Semi-linear regression problems with an application in pharmacokinetics

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Content SEMI-LINEAR REGRESSION PROBLEM S
WITH AN APPLICATION IN
PHARMACOKINETICS
by
Trudy M. R ubinger
A T h e sis P r e se n te d to th e
FACULTY O F THE GRADUATE SCH O O L
UNIVERSITY O F SOUTHERN CALIFORNIA
In P a r t ia l F u lf illm e n t o f th e
R equirem ents fo r th e D egree
MASTER O F SCIENCE
(A p p lied M athem atics)
Decem ber'"198 2
UMI Number: EP54413
All rights reserved
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D lssw tâïfen: P W M isM n g
UMI EP54413
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A
This thesis, written by
. X ? i V i . * 3 Y . . - N t.. jîsê .ç ........................................................
under the direction of h^x.....Thesis Committee,
and approved by all its members, has been pre
sented to and accepted by the Dean of The
Graduate School, in partial fulfillment of the
recpuirements for the degree of
MASTER OF SCIENCE
Dean
SEPTEMBER 2 3 , 1982
THESIS COMMITTEE
" j j l 7 ) Chairman
DEDICATION
11
A C K N O W L E D G E M E N T
I would l i k e to thank Dr. A lan Schum itzky fo r h is g u id a n ce,
s u b s t a n t ia l a s s i s t a n c e , encouragem ent, and fr ie n d s h ip . I would a ls o
l i k e to thank Dr. Wlodek P roskurow ski fo r h is com plim entary remarks
and Dr. Stan Azen fo r h is c o n s tr u c tiv e c r i t i c i s m s . F in a lly , I would
l i k e to thank Dr. W illiam H a rris fo r h is f in a n c ia l and m oral su p p ort,
111
TABLE O F CONTENTS
Page
DEDICATION
i i
A C K N O W L E D G E M E N T i i i
CHAPTER
1 INTRODUCTION AN D BA C K G R O U N D 1
1 .1 N o n lin ea r L e a st-sq u a r e s 1
1 .2 B ayesian A n a ly sis 1
1 .3 S e m i-lin e a r R e g r e ssio n 2
1 .4 P h arm acok in etic M odels 2
1 .5 O verview o f T h e sis 3
2 REGRESSION PROBLEM
2 .1 The N o n lin ea r R e g r e ssio n Model 5
2 .2 S e m i-lin e a r R e g r e ssio n 5
3 ESTIMATION OF PARAM ETERS 7
3 .1 L in ea r L e a st-sq u a r e s 7
3 .2 B ayesian A n a ly sis
10
3 .3 Summary 11
4 ■ A PHARMACOKINETIC ILLUSTRATION: DISPOSITION
KINETICS OF LIDOCAINE
13
4 .1 The L id o c a in e Model
13
4 .2 C h a n g e -o f-v a r ia b le s 15
4 .3 S o lu tio n o f th e Transform ed System 18
4 .4
D is c u s s io n
21
5 SU M M A R Y AND CONCLUSIONS
23
5 .1 D is c u s s io n
23
6 REFERENCES
24
i v
CHAPTER 1
INTRODUCTION AND BA C K G RO UN D
1 .1 N o n lin ea r L e a st-sq u a r e s
A t r a d it i o n a l approach to param eter e s tim a tio n in n o n lin e a r m odels
i s b ased on le a s t - s q u a r e s p r o c e d u r e s. L e a st-s q u a r e s e s tim a te s o f th e
param eters are n ot o b ta in a b le in c lo s e d form fo r th e n o n lin e a r m odel
and, h e n ce, some n u m erica l method to m inim ize th e sum of- sq u a r es ob
j e c t i v e fu n c tio n i s u sed in ord er to d eterm in e su ch e s t im a t e s .
In t y p ic a l lin e a r r e g r e s s io n p rob lem s, le a s t - s q u a r e s e s tim a te s a re
m inim um -variance u n b iased e s tim a te s w hich a r e lin e a r in th e o b serv a
t io n s and w hich have known s m a ll sam ple d is t r ib u t io n s . H ence, appro
p r ia t e s t a t i s t i c a l in fe r e n c e s can b e made. On th e o th er hand, in non
lin e a r s t a t i s t i c a l a p p lic a t io n s , i n f e r e n t i a l p roced u res are b a sed on
a sy m p to tic th eo ry where th e d is t r ib u t io n s o f th e estim a to rs, are known
on ly in th e li m i t as th e sam ple s i z e approaches i n f i n i t y (K atz e t a l .
[9]). H ence, a n a ly s e s b ased on th e s e s t a t i s t i c s may be in a p p r o p r ia te .
1 .2 B a y e sia n A n a ly s is
An a lt e r n a t i v e approach to param eter e s tim a tio n in n o n lin e a r
m odels i s to u t i l i z e m ethods based on Bayes* theorem . For B a y esia n
a n a ly s e s , th e param eters are regarded as random v a r ia b le s r a th e r than
unknown c o n s ta n ts .
B a y esia n a n a ly s e s r e q u ir e s p e c i f i c a t i o n o f a p r io r i p r o b a b ilit y
d is t r ib u t io n s o f th e p a ram eters, w hich i s m ost a p p r o p r ia te fo r th e
problem c o n sid e r e d in t h i s p ap er.
The p o s te r io r d is t r ib u t io n s r e s u lt in g from B a y esia n a n a ly s e s con
t a in a l l o f th e in fo r m a tio n known about th e p a ra m eters, g iv e n th e d a ta .
S p e c i f i c a l l y , th e p o s te r io r d is t r ib u t io n s o f th e p aram eters w i l l be
u t i l i z e d to d eterm in e e s tim a te s o f th e p a ra m eters, g iv e n th e d a ta .
A major drawback o f th e B a y esia n approach i s th a t i t r e q u ir e s
n u m erical in t e g r a t io n over th e e n t ir e param eter sp a c e . For n o n lin e a r
m odels w ith fou r or more p a ra m eters, t h i s i s i n f e a s i b l e (K atz e t a l .
[9]).
1.3 S e m i-lin e a r R e g r e ssio n
The n o n lin e a r m odel fo r th e p h a rm acok in etic exam ple c o n sid e r e d in
t h i s paper i s r e p r e se n te d by a system w hich i s n o n lin e a r in a l l para
m e te r s. An a p p r o p r ia te c h a n g e -o f-v a r ia b le s a p p lie d to th e o r ig in a l
p aram eters o f th e n o n lin e a r model w i l l tran sform th e model in to an
e q u iv a le n t ( i . e . , same in p u t r e s u lt s in same o u tp u t) sy stem in which
th e ou tp u t i s lin e a r in some o f th e new param eters ( i . e . , s e m i- lin e a r ) .
L in ear le a s t - s q u a r e s th eo ry and B ayesian a n a ly s is can th en be used to
d eterm ine e s tim a te s o f th e tran sform ed p ara m eters. The lin e a r para
m eters are f i r s t e stim a te d e x a c t ly by m inim um -variance e s tim a tio n con
d itio n e d on th e n o n lin e a r p a ra m eters. The f i n a l e s tim a te i s d e r iv e d
u sin g B ayesian a n a ly s is w ith th e r e q u ir e d in t e g r a t io n b e in g over o n ly
th e n o n lin e a r param eter sp a c e .
1 .4 Pharm acpjklnetic Models
In th e stu d y o f a b s o r p tio n , d is t r ib u t io n , m etab olism and e x c r e tio n
o f d ru g s, m ath em atical m odels have proved to be u s e f u l fo r u n d erstan d
in g th e u n d e r ly in g k i n e t i c phenomena in v o lv e d w ith th e s e p r o c e s s e s
(D’A rgen io and Schum itzky [5]). In a d d it io n , m odels a re u sed fo r
d e sig n in g in d iv id u a l d osage regim ens to a c h ie v e s p e c i f i c th e r a p e u tic
g o a ls . T hese a p p lic a t io n s g e n e r a lly r e q u ir e th e d e te r m in a tio n o f th e
m o d el’ s param eters from in p u t-o u tp u t o b s e r v a tio n s w h ich , in m ost c a s e s ,
le a d s to a n o n lin e a r param eter e s tim a tio n problem .
In g e n e r a l, v a r io u s n o n lin e a r le a st-s q u a r e s r e g r e s s io n m ethods are
u sed to a n a ly z e p h a rm acok in etic m o d els. Some p h a rm acok in etic r e se a r c h
e r s , how ever, have u sed th e method o f maximum lik e lih o o d , b u t t h is
approach i s a ls o dependent on a sy m p to tic th eo ry fo r m ethods o f s t a t i s
t i c a l in f e r e n c e . K a tz ’ s d o c to r a l d is s e r t a t io n [8] p r e se n ts a B ayesian
approach to th e a n a ly s is o f n o n lin e a r m odels w ith a p p lic a t io n s in phar
m a c o k in e tic s .
The d is p o s it io n k i n e t i c s o f many drugs have b een d e sc r ib e d by a
f ir s t - o r d e r tw o-com partm ent open m odel w ith in tr a v e n o u s in fu s io n
(K atz [8]). In C hapter 4 , such a m odel i s a p p lie d to th e k i n e t i c s o f
th e drug lid o c a in e , th e l e v e l s o f w hich r e q u ir e c a r e f u l c o n tr o l in
c a r d ia c p a t i e n t s .
1 .5 Overview o f T h e sis
In t h i s t h e s i s , a method b a sed on li n e a r le a s t - s q u a r e s th eo ry and
B a y esia n a n a ly s is i s d ev elo p ed fo r e s tim a tin g th e param eters in a
s e m i- lin e a r r e g r e s s io n problem . In p a r t ic u la r , th e f o llo w in g t o p ic s
a re in c lu d e d :
a . sta te m e n t o f th e s e m i- lin e a r r e g r e s s io n problem
b . d e r iv a tio n o f m inim um -variance e s tim a te s fo r th e lin e a r
p aram eters u sin g lin e a r le a s t - s q u a r e s th e o r y , fo llo w e d by
e s tim a tio n o f a l l m odel param eters u t i l i z i n g B ayesian
a n a ly s is
c . tr a n sfo r m a tio n o f a n o n lin e a r p h arm a co k in etic m odel in t o a
s e m i- lin e a r model to w hich th e e s tim a tio n method d evelop ed
i s a p p lie d in ord er to o b ta in e s tim a te s o f th e tran sform ed
param eters
d. c o n c lu s io n s as to th e a p p l i c a b i l i t y o f t h i s method o f
param eter e s tim a tio n .
The r e s u lt s p r e se n te d h e r e fo llo w from th e a p p lic a t io n o f standard
lin e a r le a s t - s q u a r e s th eo ry (s e e L uenberger [11]), elem en ta ry B a y esia n
a n a ly s is (s e e H oel e t a l . [7]) , and th e r e d u c tio n o f d im e n s io n a lity
tec h n iq u e r e c e n t ly p r e se n te d in K atz e t a l . [10].
The f i n a l e s tim a te fo r th e lin e a r p aram eters i s d e r iv e d w ith o u t
th e assu m p tion o f n o r m a lity . T his e s tim a te appears to have i n t e r e s t
in g p r o p e r t ie s . For exam ple, when th e a p r i o r i p aram eters’ d istr ib u -:
t io n s a re norm al, t h i s e s tim a te i s sim p ly th e c o n d itio n a l e x p e c ta tio n
o f th e lin e a r param eter v e c t o r , g iv e n th e d a ta .
CHAPTER 2
REGRESSION PROBLEM
2 ,1 The N o n lin ea r R é g r e ssio n Model
Suppose th a t th e n o n lin e a r r e g r e s s io n model can be w r itte n as
Yi = g(t^, i = l , . . . , n ,
where g(t^, i s n o n lin e a r in th e k -d im e n sio n a l param eter v e c to r
. , . , is th e i t h o b s e r v a tio n a t tim e t^ , and i s th e
co rr esp o n d in g e r r o r term ,- i - 1 , . . . ,n. I t i s d e s ir e d to e s tim a te th e
param eters (J)^,. . . , (J)^ .
Now, assume th a t g i s s e m i- lin e a r . More p r e c i s e l y , l e t
Y. = (y^,. . . ,y^ ) *, = ( e ^ ,. . . , e^) ’ , and W =
Y = W ^ (2.1.1)
where , . . . , g^) ’ a re m unknown random p aram eters and w^^ =
w ^ j(t^ , ^ ) a re fu n c tio n s o f th e unknown random p aram eters =
(a . , . . , , a. -) ’ . N ote th a t ( f ) = ( a ’ , g ’ ) . Assume th a t g has a known
I R— m — — —
d is t r ib u t io n w ith mean g and c o v a r ia n c e R, £ has a known d is t r ib u t io n
—o —
w ith zero mean and c o v a r ia n c e Q\ _ g and e _ a re in d e p e n d e n t, and th a t
has a known d is t r ib u t io n . E stim a te s o f th e param eters and ^ a re to
be d eterm in ed .
2 .2 S e m i-lin e a r R e g r e ssio n
The method o f param eter e s tim a tio n d ev elo p ed h e n c e fo r th w i l l a llo w
fo r th e e x a c t com p u tation o f e s tim a te s o f th e sy stem param eters fo r th e
s e m i-lin e a r r e g r e s s io n m odel. As m entioned in th e p r e v io u s c h a p te r ,
th e d e r iv a tio n o f such e s tim a te s w i l l be b ased on lin e a r le a s t - s q u a r e s
th eo ry and B ayesian a n a ly s is .
To o b ta in a lin e a r m odel, i t w i l l be assumed th a t th e para
m eters a re f ix e d . Then Eq. (2.1.1) i s lin e a r in ^ and, th u s , th e
minimum-var ia n c e e s tim a te o f _ 3 can be d eterm in ed . The param eters
w i l l th en be assumed to be random and B a y esia n a n a ly s is , w ith th e
req u ir ed in t e g r a t io n b ein g over o n ly th e n o n lin e a r param eter sp a c e ,
w i l l p ro v id e e s tim a te s o f ^ and The r e s u lt in g advantage o f red u c
in g th e d im e n s io n a lity o f th e re q u ir e d I n te g r a tio n i s th e m o tiv a tio n
fo r p erform in g th e a n a ly s is as d e s c r ib e d ab ove.
(For th e rem ainder o f t h i s p a p er, l e t t e r s r e p r e s e n tin g v e c to r s
w i l l n o t be u n d e r lin e d .)
CHAPTER 3
ESTIMATION OF PARAM ETERS
3 .1 L in ear L e a st-s q u a r e s
C onsider th e model g iv e n in Eq. (2.1.1) w ith th e same a ssu m p tio n s,
ex c e p t th a t a i s known. The method to be used fo r e s tim a tin g th e un
known random param eter v e c to r 6, g iv e n th e d ata v e c to r Y, w i l l be to
fin d a m inim um -variance lin e a r e s tim a to r o f th e form
ê = B + K Y ,
where K = ( k . .) , k . . = k . . ( t . , a) c o n s ta n ts , and B = (b -, . . . ,b )'
i j mxn’ i j ].] 1 1 m
i s a c o n s ta n t v e c t o r . The c r i t e r io n fo r o p tim a lity o f th e e s tim a tio n
i s m in im iz a tio n o f th e norm o f th e e r r o r , e x p r e sse d in t h i s c a se as
11& " A .I I »
where | | | | i s th e o rd in a ry E u clid ea n m -space norm.
The fo llo w in g theorem and c o r o lla r y w i l l p r o v id e t h i s minimum-
v a r ia n c e e s tim a te .
Theorem 1 . Suppose
Y = we + e
where Y = (y ^ ,. . . , y ^ ) ' i s a known d ata v e c t o r , g = ( g^, . . . , g^)' i s an
unknown random param eter v e c t o r , e = ( e ^ , . . . , e ^ ) ’ i s an unknown random
e r r o r v e c t o r , and W=?(w..) i s a c o n sta n t m a tr ix . Assume th a t
' i j nxm
E(g) = 0, E(e) = 0, and E(gg’ ) = R, E(ee') = Q, E(eg’) = 0. A lso
assume th a t W R W ' + Q i s i n v e r t i b l e . Then th e li n e a r e s tim a te g o f g
m in im izin g E [ ||g - g | | ^ ] i s
g = KY
where
K = R W ’ [W R W * + Q]“ ^ .
P ro o f: In e s tim a tin g g , w h ile assum ing b oth Y and g have
m eans, i t i s d e s ir e d to fin d a lin e a r e s tim a te o f th e form
zero
g = K Y
where K i s a c o n sta n t m x n m a tr ix . A ccord in g to th e p r o j e c t io n theorem ,
th e unique m in im izin g v e c to r g i s th e o r th o g o n a l p r o j e c t io n o f g on th e
su b sp a c e , M, g en era te d by th e y^ * s, or e q u iv a le n t ly , th e d if f e r e n c e
v e c to r g -g i s o rth o g o n a l to M. In p a r t ic u l a r .
E [ (g-g) * KY] = 0 fo r a l l m x n m a tr ic e s K .
S o, ch o o se K^ to be an.mxn m a trix w ith 1 i n th e i , j
w h e re. Then
en tr y and 0 e l s e -
A ^
E[(g-g)* K^Y] = E[(g-g) Y*]
and, t h e r e f o r e .
E[(g-g) Y'] = 0 .
From t h is i t fo llo w s th a t
K = E(gY*) [E(YY*)]"^
S in ce
E(gY*) = E[g(Wg+e)*] = R W *
and
E(YY*) = E[(Wg+e)(Wg+e)'] = W R W * + Q ,
8
i t i s found th a t
K = R W ’ [W R W ’ + Q]“ ^ .
C o ro lla ry 1 : C onsider th e same m odel h y p o th e siz e d in Theorem 1,
e x c e p t w ith means
E(Y) = Y^, E(3) = 3^, E(c) = 0
and c o v a r ia n c e s
E[(g-g^)(g-g^)’ ] = R, E (ee’) = Q .
A g a in , assume th a t g and e a re u n c o r r e la te d and t h a t W R W ’ + Q i s non-
1 1 2
s in g u la r . Then th e a f f i n e e s tim a te g o f g m in im izin g E[ | | g - g || ] i s
g = g + K(Y - W g )
o o
where
K = R W ’ [W R W ’ + Q]“ ^ .
P roof ; S in c e , in t h is c a s e , th e means a r e n o t assumed to be z e r o ,
th e c l a s s o f e s tim a to r s i s broadened to in c lu d e th e form
= B + K Y
w here B i s an a p p r o p r ia te c o n sta n t v e c t o r . Now, s in c e E(g) = g^ and
E(Y) = y^ im p lie s E(g-g^) = 0 and E(Y-Y^) = 0 , r e s p e c t i v e ly , i t
fo llo w s from Theorem 1 th a t
" g ^ = RW ’ [W R W ’ + Q]"^ (Y - Wg^) ,
and, s in c e ’B-B = g-g , i t i s found th a t
o o
g = g + K(Y - W g )
o o
where
K = R W ' [W R W ' + Q]“ ^ .
Remark: The above theorem and c o r o lla r y f o llo w from T.nenherger [11],
C o ro lla ry 1 , p. 88, and Problem 6 , p . 99.
3 .2 B ayesian A n a ly s is ^
For B a y esia n a n a ly s is , th e param eter v e c to r a , as w e ll as g , i s
regarded as a random v e c t o r . B a y es' Theorem (H oel e t a l . [7]) s t a t e s
th a t
f(a|Y) = — (3.2.1)
w here f(a) i s th e a p r io r i d e n s ity o f a , f(Y|a) i s th e c o n d itio n a l
d e n s it y o f Y, g iv e n a , f ( a | Y) i s th e p o s te r io r d e n s it y o f a , g iv e n Y,
and f(Y) i s th e d e n s ity o f Y g iv e n by
f(Y) =J'f(a) f(Y|a) da . (3.2.2)
Now, a g a in c o n s id e r
Y = W g + £
w here g and e a re in d ep en d en t and have known d i s t r i b u t i o n s , and a i s
f ix e d . Then, s in c e Y i s th e sum o f two in d ep en d en t random v e c t o r s ,
f(YI a) can b e r e a d ily com puted.
For exam p le, assume th a t g ~ N(g^ ,R) and e N(0,Q) . Then f (Y | a)
i s norm al and c o m p le te ly d eterm ined by mean and c o v a r ia n c e ,
E(Y|a) = W g
and °
Cov(Y) = COV(Wg+e) = W R W ' + Q .
10
H e n c e ,i n g e n e r a l, f ( a |Y ) c a n be c o m p le te ly d eterm in ed , p ro v id ed
th a t g and e are assum ed to b e in d ep en d en t and have known d is t r ib u
t io n s .
The e s tim a te s o f th e p ara m eters, g iv e n th e d a ta , can now be
d eterm in ed . The fo llo w in g p r o v id e s th e s e e s tim a te s :
E(g|Y) = E(g^ + K(q)Y|Y)
im p lie s
E(e|Y) = + f n ( a ) f(a|Y)Y da , (3.2.3)
and
E(:a|Y) = f(a|Y) da . (3.2.4)
N ote th a t th e in t e g r a t io n i n Eqs. (3.2.3) and (3.2.4) i s o n ly o v er th e
n o n lin e a r param eter s p a c e . T h is i s a n alogou s to th e r e d u c tio n o f
d im e n s io n a lity tech n iq u e p r e se n te d in K atz e t a l . [10].
In th e s p e c ia l c a se w here g and e a re a ls o assumed to be norm al,
i t can be shown th a t E(g |Y) = E(g|Y) (Sim s and L a in io t is [12]).
3.3 Summary
In summary, th e a n a ly s is o f param eter e s tim a tio n p r e se n te d h ere
s u g g e s ts th e fo llo w in g m eth od ology: f i r s t , assume a to be f ix e d in
ord er to d eterm ine th e c o n d it io n a l d e n s ity o f Y, g iv e n a , under th e
assu m p tion th a t g and e are in d ep en d en t and have known d is t r ib u t io n s .
The p o s t e r io r d e n s ity o f a , g iv e n Y, can th en be d eterm ined from Eq.
(3 .2 .1 ), s in c e th e d e n s it y o f a i s assumed to be known, and th e d e n s ity
o f Y can be found u sin g Eq. (3 .2 .2 ). F in a lly , g iv e n th e d a ta , th e e x
11
p e c te d v a lu e o f th e m inim um -variance e s tim a te o f 3 and th e ex p ecte d
v a lu e o f a can be d eterm ined w ith th e n e c e s s a r y in t e g r a t io n b e in g over
o n ly th e n o n lin e a r param eter sp a c e .
12
CHAPTER 4
A PHARMACOKINETIC ILLUSTRATION
DISPOSITION KINETICS O F LIDOCAINE
4 .1 The L id o ca in e Model
The d is p o s i t io n k i n e t i c s o f th e drug lid o c a in e fo llo w in g in t r a
venous in fu s io n can be d e s c r ib e d by a f i r s t - o r d e r , two-com partm ent
open m od el. The o b s e r v a tio n m odel i s r e p r e se n te d by
^ i + e ^ , i = 1 , ,n (4.1.1)
w here ÿ(t^) i s th e e x a c t v a lu e o f th e c o n c e n tr a tio n in th e f i r s t com
partm ent a t tim e t _, and i s th e co rresp o n d in g a ssa y e r r o r w ith
2 2
~ N(0,a ) for a l l i = l , . . . , n , a known. The lid o c a in e m odel i s
shown in F ig u r e 4 .1 .1 , where
= amount (mg) o f drug in compartment i ( i = 1 ,2 ),
V = volum e ( l i t e r s ) o f th e c e n t r a l com partm ent,
r = in tr a v e n o u s in f u s io n r a t e (mg/min) w h ich , fo r s i m p l i c i t y ,
i s assumed to b e c o n sta n t o v er tim e ,
k i 2 > ^ 21’ ^10 ~ th e f ir s t - o r d e r r a te c o n sta n ts (/m in).
The m easurem ent i s Y = X^/V = c o n c e n tr a tio n (p g /m l) i n th e f i r s t com
partm ent .
The two com partm ents co u ld be regarded as th e plasm a and p e r i
p h e r a l o rg a n s, r e s p e c t i v e ly . The r a te c o n sta n t k^Q i s g e n e r a lly th e
e lim in a t io n r a te c o n s ta n t, and Y , th e c o n c e n tr a tio n o f th e drug in th e
p lasm a, i s d eterm in ed by an a p p ro p ria te a ssa y p roced u re (K atz [8]).
The f i r s t - o r d e r d i f f e r e n t i a l e q u a tio n s w hich d e s c r ib e th e k i n e t i c s o f
13
N /
^12
X
L ^2
V
>
^10
f
5
dt
dX^
d t
Y
^ 12^ 1 ^21^2
X i/v
FIGURE 4 .1.1
Two-Compartment Model fo r th e
D is p o s it io n K in e tic s o f L id o c a in e
14
th e model are a ls o g iv e n in F igu re 4 .1 .1 . The s o lu t io n o f th e s e
e q u a tio n s i s n o n lin e a r in a l l fo u r p ara m eters. An a p p r o p r ia te tr a n s
fo r m a tio n , how ever, w i l l be found th a t w i l l r e s u lt in a new sy stem
w hich i s lin e a r in two o f th e p a ra m eters. T h is tran sform ed system w ill
th en be s o lv e d u sin g elem en ta ry m ethods o f o rd in a ry d i f f e r e n t i a l
e q u a tio n s .
The n e x t two s e c t io n s fo llo w th e a n a ly s is p r e se n te d in Katz [8].
4.2 C h a n g e -o f-v a r ia b le s
The e q u a tio n s fo r th e model d e p ic te d in F ig u re 4.1 .1 can be
w r itte n as th e system
(4.2.1)
o).
I t i s d e s ir e d to fin d an i n v e r t i b le tr a n sfo r m a tio n
/ h i h z \ ^
\ ^00 /
T =
*^21 "22
such th a t
Z = TX (4.2.2)
r e s u lt s in a new system o f th e form
Z = C Z + Er
(4.2.3)
Y = FZ, where F = ( g ^ , '
and Z depends o n ly on two n o n lin e a r param eters and Y i s
15
li n e a r in th e param eters and The r e la t io n s h ip betw een in p u t r
and o u tp u t Y i s th e same fo r S* as fo r S.
From Eq. (4 .2 .2 ), i t fo llo w s th a t X = T Z. A pp lyin g t h i s to
Eq. ( 4 .2 .1 ), i t i s found th a t
, - l
, -1 ,
or
-1
-1
Thus, i t i s r e q u ir e d th a t
-1
D = TAT
TB
-1
C T
w ith A, B, and C d e fin e d i n Eq. (4 .2 .1 ).
Suppose
and
-a -a
Then S ’ w i l l have th e d e s ir e d p r o p e r t ie s .
By e q u a tin g a p p r o p r ia te term s, T can be d e r iv e d and, th u s , th e
a ’ s and B’s o b ta in e d in term s o f th e o r i g in a l p a ra m eters. S in ce
11 1 1
'2 1
21 22
i t fo llo w s th a t
and - t ,
22
~ t
12
12
Now, s in c e
-1
TAT
-a. - a
22
12 21 12
(4.2.4)
22 21
12
i t i s found th a t
-k
and t
22
12
So
, - l
21
and T
12 12
S u b s t it u t io n in t o Eq. (4.2.4) y ie ld s
10 21
(4.2.5)
+ k
10 21
17
A lso
, - l 21
C T
12
so th a t
21
(4.2.6) and
So, th e tran sform ed sy stem S* i s
D Z + Er
(4.2.7)
FZ
w here
-a.
4 .3 S o lu tio n o f th e Transform ed System
The d i f f e r e n t i a l e q u a tio n in (4.2.7) i s s o lv e d as fo llo w s :
Find th e e ig e n v a lu e s o f D:
— X
- a
w hich im p lie s th a t th e e ig e n v a lu e s o f D are
where
~ 4ot
18
b . N e x t, s o lv e th e i n i t i a l v a lu e problem :
/« i N
Z = DZ and Z(0) = 6 = I . I . (4.2.8)
So,
„ Dt Dt
Z = e * 6 w here e
C : ) '
= (D - A ^ I ) - e 1 (D - A ^ D j .
Thus,
> \ / ^ 2 \ \
, . , 5 g - A ..Ô n V . . / ô « “ A „ 6
. " " - i P - I
H ence, Z^ = - ^
2” ^1*^1^^ ” (*^2""^2^1^^ J
1 r ^2^ 1
and = — I (-a^ 6^ -(u ^ + A ^ )6^ )e - (-a^ô^-Ca^+A^) e J .
(4.2.9)
T his r e p r e s e n ts th e p r o c e ss fo llo w in g th e i n i t i a l l e v e l s 6 w ith o u t th e
e f f e c t o f any a d d it io n a l in f u s io n r .
c . Now s o lv e
Z = D Z + Er , (4.2.10)
w hich d e s c r ib e s th e system d u rin g a c o n sta n t in f u s io n , ig n o r in g th e
e f f e c t o f drug in th e sy stem a t th e s t a r t o f th e in f u s io n p e r io d .
Eq. (4.2.10) can be r e w r itte n as
Z i = z^
Z^ = ~ ^2^2 ^ ^ 3’ w here Z^ = r (4.2.11)
Z3 = 0 .
19
Let
and -a -a
Then th e problem to be s o lv e d i s
A p p lyin g th e same method used to s o lv e Eq. ( 4 .2 .8 ), th e fo llo w in g
i s o b ta in ed : The e ig e n v a lu e s o f Q a re X = -a -b
so th a t
Qt _
-X
— X a -a .
-a
-a
( x „ - x ^ ) x
20
• Z(0) im p lie s and Z
(4.2.12)
H ence, th e s o lu t io n to th e g e n e r a l system g iv e n in (4.2.7) is
Y(t) =
are th e "com plete" s o lu t io n s to th e nonhomogeneous where Z- and
sy ste m , found by adding th e s o lu t io n s i n (4.2.9) and (4 .2 .1 2 ).
4.4 D is c u s s io n
The m easured c o n c e n tr a tio n in th e c e n t r a l compartment corresp on d
in g to Eq. (4 .1 .1 ), i s now r e p r e se n te d by th e s e m i- lin e a r e q u a tio n
w here
z f ( t , ,a) z = ( t „ a )
21
3 = (3^, , and e = . . . , ' .
N o tic e th a t Y i s now n o n lin e a r in and but li n e a r in 3^ and 3g «
Thus, th e method d e sc r ib e d in C hapter 3 can be used to fin d e s tim a te s
o f t h e tran sform ed param eters a^, a^, 3^^, and S in ce th e r e
i s a o n e -to -o n e co rresp o n d en ce b etw een th e tran sform ed p aram eters and
th e o r i g in a l o n e s , i f i t was d e s ir e d to d eterm in e th e e s tim a te s o f
th e o r i g in a l p a ra m eters, t h i s c o u ld be acco m p lish ed by an a p p r o p r ia te
c h a n g e -o f-v a r ia b le s in param eter sp a ce a p p lie d to Eqs. (4.2.5) and
(4 .2 .6 ). Many a p p lic a t io n s in p h a r m a c o k in e tic s, how ever, in v o lv e
p r e d ic tio n and c o n t r o l. Thus, th e m odel can j u s t a s w e ll be d e sc r ib e d
in term s o f th e tran sform ed param eters a^, a^, 3^^, ^ 2 s-s in th e o r i
g in a l param eters k^^' ^ 2 1 ’ ^ 1 0 ’
22
CHAPTER 5
SU M M A R Y AND CONCLUSIONS
5 .1 D is c u s s io n
A m ethod b ased on li n e a r le a s t - s qua r e s th eo ry and B a y esia n a n a ly
s i s has b een p r e se n te d fo r param eter e s tim a tio n in s e m i- lin e a r m o d els.
As an exam p le, a fo u r -p a r a m e te r , tw o-com partment p h a rm a co k in etic m odel
o f lid o c a in e d is p o s i t io n was a n a ly ze d u sin g a c h a n g e -o f-v a r ia b le s in
o rd er t o o b ta in a m odel th a t was li n e a r in two p a ra m eters.
A lthough t h i s method has b een a p p lie d to a tw o-com partm ent m od el,
i t can be a p p lie d to any m odel th a t i s r e p r e se n te d by a lin e a r dynami
c a l sy stem w ith s i n g l e in p u t and s i n g l e o u tp u t.
I f a sy stem o f th e typ e d e sc r ib e d in Eq. (4.2.1) i s of d im en sion
n , th en a t m ost 2n param eters can b e i d e n t i f i e d (K atz e t a l . [10]).
There i s alw ays a lin e a r tr a n sfo r m a tio n th a t r e s u l t s in an e q u iv a le n t
sy stem in w hich th e ou tp u t i s li n e a r in n o f th e new p a ra m eters.
H ence, th e r e s u lt o b ta in ed h e r e now a llo w s fo r th e e x a c t com p u tation of
e s tim a te s o f sy stem p aram eters fo r a la r g e c la s s o f im p ortan t m o d els.
23
REFERENCES
1 . A f i f i , A. A. and A zen, S. P. (1979). S t a t i s t i c a l A n a ly s is ; A
Computer O rien ted A pproach. 2nd Ed.. Academ ic P r e s s , New York.
2 . A rn old , V. I . (1973). O rdinary D i f f e r e n t i a l E q u a tio n s. MIT P r e s s ,
Cam bridge, M a ssa c h u se tts.
3. Bard, Y. (1974). N o n lin e a r Param eter E s tim a tio n . Academic P r e s s ,
New York.
4 . D 'A rg en io , D. Z. and Schum itzky, A. (1979). A Program Package fo r
S im u la tio n and Param eter E stim a tio n in P h arm acok in etic S ystem s.
Computer Programs in B io m ed icin e 9 , 1 1 5 -1 3 4 .
5 . D eu tsch , R. (1965). E stim a tio n T heory. P r e n t ic e - H a ll, New J e r s e y .
6 . G elb, A ., e d it o r (1974). A p p lied O ptim al E s tim a tio n . MIT P r e s s ,
Cam bridge, M a ssa c h u se tts.
7. H o el, P. G ., P o r t, S. G ., and S to n e , C. J . (1 9 7 1 ). In tr o d u c tio n to
P r o b a b ilit y T heory. Houghton M if f lin Company, B o sto n ,
M a ss a c h u s e tts .
8, Katz, D. (1980). A B a y esia n Approach to th e A n a ly s is o f N o n lin ea r
M odels w ith A p p lic a tio n s in P h a rm a co k in e tics. Ph.D. D is s e r t a
t i o n , U n iv e r s ity o f Southern C a lif o r n ia , Los A n g e le s , C a lif
o r n ia .
9. K atz, D ,, A zen, S. P ., and Schum itzky, A. (1 9 8 1 ). B ayesian
Approach to th e A n a ly s is o f N o n lin ea r M odels : Im plem en tation
and E v a lu a tio n . B io m e tr ic s 3 7 , 1 3 7 -1 4 2 ,
24
1 0 . K atz, D ., S ch u m itzk y, A ., and A zen, S . P. (1982). R ed u ction o f
D im e n sio n a lity in B a y esia n N o n lin ea r R e g r e ssio n w ith a Pharma
c o k in e t ic s A p p lic a tio n . M ath em atical B io s c ie n c e s 5 9 , 4 7 -5 6 .
1 1 . L u en b erger, D. G. (1969). O p tim iz a tio n by V ecto r Space M ethods.
John W iley and S o n s, New Y ork.
1 2 . S im s, F. L. and L a i n i o t i s , D. G. (1969). R ecu rsiv e A lgorith m fo r
th e C a lc u la tio n o f th e A d a p tiv e Kalman F i l t e r W eigh tin g Co
e f f i c i e n t s . IEEE T r a n sa c tio n s on A utom atic C o n tr o l, 2 1 5 -2 1 6 .
25

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Core Title Semi-linear regression problems with an application in pharmacokinetics

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