Chemotherapy and mathematical modeling
Chemotherapy is a medical treatment that utilizes chemical drugs to destroy cancerous cells in the body. However, traditional chemotherapy approaches often do not differentiate between healthy and cancerous cells, leading to a range of side effects, including hair loss and damage to organs. To mitigate these effects, chemotherapy dosing protocols consider various factors such as dosage, treatment frequency, and overall treatment duration. Historically, these protocols relied on clinical trial data, which can be resource-intensive and ethically complex.
Mathematical modeling is becoming an increasingly important aspect of chemotherapy, as it provides a quantitative framework to understand and optimize treatment strategies. For example, insights from mathematical models have revealed that tumors grow in different patterns, such as Gompertzian growth, which has influenced how treatment regimens are scheduled. This understanding allows for more effective timing of chemotherapy treatments, taking advantage of the tumor's growth characteristics.
Ongoing research in mathematical modeling, particularly through optimal control theory, aims to refine chemotherapy protocols further, potentially tailoring treatment plans to individual patients and specific types of cancer. This innovative approach signifies a promising direction in enhancing the efficacy and safety of cancer treatments.
Chemotherapy and mathematical modeling
Summary: Mathematical modeling has improved chemotherapy protocols and saved patients’ lives.
Chemotherapy is the use of chemical drugs to kill cancerous cells in the body. Although cancerous cells are the target of chemotherapy, traditional chemotherapies do not distinguish between “good” and “bad” cells. Hence, chemotherapy often results in side effects, such as hair loss and toxicity damage to body organs. Because of these chemotherapy side effects, chemotherapy protocols attempt to kill as much of the tumor as possible while incurring as little damage to the patient as can be managed. Thus, chemotherapy regimens are managed according to different variables, including how much drug is given in a treatment, how frequently treatments are given, and the total number of treatments given. Historically, chemotherapy protocols were designed only through experimental data from clinical trials and practice. However, such experiments can be costly or even pose ethical dilemmas. Chemotherapy variables are quantitative—each lends itself to a mathematical understanding and description that can be used to model and simulate treatment experiments, adding to the information gained in clinical settings.
Mathematics in Cancer Chemotherapy News
Mathematics is becoming an increasingly powerful tool in cancer chemotherapy treatments, especially in the dosing and management of chemotherapy protocols. For example, in 2004, Dr. Larry Norton received the American Society of Clinical Oncology’s David A. Karnofsky Award, which is given for an outstanding contribution to progression in cancer treatment. Norton’s award is notable because of his quantitative contribution to the field of chemotherapy dosing. The National Cancer Institute has a Center for Bioinformatics that addresses the issue of systematically studying the vast amounts of data associated with cancer growth and treatment response.
Cancer Geometry and Treatment
Cancer cells appear visibly different in shape and structure than normal healthy cells. This fact helps practitioners identify unhealthy cells. Quantitative measurements are associated with the geometry and the complexity of cancer cells. These measurements are related to fractal geometry. Tumor fractal dimensions reflect more complex structures generally because of the arrangement of blood vessels in the tumor. Abnormal blood vessel arrangements inhibit the tumor’s uptake of therapeutic drugs. This understanding has led to the use of anti-angiogenic drugs that inhibit the production of new blood vessels and lower the measurement of the tumor’s complexity. These drugs can now be used in concert with other cancer treatments in order to create a more effective cancer-fighting regimen.
Cancer Growth and Chemotherapy Treatment
Historically, it was believed that cancer cells grew in an exponential manner over the entire period of a tumor’s growth. In exponential growth, the doubling time of a population is constant. This belief affected the way that chemotherapy was delivered, since chemotherapy works by attacking rapidly dividing cells. If a tumor’s growth rate were constant, there would be no difference in how many cells were killed during any chemotherapy treatment, regardless of the size of the tumor. This is the “log-kill” model of tumor growth.
However, in the mid-twentieth century, it was experimentally discovered that many tumors exhibited a different kind of population growth: “Gompertzian growth,” named for Benjamin Gompertz. When populations grow in a Gompertzian fashion, they grow very rapidly at first—when the population is small. As the population size increases, the growth rate of the population slows. Thus, many tumors would have a smaller doubling time when smaller, and a larger doubling time when larger. Because chemotherapy attacks the most rapidly dividing cells, smaller tumors would be more susceptible to chemotherapy treatments. Thus, if a tumor has been reduced in size by one chemotherapy treatment, it would be better to give a second chemotherapy treatment as soon as possible without costing the patient in terms of healthy cell function. This Norton–Simon hypothesis, named for Larry Norton and Richard Simon, has led to a change in the frequency of standard chemotherapy regimens—the time between treatments was decreased in order to take advantage of the more rapid growth rate in the smaller tumor that had resulted from the previous treatment. This change in treatment timing has increased the survival time of patients undergoing chemotherapy treatments.
Looking Ahead
Although the Norton–Simon hypothesis is a prominent example of how mathematics has helped improve cancer chemotherapy treatments, there are ongoing studies by mathematicians to further improve treatment of cancer. Using a field of mathematics known as optimal control, some mathematicians study how to make chemotherapy treatments as ideal as possible. Although practitioners can make (and have made) use of the Norton–Simon hypothesis, the increase of chemotherapy treatments for a patient, while better, is not necessarily best. Using optimal control theory on mathematical models of cancer and cancer treatment, researchers can investigate the best timing and dosing strategies for chemotherapy based on the variables mentioned above. This work may even lead to determining cancer treatment plans based on a particular individual or a particular kind of cancer in the future.
Bibliography
Dildine, James. “Cancer and Mathematics.” http://mste.illinois.edu/dildine/cancer/cancer.html.
Laird, Anna. “Dynamics of Tumour Growth: Comparison of Growth Rates and Extrapolation of Growth Curve to One Cell.” British Journal of Cancer 19, no. 2 (1965).
Martin, R., and K. Teo. Optimal Control of Drug Administration in Cancer Chemotherapy. Singapore: World Scientific Publishing, 1966.
National Cancer Institute (NCI). NCI Cancer Bulletin 3, no. 18 (2006).
Schmidt, Charles. “The Gompertzian View: Norton Honored for Role in Establishing Cancer Treatment Approach.” Journal of the National Cancer Institute 96, no. 20 (2004).
Piccart-Gebhart, Martine. “Mathematics and Oncology: A Match for Life?” Journal of Clinical Oncology 21, no. 8 (2003).