Amiodarone is an antiarrhythmic drug that exhibits highly complex and non- exponential dynamics whose controlled administration has important implications for its clinical use especially for long-term therapies. Its pharmacokinetics has been accurately modelled using a fractional-order compartmental model. In this paper we design a fractional-order PID controller and we evaluate its dynamical characteristics in terms of the stability margins of the closed loop and the ability of the controlled system to attenuate various sources of noise and uncertainty.

1. Controlled
Administra/on
of
Amiodarone
using
a
Frac/onal-­‐Order
Controller
Abstract:
Amiodarone
is
an
an/arrhythmic
drug
that
exhibits
highly
complex
and
non-­‐
exponen/al
dynamics
whose
controlled
administra/on
has
important
implica/ons
for
its
clinical
use
especially
for
long-­‐term
therapies.
Its
pharmacokine/cs
has
been
accurately
modelled
using
a
frac/onal-­‐order
compartmental
model.
In
this
paper
we
design
a
frac/onal-­‐order
PID
controller
and
we
evaluate
its
dynamical
characteris/cs
in
terms
of
the
stability
margins
of
the
closed
loop
and
the
ability
of
the
controlled
system
to
aCenuate
various
sources
of
noise
and
uncertainty.
Frac*onal
Dynamics:
One
of
the
most
exo/c
proper/es
of
non-­‐integer
order
deriva/ves
is
that
they
are
non-­‐local
operators.
They
come
as
generalisa/ons
of
classical
operators.
For
instance,
using
the
Cauchy
formula
for
the
definite
integral
operator:
(In
f)(t) =
1
(n 1)!
Z t
0
(t ⌧)n 1
f(⌧)d⌧, t 0.
P.
Sopasakis1
&
H.
Sarimveis2
1
IMT
Ins/tute
for
Advanced
Studies
Lucca,
Piazza
San
Ponziano
6,
Lucca
55100,
Italy
(Tel:
+39
0583
4326
710;
e-­‐mail:
pantelis.sopasakis@imtlucca.it).
2
School
of
Chemical
Engineering,
Na/onal
Technical
University
of
Athens,
9
Heroon
Polytechneiou
Street,
15780
Zografou
Campus,
Athens,
Greece
(Tel:
+30
210
7723237,
e-­‐mail:
hsarimv@central.ntua.gr)
Using
the
fact
that
the
Gamma
func/on
intercepts
the
factorial
on
the
set
of
natural
numbers,
we
extend
the
above
integral
to
introduce
the
Riemann-­‐Liouville
frac1onal-­‐order
integral:
(I↵
f)(t) =
1
(↵)
Z t
0
(t ⌧)↵ 1
f(⌧)d⌧, t 0.
We
now
define
the
Caputo
frac1onal-­‐order
deriva1ve
as
follows:
(D↵
f)(t) = Im ↵ dm
f(t)
dtm
, where m = d↵e
L [D↵
f] (s) = s↵
F(s)
m 1X
k=0
s↵ k 1 dk
f
dtk
0
,
where F(s) = (Lf)(s)
It
is
of
fundamental
importance
that
it
is
possible
to
have
an
analy/cal
expresion
for
the
Laplace
transforma/on
of
the
Caputo
frac/onal-­‐
order
deriva/ve:
This
enables
us
to
represent
frac/onal-­‐order
dynamical
systems
in
the
Laplace
domain
using
transfer
func/ons
and
design
controllers
using
frequen/st
criteria
(such
as
the
Bode
stability
criterion).
In
this
study
we
consider
the
compartmental
pharmacokine/c
model
for
the
distribu/on
of
Amiodarone,
an
an/arrhythmic
agent.
The
compartmental
topology
is
shown
in
the
figure
below
[1]:
We
consider
that
Amiodarone
is
administered
to
the
pa/ent
intravenously
and
con/nuously,
the
controller
has
access
to
plasma
measurements
of
the
concentra/on
of
Amiodarone
and
that
the
administra/on
rate
can
be
adjusted
in
real
/me
by
the
controller.
We
use
a
frac/onal-­‐PID
feedback
controller
to
control
the
concentra/on
of
Amiodarone
in
the
pa/ent’s
plasma.
The
trea/ng
doctor
can
modify
the
set
point
in
real
/me
to
achieve
the
desired
therapeu/c
effect.
The
controller’s
dynamics
is
given
by
the
following
transfer
func/on:
Gc(s) = Kp +
Ki
s
+ Kdsµ
Jitae =
Z 1
0
⌧✏(⌧)d⌧
In
order
to
tune
the
controller
we
selected
those
parameters
that
minimise
the
Integral
Time
Absolute
Error
(ITAE)
index
following
the
excita/on
of
the
closed-­‐loop
system
with
a
step
pulse.
References
[1]
A.
Dokoumetzidis,
R.
Magin,
and
P.
Macheras.
Frac/onal
kine/cs
in
mul/-­‐compartmental
systems.
Journal
of
Pharmacokine/cs
and
Pharmacodynamics,
37:507–524,
2010a.
G(s) =
1
k10
⇣
1
k21
sa
+ 1
⌘
1
k10k21
sa+1 + 1
k10
s + k10+k12
k10k21
sa + 1
✏ysp
y
The
op/mal
tuning
parameters
are
given
in
the
table
below.
The
phase
margin
of
the
system
was
found
to
be
98deg
and
its
gain
margin
is
43.9db!
The
closed-­‐loop
is
therefore
stable
and
can
aCenuate
delays
as
high
as
3.3
days.
In
the
figure
below
we
see
how
the
system
responds
to
a
change
of
its
set-­‐point.
Tuning
Parameter
Value
Kp
50.52
Ki
151.05
Kd
0.0756
λ
0.917
μ
0.759
n! = (n + 1),
8n 2 N
The
controller
needs
to
compensate
parametric
uncertain/es
and
fluctua/ons
and
modelling
errors
or
/me-­‐varying
dynamics.
A
measure
for
the
resilience
of
the
closed-­‐loop
under
such
uncertain
condi/ons
is
quan/fied
by
the
slope
of
the
argument
of
the
open-­‐loop
func/on
at
the
cross-­‐over
frequency
of
the
system,
i.e.,
Mz =
d
d!
arg (Gol(ı!))
!=!co
= 0.5deg · rad 1
· day
Stability
Margin
Value
Phase
Margin
98deg
Gain
Margin
43.9db
The
gain
of
the
closed-­‐loop
transfer
func/on
at
high
frequencies
is
less
than
-­‐60db
which
suggests
that
the
controller
can
reject
high-­‐frequency
noise
in
the
closed
loop
and
noise
that
accompanies
the
set-­‐point.