Controlled drug administration by a fractional PID

Pantelis Sopasakis*,** and Haralambos Sarimveis** * IMT Institute for Advanced Studies Lucca ** School of Chemical Engineering, NTU Athens

Motivation

About: Amiodarone is an antiarrhythmic agent exhibiting highly nonlinear and complex dynamics. Scope: Design a feedback controller to adjust the concentration of amiodarone to desired levels when administered intra-venously.

Pulmonary fibrosis

Skin pigmentation

Side-effects due to accumulation

A. Dokoumetzidis & P. Macheras. IVIVC of controlled release formulations: Physiological-dynamical reasons for their failure. J. Control. Release, 129(2): 76–78, 2008.

The Caputo Fractional Derivative

The Cauchy formula gives the n-th order integral of a function:

(Inf)(t) =1

(n� 1)!

Z t

0(t� ⌧)n�1f(⌧)d⌧

The Caputo Fractional Derivative

which can be generalized for any positive real order α as follows:

(I↵f)(t) =1

�(↵)

Z t

0(t� ⌧)↵�1f(⌧)d⌧

The Cauchy formula gives the n-th order integral of a function:

(Inf)(t) =1

(n� 1)!

Z t

0(t� ⌧)n�1f(⌧)d⌧

by means of which we introduce the Caputo fractional-order derivative:

(D↵f)(t) = Im�↵ dmf(t)

dtm

with m = b↵c

The Caputo Fractional Derivative

which can be generalized for any positive real order α as follows:

(I↵f)(t) =1

�(↵)

Z t

0(t� ⌧)↵�1f(⌧)d⌧

The Cauchy formula gives the n-th order integral of a function:

(Inf)(t) =1

(n� 1)!

Z t

0(t� ⌧)n�1f(⌧)d⌧

Interesting properties The Caputo fractional derivative:

• Extends the standard integer-order derivative to real orders,

• Preserves analiticity & is a linear operator,

• Has the semigroup property and

•  Is a non-local operator i.e., it takes into account the whole history of the function.

R. Hilfer. Applications Of Fractional Calculus In Physics. World Scientific, 2000. ISBN 978-981-02-3457-7.

Interesting properties The Laplace transform of fractional derivatives is remniscient of the one for integer-order derivatives:

L[D↵f ](s) = s↵F (s)�

m�1X

k=0

s↵�k�1f (k)(0)

R. Hilfer. Applications Of Fractional Calculus In Physics. World Scientific, 2000. ISBN 978-981-02-3457-7.

Fractional Dynamical Systems Fractional derivatives and integrals give rise to fractional dynamical systems:

H(D↵1, . . . , D

↵n)x = T(D�1, . . . , D

�m)u

Applying the Laplace transform, l inear fractional systems can be described by a transfer function:

where P and Q are fractional polynomials.

G(s) =X(s)

U(s)=

P (s)

Q(s)

BIBO Stability: The Bode stability c r i te r ion appl ies to f ract ional dynamical systems as is!!!

Fractional PID A fractional PID controller is a fractional system with transfer function:

Gc(s) = Kp +Ki

s�+Kds

µ

Tuning following the method of Valerio and da Costa Minimisation of ITAE:

J?itae = min

Kp,Ki,Kd,�,µJitae

Jitae =

Z 1

0⌧ |✏(⌧)|d⌧

D. Valerio and J.S. da Costa, Tuning of fractional PID controllers with Ziegler-Nichols-type rules, Sign. proc., 86(1), 2006

Fractional PID

Tuning following the method of Valerio and da Costa Minimisation of ITAE:

J?itae = min

Kp,Ki,Kd,�,µJitae

Jitae =

Z 1

0⌧ |✏(⌧)|d⌧

A fractional PID controller is a fractional system with transfer function:

Gc(s) = Kp +Ki

s�+Kds

µ

Subject to additional constraints so that: •  The closed loop is BIBO-stable •  In-loop and external noise is attenuated •  Modelling errors are rejected •  Low frequency output disturbances are dumped •  The gain and phase margins have desired values

D. Valerio and J.S. da Costa, Tuning of fractional PID controllers with Ziegler-Nichols-type rules, Sign. proc., 86(1), 2006

Fractional PID – Tuning

The following constraint entails noise rejection in the closed loop:

The closed-loop transfer function

Some high frequency

Mh = |Gcl(ı!h)| < ⌘

Design parameter

Fractional PID – Tuning

The following constraint entails attenuation of the effect of modelling errors:

The open-loop transfer function

Design parameter

Mz =

�����d

dzarg(G

ol

(◆!))

����!=!

co

����� < ⇣

Fractional PID – Tuning

The following constraint is imposed so that low-frequency disturbances are rejected:

M` = |Gsens(ı!`)| < #

Design parameter The sensitivity transfer function defined as:

Gsens

(s) =1

1 +Gol

(s)

Some low frequency

Simulations Simulations were carried out using the Oustaloup Filter: Fractional-order transfer functions are approximated by rational in a range of frequencies.

A. Oustaloup, F. Levron, B. Mathieu and F.M. Nanot, Frequency-band complex non-integer differentiator: characterization and synthesis, IEEE Transactions on Circuits & Systems I: Fund. Theory and Appl. 47(1): 25-39, 2000.

Amiodarone pharmacokinetics

Amiodarone is known for its complex dynamics, non-exponential PK profiles and singular long-term accumulation patterns.

A. Dokoumetzidis, R. Magin, P. Macheras. Fractional kinetics in multi-compartmental systems. J. Pharmacokin. Pharmacodyn., 37:507–524, 2010.

dA1

dt= �(k12 + k10)A1 + k21 ·D1�aA2 + u,

dA2

dt= k12A1 � k21 ·D1�aA2

Fractional pharmacokinetics:

Amiodarone administration

A fractional PID was tuned with the above method and lead to:

• A phase margin of 98° and a gain margin of 43.9db

•  The closed-loop transfer function gives < -60db at high frequencies (higher than 100rad/day)

•  The sensitivity function gives < -20db at low frequencies (lower than 0.1rad/day)

•  The open-loop function has a low derivative at the crossover frequency (Mz = 0.0087).

Amiodarone administration

0.05 0.1 0.15 0.2 0.25 0.30

0.02

0.04

0.06

0.08

0.1

0.12

Kp=20

Kp=95

Kp=K

popt

Time (days)

Am

ount

A1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

1.5

2

2.5

3

3.5

4

Time (days)

Am

ioda

rone

Adm

inis

trat

ion

Amiodarone administration

�100

�50

0

50

Mag

nitu

de (d

B)

10�2 10�1 100 101 102�225

�180

�135

�90

�45

0

Phas

e (d

eg)

Bode Diagram

Frequency (rad/day)

Almost flat

cro

sso

ver

fre

q.

Amiodarone administration

�80

�60

�40

�20

0

Mag

nitu

de (d

B)

10�2 10�1 100 101 102�225

�180

�135

�90

�45

0

Phas

e (d

eg)

Bode Diagram

Frequency (rad/day)

Very low gain at high

frequencies

Amiodarone administration

−25

−20

−15

−10

−5

0

5

Mag

nitud

e (d

B)

102 103 104 105 106 107−30

0

30

60

Phas

e (d

eg)

Bode Diagram

Frequency (rad/day)

Very low gain at low

frequencies

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.550

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

Time (day)

Amio

daro

ne (n

g)

Effect of modelling uncertainty

Effect of modelling uncertainty: •  Kinetic constants are assumed to

follow the normal distribution with CV 20%

•  The closed-loop looks insensitive to such perturbations

Effect of modelling uncertainty

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.550

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

Time (day)

Amio

daro

ne in

pla

sma

(ng)

The fractional exponent is assumed to follow the normal distribution with CV 20%.

Recapitulation ¡ Amiodarone exhibits fractional pharmacokinetics,

¡ A fractional PID was designed for the feedback control of amiodarone,

¡ The controlled system was stable, resilient to disturbances, able to filter out noise and insensitive to modelling errors.

Thank you for your attention!

This work was partly funded by project 11SYN.10.1152, which was co-financed by the European Union and Greece, Operational Program “Competitiveness & Entrepreneurship”, NSFR 20072013 in the context of GSRT-National action “Cooperation”.