D'Onofrio chemotherapy model

D'Onofrio chemotherapy model
State dimension:	1
Differential states:	4
Continuous control functions:	2
Path constraints:	4

This cancer chemotherapy model is based on the work of d'Onofrio. The corresponding dynamic describes the effect of two different drugs administered to the patient. An anti-angiogetic drug is used to suppress the formation of blood vessels from existing vessels and thereby starving the tumors supply of proliferating vessels. In addition a cytostatic drug effects the proliferation of the tumor cells directly. The dynamic of the problem is given by an ODE model.

Mathematical formulation

For $t\in [t_0,t_f]$ the optimal control problem is given by

$\begin{array}{llcl}\underset{x,u}{\min }& x_0(t_f)& +& \alpha \underset{t_0}{\overset{t_f}{\int }}{u}_0(t{)}^2\text{d}t\\ \text{s.t.}& {\dot{x}}_0& =& -\zeta x_0\text{ln}\left(\frac{x_0}{x_1}\right)-F{x}_0{u}_1,\\ & {\dot{x}}_1& =& b{x}_0-\mu x_1-d{x}_0^{\frac{2}{3}}{x}_1-G{u}_0{x}_1-\eta x_1{u}_1,\\ & {\dot{x}}_2& =& u_0,\\ & {\dot{x}}_3& =& u_1,\\ [1.5ex]& u_0& \in & [0,u_0^{max}],\\ & u_1& \in & [0,u_1^{max}],\\ & x_2& \le & x_2^{max},\\ & x_3& \le & x_3^{max}.\end{array}$

where the control $u_0$ denotes the administered amount of anti-angiogetic drugs and $u_1$ the amount of cytostatic drugs. The state $x_0$ describes the volume of tumor and $x_1$ the volume of neighboring blood vessels. The remaining states $x_2$ and $x_3$ are used to constraint the maximum amount of drugs over the duration of the therapy.

Parameters

In the model these parameters are fixed.

$\begin{array}{rcl}{t}_0& =& 0,\\ (\zeta ,b,\mu ,d,G)& =& (0.192,5.85,0.0,0.00873,0.15),\\ (x_2(0),x_3(0),u_0^{max},x_2^{max})& =& (0,0,75,300).\end{array}$

The parameters $(x_0(0),x_1(0),u_1^{max},x_3^{max})$ can be taken from the parameter sets shown in the following section. To the remaining parameters $(F,\eta )$ exists no experimental data.

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. The unknown parameters are chosen from the following parameter sets

Parameter set 1

$\begin{array}{rclrcl}{x}_0(0)& =& 12000,& x_1(0)& =& 15000,\\ u_1^{max}& =& 1,& x_3^{max}& =& 2.\\ \end{array}$

Parameter set 2

$\begin{array}{rclrcl}{x}_0(0)& =& 12000,& x_1(0)& =& 15000,\\ u_1^{max}& =& 2,& x_3^{max}& =& 10.\\ \end{array}$

Parameter set 3

$\begin{array}{rclrcl}{x}_0(0)& =& 14000,& x_1(0)& =& 5000,\\ u_1^{max}& =& 1,& x_3^{max}& =& 2.\\ \end{array}$

Parameter set 4

$\begin{array}{rclrcl}{x}_0(0)& =& 14000,& x_1(0)& =& 5000,\\ u_1^{max}& =& 2,& x_3^{max}& =& 10.\\ \end{array}$

Furthermore in the objective function $\alpha =0$ is chosen.

• Reference solution plots
• 
  Optimal controls and states computed with a multiple shooting approach on the same discretization grid with 100 nodes and parameter set 1.
• 
  Optimal controls and states with parameter set 2.
• 
  Optimal controls and states with parameter set 3.
• 
  Optimal controls and states with parameter set 4.

Variants

• a variant where partial outer convexification is applied on the control and the continous control is replaces by binary controls, see also D'Onofrio model (binary variant),

Source Code

• Muscod code at D'Onofrio chemotherapy model (Muscod)

References

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