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Optimal control problems for differential equations applied to tumor growth: state of the art

Clara Rojas

Clara Rojas

Department of Mathematics and Mathematical Oncology Laboratory, University of Castilla-La ManchaCiudad Real, Spain
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Rojas, Clara
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Juan Belmonte-Beitia

Juan Belmonte-Beitia

Department of Mathematics and Mathematical Oncology Laboratory, University of Castilla-La ManchaCiudad Real, Spain
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Belmonte-Beitia, Juan
  
15 lug 2018
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Pubblicato online: 15 lug 2018
Pagine: 375 - 402
Ricevuto: 09 feb 2018
Accettato: 15 lug 2018
DOI: https://doi.org/10.21042/AMNS.2018.2.00029
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© 2018 C. Rojas, published by Sciendo
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Fig. 1

A 2-compartment model with G2/M-specific cytotoxic agent.
A 2-compartment model with G2/M-specific cytotoxic agent.

Fig. 2

Examples of locally optimal controls (left) and their corresponding trajectories (right) for T = 7 (top), T = 21 (middle) and T = 60 (bottom) for the parameter values given in Table 1.
Examples of locally optimal controls (left) and their corresponding trajectories (right) for T = 7 (top), T = 21 (middle) and T = 60 (bottom) for the parameter values given in Table 1.

Fig. 3

A 3-compartment model with cytotoxic and recruiting agent.
A 3-compartment model with cytotoxic and recruiting agent.

Fig. 4

Examples of locally optimal controls u (citotoxic agent, top), w (recruiting agent, middle) and their corresponding trajectories (bottom) for weights s2 = 0 (left) and s2 = 0:1 (right). The parameter values are given in Table 2.
Examples of locally optimal controls u (citotoxic agent, top), w (recruiting agent, middle) and their corresponding trajectories (bottom) for weights s2 = 0 (left) and s2 = 0:1 (right). The parameter values are given in Table 2.

Fig. 5

A 3-compartment model with cytostatic agent v and cytotoxic agent u.
A 3-compartment model with cytostatic agent v and cytotoxic agent u.

Fig. 6

Examples of locally optimal controls u (cytotoxic agent) (top), v (cytostatic agent) (middle) and their corresponding trajectories (bottom) for different weights r = (1,1,1) and q = (1,1,1) on the left and r = (8:25,8:25,8.25) and q = (0.1,0.1,0.1) on the right. The parameter values are given in Table 3.
Examples of locally optimal controls u (cytotoxic agent) (top), v (cytostatic agent) (middle) and their corresponding trajectories (bottom) for different weights r = (1,1,1) and q = (1,1,1) on the left and r = (8:25,8:25,8.25) and q = (0.1,0.1,0.1) on the right. The parameter values are given in Table 3.

Fig. 7

Optimal solution for functional Jw(u) in (29) with w = 0 and total dose M = 5, the time horizon is 30 days. Top row: a) proliferative cells P, b) nonproliferative quiescent cells Q, Bottom row: c) damaged quiescent cells Qp, d) optimal control u*(t) and switching function ϕ.
Optimal solution for functional Jw(u) in (29) with w = 0 and total dose M = 5, the time horizon is 30 days. Top row: a) proliferative cells P, b) nonproliferative quiescent cells Q, Bottom row: c) damaged quiescent cells Qp, d) optimal control u*(t) and switching function ϕ.

Fig. 8

Optimal solution for functional Jw(u) in (29) with w = 1 and total dose M = 5, the time horizon is 30 days. Top row: a) proliferative cells P, b) nonproliferative quiescent cells Q, Bottom row: c) damaged quiescent cells Qp, d) optimal control u*(t) and switching function ϕ.
Optimal solution for functional Jw(u) in (29) with w = 1 and total dose M = 5, the time horizon is 30 days. Top row: a) proliferative cells P, b) nonproliferative quiescent cells Q, Bottom row: c) damaged quiescent cells Qp, d) optimal control u*(t) and switching function ϕ.

Fig. 9

Optimal solution for functional Jw(u) in (29) with w = ws = 0:004067 and total dose M = 5, the time horizon is 30 days. (left) optimal control u(t) which is totally singular on [0,1], (right) proliferative cells P.
Optimal solution for functional Jw(u) in (29) with w = ws = 0:004067 and total dose M = 5, the time horizon is 30 days. (left) optimal control u(t) which is totally singular on [0,1], (right) proliferative cells P.

Fig. 10

Optimal solution for functional Jw(u) in (29) with w = 0 and total dose M = 5, the time horizon is T = 30 months. Top row: a) proliferative cells P, b) nonproliferative quiescent cells Q, Bottom row: c) damaged quiescent cells Qp, d) optimal control u*(t) and switching function ϕ, the switching function ϕ(t) is zero on the singular arc [t1,t2].
Optimal solution for functional Jw(u) in (29) with w = 0 and total dose M = 5, the time horizon is T = 30 months. Top row: a) proliferative cells P, b) nonproliferative quiescent cells Q, Bottom row: c) damaged quiescent cells Qp, d) optimal control u*(t) and switching function ϕ, the switching function ϕ(t) is zero on the singular arc [t1,t2].

Fig. 11

Optimal solution for functional Jw(u) in (29) with w = 1 and total dose M = 5, the time horizon is T = 30 months. Top row: a) proliferative cells P, b) nonproliferative quiescent cells Q, Bottom row: c) damaged quiescent cells Qp, d) optimal control u*(t) and switching function ϕ, the switching function ϕ(t) is zero on the singular arc [t1, t2].
Optimal solution for functional Jw(u) in (29) with w = 1 and total dose M = 5, the time horizon is T = 30 months. Top row: a) proliferative cells P, b) nonproliferative quiescent cells Q, Bottom row: c) damaged quiescent cells Qp, d) optimal control u*(t) and switching function ϕ, the switching function ϕ(t) is zero on the singular arc [t1, t2].

Fig. 12

[Color Online]. a) Evolution of the solutions to Eqs. (52), (53) and (54), p (solid line), s (dotted line), and q (solid-dot line) with the initial conditions (p0,s0,q0) = (12000,5000,14000) b) Optimal control u(t) (solid line) and switching function ϕ1 (dotted line) for M = 300 doses and uM = 75 c) Optimal control v(t) (solid line) and switching function ϕ2 (dotted line) for N = 5 doses and vM = 1.
[Color Online]. a) Evolution of the solutions to Eqs. (52), (53) and (54), p (solid line), s (dotted line), and q (solid-dot line) with the initial conditions (p0,s0,q0) = (12000,5000,14000) b) Optimal control u(t) (solid line) and switching function ϕ1 (dotted line) for M = 300 doses and uM = 75 c) Optimal control v(t) (solid line) and switching function ϕ2 (dotted line) for N = 5 doses and vM = 1.

Fig. 13

[Color Online]. a) Evolution of the solutions to Eqs. (52), (53) and (54), p (solid line), s (dotted line), and q (solid-dot line) with the initial conditions (p0,s0,q0) = (12000,5000,5000) b) Optimal control u(t) (solid line) and switching function ϕ1 (dotted line) for M = 300 doses and uM = 75 c) Optimal control v(t) (solid line) and switching function ϕ2 (dotted line) for N = 5 doses and vM = 1.
[Color Online]. a) Evolution of the solutions to Eqs. (52), (53) and (54), p (solid line), s (dotted line), and q (solid-dot line) with the initial conditions (p0,s0,q0) = (12000,5000,5000) b) Optimal control u(t) (solid line) and switching function ϕ1 (dotted line) for M = 300 doses and uM = 75 c) Optimal control v(t) (solid line) and switching function ϕ2 (dotted line) for N = 5 doses and vM = 1.

Numerical values for the coefficients and parameters used in numerical computations for the optimal control problem 1, extracted from [41]_

CoefficientInterpretationNumerical value
a1Inverse transit time through G0/G1+ S0.197
a2Inverse transit time through G2/M0.356
N10Initial condition for N10.7012
N20Initial condition for N20.2988
umaxMaximum dose rate/concentration effectiveness of the drug0.90
sPenalty/weight for the total dose of cytotoxic agent0.50
q1Penalty/weight in the objective for the average number of cancer cells in G0/G1+ S during therapy0.10
q2Penalty/weight in the objective for the average number of cancer cells in G2/M during therapy0.10
r1Penalty/weight in the objective for the average number of cancer cells in G0/G1+ S at the end of therapy3
r2Penalty/weight in the objective for the average number of cancer cells in G2/M at the end of therapy3
TTherapy horizonT = 7,21,60

Values of the biological parameters in the model of LGG evolution_

VariableValue (Units)
P00.924 mm
Q042.3 mm
Qp00 mm
K100 mm
cp0.114 mo -1
cPQ0.0226 mo-1
cQP0.0045 mo-1
δQP0.0214 mo-1
γ0.842

Values of the biological parameters in the model_

VariableDescriptionValue (Units)
ξTumor growth parameter0.084 day-1
bTumor-induced stimulation parameter5.85 day-1
dTumor-induced inhibition parameter0.00873 mm-2 day-1
μLoss of vascular support0.02 day-1
χMaduration of unstable vessels parameter0.025 day-1
γAnti-angiogenic elimination parameter0.15 kg/mg day-1
φCytotoxic killing parameter for the tumor0.1 kg/mg day-1

Numerical values for the coefficients and parameters used in computations for the optimal control problem 3 with cytostatic and cytotoxic agents, extracted from [41]_

CoefficientInterpretationNumerical value
a1Inverse transit time through G1/G00.197
a2Inverse transit time through S0.395
a3Inverse transit time through G2/M0.107
N1(0)Initial condition for N10.3866
N2(0)Initial condition for N20.1722
N3(0)Initial condition for N30.4412
umaxMaximum dosage/concentration/effectiveness of cytotoxic agent0.95
vmaxMaximum blocking effect of cytostatic agent0.30
s1Penalty/weight at the cytotoxic agent1
s2Penalty/weight at the cytostatic agent0.01
q1Penalty/weight in the objective for the average number of cancer cells in G1/G0 during therapy1, resp. 0.1
q2Penalty/weight in the objective for the average number of cancer cells in S during therapy1, resp. 0.1
q3Penalty/weight in the objective for the average number of cancer cells in G2/M during therapy1, resp. 0.1
r1Penalty/weight in the objective for the average number of cancer cells in G1/G0 at the end of therapy1, resp. 8.25
r2Penalty/weight in the objective for the average number of cancer cells in S at the end of therapy1, resp. 8.25
r3Penalty/weight in the objective for the average number of cancer cells in G2/M at the end of therapy1, resp. 8.25
TTherapy horizonT = 21

Numerical values for the coefficients and parameters used in numerical computations for the optimal control problem 2 with cytotoxic and recruitment agents, extracted from [41]_

CoefficientInterpretationNumerical value
a0Inverse transit time through G00.05
a1Inverse transit time through G10.5
a2Inverse transit time through S +G2/M1
p0Probability that cells enter G00.9
p1 = 1– p0Probability that cells enter G10.1
N0(0)Initial condition for N00.8589
N1(0)Initial condition for N10.0954
N2(0)Initial condition for N20.0456
umaxMaximum dose rate/concentration effectiveness0.95
wmaxMaximum dose rate/concentration effectiveness6
s1Penalty/weight at the cytotoxic agent1
s2Penalty/weight at the recruiting agent0, 0.1
q0Penalty/weight in the objective for the average number of cancer cells in G0 during therapy3
q1Penalty/weight in the objective for the average number of cancer cells in G1 during therapy1
q2Penalty/weight in the objective for the average number of cancer cells in S +G2/M during therapy1
r0Penalty/weight in the objective for the average number of cancer cells in G0 at the end of therapy3
r1Penalty/weight in the objective for the average number of cancer cells in G1 at the end of therapy1
r2Penalty/weight in the objective for the average number of cancer cells in S +G2/M at the end of therapy1
TTherapy horizonT = 21
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