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In this video, we'll discuss
how radiation therapy can

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be framed as an
optimization problem.

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The data's collected in the
treatment planning process,

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which starts from a CT scan,
like the one you see here,

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on the right.

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Using a CT scan, a
radiation oncologist

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contours, or draws
outlines around the tumor

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and various critical structures.

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In this image, the
oncologist would

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contour structures like
the parotid glands,

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the largest of the saliva
glands, and the brain.

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Then, each structure
is discretized

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into voxels, or volume
elements, which are typically

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four millimeters in dimension.

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The second image here shows
a closer view of the brain.

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You can see the small
squares, or voxels.

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Here, they're two-dimensional,
but in reality they

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would be three-dimensional.

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Now, we can compute how
much dose each beamlet,

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or piece of the beam,
delivers to each voxel.

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We'll start with
a small example.

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Suppose we have nine
voxels and six beamlets.

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Our voxels can be
categorized into three types:

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the tumor voxels, which
are colored pink here;

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the spinal cord voxel,
colored dark green;

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and other healthy tissue
voxels, colored light green.

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So we have four tumor voxels,
one spinal cord voxel,

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and four other
healthy tissue voxels.

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We have two beams that are
each split into three beamlets.

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Beam 1 is composed of
beamlets 1, 2, and 3,

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and comes in from the right.

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Beam 2 is composed of
beamlets 4, 5, and 6,

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and comes in from the top.

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Our objective is to minimize the
total dose to healthy tissue,

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both to the spinal cord and
to the other healthy tissue.

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We have two types
of constraints.

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The first is that the
dose to the tumor voxels

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must be at least
7 Gray, which is

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the unit of measure
for radiation.

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Our second constraint is that
the dose to the spinal cord

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voxel can't be more
than 5 Gray, since we

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want to be careful to
protect the spinal cord.

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We know the dose
that each beamlet

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gives to each voxel
at unit intensity.

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This table shows the dose
that each beamlet in Beam 1

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gives to the voxels.

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Remember that this
is at unit intensity.

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If we double the
intensity of the beamlet,

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we double the doses.

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The dose to each
voxel can depend

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on how far the beamlet has to
travel, or the type of tissue

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that the beamlet has
to travel through.

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Similarly, we know the dose
that each beamlet in Beam 2

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gives to each voxel,
again at unit intensity.

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The dose depends on the
direction of the beam

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and what it travels through.

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Putting these
tables together, we

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can write out our
optimization problem.

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Our decision variables are the
intensities of each beamlet.

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We'll call them x_1, the
intensity for beamlet 1, x_2,

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the intensity for
beamlet 2, x_3,

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the intensity for
beamlet 3, etc.,

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all the way up through x_6.

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As we mentioned
before, our objective

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is to minimize the total dose
to the healthy tissue, including

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the spinal cord.

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So we want to minimize the total
dose beamlet 1 gives to healthy

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tissue, which is (1 + 2)*x_1,
plus the total dose beamlet 2

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gives to healthy tissue,
which is (2 + 2.5)*x_2,

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plus the total dose beamlet
3 gives to healthy tissue,

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which is 2.5*x_3.

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Now for beamlets 4, 5, and 6,
beamlet 4 just gives one dose

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to healthy tissue, beamlet
5, 2*x_5, and then beamlet 6,

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we have (1 + 2 + 1)*x_6.

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Now for our constraints.

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First, we need to make sure
that each voxel of the tumor

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gets a dose of at least 7.

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Let's start with the first
tumor voxel in the top row.

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So 2*x_1 + x_5 needs to be
greater than or equal to 7.

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Now the tumor voxel
in the second row,

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we have x_2 + 2*x_4, also
greater than or equal to 7.

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Now for the two tumor
voxels in the bottom row,

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we have 1.5*x_3 + x_4,
greater than or equal to 7.

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And 1.5*x_3 + x_5, greater
than or equal to 7.

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Then for the spinal cord, we
need to make sure that 2*x_2 +

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2*x_5 is less than
or equal to 5.

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And lastly, we just
need to make sure

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that all of our decision
variables are non-negative.

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So they should all be
greater than or equal to 0.

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Now that we've set up
our optimization problem,

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we'll solve it in LibreOffice
in the next video.