Identi ability and interval identi ability of mammillary and catenary compartmental models with some known rate constants

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 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 98195-2255, USA. 0025-5564/00/$ - see front matter Ó 2000 Elsevier Science Inc. All rights reserved. PII: S0025-5564(00)00035-3

146 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 k ij for these two model classes, given that no k ij are known or otherwise xed. We call these intervals the unconstrained bounds on the k ij, 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 k ij are known, a likely situation in applications of either of these two model classes, and rst develop conditions under which the other k ij are identi able. For example, a priori information often indicates that, for some i, compartment i has no leak: in this case, k 0i 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 k ij ) of a model M is unidenti able if there exist an uncountably in nite number of solutions for k ij 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. k ij > 0; i 6ˆ j). The parameter k ij 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 k ij is unidenti able; and identi able/uniquely identi able if all k ij 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

P. Vicini et al. / Mathematical Biosciences 167 (2000) 145±161 147 compartment 1, and for all unknown k ij, with any number of k ij 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 ; q 0 ˆq dt 0 y t ˆcq t ; 1 where q is a n-dimensional vector of compartment masses, a ˆ 10...0Š T, where the superscript T indicates transpose, c ˆ 1=V 1 0...0Š; K ˆ k ij Š is the matrix of rate constants, with k ii ˆ Pn jˆ0 k ji; i ˆ 1;...; n; k ij (t 1 units) designates transfer to compartment i from compartment j j 6ˆ i, and V 1 is the volume of distribution of compartment 1. The unit impulse response for either model is a sum of n distinct exponential terms, where A i > 0 and k i < 0 8 i Fig. 1. General catenary (upper) and mammillary (lower) compartmental models with both input and output in compartment 1.

148 P. Vicini et al. / Mathematical Biosciences 167 (2000) 145±161 y t ˆXn iˆ1 A i e k it ; t P 0; when k 1i 6ˆ k 1j 8 i; j : i 6ˆ j 6ˆ 1, and k 1i k i1 > 0 8i for the mammillary model, [10, p. 62]; and k i;i 1 k i 1;i > 0 8i for the catenary model [10, p. 63], and all k 0i P 0, with k 0j > 0 for at least one j for both models. In our case, the input/output transfer function is the Laplace transform of (2): H s ˆXn A i ˆ bns n 1 b n 1 s n 2 b 2 s b 1 ; 3 s k iˆ1 i s n a n s n 1 a 2 s a 1 where all of the a i and b i can be evaluated directly from (1) in terms of the k ij, as uniquely identi able parameter combinations, or structural invariants of the model [12]. For the catenary model, these are: k 11 ˆ k 01 k 21 ; k ii ˆ k 0i k i 1;i k i 1;i ; 1 < i < n; 4 k nn ˆ k 0n k n 1;n ; c i ˆ k i 1;i k i;i 1 ; i ˆ 2; 3;...; n: For the mammillary model, the structural invariants are: 2 k 11 ˆ k 01 k 21 k 31 k n1 ; k ii ˆ k 0i k 1i ; 1 < i 6 n c i ˆ k 1i k i1 ; i ˆ 2; 3;...; n: 5 As shown in [3,4], these invariants can be algorithmically derived from the output in three steps y t ; t 2 0; T Š)fA i ; k i g)fa i ; b i g)fk ii ; c i g: In [2±4], nite parameter bounds 8 k ij were determined for both model types, with all k ij > 0 and unknown, and were implemented in the programs MAMPOOL and CATPOOL [3±6]. When some k ij 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 V 1 ˆ q 0 =y 0, e.g. V 1 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 k ij when none are given a priori).

3. Constraints must be independent. As a simple example, the following set of equations for the catenary model: k 11 ˆ k 21 k 01 ; c 2 ˆ k 12 k 21 P. Vicini et al. / Mathematical Biosciences 167 (2000) 145±161 149 provides unique solutions for all three parameters k 12, k 21, and k 01, 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 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 2 unknowns and 2n 1 equality relations among the parameters. This leaves at most n 1 degrees of freedom in the model, and thus there cannot be more than n 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 k 12 ˆ 1 and k 21 ˆ 24 are given, but k 12 k 21 ˆ c 2 ˆ 0:5 is estimated from the input±output data, then we have a contradiction. Constraints can also be redundant, e.g. k 12 ˆ 1 and k 21 ˆ 0:5 and k 12 k 21 ˆ c 2 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 ji j \ Qj 6 1 8j : 1 6 j 6 n 1orjJ j \ Qj 6 1 8j : 1 6 j 6 n 1 or both, where I j ˆ k 0j ; k j 1 ; j; k j;j 1 8j : 1 6 j 6 n 2; I n 1 ˆ k 0;n 1 ; k 0;n ; k n 1;n ; k n;n 1 and J 1 ˆ fk 01 ; k 02 ; k 12 ; k 21 g; J j ˆ k 0;j 1 ; k j;j 1 ; k j 1;j 8j : 2 6 j 6 n 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 1 independent constraints.

150 P. Vicini et al. / Mathematical Biosciences 167 (2000) 145±161 Proof. We use induction on the following proposition: For 1 < j 6 n 1, with n 1 independent parameter constraints, parameters in S j iˆ1 I i are uniquely identi able. Clearly, this proposition depends on the value of j, where 1 6 j 6 n 1. Let us refer to it as P j. Furthermore, P n 1 implies that the theorem is true. We show P n 1 is true by induction on j. Basis: P 1. Let Q be the set of constrained parameters. Then jqj ˆn 1. By the pigeon hole principle [11], ji j \ Qj ˆ1 8j. In particular, ji 1 \ Qj ˆ1, which implies that one of the k 01, k 12 and k 21 is constrained. Thus, k 01, k 12 and k 21 can be solved from k 11 ˆ k 01 k 21 ; c 2 ˆ k 12 k 21 : Induction step: P j )P j 1 for 1 6 j 6 n 3. By P j, parameters in S j iˆ1 I i are uniquely identi able. In particular, k j;j 1 is uniquely identi able. Since jq \ I j 1 jˆ1, one of k 0;j 1, k j 2;j 1, k j 1;j 2 is constrained. So we can solve for k 0;j 1, k j 2;j 1 and k j 1;j 2 from k j 1;j 1 ˆ k 0;j 1 k j;j 1 k j 2;j 1 ; c j 2 ˆ k j 1;j 2 k j 2;j 1 : By induction, P n 2 is true. To prove P n 1, the same arguments can be applied. By P n 2, parameters in S n 2 iˆ1 I i are uniquely identi able. Speci cally, k n 2;n 1 is uniquely identi- able. Since jq \ I n 1 jˆ1, one of k 0;n 1 ; k 0;n ; k n;n 1 ; k n 1;n is constrained. So k 0;n 1 ; k 0;n ; k n;n 1 ; k n 1;n can be solved via k n 1;n 1 ˆ k 0;n 1 k n 2;n 1 k n;n 1 ; k n;n ˆ k 0;n k n 1;n ; c n ˆ k n 1;n k n;n 1 : Thus, P n 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 1 constraints. We label the peripheral compartments such that k 22 > k 33 > > k nn ; where the k ii 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 ji j \ Qj 6 1, where I 1 ˆ fk 01 ; k 02 ; k 12 ; k 21 g; I j ˆ k 0;j 1 ; k 1;j 1 ; k j 1;1 8j : 2 6 j 6 n 1:

P. Vicini et al. / Mathematical Biosciences 167 (2000) 145±161 151 Note that n 1 di erent partitioning schemes are obtained for P by grouping k 01 with any one of the I j. We only group k 01 with I 1 here. The proofs for other partitioning schemes are similar. Theorem 2. The n-compartment mammillary model is uniquely identifiable under n 1 independent constraints. Proof. Let Q be the set of constrained variables. Then jqj ˆn 1. By the pigeon hole principle [11], ji j \ Qj ˆ1 8j. For j P 2, this implies that one of the k 0;j 1 ; k 1;j 1, and k j 1;1 is constrained in I j. Thus, these three variables can be found uniquely from k j 1;j 1 ˆ k 0;j 1 k 1;j 1 ; c j 1;j 1 ˆ k 1;j 1 k j 1;1 : Since this is true 8j P 2, all of the parameters except the ones in I 1 are uniquely identi able. Since ji 1 \ Qj ˆ1, it follows that one of the k 01, k 02, k 12, and k 21 is constrained. If k 01 is not constrained, then parameters k 02, k 12, and k 21 can be obtained from k 22 ˆ k 02 k 12 ; c 2 ˆ k 12 k 21 : Since all of the k i1, i 6ˆ 0, are uniquely identi able, one can solve for k 01 from k 11 ˆ k 01 k 21 k n1 : If k 01 is constrained, then it is possible to solve these equations in reverse order. k 21 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 k 0j is zero [4]. However, no more than n 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: ( k 11 k 01 ; i ˆ 1; k i 1;i ˆ k ii c i k 0i ; i ˆ 2;...; n 1; 6 k i;i 1 for the k ij elements in the lower diagonal of the system matrix K in Eq. (1), and ( k i 1;i ˆ k nn k 0n ; i ˆ n; k ii c i 1 k i;i 1 k 0i ; i ˆ n 1;...; 2; for the k ij elements in the upper diagonal. Then, with the leaks k 0i recursively set to zero, the upper bounds on the k i 1;i and k i 1;i were obtained for all i [4]. 7

152 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 k ij, consistent with the model structure and output data. Then all equality constraints for the k ij 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 k 11 ˆ k 01 k 21 ; c 2 ˆ k 21 k 12 ; for the rst compartment, and 8 k nn ˆ k 0n k n 1;n c n ˆ k n 1;n k n;n 1 ; 9 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 k 0i 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): i 1;i ˆ ci i;i 1 ; i ˆ 2;...; n; i 1;i ˆ ci 1 ; i ˆ 1;...; n 1: ki;i 1 max 4. The new upper bounds on the leaks are then found from 01 ˆ k 11 21 ; 0i 0n ˆ k ii i 1;i kmin i 1;i ; i ˆ 2;...; n 1; ˆ k nn n;n 1 : 5. The new lower bound for each unconstrained leak remains zero, unless the model is uniquely identi able, in which case, the k 0i s are uniquely determined from (11). Remark 1. If there are n 1 independent constraints, then the model is uniquely identi able (Theorem 1) and the algorithm gives kij min kij max 8 i and j. Remark 2. The fairly common case of n 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 k 0j to its maximum value is equivalent to setting all other k 0i to zero, and then one equality constraint is enough to make the model uniquely identi able, with all k ij attaining their minimum or maximum values (Fig. 2). 10 11

P. Vicini et al. / Mathematical Biosciences 167 (2000) 145±161 153 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 k ii ˆ k 0i k 1i ; 12 c i ˆ k 1i k i1 : 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 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 k ij s are tested for feasibility. If infeasible, the algorithm is terminated. 2. If any parameter in the set k 0i ; k 1i ; k i1 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 k1i max calculated from (12) by setting k 0i to zero. 4. i1 Fig. 2. Unique identi ability of the model is achieved when k 0i ˆ k0i max. In this case, k 0j ˆ k0j min ˆ 0 8 i 6ˆ j. and i1, i > 1, is calculated from k 11 ˆ k 01 k 21 k n1 by setting all parameters except k i1 to their minimum, i.e. i1 ˆ k 11 X j1 ; 13 j6ˆi;j6ˆ1 for i > 1. If any k j1 are known 1 6ˆ j 6ˆ i, then their values are simply substituted into (13) to obtain ki1 max. 5. For i > 1, k1i min and k0i max are calculated from structural invariants (12) by setting k i1 to its maximum. 6. If k 01 is unknown, then k01 min ˆ 0 and k01 max is found from 01 ˆ k 11 X j1 : 14 j>1 are Remark 3. If there are n 1 independent constraints, the model is uniquely identi able (Theorem 2), and the algorithm gives kij min kij max 8 i; j.

154 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 t i ˆy t i e t i. An asymptotic covariance matrix for the catenary model bound h i T b ˆ 01... n;n 1 kmax 01... n;n 1 15 was computed in terms of the vector of (known) structural invariants p as COV b d ˆob T op COV p ob 16 op 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 ob i op ˆ 0T 17 for each known constraint b i ˆ b min i ˆ b max 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 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 k 12 ˆ 0:28 min 1, 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, k 22 > k 33 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 k 12 is constrained (given) and nonzero. Then, by Eqs. (13) and (14), either k 01, k 21 and k 02 are uniquely identi able and nonzero, or k 03, k 21 and k 02 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

156 P. Vicini et al. / Mathematical Biosciences 167 (2000) 145±161 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 k ij : k 43 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 k 34 and k 04 become uniquely identi able and the new mean parameter bounds for the remaining k ij s are narrower than the unconstrained ones. Note, however, that some parameter-bound variabilities (%CVs) increase, e.g. all %CVs for the k 0i 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 k 14 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 k ij, 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 k 0i ). Fig. 4. The (a) unconstrained, i.e., all rate constants unknown, and (b) constrained, k 43 ˆ 0:006 min 1, 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 (min 1 ) ranges are shown by arrows and asymptotic %CVs of estimated bounds are given in parentheses. A single number designates a uniquely identi able k ij, with %CV assumed zero when k ij is xed beforehand (as for k 43 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, k 14 ˆ 0:0014 min 1, 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 (min 1 ) ranges are shown on the arrows, and asymptotic %CV of estimated bound are given in parentheses. A single number designates a uniquely identi able k ij, with %CV assumed zero when k ij is xed beforehand (as for k 14 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 k ij and k 0j > 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 k 12 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 k 12 was xed in Examples 3±5. In fact, the increase was severalfold for many of the bounds shown in Tables 1±3.

158 P. Vicini et al. / Mathematical Biosciences 167 (2000) 145±161 Table 1 Computed parameter bounds and variabilities for the unconstrained and constrained (k 12 ˆ 0:28 min 1 ) 3-compartment mammillary model reported in [6, p. 186] with 5% Gaussian measurement errors Bound Unconstrained case Constrained case Value (min 1 ) SD (min 1 ) CV (%) Value (min 1 ) SD (min 1 ) CV (%) 01 0.60567 0.05190 8.57 0.40014 0.06310 15.8 02 0.29592 0.01640 5.55 0.17583 0.03570 20.3 03 0.02022 0.00089 4.41 0.01857 0.00095 5.11 12 0.45583 0.03570 7.84 0.28000 ± ± 12 0.15991 0.02210 13.8 0.28000 ± ± 13 0.02466 0.00114 4.67 0.02446 0.00140 4.7 13 0.00424 0.00031 7.19 0.00589 0.00086 14.7 21 0.93297 0.09830 10.5 0.53283 0.11400 21.3 21 0.32700 0.05210 15.9 0.53283 0.11400 21.3 31 0.73300 0.05950 8.11 0.52715 0.06460 12.3 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 (k 12 ˆ 0:28 min 1 ) 3-compartment catenary model of the impulse response reported in [6, p. 186] with 5% Gaussian measurement errors Bound Unconstrained case Constrained case Value (min 1 ) SD (min 1 ) CV (%) Value (min 1 ) SD (min 1 ) CV (%) 01 0.60567 0.05191 8.6 0.51605 0.06457 12.5 02 0.19154 0.01354 7.1 0.05524 0.03287 59.5 03 0.02100 0.00096 4.6 0.01100 0.00411 37.3 12 0.33524 0.03287 9.8 0.28000 ± 12 0.14367 0.01990 13.9 0.28000 ± 21 1.05997 0.10581 10.0 0.54393 0.11407 21.0 21 0.45431 0.05902 13.0 0.54393 0.11407 21.0 23 0.03326 0.00194 5.8 0.03326 0.00194 5.8 23 0.01226 0.00120 9.8 0.02226 0.00479 21.5 32 0.30335 0.01843 6.1 0.16703 0.03619 21.7 32 0.11180 0.00823 7.4 0.11180 0.00823 7.4

P. Vicini et al. / Mathematical Biosciences 167 (2000) 145±161 159 Table 3 Computed parameter bounds and variabilities for the unconstrained and constrained (k 12 ˆ 0:28 min 1 ) 3-compartment catenary model of the impulse response reported in [6, p. 186] using 5% lognormal measurement errors Bound Unconstrained case Constrained case Value (min 1 ) SD (min 1 ) CV (%) Value (min 1 ) SD (min 1 ) CV (%) 01 0.60570 0.05193 8.60 0.51683 0.06457 12.5 02 0.19136 0.01359 7.10 0.05480 0.03293 60.1 03 0.02100 0.00096 4.60 0.01095 0.00414 37.8 12 0.33480 0.03295 9.80 0.28000 ± ± 12 0.14344 0.01993 13.9 0.28000 ± ± 21 1.05972 0.10580 10.0 0.54288 0.11406 21.0 21 0.45402 0.05901 13.0 0.54288 0.11406 21.0 23 0.03327 0.00194 5.80 0.03327 0.00194 5.8 23 0.01226 0.00120 9.80 0.02232 0.00482 21.6 32 0.30310 0.01851 6.10 0.16654 0.03629 21.8 32 0.11174 0.00823 7.40 0.11174 0.00823 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 k ij 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 k ij, some parameterbound variabilities increased, especially for the k 0j (leaks). This might be anticipated with variabilities expressed as 100 SD/mean, for k ij 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

160 P. Vicini et al. / Mathematical Biosciences 167 (2000) 145±161 some elements of the parameter-bound variances, VAR(b), are zero (for xed k ij s), 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 max and k12 min are approximately uncorrelated. Then, for k 22 ˆ k02 max k12 min in Eq. (12), VAR k 22 VAR k02 max VAR k12 min. Also, let VAR k 22 ˆ2 and VAR k02 max ˆVAR kmin 12 ˆ1in the unconstrained case. Then, if k 02 is known, VAR k02 max ˆ0 and thus VAR kmin 12 increases from 1 to 2. If kmax 02 and k12 min 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 k ij s, i.e. ^ ij ^k max ij < k ij < ; i 6ˆ j; j 6ˆ 0 (e.g. con dence limits, etc.), and this range is smaller than the unconstrained range, i.e., if ij < ^k min ij < k ij < ^k max ij < ij ; i 6ˆ j; j 6ˆ 0: 18 In this case, new, approximate range estimates for all parameters other than k ij might be found by min max successively using ^k ij and ^k ij 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 k ij 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. References [1] J. Eisenfeld, Partial identi ability of underdetermined compartmental models: a method based on positive linear Lyapunov functions, Math. Biosci. 132 (1996) 111. [2] J.J. DiStefano III, Complete parameter bounds and quasiidenti ability of some unidenti able linear systems, Math. Biosci. 65 (1983) 51. [3] E.M. Landaw, B.C-M. Chen, J.J. DiStefano III, An algorithm for the identi able parameter combinations of the general mamillary compartmental model, Math. Biosci. 72 (1984) 199. [4] B.C-M. Chen, E.M. Landaw, J.J. DiStefano III, Algorithms for the identi able parameter combinations and parameter bounds of unidenti able catenary compartmental models, Math. Biosci. 76 (1985) 59. [5] J.J. DiStefano III, B.C. Chen, E.M. Landaw, Pool size and mass ux bounds and quasiidenti ability conditions for catenary models, Math. Biosci. 88 (1988) 1.

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