Physiologically Based Modelling and Predictive Control

P . S o p a s a k i s , P . P a t r i n o s , S . G i a n n i k o u , H . S a r i m v e i s .

P r e s e n t e d i n t h e 2 1 E u r o p e a n S y m p o s i u m o n C o m p u t e r - A i d e d P r o c e s s E n g i n e e r i n g

Physiologically Based Pharmacokinetic Modeling and

Predictive ControlAn integrated approach for optimal drug administration

Drug administration strategies

Open loop drug administration based on

average population pharmacokinetic studies

Evaluation:• No feedback • Suboptimal drug administration• The therapy is not individualized• High probability for side effects!

Toxicity Alert!

W. E. Stumpf, 2006, The dose makes the medicine, Drug Disc. Today, 11 (11,12), 550-555


Patients for which

the therapy works

beneficially

Patients prone to

side-effects


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


Computed-Aided scheduling of drug

administration

Evaluation: • Optimal drug administration • Constraints are taken into account• Systematic/Integrated approach• Individualized therapy

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.

Problem Formulation

Problem: Control the concentration of DMA in thekidneys of mice (set point: 0.5μg/lt) while the i.v. influxrate does not exceed 0.2μg/hr and the concentration in theliver does not exceed 1.4μg/lt.

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.

Tools employed: MPC

J.M. Maciejowski, 2002 , Predictive Control with Constraints, Pearson Education Limited, 25-28.

Why Model Predictive Control ?

• Stability & Robustness

• Optimal control strategy

• System constraints are systema-

tically taken into account

Step 1 : Modeling

, , ,

, , ( )

plasma

plasma skin v skin lung v lung kidney v kidney

blood v blood residual v residual RBC RBC plasma plasma C plasma

dCV Q C Q C Q C

dt

Q C Q C u C C Q C

Mass balance eq. in the plasma compartment:

RBCplasma plasma plasma RBC RBC

dCV C C

dt

Mass balance in the RBC compartment:

And for the kidney compartments :

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 .

, ,

kidney kidney

kidney kidney Arterial v kidney kidney v kidney kidney kidney

kidney

dC CV Q C C C k A

dt P

,

,

v kidney kidney

kidney kidney v kidney

kidney

dC CV C

dt P

Step 2 : Model Discretization

( 1) ( ( ), ( ))

( ) ( ( ))

( ) ( )

m m

m m

m

t f t t

t g t

t t

x x u

y x

z Hy

m t t Ex Lu M

Discretized PBPK model:

Subject to :

( 1) ( ) ( )

( ) ( )

t t t

t t

x Ax Bu

y Cx

Linearization

Step 3 : Observer Design

G. Pannocchia and J. B. Rawlings, 2003, Disturbance models for offset-free model predictive control, AlChE Journal, 426-437.

Augmented system:

( 1) ( ) ( ) ( )

( 1) ( )

( ) ( ) ( )

d

d

t t t t

t t

t t t

x Ax Bu B d

d d

y Cx C d ( 1) ( ) ( )

( ) ( )

t t t

t t

x Ax Bu

y Cx

( 1) ( ( ), ( ))

( ) ( ( ))

( ) ( )

m m

m m

m

t f t t

t g t

t t

x x u

y x

z Hy

Step 3 : Observer Design (cont’d)

G. Pannocchia and J. B. Rawlings, 2003, Disturbance models for offset-free model predictive control, AlChE Journal, 426-437.

Augmented system:

( 1) ( ) ( ) ( )

( 1) ( )

( ) ( ) ( )

d

d

t t t t

t t

t t t

x Ax Bu B d

d d

y Cx C d ( 1) ( ) ( )

( ) ( )

t t t

t t

x Ax Bu

y Cx

( 1) ( ( ), ( ))

( ) ( ( ))

( ) ( )

m m

m m

m

t f t t

t g t

t t

x x u

y x

z Hy

This system is observable iff (C, A) isobservable and the matrix

is non-singular

d

d

A I B

C C

Step 3 : Observer Design (cont’d)

K. Muske & T.A. Badgwell, 2002, Disturbance models for offset-free linear model predictive control, Journal of Process Control, 617-632.

Augmented system:

( 1) ( ) ( ) ( )

( 1) ( )

( ) ( ) ( )

d

d

t t t t

t t

t t t

x Ax Bu B d

d d

y Cx C d ( 1) ( ) ( )

( ) ( )

t t t

t t

x Ax Bu

y Cx

( 1) ( ( ), ( ))

( ) ( ( ))

( ) ( )

m m

m m

m

t f t t

t g t

t t

x x u

y x

z Hy

ˆ ˆ( 1) ( )

ˆ ˆ( 1) (ˆˆ( ) ( ( )

)) ( )

xd

m d

d

t t t tt t

t t

LA B Bu y Cx C d

L0 I

x

d d 0

x

Observer dynamics:

Step 4 : MPC design

U. Maeder, F. Borrelli & M. Morari, 2009, Linear Offset-free Model Predictive Control, Automatica, Elsevier Scientific Publishers , 2214-2217.

ˆˆ

ˆ

d

d

B dxA - I B

uHC 0 r HC d

Maeder et al. have shown that:

12 2 2

(0),..., ( 1)0

min ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) , 0...,

( 1) ( ) ( ) ( ), 0,...,

( 1) ( ), 0,...,

ˆ(0) ( )

ˆ(0) ( )

N

Nk

d

N t k t k t

k k k N

k k k k k N

k k k N

t

t

P Q Ru ux x x x u u

Ex Lu M

x Ax Bu B d

d d

x x

d d

The MPC problem is formulated as follows:

Step 4 : MPC design


ˆˆ

ˆ

d

d

B dxA - I B

uHC 0 r HC d


Deviation from the set-point

Terminal Cost

Model

12 2 2

(0),..., ( 1)0

min ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) , 0...,

( 1) ( ) ( ) ( ), 0,...,

( 1) ( ), 0,...,

ˆ(0) ( )

ˆ(0) ( )

N

Nk

d

N t k t k t

k k k N

k k k k k N

k k k N

t

t

P Q Ru ux x x x u u

Ex Lu M

x Ax Bu B d

d d

x x

d d


Constraints

Step 4 : MPC design


ˆˆ

ˆ

d

d

B dxA - I B

uHC 0 r HC d


12 2 2

(0),..., ( 1)0

min ( ) ( ) ( ) ( )

( ) ( ) , 0...,

( 1) ( ) ( ) ( ), 0,...,

( 1) ( ), 0,...,

(

ˆ(0) ( )

)

(0 ( )

)

ˆ)

(N

Nk

d

t tN t k k

k k k N

k k k k k N

k k k N

t

t

P Q Ru ux x x u

Ex Lu M

x Ax Bu B d

d

d

x u

d

x x

d


ˆ ( )

ˆ(( ) ) (

( )

)

d

d

tt

t t t

B dA - I B

HC 0 HC du

x

rWhere:

Step 4 : MPC design


Where:

1( )( ) ( )T T T T P A PA A PB B PB R B PA Q

ˆ ( )

ˆ(( ) ) (

( )

)

d

d

tt

t t t

B dA - I B

HC 0 HC du

x

r

P is given by a Riccati-type equation:

12 2 2

(0),..., ( 1)0

min ( ) ( ) ( ) ( )

( ) ( ) , 0...,

( 1) ( ) ( ) ( ), 0,...,

( 1) ( ), 0,...,

(

ˆ(0) ( )

)

(0 ( )

)

ˆ)

(N

Nk

d

t tN t k k

k k k N

k k k k k N

k k k N

t

t

P Q Ru ux x x u

Ex Lu M

x Ax Bu B d

d

d

x u

d

x x

d

Overview

ObserverplC

Model Predictive Controller

Estimated states

Measured Plasma Concentration

Therapy

( )

( )

tt

t

xr

u

tr

Overview

ObserverplC

Model Predictive Controller

Measured Plasma Concentration

Therapy

( )

( )

tt

t

xr

u

tr

//

/ /

ˆ ˆ ˆ ˆˆ ˆˆ

ˆ ˆ ˆ ˆ

skin skin bl

skin s

lung lung bl

lung lung bl kin blood ood

C C C C

d d d d

x C dReconstructed state vector :

Estimated states

Results: Assumptions

Assumptions: Intravenous administration of DMA tomice with constant infusion rate (0.012lt/hr). PredictionHorizon was fixed to N=10 and the set point was set to0.5μg/lt in the kidney.

Additional Restrictions: The i.v. rate does not exceed0.2μg/hr and the concentration in the liver remains below1.4 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 .

Results: Simulations without constraints

Constraints are violated

Results: Simulations

The constraint is active

Requirements are fulfiled

Stability is guaranteed &

set-point is reached

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.

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.