Numerical Simulation of a Tumor Growth Dynamics Model Using Particle Swarm Optimization

An in silico exploration of combining Interleukin-12 with Oxaliplatin to treat liver-metastatic colorectal cancer

Background

Combining anti-cancer therapies with orthogonal modes of action, such as direct cytotoxicity and immunostimulatory, hold promise for expanding clinical benefit to patients with metastatic disease. For instance, a chemotherapy agent Oxaliplatin (OXP) in combination with Interleukin-12 (IL-12) can eliminate pre-existing liver metastatic colorectal cancer and protect from relapse in a murine model. However, the underlying dynamics associated with the targeted biology and the combinatorial space consisting of possible dosage and timing of each therapy present challenges for optimizing treatment regimens. To address some of these challenges, we developed a predictive simulation platform for optimizing dose and timing of the combination therapy involving Mifepristone-induced IL-12 and chemotherapy agent OXP.

Methods

A multi-scale mathematical model comprised of impulsive ordinary differential equations was developed to describe the interaction between the immune system and tumor cells in response to the combined IL-12 and OXP therapy. An ensemble of model parameters were calibrated to published experimental data using a genetic algorithm and used to represent three different phenotypes: responders, partial-responders, and non-responders.

Results

The multi-scale model captures tumor growth patterns of the three phenotypic responses observed in mice in response to the combination therapy against a tumor re-challenge and was used to explore the impacts of changing the dose and timing of the mixed immune-chemotherapy on tumor growth subjected to a tumor re-challenge in mice. An increased ratio of CD8 + T effectors to regulatory T cells during and after treatment was key to improve tumor control in the responder cohort. Sensitivity analysis indicates that combined OXP and IL-12 therapy worked more efficiently in responders by increased priming of T cells, enhanced CD8 + T cell-mediated killing, and functional inhibition of regulatory T cells. In a virtual cohort that mimics non-responders and partial-responders, simulations show that an increased dose of OXP alone would improve the response. In addition, enhanced IL-12 expression alone or an increased number of treatment cycles of the mixed immune-chemotherapy can barely improve tumor control for non-responders and partial responders.

Conclusions

Overall, this study illustrates how mechanistic models can be used for in silico screening of the optimal therapeutic dose and timing in combined cancer treatment strategies.

Simulation Study on Effects of Order and Step Size of Runge-Kutta Methods that Solve Contagious Disease and Tumor Models

Biological processes such as contagious disease spread patterns and tumor growth dynamics are modelled using a set of coupled differential equations. Experimental data is usually used to calibrate models so they can be used to make future predictions. In this study, numerical methods were implemented to approximate solutions to mathematical models that were not solvable analytically, such as a SARS model. More complex models such as a tumor growth model involve high-dimensional parameter spaces; efficient numerical simulation techniques were used to search for optimal or close-to-optimal parameter values in the equations. Runge-Kutta methods are a group of explicit and implicit numerical methods that effectively solve the ordinary differential equations in these models. Effects of the order and the step size of Runge-Kutta methods were studied in order to maximize the search accuracy and efficiency in parameter spaces of the models. Numerical simulation results showed that an order of four gave the best balance between truncation errors and the simulation speed for SIR, SARS, and tumormodels studied in the project. The optimal step size for differential equation solvers was found to be model-dependent.