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 β cytokine (TGF−β). 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. Electronic supplementary material The online version of this article (doi:10.1186/s12976-015-0007-0) contains supplementary material, which is available to authorized users.

In another vein, observing the biological results, the mice died when the tumor reached a diameter of 4.3 cm (approximately 1.6 * 10 10 tumor cells); for that reason this population was taken as a limit of the population tumor cells in the simulation and was considered constant from that moment.
Threshold in DC density for half-maximal proliferation rate of CTLs, "θ D ". Set on 212 cells and determinated by De Pillis et al. [3].
Threshold in "C a " density for half-maximal proliferation rate of CTLs, "θ A ". Found in a interval search between [1,300] and calculated at 10.
The proposed model and the experimental data of inmunotherapy in mice with melanoma is used to find the "a T ", "τ ", "r E " and "r A " parameters performing an exhaustive research and minimizing method; the minimum mean square error value is taken as a reference value to determine the optimum value of the parameters. The minimum mean square error is a result of a comparison between the tumor cells population of the biological experiments and the tumor cell population generated by the model. The parameters value found is: Maximal efficiency of cytotoxic cells, "a T ". This parameter represents the efficiency at which the cytotoxic cells kill the tumor cells. Its estimation is carried out through an exhaustive search in an interval search between 10 −13 and 10 3 the value obtained is 6 * 10 −11 h −1 .
Activation rate of CTLs, "r a ". A interval search between 0.00001 and 100 is used to find the most optimal, estimated at 61h −1 . The data used in the models proposed by Ludewing et al. [4] and Handel et al. [5] is considered to determine the interval search.
Expansion rate of activated CTL, "r e ". The value of "r e " parameter is set at 6500h −1 which is the result of the estimation using an exhaustive search in an interval search between 10h −1 and 7000h −1 Delay in the time, "τ ". Indicates the time when the therapy is becoming effective, considering that this is effective between 100 and 300 hours after injecting the first immunotherapy, and taking into account the data of immunotherapy biological tests the value of "τ " is determined in 265 hours by an exhaustive search.
The parameters listed below are taken for the Kronik et al. [6] study.
Maximal reduction effect of "T GF − β" on CTLS efficiency, "a T,β ". Using the reports about the CTL efficacy the value determinated by Kronik et Dependence of CTLs efficiency on T GF − β, "e T,β ". Estimated at 10 4 pg.

DCsize
Diameter of Dendritic Cell