Unidentifiable Systems: An Approach to Structural Parameter Bounds

Copyright © I FAC Modelling "nd Control in Biomedir,,1 Systellls, V,'nirc, 1t"II', I'IHH

UNIDENTIFIABLE SYSTEMS: AN APPROACH TO STRUCTURAL PARAMETER BOUNDS L. D' Angio and S. Audoly* * * / k/}(/I'IIIII' III "rMa lfll'll/al i(,I. Caglial'i (/ lI iJlf'l'.Iil\', Via O.l/N'dail' , 72, 0<) /I)() Caglial'i, Ilal\,

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Abstract. In this paper some aspects of unidentifiable compartmental systems are consi dered . In particu l ar a broad class of unidentifiable models is investigated , and the fundamental role of the number of excre ti ons in unidentifiabili ty is stressed. The class considered here is that of unidentifiable systems t hat become iden tif iabl e if the num be r of exc r etions is sui tably reduced . Models of t hi s kind have been studied for example by Audoly & D' Angio (1983) , Cobell i & Toffolo (1984), an d di Stefano (1983) . The peculiarity of t hi s class e nables us to const ruct parameter bounds by e xpl oiting the properties of a n associated system with no excretions , The relations employed are those stated by D' Angio (1985) which express the transfer constant by means of cycles and l oops. By using these equatio ns parameter bounds are provided for models with any number of exc r etions for st r ongly connected systems . Mor eover , not strong ly connected systems a r e investigated and analogous bounds are p r opose d for every strongly connecte d subsystem whose cyc les a n d l oops are identifiable . Key wor ds . Biomedical; parameter bounds.

compartmental

systems ;

INTRODUCTION

ident ification ;

parame t er

estimation ;

for such systems cycles and loops are identifiab l e although parameters are not. We here de n ote proper cycles and loops by " cycles ".

In the re cent literature concerning structural identifiability of dynamic systems (Chao Min-Ch en , Lan daw , di Stefano (1983) , Cobelli , Lepschy, Romanin Jacur (1979); Cobel li & Toffolo (1984); Cobel l i (1985); di Stefano (1983); Vaida (1985) ; WaI ter & LeCourtier (1981» , conc epts such as " interval identification ", "quas i identification" and " identifiable paramete r combination" have been considered, in o rd er to invest i gate what leve l of information can be obtai n ed from unidentifiable systems .

Finding mode l s of this kind is not accidenta l. In fact, as shown by Audoly & D' Angio (1983), the transfer constants always appear in the o utput, grouped in suc h a way t hat they evidence cycles and p aths . Hence , frequently , cyc l es alone are identifiable, or cycles an d some paths, but not in a sufficient number for parameter identifiabi l i ty. For example di Stefano considers an n-pole catenary system with I/O in a n extremal compartment , and provides an algorithm for calcu lating all the cycles from the transfer function coefficien ts. Audoly & 0 ' Angio (1 983) also presen t ed a procedure for calcu lating the cyc le s for an n - c ate nary system , including t h e case of an experiment

Desp i te the fact that t h e nomenc l ature recalled above is s t ill qui te varied, and a pparent ly redun dant, see Cobe lli & Toffo l o (1984), a n ew classi fica tion is proposed in thi s paper for singling out a nd stu dying some unidentifiable sys tem s in order to unify and genera lize some of t he different approac h es a lready proposed for catenary and mamil lary systems (Chao Min- Chen, Landaw, di Stefano (1985 ) ; Cobelli &Toffolo (1984); di Ste fano (1983» .. In the majority of case s to be found in the literature "parameter bounds " have be en proposed for systems that would have the fo ll owing common fe at ure: t hey wo uld be identi fiable if they had a l es ser number of ex cretions . In the cases described by Chao Min-Chen, Landaw, di Stefano (1985) and Cobelli & Toffolo (1984), and di Stefano ( 1983) , referring to unidentifiable sys tems , the authors point , out t hat all the parameter combinati ons that appear in the transfer func tion coeffic i ents - i.e. cyc les and l oops - are identifiable. Hence,

in an internal compa r tme nt. Further examples of systems whose cycles are iden t ifiable for any number of excretions are given by Audoly & D' Angio (1983) e . g. t h e model of "body- ketone metabolism". The above observation suggests the idea of gene ralizing the approach relating to catenary and mamillary systems proposed by Chao Min-Chen, Landaw and di Ste fa no ( 1 985) and di Stefan o ( 1983) by defini ng t h e class of: " cyc le- ide ntifiable systems" . The layout o f the paper is as f ol l ows: in s ection 2 some pre liminary resul ts are re ca lled. In section 3 the class of cycl e-identifiable systems is introduced by mean s of an example and some o f its peculiarities are illustrated. In section 4 par a-

11 " C 8 8-Da

H7

88

L. O'Angio and S. Audoly

meter bounds are provided for strong ly connec ted systems and are extended to not strongly connected systems in section 5. Some examples belonging to the applied area are given in the Appendix. We call "cycle-identifiable" a system in which all the cycles and loops are identifiable .

det A a

Ok

Ajk being the minor of the matrix A. aij'

related to

and det A

PRELIMINARY RESULTS

A

The evolution equations of a linear. time invari ant compartmental system. with the usual notation. are written as follows: )

~

=

Ax + Bu

y(t)

1,2 , ... ,n

J

From the last equation we deduce that Ajk does not c hange with j. Let us denote by A* the ma tri x A. with aii replaced by

x ( 0) = Xo

I

v.

jk

Cx n

where x is the state vector. u and y input and output vectors and A. B. C constant matrices of suitable dimensions. It is assumed furthermore that u is the unit impulse and therefore x(o ) = O. The elements of matrix A have the usual meaning adopted in compartmental models . We suppose the system to be observable and controllable in the sense of Kalman. that b separate inputs and c separate outputs take place therein and the structure of matri ces A. B. C is known. Let us denote the matrix of a strongly connected compartmental system by A E Rn 2 • and by GN N = ! 1.2 •...• n 1. the associated graph evaluated by assigning to each arc (ij) the value of a j i . The cycles and paths of GN will be denoted as f ol lows:

[: a .. j=l . i f j J1

1

i

1 2

... i

a .. a .. 1 1 1 1 2 1 3 2

k

a. 1

.

1

1 k

a ..

p.. . 1 1 . . .1 1 2 h

1

C

a

i. J

i .i . J J

1

h h-1

tem with the loops Fi. Denoting the minor of the matrix A* corresponding to aik by A*ik • we get: Vi.k = 1.2 • . . . • n Due to the strong con ne ction of GN • Akk is structurally different from zero; hen ce: A*

-a

ik

common nodes, containing r arcs; in particular:

(1)

f 1

n a

= ii

L k=O

a

ki

i~k

th b) a reduced form ~ij as a vector whose r component is the sum of all products. containing r arcs. of paths from i to j with the cycles and cycle products having no common nodes.

$lj)

If we suppose the system to be almost closed. i.e. a Ok ~ 0 • aOi = 0 \j i ~ k • D'Angio (1985) shows that det A hence:

kk

if a

a ik

~

ik

0

More over , if we construct the "forms" and "reduce d

forms" using the matrix A* and we denote them by FN ten:

(i)

and

~

~

ik

(n- 1 )

ik

• the above equation can be wri t (n)

~

FN (n-1)

•

(1 )

FN_!kl ~

To each subgraph related to a set N c N we associate : N th (r) a) a form fN E R whose r component FN is the sum of all the cycles and cycle products wi t hout

f

A*

ik

(n- 1 ) F N_! kl

is a loop .

1

A* can be interpreted as a matrix of a closed sys -

(i)

C

F.

(n)

where N- !kl is the set N expropriated by k and FN is the sum of the terms of FN (n) containing the cycles with the arc a ik .

It should be pointed out that the right hand side of the above equation is known if we know: 1) the cyc les of A. 2) the excretion parameters whi ch allow to calculate Fi from f i . As already stated by 0' Angio (1985). it follows that the matrix A can be uniquely calculated given both the cycles and the excretion parameters . Thus the identifiability of a system whose cycles are identifiable depends on the possibility of ca l culating the excretion parameters . To clarify this obse rvation. in the following section .

an example is shown

CYCLE IDENTIFIABLE SYSTEMS We consider the compartmental systems of Fig. 1:

89

Unidentifiable Systems

Note that a) and b) can be identified only from the cycles. 2)For system c) the equations are the following: f f f

Fig. 1. Four compartmental systems belonging to the same class We denote by S the generic system, and by A, B, C its related matrices. We know Band C, and the structure of the matrix A, i.e. which of the elements aij are zero. Wi th the same structure of matrix A, Band C we can consider as many compartmental systems as the possible distributions of the excretions, i.e.:

(~ ) (~) (~) (~) +

+

+

3 Let (ABC) be the class of such 2 different compartmental systems; four of these systems can be seen in Fig. 1. The identification equations in the cycles space are expressed for the whole clsss (ABC) as follows fl + f2 + f3

b

fl f2 + fl f3 + f2 f 3 + c 12

b

f 1 f2 f 3 + c 12 f 3 + c 123

b 3

f2 + f3

b

f2 f 3

b

P12

b 6

f P12 3

b

1 2 3

-(a -(a -(a

21 12 13

+a +a

01 32

)

c

)

c

+a ) 03

12 123

P12

-a a -a

21 21

a a

12 32

a

13

21

Obviously, from the above equations we can deduce that it is necessary to know a path in order to identify c). Conversely, d) is not identifiable without further information. In this paper we will provide parameter bounds for the systems belonging to a given class (ABC), cycle - identifiable but not parameter-identifiable. From the theorem stated by D'Angio (1985, p.215), it follows that a cycle - identifiable system with zero or one excretion is always identifiable. Moreover, if ~ is the number of excretions of a cycleidentifiable system and ~ that of experiments with I/O in different compartments, the set of parameters which can be identified has at least a number of degrees of freedom equal to ~-~-1. In the following section, we use Eq.(4) stated by D'Angio (1985) for such unidentifiable models. These relations uniquely express the transfer constants by means of the cycles, for systems with zero or one excretion. If the system is not identifiable and more excretions occur, the relations may provide parameter bounds in terms of cycles, with or without the help of possibily identifiable paths.

l PARAMETER BOUNDS

2

We suppose that the cycles and loops of the system are identifiable. We will show that:

4 5

7

Hence f , f , f , c , c (and P12) are easily 123 12 1 2 3 calculable. Thus the system is "cycle identifiable". The next important problem to be investigated is whether the whole class is psrameter identifiable. The answer is negative. In fact: 1) for system b) the excretion parameter is easily calculable from the cycles as will be clear from the following equations:

a

ij

.->

fin) N f(n-l) N-{j }

(2)

-in) (n-l) -in) (n-1) fN and fN_{j} being similar to FN and FN_{j} but constructed starting from fi instead of Fi· Relations (2) are valid for any number of excre tions, the equality holding only if there is only one excretion and/or if this excretion occurs in the compartment in which simplification causes the related fi to disappear. In fact, taking the structu re of the matrix A into consideration, it is easy to verify that: A ..

2.J. ..:. 1 A jj

hence

lJ

i,j

L. D'Angio and S. Audoly

90

a. Ij

A·. A. a. 2.J. > a. -2l Ij A· Ij A jj jj

~(n)

f1-l

f(n-1) N-IJ}

~(n) (n-1) Since f and fN_Ij} are expressible by means of cycles, the relations (2) provide a lower bound for aij'

It follows that by letting: ~(n)

In fact, if we say that "parameters are structurally identifiable", we refer to a set of parameters belonging to a minimization procedure whose solution is either globally or l ocally unique. Conversely, if only cyc les are identifiable and parameters are not, the minimization routine used to construct parameters, leads to one of the infinite number of solutions although they a ll provide the same set of cycles . Consequently, in finding one of the solutions great numerical inconveniences may arise (see e.g. Bellman & Astrom (1970».

fN a .. > IJ

f

(n-l) N-{j }

we get a complete set of lower bounds for the parameters. Several different strategies can be used to find upper bounds. In general, the closest upper bound can be found by j ntroducing the lower bounds into the loop expression, but if some a ij appear in a cycle of the 2nd order, it can be better bounded by this cycle, as verified in example 1 of the Appendix. In example 2, bounds are calculated for a catenary system, using both our method and di Stefano' s, and the results are compared. This example clearly shows that the method proposed by di Stefano is a particular case of a general method valid for the whole class of cycle-identifiable systems. A similar sequence to that adopted by di Ste fano for catenary and mamillary systems can be used for each strongly connected system with only 2nd order cyc les (Example 3).

NOT STRONGLY CONNECTED SYSTEMS Let S be a not strongly connected system, and let us suppose cycles and loops to be identifiable. Let Sl' S2"" Sk be the strongly connected subsystems with the maximum number of arcs, such that SI V S2 V ... SkV" S. It is easily verified that the relations (2) still hold for each Si, if the arcs leaving the subsys tem are considered as excretions (see examples).

CONCLUSIONS In this paper, structural upper and lower bounds are proposed in terms of cycles for the class of "cycle-identifiable systems". It is obvious that if also some paths are identifiable, then the closest bounds can be found by using known paths. Models in which not all the cycles are identifiable can also be studied by means of equation (2) and by using some parameter bounds already calculated. This can be done by means of known cycles, or possibly identifiable paths. The aim of the paper is restricted to finding a method for calculating parameter bounds, supposing that the cycles are identifiable. The authors point out that caution should be exercised when taking the practical applicabli ty of the method into consideration.

It is the authors' belief that the method outlined provides a contribution to the solution of problems related to unidentifiable systems, where the lack of known, biological constraints leaves parameter bounds as the only information obtainable from the experimental data.

REFERENCES Audoly, S ., and L. D'Angio (1983). On the identifiability of linear compartmental systems: a revisited transfer function approach based on topological properties. Math. Biosci., 66, 201-228. Bellman, R., and N.J. Astrom (]970) . On structural identifiability. Math. Biosci., Z, 329-339 . Chao Min-Chen, B., E.M. Landaw, and J. di Stefano III (] 985). Algorithms for the identifiable parameter combinati ons and parameter bounds of unidentifiable catenary compartmental models. Math. Biosci., 76, 59-68. Cobelli, C., A. Lepschy, and G. Romanin Jacur (]979). Identifiability of compartmental systems and related structural properties. Math. Biosci., 44, ]-]8. Cobelli, C., and G. Toffolo (1984). Identifiabilifrom parameter bounds. Structural and numerical aspects. Math. Biosci., 71, 237-243 . Cobelli, C. (1985). Identification of endocrinemetabolic and pharmaco-kinetic systems. IFAC Identification and System Parameter Estimation, York. D'Angio, L. (1985). On some topological properties of a strongly connected compartmental system with application to the identifiability problem. Math. Biosci., 76 , 207-220. Di Stefano III, J . J . (] 983). Complete parameter bounds and quasi-identifiability conditions for a class of unidentifiable linear systems. Math. Biosci., 65, 51-68. Godfrey, K.R., and Di Stefano III, J.J. (1985). Identifiabili ty of model parameters. In J. Eisenfeld and M. Witten (Eds.), Modeling of Biological and Biomedical Systems, North Holland. Saccomanni, M.P., C. Cobelli, L. Luzzi, D. Matthews, and P. Testari (1988). Modeling leucine metabolism in man. Preprint IFAC-BME-88 Symposium. Vaida, S. (1985). Identifiability of polynomial and rational systems: structural and numerical aspe cts . In J. Eisenfeld and M. Wi tten (Eds.) Modeling of Biological and Biomedical Systems, North Holland. Wal ter, E., and Y. LeCourtier (1981). Unidentifiable compartmenta1 models: what to do? Math. Biosci., 56 , 1-25.

Unidentifiable Systems

91

APPENDIX

& 7

Example 1 Consider the compartmental system of Fig. 2 with the indicated values of the transfer constants:

Fig. 3. A 4-catenary compartmental system the following values: f

1

-4;

f

-9;

2

f

3

C = -10; 23

C = -2; 12

= -6;

f

C 34

-21

4

-11

By applying (2) we get: C

a

a

12

32

f(2)

>-~= 0.50;

a

> 2.44

a

f(3) 134

Fig. 2. A cycle-identifiable 4-compartmental system

1

::;: -3;

C = -6; 23

-5;

f2

::; 6; C 132

f

-9;

f3

6.55;

C = 30; 342

a

-6

4

C 1342

a

12

43

a

> 4.35;

a

32

< 8.50

a a

21

< 4.00;

a

34

< 11.00;

a

23

43

> 1.17

> 1.90

23 43

<

4.09;

< 4.82;

6.05

-30

From Eqs. (1) and (2) we obtain:

a

34

> 0.30;

i=l,2,3,4.

Suppose that only cycles and loops are identifiable; let their values be the following: f

21

and so on.

C F + C C f + C 132 4 1342 1342 > 132 4 F(3) f(3) 134 132

0.43

Similar method.

results

are

obtained

with di

Stefano's

Example 3

C F + C C f + C 1342 234 1 1342 234 1 > F(3) f(3) 124 124

1. 33

Consider the compartmental system of Fig. 4. It is a model of leucine metabolism studied by Saccomanni and coworkers (1988).

and with a similar procedure: a

31

> 0.32;

a

24

> 1. 08;

a

23

> 1.60;

a

32

> 1.22

i=1,2,3,4.

From cycles and loops, min with aij , we get:

< -

4.90

denoting the lower bounds

a

43

< If

3

+

aminl 23

,,-

7.40

and so on Fig. 4. A compartmental model describing leucine metabolism in the human body

2.68 and so on. Note that closer bounds could be each compartment not open.

obtained were

Example 2 Consider the catenary system of Fig. 2. We

suppose cycles and loops are identified wi th

Compartments 1 and 2 are both injected and observed. We suppose first only two excretions take place in the system: a and a 04 03 The system is cycle-identifiable: in fact we know: f

N

f

N-{l }

f

N-{2}

~12

L. D'Angio and S. Audoly

92 from '12 = P1342

f~~;12341

1)

k=0,1,2,3

a

0,10

< f

10

min -a -a 9,10 3,10

4 - 2 - 0.59 < 1.41

f(k) we know P1234 and N-!12341 a It follows that f f f f

N N

~

f

= f

N-!lI N-!21

N-!131 1 ~-!241

= f = f

2 1

. we know f + V f1 ,fN_!131' C13 1 l/N-!lI'

It

It

. we know f + V f 2 , f N_!241' C24 2 2/N-!21'

It

f

It

f

N-!1231+

V . we know f 2/N-!1231' N-!1231

+

V . we know f _!1241 1/N-!1241' N

N-!1241

Hence from f N_!1241' f N_!1231 and f N-!12341 catenary sequences can be performed leading to other cycles and loops (Audoly & D'Angio, 1983) In addition we know the path P1342' and since P13=f , P24=f , l 2 we get P34 on the strength of prop. 2 stated by D'Angio (1985, p.216), the system is uniquely identifiable. We suppose now that an additional excretion takes place in compartment 10. In this case the system is not parameter-identifiable, though still cycleidentifiable; in fact parameters of excretion do not affect identification equations written by cycles and paths. By applying the relations between cycles and transfer constants (D'Angio, 1985), it is easily verified that a9,10,a10,9,a12a21 ,a 24 , a 42 ,a4 ,11 ,all ,4 are uniquely identified. We will provide parameter bounds for the remaining a . ij (The values of the transfer constants indicated in Fig. 4 have only a mathematical meaning). We obtain: a

min

a

0)0

a 3 ,10

a 10 ,3

-f

10

- a

min 04

0,10

°

- a

>1 C3max ,10] = ~ 2

9,10

<4-2=2

= 2.5

a

max 3,10

a

min

10,3

2.0

2.5

a 3 ,10 7.5

a

34

>1 :::x I

8 7.5

1.07

a

min

34

1.07

43 a

a

a

a

34

43

< -f - 3 - 1 4

I I

34 >C -

10,3

3,10

amax 34

~

5

5.0

1.6

a

max 34

5.0

a

min 43

1.6

max = -f -a -a -a <12-2-1.6=8.4 a 3 13 43 03 10,3

>I

C

3,101 max a 10,3

5

8.4

0.59

a

min

3,10

8.4

0.59

min

min

a 03 < -f3-al0,3-a13-a43

max 0,10

12-1.6-2.5-2

1.41

5.9

5.9 min

a 04 < -f4-a24-a34 -a 11 ,4

9 - 3 - 1 - 1.07= 3.93

3.93