Models for the Estimation of Glucose Fluxes in Non-Steady State from Concentration Measurements

CopH i)(h t © 1f..\ C 10 th Trienn ial \ \'odd Con)(ress. Munich. FR(;. 19H7

2.1 - 1 CO\:TROL OF

E"J DOeRl :-': E-~ (E L\HOUC: SYSTEI\(S

MODELS FOR THE ESTIMATION OF GLUCOSE FLUXES IN NON-STEADY STATE FROM CONCENTRATION MEASUREMENTS C. Cobelli and A. Mar i Departlllmt of Electrical Engineering, University of Padova, and I llstitute for Systelll Dywlllics anil Biol'llgilleerillg. Par/ol'([ , /taly

8~§!c~£!,

Glucose concentration -in blood is the final result of the balance of the rate of its release into the circulation and its removal by tissues, These rates need to be measured in order to characterize the glucoregulatory system. IJnfortunately, these flu:·~ es are not directly accessible and indirect modelbased measurement must be employed , This is a difficult task , especially in the so-called non-steady state ! 1,e. when a pert urbation induc2s changes in the masses and flu:
Medical systems; System analysis; Time-varying 1 dent if i cat .i. on.; F' .3r.~!lleter· e·E· t i m.~l t i (":In; T rac·:-?·c· e~<~·ef· i rn~·nt ,

I

IITF:ODUCT I OH

The concentration of glu cose in blood is determined by ttle rate at which glucose is released into the circulation and removed therefrom. Glucose homeostasis in the body i.s realized through the precise control of these flu }
i n

·~·O l rle

these diabetes, Unfortunatel')", flul~es are not directly measurable and indil~ect model-based measurement must be form

of

':!fnp If)'y'ed.

The scope of this paper is to l?:-::per i ment ·31 methods .::J IIU mode 1·3 for ( je~ermining glucose flu :{2 s in the intact or·ganism out of steady state After all overview of the physiology of the glucoregulatory systenl we briefly s how (jirect measurement of glucose flu :· ~e s D(,t possible , The role of models discussed ne:
the numerical pl~oblem, Then we present Q l·lovel ~pproach based 0)1 physiolog iC 31 modeling of the zysterll and multi p le '- te 3d',I- =t 3+e tr·.:1C2r e: :per i fl,en t s , A specific problem of physiological interest :is used as case study to compare non -

:!~:~~~s~~;:os~r:~~~~~~::t~C~~!:~~ ~~d !~: physiologically based models, Finally the potentiality of a 110n-convent~onal e:·~perimental appr·oach which minimIzes mod e ling errors in the non-steady state is b\~iefly mentioned ancj plJt in proper p~:r',=.pect .i \.l 2-,

GLUCOSE FLUXES RND HORMONAL CONTROL glucOr?9'Jlatory syst21n ar~ schem3tically ·3 ho l",ln in Fi;J , 1, In the fJlucose sub ·=.y ·:=:tern the flu ~·:es of glucose 3cross the circulation as well as those related to th..:: f:)r· ecu·(··~Ol-··=. of f;!luc:os2 {~, Q , :;l.:'J·cline ! lactate, pyruvate and gl ycerol) .3nd t c' ~Jther chemical forms of glu cose (e,g . glycogerl) are e vidence d, These flu :·~ es, which determine glucose .:oncentra tion in t~,e blood! J eper,d on ~ he le vels c, f glucose it·=.21f other i nte r·med i .::'lte=-. th~· in ::::' a t· h 'y' -=. of glucose metabolism under stric t 211docrine (~nd n ef ::,u:;::··) control On the other hand ~i-·:-:-:

1",1 .:1

l

gluco\~eg!Jlatcl l~ y

~!ormonesl

ion 3nd remo\! .31 ,j eter·m i ne the hor·m one concerltration in f)lasroa ~re controll~d mainly by gluc l)se concentration, These interactions b2t~een rate of prod1Jction and utilization and plasma concentration s cons t itute th e feejback loops on which glucoregulatioD i~ based. In an organism at rest all flu: ·~ es and concentrations are constant (steady state); when a stimulus perturbes the ·=2i~f·et

J

C. Cobelli and A. Mari

2

ENDOCRINE SVSTEM

GLUCOSE

SVSTEM

LIVER GLYCOGEN GUCOQENO · LY ...

GLYC_NO-" " • YNT. . . . .

I

QL YCOL. YSI.

~AR80-

GLUCOIIE_...

HYDRATE Su.sTRATES

GLUCOSE -I-PHOSPHATE

f I

GLAND

~ _CRETION

-,.ROOUCTION

UTILIZATION

(~~~n

()

PLASMA

1

DEGRADAi TION

I 1

MUSCLES ADIPOSE TISSUE

TISSUES

()

GLUCOSE

UTILIZATION

1 BRAIN

Fig . i . Sch ematic diagram of the glucoregulatory system, the feedback loops guarante e the restoration of a steady s tate (t he old or a new ODe depending on the pe rt urb a tion ). During this transition ( non-stead y state) hormona l con t ro ls are activa t e d to modulate glucos e flu :< es until the system re a ch es the new equilib r ium point. systenl

J

In

the

following

discussion

to

the

we

will

confine

measurement

in

our

non-

s teady state of glu c ose production by the li ve r and utilizat ion by the v a~ious tissues ( li v er muscles} etc .) , Ott-Ier important aspects , e,g. the assessment of the relative contr ibutio n of gl yco genoly si s and gluconeogene si s to total glu c ose p r oduction , or t he evaluation of the ro l e of g l yco genosynthesis and glycolysis in li v er glucose uptake are outside the scope of this revie 1...1 • j

THE DIRECT REGI ONA L BALANCE APPROACH The c lassi c al Fi ck principle is a natural candidate to measure the net balan c e of a sub s tance across an o r gan. For instance, by p la ci ng a catheter in one of th e hepati c v eins a nd a no ther in an arter-y and by estimating hepatic blood flow (e.g. with a tracer dilution method) the net splanchnic glucose o ut pu t ( NSGO ) c an be computed as th e product of hepatic blood f 1 01,.,1 (0) a n d the arte)-- i o- v en ous ';I: lu cos e concent r ation diff er en ce (G~~ -GI·iV) (Fig.

I,.,lhen difficulties te c hnical sev e r.:)l ap p lied to the intact human organism. In fact, blood samp li ng in the deep v eins and arteries is in v asi v e and techni c ally comp le :< , and the relative er r o r of the arterio-venous concentr a tion difference can be v er y high for sm all d iffe rences, unles s v er y precise measurement techniques ar e .O l,jopted. Moreo v er, the rate of blood flow estimated by indire c t methods is of limited precision. Of note is also that th i s approach can onl y measu r e D~! flu :< es , i.e. differences between glucose production and uptake , unles s a trace r is used . Fo r' instance equation 1 ( Fig . 2) measures the differen c e between glucose production b y the li v er and its u pta ke by the sp lan chn i C ti ss u e. In absence of a direct approach for measuring glu c ose flu :< es in non-steady state , indi rec t model based methods must be u s ed.

SPLANCHNtC

REGION

I

2)

I

HSGO

=

0 - (Gn-G.o",,;)

G.

~---r---u-p~

(1)

Howe v er, this mass balance calc ulation is va lid on l y at ~i~~~~ ~ i~ig , and i s not applicable to non-steady-state conditions, i,e . when flow or concentrations are c hangi ng ( Zierler, 1961 ), If this basic requirement is not met l as unfortunately happens frequentl y in the lite r atu r e, unknown systematic er ror s will affect the estimated flu )·( es. Furthermore , e v en in steady state this app r oach en coun te r s

Fig . 2 . Schematic fj i.~ I;l:r·.~m splanchnic region.

of

the

MODELS OF GLUCOSE- HORMONE CONTROL The glu co regulatory com pletel y understood,

would be s y stem and the non-s tead y -

Models for the Estimation of Glucose Fluxes

3

1983) .

deve loped (Bel'gman and others, 1979). The rationale of this modeling strategy is to de ve lop the simplest possible model retaining the fundamental aspects of the physiological system so that the parameters of the resulting model are both physiologically meaningful and identifiable from an input-output experiment in a single subject.

Such models are usually described by a set of nonlinear differential equations of the type

A model of this type is illustrated in Fig . 4. It is described by the equations

Q(t)=f(Q(t) ,u(t) ,t,p)

(2)

~(t)=[P.-X(t ) JG(t )+Pa+U(t)

G(D)=G,. ...

(4)

yet)

(3)

X(t)

:< (0) =0

( 5 )

state problem solved, if an isomorphic model could be pro vi ded. Attempts to construct EQ~Qr~b~D§i~~ models of glucose homeostasis have been made, usually adopting "bottom-up" strategies and comb ining information coming from different sour,~es (e.g. Srinivasan, Kadish and Shridar, 1970; Cobelli and Mari,

g(Q(t) ,t,p)

where Q ( t) is the state v ector (usua ll v masses ) whose va lue at the initial time t~ is QD; u(t), yet) are the input and output vectors ( usuall y exogenous material flu x es and concentrations respectively); f and g vec tors of non-linear functions that embod y the model structure and measurement configuration respectively ; and p a vector of model parameters. These models are necessarily based on some theories (prejudices) about the functioning of the s y stem and are intrinsically rather comple x, i.e . they are described by many differential equations , usually nonlinear, with a large number of parameters: are

not

suitable

I

Consequently

for

measuring

I

of

u

(t) •

the y

glucose

flu x es in an individual , because it would be impossible to determine the values of all the model parameters from an in vivo e x periment in a single subject. For e :·, ample, the model of Fig . 3, which already neglects some aspects of the glucoregulator y system of Fig. 1, clearly cannot be identified from the measured time-courses

where G(t) is plasma glucose (G •• is the steady-state va lue ), I (t) is the de viation of plasma insulin from its steady-state value, X(t) is a state variable which represents insulin action , u(t) is the impulsive test input and pa,p.,pz and 5. a re the uniquely identifiable constant parameters of the microscopic parametrization of Fig. 4 and are functions of the k.'s which represent both transport and control action. The model parameters are identifiable from the time course of plasma glucose and insulin concentration, G(t ) and I (t) , measured after the impulsi ve glucose load given at time 0,

glucose,

insulin

and

glucagon concentration perturbation.

after

a

PLASMA INSULIN I (t )

.

ke r

~I

L,ver

REMOTE INSULIN

...C.:....:.O.:.;.M.:.;.P.:.A.:.;.R.:..T:...:M.:.;.E~N:..:...:.T.r ., k4

Ll-~~rip...r.1

GLUCOSE SPACE

~

T,ssue.

k5

.. EASUREMENT SEC"ETION

Fig . 4.

The minimal model of Bergman others (1 '7'79) .

and

This model has been de v eloped t o provide from a simple test the insulin sensitivity inde }{ S~ I defined as the quantitati v e in fluence of insulin in enhancing g~ucose disappearance, but in principle it can also be used to compute glucose flu xe s in non-steady state in an individual. In fact, glucose flu :< es can be calculated from the model equations and the identified parameters in a given subject and e x periment. Howe v er since the model does not allow the separation of production and utilization flu :·~ es} the reconstr uctable flu :·: es necessarily bear some ambiguit y (Cob elli and othe r s • .1986 ) .

QLUCOSE su.SYITE ..

TRACERS AND GLUCOSE MODELS Fig.

3.

An e:
Continuous

material flu x es, control actions .

a

com~rehensive

glucoregulatory lines repres e nt

dashed

lines

To overcome some of the problems mentioned above, minim@! (or parsimonious) as

opposed

to comprehensive models have been

Tracers allow the s tud y of the dynamics of glucose in view of their indistinguishabilit y with the nati v e sUb stance (the tracee ) and in the nonsteady state the dynamics of both tracer and tracee are in fact described by the same linear time-varying model , commonly assumed to be compartmental ( Jacquez, 19 8 5) . Let 's consider for sake of I

C. Cobelli and A. I\fari

4

simplicity a single input-single output e x periment taking place in the first compartment (the accessible compartment, usu .~lly bloo,j or plasma). Under this h y pothesis the system equations are

Q( t )

o A(t ) Q(t )

+

Ra(t )

(6)

o Cl /V,

0

...

OJQ ( t )

(7)

GENERAL PURP OSE APPR OX IM ATE TRACER MODELS

1

o A(t ) Q~ ( t)

+

Ra'"

(8 )

( t)

o c~

( t)

o . ..

Cl / V,

OJ

Q~ ( t )

(9)

where the star indicates the t r acer, Q(t ) and Q*( t) are n- v ectors of tracee and tracer

compartment masses Ra ( t ) tracee and tracer rate ~:I\::.pea)-'ance (the s'y stem inputs) C (t) I

RaM(t)

I

and

of and

C* ( t ) tracee and tra cer con c entrations (the system outputs). The n x n compartmental matri x A(t) Ca 'J( t ) J ha s the structure k :c.J

I ;i.J

(t )

( 11)

k .. ", ( t )

,1 =0

J;iI where k.J ( t) is the transfer rate coef fi c ient (min -' 1 ) f ro m compartment J to I. In the output vector (eqs. 7 and 9) V, is the v olume of the first co mpa rt ment , assumed to be time-in var iant . The input-output relationships system of equations 6-11 are

C(

h ( t,

t )

rt

J

y )

Ra er)

.j 'l

h(t,y)Ra- ( y ) dy

where h(t,~) is the impulse response the time- v arying s ys tem .

Basically three specific models fo r nonsteady -state anal y sis ha v e been proposed, with the aim to pro v ide tool s of general IJse , i ,e . applicable to .~ll non-steady s tate situations . Two of these me thods are ba sed on a model that e ~·~ hibits only a single tim e-varyin g pa ra meter so th a t it is p oss ible to recons tru c t its time-course from tra c er data [R a~(t) and C*(t) ] using the system eq ua tio 11s ( equations 8-11 or equ .3t io r) 13 ), Once tt-lis paramete,-· h ·?Js been ,jeter·mined , h (t,·Y) is· co m~.l e tel y !·.-:. no ',..' n in the e x pe rim e nt and can be used to compute Ra ( t) from C(t) by means of equation 1? This computation can be performed with a direc t formul a (Steele, 195 9 ) or by computer simulation ( Radziuk , Nor',.,li,:::h .~nd '..))-·.3n i c , 1'?78 ),

(10)

n

:E

T a nd not o n l y of th e difference t- T, and c an no t be identified from the da ta unless h y potheses a r e made on the structure of the s y stem , Unf or tunatel y, many different c hoi c es fo r h ( t , T) are possible for s olving equation 13, but each cho i ce will y ield a diffe r ent solution when equatio n 12 is sol ved for Ra ( t ) I n other words , different a ssump tion s on the st ruc tu re of h ( t , T) ( i . e. on the s.':./s tem ) 1".,li ll in general , lead to different predictions o f Ra ( t ), Ob v iousl y, the true structu r e of the impulse r esponse is needed to o btain c.Ol--r·e ct pr·edictions of R .~ ( t ), but it is , in general, un known,

(12 )

these models ha ve been de v eloped as All c om pu tati on al t ools, and are not intended to be a ccwrate descriptions o f the sys teln , The y are only a pI)ro ~·~ imateJ ei ther beca u se the n umbe r of compa r tme nts i s inadequate, as in Steele's mod el, or because the s tructure of the time- vary ing paramete r i s simp listic,

Th i s model (Steele, 1959) assumes that non -stead y-state kinetics ma y be described by 3 a ne-cQmp~rtnten t t im e- v ~r y ing mod e l , The model st rtJctu re is illustra t ed in F ig, S an d the cor r esp onding equations for tracee and t racer are respectively Q (t)

-k

(t)

c ( t)

Q(

t )

of

I~~

(t )

+

Ra

(t)

/ 'J ".

(t ) IJ~·

The e s tim ation of the glucose rate of appearance, Ra(t ), is thus an input estimation pr oblem , and the r a tionale for its solution is to identify h ( t ,T) from tra c er data [Ra* ( t ) and C*(t)] 3nd then t o Ra ( t ) from h ( t ,T) and tr .~ ce e r ec:ons truct dat a [C ( t ) ]. The solution is rather s t raig htforward in the s tead y st ate , i . e. when A ( t ) is cons ta n t , because h(t,T) is the impul se response of a time-in va riant syst em and depe nds only on the diffe r ence t-T, In this case Ra ( t) can be estimated without resorting to an e :{ p li cit model of the system, being the impulse response h et) easil y determinable from an input~u tput trace r e :< pe r iment. Quite to t he in c ontr.:,r"y , the non-steady state h ( t,T) i s the impulse response of a !im~:~~[~iD9 l inear s ',/s tem and does not ha v e a precise ; tructure. It is a function of both t and I

Cl

(13)

( t ) ./ t.)~ :,

where k ( t ) i s the ti me - va rying elimination r a t~ and V ~ the vo lu me of the ~ompartm e nt ,

•

Ra·(t)

C(t) ,C (t)

/

Ra

(t)

/

?

k

(t)

S , The model of Steele

Models for the Estimation of Glucose Fluxes

5

This monocompartmental description allows easy computation of Ra ( t ) from equation 14,

k

once

tracer

(t)

data

has

been

and equations

computed

16-17,

result, the following commonly expression for Ra(t) is obtained

Ra~

Ra

(t)

',)",C

from

As

a

used

(t)

(18)

(t)

a

aCt)

(t)

i.e, -...-here ·3 ( t ) is the specific activity, a ( t ) =Cw ( t ) IC ( t ) , In these calculations V. is assumed to be known, and i t is usually expressed as a fraction p (O ( p ( 1) of the total distribution volume of the substance,

B)

(19)

More recently, varying model Radziuk, Norwich evaluation of k. inetics.

a two-compartment timehas been proposed by and Vranic (1978) for the non-stEad y -state glucose

This model

is descT' ibed b y

the

FiQ , 6,

equations :

The mo dels of Radziuk , Vranic (1'7'78)

(20)

C (t)

k::" "Q", ( t ) -k:.,:;" ( t ) Q:;" ( t )

( 21)

Q"

(22)

(t)

I V :"

(23)

The model b':/ Is s ekutz and o ther's (1974 ) 15· an e }·: tension of Steele's model a rId e }~ hibits a time- v arying v olume vet) in addition to the time-varying elimination l-'ate k (t) , Tr·acer· and tracee model equations are thu s

(24 )

-k

C-(t)

=

(t) Q (t)

+R ,=>

(30)

(t)

Q.-(t) / V. C

(311

Q ( t ) / ',) ( t )

(t)

The may

time dependence of k,. :l(t) and k 2 2 (t) be of one of the two following forms, which correspond to the diagrams of Fig, 6a and 6b respectively,

( 32) (33)

Issekutz ' s

k' ~-;~ :l+ko :1. ( t )

k:.,, ;.•,

(27)

(t)

or' k:I__l et)

k~·.: l. +ku ( t )

k:.""" ( t )

k :t:~+k.[;:t ( t )

method emplo y s a double t \~acer ' to comput e tl1 € t im e-cou rs e o f the two parameters k ( t) and V(t) , Equations 32-33 are sol v ed for k(t) and V(t) given the two input-output tracer p .:;::!} 'f'S R,3.l* ( t)! (: :1. ')(- ( t ) ~nd R a~,;~ ' )(' ( t ) , (: :;;,':0(' ( t ) The time- v a ry in g paramet er s a0e th eft substit u te d in t o e q uatio n s 3 0-31 to compute Ra(t-), As a result , the following equation is obtained 2 : ~ p~riN!ent

(29)

The fir'st for'rf! ( eq'=. , 2 ,~, -"27 , Fi/~, 6a ) a ssumes that there is no irre v ersibl ~ los~ from the second compartment, while the ::·ec ()nd f Clrm (eqs, 2:3-29, F i ';I, 6b) ·~sSUr(p?E­ equal irreversible losses from both c ompartments , Equations 26-27 or 28-29 a r2 necessary in or d er to reduce the number of the time- v arying parameters to one, This .~llo'...!s, by t .3 kin f;} ~;. :I. :'-:~ .~nd k;", ) =..s knot,.,!n, t o compute the time-colJrse of the timev ar y ing parameter [k(:l1(t ) or kr:, (t ) ] from tracer data , and to predict Ra(t) from tracee data, In this case, 3n e l'(plicit formula for Ra(t ) (such as eq. 18) does not e ~< ist, and the computations are performed by computer simulation.

Ra :!. ':":' ( t ) .~:.-:, ( t ) -R.;1 :~, ·)(- ( t ) '~ 'I ( t )

Ra

.

( t)

a :1. ( t ) a::.~ ( t ) -.;1 :;;: ( t )

( 34 )

!",'here a: 1 ( t ) El'':: t i './ i tie;:·

and a :.-:~ ( t) ar' e the "",pecific of the t",o trac~er"=- , The e::.pression o f the tirne- \!.;) r ':l in l~ '·/ c Iume i s 1

F.: -~ :I.~. ( t ) c ::.~ ' '''· ( t ) -Ra ::.: ·~ ( t )

(::1_')<'

C:I.N' ( t ) (:~:;., "" ( t ) -,2:.-;~ ·-.c- ( t ) C :

. ,,.. 1

( t )

(t )

Among the general purpose models 5teele's model is by in large the most widel y used , being that it prOVid~t:_ ~,,_ ,_,,~h.~, ~ ,~, 1 of R .~ ( t ) ! i , e, I 3n e ;·~ plicit formula, Radziuk's mo de l.s } which like Steele's model necessitate the use of only one tracer, require in fact

the

reason

e-~ :::. iest

In the following discussion, these mo d els wi 11 be called Radziuk's model A (Fi';J, t,a) and 10, (Fig, 6b),

.

a :1. ( t )

corilput .~tion

...J_

C. CoiJelli an d A. !\[a r i compute~ simulation . Issekutz ' s model has been practicall y ignored like l y due to the requirement of a double tracer e :·~ periment. Of note is that Steele's mode l is becoming of universal use , i . e. for investigating

non-stEady-state

turnover also

substan c es such as ketone fatt y acids and leucine,

ERROR

of

other

bodies ,

free

ANAL YSIS OF APPRO XIMATE TRACER MODELS

It is possible to evaluate theoreticall y the degree of app r o : ·~ imation of the models des cr ibed in the pre v ious section by assuming that the true s y stem is described by a general n-compartment model ( equ .~tions .:'.-11) Fig, 7) .~nd c .3 1culatin'J the difference between the true Ra(t) and that predicted by the appro :,: imate model, is place,j In this E.ection focus on "::. t r' u c t t.Ji-··3 1 er \-'o r s i ,e . data onl y, as sum e d to be e r ror-free , continuous function s of time. The effects of the in the measurement of tracer and tra c ee c oncentration will be discussed in the ne ~·~ t section.

The erro r

equation 38 it can be seen that it depends both on the structure of the real system and an the e :< perimental situation, It does not depend on S te ele ' s v olume V ~; , The term ev(t) arises because Steele's volume V~~i is, in general, different from V 1. , the v olume of the accessible compartment . For this reason , ev(t) will be termed ~Q!~m~ ~CCQC ' Eq , 39 shows that ev(t) depends on the obser v ed tracer and tracee concentrations, and thus on the e )< perimenta l situation! and on the d ifference V.~ -V~, It does not depend on the structure of the rea l system . 1

Unlike e'J(t ) . e b (t ) depends on unmeasu r able quantities. In some cases , howe v e r, by spe c ifying the structure of the real system, an analytica l e}·~ pression relating the structure error to the obser v ed concentrations C(t) and C*(t) and to the rate parameters k X ~ j(t) can be deri v ed . We will c onsider he r e a gene r al two-compartment time- v a ry ing model ( Fig . 8 ) f or the real systeni, a st'-' ur::ture that appears to fit many situations, in particular g l ucose kinetics, Assuming that tracer and tracee are in steady state before the perturbation ( sta r ting at time 0) the : .tructur e e r- c or e~ :; ( t ) is. (~i'·.,.. en b y' :

of Steele's model can be defined t

e

~~a ( t )

(t)

,..

-H .3

( 36)

(t )

1') :1

C·...,·

I'

( t ) (:{

-\

j't- f:,

( ry- )

de;

0-

( t;. ,.I

e

2 :1. e T)

d-r

(40 )

Cl

where the hat denotes Steele's estimate of the r a te of appea r ance. It has been shown ( Cobelli , rla r · i. and Fer )-'annini , l'7':37a ) th .3 t e( t ) c an be gi v en the followi n g e ~< press i on v

e(t ) I~i

=

(37)

e". (t)+e .... ( t)

time.:=. n d I~' ( t ) ·~"-· e posi ti ") ;? ), ,;,t ·? v ar y in g coeffi c i en ts '-'elat e d to the paramete 'f ' S k 1. ••1 ( t ) .~ nd to t r .:=. cer concentl-'ation .;1.

( 41 )

(t)

(4 2 )

th n 1, 2 :1. ( t ) - 2 :( ( t )

0 :1. :[-1<'

with

(t)

1=2

ev

(

t ) =

-

c: .~. ( t

)

(l.) ~:> - \J .1. )

Z:I

(t)

( 3'7' )

Q(

t )

whe r e 0:t ~ is the tracer f u :< from c o mpa r tment I to 1 and ~ ~( t is the in v erse of the specific act vit y in compartment I , i , e, 2 :[ ( t ) =l /·:=' :t: ( t ) , IJ ~:. i ·.=: th e volume of St e ele ! s model (equations 15 and 17) and V the volume of the first compartment o f the true s y stem (equations

( 43 )

,r

r

k :, ,~ .I . ( T)

C·)
The gene r al time- v ar y inq compartme n t model ,

tt,.,IO-

j .

7

,~n-.{j

9) .

,

•

C(t).C(t)

/

/

Ra (t)

The ge n eral time- v ar y ing c o mpa r tme n t model ,

n-

the ar·i:::.es becaus.e The term true s y stem is stru c ture of the D2! For th i s re .~son, e ,,; ( t ) mo nocompartmental, F r oHI ha s been te r med ~t~~~!~~~ ~[[2[ '

i

Models for the Estimation of Glucose Fluxes 40-43

Eqs. First ,

pro v ide considerable insight,

both ev (t)

and e s ( t )

depend on the ~1(t ),

observed specific acti v ity through

Since the specific activity depends on the rate of tracer infusion , the magnitude of

Steele's

error depends on

the

bQ~

tracer

is infused , If the tracer is infused that the specific acti v it y is constant ~]. ( t)=O)!

so (or

Steele's error is zero.

Second, evCt) and e~~ ( t) ma y, under certain circums tances, partly bal ance each other,

In

fa c t,

e ~>( t)

and ev(t)

ha v e

opposite

sign at" least if ~ :L(t) alwa y s keeps the same sign, i .e . if the specific activity is always increasing or decreasing , provided that V~5 is la rger than V 1 • On th e o the r hand , if ~ ~( t) changes, for instance, f rom positi v e to n egati v e, ev ( t) will become positi v e while e~~ ( t ) will not change to negative immed ia tel y . In thi s case, the tw o erro rs will add up.

The error anal y sis for Steele ' s model can also be e :·~ tended ~n I ssekut z's model . Equations 36-43 are st ill v alid , pro v ided

that V (t)

V. is replaced b y the time- v a ryin g ( equations 31 and 33 ). In this case,

hOl,.,lever J the dependence of the error on the time-course of tracer concen t r ation s is not as simple as for Steele ' s model because v e t ) is no l on g er constant .

ILL CONDITIONING OF GENERAL PURP OSE TRACER 110DEL'::. The gene r al pu rpose tracer models of the pre v ious section do not take into prope r account that the non-stead y -st ate problem , requiring the solution of the integ r al equa tions 12-13 , is int r insic~ll y illcon ditioned. In othe r words , these method s are v er y sensiti v e to data errors } i.e . v er y small errors in the measurements can produce very high errors in the p redic ted rate of appearan ce , This is clear-Iy !·? v ident for Steele 's 1)'Iodel ( equ .: Jtion 1 :::::) which requires the co mputation of a derivative} that is a well known illconditioned problem. For the same reason 1

Th i r1j, s· i nce e~:> (t) does not Ijepend on a suitably chosen V,:. ma y balance ev(t) e!:.( t). It is possible , for instance , choose Vs such that the integral of tota l error-, e(t ), i s zero. Ho,...'e v er, v.~lue

of V,,,

( or ,

equi v alentl'y',

I.J!:. I and to the the

the pool-

f)--ac:t i on p ) is e :< per i ment-dependent , as both e rror components depend on the particular e ~< perimental s ituation. In an y cas e , the compensation be tw e en ev(t) and e ,:.( t) may be poor be ca use of thei r different time cou r ses . Fourth, e s( t) depends on ~bi£b rate pa)-' .: Jmerters k :1; J ( t) are time- va r-ying and b2~ the y v ary. Different time-varying structures may in fact generate the same tracer and tracee data , but will yield ,j if f erent e ~:j. (t), HO'..Je v er, the determ i nants of e ,:. (t) are (':( ( t ) anlj ( t) and not the indi v idual k :[ .. , ( t), Equation 40 shows that the I;lr' e .~ter (:( (t) and sma 11 er r:. (t), the greater e s( t), and v ice v e r sa . Note also

e.

that,

although

I

the time courses of

.( t )

and B ( t) are undetermined , their stead ystate va lues are perfectl y known f rom stead y -state 3na1 y sis.

The theory of the pre v ious paragraph makes it also possible t o e v aluate the error of Radziuk!; models with respect to the general time- v ar-ying two-compartment model of Fig , 8. In fa ct, the total error e ( t ) can be computed for both models ( equation s the d i ffe r ence between these and two e(t) e ~ < a c tl y equals the di ffe r- en c e be tween the r-especti v e r3t e of appearance, If the v ol u mes of the fir s t com pa r-tmen t ar-e the same, the vo lume e rr o r s , being in depe n d~nt CIf model struc tu re, 3re the s ame i n each e (t) and will disappear f ro m the differen c e . The f' ates of appearan ce will thus differ onl y because of the different structure errors of the tw o models , The model with the g r eater e ~:. (t ) will gi v e the smalle r prediction of Ra(t ), Since ~( t ) and e ( t ) are the dete rm inants of e s( t) , and trlus o f Ra ( t ), differ·ent models can , in some circumstances , be co mpar-ed dif'ectl y on the basis of ~( t ) and e. ( t ) , F o 'c' e ;:.:: am p le , it has been sho',.,.'D (C obelli l 1'·lar· i .3 rllj Fe r r .3nnini , lS'::::7 .~) that in a comm on e :< perimental condition

Radziuk's

model

I; )reate r' Ra(t )

A

alwa y s

trl .: ln model B

predicts

( Fi l;l,

.:',)

a

Issekutz's

approach

is

also

ill-

cond itioned ; mo reover, equatio n 34 has 31so the drawbac k that the diffe rence in the denominator can be small and thus affected b y a v e ry large r-elati v e er- r o r, On the othe r hand , Radzi uk's approa c h features two ill- con ditio n ed steps} both essentiall y requ irin g the e v aluation of a deri v ati v e. The first is the c alculation of the time- varying parameter (k(~1(t) or kD ( t ) , equat ions 2t.-2·"7') f'f'o m tracer dat a ) and the second is the calculation of Ra (t) 'from tracee data as a solution of th e integral equation 12 , Wo ot t on Flecknell ;EJnd .John (lS'82',) ha v e s.tudied v ar-ious as p ec ts of the ill-conditioning of Radziukls models b y simulation, J

It is useful to e :< e mplif y the illcon ditioning of ge neral purpose models by using Stee1e's model on a specific p robl em

whi c h ne :·~ t

will

be furthe r

dis c ussed

in

the

section , Sup pose that the meas lJr-ed tra c er conc ent ration is that of Fig . 9a, that the tracee concentration is co nstant , and that the true , error-free time-cQurse of tracer concentr-ation i s represented by the cont inuous li n e of Fig, 9a, ASSIJ fne further that we can pro v ide 3 slightly different sm oot h ing of tr-acer data by means of spline interpolation , such as that of the dotted line of Fig , 9a, These two smooth t racer time-courses are v i r tu a ll y indi s tinglJ ishable , and i n fact the relati v e diff erence ne ver- e :< ceeds 3% , If the ra te of appearan ce is nQW comp u te d with Steele equ a tion using both t racerprofil es , the differen ce of the rates of ap pea r anC E, rep resen te d i n perce n t of th e basal v~ lue, is that shown in F ig. 9b A v er y small diffe r ence in the interpo lat ion of tr ac er data results in a large diffe r ence of the c omputed rates of appearance, It should be noted that if the difference is conlputed in percent of the r·3te of appearance , whi ch incr-eases with time , the relative de v iation is sm aller than that of Fig , 9b , However , as e :< plai ned in the ne :
8

C. Cobelli and A. Mari

80 60

In order to evalua te the role of insulin per se, i. e . indepe ndent 1 y 0 f change s in plasma glucos e concentra~ion on glucos e produc tion the so-cal ied I euglyc emic insulin clamp techniq ue (De Fronzo , Tobin an,j Andres , 1979) is widely use,j. In spite of elevat ed insulin levels produc ed by an e :,~ ogenous infusio n of insulin , glucos e concen tration is clampe d at its basal steady -state level throug h an exogen ous closed -loop glucos e infusio n (Fig. 10). A primed -contin uous infusio n of tracer allows the genera tion of tracer data throug hout the observ ation inter v al.

~

40 20

40

-40

SO

1~0

90

TIME IIIln

Fig.

9.

Relati ve deviat ion of the rate of appear ance comput ed with Steele equ.ati on (lOl.,'er p.,mel) for t 10.' 0 differ ent interp olation s of tracer data (contin uous and dotted line, upper pane l) .

differe nce of glucos e produc tion would be even greate r than that of 9b.

rate Fig.

The reason of this large magni ficatio n of the tracer errors lies in the fact that it is not only the magnit ude of the pertur bation of the tracer profil e which affect s the ill-con dition ed compu tation, but also its freque ncy. As the frequen cy increa ses, the error increa ses with no limits , as shown in essenc e by Phillip s

(1962) , Thus I even v er'Y SrII ·;J 11 osc i 11 at i oos of the smoot~ling curve can genera te a substa ntial altera tion of the calcul ated rate 0 f appeal' ance. With th i 5 respec t, it should be noted that common ly used smooth ing techni ques, such as polino mial or spline interp olation , v ery often e x hibit spurio us oscill ations .

PHYSIO LOGICA L MODELING OF TRACER DATA

The abo v e analys is shows that the oonsteady state problem is too comple ;< to be solved with simple recipe s of genera l applic ability . Simpl istic models are potent ially misle a ding : at their best the y c an pro v ide qualit ative or semiquanti tati v e answer s} but signif icant artifa cts can ~. ~ genera ted due to differ ent errors of the model in differ ent e ~,( perimental situat ions. Non-st ead y -state ffiodeli ng of glucos e kineti cs should be based as mu c h as possib le on the knowle dge of the structu re and functio ning of the system . In partic ular} if the system in questio n is poorl y unders tood basic invest igation ! e.g . studie s in differ ent steady -state c onditi ons } should be undert aken before approa ching the noosteady -state problem . Below we discus s some result s obtain ed b y adopti ng this modeli ng strateg y on a specif ic problem of b
..

~

1&1 z: .l: (/)0 ..

Ot;;

g~ ..J'" l!)!;

=

J.

E

Q

E

t

~

b

~GLUCOSE INFUSION

U N

-/20

INSULIN INFUSION

6

30

SO

90

1~0

TIME .In

Fi ,~.

10 . E;': perime ntal P'-'otoc ol of the euglyc emic insulin clamp (from Cobel li, Mari and Ferran nini, 1987 .3 ) .

The first attemp t to use ph y siolog ically based compar tmenta l models to estima te from tracer data the time course of glucos e p r oducti o n induced b y insulin per se is th .at b'y Insel .: .nd othel's (1975) . A three compar tment model of the glucos e subs ys tem was de v eloped and couple d with a prevj,o usly publist lsd (Sherw in and others 1974 ) three compar tment model of th~ insulin subs y·:;;tem ( Fi9' 11) . The .::ontl'o l of insulin on glucos e kineti cs was assume d to be linea r and origin ating from the slow insulin compar tment, The glucos e submod el feature d c onstan t interco mpartm ental e :,~ change rates, a consta nt glucos e utiliza tion from the contra I compar tment ( the insulin -indep endent utiliza tion) and an insulin - c ontrol led glucos e loss. Whethe r this loss should take place in the fast or slow glucos e compar tment has been discus sed b y the author s. Althou gh they antlclp ated that on physio logica l gl'ound s 1

Models for the Estimatioll of l;tllcosc Flllxcs EXOGENOUS INSULIN

The double steady-state analysis (Cob elli , Taffolo and Fe r r annini 1984 ; Ferrannini and others, 1985 ) has been confirmatory of the study of Insel and others (1975) in terms of number of compartments, but not i n terms of compartments 1 structure, The major differences of the new model (Fig. 12) are that insulin-controlled gluco se utilization takes pla ce in the slowl y e :·~ c hangeable co mpa r tment, as e :·~ pected on physiolo,;!ical gr'oun ds) ·:=in d that intercompartmental e :·( change parameter s change under insulin effect, Insulin in cr eases the irre ver sible loss parameter k(:' 3 and the e:{change parameter k3J (a 6 fold and 3 fold in cr ease respectively), and decreases the parameter k ·•. 3 (a 2 fold decre .:=ise ) , As a resu It , the s 1 0'...' 1 y compartm e nt is le ss slow than e :{ changing that of the model of Insel and others ( 1'7'75 ) '..lhen in su lin is ele v.=:Jted,

ENDOGENOUS INSULIN

TRACE!

TRACER GLUCOSE

TRACER

TRACER INPUT

Fi9'

11.

The fIIodel ( 19 75) ,

of

Insel

and

MEASUREMENT

others

the control should take place in the slowly e:·~changing compartment, they foun d that in subjects displaying a large insulin effect the model-based predictions

were

not in accordance with the

observed

data. Thus, the insulin -d ependent glucose flu: < was placed on the rapidly e:
INSULlN DEPENDENT UTILIZATION

INSULlN INDEPENDENT UTILIZATION

rno{jel

of

others The th ree -compa rtment model of Fig , 12 can be cond e nsed into a two-compartment model ( Fig, 13 ) by aggregatin g the plasma and fast-e :{ changing compartments, as detailed bv Cobe lli , Toffolo and Fe,-. ,-.annini ( 19B4 ), ....'hene v er the f·:Jstest mode ( ~J] , 6 mi n) c·un be neglected , This model retains the bas i c feattJ ·. -·es of the thr· ee-cljmp .~rtment '. . ' er·.=:ion , particular l y fo r what concerns the effects of insuli n , and ha s p ract i ca l relevance in se v era l C i rCLJm stances, Of note is that the

~~~~leth~:ea~~~:~a:~o::a!~:~st~:rr~:~er~~~ purpose Radziuk

two-compartmental models of and others (1978) suffer by h eavy appro :·~ imations, Model A ( Fig, 6a) does not

TRACER INPUT

MEASUREMENT

INSULININDEPENDENT UTILIZATION 13,

INSULIN DEPENDENT UTILIZATION

of

10

C. Cobelli and A. Mari

in c lude an y gluco s e utilization from the slow compartment; model B (Fig. 6b) has no physiologi c al basis for assuming equal losses from both compartments; and, more important, the intercompartmental e xc hange parameters (k 2 1 and k 1 2 ) are assumed not to be influenced by in s ulin.

-

3

Z:IQ

w8~ cn ... _

2

°UI'

gj3 E ..Jo ••

U)~

ILQ &

The two-compartment model of Fig. 13 constitutes the core of non-stead y -state analysis a s car ried out by Cobelli, Mari and Ferrannini (1987a). In f .act, the steady-state studies determine the initial and final configuration of the non-steadystate model and demonstrate that in the tr ans ition from the basal s tate to the normoglycemic hyperinsulinemic steady state (F ig. 10) the model p .a \'.ameter-s k"" ,_ and k(;)2 increase , k~.~ decreases I while k(:J ~, which represents in s ulin-independent g l ucose utilization, i s constant, Thus, what is needed i s to specify the timecourse of the model parameters between their steady-state values, To preser ve ph ys iologi ca l plausibilit y of the model, a siQmoida l time-cou rs e was adopted which is co~sistent with a con t r ol by insulin from a compartment remote from plasma. The ,model equations are thus

Qj_( t) =-kJ.].

Q,,, ( t )

( t ) Q,_ (t) +k",-, (t) (h

= k:.,,,_( t ) Q,_ ( t ) -k:."",

(t)


Q:;" ( t )

)

0 .In

Fig.

14.

Glu c ose production com p uted with the two-compa r tment ph ys iological model.

and

Radziuk's models, Glucose production I.'ith these models (Fi,;). 1~, ) t -",~_es on physically unrealizable negati ve va lues either after 40 min from the beginning of the e x periment (Steele's model) or in the l ast 20-40 min (R -a dziuk's mode l A) . Steele 's model predi c ts a rap id (within 20 min) and total inhibition of glu c ose p r oduction, while Radziuk's model, although definitely more accurate, predicts total suppre ss i on during the 2 r ,d ho u r. C(HYI~. uted

( 44 )

3

(45)

z:~

w8~ cn ... _ ( 47 )

2

°UI' E ga

.

~ al° • IL:

0

-I C*

(t )

k u_

=

Q,_ ~ ( t )

/1,),

6

(49)

(t)

( 50)

k :;":;,, (t)

(51)

o (t)

( k.~-k.~)

( 52 ) .J

J

~o

1~0

.In

15.

Gl ucose production computed with Steele equation (con tinuou s l ine) and Radziuk's model A (dashed 1 i ne) .

------------------- + 1I ;1: ..J-!J :1.: •.1

+ k :I

to TIME

Fig.

k" ...

30

I.J

2'1112 . 02

o k x ..1 and kz~ are the initial and fin~l steady-state va lues and !J1: ..~ and 'V :1:. J are re l ated to the d y nami cs of insu li n corltr o l . where

Modeling analysis has been perf o rmed by Cob'?lli, Mari -an d Fel'l' annini (1987 .",) ()n the mean data of Fig. 10 and with the mean steady-state parameter va lues (Co belli, Toffolo .: Jnd Ferr' annini 1984), I

Glucose production, as predicted by the model (Fig, 14 ), declines slowl y and prog ress i ve l y and re a c hes a va lue close to o only toward the end of the 2-h study; at no time point is a negati v e value ca l cul a ted. Glucose production in the 2",d hou r is still nonze ro and is suppres s ed on averag e to 1 7% of its basal va lue . This model provides a picture of the physiological system which is quite different from that obtained with Steele's

The r easons of these di scre pancie s a re discu sse d by Cobelli, Mari and Ferrannini (1987a), wi th the a id of the theor y outlined abo ve , The y s how that} although a direct unequi voc al model va lidation is not possible, the model of F ig. 13 represents an impro v ement upon previous approaches. While in fa ct there is no question that Steel e's a pproach i s too s impli s tic to be a ccura te} th e ad v ance o v er the models of Radziuk and others (1978) and Inset and others (1975) lies i n the fa c t that these model s could not include the results of the s econd stea d y -state anal ys is. Thus, the intercompartmental e :
11

Models for the Estimation of Glucose Fluxes 3ccura e modeling analysis can overcome the nconsistencies encountered with Steele sand Radziuk's models.

The validity of this approach has also been confirmed in a similar situation by Cobelli and othel-'s (1987b). This wOI'k shows that the limitation of the use of mean

data in the calculations,

paper

by

Cobelli,

Hari

and be overcome by

as in the

Ferrannini

(1987a) can performing both steady and non-steady-state studies I

in the some e x perimental protocol. The results} although based on a singl~ case) confirm the large discrepancy with Steele's model and indicate that glucose production is not promptly suppressed by insulin. THE

GLUCOSE SPECIFIC ACTIVITY CLAMP : GUIDELINES FOR THE EXPERIMENTER FROI'1 THE THEORY

An interesting 3spect of the theory de v eloped in Cobellil Mari and Fer ~ annini (19873) and briefly Qutlirled in a previous section, is the e x plicit relation between the specific activity and the error of an appro x imate model. Equations 36-40 show that if the tracer administration is adjusted so as to induce no change In specific activity over time, i,e, ~(t)=O, the error of Steele1s mod el is zero and the rate of appearance is given by Ra

(t)

Ra~ ( t )

la

(53)

(t)

It may similarly be shown that also for Radziuk's models, and for any other compartmental model, the errop is zero '..Ihen .~(t)=O, and R." ,(t) is again gi v en by equation 53. A theoretical proof that the rate of appearance i s predicted correctly in non-steady by equation 53 if the specific activity is e xac tly constant has been given in Norl,o.,lich (1973) fDr-' a distributed system of convectiondiffusion-reaction, The formulas developed by Cobelli, Mari and Ferrannini (1987a) not only prOVide the same reslJlt for compartmental models, but also show that a reduction of the magnitude of i Ct) will improve the accuracy of the appro }{ imate models even if

.3 (t);tO. It seems therefore reasonable for the investigator to try to clamp specific activity at a constant va lue by varying Ra-(t) in a suitable way, In fact even if the guessed format of tracer administration is not e~·: act , so that the specific activity is e :·: a c tly constant, a diminution of magnitude of i(t) with respect to a lib I i nd tr .3 cer adm i n i stro3t i on is likely to be obtained and the calculation of Ra(t) will be less dependent on the validity of the chosen model. Of note is that the idea· of reducing gradients in the specific 3ctivit y wa s originally proposed by Steele and others (1965) and unfortunately followed by onl y a few investigators such as Cowan .",nd Heteny i (1971) , I nse l a n d others (1975), Wolfe (1984). I1

This intelligent tracer administration may sound as circular reasoning because adjusting the tracer infusion rate presupposes knowing how the specific

acti v it y would change, i . e. how the system behaves during the particular perturbation in question. Howe v er, some a priori informa~ion about the behavior of Ralt) when the system is displaced out of s taedy - state is often available, and can be used to design the format of tracer a dmini s tration that will smooth out brisk or large changes in specific activity. This has been called the method of Ilverified guesses ll by Norwich (1973) the first guess can later be verified by measuring specific activities and if one misses the goal the repetition of the identical e x periment will improve the subsequent guess, An experimental exemplification of the method is contained in the original reference. Another possibility concerns the development of measurement methods of tracer and tracee which are rapid enough to allow t r acer administration in a closed loop scheme through appropriate algorithms ( Ncn-· I,..'ich! "1'7'7 3 ). This is not an easy task for glucose tracers even if clearly the slower the e x pected changes in Ra(t) the less rapidly specific activity needs to be measured. CONCLU5 I ONS-. This paper has surveyed e Kperimental methods and models for determining glucose flU Kes in the intact organism out of ste~dy state. The physiological basis of the problem, its importance as well as the necessity of indirect} model-based measurements ha v e been discu s sed, Tracer e){periments need to be carried out in order to so lve the non-steady-state problem which can be viewed as an input estimation problem. However their interpretation is difficult since in nonsteady-state linear time-varying compartmental models are required for des cr ibing their kinetics. This has favoured the development of simple approximate models whose use has unfortunately become common practice in physiological and clinical investigations, These models h~ve been critically reviewed in the paper and their structural errors as well as the ill-conditioning of the numerical problem have been outlined. The prin c iples of an alternati v e approach based on phYSiological modeling of the system and ffiultiple steady-state tracer e l
l~

C. Cohclli and :.... !\lari equations

FEFEF:EIICES 2.er

rn ·:Jn

R, N,

l

I

(=:,

Idee!

. ':"',

~Q!:!:!~~:!;:~

R,

E.o'",lden,

and

obe li (1979 ) , QIJantitati v e estimation I)f ftSU in sensitivity. 8m~ ~~ Eb~§iQl~ 236 : E66 -E677. (19B3) , mc,de 1 s s'y' :::. tems. A

A, Cobelli, -3nd C" l)a1 id .3tion of m .~thematiC: -31 c: omple ;'endoc:rine-metabolic

CBse study on a model of glucose regulation ~~~~ ~iQ!~ ~D9~ ~QmQ~t~ 21 : 3903':';"i,

Cobelli,

C"

t 1af'i .

H,

and E,

v

Fer!" .3nnini

(1987), Non-stead'y' state : el-T'e))"' .;'lnal'y'sis of Steele's model and developments for ':dlucose kinetics, 8m.!. d.!. Eb~~ig!.!. (in p'-'ess) , Cobelli,

C"

A,

I-\.="'i ,

G,

ToffolD,

Cobelli,

C"

!E8~

G,

~Q~!~

Pacini,

G,

~QD9C~§§'

Toffolo,

Cobelli, C'I G, T':lffDlo, and E, Ferrannini (19:34) . A model of glucose kinetics and their control by insulin, compartmental a nd noncompartmental 3pproaches, U~tb~ §ig§£i~ 72 : 291-315, Co·.... an, ,J, 5.} and G. Heten'y'i , ,J,-, (1 '3'71) , Glucoregulatory responses in normal and diabetic dogs recorded by a ne'...' tr·a c er· method. ~§t~~Q!i§m 20 : 360-372, DeFronzo f;:~. A,} ,J, D . Tobin, and ~~, Andres (1979 ) . Glucose clamp techn i ques a method for quantifying insulin secretion and resistance, 8m~ 1~ Eb~§iQ!~ 237 E214-E223, j

Fer-r·annini, E., ,J,D, Smith} C, Cobelli, G, Toff010., A, Pilo . and R,A, DeFronzo (1'7'85), Effect of insulin on the distribution and disposition of glucose in man. 1~ ~liD~ !D~§§t~ 76 357-364, Inse1, P.A.! J,E. LilJenquist . J,D, Tobin} Ft' ,5. Sher '...' in.. P . 1,l,1·3t kin:=-. . R, Andres ., an,j r·'1, germ.3n (1':';'75) , In:=-.ulin control of glucose metabolism in man. A new kinetic analysis, ~~ ~!iD~ !D~~~t~ 55:1057-1066. T,e "

e ,

Issekutz,

,Jr,

and D,

Elahi ( 1';'74), E:=.timation of hepati,::: glucose output in non-steady state . The 3 simultaneous IJse of 2- H-glucose and :L~· C-glucose in the dog, ~~n~ 4~ Eb~§iQ!~ Eb~cm~£Q!~ 52 · 215-224, Jacquez , 8D~!Y§i§

ed,).

The

J,A, in

(1985)

~iQ!99i

§Dd

~gm~~[tm~nt§l

~§~i£iD~

University of Michigan

I
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