controlled administration of amiodarone using a fractional-order controller

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:  

(Inf)(t) =1

(n� 1)!

Z t

0(t� ⌧)n�1f(⌧)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� ⌧)↵�1f(⌧)d⌧, t � 0.

We  now  define  the  Caputo  frac1onal-­‐order  deriva1ve  as  follows:  

(D↵f)(t) = Im�↵ dmf(t)

dtm, where m = d↵e

L [D↵f ] (s) = s↵F (s)�m�1X

k=0

s↵�k�1 dkf

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�+Kds

µ

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) =

1k10

⇣1

k21sa + 1

⌘

1k10k21

sa+1 + 1k10

s+ k10+k12k10k21

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 (G

ol

(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.