Understanding brain micro-structure using diffusion magnetic - PowerPoint PPT Presentation

Modeling and simulation of brain diffusion MRI Understanding brain micro-structure using diffusion magnetic resonance imaging (dMRI) 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 CEMRACS, 8/2015 1

Modeling and simulation of brain diffusion MRI 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) CEMRACS, 8/2015 2

Modeling and simulation of brain diffusion MRI Timeline of our work on brain diffusion MRI DMRI for tissue widely used 1990/2000-present, simple models 2008-2010 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 2000/2010 2013-present IVIM experiments to characterize brain micro-vessels 2015 Simulation and modeling of dMRI signal due to micro-vessels CEMRACS, 8/2015 3

Modeling and simulation of brain diffusion MRI 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) 4. DMRI signal due to micro-vessels CEMRACS, 8/2015 4

Modeling and simulation of brain diffusion MRI Large-scale Electron Micrograph Pink: blood vessels Yellow: nucleoli, oligodendrocyte nuclei, and myelin Aqua: cell bodies and dendrites. Scale bars: a, b, 100 m m; c – e, 10 m m; f, 1 m m. Bock et al. Nature 471 , 177-182 (2011) CEMRACS, 8/2015 5

Modeling and simulation of brain diffusion MRI Magnetic resonance imaging (MRI) Non-invasive, in-vivo MRI signal: water proton magnetization over a volume called a voxel. To give image contrast, MRI magnetization is weighted by some quantity of the local tissue environment. Contrast: (tissue structure) 1. Spin (water) density 2. Relaxation (T1,T2,T2*) Spatial resolution: 3. Water displacement (diffusion) One voxel = O(1 mm) Much bigger than micro-structure in each voxel CEMRACS, 8/2015 6

Modeling and simulation of brain diffusion MRI 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. CEMRACS, 8/2015 7

Modeling and simulation of brain diffusion MRI Diffusion MRI Diffusion MRI can measure average incoherent displacement of water in a voxel during 10s of milliseconds Displacement of water can tell us about cellular structure Understanding of biomechanics of cells, structure of brain Jonas: Mosby's Dictionary of Potential clinical value Complementary and Alternative o Structure change in Medicine. (c) 2005, Elsevier. diseases CEMRACS, 8/2015 8

Modeling and simulation of brain diffusion MRI 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. CEMRACS, 8/2015 9

Modeling and simulation of brain diffusion MRI DMRI measures incoherent water motion during “diffusion time” between 10 -40ms. Root mean squared displacement: 6-13 m m Voxel : 2mm x 2mm x 2 mm. CEMRACS, 8/2015 10

Modeling and simulation of brain diffusion MRI 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 Dendrite radius Soma diameter DMRI voxel space thickness 0.5-0.9 m m 1-10 m m 2mm 10-30nm 3. Cell membranes are permeable to water. Cells must be coupled together. CEMRACS, 8/2015 11

Modeling and simulation of brain diffusion MRI Simple (original) model of dMRI Brain: 70 percent water Brownian motion of water molecules Mean-squared displacement Can be obtained by dMRI   2  x x   0        2   MSD u ( x t , | x ) x x dx 2 dDt  4 Dt e   , 0 0  u ( x t , | x ) , 0 d  ( 4 Dt ) 2 CEMRACS, 8/2015 12

Modeling and simulation of brain diffusion MRI How diffusion MRI assigns contrast to displacement Water 1 H (hydrogen nuclei), spin ½ Precession Larmor frequency: 𝐶 𝐲, 𝑢 = 𝑔 𝑢 𝐡 ⋅ 𝐲 𝛿𝐶 𝐲, 𝑢 𝑒𝑢 𝑢 Proton: g /2  = 42.57 MHz / Tesla TE D Diffusion time d Gradient duration d g g RF180 Echo f(t) Pulsed gradient spin echo (PGSE) sequence (Stejskal-Tanner-1965) CEMRACS, 8/2015 13

Modeling and simulation of brain diffusion MRI 𝑢 = 0: 𝑁 𝜀 = 𝑁 0 𝑓 −𝑗𝛿𝜀𝐡⋅ 𝐲 0 𝑢 = Δ + 𝜀, 𝑁 Δ+𝜀 = 𝑁 0 𝑓 𝑗𝛿𝜀𝐡⋅ 𝐲 𝚬+𝜺 −𝐲 0 CEMRACS, 8/2015 14

Modeling and simulation of brain diffusion MRI 𝑓 −| 𝐲−𝐲0| 2 4𝐸𝑢 𝑣 𝐲, 𝑢, |𝐲 0 = 3 4𝜌𝐸𝑢 2 𝑣 𝐲, Δ + 𝜀|𝐲 0 𝑓 𝑗𝛿𝜀𝐡⋅ 𝐲 𝛦+𝜀 −𝐲 0 𝑒𝐲𝑒𝐲 𝟏 𝑇 𝑐 = 𝐲∈𝑊 𝐲 0 ∈𝑊 −𝐸 𝛿 2 𝜀 2 𝐡 2 Δ−𝜀 Experimental 3 = 𝑓 parameters 𝑐 𝐡, Δ, 𝜀 ≡ 𝛿 2 𝜀 2 𝐡 2 Δ − 𝜀 3 , g D , d can be varied MSD/(2 D ) = ADC d 𝐵𝐸𝐷 ≡ − db log (𝑇 𝑐 ) : Brain gray matter: ADC around10 -3 mm²/s “apparent diffusion Root MSD: 6-13 m m coefficient” Fitted at every voxel CEMRACS, 8/2015 15

Modeling and simulation of brain diffusion MRI Diffusion is not Gaussian in biological tissues (In each voxel) S   ( ADC ) b e Log plot not a straight line. Human visual cortex S (Le Bihan et al. PNAS 2006). 0 Simple model is “wrong” 5 Physicists try a different 4.5 simple model Free diffusion: 4 ln(signal) S ln(S/S0) = -bD     D b D b f e f e . fast slow 3.5 fast slow S 0 3 f fast = 65.9%, f slow = 34.1% 2.5 D fast = 1.39 10 -3 mm²/s, 0 1000 2000 3000 4000 b value D slow = 3.25 10 -4 mm²/s CEMRACS, 8/2015 16

Modeling and simulation of brain diffusion MRI Reference model: Bloch-Torrey PDE 𝜖𝑁 𝑘 𝐲, 𝑢 𝐡 𝐡 ⋅ 𝐲 𝑁 𝑘 𝐲, 𝑢 𝐡 + 𝛼 ⋅ D j 𝛼𝑁 𝑘 𝐲, 𝑢 𝐡 , 𝐲 ∈ Ω 𝑘 . = 𝑗 𝛿𝑔 𝑢 𝜖𝑢 PDE with interface condition between cells and the extra-cellular space 𝐸 𝑘 𝛼𝑁 𝑘 𝐲, 𝑢 𝐡 ⋅ 𝐨 j (𝐲) = −𝐸 𝑙 𝛼𝑁 𝑙 𝐲, 𝑢 𝐡 ⋅ 𝐨 k (𝐲), 𝐲 ∈ Γ 𝑘𝑙 , 𝐸 𝑘 𝛼𝑁 𝑘 𝐲, 𝑢 𝐡 ⋅ 𝐨 j 𝐲 = 𝝀 𝑁 𝑘 𝐲, 𝑢, 𝐡 − 𝑁 𝑙 𝐲, 𝑢 𝐡 𝐲 ∈ Γ 𝑘𝑙 , , 𝑁 𝑘 𝐲, 𝑢 𝐡 𝑒𝐲 ≈ exp 𝑇 𝐡, 𝑈 𝑓𝑜𝑒 = (−𝐵𝐸𝐷 𝑐 𝑓𝑦𝑞𝑓𝑠𝑗 ). 𝐲∈Ω 𝑘 𝑘 M: magnetization g: magnetic field gradient T end : diffusion time From signal, want to quantify cell geometry and membrane permeability. Ω 𝑗 , 𝐸 𝑗 𝜆 𝑗𝑓 Ω 𝑓 , 𝐸 𝑓 CEMRACS, 8/2015 17

Modeling and simulation of brain diffusion MRI 1. Numerical simulation of diffusion MRI signals using an adaptive time- stepping method, J.-R. Li, D. Calhoun, C. Poupon, D. Le Bihan. Physics in Medicine and Biology, 2013. 2. 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, 2014. M 𝐲, 𝑢 𝐡 𝐸 𝑓 𝐸 𝑗 CEMRACS, 8/2015 18

Modeling and simulation of brain diffusion MRI On-going work (2013  ) Mathematical analysis 2012: Obtained macroscopic (ODE) model using homogenization Valid in long diffusion time regime. More relevant to brain dMRI: 2013: Look for macroscopic model valid at wide range of diffusion times PhD Simona Schiavi 2013-present (co-directed w. H. Haddar) CEMRACS, 8/2015 19

Modeling and simulation of brain diffusion MRI Timeline of our work on brain diffusion MRI (DMRI for micro-vessels started to be used 2000/2010, simple models) Intra-voxel incoherent motion (IVIM) 2013-present DMRI experiments to characterize brain micro-vessels 2015 Simulation and modeling of dMRI signal due to micro-vessels CEMRACS, 8/2015 20

Modeling and simulation of brain diffusion MRI The cerebro-vasculature Dragos A. Nita Neurology 2012;79:e10 CEMRACS, 8/2015 21

Modeling and simulation of brain diffusion MRI The cortical angiome: an interconnected vascular network with noncolumnar patterns of blood flow Blinder et al. Nature Neuroscience 2013 CEMRACS, 8/2015 22

Modeling and simulation of brain diffusion MRI The cortical angiome: an interconnected vascular network with noncolumnar patterns of blood flow Blinder et al. Nature Neuroscience 2013 CEMRACS, 8/2015 23