{ "metadata": { "name": "project_overview" }, "nbformat": 3, "nbformat_minor": 0, "worksheets": [ { "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Introduction\n", "\n", "- This study aims to improve the acquisition time and/or quality of reconstruction of tractographies based on diffusion weighted MRI data.\n", "- Current approaches suffer from:\n", " - Low resolution sampling but high resolution desired outcome (2-3mm scan resolution, good for major white matter pathways, but not for micrometer axons)\n", " - Long acquisition times\n", " - Reconstruction inaccuracies due to uncertainty in gradient directions, dropped directions, low SNR, model unknowns (# directions etc.)\n", "- This project:\n", " - Single-shell HARDI imaging: at each voxel, presented with values sampled at a constant b-value but in varying directions\n", " - Q-space is sampled sparsely: saves acquisition time\n", " - Alternative approach: diffusion spectrum imaging [Eleftherios] or QBI\n", " - In which gradient directions should Q-space be sampled? Determined by the gradient directions, and controlled by us. Typical grids:\n", " - Minimum energy grid\n", " - Random grid\n", " - Optimal quadrature points (introduced by this study)\n", " - Random grid: advantage of arbitrary # of points, points can be dropped, variable scan duration\n", " - Optimal quadrature points: \"optimal\" representation, as far as integration over the sphere goes\n", " - Integration over the sphere is important when computing coefficients of spherical harmonics via inner product\n", "\n", "---\n", "- Higher b-value, lower SNR, higher directional selectivity\n", "- ^ Also, error in control of b-directions due to D2A conversion\n", "- Typical b-values of around 2000\n", "- SNR loss with increased b: b proportional to G^2 t^3. We'd like short t, have to generate stronger G. b of 2000 (relatively strong), t_E around 110ms...signal already fairly decayed.\n", "- Prefer to jump around in gradient directions. Heat induction in coils, try not to hit same parts each time to allow optimal cooling.\n", "- Eddy currents induced around activated coil. Oxford group, Jesper." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Mathematical formulation\n", "\n", "\n", "\n", "- Under certain moderate assumptions, Stejskal and Tanner [?] shows that signal attenuation is related to the diffusion PDF by the 3-D Fourier Transform:\n", "\n", "$\\frac{S(\\mathbf{q}, \\tau)}{S_0} = \\int_\\mathcal{R^3} P(\\mathbf{r} | \\mathbf{r_0},\\tau)\\exp (-2 \\pi i \\mathbf{q}^\\mathrm{T}\\mathbf{R})\\mathrm{d}\\mathbf{r} = \\mathcal{F}[ P(\\mathbf{r} | \\mathbf{r_0}, \\tau) ]$\n", "\n", "with\n", "\n", "$\\mathbf{q}=\\gamma \\delta \\mathbf{G} / 2 \\pi$, $\\delta$ the gradient pulse duration, $\\gamma$ the nuclear gyromagnetic ratio of water protons, $\\mathbf{G}$ the applied gradient vector, and $S_0$ the baseline image acquired without any gradient ($b=0$). $P$ is the diffusion PDF.\n", "\n", "Discuss PDF structure $P(\\mathbf{r} | \\mathbf{r_0},\\tau)$\n", "\n", "- An important special case occurs when the PDF is assumed to be Gaussian:\n", "\n", "$S(\\mathbf{q}, \\tau) = S_0 e^{-\\tau \\mathbf{q}^\\mathrm{T} D \\mathbf{q}}$\n", "\n", "or, with normalized $\\mathbf{q}$ and $b = \\tau |\\mathbf{q}|^2$,\n", "\n", "$S(b, \\mathbf{g}) = S_0 e^{-b \\mathbf{g}^\\mathrm{T} D \\mathbf{g}}.$\n", "\n", "We use this relationship in construction artificial data-sets for our simulations (diffusion tensor model).\n", "\n", "- We are interested in the Orientation Distribution Function, related to the diffusion PDF as follows\n", "\n", " $\\mathrm{ODF}(\\hat{u}) d\\Omega = \\int_{r=0}^{r=\\infty}P(r\\hat{u}) dv = \\int_{r=0}^{r=\\infty}P(r\\hat{u})r^2 dr d\\Omega$\n", "\n", "\n", "\n", "i.e. PDF over directions (surface of unit sphere) of diffusion, s.t.\n", "\n", "$\\mathrm{ODF}(\\hat{u}) = \\int_0^\\infty P(r\\hat{u})r^2 dr$\n", "\n", "The above $r^2$ term is important to include; was neglected in the widely cited paper by Tuch *et al*.\n", "\n", "- Based on the 3-D FFT relationship, we will proceed to model the reconstruction as a linear algebra problem, based on discretization of spherical harmonics." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Signal model\n", "\n", "Aganj shows that the ODF, with the above $r^2$ factor taken into account, can be expressed in terms of $P$'s 3D Fourier transform $E$, as\n", "\n", "$\\mathrm{ODF}(\\hat{u}) = \\frac{1}{4\\pi} - \\frac{1}{8\\pi^2} \\int_0^{2\\pi} \\int_0^\\infty \\frac{1}{q}\\nabla_b^2 E(\\mathbf{q}) dq d\\phi$\n", "\n", "where $E(q_0 \\hat{u}) = S(\\hat{u}) / S_0$ and $|\\mathbf{q}| = q_0$.\n", "\n", "Since we only have values for $E$ on a sphere in Q-space, we need to model values elsewhere. If we assume the model\n", "\n", "$E(\\mathbf{q}) \\approx E(q_0 \\hat{u})^{q^2 / q_0^2} = \\tilde{E}(\\hat{u})^{q^2 / q_0^2}$\n", "\n", "(useful because correctly has $E(0) = 1$)\n", "\n", "we can show that\n", "\n", "