{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Sphere Models"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In this section, we describe restricted sphere models, which have been used to represent (tumor)-cells.\n",
    "\n",
    "Sphere's by their nature have no orientations. Their only parameter is their diameter.\n",
    "\n",
    "Similarly to Cylinder models, Spheres have different model approximations that make different assumptions on the acquisition scheme. We start with describing the simplest Dot model (S1), and more towards more general models (S4)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Dot: S1\n",
    "The Dot model represents a non-diffusing component, which could represent trapped water in glial cells*(Stanisz et al. 1997)*, or axons with a different orientation to the main bundle*(Panagiotaki et al. 2009)*.\n",
    "Notably, *(Alexander et al. 2010, Veraart et al. 2016)* mentions that the contribution of the Dot model is negligible in \\emph{in-vivo} acquisitions. \n",
    "The signal of a Dot is described by a sphere with a diameter of zero, or equivalently an isotropic Gaussian compartment with $\\lambda_{iso}$ set to zero. In other words, it's just a function returning one no matter the input:\n",
    "\n",
    "\\begin{equation}\n",
    " E_{\\textrm{dot}}=1.\n",
    "\\end{equation}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from dmipy.signal_models import sphere_models\n",
    "dot = sphere_models.S1Dot()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Soderman Sphere: S2\n",
    "To keep naming consistency with the cylinder model with the same acquisition assumptions we call this the Soderman Sphere, but the equation is given by *(Balinov et al. 1993)*. The radius $R$ now corresponds to the radius of the sphere."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\\begin{equation}\n",
    "E(q,R|\\delta\\rightarrow0,\\Delta\\gg R^2/D)=\\left(\\frac{3}{(2\\pi q R) ^ 2}\\left(\\frac{\\sin(2\\pi q R)}{(2\\pi q R)} - \\cos(2\\pi q R)\\right)\\right) ^2\n",
    "\\end{equation}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
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LAU8hxFeoZhmJGoF0B8rAzDORLS8tLBBCfIZKkOullCeEEE+ghvL/KYRYhxod2AbV2X0A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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f57d30fac50>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import numpy as np\n",
    "from dmipy.core.acquisition_scheme import acquisition_scheme_from_qvalues\n",
    "sphere_stejskal_tanner = sphere_models.S2SphereStejskalTannerApproximation()\n",
    "\n",
    "Nsamples = 100\n",
    "bvecs = np.tile(np.r_[0., 1., 0.], (Nsamples, 1))  # doesn't matter it has no orientation\n",
    "qvals = np.linspace(0, 3e5, Nsamples)\n",
    "delta = 0.01\n",
    "Delta = 0.03\n",
    "scheme = acquisition_scheme_from_qvalues(qvals, bvecs, delta, Delta)\n",
    "\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline\n",
    "for diameter in np.linspace(1e-6, 1e-5, 5):\n",
    "    plt.plot(qvals, sphere_stejskal_tanner(scheme, diameter=diameter),\n",
    "             label=\"Diameter=\"+str(1e6 * diameter)+\"$\\mu m$\")\n",
    "plt.legend(fontsize=12)\n",
    "plt.title(\"Stejskal-Tanner attenuation over sphere diameter\", fontsize=17)\n",
    "plt.xlabel(\"q-value [1/m]\", fontsize=15)\n",
    "plt.ylabel(\"E(q)\", fontsize=15);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Callaghan Sphere: S3\n",
    "*(Callaghan 1995)*. Coming Soon..."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Gaussian Phase Sphere: S4\n",
    "*(Balinov et al. 1993)* derived the formulation of the Gaussian-Phase approximation for spheres, which models the signal attenuation for finite pulse duration $\\delta$ and pulse separation $\\Delta$. This approximation has been used for the VERDICT model for tumor characterization *(Panagiotaki et al. 2014)*."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\\begin{equation}\n",
    "\\ln\\left[E_\\perp(G,\\delta,\\Delta,R)\\right]=\\frac{2\\gamma^2 G^2}{D}\\sum_{m=1}^{\\infty}\\frac{a_m^{-4}}{a^2_mR^2-2}\\times\\left[2\\delta-\\dfrac{2 + e^{-a_m^2D(\\Delta-\\delta)} - 2e^{-a_m^2D\\delta} - 2e^{-a_m^2D\\Delta}+e^{-a_m^2D(\\Delta+\\delta)}}{a_m^2D}\\right]\n",
    "\\end{equation}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": true
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   "outputs": [],
   "source": [
    "gaussian_phase = sphere_models.S4SphereGaussianPhaseApproximation()"
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   "source": [
    "## References\n",
    "<div class=\"cite2c-biblio\"></div>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- Balinov, Balin, et al. \"The NMR self-diffusion method applied to restricted diffusion. Simulation of echo attenuation from molecules in spheres and between planes.\" Journal of Magnetic Resonance, Series A 104.1 (1993): 17-25.\n",
    "- Callaghan, Paul T. \"Pulsed-gradient spin-echo NMR for planar, cylindrical, and spherical pores under conditions of wall relaxation.\" Journal of magnetic resonance, Series A 113.1 (1995): 53-59. \n",
    "- Stanisz, Greg J., et al. \"An analytical model of restricted diffusion in bovine optic nerve.\" Magnetic Resonance in Medicine 37.1 (1997): 103-111.\n",
    "- Panagiotaki, Eleftheria, et al. \"Two-compartment models of the diffusion MR signal in brain white matter.\" International Conference on Medical Image Computing and Computer-Assisted Intervention. Springer, Berlin, Heidelberg, 2009.\n",
    "- Alexander, Daniel C., et al. \"Orientationally invariant indices of axon diameter and density from diffusion MRI.\" Neuroimage 52.4 (2010): 1374-1389.\n",
    "- Veraart, Jelle, Els Fieremans, and Dmitry S. Novikov. \"Universal power-law scaling of water diffusion in human brain defines what we see with MRI.\" arXiv preprint arXiv:1609.09145 (2016).\n",
    "- Panagiotaki, Eletheria, et al. \"Noninvasive quantification of solid tumor microstructure using VERDICT MRI.\" Cancer research 74.7 (2014): 1902-1912."
   ]
  }
 ],
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