exercises/Ex2-Multiband-Pulse-Design.ipynb

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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Exercise 2: Multiband Pulse Design\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Building on the SLR designs of exercise 1, we will now design multibanded RF pulses to excite several slices of spins simultaneously.\n",
    "\n",
    "Again, documentation for all pulse design tools can be found [here](https://sigpy.readthedocs.io/en/latest/mri_rf.html).\n",
    "\n",
    "We again begin by importing our packages:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "%matplotlib notebook\n",
    "\n",
    "# typical sigpy and numpy imports\n",
    "import sigpy\n",
    "import numpy as np\n",
    "import sigpy.mri.rf as rf # import for our RF pulse design tools \n",
    "import sigpy.plot as pl\n",
    "\n",
    "import matplotlib.pyplot as pyplot"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Problem 2a: design an SLR excitation pulse"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This problem is a review of the previous exercise! Design an SLR excitation pulse using the dzrf() function, and verify it produces transverse magentization. \n",
    "\n",
    "* Using [dzrf()](https://sigpy.readthedocs.io/en/latest/generated/sigpy.mri.rf.slr.dzrf.html#sigpy.mri.rf.slr.dzrf), design a least-squares excitation pulse with 512 samples.\n",
    "* Again, use a time-bandwidth product of 8.\n",
    "* Use relative maximum ripple levels in both passband and stopband of 0.01\n",
    "\n",
    "Once you have designed and plotted the pulse, simulate it using [rf.sim.abrm()](https://sigpy.readthedocs.io/en/latest/generated/sigpy.mri.rf.sim.abrm.html#sigpy.mri.rf.sim.abrm), as in the previous exercise. This time, look at the Mxy profile instead of Mz. The conversion from SLR polynomials to magnetization response for an excitation pulse is :\n",
    "\n",
    "\n",
    "<center>M<sub>XY</sub> = 2$\\alpha\\beta^*$</center>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
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   "outputs": [],
   "source": [
    "# assign values to the variables below\n",
    "tb = \n",
    "N = \n",
    "d1 = \n",
    "d2 = \n",
    "p_type = \n",
    "f_type = "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "pulse = # use the dzrf() function again to design a pulse"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# plot the pulse: \n",
    "pl.LinePlot(pulse, mode='r')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "[a, b] = # find the a and b polynomials using abrm().\n",
    "Mxy = # convert from a and b to Mxy using the equation above for Mxy\n",
    "pl.LinePlot(Mxy)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The Mxy profile that you plot should consist of a middle segment with value of approximately 1, corresponding to a band of excited spins, bordered on either side by segments with value of approximately 0, corresponding to spins that are not excited by our SLR pulse."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Problem 2b: Multiband the excitation pulse\n",
    "\n",
    "Now that we have a simple SLR excitation pulse designed, we can perform multibanding in order to have the pulse excite several slices simultaneously.\n",
    "\n",
    "* Use the [*rf.multiband.mb_rf()*](https://sigpy.readthedocs.io/en/latest/generated/sigpy.mri.rf.multiband.mb_rf.html#sigpy.mri.rf.multiband.mb_rf) function to create a pulse that excites 5 slices.\n",
    "* Separate the pulses by 3 slice widths (*Hint: you won't simply set band_sep = 3, but 3 times some quantity*)\n",
    "* Experiment with designing the pulse with all 3 of the provided phase schedules. For this example, the differences should be minimal."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# assign values to the variables below\n",
    "n_bands = \n",
    "phs_type =  # 'phsMod', 'ampMod', or 'quadMod' - a string\n",
    "band_sep =  # separate by 3 slice widths - 3 times some quantity...\n",
    "mb_pulse = # function to multiband the SLR pulse\n",
    "pl.LinePlot(mb_pulse, mode='r')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# abrm() provided to simulate: \n",
    "[a, b] = rf.sim.abrm(mb_pulse, np.arange(-20*tb, 20*tb, 40*tb/2000), True)\n",
    "Mxy = # convert from a and b to Mxy\n",
    "pl.LinePlot(Mxy)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Problem 2c: PINS Pulse Design\n",
    "\n",
    "Now, let's try to directly design a multiband pulse using the PINS (Power Independent Number of Slices) method. PINS interleaves a train of nonselective rectangular RF pulses that sample a slice-selective envelope with small slice gradient blips (in the Fourier domain, equivalent to convolving a single-slice profile with a train of delta functions). Unlike the previous examples, we will design this pulse with specific instead of arbitrary units in mind: \n",
    "\n",
    "* Given:\n",
    "    * A time bandwidth product of 8\n",
    "    * A maximum gradient amplitude of 4 gauss/cm\n",
    "    * A maximum gradient slew rate of 18000 gauss/cm/s\n",
    "    * A hardware dwell time of 4e-6 seconds\n",
    "    * A maximum b1 amplitude of 0.18 gauss.\n",
    "    * d1 and d2 of 0.01\n",
    "    * ptype = 'ex' (excitation)\n",
    "    * ftype = 'ls' (least squares)\n",
    "\n",
    "\n",
    "* Using [*rf.multiband.dz_pins*](https://sigpy.readthedocs.io/en/latest/generated/sigpy.mri.rf.multiband.dz_pins.html#sigpy.mri.rf.multiband.dz_pins), design both rf waveforms and gradients necessary to excite a series of slices that are:\n",
    "    * Separated by 2.5 cm\n",
    "    * With a slice thickness of 0.3 cm\n",
    "\n",
    "* Simulate the transverse magnetization profile using the rf.sim.abrm() simulator. \n",
    "\n",
    "Skeleton code for plotting with the appropriate units and labels is provided below."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
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   "outputs": [],
   "source": [
    "# fil in the variables below\n",
    "tb = \n",
    "sl_sep =  # cm\n",
    "sl_thick =  # cm\n",
    "g_max = # gauss/cm\n",
    "g_slew = # gauss/cm/s\n",
    "dt =  # seconds, dwell time\n",
    "b1_max = # gauss\n",
    "d1 = \n",
    "d2 =\n",
    "ptype = \n",
    "ftype = \n",
    "[rf_pins, g_pins] = # insert dz_pins() function to design the multiband PINS pulse"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Plotting functions provided: "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "t = np.arange(0, np.size(rf_pins)) * dt\n",
    "pyplot.figure()\n",
    "pyplot.plot(t * 1000, np.real(rf_pins))\n",
    "pyplot.title('PINS RF')\n",
    "pyplot.ylabel('Gauss')\n",
    "pyplot.xlabel('milliseconds')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "pyplot.figure()\n",
    "pyplot.plot(t * 1000, g_pins)\n",
    "pyplot.title('PINS Gradient')\n",
    "pyplot.ylabel('Gauss/cm')\n",
    "pyplot.xlabel('milliseconds')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# simulate it with Bloch simulator\n",
    "x = np.reshape(np.arange(-1000, 1000), (2000, 1)) / 1000 * 12 # cm\n",
    "[a, b] = rf.sim.abrm_nd(2 * np.pi * dt * 4258 * rf_pins, x, \n",
    "                        np.reshape(g_pins, (np.size(g_pins), 1)) * 4258 * dt * 2 * np.pi)\n",
    "Mxy = 2 * a * np.conj(b)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "pyplot.figure()\n",
    "pyplot.plot(x, np.abs(Mxy))\n",
    "pyplot.ylabel('|Mxy|')\n",
    "pyplot.xlabel('cm')"
   ]
  }
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