σ ω 1 = 0 in Y 0, δ σ c ω c 1 ν = tσ c e 1 ν on Γ 0, δ ω 1 c, 0 = 0 in Y c ω 1, t = ω 1 (, Δ) SL, t Δ p 11 t = δ + δa n + b n h in e λ nσ(δ δ) n=1 4 3 π σ SA x VOL t3 2 French-Vietnam Master in Applied Mathematics, Sept 23,

31 D eff 11 σ 1 b n SL, δ φ n Y SA x VOL 1 Y δ I F t p il t 0 Δ II F t p il t δ III = δ3 2 + δ2 2 n=1 Homogenized model for Deff n x 2 Γ = French-Vietnam Master in Applied Mathematics, Sept 23, δ 2 Δ δ π σ SA x VOL δ7 2 δa n + b n h in (e λ nσ Δ δ ) + I + II + III = δ 2 1 Δ δ + δ (δa n + b n )h in λ n σ (1 e λnσ(δ δ) ) n=1 h in 1 Y φ n x a n are the first moment of the eigenfunctions. 4 3 π σ SA x VOL 2 5 δ δ7 2

32 Numerical results French-Vietnam Master in Applied Mathematics, Sept 23,

33 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. French-Vietnam Master in Applied Mathematics, Sept 23,

34 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. French-Vietnam Master in Applied Mathematics, Sept 23,

35 o Segment high resolution T2 image of buccal ganglia. 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 French-Vietnam Master in Applied Mathematics, Sept 23,

36 Thank you for your attention! French-Vietnam Master in Applied Mathematics, Sept 23,