J. theor. Biol. (1995) 174, 341–353

0022–5193/95/110341+13 $08.00/0 7 1995 Academic Press Limited

Calculation of Organ and Whole-body Uptake and Production with the Impulse

Response Approach

A M†

Institute of Systems Science and Biomedical Engineering, National Research Council,Padova, Italy

(Received on 31 August 1994; Accepted on 3 November 1994)

A general theory for the calculation of uptake and production in the non-ready state in both a single organand in the whole body is proposed. The theory is based on an extension of Zierler’s impulse responseapproach for organ kinetics and on its application to whole-body kinetics through circulatory models.It describes the accessible (inlet-outlet) and non-accessible (inlet-uptake, production-outlet, andproduction-uptake) metabolic paths of an organ with four impulse responses. The mathematicalformalization gives the factors that must be known with sufficient accuracy to ensure a well-foundedcalculation of uptake and production, and provides a basis for the evaluation of the currently usedapproximate approaches. The theory shows that if uptake and production occur simultaneously inorgans—e.g. for amino acids—the major problem, which cannot easily be overcome, is the lack ofknowledge of the non-accessible impulse responses. If uptake and production do not occur simultaneously(e.g. for glucose), substitutes of uptake and production (denoted as inlet-equivalent uptake andoutlet-equivalent production) can be calculated from the knowledge of the inlet-outlet impulse responseonly. Qualitatively, uptake is delayed inlet-equivalent uptake and outlet-equivalent production is delayedproduction. The delays are the mean transit times of the inlet-uptake and production-outlet paths.Inlet-equivalent uptake and outlet-equivalent production can be calculated without hypotheses on theorgan (or whole-body) total mass, which is known to be indeterminable with classical tracer experiments.If independent information on the mass is available, a rigorous qualitative prediction of the fluxes canbe attempted.Forwhole-bodykinetics, this approachhas better foundations than compartmental analysisthat necessarily requires assumptions on the total mass. Whole-body glucose kinetics in a specificnon-steady-state condition are discussed as a relevant application of the theory.

Introduction

The calculation of uptake and production with tracers,for both an isolated organ and the whole body, is anintriguing problem—as shown by a recent debate onthe subject (Katz, 1992; Sacca et al., 1992; Norwich,1992). Surprisingly, no general theory has beenproposed that could, if not solve, at least focus on theproblem appropriately. In this work, the impulseresponse approach developed by Zierler (Meier &Zierler, 1954; Zierler, 1961), with appropriate

extensions, is proposed as a suitable theory. Thistheory, originally developed to study organ kinetics,applies also to whole-body kinetics if coupled with acirculatory model of the body, and thus provides anunifying approach for both the problems. The theoryis valid in both the steady and the non-steady state, andpart of it has been already developed for thesteady-state case (Mari, 1993).

The impulse response approach, with the greatadvantage of being dependent on few assumptions,provides rigorousmethods for the studyof the problemof uptake and production in the general case. Themathematical formalization elucidates the factors thatmust be known with sufficient accuracy to ensure awell-founded solution to the problem. The validity of

† Address correspondence to: Andrea Mari, LADSEB-CNR, corso Stati Uniti 4, 35127 Padova, Italy. E-mail:mari.ladseb.pd.cnr.it

341

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the approximate solutions that have been proposedcan thus be evaluated with the aid of a model-independent theory.

When uptake and production occur simultaneouslyin organs (e.g. for amino acids), the major problem,which cannot easily be overcome, is the lack ofknowledge of the system under study. When uptakeand production do not occur simultaneously to asignificant extent (e.g. for glucose), the currently usedsolutions to the problem are appropriate in thesteady-state but not in the non-steady state. Inparticular, compartmental models of whole-bodykinetics are not appropriate for calculating uptake inthe non-steady state because they are not adequate forquantifying the total whole-body mass (Mari, 1993).The theory presented here shows that in this situationonly partial information onuptake andproduction canbe obtained with classical tracer experiments. In somecases uptake and production can be qualitativelypredicted using independent experimental infor-mation. Glucose kinetics are an interesting applicationof these results and are discussed in detail.

Organ Kinetics

The methods for studying organ kinetics in thegeneral non-steady-state case and in presence ofsubstrate uptake andproduction are based on theworkof Zierler (Meier & Zierler, 1954; Zierler, 1961).Zierler’s impulse response method is based on a linearrepresentation of the system, that is possible eitherbecause the system is intrinsically linear or becausetracers are used to linearize the system in a particularexperiment. Linearity of the kinetics of the mothersubstance (the tracee) is thus not required.

If uptake and production occur simultaneously in anorgan, four impulse responses are necessary to describe

the kinetics of the four metabolic paths (Fig. 1). In thenon-steady state, these impulse responses are functionsof two variables because the system is non-stationary.The relationships between influx [8in(t)], outflux[8out(t)], uptake [8upt(t)] and production [8prod(t)] aregiven by the integral equations

8out(t)=gt

−a

r(t, x)8in(x) dx+

gt

−a

r((t, x)8prod(x) dx (1)

8upt(t)=gt

−a

r(t, x)8in(x) dx+

gt

−a

r((t, x)8prod(x) dx, (2)

where r(t, x), r(t, x), r((t, x), and r((t, x) are theinlet-outlet, inlet-uptake, production-outlet and pro-duction-uptake impulse responses, respectively. Eachintegral in eqns (1) and (2) is a separate flux termoriginating from a different source (the inlet or theproduction sites). Equations (1) and (2) are valid fortracer and tracee, although for the tracer 8prod(t)00since the tracer is not produced in the organ. Causalityimplies that r(t, x)=0 for tQx, and similarly for theother responses. Equations (1) and (2) relate materialfluxes rather than concentrations because for 8upt(t)and 8prod(t) concentration is not a meaningful variable.For influx and outflux, material fluxes and concen-trations are related by the equation of convective flux,e.g. 8in(t)=F(t)Cin(t), where F(t) is blood flow andCin(t) inlet concentration.

The inlet-outlet impulse response is determinable ina model-independent way in the steady state. If a tracerbolus is administered at the inlet, and the organ flowis known, r(t) can be simply calculated from themeasured outlet concentration. In the non-steadystate, with a bolus injected at time t0 only r(t, t0), withfixed t0, can be calculated from the measured outletconcentration. Model-independent determination ofr(t, x) would require an ideal experiment in which Ndistinguishable tracers of the same substance areinjected at the inlet at closely spaced time instants,t0, t1, . . . tN−1, and outlet tracer concentration and flowrecorded. With this experiment, r(t, x) is obtained ina discretized form: r(t, t0), r(t, t1), . . . r(t, tN−1). Theimaginary character of this approach clarifies that thedetermination of r(t, t0) for all t0 is practicallyimpossible without a model. The

F. 1. The four metabolic paths of an organ. Symbols areaccording to Appendix E.

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situation isworse for the other three impulse responses,because even the experiment above does not conveydirect information on r(t, x), r((t, x), and r((t, x).These impulse responses are determinable onlyresorting to a physical model of the organ because theuptake and production sites are in general spatiallydistributed and not available to sampling.

The foregoing considerations emphasize the funda-mental difference that exists between the determinationof the impulse response of the accessible path (r(t, x))and those of the non-accessible paths (r(t, x), r((t, x),and r((t, x)). The difficulties of the determination ofr(t, x) are due to the limited amount of informationthat can be obtained in practice from a tracerexperiment. With more complex tracer experiments,but still with inlet-outlet sampling, r(t, x) can beestimated more accurately, as in the ideal tracerexperiment outlined above. However, inlet-outlettracer experiments, of any complexity, are totallyinsufficient for a model-independent determination ofr(t, x), r((t, x), and r((t, x), even in the steady state.

In the non-steady state, the transit time densityfunction and the kinetic parameters (transmission,extraction, and mean transit time) require a newdefinition and their properties are different from thesteady state. The existence of four impulse responsesmust also be considered.

The parameter j(t), defined by the equation

j(t)=g+a

t

r(x, t) dx (3)

is called inlet-outlet transmission. It is the fraction ofthe unit dose injected at time t at the inlet recovered atthe outlet from t onward. The inlet-outlet extraction,h(t), is defined as in the stationary case by the equation

h(t)=1−j(t) (4)

The law of conservation of mass implies that

h(t)=g+a

t

r(x, t) dx (5)

Transmission and extraction are not constant intime in the non-steady state. The outflux to a constantinflux is also not constant in time, and outflux is notthe product of influx and transmission, as defined byeqn (3). Analogous considerations hold for extractionand uptake.

The inlet-outlet transit time density function isdefined as

p(t, x)=r(t, x)j−1(x), (6)

and has the property that

g+a

x

p(t, x) dt=1 (7)

for all x, as in the steady-state case. In the non-steadystate, the transit time density function p(t, x) is afunction of not only the time elapsed after the injection,t−x, but also of the time of injection, x. The meaninlet-outlet transit time is defined as

t(x)=g+a

0

tp(t+x, x) dt, (8)

and also depends on the time of injection, x. Notethat eqn (8) becomes the classical equation forthe mean transit time in the steady-state, sincep(t, x)=p(t−x).

In the non-steady state, the mean inlet-outlet transittime can be qualified as a delay between influx andoutflux, as in the steady state. In the steady state, thisidea is expressed by the property that the center ofgravity of the outflux–time curve is the center of gravityof influx–time curve shifted by the mean transit time(Lassen & Perl, 1979). In the non-steady state, thisprinciple is expressed by the equation (see AppendixA).

g+a

0

t8out(t) dt

g+a

0

8out(t) dt

=g

+a

0

tj(t)8in(t) dt

g+a

0

j(t)8in(t) dt

+g

+a

0

t(t)j(t)8in(t) dt

g+a

0

j(t)8in(t) dt

, (9)

where the left-hand side is the center of gravity of theoutflux, the first term of the right-hand side the centerof gravity of the product of transmission and influx,and the mean transit time t(t) appears weighted insidethe integral of second term of the sum. Equation (9)requires that fluxes are zero before time zero, and thatall integrals exist and are finite. If t(t)=t, a constant,eqn (9) states that the center of gravity of 8out(t) is thesum of the center of gravity of j(t)8in(t) and the meantransit time t. Note that the center of gravity ofj(t)8in(t) is not the center of gravity of 8in(t), unlessj(t) is constant. If the mean transit time is not constantbut is bounded from below and above, the time shift

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of the center of gravity is a quantity in betweenthese bounds. In summary, eqn (9) supports also inthe non-steady state the qualitative idea that 8out(t)is delayed with respect to j(t)8in(t). Another way torepresent this principle is given by Result 3 ofAppendix A.

The definitions and equations developed abovefor the inlet-outlet impulse response apply to theimpulse responses of the non-accessible paths in asimilar way. For the production-outlet and pro-duction-uptake paths described by the impulseresponses r((t, x) and r((t, x), a production-outlettransmission, j((t), and extraction, h((t), aredefined with equations analogous to eqns (3) and(4). For the inlet-outlet and inlet-uptake paths,conservation of mass implies that

h((t)=g+a

t

r((x, t) dx. (10)

Equations (3–5) and (10) define two independentextraction parameters for the four organ paths, i.e.h(t) and h((t). Equations (6) and (8) define fourindependent transit time density functions and meantransit times.

The kinetic parameters can be determined frominlet-outlet experiments under the same conditionsof the impulse responses. This means for instancethat, even in the steady state, the production-outlettransmission can be determined only resorting toa physical model of the system. By no meansone can for instance assume that h equals h(:hcan be either greater or smaller than h(, dependingon the physico-chemical characteristics of thetissues. Unfortunately, one cannot infer h( fromh without considering the fine details of theorgan structure. This basic indeterminacy is themajor obstacle in determinating uptake andproduction, both in the steady and non-steadystate.

The essential indeterminacy of r(t), r((t), andr((t), makes calculation of uptake and productionimpossible in the general case when only inlet andoutlet are sampled. However, calculable substitutesfor uptake and production can be defined thatretain all information on uptake and productioncontained in inlet-outlet experiments. These substi-tutes are denoted as inlet-equivalent uptake [8 upt(t)]and outlet-equivalent production [8 prod(t)], and aredefined by the equations

8 upt(t)=h(t)8in(t) (11)

8 prod(t)=8out(t)−gt

−a

r(t, x)8in(x) dx (12)

These fluxes are related to uptake and productionby the equations [see eqns (1–2) (11–12) and thedefinition of the kinetic parameters)]:

8upt(t)=gt

−a

p(t, x)8 upt(x) dx+

gt

−a

p((t, x)h((x)8prod(x) dx (13)

gt

−a

p((t, x)j((x)8prod(x) dx=8 prod(t). (14)

Equations (13) and (14) indicate that the lackof knowledge of the impulse responses of thenon-accessible paths prevents the calculation ofuptake and production from inlet-equivalent uptakeand outlet-equivalent production. In otherwords, once inlet-equivalent uptake and outlet-equivalent production are calculated, any calcu-lation of uptake and production requires infor-mation on the impulse responses of the non-accessi-ble paths.

Equations (13) and (14) show that 8 upt(t)=8upt(t)if and only if p(t, x)=d(t−x) and h((t)=0,and 8 prod(t)=8prod(t) if and only if h((t)=0and p((t, x)=d(t−x). In words, inlet-equivalentuptake is real uptake if and only if there is nouptake of newly produced substance (h((t)=0)and uptake occurs at the inlet (p(t, x)=d(t−x)).Outlet-equivalent production is real production ifand only if there is no uptake of newly producedsubstance and production occurs at the outlet(p((t, x)=d(t−x)).

In the general case 8 upt(t) and 8 prod(t) differ fromuptake and production for two distinct reasons. Thefirst is that these fluxes do not account for theportion of the production flux that is taken upbefore reaching the outlet. This problem is presentboth in the steady and non-steady state. The secondis that delays exist between inlet-equivalent uptakeand uptake (or between production and outlet-equivalent production) in the non-steady state.When production and uptake occur simultaneously,the major deficiency of eqns (11) and (12) both inthe steady and in the non-steady state, is that theymiss a part of production and uptake. However,even when there is no uptake of newly producedsubstance (r((t, x)00), in the non-steady stateeqns (11) and (12) cannot reconstruct the time-

345

course of production and uptake, nor when r(t, x) isprecisely known, since p(t, x) and p((t, x) remainunknown. In other words, in the non-steady state, notonly the fraction of produced substance taken upbefore reaching the outlet (h() is undetermined, butalso the delays from inlet to uptake and fromproduction to outlet (t� and t().

However, inlet-equivalent uptake and outlet-equivalent production are clearly related to uptakeand production, at least qualitatively. Uptake isinlet-equivalent uptake delayed [in the sense of eqn (9)]by a quantity related to the mean inlet-uptaketransit time, plus the flux of uptake from theproduction sites. Hence, the flux of uptake from theinflowing substance, i.e. total uptake minus uptakefrom the production sites, equals delayed inlet-equival-ent uptake. If the flux of uptake from the productionsites is a small fraction of uptake, the major differencebetween uptake and inlet-equivalent uptake is thedelay. Outlet-equivalent production is the fractionj((t) of production, delayed [in the sense of eqn (9)]by a quantity related to the mean production-outlettransit time. Outlet-equivalent production is theflux of newly produced substance released in thevein. If h((t) is small, the major difference betweenproduction and outlet-equivalent production is thedelay.

The physiological significance of inlet-equivalentuptake and outlet-equivalent production depend onthe substance. For substances that are simultaneouslyproduced and taken up, such as amino acids, thesefluxes are of modest relevance principally because theyunderestimate uptake and production of an unknownamount. If uptake and production do not occursimultaneously to a significant extent, as for glucose,the major limitation is that there is an unknown delaybetween uptake and production and inlet-equivalentuptake and outlet-equivalent production in thenon-steady state.

Equations (11) and (12) can be used to calculateinlet-equivalent uptake and outlet-equivalent pro-duction, once r(t, x) is determined with the tracer.Since the determination of r(t, x) requires a model,the calculation of 8 upt(t) and 8 prod(t) is model-dependent in the non-steady state. A special situationoccurs when inlet specific activity is constant. In thiscase, the calculation of 8 prod(t) does not require r(t, x),and is thus also model-independent in the non-steadystate (see Appendix B). However, the calculation of8 upt(t) remains model-dependent even when specificactivity is constant, as it requires the knowledge ofr(t, x).

In the steady state, eqns (11) and (12) becomethe classical equations of arterio-venous differences

(see Appendix C), and eqns (13) and (14) become

8upt=8 upt+h(8prod=8 upt+h(

1−h( 8 prod (15)

8prod=(1−h()−18 prod. (16)

Equations (15) and (16) evidence that in the steadystate only if h( is known with sufficient accuracy onecan obtain a reliable estimate of uptake andproduction from inlet-outlet experiments. Note thatthe knowledge of the inlet-outlet extraction, h, does nothelp to solve eqns (15) and (16), since the relationshipbetween h and h( is strongly dependent on the organstructure. By no means one can assume h(=h. Indeed,as shown in Appendix C, this assumption is implicit inthe modified form of the classical equations ofarterio-venous differences that uses venous (in place ofarterial) specific activity for calculating tracee uptakefrom tracer uptake (see Wolfe, 1992, for the method).The assumption h(=h cannot be considered a goodapproximation in the general case.

In summary, from inlet-outlet experiments theinvestigator can, at most, calculate inlet-equivalentuptake and outlet-equivalent production both in thesteady and in the non-steady state. In the non-steadystate, the calculation of 8 upt(t) is model-dependent,while the determination of 8 prod(t) is model-indepen-dent if inlet specific activity is kept constant. Thedetermination of uptake and production requires aphysical model of the system. Without such a model,uptake and production are, in the general case, subjectto conjecture. When uptake and production do notoccur simultaneously to a significant extent, in thenon-steady state there exists a delay between 8 upt(t)and uptake and between production and 8 prod(t). Thisdelay, related to the mean inlet-uptake and pro-duction-outlet transit times, cannot be determinedfrom inlet-outlet experiments without a physicalmodel. If the transit times can be estimated withindependent experiments, a rigorous qualitativeprediction of uptake and production from inlet-equivalent uptake and outlet-equivalent production ispossible.

The most important difficulty in the practicalapplication of the theory developed above is thedetermination of p(t, x). In the non-steady state, thetransit time density is no longer a function of onevariable, as it is in the steady state, and cannot bedetermined without a model. However, the difficultiesof developing a suitable model depend on theexperimental conditions. The simpler situation is that

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in which the extraction changes over time during thenon-steady state, but the transit time density functionremains very similar to that observed in steady-stateconditions. In this case, the impulse response r(t, x)can be approximated with the function p(t−x)j(x),p(t) can be determined in the steady state, and j(t) canbe calculated with reasonable accuracy from inlet andoutlet concentration and blood flow with eqn (1). Atracer is obviously required in the steady state and ifproduction is present.

The error on h(t) associated with a given model ofp(t, x) can hardly be evaluated in practice because theerror on p(t, x) is unknown. Some information on theerror on h(t) can be however obtained from theequation (see Appendix D)

gt

−a

p(t, x)[h(x)−h(x)]8in(x) dx

=gt

−a

[p(t, x)−p(t, x)]j (x)8in(x) dx, (17)

where the hat denotes the functions of a given model,that are different from the true ones (p(t, x) and h(x)),and 8in(x) is typically tracer influx.

A first consequence of eqn (17) is that the integralfrom zero to infinity of (h(t)−h(t))8in(t), if it exists, iszero, i.e. on average h(t)=h(t) if 8 in(t) is constant(Appendix D). This property may be useful in somecircumstances, although a zero integral error does notimply that the error on h(t) is small.

Two rules can then be deduced from eqn (17)concerning the choice of the pattern of 8in(t) thatpotentially minimizes the error on h(t). The first is thatbrisk changes of 8in(t) may have a negative influenceon the calculation of extraction. Rapid changes of8in(t) may produce rapid changes in the r.h.s. ofeqn (17), and since h(t)−h(t) is the solution of adeconvolution-like problem, extraction is subject to alarge error in this instance. The second rule is that thereis no 8in(t) that makes the difference h(t)−h(t)identically zero when p(t, x)$p(t, x). Unlike the caseof outlet-equivalent production, in which it is ideallypossible by a suitable tracer infusion reduce the errorof any model to zero, this is not so for inlet-equivalentuptake. Furthermore, although different 8in(t) pat-terns lead to different extraction errors, a best choiceis impossible due to the lack of information onp(t, x)−p(t, x).

It is worthwhile to stress that empirical models canbe used to represent p(t, x). It is for this reason that thecalculation of inlet-equivalent uptake and outletequivalent production, although model dependent,can be considered a simpler problem than the

determination of uptake and production, whichrequire physical models.

Whole-body Kinetics

The theory developed above can be applied towhole-body kinetics when a circulatory model isadopted for describing the intact body. The leftventricular valve or the aortic arch can be taken as theinlet of all the body tissues except the heart chambersand lungs, and the right atrium as the outlet. The bodytissues except the heart chambers and lungs can thusbe regarded as a single-inlet single-outlet organ block(the periphery block). If arterial blood is sampled, as istypically done in metabolic studies, inlet concentrationis measured. The heart–lung system can also beregarded as a single-inlet single-outlet organ block (thelung block). For many substances, the heart–lungsystem has little metabolic relevance (the heart tissues,nourished by the coronary circulation, are included inthe periphery). In this case, the lung block does nottake up nor produce the substance, and is denoted asinactive.

The outlet concentration of the periphery block, i.e.a mixed-venous concentration (Cv(t)), is usually notmeasured but is related to arterial concentration, if thelung block is inactive, by the equation

F(t)Ca(t)=gt

−a

f(t, x)[8inf(x)+F(x)Cv(x)] dx,(18)

where Ca(t) is arterial concentration, F(t) is cardiacoutput, 8inf(t) is the exogenous infusion of substance atthe venous site (if present), and f(t, x) is the impulseresponse of the lung block. Equation (18) is valid fortracer and tracee and can be used, at least ideally, tocalculate Cv(t).

If Ca(t) is measured, Cv(t) measured or calculatedwith eqn (18), and the cardiac output F(t) is known,for the periphery block inlet and outlet concentrationand blood flow are known. In this case, whole-bodykinetics, i.e. the kinetics of the periphery block, can bedescribed with the same methods presented above. Theequation for the periphery block is [see eqns (1) and(12)]

F(t)Cv(t)=gt

−a

g(t, x)F(x)Ca(x) dx+8 prodg(t),(19)

where g(t, x) is the impulse response of the peripheryblock, and 8 prodg(t) is its outlet-equivalent production.Equation (19) is valid for tracer and tracee, with8 prodg(t)00 for the tracer.

The conceptual identity between whole-body and

347

organ kinetics demonstrates that the analysis oftracer curves in a classical whole-body kineticexperiment contains information only on the artery-vein impulse response of the body tissues. A physicalmodel would be necessary to determine the artery-uptake, production-vein, and production-uptakeimpulse responses from the artery-vein impulseresponse, but a physical whole-body model cannotbe realistically developed. Uptake and productionare thus not determinable in general. As in the organcase, the calculation of inlet-equivalent uptakeand outlet-equivalent production rather than uptakeand production is a way to circumvent the problem.For whole-body kinetics, these fluxes can be termedarterial-equivalent uptake and venous-equivalent pro-duction, and their relationships with uptake andproduction are those previously described. As forthe organ case, the physiological significance ofarterial-equivalent uptake and venous-equivalentproduction depends on the substance.

In the steady state, eqns (15–16) apply also towhole-body kinetics. As for a single organ,the knowledge of the inlet-outlet extraction of theperiphery block, which can be calculated from thewhole-body clearance and the cardiac output ifthe lung block is inactive, is not sufficient to determinethe production-outlet extraction. In particular,equality between the two extractions cannot beassumed.

The calculation of arterial-equivalent uptakeand venous-equivalent production requires only asufficiently accurate description of the transit timedensity function of the periphery block, and thepotential error is of the same type of that discussed fororgan kinetics. In the non-steady state, the calculationof venous-equivalent production becomes model-inde-pendent if arterial specific activity is kept constant,while arterial-equivalent uptake is always modeldependent.

In the non-steady state, eqns (18–19) cannot besolved with the transfer function approach intro-duced by Cutler (1979) for studying the steady-statecase. Other approaches can, however, be adopted,such as a description of the impulse responses withtime-varying state-space models. With some assump-tions that depend on the experimental configuration,it is possible to represent in a state-space form theintact-body impulse response that relates 8inf(t) tothe measured Ca(t), and to obtain from theobservations an estimate of the impulse response ofthe periphery block. As in the steady state (Mari,1993, 1995), circulatory models are not onlytheoretically relevant, but also suitable for theanalysis of real data.

Discussion

The advantage of the theory presented in this work,which is based on Zierler’s ideas (Meier & Zierler,1954; Zierler, 1961) and on their application towhole-body kinetics through circulatory models, isthat it requires only a few very basic assumptions. Theproblem of the calculation of uptake and productioncan thus be studied in the non-steady state with amodel-independent approach, and the same theoreti-cal framework is valid for both an isolated organ andthe whole body.

A basic theoretical result is that uptake andproduction cannot be calculated from experimentswith a single tracer and sampling in blood without aphysical model of the system. In fact, an attempt todetermine the non-accessible impulse responses fromthe inlet-outlet impulse response only has some chanceof success with a physical description of the system.Approximations may be acceptable, but empiricalapproaches that have no clear relationship with thephysical system may be misleading. Although thisresult is not new, the importance of appropriatemodeling seems not generally recognized, especiallyfor whole-body kinetics, as it is demonstrated by thepredominance in this field of compartmental modelsthat are far from being physical models.

The mathematical formalization of the core of theproblem, represented in the steady state by eqns (15)and (16), is a step beyond this negative result. Thetheory offers a rigorous framework for evaluating theadequacy of the approaches that are proposed asapproximate solutions to the problem. This isparticularly desirable when the distinction betweengood or bad approximations, rather than betweentheoretically right or wrong approaches, is a crucialaspect of the problem and one of the reasons of thedivergence of opinions (Katz, 1992; Sacca et al., 1992;Norwich, 1992). For instance, the modified arterio-venous difference method discussed in this work(Appendix C) has been either considered theoreticallyincorrect (Sacca et al., 1992) or regarded as areasonable approximation in various circumstances(Katz, 1992). This work demonstrates that this methodimplicitly assumes that the production-outlet extrac-tion (h() equals the inlet-outlet extraction (h). Theerror of the method thus depends on how closely h

approximates h(, and for a given error on h( the erroron uptake and production can be quantified witheqns (15) and (16). This way of looking into theproblem allows one to evaluate the influence that aspecific organ structure can have on the differencebetween h and h(. The adequacy of the assumptionh=h( can thus be evaluated in specific cases. For

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example, if the organ is regarded as a parallelarrangement of identical capillaries with the sites ofuptake andproduction uniformly distributed along thecapillary length, it is not difficult to demonstrate thath(Qh, and give a mathematical expression of therelationship between h( and h.

The mathematical formalization is also the basis fora rational guess of the value of the production-outletextraction in the steady state. The importance ofrational guesses for a problem not amenable to aclear-cut solution seems not to be properly recognized.The development of the modified arterio-venousdifference method discussed above and its whole-bodycounterpart (the so-called A-V mode, see Wolfe, 1992)has required a considerable effort, even if the essenceof thismethod is just the conjecture of equality betweeninlet-outlet and production-outlet extraction. Thisconjecture is simple but not rational, particularly forwhole-body kinetics: more appropriate guesses can bemade by considering the structure of the system understudy.

Accurate calculation of uptake and production isprevented when reliable expressions for the impulseresponses of the non-accessible paths (or of h( in thesteady-state case) cannot be obtained in practice. Thetheory presented in this work circumvents the problembydefining substitutes for uptake andproduction—de-noted as inlet-equivalent uptake and outlet-equivalentproduction—which extend to the non-steady state theclassical calculation of the arterio-venous differencemethod across an organ. This approach has theconceptual advantage that the calculations do notrequire any hypothesis on the organ structure, whichis desirable when there is insufficient quantitativeinformation on the non-accessible paths. Hypothesesbecome necessary when the investigator attempts topredict uptake and production from inlet-equivalentuptake and outlet-equivalent production. The calcu-lation of uptake and production is thus logicallydivided into two steps, so that the role of thehypotheses is clearly evident. This two-step calculationallows a rigorous qualitative prediction of uptake andproduction from inlet-equivalent uptake and outlet-equivalent production, if sufficient information on thenon-accessible paths is available. In practice, whenuptake and production occur simultaneously to asignificant extent, such as for amino acids, inlet-equiv-alent uptake and outlet-equivalent production are ofmodest physiological significance, and the possibilityof predicting qualitatively uptake and production isremote. However, when uptake and production do notoccur simultaneously, the theory has a practicalrelevance—as discussed in greater detail below for theglucose case.

Whole-body kinetics has been most often studiedwith compartmental models, while the application ofZierler’s ideas to whole-body kinetics in non-steadystate has not been proposed before. Compartmentalmodels have been shown to be generally inadequate toquantify the total whole-body mass of a substance,even in the simpler situation of no simultaneous uptakeand production (Mari, 1993). The reasons for thisinadequacy has been identified in the inability ifcompartmentalmodels, which are not physicalmodels,to quantify correctly the kinetic parameters of thenon-accessible paths of the body organs. Asa consequence, compartmental models cannot beexpected to give reliable estimates of uptake, that isclosely related to the mass.

The quantification of total whole-body mass is anintrinsic feature of compartmental analysis, i.e. theinvestigator is forced to make hypothesis on the totalmass when a compartmental model is developed. Thearbitrary assumptions that are introduced more or lessexplicitly may be incorrect and may have a remarkableinfluence on the calculation of uptake. However, thecalculation of arterial-equivalent uptake with thetheory presented here does not require hypotheses onthe total mass. Assumptions can be made at a secondstage, when the investigator attempts to predict thetime-course of uptake using some estimate of the meanartery-uptake transit time. In other words, whilecompartmental models typically postulate in a hiddenway a structure for the non-accessible paths, the theorypresented here makes the assumptions explicit if thecalculation of uptake from arterial-equivalent uptakeis attempted. The advantages of this approach arediscussed more extensively below for the glucose case.

The calculation of production, in the situation of nosimultaneous uptake and production, is not necessarilyas dependent on the model used as the calculation ofuptake.As has beenknown long ago (see the discussionby Cobelli et al., 1987), production can be calculatedin a model independent way if arterial specific activityis kept constant over time. This result is confirmed bythe present analysis, that also shows that the calculatedflux is venous-equivalent production in the case ofmetabolically inactive lungs. When specific activity isnot constant, the key factor for an accurate calculationof production is a correct specification of theintact-body impulse response. Since direct experimen-tal information on the intact-body impulse response isavailable (the intact-body impulse response is modelindependent in the steady state), the model error withcompartmental and circulatory models may not bevery different. Although compartmental models havethe defect that they do not correctly describe thephysical structure of the system, an unavoidable error

349

also affects the transit time density function of theperiphery block of a circulatory model. Since theeffects of these errors on the intact-body impulseresponse—and more importantly on the calculation ofproduction—is difficult to predict, the advantage ofcirculatory models is not as clear for the calculation ofproduction as it is for uptake.

The calculation of inlet-equivalent uptake andoutlet-equivalent production is affected by an errordue to the limited accuracy with which the inlet-outlettransit time density function can be determined inpractice from tracer experiments. In the non-steadystate, the transit time density function changes, and thepractical success of this approach thus depend on thepossibility of describing these changes with sufficientaccuracy. If the major effect of the non-steady-stateperturbation is a change of extraction, it may bepossible to keep the error small. If, however, theperturbation markedly affects the mean inlet-outlettransit time, such as when blood flows variesconsiderably, it may be difficult to obtain acceptableaccuracy. As discussed, the errors on the inlet-outletimpulse response are more relevant for the calculationof inlet-equivalent uptake, because this error cannot bereduced by a suitable tracer infusion as that onoutlet-equivalent production.

The usefulness of the theory presented here can beparticularly appreciated in relation to whole-bodyglucose kinetics during a euglycemic clamp exper-iment. The euglycemic glucose clamp is a widelystudied experimental situation in which insulin isinfused at a constant rate to raise insulin concentrationand glucose is infused at a variable rate to keep arterialglucose concentration constant in spite of the increaseof glucose uptake. This experiment creates a transitionbetween a basal state (normoglycemic and normoinsu-linemic) and an insulin-stimulated state (hyperinsu-linemic, but still normoglycemic).

For glucose, uptake and production do not occursimultaneously to a significant extent in the organs ofthe body. Glucose is almost exclusively produced bythe liver, that has a small inlet-outlet fractionalextraction under many circumstances, and is taken upby the other tissues. To be precise, uptake andproduction do occur simultaneously in the liver,but according to the current thinking (Pagliasotti &Cherrington, 1992) one can speculate that theproduction-outlet extraction in the liver does notexceed 10% under most circumstances. Hence, in thesteady state, for the liver outlet-equivalent productionunderestimates production by 10% at most. Further-more, in the non-steady state the expected delaybetween production and venous-equivalent pro-duction is quite short (less than 1 min), the liver being

a highly perfused organ. The ultimate consequence isthat for glucose venous-equivalent production almostcoincides with production.

Glucose venous-equivalent production during thenon-steady state transition of the glucose clamp hasbeen reliably calculated with the application of theprinciple of keeping arterial specific activity constant(Bergman et al., 1992). Due to the favorable situationof glucose, this calculation closely approximatesproduction. However, the methods currently used forcalculating the whole-body glucose mass and glucoseuptake are not reliable because they are based oncompartmental models. The pitfalls in the calculationof the total mass with compartmental models in thisexperimental conditions have been already discussed(Mari, 1993), and more recently the importance of acorrect evaluation of the whole-body glucose mass fora correct calculation of glucose uptake has beenstressed (Mari, 1994).

The theory presented here can be used for theanalysis of euglycemic glucose clamp experiments. Thetransit time density function of the periphery can bedetermined in the basal and insulin-stimulated steadystate from tracer experiments similar to those ofFerrannini et al. (1985) with some assumptions oncardiac output and on the dynamics of the lung block(Mari, 1995). Since it has been suggested that thechanges of the transit time density function during aeuglycemic clamp are modest (Mari, 1993), it can beargued that the calculation of arterial-equivalentuptake is practically feasible with a circulatory modeland sufficiently reliable. The time-course of uptake canthen be guessed on the basis of independentexperimental information as follows.

During a euglycemic clamp it is expected that theglucose mass can change only in the insulin-dependenttissues, which are essentially muscle, and in muscle theglucose mass does not vary appreciably in theseexperimental conditions (Katz et al., 1988). The overallchange of total whole-body glucose mass during theclamp is thus expected to be modest. If cardiac outputis constant, as is presumable in a glucose clamp study,the simplest condition under which the whole bodyglucose mass does not change is the similarity of themean artery-uptake and artery-vein transit time (seethe expression of the total whole-body mass in Mari,1993). It can thus be hypothesized that the meanartery-uptake transit time is comparable with the meanartery-vein transit time, which has been estimated to beabout 2.5–3 min (Mari, 1993). The time-course ofglucose uptake can thus be qualitatively predicted asthe time-course of arterial-equivalent uptake delayedby 2.5–3 min. This is probably the best prediction ofglucose uptake during a euglycemic clamp that can be

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done on the basis of the available knowledge onglucose kinetics.

REFERENCES

B, R. N., S, G. M., B, D. C. & W, R. M.(1992). Modeling of insulin action in vivo. A. Rev. Physiol. 54,

861–883.C, C., M, A. & F, E. (1987). Non-steady state:

error analysis of Steele’s model and developments for glucosekinetics. Am. J. Physiol. 252, E679–E689.

C, D. J. (1979). A linear recirculation model for drugdisposition. J. Pharmacokin. Biopharm. 7, 101–116.

F, E., S, J. D., C, C., T, G., P, A. &DF, R. A. (1985). Effect of insulin on the distribution anddisposition of glucose in man. J. Clin. Invest. 76, 357–364.

K, A., N, B. L. & B, C. (1988). No accumulationof glucose in human skeletal muscle during euglycemichyperinsulinemia. Am. J. Physiol. 225, E942–E945.

K, J. (1992). On the determination of turnover in vivo withtracers. Am. J. Physiol. 263, E417–E424.

L,N.A.&P,W. (1979). Tracer Kinetic Methods in MedicalPhysiology. New York: Raven Press.

M, A. (1993). Circulatory models of intact-body kinetics andtheir relationship with compartmental and noncompartmentalanalysis. J. theor. Biol. 160, 509–531.

M,A. (1994). On the calculation of glucose rate of disappearancein nonsteady state. Am. J. Physiol. 266, E825–E826.

M, A. (1995). Determination of the single-pass impulse responseof the body tissues with circulatory models. IEEE Trans. Biom.Eng. 42, 304–312.

M, P. & Z, K. L. (1954). On the theory of theindicator-dilution method for measurement of blood flow andvolume. J. appl. Physiol. 6, 731–744.

N, K. H. (1992). Sites of infusion and sampling formeasurement of rates of production in steady state. Am. J.Physiol. 263, E817–E822.

P, M. J. & C, A. D. (1992). Regulation of nethepatic glucose uptake in vivo. A. Rev. Physiol. 54, 847–860.

S, L, T, G. & C, C. (1992). V-A and A-V modesin whole body and regional kinetics: domain of validity from aphysiological model. Am. J. Physiol. 263, E597–E606.

W, R. R. (1992). Radioactive and Stable Isotope Tracers inBiomedicine: Principles and Practice of Kinetic Analysis. NewYork: Wiley-Liss.

Z, K. L. (1961). Theory of the use of arteriovenousconcentration differences for measuring metabolism in steady andnon-steady states. J. Clin. Invest. 40, 2111–2125.

APPENDIX A

Some results that concern the impulse response inthe non-steady state are given. The following equationis considered:

8out=gt

−a

r(t, x)8in(x) dx. (A.1)

In eqn (A.1), 8out(t) represents the component of theoutflux that is sustained by the influx. Equation (A.1)coincideswith eqn (1) in themain text for the tracer andfor the tracee if the substance is not produced in theorgan. All terms of eqns (1) and (2) have an expressionanalogous to eqn (A.1).

Result 1

If 8in(t)=0 for tQ0 and the integral from zero toinfinity of 8in(t) exists and is finite, the followingequation holds:

g+a

0

8out(t) dt=g+a

0 $gt

0

r(t, x)8in(x) dx% dt

=g+a

0 $g+a

0

r(t, x) dt%8in(x) dx

=g+a

0

j(x)8in(x) dx (A.2)

The property of causality of r(t, x) is used to extend theintegration limit of the inner integral of eqn (A.2).Equation (A.2) is the non-steady-state form of thesteady-state property stating that the integral of theoutflux equals the integral of the influx times thetransmission.

Result 2

This result proves eqn (9) in the main text. If8in(t)=0 for tQ0 and the integral from zero to infinityof t8out(t) exists and is finite the following equationholds (as in Result 1 causality of r(t, x) is used):

g+a

0

t8out(t) dt=g+a

0

t$gt

0

r(t, x)8in(x) dx% dt

=g+a

0 $g+a

0

tr(t, x) dt%8in(x) dx

(A.3)

The inner integral of the r.h.s. of eqn (A.3) can bedeveloped as

g+a

0

tr(t, x) dt=g+a

−x

(z+x)r(z+x, x) dz

=g+a

0

zr(z+x, x) dz+

xg+a

0

r(z+x, x) dz

=t(x)j(x)+xj(x) (A.4)

In eqn (A.4), the change of variable t=z+x, causalityof r(t, x), and the definition of the kinetic

351

parameters [eqns (3), (6), (8)] are used. By substitutingeqn (A.4) into (A.3) and using Result 1, eqn (9) in themain text is obtained.

Result 3

This result is relevant for non-steady-state exper-iments that create a transition between two differentsteady states. In this situation, the system can beconsidered time-invariant up to time 0 and after sometime T. The system is time-varying between 0 and T.The result concerns fluxes [8x (t) and 8y (t)] that arerelated by the equation

8y (t)=gt

−a

q(t, x)8x (x) dx (A.5)

where q(t, x) is a transit time density function.Equation (A.5) is relevant for various fluxes and transittime density functions related to eqns (1–2) and (11–12)in the main text.

It is assumed that in the transition experiment 8x (t)and 8y (t) are constant for all tQ0 and tqT. For 8x (t),these steady-state fluxes are denoted as 8x (0−) and8x (T+). Owing to the hypotheses made on the system,8x (t)=8y (t) for all tQ0 and tqT. Result 3 is expressedby the equation

gT

0

[8y (t)−8x (t)] dt=8x (0−)t(0−)−8x (T+)t(T+),

(A.6)

where t(0−) and t(T+) are the mean transit timesof the stationary q(t) in the initial and final steadystates. In words, Result 3 states that the area betweenthe two flux curves 8x (t) and 8y (t) in a transitionexperiment equals the difference of the productsof flux and mean transit time in the two steadystates. If the mean transit time is constant, thisarea equals the product of the change of flux betweenthe two steady states and the mean transit time.This result gives an integral interpretation of thedelay principle for the mean transit time discussedabove.

Result 3 is proved by considering an imaginaryexperiment in which 8x (t) is first brought instan-taneously from zero to the steady-state value 8x (0−) atsome time T0�0 (so that a steady-state situation isattained at time 0), and successively dropped again tozero from its steady-state value 8x (T+). In thishypothetical experiment

g+a

T0

[8y (t)−8x (t)] dt

=g0

T0

[8y (t)−8x (t)] dt

+gT

0

[8y (t)−8x (t)] dt

+g+a

T

[8y (t)−8x (t)] dt=0 (A.7)

since the integrals from T0 to infinity of 8x (t) and 8y (t)are equal because of Result 1. To prove eqn (A.6), notethat the second term of the right member of (A.7)coincides with the integral of eqn (A.6), and that thefirst and third term are integrals of the steady-statebuildup and washout curves at the outlet of an idealnon-extracting organ defined by eqn (A.5). Theabsolute value of these integrals equals the product ofthe steady-state value of 8x (t) and the mean transittime, and the sign is positive for the washout andnegative for the buildup curve.

APPENDIX B

This appendix considers the calculation of outlet-equivalent production in the situation of constant inletspecific activity. If inlet specific activity (ain) is constant,8*in (t)=ain8in(t), and eqn (12) in the main text becomes

8 prod(t)=8out(t)−a−1in g

t

−a

r(t, x)8*in (x) dx

=8out(t)−a−1in 8*out(t)

=8out(t)−8in(t)+a−1in [8*in (t)−8*out(t)],

(B.1)

where 8*in (t) and 8*out(t) are tracer influx and outflux,respectively. Since eqn (B.1) expresses8 prod(t) onlywithmeasurable quantities, the calculation of outlet-equiv-alent production does not require a model in this case.Equation (C.1) is formally identical to the equations ofthe steady-state method [Eqs. (C.1)–(C.3)], althoughthe difference 8*in (t)−8*out(t) is only apparent traceruptake in the non-steady state.

APPENDIX C

In the steady state, the equations for inlet-equivalentuptake and outlet-equivalent production [Eqns (11–12)] coincide with the equations for uptake and

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production of the classical arterio-venous differencemethod. Furthermore, the modified arterio-venousdifferencemethod that uses venous (in place of arterial)specific activity for calculating tracee uptake fromtracer uptake (see Wolfe, 1992) implicitly assumes thatthe production-outlet extraction equals the inlet-outletextraction.

The steady-state arterio-venous difference method isbased on mass balances of tracer and tracee. Traceruptake (8*upt) is calculated as

8*upt=F(C*in−C*out)=C*in−C*out

C*in8*in=h8*in , (C.1)

where C*in and C*out are inlet (arterial) and outlet(venous) tracer concentrations, and F is bloodflow. Equation (C.1) also relates the calculationof tracer uptake to the classical calculation offractional extraction (h). Tracee uptake (8x upt) iscalculated as tracer uptake divided by inlet specificactivity (ain):

8x upt=a−1in 8*upt=

C*in−C*out

C*in8in=h8in. (C.2)

Production (8x prod) is calculated as

8x prod=8out−8in+8x upt

=8out−8in+h8in=8out−j8in. (C.3)

By comparing eqns (C.2–C.3) with eqns (11–12), itis evident that uptake and production calculatedwith the classical arterio-venous difference method areinlet-equivalent uptake and outlet-equivalentproduction in steady state, i.e. 8x upt=8upt and8x prod=8prod.

The modified arterio-venous difference method usesoutlet (venous) specific activity (aout) for calculatingtracee uptake from tracer uptake according to theequation

8upt=a−1out 8*upt, (C.4)

where 8upt is tracee uptake according to this method.Using 2the identities aout=8*out/8out and j=8*out/8*in ,eqn (C.4) becomes

8upt=8out

8*outh8*in=8outhj−1 (C.5)

By using eqn (C.3) to express 8out and taking intoaccount that h8in=8x upt=8upt and 8x prod=8prod,eqn (C.5) becomes

f� upt=[j8in+8prod]hj−1

=h8in+hj−18prod=8upt+hj−18prod. (C.6)

Comparison of (C.6) with eqn (15) in the main textshows that the modified arterio-venous differencemethod calculates uptake under the assumption thatthe production-outlet extraction h( equals theinlet-outlet extraction h. The same conclusion appliesto production, since for all methods the differencebetween uptake and production equals the differencebetween influx and outflux.

APPENDIX D

Here the influence of the modeling errors on thecalculation of extraction is considered. In thenon-steady state, due to modeling errors the trueimpulse response, r(t, x)=p(t, x)j(x), is representedby the approximate function r(t, x)=p(t, x)j (x). In agiven experiment, transmission is calculated with theequation

8out(t)=gt

−a

p(t, x)j (x)8in(x) dx, (D.1)

where 8in(t) and 8out(t) are typically tracer influx andoutflux. Since outflux is the samewith both the true andthe approximate impulse response, the equation thatrelate the two impulse responses is

gt

−a

p(t, x)j(x)8in(x) dx

=gt

−a

p(t, x)j (x)8in(x) dx (D.2)

Equation (17) in the main text is readily obtained bysubtracting to both members of eqn (D.2) the term

gt

−a

p(t, x)j (x)8in(x) dx,

and recalling that j(x)−j (x)=h(x)−h(x) [eqn (4)].In eqns (D.1–D.2) and (17), 8in(t) is usually tracer

influx, and its integral over the whole experimentalperiod (say from zero to infinity) is finite. Since theintegral �+a

x [p(t, x)−p(t, x)] dt is zero for all xbecause of eqn (7), a consequence of Result 1 ofAppendix A is that the integral from zero to infinity ofthe right member of eqn (17) (that is a function of t thatvanishes for tQ0) is zero. The integral from zero toinfinity of the l.h.s. of eqn (17) is thus also zero. Byapplying Result 1 of Appendix A to the l.h.s. ofeqn (17) the integral from zero to infinity of[h(t)−h(t)]8in(t) is zero.

353

APPENDIX E

A List of the Principal Symbols used in the Article

Concentrations, Flow, FluxesF blood flowCin(t) inlet concentration†Cout(t) outlet concentration†8in(t) influx†8out(t) outflux†8upt(t) uptake†8prod(t) production8upt(t) inlet-equivalent uptake8prod(t) outlet-equivalent production

Impulse Responses and Kinetic Parametersr(t, x), p(t, x) inlet-outlet impulse response and

transit time density functionr(t, x), p(t, x) inlet-uptake impulse response and

transit time density functionr((t, x), p((t, x) production-outlet impulse response

and transit time density functionr((t, x), p((t, x) production-uptake impulse and

transit time density functionj(t) inlet-outlet transmissionh(t) inlet-outlet extractionj((t) production-outlet transmissionh((t) production-outlet extractiont(t) mean inlet-outlet transit timet(t) mean inlet-uptake transit timet((t) mean production-outlet transit timet((t) mean production-uptake transit

time

† The same symbol is used for tracer and tracee concentrationsand fluxes if the distinction is not essential. If necessary, tracer isdistinguished from tracee with starred symbols.