On the multiple model adaptive control of pharmacokinetic systems

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Content ON THE MULTIPLE MODEL ADAPTIVE CONTROL
OF PHARMACOKINETIC SYSTEMS
by
David W a lte r l i e s
A T h e s is P r e s e n te d t o th e
FACULTY OF THE GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In P a r t i a l F u l f i l l m e n t of th e
R e q u irem en ts f o r th e Degree
MASTER OF SCIENCE
(A pplied M athem atics)
J u ly 1984
UMl Number: EP54418
All rights reserved
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UMI EP54418
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UNIVERSITY O F S O U T H E R N CALIFO RNIA
TH E GRADUATE SCHOO L
UNIVERSITY PARK
LO S A N G ELES. CA LIFO R N IA 9 0 0 0 7
This thesis, written by
Dayid W a lte r l i e s
under the direction of h3.?...Thesis Committee,
and approved by all its members, has been pre
sented to and ajccepted by the Dean of The
Graduate School, in partial fulfillment of the
requirements for the degree of
Master...of ..Science
Dean
D ate.....
THESIS COMMITTEE
Chairman
UtllVERSITY OF SOUTHERN CALIFORNIA
DEDICATION
To my parents
11
ACKNOWLEDGEMENTS
I w ould l i k e to thank P r o f e s s o r Alan S chum itzky f o r
a l l th e guidance^ en couragem ent and g e n e r a l s u p p o rt he has
g iv e n me in th e l a s t two y e a r s .
My th a n k s t o P r o f e s s o r F ra n ce sc o P a r i s i - P r e s i c c e and
P r o f e s s o r David D'Argenio f o r t h e i r h e lp a s members of my
t h e s i s g u id an c e com m ittee^ and to Dr. Roger J e l l i f f e f o r
h is s u p p o r t d u rin g th e f i n a l s t a g e s of t h i s work.
I would a l s o l i k e to th an k th e M a th em atics D epartm ent
of th e U n i v e r s i t y of S o u th e rn C a l i f o r n i a f o r t h e i r
f i n a n c i a l s u p p o rt over th e p a s t two y e a r s .
F i n a l l y , t h i s r e s e a r c h was s u p p o rte d in p a r t by th e
N a tio n a l I n s t i t u t e s of H e a lth under G ra n t RR01629.
I l l
TABLE OF CONTENTS
Page
DEDICATION Ü
ACKNOWLEDG EMENTS i i i
LIST OF FIGURES V
C hapter
1 INTRODUCTION 1
2 BACKGROUND 3
2 .1 The o p tim a l c o n t r o l l e r 3
2 .2 The l i n e a r q u a d r a t i c g a u s s ia n r e g u l a t o r 5
problem
2 .3 The t r a c k i n g problem 9
3 THE MULTIPLE MODEL ADAPTIVE CONTROL 10
ALGORITHM
4 APPLICATIONS TO PHARMACOKINETIC SYSTEMS 13
4 .1 The one com partm ent model 13
4 .2 Example 1 16
4 .3 The two com partm ent model 18
4 .4 Example 2 21
5 SIMULATIONS OF A PHARMACOKINETIC SYSTEM 24
5 .1 C o n tro l of th e one compartm ent model 24
6 SUMMARY 29
REFERENCES 31
IV
LIST DE FIGURES
F ig u re Page
4 .1 .1 The one com partm ent model 13
4 .3 .1 The two com partm ent model 18
5 .1 .1 R a tio of d a i l y c o s t f o r th e M M AC c o n t r o l l e r 27
t o d a i l y c o s t f o r th e o p tim a l c o n t r o l l e r
5*1.2 R a tio of d a i l y c o s t f o r th e open lo o p 27
c o n t r o l l e r to d a i l y c o s t f o r th e o p tim a l
c o n t r o l l e r
5*1.3 T o ta l c o s t s f o r th e MMAC, open lo o p and 28
o p tim a l c o n t r o l l e r s
V
CHAPTER 1
INTRODUCTION
T h is t h e s i s ex am in es t h e b e h a v io u r of an a d a p t iv e con
t r o l a lg o r i th m , d e s c r i b e d by Deshpande, Upadhyay and
L a i n i o t i s [1 ], when a p p l i e d t o a s i m u la t e d p h a r m a c o k in e tic
system .
An a d a p t iv e c o n t r o l a lg o r ith m i s an a lg o r i th m to f i n d
c o n t r o l s f o r a sy ste m w ith some unknown or ch an g in g p a r
a m e te rs . I t u se s m ea su rem en ts of th e s y s te m 's re s p o n se to
im prove i t s knowledge of th e sy ste m , e n a b lin g 'b e tte r *
c o n t r o l s t o be a p p l i e d in t h e f u t u r e .
I n t h i s t h e s i s we o n l y lo o k a t t h e a d a p t i v e c o n t r o l o f
l i n e a r sy s te m s w here some p a r a m e te r s of th e system a re
unknown.
An o u t l i n e of t h i s t h e s i s i s now g iv e n .
( i) In c h a p t e r tw o , we p r e s e n t an o u t l i n e o f
th e v a r i o u s c l a s s e s of c o n t r o l a lg o r i th m s ,
and a rev ie w of th e L in e a r Q u a d r a tic
G a u ssia n (LQG) r e g u l a t o r c o n t r o l problem
and i t s a d a p tio n to a t r a c k i n g problem .
( i i ) In c h a p t e r t h r e e , we p r e s e n t a r e v i e w o f
th e m u l t i p l e model a d a p t iv e c o n t r o l (MMAC)
a lg o r i th m .
( i i i ) In c h a p t e r f o u r , we p r e s e n t a r e v i e w o f t h e
one and two com partm ent p h a rm a c o k in e tic
m odels and t h e c o rre s p o n d in g d i s c r e t e
system dynam ics. C h a p te r fo u r a ls o shows
how th e MMAC a lg o r i th m can be a p p l i e d t o
such sy ste m s.
(iv) In c h a p te r f i v e , we p r e s e n t th e r e s u l t s of
s e v e r a l Monte C a rlo s i m u l a t i o n s u sin g th e
MMAC, open lo o p and o p tim a l c o n t r o l l e r s on
th e one co m p artm en t model.
(v) I n c h a p t e r s i x , we p r e s e n t a sum m ary of t h e
r e s u l t s and reco m m en d atio n s f o r f u r t h e r
work.
CHAPTER 2
BACKGROUND
2 .1 The O ptim al C o n t r o l l e r
A lg o rith m s f o r f i n d i n g an o p tim a l c o n t r o l l e r , when th e
system has known l i n e a r dynam ics, a re w e l l known. However,
even when t h e system can be m o d elled u s in g l i n e a r dynam i
c a l e q u a t io n s , th e p a ra m e te r v a lu e s a re o f t e n no t known.
In t h i s c a se one m ust n o t only c o n t r o l th e sy ste m , b u t, in
o rd e r to c o n t r o l i t more a c c u r a t e l y in th e f u t u r e , one
m ust a ls o e s t i m a t e th e unknown p a r a m e te r s .
The a p p l i e d c o n t r o l s n o t on ly c o n t r o l th e system
to w a rd s some o b j e c t i v e b u t, from th e s y s te m 's re s p o n s e ,
p ro v id e i n f o r m a ti o n f o r e s t i m a t i n g th e s y s te m 's
p a r a m e te r s .
An o p tim a l s t r a t e g y may be to i n i t i a l l y c o n t r o l th e
sy stem w ith an em p h a sis on i d e n t i f y i n g i t s p a r a m e te r s
r a t h e r th a n on a c h ie v in g some p a r t i c u l a r s t a t e . The e x t r a
c o s t s i n c u r r e d i n i t i a l l y w i l l be o f f s e t by lo w er c o s t s a t
l a t e r s t a g e s , b e ca u se u s in g b e t t e r e s t i m a t e s of th e system
p a r a m e t e r s , ' b e t t e r ' c o n t r o l s can be a p p lie d .
In g e n e r a l t h e r e a r e t h r e e c l a s s e s of c o n t r o l
a lg o r ith m ;
(i) The open lo o p c l a s s . In t h i s c ase no m easurem ent
i n f o r m a ti o n i s a v a i l a b l e . The c o n t r o l i s b ased
on an i n i t i a l e s t i m a t e of th e system p a ra m e te r s ,
( i i ) The fee d b ac k c l a s s . In t h i s c a se th e m ea su re
m ents of th e s y s te m 's re s p o n se to d a te a r e
a v a i l a b l e t o th e c o n t r o l l e r , so e s t i m a t e s of th e
sy stem s t a t e and p a r a m e t e r s may be u p d ated . The
c o n t r o l c a l c u l a t e d a t each s ta g e does n o t assum e
t h a t more m ea su rem en ts w i l l be a v a i l a b l e a t a
l a t e r s ta g e .
( i i i ) The c lo s e d lo o p c l a s s . In t h i s c ase n o t o n ly a re
m ea su re m e n ts a v a i l a b l e t o th e c o n t r o l l e r , b u t
th e c o n t r o l l e r can use th e f a c t t h a t f u t u r e
m ea su re m e n ts w i l l be a v a i l a b l e . Thus th e con
t r o l l e r can a f f o r d t o d i s t u r b th e system in
i t i a l l y , p o s s i b l y i n c r e a s i n g s h o r t term c o s t s ,
t o o b t a i n more knowledge of th e s y s te m 's r e
sp o n se. The r e s u l t i n g d i s t u r b a n c e s w i l l be
m easured and may e n a b le b e t t e r e s t i m a t e s t o be
made of th e sy stem p a ra m e te r s , th e re b y re d u c in g
lo n g te rm c o s t s .
In g e n e r a l th e o p tim a l c o n t r o l l e r i s in th e c lo s e d
lo o p c l a s s . However t h e r e a re some s t o c h a s t i c c o n t r o l
p ro b le m s f o r w hich th e o p tim a l c o n t r o l l e r i s a fee d b ac k
c o n t r o l l e r . In th e s e p ro b le m s, i n f o r m a ti o n a b o u t th e f u
t u r e sa m p lin g p la n has no e f f e c t on t h e o p tim a l c o n t r o l .
T h is i s th e c a s e f o r sy s te m s w ith th e c e r t a i n t y e q u iv a l
ence p r o p e r t y .
A system has t h e c e r t a i n t y e q u iv a le n c e p r o p e r t y i f th e
o p tim a l c o n t r o l i s th e same a s th e o p tim a l c o n t r o l f o r th e
d e t e r m i n i s t i c sy stem w here a l l th e unknown p a r a m e te r s have
been r e p la c e d by t h e i r e x p e c te d v a lu e s . Thus th e p ro b lem s
of e s t i m a t i o n and c o n t r o l a r e s e p a r a t e . One problem w hich
h as th e c e r t a i n t y e q u iv a le n c e p r o p e r t y i s t h e LQG (L in e ar
Q u a d r a tic G aussian) p ro blem .
2 .2 The L in e a r Q u a d r a tic G a u ssia n R eggie,tor Problem
T h is i s t h e problem of c o n t r o l l i n g a l i n e a r system
w i t h q u a d r a t i c c o s t and G a u ssia n n o is e .
The s t a t e e q u a tio n s a r e g iv e n by :
x (k + l) = (D(k+l,k) x(k) + G(k) u(k) + w (k ), k = 0 , l , . . . , N
where
x(k) i s t h e n d im e n sio n a l s t a t e v e c t o r ,
u(k) i s th e r d im e n sio n a l c o n t r o l v e c t o r ,
$ ( k + l,k ) i s t h e n x n s t a t e t r a n s i t i o n m a t r i x ,
G(k) i s th e n x r d im e n s io n a l c o n t r o l m a t r i x ,
w(k) i s a z e ro mean n d im e n s io n a l G au sian w h ite
n o i s e v e c t o r .
and
and
E(w(k) w( j) *)=Q(k)gj^j.
The o b s e r v a t i o n e q u a t io n s a re g iv en by ;
y (k ) = H(k) x(k) + v ( k ) , k = l , 2 , . . . , N
where
y(k) i s th e m d im e n sio n a l o b s e r v a t i o n v e c t o r ,
v(k) i s a m d im e n s io n a l z e ro mean G au sian w h ite
n o is e v e c t o r ,
H(k) i s t h e m X n d im e n sio n a l o b s e r v a t i o n m a t r ix ,
E (v(k) v( j) *) = R(k)Sk j .
The c o s t i s g iv e n by
x"^ (N) C (N) X (N) + ^ ( (x^ (k) A(k) X (k)+u"^(k)B(k)u (k) )
k = - 1
F i n a l l y , th e m a t r i c e s A, B, C, Q and R a re a l l assum ed to
be p o s i t i v e s e m i d e f i n i t e .
The o p tim a l c o n t r o l , i.e . th e c o n t r o l w hich m in im is e s
th e e x p e c te d v a lu e of th e c o s t , can be computed e x p l i c i t l y
when H (k ), G(k) and 0 ( k + l , k ) a r e known.
T his system h as th e c e r t a i n t y e q u iv a le n c e p r o p e r t y
[ 4 ,5 ] . The o p tim a l c o n t r o l i s th e same a s th e o p tim a l
c o n t r o l f o r th e d e t e r m i n i s t i c system u s in g th e b e s t e s t i
m ate of th e s t a t e . The a l g o r i t h m s f o r th e o p tim a l e s t i m a t e
and f o r th e o p tim a l d e t e r m i n i s t i c c o n t r o l a re now
The o p tim a l e s t i m a t i o n problem fo r a l i n e a r sy stem
w i t h l i n e a r dynam ics, and q u a d r a t i c c o s t s was so lv e d in
1960 i n a p a p e r by R. E. K alm an [ 3 ] . T h is r e s u l t h a s
become known a s th e Kalman f i l t e r .
The l i n e a r system i s d e s c r i b e d a s b e f o r e :
x (k + l) = 0 ( k + l ,k ) x(k) + G(k) u(k) + w(k)
y (k) = H(k) X (k) + v(k)
The Kalman f i l t e r e q u a t io n s a re :
x (k + lIk + 1 ) = x (k + lIk )+ F (k + 1 )[y (k + 1 )-H (k + 1 )x (k + 1 1 k )]
x ( k + l l k ) = 0 ( k + l ,k ) x ( k |k ) + G ( k ) u ( k )
F ( k + l) = P ( k + l|k ) H ^ ( k + l) [ H(k+1) P(k+11 k) (k+1)+R(k+1) ] "^
P(k+ l|k)=(|)(k+l,k)P(k|k)(l>''’(k+ l,k)+ Q (k )
P ( k + l|k + l)= [ I - P ( k + 1 ) H ( k + 1 ) ] P ( k + lI k )
where
P (0 |0 ) and x(0) a r e g iv e n .
The o p tim a l c o n t r o l l e r f o r a d e t e r m i n i s t i c l i n e a r
sy stem w ith q u a d r a t i c c o s t i s :
u (k) =K (k)X (k)
where
K (k)= -B "^(k)G ^(k) [S ~ ^(k + l)+ G (k)B "^(k)G ’’( k ) ] " l 0 ( k + l , k )
and
S(k)=(p’’(k + l,k ) [S " ^ (k + l)+ G (k )B " l(k )G ^ (k )] ~^(p(k+l,k)+A(k)
where
S(N) =C(N) .
These fo rm u la e a r e n o t v a l i d when B i s s i n g u l a r , e.g.
when t h e r e i s no c o n t r o l c o s t. However th e y can be r e
w r i t t e n i n a fo rm w h ic h i s v a l i d when B i s s i n g u l a r u s i n g
th e f o l lo w in g two m a t r i x i d e n t i t i e s :
(i) A(I+BA)"^ = (I+AB)“ ^A
( i i ) (A“ ^ + BC“ ^ B ^ )“ "^ = A - AB(b '^AB + C )“ ^B^A
A pplying (i) t o t h e e q u a tio n f o r K(k) we g e t :
K(k) =-B“ l(k)G '^(k) [S“ ^(k + l)+ G (k )B “ ‘^(k)G'^(k) ] “ ^(D(k+l,k)
=-B“ ^ (k )G ^ (k )[I+ S (k + l)G (k )B “ l( k ) G ^ ( k ) ]
S (k+1) (J ) (k+1, k)
= -B " l(k ) [I+G'’’( k ) S ( k + l)G (k ) B " l( k )] " ^ G ’^(k)
S(k+l)(D (k+l,k)
= - [B+G'^(k) S(k+1) G(k) ] ~^G'‘’(k) S(k+1) d) (k + l,k )
which i s v a l i d f o r s i n g u l a r B.
A pplying ( i i ) t o th e e q u a tio n fo r S(k) we g e t :
S(k)=(D^(k+l,k) [ S - l( k + l) + G (k ) B -l( k )G '^ ( k )] - ^ $ (k + l,k ) + A ( k )
=(t)'*^(k+l,k) IS (k + l) -S (k + l)G (k ) (G^(k) S(k+1) G(k)+B(k) ) “^
G ^ (k )S (k + 1 )]$ (k + l,k )+ A (k )
w hich i s a ls o v a l i d f o r s i n g u l a r B.
2 .3 The T ra c k in g Problem
The l i n e a r q u a d r a t i c g a u s s ia n (LQG) s o l u t i o n m in im iz e s
a q u a d r a t i c c o s t . Thus t h e aim i s t o m ake a l l e l e m e n t s of
th e s t a t e z e ro w ith th e w e ig h tin g m a t r i c e s A and C d e t e r
m in in g t h e r e l a t i v e im p o rta n c e of th e d i f f e r e n t com ponents
o f t h e c o s t .
A t r a c k i n g prob lem i s one w here th e d e s i r e d s t a t e i s
n o n -z e ro and may change from tim e t o tim e . T his becom es an
LQG problem i f th e s t a t e i s augm ented by a c o n s t a n t s t a t e .
Thus :
x^(k) = (x^(k) I 1)
and
and
( A(k) I -A(k) x_(k) )
(-------- I------T----------- — ------)
( -%o I Xo^(k) A(k) Xq )
( C(N) I -C(N) Xq (N) )
Ci(N) = (------ I------------------------------ )
( -Xq^ C(N) I XqT (n) C(N) Xq )
w here th e Xg (k) a re th e t a r g e t s t a t e s a t each s t a g e and
th e A(k)*s and C(N) a r e v /e ig h tin g m a t r i c e s g iv in g th e
r e l a t i v e im p o rta n c e of th e s t a t e s .
Note t h a t
(x'^(k) 11) A^ (k) (x^(k) 11)= ( x ( k ) - x ^ ( k ) )"^A(k) (x ( k )-x ^ (k ) )
so t h e new c o s t i s a q u a d r a t i c f u n c t i o n of t h e d i f f e r e n c e
betw een th e a c t u a l s t a t e and t h e d e s i r e d s t a t e .
CHAPTER 1
THE MULTIPLE MODEL ADAPTIVE CONTROL ALGORITHM
A lthough a sy stem may be a d e q u a te ly m o d elled u s in g
l i n e a r dyn am ics, th e m a t r i c e s G(k), H(k), 0 ( k + l|k ) a re
f r e q u e n t l y n o t known in advance. The o p tim a l c o n t r o l fo r
t h i s c a s e i s n o t known.
A s u b o p tim a l s o l u t i o n t o t h i s problem was p r e s e n t e d by
D eshpande, Upadhyay and L a i n i o t i s [1 ]. As im p lem en ted f o r
th e p h a rm a c o k in e tic sy stem i t u se s a number of p o s s i b l e
m odels of th e sy ste m and com bines th e o p tim a l c o n t r o l s f o r
each model u s in g th e a - p o s t e r i o r i p r o b a b i l i t i e s to c a l c u
l a t e th e c o n t r o l . In t h i s t h e s i s , t h i s c o n t r o l a lg o r i th m i s
r e f e r r e d to a s t h e m u l t i p l e model a d a p t iv e c o n t r o l (MMAC)
a lg o r ith m .
The MMAC a l g o r i t h m i s a s u b o p tim a l c o n t r o l l e r of th e
feed b ack c l a s s , i .e . a t each s t a g e i t u se s th e m easu re
m ents of th e s y s te m 's re s p o n se up to t h e p r e s e n t s ta g e in
c a l c u l a t i n g th e c o n t r o l to be a p p l i e d t o th e sy ste m , b u t
i t does n o t assum e t h a t more m ea su rem en ts w i l l be a v a i l
a b le in th e f u t u r e .
10
I t i s assum ed t h a t t h e m u l t i p l e m odels a r e p a r a m e te r
iz e d by a f i n i t e d im e n s io n a l v e c t o r Ô. The a l g o r i th m
c o n s i d e r s a l l p o s s i b l e w here i = l to M, and
a s s i g n s t o each one an a - p r i o r i p r o b a b i l i t y .
At each s ta g e :
(i) t h e o p tim a l s t a t e e s t i m a t e g iv e n each ©j^,
x ( k |k - l f © i ) , i s c a l c u l a t e d u sin g th e Kalman
f i l t e r f o r th e l i n e a r model c o rre s p o n d in g
t o 0
( i i ) t h e c o n d i t i o n a l p r o b a b i l i t i e s of th e ©j^,
g iv e n th e o b s e r v a t i o n s , yj^, a re u p d a te d .
The p r o b a b i l i t i e s , P(G ily^) a r e g iv e n by
P(© ilyic)=L (k|e i)p(eilY k_i)/ ^ L(k | 8 j ) p ( 8 j ly k _ i)
J=!
and
L ( k | 8 i ) = l P y ( k | k - l , e ) r ^ / 2
where
and
e x p (1/2 I|y (k Ik -1 ,8^) I Ip ( k | k - l , 8 ) )
L ( 0 |8 i)= l
ÿ ( k | k - l , 8 i ) = y ( k ) - H ( k , 8 i ) x ( k | k - l , e £ )
Py(k | k - l , 8 i ) = H ( k , e ^ ) P ( k | k - l , 8 i ) (k , 8 ^ ) +R(k)
( i i i ) t h e o p tim a l c o n t r o l , u(k |k,©j^), g iv e n each
of th e m odels i s c a l c u l a t e d . The o p tim a l
c o n t r o l fo r each model i s j u s t th e s o l u t i o n
of a LQG problem d e s c r i b e d e a r l i e r . Each
11
model i s a l i n e a r d y n am ical system w ith
G a u ssia n n o is e and q u a d r a t i c c o s t. The
c o n t r o l a c t u a l l y a p p li e d t o t h e system i s
th e a v e ra g e of th e c o n t r o l s f o r a l l th e
m odels w e ig h te d by th e p r o b a b i l i t y of each
m odel. T hat i s , th e c o n t r o l to a p p ly to th e
s y s te m i s t h e n g i v e n by
%
u ( k ) = ^ u(k |k , 9 i ) p(©ily]ç)
/ =/
w h e r e yj^ i s t h e s e t o f o b s e r v a t i o n s up t o
s t a g e k.
A lthough l a r g e num bers of m odels may be needed, e s
p e c i a l l y when d i s c r e t i z i n g a m u lt i d im e n s i o n a l c o n tin u o u s
sam ple sp a c e , t h e a l g o r i th m i s i d e a l fo r p a r a l l e l p r o
c e s s i n g , s i n c e th e MMAC c o n t r o l l e r i s j u s t a number of
l i n e a r c o n t r o l l e r s and e s t i m a t o r s ru n n in g in p a r a l l e l .
The a lg o r ith m i s e a s i l y im p lem en ted to s u i t many
l i n e a r sy s te m s w ith d i f f e r e n t d im e n s io n s , unknown p a r a
m e te r s e tc . For t h i s t h e s i s th e a l g o r i th m was im plem ented
in a form w hich co u ld h a n d le , s u b j e c t to s t o r a g e r e q u i r e
m en ts, any l i n e a r sy ste m w ith a r b i t a r y s t a t e t r a n s i t i o n ,
o b s e r v a t i o n and c o n t r o l m a t r i c e s and a r b i t a r y unknown
p a r a m e te r s in t h e s e m a t r i c e s .
12
CHAPTER i
APPLICATIONS TO PHARMACOKINETIC SYSTEMS
F r e q u e n tly used m od els of p h a rm a c o k in e tic s y s te m s have
o n ly a few unknown p a r a m e te r s . Thus, th e MMAC c o n t r o l l e r
i s c o m p u t a t i o n a l l y f e a s i b l e f o r th e s e sy ste m s.
The im p le m e n ta tio n of th e c o n t r o l l e r i s d e s c r i b e d f o r
tw o m o dels,
(i) a one com p artm ent m odel,
( i i ) a two com p artm en t model.
4.1 The One Compartment Model
In th e one co m partm ent m odel ( f i g . 4.1.1) t h e human
body i s m o d elled a s a s i n g l e com partm ent.
F ig u r e 4 .1 .1
The one com partm ent
model
13
The model has th e f o l lo w in g p a r a m e t e r s :
(i) V, th e e f f e c t i v e body volum e,
( i i ) kg, th e r a t e c o n s t a n t f o r th e r a t e a t w hich
th e drug i s removed from th e c o m p artm en t,
and th e q u a n t i t i e s :
(i) u, t h e r a t e of drug i n f u s i o n ,
( i i ) q, th e drug c o n c e n t r a t i o n in th e body.
The dynam ics of th e system a r e :
q=-k^q + u/V
The s o l u t i o n , g iv e n c o n s ta n t u, i s :
q ( t ) = e x p ( - k g ( t - t o ) ) q ( t g ) + u ( 1 - e x p ( - k g ( t - t ^ ) ) ) /(kgV)
T h is c o n t i n u o u s l i n e a r s y s te m can be l o o k e d a t a s a
d i s c r e t e tim e sy ste m . In a h o s p i t a l s i t u a t i o n t h e problem
i s e s s e n t i a l l y a d i s c r e t e tim e one. Drug d o sa g es a re g iv e n
in t r a v e n o u s l y f o r a f i x e d l e n g t h of tim e a t a c o n s t a n t
r a t e . M easurem ents of th e drug c o n c e n t r a t i o n a r e made a t
d i s c r e t e t im e s by t a k i n g b loo d sa m p le s. The t r u e c o s t
f u n c t i o n i s c o n t i n u o u s i n t h a t t h e aim i s t o m a i n t a i n a
c e r t a i n drug c o n c e n t r a t i o n over a l e n g t h of tim e , n o t j u s t
a t d i s c r e t e tim e s . However th e c o n c e n t r a t i o n changes on ly
s l o w l y and i n an e x p o n e n t i a l m a n n e r, so t h e r e i s no r i s k
of th e c o n c e n t r a t i o n v a r y in g w i l d l y betw een th e o b j e c t i v e
tim e s . By a s u i t a b l e c h o ic e of th e o b j e c t i v e tim e s a
d i s c r e t e c o s t f u n c t i o n a p p ro x im a tin g th e c o n tin u o u s c o s t
f u n c t i o n can be o b ta in e d .
14
The c o s t th e n i s a f u n c t i o n of th e drug c o n c e n t r a t i o n s
a t t h e o b j e c t i v e t i m e s , and i t m u s t be e x p r e s s i b l e a s a
f u n c t i o n of th e s t a t e . Thus an a p p r o p r i a t e s t a t e v a r i a b l e
f o r th e l i n e a r model i s a v e c t o r whose com ponents a r e th e
drug c o n c e n t r a t i o n s a t th e o b j e c t i v e tim e s . S ince t h i s i s a
t r a c k i n g p ro b lem , t h i s v e c t o r i s augm ented w ith a c o n s t a n t
s t a t e component a s e x p la in e d e a r l i e r in s e c t i o n 2.3.
In th e p h a rm a c o k in e tic p roblem t h e r e a re d e la y e d
m easurem ents. T y p ic a l l y th e r e s u l t s of th e b lood sa m p le s
a re n o t a v a i l a b l e u n t i l th e n e x t day, i .e . f o r t w e n ty - f o u r
h o u rs .
T h is problem can be so lv e d by ru n n in g t h e Kalman
f i l t e r and u p d a tin g th e p r o b a b i l i t i e s a s m ea su re m e n ts
become a v a i l a b l e . The c o n t r o l s a r e c a l c u l a t e d u s in g th e
s t a t e e s t i m a t e o b ta in e d by e x t r a p o l a t i n g from th e s t a t e
e s t i m a t e a t th e l a s t tim e f o r w hich m ea su rem en ts w ere
a v a i l a b l e .
An a l t e r n a t i v e s o l u t i o n , w h ic h i s t h e one we u s e d i n
c h a p t e r f i v e , i s t o grou p th e s t a t e s i n t o t w e n ty - f o u r hour
b a tc h e s . The s t a t e i s th e n a v e c t o r whose com ponents a r e
th e drug c o n c e n t r a t i o n s a t a l l th e o b j e c t i v e tim e s in a
t w e n ty - f o u r hour p e r i o d , augm ented by a c o n s t a n t . A ll th e
m ea su re m e n ts f o r th e p r e v io u s t w e n ty - f o u r hour p e r i o d w i l l
be a v a i l a b l e to c a l c u l a t e t h e c o n t r o l s (drug d o sa g es) f o r
th e n e x t t w e n ty - f o u r hour p e r io d . Thus, th e m easu re m e n ts
15:
a re no lo n g e r d e la y e d , th e y a r e a v a i l a b l e i n tim e to
c a l c u l a t e th e c o n t r o l s f o r th e n e x t s te p .
4 .2 Example 1
A t y p i c a l drug dosage and o b s e r v a t i o n reg im e i s (tim e s
in h o u rs from th e s t a r t of th e c y c le ) :
Drug d o s a g e t i m e s 0 16 24
each dosage l a s t i n g one hour
O b s e rv a tio n tim e s 1 3 8
O b je c tiv e tim e s 1 8 9 16 17 24
The r e s u l t i n g s t a t e v e c t o r i s :
x(k)'^=(qj^(l) ,qj^(8) ,q;^(9) ,q;^(16) ,q k(17) ,qj^(24) ,1)
where
q]^(t) i s t h e c o n c e n t r a t i o n a t tim e t on t h e kth
day.
The r e s u l t i n g s t a t e c o n t r o l v e c to r i s :
u(k)'^=(U|^(0) ,U]^(8) ,U|^(16))
where
U]^(t) i s t h e i n tr a v e n o u s i n f u s i o n r a t e s t a r t i n g a t
t i m e t on t h e k t h day.
The s t a t e e q u a t io n s a re :
X (k+ 1)= 0 (k + lIk )X (k) + G (k)u(k) + w(k)
and th e o b s e r v a t i o n s a re g iv e n by :
y (k )= H (k )x (k ) + v(k)
where
16
(p(k+lik) =
H(k) =
E (v (k )v (j)'^ )= R (k )£ j^j
E(w(k) w( j)'^)=Q(k)S}^j
0 0 0 0 0 e x p (-k g (l+ 2 4 -2 4 )
0 0 0 0 0 e x p (-k _ (8 + 2 4 -2 4 )
where
0 0 0 0 0 e x p ( - k g (9+24-24) 0
0 0 0 0 0 e x p (-k g (1 6 + 2 4 -2 4 ) 0
0 0 0 0 0 e x p ( - k g (17+24-24) 0
0 0 0 0 0 e x p (-k g (2 4 + 2 4 -2 4 ) 0
0 0 0 0 0 0 1
0 0 0 0 0 0
e x p ( - k g e x p (3 -1 )) 0 0 0 0 0
0 1 0 0 0 0
90(1) 0 0)
90(8) 0 0)
90(9) 9g(9) 0)
G(k)=( 90(16) 90(16) 0)
90(17) 90(17) 916(17) )
90(24) 90(24) 916(24) )
0 0 0 )
9 y (x)=u (y) (1 -ex p (-kg (1) ) exp (-kg (x-y) ) / (kgV)
17
4 .3 The Two Compartment Model
In t h e two com partm ent model ( f i g 4.3.1) t h e body i s
m o d e lle d a s two co m p a rtm e n ts. The f i r s t co m partm en t i s th e
same a s in t h e one com partm ent model* The second com
p a r tm e n t i s p e r i p h e r a l t o th e f i r s t w ith two r a t e con
s t a n t s d e te r m in in g th e r a t e a t w hich th e drug p a s s e s t o
and from th e second co m partm en t.
u
F ig u re 4 .3 .1
V
^pc ^ ^2
The two com partm ent
model
c
ll ■
..... ...
18
In t h i s model t h e p a ra m e te r s a r e :
(i) Vf th e e f f e c t i v e body yolum ef
( i i ) kgf t h e r a t e c o n s t a n t f o r th e r a t e a t w hich
t h e d r u g i s rem o v e d fro m t h e bodyf
( i i i ) k^pf t h e r a t e c o n s t a n t f o r th e t r a n s f e r of
th e d ru g from th e f i r s t com partm ent t o th e
secondf
( iv) kpcf t h e r a t e c o n s t a n t f o r th e t r a n s f e r of
th e d ru g from th e second co m partm en t to th e
f i r s t f
and th e q u a n t i t i e s a re :
(i) <31 r th e drug c o n c e n t r a t i o n in t h e f i r s t
com partm entf
( i i ) q 2 f a r e l a t i v e drug c o n c e n t r a t i o n in th e
second co m p artm en t,
( i i i ) Uf t h e r a t e of drug i n j e c t i o n o ver tim e t^ .
The dynam ics of th e system a re
^2 = kcpSl - kpcS2
The s o l u t i o n of t h i s s e t of c o u p le d d i f f e r e n t i a l
e q u a t io n s i s :
( q%(t) ) = A (t) ( q i ( t o ) ) + B (t) u
( 9 2 (t) ) ( 9 2 (to) )
19
where
A (t) = ( a i i ( t ) a i 2 ( t ) )
( a 2 i ( t ) 3 2 2 ( t ) )
w ith
a i i ( t ) = [ (k^^-kp^) 6 K P ( - k i t ) - ( k 2 -kpç) exp ( - k 2 t) ] /
(ki«k2)
a i 2 ( t ) = k p (,[ex p (“ k 2 t)-e x p (-k j^ t)]/(k 3 ^ -k 2 )
3 2 i ( t ) = k c p [ e x p ( - k 2 t ) - e x p ( - k i t ) ] / ( k i - k 2 )
322 ( t) = [ (k]^“ kpç) exp ( - k 2 t) “ (k 2 “ kp(,) exp ( - k j t ) ] /
( k i “k2)
and
B (t) = ( b i ( t ) )
( b 2 ( t) )
w ith
b ^ { t ) = [ ( k ^ - k p c ) ( l - e x p ( “ k i t ) ) / k j +
(kpc“ k 2 ) ( 1 - e x p ( - k 2 t ) / k 2 ] / [ V ( k ^ - k 2 )]
b2(t) = “k ^ p [ d - e x p ( - k 2 t ) ) / k 2 - ( l - e x p ( - k 2 t ) ) / k 2 ] /
[V (k i-k 2 )l
and
k Q ^ = [ ( k g + k ç p + k p ç ) + ( ( k g + k ç p + k p ç ) 4 k p ç k g ) ^ ] / 2
k 2 = [ ( k e + k ^ ^ p + k p ç ) — ( ( k g + k ç p + k p ç ) “ ^ k p ^ k g ) ^ ] / 2
The d i s c r e t e f o r m u l a t i o n of t h i s problem i s s i m i l a r to
th e one com partm ent c a s e w ith th e s t a t e s b e in g grouped
i n t o t w e n ty - f o u r hour b a tc h e s .
20
S in ce t h e r e i s no c o s t on t h e c o n c e n t r a t i o n in th e
second c o m p artm en t, one n e ed s o n ly one s t a t e per c y c le f o r
t h e c o n c e n t r a t i o n in t h e second co m partm ent. However, t o
a v o id th e o b s e r v a t i o n s b e in g f u n c t i o n s of th e c o n t r o l , th e
c o n c e n t r a t i o n s of th e drug i n t h e f i r s t co m partm ent a t
each o b s e r v a t i o n tim e m ust be e le m e n ts of th e s t a t e
v e c t o r #
4 .4 Example 2
The f o l lo w in g i s t h e l i n e a r d y n am ical sy stem fo r th e
two com p artm en t model w ith th e same dosage and o b s e r v a t i o n
reg im e a s f o r th e one com partm ent exam ple.
Drug d o s a g e t i m e s 0 16 24
each dosage l a s t i n g one hour
O b s e rv a tio n tim e s 1 3 8
O b je c tiv e tim e s 1 8 9 16 17 24
The r e s u l t i n g dynam ical system i s :
x(k)'^=(qi^lç(l) rqi,k(3) ,qi,k(8) fqi,k(9) rqi,k(16) ,
^1,k( f 9 i ^ k ( 2 4 ) f 9 2 , k f D
w here
qj^j^(t) i s t h e c o n c e n t r a t i o n i n co m partm ent j
a t t i m e t on t h e k t h d a y .
21
where
and
w ith
u(k)^=(U|^(0) ,Uj^(8) ,ujç(16)
Uj^(t) i s t h e in tr a v e n o u s i n f u s i o n r a t e s t a r t i n g
a t t i m e t on t h e k t h day.
x (k + l)= (I)(k + l|k )x (k ) + G (k)u(k) + w(k)
y (k)=H (k)X (k) + v(k)
T.
E(v(k) v( j)-^)=R(k)Sj^j
E(w(k) w( j)*^) =Q(k)^j^j
(0 0 0 0 0 0 a ^ i ( 1+24— 24)
®12
1+24-24) 0
(0 0 0 0 0 0 a ^ i ( 3+24— 24)
®12
3+24— 24) 0
(0
0 0 0 0 0 a ^ i ( 8+24— 24)
^12
8+24-24) 0
(0 0 0 0 0 0 a ^ i ( 9+24— 24)
^12
8+24-24) 0
(D(k+l|k) = (0 0 0 0 0 0 ( 16+24— 24)
®12
16+24— 24) 0
(0 0 0 0 0 0 a ^ i (17+24— 24) a i2
17+24-24) 0
(0 0 0 0 0 0 a ^ i (24+24— 24)
®12
24+24-24) 0
(0 0 0 0 0 0
^21(24+24— 24)
®22
24+24-24) 0
(0 0 0 0 0 0 0 0 1
w ith th e a ^ j (t ) a s above.
( 1 0 0 0 0 0 0 0
H(k)=( 0 1 0 0 0 0 0 0
( 0 0 1 0 0 0 0 0
0 )
0 )
0 )
22
where
g i ( l ) 0 0
9 i (3) 0 0
g%(8) 0 0
g i ( 9 ) gg(9) 0
G(k) = ( g i(1 6 ) g g d ô ) 0
g i ( i 7 ) g g d ? ) g i g d ? )
gi(24) gg(24) gig(24)
gî(24) gg(24) g{g(24)
9y(x) = u(y) [b2d)a2i(x-y)+b2(l)a22(x-y) ]
9y (x) = u (y) [b 2 (1) ^22 (^"V) "*"^1 ( ^) ^21 (^"y) ^
w ith
bj^(t) and a ^ j (t) a s above
23
CHAPTER 1
SIMULATIONS OF A PHARMACOKINETIC SYSTEM
5 .1 C o n tr o l of th e One Compartment Model
In th e one co m p artm en t model th e unknown p a r a m e te r s
a r e :
(i) kg, th e r a t e c o n s t a n t f o r th e r a t e a t w hich
th e drug i s removed from th e body. T h is i s
g iv e n by
kg=a+b*ccr
The unknow n p a r a m e t e r i s b, a and c c r h a v e
known v a lu e s .
( i i ) V, t h e e f f e c t i v e body volume. T h is i s g iv en
by
V=BW*VBW
w h e re VBW i s t h e r a t i o o f V t o t h e body
w e i g h t , BW. The body w e i g h t i s know n, VBW
i s t h e unknown p a r a m e te r .
Thus th e unknown p a ra m e te r sp a ce i s two d im e n s io n a l.
From p r e v i o u s l y c o l l e c t e d d a ta [ 6 ] , i t i s known t h a t th e
two p a r a m e t e r s have s t a t i s t i c s :
VBWg^ = 0,307 06
VBWstd dev = 0.10454
24
bav = 0.0036022
b s t d dev = 0.0014079
Assuming th e unknowns a r e n o r m a lly d i s t r i b u t e d , th e
s a m p le s p a c e w as d i s c r e t i z e d u s i n g a 9 x 9 g r i d t o g i v e 81
p o i n t s in t h e sam p le sp a ce . For each v a r i a b l e t h e v a lu e s
a t t h e 1 0 , 2 0 , . . . , 80 a n d 9 0 - t h p e r c e n t i l e p o i n t s w e re
u s e d , i . e . t h e a v e r a g e +. ( 0 .0 0 ,0 . 2 5 , 0 . 5 2 , 0 . 8 4 , 1 . 2 8 ) s t a n
d a rd d e v i a t i o n s .
S e v e ra l Monte C a rlo s i m u l a t i o n s w ere c a r r i e d o u t u sin g
t h i s c o n t r o l l e r on a one com partm ent model w i t h v a ry in g
t r u e v a lu e s of th e p a r a m e te r s VBW and kg.
The one co m p a rtm e n t model was th e one g iv e n in exam ple
one w ith :
BW = 70
ccr= 58
a = 0.01
B (k )= 0 , i . e . no c o n t r o l c o s t ,
Q(k)=0, i . e . no s t a t e n o i s e ,
R(k) = ( y i ( k ) V l00)«S i
w here
11 ' ^ 1 ' " ' '
y^(k) i s t h e i t h o b s e r v a t i o n in c y c le k,
i . e . th e s t a n d a r d d e v i a t i o n of th e o b s e r v a t i o n
n o is e i s one t e n t h of th e m easurem ent v a lu e .
25
A(k)=C(N)
1 0 0 0 0 0 — 8
)
0 1 0 0 0 0 -2
)
0 0 1 0 0 0 — 8
)
0 0 0 1 0 0 -2 )
0 0 0 0 1 0 — 8
)
0 0 0 0 0 1 -2
)
-8 — 2 — 8 — 2 -8 -2 204
)
N = 7, i . e . th e system ru n s f o r seven days ( c y c l e s ) .
For each of f i v e s e t s of t r u e p a ra m e te r v a lu e s , th e
f o llo w in g ite m s were computed :
(i) t h e a v e ra g e c o s t s and t h e i r s t a n d a r d d e v i
a t i o n f o r each of th e seven days o ver
tw e n ty Monte C a rlo t r i a l s u s in g t h e MMAC
c o n t r o l l e r ,
( i i ) t h e c o s t f o r th e open lo o p c o n t r o l l e r i .e .
th e c o n t r o l l e r w hich would be o p tim a l i f
t h e t r u e p a ra m e te r v a lu e s w ere th e a v e ra g e
p a r a m e te r v a lu e s ,
( i i i ) t h e c o s t f o r th e o p tim a l c o n t r o l l e r u sin g
th e t r u e p a r a m e te r v a lu e s .
F ig u r e 5.1.1 shows t h e d a i l y c o s t s a v e ra g e d over th e
Monte C a rlo t r i a l s f o r th e MMAC c o n t r o l l e r . F ig u r e 5.1.2
shows t h e d a i l y c o s t s f o r th e open lo o p c o n t r o l l e r . F ig u re
5*1#3 com pares th e t o t a l c o s t s over th e seven days f o r
each of th e c o n t r o l l e r s f o r each of th e p a ra m e te r s e t s .
26
1004
O
&
O
S
<
i
è
O
t—
< c
cr
jsn
X Model 0
o 1
T
F ig u re 5 .1 .1 R a tio of d a i l y c o s t f o r th e M M AC c o n t r o l l e r
to d a i l y c o s t fo r th e o p tim a l c o n t r o l l e r .
!0 &- -A —
1 /Î
O
L J
2
S
o
! /T l
O
O
u_
o
<
ac
m '
X Model 0
a 1
2
o 3
o 4
1 3 -
~ r --------r-
4 5
D A Y
F ig u re 5 .1 .2 R a tio of d a i l y c o s t f o r th e open lo o p
c o n t r o l l e r to d a i l y c o s t fo r th e o p tim a l
c o n t r o l l e r .
These r e s u l t s i n d i c a t e t h a t th e MMAC c o n t r o l l e r p e r
fo rm s s u b s t a n t i a l l y b e t t e r th a n th e open lo o p c o n t r o l l e r
i n t h e c a s e w h e re t h e t r u e p a r a m e t e r v a l u e s a r e a r e a s o n
a b le d i s t a n c e from t h e i r a v e ra g e v a lu e s . The s te a d y
d e c l i n e in d a i l y c o s t ( f i g 5.1.1) fo r th e MMAC c o n t r o l l e r
i s a r e s u l t of th e c o n t r o l l e r u s in g th e m ea su rem en ts on
th e sy stem to l e a r n a b o u t t h e sy ste m . T h is d e c l i n e in
c o s t s i s in c o n t r a s t t o th e open lo o p c o n t r o l l e r
(f i g 5.1.2) whose p e rfo rm a n c e does n o t im prove.
The open lo o p c o n t r o l l e r does b e t t e r o n ly f o r model
z e ro , w here th e t r u e p a ra m e te r v a lu e s a r e th o s e assum ed by
th e open lo o p c o n t r o l l e r . The MMAC c o n t r o l l e r s t i l l g iv e s
a lo w c o s t c o m p a re d w i t h t h e c o s t s f o r t h e o t h e r m o d e ls .
F ig u re 5 . 1 . 3 : A c o m p a r i s o n o f t h e p e r f o r m a n c e o f t h e
MMAC, open lo o p (OL) and o p tim a l
c o n t r o l l e r s in c o n t r o l l i n g a s i n g l e
com partm ent sy stem .
Model
nos
p a ra m e te r |
v a lu e s I M M AC
b I VBW I a v e r a g e |s t d dev
C o n tro l c o s t s
OL I o p tim a l
0 1 .00360221 .307 061 3.650 2.341 1.485 1. 485
1 1.00180001 .173081 415.2 4 .9 4266 44 .28
2 1 .00180001 .441041 49.16 2 .3 5 58.31 44 .28
3 1 .00540001 .173081 85.52 2 .0 2 38 7 .5 28 .47
4 1 .0054000 1.441041 61.69 1.89 263.3 28 .47
28
CHAPTER 1
SUMMARY
The MMAC a lg o r i th m i s n o t o p t im a l, b u t t h e r e i s no
p r a c t i c a l a lg o r i th m c u r r e n t l y known w hich w i l l g iv e th e
o p tim a l c o n t r o l l e r f o r a g e n e r a l l i n e a r sy stem w ith un
known p a ra m e te r s . T here a re a p p ro x im a tio n s to th e o p tim a l
c o n t r o l l e r b a sed on a l i n e a r i z a t i o n of th e p ro b lem , e.g.
th e a lg o r ith m d e s c r i b e d by B ar-S halom and Tse [5 ]. The
MMAC a lg o r i th m c a n n o t a c h ie v e a c o s t as low , in g e n e r a l ,
a s th e a lg o r ith m d e s c r i b e d by B ar-S halom and T se, how ever
th e MMAC a lg o r i th m i s e asy to im p lem en t f o r a v e ry wide
ran ge of s y s te m s , u n l ik e th e a lg o r i th m d e s c r i b e d by B a r-
Shalom and Tse.
For th e MMAC a l g o r i t h m a g e n e r a l program f o r a d i g i t a l
com puter i s easy t o g e n e r a t e . A lthough a l a r g e number of
p o s s i b l e m odels, as i s needed when t h e number of unknown
p a r a m e te r s i s g r e a t e r th a n a b o u t t h r e e , may make com puting
t im e s l a r g e , th e p roblem i s i d e a l l y s u i t e d t o p a r a l l e l
p r o c e s s in g . Thus a p a r a l l e l p r o c e s s in g sy stem co u ld be
used t o red u c e th e co m puting tim e s i g n i f i c a n t l y .
29
In t h i s t h e s i s i t was shown t h a t th e MMAC a lg o r i th m
p e rfo rm e d s i g n i f i c a n t l y b e t t e r th a n an open lo o p c o n t r o l
l e r , i .e . a n o n a d a p tiv e c o n t r o l l e r , when th e y w ere used t o
c o n t r o l a s i n g l e co m p artm en t p h a r m a c o k in e tic sy ste m .
T here a r e s e v e r a l e x te n s i o n s t o t h e work p r e s e n t e d in
t h i s t h e s i s which sh o u ld be p u rsu e d :
(i) u s in g a Monte C a rlo s i m u l a t i o n over th e
p a ra m e te r sp a c e an e s t i m a t e of th e e x p e c te d
v a lu e of t h e MMAC c o s t f o r t h e s i n g l e
co m p artm en t model sh o u ld be o b t a i n e d ,
( i i ) t h e e f f e c t of d i f f e r e n t num bers of m odels
in th e sam ple sp ace on th e p e rfo rm a n c e of
th e M M AC a lg o r ith m sh o u ld be s t u d i e d ,
( i i i ) th e p e rfo rm a n c e of th e c o n t r o l l e r when used
t o c o n t r o l th e tw o com partm ent model sh o u ld
be s t u d i e d .
30
REFERENCES
[1] D e s h p a n d e , J.G*, ü p a d h y a y , T.N., and L a i n i o t i s , D.G.
(1973) A d a p tiv e c o n t r o l of l i n e a r s t o c h a s t i c sy s te m s.
A uto m atica S., 107-115.
[2] D 'A rgenio, D.Z. and S ch u m itzk y , A. (1979) A program
f o r s i m u l a t i o n and p a ra m e te r e s t i m a t i o n in
p h a r m a c o k in e tic sy s te m s. Computer P ro gram s I n
B io m ed icin e 2.r 115-134 .
[3] Kalman, R.E. (1960) A new ap p ro ach t o l i n e a r
f i l t e r i n g and p r e d i c t i o n t h e o r y . ASME T ran s, of
B a s ic E n g in e e rin g &2, 3 4-45.
[4] G u n c k e l, T. an d F r a n k l i n , G. (1963) A g e n e r a l
s o l u t i o n f o r l i n e a r sam pled d a ta c o n t r o l . ASME T ran s.
J . of B a sic E n g in e e rin g £5./ 1 9 7-203 .
[5] B ar-S h alom , Y. and T se, E. (1976) C o ncep ts and
m ethods in s t o c h a s t i c c o n t r o l , in C o n tro l and Dynamic
Sy ste m s 1 2 . (Leondes, C.T., e d .) . Academ ic P r e s s , New
Y o rk . 9 9 -1 7 2 .
[6] K a t z , D. and D 'A r g e n io , D.Z. ( t o a p p e a r ) D i s c r e t e
a p p ro x im a tio n of m u l t i v a r i a t e d e n s i t i e s w ith
a p p l i c a t i o n s t o B a y sian e s t i m a t i o n . ^ Pharmacokln.^
B io p h a rm ..
31

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Creator Iles, David Walter (author)

Core Title On the multiple model adaptive control of pharmacokinetic systems

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Degree Master of Science

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