PROPT Optimal Drug Scheduling for Cancer Chemotherapy - Revision history

Per: /* Plot result */

2012-02-14T08:59:24Z

Plot result

← Older revision		Revision as of 04:59, 14 February 2012
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	[[File:drugScheduling_01.png]]		[[File:drugScheduling_01.png]]

			[[Category:PROPT Examples]]

Mbot at 12:07, 9 November 2011

2011-11-09T12:07:32Z

← Older revision		Revision as of 08:07, 9 November 2011
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	<math> J = x_1(t_f) </math>		<math> J = x_1(t_f) </math>


	subject to:		subject to:

	<math> \frac{dx_1}{dt} = -k_1x_1+k_2(x_2-k_3)*H(x_2-k_3) </math>		<math> \frac{dx_1}{dt} = -k_1x_1+k_2(x_2-k_3)*H(x_2-k_3) </math>

	<math> \frac{dx_2}{dt} = u-k_4*x_2 </math>		<math> \frac{dx_2}{dt} = u-k_4*x_2 </math>

	<math> \frac{dx_3}{dt} = x_2 </math>		<math> \frac{dx_3}{dt} = x_2 </math>


	where the tumor mass is given by N = 10^12 * exp (-x1) cells, x2 is the drug concentration in the body in drug units [D] and x3 is the cumulative effect of the drug. The parameters are taken as k1 = 9.9e-4 days, k2 = 8.4e-3 days-1 [De-1], k3 = 10 [De-1], and k4 = 0.27 days-1. The initial state considered is:		where the tumor mass is given by N = 10^12 * exp (-x1) cells, x2 is the drug concentration in the body in drug units [D] and x3 is the cumulative effect of the drug. The parameters are taken as k1 = 9.9e-4 days, k2 = 8.4e-3 days-1 [De-1], k3 = 10 [De-1], and k4 = 0.27 days-1. The initial state considered is:

	<math> x(t_0) = [log(100) \ 0 \ 0]' </math>		<math> x(t_0) = [log(100) \ 0 \ 0]' </math>


	where,		where,
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	<math> u >= 0 </math>		<math> u >= 0 </math>


	There are the following path constraints on the state variables:		There are the following path constraints on the state variables:

	<math> x_2(t) <= 50 </math>		<math> x_2(t) <= 50 </math>

	<math> x_3(t) <= 2.1e3 </math>		<math> x_3(t) <= 2.1e3 </math>


	Also, there should be at least a 50% reduction in the size of the tumor every three weeks, so that the following point constraints must be considered:		Also, there should be at least a 50% reduction in the size of the tumor every three weeks, so that the following point constraints must be considered:

	<math> x_1(21) >= log(200) </math>		<math> x_1(21) >= log(200) </math>

	<math> x_1(42) >= log(400) </math>		<math> x_1(42) >= log(400) </math>

	<math> x_1(63) >= log(800) </math>		<math> x_1(63) >= log(800) </math>


	State number 3 is converted to an integral constraints in the formulation.		State number 3 is converted to an integral constraints in the formulation.

Mbot at 13:05, 8 November 2011

2011-11-08T13:05:04Z

← Older revision		Revision as of 09:05, 8 November 2011
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	<pre>		<pre>
	Problem type appears to be: lpcon		Problem type appears to be: lpcon
	Time for symbolic processing: 0.~~099354~~ seconds		Time for symbolic processing: 0.087433 seconds
	Starting numeric solver		Starting numeric solver
	===== * * * =================================================================== * * *		===== * * * =================================================================== * * *
	TOMLAB - TOMLAB Development license 999007. Valid to 2011-12-31		TOMLAB - TOMLAB Development license 999007. Valid to 2011-12-31
	=====================================================================================		=====================================================================================
	Problem: --- 1: Drug Scheduling - Trial 26 f_k -17.121324912680684000		Problem: --- 1: Drug Scheduling - Trial 66 f_k -17.121324912680684000
	sum(\|constr\|) 0.~~000000000000627527~~		sum(\|constr\|) 0.000000000000846500
	f(x_k) + sum(\|constr\|) -17.~~121324912680056000~~		f(x_k) + sum(\|constr\|) -17.121324912679839000

	Solver: multiMin with local solver snopt. EXIT=0. INFORM=0.		Solver: multiMin with local solver snopt. EXIT=0. INFORM=0.
	Find local optima using multistart local search		Find local optima using multistart local search
	Did 110 local tries. Found 1 global, 98 minima. TotFuncEv 110. TotConstrEv ~~3078~~		Did 110 local tries. Found 1 global, 97 minima. TotFuncEv 110. TotConstrEv 3032

	FuncEv 110 ConstrEv ~~3078~~ ConJacEv 27 Iter ~~1392~~		FuncEv 110 ConstrEv 3032 ConJacEv 18 Iter 1369
	CPU time: 2.~~246414~~ sec. Elapsed time: 2.~~418000~~ sec.		CPU time: 1.965613 sec. Elapsed time: 1.956000 sec.

	</pre>		</pre>
Line 136:		Line 136:
	<pre>		<pre>
	Problem type appears to be: lpcon		Problem type appears to be: lpcon
	Time for symbolic processing: 0.~~10026~~ seconds		Time for symbolic processing: 0.087512 seconds
	Starting numeric solver		Starting numeric solver
	===== * * * =================================================================== * * *		===== * * * =================================================================== * * *
	TOMLAB - TOMLAB Development license 999007. Valid to 2011-12-31		TOMLAB - TOMLAB Development license 999007. Valid to 2011-12-31
	=====================================================================================		=====================================================================================
	Problem: --- 1: Drug Scheduling - Trial 51 f_k -17.~~267378276475135000~~		Problem: --- 1: Drug Scheduling - Trial 86 f_k -17.267378276475142000
	sum(\|constr\|) 0.~~000000000002346713~~		sum(\|constr\|) 0.000000000001722658
	f(x_k) + sum(\|constr\|) -17.~~267378276472787000~~		f(x_k) + sum(\|constr\|) -17.267378276473419000

	Solver: multiMin with local solver snopt. EXIT=0. INFORM=0.		Solver: multiMin with local solver snopt. EXIT=0. INFORM=0.
	Find local optima using multistart local search		Find local optima using multistart local search
	Did 90 local tries. Found 1 global, 88 minima. TotFuncEv 90. TotConstrEv ~~3593~~		Did 90 local tries. Found 1 global, 90 minima. TotFuncEv 90. TotConstrEv 3521

	FuncEv 90 ConstrEv ~~3593~~ ConJacEv 24 Iter ~~1594~~		FuncEv 90 ConstrEv 3521 ConJacEv 18 Iter 1570
	CPU time: 5.~~132433~~ sec. Elapsed time: 5.~~201000~~ sec.		CPU time: 4.570829 sec. Elapsed time: 4.575000 sec.

	</pre>		</pre>
Line 156:		Line 156:
	<pre>		<pre>
	Problem type appears to be: lpcon		Problem type appears to be: lpcon
	Time for symbolic processing: 0.~~094742~~ seconds		Time for symbolic processing: 0.090814 seconds
	Starting numeric solver		Starting numeric solver
	===== * * * =================================================================== * * *		===== * * * =================================================================== * * *
	TOMLAB - TOMLAB Development license 999007. Valid to 2011-12-31		TOMLAB - TOMLAB Development license 999007. Valid to 2011-12-31
	=====================================================================================		=====================================================================================
	Problem: --- 1: Drug Scheduling - Trial 6 f_k -17.~~407138702212308000~~		Problem: --- 1: Drug Scheduling - Trial 6 f_k -17.407138702212297000
	sum(\|constr\|) 0.~~000000000018677708~~		sum(\|constr\|) 0.000000000017636815
	f(x_k) + sum(\|constr\|) -17.~~407138702193631000~~		f(x_k) + sum(\|constr\|) -17.407138702194661000

	Solver: multiMin with local solver snopt. EXIT=0. INFORM=0.		Solver: multiMin with local solver snopt. EXIT=0. INFORM=0.
	Find local optima using multistart local search		Find local optima using multistart local search
	Did 10 local tries. Found 1 global, 10 minima. TotFuncEv 10. TotConstrEv ~~956~~		Did 10 local tries. Found 1 global, 9 minima. TotFuncEv 10. TotConstrEv 937

	FuncEv 10 ConstrEv ~~956~~ ConJacEv 76 Iter ~~401~~		FuncEv 10 ConstrEv 937 ConJacEv 52 Iter 390
	CPU time: 19.~~500125~~ sec. Elapsed time: 15.~~070000~~ sec.		CPU time: 36.987837 sec. Elapsed time: 10.474000 sec.

	</pre>		</pre>

Mbot at 21:29, 2 November 2011

2011-11-02T21:29:33Z

← Older revision		Revision as of 17:29, 2 November 2011
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	where,		where,

			<pre>
	H(x2-k3) = 1 if x2 >= k3 or 0 if x2 < k3		H(x2-k3) = 1 if x2 >= k3 or 0 if x2 < k3
			</pre>

	and the final time t_f = 84 days. The optimization is subject to the following constraints on the drug delivery (control variable):		and the final time t_f = 84 days. The optimization is subject to the following constraints on the drug delivery (control variable):

Mbot at 18:23, 2 November 2011

2011-11-02T18:23:02Z

← Older revision		Revision as of 14:23, 2 November 2011
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	{{Part Of Manual\|title=the PROPT Manual\|link=[[PROPT \|PROPT Manual]]}}		{{Part Of Manual\|title=the PROPT Manual\|link=[[PROPT\|PROPT Manual]]}}

	Dynamic optimization of bioprocesses: efficient and robust numerical strategies 2003, Julio R. Banga, Eva Balsa-Cantro, Carmen G. Moles and Antonio A. Alonso		Dynamic optimization of bioprocesses: efficient and robust numerical strategies 2003, Julio R. Banga, Eva Balsa-Cantro, Carmen G. Moles and Antonio A. Alonso
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	title('Drug Scheduling control');		title('Drug Scheduling control');
	</source>		</source>

			[[File:drugScheduling_01.png]]

Mbot: Created page with "{{Part Of Manual|title=the PROPT Manual|link=PROPT Manual}} Dynamic optimization of bioprocesses: efficient and robust numerical strategies 2003, Julio R. Banga, Ev..."

2011-11-02T15:29:11Z

Created page with "{{Part Of Manual|title=the PROPT Manual|link=PROPT Manual}} Dynamic optimization of bioprocesses: efficient and robust numerical strategies 2003, Julio R. Banga, Ev..."

New page

{{Part Of Manual|title=the PROPT Manual|link=[[PROPT |PROPT Manual]]}}

Dynamic optimization of bioprocesses: efficient and robust numerical strategies 2003, Julio R. Banga, Eva Balsa-Cantro, Carmen G. Moles and Antonio A. Alonso

Case Study III: Optimal Drug Scheduling for Cancer Chemotherapy

==Problem description==

Many researches have devoted their efforts to determine whether current methods for drugs administration during cancer chemotherapy are optimal, and if alternative regimens should be considered. Martin (1992) considered the interesting problem of determining the optimal cancer drug scheduling to decrease the size of a malignant tumor as measured at some particular time in the future. The drug concentration must be kept below some level throughout the treatment period and the cumulative (toxic) effect of the drug must be kept below the ultimate tolerance level. Bojkov et al. (1993) and Luus et al. (1995) also studied this problem using direct search optimization. More recently, Carrasco and Banga (1997) have applied stochastic techniques to solve this problem, obtaining better results (Carrasco & Banga 1998). The mathematical statement of this dynamic optimization problem is: Find u(t) over t in [t0; t_f ] to maximize:

<math> J = x_1(t_f) </math>

subject to:

<math> \frac{dx_1}{dt} = -k_1*x_1+k_2*(x_2-k_3)*H(x_2-k_3) </math>
<math> \frac{dx_2}{dt} = u-k_4*x_2 </math>
<math> \frac{dx_3}{dt} = x_2 </math>

where the tumor mass is given by N = 10^12 * exp (-x1) cells, x2 is the drug concentration in the body in drug units [D] and x3 is the cumulative effect of the drug. The parameters are taken as k1 = 9.9e-4 days, k2 = 8.4e-3 days-1 [De-1], k3 = 10 [De-1], and k4 = 0.27 days-1. The initial state considered is:

<math> x(t_0) = [log(100) \ 0 \ 0]' </math>

where,

H(x2-k3) = 1 if x2 >= k3 or 0 if x2 < k3

and the final time t_f = 84 days. The optimization is subject to the following constraints on the drug delivery (control variable):

<math> u >= 0 </math>

There are the following path constraints on the state variables:

<math> x_2(t) <= 50 </math>
<math> x_3(t) <= 2.1e3 </math>

Also, there should be at least a 50% reduction in the size of the tumor every three weeks, so that the following point constraints must be considered:

<math> x_1(21) >= log(200) </math>
<math> x_1(42) >= log(400) </math>
<math> x_1(63) >= log(800) </math>

State number 3 is converted to an integral constraints in the formulation.

<source lang="matlab">
% Copyright (c) 2007-2008 by Tomlab Optimization Inc.
</source>

==Problem setup==

<source lang="matlab">
toms t

nn = [20 40 120];

for i = 1:length(nn)
</source>

<source lang="matlab">
n = nn(i);
p = tomPhase('p', t, 0, 84, n);
setPhase(p);

tomStates x1 x2
tomControls u

% Initial guess
if i==1
x0 = {icollocate(x2 == 10)
collocate(u == 20)};
else
x0 = {icollocate({x1 == x1opt; x2 == x2opt})
collocate(u == uopt)};
end

% Box constraints
cbox = {
0 <= mcollocate(x1)
0 <= mcollocate(x2) <= 50
0 <= collocate(u) <= 80};

% Boundary constraints
cbnd = initial({x1 == log(100); x2 == 0});

% ODEs and path constraints
k1 = 9.9e-4; k2 = 8.4e-3;
k3 = 10; k4 = 0.27;
ceq = {collocate({
dot(x1) == -k1*x1+k2*max(x2-k3,0)
dot(x2) == u-k4*x2})
% Point-wise conditions
atPoints([21;42;63],x1) >= log([200;400;800])
% Integral constr.
integrate(x2) == 2.1e3};

% Objective
objective = -final(x1);
</source>

==Solve the problem==

<source lang="matlab">
options = struct;
options.name = 'Drug Scheduling';
options.solver = 'multiMin';
options.xInit = 130-n;
solution = ezsolve(objective, {cbox, cbnd, ceq}, x0, options);

x1opt = subs(x1, solution);
x2opt = subs(x2, solution);
uopt = subs(u, solution);
</source>

<pre>
Problem type appears to be: lpcon
Time for symbolic processing: 0.099354 seconds
Starting numeric solver
===== * * * =================================================================== * * *
TOMLAB - TOMLAB Development license 999007. Valid to 2011-12-31
=====================================================================================
Problem: --- 1: Drug Scheduling - Trial 26 f_k -17.121324912680684000
sum(|constr|) 0.000000000000627527
f(x_k) + sum(|constr|) -17.121324912680056000

Solver: multiMin with local solver snopt. EXIT=0. INFORM=0.
Find local optima using multistart local search
Did 110 local tries. Found 1 global, 98 minima. TotFuncEv 110. TotConstrEv 3078

FuncEv 110 ConstrEv 3078 ConJacEv 27 Iter 1392
CPU time: 2.246414 sec. Elapsed time: 2.418000 sec.

</pre>

<pre>
Problem type appears to be: lpcon
Time for symbolic processing: 0.10026 seconds
Starting numeric solver
===== * * * =================================================================== * * *
TOMLAB - TOMLAB Development license 999007. Valid to 2011-12-31
=====================================================================================
Problem: --- 1: Drug Scheduling - Trial 51 f_k -17.267378276475135000
sum(|constr|) 0.000000000002346713
f(x_k) + sum(|constr|) -17.267378276472787000

Solver: multiMin with local solver snopt. EXIT=0. INFORM=0.
Find local optima using multistart local search
Did 90 local tries. Found 1 global, 88 minima. TotFuncEv 90. TotConstrEv 3593

FuncEv 90 ConstrEv 3593 ConJacEv 24 Iter 1594
CPU time: 5.132433 sec. Elapsed time: 5.201000 sec.

</pre>

<pre>
Problem type appears to be: lpcon
Time for symbolic processing: 0.094742 seconds
Starting numeric solver
===== * * * =================================================================== * * *
TOMLAB - TOMLAB Development license 999007. Valid to 2011-12-31
=====================================================================================
Problem: --- 1: Drug Scheduling - Trial 6 f_k -17.407138702212308000
sum(|constr|) 0.000000000018677708
f(x_k) + sum(|constr|) -17.407138702193631000

Solver: multiMin with local solver snopt. EXIT=0. INFORM=0.
Find local optima using multistart local search
Did 10 local tries. Found 1 global, 10 minima. TotFuncEv 10. TotConstrEv 956

FuncEv 10 ConstrEv 956 ConJacEv 76 Iter 401
CPU time: 19.500125 sec. Elapsed time: 15.070000 sec.

</pre>

<source lang="matlab">
end
</source>

==Plot result==

<source lang="matlab">
subplot(2,1,1)
ezplot([x1;x2]);
legend('x1','x2');
title('Drug Scheduling state variable');

subplot(2,1,2)
ezplot(u);
legend('u');
title('Drug Scheduling control');
</source>