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Health Care Application: How Mathematical Models Can Help Improve Cancer Treatments Plans

May/June 2010

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Breakthrough project combines expertise in two different area — radiation physics and optimization.

Gino J. Limby Gino J. Lim

Cancer is the second leading cause of death in the United States. According to the American Cancer Society’s Cancer Facts and Figures, 1,479,350 Americans are estimated to have been diagnosed with cancer in 2009. The National Cancer Institute (NCI) estimates that more than 1,540 people are expected to die of cancer every day. We are losing one innocent life every minute due to cancer in this country alone.

A NCI report recently announced that rates of new diagnoses and rates of death from all cancers combined has significantly declined in recent years. NCI Director John E. Niederhuber, M.D, says that the reasons for this decline are thanks to “the aggressive efforts to reduce risk in large populations, to provide for early detection, and to develop new therapies that have been successfully applied in this past decade.” He continues, “Yet we cannot be content with this steady reduction in incidence and mortality. We must, in fact, accelerate our efforts to get individualized diagnoses and treatments to all Americans, and our belief is that our research efforts and our vision are moving us rapidly in that direction.”

When a patient is diagnosed with cancer, treatment methods are determined based on the cancer type, its location and its stage. The most common cancer treatments are surgery, radiation therapy and chemotherapy. Other types include immuno, targeted and photodynamic therapy. Physicians often use a combination of these treatments to obtain the best results. Ideally, cancer treatments should be designed to kill all cancerous cells without damaging a patient’s healthy organs. However, it is practically impossible to completely spare healthy organs during the treatments because the organs are often located in very close proximity to the tumor. Designing a cancer treatment plan often involves a series of optimization problems with different kinds of objectives and constraints. Researchers in the optimization community have made significant contributions in improving the quality of various treatment plans for cancer patients such as surgery, radiation therapy and chemotherapy.

As the oldest form of cancer treatment, surgery is often used to remove localized tumors. In cases where the cancer has not metastasized, surgery may be performed with curative intent. Radiation therapy uses different types of radiation particles such as photons, electrons and protons to sterilize the tumor or prevent cancerous cells from proliferating. Two types of radiation delivery methods are often used for radiation therapy. The most common radiation delivery machines belong to the external radiation therapy or teletherapy. In teletherapy, the radiation source is positioned at some distance from the patient. Popular three-dimensional conformal radiation therapy methods belong to this category, and they include Gamma Knife radiosurgery, intensity modulated radiation therapy (IMRT), traditional three-dimensional conformal radiation therapy (3DCRT), tomotherapy, intensity modulated arc therapy (IMAT) and intensity modulated proton therapy (IMPT). The other radiation delivery method is called brachytherapy. Brachy means “short distance” in Greek. Therefore, brachytherapy is a term of internal radiation therapy where radioactive substances are placed inside or close to the tumor.

Brachytherapy is often used for treating patients with prostate, cervical or breast cancer. Effective plans of radiation treatments require delivery of high dose of radiation to the tumor while minimizing the impact on normal cells as much as possible.

Drug therapies such as chemotherapy use medicines to control cancer systemically. Such systemic therapy may be administered alone or in combination with other treatments.

Finding a cure for cancer has been an exciting goal in research communities. Researchers now have a much better understanding of cancer. Many innovative drugs are being developed from all over the world to block cancer cell growth by inhibiting specific molecule pathways needed for cells to proliferate. Radiation therapy techniques have become very precise in achieving the treatment goals. Engineers work tirelessly to develop cutting-edge devices to aid cancer treatments. Powerful and sophisticated cancer scanning devices have helped improve the detection of cancer. Robotic arms are being used in surgeries. Government entities and private foundations have increased awareness of cancer, while public health initiatives have helped induce life style changes that reduce the risk of cancer.

Research in Radiation Therapy Planning

I have been fortunate to work in this exciting and rewarding area of research. As a graduate student in optimization at the University of Wisconsin-Madison, my thesis advisor Michael Ferris introduced me to a radiation therapy planning problem for brain tumor patients. The problem was originally presented to him by David Shepard, a graduate student in Professor Rockwell Mackie’s group in the Radiation Physics department. Shepard’s initial modeling efforts gave us the groundwork for our research.

The project proved to be an excellent example of how expertise in two separate areas — radiation physics and optimization — can lead to results that could not be achieved with expertise in one area alone. From the optimization perspective, the primary goal of a treatment plan is to deliver the prescribed dose of radiation to the tumor while minimizing the dose to non-tumor cells. The total amount of radiation for a complete radiation treatment varies depending on the tumor type and the radiation delivery device being used. For a typical prostate cancer treatment using IMRT, two gray (Gy) of radiation are administered each day and the treatment continues up to 40 days. The reason for this fractionated radiation delivery is based on cell growth in the body. Cancer cells reproduce much faster than normal cells. Radiation therapy specifically targets them by impairing or delaying this cell reproduction process. Normal cells can quickly recover from radiation damage overnight when they are exposed to a low radiation dose. But, if they are exposed to a high radiation dose, it may cause long-term damage. Cancer cells, however, are very sensitive to radiation exposure and two Gy per day can effectively stop them from replicating.

Radiation treatment planning begins with a set of images of the organs of interest. Such images can be obtained by diagnostic imaging tools such as computerized tomography (CT), magnetic resonance imaging (MRI) or positron emission tomography (PET) scans. Those diagnostic images then help physicians determine the three-dimensional shape, location and size of the tumor and surrounding normal tissues. Delineating treatment volume on the images is often a difficult task to perform. A well-trained Figure 3. Varian Medical radiation oncologist can identify which portion belongs to the gross tumor volume (GTV). But it is often difficult to draw a clear line between the tumor and the normal cells. Therefore, a clinical target volume (CTV) is defined, and it includes GTV and its surrounding suspicious area. Finally, planning target volume (PTV) is constructed for treatment planning. PTV adds margins to CTV to account for movement of organs during treatment and the movement of the patient between treatments.

The images are further divided into a collection of three-dimensional cubes (or voxels) for optimizing treatment parameters. This is where optimization models can play a key role in treatment planning. The type of radiation delivery device determines what parameters need to be optimized. My colleagues and I have developed optimization models for Gamma Knife radio-surgery, IMRT and IMPT.

Gamma Knife Radio-surgery

The Gamma Knife is a stereotactic radiation delivery device for treating tumors or vascular malformations located in a patient’s head. It can deliver a single beam of high dose radiation by 201 Cobalt-60 unit sources. All 201 beams simultaneously intersect at the same location to form an approximately spherical region that is typically termed a shot of radiation. Multiple shots are often used in a treatment using a Gamma Knife due to the irregularity and size of tumor shapes and the fact that the focusing helmets (or collimators) are only available in four different sizes in a traditional treatment planning. Therefore, treatment parameters include a set of radiation shot center locations, a discrete set of collimator sizes and radiation exposure time for each radiation shot.

We formulated this problem as a mixed integer nonlinear programming model and solved it as a variant of the sphere-packing problem where the PTV can be viewed as an empty container. Next, our job was to find a mix of spheres to fill the container while minimizing the total space that was not covered. Another goal: minimize the number of spheres to do the job. We have successfully developed a computer program, tested it and used it in radiation planning for patients at the University of Maryland hospital in Baltimore. The reliability of this treatment-planning tool has been proven by clinical trials at the hospital. In eight out of 10 cases, the tool produced plans that are better than the ones made by a physician. Both the speed of the treatment plan generation and the treatment quality are the main contributions of this decision-making tool.

Figure 1. (Left) Multiple=

Figure 1. Multiple angles are used to achieve a high radiation dose on the tumor while minimizing radiation exposure to surrounding normal cells.

Intensity Modulated Radiation Therapy

IMRT is a radiation delivery method under the umbrella of three-dimensional conformal radiation therapy. 3DCRT utilizes 3-D imaging data of the tumor and surrounding anatomy to achieve a high conformal dose of radiation delivered directly to the tumor. This treatment can be administered for a variety of cancers. In 3DCRT, multiple external radiation beams are directed from different angles into the tumor (see Figure 1). Each beam can be shaped by a computerized multi-leaf collimator mounted on a gantry to conform the beam’s-eye-view of the tumor (see Figure 2). Hence, a lethal dose of radiation can be precisely delivered to a tumor. Photon beams are used for 3DCRT.

As an advanced form of 3DCRT, IMRT utilizes an advanced multileaf collimator that can move its leaves back and forth to block the portion of each beam’s radiation dose (Figure 3). This enables a high degree of flexibility in delivering radiation from each gantry angle. Typically, between five to nine angles around the 360-degree circumference are selected for the treatment. Radiation is delivered at a fixed angle, and it continues until all the selected angles are covered. This treatment-planning problem is often formulated as a mixed integer linear programming (MIP) model, in which a set of gantry angles and their radiation intensities are optimized. In our IMRT treatment-planning tool, the MIP model is solved using the Branch-and-Bound (B&B) technique, and the linear programming relaxation models are solved using an interior point method. Since the MIP model contains more than one million continuous variables and several thousand binary variables and more than one million constraints, solving the problem optimally using B&B can take several weeks using a fast workstation PC. Therefore, heuristic approaches such as local neighborhood search, simulated annealing and genetic algorithms are often developed to expedite the treatment planning process.

Figure 2. (Right) Beam’s-eye-view is constructed for each angle.

Figure 2. Beam’s-eye-view is constructed for each angle.

Intensity Modulated Proton Therapy

Protons were first proposed for radiation therapy by Robert Wilson at U.C.-Berkeley in 1946. Since then, thanks to more than five decades of continuous research and development, we now have a potentially very powerful radiation delivery machine that uses proton beams. It has been reported that proton therapy may be more precise than other forms of radiation treatment in certain types of cancer. This is because the dose distribution from a mono-energetic beam of protons increases slowly, rising to a sharp peak, called a Bragg Peak, at the target. By superimposing beams of different energies, IMPT does only moderate damage to tissues on its way to a tumor and almost no damage thereafter.

The potential of IMPT is only now being fully understood, and optimization is an important part of realizing IMPTs full potential. I am presently working in collaboration with the University of Texas MD Anderson Cancer Center in Houston on optimizing the delivery of proton therapy. MD Anderson Cancer Center started treating certain types of cancer patients at their Proton Center in 2006.

Conclusions

Figure 3. Varian Medical System’s multileaf collimator.

Figure 3. Varian Medical System’s multileaf collimator.

Cancer is an abhorrent disease, but it’s a disease we continue to make strides against every day. One way is through cross-disciplinary efforts, such as bringing the powerful tools of optimization to bear on problems in radiation therapy. But the application of optimization, and analytics in general, isn’t limited to cancer treatment. It can be applied to the design of new drugs, imaging and genetic sequencing, not to mention hospital planning, scheduling and administration. Medical practitioners from all walks can benefit by adopting the many powerful tools analytics has to offer.

Gino J. Lim (ginolim@uh.edu) is an assistant professor of Industrial Engineering at the University of Houston. The author acknowledges Andy Boyd for his valuable comments and suggestions in preparing this article.

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