Team Number Adewale Adeniran, Thomas Cuttino, Brian Lane Paper Submitted to the 2003 Mathematics Competition in Modeling

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1 Team Number Adewale Adeniran, Thomas Cuttino, Brian Lane Paper Submitted to the 2003 Mathematics Competition in Modeling

2 Team Number Introduction: Gamma Knife Treatment Plan Stereotactic Radiosurgery is the latest way of fighting cancer. Ionizing radiation is applied to a well-defined tumor without delivering any amount of radiation to the surrounding tissue. The three common modalities used in Stereotactic Radiosurgery are: the gamma knife unit, heavy charged particle beams, and external high- energy photon beams from linear accelerators. The gamma knife unit delivers its dose using 201 cobalt-60 unit sources through a heavy helmet. All 201 beams simultaneously intersect, resulting in a spherical dose distribution. 4 inter-changeable collimator helmets with beam channel diameters of 4, 8, 14, and 18 are available for use. The most common tumors normally take between 1 and 15 shots or doses. The target area is a bounded three-dimensional digital image with millions of points. The goal of radio-surgery is to deplete the tumor cells while still preserving the normal structures of the brain. A treatment plan is needed that can limit the variance of biological uncertainties and physical limitations. The basic goals are to minimize the number of hotspots caused by overlapping, match specified isodose contours with the tumor, prohibit shots from damaging tissue outside of the tumor or touching each other which would cause hot-spots that would damage more tissue, treat a minimum of 90% of the tumor, all while using the fewest shots possible. The Problem: The problem at hand is that of the gamma knife treatment planning in which we are assigned to model a situation in which 201 cobal-60 unit sources target a spherical region within the brain of a patient diagnosed with a tumor. The available sizes of radiation doses are as follows: 4mm diameter, 8mm diameter, 14mm diameter, and 18mm diameter. The main objective of our modeling is to formulate a series of shots in which no two shots overlap so as to cause a hot-spot in the brain, and the damage outside of the cancerous region is minimized, while obtaining a minimum of 90% destruction of cancerous regions. The final requirement is to use the least possible number of shots, which is 1 to 15 for normal tumors. Assumptions We assume that the tumor has grown equally in all directions, and is thus roughly spherical ( We assume that the smallest tumor we will be working with is 2 mm in diameter (since the minimum number of shots required is 1), giving us a minimum volume of mm 3. We assume the most efficient method of packing spheres to be that of a face-centered cubic lattice, as described by Hales. We assume that we are able to hit the tumor in the center since the image of the tumor is radiographically well-defined.

3 Team Number We assume the tumor is completely surrounded by the brain, since it is intracranial. Algorithm development: We began developing our treatment plan by setting our goals. Our primary goal was to irradiate as much of the tumor as possible. Consequently, we set out to minimize the amount of empty space between the spherical shots of radiation. Our research indicated that a face-centered cubic lattice (see Appendix 2) has been proven to be the most efficient way of packing spheres, as proved by Hales. We determined that it would be best to begin setting up the lattice in the center of the tumor and then working the pattern outward. Having our dosage structure and process determined, we set out to calculate the nonirradiated volume left over by each shot, µ. We attempted to calculate µ by drawing each shot inscribed in a cube. This, however, created overlap as the shots protruded into their neighbor s cubes. We next determined that inscribing each shot in a regular rhombic dodecahedron might be a better option. This, also, proved to be too complicated as we tried to create the face-centered cubic lattice. After much frustration with all our attempts to work from the center of the tumor outward, we decided to work from the outside in. We visualized destroying the tumor by removing layers at a time, somewhat like peeling away the layers of an onion. After one layer had been taken away, we could then approximate the remaining tumor as another sphere, and repeat the process of removing the outmost layer. Fleshing the process out in flowcharts for implementation in a FORTRAN program, we decided the algorithm would involve the user putting in the tumor s radius, calculate the tumor s volume, and then determine how many shots it would take for spheres of each of the four radii to take out the first layer of the tumor. Our next task became determining how many shots would fit into the first layer of the sphere. To determine this, we consider that the centers of the shot spheres create an inner sphere (illustrated in Figure ). This inner sphere is of radius R r, where R is the tumor s radius and r is the radius of the shots, and has a surface area of 4π(R-r) 2. Each shot intersects this surface area, cutting out a shape of area A. We found this area by integrating, but first we needed to find our bounds of integration. As shown in Appendix 2, for each shot, the radius of the inner sphere subtends an angle 2θ, and θ = arctan(r/(r-r)). Thus, if we multiply our double integral by 4, our bounds of integration are 0<φ<arctan(r/(R-r)) and 0<θ<arctan(r/(R-r)). Thus our area A = r arctan R r 4 ( R r ) 2 sin( θ ) dθ dφ 0 r arctan R r 0

4 Team Number Evaluating this integral and using it to divide 4π(R-r) 2, we may determine the number of shots of radius r that will fit into the first layer of the tumor, N r = π ( R r ) 2 R 2 2 R r + 2 r 2 R 2 2 R r r 2 R 2 2 R r + 2 r R 2 R 2 2 R r + r 2 R 2 2 R r + r 2 R 2 2 R r + 2 r 2 2 R r + R 2 2 R r + r 2 R 2 2 R r + 2 r 2 r 2 arctan R 2 2 R r + r 2 r R r The program we designed would calculate this number for every radius (2, 4, 7, and 9), then calculate the volume of cancerous material that would be destroyed by each radius given the number of spheres. V r = N r (4/3 π r r^3) The program would then determine which radius destroyed the greatest amount of cancerous material and then select that shot size to be used to take out the first layer. The subsequent layers proved to be more problematic. In order to set up the lattice of shots, we had to determine the amount of overlap between one layer of spheres and another i.e., we could not simply add their radii together to determine what size of tumor we were now dealing with. We derived a formula that related the overlap, χ, to the radius of our first shot radius (illustrated in Appendix 2). This overlap χ = r/(tan(67.5) 2). Thus, our second layer of shots forms a sphere of radius R 1 2r 1 r 2 + χ (see Appendix 2), where R 1 is the original radius of the tumor, r 1 is the shot size used to destroy the first layer, and r 2 is the shot size used to destroy the second layer. Using the same process, the program calculates this radius and the number of shots that would be able to fit in the second layer, selecting the one radius that would destroy the greatest amount of cancerous material. Again, the program calculates the surface area of the inner sphere and divides this by the surface area intersected by each sphere, which this time is A = r2 arctan R1 2 r1 r2 + χ 4 ( R1 2 r1 r2 + χ) 2 sin( θ ) dθ dφ 0 r2 arctan R1 2 r1 r2 + χ 0

5 Team Number This gives us the number of spheres for each radius to be N r = π ( R1 2 r1 r2 + χ) 2 R R1 r1 + 2 R1 r2 2 R1 χ 4 r1 2 4 r1 r2 + 4 r1 χ r r2 χ χ 2 + R1 2 4 R1 r1 2 R1 r2 + 2 R1 χ + 4 r r1 r2 4 r1 χ + r2 2 2

6 Team Number r2 χ + χ 2 arctan r2 R1 2 r1 r2 + χ Had this model worked, it would have given us higher percentages of destroyed cancerous material. However, we abandoned this method for two reasons: first, due to the complexity of the above equation, errors were made in the FORTRAN code, and second, we made an error in our assumptions about the lattice structure (discussed below). Results and Conclusions. We conclude that the layering and latticing method was not completely successful. In taking out layers, the shots leave behind too many cells not irradiated, while the lattice structure does not fit exactly, leaving even more space not irradiated. We believe the concept of layering may be effective as a first strike against the tumor decimating a sizeable portion of it while leaving sparse amounts of cancerous material behind making way for more precise eradication of the remaining cancer cells. Error Analysis As shown on the graph of percentage irradiated versus radius of the tumor, the results from the iterations of our program were quite unexpectedly low. The best explanation for this anomaly is the presence of programming mistakes due to lack of experience and practice with FORTRAN. We were able to better these results slightly by correcting some of our more obvious programming mistakes. We had no time to correct the rest of them. Our second major error involved the creation of the lattice. We found that when we drew a picture of a sphere and tried to create a lattice within it, we were unable to line up the second row of spheres to touch the primary spheres and each other, also. Thus, it was impossible to achieve a true lattice configuration. Future Developments Had we had more time, we would have corrected whatever programming mistakes we had made. We would have liked to have built scale models to help us better visualize our problem and solution. We would have done deeper research into the dodecahedron model.

7 Team Number Alternate Situations This problem would have been much easier to solve had we had a cubic tumor. Since tumors tend to grow toward blood concentrations, it is possible in a few parts of the body for a cubic tumor to be developed. The problem would also have been easier had the doses been cubical or if partial doses were possible.

8 Team Number Variable Names 1. R dose - radius of dose. 2. R n - radius of sets of spheres 3. R p - radius of primary layer 4. Rmax- maximum radius of tumor 5. Vmax- maximum volume of tumor Appendix 1 6. V n - amount of volume destroyed for layer with radius n. 7. V dest - total volume destroyed 8. V 1 volume of tumor left to be destroyed(initial volume- any irradiated layers) 9. V I - initial volume of tumor 10. N n - number of shots 11. N- total number of shots. 12. S- lenrth of a side 13. χ- difference in lattice structure 14. Perd- Percent destroyed 15. W, X, Y- used to prepare calculations in program 16. µ - cells not irradiated left over by each shot Formulas 1. Volume of a sphere- 4/3 π r 3 2. Formula for calculating new perimeter = R1 - Rdose - sin (45 0 ) *Rprev. 3. Formula for volume of dodecahedron- 2 s 3.

9 Team Number X = π ( R1 Rdose ) 2 Rdose arctan R1 Rdose 5. W = R1 2 2 R1 Rdose + Rdose 2 6. Y = ( R1 Rdose ) 3 1 R1 2 2 R1 Rdose + 2 Rdose 2 7. N n = X/ (W+Y)

10 Team Number Appendix 2 Charts and Figures Tumor of radius Ri Shots of radius r Inner sphere created by centers of shots Surface area A created by intersection of shots and inner sphere

11 Team Number r θ Shot Inner sphere R-r r

12 Team Number New inner sphere of radius R1-2r1-r2+χ

13 Team Number Bibliography