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General linear compartment model with zero input: I. Kinetic equations
BioSystems 36 (1995) 121-133
General linear compartment model with zero input: I. Kinetic equations R. Varch* a, M.J. Garcia-Meseguera,
F. Garcia-Chovasb,
B. Havsteenc
aDepartamento de Quimica-Fisica. Universidad de Castilla-Lu Man& Albacete. Spain bDepartamento de Bioquimica y Biologia Molecular, Utu’versidadde Murcia, Murk. Spain cBiochemischesInstitut. Christian-Albrechts-Universitiit su Kiel, Kiel, Germany Received 4
April 1995;accepted18April 1995
Ah&act
The derivation of kinetic equations is described for n-compartment linear models, in which the substance may be simultaneouslyintroduced into one or more compartments at t = 0 and eliminated from any compartment. For a given zero-input, general formulas are derived which describe the amount of tracer in any of the compartments as a function of time and the model parameters. New algorithms have been developed which allow the expression of the kinetic equations. Keywordr:
Compartment systems; Enzyme kinetics; Kinetics equations; Linear models; Tracer
1. Introduction
Compartment systems are important for the description of many aspects of biology, engineering, economics and several other sciences. Standard and very complete references on compartment modelling and analysis are those of Rescigno (1956), Godfrey (1983), Anderson (1983) and Jacquez (1985). Some areas of interest related to studies of compartment systems are the tracer ??Corre-spondiig author, Departamento de Quimica-Fisica, Escuela Universitaria Polit&nka, Campus Universitario, E02071 Albacete, Spain. Tel.: +34 67599200, ext. 2480; Fax: +34 67599224.
isotope methods, the parameter identification and the evaluation of mean pharmacokinetics parameters. Tracer isotope labelling methods are essential for the elucidation of many aspects of metabolism and transport processes in biochemistry (Manara, 1971; Netter, 1969). Some of these tracers, e.g. fluorescent tags, are permanently linked to a substance, the fate of which is to be followed. Such labels faithfully track the course of its carrier across membranes, along guiding filaments, through tubuli and during carrier rides. These tracers aid the definition of the compartment structure. The identification of parameters is one of the
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R. Varbn et al. / BioSystems 36 (1995) 121-133
most dynamic areas of systems theory. Many contributions treat theoretical aspects or applications, e.g. to biosciences, engineering and economics. The literature on the subject is complex. Several efforts have been made to define, classify and unify the concepts (Astrom and Eykhoff, 1971; Ngugen and Wood, 1982). The definitions of identifiability differ, but several necessary and sufficient conditions for identifiability have been found (Walter and Le Cardinal, 1976; Pohjanpalo and Wahlstrom, 1977; Delforges, 1981; Vadja, 1981; Busenberg, 1987). However, most authors focussed their efforts on uniqueness criteria, i.e. on the determinancy of the problem. The calculation of the model parameters from input-output measurements has so far not been described, except for l-, 2-, 3-compartment models (Rubinow, 1975), for particular cases of n-compartment models (Cobelli et al., 1979; Travis and Haddock, 1981) or for an n-compartment, catenary model (Chau, 1985). Benet (1972) empirically derived a formula giving the concentration-time curve in the central compartment of a general n-compartment, mamillary model. The concentration being expressed as a function of the model parameters. Another problem of the compartment systems is the evaluation of the mean parameters (Jacquez, 1985) and their relation to enzyme kinetics (Sines and Hackney, 1987; Schuster and Heinrich, 1987) as well as for basic pharmacokinetic processes, e.g. drug absorption, drug distribution, drug elimination, metabolites, dosing times, etc. (VengPedersen, 1989). All such studies require the derivation of the kinetic equations for the compartment system as a function of its parameters. Rescigno (1956) derived equations for a system composed of n components assuming that at t = 0 the input only consisted of one component. The equations obtained required the expansion of determinants and the use of certain relations between determinants which became very complex if the system had many compartments. Later, Rescigno and Segre (1965) developed a graphic method which facilitated the task. Graphic methods also have been developed for other compartment systems (Mason and Zimmerman, 1960; Chou, 1989, 1990). However, for systems of some complexity,
the graphic methods are laborious and prone to human errors. Recursive methods facilitate the derivation of the kinetic equations of a linear or more complicated compartment systems (La1 and Anderson, 1990). However, these methods require the handling of matrices and determinants. In this contribution we present an analysis of a general n-compartment model which circumvents many difftculties. We develop algorithms which facilitate the derivation of the kinetic equations. The expressions for the coefftcients in these equations can easily be formulated. This procedure requires neither expansion of determinants nor operations with matrices or graphic methods. 2. Model-notation and deftitions The model consists of a closed n-compartment system with zero input, i.e. substance may at t = 0 simultaneously be injected into one or more of the compartments, which may be strongly connected (i.e., without traps and with an irreducible matrix of the system) or not (i.e. with one or more traps and with a reducible matrix of the system). We
i
i
‘I! Fig. 1. Scheme representing a 9-compartment system by a directed graph. The nodes correspond to the compartments and the lines to the direct connections between the compartments. In this scheme examples of different connections are given. Compartment XI is directly connected with the compartment X2 and indirectly connected with the compartments X3-X9, i.e. compartment X, is connected with the compartments X2-q. Compartment X2 is directly connected to compartments X, and X3 and indirectly connected with the compartments X,-X,, etc. Note that between the compartments XT and X, both a direct and an indirect connection exists. No connection exists between the compartment X3 and X2, Xs and &, etc. The compartment X, is terminal, because it is not connected with any other compartment.
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R. Varbn et al. / BioSystem 36 (1995) 121-133
also assume that the non-null eigenvalues of the system are distinct. We arbitrarily label the compartments: X1, Xz, .... X,. The non-zero fractional transfer coefficients are Kij (Kii denoting transfer from compartment Xi to compartment Xj). The connectivity diagram of our system is a directed graph. The model also includes open systems (i.e. systems with excretion from any compartment) without inputs. If the environment is treated as a single compartment which receives all excretions, the resulting closed system (Jacquez, 1985) tits our model. In this paper the standard definitions and the usual nomenclature in the literature on compartment systems are used. However, some additional definitions, which are necessary in our analysis, are introduced. Fig. 1 shows a directed graph which supports our task. xp (i = 1,2,...,n): Initial amount of the substance in the compartment Xi at t = 0. One or more of the quantities Xi0 may be zero, but at least one of them must be non-zero. Xi (i =
1,2,...$):
The amount of the substance in the compartment xi at time t.
Sum extended to all the elements of the set fl, i.e. k assumes each of the values of the elements of fl.
@: Set of all compartments of the system, i.e. @ = [X,, x2, ...) X,) . In our example, Cp= 1x1, x2,
x4,
x3,
x5,
x6,
xl,
x8,
x9
1.
0’:
Set of all the non-zero fractional transfer coefficients of the system. In the system in Fig. 1: @’=
I&,2,
K4.7,
a(k.l?
K2.1,
K~,E,
K1,9,
K2,3~ K3,4, KS.9 1
K4,3,
h.5,
K
5.6, K6,5, (1)
:
Set of the compartments of the system which simultaneously belong to the set of compartments which are reachable from the compartment Xk, R&), and to the antecedent set of the compartment Xi, Q(Xi), i.e.,
Q(XJ
a(kA = R(Xk) n
(2)
If in Fig. 1 we choose X5 as Xi, then, with Eq. (2) as well as W,) = I x3, &, x5, x6, x7, x8, x9 1, = (X5, x6 1 and QW = 1XI, X2, X3, &, x5, ?&I, we obtain ~(3,s) = (X3, &, X5, X6) and
R(X6)
Cl:
a(V) =
The set of subindeces in the notation of the compartments which contain the substance at t = 0, e.g. if the substance only is at t = 0 present in the compartments X, and X6 of the system in Fig. 1, then Cl= (341. k c Q:
(x5,
x6
1.
Z(fl,i):
Set resulting from the union of all the sets u(k,i) in which k belongs to n, i.e.: E(fl,i) =
u
u(k,i)
(3)
ken
The index k defines one of the elements in the set
n.
In the example above we have C@,5) = (X3, X4, x5,
c
keO
x6).
k-dependent expression: E(Q,i)
’:
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R. Van% et al. / BioSystems
Set of all the non-zero fractional transfer coefficients KVJwhich pertain to the set E(Q,Q. In the example:
36 (1995) 121-133
some enzyme reactions, in which the enzyme species may be considered as compartments. K,, (i = 1,2,..., n):
B(B,5) =
(K3,4,
K4,3 K4,5, kc5
&,s
K4,11
(4)
2.1. Connections between two compartments Between the compartments Xi and Xj exists a direct connection, if a path of the length 1 connects Xi with Xj The compartments Xi and Xj are indirectly connected, if a path connecting Xi with Xj of a length greater than 1 exists. If the compartment Xi directly, indirectly or both is connected with the compartment Xj, then a connection between the compartment Xi and the compartment Xj exists. If a’compartment is not connected to any other compartment, then it is a terminal compdrtment. A terminal compartment is a trap consisting of only one compartment. For any two compartments, Xi and Xj, of a SYStern, one of the following situations exists: (a) The compartment Xi is only directly connected to the compartment Xj; (b) the compartment Xi is only indirectly connected to the compartment Xj; (c) the compartment Xi is both directly and indirectly connected to the compartment Xj; (d) neither a direct nor an indirect connection exists between the compartment Xi and the compartment Xj (see Fig. 1). Kij (i,j = 1,2,..., n; i # 1): Fractional transfer coefficients corresponding to the direct connection between the compartments Xi and Xi. Kij may be zero (if no direct connection exists between the compartments Xi and Xi> or positive. We admit the possibility that two compartments, Xi and Xj, are directly connected in more than one way. In that case, we denote these fractional transfer coefficients as K’, K”ij, . . . Hence: Kij = KG + KG + . . . In the corresponding directed graph only one line segment exists which connects Xi to Xi and possesses a fractional transfer coefficient equal to Ki$ Examples of such a situation are found in
Fractional excretion coefficient when excretion occurs from compartment Xi to the environment.
3. Kinetic analysis The analysis presented here is valid for any linear system with zero input, i.e. without input. It forms the expression for the instantaneous amount of substance, Xi, in any compartment Xi, when at t = 0 substance is injected into one or more compartments. Since xi only influences the fractional transfer coefficients belonging to the set E(Q,z)‘, the original directed graph may be reduced by sup pressing all of the direct connections corresponding to the fractional transfer coefficients of the set *‘-E/(n,i)’ and arbitrarily renumbering the remaining compartments and their connections from XI to X,. That facilitates the task without loss of generality and gives the expression of xi a simple form. If for example, we derive the time-course equation for the amount of substance in compartment X5 of the system in Fig. 1, we have, with the Eqs. (1) and (4) that +’ - (QS)’ = (K1,2, K2,,, the reduced K2,3, Ku, K1.9, Ks,9) . Therefore, directed graph corresponding to X5 is obtained by suppressing in Fig. 1 the direct connections between the compartments Xi and X2, X2 and Xs and X9 (indicated with dashed line in Xl,..., Fig. 2a). The result is the directed graph in Fig. 2b, in which the five remaining’compartments and their connections have been renumbered (note that the new notation for X5 in Fig. 2a is X, in Fig. 2b). By renumbering, the subscripts belonging to D will usually change. In this case, set Q becomes ( 1,5). In the following, we consider the reduced directed graph which allows the derivation of the simplest expression of xP We begin with the analysis of closed linear compartment systems with zero input and then we extend the results obtained to open systems. The system of differential equations describing the evolution of the amount of the substance in each compartment of the system is:
R. Vardn et al. / BioSystems 36 (1995) 121-133
A.3 = Kl,jXI + dt
K~~xz + ... + &jxn (j = 1,2, . . . ,n)
(6)
125
following two relations summarize the possible values of the constants in the system of differential equations:
where KjJ 0’= 1,2,...,n) is defined as: KVJ z O(vj= Kj,j = -
i j=
Kj y
(7)
1,2,...,n;j
# v) (8)
KjjI Oo'= 1,2,...,n)
I
j#v
Note that KjJmay only be zero or negative. The
-*- ......._
‘$!r?
Xl{’ d.--‘-
)
”
.xs.
7
The set of Eqs. (6) is a homogeneous system of first order, linear differential equations with constant
4
7 ‘._ ‘...
...*
\.,
. .. . x;’ .:’ .t .’
....
”
‘..a.’ x9
Fig. 2. (a) Directed graph of Fig. 1 in which the arms corresponding to the fractional transfer coeffkients, that do not influence in the instantaneous amount of substance in X, (i.e. x5) when U = [3,6 ) , are denoted by dashed lines. (b) Reduced directed graph obtained from Fig. 1 corresponding to the compartment X5 when 0 = ( 3,6 1. With the arbitrary numbering used for this reduced directed graph, the compartmtnts X3, Xg, XT, Xs and X, in the original directed graph in (a) are denoted X,, X2, X,, X4 and X,, respectively. Hence, the set Clin (a) hecomes ( 1,5) in (b).
126
R. Varh et al. /BioSystems 36 (1995) 121-133
coefficients. The Laplace transform of any of the Eqs. (6) yields the following expression for the Laplace transform Of Xi (i = 1,2, . . . JZ):
(-l)‘+ ’ c L(Xi)
(-l)kDk.J(s)X”k
kotl
=
D(s)
(i=
1,2,...,n)
(10). This procedure becomes very tedious when the compartment system is complex. In Appendix A we show how the coefficients Fq can easily be obtained. The polynomial D(S) always has one, but it may have c (1 I; c < n) null roots. In the latter case, according to the polynomial theory, F, = F,_, = . . . = 4-c+l = 0. The pofynomial D(6) may be expressed as:
(9) D(6) = (-l)‘W
where D(6) is:
i
Fq6”-q
(13)
q=o
or: 4.1
-
K2.1
6
K2.2
4.2
-
6
. . *
41
. *’ . . .
&,2
D(6) = (-1)“P
.
D(6) =
fi
(6 - 6,)
(14)
h=l
...
K2,,
4,”
...
. K,,n - 6
(10) and ok,@) (k E %i = 1,2,...,n) is the minor of order (n-l) of D(6) obtained by removal of the k-th row and i-th column from D(6). The expansion of the determinant D(6) gives the polynomial:
where tl is equal to n-c, i.e to the number of nonnull roots of the polynomial D(6). Because of the characteristics of the elements of the determinant D(6) (Eq. (7)), its non-null roots 6,, S2,..., a,, are negative or complex with a negative real part (Hearon, 1963). According to Eq. (13) these nonnull roots coincide with the roots of the polynomial:
T(6)=
i
Fq&‘-q
(F,=l)
(15)
q=o
D(6) = (-1)”
k
FqP- q
(F, = 1)
(11)
q=o
In each column of the determinant D(0) (6 = 0) the sum of the terms in the column is zero. Hence, D(0) = 0, i.e. F,, = 0 and therefore the polynomial D(6) has at least one null-root. Hence, Eq. (11) may then be rewritten as:
In Appendix B we obtain an expression for the minors Ok,,@)(k E Q; i = l,2 ,..., n) (Eq. (B23)). The insertion of the expressions for D(6) and Dk,,@), given by Eqs. (14) and (B23), into Eq. (9), yields:
(-l)n+i+l C (-1)*X$( L(XiJ =
krO u
i
(&i)q6ubq)
q=o
n-l
D(6) = (-1)” 6 c
Fq6”-q-l
(F, = 1)
(12)
q=o
In the following, D(6) will either be referred to as the determinant D(S) or as the polynomial D(6). The coefficients F,(q = O,l,...,n-1) in Eq. (12) may be obtained by expansion of the determinant
h=l
(i = 1,2,. . . ,n)
(16)
The right-hand side of Eq. (16) may be decomposed into simple fractions. If we assume that the u non-null roots of D(6) are simple, i.e. that they are not repeated, then:
R. Varbn et al. / BioSystems 36 (1995)
(i = 1,2,
‘**’
4 (17)
From Eqs. (16) and (17) one easily derives that:
(- 1)‘+ ’+ ’ C
(- l)k(Uk,i)&{
kcfA
A,, =
Fu (i = 1,2,. . . , n)
(-l)‘+c c Ai.k =
(-l)kx”k ( c
(18)
(ok,i&i#-q)
=o
kcD u p=l p*h
(i=
1,2,.. . Jr; h = 1,2,. . . $4) (19)
(if u = 1, then the denominator on the right side of this equation is equal to 6,, which in this case is equal to -F,) The inverse Laplace transform of Eq. (17) yields: U (i=1,2,...,n) (20) Xi = Ai,o + C Ai,g*” h=l
A,, and some of the coefficients Ai,h may be zero
for some connections between the compartments. One or more pairs of roots ak (h = 1,2,...,4 may be conjugate complex. In this case Eqs. (18)-(20) remain valid, but the two exponential terms in Eq. (20) which contain a pair of conjugate complex roots, e.g. 6, and 6, (6, = a+bi and 6, = a-bi, a being a negative quantity) are preferably recast using Euler’s relations:
121
121-133
The information which we need to obtain the kinetic Eqs. (18)-(20) for the amount of substance in any compartment, Xi (i = 1,2,..., n), at time t, is the following: (a) The compartments into which the substance is injected at t = 0; (b) The coefficients Fo, F,,..., F,, i.e., the non-null coefficients of the polynomial D(6) corresponding to the scheme of the system. These coefftcients may be obtained by expansion of the determinant D(6), but in the Appendix A we outline a simpler procedure that avoids the inconvenient expansion; (c) The number of non-null and null roots, u and c, resp., of the polynomial D(6). The number of non-null roots, U, coincides with the subindex of the last non-null coefficient 4 0 = 1,2,...,n-1) and is simultaneously obtained with F,. Then, c = n-u is known; (d) The expressions for the coefficients (&j)q (k E 62; q = O,l,..., u). These quantities could be obtained by expansion of the determinants D&j) (k e Q), because they are the coefficients &‘-q of the resulting polynomials. However, in the Appendix B we outline an easier method which uses the expressions for the coefficients Fq (q = O,l,...,u); (e) 6,, i.e. the non-null roots of the poly61, 82,..., nomial T(6). The knowledge of the coefficients F,, F, gives, due to Eq. (15), all the informaF2, ---, tion needed to evaluate these roots. 3.1. Example Next we obtain the expression for the instantaneous amount of substance in the compartment X4 in Fig. 2b (X, in Fig. l), i.e. the expression for x4. As mentioned above, in this case fl = ( 15 ) . At first, we obtain the expressions for the coefftcients of the polynomial D(6) associated with the directed graph in Fig. 2b. According to Eq. (12) D(S) is for this scheme: D(6) = 6(a4 + F,S3 + F2S2 + F36 + F4)
Aj,&’ + Aj,we6wr= (B~osbt + Csinbt)eof
(24)
(21)
since F. = 1. According to Appendix A we have: where: F, =
B = Aj,y + Aj,, C = i(Aj,” - Aj,,)
K1,2
+ K2.1
+ K2,3
+ K2,4
+ K4,5
+ K5,4
(25)
(22) (i=Jij
(23)
The procedure in the Appendix A generates the following terms for the derivation of F2 from F,:
q&R 128
R. Varbn et al. / BioSystems 36 (1995) 121-133
F2 = &,2K2,3
+ &,2K2,4
+ K2,1&,5
+ K1.2K4.5 + 4,2
+ K2,1&,4
+K2,4&,5
+ K2,3&,5
c0aicients h,4h
Ksp
+ K2,3K5,4
+ K2,4&,4
(26)
The analogous process, applied to the terms of F2 in Eq. (24), yields:
h,4>1,2, (al.413, (a5,4h
(a&, (a& and (a& in Eqs. (31) or/and (32) can be obtained as outlined in Appendix B for a general model (see Eq. (B21)). As an example, we explain how the expression for (as,4)2is obtained. According to Eq. (B21) and using the values of n, i, k and q (5, 4, 5 and 2, resp.), we have: (33)
(a5,4)2 = r;;4,5 f’3 = Kl,2K2,3&,s
+ Kl,zKzzK5,4
+ Kl,2K2,&,s
Kl,z&,&,4
+
(27)
Since from the terms of F3 it is not possible, by adding any of the fractional transfer coefftcients to these terms, to generate any term which does not contain symmetric or circular combinations of K’s, we have: F4 = 0
where F’2,1,S is the sum of the terms of F2, in which &,s is not a factor and in which a connection exists between the compartments XS (X,) and X4 (X3through combinations of the constants. If we, in the expression for F2 given by Eq. (26), suppress the terms which either contain & or do not represent a connection between the compartments XS and &, then, with Eqs. (26) and (33) we obtain:
(28) (u5,4)2 = - KzK5.4
+ K2,&5,4
+ K2,3%,4
Therefore Eq. (24) becomes: (34)
+ K2,&5.4)
D(6) = 82(83 + FIS2 + F26 + F3)
(29)
Because F3 is the last non-null coefficient, u = 3 and c = 5-3 = 2 for this scheme. Then, according to Eqs. (18)-(20) and with the values of n,i and c (5, 4 and 2, resp.), we have:
Analogously, cients:
one obtains for the other coeffi-
= 0
(35)
(a,,,),
= (a5,4)0 = (q4h
(al,4)2
= - Kl.2K2.4
(36)
(Q,4)3
= - Kl,zK2,4K5,4
(37)
3 x4 = A.0
+
c
A,, e’h’
(30)
h=l h,dGxp
(38)
(~5,4)1 = - K5,4
+ @5,4)3x!
A4,o = F3
(05,4)3 k - (4,2K2,3K5,4
+ 4,2K2,4&,4)
(39)
and 3 xy
c
3 (a1,4)&4
+ x!
q=o
c
(a5,4)qc4
q=o
&,h = Ah fi
@p
-
6,)
p=l p*h
(32) 6i, &, a3 in Eqs. (30) and (32) are the roots of the polynomial 63 + F,S2 + F28 + F3 = 0. Hence, the
3.2. Extension of the analysis to linear open compartment systems The kinetic analysis has been carried out assuming that the system is a linear closed compartment without input. Nevertheless, this analysis is also applicable to linear compartment systems, in which the excretion of substance from any compartment is allowed, i.e. to open systems. For these systems it must merely be assumed that the substance excreted from a compartment moves to a hypothetical terminal compartment which must be
R. Vardn et al. /BioSystems 36 (1995) 121-133
included in the system. If the substance is removed from more than one compartment, then we introduce a hypothetical terminal compartment, in which all excreted substances are collected. If our analysis is to be applied to an open compartment system, it is convenient to denote its compartments as Xi, X2, .. .. Xn-1, and the hypothetically added compartment as X,. The fractional excretion coefficient, Ki,*, from any compartment Xi (i = 1,2, ....n-1) must then be replaced by Ki,,.
4. Results aal ditmssion This contribution describes the derivation of the kinetic equations for a general linear ncompartment model without input in terms of the model parameters, i.e. the initial amount of substance in each compartment and the fractional transfer coefficients corresponding to direct connections between the compartments. These equations were obtained by the Laplace transform. Several procedures exist which yield numerical solutions of any system of linear differential equations from a set of numerical values of the parameters, but an analysis of the compartment models in cases as those mentioned in the introduction requires analytical equations which are functions of the kinetic parameters of the model. The kinetic equations obtained here assume a very simple form, since in all polynomials we have eliminated the coefficients, which are null. The evaluation of the remaining coefficients neither requires the expansion of determinants, nor the use of graphical procedures, because they are easily obtained using the algorithms outlined in the Appendices A and B. The model to which they are applicable is very general. Hence, other compartment models, the equations of which have previously been obtained, may be considered as its particular cases. Examples are cases in which the input of substance only occurs in one compartment (Rescigno, 1956), such in which the analysis refers to models with a limited number of compartments (Rubinow, 1975; Cobelli, 1979; Travis and Haddock, 1981) and such in which the model is catenary (Chau, 1985). The kinetic equations obtained here are valid for all compartment systems which tit the propos-
129
ed model. They can also be applied to other systems which are described by a system of ordinary, linear, first order differential equations with constant coefficients, if the elements of the corresponding determinant D(6) satisfy the conditions (8) and Eq. (5). The equations for the compartment systems are also applicable to enzyme systems. The latter application is convenient, if the time dependence of the concentration of any of the enzyme species is desired. The kinetic equations obtained are valid for the whole course of the process of partitioning of the substance between the compartments, i.e. for the transient phase and the steady state. In the steady state (t - oo), Eq. (20) is simplified into: xi = Ai,o
(i = 1,2, ....n)
(40)
The kinetic analysis suggested here can also be ap plied to open linear compartment systems without input. In that case, it must merely be assumed that the substance excreted from any compartment is collected in a single hypothetical terminal compartment which must be included in the system. Hence, the system is closed and fits our model. Since the compartment added has no influence on the other compartments, the kinetic equations obtained for these compartments are the same as those for the compartments in the original open system (Jacquez, 1985). Our method is limited to the kinetic equations of systems with zero input. The analysis of systems with non zero input must begin with a solution of the system with zero input (Anderson, 1983; Jacquez, 1985). Therefore, our method both contributes to the solution of compartment systems with and without input. The formulation of the expressions of the coefticients Fq (q = 0,1,2 ,..., U) and (Uk.3, (k E B; i = 1,2,..., n; q = 0,1,2 ,..., U) in the Eqs. (18)-(20) may, due to the sequential, recurrent and systematic nature of the algorithms developed here, be performed by a simple, efficient computer program to which we devote the part II of this series. Acknowledgements
This work has been partially supported by a grant number PB92-0988 of the Direction General
R. Vardn et al. /BioSystems 36 (1995) 121-133
130
de InvestigacGn Cientifica y Tkcnica (Spain) and by the Biochemisches Institut of the University of Kiel (Germany).
s n-l exist, for which F, = 0, then m = n, because F, always is zero. Appendix B
AppendixA Algorithm for the derivation of the coefficients of the polynomial D(6) The coefficients F4 (q = 1,2,...,n-1) of Eq. (12) are the sum of terms which are the products of q K’s with different first index. Products containing symmetric or circular combinations of K’s are excluded. Symmetric combinations of K’s are those of the K~,.K&type and circular combinations of K b.c&.&d,e K’s are those of the Kb,&dKd.b-3 K c,b- ,..., types (Rescigno and Segre, 1965). The coefficients F4 (q = O,l,...., n-l) of the polynomial D(6) are found without expansion of the determinant in Eq. (10) as follows: F. is always equal to unity and F1 is the sum of all the transfer constants Kid (i#j) which are not zero. We denote the number of the non-null constants with R. Then, the terms of 4 (j = 2,3 ,..., n-l) are generated from the terms of Fbl (j = 2,3,...,n-1) in the following way: R new terms are formed from each term of Fbl: Successively, each of the non-null KiJ constants is written on the right hand side of the corresponding Fj_1-term. Such terms are then suppressed, in which the subindex i of Kij is 1~;the first subindex, a, of Kab located on the right hand side in the corresponding term of Fj_1. If, for the reason mentioned, any of the terms generated from those of the 4-1 is not suppressed, but contains symmetric or circular combinations of K’s, then it must be removed. Thus, the terms, which after this process remain, all contribute to Fj G= 2,3,...,n-1). If in the determination of a particular 4 (j = 2,3 ,..., n- l), e.g. F,, all the terms are suppressed, then, F, = 0. Moreover, if F,,, = 0, then Fm+,, Fn,+2, . . . . F,_l are also zero, i.e.:
Ftll=impliesF,=F,+,=F,,,+2=... =Fn_,
=O
(Al)
Therefore, if a F,,, = 0, then the process is ended, because the relation (Al) is satisfied. If no m
Algorithm for the abivation of the coefficients of any of the polynomial D&S) (k E n; i = 1,2,. . . ,n) Since the expansion of D(S) is a polynomial of degree n, the development of any determinant D&S) is a polynomial, the degree of which is I n- 1. The degree (n- 1) corresponds to the case in which k = i. Therefore, the expression for the polynomial D&6) is: n-l
c
Dk,i(8 =
(ak,i)&jn - ’ - ’
q=o
(-l)“-
((ak,t)o =
’ if k = i;
(ak,i )o = 0 if k # i)
W)
We distinguish between two different cases: k = i and k # i k=i We denote the principal minor resulting from the suppression in determinant D(6) of the i-th row and the i-th column (i = I,2 ,..., n) as Di,i. If, in the determinant D(6), we set: Kij = 0
(j = 1,2,..., n; j + i)
(B2)
then we obtain: i-th column 1
K,,,-6
o(qp=
42
K2,, . . . 0 K2,2- 6 . . . 0
. .
. .
::::::::
;C,,i . _ . .
iY2.i . . .
: . . .
KlJi
Kzr
. . .
: . . .
: . . .
. . . . . .
K,,, K,,,
‘-S . . .
: . . .
: . . .
K,,J . . .
0
. . .
K,,+
: . . .
(B3)
131
R. Varh et al. /BioSyslems 36 (1995) 121-133
where D(Qio denotes D(S) after insertion of Eq. (B2). The determinant in Eq. (B3) is then expanded by the elements of the i-th column: W)P = -6 Qi(@
W)
From Eqs. (B4) and (8) it follows that:
(k E 0; k # i)
n-1 Qi (6) = (-1)” -
’ C
l$.i
h-
9-
1
(W
q=o
where F$ (q = 0,1,2 ,..., n-l) is Fq after the insertion of Eq. (B2), i.e.F$ is the sum of the terms of Fq which do not contain &I, Ki.2 ,..., Kifi. The expressions for Ft,i depend on the i-value of the Eqs. (B2). Since F. = 1, it follows that Fz,i = 1. From Eqs. (Bl) and (B5), we obtain: (ai,i)q= (-1)” - ‘Fi,i
(4 = 192, . . . J - 1)
(B6)
kfi
The coel&ients of the polynomials &,(b) (k c 0; k # i) are gained, if in the determinant D(S) we set: Kij = 0
where D(6)‘,k denotes that the changes in D(6) indicated by Eqs. (B7) have been made. The determinant D(6)‘j,k is then developed Using the &mentS of the i-th column:
(B9)
i.e.: K&J6)
= (-l)k + i
[&@; + (Ki,k+ Wi,i(Ql
(kcn;k#i)
WO)
In the following we deduce from Eq. (BlO) a general expression for the derivation of the coefficients of any of the polynomials &@) (k e Q k # i) from the coefficients Fq (q = O,l,...,n-1) of the polynomial D(S). According to Eq. (12), the polynomial expression for B(6)*,k (k e $2; k # i) is: n-1
D(& = (-1)“6 c
F;,,& P-q-
’
q=o
(j = 1,2, . . . , n; j # k,i)
(B7)
(k E Q; k f
i)
(Bll)
Then, we obtain: OcGk.- =
where F*q,i,k is the coefficient Fq after insertion of Eq. (B7). Since F, = 1, it follows that F*o,j,k = 1 (k
i-ttl columm
4.1-S
K2.1
4.2
K2,2- 6 . . .
. . .
. . .
... ... ...
I 0 0 . . .
. . .
. . .
... ... ...
. . .
.... .... ....
. . .
. . .
... ... ...
. . .
.... .... ....
. . .
. . . . . . .... .... ....
e Q; k f K 41 K 42
i).
From Eqs. (B9), (BlO) and (B5), we obtain: n-l KikDk,i(6) = C (-l)n+i+k-’ q=o (-Fi + 1,i,k
(J’i,ik
= c,i
+ Ki,kQ
= 0;
+ q
+ 1.i )6” - ’ - ’
k E Q; k z i)
(BW
A comparison of Eqs. (B 12) and (B 1) finally yields: Kik(Uk,i)q= (-I)” (-F;
+ i+k - ’
+ I,~JC + &,d$,i + q + 1.i)
R, Varbn et al. /BioSystems 36 (1995) 121-133
132
circular combinations of K’S, but the terms of Ki,k G,i might contain them. In that case, a connec-
(q = O,l, . . . , n - 1; k E Q; k # i; FL
= fi,, = 0)
(B13)
Eq. (B13) is a simple relation between the coeflicients of the polynomials Dk,@) (k E fl) and those of the polynomial D(6). We consider the meaning of the sum: -F; + i,ik + K$$,i
(B14)
of Eq. (B13) to facilitate the derivation of the coefficients (ck,i)q The definitions of F*q+,,i,k and P q+l.t show that F*q++l.t,kcontains the same terms as those of the coefficients pq+,,i. Besides, it in. . eludes all terms whrch m Fq+l contain Ki,k as a factor. The terms which in Fq,! contain the constant Ki,k as a factor, have the form:
. . . K aq+ Ipbq+1
VW
where:
Ui+]
C * *. C
Uq+]
I
fl
1 5 bj; bj = uj ; j = 1,2, . . . , q + 1 is devoid of symetric or circular combinations of K’s
(Jw 1
1,i.k + q+
I,i=-
c&,,b,
&,a . . . &b
. . . Kai_ *,bi_,
(B17) where the sum is extended to all the terms in which the conditions (B16) are fulfilled. The terms of $,i do not contain symmetric or
. . . Kb,iyt?tC
(Bl@
The terms of the right hand side of Eq. (B17) coincide, with changed sign, with the terms of Ki.kF0q.i which do not contain symmetric or circular combinations of K’s. Hence: -J’; + 1,i.k +
&,kq,i
(kC8I;k
#i)
+ q
+ l,i = &,kF&,k
0319)
where F’q,i,k is the sum of the terms Fq which do not involve any of the constants Ki,l, Ki.2)..., Ki,, and in which there is a connection between the compartments Xk and Xi, i.e. these terms contain some of the combinations (B18). Since F. = 1, the F/ O,i,k= 0 (k E a; k # i). From Eqs. (B13) and (B19) we have: (Uk,i)q = (-1) (q =
Evidently, the constant Ki,k may in the terms of (B15) occupy the first place (if i = aI) or the last (if i = u,+l). Since (-F’q+,,,k + Pq+,,i) is the sum of all the possible terms in the form of (B15), it follows that: - <+
&,iy &,&z,i, &,o * - * &,iy &,a&,d&
+ q + 1.i
(q = 0,1,2, . . . ,n - 1; k E Q; k #i)
C i C
tion betWet% the compartments Xk and Xi exists in the corresponding term of c,i, i.e. one of the following combinations of constants is found in this term:
n+i+k-
IF;,~&
1,2, . . ., n - 1; k E Q; k # i)
WO)
The above equation is only valid for k ;f i.
A single equation for the coefficient (ak,Jq may exist which also is valid for k = i. If we set F/s,,,, = Pq,i (q = O,l,..., n-l) then, according to Eqs. (B6) and (B20): (Uk,i)q = (-1) (q =
n + i + k - IF;,i,k
0,1,2, . . ., n - 1; k c Q)
Wl)
It follows from Eq. (B21), that if Fq = 0 (Flq,,,k = 0), then (uk,ijq = 0. Therefore, if F, = 0, Eq. (Al) requires that:
R. Vardn et al. /BioSysrems 36 (1995) 121-133
(“&,i)m = (“k,ih
+ 1 =
* * * =
t”k,ih - 1 = O
(B22)
If Eq. (B22) is entered into Eq. (Bl) and we let u = m - 1 and c = n - u, then we obtain:
((Uk,i)o
=
(-
1)”- ’ if k = i;
&),-, = 0 if k # i)
(~23)
References Anderson, D.H., 1983, Compartmental Modeling and Tracer Kinetics (Springer Verlag, Berlin). Astrom, K.J. and Eykhoff, P., 1971, System identification. A survey. Automatica 7, 123-167. Benet, L.Z., 1972, General treatment of linear mamillary models with elimination from any compartment as used in pharmacokinetics. J. Pharm. Sci. 61, 536-540. Busenberg, S.N., Lenhart, S. and Travis, C.C., 1987, IdentiEability of time-dependent linear systems. Math. Biosci. 87, 63-71. Chau, N.P., 1985, Linear n-compartment catenary models: Formulas to describe tracer amount in any compartment and identification of parameters from a concentration-time curve. Math. Biosci. 76, 185-206. Chou, K.C., 1990, Applications of graph theory to enzyme kinetics and protein folding kinetics. Steady and non-steady state systems. Biophys. Chem. 35, l-24. Chou, K.C., 1989, Graphic rules in steady and non-steady state enzyme kinetics. J. Biol. Chem. 264, 12074-12079. Cobelli, C. Lefschy, A. and Jacur, R., 1979, Identifiability results on some constrained compartment systems. Math. Biosci. 47, 173-195. Delforges, J., 1981, Necessary and sufficient structural conditions for local identifiability of a system with linear compartments. Math. Biosci. 54, 159-180. Godfrey, K.R., 1983, Compartmental Models and their Application. (Academic Press, New York).
133
Hearon, J.G., 1963, Theorems on linear systems. Ann. New York Acad. Sci. 108, 36-68. Jacquez, J.A., 1985, Compartmental Analysis in Biology and Medicine, 2nd edn. (The University of Michigan Press, Michigan). Lal, R. and Anderson, D.H., 1990, Calculation and Utilization of Component Matrices in Linear Bioscience Models. Math. Biosci. 99, 11-29. Manara, L., 1971, Isotopic Methods and Activation Analysis, in: Fundamentals of Biochemical Pharmacology, Z.M. Bacq, R. Capek, R. Paoletti and J. Renson (eds). (Pergamon Press, Oxford). Mason, S.J. and Zimmerman, H.J., 1960, Electronic Circuits, Signal and System. (John Wiley and Sons, New York). Netter, H., 1969, Theoretical Biochemistry. Physico-Chemical Principles of Vital Processes. (Oliver and Boyd, Edinburgh). Nguyen, V.V. and Wood, E.F., 1982, Review and unification of linear identifiability concepts. SIAM Rev. 24, 35-51. Pohjanpalo, H and Wahlstrom, B., 1977, On the uniqueness of linear compartment systems. Intemat. J. Systems Sci. 8, 619-632. Rescigno, A., 1956, A contribution to the theory of tracer methods. Part II. Biochim. Biophys. Acta 21, 111-I 15. Rescigno, A. and Segre, G., 1965, On some metric properties of the systems of compartments. Bull. Math. Biophysics 27, 315-323. Rubinow, S.I., 1975, Introduction to Mathematical Biology. (Wiley, New York). Sines, J. and Hackney, D., 1987, A residence-time analysis of enzyme kinetics. B&hem. J. 243, 159-164. Schuster, S. and Heinrich, R., 1987, Time hierarchy in enxymatic reaction chains resulting from optimahty principles, J. Theor. Biol. 129, 189-209. Travis, CC. and Haddock, G., 1981, On structural identitication. Math. Biosci. 56, 157-173. Vadja, S., 1981, Structural equivalence of linear systems and compartmental models. Math. Biosci. 53, 39-64. Veng-Pedersen, P., 1989, Mean time parameters in pharmacokinetics. Definition, computation and clinical implications. Part II. Clin. Phannacokinetics 16, 424-440. Walter, E. and Le Cardinal, G., 1976, On the identifiability of linear state systems. Math. Biosci. 31, 131-141.
General linear compartment model with zero input: I. Kinetic equations R. Varch* a, M.J. Garcia-Meseguera,
F. Garcia-Chovasb,
B. Havsteenc
aDepartamento de Quimica-Fisica. Universidad de Castilla-Lu Man& Albacete. Spain bDepartamento de Bioquimica y Biologia Molecular, Utu’versidadde Murcia, Murk. Spain cBiochemischesInstitut. Christian-Albrechts-Universitiit su Kiel, Kiel, Germany Received 4
April 1995;accepted18April 1995
Ah&act
The derivation of kinetic equations is described for n-compartment linear models, in which the substance may be simultaneouslyintroduced into one or more compartments at t = 0 and eliminated from any compartment. For a given zero-input, general formulas are derived which describe the amount of tracer in any of the compartments as a function of time and the model parameters. New algorithms have been developed which allow the expression of the kinetic equations. Keywordr:
Compartment systems; Enzyme kinetics; Kinetics equations; Linear models; Tracer
1. Introduction
Compartment systems are important for the description of many aspects of biology, engineering, economics and several other sciences. Standard and very complete references on compartment modelling and analysis are those of Rescigno (1956), Godfrey (1983), Anderson (1983) and Jacquez (1985). Some areas of interest related to studies of compartment systems are the tracer ??Corre-spondiig author, Departamento de Quimica-Fisica, Escuela Universitaria Polit&nka, Campus Universitario, E02071 Albacete, Spain. Tel.: +34 67599200, ext. 2480; Fax: +34 67599224.
isotope methods, the parameter identification and the evaluation of mean pharmacokinetics parameters. Tracer isotope labelling methods are essential for the elucidation of many aspects of metabolism and transport processes in biochemistry (Manara, 1971; Netter, 1969). Some of these tracers, e.g. fluorescent tags, are permanently linked to a substance, the fate of which is to be followed. Such labels faithfully track the course of its carrier across membranes, along guiding filaments, through tubuli and during carrier rides. These tracers aid the definition of the compartment structure. The identification of parameters is one of the
0303-2647/95/%X50 0 1995 Ekvier Science Ireland Ltd. All rights reserved SSDI 0303-2647(95)01533-Q
122
R. Varbn et al. / BioSystems 36 (1995) 121-133
most dynamic areas of systems theory. Many contributions treat theoretical aspects or applications, e.g. to biosciences, engineering and economics. The literature on the subject is complex. Several efforts have been made to define, classify and unify the concepts (Astrom and Eykhoff, 1971; Ngugen and Wood, 1982). The definitions of identifiability differ, but several necessary and sufficient conditions for identifiability have been found (Walter and Le Cardinal, 1976; Pohjanpalo and Wahlstrom, 1977; Delforges, 1981; Vadja, 1981; Busenberg, 1987). However, most authors focussed their efforts on uniqueness criteria, i.e. on the determinancy of the problem. The calculation of the model parameters from input-output measurements has so far not been described, except for l-, 2-, 3-compartment models (Rubinow, 1975), for particular cases of n-compartment models (Cobelli et al., 1979; Travis and Haddock, 1981) or for an n-compartment, catenary model (Chau, 1985). Benet (1972) empirically derived a formula giving the concentration-time curve in the central compartment of a general n-compartment, mamillary model. The concentration being expressed as a function of the model parameters. Another problem of the compartment systems is the evaluation of the mean parameters (Jacquez, 1985) and their relation to enzyme kinetics (Sines and Hackney, 1987; Schuster and Heinrich, 1987) as well as for basic pharmacokinetic processes, e.g. drug absorption, drug distribution, drug elimination, metabolites, dosing times, etc. (VengPedersen, 1989). All such studies require the derivation of the kinetic equations for the compartment system as a function of its parameters. Rescigno (1956) derived equations for a system composed of n components assuming that at t = 0 the input only consisted of one component. The equations obtained required the expansion of determinants and the use of certain relations between determinants which became very complex if the system had many compartments. Later, Rescigno and Segre (1965) developed a graphic method which facilitated the task. Graphic methods also have been developed for other compartment systems (Mason and Zimmerman, 1960; Chou, 1989, 1990). However, for systems of some complexity,
the graphic methods are laborious and prone to human errors. Recursive methods facilitate the derivation of the kinetic equations of a linear or more complicated compartment systems (La1 and Anderson, 1990). However, these methods require the handling of matrices and determinants. In this contribution we present an analysis of a general n-compartment model which circumvents many difftculties. We develop algorithms which facilitate the derivation of the kinetic equations. The expressions for the coefftcients in these equations can easily be formulated. This procedure requires neither expansion of determinants nor operations with matrices or graphic methods. 2. Model-notation and deftitions The model consists of a closed n-compartment system with zero input, i.e. substance may at t = 0 simultaneously be injected into one or more of the compartments, which may be strongly connected (i.e., without traps and with an irreducible matrix of the system) or not (i.e. with one or more traps and with a reducible matrix of the system). We
i
i
‘I! Fig. 1. Scheme representing a 9-compartment system by a directed graph. The nodes correspond to the compartments and the lines to the direct connections between the compartments. In this scheme examples of different connections are given. Compartment XI is directly connected with the compartment X2 and indirectly connected with the compartments X3-X9, i.e. compartment X, is connected with the compartments X2-q. Compartment X2 is directly connected to compartments X, and X3 and indirectly connected with the compartments X,-X,, etc. Note that between the compartments XT and X, both a direct and an indirect connection exists. No connection exists between the compartment X3 and X2, Xs and &, etc. The compartment X, is terminal, because it is not connected with any other compartment.
123
R. Varbn et al. / BioSystem 36 (1995) 121-133
also assume that the non-null eigenvalues of the system are distinct. We arbitrarily label the compartments: X1, Xz, .... X,. The non-zero fractional transfer coefficients are Kij (Kii denoting transfer from compartment Xi to compartment Xj). The connectivity diagram of our system is a directed graph. The model also includes open systems (i.e. systems with excretion from any compartment) without inputs. If the environment is treated as a single compartment which receives all excretions, the resulting closed system (Jacquez, 1985) tits our model. In this paper the standard definitions and the usual nomenclature in the literature on compartment systems are used. However, some additional definitions, which are necessary in our analysis, are introduced. Fig. 1 shows a directed graph which supports our task. xp (i = 1,2,...,n): Initial amount of the substance in the compartment Xi at t = 0. One or more of the quantities Xi0 may be zero, but at least one of them must be non-zero. Xi (i =
1,2,...$):
The amount of the substance in the compartment xi at time t.
Sum extended to all the elements of the set fl, i.e. k assumes each of the values of the elements of fl.
@: Set of all compartments of the system, i.e. @ = [X,, x2, ...) X,) . In our example, Cp= 1x1, x2,
x4,
x3,
x5,
x6,
xl,
x8,
x9
1.
0’:
Set of all the non-zero fractional transfer coefficients of the system. In the system in Fig. 1: @’=
I&,2,
K4.7,
a(k.l?
K2.1,
K~,E,
K1,9,
K2,3~ K3,4, KS.9 1
K4,3,
h.5,
K
5.6, K6,5, (1)
:
Set of the compartments of the system which simultaneously belong to the set of compartments which are reachable from the compartment Xk, R&), and to the antecedent set of the compartment Xi, Q(Xi), i.e.,
Q(XJ
a(kA = R(Xk) n
(2)
If in Fig. 1 we choose X5 as Xi, then, with Eq. (2) as well as W,) = I x3, &, x5, x6, x7, x8, x9 1, = (X5, x6 1 and QW = 1XI, X2, X3, &, x5, ?&I, we obtain ~(3,s) = (X3, &, X5, X6) and
R(X6)
Cl:
a(V) =
The set of subindeces in the notation of the compartments which contain the substance at t = 0, e.g. if the substance only is at t = 0 present in the compartments X, and X6 of the system in Fig. 1, then Cl= (341. k c Q:
(x5,
x6
1.
Z(fl,i):
Set resulting from the union of all the sets u(k,i) in which k belongs to n, i.e.: E(fl,i) =
u
u(k,i)
(3)
ken
The index k defines one of the elements in the set
n.
In the example above we have C@,5) = (X3, X4, x5,
c
keO
x6).
k-dependent expression: E(Q,i)
’:
124
R. Van% et al. / BioSystems
Set of all the non-zero fractional transfer coefficients KVJwhich pertain to the set E(Q,Q. In the example:
36 (1995) 121-133
some enzyme reactions, in which the enzyme species may be considered as compartments. K,, (i = 1,2,..., n):
B(B,5) =
(K3,4,
K4,3 K4,5, kc5
&,s
K4,11
(4)
2.1. Connections between two compartments Between the compartments Xi and Xj exists a direct connection, if a path of the length 1 connects Xi with Xj The compartments Xi and Xj are indirectly connected, if a path connecting Xi with Xj of a length greater than 1 exists. If the compartment Xi directly, indirectly or both is connected with the compartment Xj, then a connection between the compartment Xi and the compartment Xj exists. If a’compartment is not connected to any other compartment, then it is a terminal compdrtment. A terminal compartment is a trap consisting of only one compartment. For any two compartments, Xi and Xj, of a SYStern, one of the following situations exists: (a) The compartment Xi is only directly connected to the compartment Xj; (b) the compartment Xi is only indirectly connected to the compartment Xj; (c) the compartment Xi is both directly and indirectly connected to the compartment Xj; (d) neither a direct nor an indirect connection exists between the compartment Xi and the compartment Xj (see Fig. 1). Kij (i,j = 1,2,..., n; i # 1): Fractional transfer coefficients corresponding to the direct connection between the compartments Xi and Xi. Kij may be zero (if no direct connection exists between the compartments Xi and Xi> or positive. We admit the possibility that two compartments, Xi and Xj, are directly connected in more than one way. In that case, we denote these fractional transfer coefficients as K’, K”ij, . . . Hence: Kij = KG + KG + . . . In the corresponding directed graph only one line segment exists which connects Xi to Xi and possesses a fractional transfer coefficient equal to Ki$ Examples of such a situation are found in
Fractional excretion coefficient when excretion occurs from compartment Xi to the environment.
3. Kinetic analysis The analysis presented here is valid for any linear system with zero input, i.e. without input. It forms the expression for the instantaneous amount of substance, Xi, in any compartment Xi, when at t = 0 substance is injected into one or more compartments. Since xi only influences the fractional transfer coefficients belonging to the set E(Q,z)‘, the original directed graph may be reduced by sup pressing all of the direct connections corresponding to the fractional transfer coefficients of the set *‘-E/(n,i)’ and arbitrarily renumbering the remaining compartments and their connections from XI to X,. That facilitates the task without loss of generality and gives the expression of xi a simple form. If for example, we derive the time-course equation for the amount of substance in compartment X5 of the system in Fig. 1, we have, with the Eqs. (1) and (4) that +’ - (QS)’ = (K1,2, K2,,, the reduced K2,3, Ku, K1.9, Ks,9) . Therefore, directed graph corresponding to X5 is obtained by suppressing in Fig. 1 the direct connections between the compartments Xi and X2, X2 and Xs and X9 (indicated with dashed line in Xl,..., Fig. 2a). The result is the directed graph in Fig. 2b, in which the five remaining’compartments and their connections have been renumbered (note that the new notation for X5 in Fig. 2a is X, in Fig. 2b). By renumbering, the subscripts belonging to D will usually change. In this case, set Q becomes ( 1,5). In the following, we consider the reduced directed graph which allows the derivation of the simplest expression of xP We begin with the analysis of closed linear compartment systems with zero input and then we extend the results obtained to open systems. The system of differential equations describing the evolution of the amount of the substance in each compartment of the system is:
R. Vardn et al. / BioSystems 36 (1995) 121-133
A.3 = Kl,jXI + dt
K~~xz + ... + &jxn (j = 1,2, . . . ,n)
(6)
125
following two relations summarize the possible values of the constants in the system of differential equations:
where KjJ 0’= 1,2,...,n) is defined as: KVJ z O(vj= Kj,j = -
i j=
Kj y
(7)
1,2,...,n;j
# v) (8)
KjjI Oo'= 1,2,...,n)
I
j#v
Note that KjJmay only be zero or negative. The
-*- ......._
‘$!r?
Xl{’ d.--‘-
)
”
.xs.
7
The set of Eqs. (6) is a homogeneous system of first order, linear differential equations with constant
4
7 ‘._ ‘...
...*
\.,
. .. . x;’ .:’ .t .’
....
”
‘..a.’ x9
Fig. 2. (a) Directed graph of Fig. 1 in which the arms corresponding to the fractional transfer coeffkients, that do not influence in the instantaneous amount of substance in X, (i.e. x5) when U = [3,6 ) , are denoted by dashed lines. (b) Reduced directed graph obtained from Fig. 1 corresponding to the compartment X5 when 0 = ( 3,6 1. With the arbitrary numbering used for this reduced directed graph, the compartmtnts X3, Xg, XT, Xs and X, in the original directed graph in (a) are denoted X,, X2, X,, X4 and X,, respectively. Hence, the set Clin (a) hecomes ( 1,5) in (b).
126
R. Varh et al. /BioSystems 36 (1995) 121-133
coefficients. The Laplace transform of any of the Eqs. (6) yields the following expression for the Laplace transform Of Xi (i = 1,2, . . . JZ):
(-l)‘+ ’ c L(Xi)
(-l)kDk.J(s)X”k
kotl
=
D(s)
(i=
1,2,...,n)
(10). This procedure becomes very tedious when the compartment system is complex. In Appendix A we show how the coefficients Fq can easily be obtained. The polynomial D(S) always has one, but it may have c (1 I; c < n) null roots. In the latter case, according to the polynomial theory, F, = F,_, = . . . = 4-c+l = 0. The pofynomial D(6) may be expressed as:
(9) D(6) = (-l)‘W
where D(6) is:
i
Fq6”-q
(13)
q=o
or: 4.1
-
K2.1
6
K2.2
4.2
-
6
. . *
41
. *’ . . .
&,2
D(6) = (-1)“P
.
D(6) =
fi
(6 - 6,)
(14)
h=l
...
K2,,
4,”
...
. K,,n - 6
(10) and ok,@) (k E %i = 1,2,...,n) is the minor of order (n-l) of D(6) obtained by removal of the k-th row and i-th column from D(6). The expansion of the determinant D(6) gives the polynomial:
where tl is equal to n-c, i.e to the number of nonnull roots of the polynomial D(6). Because of the characteristics of the elements of the determinant D(6) (Eq. (7)), its non-null roots 6,, S2,..., a,, are negative or complex with a negative real part (Hearon, 1963). According to Eq. (13) these nonnull roots coincide with the roots of the polynomial:
T(6)=
i
Fq&‘-q
(F,=l)
(15)
q=o
D(6) = (-1)”
k
FqP- q
(F, = 1)
(11)
q=o
In each column of the determinant D(0) (6 = 0) the sum of the terms in the column is zero. Hence, D(0) = 0, i.e. F,, = 0 and therefore the polynomial D(6) has at least one null-root. Hence, Eq. (11) may then be rewritten as:
In Appendix B we obtain an expression for the minors Ok,,@)(k E Q; i = l,2 ,..., n) (Eq. (B23)). The insertion of the expressions for D(6) and Dk,,@), given by Eqs. (14) and (B23), into Eq. (9), yields:
(-l)n+i+l C (-1)*X$( L(XiJ =
krO u
i
(&i)q6ubq)
q=o
n-l
D(6) = (-1)” 6 c
Fq6”-q-l
(F, = 1)
(12)
q=o
In the following, D(6) will either be referred to as the determinant D(S) or as the polynomial D(6). The coefficients F,(q = O,l,...,n-1) in Eq. (12) may be obtained by expansion of the determinant
h=l
(i = 1,2,. . . ,n)
(16)
The right-hand side of Eq. (16) may be decomposed into simple fractions. If we assume that the u non-null roots of D(6) are simple, i.e. that they are not repeated, then:
R. Varbn et al. / BioSystems 36 (1995)
(i = 1,2,
‘**’
4 (17)
From Eqs. (16) and (17) one easily derives that:
(- 1)‘+ ’+ ’ C
(- l)k(Uk,i)&{
kcfA
A,, =
Fu (i = 1,2,. . . , n)
(-l)‘+c c Ai.k =
(-l)kx”k ( c
(18)
(ok,i&i#-q)
=o
kcD u p=l p*h
(i=
1,2,.. . Jr; h = 1,2,. . . $4) (19)
(if u = 1, then the denominator on the right side of this equation is equal to 6,, which in this case is equal to -F,) The inverse Laplace transform of Eq. (17) yields: U (i=1,2,...,n) (20) Xi = Ai,o + C Ai,g*” h=l
A,, and some of the coefficients Ai,h may be zero
for some connections between the compartments. One or more pairs of roots ak (h = 1,2,...,4 may be conjugate complex. In this case Eqs. (18)-(20) remain valid, but the two exponential terms in Eq. (20) which contain a pair of conjugate complex roots, e.g. 6, and 6, (6, = a+bi and 6, = a-bi, a being a negative quantity) are preferably recast using Euler’s relations:
121
121-133
The information which we need to obtain the kinetic Eqs. (18)-(20) for the amount of substance in any compartment, Xi (i = 1,2,..., n), at time t, is the following: (a) The compartments into which the substance is injected at t = 0; (b) The coefficients Fo, F,,..., F,, i.e., the non-null coefficients of the polynomial D(6) corresponding to the scheme of the system. These coefftcients may be obtained by expansion of the determinant D(6), but in the Appendix A we outline a simpler procedure that avoids the inconvenient expansion; (c) The number of non-null and null roots, u and c, resp., of the polynomial D(6). The number of non-null roots, U, coincides with the subindex of the last non-null coefficient 4 0 = 1,2,...,n-1) and is simultaneously obtained with F,. Then, c = n-u is known; (d) The expressions for the coefficients (&j)q (k E 62; q = O,l,..., u). These quantities could be obtained by expansion of the determinants D&j) (k e Q), because they are the coefficients &‘-q of the resulting polynomials. However, in the Appendix B we outline an easier method which uses the expressions for the coefficients Fq (q = O,l,...,u); (e) 6,, i.e. the non-null roots of the poly61, 82,..., nomial T(6). The knowledge of the coefficients F,, F, gives, due to Eq. (15), all the informaF2, ---, tion needed to evaluate these roots. 3.1. Example Next we obtain the expression for the instantaneous amount of substance in the compartment X4 in Fig. 2b (X, in Fig. l), i.e. the expression for x4. As mentioned above, in this case fl = ( 15 ) . At first, we obtain the expressions for the coefftcients of the polynomial D(6) associated with the directed graph in Fig. 2b. According to Eq. (12) D(S) is for this scheme: D(6) = 6(a4 + F,S3 + F2S2 + F36 + F4)
Aj,&’ + Aj,we6wr= (B~osbt + Csinbt)eof
(24)
(21)
since F. = 1. According to Appendix A we have: where: F, =
B = Aj,y + Aj,, C = i(Aj,” - Aj,,)
K1,2
+ K2.1
+ K2,3
+ K2,4
+ K4,5
+ K5,4
(25)
(22) (i=Jij
(23)
The procedure in the Appendix A generates the following terms for the derivation of F2 from F,:
q&R 128
R. Varbn et al. / BioSystems 36 (1995) 121-133
F2 = &,2K2,3
+ &,2K2,4
+ K2,1&,5
+ K1.2K4.5 + 4,2
+ K2,1&,4
+K2,4&,5
+ K2,3&,5
c0aicients h,4h
Ksp
+ K2,3K5,4
+ K2,4&,4
(26)
The analogous process, applied to the terms of F2 in Eq. (24), yields:
h,4>1,
(a&, (a& and (a& in Eqs. (31) or/and (32) can be obtained as outlined in Appendix B for a general model (see Eq. (B21)). As an example, we explain how the expression for (as,4)2is obtained. According to Eq. (B21) and using the values of n, i, k and q (5, 4, 5 and 2, resp.), we have: (33)
(a5,4)2 = r;;4,5 f’3 = Kl,2K2,3&,s
+ Kl,zKzzK5,4
+ Kl,2K2,&,s
Kl,z&,&,4
+
(27)
Since from the terms of F3 it is not possible, by adding any of the fractional transfer coefftcients to these terms, to generate any term which does not contain symmetric or circular combinations of K’s, we have: F4 = 0
where F’2,1,S is the sum of the terms of F2, in which &,s is not a factor and in which a connection exists between the compartments XS (X,) and X4 (X3through combinations of the constants. If we, in the expression for F2 given by Eq. (26), suppress the terms which either contain & or do not represent a connection between the compartments XS and &, then, with Eqs. (26) and (33) we obtain:
(28) (u5,4)2 = - KzK5.4
+ K2,&5,4
+ K2,3%,4
Therefore Eq. (24) becomes: (34)
+ K2,&5.4)
D(6) = 82(83 + FIS2 + F26 + F3)
(29)
Because F3 is the last non-null coefficient, u = 3 and c = 5-3 = 2 for this scheme. Then, according to Eqs. (18)-(20) and with the values of n,i and c (5, 4 and 2, resp.), we have:
Analogously, cients:
one obtains for the other coeffi-
= 0
(35)
(a,,,),
= (a5,4)0 = (q4h
(al,4)2
= - Kl.2K2.4
(36)
(Q,4)3
= - Kl,zK2,4K5,4
(37)
3 x4 = A.0
+
c
A,, e’h’
(30)
h=l h,dGxp
(38)
(~5,4)1 = - K5,4
+ @5,4)3x!
A4,o = F3
(05,4)3 k - (4,2K2,3K5,4
+ 4,2K2,4&,4)
(39)
and 3 xy
c
3 (a1,4)&4
+ x!
q=o
c
(a5,4)qc4
q=o
&,h = Ah fi
@p
-
6,)
p=l p*h
(32) 6i, &, a3 in Eqs. (30) and (32) are the roots of the polynomial 63 + F,S2 + F28 + F3 = 0. Hence, the
3.2. Extension of the analysis to linear open compartment systems The kinetic analysis has been carried out assuming that the system is a linear closed compartment without input. Nevertheless, this analysis is also applicable to linear compartment systems, in which the excretion of substance from any compartment is allowed, i.e. to open systems. For these systems it must merely be assumed that the substance excreted from a compartment moves to a hypothetical terminal compartment which must be
R. Vardn et al. /BioSystems 36 (1995) 121-133
included in the system. If the substance is removed from more than one compartment, then we introduce a hypothetical terminal compartment, in which all excreted substances are collected. If our analysis is to be applied to an open compartment system, it is convenient to denote its compartments as Xi, X2, .. .. Xn-1, and the hypothetically added compartment as X,. The fractional excretion coefficient, Ki,*, from any compartment Xi (i = 1,2, ....n-1) must then be replaced by Ki,,.
4. Results aal ditmssion This contribution describes the derivation of the kinetic equations for a general linear ncompartment model without input in terms of the model parameters, i.e. the initial amount of substance in each compartment and the fractional transfer coefficients corresponding to direct connections between the compartments. These equations were obtained by the Laplace transform. Several procedures exist which yield numerical solutions of any system of linear differential equations from a set of numerical values of the parameters, but an analysis of the compartment models in cases as those mentioned in the introduction requires analytical equations which are functions of the kinetic parameters of the model. The kinetic equations obtained here assume a very simple form, since in all polynomials we have eliminated the coefficients, which are null. The evaluation of the remaining coefficients neither requires the expansion of determinants, nor the use of graphical procedures, because they are easily obtained using the algorithms outlined in the Appendices A and B. The model to which they are applicable is very general. Hence, other compartment models, the equations of which have previously been obtained, may be considered as its particular cases. Examples are cases in which the input of substance only occurs in one compartment (Rescigno, 1956), such in which the analysis refers to models with a limited number of compartments (Rubinow, 1975; Cobelli, 1979; Travis and Haddock, 1981) and such in which the model is catenary (Chau, 1985). The kinetic equations obtained here are valid for all compartment systems which tit the propos-
129
ed model. They can also be applied to other systems which are described by a system of ordinary, linear, first order differential equations with constant coefficients, if the elements of the corresponding determinant D(6) satisfy the conditions (8) and Eq. (5). The equations for the compartment systems are also applicable to enzyme systems. The latter application is convenient, if the time dependence of the concentration of any of the enzyme species is desired. The kinetic equations obtained are valid for the whole course of the process of partitioning of the substance between the compartments, i.e. for the transient phase and the steady state. In the steady state (t - oo), Eq. (20) is simplified into: xi = Ai,o
(i = 1,2, ....n)
(40)
The kinetic analysis suggested here can also be ap plied to open linear compartment systems without input. In that case, it must merely be assumed that the substance excreted from any compartment is collected in a single hypothetical terminal compartment which must be included in the system. Hence, the system is closed and fits our model. Since the compartment added has no influence on the other compartments, the kinetic equations obtained for these compartments are the same as those for the compartments in the original open system (Jacquez, 1985). Our method is limited to the kinetic equations of systems with zero input. The analysis of systems with non zero input must begin with a solution of the system with zero input (Anderson, 1983; Jacquez, 1985). Therefore, our method both contributes to the solution of compartment systems with and without input. The formulation of the expressions of the coefticients Fq (q = 0,1,2 ,..., U) and (Uk.3, (k E B; i = 1,2,..., n; q = 0,1,2 ,..., U) in the Eqs. (18)-(20) may, due to the sequential, recurrent and systematic nature of the algorithms developed here, be performed by a simple, efficient computer program to which we devote the part II of this series. Acknowledgements
This work has been partially supported by a grant number PB92-0988 of the Direction General
R. Vardn et al. /BioSystems 36 (1995) 121-133
130
de InvestigacGn Cientifica y Tkcnica (Spain) and by the Biochemisches Institut of the University of Kiel (Germany).
s n-l exist, for which F, = 0, then m = n, because F, always is zero. Appendix B
AppendixA Algorithm for the derivation of the coefficients of the polynomial D(6) The coefficients F4 (q = 1,2,...,n-1) of Eq. (12) are the sum of terms which are the products of q K’s with different first index. Products containing symmetric or circular combinations of K’s are excluded. Symmetric combinations of K’s are those of the K~,.K&type and circular combinations of K b.c&.&d,e K’s are those of the Kb,&dKd.b-3 K c,b- ,..., types (Rescigno and Segre, 1965). The coefficients F4 (q = O,l,...., n-l) of the polynomial D(6) are found without expansion of the determinant in Eq. (10) as follows: F. is always equal to unity and F1 is the sum of all the transfer constants Kid (i#j) which are not zero. We denote the number of the non-null constants with R. Then, the terms of 4 (j = 2,3 ,..., n-l) are generated from the terms of Fbl (j = 2,3,...,n-1) in the following way: R new terms are formed from each term of Fbl: Successively, each of the non-null KiJ constants is written on the right hand side of the corresponding Fj_1-term. Such terms are then suppressed, in which the subindex i of Kij is 1~;the first subindex, a, of Kab located on the right hand side in the corresponding term of Fj_1. If, for the reason mentioned, any of the terms generated from those of the 4-1 is not suppressed, but contains symmetric or circular combinations of K’s, then it must be removed. Thus, the terms, which after this process remain, all contribute to Fj G= 2,3,...,n-1). If in the determination of a particular 4 (j = 2,3 ,..., n- l), e.g. F,, all the terms are suppressed, then, F, = 0. Moreover, if F,,, = 0, then Fm+,, Fn,+2, . . . . F,_l are also zero, i.e.:
Ftll=impliesF,=F,+,=F,,,+2=... =Fn_,
=O
(Al)
Therefore, if a F,,, = 0, then the process is ended, because the relation (Al) is satisfied. If no m
Algorithm for the abivation of the coefficients of any of the polynomial D&S) (k E n; i = 1,2,. . . ,n) Since the expansion of D(S) is a polynomial of degree n, the development of any determinant D&S) is a polynomial, the degree of which is I n- 1. The degree (n- 1) corresponds to the case in which k = i. Therefore, the expression for the polynomial D&6) is: n-l
c
Dk,i(8 =
(ak,i)&jn - ’ - ’
q=o
(-l)“-
((ak,t)o =
’ if k = i;
(ak,i )o = 0 if k # i)
W)
We distinguish between two different cases: k = i and k # i k=i We denote the principal minor resulting from the suppression in determinant D(6) of the i-th row and the i-th column (i = I,2 ,..., n) as Di,i. If, in the determinant D(6), we set: Kij = 0
(j = 1,2,..., n; j + i)
(B2)
then we obtain: i-th column 1
K,,,-6
o(qp=
42
K2,, . . . 0 K2,2- 6 . . . 0
. .
. .
::::::::
;C,,i . _ . .
iY2.i . . .
: . . .
KlJi
Kzr
. . .
: . . .
: . . .
. . . . . .
K,,, K,,,
‘-S . . .
: . . .
: . . .
K,,J . . .
0
. . .
K,,+
: . . .
(B3)
131
R. Varh et al. /BioSyslems 36 (1995) 121-133
where D(Qio denotes D(S) after insertion of Eq. (B2). The determinant in Eq. (B3) is then expanded by the elements of the i-th column: W)P = -6 Qi(@
W)
From Eqs. (B4) and (8) it follows that:
(k E 0; k # i)
n-1 Qi (6) = (-1)” -
’ C
l$.i
h-
9-
1
(W
q=o
where F$ (q = 0,1,2 ,..., n-l) is Fq after the insertion of Eq. (B2), i.e.F$ is the sum of the terms of Fq which do not contain &I, Ki.2 ,..., Kifi. The expressions for Ft,i depend on the i-value of the Eqs. (B2). Since F. = 1, it follows that Fz,i = 1. From Eqs. (Bl) and (B5), we obtain: (ai,i)q= (-1)” - ‘Fi,i
(4 = 192, . . . J - 1)
(B6)
kfi
The coel&ients of the polynomials &,(b) (k c 0; k # i) are gained, if in the determinant D(S) we set: Kij = 0
where D(6)‘,k denotes that the changes in D(6) indicated by Eqs. (B7) have been made. The determinant D(6)‘j,k is then developed Using the &mentS of the i-th column:
(B9)
i.e.: K&J6)
= (-l)k + i
[&@; + (Ki,k+ Wi,i(Ql
(kcn;k#i)
WO)
In the following we deduce from Eq. (BlO) a general expression for the derivation of the coefficients of any of the polynomials &@) (k e Q k # i) from the coefficients Fq (q = O,l,...,n-1) of the polynomial D(S). According to Eq. (12), the polynomial expression for B(6)*,k (k e $2; k # i) is: n-1
D(& = (-1)“6 c
F;,,& P-q-
’
q=o
(j = 1,2, . . . , n; j # k,i)
(B7)
(k E Q; k f
i)
(Bll)
Then, we obtain: OcGk.- =
where F*q,i,k is the coefficient Fq after insertion of Eq. (B7). Since F, = 1, it follows that F*o,j,k = 1 (k
i-ttl columm
4.1-S
K2.1
4.2
K2,2- 6 . . .
. . .
. . .
... ... ...
I 0 0 . . .
. . .
. . .
... ... ...
. . .
.... .... ....
. . .
. . .
... ... ...
. . .
.... .... ....
. . .
. . . . . . .... .... ....
e Q; k f K 41 K 42
i).
From Eqs. (B9), (BlO) and (B5), we obtain: n-l KikDk,i(6) = C (-l)n+i+k-’ q=o (-Fi + 1,i,k
(J’i,ik
= c,i
+ Ki,kQ
= 0;
+ q
+ 1.i )6” - ’ - ’
k E Q; k z i)
(BW
A comparison of Eqs. (B 12) and (B 1) finally yields: Kik(Uk,i)q= (-I)” (-F;
+ i+k - ’
+ I,~JC + &,d$,i + q + 1.i)
R, Varbn et al. /BioSystems 36 (1995) 121-133
132
circular combinations of K’S, but the terms of Ki,k G,i might contain them. In that case, a connec-
(q = O,l, . . . , n - 1; k E Q; k # i; FL
= fi,, = 0)
(B13)
Eq. (B13) is a simple relation between the coeflicients of the polynomials Dk,@) (k E fl) and those of the polynomial D(6). We consider the meaning of the sum: -F; + i,ik + K$$,i
(B14)
of Eq. (B13) to facilitate the derivation of the coefficients (ck,i)q The definitions of F*q+,,i,k and P q+l.t show that F*q++l.t,kcontains the same terms as those of the coefficients pq+,,i. Besides, it in. . eludes all terms whrch m Fq+l contain Ki,k as a factor. The terms which in Fq,! contain the constant Ki,k as a factor, have the form:
. . . K aq+ Ipbq+1
VW
where:
Ui+]
C * *. C
Uq+]
I
fl
1 5 bj; bj = uj ; j = 1,2, . . . , q + 1 is devoid of symetric or circular combinations of K’s
(Jw 1
1,i.k + q+
I,i=-
c&,,b,
&,a . . . &b
. . . Kai_ *,bi_,
(B17) where the sum is extended to all the terms in which the conditions (B16) are fulfilled. The terms of $,i do not contain symmetric or
. . . Kb,iyt?tC
(Bl@
The terms of the right hand side of Eq. (B17) coincide, with changed sign, with the terms of Ki.kF0q.i which do not contain symmetric or circular combinations of K’s. Hence: -J’; + 1,i.k +
&,kq,i
(kC8I;k
#i)
+ q
+ l,i = &,kF&,k
0319)
where F’q,i,k is the sum of the terms Fq which do not involve any of the constants Ki,l, Ki.2)..., Ki,, and in which there is a connection between the compartments Xk and Xi, i.e. these terms contain some of the combinations (B18). Since F. = 1, the F/ O,i,k= 0 (k E a; k # i). From Eqs. (B13) and (B19) we have: (Uk,i)q = (-1) (q =
Evidently, the constant Ki,k may in the terms of (B15) occupy the first place (if i = aI) or the last (if i = u,+l). Since (-F’q+,,,k + Pq+,,i) is the sum of all the possible terms in the form of (B15), it follows that: - <+
&,iy &,&z,i, &,o * - * &,iy &,a&,d&
+ q + 1.i
(q = 0,1,2, . . . ,n - 1; k E Q; k #i)
C i C
tion betWet% the compartments Xk and Xi exists in the corresponding term of c,i, i.e. one of the following combinations of constants is found in this term:
n+i+k-
IF;,~&
1,2, . . ., n - 1; k E Q; k # i)
WO)
The above equation is only valid for k ;f i.
A single equation for the coefficient (ak,Jq may exist which also is valid for k = i. If we set F/s,,,, = Pq,i (q = O,l,..., n-l) then, according to Eqs. (B6) and (B20): (Uk,i)q = (-1) (q =
n + i + k - IF;,i,k
0,1,2, . . ., n - 1; k c Q)
Wl)
It follows from Eq. (B21), that if Fq = 0 (Flq,,,k = 0), then (uk,ijq = 0. Therefore, if F, = 0, Eq. (Al) requires that:
R. Vardn et al. /BioSysrems 36 (1995) 121-133
(“&,i)m = (“k,ih
+ 1 =
* * * =
t”k,ih - 1 = O
(B22)
If Eq. (B22) is entered into Eq. (Bl) and we let u = m - 1 and c = n - u, then we obtain:
((Uk,i)o
=
(-
1)”- ’ if k = i;
&),-, = 0 if k # i)
(~23)
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