wilkens.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Wilkens LET model\n",
    "Wilkens [1] described analytical model for LET calculation. The model is able to predict dose- and track-averaged values in the pencil proton beamlets. The calculations are limited to center-of-axis values and to liquid water.\n",
    "\n",
    "The model is based on the Bortfeld analytical approximation of proton stopping power in water:\n",
    "\n",
    "$$\n",
    "\\tilde S(r)  = \\frac{r^{1/p - 1}}{p \\alpha^{1/p}}\n",
    "$$\n",
    "\n",
    "where $r$ is the residual range ($R_0-z$).\n",
    "\n",
    "Bragg-Kleeman rule which relates range of protons with its initial kinetic energy employs the same constants $p$ and $\\alpha$:\n",
    "\n",
    "$$\n",
    "R_0(E) = \\alpha E^p\n",
    "$$\n",
    "\n",
    "Another assumption is energy distribution of proton beamlet follows gaussian distribution at every point of the beam. Nuclear reactions are neglected in this model.\n",
    "\n",
    "The model provides $LET$ as a function of: \n",
    "  - $z$ - depth in water (in cm)\n",
    "  - $E$ - kinetic energy of proton entering water target (in MeV)\n",
    "  - $\\sigma_E$ - Gaussian energy spread (in MeV)\n",
    "\n",
    "\n",
    "For the reference, see: \n",
    "\n",
    "[1] Wilkens JJ, Oelfke U. \"Analytical linear energy transfer calculations for proton therapy\" Med Phys. 2003 May;30(5):806-15. (DOI: 10.1118/1.1567852)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Energy, range and stopping power"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "# jupyter hacks to print some command output (i.e. %psource) in the cell, instead of in popup\n",
    "from __future__ import print_function\n",
    "from IPython.core import page\n",
    "page.page = print\n",
    "\n",
    "# first some necessary python imports\n",
    "import numpy as np\n",
    "import matplotlib \n",
    "%matplotlib inline\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\u001b[0;32mclass\u001b[0m \u001b[0mERSCalc\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mobject\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\n",
      "\u001b[0;34m\u001b[0m    \u001b[0;34m\"\"\"\u001b[0m\n",
      "\u001b[0;34m    A 'calculator' class for stopping power and range of protons\u001b[0m\n",
      "\u001b[0;34m    \"\"\"\u001b[0m\u001b[0;34m\u001b[0m\n",
      "\u001b[0;34m\u001b[0m    \u001b[0malpha_cm_MeV\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;36m0.0022\u001b[0m\u001b[0;34m\u001b[0m\n",
      "\u001b[0;34m\u001b[0m    \u001b[0mp\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;36m1.77\u001b[0m\u001b[0;34m\u001b[0m\n",
      "\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\n",
      "\u001b[0;34m\u001b[0m    \u001b[0;34m@\u001b[0m\u001b[0mclassmethod\u001b[0m\u001b[0;34m\u001b[0m\n",
      "\u001b[0;34m\u001b[0m    \u001b[0;32mdef\u001b[0m \u001b[0mp_alpha_cm_MeV\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mcls\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\n",
      "\u001b[0;34m\u001b[0m        \u001b[0;34m\"\"\"\u001b[0m\n",
      "\u001b[0;34m        Helper variable p*alpha^(1/p)\u001b[0m\n",
      "\u001b[0;34m        \"\"\"\u001b[0m\u001b[0;34m\u001b[0m\n",
      "\u001b[0;34m\u001b[0m        \u001b[0;32mreturn\u001b[0m \u001b[0mcls\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mp\u001b[0m \u001b[0;34m*\u001b[0m \u001b[0mcls\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0malpha_cm_MeV\u001b[0m \u001b[0;34m**\u001b[0m \u001b[0;34m(\u001b[0m\u001b[0;36m1.0\u001b[0m \u001b[0;34m/\u001b[0m \u001b[0mcls\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mp\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\n",
      "\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\n",
      "\u001b[0;34m\u001b[0m    \u001b[0;34m@\u001b[0m\u001b[0mclassmethod\u001b[0m\u001b[0;34m\u001b[0m\n",
      "\u001b[0;34m\u001b[0m    \u001b[0;32mdef\u001b[0m \u001b[0mrange_cm\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mcls\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0menergy_MeV\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\n",
      "\u001b[0;34m\u001b[0m        \u001b[0;34m\"\"\"\u001b[0m\n",
      "\u001b[0;34m        Bragg-Kleeman rule for energy-range relationship\u001b[0m\n",
      "\u001b[0;34m        Equation (8) in [1]\u001b[0m\n",
      "\u001b[0;34m        \"\"\"\u001b[0m\u001b[0;34m\u001b[0m\n",
      "\u001b[0;34m\u001b[0m        \u001b[0;32mreturn\u001b[0m \u001b[0mcls\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0malpha_cm_MeV\u001b[0m \u001b[0;34m*\u001b[0m \u001b[0menergy_MeV\u001b[0m \u001b[0;34m**\u001b[0m \u001b[0mcls\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mp\u001b[0m\u001b[0;34m\u001b[0m\n",
      "\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\n",
      "\u001b[0;34m\u001b[0m    \u001b[0;34m@\u001b[0m\u001b[0mclassmethod\u001b[0m\u001b[0;34m\u001b[0m\n",
      "\u001b[0;34m\u001b[0m    \u001b[0;32mdef\u001b[0m \u001b[0mstop_pow_MeV_cm\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mcls\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mresid_range_cm\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\n",
      "\u001b[0;34m\u001b[0m        \u001b[0;34m\"\"\"\u001b[0m\n",
      "\u001b[0;34m        Bortfeld approximation for stopping power of protons\u001b[0m\n",
      "\u001b[0;34m        \"\"\"\u001b[0m\u001b[0;34m\u001b[0m\n",
      "\u001b[0;34m\u001b[0m        \u001b[0;32mreturn\u001b[0m \u001b[0;34m(\u001b[0m\u001b[0mresid_range_cm\u001b[0m \u001b[0;34m**\u001b[0m \u001b[0;34m(\u001b[0m\u001b[0;36m1.0\u001b[0m \u001b[0;34m/\u001b[0m \u001b[0mcls\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mp\u001b[0m \u001b[0;34m-\u001b[0m \u001b[0;36m1.0\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;34m/\u001b[0m \u001b[0mcls\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mp_alpha_cm_MeV\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\n"
     ]
    }
   ],
   "source": [
    "# most of the code is stored in models.py file\n",
    "# we import it here and display the source code of the stopping power & range calculator class\n",
    "from models import ERSCalc\n",
    "%psource ERSCalc"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Range of protons with energy 70.0 MeV is 4.06 cm\n",
      "Stopping power at energy 70.0 MeV and depth 2.00 cm is 13.1 MeV/cm\n"
     ]
    }
   ],
   "source": [
    "# take as example some interesting proton beam energy\n",
    "E_MeV = 70.0\n",
    "# let's calculate range\n",
    "range_cm = ERSCalc.range_cm( E_MeV )\n",
    "# then at some specific depth lets obtain residual range\n",
    "z_cm = 2.0\n",
    "res_range_cm = range_cm - z_cm\n",
    "stop_power_MeV_cm = ERSCalc.stop_pow_MeV_cm( res_range_cm )\n",
    "\n",
    "# finally the printouts\n",
    "print(\"Range of protons with energy {:3.1f} MeV is {:3.3} cm\".format(E_MeV, range_cm))\n",
    "print(\"Stopping power at energy {:3.1f} MeV and depth {:3.2f} cm is {:3.3} MeV/cm\".format(E_MeV, z_cm, stop_power_MeV_cm))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7fb93b2e26a0>]"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# some plots of range\n",
    "E_MeV = np.linspace(start=60, stop=250, num=100)\n",
    "range_cm = ERSCalc.range_cm( E_MeV )\n",
    "plt.xlabel(\"Energy [MeV]\")\n",
    "plt.ylabel(\"Range [cm]\")\n",
    "plt.plot(E_MeV,range_cm)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7fb93b208e10>]"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# some plots of stopping power\n",
    "stop_power_MeV_cm = ERSCalc.stop_pow_MeV_cm( range_cm )\n",
    "plt.xlabel(\"Energy [MeV]\")\n",
    "plt.ylabel(\"Stopping power [MeV/cm]\")\n",
    "plt.plot(E_MeV,stop_power_MeV_cm)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Wilkens LET model\n",
    "\n",
    "Wilkens LET model is based on following mathematical formalism. Dose averaged LET $L_d$ is described as:\n",
    "\n",
    "$$\n",
    "L_d(z) = \\frac{\\langle S^2\\rangle_z}{\\langle S\\rangle_z}\n",
    "$$\n",
    "\n",
    "while track averaged LET $L_t$ is equal to:\n",
    "\n",
    "$$\n",
    "L_t(z) = \\frac{\\langle S\\rangle_z}{\\Phi_z}\n",
    "$$\n",
    "\n",
    "\n",
    "where $\\langle S \\rangle$ denotes stopping power averaged over an spectrum of protons with different residual range $r$. Following the formulas (A14), (A12) and (A8) from [1] we get:\n",
    "\n",
    "$$\n",
    "\\Phi_z = \\frac{\\Phi_0}{\\sqrt{2 \\pi}} e^{-\\zeta^2 / 4} D_{-1}(\\zeta)\n",
    "$$\n",
    "\n",
    "$$\n",
    "\\langle S\\rangle_z = \\frac{\\Phi_0}{\\sqrt{2 \\pi} \\sigma R \\alpha^{1/p}} \\Bigg[ \\sigma^{1 + 1/p} \\Gamma\\Big( 1 + \\frac{1}{p}\\Big) \\tilde D_{1+1/p}(\\xi, \\zeta) - R \\Big( \\frac{R}{2}\\Big)^{1/p} e^{-(\\xi + \\zeta)^2/8} \\Bigg]\n",
    "$$\n",
    "\n",
    "$$\n",
    "\\langle S^2\\rangle_z = \\frac{\\Phi_0}{\\sqrt{2 \\pi} \\sigma R \\alpha^{2/p} p (2-p)} \\Bigg[ \\sigma^{2/p} \\Gamma\\Big( \\frac{2}{p}\\Big) \\tilde D_{2/p}(\\xi, \\zeta) - 2 R \\Big( \\frac{R}{2}\\Big)^{2/p} e^{-(\\xi + \\zeta)^2/8} \\Bigg]\n",
    "$$\n",
    "\n",
    "after simplifying we obtain:\n",
    "\n",
    "$$\n",
    "L_d(z, E, \\sigma_E) = \\frac{1}{\\alpha^{1/p} p (2-p)} \\frac{\\sigma^{2/p} \\Gamma\\Big( \\frac{2}{p}\\Big) \\tilde D_{2/p}(\\xi, \\zeta) - 2 R \\Big( \\frac{R}{2}\\Big)^{2/p} e^{-(\\xi + \\zeta)^2/8}}{\\sigma^{1 + 1/p} \\Gamma\\Big( 1 + \\frac{1}{p}\\Big) \\tilde D_{1+1/p}(\\xi, \\zeta) - R \\Big( \\frac{R}{2}\\Big)^{1/p} e^{-(\\xi + \\zeta)^2/8}}\n",
    "$$\n",
    "\n",
    "$$\n",
    "L_t(z, E, \\sigma_E) = \\frac{1}{\\sigma R \\alpha^{1/p}} \\frac{\\sigma^{1 + 1/p} \\Gamma\\Big( 1 + \\frac{1}{p}\\Big) \\tilde D_{1+1/p}(\\xi, \\zeta) - R \\Big( \\frac{R}{2}\\Big)^{1/p} e^{-(\\xi + \\zeta)^2/8}}{ e^{-\\zeta^2 / 4} D_{-1}(\\zeta) }$$\n",
    "\n",
    "In this formula we have following variables:\n",
    " - $\\alpha = 0.0022 \\ \\textrm{cm MeV}^{-p}$ \n",
    " - $p = 1.77$\n",
    " - $R = 2 \\mu m$ - regularization factor\n",
    " - $\\sigma = \\sqrt{\\sigma_{\\textrm{mono}}^2 + \\sigma_r^2}$ - describes Gaussian spread of range (in cm)\n",
    " - $\\sigma_{\\textrm{mono}} = 0.012 R_0^{0.935}$ - accounts for range straggling of monoenergetic protons (in cm)\n",
    " - $\\sigma_r = \\sigma_E \\alpha p E^{p-1}$ - accounts for initial energy spread translated into range spread (in cm)\n",
    " - $R_0 = \\alpha E^p$ - range of protons with kinetic energy $E$ (in cm)\n",
    " - $\\zeta = (z - R_0) / \\sigma$ - helper variable\n",
    " - $\\xi = (z - R_0 - R) / \\sigma$ - helper variable\n",
    " - $\\Gamma$ - gamma function\n",
    " - $\\tilde D_\\nu(x,y) = e^{-x^2/4} D_{-\\nu}(x) - e^{-y^2/4} D_{-\\nu}(y)$ - helper function\n",
    " - $D_\\nu$ - parabolic cylinder function"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\u001b[0;32mclass\u001b[0m \u001b[0mWilkensModel\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mGeneralModel\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\n",
      "\u001b[0;34m\u001b[0m    \u001b[0;34m\"\"\"\u001b[0m\n",
      "\u001b[0;34m    Analytical LET model implementation following\u001b[0m\n",
      "\u001b[0;34m    [1] Wilkens JJ, Oelfke U. \"Analytical linear energy transfer calculations for proton therapy\"\u001b[0m\n",
      "\u001b[0;34m    Med Phys. 2003 May;30(5):806-15. (DOI: 10.1118/1.1567852)\u001b[0m\n",
      "\u001b[0;34m    \"\"\"\u001b[0m\u001b[0;34m\u001b[0m\n",
      "\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\n",
      "\u001b[0;34m\u001b[0m    \u001b[0;32mdef\u001b[0m \u001b[0m_d_tilde\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mnu\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\n",
      "\u001b[0;34m\u001b[0m        \u001b[0;34m\"\"\"helper function defined by equation (10) in [1]\"\"\"\u001b[0m\u001b[0;34m\u001b[0m\n",
      "\u001b[0;34m\u001b[0m        \u001b[0mpart1\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mExactSpecialFunction\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mparabolic_integral\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mnu\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mxi\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\n",
      "\u001b[0;34m\u001b[0m        \u001b[0mpart2\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mExactSpecialFunction\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mparabolic_integral\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mnu\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mzeta\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\n",
      "\u001b[0;34m\u001b[0m        \u001b[0;32mreturn\u001b[0m \u001b[0mpart1\u001b[0m \u001b[0;34m-\u001b[0m \u001b[0mpart2\u001b[0m\u001b[0;34m\u001b[0m\n",
      "\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\n",
      "\u001b[0;34m\u001b[0m    \u001b[0;32mdef\u001b[0m \u001b[0mlet_d_MeV_cm\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\n",
      "\u001b[0;34m\u001b[0m        \u001b[0;34m\"\"\"\u001b[0m\n",
      "\u001b[0;34m        dose averaged LET in [MeV/cm]\u001b[0m\n",
      "\u001b[0;34m        \"\"\"\u001b[0m\u001b[0;34m\u001b[0m\n",
      "\u001b[0;34m\u001b[0m        \u001b[0;31m# main part of <S>_z as in quation (10) or (A12) in [1]\u001b[0m\u001b[0;34m\u001b[0m\n",
      "\u001b[0;34m\u001b[0m        \u001b[0mq\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;36m1.0\u001b[0m \u001b[0;34m+\u001b[0m \u001b[0;36m1.0\u001b[0m \u001b[0;34m/\u001b[0m \u001b[0mERSCalc\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mp\u001b[0m\u001b[0;34m\u001b[0m\n",
      "\u001b[0;34m\u001b[0m        \u001b[0mmean_s_z_part\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0msigma_cm\u001b[0m \u001b[0;34m**\u001b[0m \u001b[0mq\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;34m*\u001b[0m \u001b[0mfloat\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mmp\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mgamma\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mq\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;34m*\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_d_tilde\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mq\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\n",
      "\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\n",
      "\u001b[0;34m\u001b[0m        \u001b[0mmean_s_z_part\u001b[0m \u001b[0;34m-=\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mr_cm\u001b[0m \u001b[0;34m*\u001b[0m \u001b[0;34m(\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m0.5\u001b[0m \u001b[0;34m*\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mr_cm\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;34m**\u001b[0m \u001b[0;34m(\u001b[0m\u001b[0;36m1.0\u001b[0m \u001b[0;34m/\u001b[0m \u001b[0mERSCalc\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mp\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;34m*\u001b[0m \u001b[0mExactSpecialFunction\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mexponent_square\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m\u001b[0m\n",
      "\u001b[0;34m\u001b[0m            \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mzeta\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mxi\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\n",
      "\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\n",
      "\u001b[0;34m\u001b[0m        \u001b[0;31m# main part of <S2>_z as in quation (10) or (A14) in [1]\u001b[0m\u001b[0;34m\u001b[0m\n",
      "\u001b[0;34m\u001b[0m        \u001b[0mr\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;36m2.0\u001b[0m \u001b[0;34m/\u001b[0m \u001b[0mERSCalc\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mp\u001b[0m\u001b[0;34m\u001b[0m\n",
      "\u001b[0;34m\u001b[0m        \u001b[0mmean_s2_z_part\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0msigma_cm\u001b[0m \u001b[0;34m**\u001b[0m \u001b[0mr\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;34m*\u001b[0m \u001b[0mfloat\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mmp\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mgamma\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mr\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;34m*\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_d_tilde\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mr\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\n",
      "\u001b[0;34m\u001b[0m        \u001b[0mmean_s2_z_part\u001b[0m \u001b[0;34m-=\u001b[0m \u001b[0;36m2.0\u001b[0m \u001b[0;34m*\u001b[0m \u001b[0;34m(\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m0.5\u001b[0m \u001b[0;34m*\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mr_cm\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;34m**\u001b[0m \u001b[0mr\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;34m*\u001b[0m \u001b[0mExactSpecialFunction\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mexponent_square\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mzeta\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mxi\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\n",
      "\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\n",
      "\u001b[0;34m\u001b[0m        \u001b[0;31m# factor part of <S2>_z divided by <S>_z\u001b[0m\u001b[0;34m\u001b[0m\n",
      "\u001b[0;34m\u001b[0m        \u001b[0mconst_factor_MeV_cm\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;36m1.0\u001b[0m \u001b[0;34m/\u001b[0m \u001b[0;34m(\u001b[0m\u001b[0mERSCalc\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mp_alpha_cm_MeV\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;34m*\u001b[0m \u001b[0;34m(\u001b[0m\u001b[0;36m2.0\u001b[0m \u001b[0;34m-\u001b[0m \u001b[0mERSCalc\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mp\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\n",
      "\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\n",
      "\u001b[0;34m\u001b[0m        \u001b[0;31m# result\u001b[0m\u001b[0;34m\u001b[0m\n",
      "\u001b[0;34m\u001b[0m        \u001b[0mresult\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mconst_factor_MeV_cm\u001b[0m \u001b[0;34m*\u001b[0m \u001b[0mmean_s2_z_part\u001b[0m \u001b[0;34m/\u001b[0m \u001b[0mmean_s_z_part\u001b[0m\u001b[0;34m\u001b[0m\n",
      "\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\n",
      "\u001b[0;34m\u001b[0m        \u001b[0;31m# filling nonsense values (outside model domain) with np.nan\u001b[0m\u001b[0;34m\u001b[0m\n",
      "\u001b[0;34m\u001b[0m        \u001b[0;31m# zeta > 0 is equivalent to z_cm > r_cm (points behind the range)\u001b[0m\u001b[0;34m\u001b[0m\n",
      "\u001b[0;34m\u001b[0m        \u001b[0mresult\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mzeta\u001b[0m \u001b[0;34m>\u001b[0m \u001b[0;36m0\u001b[0m\u001b[0;34m]\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mnp\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mnan\u001b[0m\u001b[0;34m\u001b[0m\n",
      "\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\n",
      "\u001b[0;34m\u001b[0m        \u001b[0;32mreturn\u001b[0m \u001b[0mresult\u001b[0m\u001b[0;34m\u001b[0m\n",
      "\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\n",
      "\u001b[0;34m\u001b[0m    \u001b[0;32mdef\u001b[0m \u001b[0mlet_t_MeV_cm\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\n",
      "\u001b[0;34m\u001b[0m        \u001b[0;34m\"\"\"\u001b[0m\n",
      "\u001b[0;34m        track averaged LET in [MeV/cm]\u001b[0m\n",
      "\u001b[0;34m        \"\"\"\u001b[0m\u001b[0;34m\u001b[0m\n",
      "\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\n",
      "\u001b[0;34m\u001b[0m        \u001b[0;31m# main part of <S>_z as in equation (10) or (A12) in [1]\u001b[0m\u001b[0;34m\u001b[0m\n",
      "\u001b[0;34m\u001b[0m        \u001b[0mq\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;36m1.0\u001b[0m \u001b[0;34m+\u001b[0m \u001b[0;36m1.0\u001b[0m \u001b[0;34m/\u001b[0m \u001b[0mERSCalc\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mp\u001b[0m\u001b[0;34m\u001b[0m\n",
      "\u001b[0;34m\u001b[0m        \u001b[0mmean_s_z_part\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0msigma_cm\u001b[0m \u001b[0;34m**\u001b[0m \u001b[0mq\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;34m*\u001b[0m \u001b[0mfloat\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mmp\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mgamma\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mq\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;34m*\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_d_tilde\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mq\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\n",
      "\u001b[0;34m\u001b[0m        \u001b[0mmean_s_z_part\u001b[0m \u001b[0;34m-=\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mr_cm\u001b[0m \u001b[0;34m*\u001b[0m \u001b[0;34m(\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m0.5\u001b[0m \u001b[0;34m*\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mr_cm\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;34m**\u001b[0m \u001b[0;34m(\u001b[0m\u001b[0;36m1.0\u001b[0m \u001b[0;34m/\u001b[0m \u001b[0mERSCalc\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mp\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;34m*\u001b[0m \u001b[0mExactSpecialFunction\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mexponent_square\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m\u001b[0m\n",
      "\u001b[0;34m\u001b[0m            \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mzeta\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mxi\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\n",
      "\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\n",
      "\u001b[0;34m\u001b[0m        \u001b[0;31m# main part of Q_z as in equation (10) or (A8) in [1]\u001b[0m\u001b[0;34m\u001b[0m\n",
      "\u001b[0;34m\u001b[0m        \u001b[0mq_z_part\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mExactSpecialFunction\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mparabolic_integral\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m1\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mzeta\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\n",
      "\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\n",
      "\u001b[0;34m\u001b[0m        \u001b[0;31m# factor part of <S2>_z divided by <S>_z\u001b[0m\u001b[0;34m\u001b[0m\n",
      "\u001b[0;34m\u001b[0m        \u001b[0mconst_factor_MeV_cm\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;36m1.0\u001b[0m \u001b[0;34m/\u001b[0m \u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0msigma_cm\u001b[0m \u001b[0;34m*\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mr_cm\u001b[0m \u001b[0;34m*\u001b[0m \u001b[0mERSCalc\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0malpha_cm_MeV\u001b[0m \u001b[0;34m**\u001b[0m \u001b[0;34m(\u001b[0m\u001b[0;36m1.0\u001b[0m \u001b[0;34m/\u001b[0m \u001b[0mERSCalc\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mp\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\n",
      "\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\n",
      "\u001b[0;34m\u001b[0m        \u001b[0;31m# result\u001b[0m\u001b[0;34m\u001b[0m\n",
      "\u001b[0;34m\u001b[0m        \u001b[0mresult\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mconst_factor_MeV_cm\u001b[0m \u001b[0;34m*\u001b[0m \u001b[0mmean_s_z_part\u001b[0m \u001b[0;34m/\u001b[0m \u001b[0mq_z_part\u001b[0m\u001b[0;34m\u001b[0m\n",
      "\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\n",
      "\u001b[0;34m\u001b[0m        \u001b[0;31m# filling nonsense values (outside model domain) with np.nan\u001b[0m\u001b[0;34m\u001b[0m\n",
      "\u001b[0;34m\u001b[0m        \u001b[0;31m# zeta > 0 is equivalent to z_cm > r_cm (points behind the range)\u001b[0m\u001b[0;34m\u001b[0m\n",
      "\u001b[0;34m\u001b[0m        \u001b[0mresult\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mzeta\u001b[0m \u001b[0;34m>\u001b[0m \u001b[0;36m0\u001b[0m\u001b[0;34m]\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mnp\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mnan\u001b[0m\u001b[0;34m\u001b[0m\n",
      "\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\n",
      "\u001b[0;34m\u001b[0m        \u001b[0;32mreturn\u001b[0m \u001b[0mresult\u001b[0m\u001b[0;34m\u001b[0m\n",
      "\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\n",
      "\u001b[0;34m\u001b[0m    \u001b[0;34m@\u001b[0m\u001b[0mclassmethod\u001b[0m\u001b[0;34m\u001b[0m\n",
      "\u001b[0;34m\u001b[0m    \u001b[0;32mdef\u001b[0m \u001b[0mlet_d_keV_um\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mcls\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0menergy_MeV\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0msigma_energy_MeV\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mz_cm\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\n",
      "\u001b[0;34m\u001b[0m        \u001b[0;34m\"\"\"\u001b[0m\n",
      "\u001b[0;34m        dose averaged LET in [keV/um]\u001b[0m\n",
      "\u001b[0;34m        \"\"\"\u001b[0m\u001b[0;34m\u001b[0m\n",
      "\u001b[0;34m\u001b[0m        \u001b[0mmodel\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mWilkensModel\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0menergy_MeV\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0msigma_energy_MeV\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mz_cm\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\n",
      "\u001b[0;34m\u001b[0m        \u001b[0;32mreturn\u001b[0m \u001b[0mmodel\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mlet_d_MeV_cm\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;34m*\u001b[0m \u001b[0;36m0.1\u001b[0m\u001b[0;34m\u001b[0m\n",
      "\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\n",
      "\u001b[0;34m\u001b[0m    \u001b[0;34m@\u001b[0m\u001b[0mclassmethod\u001b[0m\u001b[0;34m\u001b[0m\n",
      "\u001b[0;34m\u001b[0m    \u001b[0;32mdef\u001b[0m \u001b[0mlet_t_keV_um\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mcls\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0menergy_MeV\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0msigma_energy_MeV\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mz_cm\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\n",
      "\u001b[0;34m\u001b[0m        \u001b[0;34m\"\"\"\u001b[0m\n",
      "\u001b[0;34m        track averaged LET in [keV/um]\u001b[0m\n",
      "\u001b[0;34m        \"\"\"\u001b[0m\u001b[0;34m\u001b[0m\n",
      "\u001b[0;34m\u001b[0m        \u001b[0mmodel\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mWilkensModel\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0menergy_MeV\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0msigma_energy_MeV\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mz_cm\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\n",
      "\u001b[0;34m\u001b[0m        \u001b[0;32mreturn\u001b[0m \u001b[0mmodel\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mlet_t_MeV_cm\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;34m*\u001b[0m \u001b[0;36m0.1\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\n"
     ]
    }
   ],
   "source": [
    "# lets use again models.py file and display the implementation of Wilkens model\n",
    "from models import WilkensModel\n",
    "%psource WilkensModel"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x7fb93b1a0c18>"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "E_MeV = 160\n",
    "sigma_E_MeV = 0.0\n",
    "z_cm = np.linspace(start=0.0, stop=20.0, num=100)\n",
    "let_t_keV_um = WilkensModel.let_t_keV_um(E_MeV, sigma_E_MeV, z_cm)\n",
    "let_d_keV_um = WilkensModel.let_d_keV_um(E_MeV, sigma_E_MeV, z_cm)\n",
    "\n",
    "plt.xlabel(\"z [cm]\")\n",
    "plt.ylabel(\"LET [keV/um]\")\n",
    "plt.plot(z_cm, let_d_keV_um, label=\"dose-averaged LET\")\n",
    "plt.plot(z_cm, let_t_keV_um, label=\"track-averaged LET\")\n",
    "plt.legend(loc=0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "# note !!\n",
    "\n",
    "# method parabolic_integral in WilkensLET relies on mpmath package, \n",
    "# instead of scipy to calculate parabolic cylinder functions,\n",
    "# \n",
    "# scipy could be used, but it fails to calculate exp*Dv product\n",
    "# when exponent is very small and parabolic cylinder function very large\n",
    "# mpmath calculation are exact, i.e. \n",
    "#\n",
    "# >>> x = -200\n",
    "# >>> a = mp.exp(-(x**2)/4)\n",
    "# >>> b = mp.pcfd(-1/1.77, x)\n",
    "# >>> a\n",
    "# mpf('1.1354838653147362e-4343')\n",
    "# >>> b\n",
    "# mpf('1.3980110163795251e+4342')\n",
    "# >>> a*b\n",
    "# mpf('0.15874189526312063')\n",
    "#\n",
    "#  numpy+scipy fails to calculate due to lack of numerical precision\n",
    "#\n",
    "# >>> x = -200\n",
    "# >>> a = np.exp(-(x**2)/4)\n",
    "# >>> b = sp.special.pbdv(-1/1.77, x)[0]\n",
    "# >>> a\n",
    "# 0.0\n",
    "# >>> b\n",
    "# inf\n",
    "# >>> a*b\n",
    "# __main__:1: RuntimeWarning: invalid value encountered in double_scalars\n",
    "# nan\n",
    "#"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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