Physiologically Based Modelling and Predictive Control
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This document describes a physiologically based pharmacokinetic (PBPK) modeling and model predictive control (MPC) approach for optimal drug administration. It involves 4 main steps: 1) Developing PBPK models using mass balance equations to describe drug distribution in compartments like plasma, red blood cells, and kidneys. 2) Discretizing the PBPK models and designing an observer to estimate unmeasured states using the measured plasma concentration. 3) Formulating an MPC problem to determine optimal drug administration inputs over a prediction horizon while satisfying constraints like toxicity limits. 4) Solving the MPC problem online to determine the optimal control action and adjust the drug dosage based on the
Physiologically Based Modelling and Predictive Control
- 1. Physiologically Based Pharmacokinetic Modeling and Predictive Control An integrated approach for optimal drug administration P. Sopasakis, P. Patrinos, S. Giannikou, H. Sarimveis. Presented in the 21 European Symposium on Computer-Aided Process Engineering
- 2. Drug administration strategies Open loop drug administration based on average population pharmacokinetic studies Evaluation: Toxicity Alert! • No feedback • Suboptimal drug administration • The therapy is not individualized • High probability for side effects! W. E. Stumpf, 2006, The dose makes the medicine, Drug Disc. Today, 11 (11,12), 550-555
- 3. Drug administration strategies Patients for which the therapy works beneficially Patients prone to side-effects
- 4. Drug administration strategies The treating doctor examines the patient regularly and readjusts the dosage if necessary Evaluation : • A step towards therapy individualization • Again suboptimal • Again there is a possibility for side effects • Empirical approach
- 5. Drug administration strategies Computed-Aided scheduling of drug administration Evaluation: • Optimal drug administration • Constraints are taken into account • Systematic/Integrated approach • Individualized therapy
- 6. What renders the problem so interesting? Input (administered dose) & State (tissue conce- ntration) constraints (toxicity). Only plasma concentration is available (need to design observer). The set-point value might be different among patients and might not be constant.
- 7. Problem Formulation Problem: Control the concentration of DMA in the kidneys of mice (set point: 0.5μg/lt) while the i.v. influx rate does not exceed 0.2μg/hr and the concentration in the liver does not exceed 1.4μg/lt.
- 8. Tools employed: PBPK modeling About : PBPK refers to ODE-based models employed to predict ADME* properties of chemical substances. Main Characteristics : • Attempt for a mechanistic interpretation of PK • Continuous time differential equations • Derived by mass balance eqs. & other principles of Chemical Engineering. * ADME stands for Absorption Distribution Metabolism and Excretion R. A. Corley, 2010, Pharmacokinetics and PBPK models, Comprehensive Toxicology (12), pp. 27-58.
- 9. Tools employed: MPC Why Model Predictive Control ? • Stability & Robustness • Optimal control strategy • System constraints are systema- tically taken into account J.M. Maciejowski, 2002 , Predictive Control with Constraints, Pearson Education Limited, 25-28.
- 10. Step 1 : Modeling Mass balance eq. in the plasma compartment: dC plasma Vplasma QskinCv , skin Qlung Cv ,lung Qkidney Cv ,kidney dt Qblood Cv ,blood Qresidual Cv ,residual u ( RBC CRBC plasma C plasma ) QC C plasma Mass balance in the RBC compartment: dCRBC Vplasma plasmaC plasma RBC CRBC dt And for the kidney compartments : Qkidney C Arterial Cv ,kidney kidney Cv ,kidney dCkidney Ckidney Vkidney kkidney Akidney dt Pkidney dCv ,kidney Ckidney Vkidney kidney Cv ,kidney dt Pkidney M. V. Evans et al, 2008, A physiologically based pharmacokin. model for i.v. and ingested DMA in mice, Toxicol. sci., Oxford University Press, 1 – 4 .
- 11. Step 2 : Model Discretization Discretized PBPK model: x m (t 1) f (x m (t ), u(t )) y m (t ) g (x m (t )) x(t 1) Ax(t ) Bu(t ) Linearization z (t ) Hy m (t ) y (t ) Cx(t ) Subject to : Exm t Lu t M
- 12. Step 3 : Observer Design x m (t 1) f (x m (t ), u(t )) Augmented system: y m (t ) g (x m (t )) z (t ) Hy m (t ) x(t 1) Ax(t ) Bu(t ) B d d(t ) d(t 1) d(t ) y (t ) Cx(t ) Cd d(t ) x(t 1) Ax(t ) Bu(t ) y (t ) Cx(t ) G. Pannocchia and J. B. Rawlings, 2003, Disturbance models for offset-free model predictive control, AlChE Journal, 426-437.
- 13. Step 3 : Observer Design (cont’d) x m (t 1) f (x m (t ), u(t )) Augmented system: y m (t ) g (x m (t )) z (t ) Hy m (t ) x(t 1) Ax(t ) Bu(t ) B d d(t ) d(t 1) d(t ) y (t ) Cx(t ) Cd d(t ) x(t 1) Ax(t ) Bu(t ) y (t ) Cx(t ) This system is observable iff (C, A) is observable and the matrix A I Bd C Cd is non-singular G. Pannocchia and J. B. Rawlings, 2003, Disturbance models for offset-free model predictive control, AlChE Journal, 426-437.
- 14. Step 3 : Observer Design (cont’d) x m (t 1) f (x m (t ), u(t )) Augmented system: y m (t ) g (x m (t )) z (t ) Hy m (t ) x(t 1) Ax(t ) Bu(t ) B d d(t ) d(t 1) d(t ) y (t ) Cx(t ) Cd d(t ) x(t 1) Ax(t ) Bu(t ) y (t ) Cx(t ) Observer dynamics: x(t 1) A Bd x(t ) B ˆ ˆ L ˆ ˆ 0 ˆ u(t ) x y m (t ) Cx(t ) Cd d(t ) ˆ d(t 1) 0 I d(t ) L d K. Muske & T.A. Badgwell, 2002, Disturbance models for offset-free linear model predictive control, Journal of Process Control, 617-632.
- 15. Step 4 : MPC design Maeder et al. have shown that: ˆ A - I B x B d d ˆ HC 0 u ˆ r HCd d The MPC problem is formulated as follows: N 1 x( N ) x (t ) P x(k ) x (t ) u(k ) u(t ) 2 2 2 min Q R u (0),...,u ( N 1) k 0 Ex(k ) Lu(k ) M, k 0..., N x(k 1) Ax(k ) Bu(k ) B d d(k ), k 0,..., N d(k 1) d(k ), k 0,..., N x(0) x(t ) ˆ ˆ d(0) d(t ) U. Maeder, F. Borrelli & M. Morari, 2009, Linear Offset-free Model Predictive Control, Automatica, Elsevier Scientific Publishers , 2214-2217.
- 16. Step 4 : MPC design Maeder et al. have shown that: ˆ A - I B x B d d ˆ Terminal HC 0 u ˆ Cost r HCd d Deviation from the set-point The MPC problem is formulated as follows: N 1 x( N ) x (t ) P x(k ) x (t ) u(k ) u(t ) 2 2 2 min Q R u (0),...,u ( N 1) k 0 Ex(k ) Lu(k ) M, k 0..., N Constraints x(k 1) Ax(k ) Bu(k ) B d d(k ), k 0,..., N d(k 1) d(k ), k 0,..., N x(0) x(t ) ˆ Model ˆ d(0) d(t ) U. Maeder, F. Borrelli & M. Morari, 2009, Linear Offset-free Model Predictive Control, Automatica, Elsevier Scientific Publishers , 2214-2217.
- 17. Step 4 : MPC design Maeder et al. have shown that: ˆ A - I B x B d d ˆ HC 0 u ˆ r HCd d The MPC problem is formulated as follows: N 1 x( N ) x (t ) P x(k ) x (t ) u(k ) u(t ) 2 2 2 min Q R u (0),...,u ( N 1) k 0 Ex(k ) Lu(k ) M, k 0..., N x(k 1) Ax(k ) Bu(k ) B d d(k ), k 0,..., N d(k 1) d(k ), k 0,..., N x(0) x(t ) ˆ ˆ A - I B x(t ) B d d(t ) ˆ d(0) d(t ) Where: HC 0 u(t ) ˆ (t ) r (t ) HCd d U. Maeder, F. Borrelli & M. Morari, 2009, Linear Offset-free Model Predictive Control, Automatica, Elsevier Scientific Publishers , 2214-2217.
- 18. Step 4 : MPC design P is given by a Riccati-type equation: P AT PA (AT PB)(BT PB R)1 (BT PA) Q N 1 x( N ) x (t ) P x(k ) x (t ) u(k ) u(t ) 2 2 2 min Q R u (0),...,u ( N 1) k 0 Ex(k ) Lu(k ) M, k 0..., N x(k 1) Ax(k ) Bu(k ) B d d(k ), k 0,..., N d(k 1) d(k ), k 0,..., N x(0) x(t ) ˆ ˆ A - I B x(t ) B d d(t ) ˆ d(0) d(t ) Where: HC 0 u(t ) ˆ (t ) r (t ) HCd d U. Maeder, F. Borrelli & M. Morari, 2009, Linear Offset-free Model Predictive Control, Automatica, Elsevier Scientific Publishers , 2214-2217.
- 19. Overview Measured Plasma Concentration Observer r t C pl Estimated states x(t ) u(t ) r t Model Predictive Therapy Controller
- 20. Overview Reconstructed Cˆ ˆ ˆ ˆ Clung / bl Cskin Cskin / bl ˆ d lung x C ˆ ˆ state vector : ˆ ˆ ˆ ˆ dlung dlung / blood d skin d skin / blood Measured Plasma Concentration Observer r t C pl Estimated states x(t ) u(t ) r t Model Predictive Therapy Controller
- 21. Results: Assumptions Assumptions: Intravenous administration of DMA to mice with constant infusion rate (0.012lt/hr). Prediction Horizon was fixed to N=10 and the set point was set to 0.5μg/lt in the kidney. Additional Restrictions: The i.v. rate does not exceed 0.2μg/hr and the concentration in the liver remains below 1.4 μg/lt. M. V. Evans et al, 2008, A physiologically based pharmacokin. model for i.v. and ingested DMA in mice, Toxicol. sci., Oxford University Press, 1 – 4 .
- 22. Results: Simulations without constraints Constraints are violated
- 23. Results: Simulations Requirements Stability is are fulfiled guaranteed & set-point is reached The constraint is active
- 24. Conclusions Linear offset-free MPC was used to tackle the optimal drug dose administration problem. The controller was coupled with a state observer so that drug concentration can be controlled at any organ using only blood samples. Constraints are satisfied minimizing the appearance of adverse effects & keeping drug dosages between recommended bounds. Allometry studies can extend the results from mice to humans. Individualization of the therapy by customizing the PBPK model parameters to each particular patient. Next step: Extension of the proposed approach to oral administration.
- 25. References 1. R. A. Corley, 2010, Pharmacokinetics and PBPK models, Comprehensive Toxicology (12), pp. 27-58. 2. M. V. Evans, S. M. Dowd, E. M. Kenyon, M. F. Hughes & H. A. El-Masri, 2008, A physiologically based pharmacokinetic model for intravenous and ingested Dimethylarsinic acid in mice, Toxicol. sci., Oxford University Press, 1 – 4 . 3. J.M. Maciejowski, Predictive Control with Constraints, Pearson Education Limited 2002, pp. 25-28. 4. Urban Maeder, Francesco Borrelli & Manfred Morari, 2009, Linear Offset-free Model Predictive Control, Automatica, Elsevier Scientific Publishers , 2214-2217. 5. D. Q. Mayne, J. B. Rawlings, C.V. Rao and P.O.M. Scokaert, 2000, Constrained model predictive control:Stability and optimality. Automatica, 36(6):789–814. 6. M. Morari & G. Stephanopoulos, 1980, Minimizing unobservability in inferential control schemes, International Journal of Control, 367-377. 7. K. Muske & T.A. Badgwell, 2002, Disturbance models for offset-free linear model predictive control, Journal of Process Control, 617-632. 8. G. Pannocchia and J. B. Rawlings, 2003, Disturbance models for offset-free model predictive control, AlChE Journal, 426-437. 9. L. Shargel, S. Wu-Pong and A. B. C. Yu, 2005, Applied biopharmaceutics & pharmacokinetics, Fifth Edition, McGraw- Hill Medical Publishing Divison,pp. 717-720.