PROPT Optimal Drug Scheduling for Cancer Chemotherapy

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This page is part of the PROPT Manual. See 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:


subject to:


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,

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):


There are the following path constraints on the state variables:


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:


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

% Copyright (c) 2007-2008 by Tomlab Optimization Inc.

Problem setup

toms t

nn = [20 40 120];

for i = 1:length(nn)


    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);

Solve the problem

    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);
Problem type appears to be: lpcon
Time for symbolic processing: 0.087433 seconds
Starting numeric solver
===== * * * =================================================================== * * *
TOMLAB - TOMLAB Development license  999007. Valid to 2011-12-31
=====================================================================================
Problem: ---  1: Drug Scheduling - Trial 66     f_k     -17.121324912680684000
                                       sum(|constr|)      0.000000000000846500
                              f(x_k) + sum(|constr|)    -17.121324912679839000

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

FuncEv  110 ConstrEv 3032 ConJacEv   18 Iter 1369 
CPU time: 1.965613 sec. Elapsed time: 1.956000 sec. 

Problem type appears to be: lpcon
Time for symbolic processing: 0.087512 seconds
Starting numeric solver
===== * * * =================================================================== * * *
TOMLAB - TOMLAB Development license  999007. Valid to 2011-12-31
=====================================================================================
Problem: ---  1: Drug Scheduling - Trial 86     f_k     -17.267378276475142000
                                       sum(|constr|)      0.000000000001722658
                              f(x_k) + sum(|constr|)    -17.267378276473419000

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

FuncEv   90 ConstrEv 3521 ConJacEv   18 Iter 1570 
CPU time: 4.570829 sec. Elapsed time: 4.575000 sec. 

Problem type appears to be: lpcon
Time for symbolic processing: 0.090814 seconds
Starting numeric solver
===== * * * =================================================================== * * *
TOMLAB - TOMLAB Development license  999007. Valid to 2011-12-31
=====================================================================================
Problem: ---  1: Drug Scheduling - Trial 6      f_k     -17.407138702212297000
                                       sum(|constr|)      0.000000000017636815
                              f(x_k) + sum(|constr|)    -17.407138702194661000

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

FuncEv   10 ConstrEv  937 ConJacEv   52 Iter  390 
CPU time: 36.987837 sec. Elapsed time: 10.474000 sec. 


end

Plot result

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');

DrugScheduling 01.png