Mathematical models analyzing tumor-immune interactions provide a framework by which to address specific scenarios in regard to tumor-immune dynamics. Important aspects of tumor-immune surveillance to consider is the elimination of tumor cells from a host’s cell-mediated immunity as well as the implications of vaccines derived from the synthetic antigen. In present studies, our mathematical model examined the role of synthetic antigen to the strength of the immune system. The constructed model takes into account accepted knowledge of immune function as well as prior work done by de Pillis et al. All equations describing tumor-immune growth, antigen presentation, immune response, and interaction rates were numerically simulated with MATLAB. Here, our work shows that a robust immune response can be generated if the immune system recognizes epitopes that are between 8 to 11 amino acids long. We show through mathematical modeling of how synthetic tumor vaccines can be utilized to mitigate a developing cancer
One effective way to cure disease is to prevent the development of it all together. One modality to combat disease is cancer vaccines that would “program” an individual’s immune system to recognize foreign antigens by stimulating cytotoxic T lymphocytes (CTL) to attack cancer cells expressing a certain tumor antigen [1-5]. Current vaccine strategies to combat cancer include vaccines consisting of lymphocytes, which include: helper T lymphocytes (Th), dendritic cells (DC), macrophages, or reprogrammed oncolytic viruses [1,2]. Such vaccines may help deter cancer growth through stimulation of an individual’s immune system or by directly attacking a cancer growth [1].
Important questions arise when dealing with the idea of preventative cancer vaccines such as the practicality of utilizing vaccines to prevent the development of cancer as well as how many memory CTL’s need to be produced to provide a sentinel within an individual [1,6,7]. Cancer poses many issues to the vaccine development process as it displays the ability of antigen mimicry, a process by which tumor cells produce antigens with specific patterns of the host that can help cancer evade immune processing and development. Tumor antigen mimicry with self-antigen occurs since tumor-specific antigens (TSA) and tumor- associated (TAA) antigens are either mutated or overexpressed self- proteins, respectively (P53 and CEA). This results in active Th cells having a difficult time selecting for self from non-self. In addition, cancer growth displays variation; it may more rapid or slower than that of other disease processes. Such properties can result in a weak immune response. The multitude of complexities associated with cancer as well as its ability to deter host defenses has challenged researchers to seek for alternative therapies to chemotherapeutics due to their harmful side effects upon a host. One approach to treating cancer began in 1909 when the German scientist Paul Ehrlich proposed the “cancer immunosurveillance” hypothesis, which is the idea that the immune system can suppress an overwhelming number of carcinomas [4,8].
This approach was not tested until the 1950’s when the field of Immunology advanced. Experiments attempting to show support utilized mice that were inoculated with chemically-induced cancer cells; such cells lacked the capability to metastasize within a host. Over time, this led to the development of cancer-specific immunity in the recipient mice. This discovery provided the evidence needed for Ehrlich’s hypothesis. Such experiments demonstrated that it is essential to have the presence of an antigen to elicit an immune response in the host, because if no distinctive structures exist, then no recognition would be established [9]. F. Macfarlane Burnet and Lewis Thomas, well-known immunologists during the 20th century, hypothesized that for immunosurveillance to exist, lymphocytes would need to act aggressively akin to sentinels to recognize and eliminate a cancer threat. The cancer immunosurveillance theory revolves around three transitions states, denoted as “E’s” [9]:
This hypothesis served as a foundation for a previous project inspired by de Pillis et. al. which studied the interactions of cancer and the immune system utilizing mathematical biology [9]. Mathematical Biology is a field of research that draws aspects from both mathematics and the biological sciences to represent, model, and analyze complex biological processes through techniques such as numerical simulations or phase plane analysis [7,10-13]. Mathematical modeling provides insight and validity to a complex biological system for clinical research without the utilization of human or animal models, entirely bypassing ethics boards completely [7,10,12]. Generated data can have similar validity to that of data obtained from human or animal experiments. Describing systems in qualitative and quantitative manners means that behaviors can be simulated and new behaviors that aren’t evident to human/animal experimentation can be discovered. Differential equations, for example, can predict how populations can behave by analyzing variables such as time (ordinary differential equations-ODE) or space (partial differential equations) [11,13]. The probability of events can also be utilized within mathematical models through Monte- Carlo Simulations [14-18]. The development of such mathematical models has a wide range of implications including the possibility of discovering hidden behaviors within systems and determining long- term goals of a system. For the scope of this paper, we attempt to illustrate how mathematical modeling can be utilized to predict the strength of a host’s immune response to lung cancer using a coupled Monte-Carlo/ordinary differential equation model. Our work is an expanded mathematical model that is based on a previous validated by a prior mathematical model by de Pillis et al. [9]. Her prior work explored the dynamics of tumor rejection, the roles NK and CD8+ T cells, as well as the development of protective immunity to subsequent tumor re-challenges. Her model was validated through comparison of mouse and human data to determine tumor growth and lysis rates. Her model further underwent a sensitivity analysis to determine sensitive aspects that could be patient specific that could be applied to a clinical setting. Her variable analysis suggests which patients could respond to treatment. Our model expands through the incorporation of additional cellular lines; macrophages are introduced to complete the innate immune system perspective and humoral immunity has been expanded upon through the introduction of CD4+, CD8+, and CD4+ T regulatory cell lines in both their dormant and active transitional states. In addition, Interleukin-2 is introduced to see how cytokines impact the immune response. Antigen presenting cells, such as dendritic cells, have also been introduced to see how antigen presentation plays a role in cancer immunosurveillance. While B cells play an important part in the adaptive immune response, this cell line has been excluded for the purposes of this model due to the focus on T cell response and the complexity of the model. We also show how this model can be utilized in a clinical setting to predict the long-term consequences of a patient’s cancer status if injected with a vaccine composed of different lung cancer tumor epitopes [13,17,18]. The development of this model focused on first on establishing conditions in which the cancer immunosurveillance hypothesis “exists” through parameter estimations and bifurcation diagrams relating certain parameter families. For details on this work, please refer to the references section. This model then focused on validating which cell lines were the principal cell line in the innate immune, antigen presentation, and cell-mediated responses; of which, NK cells, dendritic cells, and CD8+ cells were key in the immune response against cancer. While not much insight present, validation of theoretical knowledge confirms that the development of the model is the right step. The next step of the model was to introduce “randomization” of the immune response via the introduction of Monte-Carlo simulation processes. Two variables of the model were introduced as extra equations in the model to simulate the strength of a tumor epitope vaccine that influences the strength of the immune response based on the size of the epitope. The randomness of the model can eventually be utilized in a clinical setting to allow clinicians to prognosticate the long-term health status of a patient after a tumor vaccine is utilized. We developed a mathematical model of tumor dynamics in response to a vaccine injection composed of lung cancer epitopes (Survivin, Kita-Kyushu lung cancer antigen 1 (KKLC1), and epidermal growth factor receptor (EGFR)) of different fragment sizes (8-12 amino acids (aa) long) with the goal of determining which epitopes produce a strong immune response.
The dynamics of the mathematical model, as well as parameter values, are borrowed from assertions, prior mathematical models, as well as through parameter estimation through numerical simulations. Our model is based on a previous model published by de Pillis et al. [9], but expanded to include simplified T cell development and more cell populations to better depict the immune response to cancer. No patient data was integrated into this model yet as this model is in its infancy; a literature review shows no prior model with an integrated Monte- Carlo simulator. Generated data now is theoretical but has applicability to the clinical setting.
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