optimization of spatiotemporal control for systems...
TRANSCRIPT
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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
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Outline
• Inspiration
• Adjoint sensitivity analysis • Adjoint sensitivity analysis for cellular automata • Numerical results
• Conclusion
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Inspiration
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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
…=∂∂
=
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Sensitivity model
( )( )
Tttutxgtytutxftx
,,1,0)(),()()(),()1(
…=⎩⎨⎧
=
=+
)()(
)()(
)()(
)()(
)()()(
)()()(
)()()(
)()()(
txtu
txtu
txtu
txtu
nomnom
nomnom
nomnom
nomnom
tugtD
txgtC
tuftB
txftA
∂⋅∂
=∂
⋅∂=
∂⋅∂
=∂
⋅∂=
:M
TttutDtxtCtytutBtxtAtx
,,1,0,)()()()()()()()()()1(
…=⎩⎨⎧
+=
+=+:M
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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
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)(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
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Forward vs. Adjoint Sensitivity Analysis
t0 local perturbation
Forward Model
tn
influenced area
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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
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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!