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Doctoral Thesis Defense Kedar Kulkarni Date: 10/04/2007 Advisor: Prof. Andreas A. Linninger

Mathematical modeling, problem inversion and design of distributed chemical and biological systems. Doctoral Thesis Defense Kedar Kulkarni Date: 10/04/2007 Advisor: Prof. Andreas A. Linninger Thesis committee: Prof. Royston, Prof. Yang Dai, Prof. Hui Lu and Prof. Ali Cinar (IIT)

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Doctoral Thesis Defense Kedar Kulkarni Date: 10/04/2007 Advisor: Prof. Andreas A. Linninger

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  1. Mathematical modeling, problem inversion and design of distributed chemical and biological systems Doctoral Thesis Defense Kedar Kulkarni Date: 10/04/2007 Advisor: Prof. Andreas A. Linninger Thesis committee: Prof. Royston, Prof. Yang Dai, Prof. Hui Lu and Prof. Ali Cinar (IIT) Laboratory for Product and Process Design, Department of Bioengineering, University of Illinois, Chicago, IL 60607, U.S.A.

  2. Motivation: Why Inversion? Data available: Chemical Reactor: (at specific locations) Characteristics:- Partial information available- The system cannot be studied as a whole To determine: Reaction: • Kinetic constants: • Rate of the reaction (k1) • Other constants • Thermal constants: • Thermal conductivity (k) • Porosity (ε) • Diffusivity (D) Microbial growth: Heat transfer: What are the values of k1, μmax, Ks and S0 ?

  3. Brain Tissue Blood Circulation System [18F] Methyl F-DOPA [18F] Methyl F-DOPA Methylation Methylation INJECTION [18F]FDA [18F]FDOPA [18F]FDOPA NH 2 | CH – C – CO H OH 2 2 | CH 3 OH Motivation: Why Inversion? Data available: Drug distribution in the human brain : Characteristics:- Partial information available- Biological system cannot be decomposed into parts To determine: • Kinetic constants: • Metabolic rates of the reaction • Transport constants: • Diffusivity (D), Hydraulic conductivitiy (K)

  4. Distributed system data Distributed system data Least squares data-fitting Black box parameters Transport and Kinetic Inversion (TKIP) Physical properties BLACK BOX MODELS y = A x + b FIRST PRINCIPLES DISTRIBUTED MODELS Filtering/ Data reconciliation STRUCTURED KNOWLEDGE ABSTRACT INFORMATION First principles distributed models vs. black box models Model identification: 1st principles models vs. black box models No need for filtering/ data reconciliation for 1st principles model inversion!

  5. Motivation: Why consider design under uncertainty ? Over-design to handle uncertainty System data D D GUARANTEE Feasible operation Rigorously account for parametric uncertainty in Design optimization Model building Design for nominal condition L Classical process design:A) COLLECT DATA and BUILD A MODEL:- Based on first principles; data fittingB) DESIGN OPTIMIZATION:- Solve for the optimal design (e.g. minimizing the cost)- Assume perfect information about model parametersC) DESIGN FLEXIBILITY:Empirical over-design to accommodate uncertainty System data Feasible operation?? Model building Design optimization Arbitrary overdesign

  6. PRESENTATION OUTLINE

  7. DISTRIBUTED SYSTEM DATA • Problem formulation • Previous studies/shortcomings • Proposed methodology • - Case studies (Chp 2) PROBLEM INVERSION Part I • Covariance matrix/ confidence intervals • Design flexibility under uncertainty • - Case study (Chp 2, 5) UNCERTAINTY QUANTIFICATION/ RISK ANALYSIS Part II Part III • Cost analysis and sampling • - Case study (Chp 3, 4) DESIGN UNDER UNCERTAINTY DESIGN OF BIOLOGICAL SYSTEMS Part IV • Math modeling/ problem inversion • - Case study (Chp 6)

  8. DISTRIBUTED SYSTEM DATA • Problem formulation • Previous studies/shortcomings • Proposed methodology • - Case studies (Chp 2) PROBLEM INVERSION Part I • Covariance matrix/ confidence intervals • Design flexibility under uncertainty • - Case study (Chp 2, 5) UNCERTAINTY QUANTIFICATION/ RISK ANALYSIS Part II Part III • Cost analysis and sampling • - Case study (Chp 3, 4) DESIGN UNDER UNCERTAINTY DESIGN OF BIOLOGICAL SYSTEMS Part IV • Math modeling/ problem inversion • - Case study (Chp 6)

  9. Problem Inversion-parameter estimation- finite volume discretization- trust region methods Challenges:1) How to handle PDE constraints ?2) Large state variable space3) Efficient algorithmic procedure for inversion

  10. Direct problem Inverse problem Known Postulated PHYSICAL PROPERTIES (e.g. Diffusivity, thermal conductivity) Known FIRST PRINCIPLES DISTRIBUTED MODELS SYSTEM DATA (e.g. concentrations, temperatures) FIRST PRINCIPLES DISTRIBUTED MODELS Postulated: e.g. Navier-Stokes eqns Heat conductivity eqns PROCESS SIMULATION To find PHYSICAL PROPERTIES (e.g. Diffusivity, thermal conductivity) OBJECTIVE: TO FIND PHYSICAL PROPERTIES OF THE SYSTEM

  11. Given:(Data) Temperatures/Concentrations at specific locations (from experimental data e.g. MRI) Choose: First principles distributed model Find: s.t. Mathematical Problem Formulation for Inversion

  12. Previous studies in inversion: Computational methods - Methods to handle optimization problems with PDE and/or algebraic constraints (Caracotsios and Stewart, 1995)- Least squares optimization problem with PDE constraints (Biegler et al, 2003) Applications • Inverse source problems in water pollution (Badia and Hamdi, 2007) • Inversion of kinetic and mass transfer parameters for cellulose nitration (Barbosa et • al., 2006) • Parameter estimation for reactive transport using Monte Carlo approach (Aggarwal • and Carrayraou, 2006) • Inversion in multi-phase flows (Vikhansky et al., 2006) • Reaction mechanisms in vapor decomposition processes (Hwang et al, 2005 ) • Inversion in Convective-Diffusive transport problems (Akcelika et al, 2003) • Data assimilation for air-pollution models (Hakami et al, 2003) • Reconstruction of tissue-optical property maps (Roy and Sevick Muraca, 1999) • Shape optimization (Borgaard and Burns, 1997 )

  13. Distributed models Previous studies in inversion: MOST OF THEM USED LUMPED MODELS! Lumped models • Lumped models CANNOT account for transport • Using distributed models allows us to study reaction and transport simultaneously

  14. True error surface Starting point  p1 p2 Transport and Kinetic Inversion (Proposed methodology) Step-1: Discretize PDE constraints • Finite volume methods Step-2: Reduced-space Trust Region Method: (parameters only) • Avoids update of state variables • Robust: Steepest descent • Fast: Quadratic convergence of the Newton • Globally convergent • Robust to ill-conditioning

  15. APPLICATION I: PLUTONIUM STORAGE

  16. Obtain values of: - The heat generation (q) - The wall heat transfer coefficient (U) - The overall thermal conductivity (k) Quantify parametric uncertainty through effective use of the covariance information Use flexibility analysis/feasibility test concepts to identify the worst case temperature hotspot in the system Plutonium Storage Background and motivation: • DOE safety requirements regarding the safe storage of Plutonium dioxide powder for 50 years • Temperature increase could leads to an overall pressure increase: could lead to explosion Are we safe ?! Objectives:

  17. Experimental setup Radial temperature measurements available 5.7 cm from the bottom of the vessel Actual cylinder (Bielenberg et al, 2004) Schematic

  18. Inversion problem: 1) Obtain experimental data: Temperature measurements at 6 radial points 5.7 cm from the bottom of the cylinder T (K) 2) Solve the math program: r = 0 r = R

  19. Problem formulation To determine: a) Nuclear heat generation, q, W/m3 b) Effective thermal conductivity, k, J/(K. s.m2) c) Wall heat transfer coefficients, U, W/m.K Math program: s.t. BC’s:

  20. Finite Volume Discretization: Use FVM to render PDE constraints to algebraic equations FVM

  21. Results - Simulation model with the estimated parameters: Model Boundary conditions Optimal Parameters Experimental data and simulated radial temperature-profiles at different pressures at height z = 5.7 cm (12 radial and 10 vertical nodes).

  22. Publications: Archival journals: • Kedar Kulkarni, Libin Zhang, Andreas A Linninger, “Model and Parameter uncertainty in Distributed systems”, Industrial and Engineering Chemistry Research, 2006,45, 7832-7840. • Libin Zhang, Kedar Kulkarni, MahadevaBharath R. Somayaji, Michalis Xenos and Andreas A Linninger, “Discovery of Transport and Reaction Properties in Distributed Systems”, AIChE Journal, 2007, 53(2), 381-396. • Libin Zhang, Andres Malcolm, Kedar Kulkarni, Celia Xue, Andreas A Linninger , “Distributed System Design Under Uncertainty”, Industrial and Engineering Chemistry Research, 2006,45, 8352-8360. Conference papers and presentations: • Kedar Kulkarni, Libin Zhang and Andreas A. Linninger, “Model and Parameter Uncertainty in Plutonium Storage”, AIChE Chicago Section Symposium, Northwestern University, Chicago, IL, USA, April 12, 2006. • Libin Zhang, Kedar Kulkarni, Andres Malcolm, “Modeling Uncertainty Analysis in Distributed Systems”, Design of Integrated Chemical and Biological Systems,AIChE 2005 Annual Meeting, Cincinnati, Ohio, USA, October 30-November 4, 2005. • Libin Zhang, MahadevaBharath R. Somayaji, Michalis Xenos, Kedar Kulkarni, Andreas A. Linninger, “Large Scale Transport and Kinetic Inversion Problem for Drug Delivery into the Human Brain”, “Chemical Engineering at the Cross Roads of Technology”, AIChE April 2005 Symposium, Illinois Institute of Technology, Chicago, IL, USA, April 19 -20, 2005.

  23. APPLICATION II: DISCOVERY OF UNKNOWN TRANSPORT AND KINETIC PROPERTIES OF F-DOPA

  24. Motivation: Drug distribution in the human brain revisited Brain Tissue Blood Circulation System [18F] Methyl F-DOPA [18F] Methyl F-DOPA Methylation Methylation INJECTION [18F]FDA [18F]FDOPA [18F]FDOPA NH 2 | CH – C – CO H OH 2 2 | CH 3 OH F-DOPA transport Data available: To determine: Y • Transport properties: • - Diffusivities • - Clearance constants X • Reaction properties

  25. NH 2 | CH – C – CO H OH 2 2 | CH 3 OH Discovering metabolic properties for F-Dopa Brain Tissue Blood Circulation System [18F] Methyl F-DOPA [18F] Methyl F-DOPA Methylation Methylation INJECTION [18F]FDA [18F]FDOPA [18F]FDOPA L-Dopa Methyl Dopa L-Dopa Dopamine To determine: Blood-tissue clearance K1 Tissue-blood clearance k2 The diffusivities DD , DM s. t. F-dopa clearance Methyl F- dopa clearance

  26. 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0 Clinical concentration field of F-dopa Unstructured computational grid s. t. Fdopa clearance Methyl - Fdopa clearance Results: Discovering metabolic properties for F-Dopa Putamen Predicted F-Dopa concentration field with optimal parameters ( Gjedde, A. et. al. Neurobiology, 88, 2721-2725, 1991 ) Best-fit Parameters:

  27. Publications: Archival journals: • Libin Zhang, Kedar Kulkarni, MahadevaBharath R. Somayaji, Michalis Xenos and Andreas A Linninger, “Discovery of Transport and Reaction Properties in Distributed Systems”, AIChE Journal, 2007, 53(2), 381-396. Conference papers and presentations: • Libin Zhang, MahadevaBharath R. Somayaji, Michalis Xenos, Kedar Kulkarni, Andreas A. Linninger, “Large Scale Transport and Kinetic Inversion Problem for Drug Delivery into the Human Brain”, “Chemical Engineering at the Cross Roads of Technology”, AIChE April 2005 Symposium, Illinois Institute of Technology, Chicago, IL, USA, April 19 -20, 2005. • Kedar Kulkarni, Jeonghwa Moon, Libin Zhang and Andreas Linninger, “Solution Multiplicity of Inversion Problems in Distributed Systems”, Applied Mathematics in Bioengineering, AIChE 2006 Annual Meeting, San Fransisco, California, USA, November 12 – 17, 2006 (Abstract accepted).

  28. APPLICATION III: DISCOVERY OF UNKNOWN TRANSPORT AND KINETIC PROPERTIES OF A CATALYTIC PELLET REACTOR

  29. Cooling Outlet Multi-scale Model B A Tubular Reactor Cooling inlet Packed Catalyst Pellet Bed Catalyst Pellets Micro Pores of Catalysts Discovery of unknown transport and kinetic properties of a catalytic pellet reactor BULK (MACROSCOPIC) MODEL Mass and energy balances PELLET (MICROSCOPIC) MODEL Coupled mass and energy balance

  30. Problem formulation To determine: a) Bulk diffusivity of species A, DA, m2/s b) Reference reaction constant, kref, 1/s c) Activation energy, E, J/mol Math program: BULK reaction/transport s. t. PELLET reaction/transport BC’s:

  31. Parameters determined: DA = 0.732  10-4 (m2/s) ; kref= 0.8 10-4 (1/s); E = 5268.62 (J/mol) Inversion problem (Results) Solution concentration profile Solution temperature profile C T Length of the reactor Length of the reactor

  32. Publications: Archival journals: • Libin Zhang, Kedar Kulkarni, MahadevaBharath R. Somayaji, Michalis Xenos and Andreas A Linninger, “Discovery of Transport and Reaction Properties in Distributed Systems”, AIChE Journal, 2007, 53(2), 381-396. • Libin Zhang, Andres Malcolm, Kedar Kulkarni, Celia Xue, Andreas A Linninger , “Distributed System Design Under Uncertainty”, Industrial and Engineering Chemistry Research, 2006,45, 8352-8360. Conference papers and presentations: • Kedar Kulkarni, Libin Zhang and Andreas A. Linninger, “Model and Parameter Uncertainty in Plutonium Storage”, AIChE Chicago Section Symposium, Northwestern University, Chicago, IL, USA, April 12, 2006. • Libin Zhang, Kedar Kulkarni, Andres Malcolm, “Modeling Uncertainty Analysis in Distributed Systems”, Design of Integrated Chemical and Biological Systems,AIChE 2005 Annual Meeting, Cincinnati, Ohio, USA, October 30-November 4, 2005. • Libin Zhang, MahadevaBharath R. Somayaji, Michalis Xenos, Kedar Kulkarni, Andreas A. Linninger, “Large Scale Transport and Kinetic Inversion Problem for Drug Delivery into the Human Brain”, “Chemical Engineering at the Cross Roads of Technology”, AIChE April 2005 Symposium, Illinois Institute of Technology, Chicago, IL, USA, April 19 -20, 2005. • Kedar Kulkarni, Jeonghwa Moon, Libin Zhang and Andreas Linninger, “Solution Multiplicity of Inversion Problems in Distributed Systems”, Applied Mathematics in Bioengineering, AIChE 2006 Annual Meeting, San Francisco, California, USA, November 12 – 17, 2006. • Kedar Kulkarni, Jeonghwa Moon, Libin Zhang and Andreas Linninger, “Multi-scale Modeling and Solution Multiplicity in a Fixed Bed Catalytic Pellet Reactor”, (accepted), Complex and Networked Systems II, AIChE 2007 Annual Meeting, Salt Lake City, Utah, USA, November 4 – 11, 2007.

  33. DISTRIBUTED SYSTEM DATA • Problem formulation • Previous studies/shortcomings • Proposed methodology • - Case studies (Chp 2) PROBLEM INVERSION Part I • Covariance matrix/ confidence intervals • Design flexibility under uncertainty • - Case study (Chp 2, 5) UNCERTAINTY QUANTIFICATION/ RISK ANALYSIS Part II Part III • Cost analysis and sampling • - Case study (Chp 3, 4) DESIGN UNDER UNCERTAINTY DESIGN OF BIOLOGICAL SYSTEMS Part IV • Math modeling/ problem inversion • - Case study (Chp 6)

  34. Uncertainty quantification/Risk analysis-Covariance matrix/ confidence regions - Design performance evaluation under uncertainty Challenges:1) How to quantify uncertainty ?2) How to handle parameter correlation ?3) How to quantify design flexibility to handle uncertainty ?

  35. (qN – s, qN + s) (qN – s, qN + s) Nominal value Nominal value qN Probability TIGHT BOUNDS LOOSE BOUNDS qN Plutonium storage example: Good data and models Bad data and models

  36. Parameter value P(), Probability function   = observations - * + Confidence Interval Uncertainty quantification: • How good are the parameter estimates ?- Relax the assumption that parameters are point values- Instead, they are the mean values of a probability distribution function Covariance matrix: The covariance matrix of the parameters contains information regarding the accuracy of their estimates. (Donaldson and Schnabel, 1987)

  37. 2 P(), Probability function Nominal point 2+   2* 2- 1 1- 1* 1+ = observations 2 2 2 - * + 1 1 Parameter value 1 Confidence Interval ij > 0 ij  0 ij < 0 : Uncertainty quantification (cont’d): Individual confidence regions (ICR) Joint confidence regions (JCR) Individual confidence regions (ICR) Joint confidence regions (JCR)

  38. Results for confidence bounds in Plutonium storage: ICR and JCR for q and U and k1:

  39. D q – D q + Design constraint (e.g T < Tmax) Wall heat transfer coefficient (W/m2.K) D U + U N (q N, U N) D U – Feasible region q N Heat generation rate (W/m3) Risk analysis: Design performance evaluation Given the design (d), is its performance feasible even when parameters are varying ? Plutonium storage example: Does the rectangle lie wholly inside the feasible region ?? Risk a

  40. Various approaches to quantify design flexibility • Flexibility Analysis (Halemane and Grossmann, 1983, Swaney and Grossmann, 1985): Considers the problem of uncertainty with mathematical rigor - Feasibility Test - Flexibility Index • Optimal Retrofit Design (Pistikopolous and Grossmann, 1988): Optimally redesigning an ‘existing’ process design. • Resilience index (Saboo et al, 1983) • Design reliability (Kubic and Stein, 1988) • Stochastic Flexibility index (Straub and Grossmann, 1983, Pistikopoulos and Mazucchi, 1990) • Convex Hull methods (Ierapetritou et al, 2001) • Flexibility analysis accounting for accuracy of plant data (Ostrovsky et al, 2004, 2006)

  41. Feasibility Test * x(d)  0 => feasibility ensured q  T * x(d) > 0 => design infeasible for at least one value of q Flexibility Index • F 1 => Flexibility target (F = 1) satisfied • F< 1 => Target (F = 1) not achieved. Fractional F quantifies the maximum tolerable deviation from qN

  42. (W/m2.K) (W/m3) Identification of worst case hotspot in Plutonium storage: Temperature Contours (isotherms) Solution: (a) JCR worst case T – 704 K (b) ICR worst case T – 711 K Locus of the temperature hotpot

  43. DISTRIBUTED SYSTEM DATA • Problem formulation • Previous studies/shortcomings • Proposed methodology • - Case studies (Chp 2) PROBLEM INVERSION Part I • Covariance matrix/ confidence intervals • Design flexibility under uncertainty • - Case study (Chp 2, 5) UNCERTAINTY QUANTIFICATION/ RISK ANALYSIS Part II Part III • Cost analysis and sampling • - Case study (Chp 3, 4) DESIGN UNDER UNCERTAINTY DESIGN OF BIOLOGICAL SYSTEMS Part IV • Math modeling/ problem inversion • - Case study (Chp 6)

  44. APPLICATION :CATALYTIC PELLET REACTOR Design under uncertainty-Cost analysis/ sampling techniques

  45. Design under uncertainty Objective: To obtain optimal design and cooling policy by minimizing expected costConstraints: (i) Product quality constraint (ii) Safety constraint (iii) Species and energy balance equations (iv) The parameters lie within confidence regionsMethod: Sampling in the individual and joint confidence regions of the uncertain parameters OVERDESIGN PERFORMED RIGOROUSLY as opposed to INTUITIVE OVERDESIGN!

  46. Problem formulation s. t. Maximum admissible temperature Conversion threshold ICR The parameters are inside the confidence regions OR JCR

  47. Results Nominal value ICR JCR PARAMETERS ~ 2x ~ 1.4x

  48. DISTRIBUTED SYSTEM DATA • Problem formulation • Previous studies/shortcomings • Proposed methodology • - Case studies (Chp 2) PROBLEM INVERSION Part I • Covariance matrix/ confidence intervals • Design flexibility under uncertainty • - Case study (Chp 2, 5) UNCERTAINTY QUANTIFICATION/ RISK ANALYSIS Part II Part III • Cost analysis and sampling • - Case study (Chp 3, 4) DESIGN UNDER UNCERTAINTY DESIGN OF BIOLOGICAL SYSTEMS Part IV • Math modeling/ problem inversion • - Case study (Chp 6)

  49. APPLICATION :HEPATOTOXIC GENE INTERACTION NETWORKS Problem inversion in biological systems-Mathematical modeling, problem inversion and design

  50. Motivation The harmful impact of drugs is not fully understood PROBLEM: Rats affected with harmful conditions (e.g. damaged liver) Chronic biological outcomes Toxic drugs Rats ??

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