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DRUGDIS.SIF
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***************************
* SET UP THE INITIAL DATA *
***************************
NAME DRUGDIS
* Problem :
* *********
* A control problem based on the kinetic model of Aarons and Rowland for
* DRUG DISplacemnt, which simulates the interaction of the two drugs
* (warfarin and phenylnutazone) in a patient bloodstream.
* The state variable are the concentrations of unbound warfarin (w) and
* phenylbutazone (p). The problem is to control the rate of injection (u)
* of the pain-killing phenylbutazone so that both drugs reach a specified
* steady-state in minimum time and the concentration of warfarin does not
* rise above a given toxicity level.
* The problem is discretized using the trapezoidal rule. It is non-convex.
* The problem can be made harder by diminishing the value of the lower bound
* on the final time TF (while maintaining it strictly positive).
* Source:
* H. Maurer and M. Wiegand,
* "Numerical solution of a drug displacement problem with bounded state
* variables",
* Optimal Control Applications and Methods 13, pp. 43-55, 1992.
* SIF input: Ph. Toint, Nov 1993.
* classification LOR2-MN-V-V
* Discretization: specify the number of interior points + 1
*IE NI 10 $-PARAMETER n= 34, m= 20
*IE NI 50 $-PARAMETER n= 154, m=100
*IE NI 100 $-PARAMETER n= 304, m=200 original value
*IE NI 200 $-PARAMETER n= 604, m=400
*IE NI 500 $-PARAMETER n=1504, m=1000
*IE NI 1000 $-PARAMETER n=3004, m=2000
IE NI 2000 $-PARAMETER n=6004, m=4000
* Problem parameters
RE TOXIC 0.026 $-PARAMETER warfarin toxicity level
RE WSS 0.02 $-PARAMETER initial/final warfarin levels
RE UMAX 8.0 $-PARAMETER maximal injection rate
RE PSTART 0.0 $-PARAMETER initial phenybutazone level
RE PFINAL 2.0 $-PARAMETER final phenylbutazone level
* Define useful parameters
R+ AVP PSTART PFINAL
RM AVP AVP 0.5
IA NI-1 NI -1
RI RNI NI
RD -1/2NI RNI -0.5
IE 0 0
VARIABLES
* final time
TF 'SCALE' 200.0
* warfarin concentration
DO I 0 NI
X W(I) 'SCALE' 0.02
OD I
* phenylbutazone concentration
DO I 0 NI
X P(I)
OD I
* injection rate (control)
DO I 0 NI
X U(I)
OD I
GROUPS
N TFINAL TF 1.0
N TFINAL 'SCALE' 100.0
* state equations
DO I 0 NI-1
IA I+1 I 1
* warfarin concentration dynamics
XE EW(I) W(I+1) 1.0 W(I) -1.0
XE EW(I) 'SCALE' 0.02
* phenylbutazone concentration dynamics
XE EP(I) P(I+1) 1.0 P(I) -1.0
OD I
BOUNDS
* All variables are non-negative
* Injection takes at least 200 seconds.
* The problem can be made harder by diminishing this value while
* maintaining it strictly positive.
LO DRUGDIS TF 200.0
* Impose the bound on warfarin concentration corresponding to
* the toxicity level
DO I 0 NI
ZU DRUGDIS W(I) TOXIC
OD I
* Impose the bound on the maximum phenylbutazone injection rate
DO I 0 NI-1
ZU DRUGDIS U(I) UMAX
OD I
* Fix the initial and final concentrations of warfarin and phenylbutazone
ZX DRUGDIS W(0) WSS
ZX DRUGDIS W(NI) WSS
ZX DRUGDIS P(0) PSTART
ZX DRUGDIS P(NI) PFINAL
START POINT
R- DP PFINAL PSTART
RD DP/NI DP RNI
DO I 0 NI-1
RI RI I
R* IDP/NI RI DP/NI
ZV DRUGDIS P(I) IDP/NI
ZV DRUGDIS W(I) WSS
ZV DRUGDIS U(I) UMAX
OD I
XV DRUGDIS TF 240.0
ZV DRUGDIS W(NI) WSS
ZV DRUGDIS P(NI) PFINAL
ELEMENT TYPE
* Warfarin dynamics
EV EW T W
EV EW P U
* Phenylbutazone dynamics
EV EP T W
EV EP P U
ELEMENT USES
DO I 0 NI
XT WA(I) EW
ZV WA(I) T TF
ZV WA(I) W W(I)
ZV WA(I) P P(I)
ZV WA(I) U U(I)
XT PA(I) EP
ZV PA(I) T TF
ZV PA(I) W W(I)
ZV PA(I) P P(I)
ZV PA(I) U U(I)
OD I
GROUP USES
DO I 0 NI-1
ZE EW(I) WA(I+1) -1/2NI
ZE EW(I) WA(I) -1/2NI
ZE EP(I) PA(I+1) -1/2NI
ZE EP(I) PA(I) -1/2NI
OD I
OBJECT BOUND
LO DRUGDIS 200.0
* Solution
*LO SOLTN(10) 3.82432
*LO SOLTN(50) 4.19953
*LO SOLTN(100) 4.23934
*LO SOLTN(200) 4.25762
*LO SOLTN(500)
*LO SOLTN(1000)
*LO SOLTN(Maurer) 2.62637
ENDATA
***********************
* SET UP THE FUNCTION *
* AND RANGE ROUTINES *
***********************
ELEMENTS DRUGDIS
TEMPORARIES
R Z
R ZZ
R D
R DD
R DD1
R DD2
R A
R AW
R B
R BP
R C
R CW
R CP
R CWW
R CWP
R CPP
R F
R FW
R FP
R FWW
R FWP
R FPP
R G
R GW
R GP
R GU
R GWW
R GWP
R GPP
R GUU
R H
R HW
R HP
R I
R IP
R WSS
R DCOEFF
R ABCNST
GLOBALS
A Z 46.4
A WSS 0.02
A DCOEFF 0.2
A ABCNST 232.0
A ZZ Z * Z
INDIVIDUALS
T EW
A D 1.0 + DCOEFF * ( W + P )
A DD D * D
A DD1 2.0 * DCOEFF * D
A DD2 2.0 * DCOEFF * DCOEFF
A A DD + ABCNST + Z * W
A AW DD1 + Z
A B DD + ABCNST + Z * P
A BP DD1 + Z
A C A * B - ZZ * W * P
A CW AW * B + A * DD1 - ZZ * P
A CP DD1 * B + A * BP - ZZ * W
A CWW DD2 * B + 2.0 * AW * DD1 + A * DD2
A CWP DD2 * B + AW * BP + DD1 * DD1 + A * DD2
A+ - ZZ
A CPP DD2 * B + 2.0 * DD1 * BP + A * DD2
A F DD / C
A H DD1 - F * CW
A I DD1 - F * CP
A FW H / C
A FP I / C
A HW DD2 - CW * FW - F * CWW
A HP DD2 - CW * FW - F * CWP
A IP DD2 - CP * FP - F * CPP
A FWW ( HW - FW * CW ) / C
A FWP ( HP - FW * CP ) / C
A FPP ( IP - FP * CP ) / C
A GU Z * W
A G A * ( WSS - W ) + GU * ( U - 2.0 * P )
A GW AW * ( WSS - W ) - A
A+ + Z * ( U - 2.0 * P )
A GP DD1 * ( WSS - W ) - 2.0 * GU
A GPP DD2 * ( WSS - W )
A GWW GPP - 2.0 * AW
A GWP GPP - DD1 - 2.0 * Z
F T * F * G
G T F * G
G W T * ( FW * G + F * GW )
G P T * ( FP * G + F * GP )
G U T * F * GU
H T W FW * G + F * GW
H T P FP * G + F * GP
H T U F * GU
H W W T * ( FWW * G + 2.0 * FW * GW + F * GWW )
H W P T * ( FWP * G + FW * GP
H+ + FP * GW + F * GWP )
H W U T * ( FW * GU + F * Z )
H P P T * ( FPP * G + 2.0 * FP * GP + F * GPP )
H P U T * FP * GU
T EP
A D 1.0 + DCOEFF * ( W + P )
A DD D * D
A DD1 2.0 * DCOEFF * D
A DD2 2.0 * DCOEFF * DCOEFF
A A DD + ABCNST + Z * W
A AW DD1 + Z
A B DD + ABCNST + Z * P
A BP DD1 + Z
A C A * B - ZZ * W * P
A CW AW * B + A * DD1 - ZZ * P
A CP DD1 * B + A * BP - ZZ * W
A CWW DD2 * B + 2.0 * AW * DD1 + A * DD2
A CWP DD2 * B + AW * BP + DD1 * DD1 + A * DD2
A+ - ZZ
A CPP DD2 * B + 2.0 * DD1 * BP + A * DD2
A F DD / C
A H DD1 - F * CW
A I DD1 - F * CP
A FW H / C
A FP I / C
A HW DD2 - CW * FW - F * CWW
A HP DD2 - CW * FW - F * CWP
A IP DD2 - CP * FP - F * CPP
A FWW ( HW - FW * CW ) / C
A FWP ( HP - FW * CP ) / C
A FPP ( IP - FP * CP ) / C
A G B * ( U - 2.0 * P )
A+ + Z * P * ( WSS - W )
A GW DD1 * ( U - 2.0 * P ) - Z * P
A GP BP * ( U - 2.0 * P ) - 2.0 * B
A+ + Z * ( WSS - W )
A GWW DD2 * ( U - 2.0 * P )
A GWP GWW - 2.0 * DD1 - Z
A GPP GWW - 4.0 * BP
F T * F * G
G T F * G
G W T * ( FW * G + F * GW )
G P T * ( FP * G + F * GP )
G U T * F * B
H T W FW * G + F * GW
H T P FP * G + F * GP
H T U F * B
H W W T * ( FWW * G + 2.0 * FW * GW + F * GWW )
H W P T * ( FWP * G + FW * GP
H+ + FP * GW + F * GWP )
H W U T * ( FW * B + F * DD1 )
H P P T * ( FPP * G + 2.0 * FP * GP + F * GPP )
H P U T * ( FP * B + F * BP )
ENDATA