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Using Optimization to cure Cancer

High doses of radiation can kill cells or prevent them from growing and dividing. This is true for cancer cells and normal cells, but radiation therapy is attractive as a cancer treatment because the repair mechanisms for cancer cells are less efficient than for normal cells. This observation has inspired the idea of radiation therapy, where high energy radiation is beamed at a patient. Recent advances in imaging and radiation delivery have enabled radiation therapy as an effective form of cancer treatment. On the imaging front, advances in computed tomography (CT) and magnetic resonance imaging (MRI) provide a means for accurately visualizing the details of tumorous regions. On the delivery front, intensity-modulated radiation therapy and tomotherapy enable a machine to deliver large doses of radiation beamed from different angles into the target area (e.g. the brain or other parts of the body for other types of cancer).

As a beam of radiation enters a target area, the dosage of radiation dissipates as it passes through the target region, such that the largest dose is delivered where the beam enters and the smallest dose delivered when the beam leaves. Figure 1a shows the relative intensity of a beam delivered from the upper left side of the target region whose dosage has decreased somewhat when the beam hits the tumor. By using multiple beams aimed from different directions, a large dose can be delivered to the tumor as multiple beams intersect over the tumorous region. Figure 1b offers a simple example demonstrating the significance of having multiple beams, wherein the largest dose is now at the center of the target region where the tumor resides.

In conventional radiotherapy, only a small number of beamlets (e.g. 4 to 7) are used. The radiation oncologist and physicist work together to determine the set of beam angles and beam intensities, usually by a manual trial and error process. Determining the optimal beamlets is extremely difficult to solve by hand, as there are thousands of possible choices. Furthermore, a target area (see Figure 1c) is likely to contain not only tumorous areas but also critical areas where only a small amount of radiation (or none at all) is allowed. With a small number of beamlets, many beams may need to pass through the critical area to reach the tumorous region, making it dicult to deliver a high dosage over the tumorous region and low dosages over critical areas. This problem is particularly severe in the brain, where every non-cancerous cell is critical.

The imaging and radiation delivery technologies are now at a point where a much larger number of beamlets can be used and radiation can be delivered accurately. The only missing component is an automated process to figure out the radiation delivery so as to maximize the radiation dosage to the tumorous area while minimizing the dosage to the critical area. Here is where optimization comes in.

The radiotherapy oncologists have provided us with sample 2D images of the tumorous and critical areas, along with a set of possible beamlet origins and angles. To reduce complexity, we have transformed the data into a more friendly format. Each provided image has been transformed into a binary matrix where each element of the matrix corresponds to a pixel in the original image, where a 1entry indicates the presence of the attribute shown in the image. For example, the binary matrix that corresponds to the tumor area will have a 1 entry if a tumor was present at the corresponding pixel and 0 otherwise. Note that the tumorous and critical areas need not be rectangular, and multiple tumorous regions may exist within the considered target region.

The beamlet data has been converted to a set of relative intensity matrices, where each matrix corresponds to the relative intensity delivered by a particular beam, normalized between 0 and 1 inclusively. This is the data that captures the variation in the intensity of radiation as the beam passes through a region and also the imprecision in a particular beam. Each entry of a matrix again corresponds to a pixel in the image of the target area, such that an entry in the matrix is nonzero only if the beam passes through the corresponding pixel. Note that each beam matrix corresponds to a beam origin-angle combination; it is possible for there to be multiple beams in the set of possible beamlets with the same origin but aimed at different angles.

We now formulate a basic LP model for the problem proposed here by the Oncologist which will satisfy the upper limit on the critical cells and lower limit on tumor cells while maximizing radiation on tumor cells and minimizing the same on critical cells. We decide to set the function to minimize the radiation on critical cells and adding a penalty function for radiations on the tumor cells.

We define the various sets and parameters for the formulation as follows-

Let,

B — set of matrices for beamlet intensity

R — set of rows

C — set of columns

U — the upper limit of radiation on critical cells

L — the lower limit of radiation on tumor cells

bijk — parameter indicating beam intensity from beamlet i on row j and column k

ajk — parameter indicating presence or absence of critical cell on row j and column k

ejk — parameter indicating presence or absence of tumor cell on row j and column k

Xi — decision variable indicating intensity to be set for beam i for treating

This model, when formulated in AMPL, did not give 0 as the solution for both small and actual examples. Hence, we reformulated the objective function so as to only minimize the radiations in the critical area. Even with this formulation, we were able to find a feasible solution for the small example, but no feasible solution was found for the actual example.

As we did not get feasible solution in the first formulation, we now try to relax our constraints to get a feasible solution for the actual example and we can also see the corresponding changes in the small example. We will introduce slack variables to the constraints from the previous formulation and change our objective to minimize the sum of these slack variables.

This formulation will give us the values for slack variables and this would help the oncologist in adjusting the limits on the critical and tumor areas to obtain a feasible solution using the first model.

visualizing beam intensities on the Tumor area of the actual case with Model 2

For penalizing the radiations that are incident at the bordering regions of the non-critical region, we use rolling indexes m and n to which range from -1 to 1 to locate the bordering regions of the critical region and we then penalize our objective function using multipliers on these bordering binary indexes.

visualizing beam intensities on the Tumor area of the actual case with model 3

Weighted Slacks — From model 2, we see that the value of the slack variables can be used by the oncologist as a reference for the range of adjustments to the upper and lower limit on critical and tumor regions respectively. By intuition, we can compromise on changing the limits on the tumor region, but we would not want to change the limits on tumor region. So, we propose adding weights to the critical and tumor slacks in the objective function such that the sum of slacks corresponding to the critical region are minimised and we can make minimum adjustments in the limits of the critical region.

For implementing this, we would not need additional data. The only additional task is to figure out the weights that will give us the most effective solution.

Minimizing total Radiation — As we know, no radiation is beneficial to the human body whether it is on the tumor area, critical area or the non-critical area. The goal is to enhance the quality of the life of the patient and by minimizing the total radiation sent over all the cells would be a decent objective with the same limits on tumor and critical regions.

To implement this, we use the same formulation as in model 2. Here, we update our objective function by including penalties on radiations passing through both critical and tumor regions. Also, the model will automatically try to send zero or no radiations in the non-critical regions.

Again, there is no additional data needed for implementing this model. Only additional task is to put a weight on the part of the objective with the sum of slacks so that there is enough room for the penalties in the objective function.

Including cell recovery data — It is obvious that no tumor cell can be treated with one session of radiation therapy. We can obtain past data and analyse the recovery time for certain critical cells in the body. Doing this we might be able to up the minimum limit on critical area. We can run the model before every session of therapy with updated limits on critical and tumor regions.

We can also think of a future advancement where passing a negative beam intensity will heal the cell and not damage. Implementing this would not be difficult as we would set out X variable free so that it can also take negative values. Hence, with the set of positive intensity beams which damage the tumor cells, we will have beams with negative intensities that will heal the critical cells.

Weighted Slacks

Where pjk and qjk are slack variable to minimize and wt and wc are the weights for corresponding slack variables for tumor and critical regions respectively.

For calculation of slacks, we tried different combinations of wt and wc including complete elimination of critical slacks. We observed that the model was maximizing slacks for the tumor region while adjusting radiations in the allowed range of the critical areas. So we came up with a logic to get the weights. We used the ratio of total critical cells to that of total tumor cells and used this ratio as the weights.

This ratio worked because we have less area of critical region as compared to tumor region in the current instance. There might be instances where the critical area is equal or even larger. For this, one might have to use different combinations to come u with the best solution.

visualizing beam intensities on the Tumor area of the actual case with model 4

We can see a significant reduction in the amount of radiations in the critical area as compared to previous models.

Minimizing total Radiation

After building all these models, we were curious to know which model performed the best. For this, we calculated total radiations on critical and tumor regions separately.

The following table summarises the total radiations obtained from all models on the actual example (excluding model 1)

We can observe that model 3 gives us minimum radiation on the critical area whereas model 5 gives maximum on the tumor area. Additionally, model 5 was also derived from model 3 with a few adjustments in the objective function.

Hence, we dived in deep to visualize the radiations on specific areas. We took help of the color scaling option in excel to visualize the data from the radiations.

From above color scale visuals, we can see that while model 3 has minimum total radiations on the critical region, it violates upper limits only on a few critical cells (highlighted in green). Doing this, many tumor cells (highlighted in red) in red have been left out without any radiation incident on them.

On the other hand, model 5 has more upper limits violated for critical cells (highlighted in green) than model 3 but it is able incident the required amount of radiation on the tumor cells (highlighted in red)

One thing that catches our attention here is that while model 5 violates the upper limits on the critical cells, it violates those with large intensities incident on them. This means, that a few of the critical cells have high chances of damaging completely which may not be desirable. Model 3, on the other hand violates only a few critical cell upper limits and that too with smaller impact. Yes, model 3 leaves out quite a good number of tumor cells untreated but we can't afford to lose critical cells for the cost of damaging a few extra tumor cells.

We conclude and suggest model 3 to be used by the Oncologist and treat the patient in 2 stages. We can have a new and reduced tumor region after the first stage and a similar model can be run for treatment in the second stage. The total radiations on the critical region are such small that we can afford to have two radiation sessions in the same setup which will double the radiations in the critical region which would still be much lesser total radiations from other models.

How Donald Trump Ruined My Hotel Stays

Using Optimization to cure Cancer

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