Page 106 - Control Theory in Biomedical Engineering
P. 106
Modeling and optimal control of cancer-immune system 93
characterization together with the optimal control. Therefore, the optimal
system is given as follows:
∗
∗
dE ∗ ρE ðt τÞT ðt τÞ u ∗
∗
∗
∗
∗
∗
¼ w σ + μE ðt τÞT ðt τÞ δE a 1 ð1 e ÞE ,
∗
dt η +T ðt τÞ
dT ∗ u ∗
∗
∗ ∗
∗
∗
∗
∗
¼ r 2 T ð1 βT Þ μE ðtÞT c 1 N T a 2 ð1 e ÞT ,
dt
dN ∗ u ∗
∗
∗
∗
∗
∗
¼ r 3 N ð1 β N Þ c 2 T N a 3 ð1 e ÞN ,
2
dt
du ∗
∗
∗
¼ v d 1 u ,
dt
u ∗ ρT ∗
∗
∗
0
λ ðtÞ¼ 1+ λ 1 ðtÞ δ +a 1 ð1 e Þ + λ 2 ðtÞnT + λ 1 ðt+ τÞχ μT ,
1 ½0, t f τ
η +T ∗
∗
∗
∗
0
Þ + λ 3 c 2 N
λ ðtÞ¼ 1+ λ 2 r 2 +2r 2 βT +nE +c 1 N +a 2 ð1 e u ∗ ∗
2
" #
∗ ∗
ρE T ρE ∗
+ χ ½0, t f τ 1 ðt+ τÞ + μE ∗ ,
λ
∗ 2
ðη +T Þ η +T ∗
u ∗
∗
∗
∗
0
λ ðtÞ¼ λ 2 c 1 T λ 3 r 3 2r 3 β N c 2 T a 3 ð1 e Þ γ,
2
3
∗
0
∗
∗
λ ðtÞ¼ λ 1 ðtÞa 1 e u ∗ E + λ 2 ðtÞa 2 e u ∗ T + λ 3 ðtÞa 3 e u ∗ N + λ 4 ðtÞd 1 ,
4
λ 4 ∗ λ 1 s 1
∗
v ¼ min v max , ,w ¼ min w max , :
B v B w
∗
∗
∗
E ðθÞ¼ ψ ðθÞ,T ðθÞ¼ ψ ðθÞ,N ðθÞ¼ ψ ðθÞ,uðθÞ¼ ψ ðθÞ, θ ½ τ,0,
1
3
4
2
λ i ðt f Þ¼ 0, i ¼f1,2,3,4g: ð13Þ
□
4 Numerical simulations
The numerical approximations of the OCP are carried out using forward
and backward Euler methods. Starting with an initial guess for the value
of the controls on the time interval [0, t f ], we solve the state system with
control variables (3) using forward Euler’ scheme. Meanwhile, the adjoint
system is solved using the solutions of the state system and the transversality
conditions (9) backward in time. A Pontryagin-type maximum principle is
derived, for retarded OCPs with delays in the state variable when the
control system is subject to a mixed control state constraint, in order to
minimize the cost of treatment, reduce the tumor cell load, and keep the
number of normal cells greater than 75% of its carrying capacity; see the
“Appendix” section for the Matlab program.
