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Mathematical Biosciences 167 (2000) 145±161
www.elsevier.com/locate/mbs

Identi®ability and interval identi®ability of mammillary and
catenary compartmental models with some known rate
constants
Paolo Vicini 1, Hsiao-Te Su, Joseph J. DiStefano III *
Biocybernetics Laboratory, Departments of Computer Science and Medicine, Boelter Hall 4531 K,
University of California at Los Angeles, Los Angeles, CA 90095-1596, USA
Received 12 March 1999; received in revised form 13 June 2000; accepted 3 July 2000

Abstract
The identi®ability problem is addressed for n-compartment linear mammillary and catenary models, for
the common case of input and output in the ®rst compartment and prior information about one or more
model rate constants. We ®rst de®ne the concept of independent constraints and show that n-compartment
linear mammillary or catenary models are uniquely identi®able under n ÿ 1 independent constraints.
Closed-form algorithms for bounding the constrained parameter space are then developed algebraically,
and their validity is con®rmed using an independent approach, namely joint estimation of the parameters of
all uniquely identi®able submodels of the original multicompartmental model. For the noise-free (deterministic) case, the major e€ects of additional parameter knowledge are to narrow the bounds of rate
constants that remain unidenti®able, as well as to possibly render others identi®able. When noisy data are
considered, the means of the bounds of rate constants that remain unidenti®able are also narrowed, but the

variances of some of these bound estimates increase. This unexpected result was veri®ed by Monte Carlo
simulation of several di€erent models, using both normally and lognormally distributed data assumptions.
Extensions and some consequences of this analysis useful for model discrimination and experiment design
applications are also noted. Ó 2000 Elsevier Science Inc. All rights reserved.
Keywords: Identi®ability; Interval identi®ability; Parameter bounds; Mammillary; Catenary; Compartmental model

*

Corresponding author. Tel.: +1-310 825 7482; fax: +1-310 794 5057.
E-mail address: joed@cs.ucla.edu (J.J. DiStefano III).
1
Present address: Department of Bioengineering, P.O. Box 352255, University of Washington, Seattle, WA 981952255, USA.
0025-5564/00/$ - see front matter Ó 2000 Elsevier Science Inc. All rights reserved.
PII: S 0 0 2 5 - 5 5 6 4 ( 0 0 ) 0 0 0 3 5 - 3

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P. Vicini et al. / Mathematical Biosciences 167 (2000) 145±161

1. Introduction

A problem that commands continuing interest [1] is that of bounding the parameter space for
unidenti®able, linear, time-invariant multicompartmental models, thereby providing ®nite ranges
for otherwise unidenti®able rate constants based on information inherent in the model structure.
DiStefano [2] introduced the notions of interval identi®ability and quasiidenti®ability and derived
explicit, computable expressions for ®nite bounds on the rate constants of general mammillary
models of any order, and with all possible rate constants present, for the most common case of
input and (noise-free) output in the central compartment. This work was extended, ®rst for
computing the uniquely identi®able parameter combinations of the same model class [3], then to
similarly general catenary models with input and output in the same compartment [4,5], and then
to the noisy data case, for both mammillary and catenary models [6]. Together [2±6] provide
closed-form algorithmic solutions for the ®nite intervals of all kij for these two model classes,
given that no kij are known or otherwise ®xed. We call these intervals the unconstrained bounds on
the kij , for reasons clari®ed below. Other contributions to this problem area are found in [7],
where parameter bounds for two and three compartment mammillary models with input and
output in compartment 1 are derived in terms of the parameters of the sum of exponentials responses; more general methods are presented in [8] for localizing parameters of linear compartmental models within bounded regions, and an alternative approach based on Lyapunov
functions recently has been proposed [1].
In the current work, we address the common case in which one or more rate constants kij are
known, a likely situation in applications of either of these two model classes, and ®rst develop
conditions under which the other kij are identi®able. For example, a priori information often
indicates that, for some i, compartment i has no leak: in this case, k0i ˆ 0, but this does not

necessarily assure (structural) identi®ability of the model. In brief, we use the following de®nitions
of identi®ability notions: a parameter (e.g. rate constant kij ) of a model M is unidenti®able if there
exist an uncountably in®nite number of solutions for kij from the model equations, with inputs
and outputs speci®ed in these equations; it is interval-identi®able if it is unidenti®able but can be
bounded within a ®nite interval from the same model equations and inequality relations on the
parameters (e.g. kij > 0; i 6ˆ j). The parameter kij is identi®able if one or more distinct solutions
exist from the same equations; it is uniquely identi®able if only one such solution exists. The whole
model M is unidenti®able if any one kij is unidenti®able; and identi®able/uniquely identi®able if
all kij are identi®able/uniquely identi®able.
In general, in the absence of noise in the data, the original (unconstrained) bounds shrink when
one or more constraints are applied, thereby re¯ecting the increased available information,
consistent with intuition. The reduced range or bounds in this case are termed the constrained
range or bounds. For the noisy data case, our results are consistent for the mean of the bounds, but
± interestingly ± not for the variance, using conventional assumptions of normality or log-normality of measurements. We have con®rmed the validity of our algorithms, and these counterintuitive results, by both Monte Carlo simulation and also using an independent submodel
approach [8,9]. The latter applies to a wider class of models, but unfortunately does not provide
closed form solutions.
The current results provide closure on the deterministic part of the bounding problem for open
linear mammillary and catenary compartmental models of any order, with input and output in

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147

compartment 1, and for all unknown kij , with any number of kij known a priori. The overall
solution is closed-form and algorithmic and thus can be readily programmed.

2. Catenary and mammillary models
Linear catenary and mammillary compartmental models with n compartments and scalar input
u…t† and scalar output y…t† in the ®rst compartment are shown in Fig. 1. They can be described in
terms of mass ¯ows by
dq…t†
ˆ Kq…t† ‡ au…t†;
dt
y…t† ˆ cq…t†;

q…0† ˆ q0

…1†
T

where q is a n-dimensional vector of compartment masses, a ˆ ‰1 0 . . . 0Š , where the superscript T
indicates
P transpose, c ˆ ‰1=V1 0 . . . 0Š; K ˆ ‰kij Š is the matrix of rate constants, with
kii ˆ njˆ0 kji ; i ˆ 1; . . . ; n; kij (tÿ1 units) designates transfer to compartment i from compartment
j …j 6ˆ i†, and V1 is the volume of distribution of compartment 1. The unit impulse response for
either model is a sum of n distinct exponential terms, where Ai > 0 and ki < 0 8 i

Fig. 1. General catenary (upper) and mammillary (lower) compartmental models with both input and output in
compartment 1.

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P. Vicini et al. / Mathematical Biosciences 167 (2000) 145±161

y…t† ˆ

n
X

Ai eki t ;

…2†

t P 0;

iˆ1

when k1i 6ˆ k1j 8 i; j : i 6ˆ j 6ˆ 1, and k1i ki1 > 0 8i for the mammillary model, [10, p. 62]; and
ki;i‡1 ki‡1;i > 0 8i for the catenary model [10, p. 63], and all k0i P 0, with k0j > 0 for at least one j for
both models.
In our case, the input/output transfer function is the Laplace transform of (2):
n
X
Ai
b snÿ1 ‡ bnÿ1 snÿ2 ‡


‡ b2 s ‡ b1
ˆ n n
;
…3†
H …s† ˆ
s ÿ ki

s ‡ an snÿ1 ‡


‡ a2 s ‡ a1
iˆ1

where all of the ai and bi can be evaluated directly from (1) in terms of the kij , as uniquely
identi®able parameter combinations, or structural invariants of the model [12]. For the catenary
model, these are:
ÿ k11 ˆ k01 ‡ k21 ;
ÿ kii ˆ k0i ‡ kiÿ1;i ‡ ki‡1;i ;

1 < i < n;

ÿ knn ˆ k0n ‡ knÿ1;n ;
ci ˆ kiÿ1;i ki;iÿ1 ;

…4†

i ˆ 2; 3; . . . ; n:

For the mammillary model, the structural invariants are:
ÿ k11 ˆ k01 ‡ k21 ‡ k31 ‡


‡ kn1 ;

ÿ kii ˆ k0i ‡ k1i ;
ci ˆ k1i ki1 ;

1 < i6n

…5†

i ˆ 2; 3; . . . ; n:

As shown in [3,4], these invariants can be algorithmically derived from the output in three steps
y…t†;

t 2 ‰0; T Š ) fAi ; ki g ) fai ; bi g ) fkii ; ci g:

In [2±4], ®nite parameter bounds 8 kij were determined for both model types, with all kij > 0 and
unknown, and were implemented in the programs MAMPOOL and CATPOOL [3±6]. When
some kij are known or prior estimates are available, the dimensionality of the space of unknown
parameters is reduced and the new bounds must satisfy a modi®ed set of structural relations, as
shown below. In addition, prior parameter values or estimates must be feasible, i.e. they must be
consistent with the model structure and output data y…t†, as described next.

3. The feasible subspace for equality constraints
Equality constraints on the rate constants of catenary and mammillary models must satisfy the
following conditions:
1. No additional equality constraints are possible for the 1-compartment model, because its single
rate parameter is determined uniquely by the available (noise-free) output data y…t†, for nonzero input and output, as is V1 ˆ q…0†=y…0†, e.g. V1 ˆ 1=y…0† for a unit impulse input.
2. Speci®ed parameter values (or estimates) must fall within their corresponding unconstrained
bounds (i.e. bounds for the kij when none are given a priori).

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149

3. Constraints must be independent. As a simple example, the following set of equations for the
catenary model:
ÿ k11 ˆ k21 ‡ k01 ;
c2 ˆ k12 k21
provides unique solutions for all three parameters k12 , k21 , and k01 , when any one of them is
given. If there are two con¯icting constraints among these three variables, then no solution
exists. This condition is called independence, and it is formalized below for both model classes.

4. There cannot be more than n ÿ 1 independent parameter equality constraints for n-compartment catenary or mammillary models. This is stated and proven as two theorems below. The
proofs are based on the fact that there are at most 3n ÿ 2 unknowns and 2n ÿ 1 equality relations among the parameters. This leaves at most n ÿ 1 degrees of freedom in the model, and
thus there cannot be more than n ÿ 1 constraints without violating the independence condition.
3.1. The catenary model
Constraints may be infeasible for a given set of data and thus may yield no solution, e.g., if
k12 ˆ 1 and k21 ˆ 24 are given, but k12 k21 ˆ c2 ˆ 0:5 is estimated from the input±output data, then
we have a contradiction. Constraints can also be redundant, e.g. k12 ˆ 1 and k21 ˆ 0:5 and
k12 k21 ˆ c2 ˆ 0:5. This motivates the notion of independent constraints.
De®nition 1. Let P be the set of all of the rate constants in the n-compartment catenary model. Let
Q be the set of the constrained rate constants. Then the elements of Q are independent in the
catenary model if jIj \ Qj 6 1 8j : 1 6 j 6 n ÿ 1 or jJj \ Qj 6 1 8j : 1 6 j 6 n ÿ 1 or both, where


Ij ˆ k0j ; kj‡1 ; j; kj;j‡1
8j : 1 6 j 6 n ÿ 2;


Inÿ1 ˆ k0;nÿ1 ; k0;n ; knÿ1;n ; kn;nÿ1

and

J1 ˆ fk01 ; k02 ; k12 ; k21 g;


Jj ˆ k0;j‡1 ; kj;j‡1 ; kj‡1;j

8j : 2 6 j 6 n ÿ 1:

Notice that two slightly di€erent schemes, I and J, are used to partition the P set, and that
constraints are independent if they satisfy either or both schemes. The proof presented below
assumes that constraints satisfy scheme I. The case where scheme J is satis®ed can be constructed
in a similar manner.
Theorem 1. The n-compartment catenary model is uniquely identifiable under n ÿ 1 independent
constraints.

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P. Vicini et al. / Mathematical Biosciences 167 (2000) 145±161

Proof. We use induction on the following
Sj proposition: For 1 < j 6 n ÿ 1, with n ÿ 1 independent
parameter constraints, parameters in iˆ1 Ii are uniquely identi®able. Clearly, this proposition
depends on the value of j, where 1 6 j 6 n ÿ 1. Let us refer to it as P …j†. Furthermore, P …n ÿ 1†
implies that the theorem is true. We show P …n ÿ 1† is true by induction on j.
Basis: P …1†. Let Q be the set of constrained parameters. Then jQj ˆ n ÿ 1. By the pigeon hole
principle [11], jIj \ Qj ˆ 1 8j. In particular, jI1 \ Qj ˆ 1, which implies that one of the k01 , k12 and
k21 is constrained. Thus, k01 , k12 and k21 can be solved from
ÿ k11 ˆ k01 ‡ k21 ;
c2 ˆ k12 k21 :

S
Induction step: P …j† ) P …j ‡ 1† for 1 6 j 6 n ÿ 3. By P …j†, parameters in jiˆ1 Ii are uniquely
identi®able. In particular, kj;j‡1 is uniquely identi®able. Since jQ \ Ij‡1 j ˆ 1, one of k0;j‡1 , kj‡2;j‡1 ,
kj‡1;j‡2 is constrained. So we can solve for k0;j‡1 , kj‡2;j‡1 and kj‡1;j‡2 from
ÿ kj‡1;j‡1 ˆ k0;j‡1 ‡ kj;j‡1 ‡ kj‡2;j‡1 ;
cj‡2 ˆ kj‡1;j‡2 kj‡2;j‡1 :
By induction, P …n ÿ 2† S
is true. To prove P …n ÿ 1†, the same arguments can be applied. By
P …n ÿ 2†, parameters in nÿ2
iˆ1 Ii are uniquely identi®able. Speci®cally, knÿ2;nÿ1 is uniquely identi®able. Since jQ \ Inÿ1 j ˆ 1, one of k0;nÿ1 ; k0;n ; kn;nÿ1 ; knÿ1;n is constrained. So k0;nÿ1 ; k0;n ; kn;nÿ1 ; knÿ1;n
can be solved via
ÿ knÿ1;nÿ1 ˆ k0;nÿ1 ‡ knÿ2;nÿ1 ‡ kn;nÿ1 ;
ÿ kn;n ˆ k0;n ‡ knÿ1;n ;
cn ˆ knÿ1;n kn;nÿ1 :
Thus, P …n ÿ 1† is true and the theorem is proven.
3.2. The mammillary model
Since mammillary models are structurally identical if peripheral compartments are exchanged
or renumbered, some restrictions must be imposed on the labeling before we can prove that a
generic model is uniquely identi®able under n ÿ 1 constraints. We label the peripheral compartments such that
ÿk22 > ÿk33 >


> ÿknn ;
where the kii are generically distinct. As in the proof for the catenary model, we begin with the
notion of independence.
De®nition 2. Let P be the set of all rate constants in the n-compartment mammillary model. Let Q
be the set of constrained parameters. Then the elements of Q are independent in the mammillary
model if jIj \ Qj 6 1, where
I1 ˆ fk01 ; k02 ; k12 ; k21 g;


Ij ˆ k0;j‡1 ; k1;j‡1 ; kj‡1;1

8j : 2 6 j 6 n ÿ 1:

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151

Note that n ÿ 1 di€erent partitioning schemes are obtained for P by grouping k01 with any one of
the Ij . We only group k01 with I1 here. The proofs for other partitioning schemes are similar.
Theorem 2. The n-compartment mammillary model is uniquely identifiable under n ÿ 1 independent
constraints.
Proof. Let Q be the set of constrained variables. Then jQj ˆ n ÿ 1. By the pigeon hole principle
[11], jIj \ Qj ˆ 1 8j. For j P 2, this implies that one of the k0;j‡1 ; k1;j‡1 , and kj‡1;1 is constrained in
Ij . Thus, these three variables can be found uniquely from
ÿ kj‡1;j‡1 ˆ k0;j‡1 ‡ k1;j‡1 ;
cj‡1;j‡1 ˆ k1;j‡1 kj‡1;1 :
Since this is true 8j P 2, all of the parameters except the ones in I1 are uniquely identi®able. Since
jI1 \ Qj ˆ 1, it follows that one of the k01 , k02 , k12 , and k21 is constrained. If k01 is not constrained,
then parameters k02 , k12 , and k21 can be obtained from
ÿ k22 ˆ k02 ‡ k12 ;
c2 ˆ k12 k21 :
Since all of the ki1 , i 6ˆ 0, are uniquely identi®able, one can solve for k01 from
ÿk11 ˆ k01 ‡ k21 ‡


‡ kn1 :
If k01 is constrained, then it is possible to solve these equations in reverse order. k21 can be solved
for from the last equation above, then one can substitute its value into the previous equation and
the remaining rate constants can be solved for recursively.

4. The constrained parameter-bounding algorithms
4.1. The catenary model algorithm
For the unconstrained case, the minimum possible value for any k0j is zero [4]. However, no
more than n ÿ 1 leaks can be zero at the same time, and this was used in the derivation of the
unconstrained algorithm, which used the following recursive relations:
(
ÿk11 ÿ k01 ;
i ˆ 1;
…6†
ki‡1;i ˆ ÿk ÿ ci ÿ k ; i ˆ 2; . . . ; n ÿ 1;
0i
ii
ki;iÿ1
for the kij elements in the lower diagonal of the system matrix K in Eq. (1), and
(
ÿknn ÿ k0n ;
i ˆ n;
kiÿ1;i ˆ ÿk ÿ ci‡1 ÿ k ; i ˆ n ÿ 1; . . . ; 2;
ii
0i
ki;i‡1

…7†

for the kij elements in the upper diagonal. Then, with the leaks k0i recursively set to zero, the upper
bounds on the ki‡1;i and kiÿ1;i were obtained for all i [4].

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P. Vicini et al. / Mathematical Biosciences 167 (2000) 145±161

Now, if one or more rate constants is known, the above equations are still applicable, but
algorithmic solution must account for all available information. This is done as follows in the new
algorithm:
1. The algorithm for unconstrained catenary models [4] is applied ®rst, to ®nd the largest feasible
ranges for the kij , consistent with the model structure and output data. Then all equality constraints for the kij s are tested for feasibility. If infeasible, the algorithm is terminated. If any
parameter in the ®rst or last compartment is known, the values of the other two are evaluated
from
ÿ k11 ˆ k01 ‡ k21 ;

…8†

c2 ˆ k21 k12 ;
for the ®rst compartment, and
ÿ knn ˆ k0n ‡ knÿ1;n

…9†

cn ˆ knÿ1;n kn;nÿ1 ;

for the n-th compartment.
2. Eqs. (6) and (7) are solved recursively, substituting the values of known parameters wherever
they appear and setting the unconstrained leaks k0i to zero. This gives the new upper bounds on
all rate constants but the leaks.
3. The new lower bounds on all but the leak parameters are then found, as for the unconstrained
case, from the structural invariant relations (4):
ci
min
; i ˆ 2; . . . ; n;
ˆ max
kiÿ1;i
ki;iÿ1
…10†
ci‡1
min
; i ˆ 1; . . . ; n ÿ 1:
ˆ max
ki‡1;i
ki;i‡1
4. The new upper bounds on the leaks are then found from
min
max
;
ˆ ÿk11 ÿ k21
k01
min
min
k0imax ˆ ÿkii ÿ kiÿ1;i
;
ÿ ki‡1;i
max
k0n

ˆ ÿknn ÿ

i ˆ 2; . . . ; n ÿ 1;

…11†

min
:
kn;nÿ1

5. The new lower bound for each unconstrained leak remains zero, unless the model is uniquely
identi®able, in which case, the k0i s are uniquely determined from (11).
Remark 1. If there are n ÿ 1 independent constraints, then the model is uniquely identi®able
(Theorem 1) and the algorithm gives kijmin  kijmax 8 i and j.
Remark 2. The fairly common case of n ÿ 1 leaks set to zero presents some points of interest. The
remaining unconstrained leak then attains its maximum value and all parameters attain their
minimum or maximum values, as shown in Fig. 2. Conversely, setting any one k0j to its maximum
value is equivalent to setting all other k0i to zero, and then one equality constraint is enough to
make the model uniquely identi®able, with all kij attaining their minimum or maximum values
(Fig. 2).

P. Vicini et al. / Mathematical Biosciences 167 (2000) 145±161

153

min
Fig. 2. Unique identi®ability of the model is achieved when k0i ˆ k0imax . In this case, k0j ˆ k0j
ˆ 0 8 i 6ˆ j.

4.2. The mammillary model algorithm
This algorithm is somewhat simpler, because the peripheral compartments of the mammillary
model are not directly connected with each other. The structural invariant equations associated
with peripheral compartment i > 1 are
ÿ kii ˆ k0i ‡ k1i ;
ci ˆ k1i ki1 :

…12†

By inspection, (12) can be solved for all three variables if any one is known. Furthermore, they do
not directly a€ect the parameters in other peripheral compartments unless there are n ÿ 1 constraints. These observations motivate the following algorithm:
1. The algorithm for unconstrained mammillary models [2,3] is applied ®rst, to ®nd the largest
feasible ranges for the parameters consistent with the model structure and output data. Then
all equality constraints for the kij s are tested for feasibility. If infeasible, the algorithm is terminated.
2. If any parameter in the set k0i ; k1i ; ki1 is known (given), the values of the other two are evaluated
from (12). More than one equality constraint supplied by a user could be inconsistent with each
other and relations (12). In this case, the algorithm is terminated.
3. If the parameters associated with any compartment i are all unknown, then k1imax and ki1min are
calculated from (12) by setting k0i to zero.
4. ki1max , i > 1, is calculated from ÿk11 ˆ k01 ‡ k21 ‡


‡ kn1 by setting all parameters except ki1 to
their minimum, i.e.
X
min
;
…13†
kj1
ki1max ˆ ÿk11 ÿ
j6ˆi;j6ˆ1

for i > 1. If any kj1 are known …1 6ˆ j 6ˆ i†, then their values are simply substituted into (13) to
obtain ki1max .
5. For i > 1, k1imin and k0imax are calculated from structural invariants (12) by setting ki1 to its maximum.
min
max
ˆ 0 and k01
is found from
6. If k01 is unknown, then k01
X
min
max
k01
:
…14†
ˆ ÿk11 ÿ
kj1
j>1

Remark 3. If there are n ÿ 1 independent constraints, the model is uniquely identi®able (Theorem
2), and the algorithm gives kijmin  kijmax 8 i; j.

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P. Vicini et al. / Mathematical Biosciences 167 (2000) 145±161

5. Noisy data and parameter equality constraints: algorithmic approach
E€ects of noisy output data on unconstrained parameter bounds of catenary and mammillary
models were treated in [6], for the measurement model z…ti † ˆ y…ti † ‡ e…ti †. An asymptotic covariance matrix for the catenary model bound
h
iT
max
max
min
min
. . . kn;nÿ1
k01
. . . kn;nÿ1
…15†
b ˆ k01
was computed in terms of the vector of (known) structural invariants p as
 T
ob
ob
d
C OV…b† ˆ COV…p†
op
op

…16†

in [6], and similarly for the mammillary model. Now, if a parameter is ®xed, the same procedure
applies. Eq. (16) gives the covariance of the bounds on the remaining rate constants, established
by reevaluating the elements of the matrix ob=op from these same relationships established in [6]
for both mammillary and catenary compartmental models. However, in this case, the element
values are determined using the constrained bounds, which are generally di€erent than the unconstrained bounds. Also, the covariance matrix is reduced in dimension by two for each known
parameter, because
obi
ˆ 0T
op

…17†

ˆ bmax
and structural invariants p.
for each known constraint bi ˆ bmin
i
i

6. Joint submodel parameters: another approach
In principle, parameter bounds of unidenti®able compartmental models can also be computed
using an identi®able submodels approach [8,9]. We have exploited this idea further here, to lend
validity to our primary algorithms. By this approach, the ranges for each of the parameters of the
original (unconstrained) unidenti®able model are established from the uniquely identi®able parameters of particular submodels, which are either minimum or maximum values of the range for
a given parameter [9]. Logically, then, the ranges for a constrained model should be determined
similarly by joint estimation of the uniquely identi®able parameters of the submodels of the
constrained structure.
Uniquely identi®able submodels of unidenti®able structures are typically generated by systematically eliminating parameters of the original model until it becomes uniquely identi®able
[12]. In our case, every unconstrained n-compartment mammillary or catenary model has n
uniquely identi®able submodels, each obtained by setting all leaks but one to zero, as described
earlier, and illustrated for the catenary model in Fig. 2. When parameters are constrained, the
number of such submodels is reduced, because leaks then become uniquely identi®able and
generally di€erent from zero. For this reason, the joint regions or ranges of values for the constrained model are typically reduced as well.

P. Vicini et al. / Mathematical Biosciences 167 (2000) 145±161

155

Fig. 3. The uniquely identi®able submodels of the (a) unconstrained and (b) constrained, with k12 ˆ 0:28 minÿ1 ,
3-compartment mammillary model reported in [6, p. 186]. Numerical estimates for the rate constants are reported in
Table 1.

To illustrate this, Fig. 3(a) includes all of the uniquely identi®able submodel structures of the
unconstrained 3-compartment mammillary model with input and output in compartment 1 only.
As noted earlier, ÿk22 > ÿk33 for these con®gurations, thereby maintaining unique identi®ability.
It is well known that each submodel has a single leak from only one of the three compartments, as
noted [8,10]. In Fig. 3(b), suppose k12 is constrained (given) and nonzero. Then, by Eqs. (13) and
(14), either k01 , k21 and k02 are uniquely identi®able and nonzero, or k03 , k21 and k02 are uniquely
identi®able and non-zero, both as shown. These are the two uniquely identi®able submodels of the
constrained structure, and parameter bounds of the original constrained structure can therefore
be obtained by quantifying these submodels, e.g. by direct estimation with weighted nonlinear
least squares.

7. Numerical examples
We exercised our new algorithms and compared the results with the corresponding results of
Monte Carlo (MC) simulations and the joint submodel parameters approach, applied to numerous mammillary and catenary model examples. MC simulations were implemented in
MATLAB [14] with 1000 simulations for each example, using the statistics and distributions
noted for each. The uniquely identi®able submodel parameters and their statistics were evaluated
using the kinetic analysis software SAAM II [15].
Results for four di€erent model structures, with two di€erent data error types, are given below.
For each example case study, analytical, simulated and direct parameter estimation gave results

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within a few percent of each other and therefore only one set is given in the ®gures or tables for
each example.
7.1. Four-compartment models with Gaussian measurement errors
Example 1. We reexamine the 4-compartment catenary model example presented in [4], previously
evaluated with the unconstrained parameter algorithm implemented in the program CATPOOL
[4], as shown in Fig. 4(a). Numbers in parentheses are %CVs ( ˆ 100 SD/mean) on the parameter
bounds, computed as in [6]. When we constrained one of the kij : k43 ˆ 0:006, and analyzed the
resulting model using the new algorithm, we got the results depicted in Fig. 4(b). The constrained
model remains unidenti®able overall, but k34 and k04 become uniquely identi®able and the new
mean parameter bounds for the remaining kij s are narrower than the unconstrained ones. Note,
however, that some parameter-bound variabilities (%CVs) increase, e.g. all %CVs for the k0i s.
Example 2. An unconstrained 4-compartment mammillary model is shown in Fig. 5(a), the same
model analyzed in [3]. The new algorithms with k14 ˆ 0:0014 given yielded Fig. 5(b). As in the
catenary example, the model remains unidenti®able overall, but with narrower mean bounds on
all interval identi®able kij , and all of the parameters associated with compartment 4 are uniquely
identi®able. However, as with the catenary model example above, some parameter-bound variabilities increase (e.g. all of the k0i ).

Fig. 4. The (a) unconstrained, i.e., all rate constants unknown, and (b) constrained, k43 ˆ 0:006 minÿ1 , parameter
bounds in a 4-compartment catenary model (lower bounds below or on the left of the arrow, upper bounds above or to
the right). Rate constant (minÿ1 ) ranges are shown by arrows and asymptotic %CVs of estimated bounds are given in
parentheses. A single number designates a uniquely identi®able kij , with %CV assumed zero when kij is ®xed beforehand
(as for k43 in (b)). The sampling schedule for Monte Carlo simulation and parameter estimation for this example was: 0,
0.3, 0.5, 1, 3, 4, 5, 7, 10, 13, 20, 50, 200, 250, 500, 800, 900, 1200 min, in triplicate, and data measurement errors were 5%
(CV).

P. Vicini et al. / Mathematical Biosciences 167 (2000) 145±161

157

Fig. 5. The (a) unconstrained, i.e., all rate constants unknown, and (b) constrained, k14 ˆ 0:0014 minÿ1 , parameter
bounds in a 4-compartment mammillary model (lower bounds below or on the left of the arrow, upper bounds above or
to the right). Rate constant (minÿ1 ) ranges are shown on the arrows, and asymptotic %CV of estimated bound are given
in parentheses. A single number designates a uniquely identi®able kij , with %CV assumed zero when kij is ®xed beforehand (as for k14 in (b)). The sampling schedule for Monte Carlo simulation and parameter estimation for this
example is: 0, 0.3, 0.5, 1, 3, 4, 5, 7, 10, 13, 20, 50, 200, 250, 500, 800, 900, 1200 min, in triplicate, and data measurement
errors were 5% (CV).

7.2. Three-compartment models with Gaussian or lognormal errors
We applied the new algorithm to two unconstrained 3-compartment mammillary and catenary
model examples published earlier in [6, p. 186], each with all kij and k0j > 0.
Examples 3 and 4. Table 1 is a summary of results for the mammillary model, Table 2 for the
catenary model, each unconstrained versus constrained by k12 ˆ 0:28. For both, measurement
errors were Gaussian, with 5% (%CV) errors.
Example 5. The catenary model was also run assuming lognormal measurement errors. The results
are shown in Table 3.
As with Examples 1 and 2, some bound variabilities (shown in %) increased when k12 was
®xed in Examples 3±5. In fact, the increase was severalfold for many of the bounds shown in
Tables 1±3.

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Table 1
Computed parameter bounds and variabilities for the unconstrained and constrained (k12 ˆ 0:28 minÿ1 ) 3-compartment
mammillary model reported in [6, p. 186] with 5% Gaussian measurement errors
Bound

max
k01
max
k02
max
k03
max
k12
min
k12
max
k13
min
k13
max
k21
min
k21
max
k31
min
k31

Unconstrained case

Constrained case

Value (minÿ1 )

SD (minÿ1 )

CV (%)

Value (minÿ1 )

SD (minÿ1 )

CV (%)

0.60567
0.29592
0.02022
0.45583
0.15991
0.02466
0.00424
0.93297
0.32700
0.73300
0.12701

0.05190
0.01640
0.00089
0.03570
0.02210
0.00114
0.00031
0.09830
0.05210
0.05950
0.00877

8.57
5.55
4.41
7.84
13.8
4.67
7.19
10.5
15.9
8.11
6.91

0.40014
0.17583
0.01857
0.28000
0.28000
0.02446
0.00589
0.53283
0.53283
0.52715
0.12701

0.06310
0.03570
0.00095
±
±
0.00140
0.00086
0.11400
0.11400
0.06460
0.00877

15.8
20.3
5.11
±
±
4.7
14.7
21.3
21.3
12.3
6.91

Table 2
Computed parameter bounds and variabilities for the unconstrained and constrained (k12 ˆ 0:28 minÿ1 ) 3-compartment
catenary model of the impulse response reported in [6, p. 186] with 5% Gaussian measurement errors
Bound

max
k01
max
k02
max
k03
max
k12
min
k12
max
k21
min
k21
max
k23
min
k23
max
k32
min
k32

Unconstrained case

Constrained case

Value (minÿ1 )

SD (minÿ1 )

CV (%)

Value (minÿ1 )

SD (minÿ1 )

CV (%)

0.60567
0.19154
0.02100
0.33524
0.14367
1.05997
0.45431
0.03326
0.01226
0.30335
0.11180

0.05191
0.01354
0.00096
0.03287
0.01990
0.10581
0.05902
0.00194
0.00120
0.01843
0.00823

8.6
7.1
4.6
9.8
13.9
10.0
13.0
5.8
9.8
6.1
7.4

0.51605
0.05524
0.01100
0.28000
0.28000
0.54393
0.54393
0.03326
0.02226
0.16703
0.11180

0.06457
0.03287
0.00411
±
±
0.11407
0.11407
0.00194
0.00479
0.03619
0.00823

12.5
59.5
37.3

21.0
21.0
5.8
21.5
21.7
7.4

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P. Vicini et al. / Mathematical Biosciences 167 (2000) 145±161

Table 3
Computed parameter bounds and variabilities for the unconstrained and constrained (k12 ˆ 0:28 minÿ1 ) 3-compartment
catenary model of the impulse response reported in [6, p. 186] using 5% lognormal measurement errors
Unconstrained case

Bound

max
k01
max
k02
max
k03
max
k12
min
k12
max
k21
min
k21
max
k23
min
k23
max
k32
min
k32

Constrained case

Value (minÿ1 )

SD (minÿ1 )

CV (%)

Value (minÿ1 )

SD (minÿ1 )

CV (%)

0.60570
0.19136
0.02100
0.33480
0.14344
1.05972
0.45402
0.03327
0.01226
0.30310
0.11174

0.05193
0.01359
0.00096
0.03295
0.01993
0.10580
0.05901
0.00194
0.00120
0.01851
0.00823

8.60
7.10
4.60
9.80
13.9
10.0
13.0
5.80
9.80
6.10
7.40

0.51683
0.05480
0.01095
0.28000
0.28000
0.54288
0.54288
0.03327
0.02232
0.16654
0.11174

0.06457
0.03293
0.00414
±
±
0.11406
0.11406
0.00194
0.00482
0.03629
0.00823

12.5
60.1
37.8
±
±
21.0
21.0
5.8
21.6
21.8
7.4

8. Discussion
One or more rate constants of a compartmental model are often known, typically due to the
absence of compartment leaks. All leaks are present, in general, in the context of the most general
mammillary and catenary compartment models we treat here. When additional parameter information is available, it should be used in the overall identi®ability problem solution, and one
would anticipate intuitively that this should reduce the range of computable bounds for parameters that remain unidenti®able, just as it might render other parameters or the entire model
identi®able. This was the motivation for this work, and we found that the resulting algorithms for
incorporating parameter equality constraints for catenary and mammillary models consistently
reduce the ranges in unidenti®able rate constants, in the limit providing equal upper and lower
bounds for some kij s when the additional parameter information renders them identi®able.
Moreover, we showed that parameter constraints have to satisfy some conditions (namely, independence and feasibility) to be considered acceptable vis a vis the model structure and the
information given by the data.
Although constraints consistently reduced mean ranges in unidenti®able kij , some parameterbound variabilities increased, especially for the k0j (leaks). This might be anticipated with
variabilities expressed as 100 SD/mean, for kij with reduced mean ranges. But some SDs also
increased. We veri®ed this seemingly paradoxical result for each example, by Monte Carlo
simulation, as well as for several additional examples.
By way of explanation, we refer to Eqs. (16) and (17) for the parameter variances. First, we note
that the variance of the structural invariants, VAR(p), is established by the output data. When

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P. Vicini et al. / Mathematical Biosciences 167 (2000) 145±161

some elements of the parameter-bound variances, VAR(b), are zero (for ®xed kij s), VAR(p) must
nevertheless remain the same. Therefore, other parameter-bound variances might increase, to
max
min
and k12
are
re¯ect this conservative feature of the data. For example, suppose estimates of k02
max
min
max
approximately uncorrelated. Then, for ÿk22 ˆ k02 ‡ k12 in Eq. (12), VAR…ÿk22 †  VAR…k02
†‡
min
max
min
†. Also, let VAR…ÿk22 † ˆ 2 and VAR…k02
† ˆ VAR…k12
† ˆ 1 in the unconstrained case.
VAR…k12
max
min
max
min
† ˆ 0 and thus VAR…k12
† increases from 1 to 2. If k02
and k12
Then, if k02 is known, VAR…k02
estimates were correlated, the variance estimate increase would possibly be smaller, or nonexistent.
The parameter equality constraint algorithms developed here are readily extended to the case
where ranges of values (or estimates) are available for particular unidenti®able kij s, i.e.
k^ijmin < kij < k^ijmax ; i 6ˆ j; j 6ˆ 0 (e.g. con®dence limits, etc.), and this range is smaller than the
unconstrained range, i.e., if
kijmin < k^ijmin < kij < k^ijmax < kijmax ;

i 6ˆ j;

j 6ˆ 0:

…18†

In this case, new, approximate range estimates for all parameters other than kij might be found by
successively using k^ijmin and k^ijmax as constraints in the equality constraint algorithms.
A model discrimination application of this new algorithm is also suggested by the condition
that known values for rate constants must lie within the range determined by the unconstrained
algorithm. If the given value falls outside the range, with the data ®tted well by an output
equation like (2), the mammillary or the catenary model structure may be rejected, depending on
statistical considerations in the ®tting procedure [13].
The new algorithms presented here can be readily programmed to provide solutions for all
parameter ranges of the two classes of models treated in this paper (open mammillary and catenary models), with any number of kij ®xed, for any i and j.

Acknowledgements
This work was motivated by thyroid hormone metabolism kinetic modeling problems and was
supported in part by NIH Grant no. DK34839.

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