Difference between revisions of "De Pillis chemotherapy model"
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\begin{array}{llcl} | \begin{array}{llcl} | ||
\displaystyle \min_{x, u} & p_0 x_0(t_f) & + & \int_{t_0}^{t_f} p_1 x_0(t)^2 \mbox{d}t + \sum_{i=0}^{3} \int_{t_0}^{t_f} p_{i+2} u_i(t)^2 \mbox{d}t \\[1.5ex] | \displaystyle \min_{x, u} & p_0 x_0(t_f) & + & \int_{t_0}^{t_f} p_1 x_0(t)^2 \mbox{d}t + \sum_{i=0}^{3} \int_{t_0}^{t_f} p_{i+2} u_i(t)^2 \mbox{d}t \\[1.5ex] | ||
− | \mbox{s.t.} & \dot{x}_0 | + | \mbox{s.t.} & \dot{x}_0 & = & a x_0 (1-b x_0) -c x_1 x_0 - D x_0 - K_T (1- \mbox{e}^{- x_4}) x_0 ,\\ |
− | & \dot{x}_1 | + | & \dot{x}_1 & = & e x_3 - f x_1 + g \frac{x_0^2}{h+x_0^2}-p x_1 x_0 - K_N (1- \mbox{e}^{- x_4}) x_1, \\ |
− | & \dot{x}_2 | + | & \dot{x}_2 & = & -m x_2 + j \frac{D^2 x_0^2}{k+ D^2 x_0^2} x_2 - q x_1 x_2 + (r_1 x_1 + r_2 x_3) x_0 \\ |
& & & - v x_1 x_2^2 - K_L (1- \mbox{e}^{- x_4}) x_2 + \frac{p_I x_2 x_5}{g_I + x_5} + u_2, \\ | & & & - v x_1 x_2^2 - K_L (1- \mbox{e}^{- x_4}) x_2 + \frac{p_I x_2 x_5}{g_I + x_5} + u_2, \\ | ||
− | & \dot{x}_3 | + | & \dot{x}_3 & = & \alpha - \beta x_3 - K_C (1- \mbox{e}^{- x_4}) x_3,\\ |
− | & \dot{x}_4 | + | & \dot{x}_4 & = & - \gamma x_4 + u_0,\\ |
− | & \dot{x}_5 | + | & \dot{x}_5 & = & - \mu_I x_5 + u_1. |
\end{array} | \end{array} | ||
</math> | </math> | ||
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[[Category:MIOCP]] | [[Category:MIOCP]] | ||
[[Category:ODE model]] | [[Category:ODE model]] | ||
+ | [[Category:Bang bang]] | ||
+ | [[Category:Path-constrained arcs]] |
Latest revision as of 23:09, 27 June 2016
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.