Soutenance mémoire
Diffusion magnetic resonance imaging (dMRI) is a powerful non-invasive imaging modality that gives a measure of the displacement of water molecules. This technique has been extensively applied in materials science to investigate structural and transport properties of porous media such as sedimentary rocks, concrete, and cement. In medical and biological applications it has been used to study structural and functional properties of biological tissues in almost all organs, with the most common being the brain (for some review and survey papers, see .
To make a MRI experiment sensitive to diffusion, a diffusion-encoding gradient is applied to capture the effects that the incoherent motion of water molecules has on the MRI signal through spin dephasing and signal attenuation. In the brain, dMRI measures the spin (water proton) displacement during a diffusion time on the order of tens of microseconds. The dMRI signal is the transverse magnetization in a tissue volume (called a voxel) whose size is on the order of 1 mm. However, the dimensions of cell features in the brain are of the order of micro-meters, meaning that they cannot be individually distinguished at the MRI resolution. Because water displacement is affected (restricted or hindered) by the presence of cell membranes, the dMRI signal measured at different diffusion times, and gradient intensities and directions, is strongly dependent on the tissue microstructure. The aim of dMRI is inferring the morphological structure of a sample and characterizing the dynamics of the system from the MRI signal. In spite of numerous practical applications of dMRI and many years of intensive theoretical work, this inverse problem has not been fully solved. This constitute a strong motivation for mathematical investigations.
Diffusion magnetic resonance imaging
Diffusion magnetic resonance imaging (dMRI) is a non-invasive technique which is extensively applied in material science to investigate structural and transport properties of porous media (such as sedimentary rocks or concrete), as well as in medical and neuroscience to study anatomical, physiological and functional properties of biological tissues (for some review and survey papers, see . The original idea to make the classical MRI sensitive to diffusion, is to apply a diffusion-weighting (or diffusion-sensitizing) gradient to encode random trajectories of the water molecules (Brownian motion) in the direction specified by the applied diffusion gradient.
DMRI gives a measure of spins displacement during a diffusion time which can vary on the order of tens of microseconds. In particular, the dMRI signal is an average of the magnetization over a voxel, which, in clinical scanners, is a volume of the order of 1 mm3 . Since the dimensions of cells are of the order of micro-meters, they cannot be individually distinguished with the dMRI resolution. Measuring the signal at different diffusion times, and gradient intensities and directions, one aims at inferring the morphological structure of a sample and characterizing the dynamics of a system.
Diffusion
On a molecular level self-diffusion results from collisions between atoms or molecules in liquid or gas state and it occurs even in thermodynamic equilibrium. This translational and random motion is called Brownian motion and it occurs when temperatures are above the zero degree Kelvin. It was observed for the first time by Robert Brown, who saw the random motions of pollen grains while studying them under his microscope [27]. A few years later [52, 53], Einstein used a probabilistic framework to describe the motion of an ensemble of particles undergoing diffusion.
Some biological applications and advanced acquisition
We have seen that the dMRI signal is the transverse magnetization in a volume of tissue called a voxel. The application of diffusion-encoding gradients causes an attenuation of the magnetization due to the dephasing of spins (water protons) from the incoherent motion of water. Since in biological tissues water diffusion is not free, and is instead strongly affected by the local environment that can contain hinderance to water displacement such as cell membranes and macromolecules, dMRI can reveal information about the tissue microstructure even though the signal is collected on a macroscopic level (voxel). In biological tissues, the image contrast in water proton diffusion magnetic resonance imaging is given by the difference in the average water displacement due to the difference in diffusion between imaged tissues at different spatial positions [106]. Since the first diffusion MRI images of normal and diseased brain in [107], an early major application has been the study of acute cerebral ischemia in stroke [131, 187]. In the brain, dMRI has been also used to detect a wide range of physiological and pathological conditions, including tumors [125, 166, 174, 181], myelination abnormalities [80, 108], connectivity [105], as well as in functional imaging [110, 111]. There are multiple ways to display contrast using dMRI. Although this is not the focus of the thesis, for completeness, and in order to give the reader some basic references on the applications of this technique, here I report a short list of some of the most popular and used ways to display contrast using dMRI.
An early contrast is the simplest one, where the intensity of each pixel of the image represents the magnitude of the transverse magnetization for a certain choice of diffusion-encoding gradient strength and direction. This contrast is the basis of diffusion-weighted imaging (DWI), which was shown to be more sensitive to early cellular changes after a stroke than more traditional MRI measurements . DWI is most applicable when the tissue of interest results in isotropic water diffusion displacement, for example in the brain grey matter.
In tissues where water diffusion is not isotropic, a contrast that takes into account the anisotropy is more appropriate. Diffusion tensor imaging (DTI) uses multiple diffusion-encoding gradient directions and fit a diffusion tensor at each voxel. Typical images assign contrast or colors to pixels based on the diffusion tensor eigenvalues .
DTI is then very useful to image tissues that have an internal fibrous structure such as the axons of brain white matter or muscle fibers in the heart. The fitted diffusion tensor in DTI has been used to produce tract or fiber images in the brain white matter (for a review see cite for example [10]), in the heart [35, 36, 97, 163], as well as other tissues, such as the prostate [71, 74, 157]. In the brain, the principal direction of the diffusion tensor have been used to infer the white-matter connectivity of the brain. Recently, more advanced models of the diffusion process, that go beyond the description by a diffusion tensor, have been proposed. These include, among others, diffusion spectrum imaging (DSI) [189], q-space imaging [7, 30], high angular resolution diffusion imaging (HARDI) ([58, 59]), persistent angular structure MRI (PAS-MRI) [85], generalized diffusion tensor imaging (GDTI) , q-ball MRI [182], composite hindered and restricted model of diffusion MRI (CHARMED) [9], diffusion orientation transform (DOT) [147], diffusion kurtosis imaging (DKI) [88, 121] and multi-tensor MRI .
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Table des matières
Introduction
1 Diffusion magnetic resonance imaging
1.1 Diffusion
1.2 Physics of diffusion magnetic resonance imaging (dMRI)
1.3 Some biological applications and advanced acquisition
2 Mathematical models
2.1 The microscopic Bloch-Torrey model
2.1.1 Probabilistic interpretation
2.1.2 Important length scales
2.1.3 Solutions of Bloch-Torrey equation
2.1.4 Narrow pulse approximation
2.1.5 The apparent diffusion coefficient (ADC)
2.1.6 Gaussian phase approximation
2.2 Approximate models
2.2.1 Short-time approximation
2.2.2 Long-time approximation
2.2.3 Multi-compartments models
2.2.4 Geometric models
3 Finite pulse Kärger and Kärger Models
3.1 Description of the models
3.1.1 Finite Pulse Kärger model
3.1.2 Kärger model
3.2 Convergence of the Kärger model to the FPK model
3.2.1 Asymptotic expansion in ζ
3.2.2 Error estimates
3.3 Kurtosis formula for FPK model
3.4 Numerical results
3.4.1 Trapezoidal PGSE
3.5 Conclusions
4 New homogenized model of time-dependent ADC (H-ADC)
4.1 Problem setting
4.1.1 Bloch-Torrey equation
4.1.2 Periodicity length
4.2 An asymptotic model
4.2.1 Transformed Bloch-Torrey equation
4.2.2 Choice of scaling
4.2.3 Asymptotic model corresponding to α “ γ “ 2
4.2.4 Asymptotic dMRI signal model and its ADC
4.3 Numerical results
4.3.1 Convergence
4.3.2 Time-dependent ADC
4.4 Comparison between the new asymptotic model and the linearized model
4.4.1 Convergence
4.5 Conclusions
Conclusion
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