De Pillis chemotherapy model - Revision history

FelixMueller at 22:09, 27 June 2016

2016-06-27T22:09:32Z

← Older revision		Revision as of 22:09, 27 June 2016
Line 84:		Line 84:
	[[Category:MIOCP]]		[[Category:MIOCP]]
	[[Category:ODE model]]		[[Category:ODE model]]
			[[Category:Bang bang]]
			[[Category:Path-constrained arcs]]

FelixMueller: /* Mathematical formulation */

2016-04-10T09:28:58Z

Mathematical formulation

← Older revision		Revision as of 09:28, 10 April 2016
Line 15:		Line 15:
	\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~~(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 ,\\		\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~~(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, \\		& \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~~(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 \\		& \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~~(t)~~ & = & \alpha - \beta x_3 - K_C (1- \mbox{e}^{- x_4}) x_3,\\		& \dot{x}_3 & = & \alpha - \beta x_3 - K_C (1- \mbox{e}^{- x_4}) x_3,\\
	& \dot{x}_4~~(t)~~ & = & - \gamma x_4 + u_0,\\		& \dot{x}_4 & = & - \gamma x_4 + u_0,\\
	& \dot{x}_5~~(t)~~ & = & - \mu_I x_5 + u_1.		& \dot{x}_5 & = & - \mu_I x_5 + u_1.
	\end{array}		\end{array}
	</math>		</math>

SebastianSager at 09:51, 28 January 2016

2016-01-28T09:51:12Z

← Older revision		Revision as of 09:51, 28 January 2016
Line 81:		Line 81:
	== References ==		== References ==

			[[Category:Medicine]]
	[[Category:MIOCP]]		[[Category:MIOCP]]
	[[Category:ODE model]]		[[Category:ODE model]]

SebastianSager: Moved C to Muscod

2016-01-28T09:29:49Z

Moved C to Muscod

← Older revision		Revision as of 09:29, 28 January 2016
Line 77:		Line 77:
	==Source Code==		==Source Code==

	* [[:Category:C \| C code]] at [[De Pillis chemotherapy model (C)]]		* [[:Category:Muscod \| Muscod code]] at [[De Pillis chemotherapy model (Muscod)]]

	== References ==		== References ==

	[[Category: MIOCP]]		[[Category:MIOCP]]
	[[Category:ODE model]]		[[Category:ODE model]]

FelixMueller: /* Source Code */

2016-01-27T17:40:29Z

Source Code

← Older revision		Revision as of 17:40, 27 January 2016
Line 78:		Line 78:

	* [[:Category:C \| C code]] at [[De Pillis chemotherapy model (C)]]		* [[:Category:C \| C code]] at [[De Pillis chemotherapy model (C)]]
	~~<source lang="cpp">~~
	~~/* volume of tumor in absolute cell count*/~~
	~~rhs[0] = - ax[0] (1-bx[0]) -cx[1]x[0] - Dx[0] - K_T(1- exp(- x[4]))*x[0];~~

	~~/* volume of unspecific immune cells */~~
	~~rhs[1] = ex[3] - fx[1] + gx[0]x[0]/(h+x[0]x[0])-px[1]x[0] - K_N(1- exp(- x[4]))*x[1];~~

	~~/* volume of tumor-specific cytotoxic T-cells */~~
	~~rhs[2] = -mx[2] + jDDx[0]x[0]/(k+ DDx[0]x[0])x[2] - qx[1]x[2] + (r_1x[1] + r_2x[3])x[0] ...~~
	~~- vx[1] x[2]x[2] - K_L (1- exp(- x[4]))x[2] + p_Ix[2]*x[5]/(g_I + x[5]) + u[2];~~

	~~/* volume of circulating lymphocytes */~~
	~~rhs[3] = alpha - beta x[3] - K_C (1- exp(- x[4])) *x[3];~~

	~~/* amount of chemotherapeutic drug */~~
	~~rhs[4] = u[0] - gamma*x[4];~~

	~~/* amount of immunotherapeutic drug */~~
	~~rhs[5] = u[1] - mu_I*x[5];~~

	~~</source>~~

	== References ==		== References ==

FelixMueller: /* Source Code */

2016-01-27T17:40:16Z

Source Code

← Older revision		Revision as of 17:40, 27 January 2016
Line 77:		Line 77:
	==Source Code==		==Source Code==

	[[:Category:C \| C code]]		* [[:Category:C \| C code]] at [[De Pillis chemotherapy model (C)]]
	<source lang="cpp">		<source lang="cpp">
	/* volume of tumor in absolute cell count*/		/* volume of tumor in absolute cell count*/

FelixMueller: /* References */

2016-01-27T17:37:46Z

References

← Older revision		Revision as of 17:37, 27 January 2016
Line 102:		Line 102:
	== References ==		== References ==

			[[Category: MIOCP]]
	[[Category:ODE model]]		[[Category:ODE model]]

JenniferUebbing at 12:55, 17 October 2015

2015-10-17T12:55:24Z

← Older revision		Revision as of 12:55, 17 October 2015
Line 5:		Line 5:
	}}		}}

	The model by de Pillis combines chemotherapy with immunotherapy		The model by de Pillis combines chemotherapy with immunotherapy.

	== Mathematical formulation ==		== Mathematical formulation ==
Line 26:		Line 26:
	</p>		</p>

	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.		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.

	== Parameters ==		== Parameters ==

JenniferUebbing: Created page with "{{Dimensions |nd = 1 |nx = 6 |nu = 3 }} The model by de Pillis combines chemotherapy with immunotherapy == Mathematical formulation == For $t \in..."$

2015-10-17T12:52:58Z

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..."

New page

{{Dimensions
|nd = 1
|nx = 6
|nu = 3
}}

The model by de Pillis combines chemotherapy with immunotherapy

== Mathematical formulation ==

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

<p>
<math>
\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]
\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 ,\\
& \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, \\
& \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 \\
& & & - 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(t) & = & \alpha - \beta x_3 - K_C (1- \mbox{e}^{- x_4}) x_3,\\
& \dot{x}_4(t) & = & - \gamma x_4 + u_0,\\
& \dot{x}_5(t) & = & - \mu_I x_5 + u_1.
\end{array}
</math>
</p>

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.

== Parameters ==

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

<math>
\begin{array}{lll}
a = 4.31 \times 10^{-1} & b = 1.02 \times 10^{-9} & c = 6.41 \times 10^{-11}\\
d = 2.34 & e = 2.08 \times 10^{-7} & f = 4.12 \times 10^{-2}\\
g = 1.25 \times 10^{-2} & h = 2.02 \times 10^{7} & j = 2.49 \times 10^{-2}\\
k = 3.66 \times 10^{7} & l = 2.09 & m = 2.04 \times 10^{-1}\\
q = 1.42 \times 10^{-6} & p = 3.42 \times 10^{-6} & s = 8.39 \times 10^{-2}\\
r_1 = 1.01 \times 10^{-7} & r_2 = 6.50 \times 10^{-11} & u = 3.00 \times 10^{-10}\\
\alpha = 7.50 \times 10^{8} & \beta = 1.20 \times 10^{-2} & \gamma = 9.00 \times 10^{-1}\\
p_I = 1.25 \times 10^{-1} & g_I = 2.00 \times 10^{7} & \mu_I = 1.00 \times 10^{1}\\
K_T = 9.00 \times 10^{-1} & K_N = K_L = K_C = 6 \times 10^{-1}
\end{array}
</math>

== 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
<math>
\begin{array}{rclr}
p_0 &=& 1,&\\
p_2 &=& 1,&\\
p_i &=& 0,&otherwise.\\
\end{array}
</math>

Objective function 2
<math>
\begin{array}{rclr}
p_0 &=& -1,&\\
p_2 &=& 1,&\\
p_i &=& 0,&otherwise.\\
\end{array}
</math>

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.

<gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="2">
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.
Image:DePillis_2.png| Optimal controls and states with objective function 2.
</gallery>

==Source Code==

[[:Category:C | C code]]
<source lang="cpp">
/* volume of tumor in absolute cell count*/
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];

/* volume of unspecific immune cells */
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];

/* volume of tumor-specific cytotoxic T-cells */
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] ...
- 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];

/* volume of circulating lymphocytes */
rhs[3] = alpha - beta *x[3] - K_C *(1- exp(- x[4])) *x[3];

/* amount of chemotherapeutic drug */
rhs[4] = u[0] - gamma*x[4];

/* amount of immunotherapeutic drug */
rhs[5] = u[1] - mu_I*x[5];

</source>

== References ==

[[Category:ODE model]]

De Pillis chemotherapy model - Revision history

FelixMueller at 22:09, 27 June 2016

FelixMueller: /* Mathematical formulation */

SebastianSager at 09:51, 28 January 2016

SebastianSager: Moved C to Muscod

FelixMueller: /* Source Code */

FelixMueller: /* Source Code */

FelixMueller: /* References */

JenniferUebbing at 12:55, 17 October 2015

JenniferUebbing: Created page with "{{Dimensions |nd = 1 |nx = 6 |nu = 3 }} The model by de Pillis combines chemotherapy with immunotherapy == Mathematical formulation == For t \in..."

JenniferUebbing: Created page with "{{Dimensions |nd = 1 |nx = 6 |nu = 3 }} The model by de Pillis combines chemotherapy with immunotherapy == Mathematical formulation == For $t \in..."$