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In this video, we'll solve our
simple example in LibreOffice.

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Go ahead and open
the spreadsheet,

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IMRT_SimpleExample.

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At the top of the spreadsheet,
you should see our data.

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For each beamlet
and each voxel, we

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have the dose that
that beamlet gives

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to that voxel at unit intensity.

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So we have this data for voxel
one, voxel two, voxel three,

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all the way up to voxel nine.

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Each row corresponds to
one of the six beamlets.

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This is the data that
we saw on the slides

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in the previous video.

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Below the data, we've outlined
our decision variables,

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which are the intensities
of the beamlets.

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So for beamlets one through six,
we have one decision variable.

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These six decision variables
are outlined in yellow.

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Right now, the
decision variable cells

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are blank, because the values
will be filled in by Solver.

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Below the decision variables,
we have our objective.

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

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The healthy-tissue voxels are
voxels one, three, five, six,

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and nine.

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So let's go ahead and build
our objective in the blue cell

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here.

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So first, we want to
add up the total dose

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that each beamlet
gives to voxel one.

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So here we'll use
the function that we

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used in the previous
lecture, sumproduct.

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So type an equals sign,
and then sumproduct,

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and select all of the decision
variables, semicolon, and then

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all of the doses.

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This will add up the total
dose that beamlet one gives

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to voxel one, plus the
total dose beamlet two

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gives to voxel one,
plus the total dose

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beamlet three gives
to voxel one, etc.

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Now we want to repeat this
for voxels three, five, six,

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and nine-- the other
healthy-tissue voxels.

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So go ahead and
type a plus sign,

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and then sumproduct, again,
the six decision variables,

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semicolon, and this time, select
the dose data for voxel three.

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Now let's repeat this again,
but this time for voxel five.

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So sumproduct, and
then the decision

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variables, and the dose
data for voxel five.

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Now for voxel six, sumproduct,
the decision variables,

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semicolon, and the dose
data for voxel six.

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And lastly, we're going to add
the sumproduct of the decision

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variables, semicolon, and then
scroll over to voxel nine,

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and select the dose
data for voxel nine.

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Close the parentheses,
and hit Enter.

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You should see that the
objective has a 0 right now,

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because none of our
decision-variable values

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are filled in.

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When Solver fills in
our decision variables,

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our objective
value will be here.

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Below the objective
is our constraints.

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The first four constraints
make sure that each voxel

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of the tumor is getting
a dose of at least 7.

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The last constraint makes
sure that the spinal cord

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receives a dose
of no more than 5.

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Let's go ahead and
construct our constraints.

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For the first four
constraints, the left-hand side

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is going to be the total dose
that that voxel of the tumor

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gets.

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So for voxel two, we
have the left-hand side

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is equal to the
sumproduct of the decision

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variables, semicolon, and
then the data for voxel two.

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Close the parentheses,
and hit Enter.

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We want to make sure
that this value is

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greater than or equal to 7.

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Now let's repeat
this for voxel four.

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So equals sumproduct,
and then, in parentheses,

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select the decision
variables, semicolon,

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and the data for voxel four.

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Again, we also want this one to
be greater than or equal to 7.

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Now let's repeat
this for voxel seven.

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So sumproduct of the decision
variables, semicolon, and then

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the data for voxel seven, again,
greater than or equal to 7.

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And lastly, for voxel eight,
we want the sumproduct

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of the decision variables,
and the data for voxel eight

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this time to also be
greater than or equal to 7.

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And our last constraint,
we want to make sure

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that the total
dose to voxel five,

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the spinal cord
voxel-- so sumproduct

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of the decision variables, and
then the data for voxel five,

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is less than or equal to 5.

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The remaining
constraints we have

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are the non-negativity
constraints,

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which we'll add in
directly in the Solver.

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So now go ahead and go to the
Tools menu, and select Solver.

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The Solver window should pop up.

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First we need to tell Solver
what our objective is.

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So go ahead and delete
what's in "Target cell",

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and making sure that your
cursor is in "Target cell",

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select the blue objective cell.

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Now, we want to change "Maximum"
to "Minimum", because we're

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trying to minimize the total
dose to healthy tissue,

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and our decision variables
should be the six yellow cells.

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Now let's add in
our constraints.

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So in the first
"Cell reference" box,

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let's select the first
four constraints.

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Because they're all greater
than or equal to constraints,

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we can add them in together.

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And change the "Operator"
to greater than or equal to,

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and then in "Value", select
the four right-hand sides.

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Now let's add in the
spinal-cord constraint.

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So in the next "Cell
reference" box,

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select the spinal
cord left-hand side,

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make sure that
"Operator" is less than

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or equal to, and in
the second "Value" box,

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select the spinal
cord, right-hand side.

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Now let's add our
non-negativity constraints.

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So in the "Cell reference", just
directly pick the six decision

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variables, and make
sure the "Operator"

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is greater than or equal to, and
the "Value" should just be 0.

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Now, in the Options,
make sure you've

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selected the Linear
Solver, and click OK.

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Now go ahead and hit Solve.

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You should see a
solving result that

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says: "Solving
successfully finished.

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Result: 22.75".

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That's the optimal
objective function value.

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Go ahead and select Keep Result.

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Now let's take a
look at our solution.

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So the optimal
solution is to have

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beamlet one at an
intensity 2.25,

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beamlet two at an
intensity of 0,

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beamlet three at
an intensity of 3,

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beamlet four at an
intensity of 3.5,

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beamlet five at an
intensity of 2.5,

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and beamlet six at
an intensity of 0.

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This makes sense,
because beamlet two goes

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across the spinal cord,
and beamlet six only

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goes down healthy-tissue voxels.

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And if we look at
our constraints,

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we can double-check that each
tumor voxel is receiving a dose

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of at least 7 -- one tumor
voxel gets a dose of 8 --

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and the spinal cord is
receiving a dose of 5,

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which is the maximum
possible dose.

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In the next video, we'll see
an example of a real problem,

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and how big the problem is
on an actual tumor case.