View source for D'Onofrio chemotherapy model

{{Dimensions
|nd        = 1
|nx        = 4
|nu        = 2
}}

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 [[:Category:ODE model|ODE model]].

== Mathematical formulation ==

For <math>t \in [t_0, t_f]</math> the optimal control problem is given by

<p>
<math>
\begin{array}{llcl}
 \displaystyle \min_{x, u} & x_0(t_f) &+& \alpha \int_{t_0}^{t_f} u_0(t)^2 \text{d}t   \\[1.5ex]
 \mbox{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]
             & 0 & \leq & u_0 \quad \leq  u_0^{max},  \\
             & 0 & \leq & u_1 \quad  \leq u_1^{max}, \\
             & x_2 & \leq & x_2^{max},  \\
             & x_3 & \leq & x_3^{max}.
\end{array} 
</math>
</p>

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

== Parameters ==

In the model these parameters are fixed.

<math>
\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}
</math>

The parameters <math>(x_0(0), x_1(0), u_1^{max}, x_3^{max})</math> can be taken from the parameter sets shown in the following section. To the remaining parameters <math>(F, \eta)</math> 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

<math>
\begin{array}{rclrcl}
x_0(0) &=& 12000,& x_1(0) &=& 15000,\\
u_1^{max} &=& 1,& x_3^{max} &=& 2.\\
\end{array}
</math>

Parameter set 2

<math>
\begin{array}{rclrcl}
x_0(0) &=& 12000,& x_1(0) &=& 15000,\\
u_1^{max} &=& 2,& x_3^{max} &=& 10.\\
\end{array}
</math>

Parameter set 3

<math>
\begin{array}{rclrcl}
x_0(0) &=& 14000,& x_1(0) &=& 5000,\\
u_1^{max} &=& 1,& x_3^{max} &=& 2.\\
\end{array}
</math>

Parameter set 4

<math>
\begin{array}{rclrcl}
x_0(0) &=& 14000,& x_1(0) &=& 5000,\\
u_1^{max} &=& 2,& x_3^{max} &=& 10.\\
\end{array}
</math>

Furthermore in the objective function <math>\alpha =0</math> is chosen.

<gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="2">
 Image:d'Onofrio 1.png| Optimal controls and states computed with a multiple shooting approach on the same discretization grid with 100 nodes and parameter set 1.
 Image:d'Onofrio 2.png| Optimal controls and states with parameter set 2.
 Image:d'Onofrio 3.png| Optimal controls and states with parameter set 3.
 Image:d'Onofrio 4.png| Optimal controls and states with parameter set 4.
</gallery>

==Source Code==

[[:Category:Muscod | Muscod code]] 
<source lang="cpp">
/* volume of tumor */
rhs[0] = - zeta*x[0]*log(x[0]/x[1])-F*x[0]*u1_;
/* volume of neighboring blood vessels */
rhs[1] = b*x[0] - (mu + d*pow(x[0],2.0/3.0))*x[1] - G*u0_*x[1] - eta*x[1]*u1_;
/* amount of anti-angiogetic drug */
rhs[2] = u0_;
/* amount of cytostatic drug */
rhs[3] = u1_;
</source>

== References ==
<biblist />

[[Category:MIOCP]]
[[Category:Medicine]]
[[Category:ODE model]]
[[Category:Bang bang]]
[[Category:Path-constrained arcs]]