A Model for Glucose Metabolism in M a n Jordan L. Spencer DepartTiient o j Clieniical Engiiieering a i d A4pplied Chemistry, Colicuibia I.nicersity, Sew Ivork, .V. Y . 10027
Calvin L. Long and John M. Kinney Departnient of Sitrgery of the College o j Physicians a n d Surgeons, Coliriubin I*niversity, .\-ew
I'ork,
.v.y . 10027
The development of a three-compartment model for glucose metabolism in man is discussed. A nonlinear regression method was used to determine the parameters of the model by simultaneously fitting typical blood glucose and breath CO? specific activity data obtained after intravenous administration of C'*-tagged glucose. The three-compartment model i s compared with a two-compartment model on the basis of ability to fit the data, confidence limits on the best parameter values, strength of parameter correlation, and sensitivity to changes in certain fixed parameters. The three-compartment model fits the data better than the two-compartment model, and in most but not all cases has reasonably well-determined parameter values.
T l i e I i u i i i n i i iiotly cnii lie v i e w ( { i calieniicxl iwctor cloyely liiihcil t o a coiitiol -all Iich:ir-ior i i go~-rriied by iiitcractiiig 1ir 1-iiig chemical mictioii::, uid flon-, :iiitl :i v:iricty of t'ralisport of 1ii:iss : 11 a i i t l inechniiic:il pheiioniena. T h e complexity of teiii i-: so great that :iquaiititativc tre:itinciit of the otly i q w r y tlifficwlt. I n ntlditioii, it i-: oftcii iiiil)os>il)lc to i.:olntc dl*y3tetii< \vliirli c x i i lie >tutiied iiitlcpciitleiitly. Severtheless;, rerelit ,clopiiieiits in mniiy :irc:is, :itid especially iii 1ii:ithcmati i i i d c'oiiilxiters, havc ninde the complexity of the Iiocly *e le>>forinitl:iblr, aiid tletailetl q i i n i i titntive iiivcstigiitioiis :ire iiow becoriiiiig possilile. ('1icniie:iI eiigiiieers have coiitril)iited :ind iiicwnsiiigly will coiitriliiite to this work, licc:iu*c their training equips them well t o iuitlerstatid :iiitl iiitcgrat? the liroeews that are cseiiti:il t o the operatioii of the liody. 111 retiirii, t h e method. developed hy p1iysiti;iiis; physiolo , aiid Iiioc~liemi~tz foi, the ,study of the hotly ni:iy \w11 li:ivc :ipplic:ition to the cotiiples iioiiliviiig systems oftcii ciicoiuitcrctl hy ciiginccr*. 111thi.: piper we shonlion. ni:itlicniaticnl a i i d >t:iti-tic:il nictliotl*, of the kiiirl ofteii used by eiigiiieci,s i i i htu(lies of rcactioii kiiietich :tiid reactor performuiice, cnii lie ii>ed t o study tlic kiiietics of glucose metnboliwi iii niaii. C14--glucoseha. h i i i i w l :is :I tr:iecr i n m ~ r : i htudicq l of glucose iiictnboli3ni i i i iiori1i:iI (13nkcr et XI.) 1954; Rcichnrd et al., 1961), di:il)ctic (Slirccvc et iil.! 19X), critically ill (Long ct nl., loll), :ieroiiirg:ili(~ (1Iaiioiigi:iii et :iL, 1964), and esercisiiig (Youiig e t al.. 1967) i w i i i . 'Thc lxisic experiineiit h n i iismilly iiivolved injecting iiitrnveiioii4y :i kiion-11 amount of uiiifomly lalit~letl ('l4-g1wose, although Kegal et, :i1. (1861) iihcd glucohe labeled i i i sliecific carbons. .liter the injection, lilootl and hrenth s:iiiililes are takeii :daiialyzed to deteriniue the ipecific :ictivity of I~loodglucose x i i d liimth CO,. The rate at x1iic-h ('02 is esliirecl i q also dcteriniiicd, so that the conipletc d:itti from :i run coiiqist of tlie niiiouiit of nctivity injected, the nvrrnpe rate of CO, productioii. :iiid vnliies of lilood g l w o v ant1 1ire:ith CO, specific activity :it n series of tiiiies followiiig iiijection. 'These dat:i eontiiiii roii.itlei.:ible inforiiiatioii oil the 3t:itr of roee::+s of gluco.;e inetnliolisii it require fiirtlici n i l before hpeeifir \--aria\ile> wrii lueosc oxiclntioii rate. gluco-:r turnover rzitc, :iiitl tlie nmoiint of glucose i i i tlic liotly 2
Ind. Eng. Chem. Fundam., Vol. 10, No. 1 , 1971
be detemiiied. I n this :iii:iIyGis it is 1iecehs:lry t o use a iiiutheni:itical iiiodel for g l u c o ~inet:iboliom. 'The lnodel relate> the iiiensurahle vnrinbles to the variables of interest, a i i d t h e finid results depend for their +qiifieniicr on the validity of the iiiodel chosen. Aifurther p u r p o s of this pnper is to prc>ciit :I iielv model for gluco.;e inetnbolism aiid to give a thorough cli>cus-:ioiiof eertniii featiiwh of the niotlel nnd its II.SC, iiicludiiig the method of pnraincter clcteriniii:ition, the >t:iti>tic:il wliabilit'y of the results, the ability of the niodel to fit xvailalilc data, niid the effect oii the rehillti of possible orrors iii the iiie:i>iireiiwiit of ~ c quniititics h :is the nrnouut of iictivity iiijectccl :iiitl the rntc of ('Os 1)rocliirtioii. (xii
Model Development
'The gciicr:il prolileiii of inotlel dcveloliiiieiit (or syoteiii idciitificxtioii) c a i i be stated n h fol1on.h: Given :I set of data, both qun1it:itive aiid qii:iiititntive, t o fiiitl t h e hiinlile>t niodel consistent with tlie giveii d:ita. Tlic quiilit:itive data niay coiisixt of the kiiowlcdge that eert:iiii proeesich c:in occiir niid that other.: :ire foi~hitltleii-for cxaiiilile! while glucose c a n be ositlizcd to earlioii dioxide, i i i ni:imni:il- this l>rocess is irreversible. \l-e label :is rcnsoiinhle models that iiiclude oiily prores>e-: that actually can occur. Froin the >et of all renson:ible niodels we cnii iiiiiiiedintely rciiioi~e,a:: beiiig too simple>those thnt : i i ~iiicap:ilile of fitting thc qmiiititativc data. I{iit thew i w y still reniaiii reasoiinlile inodcls n-liicsli xi'c cs 'Tlie*c model- fit t.he qwiititativc d;it:i very well. but :ire rcvcalctl by the fnct thnt their lxir:iiiictcrs :ire oiily pooi~ly deteriiiiiicd, i i i the sense t h a t iiiaiiy coiiil)iiiatioiis of 1)nr:iiiietcr v:ilucs gil-e :iliiio>t equ:illy good fit. to tlie d a t u . If the overly coriiplcs inodela nre esctludetl. the motlcl de\,clopnieiit pi'occ~:: should tcriniii:ite i i i a rnodel (Figure 1) it-liicah involves oiily real Iirocesses, fits the qri:iiitit:itivr tl:it:i w l l . and has well tlcteriniiieti p:ir:iiiictcr values;. lye x i l l attempt to develop ~ w liii inotlcl for gluc~o>e iiiet:iboli-ni i i i i i i a i i . IIost niodcls for gIiico>e iiiet:iliolim rcliorted to k i t e have Iieeii 1i:i-ed oil the :ishiiiiil)tioii tlixt the ('11 iiijccted intravciiously glu(whe tr:lvcr.cs oiie or n i o ~ ecolii1):irtiiieiitx or liools befoi,c lwviiig iii t h e 1irr:ith t i s ( ' 1 4 0 2 . .l 1)ool maJ be tlcfiiiecl :I- :i collectioii of tr:~cer-c,ont:iiiiiiigiii:ttwi:iI a i c h that the ,+ecifir :ictivity of i i i i t r r i a l Icaving thc pool is equal a t
all times to the average specific activity of material within the pool. The inaterial within the pool is usually aswnietl to have uniform specific activity-i.e., t o be well mixed. The differential equation w h i c h deqcribes the tmnsietit response of a single pool is
PHYSIOLOGICALLY REASONABLE M O D E L S 7
y M O D E L S THAT FIT T H E DATA TOO COMPLEX
-
-
NOT REASONABLE
FINAL MODEL
where -1 i q the amount of tracer or laliel in the pool, S ! the pool zizc, is the total :imouiit of tmced material in the pool, aiitl R ih the rate a t which traced inaterial leave3 the 11001. The ratio . I S is ralled the specific8 activity of Inaterial i i i (or leaviiig) the i)ool, denoted by y. Equ:itioii 1 hold. evcii irlirn S i i not miistatit, but when S is con-taiit it c a n be rewritten to give the b a > k equation for a pool d!J
-
dt
=
1
s ~
[(tracer input rate)
PARAMET E RS
Figure 1 .
Model development criteria II
T w o - P o o l Model - M o d e l
R,tR3
{
2I :
Glucose
I 4 -1 R,
Oxidation
- Ry]
I n :ill that follow> we w-huliie that the g1uco.e met:iboli-:m teni, :i-h revenled ill ("l-glucose experiments, is describable i y n:ultil)ool niodcl; n.ith conitatit pool size-; and transfer 1,:itee. Thi. i q i r i r c s fiilzt that the teni be liiiear aiid timeiiivatkiiit. 'The as-uni1)Tioii of liiie y i. ncll ju>tifird. +ice i3 coii~itlciecl :I perfect tracer for C'I?. However, the unii'tioll of tiriic iiivariatice is not cwctly correct .inre re i- iio rc:i-ott to cspect the r:ite of gI:ico-e osiclntion 01' the blootl gliicwe level to remaiii coii$tant for so long a - 6 hour.. Sevcrthcle.i, thp :i is that tlicrc is a rcgion i i i 1i:ir:iiiieter s p c e around the best 1iar:iiiictcr cstini:itc.;, b k , t1i:it coiitain*, with a cwtniir prolia1)ility (*ny !Xi%,), the true 1x1rameter values. This region i-: liounrled by n surface (contour) on which the slim of sqiiarcs function 0 has :i wrtaiii fixed value correspoiidiiig to a fr;ictionnl iiicrc:isc in Q of 6. For 9570 confidence limit5 6 is given by (Box, 1960)
+
n-here I.',(P, JI A' - 1') i* the cy sigiiific:iitcac poitit for t h e I.' distribution \\it11 1' :itid JI S - P drgrccs of frcctlom. I n fiittliiig ixiranieter coiifitleiice limits, we esscnti:illy waiit to explore in c e r t a i n tlircctioiis in I'-diineii~ioii:il S~J:ICC :II'(JIllld the niiiiimiiiii of $, e.tiriiatiiig or calciil:itiiig tlirecatly t h e tlist:tiice Ivhicli ivill iiicrcase Q 11s the -:i)ccificd ariiouiit 69. Eqilorntion :doiig the axch of pnranieter space correslionds to varying oiie par:imeter a t a time, and will lead to iiiirc:ilihtically Ion- coiifideiice limits if parameter correhtioii is prcseiit, since the remnitiiiig p:irnnieter.j :ire not allon-ed to ncljiist to the c1i:iiige iti the varied parameter. Thus we calculate what nre cdled hripport plane coiifitleiice limits. Let b denote the vector of hest pr:imeter e*timxtes. The contours of 4 corrralinnd approsimately to a hyperellipsoid centered :it b, with priiicipal :isw defined by the eigenvectors of B (or C). For the kth parameter n-e can define two support planes as thoie pliities, tangent to the hyperellipsoid, which nre normal to the k-axis. The iiitcrscctions of the wpport p1:inw with the k-axis :ire the upper : i i i d lower support pkiiir coiifideiice litnits oii the kth parameter b k . The tliat:iiicc from bi; to the support plane is given by [ 6 ~ C k t ] ~ 'where * , C i i the inverse of B atid 6 i.i g i v i i by Equation 10. Tlius n-e c:iii w y 6 Ind.
+
Eng. Chem. Fundom., Vol. 10, No. 1, 1971
that, with prolmbility 1 sati4ea
CY,
the true Ixiranicter value bn*
Table I.
Parameter Values and 95% Support Plane Confidence limits for Models I1 and 111. M o d e l II
Parameter
R1
Rz RB
M o d e l Ill
Data from Subject A1.C. ( F = 6.70, D = 27.28 Microcuries) 10.02 f 100 3(-53 9, +123.2)% 1.87 f 1 0 . 2 ( - 8 . 1 , $9 21% 1 . 9 3 dr 1 6 . 7 ( - 1 3 . 5 , + 1 6 , 0 ) % 1 . 6 9 + 1 1 . 7 ( - 9 . 0 , +7 91% 1 . 9 0 f 16.7(--14.8, + 1 2 . 5 ) %
s 1
sz
s 3
+ RB + sz RhIS error for
336 i 4 . 5 ( - 4 . 5 , + 4 . 6 ) % 681 f 31.4(-29.3, + 2 4 . 8 ) %
124 f 53.3(-28.5, 224 f 39.0( -48.6, 666 f 20.5(-12.8,
2.95
x
10-3
3.56 + 4.570 348 f 7 . 7 % 1 . 2 3 x 10-3
blood data RXS error for breath data
o
48
x
10-3
0.45
x
52
1.56
x
0.44
x
Rz
3 . 8 3 rt 5 . 8 %
SI
10-3
Data from Subject 1I.C. ( F = 8.57, D = 19.55 Microcuries)
+ R3 + sz Rills error for
10.31 f 18 270 4 . 3 8 dr 37.5%
15 76 f 21 1% 5.38 f 23 9% 2.54 dr 84.3%
821 Z!Z 1 3 . 2 % 3197 f 25.970
1041 f 61.7% 550 i 1 8 . 3 % 1573 + 39.07,
2.02
x
10-3
7 . 9 2 f 38.670 1591 =!= 43.6% 0 . 7 1 6 x 10-3
blood data RNS error for breath data
2.17
x
10-3
0 . 2 2 3 x 10-3
52
0.551
Rz
14.69
SI
a
f
17.070
x
95% eigenvector exploration limits shown in parentheses. Units: R
Model I1 and Model 111 have been used, so that differences between the models can be demonstrated. Unless otherwise noted, in the parameter determination calculations equal = 1.0) were used for blood niid breath d a h , weights (iL7Q/li7G and this produced for Model I11 a n R l I S (root mean square) residual error (based on t8henumber of data points) for the blood data two or three times larger t'han the R l I S reridual error for the brenth data. While values of \ l T g / \ i T of C 2 t,o 3 corresponded to the theoretically desirable case of equal RbIS errors for the weighted blood and h e a t h data, it will be shown later that the parameter estimates aiid confidence limits are relatively insensitive to the choice of weights. The first data used w r e obtained i i i a C14-glucose study on a normal subject. The details of the experiment' and a desc,ription of the subject' (11.C.) are given by Long et a1. (1971). The study was selected as being typical of the 13 studies on normal subjects presented by Long. I n Table I are shown the best parameter values for subject M.C. obt'ained using Model I1 (tivo pools) aiid l l o d e l 111 (three pools). The calculated blood and breath curve.; for bot'h models are hhown wit,h the experimental points in Figures 3 and 4. With the exception of a single perhaps questionable blood specific activity point a t 7 5 minutes, 1Iodel I11 fits both the blood and breath data very well. It is clear that Xodel I1 does not, fit nearly so well, maiiily because the single exponential decay for blood glucose specific nctil-ity
=
0.114 milliatomi C /min, S
x =
lo-' milliatomi C.
predicted by Model I1 c:iiinot fit the blood (lata which, when plotted senii1og:iritliniic:illy~ show a c u r v a t w e corwspoiiding t o :It 1c:iht two espoiiciiti:il5;. I n f:ictt tlii.: cui.vatiire, noted also by Rcic8h:trtl et :d. (1961) :\lit1 8eg:il et :rl. (1961), is a major factor i i i showing the need for :i m ~ o n t gliwohe l pool. 111general, lack of fit due to :iii iiicoriwt or o\~rrsinililificd niodcl should lie revenled by nn inipi,ol):iljle o i ' iioi1r:iiidoni distriinitioii of iesitiu:il.. 111paiticul:ir, with t1:it:i ol)t:iiiicd sequentially, we n-ould expect :I ixiitloni occwrciirc of po-itive niitl iicgativc rcsiduiils. This r:intloniuc.:s ( x i i lie te>tctl for i)y using a riiii.: test, :i3 descriiied by Sicgel (1956). .i r i i n is a group of one or niorc consecutive rcsitliials of the s:\nic >ign, It c:iii be seeu iu Figiiw 2 ttint for both n m l c l i the iiiiniljer of ruus for the lilootl c1:Lt:i i iccoidiiig to the riiiis t w t , \rith 95% proliability the iiunilicr of i u n h shoultl lie 1)i~tn.ccii2 9. At thi. qiguificxnce level 110th models gi1.c :t tli.:ti,iliiition of residiinls JThich could well lie r:iiidoni. Loiwring the ~ the quentioiinhly high blood value at 7 5 ~ i i i i i u t riwliicc. iiumlier of r u n h for l\Iotlel I1 from 5 t o 3, c>loscto h u t not pi-t the critical value for the ruiis test :it the 9SyGli~rcl.' h i i s the runs test is iiot quite alile to dcnionstr:ite hat is c r i t l i l i i t t o the eye, liec:iu>e the i i i i i i i h r of lilootl t1at:i iioiiits is -0 m : i l l that :ilino>t aiiy iiumlier of ru1i.j has :I rensomiljly 1:irgc 1)1,ol)ability. 1311th niodeli fit the lireath &ita so ~ v c l lt h a t tlic residu:ils occwrred rniidornly, a t least as judged by tlic riiiis teht a t the 957Glevel. Ind. Eng. Chem. Fundam., Vol. 10, No. 1 , 1971
7
0 IO
1
E
0
e
:
v)
W
u) 0
0.02(I)
n
0
m
I
0
Figure 3.
1
40
0
80 TIME
I
I
-
200 200
160
120
MINUTES
Best fits to blood glucose specific activity data from subject
M.C.
N
8
0.008
(Y
0
E E
2
=L
0.006
I
>.
k 0.004 0
a 0 LL
p"
0.002
VI
c" 0 0
40
80
120
TIME
Figure 4.
160
200
240
280
: 0
- MINUTES
Best fits to breath COn specific activity data from subject M.C.
0
50
0
100
150
200
TIME - M I N U T E S
Figure 5.
Best fit to blood glucose specific activity data from subject
M.G. 8
Ind. Eng. Chem. Fundam., Vol. 10, No. 1 , 1971
One me:isurr of h c k of fit' is the viiriaiice cstiinat'e s2,where q,'(-ll S - 1'). The variance estimate for Model 11, s? = 1.56 x 10-6, is larger l)y a factor of 3.68 thaii the correspoirdiiig value for Model 111, s2 = 0.43 X This is due maiiily to the fact that, for Xodel 11, the KAIS r e d u a l error for the tilood (lata ii: 2.39 times larger than the correspoiiding value for ;\lode1 I l l , while the TU18 errors for the breath dat'a are :ilmost rqual. The lack of a good estimate of variance due to cxliri,imciitnl erroi' mnkcs it difficult to give a statiqtical tre:itmcnt' of the differelice betn.ecn models. However, if we ex:iminc thc wikince ratio of 3.58 using the F test' with 56 and 58 degree- of freedom (corresponding to the tot'al number of expcrinictit:il poilit5 less the iiunilier of parameters), we fiiid that Notlcl I1 is sigiiificantly less good than Xodel 111. Finally, t h t RLIS eri'or for the lilood data, for Model 11, is 3.So/G of thc inasimiini 1)lood value, iind thus somewhat l a r g c ~th:iii n.oulrl be expected from cxperimeiital error. ;ill the evit1eirc.c~coiiihiiics to show that Model I11 fits typical data better than 11odel 11, and fits so ncll that a more complex model ic pi,ol)aI~lyiiot justificd. To drnionstixtc how the two models differ, and to show clearly that 1Iodcl I1 is in some c a w ? uiialjle to fit data from C14-glu~osc i'un.:, :isecond study (sutiject' 1I.G.) was selected fi.oni tlie 30 htudieq iq)orted hy Long, 011 the hasis of the largeat tliff'ci~cncri i i cwinpaixlile parameter values from the two motlcls. T h e 11:ilanicter estimites and the best fit curves are .showti i i i 'Yal~lcI :inti Figureh 5 aird 6, rehpectively. I t is clear that XIodcl 111 gives the better fit to the blood d a h , while 110th niodrlc fit thc breath data about equally well. The ratios of R I I S rcsidunl errors for the blood and breath data :ire 2.83 : i t i d 0.97. i,ry)crtively, and thc ratio of variance estimates, 1):wcd oil all data, is 4.85. Xgain u+ig the I' statistic to test tlic vaii:iiice ratio, we reject the iiull hypothesis that Xlodrl I1 fits :I* well as Model 111, rveii at, the 99.8% signifiraiicr level. I t iq iiitciwting to compare the 1):ir:inictc.r values obtained uainp tlic tv-o niotlrls. Table I1 containh pai,:inieter values for subjects 1 l . C ' . ( ~ r o r i n a l )and lLI.G. (critically ill), with t.he sum SI SI2 froin Xlodcl I11 compared to the value of S?as computed u h p AIodel 11. For normal subject M.('. none of the comp:ir>ibleixtes or pool size' except S2changes by more than l I y G i i i going f i w n AIodcl I1 to hlotirl 111, and the sum S1 Sz for l l o d r l I11 agrees well \\-ith the compar:il)le value of Sz f i , o n i Alodcl 11. However, ;\lode1 I1 give.: :I much larger value for Ss thaii Nodel 111. If S pis intcrliretcd a s the glucose iii extlacellulai, fluid and uqed to calculat,e t,he volume of extracellular fluid from the blood glucose level, the use of RIodcl I1 could lead to a major overestimate of this volume. I n t h t c : i v of subject XI.G., the agreement lietween compai:itilc pnr:imcter valurs is much leqs good, although it should t x eniphasizcd that' this case wis choxni especially to denionstrate how the two models c:in give different, result's. On thc assumptioil th:it, the parameter values from Model I11 are correct, we see that an analysis based on Model I1 can seriously overestimate Rz, RI, S2,and Sa,while underestimat'iiig the amount of glucoie or glucose equivalent substances prescnt. Confidence Limits and Parameter Correlation. It, is useful to kiiow hoc7 r,eliahle the parameter estimates are, and this reliability i i b e s t expressed in terms of confidence limits for the paramctcr vnlws. Confidence limit,s, calculated in several ways, are shown i n Tahlc I for subject 11.G The 95% cigeiivector explorat'ion limit.: (found by exploring along the eigenvectors of parameter correlation mat,rix) are prolmhly the mort realistic limits, since they take into
+
s2 =
-+
+
Table 11.
Comparison of Parameter Values from Models I1 and 111 Rates
Pool Sizes
-
---
Subject 1I.C
Rz
Xodel I1 1 93 1 90 Model I11 1 87 1 69 yc diffeiencea -4 -11
sz or
+ Rs
R2
Rs
+
Si S2 336 348 -5
3 83 3 56 -7
SP 336 224 -33
Ss 681 666 -5
Subject 1LI.G 821 821 3197 Model I1 10 31 4 38 14 69 550 1573 1591 Nodel 111 5 38 2 54 7 92 -40 -68 -64 -60 % difference -63 - 5 2 a Baaed on average parameter value Unit< X = milllatomi C/min, S = milllatoms C
9 """, 0"
0"
.
E 0003
U
I
;0 0 0 2 ? + U
O.OO'if ON
U
O
0
I
I
I
I
I
80
160
240
320
400
TIME
-
480
MINUTES
Figure 6. Best fits to breath COZ specific activity data from subject M.G.
account both nonlinearity a i i d p:iranictt,r coi~relatioii.We note first that the eigenvector exploixtion limit.: ai'e close to the 95% support plaiie limit's n-heii both are closely sp:ic*ed, but caii be c o n s i d e i ~ a l differciit ~l~ when the limits we wide. Thus it can he of value t o calculate 110th kinds of limit when the possiliilit'y of pooi~lydefined p:irameters exists. I t is :il.:o evident that' the upper and lower eigenvector esp1or:ition limits are not aliwys equidistant from the best pnr:mieter vtilues, which indicates that no~ilincxiity is present. The eigenvector exploration limits were found to be f:iii.ly sensitive to small changes in the parameter values (011 the order of O.lyc)such as those due to converging from different initial guesses. For Models I1 and 111 the 95% eigenvector exploration limits are about' the samc for the eqiiivaleiit parameters Rz, R a ,aiid Sa,with Model I11 giving slightly n m w v e r limits in spite of its greater complexity, because both lack of fit, as measured by the variance estimate s2>atid parameter correlation contribut'e to wide confidence limits. Model I11 fits bet'ter t,hnn Model I1 but has stronger parameter correlntioii, aiid these effect's tend to cancel for some of the paranieters. Xs showi by Rosenbrock and Storey (1966), the support plane confidence limits for any linear combination of parameters can be estimated when the C matrix is known-for example, for bi b, we use C i i 2CZj Cj, in place of C l ; k in Equation 11. Table I shows the 95y0 support plane confidence
+
+
+
Ind. Eng. Chem. Fundam., Vol. 10, No. 1 , 1971
9
Table 111. Parameter Correlation Matrices Parameter
Ri
R1
SI
R3
RP R3
0.11 -0.83 0.69 0.08 1. O O - 0 . 1 9 1 .00
1 .00
sz
53
S2
SUBJLCT 1I.C. Xodel 11 1 . 0 0 -0 76
s 3
Model 111
RI R2 R8
0.30 0.01 - 0 . 0 3 0.95 -0.98 0.79 1 . 0 0 -0 67 - 0 . 0 3 -0.01 0 . 0 7 -0.57 1.00 - 0 . 1 3 0.25 1 . 0 0 -0.97 1.00 -0.31 1 .oo
1.00
S1 SS s 3
SUBJECT 1I.G. illode1 II 1.00 0.00 1.00
RP R3
0.31 0.21 1 .oo
SZ s 3
-0.48 0.47 -0.17 1.OO
Model 111
Ri R2
1.00
-- 0
49 1 00
R3
-0 23 0 56 1 00
s1
sz
0 22 -0 68 -0.94 1 00
S3
-0.51 -0.35 0.91 -0.17 -0.41 0.55 0 . 4 4 -0.71 1 .00 - 0 . 3 2 1 .00
+
limits on the gluco.:~turnover rate R, R a aiid the glucose system size S1 S 2 , as calculated for subject 1l.C. using hlodel 111. Khile Si aiid S2 :ire deteimiiiied only within 53 and 397,, respectively, t h r sum S1 SSis deteriiiiiied withiii 7 . 7 7 , , almo
