Identification of Endocrine-Metabolic and Pharmacokinetic Systems

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IDENTIFICATION OF ENDOCRINE· METABOLIC AND PHARMACOKINETIC SYSTEMS C. Cobelli /) (,/lfIrl llll' lIl

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A survey on modeling and identification of endocrine-metabolic and pharmaco-

kinetic systems is given. Principal uses of mode ls in physio l ogical and c linical studies are illu strated . Specific classes of parametric models are described. Aspec ts of experiment design,identification and validation are discussed. Relevant publications in some application areas illustrate the methodo l o gi c al and practical role of system identification in research , diagnosis and therapy . Keyword s . Physiological models; identifi cation ; parameter estimation; system analys i s .

care of the formati on , transformation and e l imina -

INTRODUCTION The use of modeling and identification techni q ue s as an investigative tool for s tudying

endocrine-m~

tabolic and phal.macokinetic.; sys tems is inc.r easing, and

it is gaining impor t ance and cr.edibil ity. This field has been first surveyed at the 6 th IFAC Symposium on Identification and System Parameter Estimation (Carson and coll., 19 79) . Sinc e then two events we re o rganized inside IFAC , a Session on - Identi fication of Metabolic Systems in Physiology and Clinica l Science " at the 6th Symposium on Identification and System Parameter Estimation (Bekey and Saridis, 1982) and a Colloquium on "Control of End ocri ne Glands" at the 9th IFAC Wor ld Congress (Gertler and Keviczky , 1984). In these la st years advances have been made both on theoretical aspects of modeling , identifi cat i on and validation of endocrine - metabo l ic systems as well as in terms of physio logical and clinical applications . This is well reflected in the specific literature but als o by the se veral texts, monographs , r ev iews which have appeared since 1979 (e.g. Carson and Jones , 1979 ; Brown, 1980; McIntosh and McIntosh, 1980; Carson and co ll. , 1981; Cobelli and Bergman , 1981; Endrenyi, 1981; Zierler, 1981 ; Berman, 198 2 ; Berman and co ll . , 1982; Cobelli and coll., 1982b;Cramp, 1982 ; Gibaldi and Perrier , 1982 ; Jacq uez , 1982; Katona , 1982 ; Molino and Milanese , 1982 ; Radziu k and Hetenyi , 1982; Stefan elli and coll., 1982 ; Walter, 1982; Anderson, 1983 ; Carson a nd col l. , 1983; Godfrey , 1983 ; Jones and co ll. , 1983 ; Belfo r'te and coll., 1984; Cobel l i , 1984; Cobelli and coll ., 19 8 4a;Di S t~ fano and Landaw, 198 4; Jacquez, 198 4 ; Landaw and Di Stefano , 1984; Swan , 1984) . The scope of this paper is to s ummarize the state of the art of mode ling and identification of endocr ine - metabolic sys t e ms. Both theoretical aspects and practical applications in researc h , diagnosis and therapy are discussed.

ENDOCRINE-METABOLIC SYSTEMS In living organisms the metabolic processes take

tion of materials . The blood c ir c ulation constitutes the transportation system between the sites of production of materials and the ce lls that particl pate in the metabolic system. The many comp l ex exc hange-, transport- and metabolism p.ocesses consti tute a l arge, integrated system with a well - deve l oped control. The processes are se l fregu l at<>ry , with reaction speeds that depend on concentrations , thus c reating a negative feedback , but t h e overall control involves the endocrine - system. Endocri ne

glands secrete hormones which control the chemical proc esses through various modes of action , e.g. modulating enzyme (bio l ogical catalysts) concentrat ion , within the cell . The goal of these mechanisms is the creation and maintaining of a constant, interna l environment needed for the survival of the organ ism, in spite of wide fluctuations in the external environment.

USES OF MODELS As a l ways in modeling activities the types of models and the way in which they are being built are s trong l y dependent on the ultimate use o f the model. This may be used as the ba s is for one type of r eviewing scope of modeling in a specific field. Other types of classification are possib l e. A cla~ sical framework is that of models for insight, si mulatio n , prediction and contro l. Another type of classification i s given by the spec ific metabolic system to which the modeling is applied. I n endocrinology and metabolism one may recognize the following uses

or purposes of mode l s:

- ident ifi cation of system structure , i . e . models to examine different hypotheses regarding the nature of specific physiological mechanisms. This use finds elective application both at the whole-or ganism (e.g. Berman , 1982) and organ level (e.g. Gr odsky, 1972); - estimation of unmeasur'able quanti ties, i . e. estimating internal parameters/variables of physiol og i ca l

interest . Mea s u rement via parameter esti-

46

C. Cohcl li mation finds extensive application both in

physi~

l ogical and c lini cal studies (e.g. Bergman and coil., 1979a , 1981; DiStefano and coil. ,

1982a;FeE.

rannini and coli ., 1985; Molino and Milanese , 1978) ; - simulation of the intact system behavior where ethical or techn ical reasons do not allow direct experimentation on the sys tem itself (e.g. Cobe! li and Ruggeri , 1983); - pred i cti on/control of physio l ogica l VaJ iables by administration of therapeutic agents, i.e. mode l s to predict optimal-dose administration of drug in or·der to keep one or more physiological var iables wi thin des i rable limits. These models are increasingly becoming part of open -and closed-loop drug de l ivery systems in patient therapy (e.g. Shei ner and coi l., 1972 , 197 5) ; - optimize cost/effectiveness of dynamic c linic a l tests, i . e. models to obtain maximum of information from the minimum number of blood sa mp les withdrawn from a patient (e.g. Cobelli and Ruggeri , 1985); - diagnos i s , i.e. models to augment quantitative

i nformation from laborato ry tests and clinica l symptoms, thus improving the rel iab i lity of diagnOSis (e . g. Groth, 1984) ; - teaching , i.e. models to simulate a wide var.iety of s i tuation provide a significant aid in the teac hi n g of many aspect of physiology, c linical medicine and pharmacokinetics (e.g . Bloch and coi l. , 1980) . I n addition to purpose , the level at which models are deve l oped strong l y influences the modeling/ide~ ti f ication process . Models c an be developed at var i ous leve l s : from sub-cellular to organ models, to mode l s encompassing the whole organism , i . e .

gl~

b al mode l s . Globa l models are of particular interest here. These models are wide l y used for simul at ion /p r edict i on o f the intact system behavior in various pathologi -

cal , experimental etc . situations. Often a "bottomu p " modeling is adopted and l arge - sca le, differen tial equation descriptions are developed. Thi s is n ot withou t difficul tie s as often it is impossible t o obtain good, r"e l iabl e information on the many s u bsystems that together from the g l obal structure . The heuristic potentia l of these models is however e normous b u t severe problems of validation are posed as usually only a few variables are directly measurable . Globa l mod els can also be simple. This is the case when they are used to measure unaccess ibl e parameters e.g . in clinical studies.

In this context pa rsimonious (minimal) models, i.e. models of l ow or d er with few parameter s , are needed as input/output

da ta for i dentification in e ach individual a1.e sev~ re l y l imi ted in type and number. Here a high degree of aggregation is required in the model of the mol~ cul a r, cel l ular and organ processes occ uring within

the organism. The choice of model str ucture is a cri tica l issue in "top - down 11 modeling . The issue of complex vs simple global models is rather controversial and mutual l y e x~l us iv e viewpo ints are often taken (Garfinke l , 1982 ; Guy ton , 1979 ; Ya tes , 1979). Both kind of models are however. u sef u l as they serve essentially different purposes , a n d in fact they can be easi ly reconciled if a ro -

bust validation prog ram i s called for (Carson and coli ., 1984; Cobel li, 1984; Cobelli and ~ o ll. , 19b4a) . Organ models are developed at a lower level of phi:. siological complexity , i.e. target ti ss ues such as liver, kidney , muscles etc. and endocri ne glands s uch as thyroid, pancreas etc. Organ models provi de important information for a global model and its subsystems . At the o rgan l evel fewer cons traints ale imposed on experiment des ign.

This , in addition

with an intrinsic reduction in physiological com-

plexity, allows t o use also more sophisticated tools f or analysis, e.g . stochastic (Grodsky , 197 2 ; Licko and Si lvers , 1975) or distributed (Go reski, 1983) models . Usually o rgan models are mor e testable than global mode l s. Car"e must be taken however" in integr.ating at the who le-organi sm level experimenta l and model o rga n stud ie s as these last are usually obta ined in isolation from other subsys tems. Model s at ce llular and s ub- cellular level study the ba sic proces ses of biochemistry and biophysics and s tocha stic models play an important role (e.g . Bertuzzi and coli. , 1981 ; Mohl er and coli., 1980; Segel, 19 80) .

CLASSES OF MODELS In addition to purpose and level of descr i ption, the na ture of available data heavil y conditions the approach to be adopted to model building . In some cases data become s available through measurement of ' intrinsic ' signals , whereas in other cases s uitable test s ignals have t o be applied. Usual l y the amount of informatio n through ' intrinsic ' signals

is limited and therefore has to be augmented by

me~

surement from s uitably designed input/output e xpe riments. From the preceding it i s not surprising that the whole mode ling spect rum of systems analysis is u sab l e , i.e. deterministic and s tochastic ,

linear and nonlinear , l umped and di s tr ibuted models . Here focus is placed on a mode l of data vs model of system ca tegorization (DiStefano and Landaw, 198 4) and on some specific c l asses of models of systems. Model s o f data require few if any structural hypoth eses about the sys tem fr om which data are obtained. They include mo s t statistica l mode ls; models of mathematical functions (no t necessa-

~onsisting

rily with a physical basi s)

fitted t o data , like time series mo

po l ynomia l or exponential functions;

de l s . These models are not usable i n basic physiol~ gical studies as nothing can be said about the internal mechanisms of the system , but can be very usefully employed for p r ediction and control in p~ tient therapy. Mor eover they can also ser.ve to fix from the data ove rall system characte ristics , e.g. system order in l i near kinetic studies assessed by exponential fitting, before turning to a physiologically s tructur ed model (e .g. Cobel li and coli. , 1984 b) .

f10del s of systems co ntra st with the models o f data as they a re usually based on physical principles and hy potheses about the st ructure of the s y stem and possibly also on its f un ct ioning . All available a pr i ori knowledge is used in formulating the model and state variables/parameters t e nd to have a di r.ect ~ountc lpar.t in th e syste m. 3 xamples of these

47

ElIdocrille-Metabolic alld Pharmacokilletic Systems models include those based on the law of mass ac-

ments. All de novo sources, all Lr-reversible losses

tion, on mass-balance relationships,

are assumed to take place in each accessible pooli

control-theo

ry and network thermodynamics. Models of systems

in addition material is allowed to recirculate. An

are usually described by set of differential equa-

integral analysis of data provides useful kinetic

tions. Some classes of models of systems are now

parameter"s, e.g. production, clearance r"ate, equi-

briefly examined.

valent distr"ibution volume and mean residence time

(Gurpide, 1975; Rescigno and Gurpide, 1973). Compartmental Models

Noncompartmental models are an appealing alternati-

This is one of the most widespread classes of lum-

ped parameter deterministic models employed for studying endocrine-metabolic and phdrmacokinetic stems. Standard references (Anderson,

sy

1983; Brown,

1980; Carson and coll., 1983; Godfrey, 198 3; Jacquez, 1972; Rescigno and Segre, 1966) and critiques (Zierler, 1981) are available. A compartmental model represents the system by a finite number of components, the compartments, and

by the material exchange between them. A compartment is an amount of material which acts kinetically in a homogeneous and distinctive manner; it is

characterized by its physico-chemical state, its location or both. Thus if a substance is present in a metabolic system in several distinguishable forms or locations, then all the substance in a particu-

ve to compartmental models both in terms of required modeling effort and computational demand. As there is no need to specify a definite compartmen-

tal structure this has lead to the common believe that estimated noncompar"tmental kinetic par."ameters

are structure-free or model-independent. This is not true. Noncompartmental models have severe intrinsic structural limitations due to their accessi ble pool equivalent source-equivalent sink natur"e,

which limits their practical applicability (Cobelli and Toffolo, 1984a; DiStefano, 1982; DiStefano and Landaw, 1984). Moreover they are also computationally not robust (Cobelli and Toffolo, 1985). Control-System Models Compartmental and noncompartmental models are use-

lar form, or all the substance in a particular loca

ful for studying the kinetics of substrates,

tion or all the substance in a particular form and

nes, and drugs. However in physiological control

hormo~

location are said to constitute a compartment. The

systems hormones exert a controlling influence on

flux of material between compartments corresponds

substrate kinetics and viceversa. Nonlinear control

either to physieo-ehemical conver"sion of one meta-

system models must usually be adopted. Such models

bol i te to another in the same loea tion c.t" to the

can be formulated using classical control system

transport of a substance fr"om one location to ano-

ideas, i.e. a controlled metabolic plant and a

ther without change of form.

monal controller. Often the controlled

Mass balance equations

ho~

system and

are wr"itten to describe the rate of change of mate-

the controller are individually described in com-

rial within the compartment.

partmental terms and control is exerted pararnetri-

cally (e.g. Cobelli and coll., 1982a).

Compartmental models are used extensively to study the kinetics, i.e. distribution and metabolism, of

endogenous substances and drugs. The flux of material from one compartment to another usually de-

Stochastic Compartmental Models Under cer"tain circumstances it is appropriate to iE:

pends in a linear or nonlinear manner solely upon

corporate stochasticity into the basic elements of

the material in the source compar"tment. Linear mo-

deterministic compartmental models. Common sources

dels ar"e appropriate either if the "intrinsic" dy-

are some variability among particles in a given ex-

namics is linear or" if the input/output experiment

periment, i.e. stochasticity in compartmental state

realizes a tracer or first order perturbation;

if

variables, and also some variability between repli-

the system is in a constant steady state, model

p~

cates of an experiment, i.e. stochasticity in

rameters

co~

fer rate parameters. Both stochastic generaliza-

(i.e. transfer rate parameters between

partments)

are time invariant, if not, time-varying

parameters arise (Carson and coll., 1983). Structural and stability properties of linear time invariant compartmental models are well studied (Ander·-

tions have been considered, e.g. reviews in (God-

frey, 1983; 11atis and lVehrly, 1979). A general compartmental model which combines the various forms of stochastic causal mechanism has been also deve-

son, 1983; Hearon, 1963; Smith and Mohler, 1976; Zazworsky and Knudsen, 1978) . For these models the

loped (Matis and Wehrly, 1981).

mean residence time matrix

Distributed Kinetic Models

(minus the inverse of

the state matrix) provides an important kinetic ch~ racterization of the system (Eisenfeld, 1979; Matis and coll., 1983). Some results are also available on linear time-varying (e.g. Mulholland and Keener, 1974), nonlinear (e.g. Eisenfeld, 1982; Mae da and colI., 1978; Sandberg, 1978) and time-delay (e.g. Cobelli and Rescigno 1978, Mazanov,1976; Plu~ quellec and Steimer, 1984) compartmental models. Noncompartmental Models These are linear time invariant models which are wi

tran~

In many cases the r·epresentation by a lumped parameter description is inadequate. Explicit

considel"~

tion not only of time-dependent effects but also of the distributed-in-space nature of phenomena is of considerable importance especially if fine details of kinetic pr.ocesses within an organ are needed. Partial differential equation models arise. An area

of active research is that of hepatic transport kinetics where various approaches have been suggested

(e.g. reviews by Forker and Luxon (1983) and Goreski (1983». In this context the approximations pr.':'

dely used for kinetic data analysis at the whole

vided by a lumped compartmental vs a distributed m.':'

organism level. Focus is placed on the accessible

del for analyzing disappearance curves at the in-

compartments only (usually one or two) which are

tact organism level have been analyzed (Forker and

assumed to be embedded in a network of an undeter-

Luxon, 1978).

mined and unspecified number of connected compart-

48

C. Cobelli IDENTIFICATION AND VALIDATION

Models of endocrine-metabolic systems are

fiability testing for linear and nonlinear state usually

parametric models,i.e. models of systems,as the goal is understanding or estimating physiological parameters. Severe constraints exist however for their identification and validation, especially in the intact system studies, due to problems of observability and experiment design. This also applies when models of data may be appropriate, i.e. for prediction and control. Intrinsic biochemical signals (concentration in biological fluids) carry little information, thus identification experiments needs to be performed for generating input-output data. Heavy restrictions exist however on type and number of test inputs, on number of measurable variables, on number and frequency of blood sampling etc. Moreover measurement of biochemical signals usually require complex and time consuming laboratory processes (e.g. radioimmunoassay), thus preventing their on-line use. This scenario has thus some distinctive features

from the usual one where the system identification problem is classically posed. Thus, although in pri~ ciple the whole spectrum of models and methods of system identificatiun is usable (e. g. Eykhoff, 1974,1981; Goodwin and Payne, 1977; Ljung and S6derstrom, 1983; S£ derstrom and Stoica, 1983; Young, 1984), the characteristics of the systems under discussion have sti mulated specific considerations and techniques. Along this line reference Cdn be made to recent texts, monographs, reviews such as (Anderson, 1983; Belforte and coli., 1984; Carson and coli., 1983; Cobelli, 1984; Cobelli and coli., 1984a;DiStefano, 1979; Godfrey, 1983; Landaw and DiStefano, 1984; Jacquez, 1982; Waiter, 1982). In the following some of these issues are addressed focusing upon differential equation parametric models.

space models have been discussed by Grewal and Glover, 1976; Nguyen and Wood, 1982; Pohjanpalo, 1978; Waiter and Lecourtier, 1982; Vajda, 1983; Waiter, 1983. Special effort has been devoted to study ide~ tifiability of linear c ompartmental models by taking advantage of the physical constraints which are imposed on the state matrix. For these models various specific approaches based e.g. on similarity transformation, normal-mode analYSiS, graph topology have been developed (e.g. Anderson, 1983, Sect. 22; Audoly and D'Angio, 1983; Cobelli and coli., 1979b;Delforge, 1984; Norton, 1980; Travis, 1981; Vajda, 1985; Waiter and Lecourtier, 1981), which complement the general transfer function and Markov parameter methods (e.g. Cobelli and DiStefano, 1980). Explicit identifiability resul ts are also available (e.g. Cobelli and coli., 1979a;Norton, 1982). A parameter-bound strategy for dealing with unidentifiable linear" compartlu"ntal models has been also developed and formalized (DiStefano, 1983) and discussed (Cobelli and Toffolo, 1984b; DiStefano, 1984). For nonlinear compartmental models the general methods proposed in (Pohjanpalo, 1978; Waiter and Lecourtier, 1982) have been applied to study some classes of saturative compartment models (e.g. Godfrey and Fitch, 1984a; Pohjanpalo, 1978).

Parameter Estimation Usually the input-output experiment provides a lim~ ted set of discrete-time measurements. Pr"oblems ari se in parameter estimation as a result of var-iolls types of error, e.g. measurement errors and errors

in model structure. It is usually not possible to consider explicitely errors in model structure: alternative model structures can be analyzed in order" to minimize this error. The Simplest situation is

A Priori Identifiability A priori (or structural) identifiability is a

prer~

quisite for well-posedness of experiment design and parameter estimation. A large literature has grown on the subject since the original formalization by Bellman and Astrom (1970). see reviews/books (AndeE son, 1983; Carson and coli., 1983; Cobelli and DiStefano, 1980; Jacquez, 1982; Waiter, 1982). A priori identifiability examines whether, given ideal noise-free data and an error-free model struc tUre, as well as all a pr"iori constraints available on the system, it is possible to make unique estim~ tes of all the unknown model parameters. If a model is ~niquely (globally) identifiable (or, no~unique­ ly (locally) identifiable - that is, one or more of the parameters has more than one, but a finite number of, solutions), then identification techniques can be used to estimate from the noisy data the unknown parameters. If a model is a priori unidentifiable - that is, one Or more of the parameters has

to consider the noisy real data as reflecting model output corrupted by measurement error (additive or multiplicative) only. Parameter estimates are thus obtained by nonlinear least squares or maximum 1ikellihood together with a meaSUre of their prevision (accuracy) I i.e. of a posteLiori or numerical identifiability. Relevant aspects of parameter

est~

mation are discussed in several books and reviews

(Bard, 1974; Beck and Arnold, 1977; Belforte and coli., 1983; Carson and coli., 1983; Cook and Weisberg, 1982; Endrenyi, 1981; Jennri~h and Ralston, 1979; Landaw and DiStefano, 1984). Weighted nonlinear least squares is mostly used and both direct and gradient-type search methods are implemented in estimation s c hemes. A correct knowledge of the error structure is nee ded in order to have a correct summary of the statistical properties of the estimates. This is a difficult task. Measurement errors are usually white and often a known distribution, e.g. Gaussian,is assumed. How-

parameters can be estimated using identification techniques; various strategies can be used, e.g. derivation of bounds for unidentifiable parameters;

ever many properties of least squares hold approx~ mately for wide class of distributions, if weights ar'e chosen optimally, i. e. inverse of known varia.!: ces or of their relative value s if variances are

model reparametrization (parameter aggregation); i~ corporation of additional knowledge, or design of

known up to a proportionality constant (on the cho~ ce of weights see also the extended least square

an infinite number of solutions - then not all its

more informative experiment. The problem is in

gen~

ral a large nonlinear algebraic one, and thus of difficult solution. Symbolic algebr"aic manipulative languages can be useful. Approaches to identi-

method (Peck and coli., 1984». Under these circu~ stances an asymptotically correct approKimation of the covariance matrix of parameter estimates can be used to evaluate p~e~ision of parameter estima-

49

Elldocrille-Mctabolic and I'harmacokillctic Systcms tes. If measurement errors are Gaussian, then this

approximation is the Cramer-Rao lower bound (inve£ se of the Fisher information matrix), i.e. the optimally weighted least squares estimator is also the maximum likelihood estimator. Care must be taken in not using lower bound variances as true parameter variances. General factors corrupt this v~ riance, e.g. not completely white noise, inaccurate knowledge of enor structure, limited data set. Monte Carlo studies are needed to assess robustness of Cramer-Rao lower bound in specific practical applications; published experiences tend to favor

the validity of these approximations espe-

cially if sampling designs are optimal (Metzler, 1981; Mitchell and Gupta, 1982; Landaw and DiStefa no, 1984). To examine the quality of model predictions to observed data various statistical tests on residuals are available to check for presence of systematic misfitting, nonrandomness of the errors and accor-

dance with assumed experimental noise. Model order estimation is also relevant here and criteria such as F-test, and those based on the parslmony prlncl pie lika Akaike (1974) and Schwarz (1978) criterion can be used. Optimal Experiment Design The rationale of optimal experiment design is to act on design variables such as number of test in-

puts/outputs, form of test inputs, number of samples and sampling schedule, measurement errors so as to max!. mize, according to some criterion, the precision with which model parameters can be estimated. The role of a priori identifiability (where and how many test inputs and outputs) and measurement error have been already discussed. Moderate focus has been placed on optimal input design (Kalaba and Spingarn, 1981, 1982; Cobelli and Thomaseth, 1985) while optimal design of sampling schedule, i.e. nu~ ber and location of discrete-time points at which samples are collected, has received much attention as it is the variable which is less constrained by the experimental situation. A D-optimality criterion has been usually employed in optimal sampling schedule studies, i.e. minimization of the determinant of the covariance matrix (Box, 1970; Fedorov, 1972; St. John and Draper, 1975). Model structure, measurement error and nom~ nals for the parameter values ar·e assumed (wellposed). Theoretical and algorithmic aspects have been studied for both the single- and multi-output case (e.g. Cobelli and coll., 1985; D'Argenio,1981; DiStefano, 1981, 1982; Landaw, 1982, 1985; Landaw and DiStefano, 1984; Mori and DiStefano, 1980). For a wide class of models with one output the number of optimal sample times is very often equal to the number of unknown model parameters. The important role of replicate samples has been also evidenced. Optimal sampling schedule design allows to achieve adequate or- improved accur"acy as compared to schedules designed by intuition or other convention; to optimize the cost/effectiveness of a dynamical clinical test typically by reducing the number of blood samples withdrawn from a patient without significantly deteriorating the accuracy; and to obtain less dispersion in population parameter estimates (e.g. Cobelli and Ruggeri, 1985; Cobelli and coll., 1985; D'Argenio, 1981; DiStefano, 1982a,b;

Mori and DiStefano, 1980). The concept of D-efficiency (Landaw, 1982) which express the relative amount of information per unit sample size is use-

ful for comparing a generic schedule with an optimal one.

Model Validation Validation involves to assess whether or not the model is adequate for its purpose. This is a difficult and highly subjective task, i.e. intuition, understanding of the system etc. play an important role. It is also difficult to formalize related issues such as model credibility, i.e. the use of the model outside its established validity range. Some efforts have been made however to provide some formal aids for assessing the value of models of physiological systems (Carson and coll., 1983; Cobelli and coll., 1984a) .Validity criteria have been defined, i.e. empirical, theoretical, pragmatic and he~ ristic validity,and validation strategies have been outlined for two classes of models, essentially co£ responding to simple and complex models. This operational classification is based on a priori identifiability and leads to clearly defined strategies as both complexity of model structure and extent of available experimental data are taken into account. For simple models quantitative criteria based on identification can be used in addition to physiological plausibility. For complex models validation is based on necessarily less solid grounds and should involve, increasing model testability through model simplification, improved experimental design and model decomposition; adaptive fitting based on qualitative and/or quantitative featur·e comparison and time-co'_'rse prediction; and model plausibility.

APPLICATIONS Below some relevant publications are reported grouped per system. This list is aimed to illustrate both the methodological and practical role of system identification in endocrinology, metabolism and pharmacokinetics. - Hepatobiliary system: Belforte and coll. (1983); Carson and Jones (1979); Cobelli and coll. (1983);; Forker and Luxon (1978, 1983); Goreski (1983); Jones and coll.

(1983); Hofmann and coll.

(1983);

Milanese and Molino (1975); Molino and Milanese (1ge2); Molino and coll. (1978). - Thyroid system: De La Salle and coll. (1984); DiStefano and Fisher (1976); DiStefano and coll. (1975; 1982a, b); Mori and DiStefano (1980). - Iron system: Barosi and coll. (1975); Berzuini and coll. (1978); Colli Franzone and coll. (1979); Nathanson and coll. (1984); Stefanelli and coll. (1982, 1984). - Glucose-insulin system: Bergman and coll. (1979a, b; 1981); Cobelli (1984); Cobelli and Bergman (1981); Cobelli and Mar·i (1983); Cobelli and Ruggeri (1983, 1985); Cobelli and Thomaseth (1985); Cobelli and coll. (1982a, 1984b, 1985); Ferrannini and coll. (1985); Finegood and coll. (1984); Grodsky (1972); Licko and Silvers (1975); Insel and coll. (1975); Radziuk and Hetenyi (1982); Sano and coll. (1983); Sherwin and coll. (1974); Sestoft and Vplund (1983); Toffolo and coll.

(1981).

- Ketone body system: Cobelli (1984); Cobelli and Toffolo (1984a,1985); Cobelli and coll. 1985); Landaw (1982); Nosadini and coll.

(1982c,d, (1985);

50

C. Cobclli

Wastney and colI. (1984) . - Lipoprotein system: Berman (1982); Berman and (;011.

(1982).

- Pharmacok i neti(;s: D ' Argenio (198 1 ) ; Endrenyi (1981) ; Gibaldi and Perrier ( 1982); Godfrey (1983, Ch. 11); Godfrey and Fitch (1984a , b). - Control and automated drug ther.apy: D ' Ar. genio and Khakmahd (1983); Katona (1982); Kusuoka and (;011. (1981); Powers and (;0 1 1. (1980); Sheiner and (;011. (1972, 1975) .

identifiability . Math. Biosci. ,

2,

329 - 339 .

Bergman , R.N ., Z. Ider, L. Bowden, and C. Cobelli (1979a) . Quantitative estimation of insulin sensitivity. Am. J. Physio l . , 246 , E667-E677. Bergman, R.N . , G. Borto l an , C . Cobelli, and G. Toffolo (1979b). Identifi(;ation of a mi nimal model of glucose disappearance for estimating insulin sensitivity . In I se rmann R. (Ed . ), Identification and System Parameter Estimation, vol. 2 , (5th IFAC Symp.) , Pergamon , Oxford , pp. 883- 890 . Bergman , R. N., L.S . Phill i ps, and C. Cobe ll i (1981) . Physio l ogi(; evaluation of fa(;tors controlling

CONCLUSIONS

glUCOSE;: t olerance in man. J . elin e Invest. , 68 ,

Modeling and identifi(;ation of endocrine-metabolic

1456- 1467 .

and pharmacokinetic systems has been surveyed . Both

Berman, M.

theoretica l aspects and practical applications in resear·ch, d i agnosis and therapy h ave been discus sed . The uses of models in both physio l ogica l and

theory and applications to l ipoproteins. In M. Berman , S.M. Grundy , B. Howard (Eds . ), Lipoprotein Kinetics and Modeling , Academic , New York , pp. 3 - 36 . Berman , M. , S.M . Grundy , and B . V. Howard (Eds . ) ( 1982). Lipoprote in Kineti(;s and Modeling . Aca-

c l inica l studies have been i llustrated. Emphasis has been placed on g l obal , i . e . whole - organism , models . Some specific c lasses of differen t ial equation par ametric models , e . g . compar"tmental , nonco!!!. par·tmenta l a n d contr·ol - system mode l s , have been re viewed . Some distinct i ve features of endocrine - me tabolic systems in relation to input/output experi ments and t o classica l models/methods of system identification and control have been eviden(;ed. In t h is li ght speci f ic aspects of identification/va l idation have been discussed, e . g. a prior i identi f i ab ility, p a rameter estimat ion fr:om ver.y l imi ted data , opt i ma l design of test- i nput and samp l ing schedule , formal a i ds for model validat i on . A f ew applicat i on areas have been rev i ewed (hepatobi l iary- ,thyroi d- , iron- , glucose -, ketone body- , l ipopr£ te i n-system , pharmacokine ti cs, automated drug therapy) and some re l evant p u b li cat i ons grouped per system have been reported i n order to i llu strate both the me t hodo l ogical and practical ro l e of system identifi cation in endocrinology, metabol i sm and pharmacokinetics .

(19 82) . Kineti(; analysis and mode l ing:

demic, New York. Bertuzzi , A., A. Gandolfi , and M.A. Giovenco ( 1981). Math emat i cal models of the cell (;yc l e with a

view to tumor studies. Math. Bios(;i ., 2l , 159188. Berzuini, C. , P . Colli Franzone, M. Stefanelli, and C . Viganotti (1978) . Iron kinetics: modelling and parameter estimation in normal and anemic

states . Comput. Biomed . Res ., l!, 209-227 . Bloch , R. , D. Ingram , G.D. Sweeney , K. Ahmed, and C.J . Dickinson ( 1980). Ma(;dope : a simulation of drug disposition in the human body. Mathematical (;onsiderations.J. Theor. BioI. , 87 , 211 - 236. Box, M.J. (1970) . Some experien(;es with a non l inear e x pel imental design cri ter:ion . Technometri cs , 12, 569-589. Brown , R . F .

(1980). Compartmental systems analysis:

state of the art. BME- 27 , 1-11.

IEEE Trans. Biomed. Eng .,

Carson , E.R. , and E . A. Jones (1979). The use of kiREFERENCES Akaike, H. (1974). A new l ook at the statist i (;al mode l i denti f ication . IEEE Trans . Au tom. Contr. , AC-19 , 7 16-723 . Anderson , D.H. (1983) . Compartmental Modeling and Tracer Kinetics. Spring er, Berlin. Audo l y , S ., and L . D 'Ang i o (1983) . On t h e i denti fiabi l ity of linear compart mental systems: a revis i ted transfer function approach based on topolog i ca l properties . Math. Biosci., 66 , 201 - 228. Bard , Y. ( 1974). Non l inear Parameter Estimation . Academ i c , New York. Barosi, G. , M. Cazzola , C . Berzuini, S. Quaglini , and M. Stefanelli ( 1985).

Classif i (;ation of

anemia on the basis o f ferrokinetic parameters . BL J . Haematol. , (in press). Beck, J . V. , and K. J . Arno l d ( 1977). Parameter Esti mation . Wi l ey, New York . Bekey, G . A. , and G . Saridis (Eds.) ( 1982) . I dent ification and System Parameter Estimation , vol . 2, (6th IFAC Symp.) , Pergamon, Oxford , pp. 884913 . Be l forte , G. , B. Bona , and M. Mi l anese (1984). Advanced mode l ing and identifi(;ation techniques for metabolic processes. CRC Crit. Rev. Biomed. Engng. , 10 , 275-3 16 . Be llman , R.,--;nd K.J. Astr.6m (1970) . On structural

netic ana l ysis and mathematical modeling in the quant i tat ion of metabo l ic pathways in vivo: application to hepati(; anion metabolism. New Engl. J. Med. , 300, 1016-1027 and 1078- 1086. Carson, E.R. , C . Cobelli , and L . Finkelstein (1979). The identification of metabolic systems . A review. In I sermann R. (Ed.), Identification and System Paran'eter Estimation, vol. 1, (5th IFAC Symp.), Pe r gamon , Oxford , pp. 1 5 1- 1 71. Car.·son, E.R. , C. Cobelli, and L . Finkelstein ( 198 1 ) . Modeling and identification of metabo l i(; systems. Am . J. Physiol., 240 , R120-R129. Carson , E.R. , C. Cobel l i , and L. Finkelstein (1983). Mathematical Modelling of Metabolic and Endo crine Systems. Wiley, New York . Cobelli, C . (1984). Modeling and identification of endocrine - metabolic sys tems . Theoret ical aspects and their importance in practice. Math. Biosci. ,

!l:.,

263-289 . Cobe ll i, C ., and R.N . Bergman (Eds.)

(1981). Carbo-

hydrate Metaboli s m. , Wiley , Chichester . Cobelli, C. , and J.J . DiStefano I I I (1980). Paramet er and structural identifiability concepts and ambiguities: a critica l review and ana l ysis. J . Physiol ., 239 , R7-R24.

Cobe l li , C . , and A. Mari

~

( 1983) . Validation of ma-

thematical mode l s o f complex endocrine - metabolic systems. A (;ase study on a model of glucose regulation. Med . BioI. Eng. Comput . , 1l,390-39~

51

Elldocrillc-Metaholic all d I'harlll aco killctic Systcllls Cobelli , C., and A . Rescigno (1978). Some observations on the physical realizability of compartmental models with delays. IEEE Trans . Biomed. Eng., BME-25, 294 - 295. Cobelli, C ., and A. Rugg er i (1983) . Evaluation of portal/peripheral route and of algorithms for insulin delivery in the closed -l oop control of glucose in diabetes. A modeling study . IEEE Trans. Biomed . Eng., 30 , 93 -1 03. Cobelli , C ., and A . Ruggeri (1985). A four sample schedu l e for estimating glucose kinet i cs parameters from tracer." experiments. Design and in vivo evaluation of robustness. Am. J. Physiol., (submi tted ) . Cobelli, C., and K. Thomaseth (1985). Optima l input design for identification of compartmental models. Theory and app l ication to a model cf gl~ cose kinetics. Math. Biosci ., (in press). Cobelli , C., and G. Toffolo ( 1984a) . Identifiability from par-ameter bounds. Structural and numerical aspects. Math. Biosci ., 2!, 237 -243. Cobelli, C. , and G. Toffolo (1984b) . Compartmental vs . noncompartmental modeling for two acce ss ible pools. Am. J . Physiol ., 247 , R48 8-R496. Cobelli , C., and G. Toffolo (1985). Compartmental and noncompar"tmental models as candidate classes for kinetic modeling. Theory and computational aspects. In Eisenfeld J. and DeL i si C. (Eds.) , Mathematics and Computers in Biomedi cal Applications , EI sev i er , New York, pp . 219 236 . Cobelli , C. , A. Lepschy, and G. Romanin Jacur ( 1979a) . Identif iability resu lts on some con strained compartmental systems. Math. Biosci. ,

A model of glucose kinetics and their control by insulin. Compartmental and noncompartmental approaches. Math. Biosci., ~, 291 - 315. Cobelli, c., A. Ruggeri, J.J. DiStefano Ill, and E.M. Landaw ( 198 5). Optimal design of multioutput sampling schedules: software and app li c ations to endocrine-metabolic and pharmacoki n etic models. IEEE Trans. Biomed . Eng., April is sue. Co lli Franzone, P., M. Stefanelli , and C. Viganotti (1979) . Parameter estimation of a distributed model o f ferrokinetic s : effects of data errors on parameter estimate accuracy. In Isermann R. (Eds.) Ide ntification and System Parameter Estima tion , vol. 2 , (5th IFAC Symp.), Pergamon, Oxford, pp. 827-834 . Cook, R.D., and S. Wei sberg (1982). Residuals and I nfluence i n Regress i on. Chapman & Hall, New York. Cramp , D. G., (Ed . ) (1982 ) . Quantitative Approaches to Metaboli s m. Wiley, Chichester. D'Argenio , D.Z. (1981). Optimal sampling times for pharmacokinetic experiments. J. Pharmacokin. Biopharm., ~, 739- 757. D'Argenio, D.Z., and K. Khakmahd ( 1983) . Adaptive contro l of theophylline therapy: importance of b l ood sampling times. J. Pharmacokin. Biopharm.,

.!.!.' 547-559. De La Sal le, S ., M. S . Leaning, E .R. Carson , P.R. Edwards, and L. Finkelstein ( 1984) . Control system modelling of hormonal dynamics in the management of thyroid disease. In J. Gertler and L. Keviczky (Eds.), A Bridge between Control Sc i ence and Technology, vol. 11, (9th IFAC Congr.), Pergamon, Oxford, pp. 97 -10 2 . De lforge , J. ( 1984). On local identif iability of l i

i2, 173-196. Cobelli, C. , A. Lepschy , and G. Romanin Jacur (1979b) . Iden tifiabi lity of compartmental sy-

near systems. Math. Biosci., ]£, 1-37. DiStefano Ill, J.J . (1979). Matching the model and the experiment to the goals: data limitation s ,

stems and related structura l propert i es . Math. Biosci., 48 , 1-18. Cobe lli, C., G. Feder sp il , G. Pacini, A . Sa l van,and C . Scande ll ari (1982a). An integrated mathema-

complexity and optimal experiment design for dynamic systems with biochemical s ignals. ~ Cybern. Inf. Sci., ~, 6-20. DiStefano Ill, J.J. (1981). Optimized b l ood sampling protocols and sequentia l design of kinetic ex-

tical model of the dynamics of blood glucose and its hormonal co ntrol. Math. Biosci., 58 , 27 - 60 . Cobelli , C. , L. Finkelstein , and E . R. Carson (1982b). Mathematica l modeling of endocrine and metabolic systems : model formulation, identification and validation. Math. Comput. S i mul . XX IV, 442-45 1. Cobelli , C., R . Nosadini, G. Toffolo , A. McCulloch, A. Avogaro , A. Tiengo , and K. G .M .M. Alberti ( 1982c). Model of the kinetics of ketone bodies in humans. Am. J. Physiol., ~ , R7-R17. Cobelli, C. , G. Toffolo, and R. Nosadini ( 1982d). Mathemati cal models of ketone body kinetics in the human. Experiment design and their identi fication/validation. In Bekey, G . A. and Saridis, G . N. (Eds . ) Identification and Sys tem Parameter Estimation, vol. 1, (6th IFAC Symp.), Pergamon , Oxford , pp. 222 - 227 . Cobelli , C . , J.J. DiStefano Ill, and A. Ruggeri (1983) . Minimal samp ling schedule for identification of dynamic models of metabolic systems of clinical interest: case studies for two liver function tests . Math . Biosci. , 63 , 173-186. Cobelli , C. , E.R. Carson, L . Finkelstein, and M.S. Leaning (1984a). Validation of simple and complex models in physiology and medicine . Am. J. Physiol., 246 , R259-R26 6 . Cobelli , C. , G. Toffolo, and E. Ferrannini (1984b).

periments. Am. J. Physiol., ~, R259 -R265. DiStefano Ill, J.J. (1982a). Algorithms, so ft ware and sequential optimal sampl ing schedule designs for pharmacokinetic and physiolog i cal experiments. Math. Comput . Simul ., XXIV, 53 1- 534 . DiStefano Ill, J.J.

(1982b). Noncompartmental vs .

compartmental ana l ysis : some bases for c h oice. Am. J. Physiol., 243 , R1 - R6. DiStefano Ill, J.J. (1983). Complete par"ameter bounds and quasiidentifiability conditions for a class of unident ifiable l inear systems. Math. Biosci., 65, 5 1-68 . DiStefano Ill, J.J. (1984) . Repl y to "Identif iabili ty from parameter bounds. Structural and numerical aspects " by Cobe lli C. and G. Toffolo . Math . Biosci., 2!, 245- 246. DiSte fano Ill, J.J., and D.A . Fi s her (1976) . Peripheral distribution and metabolism of the thyroid hormones: a primarily quantitative assessment. Pharmac. Ther. B, ~, 539-570. DiStefano Ill, J.J., and E .M . Landaw (1984) . Multiexponentia l, multicompartmental , and noncompar! mental modeling. I. Methodological l imitations

and physio l ogical interpretat ion s , Am. J. Physiol., 246, R651-R664 . DiStefano Ill, J.J., K.C. Wilson, M. Jang, and P.H. Mak ( 19 75) . Identifi cation of the dynam i cs thyroid hormone metabolism. Automatica ,

.!.!.'

of

52

C. C:ohclli

149- 159. DiStefano Ill , J.J. , M. Jang , T . K. Mal one , and M. Broutman (1982a). Comprehensive k i netics of

tr~

iodothyronine (T 3 ) production distribution and metabolism in blood and tissue pools of the rat using optimized blood samp l ing protocols . Endo cr i nology , ~, 198 - 213 . DiStefano Ill , J . J. , T.K. Malone , and M. Jang (1982b). Comprehensive kinetics of t hyroxine ( T4) distribution and metabo l ism in blood and tissue pool of the rat from only 6 b l ood samples : Domi nance of l arge , slowly exc h a nging

ti~

sue pools . Endocrino l ogy , ~, 108 - 1 17 . Eisenfe l d , J . (1979). Relat i onshi p between stochas ti c and differentia l models of compa r t me n ta l systems. Mat h . Biosci., ~ , 289 - 305. Eisenfe l d , J . ( 1982). On washout i n non l inear compartmental systems. Math . Biosc i., ~ , 259 - 275 . Endrenyi , L . (Ed . ) (198 1). Kine t ic Data Analysis. Plenum , New York . Eykhoff , P. ( 1974) . System I den t i f ication: Parame ter and State Estimation . Wiley , London. Eykhoff, P . (Ed . ) ( 1981) . Trends and Progr ess i n System I dentif i cation . Pergamon , Ox f ord . Fedorov , V. V.

Des i gn . Academic , New York. Ferta nni ni, E ., J. D. Smith, C . Cobe l li , G. Toffo l o , A. Pilo , and R. A. De Fronzo (1985). Effect of insu li n on the distribution a n d d i sposition of g l ucose i n man. J. Clin. I nvest. , i n press . Finegood , D.T ., G. Pac i n i, and R.N. Berg man (1984) . The insu l in sensitivity index : cOI-re l a tio n in dogs between values determi ned f rom the i ntra venous g lucose tolerance test and the euglyce ~,

362- 368.

Forker , E.L . , and B . A. Lux on (1978) . Hepat i c trans port kinet i cs and p l asma disappearance curves: dis t ributed model i ng versus con vent i ona l ach . Am . J. Ph ysio l., ~ , E648 -E660 .

mathemat i ca l modeling . J . Clin . In ves t . , ~ , 204 7 - 2059 . Groth , T. ( 1984) . The ro l e of formal biodynamic mo dels i n laboratory medicine. Scand . J . Clin .Lab. Invest. , ii, Suppl. 171, 175-193. Gurpide , E. (1975). Tracer Methods in Hormone Research , Springer, Ber l in. Guy ton , A. C . (1979) . On the value of l arge mode l s

of biological systems . J . Cybern. I nf . Sci. , 7 1- 72 . Hearon , J.Z. ( 1963) . Theorems on linear systems.

2,

Ann . N. Y. Acad. Sci. , ~ , 36 - 68. Hofmann , A. F . , G. Mo l ino , M. Mi lanese , a n d G. Bel forte ( 1983) . Description and s i mulation of a p h ysio l ogical pharmacokinetic mode l for the me tabo l ism and enterohepatic circulation of bi l e acids in man . J . Clin. Invest. , 2!, 1003 -1 022 . Insel , P . A ., J . E. Liljenquist , J . D. Tobin, R.S. Sherwin, P . Watkins , R. Andres , and M. Berman ( 1975). In s ul in control of gl ucose metabolism in man. A new kinetic analysis . J . Cl in.

(1972) . Theory of Optima l Experiment

mic g l ucose clamp. Diabetes ,

Grodsky, G.M. (1972). A treshold distribution hyp~ t h esis for packet storage of insuli n and its

appr~

Porker , E .L . , and B.A . Luxon ( 1983) . Ana l yzi n g tr~ cer d i sappearance curves to study h epatic tran~ port k i netics . Am . J . Phys i o l., 244 , G573 - G577. Garfinke l, D. (1982). Comp l ex biologica l mode l s: their construction and effective use. Math . Comput . Simul., XXIV , 425 - 429. Gert l er , J. , and L. Keviczky (Eds.) (198 4 ). A Bridge between Control, Science and Technology, vo l . II , (9th I FAC Congr . ) , pp. 75 - 120 . Giba l di, M., and D. Perrier (1982). Pha rmacokine -

I nvest. , 22, 1057- 1066. Jacquez , J . A . (1972) . Compartmental Anal ysis in Bio logy and Medicine. Elsevier, Amsterdam . Jacqu ez , J . A. ( 1982). The inverse probl e m for compartment al systems . Math. Comput. Simul . , XX I V, 452 - 459 . Jacquez , J . A. (Ed.) (1984) . Berman Memoria l I ssue , Math. Biosci., 2l , December issue . Jennrich , R . I., and M.L . Ra l ston (1979) . Fitting nonl i near mode l s to data. Annu . Rev. Biophys .

Bioeng. , ~ , 195-238. Jones, E . A., E .R. Carson , and P.D. Berk (1983). The role o f kinetic ana l ysis and mathemat i cal model i ng i n the study of bilirubin metabolism i n vivo. I n He i rwegh,K.P.M . and Brown, S .B. (Eds . ) , Bili rub i n Metabolism , vol. 2, CRC, Cl eveland , pp. 133- 172 . Ka l aba , R . E ., and K. Spingar·n (1981) . Op t ima l inpu ts for b l ood g l ucose regu l ation parameter estima tion , Part 3 . J. Optim. Theory App l . , ~ , 267285 . Ka l aba, R.E. , and K. Spingarn (1982) . Control , Iden tifica ti on and Input Optimization , Plenum , New York. Katona, P.G .

( 1982 ) . Automated contro l of

physio l~

tics. 2nd ed. , Dekker , New York . Godfrey , K. (1983) . Compartmenta l Mode l s and their

gica l variab l es and clinical therapy. CRC Crit . Rev . Biomed . Engng., ~, 281-310 . Kusuoka, H., S . Kodama , H. Maeda , M. Inoue , M. Hori,

Ap p li cati on . Academic Press , Lond o n. Godfrey , K. R ., and W. R . Fitch (1984a) . The deter·mi nistic identi f iabi l ity of nonlinear pharmacok inetic mode l s . J. Pharmacokin. Bi ophar m. , ~ ,

H . Abe , and F . Kajia (1981). Optimal contro l in compartmental systems and its app l ication to dr u g administration. Math. Biosci . , ~ , 59 - 77 . Landaw, E .M. (1982). Optimal multicompa r tmenta l sa~

177 -1 91. Godfrey , K.R ., and W.R. Fitch ( 1984b). On the iden -

pling designs for parameter estima t ion : practical aspects of the identification prob l em . Math. Comput . Simul. , XXIV , 525-530.

tification of Michae l is - Meuten e limi nation parameters from a single dose - response curve.

~

Pharma cokin . Biopharm ., ~ , 193 - 22 1 . Goodwin , G. C. , and R . L . Payne (1977) . Dynamic System Identification: Experiment Design a nd Data Ana l ysis . Academic.: Press , New Yor-k. Goresky, C . A. ( 1983). Ki netic interpretat ion of he pat i c mul t i p l e- i ndi cator dilution stud i es. Am. J . Ph ysio l., 245, G1 - GI2. Grewal , M.S. , and K. Glover (1976). I dent i fiability of li near and nonlinear dynamica l systems . I EEE Trans. Autom . Contr ., I'C - 21, 833 - 837 .

Landaw , E. M. (1985). Robust sampling desig n s for compartmental models under large prior eigen value uncertaintie s . In Eisenfe l d , J . and DeL isi, C. (Eds.) , Mathematics and Computers in Biomedical Applications, Elsevier , New Yor"k. Landaw, E. M., and J . J. DiStefano I II ( 1984) . Multi exponent ia l, multicompartmenta l , and noncompar! mental modeling . 11 . Data analysis and statist~ cal considerations. Am. J. Physio l . , 246 ,

R665-R667. Licko, V., and A. Silvers (1975). Open-loop glucose insulin control with thre s hold secretory mecha -

Endocrine-Metabolic and Pharlll
tests in man. Math. Biosci., ~ , 319-332. Ljung, L., and T. SoderstrCim (1983). Theory and Practice of Recursive Identification. MITPres s , Cambridge. Maeda, H., S . Kodarna, and Y. Ohta (1978). Asymptotic behaviour of non-linear compartmental systems: non-oscillation and stability. IEEE Trans. Circuits & Syst., CAS-25, 372-378. Matis, J.H., and T.E. Wehrly (1979). Stochastic mo dels of compartmental systems. Biometrics, 12, 199 - 220 . Matis, J.H., and T.E. Wehrly (1981). Compartmental models with multiple sources of stochastic variability: the one-compartment models with cluster ing. Bull. Math. Biol., ~, 65 1-664. Matis, J.H., T.E. Wehrly, and C.M. Metzler (1983). On some stochastic formulations and related sta tistical moments of pharmacokinetic models.

~

Pharmacokin. Biopharm., ll' 77-92 . Mazanov, A. (1976). Stability of multi-pool models with lags. J. Theoret. BioI., 59, 429-442. Mclntosh, J.E.A., and R.P. Mclntosh (1980). Mathe-

53

analysis of linear compartmental sys tems in linear stages . Math. Biosci., 50, 95-115. Norton, J.P. (1982). An investigation of the sources of non-uniqueness in deterministic identifiability. Math. Biosci., 60 , 89 -1 08 . Peck, C.C., S.L. Beal, L.B. Sheiner, and A.I. Nicho ls ( 1984). Extended least squares nonlinear regression: a possible solution to the "choice of weights" problem in analysis of individual pharmacokinetic data. J. Pharmacokin. Biopharm., g, 545-558. Plusquellec, Y., and J.L. Steimer (1984). An analytical solution for a class of delay-differential equations in pharmacokinetic compartment modeling. Math. Biosci., 70, 39-56. Pohjanpalo, H. (1978). System identifiability based on the power series expansion of the solution. Math. Biosci., il, 21-33. Powers, W.!""., P.H. Abbrecht, and D.G. Covell (1980) .. Systems and microcomputer approach to anticoagulant therapy. IEEE Trans. Biomed. Engng., BME-27, 520-523. Radziuk, J _, and G. Hetenyi Jr.

(1982). Modeling and

matical Modelling and Computers in Endocrinolo~. Springer, Berlin.

the use of tracers in the analysis of exogenous co ntrol of glucose homeostasis. In Cramp, D.G.

Metzler, D.M. (1981). Statistical properties of kinetic estimates. In Endrenyi, L. (Ed.), Kinetic Data Analy sis , Plenum, New York,pp.25-37. Milanese, M., and G. Molino (1975). Structural ide~ tifiability of compartmental models and pathophysio l ogical information from the kinetics of drugs. Math. Biosci., 26, 175-190.

(Ed.). Quantitative Approaches to Metabolism, Wiley, Chichester, pp. 73-142. Rescigno, A., and E. Gurpide (1973). Estimation of average times of residence, recycles and interconversion of blood-borne compounds using tracer methods. J. Clin. Endocrinol. Metab ., ~, 263 - 276.

Mitchell, R.R., and N.K. Gupta (1982). Variability of radionuclide clearance parameter estimates.

Rescigno, A., and G. Segre (1966). Drug and Tracer Kinetics. Blaisdell, Waltham.

Comput. Biomed. Res., ~, 387 -403 . Mo lino , G., and M. Milanese (1982). Modeling of

Sandberg, I.W. (1978). On the mathematical foundations of compartmental analysis in biology, me-

hepatobiliary transport processes: bromosulphtale in and bilirubin kinetics. In Cramp. D.G. (Ed .), Quantitative Approaches to Metabolism, Wiley, Chichester, pp. 185-217. Ma lino , G., M. Milanese, A. Villa, A. Cavanna, and G. Gaidano ( 19 78). Discrimination of hepatobiliary diseases by the evaluation o f bromosulphtalein blood kinetics. J. Lab. Clin. Med., 2!, 396-4 08. Mohler , R.R., C. Bruni, and A. Gandolfi (1980). A systems approach to immunology. Proc . IEEE, 68, 96 4- 990. Mori , F., and J.J. DiStefano III (1980). Optimal nonuniform sampling interval and test-input design for identification of physiological systems from very limited data. IEEE Trans. Automat. Contr., AC-24, 893-900. Mulholland, R.J., and M.S. Keener (1974) . Analysis of linear compartment models for eco - systems . J . Theoret. Biol., 44, 105-116. Nathanson, M.H., G.M. Saidel, and G.D. McLaren (1984). Analysis of iron kinetics: identifiabi lity, experiment design, and deterministic interpretations of a stochastic model, Math. Biosci., 68, 1-21. Nguyen, V.V., and E.F. Wood (1982). Review and unification of linear identifiability concepts. S IAM Rev.,~, 34-51. Nosadini, R., A. Avogaro, R. Trevisan, E. Duner, C. Marescotti, E. Iori, C . Cobelli , and G. Toffolo ( 1985). Acetoacetate and 3hydroxybutyrate kinetics in obese and insulin-dependent diabetic humans, Am. J. Physiol., (in press). Norton, J.P. (1980). Normal-mode identifiability

dicine and ecology. IEEE Trans. Circuits CAS-25, 273-279.

&

Syst .,

Sano, A., K. Kondo, M. Kituchi, and Y. Sakurai ( 1983) _ Adaptive control system for blood glucose regulation. Trans. Inst. Meas. Contr., ~, 207-216. Schwarz, G_ (1978). Estimating the dimension of a model. Ann. Stat., ~, 461-464. Segel, L.A_ (Ed.) (1980). Mathematical Models in Molecular and Cellular Biology. Cambridge University Press, Cambridge. Sestoft, L., and A. V~lund (1983). Description of degree and goodness of blood glucose control in diabetics and ' normal subjects. Acta Med_ Scand., Suppl. 671, 11-17. Sheiner, L.B., B. Rosenberg, and K.L. Melmon (1972). Modelling of individual pharmacokinetics for computer-aided drug dosage. Comput. Biomed. Res., 2, 441-459. Sheiner, L.B., H. Halkin, C. Peck, B. Rosenberg, and K.L. Melmon (1975). Improved computer-assisted digoxin therapy: a method using feedback of measured serum digoxin concentrations. Ann. Intern . Med., 82, 619-627. Sherwin, R.S., K.J. Kramer, J.D. Tobin, P.A. Insel, J.E. Liljenquist, M. Berman, and R. Andres (1974) . A model of the kinetics of insulin in man. J. Clin. Inve st ., 2l' 1481-1492. Smith, W.D., and R.R. Mohler (1976). Necessary and sufficient conditions in the tracer determination of compartmenta l system order. J. Theor. ~, '32, 1-21. Soder'str'om , T., and P.G. Stoica ( 1983). Instrumen-

tal Variable Methods for System Identification.

54

C. Cohelli

Springer , Berlin . Stefanelli , M., G. Baros i, and M. Cazzola (1982). Iron k i netics. In Cramp , D.G. (Ed . ) , Quantita t i ve Approaches to Metabolism, Wi l ey, Chichester , pp. 143-183. Stefanelli , M. , C. Larizza , and S . Quag li ni (1984) . Re g u l ation of i ron metabolism. In Gertler, J . a nd Ke vi czky L . (Ed s . ). A Br i d g e between Con tro l Sc i ence and Technolog y , vo l . 11, (9th IFAC Congr . ) , Pergamon , Oxfor d , pp. 1 15 - 120. St. John , R.C ., and N. R. Draper ( 1975). D- optimali ty for regression designs: a revi ew . Technometrics , .!2, 15- 23. Swan , G. W. ( 1984) . Appl ications o f Opti mal Control Theory in Biomed i cine. Dekker , New York. To f folo , G. , R.N. Bergman , D. T . Finegood , C.R. Bowden , and C. Cobelli (1980). Quantitative estimat i on of B- cell sensitivity to g l ucose i n the in t ac t organ i sm; a minima l mode l of i nsul i n kin e ti cs i n the dog. Diabetes , ~, 979 - 990. Travi s , C. C. , and G. Haddock ( 198 1). On structura l i den t ification. Math . Biosci ., 56, 157 -1 73 . Vajda , S . (1983). Structura l ident i f iab il ity of dynamical systems. I n t . J. Sys t. Sc i., 12 4 7.

l!,

1229-

Vajda, S. ( 1985). Ana l ys i s of unique structural i de nti f i abi l ity via submode l s. Math. Biosci. , (in press) . WaI t er , E. ( 1982) . I dentifiabi l ity of State Space Mode l s. Spr i nger , Berl in . Wa I ter , E., and Y. Lecou rtier (1981) . Unidentifia bl e compartme n tal models: what t o do?, Math. Bi osci. , 56 , 1- 25 . Wa I ter , E. , and Y. Lecourtier ( 1982) . Global appro~ c hes t o ident ifiab i lity testing f or linear' and nonli near state space mode l s . Math . Compu t . Simul., XX I V, 472-482. Wastney , M.E. , S . E.H . Hall , and M. Berman ( 1984). Ketone body kinetics in humans: a mathematical model . J . Lip i d. Res. , ~, 160- 174 . Yates , F.E. ( 1979) . Phys i ca l biol ogy: a basis for modeling living systems . J . Cybern . I nf . Sci. ,

3"

57 - 70 . Yo u ng , P. ( 1984). Recurs i ve Est i ma tio n and Time Ser i es Anal ys i s . Spr i nger , Berlin. Zazworsky , R.M ., and H.K. Knudsen ( 1978). Controll abi l ity and observability of l i n ear time-invariant compar·tmenta l mode l s. I EEE Trans. Automat. Contr. , AC - 23 , 872-877 . Zier l er , K. (1981). A crit i que of compartmental anal ysis. Annu.Rev . Biophys. Bioeng., 531-562.

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