(I-334) Mathematical Analysis of Differential Model for Melanoma Immunotherapy

Introduction:: While Melanoma, a form of skin cancer, accounts for only 1.3 percent of skin cancers, Melanoma causes the majority of skin cancer-related deaths and is one of the most common cancers in young adults under the age of 30 \cite{clevelandclinic2021melanoma}. In the United States alone, approximately 7990 people are predicted to die from this disease in 2023 \cite {melanomastats2023}. Among a variety of cancer treatment methods now available, immunotherapy can commonly be applied to treat Melanoma. Specifically, immunotherapy is a type of cancer treatment that helps the patient's immune system be able to better fight cancer. There are several different routes that can be taken with immunotherapy for cancer treatment; however, a particularly common form of immunotherapy utilizes monoclonal antibodies, which are proteins from the immune system that can be created in a lab to bind to specific cancer cells so that these cells can be better seen and destroyed by the body's immune system \cite{nci2019immunotherapy}. In his 2012 study on immunotherapy applications for melanoma treatment, Marzio Pennisi used OT1 chronic lymphocytes (CTLs) and Anti-CD137 antibodies to treat melanoma in mice \cite{Pennisi2012}. Pennisi used the results from the outcome of these trials to develop a model of seven differential equations that models the interactions between time, parameters, and additional independent variables to show their influence on the efficacy of melanoma treatment. The purpose of this study is to perform stability analysis on the equilibria points of an altered version of the previously mentioned model.

Materials and Methods:: The system of differential equations used to model the interactions between the treatment and melanoma cells was adapted from the models demonstrated by Pennisi. Specifically, each of the seven ordinary differential equations describes a different dependent variable in relation to time. To simplify this system for the purposes of analysis, the term representing treatment input was changed to a constant value, the melanoma’s growth was modified from logarithmic to logistic. Additionally, the delayed function representing arrival of medication to the melanoma cell location was reduced to a non-delayed constant value to further simplify the system. Following adaptation, the seven ordinary differential equations were coded into MATLAB software to provide graphical outputs that confirmed the alterations to the system did not greatly affect its overall integrity. From here, the system of differential equations will be solved, and the stability of the critical points will be analyzed using the Jacobian matrix of the seven equations. Additionally, sensitivity of each equation in relation to the 18 equation parameters will be analyzed to determine which parameters have the greatest effect on the differential equations.

Results, Conclusions, and Discussions:: There should be multiple resulting critical points. The most ideal critical points are the ones for which the value of cancer is 0 and the result is stable. This is ideal because it represents a state where the treatment is effective in eradicating melanoma cells. Once these ideal points of stability are determined, as previously mentioned, the sensitivity of the differential equations in relation to their parameters would be analyzed. It can be reasonably predicted that some parameters will have a greater affect on the outcome of successful treatment in comparison to others.
Understanding where stable points of equilibria exist can optimize real-life immunotherapy treatment for melanoma. Specifically, knowing the values expected to deliver the target state of zero cancer cells provides a suggested treatment plan most effective in harnessing the power of one’s immune system against evasive cancer. \\\\
Furthermore, findings related to the sensitivity of dependent variables in relation to parameters allow for further understanding of which factors deserve the most focus and care in therapy applications. Again, this improves the chances of success and efficiency of treatment.
While this model is adapted to be relatively accurate to simulated lab data, more complexity could be added to lessen the gap between model and reality. This could be done by exploring non-constant CTL and antibody injection rates. Alternatively, delayed-differential equations could be applied when modeling the arrival rate of CTLs and antibodies in the skin compartment. Both of these concepts are originally demonstrated in the Pennisi model. By exploring more complex models, results can be obtained which more closely fit reality and thus, provide a better picture of ideal equilibria states for cancer eradication. \\\\
While this model exists under the specific scope of using CTLs and antibodies to treat melanoma, these fundamental modeling methods can be applied to wide varieties of immunotherapies for cancer. In an ever-increasing field of immunotherapy research --- aiming to improve the outlook of cancer patients around the world --- this analysis is another step towards opening new, more effective avenues for cancer treatment.

Acknowledgements (Optional): : Dr. Maila Hallare from USAFA is our advisor for this project.

References (Optional): :

Immunotherapy for cancer. National Cancer Institute. (2019, September 24). https://www.cancer.gov/about-cancer/treatment/types/immunotherapy

Melanoma skin cancer statistics. Melanoma Skin Cancer Statistics. (2023, January 12). https://www.cancer.org/cancer/types/melanoma-skin-cancer/about/key-statistics.html

Melanoma: Symptoms, stages, diagnosis, treatment & prevention. Cleveland Clinic. (2021, June 21). https://my.clevelandclinic.org/health/diseases/14391-melanoma

Pennisi, M. (2012). A mathematical model of immune-system-melanoma competition. Computational and Mathematical Methods in Medicine, 2012, 1–13. https://doi.org/10.1155/2012/850754