{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Cylinder Models"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In this section, we describe models of intra-axonal diffusion.\n",
    "In all cases, the intra-axonal diffusion is represented using axially symmetric cylinder models with $\\boldsymbol{\\mu}\\in\\mathbb{S}^2$ the orientation parallel to the cylinder axis.\n",
    "The three-dimensional diffusion signal in these models is given as the separable product of (free) parallel and restricted perpendicular diffusion *(Assaf et al. 2004)*.\n",
    "This means that the three-dimensional signal is given by\n",
    "\n",
    "\\begin{equation}\n",
    " E_{\\textrm{intra}}(\\textbf{q},\\Delta,\\delta,\\lambda_\\parallel,R) = E_\\parallel(q_\\parallel,\\Delta,\\delta,\\lambda_\\parallel)\\times E_\\perp(q_\\perp,\\Delta,\\delta,R)\n",
    "\\end{equation}\n",
    "\n",
    "with parallel q-value $q_\\parallel=\\textbf{q}^T\\boldsymbol{\\mu}$, perpendicular q-value $q_\\perp=(\\textbf{q}^T\\textbf{q}-(\\textbf{q}^T\\boldsymbol{\\mu})^2))^{1/2}$, parallel diffusivity $\\lambda_\\parallel>0$ and cylinder radius $R>0$[mm]. The parallel signal is usually given by Gaussian diffusion as\n",
    "\n",
    "\\begin{equation}\n",
    "E_\\parallel(q_\\parallel,\\Delta,\\delta,\\lambda_\\parallel)=\\exp(-4\\pi^2q_\\parallel^2\\lambda_\\parallel(\\Delta-\\delta/3)).\n",
    "\\end{equation}\n",
    "\n",
    "The perpendicular signal $E_\\perp$ is described using various cylinder models.\n",
    "In the rest of this section, we start with describing the simplest, having the strongest tissue assumptions (C1), and more towards more general models (C4)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Stick: C1\n",
    "The simplest model for intra-axonal diffusion is the ``Stick'' -- a cylinder with zero radius *(Behrens et al. 2003)*.\n",
    "The Stick model assumes that, because axon diameters are very small, the perpendicular diffusion attenuation inside these axons is negligible compared to the overall signal attenuation.\n",
    "The perpendicular diffusion coefficient is therefore be approximated by zero, so the perpendicular signal attenuation is always equal to one as $E_\\perp=1$.\n",
    "Inserting this definition into the equation above leads to the simple signal representation\n",
    "\n",
    "\\begin{equation}\n",
    "E_{\\textrm{Stick}}(b,\\textbf{n},\\boldsymbol{\\mu},\\lambda_\\parallel)=\\exp(-b\\lambda_\\parallel(\\textbf{n}^T\\boldsymbol{\\mu})^2),\n",
    "\\end{equation}\n",
    "\n",
    "which is the same as a DTI Tensor with $\\lambda_\\parallel=\\lambda_1$ and $\\lambda_\\perp=\\lambda_2=\\lambda_3=0$.\n",
    "Despite its simplicity, it turns out approximating axons as Sticks is quite reasonable at clinical gradient strengths *(Burcaw et al. 2015)*.\n",
    "In fact, the Stick is used in the most state-of-the-art microstructure models modeling axonal dispersion *(Tariq et al. 2016, Kaden et al. 2016)*."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "data": {
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PziqMMZWHJYpiuLNnJnvyjvLulDXhDsUYY0qNJYpiaFMvkV6t0hgxZRW7D9hZhTGmcrBE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\nZmbWyoHCzMxa/Qvqi4Z3O89/ggAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fceffc4b390>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from dmipy.signal_models import cylinder_models\n",
    "from dmipy.core.acquisition_scheme import acquisition_scheme_from_bvalues\n",
    "import numpy as np\n",
    "stick = cylinder_models.C1Stick(mu=[0, 0], lambda_par=1.7e-9)\n",
    "\n",
    "Nsamples = 100\n",
    "bvecs_parallel = np.tile(np.r_[0., 0., 1.], (Nsamples, 1))\n",
    "bvecs_perpendicular = np.tile(np.r_[0., 1., 0.], (Nsamples, 1))\n",
    "bvals = np.linspace(0, 2e9, Nsamples)\n",
    "delta = 0.01\n",
    "Delta = 0.03\n",
    "scheme_parallel = acquisition_scheme_from_bvalues(bvals, bvecs_parallel, delta, Delta)\n",
    "scheme_perpendicular = acquisition_scheme_from_bvalues(bvals, bvecs_perpendicular, delta, Delta)\n",
    "\n",
    "Estick_parallel = stick(scheme_parallel)\n",
    "Estick_perpendicular = stick(scheme_perpendicular)\n",
    "\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline\n",
    "plt.plot(bvals, Estick_parallel, label=\"Stick $E_\\parallel$\")\n",
    "plt.plot(bvals, Estick_perpendicular, label=\"Stick $E_\\perp$\")\n",
    "plt.legend(fontsize=12)\n",
    "plt.title(\"Signal attenuation Stick\", fontsize=17)\n",
    "plt.xlabel(\"b-value [s/m$^2$]\", fontsize=15)\n",
    "plt.ylabel(\"Signal Attenuation\", fontsize=15);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Stejskal-Tanner Cylinder: C2\n",
    "In reality, axons have a non-zero radius.\n",
    "To account for this, different cylinder models for perpendicular diffusion have been proposed for different combinations of PGSE acquisition parameters.\n",
    "The simplest is the Stejskal-Tanner approximation of the cylinder *(Soderman and Johnson 1995)*, which has the hardest assumptions on the PGSE protocol.\n",
    "First, it assumes that pulse length $\\delta$ is so short that no diffusion occurs during the application of the gradient pulse ($\\delta\\rightarrow0$).\n",
    "Second, it assumes that pulse separation $\\Delta$ is long enough for diffusion with intra-cylindrical diffusion coefficient $D$ to be restricted inside a cylinder of radius $R$ ($\\Delta\\gg R^2/D$).\n",
    "Within these assumptions, the perpendicular, intra-cylindrical signal attenuation is given as\n",
    "\n",
    "\\begin{equation}\n",
    " E_\\perp(q,R|\\delta\\rightarrow0,\\Delta\\gg R^2/D)=\\left(\\frac{J_1(2\\pi q R)}{\\pi q R}\\right)^2,\n",
    "\\end{equation}\n",
    "\n",
    "where we use the ``$|$'' to separate function parameters from model assumptions, and $J_1$ is a Bessel function of the first kind. Taking $\\lim_{R\\rightarrow0}$ of this equation simplifies the three-dimensional Soderman model to the Stick model."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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CCEG/C7qQcbyA9KR8xzKurnR64AFKjxwhb/2GeqXxxPAnCPcN57FN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\nSZKuJB1gkiRJ0pWkA0ySJEm6kv8HyyakFtLmD24AAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fceffbfdb50>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from dmipy.core.acquisition_scheme import acquisition_scheme_from_qvalues\n",
    "stesjskal_tanner = cylinder_models.C2CylinderStejskalTannerApproximation(mu=[0, 0], lambda_par=1.7e-9)\n",
    "\n",
    "Nsamples = 100\n",
    "bvecs_perpendicular = np.tile(np.r_[0., 1., 0.], (Nsamples, 1))\n",
    "qvals = np.linspace(0, 3e5, Nsamples)\n",
    "delta = 0.01\n",
    "Delta = 0.03\n",
    "scheme_perpendicular = acquisition_scheme_from_qvalues(qvals, bvecs_perpendicular, delta, Delta)\n",
    "\n",
    "for diameter in np.linspace(1e-6, 1e-5, 5):\n",
    "    plt.plot(qvals, stesjskal_tanner(scheme_perpendicular, diameter=diameter),\n",
    "             label=\"Diameter=\"+str(1e6 * diameter)+\"$\\mu m$\")\n",
    "plt.legend(fontsize=12)\n",
    "plt.title(\"Stesjkal-Tanner attenuation over cylinder diameter\", fontsize=17)\n",
    "plt.xlabel(\"perpendicular q-value [1/m]\", fontsize=15)\n",
    "plt.ylabel(\"E(q$_\\perp$)\", fontsize=15);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Callaghan Cylinder: C3\n",
    "The ``Callaghan'' model relaxes Soderman's $\\Delta\\gg R^2/D$ assumption to allow for unrestricted diffusion at shorter pulse separation $\\Delta$ *(Callaghan 1995)*. In this case, the perpendicular signal attenuation is given as\n",
    "\n",
    "\\begin{align}\n",
    " E_\\perp(q,\\Delta,R|\\delta\\rightarrow0)&=\\sum^\\infty_k4\\exp(-\\beta^2_{0k}D\\Delta/R^2)\\times \\frac{\\left((2\\pi qR)J_0^{'}(2\\pi qR)\\right)^2}{\\left((2\\pi qR)^2-\\beta_{0k}^2\\right)^2}\\nonumber\\\\\n",
    " &+\\sum^\\infty_{nk}8\\exp(-\\beta^2_{nk}D\\Delta/R^2)\\times \\frac{\\beta^2_{nk}}{\\left(\\beta_{nk}^2-n^2\\right)}\\times\\frac{\\left((2\\pi qR)J_n^{'}(2\\pi qR)\\right)^2}{\\left((2\\pi qR)^2-\\beta_{nk}^2\\right)^2}\n",
    "\\end{align}\n",
    "\n",
    "where $J_n^{'}$ are the derivatives of the $n^{th}$-order Bessel function and $\\beta_{nk}$ are the arguments that result in zero-crossings. Taking $\\lim_{\\Delta\\rightarrow\\infty}$ of this equation simplifies the Callaghan model to the Soderman model. The Callaghan model has been used to estimate the axon diameter distribution in the multi-compartment AxCaliber approach *(Assaf et al. 2008)*. However, the authors also mention that the perpendicular diffusion is likely already restricted for realistic axon diameters ($<2\\mu$m) *(Aboitiz et al. 1992)* for the shortest possible $\\Delta$ in PGSE protocols (${\\sim}10$ms). This limits the added value of the Callaghan model over the Soderman model in axon diameter estimation."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x7fceffbe94d0>"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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Cv/uVdZHZiCRIrf4oKOkGGz7W/fAMqAW+a1fYuBE6dICE/y0g4lC8CCsG1cbF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\no9EEIVq5azQaTRCilbtGo9EEIf8fmKLFUmZtjd0AAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fceffbe9610>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "callaghan = cylinder_models.C3CylinderCallaghanApproximation(mu=[0, 0], lambda_par=1e-7)\n",
    "\n",
    "Nsamples = 100\n",
    "bvecs_perpendicular = np.tile(np.r_[0., 1., 0.], (Nsamples, 1))\n",
    "qvals = np.linspace(0, 3e5, Nsamples)\n",
    "delta = 0.001\n",
    "Delta = 0.001\n",
    "scheme_perpendicular = acquisition_scheme_from_qvalues(qvals, bvecs_perpendicular, delta, Delta)\n",
    "plt.plot(qvals, np.exp(-scheme_perpendicular.bvalues * 1.7e-9), label=\"Free Diffusion\", c='r', ls='--')\n",
    "\n",
    "for Delta in [0.001, 0.0025, 0.015]:\n",
    "    scheme_perpendicular = acquisition_scheme_from_qvalues(qvals, bvecs_perpendicular, delta, Delta)\n",
    "    plt.plot(qvals, callaghan(scheme_perpendicular, diameter=10e-6), label='Callaghan Delta='+str(1e3 * Delta)+'ms')\n",
    "plt.plot(qvals, stesjskal_tanner(scheme_perpendicular, diameter=10e-6), label=\"Soderman\", c='blue', ls='--')\n",
    "plt.legend()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For a big cylinder of 10$\\mu$ diameter, it can be seen that free diffusion and the Callaghan model are very similar for an extremely short pulse separation of 1ms. The signal is already becoming significantly restricted at 2.5ms, and at 15ms the Callaghan and Soderman approximations have converged (completely restricted).\n",
    "\n",
    "This shows the problem of using the Callaghan model for axon diameter estimation - for axons of diameter 0.1-2 $\\mu$m the diffusion is already restricted around 1 or 2 ms, meaning there is no signal contrast for intra-axonal diffusion when Delta varies. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Gaussian Phase Cylinder: C4\n",
    "The last cylinder model generalization we discuss is the \"Van Gelderen\" model *(VanGelderen et al. 1994)*, which relaxes the last $\\delta\\rightarrow0$ assumption to allow for finite pulse length $\\delta$. This model is based on the ``Neuman'' model *(Neuman 1974)*, which assumes Gaussian diffusion during the gradient pulse. In this case, the signal attenuation is given as\n",
    "\n",
    "\\begin{equation}\n",
    " E_\\perp(q,\\Delta,\\delta,R)=-8\\pi^2q^2\\sum^\\infty_{m=1}\\dfrac{\\left[2Da_m^2\\delta-2 + 2e^{-Da_m^2\\delta} + 2e^{-Da_m^2\\Delta}-e^{-Da_m^2(\\Delta-\\delta)}-e^{-Da_m^2(\\Delta-\\delta)}\\right]}{\\delta^2D^2a_m^6(R^2a_m^2-1)}\n",
    "\\end{equation}\n",
    "\n",
    "where $a_m$ are roots of the equation $J_1^{'}(a_mR)=0$, with $J_1^{'}$ again the derivative of the Bessel function of the first kind, and $D$ is the intra-axonal diffusivity.\n",
    "According to *(Neuman 1974)*, taking the double $\\lim_{(\\delta,\\Delta)\\rightarrow(0,\\infty)}$ of the equation above should simplify the Van Gelderen model to the Soderman Model, although he does not show this explicitly.\n",
    "For its generality, the Van Gelderen model has been used in most recent studies regarding in-vivo axon diameter estimation *(Huang et al. 2015, Ferizi et al. 2015, De Santis et al. 2016 )*."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "vangelderen = cylinder_models.C4CylinderGaussianPhaseApproximation()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## References"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
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    "- Assaf, Yaniv, et al. \"AxCaliber: a method for measuring axon diameter distribution from diffusion MRI.\" Magnetic resonance in medicine 59.6 (2008): 1347-1354.  \n",
    "- Assaf, Yaniv, et al. \"New modeling and experimental framework to characterize hindered and restricted water diffusion in brain white matter.\" Magnetic Resonance in Medicine 52.5 (2004): 965-978.  \n",
    "- Behrens, Timothy EJ, et al. \"Characterization and propagation of uncertainty in diffusion‐weighted MR imaging.\" Magnetic resonance in medicine 50.5 (2003): 1077-1088.  \n",
    "- Burcaw, Lauren M., Els Fieremans, and Dmitry S. Novikov. \"Mesoscopic structure of neuronal tracts from time-dependent diffusion.\" NeuroImage 114 (2015): 18-37.  \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",
    "- De Santis, Silvia, Derek K. Jones, and Alard Roebroeck. \"Including diffusion time dependence in the extra-axonal space improves in vivo estimates of axonal diameter and density in human white matter.\" NeuroImage 130 (2016): 91-103.  \n",
    "- Ferizi, Uran, et al. \"White matter compartment models for in vivo diffusion MRI at 300mT/m.\" NeuroImage 118 (2015): 468-483.\n",
    "- Huang, Susie Y., et al. \"The impact of gradient strength on in vivo diffusion MRI estimates of axon diameter.\" NeuroImage 106 (2015): 464-472.  \n",
    "- Kaden, Enrico, et al. \"Multi-compartment microscopic diffusion imaging.\" NeuroImage 139 (2016): 346-359.  \n",
    "- Neuman, C. H. \"Spin echo of spins diffusing in a bounded medium.\" The Journal of Chemical Physics 60.11 (1974): 4508-4511.\n",
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    "- Vangelderen, P., et al. \"Evaluation of restricted diffusion in cylinders. Phosphocreatine in rabbit leg muscle.\" Journal of Magnetic Resonance, Series B 103.3 (1994): 255-260."
   ]
  }
 ],
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