Directory UMM :Data Elmu:jurnal:M:Mathematical Biosciences:Vol167.Issue2.Oct2000:

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 forn-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 ofindependent constraintsand show thatn-compartment linear mammillary or catenary models are uniquely identi®able under nÿ1 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 (deter-ministic) 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 98195-2255, USA.

0025-5564/00/$ - see front matter Ó _{2000 Elsevier Science Inc. All rights reserved.}

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 nokijare known or otherwise ®xed. We call these intervals theunconstrained boundson

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 re-sponses; more general methods are presented in [8] for localizing parameters of linear compart-mental 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 constantskij 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, k0iˆ0, 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 constantkij) of a modelMisunidenti®ableif 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®ableif 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; i6ˆj). The parameterkij isidenti®able if one or more distinct solutions

exist from the same equations; it isuniquely identi®ableif only one such solution exists. The whole

model M is unidenti®able if any onekij 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-nor-mality of measurements. We have con®rmed the validity of our algorithms, and these counter-intuitive 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.

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 withncompartments and scalar input

u…t†and scalar outputy…t†in the ®rst compartment are shown in Fig. 1. They can be described in

terms of mass ¯ows by

dq…t†

dt ˆKq…t† ‡au…t†; q…0† ˆq0

y…t† ˆcq…t†;

…1†

whereqis an-dimensional vector of compartment masses,aˆ ‰1 0_{. . .}0ŠT, where the superscript T

indicates transpose, cˆ ‰1=V10. . .0Š; K ˆ ‰kijŠ is the matrix of rate constants, with

kiiˆ

Pn

jˆ0 kji;iˆ1;. . .;n;kij (tÿ1 units) designates transfer to compartmenti from compartment

j …j6ˆi†, and V1 is the volume of distribution of compartment 1. The unit impulse response for

either model is a sum ofn distinct exponential terms, whereAi >0 andki<0 8 i

y…t† ˆX

In our case, the input/output transfer function is the Laplace transform of (2):

H…s† ˆX

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:

For the mammillary model, the structural invariants are:

ÿk11ˆk01‡k21‡k31‡ ‡kn1;

ÿkiiˆk0i‡k1i; 1<i6n

c_iˆk1iki1; iˆ2;3;. . .;n:

…5†

As shown in [3,4], these invariants can be algorithmically derived from the output in three steps

y…t†; t2 ‰0;TŠ ) fAi;kig ) fai;big ) fkii;cig:

In [2±4], ®nite parameter bounds8kij were determined for both model types, with allkij>0 and

unknown, and were implemented in the programs MAMPOOL and CATPOOL [3±6]. When

somekij 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 datay…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 datay…t†, for

nonze-ro input and output, as is V1 ˆq…0†=y…0†, e.g.V1ˆ1=y…0† for a unit impulse input.

2. Speci®ed parameter values (or estimates) must fall within their corresponding unconstrained

3. Constraints must be independent. As a simple example, the following set of equations for the catenary model:

ÿk11ˆk21‡k01;

c₂ ˆk12k21

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 nÿ1 independent parameter equality constraints forn

-compart-ment 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 3nÿ2 unknowns and 2nÿ1 equality

re-lations among the parameters. This leaves at mostnÿ1 degrees of freedom in the model, and

thus there cannot be more than nÿ1 constraints without violating the independence

condi-tion.

3.1. The catenary model

Constraints may be infeasible for a given set of data and thus may yield no solution, e.g., if

k12ˆ1 andk21ˆ24 are given, butk12k21ˆc2ˆ0:5 is estimated from the input±output data, then

we have a contradiction. Constraints can also be redundant, e.g. k12ˆ1 and k21ˆ0:5 and

k12k21ˆc2 ˆ0:5. This motivates the notion of independent constraints.

De®nition 1.LetPbe the set of all of the rate constants in then-compartment catenary model. Let

Q be the set of the constrained rate constants. Then the elements of Q are independent in the

catenary model ifjIj\Qj618j:16j6nÿ1 or jJj\Qj61 8j:16j6nÿ1 or both, where

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 schemeI. The case where schemeJis satis®ed can be constructed

in a similar manner.

Proof.We use induction on the following proposition: For 1<j6nÿ1, withnÿ1 independent

parameter constraints, parameters in Sj_iˆ1Ii are uniquely identi®able. Clearly, this proposition

depends on the value ofj, where 16j6nÿ1. Let us refer to it asP…j†. Furthermore, P…nÿ1†

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 nÿ1 constraints. We label the peripheral

compart-ments 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.LetPbe the set of all rate constants in then-compartment mammillary model. LetQ

be the set of constrained parameters. Then the elements ofQare independent in the mammillary

model ifjIj\Qj61, where

I1 ˆfk01;k02;k12;k21g;

Ij ˆ k0;j‡1;k1;j‡1;kj‡1;1

Note thatnÿ1 di€erent partitioning schemes are obtained forPby groupingk01with any one of

theIj. We only groupk01 withI1 here. The proofs for other partitioning schemes are similar.

Theorem 2.The n-compartment mammillary model is uniquely identifiable undernÿ1independent constraints.

Proof.LetQbe the set of constrained variables. Then jQj ˆnÿ1. By the pigeon hole principle

[11],jIj\Qj ˆ1 8j. ForjP2, this implies that one of thek0;j‡1;k1;j‡1, andkj‡1;1 is constrained in

Ij. Thus, these three variables can be found uniquely from

ÿkj‡1;j‡1 ˆk0;j‡1‡k1;j‡1;

c_j‡1;j‡1 ˆk1;j‡1kj‡1;1:

Since this is true8jP2, all of the parameters except the ones inI₁ are uniquely identi®able. Since

jI1\Qj ˆ1, it follows that one of thek01,k02,k12, andk21is constrained. Ifk01is not constrained,

Ifk01is constrained, then it is possible to solve these equations in reverse order.k21can 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 nÿ1 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:

ki‡1;i ˆ

for thekij elements in the lower diagonal of the system matrix Kin Eq. (1), and

kiÿ1;i ˆ

for thekijelements in the upper diagonal. Then, with the leaksk0irecursively set to zero, the upper

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 thekij, consistent with the model structure and output data. Then all equality

con-straints for the kijs 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

2. Eqs. (6) and (7) are solved recursively, substituting the values of known parameters wherever

they appear and setting the unconstrained leaksk0ito 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):

k_imin_ÿ1;i ˆ ci

4. The new upper bounds on the leaks are then found from

k01maxˆ ÿk11ÿk21min;

5. The new lower bound for each unconstrained leak remains zero, unless the model is uniquely

identi®able, in which case, thek0is are uniquely determined from (11).

Remark 1. If there are nÿ1 independent constraints, then the model is uniquely identi®able

(Theorem 1) and the algorithm gives kmin

ij k

max

ij 8 iand j.

Remark 2.The fairly common case ofnÿ1 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 onek0jto its maximum

value is equivalent to setting all otherk0i 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

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 compartmenti>1 are

ÿkiiˆk0i‡k1i;

c_i ˆk1iki1:

…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 nÿ1

con-straints. 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 thekijs are tested for feasibility. If infeasible, the algorithm is

termi-nated.

2. If any parameter in the setk0i;k1i;ki1is 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 compartmenti are all unknown, thenk₁max_i and k_imin₁ are

calculated from (12) by setting k0i to zero.

4. kmax_i1 ,i>1, is calculated fromÿk11ˆk01‡k21‡ ‡kn1 by setting all parameters exceptki1 to

Remark 3.If there arenÿ1 independent constraints, the model is uniquely identi®able (Theorem

2), and the algorithm giveskmin

ij k

max ij 8 i;j.

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† ˆy…ti† ‡e…ti†. An asymptotic

co-variance matrix for the catenary model bound

bˆhk₀₁min. . .k_nmin_;nÿ1k₀₁max. . .k_nmax_;nÿ1i

T

…15†

was computed in terms of the vector of (known) structural invariants pas

COVd…b† ˆob

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 matrixo_b=o_{p 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 un-constrained bounds. Also, the covariance matrix is reduced in dimension by two for each known parameter, because

i and structural invariants p.

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 pa-rameters 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 sys-tematically 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

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>ÿk33for 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), supposek12is constrained (given) and nonzero. Then, by Eqs. (13) and

(14), eitherk01,k21and k02 are uniquely identi®able and nonzero, ork03,k21andk02are 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 nu-merous 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].

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-compartmentcatenarymodel 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 (ˆ100 SD/mean) on the parameter

bounds, computed as in [6]. When we constrained one of the kij:k43ˆ0:006, 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 remainingkijs are narrower than the unconstrained ones. Note,

however, that some parameter-bound variabilities (%CVs) increase, e.g. all %CVs for thek0is.

Example 2.An unconstrained 4-compartment mammillary model is shown in Fig. 5(a), the same

model analyzed in [3]. The new algorithms with k14ˆ0:0014 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®ablekij, and all of the parameters associated with compartment 4 are uniquely

identi®able. However, as with the catenary model example above, some parameter-bound

vari-abilities increase (e.g. all of the k0i).

Fig. 4. The (a) unconstrained, i.e., all rate constants unknown, and (b) constrained, k43ˆ0:006 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®ablekij, with %CV assumed zero whenkijis ®xed beforehand

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 allkij 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 k12ˆ0:28. For both, measurement

errors were Gaussian, with 5% (%CV) errors.

Example 5.Thecatenarymodel was also run assuminglognormalmeasurement 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.

Fig. 5. The (a) unconstrained, i.e., all rate constants unknown, and (b) constrained,k14ˆ0:0014 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®ablekij, with %CV assumed zero whenkijis ®xed

Table 1

Computed parameter bounds and variabilities for the unconstrained and constrained (k12ˆ0:28 minÿ1) 3-compartment

mammillarymodel reported in [6, p. 186] with 5%Gaussianmeasurement errors

Bound Unconstrained case Constrained case

Value (minÿ1_{) SD (min}ÿ1_{) CV (%) Value (min}ÿ1_{) SD (min}ÿ1_{) CV (%)}

kmax

01 0.60567 0.05190 8.57 0.40014 0.06310 15.8

kmax

02 0.29592 0.01640 5.55 0.17583 0.03570 20.3

kmax

03 0.02022 0.00089 4.41 0.01857 0.00095 5.11

kmax12 0.45583 0.03570 7.84 0.28000 ± ±

kmin

12 0.15991 0.02210 13.8 0.28000 ± ±

kmax

13 0.02466 0.00114 4.67 0.02446 0.00140 4.7

kmin

13 0.00424 0.00031 7.19 0.00589 0.00086 14.7

kmax

21 0.93297 0.09830 10.5 0.53283 0.11400 21.3

kmin

21 0.32700 0.05210 15.9 0.53283 0.11400 21.3

kmax

31 0.73300 0.05950 8.11 0.52715 0.06460 12.3

kmin

31 0.12701 0.00877 6.91 0.12701 0.00877 6.91

Table 2

Computed parameter bounds and variabilities for the unconstrained and constrained (k12ˆ0:28 minÿ1) 3-compartment

catenarymodel of the impulse response reported in [6, p. 186] with 5%Gaussianmeasurement errors

Bound Unconstrained case Constrained case

Value (minÿ1_{) SD (min}ÿ1_{) CV (%) Value (min}ÿ1_{) SD (min}ÿ1_{) CV (%)}

kmax

01 0.60567 0.05191 8.6 0.51605 0.06457 12.5

kmax

02 0.19154 0.01354 7.1 0.05524 0.03287 59.5

kmax03 0.02100 0.00096 4.6 0.01100 0.00411 37.3

kmax

12 0.33524 0.03287 9.8 0.28000 ±

kmin

12 0.14367 0.01990 13.9 0.28000 ±

kmax

21 1.05997 0.10581 10.0 0.54393 0.11407 21.0

kmin

21 0.45431 0.05902 13.0 0.54393 0.11407 21.0

kmax

23 0.03326 0.00194 5.8 0.03326 0.00194 5.8

kmin

23 0.01226 0.00120 9.8 0.02226 0.00479 21.5

kmax

32 0.30335 0.01843 6.1 0.16703 0.03619 21.7

kmin

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 in-formation 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 param-eters that remain unidenti®able, just as it might render other paramparam-eters 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 kijs when the additional parameter information renders them identi®able.

Moreover, we showed that parameter constraints have to satisfy some conditions (namely, in-dependence 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

parameter-bound 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 Table 3

Computed parameter bounds and variabilities for the unconstrained and constrained (k12ˆ0:28 minÿ1) 3-compartment

catenarymodel of the impulse response reported in [6, p. 186] using 5%lognormalmeasurement errors

Bound Unconstrained case Constrained case

Value (minÿ1_{) SD (min}ÿ1_{) CV (%) Value (min}ÿ1_{) SD (min}ÿ1_{) CV (%)}

kmax

01 0.60570 0.05193 8.60 0.51683 0.06457 12.5

kmax

02 0.19136 0.01359 7.10 0.05480 0.03293 60.1

kmax

03 0.02100 0.00096 4.60 0.01095 0.00414 37.8

kmax

12 0.33480 0.03295 9.80 0.28000 ± ±

kmin

12 0.14344 0.01993 13.9 0.28000 ± ±

kmax21 1.05972 0.10580 10.0 0.54288 0.11406 21.0

kmin

21 0.45402 0.05901 13.0 0.54288 0.11406 21.0

kmax

23 0.03327 0.00194 5.80 0.03327 0.00194 5.8

kmin

23 0.01226 0.00120 9.80 0.02232 0.00482 21.6

kmax

32 0.30310 0.01851 6.10 0.16654 0.03629 21.8

kmin

some elements of the parameter-bound variances, VAR(b), are zero (for ®xedkijs), VAR(p) must nevertheless remain the same. Therefore, other parameter-bound variances might increase, to

re¯ect this conservative feature of the data. For example, suppose estimates of kmax

02 and kmin12 are

approximately uncorrelated. Then, forÿk22ˆk₀₂max‡k₁₂minin Eq. (12), VAR…ÿk22† VAR…kmax₀₂ †‡

VAR…k₁₂min†. Also, let VAR…ÿk22† ˆ2 and VAR…kmax02 † ˆVAR…k

min

12 † ˆ1 in the unconstrained case.

Then, if k02is known, VAR…k02max† ˆ0 and thus VAR…k12min†increases from 1 to 2. If k02max andkmin12

estimates were correlated, the variance estimate increase would possibly be smaller, or nonexis-tent.

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 kijs, i.e.

^

In this case, new, approximate range estimates for all parameters other thankijmight be found by

successively usingk^min_ij 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

ca-tenary models), with any number of kij ®xed, for any i andj.

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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