Discovery of Cerebral Transport and Metabolic Reaction Properties by Problem Inversion

1. Discovery of Cerebral Transport and Metabolic Reaction Properties by Problem Inversion AIChE 2005 Annual Meeting Cincinnati, OH October 30-November 4 Session 10042: Systems Engineering Approaches in Biology Paper 379f Libin Zhang, MahadevaBharath R. Somayaji, Michalis Xenos and Andreas A. Linninger 11/02/2005 Laboratory for Product and Process Design, Department of Chemical Engineering, University of Illinois, Chicago, IL 60607, U.S.A.

2. Imaging Techniques CT-Shows the structure of the brain and NOT its functions MRI-It provides an anatomical view of the brain Cine MRI – Flow velocities fMRI – Used to visualize brain function (E.g. Blood Flow to pathological organs) PET- detects radioactive material that is injected or inhaled to produce an image of the brain DTI-Used to demonstrate anatomical substructures

3. Water Content in Brain Tissue: 3T-MRI (a) (c) (b) (d) (e) (f) (a) Proton-density weighted image, (b) T2 weighted image, (c) T1 weighted image, (d) T1 map generated based on images (a) and (c), (e) average T1 vs. distance from the ventricle wall, and (f) a color map of figure (e).

4. Proposition: Transport and Kinetic Inversion Problems (TKIP) Extract from distributed image data mechanistic and quantitative knowledge about biological transport and reaction phenomena

5. TKIP for Invasive Drug Delivery 1. Clinical Experiments 2. Image-based Measurements MRI, CT,FMRI, Histology • Transport and Metabolic • Reaction Mechanism. • Drug diffusivity • Reaction rates 3. Transport and Kinetic Inversion Problem (TKIP) Patient Specific Treatment Computer-aided Therapy Design Catheter Design • Penetration depths • Drug • Concentrations Injection Placement/location Catheter Geometry Injection angle Injection Policy

6. Overview • Methodology • Mathematical Problem Formulation • Proposed Solution Methodology • Recovery of Metabolic Uptakes • Case studies • Drug Transport in Invasive Drug Delivery • Recovery of Metabolic Parameters

7. Mathematical Problem Formulation Given: Concentration Field (from histological data, autoradiography and MRI) Find: Metabolic uptake and drug transport mechanism s.t.

8. Characteristics of Large-scale TKIP • Medical and biological systems are not separable -> 2D, 3D, 4D • Complex Geometry and Physiology • anisotropy, inhomogen. • Lack of detailed drug mechanism in human brain • Problem involves a large number of state variables (Velocity, Pressure, Species Concentration) • Search for optimal parameters globally (Porosity, Directional Diffusivity, Tortuosity) • Robustness of problem inversion (multiple minima, noise)

9. Components of the Solution Approach 1. Geometrical Representation • Imaging Data, physiology, study frame • Unstructured or moving computational grids 2. Conservation Laws and Constitutive Equations: Finite volume method (FVM) or finite element method (FEM) 3. Parameter Estimation Algorithm: • Response Surface: • Inexact Trust Region method to solve the data fitting optimization problem • Sparse linear solver (e.g. GMRES)

10.  Physical Plane y  N x n e w P E W s Transformed Plane S   Geometrical ComplexityCoordinate Transformation and discretization Curvilinear Transformation Discretization in Transformed Space Transport Model Integrating over a control volume Discretized Model Jacobian of transformation

11. yp) 0.6 0.5 True error surface 0.4 y(p+p) - true Solution Second order Appr. 0.3 Starting point 0.2 0.1 First order Approxim. 0 -0.1 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5  p1 p2 Response Surface Map Response Surface to parameter perturbations p Sensitivity Matrix Response Surface Approximation

12. Steepest descent direction Newton direction Dogleg direction Steepest Direction B Cauchy Point A D Newton Point C Trust Region Methods • Levenberg-Marquardt Method • Powell Dogleg Method The dogleg direction combined the Gauss-Newton direction with Steepest descent direction The most time consuming step is evaluating original function and sensitivity equations

13. Components of the Solution Approach Data Collection and Equation Preparation Parameter Estimation Algorithm • Response Surface: First and/or second order sensitivity • Inexact Trust Region method to solve the data fitting optimization problem • Sparse linear solver (e.g. GMRES) to solve large scale problem Experimental data from clinical images Construct the geometry of computational domain Optimization Initialize the unknown parameters Solve the original discretized equations and sensitivity equations Calculate the Newton and Steepest descent direction Update the unknown parameters Problem variables: 1,000 nodes in 2D problem, the number of the state variables will be 8000 and another 16,000 sensitivity variables Convergence? Output the optimal parameters

14. Part II Sample Problems Invasive Drug Delivery Design of L-Dopa Adminstration

15. Examples: Invasive Drug Delivery into Human Brain Drug Injection Simplified Geometry Real Brain Geometry • Brain tissue is pore medium • Diffusion is isotropic in gray matter. • It is extremely anisotropic in white matter due to the directionality of the myelin fiber tracts. • Easy to illustrate the concepts • Help to validate the methodology • Good for preliminary research • Save the computation time Anisotropic white matter

16. Drug Metabolism Sample Problem (simplified) R A B S Concentration Field

17. Case I: Simplified Example Efficacy in Convection-Enhanced Drug Delivery Problem Formulation: Recover Flow Field, U,V; Diffusivity D, kinetic parameters k1, k2 Pressure Field Velocity Field Continuity equation Dick’s law in pore media Reaction Mechanism, R(k1,k2) Concentration A Concentration, R K2 not shown

18. 800 0.4 0.03 700 0.35 0.025 600 0.3 0.02 500 0.25 Diffusivity 400 Reaction Rate 0.015 300 0.2 0.01 200 0.15 100 0.005 0.1 0 0 2 4 6 8 10 12 14 16 18 20 0.05 0 1 3 5 7 9 11 13 15 17 19 Iteration Case Study II: Extract Unknown Metabolic uptake from Concentration Field Unstructured grid Optimal parameters: D = 9.97 e-2 [m2/s] K = 1.30 e-3 [1/s] U(x,y) …. Invasive Drug Insertion into the Midbrain Convergence history Error Iteration

19. Case III. Metabolic Mechanism for L-DOPA (Katzenschlager et al., 2002). L- Methyl DOPA L-Dopa is the precursor to dopamine. Dopamine itself cannot pass BBB. L-Dopa id converted into dopa-decarboxylase in the Brain.

20. Metabolic Mechanism for L-DOPA Brain Tissue (Basal Ganglia) BBB Plasma/Blood Circulation system L-Dopa Decarboxylation Dopamine L-Dopa Methylation L-Dopa Methyl Dopa Methyl Dopa L-Dopa Methyl Dopa No Dopamine in Plasma Dopamine will stay in brain tissue Dopamine hydrophobic L-Dopa Dopamine L-Dopa Methyl Dopa L-Dopa Methyl Dopa

21. 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0 0.4 0.35 FDopa’s Diffusivity in Brain Tissue 0.3 0.25 0.2 0.15 0.1 0.05 0 Blood clearance -0.05 1 2 3 4 5 6 7 Case Study III: Extract Metabolic Reaction and Transport Properties for L-Dopa Computational grid Clinical concentration field of L-dopa Optimal result, 0.35 0.3 PET image of Fdopa-derived radioactivity, merged with magnetic resonance image 0.25 0.2 Error Parameters 0.15 0.1 0.05 0 1 2 3 4 5 6 7 Iteration Iteration Gjedde, A. et. al. Neurobiology, 88, 2721-2725, 1991

22. L-dopa distribution in the brain Computational grid Dopamine distribution in the brain Coronal View - Decarboxylation of L-Dopa L-dopa distribution in the human brain L-Dopa Dopamine

23. Conclusions • Transport and Kinetic Inversion Problem for extracting quantitative knowledge • Metabolic reaction rates • Transport properties • Mathematical Approach • Response Surface: sensitivity information • Trust Region method (Powell dogleg method) • Geometric Complexity and physiological consistency of tissues • Unstructured grid, metrics of transformations • Finite volume approach • Anisotropic tissue models • Future Steps: • Study in-vivo animal models and limited human data • Hindered convection and diffusion • (Electric charge effects using Nernst-Planck Eqs,…)

24. Acknowledgments Department of Neurosurgery, University of Chicago • Richard Penn, MD, Professor,Department of Neurosurgery, University of Chicago • David Zhu, PhD, Research Associate,Department of Radiology, University of Chicago LPPD Team, University of Illinois at Chicago • Dr. Michalis Xenos, Research Associate, Department of Chemical Engineering, University of Illinois at Chicago • Dr. Libin Zhang, Research Associate, Department of Chemical Engineering, University of Illinois at Chicago • Dr. Chris Zhou, Research Associate, Department of Chemical Engineering, University of Illinois at Chicago • Andres Malcolm, PhD candidate, Department of Chemical Engineering, University of Illinois at Chicago • Srinivas Kondapalli, PhD candidate, Department of Chemical Engineering, University of Illinois at Chicago • MahadevaBharath. R. Somayaji, PhD candidate, Department of Chemical Engineering, University of Illinois at Chicago • Kedar M. Kukarni, PhD candidate, Department of BioEngineering, University of Illinois at Chicago • Mick Parikh, MS candidate, Department of Chemical Engineering, University of Illinois at Chicago • Romeo Ibrahim MS candidate, Department of Chemical Engineering, University of Illinois at Chicago

25. Complex Drug Chemistry

26. 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0 Blood Circulation system Generate loss Methylation of Fdopa Brain Tissue BBB Methylation of Fdopa and decarboxylation of dopa

27. Image Based Techniques PET MRI

28. Transport and Kinetic Inversion Problem (TKIP) (drug delivery in the human brain) Image-based Measurements Gjedde, A. et. al. Neurobiology, 88, 2721-2725, 1991 MRI, CT,FMRI, Histology Clinical Experiments Transport and Kinetic Inversion Problem (TKIP) Patient Specific Treatment Computer-aided Therapy Prediction • Penetration depth • Drug Concentration • Transport and Metabolic • Reaction Mechanism. • Drug diffusivity • Reaction rates Catheter Design Computer-aided Therapy Design Injection Placement/location Catheter Geometry Injection angle Injection Policy

29. Case Study III: Assessment of Dopa Efficacy(preliminary)