Parameter Identification using δ Decisions for Hybrid Systems in Biology

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1 CMACS/AVACS Worshop Paraeter Identification using δ Decisions for Hbrid Sstes in Biolog Bing Liu Joint wor with Soonho Kong, Sean Gao, Ed Clare

2 Biological Sste Bolins et al. SIGGRAPH, 26 2

3 Coputational Modeling Building a odel Structure Calibration (paraeter estiation Validation Perforing analsis Tie

4 Coputational Modeling Building a odel Structure Calibration (paraeter estiation Validation Perforing analsis 4

5 Coputational Modeling Foraliss Differential equations Boolean networ Petri nets Rule based odels Multi ode? 5

6 Hbrid Sstes Activation of different networs at different stages E.g. Cell ccle, Cell differentiation Cell Ccle Benes et al, PNAS 29 6

7 Hbrid Sstes More eaples: E. coli cheotais, inesin waling, gene transcription, drug treatent, 7

8 Hbrid Sstes Evolve in a continuous wa in each ode Ruled b discrete transitions Hbrid autoata H!X,Q, flow,guard, reset, inv, init" w r 2 r 2 q d w v d2 v2 2 r 2 r q d w v d2 v2 v v 2 8

9 Hbrid Sstes H!X,Q, Init, Flow, Jup" A continuous space Init! Q " X Flow : initial configurations : continuous flows and a finite set of odes Each ode q is equipped with differential equations d! f (! q,t Jup : discrete X! R The sstes can be switched fro to, resetting odes and variables Q 9

10 Hurdles How to answer questions such as: Which odel structure is better? What conditions a lead to a desired state? How to control the sste to avoid bad states? Paraeter snthesis proble Analing nonlinear hbrid autoata is challenging even siple reachabilit questions can be undecidable

11 Our Approach Use δ coplete decision procedures to tacle the paraeter snthesis proble for nonlinear hbrid odels Encode a paraeter snthesis proble as a first order forula over the reals Perfor bounded odel checing Eplo an interval constrains propagation (ICP based algorith to identif the resulting paraeters

12 Delta Decisions A decision procedure using nuerical techniques (with an error bound δ: ϕ is false ϕ δ is true The delta decision proble is decidable for bounded first order forulas over arbitrar Tpe 2 coputable functions ep, sin, etc, and Lipschit continuous ODEs 2

13 δ Coplete Bounded Model Checing Practical tools: dreal and dreach DPLL(T, interval arithetic, constraint solving, reliable integration, etc.

14 Case Studies I Prostate cancer Second leading cause of cancer related deaths aong en in US Horone therap Androgen deprivation Continuous androgen suppression (CAS Interittent androgen suppression (IAS 4

15 Continuous Androgen Suppression Side effects: aneia, osteoporosis, ipotence, etc. Relapse after a edian duration of 8 24 onths, due to the proliferation of androgen independent (AI cancer cells. CAS Tuor sie Tie 5

16 Interittent Androgen Suppression Reduce side effects Ma dela the tie to relapse Avoid eergence of AI cells CAS Tuor sie IAS Tie 6

17 Interittent Androgen Suppression Clinical phase II and III trials confir its advantage in ters of qualit of life and cost For tie to relapse and cancer specific survival, its advantage depends on individual patients and the treatent schee How to design a personalied treatent schee for each individual patient? 7

18 Population of AD cells, AI cells, seru androgen concentration, PSA level (Ideta et al Model d dv d d d d τ ( ( ( ( : flow d dv d d d d τ ( ( ( ( : flow : jup 2 < d d r : jup 2 > d d r Mode (on-treatent Mode 2 (off-treatent

19 Proliferation, apoptosis, utation rates Treatent thresholds: r and r 9 Model d dv d d d d τ ( ( ( ( : flow d dv d d d d τ ( ( ( ( : flow : jup 2 < d d r : jup 2 > d d r Mode (on-treatent Mode 2 (off-treatent

20 Model Selection Hpothesis AI cells grow at the constant rate independent of the androgen level Hpothesis 2 AI cells do not grow when the androgen level is noral Hpothesis AI cells decrease when the androgen level is noral

21 Model Selection Observation When r 4 (ng l and r (ng l, cancer relapse can be avoided within das Define cancer relapse: PSA level > ng l Bounded odel checing Propert: 9 tau H: unsat H2: unsat H: sat

22 Personalied Therap Design Personalied paraeters the proliferation rate of AI cells the apoptosis rate of AI cells the utation rate fro AD to AI cells ( the initial androgen level 22

23 Personalied Therap Design Different patients a response differentl to the sae treatent schee (r 4 and r Patient# avoided (r 4 and r Patient#2 delaed (r 4 and r Patient# relapse (r 4 and r Patient# delaed (r 5 and r 6 2

24 Personalied Therap Design Appl IAS therap to a patient for 2 ccles and easure PSA tie serials Estiate personalied paraeters b fitting PSA data Given r in [,8 and r in [8,5], verif if H can reach 9 tau (i.e. no relapse False: androgen suppression does not wor True: feasible values for r and r will be returned. 24

25 Personalied Therap Design Patients dataset: 9 patients Phase II clinical trials (Bruchovs et al, Cancer, 27 Partial results: 25

26 Case Studies II Cardiac cell Minial Resistor Model (Fenton et al, J Theor Biol, 28 The electrical activit are governed b opening and closing of ion channels 26

27 27

28 Cardiac Disorders When the cardiac cell loses its ecitabilit, it will lead to disorders such as ventricular tachcardia and fibrillation 28

29 Cardiac Disorders When the cardiac cell loses its ecitabilit, it will lead to disorders such as ventricular tachcardia and fibrillation Under what circustances, a cardiac cell will fail to generate a successful AP (i.e. globall u < θ v? Related paraeter ranges identified b Grosu et al. CAV 2 (linear approiation We answer this b identifing the ranges for paraeters in the original nonlinear odel! o! (,.6"! o2! (,."6.2! o2! o #

31 Suar A delta decision based fraewor for paraeter identification Model selection Therap optiiation Diseased related conditions identification What s net More analsis for cardiac cell odel Paraeter estiation DNA daage induced cellular senescence

Parameter Identification using Model Checking for Systems Biology

Celebrating Years of Model Checing Parameter Identification using Model Checing for Sstems Biolog Bing Liu Ed s postdoc 22-23 Joint wor with P.S. Thiagarajan s group and Ed Clare s group Universit of Pittsburgh