1.Modeling and Linear
About half of all cancer patients are treated with intensity-modulated radiation therapy (IMRT),
a method invented in the early 1980s. When deciding a treatment plan with IMRT, oncologists
first start by taking a scan of a patient. Figure 1 shoes one such example; here the “Target” is
the cancerous tumor, which is surrounded by several “Critical Structures” (these could be a
kidney, liver, etc.) as well as by other healthy tissue. This scan is then divided into many small
blocks (called “voxels”), which are labeled as tumorous tissue, critical tissue, or healthy (noncritical) tissue.
Once this task is done, the next critical step is to decide the optimal treatment plan. The key
idea behind IMRT is to shine high-energy protons through many beams with adjustable
intensity (shown in Figure 1 as Beam 1, Beam 2, etc.). This allows the oncologist to give a high
dose of radiation to the target tumor, while not exposing the surrounding critical structures and
healthy tissue to extensive radiation (radiation damages any type of tissue, and thus too much
radiation on healthy organs can cause severe side-effects). The critical decisions behind the
treatment plan involves the intensities of the various beams.
In this problem, we will be deciding the treatment plan for a very simple case. More precisely, we will be looking at a patient case with 9 voxels and 6 beams, shown in Figure 2. The voxels are shaded according to the type of tissue. The white voxels (i.e., voxels 2, 4, 7 and 8) are tumor voxels, the dark blue voxel (i.e., voxel 5) is a critical structure, and the light blue voxels (i.e., voxels 1, 3, 6 and 9) are healthy nearby tissue.
When a beam is used, it gives a different radiation dose to each voxel, due to the tissue that the beam has to travel through. Figure 3 shows the radiation doses received by each voxel when each beam is used with a unit intensity (i.e., with intensity = 1). To understand this, if Beam 1 is used with intensity = 1, it will give a dose of 2 to Voxel 3, and a dose of 1 to Voxel 1. If Beam 1 were used with intensity = 1.5, each voxel would receive an extra 50% dose (for example, the total dose to Voxel 3 would be 3). When several beams are used simultaneously, their respective doses add up (for instance, when Beams 1 and 6 are each used with intensity 1, Voxel 3 will receive a total dose of 2+1 = 3). Keep in mind that the intensity of a beam can be any non-negative number.
For this patient, the oncologist would like to ensure that each tumor voxel gets a total dose of at least 7, and each critical structure voxel gets a total dose of at most 5, while the sum of the total dose delivered to the healthy tissue and the critical structure is minimized.
1. Formulate the problem of determining the optimal beam intensities for this particular patient’s case as a linear optimization problem. Be sure to clearly indicate what your decision variables, objective and constraints are. What is the optimal objective function value, and what are the optimal values of the decision variables?
Address each of the following questions separately. Make sure your final models remain linear. If you are unable to construct a model in Q1, try to describe in words what changes you would make for Q2 and Q3. For instance, what the (new) decisions variables, objective and constraints would be.
2. The oncologist realized that the precise location of the critical structure may change slightly when treatment is delivered (e.g., due to a slightly different positioning of the patient or even due to breathing). As such, in addition to the constraints already described, the oncologist would like to ensure that the total dose delivered to Voxels 2, 3, 5 and 6 is at most 20% of the total dose delivered to all the nine voxels. Update the model you constructed in 1. to address this, and then re-solve the model. Did the optimal objective function value increase or decrease? Can you explain why?
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