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https://ntrs.nasa.gov/search.jsp?R=19850017914 2020-07-10T05:43:44+00:00Z

ITIR 2114--MED-5005

FROM TOJoel I. Leonard, Ph.D. N. Cintron, Ph.D./SD4

DATE CONTRACT NO- T.O.OR A.D. REF: MIS OR OTHER NASA REF:April 22, 19851

1 NAS9-17151

SUBJECTThe Modeling and Simulation of Feedback Control Systems

(NASA-CE-11105) TEr CIUDELIbC ANC NE5 -26225SIMUTATIcb CF iif rEAC F CC b1EC I SYS'TIMS(CanageiEnt ac d icchnacal sEtvic__ Cc.)14 p HC 11L2/!fF A; 1 CSCL 99B Unclae

G3/63 w 1222

This document contains a brief description ofthe principles of mathemat;cal models and

their development. It should serve as anintroduction to those unfamiliar with this

topic. It was originally prepared as anAppendix for a NASA Reference Publica!ionand is documented here in preliminary form.

Li:-'Joe I. eon d, Ph.

/db

Attachment

Unit Approving Q . NASAManager Manager F.A. Kutyna, Ph.D. Concurrence

DISTRIBUTION NASA/JSC: MATSCO/HO: GE/H0:NASA/H S: MATSCO/H S: BE3 B.J. Jefferson R. F. Meyer T—.D. Gregory(2)

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SD3/M. Bungo

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A. Schneider J _^P

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68,11

Cq- 1-11 ^^^TECHNICAL INFORMATION RELEASE

C.I

fn•n•Q•ment and • n Ca Nrvices comparg

U

TIR 2114-MED-5005

THE MODELING AND SIMULATION OF FEEDBACK CONTROL SYSTEMS

Prepared by

Joel I. Leonard, Ph.D.

Management and Technical Services Company

Houston, Texas

Prepared for

National Aeronautics and Space Administration

Contract NAS9-17151

1985

i

Ji

M i

1►

The Modeling and Simulation of Feedback Control Systems

The purpose of this appendix is to explain thebasic vocabulary and principles of model develop-ment so that the reader may better grasp thedescription of the simulation models. Exampleswhich describe the steps leading to a computeralgorithm of a model subsystem are provided, andthe simulation techniques used to assess modelbehavior and accuracy are discussed.

DEFINITIONS OF TERMSAND CONCEPTS

Parameters and Variables

Two general types of quantities are used inmathematical models: parameters and variables.The value of a parameter is generally constant withrespect to time, whereas the value of a variablechanges with time (i.e., is rime-v-arying). Biologicalparameters often vary slowly with time but may beassumed to be constant in the mathematical model.

Parameters may be considered independent ofsystem actions. Variables may also be consideredindependent if they influence the system from theoutside. Parameters and independent variables arealso called input functions, or forcing .functions. Moreoften, variables are dependent (also called outputfunctions), since they vary according to the rela-tionships within the system. The objective of asystems analysis is to specify the state of thesystem; that is, to specify the values of all depend-ent variables at every instant of time.

Classification of Mathematical ModelSystems

Mathematical models are classified according tothe types of equations they employ (ref. A-1).

Distributed- and lumped parameter system.—Indistributed-parameter systems, the values of varia-bles depend on time as well as space coordinates,and the system must be analyzed by solving partialdifferential equations. In a lumped parameter system,each element is treated as if it were concentrated

("lumped") at one particular point, which is calleda node or a compartment. The use of lumpedparameters greatly simplifies the analysis becauseordinar y differential equations are used to describethe changes with time. To account for spatialchanges that occur in such a system (e.g.. the valueof blood pressure is different in arteries, capillaries,and veins), compartments are added for each majorspace location until one reaches the level of com-partmentalization (lumping) commensurate withthe basic purpose of the model being designed.

Generally, as subdivisions are included in amodel, the fidelity and accuracy of the model'sresponse increases. For example, in this project,two circulatory models were emp;oyed, one com-posed of 7 compartments and the other of 28 com-partments. The 7-compartment model could simu-late mean blood pressures in arteries and veins,whereas the 28-compartment model could simulatepulsatile flow and provide systolic and diastolicpressure for all anatomical portions of thevasculature (i.e., aorta, arteries, arterioles, capil-laries, venules, veins, and vena rava).

All models described in this publication arelumped-parameter systems. Whether the lumpingof components and the simplifying of input-outputrelationships are appropriate depends entirely onthe intended purpose of the model. Lumped, com-partmental modeling is in keeping with a basicfacet of the systems approach: to simplify the prob-lem and define the essentials of its solution.

Linear and nonlinear systems—In modeling,systems are generally taken to consist of compo-nents with a known (or assumed) relationship. In abroad sense, a component of a system may bethought of as transforming certain inputs into cer-tain outputs. Such relationships are often (loosely)termed transfer functions (although transfer func-tion has a rigorous meaning only in terms ofLaplace transforms).

Mathematically, the components of a systemmay be represented by several operators, whichtransform one function into another. If all theoperators for the system are linear, in the loosesense that their operation on two independentfunctions produces two independent results, thenthe system itself is said to be linear. Otherwise, the

system is said to be nonlinear. Mathematically,much more is known about the behavior of linearsystems.

Unfortunately, most biological systems—withtheir threshold and saturation behavior, sigmoidaldose-response relationship, and dead time-,arenonlinear; analysis of such systems revolvesaround numerical -ethods and the use of large com-puters. Obtaining rapid, accurate, numerical solu-tions of nonlinear system equations is absolutelyessential in biological modeling and often repre-sents a separate challenge.

Other classifications—Models can be developedfor either predictive or descriptive purposes. A pre-dictive model must only produce accurate predic-tions of the output variables in the system. For ex-ample, a singie equation may be used to predict thefractional saturation of hemoglobin with oxygen atdifferent levels of oxygen partial pressure. In thelatter case, the modeler is not concerned with exactreplication in the model of the interactions betweenthe variables in the system. The relationshipsemployed in the predictive model to generate thepredicted output need not conform ;o the mecha-nisms in the real system which lead to the sameoutputs. In contrast, descriptive modeo not onlymust generate predictions in agreement with realsystem output but also must employ intermediaterelationships that are realistic representations ofthe true processes which generate the observed out-puts.

The vast majority of biological systems are con-tinuous systems, in which values are always chang-ing with time, and they are best described bydifferential equations. These equations are typicallysolved with numerical procedures on digital com-puters by transforming them into finite-diffeerenceequations. In this process, the variables actually ap-pear to be constant for the duration of the integra-tion step sire and, therefore, more properly belongto a discrete system. This approximation of con-tinuous biological systems by discrete numericalsystems can produce inaccuracies and instabilitiesin the solution, unless appropriate care is taken andthe system is carefully verified.

The models described in t his section can also besaid to be deterministic, since the input-output rela-tionships for each component are based on simplephysical laws (i.e., flow-pressure, diffusion, massaction) and are therefore fixed, predictable, andreproducible. In reality, most biological systems arestochastic, in the sense they are subject to random

noise, and their responses can be described bystatistical laws and in terms of probabilities and ex-pected values.

Riggs (ref. A-1) makes the observation that "allnaturally-occurring systems are, in the finalanalysis, time-varying distributed-parameter, quan-tized, stochastic, nonlinear systems. That we cansometimes obtain useful information by treatingthese wayward creatures as if they were fixes;lumped-parameter, continuous, deterministic,linear systems is little short of miraculous."

Biological Control Systems

It is difficult to conceive of any biological quan-tity which is not controlled or influence: by one ormore factors. For example, blood pressure in-fluences and controls the rate of baroreceptorneural firing. In this situation, baroreceptor neuralfiring is controlled, or regulated, within narrowlimits at the expense of the other system quantities,the limits of which may vary widely. The relation-ship between blood pressure and baroreceptorneural firing would be ronstrued as an open-loopsystem for controlling baroreceptor neural tiring.Further examination of the process shows that thebaroreceptor afferent signal is processed by thecentral nervous system (CNS) and n suits in anefferent neural signal that controls peripheralresistance. This relationship between baroreceptorfiring and peripheral resistance is an open-loopsystem for controlling peripheral resistance.However, it is known that peripheral resistancedirectly regulates blood pressure. Once this control

added to the system, the complete loop of physi-cal changes (blood pressure, peripheral resistance)and information transmission (afferent andefferent signals) forms a closed-loop control system.The difference between the closed-loop and open-loop systems is obviously the presence of the rela-tionship of peripheral resistance to blood pressure.Blood pressure in the closed-loop system is calledthe feedback variable.

A feedback control system, as described pre-viously, is composed of two major elements, orcomponents: the controlled system and the con-trolling system or controller. Figure A-I is a diagramillustrating the relationships of these elements. Thepurpose of the controlled system is to transform in-puts from the outside (load inputs, disturbances, orstress stimuli) and inputs from the controller (con-

.-

pa l V1^^{ Controlled I^ Y (outpuu

I system T a

/Controlled variableor feedback variable/

concrol1hrig\ Ysigns'

contiolh,g

systemR 1 (reference

mput)

FIGURE A - I.—A generalized feedback control system. Mostbiological homeostatic systems can be reduced to these basiccomponents.

troliing signals) into responses (outputs). Thereference input of figur A-I is also referred to as asee point, and it represents a normal or desired val eof a feedback variable. Differences between the set-point and the actual feedback variable result in cor-rective action by the controller; this correction per-mits the output to revert toward normal values inthe face of the disturbance. Consequently, the pres-ence of feedback control in biological systems allowsimportant quantities to fluctuate within limitsnecessary for maintaining life in the face of manymetabolic and environmental disturbances encoun-tered by the organism.

In manmade lachnological control systems, theset-point is a physical quantity that can be varied atwill (e.g., a thermostat setting for a home heatingsystem). Biological systems, however, often haveneither a reference input nor an error detector. Theestablishment of a set-point, or an operating point, inmathematical models of biological systems is oftenincluded for convenience, only because the realsystem behaves as if it were controlled by such areference value. (See refs. A-1 and A-2 for furtherinformation.)

In terms of the example described previously,blood pressure is the controlled feedback variableand peripheral resistance can be considered thecontrolling variable. The controlled system consistsof components that transform changes in bloodpressure rto changes in peripheral resistance (i.e.,barorecep-,or afferent signals, CNS processing,efferent signals, effect of autonomies onvasculature resistance). A typical blood pressureload disturbance might be an infusion of blood into'he circulatory system (which implies that adescription of the volume-pressure relationships ofthe circulation must be included in the controlled

system) or the introduction of an upright tiltdisturbance (which implies that a description ofgravity effects on blood volume distribution mustbe included in the controlled system).

FORMULATION OF ACOMPUTER MODEL

It would be instructive to review the steps thatlead from conceptualization of a model to digitalcomputer implementation.

Model systems can be represented by some com-bination of boxes which represent the elements ofthe system. Such combinations of boxes are calledblock diagrams, which include logic diagrams,schematics, flow diagrams, and system diagrams.The first step in constructing a model of anysystem is to draw a block diagram representingthese interconnections. This type of diagram, orseries of block diagrams, becomes a "roadmap" or a"key" for taking a system apart and putting it backtogether. Some of the most common basic symbolsused in a special form of block diagram, called ananalog diagram, are shown in figure A-2. The

io

*'r-=VTTn

r r— r

= o s

Add M I on!subtract I onTransfer function(eraob in I l

r x -*r=r=1+3,-2,

^21

Tmnsfar function- t (algabralc)

Multipllutlon

K K, , r= a,+v+3w)u r

Nultlpllcatlon by w uconstant facry K Transfer function

(multiple verlables)

, r •. r-= r

Division

' nor ,u It 1pl )cat l on)s urinat Ion

FIGURE A-2.—Symbols used In block diagrams representlnQmatherne lcal operations.

/

analog diagram is an adaptation of the diagram-matic method so useful with linear systems. Themathematical operations are self-explanatory. Asan example of the loosely termed " transfer func-tion," the graphical transfer function in the figuremight represent the relationship between cardiacoutput x and right atrial pressure y. This transferfunction can then represent a measured cardiacfunction curve.

As a specific example of the programing process,the subsystem of the circulatory, fluid, andelectrolyte model that deals with angiotensin for-mation has been ch o sen. Figure A-3(a) can be con-strued as a hypothesis diagram for the open-loopcontrol of renin -angiotensin. This diagram indi-cates the major fa„tors which control angiotensinformation and the man ) , effects of angiotensin onother subsystems. It is generally agreed that two Lfthe major influences on renin release (a precursorto angiotensin formation) from the juxtaglomeru-lar cells of the kidneys are renal perfusion pressure

and the load of sodium filtered through the tubules(i.e., glomerular filtration times plasma sodiumconcentration). Renin enters the circulation andforms angiotensin, which has a slight negative feed-back effect on renin formation. (See dashed line infig. A-3(a).) The final effect of angiotensin is con-sidered to be widespread, affecting vascular resist-ance, renal resistance, aldosterone secretion rate,thirst and drinking, and the rate of tubular fluidreabsorption.

To progress to a computer model of this system,each of the pathways connecting any two variablesmust be described in as much detail as is possible,or as much as is commensurate with the objectivesof the model design. Therefore, the next level ofdetail is shown in the block diagram of figureA-3(b), in which the re !ationships between varia-bles are qualitatively organized. Functional rela-tionships are identified wherever data describinghormonal secretion rates or hormonal effects areavailable. This diagram was developed from some

fVascular resistance

Renal arterial— Renal resistancepressure

filo ation rRenin ^ f concentration

IIf

secretefiltrios rateation secretn rate

[Na +] Plasma r H f Thirst and drinking

Tubular fluidreabsorption rate

(a)

FIGURE A-3 —Formulation of a control system algorithm. (a) Hypothesis diagram showing factors which influence renin-anglolen-sin production and the Quantities which they affect. These represent the assumptions for one of the hormonal subsystems In theGuy-ton model. (b) A more detailed block diagram of the hypothesis diagram of figure A-3(a) showing the mathematical operatorsrelating each Quantity. (c) An analog computer diagram of the renin-angiotensin system. This diagram contains sufficient informa-tion !o wire s patch panel for programing an analog computer. (d) A Fortran algorithm for the renin -angiotensin control system as Itappears in the Guytca model. The symbols can be Interpreted from the preceding diagrams.

4

very p neral algebraic relationships, and the mathe-matical operations of summation, division,multiplication, and integration are indicated.

Figure A-3(c) is the same diagram, reorganizedto reflect the computer program names for eachvariable and the quantitative transfer functions foreach element. It was common at one time to simt,late models on analog computers, and figure A-3(L)is known as an analog diagram because it representsa circuit diagram for programing this segment ofthe model on an analog computer. However, adigital computer requires a different language, andfigure A-3(d) is the Fortran version of the renin-angiotensin subsys. ,m. Note that the first state-ment represents a function relating renal pressureand renin secretion. In the analog computer, thisfunction would be represented by a functiongenerator; in the digital computer, it is representedby a table of x-y values (not shown) from which thedesired operating points are interpolated. The expo-nential functions (e.g., EXP( — I/RNK)) in figureA-3(d) are numerical approximations for integra-tion, in which " I" is the integration step size.

It can be appreciated that the discipline of

translating an ordinary physiological hypothesis

diagram into a quantitative computer representa-tion can lead to a better understanding of thebiological system. Available data from diverserrources are integrated into a common framework;other data are excluded as being nonessential forthe given level of detail (i.e., importance of data canbe ranked), arad missing information, which sug-gests the need for new experiments, is quickly iden-tified.

SIMULATION TECHNIOUES

Once a model is implemented on a computerand verification procedures ensure that it is operat-ing appropriately, the model is ready to be testedfor accuracy. A model that is deemed credible canalso be used to describe the behavior of the systemin terms familiar to control engineers and can beused to predict system responses in terms that canbe tested by biological experimentation. Some ofthe more important techniques used to producemodel cret'ibility and to predict system behaviorare discussed in the following paragraphs.

Renal Ne.sureRenin effect ouannul

enmPresslte

secret10n

Effects ofPAR

Nabo:: stn

tulw tar Re ninon:

110. slip- destruction PO S"o Va scll l ar

Plasma press "t late vo1wre 1 resistant(

[Na + ] Renin effect on Renal 'yrote"s Annlo effectren,n s[cr tile" plawla colic

Renalsecretion ♦^ rate ^ — —

resntlnuTotal

_

Gbmerular GFR + Na • Imtl11 p la 11na A.,9, conee1,

1. ltrali pn plasma I Ilal,m^'otxlxnl

secle,secre!lon

rate rent" f piasn4l f

Total allylotellsm Thirst andplasma R en l" do nk InyIenln

p4v snci7­_1PI111na

colicTubular f6d

vo h,lme Auyloiens'n — reabsorptlonRenin colic

secretion

rate

Anlylolellsldestnluloa

rate

Negativefeedback

effect of

anglolerlsm AIly^otelsm

Time Lenten Ual ion "1111", coat

del,y

fill Mglo Co.

FIGURE A-3.—•Conttnued.

^ i fir` -- -

(C ) 142 0.15 360 0.15

FIGURE A-3—Contiowd.

i

Dynamic Similation

In the context of this study, obtaining the solu-tion to a model means introducing some type ofload disturbance (i.e., a parameter perturbation)and solving the model's differential equationsiteratively, using numerical techniques. This proc-ess, known as dynamic simulation, is accomplishedusing high-speed digital computers and results intime-varying values of the dependent variables.These responses are examined in qualitative termsfor their reasonableness by analysts who arefamiliar with the physiological system or, moreoften, they are compared quantitatively with ex-perimental data. In this project, simulationresponses were available from digital computers intabular and graphical form and could be comparedwith previously stored data.

.-^.—.-

CALL FUNCTN(PAR, RNS, FUN8)

RSR = 300.*REK*RNS*(1.-ANGS-V(GFR*CNA-17.75))

IF (RSR.LT.0.)) RSR = 0.0

RT = RT * (RSR- QRA*RT)*(1.-EXP(-I/RRC))

RC = RT/VP

AN1 = F<8IV+1 A *RC

AN2 = AN2+(AN1-CAA*AN2)*(1 .-EXP(-I/AA))

ANC = AN2/VP/ANCN

ANM = ANMM -AU—EXP(-ANC/ANT )

IF (ANM. LT.0.5) ANM = 0.5

ANSS = 6*(ANC-1 .)

ANGS = ANGS+tANSS-ANGS)*I/ANGT

ANK = ANM

IF (ANK. LT. 1 .;ANK = 1.

ANU = ANM

IF (ANU.LT.0.6) ANU = 0.6(d)

FIGURE A-3.-concluded.

6

All

$ensltivity Analysis

Sensitivity analysis is a rnethod for studyingsystem responses due to ,ariation in parameters(ref. A-3). The conceptual basis of sensitivityanalysis is simple; small variations are made in thevalues of the system parameters of a model, andthe effects of these changes are observed in-dividually in 'i_e solution. (See app. E.) Sensitivityfunctions which describe the observed effects maybe computed, and these are interpreted to extractinformation about the dynamic system that couldnot be obtained from simply finding solutions witha particular set of input conditions (refs. A4 andA-5). Sensitivity analysis provides the following.

1. A quantitative means of comparing the rela-tive importance of individual parameters on anysystem variable

2. A means of determining interactive effects oftwo or more parameters on model behavior

3. A tool to help assess the validity of a particu-lar model without the need to collect and use exten-sive measurements from the real system

4. Information in a form that can be easily in-terpreted by those familiar with the subjec, matterof the model but not necessarily knowledgeable ofsimulation tecFniques

5. A means of assigning relative importance toall parameters, a process that can be valuable bothto the simulater in performing parameter estima-tion or stability analysis and to the experimenter inallocating resources for data collection

6. A practical method of analyzing and compar-ing two different models designed to represent thesarne physical system

A sensitivity analysis is very useful when per-formed early in model development, before modelvalidation. This analysis is parricuiarly importantto the experimenter, who can help evaluate themodel based on the relative sensitivities of theparameters without realty knowing much about themodel. The technique also is useful for involvingthe experimenter early in the modeling ptccess,another important factor for event ial model ac-ceptance (ref. A-6). The -estdts of the sensitivityanalysis must be evaluated in the light of otherknown information about the real system. The factthat a parameter has a very significant effect on aparticular variable of the model is of little impor-tancL if it is known that the parameter in the real

system is relatively constant or !sat changes inother parameters are capable of canceling the origi-nal effects.

Although sensitivity analysis can be consideredto be a special case of dynamic simulation, there areseveral important differences between these twoprocedures.

1. Sensitivity analyses are characterized by com-paratively smf.11 perturbations.

2 Sensitivity analyses often are performed byvarying one parameter at a time.

3. Sensitivity analyses often are performed toobtain sensitivity functions rather than solutions ofthe dependent variable

4. Sensitivity analyses usually entail com-parisons between twc, or more simulation runsrath :r than between model results and experimen-tal data.Examples of sensitivity analysis are provided in thedescription.: of the erythropoietic model and of thethermoregulatt.ry model. (See app. E.)

Variation of Parameters

Once a parameter has been identified as particu-larly influential, either by sensitivity analysis orfrom direct knowledge of the real system, it is oftendesirable to determine its effect on differentparameter values. For example, the volume ofblood is known to he important in the bloodflow/pressure response to upright tilt from thesupine position. It is reasonable to ask, "How doesthe response change as more and more blood isremoved?" Another way to pose this question is,"What is the effect of hemorrhage on standing?"Documenting this effect with a simulation modelof circulatory control is relatively straightforward;the parameter representing blood volume isassigned a series of values (i.e., a percentage of thecontrol or normal value) and, at each level, adynamic simulation is performed. The resultingtime-varving responses (of, for example, heart rate,cardiac output, or blood pressure) can be plotted asoverlays on the same graph. if steady-stateresponses are desired, the graph often is con-structed with blood volume on the absc`;sa and theresponse variable on the ordinate. vee Sec. V,"Cardiovascular Subsystem.")

Error Analysis

Parameter values are never known with 100 per-cent accuracy. If the standard deviation (SD)around the mean value can be estimated for eachparameter, it is possib l e to place statistical confi-dence limits on a model's behavior. For the exam-ple discussed previously, assume an experiment isperformed in which blood volume reduction ismeasured as — 10 percent ± 1.5 percent (SD).Dynamic simulations can be performed for threevalues of blood volume: — 8.5 percent, — 10 per-cent, and — 11 . 5 percent, corresponding to themean minus SD, the mean, and the mean plus SD,respectively. The response, for example, for heartrate could also be expressed in terms of a mean anda deviation. This expression would represent a pre-diction of the minimum error interval in theresponse variable, because of the inherent design ofthe system and the uncertainty in measuring bloc:dvolume, but would not includz- any experimentalerrors that could occur in measuring the responsevariables. Conversely, if a decrease in the confi-dence interval of a response variable to a givenwidth was desired, it would be possible, using thesetechniques, to determine the minimum experimen-ta', accuracy required in measuring the independentparameter.

Error analysis becomes especially desirable inlarge -scale systems containing many parametersthat promulgate errors through the simulation, andin certain nonlinear systems in which the interac-tive effects of different parameters lead toamplification of individual errors. Unfortunately,even though the tahniques to accomplish thisanalysis are rather straightforward, there are fewexamples in the literature of physiological systems.

A related problem that has application to errorand sensitivity analysis is the effect of noise on :hebehavior of the system. Noise can be described as astatistical disturbance of a particular variable, and itis characterized by statistical properties such asmean value, probability distribution, or spectraldensity. Model output variation which resu'ts fromnoise can be found by including a distribution func-tion for each parameter or variable that exhibitsnoisy behavior. This problem becomes extremelyrelevant in parameter estimation analysis whenmodel output is compared to data having a signifi-cant noise level (ref. A-7).

Stability Analysis

It is appropriate to mention stability analysis ofdynamic systems because of the inverse relation-ship between sensitivity and stability in negativefeedback systems. In general, sensitivity to Disturb-ing factors can be jeduced by an increase in feed-back gain (in technological systems at least)However, instability occurs as a consequence ofthis gain increase. Thus, systems with high gainmay have low sensitivity to external perturbationsbut may also be operating on the borderline of in-stability. Most biological systems ere normally sta-ble, and they do exhibit low sensitivity. Whetherthey are working somewhere near the stability limitby way of high gain factors is not kno vn but shouldbe studied on a case-by-case basis (ref. A-8). Ananalysis of stability can become an iri,portantmeasure of the competence of a model, in that ifboth model and actual system can be thrown intoinstability by the same parametric changes, there isreason for having greater confidence in the mathe-matical representation.

Little practical work has been done on stabilityanalysis of complex physiological systems. Formaltechniques for investigating stability in linearsystems and in simple nonlinear systems have beenreported (refs. A-7 and A-9); but, for the most part,studying large-scale nonlinear models is a trial-and-error experience. Systems that exhibit oscillatory orperiodic behavior in normal operation (e.g., eyemovement, respiration) can often be made unsta-ble more easily than systems that behavemonotonically. Inherent instability is dependent onthe properties of the system and is normally not afunction of the specific disturbance. if the system isinherently stable, all transients will ultimately dis-appear regardless of the disturbance causing them.Conversely, any disturbance to an unstable systemwill initiate oscillations that increase in amplitudewith time.

Stability can arise from either inherent featuresof the real system or from structural features of t;iemathematical model (such as long integration in-terval). The techniques of sensitivity analysis canreveal both types, although it is not always possibleto distinguish between the two. Like sensitivity,stability is a function of the operating point;therefore, all possible operating points must betested for stability. A careful, systematic sensitivity

, D

t

y

w +t^

•

analysis may ofter, reveal not only points of in-stability out their causes as weil.

model variable, and r is time. The error criterion Eis a function of e, usually I e or e,, integrated over aspecified time interval; e.g.,

Paramteter Estimation

The object of parameter estimation (or iden-tification) analysis is to determine the value of nneor more parameters in a model. The parametersselected for estimation are usually impossible ordifficult to measure directly in the real system Inpractice, the technique involves repetitive adjust-ment of the parameter values until some objectivejudgment of acceptable correlation between modeloutput and corresponding measurements in thesystem prototype has been satisfied (See fig. A4.)Because the automatic optimization of parametershas been the object of considerable attention (ref.A-10), there is a large body of literature onparameter estimation in physiological systems andalgorithms.

The error criterion -aced in parameter estimationis usually a difference function of the form

eft) = y(r) - y'(r)

where y' is a dependent variable that has beenmeasured in the real system, y is the corresponding

'rE-

J I e I dr

0

Since y is dependent on the system parameters q,, ecan be expressed as e(f) — e(r, q l , qZ , ..., q,) Thecriterion for the best fit between data and model isachieved when E reaches a minimum value.

A more powerful use of sensitivity analysis, butused infrequently, is the determination ofparameters that could be t stimated most accuratelyby means of the curve-fitting procedure discussedpreviously. Parameter estimation is used mosteffectively on parameters exerting a strong in-fluence on a particular model variable which can beeasily measured in the real system. If sensitivityanalysis is used before parameter estimation, it ispossible to select those parameters with the highestsensitivity coefficients as the best candidates forparameter estimation analysis. When theparameter sensitivity is low, then that parametervalue -annot be estimated with certainty using thatcriterion. Tate low-sensitivity parameter should beset at a reasonable constant value determined fromother sources.

Adjust values ofsystem parameters

Model of

biological Dependent variable

Control system

Comparison:"Best fit" solutionexperimental data vs

sim,ilation response

Initial It

conditions imulationre. ponce I L. 0 0 * 0 0

1 ti

Experimentalresponse

FIGURE A4 —Simulation Procedure for parameter estimation. 7LIs metDad or fitting model output to experimental results can pro-

duct values for system parameters that are dimrull to measure directly.

9

If the problem of sensitivity analysis is ex-pressed as determining the behavior of a modelgiven all the parameter variations, then the inverseproblem would be to determine (or identify) theparameter variations capable of p—ducing a givenbehavior of the real system. Unlike sensitivityanalysis, variation of parameters, error analysis,and stability analysis, parameter estimation, re-quires data measurements from the real system.This inverse problem may not have a unique solu-tion. Nevertheless, it would be valuable to knowthe various solutions possible, since this knowledgewould be a great aid in hypothesis testing. If severaldifferent parameter perturbations could producesimilar model results, it might be possible to acceptthe most reasonable, based on physiologicalplausibility, alternatively, this information couldprovide the basis for further experimental testing.

MODEL VALIDATION

The validation process is primarily concernedwith demonstrating the accuracy and the capabilityof simulation modeis. Two general criteria must be,net before model credibility can be established.First, a quantitative variable or parameter validitycriterion must be met, and, second, a qualitative"plausibility" criterion must be met. The first con-dition refers to tests in which model output is com-pared directly to experimental data, whereas thesecond condition includes all other tests in whichonly the general behavior of the model is examinedon a more subjective basis In the first case, a highdegree of fidelity in model response is expected,whereas in the second, the model responses needonly be "reasonable." No validation procedure isappropriate for all models Rather, validation de-pends on the nature of the model and the goals andob;ectives of the modeling study.

Quantitative Tests

An important aspect of validation involves com-paring the behavior of the model's dependent varia-bles with that of their expel imental counterpartsfor the same stress. Differences between modelbehavior and experimental data can often he cor-rected or minimized either by introducing new, pre-viously omitted elements into the model's structureor Dy modifying the existing structure (i.e., adjust-

ing the value of parameters that are not wellknown).

The t:xtent to which the validation process canbe carried is often limited by data availability.Thus, if only steady-state data are available, valida-tion in the dynamic, or trans i ent, mode cannot beperformed, even thwigh the model has thatcapability. Similarly, if only a relatively small num-ber of experimental variables have been measuredduring a particular stress, then it is possible to vali-date ► t-%; model for the responses of only thosemeasured variables; simulated %, slues of all othervariables c kn be obtainer) but should be consideredas predictions requiring experimental verification.The response of the body to a given stress is almostalways related to the level or intensity of thatstress, and, more often than not, this relationship isnonlinear. Therefore, to validate the modelproperly, it is desirable to obtain data not merelyfor a given stress but for a rang!: of intensities ofthat stress. If it is important to simulate more thanone type of stress, the validation process will resultin a more accurate model if the experimentalresponse of the same variables is known for each ofthe desired stresses. Thus, an idealized set of ex-perimental data suitable for complete validation ofa complex model should include the following.

1. Steady-state data2. Transient data3. Data for a wide range of stress intensities4. Data for all major dependent variables of in-

terest or importance5. Data for a variety of stresses

In addition, since experim^ntal protocol, measure-ment techniques, and number and type of subjectsmay vary widely from one investigation to another,ever, when studying the same stress, it is desirableto obtain many of these da!a from the same experi-mental study.

The data used to validate the model should meetseveral conditions.

1. The data used in the verification process (i.e.,establishing the correctness of cc - , ter codingand ensuring that the model runs as , tended) arethe same as those used in the development of themodel. However, to ascertain model validity, datathat were not originally included in the model's for-mulation must b- used The model must be capableof predicting beyond the data from which themodel was generated.

2. The data must be of sufficient precision tomake the test meaningful The data should cover

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the range of interest and should have a minimumlevel of noise The latter condition is best assuredby including data representing a large subjectpopulation.

3. The objectives of the modeling studies shouldbe kept in perspective Not all data are appropriatefor use in validation The assumptions about thebiological 5vstem being studied are often reflectedin the experimental protocol used to gather thedata, and these - re often not the same assumptionsused in the mm(',:i.

Obtaining good agreemen , between a simulationresponse and data is especiall y important when try-ing to simulate a diverse number of variables hav-ing constantly changing values. Agreement be-tween simulation response and data is oftenimproved by adjusting parameter values. This pro-cedure is the same as that described underparameter estimation techniques. More than = ecombination of parameter values may lead to 9

good "Fit." In these casts, it is important that allchanges be reasonable and consistent with knownphysiological processes.

Qualltstive Tests

I he purpose of modeling and simulation, in thecontext of the current study is not necessarily toproduce an optimal fit betv.ren experimental dataand model output, although this result would netbe undesirable. Rather, the objectives are to helpunderstand the behavior and interactions of thesystem and its components and to assist in new ex-perimental approaches Model credibi:ity, in thiscase, can also be established by perforating fewerquantitative tests.

Often, the model analyst i's content to verify in-itially :hat the "shape" of the data and model out-put agree. This situation would occur if the :modelerwere primarily interested in the validation ofmodel dynamics as contrasted to the exact fit ofmodel output and data In this case, it is not con-sidered c:itical !hat the absolute magnitude of theresponse is in error; this type of discrepancy canoften be remedied by the adiustment of a systemparameter.

An important criterion for indicating whether amodel is good enough to be used for forming con-clusions about the real system is that a one-tinecorrespondence and similarity of form must existbetween model and real system So-called "black-

box models" (i.e., models which represent overallbehavior of systems without representing their un-derlying mechanisms) are not good models formaking predictions regarding general systembehavior, at best, they may be used as descriptionsof data. Thus, there is always some -lode[ that canfit a particular set of experimental data (usually byadjusting one cr more parameters), but only thebiological plausibility of a particular model justifiespreferring it to all others. Therefore, the inclusionof a greater number of adjustable parameters in amodel, although perhaps pro v iding a better agree-ment wit[ the data, does not necessarily add insightinto the physiological mechanisms.

The techniques Rio.- cribed earlier in this appen-dix---sensitivity analysis, error analysis, stabilityanalysis, and variation of parameters—do not re-quire extensive data sets. They can be very usefulin establishing the plausibility of a model withoutnecessitating excessive analysis of the mathematicsof the model and all the explicit and implicitassumptions. Sens i tivity analysis, in particular, canbe important in this regard by quickly andsystematically analyzing these component relation-ships witi;out the need of actua l subject data. Thiscapability is particularly usJul in comparing twodifferent models.

For a model tO contribute to a particular in-vestigative field, it must ultimately be judged 5vscientists farnili::r wi g ' Jim area. When T hese sJen-Lists are not the people who developed and vali-dated the model, it is important that lines of cotn-mimication between them be estab l ished as earl y inthe modeiirg process as possthle. Model validation,ideally, should be an interdisciplinary process Dur-ing this project, the experience has been thatgeneral simulation-,s, sensitivity analyses, orparameter variation studies performed withoutcomparison to good ex; erimental data appear tomake less of an impact on experimenters no!familial with systems analysis than the same worksupplemented with at leest a single simulationshowing reasonably good agreement with experi-mental data

REFERENCES

A-1 Riggs. D S.: Control The r) and Physiological FeedbackMechanisms Williams and Wilkins Co (Baltimore).1970

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/ 2. Milhorn, H. T., Jr.. The Application of Control Theory toPhysiological Systems. W. B. Saunders Co. (Philadel-phia),1966.

A-1 Leonard, 1 1.: The Application of Sensitivity Analysis toModels of Large Scale Physiological Systems (GeneralElectric Co., Houston, Tex , TIR 741-MED-4028, Con-tract NAS 9 . 12932 ) NASA CR-160228, 1974.

A4 Cruz, Jose' B., Jr. System Sersitivity Analysis.Benchmark Papers in Electrical Engineering and Com-puter Science Dowden, Hutchinson, and Ross(Stroudsburg, Pa.), 1973.

A-5.Tomovic, Rajko. and Vukobratovic. M. General Sen-sitivity Theory. American Elsevier (New York), 1972.

A-6. Miller, D. R. Model Validation Through SensitivityAnalysis Proceedings of the 1974 Summer ComputerSimulation Conference, Vol. I;, AFIPS Press (Montvale,N.J.), 1974, pp. 911.914.

A-7.Rosen, R. Dynamical System Theory in Biology, Vol 1Stability Theory and Its Applications Biomed Engineer-ing Series, Wiley-Interscience (New York), 1970

A-9 Jones. R W.: Principles of Biological Regulation An In-troduction to Feedback Systems Academic Press (NewYork), 1973.

A-9. Tomovic, Rajko (David Thornquist, transl ) Sensiu^,tyAnalysis of Dynamic Systems McGraw-Hill (NewYork), 1%3.

A-10 Johnson, L E Computers, Models. and Optimization inPhysiological Kinetic CRC Crit Rev Bioeng , vcl 2,Feb. 1974, pp. 1.37.

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