Drug Displacement Problem

ITERATIVE DYNAMIC PROGRAMMING, REIN LUUS

12.4.3 Example 3: The desired level of two drugs, warfarin and phenylbutazone, must be reached in a patients bloodstream in minimum time.

CHAPMAN & HALL/CRC Monographs and Surveys in Pure and Applied Mathematics

Contents

Problem Formulation

Find u over t in [0; t ] to minimize

$$ J = t_F $$

subject to:

$$ \frac{dx_1}{dt} = g_1*(g_4*(0.02-x_1)+46.4*x_1*(u-2*x_2)) $$

$$ \frac{dx_2}{dt} = g_1*(g_3*(u-2*x_2)+46.4*(0.02-x_1)) $$

$$ g_2 = 1+0.2*(x_1+x_2) $$

$$ g_3 = g_2^2+232+46.4*x_2 $$

$$ g_4 = g_2^2+232+46.4*x_1 $$

$$ g_1 = \frac{g_2^2}{g_3*g_4-2152.96*x_1*x_2} $$

$$ 0 <= u <= 8 $$

x1 is the concentration of warfarin, and x2 of phenylbutazone. The initial and final condition are:

$$ x_0 = [0.02 \ 0] $$

$$ x_{t_f} = [0.02 \ 2.00] $$

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

Problem setup

toms t
toms tf
p = tomPhase('p', t, 0, tf, 50);
setPhase(p);

tomStates x1 x2
tomControls u

% Initial guess
x0 = {tf == 300
    icollocate({
    x1 == 0.02; x2 == 2*t/tf})
    collocate(u == 8-8*t/tf)};

% Box constraints
cbox = { 1 <= tf <= 500
    0 <= collocate(u) <= 8};

% Boundary constraints
cbnd = {initial({x1 == 0.02; x2 == 0})
    final({x1 == 0.02; x2 == 2})};

% General variables
g2 = 1+0.2*(x1+x2);
g3 = g2.^2+232+46.4*x2;
g4 = g2.^2+232+46.4*x1;
g1 = g2.^2./(g3.*g4-2152.96*x1.*x2);

% ODEs and path constraints
ceq = collocate({
    dot(x1) == g1.*(g4.*(0.02-x1)+46.4*x1.*(u-2*x2))
    dot(x2) == g1.*(g3.*(u-2*x2)+46.4*(0.02-x1))});

Solve the problem

options = struct;
options.name = 'Drug Displacement';
% Objective is first parameter
solution = ezsolve(tf, {cbox, cbnd, ceq}, x0, options);
t = subs(collocate(t),solution);
u = subs(collocate(u),solution);

Problem type appears to be: lpcon
===== * * * =================================================================== * * *
TOMLAB - Tomlab Optimization Inc. Development license  999001. Valid to 2010-02-05
=====================================================================================
Problem: ---  1: Drug Displacement              f_k     221.333418113809330000
                                       sum(|constr|)      0.000000058988258573
                              f(x_k) + sum(|constr|)    221.333418172797590000
                                              f(x_0)    300.000000000000000000

Solver: snopt.  EXIT=0.  INFORM=1.
SNOPT 7.2-5 NLP code
Optimality conditions satisfied

FuncEv    1 ConstrEv   14 ConJacEv   14 Iter   10 MinorIter  259
CPU time: 0.140625 sec. Elapsed time: 0.141000 sec.

Plot result

figure(1)
plot(t,u,'+-');
legend('u');
title('Drug Displacement control');