Stochastic Hybrid Systems: Modelling Prostate Cancer and Psoriasis - PowerPoint PPT Presentation

Stochastic Hybrid Systems: Modelling Prostate Cancer and Psoriasis Fedor Shmarov School of Computing Science Newcastle University, UK 1 / 28

Introduction ◮ We use hybrid systems for modelling and verifying systems biology models ◮ Hybrid systems combine continuous dynamics with discrete state changes ◮ Parametric hybrid systems feature random and nondeterministic parameters ◮ Reachability is one of the central properties of hybrid systems ◮ Undecidable even for linear hybrid systems (Alur, Courcoubetis, Henzinger, Ho. 1993) ◮ Bounded reachability – number of discrete transitions is finite 2 / 28

Bounded Reachability Does the hybrid system reach the goal state within a finite number of (discrete) steps? ◮ Nonlinear arithmetics over the reals is undecidable (Richardson, 1968) ◮ Bounded reachability is δ -decidable ◮ δ -complete decision procedure (Gao, Avigad, Clarke. LICS 2012) ◮ Used for parameter set identification in parametric hybrid systems 3 / 28

Parametric Hybrid Systems (PHS) H = < Q , T , X , P , Y , R , jump, goal > ◮ Q = { q 0 , · · · , q m } a set of modes (discrete components of the system), ◮ T = { ( q , q ′ ) : q , q ′ ∈ Q } a set of transitions between the modes, ◮ X = [ u 1 , v 1 ] × · · · × [ u n , v n ] ⊂ R n a domain of continuous variables, ◮ P = [ a 1 , b 1 ] × · · · × [ a k , b k ] ⊂ R k the parameter space of the system, ◮ Y = { y q ( x 0 , t ) : q ∈ Q , x 0 ∈ X × P , t ∈ [0 , T ] } the continuous dynamics, ◮ R = { g ( q , q ′ ) ( x , t ) : ( q , q ′ ) ∈ T , x ∈ X × P , t ∈ [0 , T ] } ‘reset’ functions, and predicates (or relations) ◮ jump ( q , q ′ ) ( x ) defines a discrete transition ( q , q ′ ) ∈ T ◮ goal q ( x ) defines the goal state x in mode q . 4 / 28

Parametric Hybrid Systems (PHS) H = < Q , T , X , P , Y , R , jump, goal > ◮ Q = { q 0 , · · · , q m } a set of modes (discrete components of the system), ◮ T = { ( q , q ′ ) : q , q ′ ∈ Q } a set of transitions between the modes, ◮ X = [ u 1 , v 1 ] × · · · × [ u n , v n ] ⊂ R n a domain of continuous variables, ◮ P = [ a 1 , b 1 ] × · · · × [ a k , b k ] ⊂ R k the parameter space of the system, ◮ Y = { y q ( x 0 , t ) : q ∈ Q , x 0 ∈ X × P , t ∈ [0 , T ] } the continuous dynamics, ◮ R = { g ( q , q ′ ) ( x , t ) : ( q , q ′ ) ∈ T , x ∈ X × P , t ∈ [0 , T ] } ‘reset’ functions, and predicates (or relations) ◮ jump ( q , q ′ ) ( x ) defines a discrete transition ( q , q ′ ) ∈ T ◮ goal q ( x ) defines the goal state x in mode q . Bounded reachability can be encoded as the following ( π a path and B ⊆ P ): φ ( π, B ) := ∃ B p , ∃ [0 , T ] t 0 , · · · , ∃ [0 , T ] t | π |− 1 : � x t � π (0) = y π (0) ( p , t 0 ) ∧ | π |− 2 � �� � jump ( π ( i ) ,π ( i +1)) ( x t � x t π ( i +1) = y π ( i ) ( g ( π ( i ) ,π ( i +1)) ( x t π ( i ) ) ∧ π ( i ) ) , t i +1 ) i =0 ∧ goal π ( | π |− 1) ( x t π ( | π |− 1) ) 4 / 28

Parameter Set Identification (I) ◮ Parameter set identification can be encoded by the formula: φ ∀ ( π, B ) := ∀ B p , ∃ [0 , T ] t 0 , · · · , ∃ [0 , T ] t | π |− 1 : x t � � π (0) = y π (0) ( p , t 0 ) ∧ | π |− 2 � �� � jump ( π ( i ) ,π ( i +1)) ( x t x t π ( i +1) = y π ( i ) ( g ( π ( i ) ,π ( i +1)) ( x t � π ( i ) ) ∧ π ( i ) ) , t i +1 ) i =0 ∧ goal π ( | π |− 1) ( x t π ( | π |− 1) ) ◮ Problem: parameter p is quantified universally ◮ δ -complete decision procedures currently support formulae with universal quantification over a single variable only 5 / 28

Parameter Set Identification (II) ◮ We solve a series of formulae ψ j ( π, B ): ψ j ( π, B ) := ∃ B p , ∃ [0 , T ] t 0 , · · · , ∀ [0 , T ] t j : � x t � π (0) = y π (0) ( p , t 0 ) ∧ j − 1 � � � x t π ( i +1) = y π ( i ) ( g ( π ( i ) ,π ( i +1)) ( x t ∧ ¬ jump ( π ( j ) ,π ( j +1)) ( x t π ( i ) ) , t i +1 ) π ( j ) ) i =0 if j < | π | − 1 and ψ j ( π, B ) := ∃ B p , ∃ [0 , T ] t 0 , · · · , ∀ [0 , T ] t j : x t � � π (0) = y π (0) ( p , t 0 ) ∧ j − 1 � � � x t π ( i +1) = y π ( i ) ( g ( π ( i ) ,π ( i +1)) ( x t ∧ ¬ goal π ( j ) ( x t π ( i ) ) , t i +1 ) π ( j ) ) i =0 if j = | π | − 1. | π |− 1 � ¬ ψ j ( π, B ) ⇒ φ ∀ ( π, B ) j =0 6 / 28

Parameter Set Identification (III) 1 input: H - PHS, l - reachability depth, B - subset of parameter space, δ - precision; 2 output: sat / unsat / undet ; 3 Path ( l ) = get all paths ( H , l ) ; // compute all paths of length l for H 4 for π ∈ Path ( l ) do if φ ( π, B ) - δ -sat then 5 for i ∈ [0 , l ] do 6 if ψ i ( π, B ) - δ -sat then 7 return undet ; 8 return sat ; // all ψ i ( π, B ) are unsat 9 10 return unsat ; // all φ ( π, B ) are unsat ◮ sat – goal reached in l steps for all parameter values in B ◮ unsat – goal reached in l steps for no parameter values in B ◮ undet – goal reached in l steps for some parameter values in B ◮ This can also mean a false alarm due to large value of δ ◮ B can be a singleton ◮ Strengthening δ -sat answer ( δ -sat ⇔ sat ) ◮ Necessary for statistical model checking Fedor Shmarov and Paolo Zuliani, HVC 2016 7 / 28

Parameter Set Identification Application Parameter Set Synthesis ◮ Given a parametric hybrid system find a subset of parameter space satisfying the given time series data Probabilistic Bounded Reachability ◮ What is the maximum (minimum) probability that the system reaches the goal state in a finite number of discrete steps? 8 / 28

Parameter Set Synthesis ◮ Parameter Set Synthesis in continuous ( no discrete transitions) biological systems Curtis Madsen, Fedor Shmarov and Paolo Zuliani, CMSB 2015 y(p,t) y(p 5 ,t) y(p 3 ,t) B 0 y(p 2 ,t) y(p 4 ,t) y(p 1 ,t) (t 0 , y 0 ) ◮ This approach was extended to parameter set synthesis in hybrid systems 9 / 28

Parameter Set Synthesis ◮ Parameter Set Synthesis in continuous ( no discrete transitions) biological systems Curtis Madsen, Fedor Shmarov and Paolo Zuliani, CMSB 2015 y(p,t) y(p 5 ,t) y(p 3 ,t) B 0 B 1 p 1 y(p 4 ,t) y(p 2 ,t) y(p 1 ,t) (t 0 , y 0 ) t 1 ◮ This approach was extended to parameter set synthesis in hybrid systems 9 / 28

Parameter Set Synthesis ◮ Parameter Set Synthesis in continuous ( no discrete transitions) biological systems Curtis Madsen, Fedor Shmarov and Paolo Zuliani, CMSB 2015 y(p,t) y(p 5 ,t) y(p 3 ,t) B 0 B 1 B 2 p 1 p 2 y(p 4 ,t) y(p 2 ,t) y(p 1 ,t) (t 0 , y 0 ) t 2 t 1 ◮ This approach was extended to parameter set synthesis in hybrid systems 9 / 28

Parameter Set Synthesis ◮ Parameter Set Synthesis in continuous ( no discrete transitions) biological systems Curtis Madsen, Fedor Shmarov and Paolo Zuliani, CMSB 2015 y(p,t) y(p 5 ,t) y(p 3 ,t) B 0 B 1 B 2 B 3 p 1 p 2 p 3 y(p 4 ,t) y(p 2 ,t) y(p 1 ,t) (t 0 , y 0 ) t 2 t 3 t 1 ◮ This approach was extended to parameter set synthesis in hybrid systems 9 / 28

Parameter Set Synthesis ◮ Parameter Set Synthesis in continuous ( no discrete transitions) biological systems Curtis Madsen, Fedor Shmarov and Paolo Zuliani, CMSB 2015 y(p,t) y(p 5 ,t) y(p 3 ,t) B 0 B 1 B 2 B 3 B 4 p 1 p 2 p 3 p 4 y(p 4 ,t) y(p 2 ,t) y(p 1 ,t) (t 0 , y 0 ) t 2 t 3 t 4 t 1 ◮ This approach was extended to parameter set synthesis in hybrid systems 9 / 28

Parameter Set Synthesis ◮ Parameter Set Synthesis in continuous ( no discrete transitions) biological systems Curtis Madsen, Fedor Shmarov and Paolo Zuliani, CMSB 2015 y(p,t) y(p 5 ,t) y(p 3 ,t) B 0 B 1 B 2 B 3 B 4 B 5 p 1 p 2 p 3 p 4 p 5 y(p 4 ,t) y(p 2 ,t) y(p 1 ,t) (t 0 , y 0 ) t 5 t 2 t 3 t 4 t 1 ◮ This approach was extended to parameter set synthesis in hybrid systems 9 / 28

Probabilistic Bounded Reachability Formal approach ◮ Integrating probability measure ◮ Returns a list of enclosures P ( B N ) = [ P lower , P upper ] ◮ For all p N ∈ B N : Pr ( p N ) ∈ P ( B N ) ◮ P ( B N ) can be arbitrarily small (up to user defined ǫ > 0) when ◮ Probability function is continuous or ◮ Only random parameters are present ◮ Exponential complexity with respect to the number of parameters Fedor Shmarov and Paolo Zuliani, HSCC 2015 Statistical/formal approach ◮ Bayesian Estimations Algorithm (probability) ◮ Cross-Entropy Algorithm (nondeterminism) ◮ Returns a confidence interval [ P lower , P upper ] containing the maximum (minimum) probability value with the user-defined confidence c ∈ (0 , 1). ◮ Constant complexity with respect to the number of parameters Fedor Shmarov and Paolo Zuliani, HVC 2016 10 / 28

Our Software ProbReach ◮ Parameter Set Synthesis in Hybrid Systems ◮ Probabilistic Bounded Reachability in Hybrid Systems https://github.com/dreal/probreach BioPSy ◮ Parameter Set Synthesis in Continuous Systems (no discrete transitions) ◮ SBML file as input, ◮ Graphical User Interface. https://github.com/dreal/biology 11 / 28

Personalized Prostate Cancer Therapy ◮ Identification: prostate-specific antigen (PSA) – tumor marker ◮ Therapy: androgen suppression ◮ Low androgen level causes growth of castration resistant cells (CRC) ◮ Solution: alternation between treatment and rest episodes 12 / 28