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Optimization of spatiotemporal control for systems described by cellular automata
Krzysztof Fujarewicz
Silesian University of Technology Gliwice, Poland
Micro and Macro Systems in Life Sciences, Będlewo, 10.06.2015
2
Outline
• Inspiration
• Adjoint sensitivity analysis • Adjoint sensitivity analysis for cellular automata • Numerical results
• Conclusion
3
Inspiration
4
Structural sensitivity analysis for discrete-time systems
qmn RtyRtuRtx ∈∈∈ )(,)(,)(
)()(,)0( 0 tutuxx nom== )()(),()( tytytxtx nomnom ==→
Tttutxgtytutxftx
M ,,1,0))(),(()())(),(()1(
: …=⎩⎨⎧
=
=+
5
)()(),()( txtxtutu nomnom ==
Tttutxgtytutxftx
M ,,1,0))(),(()())(),(()1(
: …=⎩⎨⎧
=
=+
Ttt
tuty
tuty
tuty
tuty
S
txtum
mtytu
nomnom
,,1,0,,
)()(
)()(
)()(
)()(
21
)()(1
2
11
2
1
21
11
21
)()(
df21
…
!
"#"
!
=
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
∂
∂
∂
∂
∂∂
∂∂
=
Input-output sensitivity function
TtttutyS
txtuj
itytu
nomnom
ij
,,1,0,,)()(
21
)()(1
2)()(
df21
…=∂∂
=
6
Sensitivity model
( )( )
Tttutxgtytutxftx
,,1,0)(),()()(),()1(
…=⎩⎨⎧
=
=+
)()(
)()(
)()(
)()(
)()()(
)()()(
)()()(
)()()(
txtu
txtu
txtu
txtu
nomnom
nomnom
nomnom
nomnom
tugtD
txgtC
tuftB
txftA
∂⋅∂
=∂
⋅∂=
∂⋅∂
=∂
⋅∂=
:M
TttutDtxtCtytutBtxtAtx
,,1,0,)()()()()()()()()()1(
…=⎩⎨⎧
+=
+=+:M
7
Structural sensitivity model
Element of the original block diagram
Summing junction
Linear dynamic element )(zK
Linear static element A
Branching node
Nonlinear static element )(ufu
Element of the structural sensitivity model
)(zK
A
)(
)()(tuuu
ftH=∂
⋅∂=)(tH
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Modified adjoint system
⎩⎨⎧
+=
+=+
)()()()()()()()()()1(tutDtxtCtytutBtxtAtx
:M
ˆ ˆ ˆ( 1) ( ) ( ) ( ) ( )
ˆ ˆ ˆ( ) ( ) ( ) ( ) ( )
T T
T T
x t A T t x t C T t u t
y t B T t x t D T t u t
⎧ + = − + −⎪⎨
= − + −⎪⎩ˆ :M
9
)(zK
A
Element of the structural sensitivity model
)(tH
Element of the modified adjoint system
)(zKT
TA
)( tTHT −
Adjoint system – the structural form
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The property of the adjoint system
0
ˆˆ ˆ( ) ( , ) ( )t
y t K t uτ=
= τ τ∑0
( ) ( , ) ( )t
y t K t uτ=
= τ τ∑
1 2ˆ ( , )TK T t T t− −),( 12 ttK =
)(tu )(tyM ˆ( )u tˆ( )y t M
21
( )( ) ( , )y t
nom nomu tS u x=
The input-output sensitivity is the Green function (pulse response) of the sensitivity model OR the adjoint system
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)(tu )(tyM
)(tu j
t
)(tyi
t1t T2t1t T
1
T
1
2tT −
ˆ ( )iu t
t
ˆ ( )jy t
t1tT −T
2tT −
ˆ( )u tˆ( )y t M
The property of the adjoint system
21
( )( )ij
y tu tS
12
Forward vs. Adjoint Sensitivity Analysis
t0 local perturbation
Forward Model
tn
influenced area
13
Forward vs. Adjoint Sensitivity Analysis
t0 local perturbation
tn
local perturbance
Forward Model
Adjoint Model
tn
influenced area
t0 area of possible origin
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Applica'on of the adjoint sensi'vity analysis for different models and problems
Hybrid continuous-discrete systems Fujarewicz K, Galuszka A: Generalized backpropagation through time for continuous time neural networks and discrete time measurements. Artificial Intelligence and Soft Computing - ICAISC 2004 (eds. L. Rutkowski, J. Siekmann, R. Tadeusiewicz and L. A. Zadeh), Lecture Notes in Computer Science, 2004, 3070, p. 190-196 Parameter estimation for ODE systems Fujarewicz K, Kimmel M, Swierniak A: On fitting of mathematical models of cell signaling pathways using adjoint systems. Mathematical Biosciences and Engineering, 2005, 2(3), p. 527-534 Fujarewicz K, Kimmel M, Lipniacki T, Swierniak A: Adjoint systems for models of cell signaling pathways and their application to parameter fitting. IEEE-ACM Transactions On Computational Biology and Bioinformatics, 2007, 4(3), p. 322-335 Kumala S, Fujarewicz K, Jayaraju D, Rzeszowska-Wolny J, Hancock R: Repair of DNA Strand Breaks in a Minichromosome In Vivo: Kinetics, Modeling, and Effects of Inhibitors. Plos One, 2013, 8(1), p. 1-12
Łakomiec K, Kumala S, Hancock R, Rzeszowska-Wolny J, Fujarewicz K: Modeling the repair of DNA strand breaks caused by γ-radiation in a minichromosome . Physical Biology, 2014, 11(4), p. 1-8 Parameter estimation for systems with delays (DDE) Fujarewicz K, Łakomiec K: Parameter estimation of systems with delays via structural sensitivity analysis . Discrete and Continuous Dynamical Systems-series B, 2014, 19(8), p. 2521-2533
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Structural sensi'vity analysis for cellular automata
Cellular automaton 1D with con'nuous state (corresponds to finite difference method for PDE models)
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Difference scheme
-‐ local (in ,me and space) input
space
,me
yt+1x
= f(ytx
, ytx�1, y
t
x+1, ut
x
)
ut
x
ytx
yt+1x
x
t = 0, 1, 2, . . . , tmax
x = 1, 2, 3, . . . , xmax
17
J = g(ytmax
1 , ytmax
2 , ... , ytmax
x
max
)Objec,ve func,on
Matrix of the whole spa,otemporal control
U =
2
66666664
u11 . . . . . . . . . ut
max
1...
. . ....
... ut
x
......
. . ....
u1x
max
. . . . . . . . . ut
max
x
max
3
77777775
G = rUJ =
2
666666664
@J@u1
1. . . . . . . . . @J
@ut
max
1
.... . .
...... @J
@ut
x
...
.... . .
...@J
@u1x
max
. . . . . . . . . @J
@ut
max
x
max
3
777777775
Searched spa,otemporal gradient of the objec,ve func,on
yt+1x
= f(ytx
, ytx�1, y
t
x+1, ut
x
)
Block diagram of one cell of the cellular automaton
The sensi,vity model of one cell
yt+1x
=
✓@f
@yx
◆t
x
· ytx
+
✓@f
@yx�1
◆t
x
· ytx�1 +
✓@f
@yx+1
◆t
x
· ytx+1 +
✓@f
@u
◆t
x
· ut
x
The adjoint model of one cell
yt+1x
=
✓@f
@yx
◆t
max
�t
x
· ytx
+
✓@f
@yx+1
◆t
max
�t
x�1
· ytx�1 +
✓@f
@yx�1
◆t
max
�t
x+1
· ytx+1 + ↵t
x
↵t
x
The cellular automaton with the block calcula,ng the objec,ve func,on
J = g(ytmax
1 , ytmax
2 , ... , ytmax
x
max
)
J = J(tmax
)
The sensi,vity model
ut
x
ut1
ut
x
max
The adjoint model
↵t
x
max
↵t
x
↵t1
The spa,otemporal gradient of the objec,ve func,on.
G = rU
J =
2
66666664
�t
max
1 . . . . . . . . . �01
.... . .
...... �t
max
�t
x
......
. . ....
�t
max
x
max
. . . . . . . . . �0x
max
3
77777775
@J
@ut
x
= �t
max
�t
x
@c
@t= r · (Drc) + ⇢c
✓1� c
kt
◆�R · c
✓1� c
kt
◆
D
⇢
kt
R = (1� e�↵d��d2
)
PDE model of tumor growth (Corwin et al. 2013)
diffusion coefficient
prolifera,on coefficient carrying capacity
radia,on influence (LQ model)
@c
@t= r · (Drc) + ⇢c
✓1� c
kt
◆�R · c
✓1� c
kt
◆
D
⇢
kt
R = (1� e�↵d��d2
)
PDE model of tumor growth (Corwin et al. 2013)
diffusion coefficient
prolifera,on coefficient carrying capacity
radia,on influence (LQ model)
Discre,zed model (difference scheme)
c
t+1x
� c
t
x
�t
= D
c
t
x�1 � 2ctx
+ c
t
x+1
(�x)2+ (⇢�R)ct
x
✓1� c
t
x
k
t
◆
Op,mized performance index: number of cancer cells at final ,me
J =x
maxX
x=1
ctmax
x
Constraints
S =t
maxX
t=1
x
maxX
x=1
dtx
, S Smax
dtx
dmax
for total dose:
for local dose:
Model simula,on (without treatment)
,me t
space x
D = 18, ⇢ = 35, kt = 2.4e5
x x
t t
Op,mized spa,otemporal signal Tumor growth and regression
Op,miza,on results of the spa,otemporal irradia,on
dmax
= 2Smax
= 2.3e4J = 346
x x
t t
Op,mized spa,otemporal signal Tumor growth and regression
Op,miza,on results of the spa,otemporal irradia,on
dmax
= 1.8Smax
= 1.4e4J = 2.9e5
x x
t t
Op,mized spa,otemporal signal Tumor growth and regression
Op,miza,on results of the spa,otemporal irradia,on
dmax
= 1.4Smax
= 1.1e4J = 9.9e6
Further work
• Different objec,ve func,on • Frac,oned treatment • Other models
Further work
• Different objec,ve func,on • Frac,oned treatment • Other models
Conclusions • Effec,ve method for sensi,vity analysis w.r.t. spa,otemporal control • May be used for spa,otemporal control op,miza,on • Not limited to biological models
Acknowledgements
• Krzysztof Łakomiec – Silesian technical university PhD student
• grant DEC-‐2012/05/B/ST6/03472
Thank you!
