https://mintoc.de/api.php?action=feedcontributions&user=JenniferUebbing&feedformat=atommintOC - User contributions [en]2024-03-28T23:43:33ZUser contributionsMediaWiki 1.25.2https://mintoc.de/index.php?title=De_Pillis_chemotherapy_model&diff=738De Pillis chemotherapy model2015-10-17T12:55:24Z<p>JenniferUebbing: </p>
<hr />
<div>{{Dimensions<br />
|nd = 1<br />
|nx = 6<br />
|nu = 3<br />
}}<br />
<br />
The model by de Pillis combines chemotherapy with immunotherapy.<br />
<br />
== Mathematical formulation ==<br />
<br />
For <math>t \in [t_0, t_f]</math> and <math>D = d \frac{(x_2/x_0)^l}{s+(x_2/x_0)^l}</math> the optimal control problem is given by<br />
<br />
<p><br />
<math><br />
\begin{array}{llcl}<br />
\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]<br />
\mbox{s.t.} & \dot{x}_0(t) & = & a x_0 (1-b x_0) -c x_1 x_0 - D x_0 - K_T (1- \mbox{e}^{- x_4}) x_0 ,\\<br />
& \dot{x}_1(t) & = & 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, \\<br />
& \dot{x}_2(t) & = & -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 \\<br />
& & & - 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, \\<br />
& \dot{x}_3(t) & = & \alpha - \beta x_3 - K_C (1- \mbox{e}^{- x_4}) x_3,\\<br />
& \dot{x}_4(t) & = & - \gamma x_4 + u_0,\\<br />
& \dot{x}_5(t) & = & - \mu_I x_5 + u_1.<br />
\end{array} <br />
</math><br />
</p><br />
<br />
The states <math>x_0</math> to <math>x_3</math> are measured in absolute cell counts, where <math>x_0</math> describes the number of tumor cells, <math>x_1</math> of unspecific immune cells, <math>x_2</math> of tumor-specific cytotoxic T-cells (CD<math>8^+</math> T) and <math>x_3</math> of circulating lymphocytes. The chemotherapeutic drug concentration is given by <math>x_4</math> and the immunotherapeutic by <math>x_5</math> (Interleukin-2) respectively.<br />
<br />
== Parameters ==<br />
<br />
This set of parameters can be found as “patient 9” in [].<br />
<br />
<math><br />
\begin{array}{lll}<br />
a = 4.31 \times 10^{-1} & b = 1.02 \times 10^{-9} & c = 6.41 \times 10^{-11}\\<br />
d = 2.34 & e = 2.08 \times 10^{-7} & f = 4.12 \times 10^{-2}\\<br />
g = 1.25 \times 10^{-2} & h = 2.02 \times 10^{7} & j = 2.49 \times 10^{-2}\\<br />
k = 3.66 \times 10^{7} & l = 2.09 & m = 2.04 \times 10^{-1}\\<br />
q = 1.42 \times 10^{-6} & p = 3.42 \times 10^{-6} & s = 8.39 \times 10^{-2}\\<br />
r_1 = 1.01 \times 10^{-7} & r_2 = 6.50 \times 10^{-11} & u = 3.00 \times 10^{-10}\\<br />
\alpha = 7.50 \times 10^{8} & \beta = 1.20 \times 10^{-2} & \gamma = 9.00 \times 10^{-1}\\<br />
p_I = 1.25 \times 10^{-1} & g_I = 2.00 \times 10^{7} & \mu_I = 1.00 \times 10^{1}\\<br />
K_T = 9.00 \times 10^{-1} & K_N = K_L = K_C = 6 \times 10^{-1}<br />
\end{array}<br />
</math><br />
<br />
== Reference Solutions ==<br />
<br />
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.<br />
<br />
Objective function 1<br />
<math><br />
\begin{array}{rclr}<br />
p_0 &=& 1,&\\<br />
p_2 &=& 1,&\\<br />
p_i &=& 0,&otherwise.\\<br />
\end{array}<br />
</math><br />
<br />
Objective function 2<br />
<math><br />
\begin{array}{rclr}<br />
p_0 &=& -1,&\\<br />
p_2 &=& 1,&\\<br />
p_i &=& 0,&otherwise.\\<br />
\end{array}<br />
</math><br />
<br />
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.<br />
<br />
<gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="2"><br />
Image:DePillis_1.png| Optimal controls and states computed with a multiple shooting approach on the same discretization grid with 100 nodes and objective function 1.<br />
Image:DePillis_2.png| Optimal controls and states with objective function 2.<br />
</gallery><br />
<br />
==Source Code==<br />
<br />
[[:Category:C | C code]] <br />
<source lang="cpp"><br />
/* volume of tumor in absolute cell count*/<br />
rhs[0] = - a*x[0] (1-b*x[0]) -c*x[1]*x[0] - D*x[0] - K_T*(1- exp(- x[4]))*x[0];<br />
<br />
/* volume of unspecific immune cells */<br />
rhs[1] = e*x[3] - f*x[1] + g*x[0]*x[0]/(h+x[0]*x[0])-p*x[1]*x[0] - K_N*(1- exp(- x[4]))*x[1];<br />
<br />
/* volume of tumor-specific cytotoxic T-cells */<br />
rhs[2] = -m*x[2] + j*D*D*x[0]*x[0]/(k+ D*D*x[0]*x[0])*x[2] - q*x[1]*x[2] + (r_1*x[1] + r_2*x[3])*x[0] ...<br />
- v*x[1]* x[2]*x[2] - K_L* (1- exp(- x[4]))*x[2] + p_I*x[2]*x[5]/(g_I + x[5]) + u[2];<br />
<br />
/* volume of circulating lymphocytes */<br />
rhs[3] = alpha - beta *x[3] - K_C *(1- exp(- x[4])) *x[3];<br />
<br />
/* amount of chemotherapeutic drug */<br />
rhs[4] = u[0] - gamma*x[4];<br />
<br />
/* amount of immunotherapeutic drug */<br />
rhs[5] = u[1] - mu_I*x[5];<br />
<br />
</source><br />
<br />
== References ==<br />
<br />
[[Category:ODE model]]</div>JenniferUebbinghttps://mintoc.de/index.php?title=De_Pillis_chemotherapy_model&diff=737De Pillis chemotherapy model2015-10-17T12:52:58Z<p>JenniferUebbing: Created page with "{{Dimensions |nd = 1 |nx = 6 |nu = 3 }} The model by de Pillis combines chemotherapy with immunotherapy == Mathematical formulation == For <math>t \in..."</p>
<hr />
<div>{{Dimensions<br />
|nd = 1<br />
|nx = 6<br />
|nu = 3<br />
}}<br />
<br />
The model by de Pillis combines chemotherapy with immunotherapy<br />
<br />
== Mathematical formulation ==<br />
<br />
For <math>t \in [t_0, t_f]</math> and <math>D = d \frac{(x_2/x_0)^l}{s+(x_2/x_0)^l}</math> the optimal control problem is given by<br />
<br />
<p><br />
<math><br />
\begin{array}{llcl}<br />
\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]<br />
\mbox{s.t.} & \dot{x}_0(t) & = & a x_0 (1-b x_0) -c x_1 x_0 - D x_0 - K_T (1- \mbox{e}^{- x_4}) x_0 ,\\<br />
& \dot{x}_1(t) & = & 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, \\<br />
& \dot{x}_2(t) & = & -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 \\<br />
& & & - 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, \\<br />
& \dot{x}_3(t) & = & \alpha - \beta x_3 - K_C (1- \mbox{e}^{- x_4}) x_3,\\<br />
& \dot{x}_4(t) & = & - \gamma x_4 + u_0,\\<br />
& \dot{x}_5(t) & = & - \mu_I x_5 + u_1.<br />
\end{array} <br />
</math><br />
</p><br />
<br />
The states 0 to 3 are measured in absolute cell counts, where <math>x_0</math> describes the number of tumor cells, <math>x_1</math> of unspecific immune cells, <math>x_2</math> of tumor-specific cytotoxic T-cells (CD<math>8^+</math> T) and <math>x_3</math> of circulating lymphocytes. The chemotherapeutic drug concentration is given by <math>x_4</math> and the immunotherapeutic by <math>x_5</math> (Interleukin-2) respectively.<br />
<br />
== Parameters ==<br />
<br />
This set of parameters can be found as “patient 9” in [].<br />
<br />
<math><br />
\begin{array}{lll}<br />
a = 4.31 \times 10^{-1} & b = 1.02 \times 10^{-9} & c = 6.41 \times 10^{-11}\\<br />
d = 2.34 & e = 2.08 \times 10^{-7} & f = 4.12 \times 10^{-2}\\<br />
g = 1.25 \times 10^{-2} & h = 2.02 \times 10^{7} & j = 2.49 \times 10^{-2}\\<br />
k = 3.66 \times 10^{7} & l = 2.09 & m = 2.04 \times 10^{-1}\\<br />
q = 1.42 \times 10^{-6} & p = 3.42 \times 10^{-6} & s = 8.39 \times 10^{-2}\\<br />
r_1 = 1.01 \times 10^{-7} & r_2 = 6.50 \times 10^{-11} & u = 3.00 \times 10^{-10}\\<br />
\alpha = 7.50 \times 10^{8} & \beta = 1.20 \times 10^{-2} & \gamma = 9.00 \times 10^{-1}\\<br />
p_I = 1.25 \times 10^{-1} & g_I = 2.00 \times 10^{7} & \mu_I = 1.00 \times 10^{1}\\<br />
K_T = 9.00 \times 10^{-1} & K_N = K_L = K_C = 6 \times 10^{-1}<br />
\end{array}<br />
</math><br />
<br />
== Reference Solutions ==<br />
<br />
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.<br />
<br />
Objective function 1<br />
<math><br />
\begin{array}{rclr}<br />
p_0 &=& 1,&\\<br />
p_2 &=& 1,&\\<br />
p_i &=& 0,&otherwise.\\<br />
\end{array}<br />
</math><br />
<br />
Objective function 2<br />
<math><br />
\begin{array}{rclr}<br />
p_0 &=& -1,&\\<br />
p_2 &=& 1,&\\<br />
p_i &=& 0,&otherwise.\\<br />
\end{array}<br />
</math><br />
<br />
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.<br />
<br />
<gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="2"><br />
Image:DePillis_1.png| Optimal controls and states computed with a multiple shooting approach on the same discretization grid with 100 nodes and objective function 1.<br />
Image:DePillis_2.png| Optimal controls and states with objective function 2.<br />
</gallery><br />
<br />
==Source Code==<br />
<br />
[[:Category:C | C code]] <br />
<source lang="cpp"><br />
/* volume of tumor in absolute cell count*/<br />
rhs[0] = - a*x[0] (1-b*x[0]) -c*x[1]*x[0] - D*x[0] - K_T*(1- exp(- x[4]))*x[0];<br />
<br />
/* volume of unspecific immune cells */<br />
rhs[1] = e*x[3] - f*x[1] + g*x[0]*x[0]/(h+x[0]*x[0])-p*x[1]*x[0] - K_N*(1- exp(- x[4]))*x[1];<br />
<br />
/* volume of tumor-specific cytotoxic T-cells */<br />
rhs[2] = -m*x[2] + j*D*D*x[0]*x[0]/(k+ D*D*x[0]*x[0])*x[2] - q*x[1]*x[2] + (r_1*x[1] + r_2*x[3])*x[0] ...<br />
- v*x[1]* x[2]*x[2] - K_L* (1- exp(- x[4]))*x[2] + p_I*x[2]*x[5]/(g_I + x[5]) + u[2];<br />
<br />
/* volume of circulating lymphocytes */<br />
rhs[3] = alpha - beta *x[3] - K_C *(1- exp(- x[4])) *x[3];<br />
<br />
/* amount of chemotherapeutic drug */<br />
rhs[4] = u[0] - gamma*x[4];<br />
<br />
/* amount of immunotherapeutic drug */<br />
rhs[5] = u[1] - mu_I*x[5];<br />
<br />
</source><br />
<br />
== References ==<br />
<br />
[[Category:ODE model]]</div>JenniferUebbinghttps://mintoc.de/index.php?title=File:DePillis_2.png&diff=736File:DePillis 2.png2015-10-17T12:52:05Z<p>JenniferUebbing: </p>
<hr />
<div></div>JenniferUebbinghttps://mintoc.de/index.php?title=File:DePillis_1.png&diff=735File:DePillis 1.png2015-10-17T12:51:45Z<p>JenniferUebbing: </p>
<hr />
<div></div>JenniferUebbinghttps://mintoc.de/index.php?title=D%27Onofrio_chemotherapy_model&diff=723D'Onofrio chemotherapy model2015-10-10T11:53:17Z<p>JenniferUebbing: Created page with "{{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..."</p>
<hr />
<div>{{Dimensions<br />
|nd = 1<br />
|nx = 4<br />
|nu = 2<br />
}}<br />
<br />
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]].<br />
<br />
== Mathematical formulation ==<br />
<br />
For <math>t \in [t_0, t_f]</math> the optimal control problem is given by<br />
<br />
<p><br />
<math><br />
\begin{array}{llcl}<br />
\displaystyle \min_{x, u} & x_0(t_f) &+& \alpha \int_{t_0}^{t_f} u_0(t)^2 \text{d}t \\[1.5ex]<br />
\mbox{s.t.} & \dot{x}_0(t) & = & - \zeta x_0(t) \text{ln} \left( \frac{x_0(t)}{x_1(t)} \right) - F \; x_0(t) u_1(t), \\<br />
& \dot{x}_1(t) & = & b x_0(t) - \mu x_1(t) - d x_0(t)^{\frac{2}{3}}x_1(t) - G u_0(t) x_1(t) - \eta x_1(t) u_1(t), \\<br />
& \dot{x}_2(t) & = & u_0(t), \\<br />
& \dot{x}_3(t) & = & u_1(t), \\ [1.5ex]<br />
& 0 & \leq & u_0(t) \leq u_0^{max}, \\<br />
& 0 & \leq & u_1(t) \leq u_1^{max}, \\<br />
& x_2(t) & \leq & x_2^{max}, \\<br />
& x_3(t) & \leq & x_3^{max}.<br />
\end{array} <br />
</math><br />
</p><br />
<br />
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.<br />
<br />
== Parameters ==<br />
<br />
In the model these parameters are fixed.<br />
<br />
<math><br />
\begin{array}{rcl}<br />
t_0 &=& 0,\\<br />
(\zeta, b, \mu, d, G) &=& (0.192, 5.85, 0.0, 0.00873, 0.15),\\<br />
(x_2(0), x_3(0), u_0^{max}, x_2^{max}) &=& (0,0,75,300).<br />
\end{array}<br />
</math><br />
<br />
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.<br />
<br />
== Reference Solutions ==<br />
<br />
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 <br />
<br />
Parameter set 1<br />
<br />
<math><br />
\begin{array}{rclrcl}<br />
x_0(0) &=& 12000,& x_1(0) &=& 15000,\\<br />
u_1^{max} &=& 1,& x_3^{max} &=& 2.\\<br />
\end{array}<br />
</math><br />
<br />
Parameter set 2<br />
<br />
<math><br />
\begin{array}{rclrcl}<br />
x_0(0) &=& 12000,& x_1(0) &=& 15000,\\<br />
u_1^{max} &=& 2,& x_3^{max} &=& 10.\\<br />
\end{array}<br />
</math><br />
<br />
Parameter set 3<br />
<br />
<math><br />
\begin{array}{rclrcl}<br />
x_0(0) &=& 14000,& x_1(0) &=& 5000,\\<br />
u_1^{max} &=& 1,& x_3^{max} &=& 2.\\<br />
\end{array}<br />
</math><br />
<br />
Parameter set 4<br />
<br />
<math><br />
\begin{array}{rclrcl}<br />
x_0(0) &=& 14000,& x_1(0) &=& 5000,\\<br />
u_1^{max} &=& 2,& x_3^{max} &=& 10.\\<br />
\end{array}<br />
</math><br />
<br />
Furthermore in the objective function <math>\alpha =0</math> is chosen.<br />
<br />
<gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="2"><br />
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.<br />
Image:d'Onofrio 2.png| Optimal controls and states with parameter set 2.<br />
Image:d'Onofrio 3.png| Optimal controls and states with parameter set 3.<br />
Image:d'Onofrio 4.png| Optimal controls and states with parameter set 4.<br />
</gallery><br />
<br />
==Source Code==<br />
<br />
[[:Category:C | C code]] <br />
<source lang="cpp"><br />
/* volume of tumor */<br />
rhs[0] = - zeta*x[0]*log(x[0]/x[1])-F*x[0]*u1_;<br />
/* volume of neighboring blood vessels */<br />
rhs[1] = b*x[0] - (mu + d*pow(x[0],2.0/3.0))*x[1] - G*u0_*x[1] - eta*x[1]*u1_;<br />
/* amount of anti-angiogetic drug */<br />
rhs[2] = u0_;<br />
/* amount of cytostatic drug */<br />
rhs[3] = u1_;<br />
</source><br />
<br />
==Variants==<br />
<br />
In <bibref></bibref> can be found other approaches to solving this problem, using indirect methods and [http://en.wikipedia.org/wiki/Pontryagin%27s_minimum_principle Pontryagins maximum principle].<br />
<br />
== References ==<br />
<bibreferences/><br />
<br />
[[Category:ODE model]]</div>JenniferUebbinghttps://mintoc.de/index.php?title=File:D%27Onofrio_4.png&diff=722File:D'Onofrio 4.png2015-10-10T11:36:31Z<p>JenniferUebbing: </p>
<hr />
<div></div>JenniferUebbinghttps://mintoc.de/index.php?title=File:D%27Onofrio_3.png&diff=721File:D'Onofrio 3.png2015-10-10T11:36:17Z<p>JenniferUebbing: </p>
<hr />
<div></div>JenniferUebbinghttps://mintoc.de/index.php?title=File:D%27Onofrio_2.png&diff=720File:D'Onofrio 2.png2015-10-10T11:36:00Z<p>JenniferUebbing: </p>
<hr />
<div></div>JenniferUebbinghttps://mintoc.de/index.php?title=File:D%27Onofrio_1.png&diff=719File:D'Onofrio 1.png2015-10-10T11:34:18Z<p>JenniferUebbing: </p>
<hr />
<div></div>JenniferUebbing