# De Pillis chemotherapy model

De Pillis chemotherapy model | |
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State dimension: | 1 |

Differential states: | 6 |

Continuous control functions: | 3 |

The model by de Pillis combines chemotherapy with immunotherapy.

## Mathematical formulation

For and the optimal control problem is given by

The states to are measured in absolute cell counts, where describes the number of tumor cells, of unspecific immune cells, of tumor-specific cytotoxic T-cells (CD T) and of circulating lymphocytes. The chemotherapeutic drug concentration is given by and the immunotherapeutic by (Interleukin-2) respectively.

## Parameters

This set of parameters can be found as “patient 9” in [].

## Reference Solutions

The problem can be solved with the [multiple shooting method]. For the following solutions the control functions and states are discretized on the same grid, with 100 nodes. For the objective function parameters have been chosen from the following sets to grand the results shown in the graphics below.

Objective function 1

Objective function 2

In both objective function the amount of chemotherapeutic drugs is penalized. The objective function 2 describes the worst case scenario of the tumor growth at end time.