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A three compartment open model with two time lags
Int J Biomed Comput, 32 (1993) 269-211
269
Elsevier Scientific Publishers Ireland Ltd.
A THREE COMPARTMENT
OPEN MODEL WITH TWO TIME LAGS
Y. CHERRUAULTB and V.B. SARINb ‘Medimat, Universite Paris VI, IS, Rue de I’Ecole de Medecine. 75270, Paris, Cedex 06 (France) and bCentre for Atmospheric and Fluids Sciences, Indian Institute of Technology, Hauz Khas, New Delhi110016 (India)
(Received October Sth, 1992) (Accepted November 9th, 1992)
The present study deals with the identification of exchange parameters involved in a three-compartment open model with two time lags in which elimination occurs from the central compartment. Two different optimization methods have been used which involve the reduction of different unknowns to a single variable 0, with the help of Archimedes Spiral. Thus, the solution requires the global minimum of a functional of single variable 13.Results are compared with those obtained by the generalized least square method. Key words: Compartment;
Time lag; Optimization exchange parameters; Absorption constant
1. Introduction Although the concept of compartments has no fixed pecularities, circumstances which arise frequently in one field or another may serve to emphasize certain aspects of compartment theory. One of the least complicated instances of applications of compartment theory occurs in the physiological studies of exchange of inert gases in mammals. The phenomenon of nitrogen exchange by and elimination from the various tissues via the lung and circulation was studied experimentally in dogs and man from the view point of determining cardiac output, determinig body composition and prevention of decomposition sickness [l]. In effect, these studies used a three-compartment closed system as a model, wherein one compartment represented the body fat plus lipoid tissues, another compartment represented the aqueous spaces and the third compartment represented the expired breath and oxygen of the subject’s closed environment. The effectively infinite volume of the third compartment restricted the transport of nitrogen essentially one way. Because of the large and still increasing interest in the problems of drug kinetics, it is necessary to evolve considerable sophistication in evaluating them and in applying them to specific cases at hand. The problem of absorption and elimination by an organism through its natural processes are, in reality, quite complicated and could be analysed more easily after an explicatory formulation of the general compartment theory is presented [2,3]. The blood and all readily accessible fluids and tissues may be treated kinetically as a common homogeneous unit which is referred 0020-7101/93/SO6.00 0 1993 Elsevier Scientific Publishers Ireland Ltd. Printed and Published in Ireland
270
Y. Cherruault and V.B. Sarin
to as the central compartment. In this study, we consider a three compartment open model (Fig. 1) involving two different time lags. Cherruault and Sarin [4] have studied the pharmacokinetics of cimetidine in humans with the help of a two compartment open model. Figure 1 represents schematically a three compartment open model, with first order absorption, k,. The drug is presented in the compartment G, called the gut. A fraction J’G, of the dose D is absorbed by a first order process into the central compartment 1. The remaining fraction, 1 - FGI, of the drug goes into compartment B, by first-pass transfer process. The hepatic parenchymal tissue and the bila phase are the most likely storage areas. No assumptions are made about the type transfer process from G to B because the kinetic behaviour of the system does not depend on the rate of input into B but only on the amount of drug in B at a particular time, TB. At this time, a fraction FB, of the drug accumulated in B is released momentarily into compartment G. The drug is eliminated from the central compartment 1 and transferred into compartment B and two peripheral compartments 2 and 3, by first order processes. The basic system (Fig. 1) is described by a set of ordinary coupled differential equations for concentrations Cl(t) and Cz(t) in central compartments for two different phases respectively. The problem involves the identification of various exchange parameters k, (i = 1 to 3, j = 1 to 3, i # j), elimination rate constant klo, transfer rate constant klB, absorption rate constant k,; two time lags TL and TB, fraction of doses FG1, FB and dose DB, from a set of given observations. The solution of the basic system of equations involves two optimization methods, which are deterministic in nature and use the approximation properties of Archimedes spirals. The first method [5], designated as Alienor technique, gives a point R” realizing the global minimum of n-variable function f(xl,xz , . . . , x,,). Cherruault and Sarin have used this method to obtain the pharmacokinetics of(i) a general mammillary system [6] and (ii) a six compartment model to study the kinetics of 14C-labelled glucose [7]. The second method [8], designated as the Gabriel tech-
Fig. 1. Schematic representation of a three compartment open model with first order absorption (k,), where elimination occurs from compartment 1.
271
Three comparrment model
nique, shows improvement over the Alienor technique in terms of computational time and accuracy of obtaining global minimum. Figure 2 shows the different structures of the two methods. While the Alienor technique possesses a tree structure, the Gabriel technique shows a parallel structure. These techniques give an efficient way for drawing dense paths in R”. Furthermore, these techniques do not use derivatives as is usual in the case of classification optimization methods. 2. Mathematical Formulation The concentration C(t) of the drug in the central compartment 1 at a given instant of time t, can be divided into two parts, relevant to the two different phases. Thus, we can write CO)
= C,(t)
+ C20)
(1)
where C,(t) and C2(t) represent the concentrations of drug for the two phases. If F is the total fraction of the given dose absorbed, the amount of the drug absorbed can be written as FD = FG,D + FBDB
(2)
where D is the given dose; FGI, FB are the fraction of the doses absorbed in two phases and D, is the dose present in compartment B in the second phase. The basic set of equations describing the given system (Fig. 1) is given by the following set of differential equations: G(t) ___
dt
= -&2
+ h
+ ho
+ k3p3(t)
+ h&d~)
+ k21X2Q)
+ k,D, ewka’
(34
Second level First lewl ALIENOR x2
Xl
x3
X4
GABRIEL r-i-n Xl
x2
x3
x4
Fig. 2. Schematic representation of the Alienor and the Gabriel techniques.
212
Y. Cherruault and V. B. Sarin
d+(t) dt
dx,(t)
-
dt
= h2GO)
-
=
+ k31X3Q)
kl3W)
WO
klZX20)
(34
Here C,(t), x*(t) and x3(t) represent the concentration of drug in the central compartment 1 and compartments 2 and 3, respectively, at the instant of time, t. k, (i = 1 to 3, j = 1 to 3, i # j) are the various exchange parameters involved in the given system. k, and klo are the first order absorption and elimination rate constants respectively, while klB represents the exchange parameter from compartment 1 to compartment B. Various exchange parameters (k,, klo, k,, klB) involved in the given system are considered to constants so that the set of Eqns. 3a-3c are linear in nature. At the instant of time, t = TL, the conditions provided are as follows:
C,(t) = 0, x2(t) = 0, x3(t) = 0, x&t)
FGI = -y= D1
Here xG(t) represents the concentration of drug at time t in the compartment G and V represents the unknown distribution volume for the central compartment 1. The concentration C2(t) for the second phase is given by the set of Eqns. 3a-3c, with C,(t) being replaced by C2(t). The initial concentrations at time t = TB(TB > TL), are given as follows: C2(t) = 0, x2(t) = 0, x3(t) = 0,
FBDB x&t) = v = D2
The set of Eqns. 3a-3c along with conditions (4) and (5) for initial concentrations describe completely the basic system (Fig. 1). The problem is to identify the speed constants kJi = 1 to 3, j = 1 to 3, i #j), klB, absorption and elimination rate constants k,, kl,-,, fraction of doses FGI, FB, distribution volume V and drug dose DB from a knowledge about the system in the form of given concentrations C(t) in the central compartment 1, at different times of observations. The approach of solution involves two optimization technique (Alienor and Gabriel [8]). These methods are deterministic in nature and reduce n variable to a single variable with the help of dense parts in R”. These paths allow one to easily find the global optima (maxima or minima). 3. Method of Solution The form of basic set of Eqns. 3a-3c along with conditions (4) and (5) for initial concentrations, suggests the solution for concentration C(t) (Eqn. 1) in the following form:
k, C(t) = v
FGID
i i=l
where b = k,, Xi > 0.
ai
e%‘)-
TL) +
FBDB
i i=
ai
I
e-“i(t-
TB) 1
(6)
Three compartment
213
model
Here hi, i = 1 to 4, are the eigenvalues of the characteristic matrix of the given system (3a)-(3c) and are governed by the following set of relations:
A, + A* + A3 = hh2
+ A2h3
hh2A3
k,2
+ k2,
+ x3x,
= (k,,
= &IO
+ ‘h3
+ ho
+ k3l
+ k10
+ k,B)k21
+ klB
+ (b2
(W + ‘ho
+ ‘bB)k31
+ k21k31
(7b)
(7c)
+ klB)k21k31
The coefficients ai, i = 1 to 4 satisfy the following relations: a1
(‘b
+
a2 + a3 + a4 = 0 + x2
+ (k,
(02
+ h3)al + Xl
+ (0,
+ (k
+ X2)U3
+ 03
(W + h + (A,
+ ~2~3)al
+ 02
+ X3)a2 + X2 + A4)U4
+ (W,
+ hlh2)a3
+ kJ3
+ @,x2
= U4 = k,Dl
(8b)
+ hA3)a2
+ h2x3
+ ~3bb4
= kA(k2,
+ k3,)
X2X3kaal+ X3Xlkaa2 + XlX2k,a3 + X,X&a4 = kaDlk21k31
Xa, the amount of drug transferred from compartment Ta iS given by the following relations. xB
= k,B~i
j c(Wl,=
(86
(W
1 to compartment B at time
(9)
TB
With help of Eqns. l-5, Eqn. 9 gives Xg as follows: XB
= klBkaFG,D
+ !!_
(1
2
_
e-h3(TB
x3
and DB = XB - (1 - FdD
(1
_
e-xlfTB
- TL))
- TL))
+ !% x,
(1
_
+ $
e-hcTB
(1
-
evh2tTB
- TL))
1
- TL))
(10)
(11)
The Alienor Technique
This technique [5] involves the transformation of various unknowns (i = 1 to 4), aXi = 1 to 3), TL, TB, FB, Dg, FG,, V to a single variable 8 (0 2 0) by using properties of Archemedes spiral. The constant a4 is governed by the relation (8a). Time lag
274
Y. Cherruault and V.B. Sarin
TL is less than time lag TB and fractions of dose of drug, FB and FGI, lie between 0 and 1. The above-mentioned unknowns are identified by introducing a functional 6 as follows: N
C[
Y=
C&j)
t
j= 1
i
e-Ak(‘i -
TL)
VW
II
ak e-‘k%’- TB)
k=l zO =
ak
k=l
4
FBDB c
+
f:
FGID
cgCtj>
U2b)
j=l
6 = ;(YN)‘”
WC)
Here N is the number of given observations and a4 is given by relation @a). No exact solution exists for Eqn. 12a (Y = 0), and we seek the global minimum of the functional 6 (Eqn. 12~). In order to obtain the global minimum of 6, we introduce a single variable 8, with the help of following relations: h, = ABS
,
y=
al =
ABS
(Y6
TB = ABS
FGI
cos ‘26 ,
a2 =
a6
Q7 cos (r7
Al Q, = 81 cos PI
9
Al
9 cxj z O(i = 1 to 7)
,
a2 =
9
sin (Yg
Al a3 =
ABS
=
9
81 sin PI Al
9
(134
Three compartment model a2 cos 82
cY3 =
,
cY4=
02 sin 82
Al 83 cos 63
83 sin 03 ,
(Iy,j=
,
Al
03 =
f
Al
015 =
(Y-j =
Al
p4 ‘0, O4 ,
fli Z 0 (i = 1 to 4)
Y2cAqsY2
fi4 =
ecos e Yl =-,
275
Al
)
72 =-,
T2 y2
e sin 0
Al
)
yi
820
WI
Z
0 (i = 1 to 2)
(13c)
(134
Relations (13a)-( 13d) help us to minimise 6 over the variable 8. Here Al is a constant, sufficiently large. For numerical purposes, this value is of order 10. F, the total fraction of dose absorbed is given by Eqn. 2. Xs, the amount of drug transferred from compartment 1 to compartment B, at time TB is given by Eqn. 11 and exchange parameter, klB, identified from Eqn. 10. The exchange parameters k 2, and k3, are obtained for relations (8~) and (8d), while exchange parameters k12 and k21 and elimination rate constant klo are given by relations (7a)-(7c). The Gabriel Technique
This technique is the generalized form of the Alienor technique [5]. Various unknowns introduced ((Y,B and y values) in relations (13a)-( 13d) are rewritten by introducing a constant q , as follows:
rl =
e-
r2 = 8 -
ql,
(14)
q2
where qi depends on 8 and similiar relations hold good for CYand @values. Relations (13d) and (14) imply that
q,=e--,
8 cos
Al
e
q2=e--
8 sin
e
(15)
Al
Relation (14) gives the generalized form of the Alienor technique, with constants qi dependent on 8. The Gabriel technique involves the treatment of the constants qi as independent
276
Y. Cherruault and V.B. Sarin
of 8. The general form of the Gabriel transformation
(e‘--,qil ;; (e -
xi =
can be written as
(16)
qj)
qi being constants and independent of 0. This transformation leads to a set of points ‘dense’ in R”. By ‘dense’, we mean that in every neighbourhood of R2, one can find at least a point given by the Gabriel transformation provided that A, is sufficiently large. For the present case, the value of Ai is taken to be 32 and emax, the bound of the variable 8 is taken as
e*ax
=
64(n - 1)
(17)
where n represents the number of unknowns involved in the problem. In the case of the Alienor technique, the upper bound of 8 is given by
where S is the number of levels (see Fig. 2). The calculation time T is proportional to emax/0.2 (A0 = 0.2, the discretization step) which shows that the time is proportional to the number of variables involved in the problem. Results and Discussion Four sets of hypothetical data are tested for both the techniques (Alienor and Gabriel) and results with compared which are compared with those obtained by generalized least squares method. Table I provides the value of 6 as obtained by each method. The Gabriel technique shows improvement over the remaining two methods. The computational time is also comparatively less in the case of the Gabriel technique. The present techniques allow the deviation of the global minimum of the deviation 6 from the observed values, which in turn helps us to provide the accurate solution of Y = 0.
TABLE I DEVIATION (6) FROM THE OBSERVED VALUES Columns (a), (b) and (c) give the value of 6 obtained by the Gabriel technique, Alienor technique and the generalised least squares method. Subject
(a)
(b)
(c)
1 2 3 4
0.415 0.440 0.437 0.490
0.670 0.725 0.695 0.767
2.921 3.225 3.212 3.675
Three compartmenr model
211
References 1 2 3 4 5 6 7 8
Behnke AR, Thompson RM and Shaw LA: The rate of elimination of dissolved nitrogen in man in relation to the fat and water contents of body. Am J Physiol, 114 (1935) 137-146. Jacquez JA: Compartmental Analysis in Biology and Medicine, Elsevier, New York, 1972. Wagner JG: Fundamentals of Clinical Phannacokinetics, Drug Intelligence Publications, Hamilton, IL 1975. Cherruault Y and Sarin VB: Pharmacokinetics of Cimetidine in humans, Int .I Biomed Comput, 18 (1985) 123-130. Cherruault Y and Guillez A: A simple method for optimization, Kybernetes. 12 (1983) 59-63. Cherruault Y and Sarin VB: General treatment of linear mammillary models, Inr J Eiomed Cornput, 16 (1985) 119-126. Cherruault Y and Sarin VB: A six compartmental model to study the kinetics of 14C-labelled glucose, Kybernetes, 20(2) (1991) 29-34. Cherruault Y: A new method for global optimization (ALIENOR), Kybernetes, 19(3) (1990) 19-32.
269
Elsevier Scientific Publishers Ireland Ltd.
A THREE COMPARTMENT
OPEN MODEL WITH TWO TIME LAGS
Y. CHERRUAULTB and V.B. SARINb ‘Medimat, Universite Paris VI, IS, Rue de I’Ecole de Medecine. 75270, Paris, Cedex 06 (France) and bCentre for Atmospheric and Fluids Sciences, Indian Institute of Technology, Hauz Khas, New Delhi110016 (India)
(Received October Sth, 1992) (Accepted November 9th, 1992)
The present study deals with the identification of exchange parameters involved in a three-compartment open model with two time lags in which elimination occurs from the central compartment. Two different optimization methods have been used which involve the reduction of different unknowns to a single variable 0, with the help of Archimedes Spiral. Thus, the solution requires the global minimum of a functional of single variable 13.Results are compared with those obtained by the generalized least square method. Key words: Compartment;
Time lag; Optimization exchange parameters; Absorption constant
1. Introduction Although the concept of compartments has no fixed pecularities, circumstances which arise frequently in one field or another may serve to emphasize certain aspects of compartment theory. One of the least complicated instances of applications of compartment theory occurs in the physiological studies of exchange of inert gases in mammals. The phenomenon of nitrogen exchange by and elimination from the various tissues via the lung and circulation was studied experimentally in dogs and man from the view point of determining cardiac output, determinig body composition and prevention of decomposition sickness [l]. In effect, these studies used a three-compartment closed system as a model, wherein one compartment represented the body fat plus lipoid tissues, another compartment represented the aqueous spaces and the third compartment represented the expired breath and oxygen of the subject’s closed environment. The effectively infinite volume of the third compartment restricted the transport of nitrogen essentially one way. Because of the large and still increasing interest in the problems of drug kinetics, it is necessary to evolve considerable sophistication in evaluating them and in applying them to specific cases at hand. The problem of absorption and elimination by an organism through its natural processes are, in reality, quite complicated and could be analysed more easily after an explicatory formulation of the general compartment theory is presented [2,3]. The blood and all readily accessible fluids and tissues may be treated kinetically as a common homogeneous unit which is referred 0020-7101/93/SO6.00 0 1993 Elsevier Scientific Publishers Ireland Ltd. Printed and Published in Ireland
270
Y. Cherruault and V.B. Sarin
to as the central compartment. In this study, we consider a three compartment open model (Fig. 1) involving two different time lags. Cherruault and Sarin [4] have studied the pharmacokinetics of cimetidine in humans with the help of a two compartment open model. Figure 1 represents schematically a three compartment open model, with first order absorption, k,. The drug is presented in the compartment G, called the gut. A fraction J’G, of the dose D is absorbed by a first order process into the central compartment 1. The remaining fraction, 1 - FGI, of the drug goes into compartment B, by first-pass transfer process. The hepatic parenchymal tissue and the bila phase are the most likely storage areas. No assumptions are made about the type transfer process from G to B because the kinetic behaviour of the system does not depend on the rate of input into B but only on the amount of drug in B at a particular time, TB. At this time, a fraction FB, of the drug accumulated in B is released momentarily into compartment G. The drug is eliminated from the central compartment 1 and transferred into compartment B and two peripheral compartments 2 and 3, by first order processes. The basic system (Fig. 1) is described by a set of ordinary coupled differential equations for concentrations Cl(t) and Cz(t) in central compartments for two different phases respectively. The problem involves the identification of various exchange parameters k, (i = 1 to 3, j = 1 to 3, i # j), elimination rate constant klo, transfer rate constant klB, absorption rate constant k,; two time lags TL and TB, fraction of doses FG1, FB and dose DB, from a set of given observations. The solution of the basic system of equations involves two optimization methods, which are deterministic in nature and use the approximation properties of Archimedes spirals. The first method [5], designated as Alienor technique, gives a point R” realizing the global minimum of n-variable function f(xl,xz , . . . , x,,). Cherruault and Sarin have used this method to obtain the pharmacokinetics of(i) a general mammillary system [6] and (ii) a six compartment model to study the kinetics of 14C-labelled glucose [7]. The second method [8], designated as the Gabriel tech-
Fig. 1. Schematic representation of a three compartment open model with first order absorption (k,), where elimination occurs from compartment 1.
271
Three comparrment model
nique, shows improvement over the Alienor technique in terms of computational time and accuracy of obtaining global minimum. Figure 2 shows the different structures of the two methods. While the Alienor technique possesses a tree structure, the Gabriel technique shows a parallel structure. These techniques give an efficient way for drawing dense paths in R”. Furthermore, these techniques do not use derivatives as is usual in the case of classification optimization methods. 2. Mathematical Formulation The concentration C(t) of the drug in the central compartment 1 at a given instant of time t, can be divided into two parts, relevant to the two different phases. Thus, we can write CO)
= C,(t)
+ C20)
(1)
where C,(t) and C2(t) represent the concentrations of drug for the two phases. If F is the total fraction of the given dose absorbed, the amount of the drug absorbed can be written as FD = FG,D + FBDB
(2)
where D is the given dose; FGI, FB are the fraction of the doses absorbed in two phases and D, is the dose present in compartment B in the second phase. The basic set of equations describing the given system (Fig. 1) is given by the following set of differential equations: G(t) ___
dt
= -&2
+ h
+ ho
+ k3p3(t)
+ h&d~)
+ k21X2Q)
+ k,D, ewka’
(34
Second level First lewl ALIENOR x2
Xl
x3
X4
GABRIEL r-i-n Xl
x2
x3
x4
Fig. 2. Schematic representation of the Alienor and the Gabriel techniques.
212
Y. Cherruault and V. B. Sarin
d+(t) dt
dx,(t)
-
dt
= h2GO)
-
=
+ k31X3Q)
kl3W)
WO
klZX20)
(34
Here C,(t), x*(t) and x3(t) represent the concentration of drug in the central compartment 1 and compartments 2 and 3, respectively, at the instant of time, t. k, (i = 1 to 3, j = 1 to 3, i # j) are the various exchange parameters involved in the given system. k, and klo are the first order absorption and elimination rate constants respectively, while klB represents the exchange parameter from compartment 1 to compartment B. Various exchange parameters (k,, klo, k,, klB) involved in the given system are considered to constants so that the set of Eqns. 3a-3c are linear in nature. At the instant of time, t = TL, the conditions provided are as follows:
C,(t) = 0, x2(t) = 0, x3(t) = 0, x&t)
FGI = -y= D1
Here xG(t) represents the concentration of drug at time t in the compartment G and V represents the unknown distribution volume for the central compartment 1. The concentration C2(t) for the second phase is given by the set of Eqns. 3a-3c, with C,(t) being replaced by C2(t). The initial concentrations at time t = TB(TB > TL), are given as follows: C2(t) = 0, x2(t) = 0, x3(t) = 0,
FBDB x&t) = v = D2
The set of Eqns. 3a-3c along with conditions (4) and (5) for initial concentrations describe completely the basic system (Fig. 1). The problem is to identify the speed constants kJi = 1 to 3, j = 1 to 3, i #j), klB, absorption and elimination rate constants k,, kl,-,, fraction of doses FGI, FB, distribution volume V and drug dose DB from a knowledge about the system in the form of given concentrations C(t) in the central compartment 1, at different times of observations. The approach of solution involves two optimization technique (Alienor and Gabriel [8]). These methods are deterministic in nature and reduce n variable to a single variable with the help of dense parts in R”. These paths allow one to easily find the global optima (maxima or minima). 3. Method of Solution The form of basic set of Eqns. 3a-3c along with conditions (4) and (5) for initial concentrations, suggests the solution for concentration C(t) (Eqn. 1) in the following form:
k, C(t) = v
FGID
i i=l
where b = k,, Xi > 0.
ai
e%‘)-
TL) +
FBDB
i i=
ai
I
e-“i(t-
TB) 1
(6)
Three compartment
213
model
Here hi, i = 1 to 4, are the eigenvalues of the characteristic matrix of the given system (3a)-(3c) and are governed by the following set of relations:
A, + A* + A3 = hh2
+ A2h3
hh2A3
k,2
+ k2,
+ x3x,
= (k,,
= &IO
+ ‘h3
+ ho
+ k3l
+ k10
+ k,B)k21
+ klB
+ (b2
(W + ‘ho
+ ‘bB)k31
+ k21k31
(7b)
(7c)
+ klB)k21k31
The coefficients ai, i = 1 to 4 satisfy the following relations: a1
(‘b
+
a2 + a3 + a4 = 0 + x2
+ (k,
(02
+ h3)al + Xl
+ (0,
+ (k
+ X2)U3
+ 03
(W + h + (A,
+ ~2~3)al
+ 02
+ X3)a2 + X2 + A4)U4
+ (W,
+ hlh2)a3
+ kJ3
+ @,x2
= U4 = k,Dl
(8b)
+ hA3)a2
+ h2x3
+ ~3bb4
= kA(k2,
+ k3,)
X2X3kaal+ X3Xlkaa2 + XlX2k,a3 + X,X&a4 = kaDlk21k31
Xa, the amount of drug transferred from compartment Ta iS given by the following relations. xB
= k,B~i
j c(Wl,=
(86
(W
1 to compartment B at time
(9)
TB
With help of Eqns. l-5, Eqn. 9 gives Xg as follows: XB
= klBkaFG,D
+ !!_
(1
2
_
e-h3(TB
x3
and DB = XB - (1 - FdD
(1
_
e-xlfTB
- TL))
- TL))
+ !% x,
(1
_
+ $
e-hcTB
(1
-
evh2tTB
- TL))
1
- TL))
(10)
(11)
The Alienor Technique
This technique [5] involves the transformation of various unknowns (i = 1 to 4), aXi = 1 to 3), TL, TB, FB, Dg, FG,, V to a single variable 8 (0 2 0) by using properties of Archemedes spiral. The constant a4 is governed by the relation (8a). Time lag
274
Y. Cherruault and V.B. Sarin
TL is less than time lag TB and fractions of dose of drug, FB and FGI, lie between 0 and 1. The above-mentioned unknowns are identified by introducing a functional 6 as follows: N
C[
Y=
C&j)
t
j= 1
i
e-Ak(‘i -
TL)
VW
II
ak e-‘k%’- TB)
k=l zO =
ak
k=l
4
FBDB c
+
f:
FGID
cgCtj>
U2b)
j=l
6 = ;(YN)‘”
WC)
Here N is the number of given observations and a4 is given by relation @a). No exact solution exists for Eqn. 12a (Y = 0), and we seek the global minimum of the functional 6 (Eqn. 12~). In order to obtain the global minimum of 6, we introduce a single variable 8, with the help of following relations: h, = ABS
,
y=
al =
ABS
(Y6
TB = ABS
FGI
cos ‘26 ,
a2 =
a6
Q7 cos (r7
Al Q, = 81 cos PI
9
Al
9 cxj z O(i = 1 to 7)
,
a2 =
9
sin (Yg
Al a3 =
ABS
=
9
81 sin PI Al
9
(134
Three compartment model a2 cos 82
cY3 =
,
cY4=
02 sin 82
Al 83 cos 63
83 sin 03 ,
(Iy,j=
,
Al
03 =
f
Al
015 =
(Y-j =
Al
p4 ‘0, O4 ,
fli Z 0 (i = 1 to 4)
Y2cAqsY2
fi4 =
ecos e Yl =-,
275
Al
)
72 =-,
T2 y2
e sin 0
Al
)
yi
820
WI
Z
0 (i = 1 to 2)
(13c)
(134
Relations (13a)-( 13d) help us to minimise 6 over the variable 8. Here Al is a constant, sufficiently large. For numerical purposes, this value is of order 10. F, the total fraction of dose absorbed is given by Eqn. 2. Xs, the amount of drug transferred from compartment 1 to compartment B, at time TB is given by Eqn. 11 and exchange parameter, klB, identified from Eqn. 10. The exchange parameters k 2, and k3, are obtained for relations (8~) and (8d), while exchange parameters k12 and k21 and elimination rate constant klo are given by relations (7a)-(7c). The Gabriel Technique
This technique is the generalized form of the Alienor technique [5]. Various unknowns introduced ((Y,B and y values) in relations (13a)-( 13d) are rewritten by introducing a constant q , as follows:
rl =
e-
r2 = 8 -
ql,
(14)
q2
where qi depends on 8 and similiar relations hold good for CYand @values. Relations (13d) and (14) imply that
q,=e--,
8 cos
Al
e
q2=e--
8 sin
e
(15)
Al
Relation (14) gives the generalized form of the Alienor technique, with constants qi dependent on 8. The Gabriel technique involves the treatment of the constants qi as independent
276
Y. Cherruault and V.B. Sarin
of 8. The general form of the Gabriel transformation
(e‘--,qil ;; (e -
xi =
can be written as
(16)
qj)
qi being constants and independent of 0. This transformation leads to a set of points ‘dense’ in R”. By ‘dense’, we mean that in every neighbourhood of R2, one can find at least a point given by the Gabriel transformation provided that A, is sufficiently large. For the present case, the value of Ai is taken to be 32 and emax, the bound of the variable 8 is taken as
e*ax
=
64(n - 1)
(17)
where n represents the number of unknowns involved in the problem. In the case of the Alienor technique, the upper bound of 8 is given by
where S is the number of levels (see Fig. 2). The calculation time T is proportional to emax/0.2 (A0 = 0.2, the discretization step) which shows that the time is proportional to the number of variables involved in the problem. Results and Discussion Four sets of hypothetical data are tested for both the techniques (Alienor and Gabriel) and results with compared which are compared with those obtained by generalized least squares method. Table I provides the value of 6 as obtained by each method. The Gabriel technique shows improvement over the remaining two methods. The computational time is also comparatively less in the case of the Gabriel technique. The present techniques allow the deviation of the global minimum of the deviation 6 from the observed values, which in turn helps us to provide the accurate solution of Y = 0.
TABLE I DEVIATION (6) FROM THE OBSERVED VALUES Columns (a), (b) and (c) give the value of 6 obtained by the Gabriel technique, Alienor technique and the generalised least squares method. Subject
(a)
(b)
(c)
1 2 3 4
0.415 0.440 0.437 0.490
0.670 0.725 0.695 0.767
2.921 3.225 3.212 3.675
Three compartmenr model
211
References 1 2 3 4 5 6 7 8
Behnke AR, Thompson RM and Shaw LA: The rate of elimination of dissolved nitrogen in man in relation to the fat and water contents of body. Am J Physiol, 114 (1935) 137-146. Jacquez JA: Compartmental Analysis in Biology and Medicine, Elsevier, New York, 1972. Wagner JG: Fundamentals of Clinical Phannacokinetics, Drug Intelligence Publications, Hamilton, IL 1975. Cherruault Y and Sarin VB: Pharmacokinetics of Cimetidine in humans, Int .I Biomed Comput, 18 (1985) 123-130. Cherruault Y and Guillez A: A simple method for optimization, Kybernetes. 12 (1983) 59-63. Cherruault Y and Sarin VB: General treatment of linear mammillary models, Inr J Eiomed Cornput, 16 (1985) 119-126. Cherruault Y and Sarin VB: A six compartmental model to study the kinetics of 14C-labelled glucose, Kybernetes, 20(2) (1991) 29-34. Cherruault Y: A new method for global optimization (ALIENOR), Kybernetes, 19(3) (1990) 19-32.