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@mwaskom
Created September 15, 2012 22:39
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First MR Tutorial
{
"metadata": {
"name": "mrTutMR"
},
"nbformat": 3,
"nbformat_minor": 0,
"worksheets": [
{
"cells": [
{
"cell_type": "heading",
"level": 1,
"metadata": {},
"source": [
"Basic MR"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"**Class:** Psych 204a\n",
"\n",
"**Tutorial:** Basic MR\n",
"\n",
"**Author:** Wandell\n",
"\n",
"**Date:** 03.15.04\n",
"\n",
"**Duration:** 90 minutes\n",
"\n",
"**Copyright:** Stanford University, Brian A. Wandell\n",
"\n",
"**Checked:** \n",
"\n",
" - 2007: Rory Sayres\n",
" - 09.22.09: Jon Winawer\n",
" - 09.18.10 Jon Winawer\n",
"\n",
"Translated to Python by Michael Waskom, 09/2012"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"This tutorial explains basic principles of magnetic resonance signals.\n",
"As the first tutorial in the series, it also gives you an opportunity to\n",
"examine basic Python commands.\n",
"\n",
"In this tutorial, principles are illustrated for signals derived from a bulk\n",
"substance, such as a beaker of water. Various terms used to describe the\n",
"MR principles, including spins, parallel and anti-parallel, Boltzman distribution\n",
"are introduced.\n",
"\n",
"Also, tissue properties such as T1 (longitudinal) and T2 (spin-spin)\n",
"interactions, are explained and the way in which signal contrast depends\n",
"on these parameters is explained. \n",
"\n",
"The next tutorial, MRImaging, takes up the topic of how to make\n",
"images that measure these tissue properties in a non-uniform volume, such\n",
"as a head.\n",
"\n",
"\n",
"**References to help with this tutorial:**\n",
"\n",
"- Huettel et al. Chapters 1-3 (mainly Chapter 3)\n",
"- John Hornak [online MRI book](http://www.cis.rit.edu/htbooks/mri/index.html) (especially [Chapter 3](http://www.cis.rit.edu/htbooks/mri/inside.htm))\n",
"- McRobbie et al, MRI, From Picture to Proton, Chapter 8\n",
"\n",
"1st v. 2nd edition of text: \n",
"The course text is Huettel et al, 2nd edition. The first and second\n",
"editions are quite similar, especially in the earlier chapters.\n",
"References followed by \"ii\" are 2nd edition, \"i\" first edition. Hence p.\n",
"51i, 60ii means p 51 in the first edition, p 60 in the second. And so on.\n"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"%pylab inline\n",
"import matplotlib as mpl\n",
"mpl.rcParams[\"figure.figsize\"] = (8, 6)\n",
"mpl.rcParams[\"axes.grid\"] = True"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"\n",
"Welcome to pylab, a matplotlib-based Python environment [backend: module://IPython.zmq.pylab.backend_inline].\n",
"For more information, type 'help(pylab)'.\n"
]
}
],
"prompt_number": 1
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Spin Velocity"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"When a substance with net spin is placed in a steady magnetic field, it\n",
"precesses at a frequency that is characteristic of that substance.\n",
"We can compute the precession frequency from the following simple\n",
"formula. Note the units.\n",
"(see chapter 3 and ca. p. 51i, 60ii for a discussion of precession)."
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"B0 = 1.5 # Magnetic field strength (Tesla)\n",
"g = 42.58e6 # Gyromagnetic constant for hyrdogen (Hz / Tesla)\n",
"v = g * B0 # The resonant frequence of hydrogen, also called its Larmor frequency"
],
"language": "python",
"metadata": {},
"outputs": [],
"prompt_number": 2
},
{
"cell_type": "heading",
"level": 4,
"metadata": {},
"source": [
"Question 1"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"What are the units of the Larmor frequency, $v$, for hydrogen as expressed\n",
"above? The gyromagnetic constant of sodium is $11.27 \\times 10^6 Hz/Tesla$. Compute\n",
"the Larmor frequency of sodium in a 3T magnet.\n",
"\n",
"Ordinarily, the resonant frequencies are expressed in units of MegaHertz\n",
"(millions of hertz)"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"# Python string interpolation is accomplished through the % operator\n",
"print \"The resonant frequency of spins in hydrogen is %0.4f (MHz) at %.2f Tesla\" % (v / (10 ** 6), B0)"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"The resonant frequency of spins in hydrogen is 63.8700 (MHz) at 1.50 Tesla\n"
]
}
],
"prompt_number": 3
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Spin Energy"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The energy in precessing spin is proportional to its resonant frequency, $v$.\n",
"The constant of proportionality between energy and frequency is a famous\n",
"constant called Planck's constant.\n",
"(Extra credit: Find out something interesting about Max Planck). \n",
"\n",
"Hence, the amount of energy in a hydrogen molecular in this magnetic\n",
"field is"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"h = 6.626e-34 # Planck's constant (Js)\n",
"E = h * v\n",
"print \"E = %.4g\" % E"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"E = 4.232e-26\n"
]
}
],
"prompt_number": 4
},
{
"cell_type": "heading",
"level": 4,
"metadata": {},
"source": [
"Question 2"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"What are the units of E?\n",
"\n",
"At steady state, say when the subject enters the magnet and prior to\n",
"any measurements, the dipoles align parallel or anti-parallel to the\n",
"magnetic field. \n",
"(The textbook illustrates this with a nice metaphor of gravity in Figure\n",
"3.7ii, 3,6i.) \n",
"\n",
"The energy difference between these two states is proportional to the\n",
"mean magnetic field. This is called the Zeeman effect (see Figure 3.8i,\n",
"3.9ii in the book). "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The proportion of dipoles aligned parallel (low energy) and anti-parallel\n",
"(high energy) to the main field is described by the Boltzmann\n",
"distribution.\n",
"The formula that determines the fraction of dipoles in the low and\n",
"high energy states is in Hornak (Chapter 2)."
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"k = 1.3805e-23 # Boltzmann's constant, J/Kelvin\n",
"T = 300 # Degrees Kelvin at room temperature\n",
"dE = h * g * B0 # Transition energy\n",
"ratio_high_to_low = exp(-dE / (k * T)) # Boltzmann's formula, Hornak, Chapter 3; Huettel, p. 76ii\n",
"print \"High-low ratio: %.6f\" % ratio_high_to_low"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"High-low ratio: 0.999990\n"
]
}
],
"prompt_number": 5
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"At low temperatures (near absolute zero), very few of the dipoles enter\n",
"the high (anti-parallel) energy state.\n",
"\n",
"No, this is not important for us.\n",
"But I find the numbers interesting and can see why people might want to\n",
"work on low temperature physics for a while."
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"T = logspace(-3, 2.5, 50)\n",
"r = exp(-dE / (k*T))\n",
"plot(T, r)\n",
"semilogx()\n",
"xlabel('Temperature (K)')\n",
"ylabel('Ratio of high/low energy state dipoles')\n",
"ylim([0, 1.1]);"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "display_data",
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bNm2KY8eOOZZzc3Px8MMP4+jRo2jevDmOHDnict/g4GDExsaiQYMG6N69O3bt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\nT0RE5IE4wRMREXmg/wc8SUmXVrI9UAAAAABJRU5ErkJggg==\n"
}
],
"prompt_number": 6
},
{
"cell_type": "heading",
"level": 4,
"metadata": {},
"source": [
"Question 3"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Where is human body temperature on the graph? Given human\n",
"body temperature, what is the ratio of high/low energy?\n",
"\n",
"Would it matter if we kept the room cooler?"
]
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"T1 Tissue Contrast"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The tendency of dipoles to align (either parallel or anti-parallel) with\n",
"the B0 magnetic field imposes an opportunity to probe tissue properties\n",
"using magnetic resonance. MR signals are derived by perturbing the\n",
"ordered spins (excitation) and then measuring the emissions (reception)\n",
"as the dipoles return to their low energy state. The way the dipoles\n",
"return to their low energy state provides information about the local\n",
"tissue.\n",
"\n",
"Summing the effect of the many many dipoles within a voxel, we\n",
"obtain a measure of the dipoles within the voxel called the net\n",
"magnetization. This is represented by a single vector (see the many\n",
"examples in Hornak). Most reasoning about the MR signal is based\n",
"on understanding models of the net magnetization vector and how it\n",
"recovers after being perturbed by the radio frequency pulses in the\n",
"presence of changing magnetic fields. \n",
"\n",
"\n",
"The MR measurements that we obtain describe the net magnetization in the\n",
"direction perpendicular to the main axis (B0 field). This direction is\n",
"illustrated in the following figures.\n",
"\n",
"First, here is a 3D plot that shows the net magnetization as a red circle\n",
"in the steady-state. The black lines show the three axes."
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"def axisplot(ax):\n",
" \"\"\"Convenience function to plot axis lines in a 3D plot\"\"\"\n",
" ax.plot([-1, 1], [0, 0], [0, 0], \"k:\")\n",
" ax.plot([0, 0], [-1, 1], [0, 0], \"k:\")\n",
" ax.plot([0, 0], [0, 0], [-1, 1], \"k:\")\n",
" for axis in [\"x\", \"y\", \"z\"]:\n",
" getattr(ax, \"set_%slabel\" % axis)(axis)"
],
"language": "python",
"metadata": {},
"outputs": [],
"prompt_number": 7
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"from mpl_toolkits.mplot3d import Axes3D\n",
"ax = subplot(111, projection='3d')\n",
"ax.scatter(0, 0, 1, color=\"r\")\n",
"axisplot(ax)\n",
"az0 = 332.5\n",
"el0 = 30\n",
"ax.view_init(az0, el0)"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "display_data",
"png": 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GDRrE/r+9vR3r1q3DXXfdZeIWWRN+4UgJamZYOnSNl/O9Bw4cwD333IM77rgD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\n"
}
],
"prompt_number": 8
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The size of the MR signal we measure depends on how far the \n",
"magnetization deviates from the main z-axis. We can see this \n",
"more easily by looking at the figure straight from the top. In \n",
"initial situation, the point is at ``(0, 0)`` in the ``(x, y)`` plane."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"**TODO** Figure out clean way to show multiple views of the same plot"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Suppose we excite the tissue and place the net magnetization along the x-axis."
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"ax = subplot(111, projection='3d')\n",
"ax.scatter(1, 0, 0, color=\"g\")\n",
"axisplot(ax)\n",
"ax.view_init(az0, el0)"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "display_data",
"png": 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DBg1i/9/e3o5169bhjjvuMHGLrAm/cKQENTMsHbrGK/nezz//HHfddRduu+02\nAAcbfbz22mtMMMeOHYuxY8ey91Ndr/L7JUkqOIzZqIWz1Z4/kUgETU1NJd+3cePGgr976KGHsGvX\nLowcORK7du3C6NGjVd935JFHAgCGDh2KadOm4fHHH8e8efMq2/B/I1yy/0btwiJxVMYbI5EIZFlm\nfSeV8UYjb4h6xizxP/nkk3H55Zcb/r2AtWKY2WyWDWKm4cI0+Ju8LrFYjLWYtBKRSIT9fywWw1ln\nncXijU1NTTjxxBPZ74866ij86Ec/qur7aHi1z+dDMplEIpGom+HVyqQfLYJZjLFjx2L16tWIx+NY\nvXo1Tj311B7voWH0APDZZ59hw4YNOO+886r6XkAIZh7ZbJbVN3Z2drIektRXlY83NjY2ssw4O/VX\nFRZm9TQ0NOCkk04CcPBhMH/+fOzdu9fkraotvDiGw2F0dnaiq6sLiUQCuVyOtVXkG/FTrSIA0wXi\n5ZdfhizLbF9GjRrFHqjBYBD33Xcfe6j7fD5ccsklVX2fmlVNrfaCwSA8Hg8SiUSPRuV6YZX4pRKt\nLtliXHXVVfjggw/wpS99CR999BGuvPJKAMDHH3+MKVOmADg40u+MM87AiBEjcMkll+Daa6/VZVqR\nlBNPT0Y4HIYsy8y1KssystksKzVwAtFolK127U4mk0E4HEZra6tp25DL5fDcc8/hzDPPrLj+LpvN\noqurC21tbTpvXfmQNcg341eOcKP7gw85pFIpJJPJPOuBivkjkQh8Ph/zyvj9/povMh988EGcddZZ\nGDhwIABg7ty5uO2225ibLpvN1nQbotEoGhoain5HLpeDLMtIpVK6t9rLZrOIx+OWeHbFYjEWppox\nYwYef/xxBINBszerIkQMk4O669NFy8cnnYKwMPWFMvGIZ555Bp9++im+853vmLhV2ig235SPx9Po\npkof5mRZ8Q259RaI22+/HRdccAFGjBgB4KCA8+3XHnjggbz3W8ErRBm11KNWz+HVVrIw+W1JJpOq\nMV+7YP5VYyHUXChOExcn7ZMV9+WEE07AsGHDyv47rftRaQyT4vGUsUluVWqOAeTPN61FPJ6fnUi9\nWJUt5QpB/ZGJ6667Do899hh7fe655+ZlMc+dO9fUrOZyBIsSg4LBYMFWe07CCouVShEWZhGs+EAW\nWJtBgwaxjNp0Oo0f//jH+PGPf1zUBaW3JaCcVEP/BcBcqn6/H6FQSLeHVzn3CZ85SnkCypZyu3fv\nRiaTYclso7qMAAAgAElEQVQ3d999N4477jh885vfBHCwzIePhY0ZM0aX/TATSiJUzuDkx2NpxYoW\nphOepUIwOZQn1SknmUeSJNF9xCAymQyGDBmi6+gwZR1mIXGUJInFHAOBABvjVkuKDShX+x3fi3XT\npk3Yv38/vv71r8PlcvUQzB//+Md5f9urVy/9d0Anqn1m8MfFzsOrAfVjYRUhrwQhmBx2nn9Zj1h9\nQeP3+3HZZZex148//jh69+6N0047raLPs5I4VkI6nWbW0qZNm/Dcc89h6dKl8Hq9OOyww5BOp1l8\nc/r06VXdh1a4j6v5fn4GJ00n0jq8GrDG/vNY/V7VivXuKgvhlJPM48R9sguBQEBzwgNlqiaTybwZ\np0uXLoUsy5AkidUAG9lWUSuJRAK7du1ir7ds2YJZs2ax1yeeeCJzrwLA8OHDMXHixIrim06Gj/tK\nkoRYLMYGMtgBfjuTySS8Xq+JW1M95t9ZFkaIi0BPJk6ciFGjRgEA4vE47r333ryifqU4KhtktLa2\nIhAIWLIGuLu7G48++ih7/dFHH+H2229nr0ePHp2XpHPYYYdh8ODBPT6HnzVJfZidlqleCWrDq4st\nKKxkYerZtMBsrHG3WQSRJWsv6HzZaX/Icty/fz+6u7vZKDcAPcRRrUGGVXrJxuNxXHXVVezYu91u\nvP766+z3xx13HB5++GH2mpKNtEJxPBqZVasC/1pQS7HiFxTkwk6n05a9B/Tu8mM2QjA5iiUtOAUn\nCabVKTTKLRqNoq2tDYsXL0YwGERrayueffZZdHV1Wcpy7OrqYiKVy+UwduxY1lIuEAhgwoQJ7PfB\nYJD1XtULiuNV4o60koVVC6j5CDV3V7PErbb/4XC46kklZmP+XWlhJEkSAiPQRDFxVI5ya2lpQWNj\nIwKBACtSf/PNN/P6mxailr1kt2zZwtrFAcBXv/pV1vJPkiSsXbuWZfxKkoTZs2cXzdzk3c3VJsCU\n4440G6PEmhpCKIdXZ7NZyxwb/liEw2HbW5giS7bOcNoCwIzMZmoCQG3jqCMUjXKjOsdyCv6vvvpq\nVirR2dmJV155Ja+DUC1Yu3YtTj31VFY3um7dOvTq1Qtf+tKXABycO8hv/3HHHVfT7SmFlvrNekSZ\nURuLxeByuSxxXPhnjRNcskIwOWoxRFpgb5R9VXlxpLic3nNOP/jgA2zZskVVMMuNYfKLiTvvvBMn\nn3xy3udSKzwA+NWvfpX3t1Zz6RF8nWKhcot6vGf5VnvxeJxlU+vRaq/a7QKEYNYFThNMp+2Pnpgh\njmoMHz4cw4cPZ69ffPFFjBkzBj6fr+jfdXd3I5FI4IgjjgAA3HbbbWhra8M111wDADjnnHNw+OGH\ns/fzZR3VYrSVz1tV/IBmXhzMEgmz46dkXUqShEwmk9cxyIy5pLxLttpJJWYjBLMEQmCciVrT8Ww2\ny8TR6/UaIo6lyOVyWL16NY499lgcffTRWL58ObMy33nnHXz++eesLdzDDz+MTCaDhQsXAgAWLVqU\n15Lv5JNPNn4HagzFN71eb55w1vs9SyPX/H6/Lq329CAajeYt2OyIEMwSOE0w63F/CokjTeSwijiq\nIUkSVq9eDQD4+9//jvb2dnzwwQfo378/PvjgA7z//vtMML/3ve/l/a3dV/PloIxv5nI5S2QaWwFy\nYSuF04gYpygrcTD1EMN02v4oyWazrOCdJnJ0dnay7EGv14umpia0tbWhubkZoVAIfr/fFHeVGvxI\nqh07duCqq65ir5uamjBq1CisWbMGAHD22Wfj8ssvN3oTLQ2Jg8vlQiaTMaV+02yXrNo28MOr3W63\nKbWtThBMYWGWwOkCY2cofZ5u/Ewmg1wux9yq5H6qZpZjLUmlUti5cyeb4fjmm2/isssuQ3t7OwDg\n+OOPx+LFi9n7Bw0ahFtvvTXvM9566y2ccMIJxm20RuieMUM8qLculYWpxTfrFUoM4l3Yes8m5eEt\nfSeUlQgLU4V6EEi77WM2m2UF2mQ5dnV1sVo/n8+HpqYmtLa2orm5GcFgsOzSjlqTSqWwatUq9joW\ni+VlvQ4ePBgvvvgie93U1MRKPAi+DjMej2P+/Pma6jdrjRWsKiV8/WY9zJnkKXU+1GpbU6mU7sdG\nuGQdDK1IlT9z0g1mtYeaEprIwYvjgQMH0NXVhUQigVwuB7/fz8SR73hiJXEEDu7L3LlzkUgkAABe\nrxfvvfceK+VobW3Fk08+yd7vcrnKSshoaGjAc889x7qnfPrpp3kuXcFBqN2gUf1prbh4KATfao8W\nFbVqtReNRoVgOh2nCSZgnX2iBgDkGuItR0rc8Pv9aG5uzrMcKWHBCg+lrq6uPJGaOHEi9uzZA+Dg\ncZ42bVpet5uf/vSnVWUpKusw+WPwwAMP4KGHHqr4s52EmmjZuT9tuZQr2nRsirXaq3Y7IpGI7RPR\nRAyzBGLgsj6Q5ajMVgXAslXJRVROzNFo8d+2bRsGDBjAah1nz56NFStWYOTIkQCA1atX46ijjmLv\nnzZtmmHb9qMf/SjvWv3iiy8sPWjZDLTUb1aLnSxMJXQvUlMIl8sFv9+vS8YxZefaGWFhlsAq1pie\n1HqflJYjTeTo7u5GMplk8RN+lqPVLEfiT3/6E1599VX2esOGDXj//ffZ6yeffJKJJQAcc8wxZafr\nlzN1pVQvWXqw7d27FxdeeKHjrl29UItvWnnqRzlUuw9803u3241YLFaxNa6WrWtnhIWpQCkmQjCL\nQ5ajskMOZSpSdxzKVrUi/E39wAMPoKWlBRdddBEA9Gi5VswlaiWOOeYYvPDCC2z7otEoQqFQzb6v\n0IPR6tYWxTfT6TRzrTulP221x10to7Zca5zOv1OeoUIwFdSDYFaKFcWx3PMTDofR1dWFfv36AQDu\nuecefPrpp7jlllsAABMmTGATOQDgwgsv1HeDq6ScXrL8dPsFCxZgzpw5NW3obiVhLLdxgdIVqdaf\ntlbfrTd6L1IKdVMqVbusvC/VkirthhBMFay+Kq4WLSJDblVlzJH6VNJYIbfbbVnLETjYyPydd97B\n2WefDQB45plnsGvXLtx8880AgMsvvxyBQIC934o1jXpw//33w+/3Azh4bp0QT9IbI+KbdkZtWoyW\nMIqT8kCs+6QzCaeXlahBTceTySSi0Si6u7tx4MABRCIRyLLM3Fatra0s5milQcc8b7/9Nu688072\n+vPPP8fOnTvZ64suuoiJJQCEQiFbud8qnYdJ3W8A4OWXX8all16q52b1wM73jN3jm7Ve8FNGrc/n\nQzKZRCKRKJlRm0gk2ILNzggLswROE0zal1QqhVQqxaxHqgEkV5SVLcd4PM7cpnv27MGyZcvwxz/+\nEQDQ0tKCY489lr135MiReUk5AmD8+PEYNmwYe53NZi17rs1EGd+kmF6pBZbTPVTAoVZ7xdzYThse\nDQgLsyR2FkyyHBOJBKLRKLq6unDgwAE2+NjtdiMYDKKtrQ2tra1obGy0nOWYyWSwZcsW9nr//v0Y\nPnw4OydHH310nsV4xBFHYMaMGYZvp1GUOw+zEG1tbQAOXiOTJk3CO++8U9XnWe0e0VO0KPzg8Xhs\nUb9ppGDzGbUulwuxWIzVUDutyw8gBLMHdnXJFhLHaDTaQxx9Ph/r7mG1+Ewul8Pdd9+dN9j41ltv\nhSzLAIDevXtj9+7dbJt9Ph+GDBliyrY6AUmS8Nvf/haDBg0CUJ3wFbqOrHR9VQovDNSftlgrOSfs\nczmQ9U2t9qLRaF4bwkgkwjpS2RkhmAoKXehWEk1eHCORiKo4hkIhtLW1oaWlBY2NjXniaLVFwMKF\nC7Fv3z4AB49/JBJBLBYDcDBe8vTTT+dlfNop5qg3lcYwi3HUUUex637t2rW47bbbdPtsK11neqAl\nvmn2PpvpEqbEoGAwyIYjfPHFF+jq6nKEhSlimCUwe6VI4shnq2YyGZapSqUcViv45+nq6mKrcwC4\n+OKLsWTJEjbHcfr06XmDjm+88UbNn2018bc7M2fOxP79+83eDMtTaXyzXnC5XPD5fEilUvjlL3+J\n559/HmeccYbZm1U17mXLli0zeyOsBJVT8OJDGV61FiQSR1mWkUgkEI/HEYvF2PaQOAaDQQQCgYrH\nV5G7k7fa9GTHjh1IpVJobW0FAFx99dVoa2tjbr8zzjgDxx9/PIuTDho0qOIMOsriNWuKvF5QIlOp\n8zhhwoSabofX60VLSwuAg9f9tGnTMG3atJLnJ5VKsUUckc1m2b9aXWuFkGXZkHpg/tpLJpOs/WOt\nxmVpgSw7s+8Jypw999xz8cknn2DNmjXYvHkzTj75ZNZa0m7Y+ylTA4waIq3WVzWbzbKHjtfrtbzl\nSDz99NMIhULsYb5lyxYMHToUAwYMAHCwvypP3759Dd9GQfkEAgGsWLFCkyutnq18Zf0mcKhvqhn3\nrlWydGk73G43jj76aCxfvhyJRALXXnstnn76aUtsYyFWrVqF++67DwDQ2dmJY489Fv/3f/8nYpha\nqFYws9ksGysUiUTQ2dmJzs5OxONxtvpubGxkMcdQKAS/31+yk0alVLI/fFbgo48+yi4m4GAtI1/8\nf/XVV7NGAbWm3lyytYhhFoMvybn33nvxpz/9qeB76VxQfN3KmaS1gOKbAGxZv1lLotEoDjvsMCxe\nvLgssdy0aROGDh2KwYMHY+XKlarvueGGGzBo0CCccsop2L17ty7bu2DBAuzYsQPbtm3DMcccg2uv\nvRaAsDA1Uc5DmbccyXrMZrOsZskqlmOx/YlEIvjss89YPePvfvc7bN68Gffeey8AYOzYsXl/X2s3\nocAaXHDBBXnXLIUv6DqPxWLMS+JyuSDLMtxud10KRkNDA+uIQ/E8o+KbVrMwgfw6zHK2bdGiRVi1\nahUGDBiAyZMnY/bs2ejduzf7/datW7F582Zs374dGzZswJIlS/DEE0/otg8LFy7EV7/6VUyZMgWA\nyJLtQTkuWeWgY5rlGI/Hkcvl4PP50NTUhLa2NjQ3N9fcctSK8rv37duXd5Ft3749bzU3c+bMvNcD\nBgzAwIEDa76dgp7oVYdZLrlcDscccwz69u2LaDSKPXv2YM6cOeju7mYxcb/fn5eZTSUG5GExUjit\nIBoej4dN/KD6zXpcPACVDY/u6uoCcHBBPmDAAEyaNAkdHR157+no6MCsWbPQq1cvzJ49G7t27dJt\nmx988EHs3bsXS5cuZT8TgqkRGlelFEe6CUgc+UHHNPXA7BtXiXIBkEgk8trHnXXWWfjVr37FXlt5\nekO9uWSNgCzHZDKZN54tHA6zJKvDDjsM3/jGN9CrVy80NjbC7XazaySXy7FYntfrhdvtZolsTnfT\nqjUcp/pESZIQjUaL1m/qtQ1WeOZU27hg27ZteTXWw4YNQ3t7e957tm7dmte16vDDD6+6CQcA/O1v\nf8Mdd9yBhx9+OO/nQjBVUFqOFH+0ozhqYeDAgbj++uvN3gyBBmoRw6TrPRaLscVgd3c3K5doaGhg\ns0upG1RTUxMmT57MLMhVq1bhlVdeYaEICk1QpiQ16uc7wTgZtQYodu5PWwlGtMajjkI8ejyD77nn\nHhw4cABnn302Ro4cifnz5wMQMUxVZFlmaelUTkINh52AsMrqF2WMndypFGP3+/0IhUKq5Rh0zVDZ\nAn8NDR48GK2trUgkEvB4PHnxTZfLhWw2y8qgyhkR5TT4+k2Kb/r9fl3LX6xiYfJEIhE0NzeX9Tej\nR4/Gddddx17v3LkT5513Xt57xo4dizfeeAOTJ08GAHz22WesfK0alJn9hLAwFUiShEAggKamJgSD\nwYpn4gmMwWoPhlpTTgyTxnipZWeTp6S5uZlNoOGvdxJEfsQbbz0Ch8Y2ybKMc845B3369IEsy/jn\nP/+JX//61/B4PCyDmqxYAKwtI21bqUkXlWCWaGhdiPLxzXg87kirmz8HlcQwqR5406ZN2LNnDzZu\n3IixY8fmvWfs2LH44x//iM8//xxr167F0KFD9dn4AggLUwNOs8ictj9O2ZdqHvK8RccLG5+dTSO+\nlN9B4khCqLQe6f1KAaUpJ1Q7TJZSZ2cn+vbti1QqxZr5ezweyLKc16mKLC2ySs0s9tcTrftA8U1+\nMLPX6626v7MVLcxMJlNRI4W77roLCxYsgCzLWLhwIXr37o1Vq1YBOFj6MWbMGJx++un4yle+gl69\nemHNmjV6b3oeUs4pTxsdSSQSAA5d+DTvLRQKmblZuiHLMmKxGFvB2RmnnJsvvvgCra2tJb0Zy5cv\nx4033tij6QXfLpH/r5o4Aigqjkrrkv6ROPL/ivVeTqfTiMfjWLt2LWbMmIG+ffsyi5TKT6jsJJlM\nsg45erhpI5EIQqGQ4cKRzWYRj8cruh4psTCbzbJs+kqIxWKWSNSLRqNskXb++edj8+bNlhPychEW\npgpKC8xpFpndL1onUkx4stkssxqTySQOHDjAWrKRZVdIHNX+8d9Jr3nLkVox8l2nyk1oo+43breb\nJc4lk0n4fD74fL6876KWj5SZK8tyVQ98M+/Vaqw7t9utehzKDQlZxcKk7XDSs1MIZh3ipIvYSfui\nbATAixdZjDfffLOqBaZFHAml5QiAWXskVHrF7V0uF6677jpkMhlWvvT5559j4sSJrNykkJuWtqfS\nh78VRKNc+HNNniCv12t7dzVNSbI7QjBVcLqFKbAGvOWYy+XQ3d0NAMxypFIMpXjxYqiWsQocekDx\nwkjfQ5YjdZ8x4mFG81hjsRj27t2LWCzGBgiQm5bcvnxf1nrNpuXjm1QPqzW+aQULUznuzOzt0Qsh\nmAXgT7LTBNNp+2MHSpVzAEBTU5Nq3IoXxxUrVuBHP/pRjwWdUhypJSNZjnxSjlkPL0mScPrpp2P8\n+PFIJpOIRCJ4+umnMX36dPj9/h5uWorj6eGmNQq9xYGy9im+mU6n2QLCDtB8W77XtJ2xx1E3GLWi\nYyEw1sSK54YSXpSTaMhypM4vvHhRTV6ppBz+O7RkrFpxZU8ikM1m8cILL+Dss89Gr1692OJBOfe1\nXDet1a4HPdAa37TKvlfb5ceqCMHUgBUfytXgtP0xEz3LOWimJKGWsbpo0SJ0d3fnZayS69aK4liM\nYDCIe+65B+l0GrFYDP/85z/Rq1cvDBw4ME84K3HT2u1YaKGc+KbZ+68UzMbGRlO3Ry+EYGrAqQLj\npNiCERRKyiGrjrI9K8lY9fv9iMfjLLOU4pt6ZKxaGcqm9Xg8+Pvf/w63240jjzySZdPyixCyLq3s\npjXinqL4ptoCwor3tLAwHY7SzeE0wbTaDVUNtTo3ynIOvs2bHuUcBB9zVDYtV8tYXbFiBW688Ubd\n99dsJEnCd7/7XZZNm0wm8frrr2PcuHFwuVx4+/O30Z3oxgm9TkDQF+zhpq3HjlwulysvvinLMrxe\nr9mbBcCYPrJmIASzDKy4eqsUEhqn7E818OLIu1dJHOmBXKycA6g8Y5U+v6GhAZIksYJ/AJaynoyA\nsmk/+ugj3HnnnfjySV/G1f93NZ559xm4XW60+Frw5NefRP+W/qpuWq/Xa+p1bcZ3K+ObdD1bZQEh\nLMw6Q4iKs1CzHIHS5RxAz045ynFVemSskhBQJmkgEGDlBE60LpVIkoR+/frh0Ucfxf/b8f+w4d0N\niH8SB1qAuBzH1X+9Go/PfJwNqPb7/az8Ip1O2yaDVE8ovgkgrwzFrPpNYWHWOU6zyJziZi61H3pN\n56DXauUc/CgrvTJWKZPU6/UiHo+zvqz1ZHFKkoTdnbsRS8eA7QBGAJk+Gbz5xZvw+/09smmtbGUZ\nhSRJbAKKmXWs/H0SiUTQ2tpq2HfXEiGYKqgVcjtFYJxMJeUc/N8CxcWRfl6sx6reGatutxuhUAip\nVArRaBR33nknbr75Zscs3Erx5cO/jKAniNjkGADALblxfOvxyOVyLAFK2REJONhjmHfTGnW8zF5U\n0zXLxzcpQcro+k0+S7Z///6GfW8tEYJZAOWF7zTBtPv+kHBRs+rOzs6yyznK6bFKAmxGxqokSczt\nmMvl8ty0TudbJ34Lz7z7DP733f+Fx+VBk68J9026D5FIBH6/P683Lc2xBcC8B3w2bT26amkRV21/\n2nIRdZh1jt0FRomd9qdYOQfd+I2NjbpmrCp7rFJSjpkuPpfLhWXLlrE5krIsIxAIONrt6Ha58ci0\nR/DPA/9EJBXB0N5DEfAEWDYtuarJ5UjZonSO9WhmbifULFyz+9NWMgvTqgjBVKFe3F1WpFA5hzJh\nhsQxk8kgHA6zGjQt4qglY9WoHquVoEwKIkvLituqB5IkYXCvwXk/ozidLMuIRqPs5/zYMXLTulwu\nBINBJha1dNPmcjnLCnKx+k29jwV/HISFWYfYySLTghX2p5pyDvp7+keuOB679FgtB6rDVCYFybLs\n2KQgOsd8kg8fO/b7/ew64i0nctNS5ixdS05202qJoarFN/VuAKFM+mlubtbts83EWVdLDbGCwNgd\nPcs5eLF0uVxsWC0vkHbrsVoJyqQgn89X1UgsK8BnHau5xwt1U1Jz07pcrrzrja4zmi1aD27aQij7\n9JLVrtex4MtKRGs8B8NnRPJuPCcJZq33R81yBPQr56Cfk6WRTqeZa04pwHYWDyVqdZh8UlAikUA4\nHEZDQ4MtkoLUso55D0A57nFqekC9aalVISVnkbXpcrlq5qa1QpZsueVL5OLXM77Jb4eIYdYBoqxE\nO6XKOfx+f1XlHFoyVukGTyaTef1d6wleBKyYFEQueLWyHP5aqcYDwAuAMsZLI8RKuWmdtsjSQi3j\nm5Qf4ATq64lSBU4TzEr3h48lVTqdo5YZq7zV4PV6EQgEHPXw09JL1ipJQcqGDnytJF0rtRInPsZb\nzE2rHCFGY9YqdU1awcKsZoGkV3zT7ONQK4RglkG9CWaxco5qp3PUMmOVBCMej9vKPaknRicF8a5V\nuk6AQ4sctUbyRqDVTUveCuWCy8nZx8VQm0NaSYN7uuedcgyFYBZAKShOOeGFUJZz8OKoVs6h/NtK\nGpDXMmNVkiRLuycrpdxespQUROUXeiQFKbObleeRLHurJFdV6qYl16Sd3LR6WnbVxDeV+R9OQQim\nRpzkkiVBy2QyiMViquUcwWCwaDkHYE6P1XKhGz6RSCASidSttUkiUG5SUKmSDrIerSKOxSjXTcu7\nJimxqtSCy4muyHLjm/yzIJvNOup4CMEsgJqFaVfBVLMcKdZRaTkHocxYNarHajlIksQEgm9kbkdr\ns5p5mHxSUCKRULW6Ky3psBPlumkbGhps46atpWCXG9+UJAnhcBihUKgm22MGQjALQAJpt7KSUuUc\ngUCArRQzmQwaGhrY3+qRsWrlh6rH40FjY6Pq2Kx6gre6qUsScPC8k2uVXJNmxB2NoJSbNpvNIpVK\nsS5TRnXIsQOl4ptO7SMLCMHUjBUFU62cgxJmSpVz0N/zMUe79lgtB94tF4vFWDKMXba/UuuyUEkH\n30KOajnrSQSKuWn5EWK0kOAtrHQ6zRYVhNkuWaO+v1h8k8dJNZiAEEzNmC2YtSjnkGUZyWSSxbJq\nkbFqVdxud5616bR+rGolHbyLnC/poGshkUggm83avlNQJVTqpk0kEnl1wPWGWnyTr392UpcfQAhm\nQcxeJVZbzgEUz1ilUVGJRAKJRIL9rV17rFaC0aUXeqAWw6y2pKOapCAnUYmb1uv1IplMMgvLbMyy\ncPn4ZiKRQC6XQyqVEi7ZesGoTj/VlHPQ3wOlyzmKZazSNgSDQUvc9Eaj7MdqdWuTF0a1kg7qqVvu\n9mtJCqoHqnHTAmA/r0doIZFKpXDjjTfigw8+wOjRo83eLN2QclYLzFmETObgcGI+kH3gwAG0tbVV\n/CBV1q+plXOQBVlpOQf9vljGKj8CiaB6RSd2xymHbDaLWCwGAKZbm8VayfELqlp4AXK5HJLJJFKp\nlOUXELWEQiHxeJx5dlwuF7LZbN7cTbpOYrEY62Zk9DHL5XKIRqMIhUKmnitZlllt6/XXX4/169dj\nyZIl+MEPfoBgMGjadumBEMwCKAUTAL744ouyBFPNcgQONQcnkdSznEOZsUrfoWWbs9ksEokEy56t\nt16sBB/TM3L6R7GSDjqPRmcfZzIZxONxAOYvIMyk0AKChJM8NslkEsFgkPU0NjKblgTT7JghCWYg\nEMD9998PSZLQ0dGBzz77DM8++6yp21Yt9eVrKQO1C7yYW5biG9SO7cCBA+jq6mJumkAggJaWFrS1\ntaGpqYnFiFwuFxNDemDyIkuuVoKEPBaLIRwOIxKJIJVKAQB8Ph+amprQ3NyMUCjESki03qzkkvP7\n/YjFYiwWUW9QTK+xsRGZTAaRSISV5+gFWS4U/wqHwwiHw+x6oe+nc0mzG2+//XZdt6MU5K72+XyI\nRqN1fU0EAgGEQiGk02l2TbhcLnYfk8WZy+UQCAQQCASY54YWP7XE7Axdte2IRCI49thj8fvf/x7r\n16/X/BmbNm3C0KFDMXjwYKxcubLH759//nm0tLRg5MiRGDlyJJYvX67b9hejPk2ICiHBrKacgx42\nZvRY1QqtiuPxOCKRCILBYF1aFsoRUJW6q5WuVVoI2SXBSiQFHYJilhTvpmcCucldLpdqU3eqV6yH\nDORCdZjlDJFetGgRVq1ahQEDBmDy5MmYPXs2evfunfeeM888E3/5y1/023ANCMEsAV/OkcvlEA6H\ndZ/OYWSPVa3wYuGUwcSVoBQLaq9XzF1dqqSjmq5HldZh6kE9JgWphTxyuRyLUdK59vl8Pdy0lE1L\nmbdGND2w2v1ZSZZsV1cXAGDChAkAgEmTJqGjowNTpkzJe58Zng4hmEWIx+NIJpNMtICDcZxCwfxK\nMlaViRxWsjZ4saCes/UaxypkbQI9s1YB86d01BKrjA/Tm2J9c4vN66RSCj6b1ufz9cimJbd6NWOz\nSm2/FSCLG6hMMLdt24YhQ4aw18OGDUN7e3ueYEqShJdffhkjRozAOeecg+9973s47rjj9NmBIgjB\nLDMPhRkAACAASURBVAB1PgkGg+zmoJuHj2WW04BcLcuxljMB9cLlctmq7KJW0IOA0uYpdqxHSYdW\nquklqyd2rGFVoqx3ViZZab03izU9oE44xdy0lYzNsguRSKQsV6xWRo0ahb1798Lr9eKhhx7CokWL\n8MQTT+j+PUqEYBZBKY4AWBaqljZydumxqgVaQFBsU5ZlFqd1IqVKOgKBAMucrGbgsN1Rjg+zalmS\nMqlOGfaoNiegWNMDct2ScFKikNJNW21bQqsm/ZRrYY4ePRrXXXcde71z506cd955ee/hP/OKK67A\njTfeiGQyCb/fX8WWl0YIZgGy2SxWr16Niy++mDUop0LmbDYLr9fb44EK2LfHqlboAem0BualSjoK\nWRs+n8/Q0WFWsC6VWDEpqFQcuVZhj2JNDwq5aalbUDqd1t1Nawa8YMbj8bKnlbS0tAA4mCnbv39/\nbNy4EUuXLs17zyeffIIjjjgCkiTh8ccfx/Dhw2suloAQzIK4XC40Nzdj6tSp+MlPfoJEIoE9e/Zg\n1qxZSCaTzLLgpzo4oceqFuzewLxYIgdvbWjZH0lyzuiwajErKajU+TQjjlyumzYQCFTtprWKhamk\nkuN+1113YcGCBZBlGQsXLkTv3r2xatUqAMCCBQvw2GOP4Te/+Q08Hg+GDx+OO+64Q+/NVkU0LlDh\n448/xvr167Ft2za89NJL2LNnD4YNG4bx48fj9ttvZzVXJBT1mF5P8AXdgUCgx7QCsylV0kH/9LA2\nlMeiFpa3VWKYpahVp6BSiTl6nk+9KNX0gC9LoR6ssiyX7ablGwaYSTQaZbH8KVOmYNOmTZY5F9Ui\nLEwV9u3bhx07duC0007DwoULMWTIEPz0pz/F9u3bceDAAfTp04e56Sh71IpxGyMga5Nim3QszLCw\n6GGqTOQwapC13S1vPdErKUivxBwzqdZNS+7uUljVwrTiNlWKsDA1ksvlsHnzZixZsgTXX389zjvv\nPJb8Q5086rXAn6DpJ0ZZ3sUepkrr0WhEL9ZD8K0GiyUFlUrMKdZK0i4U6k1Lx4iy7mkRQE3dtbhp\nU6kUcrmcIbG8YkQiERa3nDJlCjZv3mzq9uiJEMwyOXDgAK6++mq0trZi+fLlaGhoyHsg1PvDEUBe\n3IZcM9WijFMpH6b0QLVaHFn0Yj0E9Srm63mLJebUqrG8FdDSm5aOg1Y3bTKZZAlYZsE3gE+n05g5\ncyaef/5507ZHb+y7VDOJtrY2rF27FmPHjsWFF16I119/Pa/3KBW200qxHvF4PCztOxwOl92HlcSR\nevNGIhF0d3czS97j8SAYDKK5uRmNjY2smYQVH66UVez1ehGNRpFMJqsqMF+xYoWOW2ccZD2ShUj9\nc6l3rt/vR1NTE5qamlg/Y6u7WquhWG9acsFSf2qyGoPBIDKZDAsDWRlJkvIsTacgYpgVIEkSLr/8\ncowfPx5z587F9OnTMX/+fFbgTyUX9ZwQJEmS5j6slZZ02AWqYbVzkX85FKthpbgdnzhH47Dsen6r\nodKmB3y3IH4Eodnuaj6OGg6HTZ+cojfCJVslqVQKy5Ytw2uvvYaVK1fiiCOOAHDILWnVQm4j4ceG\n0bEoVAJgZtzRCMwaHVZLSiXmFBtLJlzWhyjXTSvLMlKpFHPTUqzT7PpXqr1844038MADD+D+++83\nbXv0RliYVeLz+bBixQo8//zzuPjii3HjjTfi3HPPZW7Jep74wVsa9JofsGulvrlGwRf507Vhp9mj\nWjvmaF3w2KVTkBFozaalsARdR2Rt0n1lJk63MJ25jDcYSZJw9tln45lnnsEjjzyCH/7wh0gkEqyo\n3e/36xK/sjL0IKX5fxR3pHgLuZ4aGxtZsocZA5GtgnL2aDwe13RtGB3DpHNK3YzUzikfS6bZkOXA\n5wDQRCBZlmu0R9aHjmsgEEAsFkMsFkMul4PX62Wx3Uwmw44RPWMo5mlm/kS1bfGsjhBMHenVqxd+\n//vf4+STT8bUqVOxa9cuxyYE8a7FaDSaN8yaYnbNzc15SRzknqPX9TyQGOgpFLUYVF0OyqHW3d3d\neQPKa52YQ4uIhoYGJBIJx9wrlUC9aZuamuByuRCJRPKGi5O7lqxOigFTQpUVFudOFEwRw6wRu3fv\nxvz58zFr1izMmTOH1VpRjMJOCUGlSjpICMst6aB4Ry6Xq/v4FQBmnRvhliyVmGN2xxxRx5oPjRDL\nZrPMTauc1UutCCVJQiqVQiaTMTzbmO82tHbtWuRyOVx55ZWGfLcR1IWFuW7dOpx44olwu9145ZVX\nCr5v06ZNGDp0KAYPHoyVK1dW9Z1DhgzBX//6V3z44Ye49NJLsX//fhajCAaDiMfjmt1wRlJJSQe5\n4cq9Kcmi0Kvkwu6QRQFAd7ekmkcgGo1ClmW4XC4EAgHmEaAyHTPd5XzZBcU3KRZej6i5abPZLLvv\nSDTpGAUCAfj9fqRSKSa0RhMOh2sy2stM7JFpUCUnnXQS1q9fjwULFhR936JFi7Bq1SoMGDAAkydP\nxuzZs9G7d++Kv9fn8+FnP/sZnn32WcyaNQs333wzzj77bEslBCmtDBpJRlZGrUs66m1sWCmUzdwr\naWCud2KOmYikoJ54vd68khy6TymOSfFluof50q5aW+rKGKbTkn7qQjD56d2F6OrqAgBMmDABADBp\n0iR0dHTkTfmuBEmScO6552LEiBFYsGABnn32Wdx8882sENnIocy8a5V35fBTOszqfUoPxlQq5aix\nYZXi8XjQ2NjIkm2KHQ+zRlkZhRXHh9WaYosevoNWMpnM+7myxIfPpuVnb9aq7lUpmMLCdCjbtm3L\nE9Zhw4ahvb29asEkevfujXXr1uH+++/H1KlTsXLlSnzpS19iFy9lHuolWKWsDFqpW+lBylub9d68\nHChsbSoXPgDyxNGpTQDIhU+9WI0aH2YEygks1PWn1KLH7Xb3aHrAW5bKEWLUm5Zmb9by2Dkx6ccx\ngjlx4kTs27evx89vv/12TJ061YQt6onL5cKCBQtwxhlnYP78+fjmN7+J73znO1V3CCo28khtorzV\ncbvdaGxsdNyQ6nKhRQ8tdKiFWj3XsQKHrG+6PuyWFFTKeizH00PZtFSPyR8Pn8/HyoLITUvdgmrl\npuW7DQnBtDAbN26s6u9Hjx6N6667jr3euXMnzjvvvGo3S5Vhw4bh2WefxQ033IDLLrsMd999N3r1\n6sVWh6VGhqmJI3DIyggEArYRx0Iox4bVg7VZ7LzSgxQASxaj5uT1iF7jw4xAzXrk8wT0WPSUanpg\nhptWCKYDKJSF2dLSAuBgpmz//v2xceNGLF26tGbb4ff7cccdd2DDhg2YOXMmbr31Vpxxxhk9EoIa\nGhoAoGBJB2852lkgC6GM5TkldkVWBv8g1ZqY09jYyGK9drOu9MZqSUFaEq5qufAr1JtWi5uWetNW\ns+hweuOCuqjDXL9+PRYuXIj9+/ejpaUFI0eOxNNPP42PP/4Y8+bNw5NPPgkAeOGFF3DllVdClmUs\nXLgQCxcuNGT7Pv30U8ybNw+DBw/GrFmz8Oqrr+KEE07AsGHD2AVILjirTZM3EopdkVvJTsdA+RDV\nY5SV6MOaj3J8mBELq0J9dJW9ka1Uy1qoN206nUYqlWIWaCXbTHXEHo8H559/Pl544QVHXZd1IZhW\n5umnn8aLL76IrVu34qWXXkJTUxPGjBmDK6+8EmeccQYkSUI8HmcJIPXqgiP4IdXBYND03plqKBs9\nKBNz+NmdenxXKpVCMpl0TDP3aqGFlcvl0vWeUTZ7SKfTthhwXajpgXL4Ad9cJZPJVOSmjcVizEo9\n//zzsXnzZkddj9Z72tQZGzZsQEtLCxYvXoxHHnkEn3zyCa666ip8+OGHcLlcYmSYAlo4UKzXSi44\n/h//EK1lYo5ydJjZdb1WQK+koFLWI1n1VheEUm5afv+qddOSR8ypdpiwMC1IIpHAD37wA/zrX//C\nXXfdhba2NgBiZJiSXC7HOhAZNfGjUGKOVVxwThsdVi1a3dZ2tR7LpZiblvaZQgW8m5aSk0pdT9Fo\nlIVLLrjgArz44osG7ZkxCMG0KLlcDk899RRuueUWrFixAuPGjWMrNxKJerckiFr1YC2VmGPVhyj1\n6OVdcPUMv5CgawSAZWOPRqDmpgUOLsr5EAK5aVOpFNLpdEk3bSQSQSgUAgBMmTIFmzdvNmaHDEII\npsXZt28f5s2bhxNPPBE33HADc8dSj8h6z5IkeJGodCFRi8QcsyDrwKhm7laGX/jw46+svvCpNfw1\nQm5aEki6B4BDwkluWgCqbtpcLodoNMo6dl100UV47rnnDN+vWlJfV4gN6du3L/785z/jyCOPxLRp\n0/Duu+8CQN7IsGg0WrdjkAh+vqSWsWH8KKtoNJo3yqrQeDI7WRxU0M7PmDRzdJiR0LlVNponC4nv\ncEXx33oTS6D4CDFqaEBCSfFNyqFIJBIF7zFJkhxZUgKIpB9b4HK5sGjRIpx55pmYP38+5syZg9mz\nZ4uEIAV8z9F4PM7iKS6Xq2hijhXbBOoFLSSoswvfh9QJlIo9er1e1UxZn89n205BeqOl6QFfhkJl\nI9T0wOv1wuv15tVghsNhRwpm/S2rbMyIESPw7LPPYseOHbjiiivQ2dmZNzKMhu7Wu5edVs600iXr\nkfpz0igrGk9mp7aBlUKWhCRJuo8OMxKlZ0A5pqyhoaHHuVWzHsX4sJ4UGiFGCT9ut5uJZy6Xg9/v\nZxm1lFdBOHFSCSAE03Y0NDTg17/+Nb75zW9ixowZ2LJlC4BDqfTAwYu1Xm5+Sv1PJpOIxWIIh8Po\n7u5mmX3UJpD+38nNyUtBJTn84srKrnyyHpWzWSlZhVzOTU1NCIVCFZ1b6hTk8/kQjUYtOaPWSMpx\n01J/WrJOaZ5tNpt1rIUpkn5szMcff4y5c+di1KhR+MEPfsAy3ZycEFQsMadQJyS+uN+Jx6QS+AYQ\nVmlurzZ+DkDeua2lJ8CMTkFWp1A2LVmawKGkIIob//a3v8XHH3+M1tZW3HLLLWZuvu4IwbQ5mUwG\nd955J5566in8+te/xsCBAwEcvPljsRgA2HYgc7EHaCUdc8h1JLomHaJWXXFKUWjCjtq5NVrIzTom\nVkVLNi3FL9PpNN5//30sXLgQH3zwAdauXYszzzzT7F3QDfeyZcuWmb0RgspxuVwYN24cvvzlL+PK\nK69EIBDAiSeeCJfLxQLxdPNbuWZTmfpPWXjZbJb10qWYCSX2lGtt0DGhEhRJkhyZ6FMOLpeLjYGq\n5TGhxY8sy0gmkyy5JJfL5TUlp6xVMwcKKI8JAMfHuIshSRI7R8pjwtdqUpu9Pn36AABaW1vxi1/8\nAm+99RYuvPBCM3dBN+p76VQDwuEwpk+fjv79+2PGjBmIRCKq7xs4cCCGDx+OkSNHYsyYMVV/7ymn\nnIK//vWv2LJlCxYsWIDu7m5LJwRRA2i11P9CyRt6PLT4ZA+Ke1o5jmcE/DFJpVJVlykZEXusNSIp\nqCf880SWZYTDYUQiEVbGBYDFNyORCM466yy88cYb+Pa3v63p8+fMmYM+ffrgpJNOKvieG264AYMG\nDcIpp5yC3bt367Jf5SAEU2d+85vfoH///nj77bfRr18/3HfffarvkyQJzz//PHbs2IGtW7fq8t2h\nUAirVq3CzJkzMX36dPa5yoQgo+vx1BJzwuEwSyagmtLm5mZW81jrBygNqabEBrtmjeoJJcB4vV6W\n7KFlgaWWuUrHtFBWsl0se5EU1LOulRZUVK713//93+ju7kYwGMS+ffuwbt06PProo/j0008RCoUw\nbtw4Td/z3e9+F88880zB32/duhWbN2/G9u3bsWTJEixZskSvXdSMiGHqzKxZs3DTTTdhxIgReOWV\nV/CTn/wE69at6/G+Y489Ftu3b8dhhx1Wk+348MMPccUVV+DUU0/Ftddea2hCUCWJOWZi57FhtaJY\nD1Zlu0A6v3RuzYo91pp6SQriz2+xnrrJZBI7duzAAw88gCeffBKtra047bTTMH78eIwaNQojRoxg\nbfK0smfPHkydOhWvvfZaj9+tXLkSmUwGixcvBgAcd9xxeOedd3TZZ60IC1Nntm3bhiFDhgAAhgwZ\nUtB6lCQJ55xzDmbMmIG//OUvum9Hv3798NRTTyEYDGLmzJnYu3cvAP07BJXqmEPWBXXMsWLNI1ng\ndq9R1BOltRmNRvNqWmVZ7nF+7WY9lgs1gaAQhxM6bCm9P/z55UMjoVAI0WgUGzZswNKlSzFlyhRM\nnz4d//M//4Pp06fjiSeewLBhw/CPf/wDo0ePxvjx48sWy1Js3boVw4YNY68PP/xwwwVTdPqpgIkT\nJ2Lfvn09fr5ixQrN7pqXXnoJRx55JHbt2oWpU6dizJgx6Nu3r67b6Xa7cf311+Pss8/GZZddhu99\n73uYOXNmxR2CtHZVsaN1wY8Ni8fjSKfTddl/Vc07IEkSi99ZpQTFTPQaH2YGfGYr/Zf3/tAoulwu\nh7feegvt7e1ob2/HW2+9hV69euG0007DhRdeiFtuuQWhUChvn5966in8+c9/rllyIWVW8xh9zIVL\nVme+/vWv46abbsLIkSPxt7/9DT/5yU/w2GOPFf2b73//+xg6dCjmzZtXs+0Kh8NYvHgx0uk0fv7z\nn7Oi4mLuSLV2coDzJzqYMTbMDEoNuuYzkcXoMHW0jg8zg2ILXN59DgDxeByvvPIKOjo60N7eji++\n+AInnHACTjvtNJx++ukYOnSoIftWyiWbTqfxn//5nwDMcck680lgImPHjsXq1avx85//HKtXr8ap\np57a4z2xWAyZTAZNTU347LPPsGHDBnYR1IqmpiY88MADeOyxxzBt2jT88pe/xCmnnMJWy/F4HOFw\nmJWi8KOsqKdkvUx0kCQpr/+qU6Z9FIotezyekoOulX16yTPh1MWEVsh1TSEOM6+VQnXLJI58Pfan\nn36Kl19+Ge3t7fj73/8Ol8uFU045BePHj8e8efPQp08fy13vY8eOxfe//3185zvfwYYNGzB06FDD\nt0FYmDoTDodx6aWXYseOHRg1ahTWrFmDxsZGfPzxx5g3bx6efPJJvPvuu5g5cyYA4LDDDsO3vvUt\nzJkzx7BtfP/993HppZfi+OOPRygUwq5du/Dwww8zVww9PJ1oPZaLXWdLlmM9VkKtZpDaGaOTgoot\ngPhxdJlMBm+88QZzr7733nvo06cPsx5POeUUS5zD2bNn44UXXsD+/fvRp08f3HLLLSyfYMGCBQCA\nH/7wh/jDH/6AXr16Yc2aNYaLphDMOqKjowM///nP0d7eDlmW0bdvX/Tv3x+XXHIJpk2bBo/H44gO\nQXpjB3dkqYdnLTKT6yVrtFxq0SmId6/SOc5ms3niSC7TaDSKbdu2YcuWLdi6dSsikQiGDh2K8ePH\nY/z48Rg8eLC4rytECGYd8dZbb+Fvf/sbxo4di2OPPRaSJGHLli1YvHgxFi9ejGnTprF4VTKZZGN+\nxIPwIHoMqdaDWluP5ULWptNGh1UDfw9VkhRE55hP0FEbZg4cLCHbsmULOjo68I9//AOBQIBlqo4b\nNw69evUS50QnhGAK0N3djYULF8LtduOnP/0pSwcX9Yk94a1No7Ij7VDXyjdzF4usQ2hNClKrbVWr\nfZRlGa+99hra29vR0dGBvXv3ol+/fhg3bhzGjx+PESNGwO/3G7mLdYUQTAGAgw+83//+97j77rtx\nxx13YOTIkezn9ZAxWi6861rP7Ei+py6JJMWVK2k4bzT8IouadNc7/CLL6/XC7/f3KO8Aek5lAYCu\nri5s3boV7e3t2L59O+LxOE466STmXh04cKA4xgYiBFOQx3vvvYe5c+fi3HPPxTXXXMNuXHK7WTWG\nZwZ6jA1Tpv1b0XosFyuODjMTEsd0Os2GL2ezWdYCkm9g/t5777HSjjfeeAONjY0YO3YsTj/9dJx6\n6qlobm6u62NpNkIwBT1Ip9NYsWIFtmzZgnvuuQdHHnkkAGeMDKsFWseGKa1HZeKG1a3HcqnHMVla\nah9zuRx+8Ytf4NVXX8V3v/tdvP322+jo6MAnn3yCgQMHMvfq8OHDhUfHYgjBFBTkpZdewve//30s\nWbIEF1xwgUgIKgJ/XMiqUquLs7v1WC5qx8VJ+1uq9pF3r37++efMenz11VfhdruxefNmfOMb38Cy\nZcswYMAARx0bJyIEU1CUzs5OXHPNNQiFQlixYgWCwSAAkRCkhB9plUql2M+daj2Wi5U74pSDltpH\nWlhSa7mOjg689dZbaGtrY83Jx4wZg1AohDfffBNXXnklevfuXbIjmMB8hGAKSpLL5bBmzRrce++9\nuPPOOzF8+HD283pNCCoWe3S73eznwgo/hB4xXyMpp/YxkUjglVdeYc0BvvjiCwwePBjjxo0r2Vou\nl8vho48+Qr9+/YzcvZqxbt06LFu2DLt378a2bdswatQo1fdt2rQJCxYsQDqdxsKFC/Ef//EfBm9p\n+QjBFGjmnXfewdy5c3HBBRfgqquuYjEppycEFYo9KrMalTE6kTGqjlWtTWXtIzWeV6t9/PTTT7Fl\nyxbmXqXWchR/tGJrOaPYvXs3XC4XFixYgDvuuKOgYI4cORJ33303BgwYgMmTJ+PFF19E7969Dd7a\n8qgfk0BQNccddxz+93//F7feeisuvvhi3HPPPejTpw+8Xi/cbjdisRjS6bTtE4JKWY/Feq7yUJ/e\nRCJR1lQYp0P9V1OpFKLRqGkLrWK1j3zv5Ewmg127drHmAO+++y6OOOIIjBs3DhdddBF+9rOfWaK1\nnFWg8YbF6OrqAgBMmDABADBp0iR0dHRgypQpNd22ahGCKSgLr9eLW2+9FZs3b8Yll1yC66+/HpMn\nT2Yjw1KpFCKRCAKBAHw+n9mbW5JCbjd+3FE1Tecpc9br9SIWi7HC/np/uEqSBL/fD6/Xa0gzd7X6\nVuBQ7SMt+oCDreU6OjpYa7nu7m4MGzYM48aNw9KlS0VrOR3g5wYDwLBhw9De3i4EU6A/Wnz/N9xw\nA/7whz+gra0NjzzyiKZVn1YkScKECRPwzDPP4Oqrr8bGjRuxfPlyNDQ0MIGhmZJWEwe+YFyt5ZhW\n67FcPB4Pmpqa2FSYYDBYVzHfQtBQZr0nw2id+wgcai23detW/OMf/4Df78fo0aMxbtw4LF68WLSW\nU6HQTODbb78dU6dONWGLjEHcsTZk0aJFWLVqFfP9z549O8/3v3XrVmzevBnbt2/Hhg0bsGTJEjzx\nxBO6b0dbWxvWrl2Lhx56CBdeeCH+67/+CyeeeGLeyDAzx0CVsh59Pp+h9YFOHRtWLfzoMN59rfWa\nocHCvEAW8hLIsozXX3+duVf37t2Lo48+GuPGjcNll10mWstpZOPGjVX9/ejRo3Hdddex1zt37sR5\n551X7WbVHCGYNkOL77+jowOzZs1Cr169MHv2bNx000012x5JknD55Zdj/PjxmDt3LmbMmIF58+b1\nsByMiFOZZT2WC7n/KhEHJ6PV2ixU+0gCyScSUWu5jo4ObNu2jbWWGzduHH72s5+J1nI1plBOaUtL\nC4CD3rL+/ftj48aNWLp0qZGbVhHiLrUZWnz/W7duxbe//W32+vDDD8c777yD4447rmbbNXjwYGzc\nuBHLli3D7NmzsXLlShxxxBE1SwiymvVYLiQOqVTKsAWFXfB6vWxQdTgcZqKpdK9S7JHc/rlcDnv2\n7GG1jzt37mSt5c466yxcf/31aGlpEce4xqxfvx4LFy7E/v37MWXKFIwcORJPP/103kxgAP+/vbuP\nqbr8/zj+BESRUAPS0ImAOBPyBpHbA/Ql19J/1CxsI5UVImBNDGIq0zRaZDMVFavp1jRDN7xZptYg\n3UKzhAMo3uMU581UFKwheRDP4fD7w53Pj3uOCef2/dj84+DHeXn87LzPdX2u9+ti48aNpKSkoNVq\nSUtLs/gdsiAF0yYZlqhaM8WHRP/+/cnJyaG4uJh3332XFStW8MYbb/TKhiBrmT0+K8NSpGH52pzH\nhlmC1l+EDBobG5V8Wk9PTyUcoKmpicrKSqX30RAtFxkZyaJFiyRazkxmz57N7NmzO/x8xIgRSrEE\n+N///selS5dMObTnJn2YVqa+vp7Y2FhOnz4NwOLFi5k+fXqbGWZeXh46nY709HTgaTtIdXW1Scf5\n4MEDUlNT8fLyIjs7GxcXF+BpD55Go+k2Iai72WP7WDlbYQ2HVPeFznofgQ7JOcXFxSQlJZGcnIxG\no6GiooLm5maCgoKIjIwkOjqakSNH2sV7JsxHCqYVMjT8jho1iunTp3do+FWr1WRkZPDzzz9TVFTE\n7t27+2TTT0/0ej3ff/89O3bsIC8vT1lKbp8Q1DoZp7MPTcMve/gw7KtjwyxFZ9FynX0R0uv1XLly\nRZk9Xr58mZEjR3L8+HECAgLYvn27cgi6EKYiBdMKHTt2jNTUVGXtPy0tja1btwKQkpICwPLlyyko\nKMDDw4P8/HwCAgLMNt6qqiqSk5OJi4vj/fff5+rVqzQ1NTFmzBilH87wTMoWZ4/Pytoi5LrSXe9j\nd9FypaWlPHjwgDFjxijRcoGBgcpGqezsbH766SfOnz8vS67CpKRgij7V0NCgtLns3LmTuro6hgwZ\nQmpqKh9++CEODg5KWLm1JwT1ttYRctbw3nT3nLn9FyFDtFxpaSmVlZUATJkyhaioKFQqFV5eXt1+\nSXj48CGDBw82yb/LVBoaGpg3bx6nT58mODiY/Px83NzcOlzn6+vL4MGDcXJywtnZGbVabYbR2icp\nmKJPqVQqHB0diYyMJCIiAp1Ox/r161m9ejWxsbHK7kbDjMpaEoJMxVKPx+qp97F1vm5zczNVVVVK\n9qohWs7w7DEkJET6UYG1a9dy69Yt1q1bxyeffIKvry+ZmZkdrvPz86OiogIPDw8zjNK+ScEUfaql\npaXDB2FdXR3Jycn4+PiwatUqpVHcmA1B9srw3pjrMOaeeh9bn/v46NEjysvLKSkpQa1WU19fQgcI\njwAAC9NJREFUr0TLRUVFMXbsWIufLZtDXFwcK1euJCgoiFOnTrFmzRr27t3b4To/Pz/Ky8vx9PQ0\nwyjtmxRMYRZ6vZ5t27aRn59PXl4er7zyCoDSPqDVaiU+rp32s82+nIl3FUDffvcqwO3bt5Xl1bNn\nz9K/f39CQ0OJiooiKipKouWM5OPjw+XLl3FxcUGj0RAQEMCNGzc6XDd69GgGDRqEn58fiYmJzJw5\n0wyjtU/yaWQBysrKSEpKQq1Wo9PpCA8PZ8+ePQQGBpp7aH3G0dGR1NRUYmJiSE1NJT4+noSEBGUG\n1a9fP2nob8fBwQEXFxelb1On0/XKsWE9nfs4YMAApUDqdDrOnTunbM65efOmEi2XkJDA5MmTJVqu\nG11lsObk5HSZitPen3/+yfDhw7l06RIzZswgLCwMLy+v3h6q6ITMMC3Ep59+yuPHj2lsbMTb25tl\ny5aZe0gm09TURFZWFjdu3GDTpk3Ks5nWLRbWsOnFlFrPxJ/12DBjeh8N7/XDhw/bRMtpNBrGjx+v\nzB79/Pzk/6WXvPPOO6xcuZLJkydTUVHBmjVr2LdvX7d/JiMjg4CAABYuXGiiUdo3KZgWQqvVEhIS\nwsCBAzl58qTdzahaWlooKipi1apVfP7558TExMiGICPodDo0Go2SodrZffMsvY83btxQllcvXrzI\nCy+8QFhYGNHR0UREREi0XB8ybPpZu3YtmZmZ+Pn5ddj0o9FoaG5uZtCgQdTW1hIbG0thYSHe3t5m\nGrV9kYJpIe7evUtMTAwuLi6o1WpcXV3NPSSzuH//PgsXLmTs2LGsWLFCKZCyIahrhiAIw3Fqjo6O\nPfY+Gtp5KisrKS0tpaSkhJqaGnx8fFCpVKhUKoKCguQZsgl11VbSOoP12rVrvP322wB4enoyd+5c\nEhMTzTxy+yEF00LMnDmT9957j2vXrnH37l3y8vLMPSSz0ev1fPvtt+zZs4ctW7YwZswYQDYEdab1\n8qpWq0Wv1ys/N+TSGpZM//77b6U4njp1Cp1Ox6RJk5Tdq97e3vJFRIhuSMG0ADt37uTQoUPs3bsX\nvV6PSqXiq6++IjY21txDM6uzZ8+yaNEiEhISmDt3rvLBr9VqaWxstLsNQd31PrZ+9lheXk5ycjKf\nffYZDQ0NSrScu7s7ERERREVFERYWhpubm928d0L0BimYwqI9fvyYpUuXUlNTQ25uLu7u7sDTWajh\nFAtb3RDUuvex/fJq+97H9tFyAL///jsxMTHk5OQQFBRkc7m0QpiaFExh8VpaWvj111/Jzs4mJycH\nlUplkxuCeup9bL28WltbqyTntI6WMyyvenl5cf/+fZKTk7l79y4nT56UginEc5KCKXp0/PhxUlJS\n0Ol0pKWlsXjx4ja/X1xczKxZsxg9ejTw/9vje1tNTQ1JSUlMmDCB5cuXK60U1rghqKcDsFuHAxii\n5Qwnd1RXVyvRclFRUcru6q6OSrtw4QLjx483w7+yb/R0PwJkZWVRUFCAu7s7u3btanPouhD/lRRM\n0SPDcWI+Pj5Mmzatw3FixcXFbNiwgYMHD/b5WPR6PZs3b+bAgQNs2bJFKdKWviGoq2i5znofNRoN\n5eXlnDx5ErVazcOHDxk3bpxycoe9R8v1dD8ajrc7ePAgRUVF7Nq1yyzH2wnbY1mfKsLi1NfXA/Da\na68B8Oabb1JaWtrmwGrA6JSS5+Xo6MjHH39MbGwsycnJJCYmEh8fj4ODg0UlBHXX++js7Ky0f7S0\ntHD79m3l2eOZM2eUaDmVSkVaWhqenp5WMWs2BWPux9LSUuLi4vDw8CA+Pr5PVjuEfZKCKbpVVlbW\nZjkrMDCQkpKSNh9QDg4O/PXXXwQFBTF16lQ++ugj/P39+3RcQUFBHD16lMzMTI4ePUpubi5DhgzB\n2dkZJycnGhsbefTokUk2BLU/97F9tJyLi4vS+6jT6Th//rwye7x58ybDhw9HpVIxf/58NmzYINFy\n3TDmflSr1cyfP195PXToUKqrq/v8nhS2TwqmeG7BwcHcunULZ2dnfvjhB5YsWWKSJTBXV1e++eYb\nDh48yKxZs1izZg2RkZE4Ojri6urKkydP+Pfff3t9Q1D7aLn25z4OGDCgTbTciRMnKCkpaRMtp1Kp\n+PLLLyVarg8Y2m9akxm66A3yDFN0q76+ntjYWE6fPg3A4sWLmT59eoclWYOWlha8vLy4efOmSWdK\nd+7cISkpieDgYJYuXao8w3zeDUHG9D52Fi2nVqu5cOECrq6uhIeHExUVRWRkpETLPSdj7se8vDx0\nOh3p6ekA+Pv7U11dbZbxCtsiM0zRrSFDhgBPdyaOGjWKI0eOsHr16jbX3Lt3j2HDhuHg4MChQ4eY\nOHGiyZcVR4wYwaFDh8jNzeWtt95iy5Yt+Pr64uTkhJubG48fP6ahoaHHDUE99T4OHDiwQ7ScYfdq\nTU0No0aNQqVSkZyczKRJk54pFF30zJj7MTw8nIyMDBISEigqKiIgIMAcQxU2SAqm6NHGjRtJSUlB\nq9WSlpbGSy+9xNatWwFISUlh3759fPfdd/Tr14+JEyeyfv16s4zTycmJzMxMXn/9dRYsWEBKSgpz\n5szpdkNQ+9lj697H9sur//zzjxItV1FRoUTLRUZGEh8fL9FyJtLT/WgIiw8JCcHDw4P8/Hwzj1jY\nClmSFTbp0aNHpKeno9FoWLduHYMHD0av1/PkyRO0Wq3yjKulpaXT3ke9Xs/Vq1eV3atVVVW8+OKL\nREREEB0dLdFyQtghKZjCZjU2NpKbm8vWrVsZO3YslZWVfPHFF8ppD2fOnOHKlSvMmzcPrVbbJlqu\nrq4Of39/pffx1VdflaQcIeycFExhc+7cucOcOXM4c+YM48aNY+LEidTV1TFhwgSWLVuGk5MTtbW1\nHD58mI0bN9Lc3MyIESMICwtTouWGDx8us0chRBtSMIXN0Wq1nDhxgtDQUNzc3ICnu2W//vprtm3b\nxrBhwxg6dCiRkZGEhoayf/9+CgsL+fHHH4mJiTHz6IUQlkoKprAr58+fJzAwsEPv4+HDhzl37hxZ\nWVlmGlnfsJQcYCFsgRRMIWyYJeUAC2HtJGJECBvVOnfVx8dHyV1tzx6+M69evZpNmzYpr1esWMHm\nzZvNOCJhjaRgCmGjuspdba11DnBGRobNJuIkJiayc+dO4GkwfkFBQZu8WSGMIQVTCDtmyAEuKysj\nMDCQJUuWmHtIfcLHxwdPT08qKyv57bffCA4Oxt3d3dzDElZGCqYQNio0NJSqqirl9YULF4iIiGhz\nzaBBg3B1dcXZ2ZkFCxZQVlZGU1OTqYdqEklJSWzfvp0dO3aQmJho7uEIKyQFUwgb1Tp39fr16xw5\ncoTw8PA219y7d095hmmuHGBTmT17NoWFhZSXlzNt2jRzD0dYIcmSFcKGWUsOsCk4OzszdepU3N3d\nJZRC/CfSViKEsAt6vZ7g4GAOHDiAr6+vuYcjrJAsyQohbN7FixcJDAxkzpw5UizFfyYzTGGREhMT\n+eWXXxg2bBjnzp3r9JqsrCwKCgpwd3dn165dbVoohBCit8kMU1ikDz74gMLCwi5/X61W88cff1Be\nXk5mZiaZmZkmHJ0Qwh5JwRQWKSYmpts+udLSUuLi4vDw8CA+Pp5Lly6ZcHRCCHskBVNYJbVaTWBg\noPJ66NCh/zmlJjExkZdffpkJEyZ0eU1WVhajR49mypQpbXobhRD2QwqmsEotLS0dMlD/a6uALP8K\nIYwhBVNYpfDwcC5evKi8rq2tVY6oelay/CuEMIYUTGGVwsPD2b9/Pw8ePGD37t0EBAT02d/Vm8u/\nQgjrJUk/wiLFx8dz7Ngx6urq8Pb2Jjs7G61WCzxNqAkLCyM6OpqQkBA8PDzIz8/vs7H05vKvEMJ6\nSR+mEMD169eZMWNGpz2feXl56HQ60tPTAfD395cZphB2SJZkheiBKZd/hRCWS5Zkhd2zpOVfIYTl\nkiVZIYQQwgiyJCuEEEIYQQqmEEIIYQQpmEIIIYQRpGAKIYQQRpCCKYQQQhhBCqYQQghhBCmYQggh\nhBGkYAohhBBG+D9UHuX/4ibttQAAAABJRU5ErkJggg==\n"
}
],
"prompt_number": 9
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"When we change the net magnetization from the steady-state position (red\n",
"circle) to the excited position (green circle), it is like introducing a 90\n",
"deg rotation in the magnetization direction. This is usually called the\n",
"flip angle. This is one of the parameters that you select when doing MR\n",
"imaging."
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"ax = subplot(111, projection='3d')\n",
"ax.scatter(0, 0, 1, color=\"r\")\n",
"ax.scatter(1, 0, 0, color=\"g\")\n",
"axisplot(ax)\n",
"ax.view_init(az0, el0)"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "display_data",
"png": 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Bgwax/29vb8f69etx6623mrhF1oRfOFKCmhmWDl3j5Xzv/v37cfvtt+Pmm28G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\n"
}
],
"prompt_number": 10
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"As the net magnetization returns to the steady-state position, the\n",
"distance from the origin decreases, reducing the signal. This is\n",
"illustrated by the collection of green points plotted here that\n",
"illustrate the net magnetization rotating back towards the z-axis."
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"theta = (pi / 2) * (linspace(0, 10, 11) / 10)\n",
"x = cos(theta)\n",
"y = zeros_like(x)\n",
"z = sin(theta)\n",
"ax = subplot(111, projection='3d')\n",
"ax.plot(x, y, z)\n",
"axisplot(ax)\n",
"ax.view_init(az0, el0)"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "display_data",
"png": 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XbCFxjMfjfcQxEAiw7h52i8+oqoq77rorb7DxzTffDFmWAQADBgzA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\nEEKICqhgEkIIISqggkkIIYSo4P+7TkmqPMhYIwAAAABJRU5ErkJggg==\n"
}
],
"prompt_number": 11
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"After receiving an excitation pulse, this component of the MR signal\n",
"decays gradually over time. The spin-lattice decay has been measured\n",
"and, in general, it follows an exponential decay rate.\n",
"\n",
"Specifically, here is the rate of recovery of the T1 magnetization for hydrogen\n",
"molecules in gray matter."
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"T1_gray = 0.88 # Time constant units of S\n",
"t_T1 = arange(0.02, 6, 0.02) # Time in seconds\n",
"Mo = 1 # Set the net magnetization in the steady state to 1 and ignore."
],
"language": "python",
"metadata": {},
"outputs": [],
"prompt_number": 12
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"This is the exponential rate of recovery of the T1 magnetization, after\n",
"it has been set to zero by flipping the net magnetization 90 degrees."
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"MzG_T1 = Mo * (1 - exp(-t_T1 / T1_gray))"
],
"language": "python",
"metadata": {},
"outputs": [],
"prompt_number": 13
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Plotted is a graph we have the magnetization of gray matter as"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"plot(t_T1, MzG_T1)\n",
"xlabel('Time (s)')\n",
"ylabel('Transverse magnetization (T1)');"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "display_data",
"png": 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58iTR0dFs3bqV5s2bs3nzZu8XGWAz8XvvtS50MmaM05WIiEggsm0m3q9fPwYO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\noG5zAAAAAElFTkSuQmCC\n"
}
],
"prompt_number": 14
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The decay rates for various brain tissues (summarized by the parameter T1\n",
"above) differs both with the material and with the level of the B0 field.\n",
"The value T1 = 0.88 seconds above is typical of gray matter at 1.5T.\n",
"\n",
"The T1 value for white matter is slightly smaller. Comparing the two we\n",
"see that white matter recovers slightly faster (has a smaller T1):"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"T1_white = 0.64;\n",
"MzW_T1 = Mo *(1 - exp(-t_T1 / T1_white))\n",
"plot(t_T1, MzG_T1, 'black', label=\"Gray\")\n",
"plot(t_T1, MzW_T1, 'gray', linestyle=\"--\", label=\"White\")\n",
"xlabel('Time (s)')\n",
"ylabel('Transverse magnetization (T1)')\n",
"legend(loc=\"best\");"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "display_data",
"png": 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EOs1QPqtNTExw584d2NvbV3pt165d2LFjh+rJZ3XFnjip6HpPztLSErNnz8bIkSNlL+BV\n5erJkyf4/vvv0bVrV/zwww9YvXo1oqKi8NprrxltAdf13yldwlzprs8//xzPP/98hXVdunSpct3e\nvXufua9XXnmlQgE3MTFBXFycdMHWEIs4aY1CoYCNjY3cYVSSnZ2NVatWwd7eHseOHcPOnTsRHByM\nMWPG6OxQPxGJ8/Lywvnz51VntsnJySgpKcHly5dRVlamWhcbGwtPT88a71/O0QcWcQOiS09QkuOK\n85rw9vZGXl4evvjiCzg4OODatWs4efIkjhw5Ag8PD7nD0xm69Dul65gr3eXm5obi4mJcvnwZAHDm\nzBkMHToUXbt2rbDOwcEBbdq0AQCEhITA1dUVDg4O+Oabb1T72rZtG4YMGQIAqoLfq1cvNG3aFL/9\n9hsAICoqCvPnz0fHjh3x9ttvIyEhQWM/G4s4SS46OhrfffcdCgsL5Q6lSgUFBfj222/h4OCAiIgI\nBAYG4pdffkHPnj3lDo2INKBBgwZwd3dHUFAQgKczQg4ZMgSDBw9WPbgkODi4wln41q1bsWfPHvj5\n+WHFihWIjY2ttN/y90ZFRSEnJweTJk3C48eP4e3tjTFjxuDatWuwsbHBtGnTNPazsYgbELl7coWF\nhTh8+DBOnTqFCRMmwMzMTNZ4/qm4uBg//vgjunbtir179+LYsWPw8/ODk5OT3KHpLLl/p/QJc6Xb\nvLy8VEX37Nmz8PT0xJAhQ1Trzpw5Ay8vL9X2b7zxBhwdHdGnTx8MGjQIJ0+eFDrOgQMHMHHiRIwb\nNw4WFhZ49913cefOHTx69Ej6Hwos4iSR1NRUbN68GQqFAr6+vmjXrp3cIakolUr8/vvvcHZ2hp+f\nH/bt24f//e9/cHV1lTs0ItIST09PnD17FhkZGUhNTUXnzp0xcOBAnD9/HhkZGbh+/XqFM/G/fz60\nadMGSUlJQscJCAjArl27YG1tDWtra9jY2CAvLw9nzpyR/GcCONmLQZGrJ1dUVIRdu3bB29tb5wpj\nVFQU3n77bdy/fx9r1qzhxWo1xD6vOOZKTGBgoGpY+++8vLyqzOE/t69uO3UGDBiArKwsbN68WXXd\ni4WFBdq2bYsff/wRbdu2feYcENV9bpiYmFS4sG3YsGFo1qwZvv/++xrHWBss4lRnDRo0wIIFC3Rq\n+Pzhw4f473//iyNHjuDDDz/EvHnzYGpqKndYREbP29u7RkW4pttXp2HDhnBzc8OaNWvwwQcfqNYP\nHjwYa9aswciRI6t9r1KprPYK9L59+yIiIgKdO3cGAEyZMgUffvghRo0ahREjRgB4+oeIl5eXRh7o\nxOF0AyJnT05XCnhJSQnWrFmDnj17wtLSEjdv3sQbb7xRqYCzfymGeRLHXOk+Ly8vpKamYvDgwap1\nQ4YMQVpaWoWh9H+edSsUCtW6v38PAEuXLsVXX30Fa2tr7Nu3D1ZWVjhx4gROnz6Nrl27okuXLtix\nY4fGfibO2GZAAgMDjXpILzQ0FPPnz4eNjQ02btyIrl27VrutsedKFPMkjrniZ3VtcNpV0qq4uDjk\n5eXB2dlZ7lBUMjIysHz5chw+fBhff/01pk2bxr43kQz4WV1znHaVtObSpUs4cOAAmjZtKncoAJ72\nqXbt2gUnJyeYmJggOjoa06dPZwEnIqPBIm5ANNWTUyqVOH36NM6dO4c5c+bAzs5OI8epiaSkJLzw\nwgv44osvcOjQIWzcuBHW1tbC72f/UgzzJI65IjmwiNMzKZVKHD9+HLdu3cLcuXPRvHlz2ePZvn07\nevfujX79+iE8PBzu7u6yxkREJBf2xOmZMjIycOzYMYwfP172J489ePAA8+bNQ2JiIrZt24bevXvL\nGg8RVcTP6prjhW1kFA4dOgRfX1/MmzcP//3vf9GgQQO5QyKif2jWrBkyMjLkDkOvWFtbIz09vdJ6\nXthmhAyxJ5efn4/58+djyZIlOHToED799FNJCrgh5koTmCdxzBWQnp6umhiluq/Tp0+r3caYvqoq\n4DXBIk46KzIyEn379kVubi4iIyMxcOBAuUMiItIpHE4nldLSUty9excODg6yxqFUKvHDDz/gww8/\nxNq1a/HKK6/IGg8RkbaJ1j3OnU4AgLKyMhw4cAClpaXo3LmzbPda5+bmwtfXF9evX0dISIjsf1AQ\nEekyDqcbkNr25MrKynDo0CEUFhZi4sSJshXwmJgY9O/fH+bm5hov4OxfimGexDFXYpgnabGIG7ny\n+8BzcnIwZcoU1K8vz+CMn58fPD09sXTpUvz0009o2LChLHEQEekT9sSNXEhICK5cuYLZs2fLch94\nWVkZPvzwQ/zyyy84dOiQzj2PnIhIDrxPnISkpKTA3NwcFhYWWj92Tk4OZs6cifT0dOzbtw8tW7bU\negxERLqI94kbodr0mlq2bClLAY+Li8PAgQPRsmVLBAQEaL2Asy8nhnkSx1yJYZ6kxSJOWnf+/Hl4\neHhgwYIF2LRpE2dfIyKqJQ6nk1bt378fCxYswM6dOzFq1Ci5wyEi0kkcTqdKSktLcf36ddn+IFq7\ndi0WL16MEydOsIATEUmARdyAqOs1nThxAleuXNFOMH9TVlaGt956Cz/++CPOnTunE08fY19ODPMk\njrkSwzxJizO2GYlLly4hLi4Or732mlYncykuLsasWbPw4MEDnDt3DtbW1lo7NhGRoWNP3Ajcv38f\nu3fvxpw5c2BjY6O14z558gSTJk1CvXr1sHfvXtmfR05EpC/YEycAT+ci9/Pzw4svvqjVAp6dnY0x\nY8bAysoK+/btYwEnItIAFnEDUlWvqbi4GF5eXujWrZvW4nj8+DGGDx+O7t27Y8eOHTA1NdXasUWx\nLyeGeRLHXIlhnqSl0SIeHByM7t27o0uXLli/fn2l1588eYJXX30VvXv3hpeXFw4fPqzJcIyStbU1\n+vbtq7XjpaSkwMvLC8OHD8fGjRthYsK/E4mINEWjPfHevXtj3bp1sLW1xahRo3D27NkKQ7o//PAD\noqKisHHjRty7dw/Dhg3DnTt3Kl14xZ64fkhJScGwYcMwceJEfPTRR3KHQ0Skt2TviWdlZQEAPD09\nYWtri5EjR+LChQsVtrG0tEROTg6Ki4uRnp6ORo0ayfYYTKqb1NRUDB8+HBMmTGABJyLSEo0V8fDw\ncDg6OqqWnZycEBoaWmGbadOmobS0FDY2Nhg8eDB27dqlqXCMQmBgIJRKpdZHLdLS0jB8+HC89NJL\nelPA2ZcTwzyJY67EME/SkvU+8Q0bNqB+/fpITk7G1atX4ePjg3v37lXZR509ezbs7OwAAFZWVnB1\ndYW3tzeA//+XwtiXAeDixYsIDg5Gnz59tHL8x48fw93dHQMHDsQnn3wChUKhM/l41vLly5d1Kh4u\n6/9yOV2JR1eXL1++rFPx6Mpy+ffx8fGoCY31xLOysuDt7Y3IyEgAwMKFCzF69Gj4+Piotpk8eTL+\n9a9/qabgdHd3x/bt2yucwQPsiYtKT0/Hli1bMHfuXK3cTpaTk4Nhw4Zh6NChWL16NVshREQSkb0n\nbmlpCeDpFerx8fE4efIk3N3dK2wzfPhwHD16FGVlZYiLi0N6enqlAk5iysrKcOjQIXh6emqlgBcU\nFGDcuHHo06cPCzgRkUw0ev/P2rVr4evri+eeew7//ve/YWNjg02bNmHTpk0AgKlTp6JevXpwc3PD\nggULsG7dOk2GY9BCQ0Nx69atSn8oaUJJSQmmTZuGFi1aYOPGjXpZwP85BEpVY57EMVdimCdpabQn\n7uXlhZiYmArrfH19Vd9bWlqycEsgLS0N586dg4eHh8YLallZGV5//XXk5+dj7969qFevnkaPR0RE\n1ePc6QagpKQEycnJ6NChg8aPtXTpUpw/fx4nT55E48aNNX48IiJjJFr3+BQzA1C/fn2tFPB169bB\n398fZ8+eZQEnItIBnBPTgGiy13T48GGsXr0a/v7+aNasmcaOoy3sy4lhnsQxV2KYJ2nxTJzUCg8P\nx2uvvQZ/f3/VvfpERCQ/9sTpme7evQsPDw98//33GDdunNzhEBEZBdnvEyfNKSwsxM8//4yioiKN\nHiczMxM+Pj547733WMCJiHQQi7geCgwMRLNmzdCgQYNK66VSWlqKadOmYfjw4Vi0aJFk+9UV7MuJ\nYZ7EMVdimCdpsYjrmZSUFERFReG5557T6HH+85//oLCwEGvWrNHocYiIqPbYE9cjSqUS27dvh5OT\nE/r376+x4+zevRv/+c9/EB4erpUpXImIqCLeJ26Arl27hsLCQri5uWnsGJGRkVi0aBECAgJYwImI\ndByH0/VIo0aN4OPjU+WjWoG695pSUlLw8ssvY+PGjejVq1ed9qXr2JcTwzyJY67EME/S4pm4Hunc\nubPG9l1SUoIpU6bglVdewaRJkzR2HCIikg574gQAeP/99xEWFobjx4/zoSZERDJjT5yEHTt2DNu3\nb8elS5dYwImI9Ah74gakNr2mxMREzJkzB7/++itatmwpfVA6in05McyTOOZKDPMkLRZxHXflyhUU\nFxdrZN/FxcWYMmUK3nrrLXh6emrkGEREpDnsieuw+Ph4HD58GP/3f/+nkWHud955B9HR0Th69Gi1\nV7wTEZH2sSeu55RKJU6dOoWhQ4dqpID/8ccf8PPzw6VLl1jAiYj0FD+9ddStW7dQVFQEZ2dn4feI\n9ppSUlLw+uuv45dffkHz5s1rGaF+Y19ODPMkjrkSwzxJi0VcB5WVleHUqVMYNmwYFAqFpPtWKpWY\nO3cuZs+ejSFDhki6byIi0i7hnnhCQgIUCgU6dOig6ZgqMbaeeGxsLIKCgjBnzhzJi/gPP/yALVu2\n4Pz585WegkZERLpBtO5VW8QLCwvx66+/YvPmzYiLi0ObNm2gVCrx8OFDdOrUCfPmzcP06dNhZmYm\nefCVgjSyIg4ARUVFkhfZGzduYMiQITh79iy6desm6b6JiEg6onWv2uH05557Dmlpafjtt9/w8OFD\nREZG4vLly3j48CF+++03pKamavxxmMasNgX8Wb2moqIivPLKK/j0009ZwMG+nCjmSRxzJYZ5kla1\nV6efOXOm2je1b98e7777Lt59912NBEXS++ijj9CuXTv4+vrKHQoREUmkVveJ37hxA46OjpqIp0rG\nOJwupfDwcIwdOxZXrlxBq1at5A6HiIjUqPNw+rOMHDmyNm8jGRQWFmLOnDn45ptvWMCJiAxMtcPp\nCxcurPZNGRkZGgnGmEVHRyM3Nxf9+/ev9T4CAwPh7e1dYd3KlSvh4OCAqVOn1jFCw1JVrqgy5kkc\ncyWGeZJWtUV827Zt+Oqrr2BmZlbhNielUolff/1VK8EZC6VSiaCgIMkvFIyMjMSPP/6Iy5cvS36r\nGhERya/anvjQoUOxcuVKeHh4VHrNzs4O8fHxmo5NxdB74rdv38apU6fg6+srWbEtLi5Gv379sGTJ\nEsyaNUuSfRIRkXbUee70/fv3w9zcvMrXtFnAjcG5c+fg4eEh6dnyqlWr0K5dO8ycOVOyfRIRkW6p\n9sK2JUuWoFGjRtqMxSglJiYiKysLPXr0qPO+yu+/vH79OtavX49NmzZxGL0avFdVDPMkjrkSwzxJ\nq9oifuXKFW3GYbSSkpLg4eEh2ZPElEolFixYgI8//hjt27eXZJ9ERKSbqu2JOzo64tdff4VSqazy\nbK5Pnz4aD66coffEpbRt2zZs3LgRISEhGnmEKRERaV6d505v2rQp3Nzcqn3j6dOnax9dDbGIi3n8\n+DF69OgBf39/rf6RRURE0qpzEe/duzciIyMlD6w2WMTF+Pj4oEuXLli7dq3coeg83qsqhnkSx1yJ\nYZ7E1PnqdNIvZ8+eRVhYGHbv3i13KEREpCXVnon/+eefOjO9Ks/En624uBh9+vTBhx9+iEmTJskd\nDhER1VGd507/4osvsGXLFuTm5lZ6LScnB5s3b+ajSGvp6tWrCA4Olmx/a9euRbt27TBx4kTJ9klE\nRLqv2iK+f/9+5OTkYMCAAbCzs4OnpyeGDBkCW1tbDBgwALm5uThw4IA2YzUISqUSISEhaN26tST7\nS05OxurVq7F+/XoEBQVJsk9jwHtVxTBP4pgrMcyTtKrtiVtaWuKtt97CW2+9hSdPnuDOnTsAAAcH\nBzRs2FBp98AcAAAgAElEQVRrARqaxMREFBYWokuXLpLs7/3338fcuXPRpUsXJCUlSbJPIiLSD7V6\nnri2GVJPfN++fejQoQPc3d3rvK+IiAiMHTsWN2/ehIWFhQTRERGRLtDo88SpdnJzcxEbG4tevXrV\neV9KpRJvvvkmVq5cyQJORGSkWMS16OHDh+jdu3e1D5apib179yI/Px+zZ89WrWOvSRxzJYZ5Esdc\niWGepMX7xLXIwcEBDg4Odd5Pfn4+3n33XezatYtTqxIRGTG1PfHIyEisX78eISEhKCgoePomhQJx\ncXFaCbD8eIbSE5fCxx9/jOvXr8PPz0/uUIiISAPqPO1qOS8vL8ybNw9Dhw5FgwYNVOttbGzqHqUg\nFvH/X1JSElxcXHDx4kXY2dnJHQ4REWmAZBe25eXlYerUqWjbti1sbGxUXySPjz76CK+//nqVBZy9\nJnHMlRjmSRxzJYZ5kpbanvjYsWPx+uuv45VXXoG1tbVqPZ+SpX0xMTE4dOgQbt26JXcoRESkA9QO\np3t7e1f5PHE+ilRMaWkp9u3bh/Hjx8PU1LRO+xo/fjwGDhyId955R6LoiIhIF0n2FDMOfdTN7du3\nkZeXV+cCHhISgvDwcOzatUuiyIiISN+p7Ynn5OTg+++/x5gxYzBmzBj88MMPVT4UhaoWGRmJ3r17\n12kfSqUSy5Ytw8cff/zMKW/5B5c45koM8ySOuRLDPElLbRH/6quvcPXqVXzyySf4+OOPcfXqVXz5\n5ZfaiE3v5eTkICEhAT169KjTfvz9/fH48WPMmjVLosiIiMgQqO2Ju7q6IiIiAvXrPx15LykpgZub\nGy5fvqyVAAH97YmfPXsW6enpePHFF2u9j9LSUri6uuJ///tfnfZDRET6Q7JbzPr06YP9+/dDqVSi\nrKwMBw8e5JXpgu7cuVPnofRdu3bB0tISY8eOlSgqIiIyFGqL+PLly7F371507NgRtra22LNnD5Yv\nX66N2PTerFmz0L59+1q/v7i4GB9//DE+++yzKu8Q+Cf2msQxV2KYJ3HMlRjmSVpqr07v0qULDhw4\ngOLiYgCo81XWxsTEpG7Pl9m5cyfs7Ozg6ekpUURERGRIqu2J79y5EzNnzsTXX39d4SxQqVRCoVBg\nyZIl2gtST3vidVFcXIxu3bph+/btGDJkiNzhEBGRFtX5PvH8/HwAT6+wFhnKJWnt2LED9vb2LOBE\nRFQ9pRpnzpwRWleVoKAgpaOjo9LBwUH57bffVrlNWFiY0s3NTeno6Kj08vKqchuBMA1KYWGh0s7O\nTjjP5U6fPq2ZgAwQcyWGeRLHXIlhnsSI1j21TduFCxcKravK4sWLsWnTJgQEBOC7775DWlraP/+A\nwNy5c/H5558jJiYG+/btE9qvLsvLy0NERESd9rFjxw44ODhg8ODBEkVFRESGqNrh9JCQEJw/fx6p\nqalYs2aNamw+NTUVzZs3V7vjrKwsAFBdlDVy5EhcuHABPj4+qm0iIiLg4uKC5557DoB2H2+qKdeu\nXUNSUhLc3Nxq9f6ioiKsXLkSv/76a43f6+3tXatjGiPmSgzzJI65EsM8SavaM/GioiLk5OSgtLQU\nOTk5yMnJQW5uLhwdHbFjxw61Ow4PD4ejo6Nq2cnJCaGhoRW2OXHiBBQKBYYMGYKxY8fixIkTdfhR\ndENUVBRcXFxq/f5t27ahW7duGDRokIRRERGRIar2TNzLywteXl6YPXt2lc+ulkJBQQEuX76MgIAA\n5OfnY8SIEbh27VqV84P/PQ4rKyu4urqq/qIrv+9Q7uWePXsiOzsbCQkJuH//fo3fP2jQIHz22Wd4\n5513EBgYWOP3l6/TlXzo8vLly5fx5ptv6kw8urr8z98tuePR5eXydboSj64ur127Vic/v+VeLv8+\nPj4eNaF22tX09HT89ttvOHHiBDIyMp6+SaHAX3/99cwdZ2VlwdvbG5GRkQCe9tFHjx5dYTj9jz/+\nQGBgoGou9ilTpmDu3LkYNWpUxSD15Baz06dPo6ioqFL8orZv346dO3ciICCgVu8P/Fvhp2djrsQw\nT+KYKzHMkxjJpl394IMPkJWVhejoaCxevBhWVlbw8vJSu2NLS0sAQHBwMOLj43Hy5Em4u7tX2GbA\ngAEICgpCfn4+0tPTERkZCQ8PD7X71kVKpRLXr19Hz549a/X+srIyrF69uk6z4fF/DHHMlRjmSRxz\nJYZ5kpbaM/HevXsjMjISzs7OuHLlCgoKCjBkyBBcvHhR7c6DgoIwf/58FBcXY9GiRVi0aBE2bdoE\nAPD19QUAfP/991i/fj1atGiBBQsWYOrUqZWD1IMzcaVSiQcPHqBt27a1uq/+8OHDWLlyJcLCwnhf\nPhGRkROte2qL+IABAxAaGorXX38dAwcOhIODA958801cunRJsmDV0YciXhdKpRIDBw7EO++8gwkT\nJtR6PxymEsdciWGexDFXYpgnMZINp7///vvIzMzEu+++i+DgYHz66af4+uuvJQmSngoODkZGRgZe\neukluUMhIiI9ovZMPDExER06dKiwLjk5GW3atNFoYH9n6Gfio0ePxqRJk/Cvf/1L7lCIiEgHSHYm\n3qlTJ0ydOlU1lzoAPP/883WLjlQiIyNx9epVzJgxQ+5QiIhIz6gt4s7OzhgyZAg8PDxw584dbcSk\nV5RKperWu9pYvXo1lixZAjMzszrH8vf7DenZmCsxzJM45koM8yQttUUcAN544w1s2LABY8eOxdGj\nRzUdk15JTEzEnj17avXeO3fuICAgAPPmzZM4KiIiMgbCt5gBT3vhkyZNwsWLF/HkyROtBAjodk/8\n+PHjaNiwodC98//0xhtvwNraGitXrtRAZEREpK/q/Dzxcv7+/qrv27Rpg8DAQJw/f75u0RkIpVKJ\nmJgYvPLKKzV+b0ZGBnbv3o3r169rIDIiIjIG1RbxnTt3YubMmVU+TUuhUKieTmbMkpOTYWpqihYt\nWtT4vZs3b8YLL7wg6VX+vP9SHHMlhnkSx1yJYZ6kVW0RL78aPScnhzOIVSMmJgbdu3evcX6Ki4ux\nYcMGHDx4UEORERGRMVDbEz979iwGDx6sdp0m6WpPPDQ0FHZ2dmjdunWN3ufn54cNGzYgODhYQ5ER\nEZE+k2za1b9f2PasdZqkq0W8tgYNGoSlS5di/PjxcodCREQ6qM4XtoWEhOD8+fNITU3FmjVrVDtL\nTU1F8+bNpYvUyFy4cAHJyckYN26c5Ptmr0kccyWGeRLHXIlhnqRVbREvKipCTk4OSktLkZOTo1rv\n6OiIRYsWaSU4Q7Ru3TosWrQI9erVkzsUIiLSc2qH0+Pj42FnZ4cnT56gYcOG2oqrAkMZTr9//z5c\nXFxw9+5d1fPWiYiI/kmyudMzMzPh4+MDJycnAMDly5fx73//u+4RGqHvvvsOM2fOZAEnIiJJqC3i\n//vf/7B69WpYWVkBAFxdXREUFKTxwHSZv78/Hj9+XKP3PHnyBFu2bMHChQs1FBXnJK4J5koM8ySO\nuRLDPElLbRF/8OABevbsqVouLCxEo0aNNBqULisoKMCVK1dgYWFRo/f5+fmhX79+cHBw0FBkRERk\nbNQW8ZEjR+Lw4cMAgISEBHzwwQcaubJaX9y5cwe2trYwNTWt0fu+//57zJ8/X0NRPcUrPsUxV2KY\nJ3HMlRjmSVpqi/iiRYsQGRmJ0tJSjBkzBlZWVhodEtZ1t27dQteuXWv0nsjISDx48AA+Pj4aioqI\niIyR2iJubW2Njz76CFFRUbh+/Tref/99o70wq7S0FLdv365xEf/hhx8wb948jd9Wxl6TOOZKDPMk\njrkSwzxJS+1TzNLT0/H7778jJCQEBQUFAJ5e+r5161aNB6drkpKSYG1tXaN+eHZ2Nvz8/BATE6PB\nyIiIyBipvU98+vTpaNy4MYYNG6bqAysUCkyYMEErAZYfTxfuE1cqlSgoKKjR/fLfffcdgoKC4Ofn\np8HIiIjIkEg2d3qPHj1kf+a1rhTxmlIqlXB2dsa3336LYcOGyR0OERHpCckme5k6dSp++ukn1VA6\niTt37hxKSkowdOhQrRyPvSZxzJUY5kkccyWGeZKW2p746tWrkZ+fjwULFsDMzAzA078QsrOzNR6c\nviu/rYzPYyciIk1QO5yuC/RxOD01NRVdunTB3bt3YW1tLXc4RESkR+r8KNJyly5dqrTO1tbWqB5H\nWlxcjIyMDLRs2VL4Pdu2bcPLL7/MAk5ERBqjtie+cOFCuLm5YcKECZgwYQLc3Nzg7e0NDw8PhIaG\naiNG2d29exf+/v7C2yuVSmzduhWvvfaaBqOqjL0mccyVGOZJHHMlhnmSltoi3qFDB5w6dQp3797F\n3bt3cfr0afTo0QNr167Fl19+qY0YZRcbG1ujOc9DQ0NRVlaGQYMGaTAqIiIydkK3mEVGRqJBgwYA\ngKKiIri6uiI6OhrOzs64evWq5oOUuSe+fv16TJo0Ca1btxba/vXXX4eDgwOWLVum4ciIiMgQSdYT\nnzJlCmbOnImpU6cCePo0rsmTJ6OwsBDm5uZ1j1THpaeno6ioCK1atRLaPi8vD/v27UN0dLSGIyMi\nImOndjh92bJlmDhxIv744w/4+/tjwoQJeO+992BqaorTp09rI0ZZ3blzBw4ODsK3ie3btw8eHh5o\n06aNhiOrjL0mccyVGOZJHHMlhnmSltozcTMzM0yaNAmTJk2q9FqTJk00EpQuadiwIZydnYW3//nn\nn7Fo0SINRkRERPSU2p54fHw8Nm3ahBMnTiAjI+PpmxQKxMXFaSXA8uPpw33id+7cwaBBg3D//n3V\nNQREREQ1Jdm0qytWrEDv3r1RUlKCgwcP4vnnn8e8efMkCdLQbNu2DTNmzGABJyIirVBbxKOiojB5\n8mQoFArVrWW7d+/WRmx6pbS0FNu2bcPcuXNli4G9JnHMlRjmSRxzJYZ5kpbannjDhg1RWloKLy8v\nfPbZZ+jUqZNR9MJr6uTJk2jbti169uwpdyhERGQk1PbEw8LC0L17dzx58gQbN25EUlISFi5cCBcX\nF23FqBc98cmTJ2PYsGGYP3++3KEQEZGek+x54rpAjiIeGxuLrKws9OnTR+22mZmZsLW1xb1792Bl\nZaWF6IiIyJBJdmHbtWvXVPOnOzs7w9nZWatn4XKJiYlBUVGR0Lb79u3DiBEjZC/g7DWJY67EME/i\nmCsxzJO01PbEfX19MW/ePCxcuNCorrqOi4tD//79hbb95ZdfsHjxYg1HREREVJHa4XQPDw+cPn1a\n1gKu7eH0jIwM/PTTT3j77bfVztSWkJCA3r1748GDBzAzM9NShEREZMgkmzv9yy+/xIwZMzB69GhY\nWlqqdj5+/Pi6R6mj4uLiYG9vLzTV6u7duzFhwgQWcCIi0jq1PfHdu3fjypUrOHXqFH7//Xf8/vvv\nOHr0qDZik015ERexa9cuzJgxQ8MRiWGvSRxzJYZ5EsdciWGepKX2TPz48eO4fv26UfXDR44cKXRm\nHRUVhczMTAwePFgLUREREVWktic+b948vPLKK/Dy8tJWTJXo6n3iy5Ytg4mJCT7//HO5QyEiIgMi\n2X3iTk5OuHHjBtq1a6e6hUqhUCAqKkqaSAXoYhEvKyuDra0tjh07xlnaiIhIUpJd2Obv7y9JQIYm\nODgYzZo106kCHhgYCG9vb7nD0AvMlRjmSRxzJYZ5kpbaIm5nZ6eFMPSPLl3QRkRExonTrv5NcXEx\n6tWrBxOTZ1+0X1BQgLZt2yIqKgrt27fXeFxERGRcJJt21ZhERETgzz//VLvdsWPH0KtXLxZwIiKS\nlVARLyoqQnBwMAAgPz8f2dnZGg1KLvHx8ejQoYPa7fz8/DB16lQtRFQzvP9SHHMlhnkSx1yJYZ6k\npbaIHzhwAAMGDMCcOXMAAPfv38fLL7+s8cC0raysDAkJCbC1tX3mdk+ePMGxY8cMesY6IiLSD2qL\n+MaNG3HmzBlYWFgAALp27YqUlBSNB6Ztjx49QpMmTdCkSZNnbnfs2DG4ubmhRYsWWopMHK/4FMdc\niWGexDFXYpgnaakt4gqFAo0aNVItp6amonnz5hoNSg7x8fFCV+L/9ttvmDRpkuYDIiIiUkNtEZ88\neTKWLl2K/Px8bN++HVOnTsXMmTO1EZtWPXnyRO186bo+lM5ekzjmSgzzJI65EsM8SUvtfeKvvfYa\ngoKCUFRUhLCwMHzyySfw8PDQRmxaNWzYMLXb6PJQOhERGZ8a3SdeXFyMR48eaf3WKl2ZdnXatGkY\nOnQo5s2bJ3coRERkwCS7T9zLywvZ2dkoLCyEk5MTRo8ejVWrVgkFERwcjO7du6NLly5Yv359tduF\nh4ejfv36OHDggNB+5VA+lG6IV+YTEZF+UlvEMzMzYWFhgd27d+Pll1/G1atXcejQIaGdL168GJs2\nbUJAQAC+++47pKWlVdqmtLQUy5Ytw+jRo3XibLs6+jCUzl6TOOZKDPMkjrkSwzxJS20Rt7S0RFxc\nHLZv344ZM2ZAoVAgPz9f7Y6zsrIAAJ6enrC1tcXIkSNx4cKFStutX78eEydO1OniCDy9Kn3y5Mly\nh0FERKSitoj/97//xdy5c+Hh4QEXFxfExsaiS5cuanccHh4OR0dH1bKTkxNCQ0MrbJOUlITDhw9j\nwYIFAJ72ALQtNzcXd+7ceeY2+jKUzvsvxTFXYpgnccyVGOZJWs+8Or20tBRXr16tMPzRuXNn7N+/\nX5KDv/nmm1i1apWqgf+s4fTZs2er7uO2srKCq6ur6pehPL7aLMfGxuLQoUPw8vKqdvsvv/wSnTt3\nVo0W1OV4XOYyl7nMZS7/c7n8+/j4eNSE2qvT3dzccP78eTRo0KBGO87KyoK3tzciIyMBAAsXLsTo\n0aPh4+Oj2sbe3l5VuNPS0tCoUSNs3rwZL774YsUgNXh1+tGjR9GyZUu4u7tXu42+XJUeyOf0CmOu\nxDBP4pgrMcyTGNG6p/Y+8VGjRmH27NmYPn062rZtq1rfp0+fZ77P0tISwNMr1Dt27IiTJ09ixYoV\nFbaJi4tTfT9nzhyMHTu2UgHXtISEBLi5uVX7emFhIY4dO4Z169ZpMSoiIiL11Bbxc+fOQaFQ4Ouv\nv66w/vTp02p3vnbtWvj6+qK4uBiLFi2CjY0NNm3aBADw9fWtZcjSyc/PR05ODlq1alXtNn/99Rdc\nXFzQsmVLLUZWO/zrVhxzJYZ5EsdciWGepFWjyV7koqnh9Js3byI8PBwzZsyodhtfX1907doVb7/9\ntuTHJyIiqopkk708fvwYq1evVg1zR0dH46effqp7hDqgadOm6N+/f7Wvl5WV4fDhwxg3bpwWo6q9\nv18gQc/GXIlhnsQxV2KYJ2mpLeIffvghmjZtqrpirkuXLvjmm280HZdWtG3bFl27dq329QsXLqBF\nixZwcHDQYlRERERi1A6nu7u748KFC+jduzciIyOhVCrh6uqKK1euaCtG2eZOf++992BqaopPP/1U\n68cmIiLjJdlwep8+fZCYmKhaPnDgAIYMGVK36PTEoUOH9GYonYiIjI/aIv7mm2/ijTfewL179+Dg\n4IAffvgBixcv1kZssrpx4wZyc3PRt29fuUMRxl6TOOZKDPMkjrkSwzxJS+0tZt26dcORI0eQkpKC\n0tJStGnTRhtxye7QoUN46aWXZJkKloiISITanriLiwumTp2KKVOmoHPnztqKqwKpe+JKpRIHDx7E\nCy+8UO1MdAMGDMDKlSvx3HPPSXZcIiIiEZL1xI8cOYJ69eph8uTJcHNzw1dffYWEhARJgpRLVlYW\n7t69C1NT0ypfT05Oxq1bt+Dl5aXlyIiIiMSpLeJ2dnZYtmwZLl68iN27dyMqKgqdOnXSRmwac//+\nfbRv377aofIjR45gzJgx1RZ5XcVekzjmSgzzJI65EsM8SUttTxwA4uPjsXfvXvj5+aFevXr44osv\nNB2XRpUX8eocOnQIc+fO1WJERERENSd0n3hRUREmT56MKVOmwN7eXluxqUjdE9+yZQtGjBgBW1vb\nSq9lZ2ejffv2SEpKQtOmTSU7JhERkSjJnmK2Y8cOdOvWTZKgdEFJSQlSUlKqvcr+xIkT8PDwYAEn\nIiKdp7YnfuzYMWRnZwMAli1bhhEjRiA0NFTjgWmKiYkJ5s6dW+1V6X/88QdeeOEFLUclDfaaxDFX\nYpgnccyVGOZJWmqL+NatW2FhYYHz58/j8uXL+OSTT/Df//5XG7FphImJCVq3bl3la2VlZTh27Bie\nf/55LUdFRERUc2p74n379sXFixcxf/58jBgxAhMmTFDNo64t2po7PTw8HK+++iqio6M1fiwiIqLq\nSNYTHzFiBDw9PZGeno4NGzYgOzsbJiZqT+D1kr+/P8/CiYhIb6itxqtWrcK2bdtw6dIl1K9fH8XF\nxfj555+1EZvW/fHHH/Dx8ZE7jFpjr0kccyWGeRLHXIlhnqQldJ94hw4dEBkZiYKCAiiVSr2dT/xZ\nsT969Ai3bt2Ch4eHlqMiIiKqHbU98W+//RZffvklnJycKlzRffToUY0HV06qnvipU6fQqFEjDBw4\nsNJr27dvx9GjR7Fv3746H4eIiKguJOuJ//jjj4iOjjaI+6aTkpIwYMCAKl9jP5yIiPSN2p54x44d\nkZubq41YNEqpVOLBgwdo165dpdeKi4vx559/YsyYMTJEJh32msQxV2KYJ3HMlRjmSVpqz8QtLCzg\n6uqKkSNHwsrKCsDT0/xvv/1W48FJKT09Hebm5mjcuHGl10JCQmBvb280z0onIiLDoLYnvm3btspv\nUijw6quvaiqmKo9X1554VFQUbty4gcmTJ1d6bdmyZTAzM8Mnn3xSp2MQERFJQbKe+OzZs6WIR3YZ\nGRlVDqUDT/vhmzdv1nJEREREdaO2Jx4fH4/ly5ejT58+6NSpEzp16iTLk8zqysvLC4MGDaq0PiEh\nAQ8fPkS/fv1kiEpa7DWJY67EME/imCsxzJO01BbxFStWoHfv3igpKcHBgwfx/PPPY968edqITXJV\n3SPu7++P0aNHo169ejJEREREVHtqe+Ll86T36tULERERAAA3NzdcuXJFKwECmp07/cUXX8S0adMw\nbdo0jeyfiIiopiTriTds2BClpaXw8vLCZ599hk6dOqFJkyaSBCm3oqIiBAUFYevWrXKHQkREVGNq\nh9PXrVuH/Px8fPDBB1AqlThz5gy+//57bcSmcaGhoejatStsbGzkDkUS7DWJY67EME/imCsxzJO0\nnnkmXlpaCj8/P3z55Zdo2rQpPvroIy2FJa2HDx+iZcuWlZ6+9ueff2LEiBEyRUVERFQ31fbES0pK\nUL9+fbi7uyMgIEDWaVfr0hMvKSnB6tWr8e6778LU1LTCa+7u7li9ejW8vb0liJKIiEgade6J9+/f\nH5cuXYKHhwfGjh2LiRMnqmY0UygUGD9+vHTRatCjR4/QvHnzSgU8PT0dMTExVT4MhYiISB9U2xMv\n/wsgPT0dnTp1wsWLF/H777/j999/1+oTzOoqOTkZbdu2rbT+1KlTGDJkCMzMzGSISjPYaxLHXIlh\nnsQxV2KYJ2lVeyaempqKNWvWwNnZWZvxSC45ORmtW7eutP7kyZMYOXKkDBERERFJo9oiXlpaipyc\nHG3GohEPHz6Eq6trhXVKpRJ//vkn3nzzTZmi0gz29sUxV2KYJ3HMlRjmSVrVFvHWrVtjxYoV2oxF\nckqlEtbW1mjVqlWF9bdv30ZJSQm6d+8uU2RERER1p/Y+cX2mUCgwceJENGjQoML6kydPYsSIEVVO\nw6rP2GsSx1yJYZ7EMVdimCdpVVvEAwICtBmHVv3555/shxMRkd5TO3e6LpBy7vTi4mK0aNECt2/f\nRosWLSTZJxERkZRE655BD6dX5cKFC7C3t2cBJyIivWd0RdyQh9LZaxLHXIlhnsQxV2KYJ2kZbBHP\nyclBdHR0pfXlF7URERHpO4PtiV+9ehUxMTGYPHmyal1GRgY6duyI1NRUmJubSx0mERGRJIy+J/7w\n4UPVXO/lAgMDMWjQIBZwIiIyCAZbxJOTkysV8dOnT2Po0KEyRaR57DWJY67EME/imCsxzJO0DLKI\nK5XKKudMN/QiTkRExsUge+KZmZnYunUrlixZolqXmpoKBwcHPH78GPXrVzvbLBERkeyMuiduYmKC\nYcOGVVgXGBiIwYMHs4ATEZHBMMgibmFhUenJZcYwlM5ekzjmSgzzJI65EsM8Scsgi3hVjKGIExGR\ncTHInvg/JScnw8nJCWlpaahXr56EkREREUnPqHvi/xQYGAhPT08WcCIiMihGUcSNZSidvSZxzJUY\n5kkccyWGeZKWwRXxW7du4dq1axXWGUsRJyIi42JwPfE//vgDNjY2cHd3BwDcv38frq6uSElJgYmJ\nwf3NQkREBshoe+KPHj1Cq1atVMuBgYHw8vJiASciIoNjUJVNqVRWKuLGNJTOXpM45koM8ySOuRLD\nPEnLoIp4RkYGzM3N0bBhQ9U6YyriRERkXDRaxIODg9G9e3d06dIF69evr/T6rl270KtXL/Tq1QvT\np0/HrVu36nS8R48eVXjoyb1795CXlwcnJ6c67VdfeHt7yx2C3mCuxDBP4pgrMcyTtDRaxBcvXoxN\nmzYhICAA3333HdLS0iq8bm9vj+DgYFy5cgWjRo3Cp59+WqfjtW/fvsJZ9+nTp+Ht7Q2FQlGn/RIR\nEekijRXxrKwsAICnpydsbW0xcuRIXLhwocI2AwcOhKWlJQDAx8cHQUFBdTpm06ZNK5yJG9tQOntN\n4pgrMcyTOOZKDPMkLY0V8fDwcDg6OqqWnZycEBoaWu32P/74I8aOHStpDGfOnIGnp6ek+yQiItIV\nOvFczoCAAPzyyy84f/58tdvMnj0bdnZ2AAArKyu4urqqeivlf9n9fTktLQ3Z2dno3r17la9zmcvl\ndCUeXVz29vbWqXi4rP/L5et0JR5dWS7/Pj4+HjWhsclesrKy4O3tjcjISADAwoULMXr0aPj4+FTY\nLtlerbQAAA6zSURBVCoqCuPHj8fx48fh4OBQdZC1eADKnj17sGfPHhw6dKh2PwAREZFMZJ/spbzX\nHRwcjPj4eJw8eVI1i1q5hIQETJgwAbt27aq2gNfWmTNnMGTIEEn3qev+/hcdPRtzJYZ5EsdciWGe\npKXR4fS1a9fC19cXxcXFWLRoEWxsbLBp0yYAgK+vLz755BOkp6dj/vz5AABTU1OEhYXV6liBgYFo\n0aIFevToAQA4e/YsZs2aJc0PQkREpIMMZu70HTt2wMPDA507d0ZmZiY6dOiA9PR0mJqaailKIiIi\nacg+nK5tKSkpaNmyJQDg3Llz6N+/Pws4EREZNIMo4nl5eSgtLUWTJk0APB1KN7Z+OMBeU00wV2KY\nJ3HMlRjmSVoGUcTLz8LLZ2Y7c+YMBg8eLHNUREREmmUQPfELFy4gLS0NPj4+KCgogI2NDR4+fKg6\nMyciItInoj1xnZjspa769u2L4uJiAEBYWBicnJxYwImIyOAZxHB6/fr1VY8fNdZ+OMBeU00wV2KY\nJ3HMlRjmSVoGUcT/jv1wIiIyFgbREy9XWlqK5s2b4/bt22jRooUWIiMiIpKe0d0nDgBXr15FmzZt\nWMCJiMgo6H0RLy0tVX1vjPOl/x17TeKYKzHMkzjmSgzzJC29L+JHjx7FlStXALAfTkRExkXve+I/\n/vgjnn/+ebRr1w7t2rXDuXPn0KlTJy1HSEREJB2j6ImXlZUhLS0NLVq0wN27d6FQKGBnZyd3WERE\nRFqh10U8MzMTjRo1gpmZGc6fP49Bgwappl41Ruw1iWOuxDBP4pgrMcyTtPS6iKempqquRA8NDcXA\ngQNljoiIiEh79LonHhERgYyMDIwYMQJ9+/bFhg0bWMiJiEjvifbE9bqIA4BSqUR+fj5atmyJx48f\nw9zcXMvRERERScsoLmwDnv6gFy9ehLOzs9EXcPaaxDFXYpgnccyVGOZJWnpfxAEgJCSEw+hERGR0\n9H44HQBeeuklTJs2DVOmTNFiVERERJphVD3x1q1bIzw8HB07dtRyZERERNIz+J54bm4uCgoKEB8f\nj/r166NDhw5yhyQ79prEMVdimCdxzJUY5kla9eUOoLaCgoJgY2OD2NhYDBw40KgneSEiIuOkt8Pp\n27Ztg6enJ7755hvY2tpi6dKlMkVHREQkLYMfTi+frY0ztRERkbHSyyKen5+P0tJS1KtXD9HR0ejT\np4/cIekE9prEMVdimCdxzJUY5klaelnE09LSYGNjg4sXL6JHjx5o2LCh3CERERFpnV72xGNjY3Hv\n3j1EREQgKSkJ69atkzE6IiIiaRl0T7xz584YNmwY++FERGTU9LKIA08neQkJCcGAAQPkDkVnsNck\njrkSwzyJY67EME/S0tsifu/ePQCAra2tzJEQERHJQy974gCwZ88e+Pn54cCBAzJFRUREpBkG3RMH\nwKF0IiIyenpXxNPT05GSkoKwsDD0799f7nB0CntN4pgrMcyTOOZKDPMkLb2bOz0qKgqlpaWIiopC\n37595Q6HiIhINnrXE9+3bx/MzMywbNkyREdHyxwZERGR9ER74np3Jp6WloaSkhL069dP7lCIiIhk\npVc9caVSifT0dERFRbGIV4G9JnHMlRjmSRxzJYZ5kpZeFfHs7GyYm5sjLCyMRZyIiIyeXvXE09PT\ncfHiRYwbNw7p6ekwNzeXOzQiIiLJGeR94s2aNUOTJk3QvXt3FnAiIjJ6elXEASA8PJxD6dVgr0kc\ncyWGeRLHXIlhnqTFIk5ERKSn9KonDgCOjo7w8/ODi4uLzFERERFphmhPXK+KeFZWFtq1a4fMzEzU\nr693t7gTEREJMbgL27Kzs3H8+HG4urqygFeDvSZxzJUY5kkccyWGeZKW3hTx+/fv49atW+yHExER\n/T96M5weHByMvXv3YtCgQZg+fbrcIREREWmMwQ2np6en4+bNmzwTJyIi+n/0pog/fPgQDx48gIOD\ng9yh6Cz2msQxV2KYJ3HMlRjmSVp6U8TT0tJga2sLhUIhdyhEREQ6QW964itWrEBRURE+++wzucMh\nIiLSKIPriXOmNiIioopYxA0Ie03imCsxzJM45koM8yQtvSnirq6uaNeundxhEBER6Qy96YnrQZhE\nRESSMLieOBEREVWk0SIeHByM7t27o0uXLli/fn2V2yxfvhz29vbo27cvbty4oclwDB57TeKYKzHM\nkzjmSgzzJC2NFvHFixdj06ZNCAgIwHfffYe0tLQKr4eFheHMmTOIiIjA0qVLsXTpUk2GY/AuX74s\ndwh6g7kSwzyJY67EME/S0lgRz8rKAgB4enrC1tYWI0eOxIULFypsc+HCBUycOBHNmjXDtGnTEBMT\no6lwjEJmZqbcIegN5koM8ySOuRLDPElLY0U8PDwcjo6OqmUnJyeEhoZW2CYsLAxOTk6q5RYtWiA2\nNlZTIRERERkUWS9sUyqVla6+47SqtRcfHy93CHqDuRLDPIljrsQwT9LS2C1mWVlZ8Pb2RmRkJABg\n4cKFGD16NHx8fFTbrF+/HiUlJXjrrbcAAJ07d67yTJyFnYiIjI1Iea6vqYNbWloCeHqFeseOHXHy\n5EmsWLGiwjbu7u5YsmQJZs2ahRMnTqB79+5V7ov3iBMREVWmsSIOAGvXroWvry+Ki4uxaNEi2NjY\nYNOmTQAAX19f9O/fH4MHD4abmxuaNWuGX375RZPhEBERGRSdnrEtODgYvr6+KCkpwaJFi7Bw4UK5\nQ9JJc+fOxR9//IGWLVvi6tWrcoejsxITEzFr1iykpKSgRYsWmDdvHqZPny53WDqnoKAAXl5eKCws\nhLm5OaZMmaJqeVFlpaWlcHNzQ/v27XH06FG5w9FZdnZ2sLCwQL169WBqaoqwsDC5Q9JJeXl5+Pe/\n/42QkBDUr18fW7duxYABA6rdXqeLeO/evbFu3Tr8f+3dX0hTbRwH8O9EZloLSqlJWQ4nshTdoXJs\nUBcmFIGr2EWeiy2iIiQvyuzGCMyLJBZ2ZwTVRawIITDJYtofw1RGWoxY9GfUIMFF7molm7rtvXjp\nvL3plrzpe87c93O1s3Oe5/zOGPvtOef5s3nzZuzevRvPnz9HQUGB3GEpzuDgIFatWgWHw8EknkIw\nGEQwGITRaMTk5CSqq6vh9Xqh0WjkDk1xpqamkJeXh2g0iq1bt6K7uxt6vV7usBSpo6MDY2NjCIfD\n6OnpkTscxdLpdBgbG8PatWvlDkXRmpubkZubi7NnzyI7Oxvfv3+XHk/PR7HTri5knDn9bceOHViz\nZo3cYSieVquF0WgEABQUFKC8vByjo6MyR6VMeXl5AIBv375hdnYWOTk5MkekTOPj43jw4AGOHj3K\nvjsLwM/o9x49eoSWlhasWLEC2dnZKRM4oOAkvpBx5kT/ld/vh8/nQ3V1tdyhKFI8HkdVVRXWr1+P\nxsZGFBUVyR2SIp06dQpOpxNZWYr9KVUMlUqFmpoa7N+/n3cskhgfH0ckEkFDQwNMJhMuXryISCSS\nsgy/eZRxwuEwDh48iMuXL2PlypVyh6NIWVlZ8Hq98Pv96OzslIaK0j/u37+PdevWQRAEtjAXYGho\nCF6vF+3t7WhqakIwGJQ7JMWJRCJ4//49bDYbBgYG4PP50NXVlbKMYpP49u3b/7Ugis/nS/lwn2gh\nZmZmYLPZYLfbsW/fPrnDUbzi4mLs3buXj7LmMTw8jJ6eHuh0OoiiiCdPnsDhcMgdlmIVFhYCAAwG\nA6xWKzsBzkOv16OsrAx1dXXIzc2FKIp4+PBhyjKKTeI/jzMPBALo7++HyWSSOSpKZ4lEAkeOHEFF\nRQVOnjwpdziKNTk5Kc1vHQqF0NfXxz8887hw4QI+f/6MT58+4c6dO6ipqcHNmzflDkuRpqamEA6H\nAQBfv36F2+3Gnj17ZI5KmUpLS+HxeBCPx9Hb24va2tqUxy/pOPE/Nd84c5pLFEU8e/YMoVAIRUVF\naGtrw+HDh+UOS3GGhobgcrlQWVkJQRAAAO3t7fwx+cXExAQOHTqEWCwGrVaL5uZmqRVFyXFmyeS+\nfPmCAwcOAADy8/Nx+vRp9rNI4tKlS3A4HIhEIqitrUV9fX3K4xU9xIyIiIiSU+ztdCIiIkqNSZyI\niChNMYkTERGlKSZxIiKiNMUkTrSMhEIhCIIAQRBQWFiIjRs3QhAEaDQaNDY2Lsk5r1+/jitXriTd\n39XVBafTuSTnJsp07J1OtEydP38eGo0GTU1NS3oei8UCt9uddCGZ6elpWCwWvHjxgsOwiBYZW+JE\ny9iP/+gDAwOoq6sDALS2tuL48ePYuXMnSkpK0NfXh3PnzqGiogINDQ1SmXfv3klzOJ84cQKhUGhO\n/R6PBxs2bJAS+O3bt2E2m1FVVQVRFAEAarUagiCgv7///7hkoozCJE6UgTweD3p7e3Hjxg3YbDbo\n9Xq8fv0aHz58wMuXLwEAZ86cQUtLCzweD8rLy3Ht2rU59bx69QoGg0Habmtrw+PHj+H1enH16lXp\nfYPBINVLRItH0TO2EdHiU6lUsFqt0Gg0MJvNiEajqK+vh0qlgslkwsjICDZt2oTBwUFYrVYAQCwW\nQ3Fx8Zy6/H4/tmzZIm1v27YNoijCbrdLM3QBQElJCbq7u5f82ogyDZM4UQb6sTaBWq1GTk6OtF64\nWq3G9PQ0YrEY8vPzF7R62c/dalwuF4aHh+FyueB0OqWFU+LxOJ+HEy0B3k4nyjC/68uaSCSg1Wqh\n0+lw9+5dJBIJzMzM4M2bN3OOLS0tRSAQkMoFAgFYLBZ0dHRgYmIC0WgUAPDx40eUlZUt+rUQZTom\ncaJl7EfrV6VSzfv652N+3e7s7MTTp09hNBohCAJGRkbm1G80GqUlg2dnZ2G321FZWYldu3ahtbVV\nauG/fftWWnSGiBYPh5gR0R8xm81wu91YvXr1vPuj0SgsFgtGR0d5S51okbElTkR/5NixY7h161bS\n/ffu3YMoikzgREuALXEiIqI0xZY4ERFRmmISJyIiSlNM4kRERGmKSZyIiChNMYkTERGlKSZxIiKi\nNMUkTkRElKb+Ap2c4f6ISWAIAAAAAElFTkSuQmCC\n"
}
],
"prompt_number": 15
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Notice that the time to recover all the way back to Mo is fairly long, on\n",
"the order of 3-4 sec. This is a limitation in the speed of acquisition\n",
"for T1 imaging.\n",
"\n",
"The difference in the T1 component of the signal from gray matter and\n",
"white matter changes over time. This difference is plotted in the next graph."
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"plot(t_T1, abs(MzW_T1 - MzG_T1))\n",
"xlabel('Time (s)')\n",
"ylabel('Magnetization difference');"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "display_data",
"png": 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Xjcl1gwZBxYp2RyMi4j1BuRRu4EC46SZj/bMEl/h4aNTI2HdeRMQfWdJzD0Tq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\nwPx/QgZpVkJ6EecAAAAASUVORK5CYII=\n"
}
],
"prompt_number": 16
},
{
"cell_type": "heading",
"level": 4,
"metadata": {},
"source": [
"Question 4"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"If you are seeking to make a measurement that optimizes the\n",
"signal to noise ratio between these two materials, at what time would you\n",
"measure the recovery of the T1 signal? "
]
},
{
"cell_type": "heading",
"level": 4,
"metadata": {},
"source": [
"Question 5"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Look up the T1 value of cerebro-spinal fluid (CSF). Plot the T1 recovery\n",
"of CSF. At what time you would measure to maximize the white/CSF\n",
"contrast."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We can visualize this difference as follows. Suppose that we have two\n",
"beakers, adjacent to one another, containing materials with different T1\n",
"values. Suppose we make a pair of images in which the intensity of each\n",
"image is set to the T1 value over time. What would the T1 images look\n",
"like?\n",
"\n",
"The beakers start with the same, Mo, magnetization."
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"beaker1 = Mo * ones((32, 32))\n",
"beaker2 = Mo * ones((32, 32))"
],
"language": "python",
"metadata": {},
"outputs": [],
"prompt_number": 17
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"They will have different T1 relaxation values"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"T1 = (0.64, 0.88) # White, gray T1\n",
"movie_t = arange(0.001, 4, .1) # Make images at these sample times (in sec)"
],
"language": "python",
"metadata": {},
"outputs": [],
"prompt_number": 18
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Here is a movie, slowed down, showing the relative intensities of the\n",
"images over a 4 sec period, measured every 100 ms. The frames are shown\n",
"at 1/2 the real speed (i.e., every 200 ms).\n",
"\n",
"It's not possible to play a movie in the notebook, but this gif was generated using the following code and an external gif program.\n",
"\n",
" beakers = [beaker1, beaker2]\n",
" for t in movie_t:\n",
" f = figure()\n",
" for i, beaker in enumerate(beakers):\n",
" subplot(1, 2, i + 1)\n",
" img = beaker * (1 - exp(-t / T1[i]))\n",
" imshow(img, vmin=0, vmax=1, cmap=\"gray\")\n",
" grid(False)\n",
" xticks([])\n",
" yticks([])\n",
" text(8, -3, 'Time: %.2f sec' % t, fontdict=dict(size=12))\n",
" ylim([-5, 32])\n",
" title(\"Beaker %d\" % (i + 1))\n",
" f.savefig(\"pngs/beaker_%.2f.png\" % t)"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"from IPython.display import Image\n",
"Image(url=\"http://web.mit.edu/mwaskom/www/beaker1.gif\")"
],
"language": "python",
"metadata": {},
"outputs": [
{
"html": [
"<img src=\"http://web.mit.edu/mwaskom/www/beaker1.gif\" />"
],
"output_type": "pyout",
"prompt_number": 19,
"text": [
"<IPython.core.display.Image at 0x79bd490>"
]
}
],
"prompt_number": 19
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"As you can see, if we make a picture of the net magnetization around\n",
"0.6-1.0 sec during the decay, there will be a good contrast difference\n",
"between the gray and white matter. Measured earlier or later, the\n",
"picture will have less contrast.\n",
"\n",
"As a preview of further work, later in the course, we should add just a\n",
"little noise to the measurements. After all, all measurements have some\n",
"noise. Let's look again.\n",
"\n",
"The only line that differs from the code above is:\n",
"\n",
" img = beaker * (1 - exp(-t / T1[i])) + rand(*beaker.shape) * 0.05"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"Image(url=\"http://web.mit.edu/mwaskom/www/beaker2.gif\")"
],
"language": "python",
"metadata": {},
"outputs": [
{
"html": [
"<img src=\"http://web.mit.edu/mwaskom/www/beaker2.gif\" />"
],
"output_type": "pyout",
"prompt_number": 20,
"text": [
"<IPython.core.display.Image at 0x79a1730>"
]
}
],
"prompt_number": 20
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"T2 Contrast"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"\n",
"There is a second physical mechanism, in addition to the spin-lattice measurement, \n",
"that influences the MR signal. This second mechanism is called spin-spin interaction\n",
"(transverse relaxation). This signaling mechanism is particularly important for\n",
"functional magnetic resonance imaging and the BOLD signal.\n",
"\n",
"In describing the T1 signal, we treated the MR signal as a single unified\n",
"vector. In the example above, we explored what happens to the net\n",
"magnetization of the MR signal when the vector is rotated 90 deg into the\n",
"x-y plane.\n",
"\n",
"But we omitted any discussion the fact that the dipoles are assumed to be\n",
"continuously precessing around the main axis together, in unison.\n",
"Perhaps in a perfectly homogeneous environment, these rotating dipoles\n",
"would precess at the Larmor frequency in perfect synchronization and we\n",
"could treat the single large vector as we have.\n",
"\n",
"But in practice, the dipoles within a single voxel of a functional image\n",
"each experience slightly different magnetic field environments.\n",
"Consequently, they each have their own individual Larmor frequencies,\n",
"proportional to the magnetic field that they experience. An important\n",
"second mechanism of MR is a consequence of the fact that the individual\n",
"dipoles each have their own local magnetic field and the synchrony soon\n",
"dissipate.\n",
"\n",
"\n",
"Suppose we have a large sample of dipoles that are spinning together in\n",
"perfect phase. We can specify their orientation as an angle in this\n",
"plane, theta. Let's assume they all share a common angle\n"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"n_samples = 10000\n",
"theta = zeros(n_samples)"
],
"language": "python",
"metadata": {},
"outputs": [],
"prompt_number": 21
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The position of the spins in the $(x, y)$ plane will be"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"spins = [cos(theta), sin(theta)]"
],
"language": "python",
"metadata": {},
"outputs": [],
"prompt_number": 22
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"And they will all fall at the same position"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"plot(spins[0], spins[1], \"o\")\n",
"subplot(111, aspect=\"equal\")\n",
"xlim(-2, 2)\n",
"ylim(-2, 2)\n",
"xlabel(\"x\")\n",
"ylabel(\"y\");"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "display_data",
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VdxSvXcCRFixYYB599NEkFyV2vM50eJ4zZ840W7ZsMcYYs3nzZnPVVVd5/jNB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\nQmCC\n"
}
],
"prompt_number": 23
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The total magnetization, summed across all the dipoles, is the vector\n",
"length of the sum of these spins "
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"avg_pos = sum(spins, axis=1) / n_samples\n",
"net_mag = sqrt(sum(square(avg_pos)))\n",
"print \"Net magnetization: %.4f\" % net_mag"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"Net magnetization: 1.0000\n"
]
}
],
"prompt_number": 24
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Now, suppose that spins are precessing at slightly different rates. So\n",
"after a few moments in time they do not fall at exactly the same angle.\n",
"We can express this by creating a new vector theta that has some\n",
"variability in it."
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"theta = rand(n_samples) * 0.5 # Uniform random number generator"
],
"language": "python",
"metadata": {},
"outputs": [],
"prompt_number": 25
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Here is the distribution of the angles"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"hist(theta)\n",
"xlabel('Angle')\n",
"ylabel('Number of spins')\n",
"xlim(0, 2 * pi)\n",
"xticks([0, pi / 2, pi, 3 * pi / 2, 2 * pi],\n",
" [\"0\", r\"$\\frac{\\pi}{2}$\", \"$\\pi$\", r\"$\\frac{3\\pi}{2}$\", \"$2\\pi$\"],\n",
" size=14);"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "display_data",
"png": 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HDhwIfZVB4MzfXDxb3Gz0z2z0z1whfbZ/RUVFnT4YAACcmwKe+a9du1bXX3+9mjdvruXL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\n"
}
],
"prompt_number": 26
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Going through the same process we can make a plot of the spin positions"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"spins = [cos(theta), sin(theta)]\n",
"subplot(111, aspect=\"equal\")\n",
"plot(spins[0], spins[1], 'o')\n",
"xlim(-2, 2)\n",
"ylim(-2, 2)\n",
"xlabel('x'),\n",
"ylabel('y');"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "display_data",
"png": 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bVFpamnQv91gAgD98bUO9/PLLmjNnjpYvX65TTz016Z6uXbtKSrwjqr6+XqtW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\n"
}
],
"prompt_number": 27
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Now, you can see that the net magnetization is somewhat smaller"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"avg_pos = sum(spins, axis=1) / n_samples\n",
"net_mag = sqrt(sum(square(avg_pos)))\n",
"print \"Net magnetization: %.4f\" % net_mag"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"Net magnetization: 0.9895\n"
]
}
],
"prompt_number": 28
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"If the spins grow even further out of phase, say they are spread over a\n",
"full $\\pi$ radians (180 degrees), then we have a dramatic reduction in the\n",
"net magnetization"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"theta = rand(n_samples) * pi\n",
"hist(theta)\n",
"xlabel('Angle')\n",
"ylabel('Number of spins')\n",
"xlim(0, 2 * pi)\n",
"xticks([0, pi / 2, pi, 3 * pi / 2, 2 * pi],\n",
" [\"0\", r\"$\\frac{\\pi}{2}$\", \"$\\pi$\", r\"$\\frac{3\\pi}{2}$\", \"$2\\pi$\"],\n",
" size=14);"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "display_data",
"png": 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ck9m25ZrMpucP+IGePwAAiBjFHwAcRs/fTRR/AAAcQ8/fO9myXJPZtuWazLYt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\ngg==\n"
}
],
"prompt_number": 29
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"spins = [cos(theta), sin(theta)]\n",
"subplot(111, aspect=\"equal\")\n",
"plot(spins[0], spins[1], 'o')\n",
"xlim(-2, 2)\n",
"ylim(-2, 2)\n",
"xlabel('x')\n",
"ylabel('y')\n",
"avg_pos = sum(spins, axis=1) / n_samples\n",
"net_mag = sqrt(sum(square(avg_pos)))\n",
"print \"Net magnetization: %.4f\" % net_mag"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"Net magnetization: 0.6369\n"
]
},
{
"output_type": "display_data",
"png": 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NZYypV5w8z4u678SJE6vydOrUidzcXAoLC4HqH1yqtyuFJU/ltr3jpBu2T7ET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\n"
}
],
"prompt_number": 30
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"In a typical experiment, in which the spins are in an inhomogeneous\n",
"environment, the spins spread out more and more with time, and the\n",
"transverse magnetization declines. The loss of signal from this\n",
"spin-dephasing mechanism follows an exponential time constant."
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"t_T2 = arange(0.01, 0.3, 0.01) # Time in secs\n",
"T2_white = 0.08 # T2 for white matter\n",
"T2_gray = 0.11 # T2 for gray matter\n",
"MzG_T2 = Mo * exp(-t_T2 / T2_gray)\n",
"MzW_T2 = Mo * exp(-t_T2 / T2_white)\n",
"plot(t_T2, MzG_T2, 'k', label=\"Gray\")\n",
"plot(t_T2, MzW_T2, 'gray', linestyle=\"--\", label=\"White\")\n",
"xlabel('Time (s)')\n",
"ylabel('Transverse magnetization (T2)')\n",
"legend();"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "display_data",
"png": 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21qj4xVxOTEzElClTNCYeTV3+92dL6ng0efn9Ok2JR1OXly5dyn+/i1h+/+fk\n5GSoktJpbZ8/f449e/bg2LFjePHixbs3yWQ4derUB3eclpYGT09PJCQkAHjXx+/WrVuB4frDhw8j\nIiJCMRd+v379MHz4cEXxUQSp47fQFef97YpCRUREKD44H9rnwIEDoa+vj82bN5fbR9MKyRUxTyXB\nXAnDPAkj2rS2s2bNQlpaGv744w989tlnMDMzg4eHh9Idm5qaAnh3hX1ycjJOnDhR6PnpTk5OiIyM\nREZGBp4/f46EhAS4uLiU8lfRLbm5uVi3bl2Jnvgn5C+OTCbD2rVrceXKFSxdurQMEWo3/iMjDPMk\nHHMlDPMkLqVn8q1bt0ZCQgLs7Ozw+++/4+3bt3BzcxP0SNPIyEiMHTsWOTk5mDx5MiZPnozVq1cD\n+PsWvZCQEKxYsQLVq1fHuHHjEBgYWDjIcnomf/LkSbx8+RK9e/dW+b7v3r0LJycnbNmyRXHhIxER\naQZV1T2lRd7JyQlnz57FqFGj4OzsDBsbG0yZMgUXL14s88GFKq9FPicnB8HBwfD19YWNjY3S7Us6\nDBYREYHAwEDExMSodKY9bcAhQ2GYJ+GYK2GYJ2FEG66fOXMmXr58iWnTpiEqKgrfffcdFi1aVOYD\nk3IVKlSAr68vjhw5onh2gCp5enpi5syZ+OSTT/DmzRuV75+IiKSl9Ez+/v37qFu3boF1jx8/Rq1a\ntdQa2D+V1zP593799VeYm5sLnmmwJN5PSJSeno7du3dDT0/p9z4iIlIz0c7kra2tERgYqJjLHgC6\nd+9e5gOTcN26dSv0RUtVZDIZQkJCkJqaii+//FItxyAiImkoLfJ2dnZwc3ODi4sLbt26JUZM9C+V\nK1dG48aNlW73z/stS6JixYrYv38/fvvtt3JzxX1pc1XeME/CMVfCME/iUvo8eQCYMGEC7O3t4efn\nhwULFqg7JpKAubk5fvvtN7i4uKBOnToICAiQOiQiIiojwbfQAe968X369MGFCxeQmZkpSoAAe/Ji\nSkxMhJeXF/bt2wdXV1epwyEiKpdEu4Xu3xfZ5ebmIiYmRtSHnLDIi+v48eMYPHgwIiMjVf5UPCIi\nUk7tF95t2bIFALB9+3YsWrRI8Vq2bBnOnz9f5gNT6cXHx+P27duF1quq1+Xl5YX58+fDx8cHKSkp\nKtmnpmFfUBjmSTjmShjmSVzFFvn3V9O/f/rcP1/p6emiBUiFmZub4/Dhw2q5d/69oUOHYvjw4fD1\n9S3R1LqDuaZFAAAgAElEQVRERKQ5lA7XnzlzplBvtqh16sTh+sL27NkDCwsLdOzYUW3HkMvlGD16\nNB48eICDBw+iQoUKajsWERH9TbT75CdNmiRoHYmrW7duOH/+PFJTU9V2jPf30Ovp6WHcuHH8okVE\npGWKLfKxsbFYtGgRUlNTsXjxYkVPfsaMGfjoo4/EjJGKUKVKFXh4eCAsLExRfNXR6zIwMMCuXbuQ\nmJiI7777TuX7lwr7gsIwT8IxV8IwT+Iq9j757OxspKenIy8vr0APvmnTppg8ebIowdGHOTg4IC0t\nDdnZ2TAyMlLbcSpXrozDhw/D2dkZdevWxbBhw9R2LCIiUh2lPfnk5GRYWVkhMzMTxsbGYsVVAHvy\nmuHGjRvw8PDApk2b4O3tLXU4REQ6S7Se/MuXL+Hr6wtbW1sA7yZLGT9+fJkPTNqnSZMm2LdvHwYP\nHqyYIImIiDSX0iL/3//+F/Pnz4eZmRkAwN7eHpGRkWoPjEpOjF5Xhw4dsGrVKvj5+eHu3btqP566\nsC8oDPMkHHMlDPMkLqVz1z969AgtWrRQLGdlZcHExEStQZFm8/f3x4MHD+Dj44MzZ86gWrVqUodE\nRERFUNqTnzNnDuzt7fHtt9/iwIEDWLFiBUxNTTFr1iyxYmRPXqD09HRcuXIFzs7OohwvKCgI8fHx\nOH78OCpWrCjKMYmIygPRevKTJ09GQkIC8vLy4OPjAzMzM94nr6EMDQ0RHx+P69evi3K8BQsWwNLS\nEgMHDkRubq4oxyQiIuGUnslrAp7JCxMREYGGDRti165dGDNmDKpUqaL2Y2ZlZaFXr14wNzfH5s2b\noa+vr/ZjqkJERAQ8PT2lDkPjMU/CMVfCME/CqKruKe3JP3/+HGFhYYiNjcXbt28VB1+/fn2ZD06q\nV7duXTg4OODAgQMYOHAgZDKZWo9nZGSEvXv3okePHhg1ahTWrl0LPT2lA0RERCQCpWfyAwYMQKVK\nldCpUyfF3OUymQy9e/cWJcD3x+OZvHD5+flYv3497Ozs4OjoKMox37x5g27dusHOzg4rV65U+5cL\nIiJdJtrz5Js3b46rV6+W+UBlwSJfcs+fP8eff/4pWpEHgFevXqFr165wcXHBokWLWOiJiEpJtAvv\nAgMDsW7dOsVQPWmuf95/Wq1aNVELPABUrVoVR48exenTp0W9+6I0eK+uMMyTcMyVMMyTuJT25OfP\nn4+MjAyMGzdOMT+6TCbDq1ev1B4caR9zc3OcOHECnp6eMDY21vhiT0Sky3h1PalFSkoKPDw8MGrU\nKAQFBUkdDhGRVhHt6vqLFy8WWle/fn0+bpY+6OOPP8bJkyfh4eGBihUrYuLEiVKHRERU7ijtyU+a\nNAkODg7o3bs3evfuDQcHB3h6esLFxQVnz54VI0YS6EO9rpSUlALPnhdDnTp1cPLkSfz0009Ys2aN\naMcVgn1BYZgn4ZgrYZgncSkt8nXr1sXJkydx584d3LlzB6dPn0bz5s2xdOlS/PTTT2LESCpgYWGB\n+/fv4/fffxf1uFZWVggPD8ecOXOwdetWUY9NRFTeCbqFLiEhAYaGhgCA7Oxs2Nvb448//oCdnR0u\nX76s/iDZk1eJJ0+eYPPmzRgxYoToD5W5du0aOnfujGXLlqFPnz6iHpuISNuI1pPv168fBg8ejMDA\nQADA7t270bdvX2RlZfGhJFqmZs2acHV1RWhoKIYNGybqzHTNmjXD0aNH4eXlBUNDQ/Ts2VO0YxMR\nlVdK/5WfPn06AgICcPjwYRw5cgS9e/fGjBkzUKFCBZw+fVqMGEkgIb0uJycnGBoaIioqSv0B/UvL\nli1x+PBhjBo1CkePHhX9+P/EvqAwzJNwzJUwzJO4lJ7JGxkZoU+fPkUOsVauXFktQZH6yGQyfPLJ\nJ3j58qUkx2/bti3279+Pnj17YteuXejUqZMkcRARlQdKe/LJyclYvXo1jh07hhcvXrx7k0yGpKQk\nUQJ8fzz25HVLZGQk+vTpg71798LNzU3qcIiINIpo09rOnj0brVu3Rm5uLkJDQ9G9e3eMHj26zAem\n8s3DwwPbt2+Hv78/jh8/LnU4REQ6SWmRv3TpEvr27QuZTKa4dW7Hjh1ixEYlpG29ri5duiA0NBSD\nBg1CaGioqMfWtlxJhXkSjrkShnkSl9KevLGxMfLy8uDh4YEffvgB1tbW7MXroLy8POjr64t+XFdX\nVxw9ehS+vr54/fo1Bg8eLHoMRES6SmlPPj4+Hs2aNUNmZiaCg4Px8OFDTJo0CS1bthQrRvbk1ezW\nrVuIiYnBoEGDRL2t7p+uXbsGLy8vfPXVVxg/frwkMRARaQrRnievCVjk1Ss/Px/bt29HjRo14OXl\nJVkcd+7cQZcuXTBq1CjMmDFDsjiIiKQm2oV3V65cUcxfb2dnBzs7O1HP4km40va69PT00Lt3b1y/\nfl2UGQyLY21tjaioKGzevBn/93//p9YvduwLCsM8CcdcCcM8iUtpT37MmDEYPXo0Jk2apJjalnSP\nsbEx+vXrh82bN8PCwgK1atWSJI7atWsjKioK3t7eePXqFZYvXy5ZC4GISNspHa53cXHB6dOnJS3w\nHK4Xz9WrV3H+/Hl8+umnksaRlpaGHj16oEGDBli3bh0MDJR+HyUi0hmi9eRjYmKwdOlSdOvWDaam\npoqD+/v7l/ngQrHIiysnJwcVKlSQOgxkZGSgV69eqFy5MrZv3w4jIyOpQyIiEoVoPfkdO3bg999/\nx8mTJxEWFoawsDAcOnSozAcm1VNVr0sTCjwAmJiY4ODBgwCAnj17IiMjQ2X7Zl9QGOZJOOZKGOZJ\nXErHQI8ePYqrV6+yH0+SMDIywq5duzBixAh4e3sjLCxMMaJEREQfpnS4fvTo0Rg4cCA8PDzEiqkQ\nDtdTfn4+Jk+ejNjYWBw7dgwWFhZSh0REpDai9eRtbW1x/fp11K5dG2ZmZoqDX7p0qcwHF4pFXjpy\nuRyRkZFo164dKlWqJHksM2fOxIEDB3DixAlYWlpKGg8RkbqI1pM/cuQIkpKSEB0djUOHDuHQoUOK\nPilpFnX0umQyGfLy8vDrr78iPz9f5fsvaSw//PADBg8eDDc3tzI9CZF9QWGYJ+GYK2GYJ3Ep7clb\nWVmJEAZpso4dO2L79u04ceIEvL29pQ4HM2bMgJmZGVxcXBAaGgonJyepQyIi0kic1pYEyczMxJo1\na+Dh4YFWrVpJHQ4A4PDhwxg6dChCQkIQEBAgdThERCoj2nA9EfD3jHjHjx/H48ePpQ4HAODr64vj\nx49j6tSp+Omnn/hFkIjoXwQV+ezsbERFRQF4N0HJq1ev1BoUlY66e101a9ZEnz59FBdgaoLWrVsj\nNjYWW7duxbhx45CbmyvofewLCsM8CcdcCcM8iUtpkd+3bx+cnJwwbNgwAMCDBw/Qq1cvtQdGmsnK\nygrGxsZSh1FAnTp1EB0djbt378LPz49fQomI/j+lPfkuXbrgwIEDcHV1RUJCAgDAzs5O1KeVsSdP\nQuTm5mLixImIjY3F4cOHUadOHalDIiIqFdF68jKZDCYmJorl1NRUfPTRR2U+MJGqGRgYICQkBIMH\nD4azs7PiSykRUXmltMj37dsXQUFByMjIwKZNmxAYGIjBgweLERuVkBS9rtzcXDx//lz04xZHJpMh\nKCgIS5cuhbe3Nw4fPlzkduwLCsM8CcdcCcM8iUtpkR85ciT8/Pzg5eWF+Ph4zJ07FyNGjBAjNtIC\n9+7dw8aNG/HixQupQymgd+/eOHjwIEaOHIng4GCpwyEikkSJ7pPPycnBkydPRO91siev2eLj4xEX\nF4fhw4dLPvXtvyUlJcHX1xfdu3fHggULoK+vL3VIRERKidaT9/DwwKtXr5CVlQVbW1t069YN8+bN\nE7TzqKgoNGvWDI0aNcKKFSuK3e7cuXMwMDDAvn37hEdOGqN9+/Zo0aIFtm/fjuzsbKnDKaBBgwaI\niYnBxYsX0adPH5U+rpaISNMpLfIvX75E1apVsWPHDvTq1QuXL1/G/v37Be38s88+w+rVqxEeHo6V\nK1fi6dOnhbbJy8vD9OnT0a1bN56tl5GUvS5PT098/PHH2LVrF/Ly8iSLoyjm5uY4duwYqlSpgo4d\nO+LJkyfsCwrEPAnHXAnDPIlLaZE3NTVFUlISNm3ahEGDBkEmkwk6G0pLSwMAuLu7o379+vDy8kJc\nXFyh7VasWIGAgABUr169FOGTppDJZPD19YW1tbXkD7IpiqGhITZu3AhfX184OTmV6eE2RETaQmmR\n//rrrzF8+HC4uLigZcuWuH37Nho1aqR0x+fOnUPTpk0Vy7a2tjh79myBbR4+fIgDBw5g3LhxAN4V\nCio9T09PSY+vp6cHV1dXVKhQQdI4iiOTyfDNN9/ghx9+wPTp07Fjxw6pQ9J4Un+mtAlzJQzzJK4P\nPoUuLy8Ply9fLjC80rBhQ+zdu1clB58yZQrmzZunuMDgQ8P1Q4cOVTwRz8zMDPb29ooPy/v4uMxl\nIcu1atXCvHnz8PXXX+PcuXPo3r07DAwMNCY+LnOZy+Vv+f2fk5OToUpKr653cHBATEwMDA0NS7Tj\ntLQ0eHp6KiYkmTRpErp16wZfX1/FNg0aNFAU9qdPn8LExARr1qzBf/7zn4JB8up6QSIiIhQfHPqw\niIgItGrVCoMGDcLr16+xe/du1KxZU+qwNA4/U8IxV8IwT8KIdnW9t7c3hg4dirCwMFy8eFHxUsbU\n1BTAuyvsk5OTceLECTg6OhbYJikpCXfu3MGdO3cQEBCAkJCQQgWetFt2djbu3bsndRhFMjc3x6FD\nh9CxY0c4ODgUaicREWk7pWfynp6eRfbKT58+rXTnkZGRGDt2LHJycjB58mRMnjwZq1evBgCMGTOm\nwLbDhg2Dn58f/P39CwfJM3mtlZKSgi1btqBfv36oV6+e1OEU69ChQxgxYgTmzp2LMWPG8PoQIpKU\nqupeiSbDkQqLvHa7ffs2QkNDMWTIENSoUUPqcIr1559/wt/fH+3atcPKlSs17ml7RFR+iDZc/+zZ\nM8yfP18xjP7HH39g3bp1ZT4wqd4/L+DQJA0bNoS3tze2bduGly9fSh0OgKJz1ahRI5w9exaZmZlw\ndXXF3bt3xQ9Mw2jqZ0oTMVfCME/iUlrkv/nmG1SpUkVxxV+jRo2wZMkSdcdFOsbOzg7Ozs7Ytm0b\nMjMzpQ6nWJUqVcL27dsxaNAgODo64sSJE1KHRERUakqH6x0dHREXF4fWrVsjISEBcrkc9vb2+P33\n38WKkcP1OuTKlSto1qyZVswhHxERgQEDBmDy5MmYPn06+/REJBrRhuvbtGmD+/fvK5b37dsHNze3\nMh+YyqcWLVpoRYEH3l10Gh8fj/379yMgIACvXr2SOiQiohJRWuSnTJmCCRMm4O7du7CxscGqVavw\n2WefiREblRB7XcIJzVWdOnUQGRmJ6tWrw9HREdevX1dvYBqGnynhmCthmCdxfXDGOwBo0qQJDh48\niL/++gt5eXmoVauWGHERaQwjIyOsWrUK69atg5ubG5YsWYJBgwZJHRYRkVJKe/ItW7ZEYGAg+vXr\nh4YNG4oVVwHsyeuurKwsXLhwAc7OzlrR805MTET//v3h4OCAlStXomrVqlKHREQ6SLSe/MGDB6Gv\nr4++ffvCwcEBCxcu1NgZzEg73bhxAwcPHtTIp9f9m729Pc6fPw8TExO0bt2as+QRkUZTWuStrKww\nffp0XLhwATt27MClS5dgbW0tRmxUQtrY6zIyMsLAgQPx8uVL7N+/X7RCX5ZcVapUCatXr8ZPP/2E\nnj174r///S/y8vJUF5wG0cbPlFSYK2GYJ3EpLfIAkJycjPnz5yMwMBDXr1/HggUL1B0XlSOGhoYY\nMGAAMjIysHfvXq0pmP7+/rhw4QLCw8PRqVOnAnehEBFpAkH3yWdnZ6Nv377o168fGjRoIFZsCuzJ\nlw+5ubnYs2cPGjRoUOhhRposLy8PCxYswNKlSxESElLk8xeIiEpCtLnrb9y4gSZNmpT5QGXBIl9+\n5OXlQSaTQU9P0CCTRomLi8PAgQPRqVMnLFmyBJUqVZI6JCLSUqJdePfbb78pJgGZPn06unbtyouN\nNJQu9Lr09fVFKfDqyJWjoyMuXryIrKwstG3bFgkJCSo/hth04TMlFuZKGOZJXEr/NV2/fj2qVq2K\nmJgYJCYmYu7cufj666/FiI1I61StWhWbNm3CN998A29vbyxZskQr7hogIt2kdLi+bdu2uHDhAsaO\nHYuuXbuid+/einnsxcLh+vItMzMTMpkMFStWlDqUErlz5w4GDBgAU1NTbNy4ER9//LHUIRGRlhBt\nuL5r165wd3fHmTNn0LNnT7x69Uor+6WkvRISErBlyxaNfnpdUaytrREdHY327dujdevWOHLkiNQh\nEVE5o7Raz5s3Dxs3bsTFixdhYGCAnJwcbNiwQYzYqIR0tdfl7OyMevXqYfPmzcjIyFDJPsXKlYGB\nAebOnYtdu3Zh/PjxGDlyJF6+fCnKsVVBVz9T6sBcCcM8iUvQKXndunWRmJiIqKgoXLlyRav+kSLt\nJ5PJ4OXlBRsbG2zatAmvX7+WOqQSc3d3x6VLl2BoaIgWLVogLCxM6pCIqBxQ2pNfvnw5fvrpJ9ja\n2sLQ0FCx/tChQ2oP7j325AkA5HI5IiMjcfXqVQwbNgwmJiZSh1QqERERGDFiBJydnbFs2TJ89NFH\nUodERBpGtPvkW7RogdjYWFSpUqXMBystFnn6pxs3bqBx48Za8UCb4rx58wazZs3Czp07sWLFCgQE\nBEgdEhFpENEuvKtXr55WDo+WR+Wl19WkSZMyF3ipc1WpUiUsWbIEe/fuxaxZsxAQEIAnT55IGlNR\npM6TNmGuhGGexKX0efJVq1aFvb09vLy8YGZmBuDdN4zly5erPTgiXdehQwckJiZizpw5aNmyJRYv\nXowBAwZo9SgFEWkOpcP1GzduLPwmmQyffvqpumIq8ngcrqcPycrKgqGhoVYXx/Pnz2P48OGoX78+\nVq1ahdq1a0sdEhFJRLSevCZgkSdl9u7dC2NjY3Tr1k2r53HIzs7Gjz/+iJ9//hnz5s3D8OHDtfqL\nCxGVjmg9+eTkZHz11Vdo06YNrK2tYW1tLcmT6Ei58tzr8vX1xbNnz7Bz505kZWUp3V5Tc2VoaIjZ\ns2fj5MmTCA4OhpeXF5KTkyWLR1PzpImYK2GYJ3EpLfKzZ89G69atkZubi9DQUHTv3h2jR48WIzYi\nwSpWrIgBAwagatWq2LBhg+KhStqqZcuWiIuLQ+fOneHg4ICff/4ZeXl5UodFRFpG6XD9+3nqW7Vq\nhfPnzwMAHBwc8Pvvv4sSIMDhehJOLpcjNjYWcXFxGDlypKS3fqrKtWvXMGbMGLx58wbBwcFwdHSU\nOiQiUjNV1T2lV9cbGxsjLy8PHh4e+OGHH2BtbY3KlSuX+cBE6iCTydChQwdYWVnpzOe0WbNmiIyM\nxLZt29CrVy90794dP/74I6pXry51aESk4ZQO1y9btgwZGRmYNWsW5HI5oqOjERISIkZsVELsdf3N\n0tLygxesaVuuZDIZBg0ahGvXrqFKlSpo3rw5QkJC1D6Er215khJzJQzzJK4PFvm8vDzs3r0bVapU\nQY0aNfDtt99izZo1aNmypVjxEdE/mJqaYsmSJTh58iR27NiB9u3bIy4uTuqwiEhDFduTz83NhYGB\nARwdHREeHs5pbUnrpaWlwdjYuMAzGLSZXC7Htm3bMG3aNA7hE+kYtd9C1759ewCAi4sL/Pz88PPP\nP2Pv3r3Yu3cv9u3bV+YDE4ktISFBJ668f0+qIXwi0h7FFvn33yCeP38Oa2trXLhwAWFhYQgLCxP1\nCXQkHHtdH+bh4YHmzZtj3bp1+PXXX6UOR2WKGsI/e/asSvbNz5RwzJUwzJO4ir26PjU1FYsXL4ad\nnZ2Y8RCpjUwmg6urKywsLLB06VLUq1cP7dq105kZ5ezs7BAZGYnt27fD398fPj4+mDdvHofwicqx\nYs/k8/LykJ6ejtevXxf5Is3j6ekpdQhaoWnTpvjhhx+QkJCApKQkqcNRKZlMhoEDB+LatWuoWrUq\nmjdvjpUrVyInJ6dU++NnSjjmShjmSVzFXnj3fhIcTcAL70gd8vLyoKenpzNn8kW5fPkypk6divv3\n72PevHn45JNPdPr3JdIVos1dT9qDvS7hIiIioK+vr/MFz87ODidOnMDy5csxe/ZsuLm5ITY2VvD7\n+ZkSjrkShnkSV7FFPjw8XMw4iDSGro0ayWQyeHt7IyEhASNHjkTfvn0REBCAmzdvSh0aEakZHzVL\n9A+vXr3Crl278Mknn+jsBWuZmZlYtmwZFi5ciMDAQHzzzTeoUaOG1GER0T9wuJ5IDapWrQoHBwds\n3LhR1IcwicnY2BgzZszA9evXYWBgAFtbW3z//fd48+aN1KERkYqxyOsQ9rqE+1CuWrdujSFDhiA6\nOhqHDh0q9ZXpmu79rYRxcXG4fPkyGjdujLVr1yI3N1exDT9TwjFXwjBP4mKRJypCzZo1MWrUKGRn\nZ2Pz5s063S5q2LAhdu3ahX379mHLli2wt7fH4cOHdfp3Jiov2JMn+gC5XI5nz57BwsJC6lBEIZfL\nERYWhunTp6NmzZqYP3++YoprIhIPe/JEIpDJZOWmwAPvfl8/Pz9cunQJ/fv3R+/evdGjRw+cO3dO\n6tCIqBRY5HUIe13CMVcfZmBggNGjR2PdunXw8fFBr169WOyV4GdKGOZJXCzyRKVw+fJlhIeHF7hI\nTRcZGhpiwoQJuHXrFos9kRZiT56oFN68eYPDhw8jNTUVn3zyCWrXri11SKJ4+/Yt1q1bhx9//BH2\n9vaYPXs22rVrJ3VYRDpHVXWPRZ6olORyOa5evYqjR4/C3t4enp6eMDAo9sGOOoXFnki9eOEdFcJe\nl3CqyJVMJkOLFi0wduxYPH/+HKGhoWUPTMMUl6eKFStyGP9f+PdPGOZJXCzyRGVUuXJl9OnTBz16\n9JA6FNGx2BNpNg7XE5HK/HMY387ODl9++SU6duyo80/7I1I19uSJtEB2djb09PTKTa/+vaysLGzd\nuhULFy6EiYkJgoKC0KdPn3KXB6LSYk+eCmGvSzixcpWYmIg1a9bg0aNHohxP1UqbJyMjI4wYMQJX\nr17Ft99+i5CQENjY2GDZsmV4/fq1aoPUEPz7JwzzJC4WeSI1ateuHVxcXLB9+3acOnVKZx92Uxw9\nPT34+fkhKioKu3btQnR0NKytrTFz5kykpKRIHR6RzuNwPZEI0tPT8dtvv+Hx48fw9vZG06ZNpQ5J\nMrdu3cLixYuxY8cOBAQE4IsvvijX+SAqCnvyRFooKSkJDx48gLu7u9ShSC41NRXBwcEIDg6Go6Mj\nvvzyS7i6uvIiPSKwJ09FYK9LOKly1aBBA60q8OrMU/Xq1TF79mzcuXMHPj4+GDZsGJydnbF3717k\n5eWp7bjqwr9/wjBP4mKRJ9IQ5XW0ysTEBOPGjcONGzcwbdo0LFy4EDY2Npg/fz5SU1OlDo9Iq6m1\nyEdFRaFZs2Zo1KgRVqxYUejn27ZtQ6tWrdCqVSsMGDAAN2/eVGc4Os/T01PqELSGpuUqJSUFGzZs\nwOPHj6UOpQAx86Svrw9/f3/ExsZi9+7duH79Oho3bowhQ4YgLi5O478EadpnSlMxT+JSa0++devW\nWLZsGerXrw9vb2+cOXOmwLO5Y2NjYWtrC1NTU2zatAnh4eHYsmVL4SDZkycdJ5fLkZCQgFOnTqFp\n06bo1KkTTExMpA5Lcs+ePcOGDRsQEhICc3NzjB8/Hv3794exsbHUoRGplcb35NPS0gAA7u7uqF+/\nPry8vBAXF1dgG2dnZ5iamgIAfH19ERkZqa5wygX2uoTTtFzJZDK0adMGEyZMgL6+PlauXIlz584h\nPz9f0rikztNHH32EoKAg/Pnnn5g7dy727duHevXqISgoCLdv35Y0tn+TOlfagnkSl9qK/Llz5wrc\nFmNra4uzZ88Wu/0vv/wCPz8/dYVDpBWMjY3h4+ODIUOGICkpCW/fvpU6JI2gp6eH7t27IywsDHFx\ncdDT04OTkxN8fHwQFhamlRfqEYlBI+aYDA8Px9atWxETE1PsNkOHDoWVlRUAwMzMTPFoT+Dvb4Zc\n5nJJlt/TlHj+vdyvXz/J4/H09NSYfLxfvnfvHrp37445c+Zg9+7dCAoKwqhRozB16lQMHz4cV65c\n0ah4uVxw+f06TYlHU5bf/zk5ORmqpLaefFpaGjw9PZGQkAAAmDRpErp16wZfX98C2126dAn+/v44\nevQobGxsig6SPXkiBblcznvJ/+XcuXMIDg7G/v374ePjg+HDh6NTp07Q0+MNRKSdNL4n/77XHhUV\nheTkZJw4cQKOjo4Ftrl37x569+6Nbdu2FVvgSbh/fiOkD9PmXG3duhXHjx/Hmzdv1H4sbclTu3bt\nsGHDBty+fRsdOnTA9OnTYW1trbgPXwzakiupMU/iUuvX3KVLl2LMmDHo0qULxo8fDwsLC6xevRqr\nV68GAMydOxfPnz/H2LFj0bp1a7Rv316d4RDphE8++QS5ublYuXIlTp8+zb79P1SrVg0TJ07EhQsX\ncODAAbx8+RLt27dHx44dsWXLFmRkZEgdIpGoOK0tkZZ6+fIlIiMjcfPmTXTq1Alt27aVOiSNlJWV\nhUOHDmHDhg2IjY1FQEAAhg0bBicnJ7Y9SGNx7noiAgA8ffoUr169QoMGDaQOReM9fPgQW7Zswfr1\n66Gvr49hw4Zh8ODBqFWrltShERWg8T15Eh97XcLpUq4sLCzUVuB1KU8AULt2bcyYMQM3btzA2rVr\ncePGDdja2sLPzw/79u1DVlZWqfeta7lSF+ZJXCzyRDoqLy8P165d4yhYEWQyGVxcXLBu3Trcv38f\nAQEBWL58OSwtLTFixAiEh4fz3nvSCRyuJ9JRr169wu7du5GTk4MOHTqgRYsW0NfXlzosjfbgwQPs\n2tVv60AAABTtSURBVLULO3bswIMHD9C3b1/079+f/XsSHXvyRKSUXC7H7du3ERMTg2fPnsHR0RFt\n27aFkZGR1KFpvJs3b2Lnzp3YsWMH3r59i8DAQPTv3x92dnYs+KR27MlTIex1CVdeciWTyWBjY4Mh\nQ4agX79+ePToEZKSkgS/v7zkqSiNGzfGN998gz/++AOhoaHIz8+Hn58fWrRoge+//77Q3PnlOVcl\nwTyJi0WeqJywtLREQEAAmjVrJnUoWkUmk8He3h7z58/HnTt38MsvvyAlJQUdOnRA+/btsWTJEjx6\n9EjqMImKxOF6IsLbt2/x119/oV69elKHojVyc3Nx6tQp7NixAwcOHECzZs3Qq1cv9OrVCw0bNpQ6\nPNJy7MkTkcqkpKRg9+7dqFSpElxcXNCkSRP2nUsgKysLp0+fRmhoKPbv34+aNWvC398fvXr1QsuW\nLZlLKjEWeSrkn092og9jrgrLz8/H9evX8b///Q9ZWVlwdnbGy5cv0blzZ6lD0wrvP1N5eXmIjY1F\naGgoQkNDIZPJFGf4zs7O5f6hOfy7JwwvvCMildLT04OtrS1GjhyJHj164MaNG3j69KnUYWkdfX19\nuLq6YtGiRbh9+zb27duHSpUqYdy4cbC0tMSYMWNw9OhRZGdnSx0qlQM8kyciEsmtW7cUZ/jXrl1D\n9+7d0atXL3Tt2lXx5E4igMP1RCSRtLQ0XLx4EW3atGFhKoPHjx/jwIED2L9/P/73v//BwcEBvr6+\n6N69O5o1a8Y+fjnH4XoqhPefCsdcCVNUnmQyGd6+fYvVq1djx44duHnzJvLz88UPTsOU9DNVq1Yt\njB07FkePHkVKSgq++OILJCUlwcfHB9bW1hg/fjzCwsJ07vG4/LsnLhZ5IiqRqlWrwsfHB1OmTEHT\npk0RGRmJ5cuXF5ochoSrVKkSevTogeDgYCQnJ+Pw4cOwtrbGokWLULNmTfj4+GDFihXMMZUYh+uJ\nqMweP34MExMTDt+rQVpaGk6cOIEjR47gt99+Q9WqVdG9e3d0794d7u7unKJYR7EnT0Ra4fnz56hW\nrZrUYeiE/Px8JCYm4siRIzhy5AiuXLkCFxcXdOnSBZ07d0bLli3L/S16uoI9eSqEvS7hmCthypqn\nN2/eYOPGjfjll18QGxuL9PR01QSmgcT4TOnp6aFNmzaYNWsWYmJikJycjFGjRuH27dvo27cvatas\nicDAQKxduxZ37txRezylwb974jKQOgAi0l2VKlXClClTkJycjMuXLyM4OBiWlpZo3749mjRpInV4\nWq9atWrw9/eHv78/AODevXs4efIkwsPDMXPmTFSuXFlxlt+pUydYWFhIHDGJjcP1RCSanJwc3Lx5\nEwDQvHlziaPRbXK5HFevXkV4eDjCw8MRHR2Nhg0bKoq+m5sbTExMpA6TisGePBHpnPT0dFSuXJn3\niKtBTk4O4uPjER4ejpMnT+LChQto1aoV3N3d4e7uDhcXF144qUFY5KkQzgktHHMljNh52r59O54+\nfYrmzZujadOmsLS01JqCr22fqYyMDMTFxSEqKgpRUVGIj4+HjY2Noui7ubmhRo0aKj+utuVJKqqq\ne+zJE5HG6N+/Px4/foxr164hNDQU2dnZaNq0Kbp168arxlXMxMQEHTt2RMeOHQEA2dnZuHDhAqKi\norBhwwaMGDECtWrVKlD0+Shi7cMzeSLSWE+fPsXdu3fRtm1bqUMpd/Ly8nD58mXFmX5UVBRMTEzg\n7u4OV1dXODs7w9bWFvr6+lKHqpM4XE9E5VpqaioePnyIJk2awNjYWOpwdJ5cLseNGzcQFRWFmJgY\nxMbGIiUlBe3atYOTkxOcnZ3h6OjIK/hVhEWeCmGvSzjmShhNztOjR48QHR2NpKQk1K5dG02bNkWT\nJk0ku3hMk3OlLs+ePUNcXBzOnj2L2NhYxMfHo2bNmoqi7+TkBDs7OxgY/N0ZLo95Kg325ImoXLO0\ntES/fv2Qk5OD27dv4/r164iIiICXlxfs7e2lDq9c+OijjxRT7ALvhvivXbumKPo///wz7t27h7Zt\n2yqKfm5ursRRly88kycinZGfn4+8vDxUqFCh0M8yMjJ4X7gEXr58ifj4eMTGxiI2Nhbnz5+HkZER\nHBwc0LZtW8V/a9asKXWoGoXD9UREJbBmzRpkZmbCxsYGDRs2hJWVFR/uIgG5XI67d+/i/PnzuHDh\nguK/lSpVUhT994W/evXqUocrGRZ5KoS9LuGYK2F0KU9yuRx//fUXbt26hdu3b+Phw4eoXbs2Bg0a\npJLb83QpV+pUVJ7kcjnu3LlToPBfvHgRVatWLXC237p1a7Xcu6+J2JMnIioBmUyGmjVrombNmnBx\ncUF2djYeP35cZIF//4+rtkzEo+1kMhkaNGiABg0aoG/fvgDetV6SkpIUhX/evHlITExExYoV0apV\nqwKvJk2aFLi4j/7GM3kion+5e/cudu/ejfr166N+/fqwsrJCjRo1WPQlJpfLcf/+ffz+++8FXg8f\nPkSzZs0KFX8zMzOpQy41DtfT/2vvXmPaKv84gH8xhPu1jAGjm1yKjLFAO24COh1mBkNGIJhMjDPq\nYnBRZ1xmvMU4fWemRrNMQcX4AsjULMqMGgQj2yIbg7g55OaAdcgAYQ1UaBnX5/9iaf/WdnCgtIXT\n7ydp0tLztE+/eZbfnnOecw4ROZBer8e1a9dw7do1aLVaTE1NITc3F7m5ua7uGv3H5OQk2traLAp/\nW1sbFAqFueBv374dycnJuOuuu+Dl5eXqLi+JRZ6s8JigdMxKGub0fxMTE5iZmUFYWJjVe0ajEefP\nn0deXp4Lera+OGtMmXb3m4r+H3/8gfb2dvT39yM2NhbJycnmwp+cnAyVSmXzrAxX4TF5IiInCgwM\nvO17TU1NOHHiBK5evQqlUono6GgolUoEBwdzF7+L3HHHHVCpVFCpVCgpKTH//ebNm+ju7kZ7ezva\n29tRVVWF9vZ2DAwMQKVSmYu+6REfH7+uj/dzJk9EtApmZmYwODiIgYEBXL9+HQMDAygqKkJ8fLyr\nu0YSTE1Noaury1z8TY+hoSHExcUhMTHR/DBdXdGRx/y5u56IaA1bbIX+mTNnEBAQAKVSifDwcM72\n1zCj0YgrV66gu7sb3d3d6OrqMj/38/OzKPqmR2xsrN2zfxZ5ssLjp9IxK2mYk3TLyaq1tRV//fUX\nBgYGYDAYEBERgcjISOzevXtd7xqWQi5jSgiBoaEhi6Jvej48PIzY2FgkJSXh66+/XtF1GHhMnoho\nnTJd1Q24tZt4eHgYf//9t83bti4sLOD69euIiIhYF6vC3YWHhwc2bdqETZs2WS24nJqaQk9PD7Ra\n7apcaMkenMkTEa1hBoMBNTU1GB0dRVBQEKKiohAZGYno6GjExMS4unvkINxdT0TkRubn53Hjxg0M\nDw9jeHgYCwsLeOihh6y2m52dxdzcHHx9fV3QS1otLPJkRS7HupyBWUnDnKRbK1ldvXoVJ06cgJeX\nFzZu3Ijw8HBs3LgRSqVyTVz3fa3ktNbxmDwREVmJjY3FK6+8Ar1ej9HRUYyMjKC/vx8Gg8FmkTcY\nDFhYWEBAQABX+csQZ/JERG7s0qVLaGhowOzsLBQKBTZs2ICwsDAkJCQgOjra1d1zW9xdT0REq+bm\nzZvQ6XTmh1KpREJCgtV2pr0CoaGhCA0Nhbe3twt6K38s8mSFx7qkY1bSMCfp3CWry5cvo6OjA2Nj\nYxgbG4OnpydCQ0ORl5cn6ep+7pKTvXhMnoiInC4lJQUpKSkAbl0Qxmg0YmxsDMHBwTa3//HHHzE6\nOoqQkBCEhoair68PcXFxiIiI4F4AJ+BMnoiIHGZ8fBw6nQ5jY2MYHx/H+Pg4/vnnH+zevRubN2+2\n2r67uxtCCAQHByM4OBi+vr5uuSCQM3kiIlrzQkJClnUjl5GREQwMDECv10Ov12Nubg7BwcHYu3cv\nwsPDrbY3Go3w8fFx+ZXl1irO5GWEx7qkY1bSMCfpmJU0y81penoaer0eoaGhNu/3XllZicHBQfj5\n+SEwMND82LVrF/z9/Vex587FmTwREcmet7f3ohfx2b9/PxYWFjA5OYmJiQnzw9Z/CACgvLwcQggE\nBAQgICAA/v7+8Pf3R0ZGhizvDcCZPBERuY2JiQkYDAZMTk5icnISBoMBBoMBu3btuu2eAk9PT/j5\n+cHf3x9+fn7w8/NDWlqazRsKrRaeQkdERORgw8PDMBqNVo/8/HyrdQBCCBw/fhw+Pj7w8/PDjh07\nsHXr1hV9L4s8WeExQemYlTTMSTpmJY2ccxJCYGxsDAaDAUajEQqFwuZiQSl4TJ6IiGgN8fDwgEKh\ngEKhcHVXzDiTJyIiWmNWq+7xxEIiIiKZcmiRP3PmDJKSkpCQkIBjx47Z3ObVV19FXFwc0tLS0NXV\n5cjuyF5jY6Oru7BuMCtpmJN0zEoa5uRcDi3yL7zwAioqKtDQ0IDjx4/jxo0bFu9fuHABZ8+eRWtr\nKw4fPozDhw87sjuyd+nSJVd3Yd1gVtIwJ+mYlTTMybkcVuT1ej0AYOfOnbjzzjvx4IMPorm52WKb\n5uZmPPzww1AoFCgtLUVnZ6ejuuMWxsfHXd2FdYNZScOcpGNW0jAn53JYkW9pabE4P3Dbtm04f/68\nxTYXLlzAtm3bzK/Dw8PR29vrqC4RERG5FZcuvBNCWK0edMe7Da0WrVbr6i6sG8xKGuYkHbOShjk5\nl8NOodPr9bj//vtx8eJFAMDzzz+P/Px8FBQUmLc5duwY5ubm8OKLLwIA4uPjbc7kWfiJiMjdrOmL\n4QQHBwO4tcJ+y5YtqK+vx5tvvmmxTVZWFg4dOoTHH38cdXV1SEpKsvlZPEeeiIho+Rx6xbsPPvgA\nZWVlmJ2dxcGDB7FhwwZUVFQAAMrKypCZmYl77rkH6enpUCgUqKqqcmR3iIiI3IpDj8nfd9996Ozs\nRE9PDw4ePAjgVnEvKysDcGuWX1tbC09PTzzxxBM2Z/K3O49eyjn4cmHP9QZiYmKQkpICjUaDzMxM\nZ3XZZZbKqqurC9nZ2fDx8cF77723rLZyYk9OHFOWqqurkZqaitTUVDz66KP4888/JbeVE3tycqcx\ntVROtbW1SE1NhVqtRkFBAVpaWiS3tUm4kFqtFqdPnxZarVYkJiaK0dFRi/ebm5tFbm6u0Ol0oqam\nRhQUFEhuKyf25BQTEyN0Op2zu+wyS2U1MjIiWlpaxOuvvy7efffdZbWVE3ty4piyzKqpqUmMj48L\nIYT44osvxGOPPSa5rZzYk5M7jamlcpqcnDQ/b2xsFPfee6/ktra4bHW9PefRS2krF6txvQHhJmsa\npGQVHh6O9PR0q/tGc0xJy8mEY+r/srOzzWuQCgoKcPr0aclt5cKenEzcYUxJycnf399iex8fH8lt\nbXFZkbfnPHopbeVipTn19fUBuHVmQl5eHoqKinDq1CnndNpF7BkXHFPSfyvH1O2z+uSTT7Bnz54V\ntV3P7MkJcJ8xJTWnb775BjExMXjqqafw6aefLqvtf63pW80Knkcvia2cTH799VdERUWhs7MTe/bs\nQWZmJiIjI53cQ5ITjinbGhoaUFVVhaamJld3ZU2zlRPHlKXi4mIUFxfjyy+/RFFRkflU9JVw2Uw+\nIyPDYoFYe3s77r77bottsrKy0NHRYX49OjqKuLg4pKenL9lWLuzJCQCioqIAAElJSSgsLMR3333n\nhF67hpSsHNF2vbH3t3JMWWd1+fJlPPPMMzh16hRCQkKW1VYO7MkJcJ8xtdwxsXfvXgwODmJqamrF\ndc9lRf7f59FrtVrU19cjKyvLYpusrCycPHkSOp0ONTU15tX3psGxWFu5sCcno9GIiYkJALcKf11d\nHfLz8537A5xISlYm/93zsZy26509OXFMWWfV39+PkpISVFdXQ6VSLautXNiTkzuNKSk59fb2mv/d\n/fDDD0hLS4Ovr+/K6569KwXt0djYKLZu3Sri4+PFhx9+KIQQory8XJSXl5u3efnll0VMTIzYsWOH\n6OjoWLStXK00p97eXpGamipSU1NFXl6eqKysdEn/nWmprIaGhoRSqRRBQUEiJCREbN68WUxMTNy2\nrVytNCeOKeus9u/fLxQKhVCr1UKtVouMjIxF28rVSnNytzG1VE7vvPOOSE5OFmq1Wjz55JOira1t\n0bZLcdhlbYmIiMi1XHqDGiIiInIcFnkiIiKZYpEnIiKSKRZ5IiIimWKRJ5IRnU4HjUYDjUaDqKgo\nKJVKaDQaBAYG4rnnnnPId1ZWVuLjjz++7ftfffUVjh496pDvJqLFcXU9kUy99dZbCAwMxKFDhxz6\nPTk5Oairq0NgYKDN92dmZpCTk4OWlhZesZLIyTiTJ5Ix0//hGxsbzdcKP3LkCMrKyrBz507Ex8fj\np59+whtvvIHt27fjwIED5jbd3d04cOAAsrKy8Oyzz0Kn01l9fnNzM6Kjo80FvqamBtnZ2UhNTUVp\naSkAwMvLCxqNBvX19c74yUT0LyzyRG6oubkZ33//PT7//HOUlJRApVKhra0NV65cwW+//QYAeOml\nl/Daa6+hubkZycnJ+Oyzz6w+5+LFi+YrLALA22+/jZ9//hm///47KioqzH9PSkoyfy4ROc+avkEN\nEa0+Dw8PFBYWIjAwENnZ2ZiensYjjzwCDw8PZGVl4dy5c9iyZQvOnj2LwsJCAMD8/DxiYmKsPqun\np8fiDojp6ekoLS3Fvn37UFxcbP57fHw8vv32W4f/NiKyxCJP5IZM19D28vKCt7c3vL29za9nZmYw\nPz+PsLAwSXe/+veyHtPdxaqqqnD06FHz/a4XFhZ4PJ7IBbi7nsjNLLXWVgiByMhIxMbG4uTJkxBC\nYHZ21uJOhyYJCQnQarXmdlqtFjk5OXj//fcxNDSE6elpAEBfXx8SExNX/bcQ0eJY5IlkzDR79vDw\nsPn839v89/VHH32EX375BWq1GhqNBufOnbP6fLVabb795dzcHPbt24eUlBQ88MADOHLkiHkPQVdX\nFzQazer/QCJaFE+hIyK7ZGdno66uDkFBQTbfn56eRk5ODlpbW7nLnsjJOJMnIrs8/fTTqK6uvu37\ntbW1KC0tZYEncgHO5ImIiGSKM3kiIiKZYpEnIiKSKRZ5IiIimWKRJyIikikWeSIiIplikSciIpIp\nFnkiIiKZ+h98s+6VbcdyjgAAAABJRU5ErkJggg==\n"
}
],
"prompt_number": 31
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Experimental measurements of spin-spin decay shows that it occurs at a\n",
"much faster rate than spin-lattice. Comparison of T1 (spin-lattice) and\n",
"T2 (spin-spin) decay constants at various B0 field strengths are:\n",
"\n",
"<table>\n",
" <tr>\n",
" <td></td>\n",
" <td colspan=3><b>T1</b></td>\n",
" <td colspan=3><b>T2</b></td>\n",
" </tr>\n",
" <tr>\n",
" <td><b>Field</b></td>\n",
" <td><i>1.5T</i></td>\n",
" <td><i>3.0T</i></td>\n",
" <td><i>4.0T</i></td>\n",
" <td><i>1.5T</i></td>\n",
" <td><i>3.0T</i></td>\n",
" <td><i>4.0T</i></td>\n",
" </tr>\n",
" <tr>\n",
" <td><i>White</i></td>\n",
" <td>0.64</td>\n",
" <td>0.86</td>\n",
" <td>1.04</td>\n",
" <td>0.08</td>\n",
" <td>0.08</td>\n",
" <td>0.05</td>\n",
" </tr>\n",
" <tr>\n",
" <td><i>Gray</i></td>\n",
" <td>0.88</td>\n",
" <td>1.20</td>\n",
" <td>1.40</td>\n",
" <td>0.08</td>\n",
" <td>0.11</td>\n",
" <td>0.05</td>\n",
" </tr>\n",
"</table>\n",
"\n",
"**Source:** Jezzard and Clare, Chapter 3 in the Oxford fMRI Book\n",
"\n",
"Also, notice that the peak difference occurs a very short time compared to the T1 difference."
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"plot(t_T1, abs(MzG_T1 - MzW_T1), label=\"T1\")\n",
"plot(t_T2, abs(MzG_T2 - MzW_T2), label=\"T2\")\n",
"xlabel('Time (s)')\n",
"ylabel('Transverse magnetization difference')\n",
"legend();"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "display_data",
"png": 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7j3Mf85R4POk4Pj6ehQsXAhAZGYkSrQhnz57V6tevryUlJWlHjx7N96OiQYMG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\nzDC5ExERhZn/Aeq+BZPbNwBKAAAAAElFTkSuQmCC\n"
}
],
"prompt_number": 32
},
{
"cell_type": "heading",
"level": 4,
"metadata": {},
"source": [
"Question 7"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Suppose you wanted to measure brain structure, and you were\n",
"particularly interested in gray/white differences. Based on the T1 and\n",
"T2 curves we have drawn, would you choose to distinguish these two\n",
"tissues using T1 or T2? "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The T2 difference is so short that measurement of the T2 signal would be a\n",
"problem were it not for the invention of an important measurement method\n",
"called 'Spin Echo'. We will begin the next tutorial by explaining that\n",
"idea.\n",
"\n",
"In fact, there are many essential imaging ideas that we have not yet\n",
"explored. For example, how can we measure T1 and T2 separately? Perhaps\n",
"most importantly, how can we form an image? These are the topics we will\n",
"take up in the next tutorial."
]
}
],
"metadata": {}
}
]
}
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