490 Mathematics and Computers in Simulation XXIV (1982) 490-493 North-Holland Publishing Company

SOFTWARE FOR SOLVING IDENTIFICATION AND IDENTIFIABILITY PROBLEMS, E.G. IN COMPARTMENTAL SYSTEMS

H. POHJANPALO and B. WAHLSTROM Technical Research Centre of Fin~n~ VTT/SAH, SF-02150 Espoo 1~ Fin~nd

The ident i f icat ion problem for a system with a known structure is considered. The mathematical background of the i d e n t i f i a b i l i t y problem is shortly discussed. Two minicomputer programs which have been used in several studies to solve the ident i f ia- b i t i t y and the ident i f icat ion problem for biomedical systems are described.

1. INTRODUCTION 2. IDENTIFICATION AND IDENTIFIABILITY

Compartmental systems (cf. /7/) have been used for the modelling of several types of bio- medical systems. The structure of the models are obtained using balance equations where a physical i n te rp re ta t i on of the s tate var iab les and the parameters of ten could be found. The problem of construct ing a model of the system could then be reduced to the i d e n t i f i c a t i o n of set of unknown parameters. The i d e n t i f i c a t i o n could be solved using the model reference approach where the parameters of the model are adjusted to obta in the best f i t with respect to a su i tab le ob jec t ive funct ion.

The identification problem could be divi- ded into three different problems: the identifia- bility problem, the problem of obtaining the best estimates for the parameters and the problem of judging the quality of the obtained estimates.

The identifiability problem is to deter- mine if the structure of the model allows the parameters to be determined with a proposed ex- periment. It is clear that the identifiability should be investigated before costly experiments are carried through. The identifiability has been attacked by programming the algorithms uti- lizing the Taylor series expansion of the solu- tion of the differential system /1/.

The best estimates of the parameters could be obtained using standard methods for numerical optimization where the objective function is calculated by comparing a simulated response to the response obtained in the experi- ment. The problem has been solved using a simu- lation language i~,a repetitive mode within the optimization program.

The quality of the best estimate could be judged by evaluating the parameter sensiti- vities i.e. the second order derivative matrix of the objective function with respect to the parameters. The quality judgement have not been included in the present programs but the method and examples have been described in the refe- rences /2/ and /3/.

The starting point is the differential system /4/ represented as

x(t) = f(x,u,t,O), x(O)= x (@), u(-)cU o

y(t) = g(x,O) (1)

h(O) = 0

x(t)gX, tc[O,T]

where x(-) is the state vector with values in the set of feasible states XcR n, u(-) is the control vector in the set U of feasible functions with values in R n, y(-) is the measurement vector re- corded over the time interval [O,T] with values in R p, and 0 is the vector of unknown parameters in the open set of possible values ~--Rq. The functions f, g, and h are explicitly given for each analysis. The optional function h represents auxiliary binding equations between the components of 0.

The collection of objects (f,g,h,U,X,~) is called the structure S imposed by the system (1).

Suppose first that the underlying system to be simulated obeys Eq. (1) with the parameter values 0 = 0 . The identifiability problem can be defined a~ finding out whether or not the parameter vector@ocould be uniquely solved from Eq. (1). The identification problem is to find an estimate of 0, say ~o' so that some objective function J(y,~) measuring the difference of the responses y(.) and ~(-) is minimized, where ~ is the trajectory resulting from 0 = 0 o. It should be noticed that no stochasticity is accepted in the state equation nor in the measurements.

By associating the nominal parameter values 0 with the structure we have the model (S,O). Identifiability concepts for models (S,@) are meaningless without the introduction of the experi- ments Cx~(O), ui(-)), denoted as E i for short. Let the vector of the experiments E i, i = 1 .... ,s, be called the experimentation E. Each triple (S,@,E) implies a unique vector Y or measurement trajectories Yi, i = 1, .... s (each of which is a Pi-dimensionat vector of functions [O,T]~R. By introducing the mapping F we can write this as

F(S,O,E) = Y(G) (2)

0378-4754/82/0000-0000/$02.75 © 1982 IMACS/North-Holland

H. Pohjanpalo, B. Wahlstram / Solving identification and identifiability problems 491

Thus, given any structure S and experi- mentation E the resulting vector of trajectories is a function of @, only.

Auxiliary definition 1. Model (S,@_) is identi- fTaS~e-wTth-the-experTmentation E i~ F(S,O ,E) = Y(@o ) implies 0 = 0 o.

Auxiliary definiton 2. Model (S,@ o) is locally ident~f~ab~e-at-O-w~th the experimentation E if @ has an c-neighbourhood N(O,c)~Q so that @'oN(@,c) and (S,O',E) = Y(@o ) together imply 0" = 0.

The underlying unknown 0o has to be in- volved in the definitions since the identifiabi- lity properties may vary largely with its value. This is because the numerical values of the Taylor series terms vary with @o and therefore the number of possible solutions of the corresponding identi- liability equations varies /1/, /4/. Model (S,@ o) can thus be locally identifiable, if anywhere, at the points which yield the measurement trajectories Y(@o ). From the identifier's point of view all such points are equivalent. If several such points exist the identification procedure may end up with the correct solution @o or any other point @ where Y(O) = Y(@o ).

The auxiliary definitions are useless in the sense that they are restricted to a single unknown value of the parameter vector describing the system to be identified. The modeler probably finds identifiability results based on definitions concentratinq on the structure more useful.

Definition I. The model structure S is (almost everywhere)-Tdentifiable with the experimentation E if for (almost) all @oC~ the equation F(S,@,E) = Y(@o ) implies @ = @o-

Definition 2. The model structure S is (almost everyw~ere)-~ocaLly identifiable with the experi- mentation E if for (almost) all 8oC~ there is a @c~, such that the model (S,@ o) is locally identi- fiable at ~.

(S, " ,E)

Let G(@) be a matrix function of 0. The point @'c~ is called a regular point of G if rank G(@) is at its maximum at @ = 0". The open set of regular points in Q is denoted by ~r"

Restricting to the set of regular points in~, the following mathematically equivalent formu- lation of Definition 2 can be presented. The model structure S is locally identifiable with the experimentation E if for all OoC~ the model (S,O o) iQ locally identifiable at 0 = @o-

Model structure imposed by Eq. (I) with OE~, is locally identifiable /4/ if and only if the infinite partial derivative matrix 3g(k)/80, k = 0 .... augmented by the lines 3h/~O is of rank q for some OcQ. If E is a vector the component matrices can be put one below the other.

3. EVALUATING IDENTIFIABILITY

Global identifiability can be asserted if the terms of the Taylor series of the measurement trajectory are bijective functions of the unknown parameters /1/. Computing the symbolic terms of the Taylor series by hand becomes extremely tedious and practically impossible as the order of the derivative increases, even for relatively simple models. Therefore the approach of computing the derivatives by computer has been taken. The criteria for local identifiability is straight- forward to implement on the computer once that the symbolic Taylor terms are available. Evalu- ating global identifiability with a computer still remains a dream: solving a set of highly nonlinear symbolic equations is not a task easily amenable to computerization. Attempts in this direction should be based on 'stronger' tools than general high level languages, such as LISP /8/.

The program DER has been developed for analyzing the identifiability of linear and cer- tain types of nonlinear differential models in state space. It can be used to computer the time zero derivatives of the measurements as functions of the parameters, to be used on for evaluating either local identifiability using the same prog- ram or global identifiability by hand.

Local i d e n t i f i a b i [ ~ t v : DER #

By handh (0) , / /

, / " (k )

{'~ I , k = 1 . . . } r a n k [ - - Y ~ T -] = f u l l 7

\, 3 t t = 0 \ ~ 3hl;~(~

By hand Global identi fiabl lily: each C~ i solvable ?

F~g. I. The uses of DER for evaluatlng local and global i d e n t i l i a b i l i t y .

The computer iza t ion of eva lua t i ng i d e n t i - f i a b i l i t y fo r models o f the form Eq. (1) requ i res r e s t r i c t i o n to some p a r t i c u l a r c lass of systems. In the present implementat ion the models can be as follows.

The function f(x,u,t,@) can be polynomial - in the state vector components x., i = 1 ....

i - in time t, both of these in nonnegative

powers, and - in the parameters @i' i = 1 ..... q + q~ in arbitrary combinations. Here q is the number of unknown parameters and q' is the number of parameters with a priori known values~ as an example:

-1 +I Xl = -3@~xI + @2tx~x3 - @ 3@4

x2 = "'" (3)

R3 = "'"

The model can also be specified by the matrix A with exactly the same restrictions as those posed on f, as follows

492 H. Pohjanpalo, B. WahlstrOm / Solving identification and identifiability problems

~(t) = A(x,t,O)x(t) + u(t) (4)

In both of the representations only the zero input function is allowed. The initial state, instead, is assumed to be nonzero, consisting either of constants or polynomials of the para- meter components (known or unknown). The measure- ment function g(x,@) in Eq. (1) has been restricted to allow the measurement of state components or their sums, only, that is

y(t) = Cx(t) (5)

with CE{O,1} pxn. This choise conforms to the practical situation in most of the cases in which compartmental models have been adopted in bio- sciences.

If the experimentation E is a vector, the initial state derivatives resulting from each experiment shall be merged together by hand. The same is the case for the external binding equations h(@) = O.

The Taylor series terms, computed by the program, are in symbolic form. They are inter- nally represented as vectors of integers, the successive components of which are interpreted as the coefficients of terms, component numbers and their powers, and the term and record separators, so that the direct interpretation is possible.

In the structure of the program DER there is a monitor in a central role, in a similar fashion as in the simulation program. There are monitor commands e.g. for reading in the model or ready-computed derivatives (including possible h(@)-terms), for computing derivatives, and for evaluating local identifiability. The program can be run interactively as well as from a command file, the latter possibility has been utilized e.g. for a systematic analysis of open three- compartment models with and without Langmuir-type nonlinearities /4/.

In practice the tool has proved to be use- ful for analyzing moderate size models. The prac- tical upper limit is primarily a function of the number of measured states because this determines the number of derivatives required, and in a lesser extent a functioff'of the numbers of parameters and of state components. The limiting factor is always the available memory space, in spite of the fact that nearly all memory space has been utilized, up to 83 kW of PDP 11 memory, most of which is for the data tables containing the sym- bolic Taylor series terms. For example, with linear three compartment models with five para- meters 10...18 derivatives could be computed de- pending on the model topology /4/. It is quite obvious that under a true virtual memory operating system the program would be outstandingly more useful, though the execution time with excessive disk swapping might then make the usage somewhat time consuming.

4. PARAMETER IDENTIFICATION

In a minicomputer environment simulation studies are most conveniently exercised if the tools are interactive. Then the model can be eva- luated at run time and the effect of modifications in the model can be monitored all the time. On the other hand lengthy interactive optimizations require the existence of powerful batch mode facilities. The key property required in identification studies is the flexibility of the simulation and optimi- zation tools with respect to the mode, interacti- vity versus batch: at any time it must be possible to switch from batch to interactive mode and vice versa .

Another necessary f e a t u r e needed in i den - t i f i c a t i o n s t u d i e s i s the a b i l i t y to L ink o p t i m i - z a t i o n r o u t i n e s and model s p e c i f i c r o u t i n e s to the s i m u l a t i o n program. In o t h e r words, i t must be p o s s i b e l t o c o n t r o l the s i m u l a t i o n from an o p t i - m iza t i on program, and a l so to p repare the model b e f o r e each s i n g l e s i m u l a t i o n run and a f t e r the run to e v a l u a t e an o b j e c t i v e f u n c t i o n r e q u i r e d by the o p t i m i z a t i o n r o u t i n e .

These two fundamenta l requ i rements are s a t i s f i e d by the s i m u l a t i o n language IRS / 5 / , / 6 / , des igned f o r gene ra l purpose cont inuous t ime sys - tems. I n t e r a c t i v i t y has been impLmented by a mon i to r cen te red program s t r u c t u r e accep t ing two- cha rac te r mnemonic codes f o r the d i f f e r e n t f u n c t i o n s t h a t are a v a i l a b l e . The commands, 43 in number (IRS V3.1) w i th s e v e r a l subcommands, can be used t o - c o n t r o l the s i m u l a t i o n modes: s i n g l e runs /

c o n t r o l l e d by an e x t e r n a l program~ i n t e r - a c t i v i t y / b a t c h ,

- d e f i n e and a l t e r the model, both s t r u c t u r e and parameters ,

- s p e c i f y s i m u l a t i o n parameters , e .g . t ime s tep , i n t e r g r a t i o n method (Runge-Kut ta 4 i s the d e f a u l t ) , t r a j e c t o r y l i s t i n g s , r ea l t ime control,

- control the simulations, e.g. to initialize or operate the model, to define command sequences,

- utilize several RSX-11M services.

The simulation program IRS is welt suited for model reference parameter identification. The proposed model structure, a set of Linear or non- linear differential equations must first be trans- formed into a block diagram consisting of inte- grators, adders, multipliers, e.t.c. (IRS supports 41 different component types including e.g. delays, function generators, PID-controllers, time step control, trigonometric and logical functions). At the same time the parameters to be identified are declared to the optimization interface. The standard form data file could then be prepared. Finally the logic to control the model before each run and to evaluate the objective function result- ing from the current parameter values has to be coded in a Fortran subroutine. At this phase the identification study can be started. Usually, however, it is convenient to introduce a command

H. Pohjanpalo, B. Wahlstr6m / Solving identification and identifiability problems 493

file consisting of IRS monitor commands and the associated other input to execute at least the initial phase of the identification process auto- nomously, i.e. defining the data and model files, reading in the data and the model and a first guess estimate of the parameter vector.

In the VTT Electrical Engineering Labora- tory several biomedical studies have been carried out, in the first place with linear and nonlinear compartmental models. Experience has shown that models whose parameter dimensionality is below ten can be quite comfortably identified while for those with a dimensionality around 15 - 17 (the maximum) the computational load is getting rather heavy. Especially in large problems it has turned out that the range of interactivity levels provided by IRS is of ultimate importance: in the middle of a optimization process the user can break in by requesting entrance to the monitor in order to analyze the current optimum and to control the process e.g. by directing the parameter vector, by fixing or releasing selected vector components or by changing optimization parameters such as the step length of the random search optimization algorithm. After the break-in the user can then allow the computer to continue the optimization autonomously, again.

5. REFERENCES

I. Pohjanpalo, H., System identifiability based on the power series expansion of the solution. Math. Biosci. 41(1978), pp. 21-33.

2. WahlstrSm, B., Julkunen, R., Kekki, M., On the modelling and simulation of the warfarin distribution and metabolism in rat, 5th Soviet-Finnish Symposium in Cybernetics, 1975.

3. Pohjanpalo, H., Kekki, M. & WahlstrSm, B., Estimation, resolution and solvability of linear compartmental models: choleste- rol metabolism in man. Int. J. Biomedical Computing, Vol. 11(1980).

4. Pohjanpalo, H., Identifiability of deter- ministic differential models in state space - an implementation for a computer. Techn. Rec. Centre of Finland, Research Reports 56/1982.

5. Pohjanpalo, H., IRS V3.1. VTT/S~H 9/1981 (Unpublished report).

6. Pohjanpalo, H., Interactive simulation program for minicomputers. Proc. of the Digital Equipment Computer Users Soc. (DECUS), Monte Carlo 1979.

7. Brown, R.F., Compartmental system analysis: State of the art. IEEE Trans. Biomed. Eng., Vol. BME-27, No. 1, 1980.

8. Nicol, R.L., Symbolic differentiation a la LISP. Byte, September 1981.

H. Pohjanpalo is presently at Nokia Electronics, PL 780, SF-O0101HELSINKI 10, Finland.