Modeling Pancreatic Cancer Dynamics with Immunotherapy

Abstract

We develop a mathematical model of pancreatic cancer that includes pancreatic cancer cells, pancreatic stellate cells, effector cells and tumor-promoting and tumor-suppressing cytokines to investigate the effects of immunotherapies on patient survival. The model is first validated using the survival data of two clinical trials. Local sensitivity analysis of the parameters indicates there exists a critical activation rate of pro-tumor cytokines beyond which the cancer can be eradicated if four adoptive transfers of immune cells are applied. Optimal control theory is explored as a potential tool for searching the best adoptive cellular immunotherapies. Combined immunotherapies between adoptive ex vivo expanded immune cells and TGF-$\beta$ inhibition by siRNA treatments are investigated. This study concludes that mono-immunotherapy is unlikely to control the pancreatic cancer and combined immunotherapies between anti-TGF-$\beta$ and adoptive transfers of immune cells can prolong patient survival. We show through numerical explorations that how these two types of immunotherapies are scheduled is important to survival. Applying TGF-$\beta$ inhibition first followed by adoptive immune cell transfers can yield better survival outcomes.

Appendix: Sensitivity Analysis Based on the Survival Data of Niu et al.

From Fig. 1a of Niu et al. (2013), there is one patient who survived for about 25 months with no immunotherapy and there is one patient who survived for about 30 months with four additional immunotherapies. For both groups of patients, there is one patient who survived for only about three months. These survival times are not outliers of the data sets, and there are no numerical values for the mean and standard deviation given by Niu et al. (2013). If such information were provided, then change in the parameter values based on a 90% or 95% confidence interval of the clinical data would be performed instead. We therefore increase each individual parameter by up to 3000% or until the survival time lies outside of (120, 750) days for cryotherapy and (120, 900) days for cryo-immunotherapy. For both groups of patients, only the median immunity is considered in our numerical investigation. We also decrease each baseline parameter value until either up to 90% or the survival time falls outside of the above time intervals. These results are summarized in Tables 5 and 6 for cryotherapy and cryo-immunotherapy, respectively. In addition, the numbers of survival days for the corresponding maximum percentage of changes are also provided.

Table 5 Maximum percentage of change from baseline parameter values with no immunotherapy for median immune strength

Table 6 Maximum percentage of change from baseline parameter values with four immunotherapies for median immune strength

From Table 5, one can see that parameters $\beta _3$, $k_3$, $m_3$, $r_3$, $k_4$, $k_5$ and $\mu _5$ can be increased up to 3000% and decreased up to 90% without changing the survival time of 215 days from the baseline parameters when no immunotherapy is applied. It is quite surprising to observe the lack of effect of parameter $\beta _3$ since $\beta _3$ is the tumor antigenicity. Increasing/decreasing tumor antigenicity cannot prolong/reduce a patient’s survival time in the proposed model when there is no immunotherapy. For the tumor killing rate $\delta _1$, it can be increased only up to 400% with the corresponding survival time being 529 days. Increasing $\delta _1$ further would result in tumor eradication since the number of cancer cell would be less than one. It is also clear that the survival time is sensitive with respect to two other parameters $r_1$ and $b_1$ which are tumor dependent. Increasing $b_1$ or decreasing $r_1$ beyond the percentage changes given in the table also results in tumor eradication. Further, the survival time is sensitive to the parameter $\mu _4$, the decay rate of the pro-tumor cytokines. Increasing this natural loss rate can clear off the tumor.

With four immunotherapies, the survival time is 387 days for the baseline parameter values with median immunity. Table 6 implies that the survival time is insensitive to the parameters $\beta _3, k_3, m_3, r_3, m_4, \beta _5, k_5$ and $\mu _5$ since increasing each of these up to 3000% or decreasing each up to 90% yields the same number 387 of survival days. Comparing with the above discussion of Table 5 of no immunotherapies, there are two new parameters $m_4$ and $\beta _5$ that appear here. However, their corresponding changes in Table 5 are very small which can also be viewed as insensitive when there is no immunotherapy. The survival time is sensitive to the parameters $\delta _1, r_1, b_1, \mu _4$ and $\beta _4$. The first four of these parameters are also sensitive to the survival time when there is no immunotherapy. The additional parameter $\beta _4$, the maximum activation rate of pro-tumor cytokines, is sensitive when immunotherapy is applied. In particular, the number of tumor cells is less than one and tumor eradication occurs when it is decreased 60% from its baseline value.

Since bar charts provide better visualization of the effects on survival time, we also summarize the two tables using graphs presented in Fig. 7a–b, respectively. However, it is unclear how much each individual parameter can be varied from the bar graphs, and hence, Tables 5 and 6 are also provided to reflect the maximum percentages of changes and their corresponding survival times.