{ "cells": [ { "cell_type": "markdown", "metadata": { "collapsed": true }, "source": [ "# Gaussian Models" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Gaussian models that are typically used to described extra-axonal diffusion. Briefly, we have: \n", "- Ball model: Isotropic Gaussian diffusion with diffusion coefficients $\\lambda_{iso}$.\n", "- Zeppelin Model: Axially symmetric Gaussian tensor, oriented along orientation $\\mu$ with parallel and perpendicular diffusivity $\\lambda_\\parallel$ and $\\lambda_\\perp$.\n", "- Restricted zeppeling model: Axially symmetric Gaussian tensor with time-dependent perpendicular diffusion to account for diffusion restricted in the extra-axonal space between axons. Instead of $\\lambda_\\perp$, this model has $\\lambda_{inf}$ and characteristic coefficient $A$." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Ball: G1\n", "the ``Ball'' model *(Behrens et al. 2013)* models the totallity of all extra-axonal diffusion as a Tensor with isotropic diffusivity $\\lambda_{\\textrm{iso}}$ as\n", "\n", "\\begin{equation}\n", "E_{\\textrm{iso}}(b,\\lambda_{\\textrm{iso}})=\\exp(-b\\lambda_{\\textrm{iso}}).\n", "\\end{equation}\n", "\n", "In current models, the Ball is usually only used to describe the CSF and/or grey matter compartment of the tissue, where the isotropic diffusion assumption is reasonably valid*(Alexander et al. 2010, Jeurissen et al. 2014, Tariq et al. 2016)*." ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [ { "data": { "image/png": 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JqQuETYc9X8Duz6DNt3pHf00LZqbGjGpWibdr2jJr92UWHbnBxlP+fNTSkd61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\nBl80zlcsIMLfnpI9fH/l98k+ms1t6bfxzcnfDM7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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "from dmipy.signal_models import gaussian_models\n", "from dmipy.core.acquisition_scheme import acquisition_scheme_from_bvalues\n", "import numpy as np\n", "import matplotlib.pyplot as plt\n", "%matplotlib inline\n", "\n", "samples = 100\n", "bvalues = np.linspace(0, 1e9, samples)\n", "bvectors = np.tile([0, 0, 1], (samples, 1))\n", "delta = 1e-2\n", "Delta = 3e-2\n", "scheme = acquisition_scheme_from_bvalues(bvalues, bvectors, delta, Delta)\n", "\n", "ball = gaussian_models.G1Ball()\n", "\n", "for lambda_iso in [1e-9, 2e-9, 3e-9]:\n", " plt.plot(bvalues, ball(scheme, lambda_iso=lambda_iso),\n", " label='lambda_iso=' + str(lambda_iso * 1e9) + 'e-9 m^2/s')\n", "plt.legend()\n", "plt.xlabel('b-value [s/m^2]', fontsize=13)\n", "plt.ylabel('Signal Attenuation', fontsize=13)\n", "plt.title('Signal Attenuation Ball', fontsize=15);" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The higher the diffusivity the faster the signal attenuates." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Zeppelin: G2\n", "Hindered extra-axonal diffusion, i.e. diffusion of particles in-between axons, is often modeled as an anisotropic, axially symmetric Gaussian, also known as a ``Zeppelin'' *(Panagiotaki et al. 2012)*.\n", "Using DTI notation, a Zeppelin with $\\lambda_\\parallel=\\lambda_1$, $\\lambda_\\perp=\\lambda_2=\\lambda_3$ and $\\lambda_\\parallel>\\lambda_\\perp$ is given as\n", "\n", "\\begin{equation}\n", " E_h(b,\\textbf{n},\\lambda_\\parallel,\\lambda_\\perp)=\\exp(-b\\textbf{n}^T(\\textbf{R}\\textbf{D}^h_{\\textrm{diag}}\\textbf{R}^T)\\textbf{n})\\quad\\textrm{with}\\quad\n", " \\textbf{D}^h_{\\textrm{diag}}=\n", "\\begin{pmatrix}\n", " \\lambda_\\parallel & 0 & 0 \\\\\n", " 0 & \\lambda_\\perp & 0 \\\\\n", " 0 & 0 & \\lambda_\\perp\n", " \\end{pmatrix}.\n", "\\end{equation}\n" ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [ { "data": { "image/png": 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Me+HkoWCHaoxJgfdY/3M/jxo1ipEjR7Jq1SrGjx9/wc1iRYsWPf/za6+9RoUKFVixYgXR\n0dHEx6fdaqCqTJ8+neXLl7N8+XJ27NhBgwYNAvwb5Rz5PkGA648Yd0sL9h87w0Ofr0BVoV4PuPcP\n6PR3WDkNxrWCmImQnBzscI0xXs41QU2bNo0OHToAEBcXR5UqbqT8xIkTUz02Li6OSpUqERISwqRJ\nk0hKSkrzWj179uTNN9/k3FLNy5YtC8SvkGNZgvBoEVmKx3s3YN66/Yz/davbWKAIXPk03L0AyjWA\nWffDR71gX4ozhhhjAsC3D+LRRx9Ns/zZs2dp164db7zxBq+99hoAo0eP5sYbb6Rz586ULVs21WPv\nvfdeJk6cSPv27dm4ceMFtYuUPPnkkyQkJNC0aVMaN27Mk08+mf5fMBeRc5kwN2rdurUGckU5VWXk\np8v4btVeJg9vT4faZbx3wvIpMPdJOH0U2t8DXR+FgsUCdn1jgm3dunW5qsnk3KqSaSWB/C6l91RE\nYlS19cWOtRqEFxHhP9c3pUbZooz6dBn7j53x3gktBsHIaGhxKyx6C8a1gdVfuuRhjDF5jCUIHxEF\nwxh/aytOxSdy3+SlJCT59DkUKQ19x8LweVC0HHxxO0y6Fv7cFJyAjckH+vfvf0GzU/PmzZkzZ875\n+xlM1sgpN8rlKHUrFGPMdU14YOpyxny7nqeuafjXQlVbw4ifIeoDNyT27Q7QcRRc9jAUSLsd0xiT\nPl999VWwQ8iXrAaRin7NqzC0Yw0+XLiNr5fvTrlQSCi0GwGjoqHJDbDgVXirHaz92pqdjDG5niWI\nNPzr6ga0qVGKR6evYv2+NKYciCgP/d+B27+DgsXhs9tgUn9rdjLG5GqWINIQHhrCW7e0pFihMO6a\nFEPc6YS0D6jeEe76FXr9B3Z7puyY+xScPZE9ARtjTABZgriI8sUL8b9bW7L7yGn+NnUZSckXaToK\nDYP2d8OoGGg6ABa+AeNaw6ovrNnJGJOrWILwQ6vqpXm6byPmbzjI6/PSnqvlvIjycO1bcMc89/P0\nO+Cj3rBvVdYGa0wulxPWgwiEiIgIAPbs2cMNN9yQoXNMmDCBkSNHBjKsdLFRTH66tV0kq3Yd5c2f\nNtOocnF6NfZzRdRqbeDO+bBsEvz4LIy/DFoPg8v/5YbMGmMukNfWg6hcuTJffPFFll8HAv87WYLw\nk4jwbL/GbNh/goc+W0GtchHUq+DnXdQhodBqKDTsB/P/7YbGrp7ukkSr212zlDE5zXePBr7GW7EJ\nXBW4GkFq60EcPHiQu++++/xMq6+//jqXXnrpX9aD6NGjB1999RVnz55l27Zt3HLLLTz99NMAfPLJ\nJ4wdO5b4+HjatWvH22+/TWhoKBERETzwwAPMnj2bwoUL8/XXX1OhQoXzxycmJtKrV6/zMW7fvp0+\nffqwevVqkpKS+Oc//8mcOXMQEe68805GjRp1wR3h0dHRPPzww/z8888X/K7+rnExZcqUgL2+1sSU\nDoXCQxl/aysKFwjjzo+jOXoqnetFFC4FvV+Gu3+DCo3h24ddjcLWxTbmvECsB/HAAw/w4IMPEhUV\nxfTp0xk+fPj58t7rQQAsWbKEyZMns3z5cj7//HOio6NZt24d06ZNY+HCheenAp88eTLgFg1q3749\nK1as4LLLLuO99947f8177rmHqKgoKlasmGKs7777Ltu2bWPZsmWsXLmSQYMG+f26+LvGRSBl21dX\nEekFvAGEAu+r6os++6sDHwLlgMPAraq6K7vi81fFEoUYP7glN7+7mFGfLuOjoW0IC01nnq3QCIbM\ngnUzYc4Tbl3sBn2hx3NQqkaWxG1MugXwm356BGI9iHnz5rF27drzZY4dO8bx48eBC9eDAOjevTtl\nyrh516677joWLFhAWFgYMTExtGnTBnBJq3z58gAUKFDg/PrTrVq1Yu7cuQAsXLiQ6dOnAzB48GD+\n+c9//iXWefPmcffdd59vBipd2v9mZn/XuAik7FpRLhR4C7gKaAgMFBHf25NfAT5W1abAs8CY7Igt\nI1pVL83z1zbmt01/8uJ36zN2EhHX5DRyCVz+BGyeB+Paun4KGxZrjN9SWg8iOTmZRYsWnV+3Yffu\n3RQr5pqEfWdsFZ816EUEVWXIkCHnj9+wYQOjR48GIDw8/PwxoaGhJCYmpnouX6qaYpmwsDCSPUsJ\neK9d4c3fNS4CKbuamNoCm1V1q6rGA1MB34VcGwI/en6en8L+HGVAm0iGdKjO+wu28UVMJio64YWh\nyyNuEsCG/eC3/8KbrdzMsbb2hDEXldJ6ED169GDcuHHny6RVI5k7dy6HDx/m9OnTzJgxg0svvZRu\n3brxxRdfcODAAQAOHz5MbGxsmnFceumlTJ06FeB8c5SvHj168M4775xPKocPHwZcX0pMTAzA+VqI\nL3/XuAik7EoQVYCdXs93ebZ5WwFc7/m5P1BMRMr4lEFERohItIhEHzx4MEuC9dcTfRrSoVYZHv9y\nFTGxRzJ3shJV4Pr33LDYElVgxj3wfjfY8UdggjUmlwjEehBjx44lOjqapk2b0rBhQ955551Uj+/U\nqRODBw+mefPmXH/99bRu3ZqGDRvy/PPP06NHD5o2bUr37t3Zu3dvmnG88cYbvPXWW7Rp04a4uLgU\nywwfPpzIyEiaNm1Ks2bNzvcZPP300zzwwAN07tyZ0NDQFI/1d42LQMqW9SBE5Eagp6oO9zwfDLRV\n1VFeZSoD44CawK+4ZNFIVVN+pQn8ehAZceRkPNe+vZCTZxP5emQnqpQMQDtgcjKs+hzmjYbje6DR\ndXDlaChVPfPnNiYN+W09iAkTJhAdHX1BbSOvyQ3rQewCqnk9rwrs8S6gqntU9TpVbQH8y7Mt1eSQ\nU5QqWoAPhrTmbEIyd06M5lR84sUPupiQEGg2wE0C2OWfsOE7t/bEvNFwJo05oYwxJoCyqwYRBmwE\nugG7gSjgFlVd41WmLHBYVZNF5AUgSVWfSuu8OaEGcc78DQe4Y0IUPRpW5O1BLQkJSbuzKl3idrnO\n65XT3BoUl/8LWgy2+ydMwOXUGkT//v3Ztm3bBdv+85//0LNnzyBFlHtkpgaRLZ8wqpooIiOBObhh\nrh+q6hoReRaIVtWZQFdgjIgoronpvuyILVAur1+ex3s34Plv1vHfuRt4pOclgTt5iapw3bvQ9i6Y\n8zjM/hssedcNi61zZeCuYwypj7QJJlsPImMyWwGwNakDSFV5/KtVfLpkJ/+9sRnXt6qaFRdx90/M\nfQqObHcJovtzUCGFRY2MSadt27ZRrFgxypQpk+OShEkfVeXQoUMcP378gnsmwP8ahCWIAEtISmbI\nh0uI2n6YycPb07ZmFs23lHgWlrwHv74MZ4+5JqfLH4diKd/BaYw/EhIS2LVrV6pj8U3uUqhQIapW\nrUp4ePgF2y1BBFHcqQT6v72QI6fi+ereS6lRNguXID11GH59xTU5hRaAS++HDiOhYETWXdMYk6vl\ntFFM+UqJIuF8MLQNCgybEJX+OZvSo0hp6PVvd0d2vR7w8xh4syXETICkAIyoMsbkW5YgskjNskV5\nd3Brdh05zV2TYohPzOK7okvXghsnuBvtStWEWQ/A/zrC+m9toSJjTIZYgshCbWuW5qUbmvLHtsM8\n+uXKTI8o8Eu1NjDsexgwGTQJpg50CxXtjMr6axtj8hRLEFns2hZVePDKeny5dDevz9uUPRcVgQZ9\n4N7FcPWrcGgzfHAlTBsMf27OnhiMMbmeJYhscH+3OtzQqipv/LiJz6N3XvyAQAkNhzZ3wP3LoOvj\nsOUneKstzH4Qju/LvjiMMbmSJYhsICKMua4JneqU5bEvV7Fg05/ZG0DBCOj6T5co2twBSz+GsS3g\nx+fgTI6fzcQYEySWILJJeGgIb9/akjrlI7j7kxjW7gnCnEoR5d2KdiOjoP5V8Nsr8EYz+P1NSLBx\n78aYC1mCyEbFC4Xz0e1tiCgYxu0TlrD76OngBFK6FtzwIYz4BSq3gB+ecGtQLJ1kQ2ONMedZgshm\nlUoUZsKwNpw6m8SQD5cQdyoheMFUbg6Dv4LbZkKxCjBzJPyvA6z92obGGmP8TxAiUk1ErhGRW7wf\nWRlcXnVJxeKMv60VOw6d4s6PozmTkBTcgGp1geE/woBP3PPPboN3u8LmHy1RGJOP+TXVhoiMwC3m\ncxQ46bVLVbVWFsV2UTl1qg1/zVqxh1GfLqNnowq8PagVoYGcIjyjkhLdtOI/j4G4nVC9E3R7CiLb\nBTsyY0yABHqqjSeBAapaXlVrej2ClhzygmuaVeapPg2Zs2Y/T329OntupLuY0DBoMQhGxcBVL8Gf\nG+DDHjD5Jti7MtjRGWOykb8JIkJVbUL2LDCsU03u7lKbyX/sYOyPOegmtrCC0O4ueGAFdHsadv4B\n4zvDZ0Pg4IZgR2eMyQb+JojPReTqzFxIRHqJyAYR2Swif1mFXEQiRWS+iCwTkZUi0jsz18tN/tmr\nPte3rMpr8zYyaXFssMO5UIGi0PnvLlFc9g/YPA/ebg9f3gWHtwY7OmNMFvK3D+Jj4HrgJ2Cv9z5V\nHeHH8aG4JUe749anjgIGquparzLvAstU9X8i0hD4VlVrpHXe3N4H4S0hKZm7J8Xw04YDvDmwBX2a\nVg52SCk7eQgWvubWokhKcM1Rlz0CJSODHZkxxk+B7oNIAj4D/gTCfR7+aAtsVtWtqhoPTAX6+ZRR\noLjn5xLAHj/PnSeEh4bw1qCWtK5eigenLee3TQeDHVLKipaBHs+7GkWb4bBiKoxtCd88BMfy1Vtm\nTJ6XLQsGicgNQC9VHe55Phhop6ojvcpUAn4ASgFFgStVNSat8+alGsQ5cacTGDB+ETsOn2Ly8Ha0\niCwV7JDSFrfLLVi0bBJIKLS+HTo9aCvbGZODBXzBIBGJEJEBIvKwiNwkIulZsiyl8Zu+mWkgMEFV\nqwK9gUki8pf4RGSEiESLSPTBgzn0W3YmlCgczsfD2lKuWEGGfhTFhn3Hgx1S2kpUhWted6Oemt7o\nmp7eaAZz/gUnDgQ7OmNMJviVIESkEa4P4RVc09B/gY0i0tjP6+wCqnk9r8pfm5DuwDVjoaqLgEJA\nWd8Tqeq7qtpaVVuXK1fOz8vnLuWLF+KTO9pRMCyEwR/8wY5Dp4Id0sWVqgH93nLzPDW6Dha/Da83\nddN4nMh7idyY/MDfGsTrwHggUlU7A5HA/4A3/Dw+CqgrIjVFpABwMzDTp8wOoBuAiDTAJYh8+8lS\nrXQRPhnejvikZAZ9sJh9cblkMr0ytaH//+C+KGjYFxa9Ba83sURhTC7k7yimP4FKqprgtS0c2Keq\nZfy6kBu2+joQCnyoqi+IyLNAtKrO9Ixceg+IwDU//UNVf0jrnHmxD8LX8p1HGfTeYiqVLMy0Ee0p\nE1Ew2CGlz5+b4deXYdVnEFrQTTfe8X4395MxJij87YPwN0FswXUyb/LaVhf4QVVrZirSTMgPCQJg\n8dZDDPlwCXXKRzDlzvaUKOzv4LEc5M/NbnrxldNcomh9O1z6gHVmGxMEge6kngh8IyLDRORyERkG\nzAImZCJG46f2tcrwzuBWbNx/nGETojh5NhdOyV22DvR/B0ZGQ6P+8Md410fx7T8gbnewozPGpMDf\nGkQo8A9gKK6zeScuObysqkH7tMovNYhzvlu1l/umLKVdzTJ8dHsbCoWHBjukjDu8FX57FVZ8ChIC\nLW51w2PthjtjslxAm5hyqvyWIABmLNvNg58tp3Pdcrx3WysKhuXiJAFwJBYWvAbLPgEUmt7spvYo\nUzvYkRmTZwX8PgiTM1zbogovXteEXzce5L7Jy0hISg52SJlTqrq7j+KBFdD6Dlj9BYxrDdOHw4F1\nwY7OmHwt1QQhIoe9fk4QkfiUHtkTpvE2oE0kz/ZrxLx1+7n/0zyQJABKVIHeL8HfVkHHUbD+Wzcp\n4NRBsGdZsKMzJl9KtYlJRDqp6gLPz11SO4Gq/pJFsV1Ufmxi8vb+b1t5/pt19GlaidcHNCcsNA9V\nCE8dhj/ecY8zcVC7G3R+CKp3BMkBCysZk4v528QUltqOc8nBI1ZVt6dwkeoZC88EwvDOtUhKVsZ8\nt57QEOHVm5rnjFXpAqFIabj8cegwEqI/cDfcTegN1dq7RFG3uyUKY7KYv185U1tKzOr+QXZXl9o8\n0rM+Xy/fw8OfryApOfcOOkhRoeJudNPfVkHvV+DYbphyI7zTGVZ9AclBXs/bmDzM3wTxl69qnjup\n89inUe503+V1eKh7Pb5atjsHLh+YAAAgAElEQVRvJgmA8MLQ9k4YtRT6vQ1JZ2H6HfBmK4j+CBJy\nyVQkxuQiqTYxAYjIXFwSKCgivtNeRAJLsyowkz6jutUF4L9zNwLwyo3N8k5zk7ewAm6RomYDYf1s\nN0R29t/g5zHQ/l53h3ahEsGO0pg8Ic0EAZzrh+gCLPTangzsAz7PiqBMxngniWRV/ntjs7zVce0t\nJMRNBtjgGtj2Kyx4FeY9Db/9F1oPg/b32DQexmRSmglCVZ8BEJF1qvpZ9oRkMmNUt7qEhAgvz9lA\nUrLmvdFNvkSgVhf32LMMFrwOv4910403G+gmBixbJ9hRGpMrXawGAcC55CAihXFrNIjXvh1ZE5rJ\nqPsur0NYiDDmu/Ukq/LGzS0Iz8tJ4pzKLeCmiXBoCywaB8smw9KPoUEf6PgAVGsT7AiNyVX8ShAi\nUgv4BGiXwu5cPtdD3nRXl9qEhgjPf7OOhKSljLulRe6flsNfZWpDn9eg62NuUsCo92HdLIjs4GoU\n9Xq5JipjTJr8/V8yDjdBXzPgONAUmIFbBc7kUMM71+LZfo2Yu3Y/Iz6O4UxCPhsSGlEeuj0JD66B\nXi+6WWOnDoS320HMRBv5ZMxF+Dub6yGghqoeF5GjqlpSRMoCv6hqoyyPMhX5/U5qf01dsoPHvlpF\nh1pleH9Ia4oU8KvimPckJcLaGa6PYu8KKFoO2t7lFjEqUjrY0RmTbQI9WV8ycNrz8wkRKQkcxg11\n9TegXiKyQUQ2i8ijKex/TUSWex4bReSov+c2abu5bSSv3tSMxVsPcdsHSzh2JuHiB+VFoWHQ5AYY\n8QsMmeX6LOY/D682hNl/d30Xxpjz/K1B/Aw8raq/iMh04BhwErhMVZv6cXwosBHoDuzCrVE9UFXX\nplJ+FNBCVYeldV6rQaTPt6v28sDUZdSvWIyPh7WjdNECwQ4p+A6sdx3aK6dBUgLUv8pN72FzPpk8\nLNA1iPtxNQaAR4AqQGvgLj+PbwtsVtWtqhoPTAX6pVF+IPCpn+c2furdpBLvDm7Npv0nGDB+EfuP\nWRs85S+BfuNcP8Vlj8COxW7Op3e7wsrPXNIwJp/KlgWDROQG3JrWwz3PBwPtVHVkCmWrA4uBqqr6\nl15VERkBjACIjIxsFRsbm6Wx50W/b/mT4ROjKRtRkE/uaEdkmSLBDinniD/lVrlb/D84tAmKVXZT\nfLQaav0UJs8I6IpyItIxtX2q+rsfx98I9PRJEG1VdVQKZf+JSw5/2efLmpgybvnOowz9aAkFQkP4\nZHg76lUoFuyQcpbkZNg8zzU/bfsFwgpD84HQ7h4oVy/Y0RmTKYFuYlqQwuM3z8Mfu3BrWZ9TFdiT\nStmbsealLNe8WkmmjegAwE3jF7F8p40JuEBICNTrAUNmwj2/Q5Pr3Y13b7WBT66HTfNcEjEmD/Mr\nQahqiPcD9wE/EbjRz+tEAXVFpKaIFMAlgZm+hUSkPlAKWOTneU0m1K9YjC/u7kjxQuHc8t5iftt0\nMNgh5UwVGkG/t1w/xeX/gr0rYfL17n6KqPch/mSwIzQmS2TodlJV3QM8APzHz/KJwEhgDrAO+ExV\n14jIsyLS16voQGCqZkfHiAEgskwRvri7A5GlizBsQhSzV6ZWsTNElIMu/4AHV0P/8RBeBL55CF5t\nAD88AUesP8zkLRnupPbcKLdFVYM2t7L1QQRO3OkEhk+MIjr2CM/2a8zg9rZY4EWpws4/XIf2ulmA\nQv3e0HYE1LzMhsmaHCvTS476nOxxn01FccNU52YgNpMDlSgczqQ72jFyylKenLGag8fP8uCVdRH7\nkEudCES2d4+43W5p1OiP3DoV5Rq40U/NboYCRYMdqTEZ4u8opvk+m04A0cBrqnosKwLzh9UgAi8x\nKZnHv1rFZ9G7GNg2kuf6Ncrb04UHWsIZWD0d/ngH9q2EgiWgxa1uOo8ytYMdnTFAgIe55lSWILKG\nqvLKDxt4a/4WrmxQgTcHtqBwgXwyE2ygnGt+WvIurP3arZ1d50pXq6jT3WaTNUFlCcJk2sTftzN6\n1hqaVyvJB0Pa2NQcGXV8n2t6ivkITuyHktVdjaLFYLv5zgRFoG+Uqwe8iZte44I7qlQ1aJ8aliCy\n3ner9vLAtOVUKVmYibe3tbuuMyMxHtbPgiXvw47fIawQNL4e2gyHKi2DHZ3JRwKdIH7H3ew2ATdJ\n33mq+ksGY8w0SxDZI2r7YYZPjCY8VPhgSBuaVSsZ7JByv32rXaf2immQcBIqt3S1ikbXQQFLwiZr\nBTpBHAPKqGqOmrnMEkT22XzgBEM/WsKhE/GMHdiC7g0rBDukvOFMnJsUMOp9OLgeCpWA5oOg9TAo\nWzfY0Zk8KtBTbawHymcuJJOb1SkfwVf3XkrdChHcNSmaib9vD3ZIeUOhEq7j+t7FMPRbqN0NlrwH\n41rDxGtgzVeuacqYIPC3BnEPMAR4Cdjnvc+fyfqyitUgst+p+ETu/3Q589btZ9ilNfnX1Q0IDbF7\nJQLqxAFY+rFbFjVuBxQtDy0HQ8shUMpuYDSZF+gmptRmJVNVDdr4R0sQwZGUrDz/zVo+WridKxtU\nYOzA5vl3GdOslJwEm3+E6A9h0xw3dLZON2h1O9Tr5VbIMyYDbJiryXITf9/OM7PW0LBycd6/rQ0V\nSxQKdkh5V9wuV6tYOgmO74FildwNeC1vg5J+r/xrDJBFCULcvAsVVXVvZoILFEsQwffT+v2MmrKM\niEJhfDCkDY2rBG1qrvwhKRE2/QAxE9y/4KlVDPXUKsKDGZ3JJQLdxBQBvAEMApJUtaiIXAs0U9Vn\nMh1tBlmCyBnW7T3G8InRHDp5ltcHNKdX40rBDil/OLrT1SqWfeJqFREVoPkt7gY8m9bDpCHQCWI8\nbh3qp4F5qlpKRKoAc1W1YaajzSBLEDnHweNnGTEpmmU7jvJIz/rc27W2TfSXXZIS3ep3SyfCxjmg\nSVCjs+vUbnANhFvTn7lQoBPEbqChqsaJyGFVLe3ZflRVg3bXlCWInOVMQhKPTl/JjOV76NusMi/d\n0JRC4TaHU7Y6tgeWT3Z9FUdjoVBJaDrA9VVUbBzs6EwOEej7IAQ47XOBCNysrv4G1EtENojIZhF5\nNJUyN4nIWhFZIyJT/D23yRkKhYfy2oDmPNKzPjNX7GHA+EXsizsT7LDyl+KV4bJH4P7lcNvXrn8i\n5iN451IY38XdkHfalpc1/vG3BvE5sFpVnzlXgxCRx3C1isF+HB8KbAS646bsiAIGquparzJ1gc+A\nK1T1iIiUV9UDaZ3XahA51w9r9vHgtOUULRjGO4Nb0TKyVLBDyr9OHXZ3ay+bBPtXuzmgGvZzo6Cq\nd7KZZfOhQDcxVQN+8jytDmwAwoFuqrrbj+M7AKNVtafn+WMAqjrGq8xLwEZVff+iAXlYgsjZNuw7\nzp0fR7Mv7gzP92/MTa2rBTuk/E0V9ixzndqrvoCzcW5m2eaDoPlAGy6bjwR8mKuIFAT6ADWBWGC2\nqp5O+6jzx94A9FLV4Z7ng4F2qjrSq8wMXC3jUiAUl1C+T+FcI4ARAJGRka1iY20d4Jzs6Kl4Rk5Z\nxoLNfzKkQ3We6NOQcFuAKPgSTrtlUpd9Att+AcQtk9riVk/HduFgR2iyUKBrEINVdVIK2wep6mQ/\njr8R6OmTINqq6iivMrOBBOAmoCrwG9BYVVNtMLUaRO6QmJTMi9+t5/0F22hbszRvD2pJ2YiCwQ7L\nnHMkFlZ86jq3j+6AgsWh8XWuZlG1ja2tnQcFupP6rVS2v+nn8bsA7/aFqsCeFMp8raoJqroN14xl\n01nmAWGhITzRpyGvD2jOip1HuebNBazYaR2lOUap6tD1Ubh/BQyZDZdc7fosPugOb7aCX19291yY\nfCc9o5gu3CBSA0j08/gooK6I1BSRAsDNwEyfMjOAyz3nLgvUA7b6eX6TC1zbogrT7+lIiAg3vrOI\naVE7gh2S8RYSAjU7Q/934OGN0O8tKFYRfnoeXm8CE/vCiqkQf/Li5zJ5QppNTCKSACiuTyDJZ3co\n8LZ3M1GaFxLpDbzuOe5DVX1BRJ4FolV1pmcaj/8CvTzXekFVp6Z1Tmtiyp0On4zn/k9dv8TAttUY\n3bcRBcPsfokc6/A2lxhWfOrurQgv6kZBNbvZ3ZBno6BynYD0QYhIF1zt4VvgKq9dycA+Vd2U2UAz\nwxJE7pWUrPz3hw28/fMWmlYtwduDWlK1lK2klqMlJ8OORbByKqyZAWePQfGq0PRGaDYQytUPdoTG\nT4HupO6gqotS2F5dVYM2jMgSRO43Z80+Hv5sBaGhwusDmtO1vq1LlSsknIb137iaxZaf3PQelZq7\nWkXjGyCiXLAjNGkI+JKjqlo8he3np90IBksQecP2P09y9ycxbNh/nFFX1OWBbnVtEaLc5Ph+WP0F\nrJwGe1eAhELtK9wUH5f0hgJFgx2h8RHoBHFcVYv5bAvHNTOVyXiYmWMJIu84HZ/Ek1+v5ouYXXSq\nU5bXb25uQ2FzowPrXRPUqi8gbqfrr2hwjWuGqtnVFjnKIQLVBzEX10ndFfjZZ3cksFNVu2c8zMyx\nBJH3fBa9kydnrKZE4XDeHNiCdrWC9v3DZIZ3f8Xar+FMnFs6tfF10OQmqNLS7q8IokAliKc9Pz4O\n/NtrVzJuberP07qRLatZgsib1u09xn2Tl7L90En+3r0e93atQ4g1OeVeiWfd4kYrP3PTkSedhdK1\noMmNrr+iXL1gR5jvBLqJ6SZV/SyF7eGqmpDBGDPNEkTedeJsIo9/uYqZK/bQuW5ZXr2pOeWKWZNT\nrnf6qJviY9VnsO03QKFSM5coGl8HJaoGO8J8IUvXpPbMvDoCGKKqQRt2Ygkib1NVpkbtZPTMNRQv\nHM5rNzWnU92ywQ7LBMqxvbDmK1j1OexZ6rZFdoQm10PDa6GovddZJSsm6wsHrgPuArrgpsJ4TVXf\ny0ygmWEJIn9Yv+8YI6csY8vBE9zbtTYPXlmPMJvwL285tAVWf+lGQx1c70ZC1eoKja93U38UDtq6\nZHlSwBKEiNTBU1sAigOzgB5AvYut15DVLEHkH6fiE3l21lqmRu2kZWRJ3ri5BdVK2411eY4qHFgL\nq6e7kVBHYyG0ANS50iWLer2gYESwo8z1AtVJPQ83P9Jq4CNgkqoeEpG9QDNLECa7zVqxh8e/XAXA\nC9c1oW+zykGOyGQZVdi91CWLNV/B8T0QVhjq9YBG/aFuTyhgXxIyIlAJIgk4DIzBJYeDnu2WIEzQ\n7Dx8igemLmPpjqPc0Koqo/s2IqKgja/P05KTYedi1wy1dgacPAjhRVyNolF/qNvd1rBIh0AliGrA\ncGAYUB6YjatJvA80tQRhgiUxKZmxP25i3PzNVC1VhNdvbm7LmuYXyUkQu9Ali3Uz4dQhKBAB9Xq6\nZFHnSksWFxHoYa4hQF/gTqAnbgK/McDrqvpnJmPNMEsQJmr7Yf42dTn7jp1h1BV1GHl5HevAzk+S\nEmH7b65WsXYmnD78/8miYT+o092aoVKQZcNcRaQ6LlEMBUqpatAmWrEEYQCOnUngqRmrmbF8D82r\nleS1Ac2pWdbm/8l3khJcslgzA9bPdjWL8KKuz6JhP6jbw+aF8sjS+yA8FwgFrlHVGRk6QQBYgjDe\nZq3Yw7++WkVCkvJEnwbc0jYSsekc8qdzNYt1M92NeScPug7uOt1csqjXEwqVCHaUQZPlCSK9RKQX\n8AZuwaD3VfVFn/1DgZeB3Z5N41T1/bTOaQnC+Nobd5pHPl/Jgs1/0qVeOV66oSkVihcKdlgmmJKT\nIPb3/08Wx/e6obO1ukKDvlC/NxTNX3N+5agE4altbAS649aejgIGquparzJDgdaqOtLf81qCMClJ\nTlYmLY5lzHfrKBgWynPXNrbhsMZJToZdUS5ZrJ0JcTvcTXk1LnXJ4pKroXje/1vJaQmiAzBaVXt6\nnj8GoKpjvMoMxRKECaCtB0/w989WsHznUa5uUoln+zWijE0hbs5Rhb3LYd1slzD+3Oi2V20Dl/Rx\n05SXqR3cGLNITksQNwC9VHW45/lgoJ13MvAkiDHAQVxt40FV3ZnCuUbg7uwmMjKyVWxs0Ba0M7lA\nYlIy7/62ldfnbqJYoTBe6N+EXo0rBjsskxMdWA/rZ7mEsXe521auATTo42oWlZrnmSnKc1qCuBHo\n6ZMg2qrqKK8yZYATqnpWRO4GblLVK9I6r9UgjL827DvO3z9bzpo9x7imWWWe6duI0kULBDssk1Md\n3eGWVF03G3b8Dprs1t++5Gr3qN4RQsODHWWGZTpBiEgCbrGgNKnqRf+X+dPE5FM+FDisqmkOM7AE\nYdIjISmZd37ewtifNlG8UDjPXduY3k0qBTssk9Od/NOtY7H+G9jyIySegUIl3Uio+r3dyKiCxS5+\nnhwkEAmiiz8XUtVf/AgmDNds1A03SikKuEVV13iVqaSqez0/9wf+qart0zqvJQiTEev3HeORz1ey\nanccVzWuyDP9GlG+mI10Mn6IPwlb5rtksfF7d2NeaAGo2cWtv13vKiie87905KgmJgAR6Q28jhvm\n+qGqviAizwLRqjpTRMbg7tZOxM3/dI+qrk/rnJYgTEad75uYt4nC4aE82ach17esYvdNGP8lJbr5\nodZ/Cxu+gSPb3fbKLV3Nov5VUKFRjuy3yIr1IMoAbYByuKk2AFDVjzMaZGZZgjCZteXgCR6dvpKo\n7UfoXLcs/+7fxKYRN+mn6taxWP8NbPgWdse47SWquURRrxfU6ARhOWMUXaDnYroSmA7EAyWBo55/\nt6lq0BaUtQRhAiE5WZn8Ryz/+X4DScnKQz3qMbRjDZvTyWTc8f2waY6rXWz9GRJPuzmial/hkkXd\nHhBRLmjhBTpBRAOfqup/ReSIqpYSkadwo45eDUC8GWIJwgTSnqOneXLGan5cf4DGVYozpn9TmlTN\nv9MxmACJPwXbfoWN37nO7uN7AYGqrV1Hd71eUKFxtjZFBTpBxAGlVTVJRI6qakkRKQhsUtXIAMSb\nIZYgTKCpKt+u2sfoWWs4dOIsQzvW5KEe9Shq602YQFCFvStcotj4HexZ5rYXr+rWtKjX03V4Z/EM\ntIFOEHuB2qp6SkS24FaZOwLsUdWgje+yBGGyStzpBP7z/Xqm/LGDSiUKMbpvI3o2shvsTIAd3w+b\nfnAjorb+DPEnILQg1OzsVsyr1wNK1Qj4ZQOdIKYD01V1ioi8CXQGzgJxqtoj09FmkCUIk9ViYg/z\nr69Ws37fca5sUIHRfRtStZR1YpsskHjWTSq4cY5LGoe3uO1l67k+i7rdIbIjhGX+Bs9AJ4jCQIiq\nnhSRQsBDQDHg1WCuKmcJwmSHhKRkJizczqtzN6Io93ery/BOtSgQZp3YJgsd2uISxaYfYPsCSIp3\nHd01u0DdKzN1z0WOuw8iK1iCMNlp99HTPDdrLd+v2UftckV5rl9jOtYpG+ywTH4Qf9J1dG+a6x5x\nO6D3K9D2zgydLivug+gAtMbVHM5T1X9nKMIAsARhgmH++gM8PXMNOw6f4uqmlXji6gZUKmFrIJts\nogoHN0DRchlexyLQTUyjgceB5cDJC0K9yIR6WckShAmWMwlJjP9lK2//vJkQEUZeUYfhnWtSMCw0\n2KEZc1GBThD7gL6quiQQwQWKJQgTbDsPn+LZ2WuZu3Y/NcoU4alrGnLFJRWCHZYxafI3QfjbyyaA\nfRIb46Na6SK8d1trJg5rS0iIMGxCNEM/WsKWgyeCHZoxmeZvgngfuCMrAzEmN+tSrxzfP3AZ/+rd\ngJjtR+j52q88N3stcacTgh2aMRnmbxPTPNy9DxuBvd777D4IYy508PhZ/vvDBqZF76RUkQI82L0e\nA9tUs7mdTI4R6D6Ip1Pbp6rPpDO2gLEEYXKy1bvjeG72Wv7Ydph6FSJ44uqGXFYveBO0GXOO3Qdh\nTA6gqsxZs58x360j9tAputQrx+O9G1C/Yu5agczkLYGuQVyWyq6zQKyq7vPjHL2AN3ALBr2vqi+m\nUu4G4HOgjaqm+elvCcLkFmcTk5i0KJaxP27ixNlEBrSpxoNX1qN8cVvJzmS/QCeIBFyHtvd8tN4H\n/gIMOrdkaArHh+L6L7oDu3BLjg5U1bU+5YoB3wAFgJGWIExec+RkPGN/2sQni2MJCwnhzstqMeKy\nWkTYbLEmGwV6mOsw3Lf6OkC4599puJFNDYHTuOVEU9MW2KyqW1U1HpgK9Euh3HPAS8AZP+MyJlcp\nVbQAT1/TiHl/78IVDcoz9sdNdH15PpMWbSchKTnY4RlzAX8TxLPAcM8HfJKqbgXuAp5W1Q24BNIl\njeOrADu9nu/ybDtPRFoA1VR1dlqBiMgIEYkWkeiDBw/6Gb4xOUv1MkV565aWzLjvUmqXi+DJr9fQ\n/dVfmL1yD8nJubdf0OQt/iaI4oDvYqoFgXPLbR0E0poDOaWlks7/LxCREOA13CyxaVLVd1W1taq2\nLlfORoSY3K15tZJMHdGej4a2oVB4KCOnLKPvWwv4ZeNBcvMAEpM3+JsgvgG+EpEuIlJTRLoCXwDn\nvu23A2LTOH4XUM3reVVgj9fzYkBj4GcR2Q60B2aKyEXbyIzJ7USEyy8pzzf3d+a1Ac04eiqBIR8u\nYeB7i4mJPRzs8Ew+5m8ndQQwFrgF14EcD0wB7lfVEyJSAyisqutSOT4M10ndDdiN66S+RVXXpFL+\nZ+Bh66Q2+dHZxCSmLtnJmz9t4s8T8VxxSXke6lGPRpVtfWwTGAHtpFbVE6o6DCgMVMQlg2GqesKz\nf3tqycGzPxEYCcwB1gGfqeoaEXlWRPr6E4Mx+UXBsFCGdKzBr/+4nH/0qk9M7BGuHruAeyfHsGn/\n8WCHZ/IRu1HOmBwu7nQCH/y2lQ8XbudkfCL9mlXm/m51qVUuItihmVwq0/dBiMgyVW3h+XkTF973\ncJ6q1stMoJlhCcLkJ0dOxjP+161M/H07ZxOTuLZ5FUZ1q0vNskWDHZrJZfxNEGndnfOy18/PZz4k\nY0xmlCpagEevuoThnWsy/pctTFocy4zlu7m2eRVGXlHHahQm4KyJyZhc6uDxs4z/ZQuf/BFLfGIy\nfZtVZuQVdahT3uZ5MmkLyFQbntFHoqoJXtuGAs2BX1X1ywDEmmGWIIxxieL937by8aJYziQm0btx\nJe67vA4NKxcPdmgmhwrUKKZpwO1eJ30CeBfoBEwWEVtEyJggK1esII/1bsDCR6/g3q61+WXjQXqP\n/Y07JkSxdMeRYIdncrGL1SBigQ6qusfzfD/wiKp+LCLXA4+raqvsCfWvrAZhzF/FnUpgwu/b+ej3\nbRw9lUDH2mW4t2sdLq1TBpGUJjUw+U2gmpiOqWpxz88NgGVASVU945mh9YCqlglU0OllCcKY1J08\nm8inS3bw7q9bOXD8LE2rluCeLrXp0agioSGWKPKzQDUxnfTcRQ3QGlitqudmWhXSHgVljAmiogXD\nGN65Fr/+43LGXNeEY6cTuGfyUq589Rc+XbKDMwlJwQ7R5HAXSxC/Ac+JyCW42Vu/99pXH5/1qY0x\nOU+h8FAGto3kx4e68tYtLYkoGMZjX66i03/m89b8zcSdSrj4SUy+dLEmpprAt7hksBroqqqHPfv+\nDZRX1eHZEWhKrInJmPRTVRZtOcQ7v27l140HKVIglJtaV+OOTjWpVjqtSZlNXhHoFeVKn0sMXttK\nAvGqeirjYWaOJQhjMmfd3mO899tWZi7fQ7IqVzWuxB2da9IyslSwQzNZKKAJIqeyBGFMYOyNO83E\n32OZ8kcsx84k0jKyJMM61aRXo4qEhfq7KoDJLSxBGGPS7eTZRD6P3slHv28n9tApKpcoxG0da3Bz\nm2qULFIg2OGZALEEYYzJsKRkZf76A3ywYBuLth6iUHgI/VtU5fZLa1Cvgk3lkdtZgjDGBMS6vceY\nsHA7M5bv5mxiMh1qlWFIxxpc2aC8NT/lUjkuQYhIL+ANIBR4X1Vf9Nl/N3AfkAScAEao6tq0zmkJ\nwpjsc/hkPNOidvLJ4lh2Hz1N5RKFGNS+OgPaVKNshO+S9SYny1EJwnPX9UagO2596ihgoHcCEJHi\nqnrM83Nf4F5V7ZXWeS1BGJP9EpOS+XH9AT5etJ2Fmw9RIDSE3k0qcmv76rSqXsqm88gFArEeRCC1\nBTar6lYAEZkK9APOJ4hzycGjKKksUGSMCa6w0BB6NqpIz0YV2XzgBJ8sjmV6zC5mLN/DJRWLMah9\nda5tXplihcKDHarJpOxqQKwC7PR6vsuz7QIicp+IbAFeAu7PptiMMRlUp3wEo/s2YvHj3RhzXRNC\nQ4QnZ6ym3b9/5NHpK1m562iwQzSZkF1NTDcCPc/ddS0ig4G2qjoqlfK3eMoPSWHfCGAEQGRkZKvY\n2NisC9wYky6qyopdcUz5I5ZZK/ZyOiGJRpWLc3PbSPo1r0xxq1XkCDmtD6IDMFpVe3qePwagqmNS\nKR8CHFHVEmmd1/ogjMm5jp1JYMay3Xy6ZCfr9h6jcHgoVzetxM1tqllfRZDltD6IKKCuZ26n3cDN\nwC3eBUSkrqpu8jy9GtiEMSbXKl4onNs61GBw++qs3BXHp0t2MGvFHr6I2UXtckW5qXU1+resQvli\nhYIdqklFdg5z7Q28jhvm+qGqviAizwLRqjpTRN4ArgQSgCPASFVdk9Y5rQZhTO5y8mwi36zay7So\nncTEHiE0RLi8fjluaFWNKy4pT4Ewu68iO+SoJqasYgnCmNxr84ETfB6zky+X7ubg8bOULlqAa5tX\n4fpWVWhUOc3WZZNJliCMMblCYlIyv246yBcxu5i39gDxSclcUrEY17esSr/mlSlf3JqgAs0ShDEm\n1zlyMp7ZK/fwxdLdrNh5lBCBTnXLcV2LKvRoVIEiBWwRy0CwBGGMydU2HzjBV8t2MWPZHnYfPU2R\nAqH0bFSRfs0r06lOWZsHKhMsQRhj8oTkZCVq+2FmLN/NNyv3cuxMImUjCtCnaWX6Nq9Mi2olbchs\nOlmCMMbkOWcTk5i//kBzh/YAAA4pSURBVCAzV+xm3roDxCcmU7VUYa5pVpm+zSpzScViliz8YAnC\nGJOnHT+TwJw1+5m5Yg8LN/9JUrJSp3wEVzepxDXNKlGnvK1bkRpLEMaYfOPQibN8u3ofs1fsYcn2\nw6jCJRWLcXWTSvRuWona5SKCHWKOYgnCGJMv7T92hm9X7eWblXuJjj0CuGTRu0klejepaDULLEEY\nYwx7407z3ap9fLtqLzE7jqDqZqC9qnFFejWuSMNKxfNln4UlCGOM8bL/2BnmrHHJYsm2wyQrRJYu\nQo+GFejZuCItI0sRGpI/koUlCGOMScWhE2eZt24/36/ex8LNh4hPSqZsREG6NyxP94YV6Fi7LIXC\nQ4MdZpaxBGGMMX44fiaB+RsOMmfNPn7ZcJATZxMpUiCULvXKcWWDClxxyf+1d/fBVdV3HsffHx4C\nTQhF86COEAIICDiu9amgIjBYpbYVu32SLuOypesu3Xama+tMd+m0HdtOd6u7nbrbjk3R2tpauzK1\npXVdpiqMIoRZ1FV5MEAipgGXPChPImDg2z9+J3hJD8kJuffce5Pva+bOnHvPL+d8v7k3+d7z+937\n+1VzVllJvsPMKi8QzjnXR0c7j7OhsYM/bN3LE9v2svfAUYYILh9/NvOnVTN/2jlMqior+nELLxDO\nOdcPJ04Ym/fs54mte3liWytbXz8AwPiKUuZNrWb+tGqunHA2I4YVX1eUFwjnnMuiPfve5slXWnlq\n216ebezgWOcJykqGcvUFlcy7sJp5U6s5973FMfOsFwjnnMuRt48d59md7axpaGXNK63s2X8ECN+3\nmDO1irlTqrls/FkFuwBSwRUISQuA7xNWlFthZv/Sbf/twGeBTqAN+IyZvdbTMb1AOOfyzczYvvcQ\naxpaWdvQyqZdb9J5wigrGcpVF1QyZ0oV106uoqaiNN+hnlRQBULSUGA78AGghbBG9SIz25rRZh6w\n0cwOS1oGzDWzT/V0XC8QzrlCc/DIO6xv7ODp7W2sbWhj9763AaitKGX25CpmT65k1qQKykcOz1uM\nSQtEWqtvXAnsNLMmAEkPAwuBkwXCzNZktK8HFqcUm3POZU35yOHcMONcbphxLmZGU/tbPLO9jad3\ntLPyuRYerH+NoUPEJePGcM0FlVwzuZJLxo1heAGub5FWgTgf+GPG/Rbg/T20Xwo8HrdD0m3AbQA1\nNTXZis8557JOEpOqRjGpahRLrp7Asc4TPN/8Js/saGPdjnbueWoH339yB2UlQ3n/xAqumlTB1RdU\nMvWccoYUwLe60yoQcZnG9m1JWgxcDsyJ229mdUAdhC6mbAXonHO5VjJsCDMnVjBzYgV33AD7Dh+j\nvqmDdTvbWb+zg6deaQXg7LISZk2sYOakCmZNrMjbdy/SKhAtwLiM+2OBPd0bSboOWA7MMbOjKcXm\nnHN5Maa0hAUXnceCi84Dwkdp1zd2sKGxg/WN7Tz28usAVJePYObECmZNCsWltqI0lYKR1iD1MMIg\n9XxgN2GQ+tNmtiWjzfuAlcACM9uR5Lg+SO2cG6jMjNc6DrOhqYP1jR3UN3XQdjC8b64uH8HyD01j\n4SXnn9GxC2qQ2sw6JX0eWE34mOv9ZrZF0p3AJjNbBdwFjAIeiSpjs5ndlEZ8zjlXaCRRW1lGbWUZ\ni66sOTngvbHpDeqbOjhndO6/lOdflHPOuUEm6RVE4X2uyjnnXEHwAuGccy6WFwjnnHOxvEA455yL\n5QXCOedcLC8QzjnnYnmBcM45F8sLhHPOuVhF/UU5SW1Aj4sK9aASaM9iOMXAcx4cPOfBoT85jzez\nqt4aFXWB6A9Jm5J8k3Ag8ZwHB895cEgjZ+9ics45F8sLhHPOuViDuUDU5TuAPPCcBwfPeXDIec6D\ndgzCOedczwbzFYRzzrkeeIFwzjkXa8AXCEkLJDVI2inpKzH7R0j6VbR/o6Ta9KPMrgQ53y5pq6SX\nJD0paXw+4sym3nLOaPdxSSap6D8SmSRnSZ+Mnustkh5KO8ZsS/DarpG0RtIL0ev7xnzEmS2S7pfU\nKmnzafZL0j3R7+MlSZdmNQAzG7A3wvKmjcBEoAR4EZjerc3ngHuj7VuAX+U77hRyngeURtvLBkPO\nUbty4GmgHrg833Gn8DxPBl4AzoruV+c77hRyrgOWRdvTgV35jrufOV8LXApsPs3+G4HHAQEzgY3Z\nPP9Av4K4EthpZk1mdgx4GFjYrc1C4KfR9kpgvqJFsYtUrzmb2RozOxzdrQfGphxjtiV5ngG+CXwX\nOJJmcDmSJOe/BX5gZm8CmFlryjFmW5KcDRgdbb8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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# We set the parallel diffusivity larger than the perpendicular diffusivity\n", "zeppelin = gaussian_models.G2Zeppelin(lambda_par=1.7e-9, lambda_perp=0.8e-9)\n", "E_zeppelin_par = zeppelin(scheme, mu=[0, 0])\n", "E_zeppelin_perp = zeppelin(scheme, mu=[np.pi / 2, 0.])\n", "\n", "plt.plot(bvalues, E_zeppelin_par, label='E_parallel')\n", "plt.plot(bvalues, E_zeppelin_perp, label='E_perpendicular')\n", "plt.legend()\n", "plt.xlabel('b-value [s/m^2]', fontsize=13)\n", "plt.ylabel('Signal Attenuation', fontsize=13)\n", "plt.title('Signal Attenuation Zeppelin', fontsize=15);" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Temporal Zeppelin: G3\n", "In DTI, the signal attenuation decays like a Gaussian over q-value and like an expontial over diffusion time $\\tau$. \n", "However, recent works argue that hindered diffusion is actually slower-than-exponential over $\\tau$ due to how the external axon boundaries still restrict diffusing particles*(Novikov et al. 2014)*.\n", "To account for this, *(Burcaw et al. 2015)* proposed a modification to the Zeppelin as\n", "\n", "\\begin{equation}\n", "\\textbf{D}_{\\textrm{diag}}^{\\textrm{r}}=\n", "\\begin{pmatrix}\n", " \\lambda_\\parallel & 0 & 0 \\\\\n", " 0 & \\lambda_\\perp^{\\textrm{r}} & 0 \\\\\n", " 0 & 0 & \\lambda_\\perp^{\\textrm{r}}\n", " \\end{pmatrix}\\quad\\textrm{with}\\quad\\lambda_\\perp^{\\textrm{r}}=D_{\\infty}+\\frac{A\\ln(\\Delta/\\delta)+3/2}{\\Delta-\\delta/3}\n", "\\end{equation}\n", "where perpendicular diffusivity $\\lambda_\\perp^{\\textrm{r}}$ is now time-dependent with $D_{\\infty}$ the bulk diffusion constant and $A$ is a characteristic coefficient for extra-axonal hindrance.\n", "Notice that when $A=0$ then G3 simplifies to G2." ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [ { "data": { "image/png": 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VqlVZs2YNAK1ateLy5cs4OTnh6+tL8+bN8fLyonPnzlSoUCFeWocPHyY2NpYq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\ncXfPu5ncbTJN612yGIPvXch0A9tb34e0DSC1XEmPftNqTBd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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# we just sample the (restricted) perpendicular direction now\n", "restricted_zeppelin = gaussian_models.G3TemporalZeppelin(\n", " mu=[np.pi / 2, 0.], lambda_par=1.7e-9, lambda_inf = 1e-9)\n", "\n", "for A in [0, 2e-12, 4e-12]:\n", " plt.plot(bvalues, restricted_zeppelin(scheme, A=A), label='A={} $\\mu m^2$'.format(A*1e12))\n", "plt.legend()\n", "plt.xlabel('b-value [s/m^2]', fontsize=13)\n", "plt.ylabel('Signal Attenuation', fontsize=13)\n", "plt.title('Signal Attenuation Restricted Zeppelin', fontsize=15);" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Notice that for larger characteristic coefficients the signal attenuates faster." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## References\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", "- Alexander, Daniel C., et al. \"Orientationally invariant indices of axon diameter and density from diffusion MRI.\" Neuroimage 52.4 (2010): 1374-1389.\n", "- Jeurissen, Ben, et al. \"Multi-tissue constrained spherical deconvolution for improved analysis of multi-shell diffusion MRI data.\" NeuroImage 103 (2014): 411-426.\n", "- Tariq, Maira, et al. \"Bingham–noddi: Mapping anisotropic orientation dispersion of neurites using diffusion mri.\" NeuroImage 133 (2016): 207-223.\n", "- Panagiotaki, Eleftheria, et al. \"Compartment models of the diffusion MR signal in brain white matter: a taxonomy and comparison.\" Neuroimage 59.3 (2012): 2241-2254.\n", "- Novikov, Dmitry S., et al. \"Revealing mesoscopic structural universality with diffusion.\" Proceedings of the National Academy of Sciences 111.14 (2014): 5088-5093.\n", "- Burcaw, Lauren M., Els Fieremans, and Dmitry S. Novikov. \"Mesoscopic structure of neuronal tracts from time-dependent diffusion.\" NeuroImage 114 (2015): 18-37." ] } ], "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": 1 }