{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Parameter Distributions\n", "\n", "## Gamma Distribution: DD1\n", "Histology studies show that axons do not have only one fixed diameter throughout the brain, but follow an axon diameter distribution *(Aboitiz et al. 1992)*.\n", "In Microstructure Imaging, the true axon diameter distribution is usually modeled as a Gamma distribution $\\Gamma(R;\\alpha,\\beta)$ with $R$ the axon radius as half its diameter and $\\alpha$ and $\\beta$ its scale and rate parameters *(Assaf et al. 2008)*.\n", "The probability density of a Gamma distribution is given as\n", "\n", "\\begin{equation}\n", " \\Gamma(R;\\alpha,\\beta)=\\frac{\\beta^\\alpha R^{\\alpha-1}e^{-R\\beta}}{\\Gamma(\\alpha)}\n", "\\end{equation}\n", "\n", "with $\\Gamma(\\alpha)$ a Gamma function.\n", "However, we must take the cross-sectional area of these cylinders into account to relate this Gamma distribution of cylinder radii to the signal attenuation of this distribution of cylinders.\n", "The reason for this is that it is not the cylinders themselves, but the (simulated) particles diffusing inside these cylinders that are contributing to the signal attenuation.\n", "The final perpendicular, intra-cylindrical signal attenuation for a Gamma-distributed cylinder ensemble, using any of the previously describe cylinder representations, is then given as\n", "\n", "\\begin{equation}\n", "E^{\\Gamma}_\\perp(q,\\Delta,\\delta;\\alpha,\\beta)=\\frac{\\int_{\\mathbb{R}^+}\\overbrace{\\Gamma(R;\\alpha,\\beta)}^{\\textrm{Gamma Distribution}}\\times\\overbrace{E_\\perp(q,\\Delta,\\delta,R)}^{\\textrm{Cylinder Signal Attenuation}}\\times \\overbrace{\\pi R^2}^{\\textrm{Surface Correction}} dR}{\\underbrace{\\int_{\\mathbb{R}^+}\\Gamma(R;\\alpha,\\beta)\\times \\textrm{N}(R) dR}_{\\textrm{Normalization}}}.\n", "\\end{equation}\n", "\n", "When modeling cylinders, $\\textrm{N}(R)$ is the surface function $\\textrm{N}(R)=\\pi R^2$. But, if we were to model a Gamma distribution of spheres, then it is the volume function $\\textrm{N}(R)=(4/3)\\pi R^3$. In fact, Dmipy internally checks what model parameter it is distributing, and normalizes the Gamma distribution accordingly." ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [ { "data": { "image/png": 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pM3io7UPc1+I+Fv27iF/PqH+W/xXqUomBy5cv4+1tyCWo1+tLMgXMnDmTUaNG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\nzgE180Tbd2Ef2QXZdglRk7ZqFUWZmXiPq58RA6pKa38Pnr6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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "from dmipy.distributions import distributions\n", "gamma_standard = distributions.DD1Gamma(normalization='standard')\n", "gamma_plane = distributions.DD1Gamma(normalization='plane')\n", "gamma_cylinder = distributions.DD1Gamma(normalization='cylinder')\n", "gamma_sphere = distributions.DD1Gamma(normalization='sphere')\n", "\n", "radii_std, Pstd = gamma_standard(alpha=2., beta=1e-6)\n", "radii_pln, Pstd = gamma_plane(alpha=2., beta=1e-6)\n", "radii_cyl, Pcyl = gamma_cylinder(alpha=2., beta=1e-6)\n", "radii_sph, Psph = gamma_sphere(alpha=2., beta=1e-6)\n", "\n", "import matplotlib.pyplot as plt\n", "%matplotlib inline\n", "plt.plot(radii_std, Pstd, label='Gamma Standard')\n", "plt.plot(radii_pln, Pstd, label='Gamma Plane')\n", "plt.plot(radii_cyl, Pcyl, label='Gamma Cylinder')\n", "plt.plot(radii_sph, Psph, label='Gamma Sphere')\n", "plt.legend()\n", "plt.xlabel('Radius [m]')\n", "plt.ylabel('Probability Density [-]');" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Notice that the normalization changes a lot in the apparent shape of the Gamma distribution. The distance between planes linearly with radius, the surface of cylinders grows quadratically with radius, and the volume of spheres grows cubically with radius. This is why the re-normalized distributions shift increasingly to the right.\n", "\n", "Upon initialization the Gamma distribution pre-calculates between which radii there is 99% of the volume under the distribution, and only samples there. This is why the different distributions produce samples at different positions, depending on $\\alpha, \\beta$ and the normalization." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## References" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "- Aboitiz, Francisco, et al. \"Fiber composition of the human corpus callosum.\" Brain research 598.1 (1992): 143-153.\n", "- Assaf, Yaniv, et al. \"AxCaliber: a method for measuring axon diameter distribution from diffusion MRI.\" Magnetic resonance in medicine 59.6 (2008): 1347-1354. " ] } ], "metadata": { "kernelspec": { "display_name": "Python 2", "language": "python", "name": "python2" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 2 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython2", "version": "2.7.14" } }, "nbformat": 4, "nbformat_minor": 2 }