Enhancing dendritic cell immunotherapy for melanoma using a simple mathematical model.

BACKGROUND: The immunotherapy using dendritic cells (DCs) against different varieties of cancer is an approach that has been previously explored which induces a specific immune response. This work presents a mathematical model of DCs immunotherapy for melanoma in mice based on work by Experimental Immunotherapy Laboratory of the Medicine Faculty in the Universidad Autonoma de Mexico (UNAM). METHOD: The model is a five delay differential equation (DDEs) which represents a simplified view of the immunotherapy mechanisms. The mathematical model takes into account the interactions between tumor cells, dendritic cells, naive cytotoxic T lymphocytes cells (inactivated cytotoxic cells), effector cells (cytotoxic T activated cytotoxic cells) and transforming growth factor ß cytokine (T G F-ß). The model is validated comparing the computer simulation results with biological trial results of the immunotherapy developed by the research group of UNAM. RESULTS: The results of the growth of tumor cells obtained by the control immunotherapy simulation show a similar amount of tumor cell population than the biological data of the control immunotherapy. Moreover, comparing the increase of tumor cells obtained from the immunotherapy simulation and the biological data of the immunotherapy applied by the UNAM researchers obtained errors of approximately 10 %. This allowed us to use the model as a framework to test hypothetical treatments. The numerical simulations suggest that by using more doses of DCs and changing the infusion time, the tumor growth decays compared with the current immunotherapy. In addition, a local sensitivity analysis is performed; the results show that the delay in time " τ", the maximal growth rate of tumor "r" and the maximal efficiency of tumor cytotoxic cells rate "aT" are the most sensitive model parameters. CONCLUSION: By using this mathematical model it is possible to simulate the growth of the tumor cells with or without immunotherapy using the infusion protocol of the UNAM researchers, to obtain a good approximation of the biological trials data. It is worth mentioning that by manipulating the different parameters of the model the effectiveness of the immunotherapy may increase. This last suggests that different protocols could be implemented by the Immunotherapy Laboratory of UNAM in order to improve their results.

On a mathematical model of bone marrow metastatic niche.

We propose a mathematical model to describe tumor cells movement towards a metastasis location into the bone marrow considering the influence of chemotaxis inhibition due to the action of a drug. The model considers the evolution of the signaling molecules CXCL-12 secreted by osteoblasts (bone cells responsible of the mineralization of the bone) and PTHrP (secreted by tumor cells) which activates osteoblast growth. The model consists of a coupled system of second order PDEs describing the evolution of CXCL-12 and PTHrP, an ODE of logistic type to model the Osteoblasts density and an extra equation for each cancer cell. We also simulate the system to illustrate the qualitative behavior of the solutions. The numerical method of resolution is also presented in detail.

Nonlinear stability of a heterogeneous state in a PDE-ODE model for acid-mediated tumor invasion.

This work studies a general reaction-diffusion model for acid-mediated tumor invasion, where tumor cells produce excess acid that primarily kills healthy cells, and thereby invade the microenvironment. The acid diffuses and could be cleared by vasculature, and the healthy and tumor cells are viewed as two species following logistic growth with mutual competition. A key feature of this model is the density-limited diffusion for tumor cells, reflecting that a healthy tissue will spatially constrain a tumor unless shrunk. Under appropriate assumptions on model parameters and on initial data, it is shown that the unique heterogeneous state is nonlinearly stable, which implies a long-term coexistence of the healthy and tumor cells in certain parameter space. Our theoretical result suggests that acidity may play a significant role in heterogeneous tumor progression.

On a mathematical model of tumor growth based on cancer stem cells.

We consider a simple mathematical model of tumor growth based on cancer stem cells. The model consists of four hyperbolic equations of first order to describe the evolution of different subpopulations of cells: cancer stem cells, progenitor cells, differentiated cells and dead cells. A fifth equation is introduced to model the evolution of the moving boundary. The system includes non-local terms of integral type in the coefficients. Under some restrictions in the parameters we show that there exists a unique homogeneous steady state which is stable.