\

ESTIMATION OF TRANSPORT RATESBY RADIOISOTOPE STUDIES

OF NON-STEADY-STATE SYSTEMS

earl M. Metzler, Gennard Matrone and H. L. Lucas, Jr.

This investigation was made possible with the aidof a fellowship under Public Health Service TrainingGrant Number GM-618 from the Division of GeneralMedical Sciences and computing service providedunder Public Health Service Grant Number FR-OOOllfrom the Division of Research Facilities andResources.

Institute of StatiS"tics M1meo Series No. 446~pt~r~65

.A-l~

TABLE OF CONTENTS

LIST OF TABLES •

LIST OF FIGURES

1. INTRODUCTION •

2 • REVIEW OF LITERATURE

2.1 Tracers in Biological Research2.2 Compartment Analysis2.3 Criticism of Compartment Analysis2.4 Other Methods of Analysis •2.5 Summary of Literature Review.

3. A BIOLOGICAL PROBLEM

4. MATHEMATICAL FORMULATION AND EXPERIMENTAL CONSIDERATION,SYSTEM I: THE PLASMA-RUMEN-SODIUM TRANSPORT PROBLEM •

4.1 The Problem4.2 Definitions and Assumptions4.3 Derivation of the Estimates4.4 Experimental Considerations

5. EXTENSIONS OF THE METHOD •

5.1 Two Compartment Models •5.2 Three Compartment Systems •5.3 Limitations to Application of the Method •

6. AN APPLICATION AND COMPUTER SlMULATION

6.1 An Analysis of Slyter Experiments6.2 Computer Simulation of System I .

7. DISCUSSION.

7.1 Advantages of this Method •7.2 Weaknesses of this Method •7.3 Suggestions for Further Investigation.

8. SUMMARY.

9. LIST OF REFERENCES

10. APPENDICES.

10.1 Solutions of Systems of Linear Equations10.2 Data from Slyter Experiments

iv

Page

v

vi

1

2

247

1415

16

· 20

• 20

· 22• 27• 35

39

394453

61

• 6167

82

828384

86

88

93

9396

6.1

6' r.c

LIST 01" TABLES

K+ ·."C' ::.L' J(t) from data of Sl.yter

'Io';,: {'i>U:; J' ,"::JCiiu.m transported from plasma to rumen''''U~",: ':Lth analys1s of variance

v

Page

63

64

of' sheep and weight in kilograms 64

6.4 Gl'<': .:,:!i.l' transp,Jrted per kilogram of body weightL, l::.u.':'< a1'Ce1' i.n,jection of tracer 66

69

], (t) fry! Eim;u.Jated curves.''r. ~_

75

10. j

10.2

97

98

10.3 Dn:.. '; '" 99

vi

LIST OF FIGURES

Page

", 0 Example of a transfer problem 3

1+ • .l Sys tem I

1>.2 Example of a system not satisfyi.ng Assumption 4.8

1.:: ,'.,:,.., f,l::,

System r(a):

System I(b):

a modification of System I

the two-compartment open model

27

34

39

42

SysteDl I T.....).4 System IIJ.: e. thrE'e cumpartment J modified catenary system 46

~). (; An N-compartment, one-way catenary system

)0> System 1II(a): a model for the plasma-rumen-omasum system

5.7 System III(b): a three-compartment catenary sYstem.

5. 13 System IV .)·9 System IV(a)

.~ig 10 System IV(b) .

49

52

52

54

57

59

67An analog of System I for a simulation study/ ­:.~' G L

0.)

Simulated Curves 100, 120, 125 and 130; illustrating thesteady·~state situation and the effect of changes in B

O

Simulated Curves 130) 138 and 139; showing the effect ofchanges in total rumen sodium

70

71

6.6

Simulated Curves 126, 127 and 136; illustrating the effectof changes in B

l•

Simulated Curves 130, 136 and 137; illustrating the effectof changes in B

3•

True values of B21 and average and s.d. of b21 forSeries 120 and Series 130 •

73

77

True values of B21 for Curve 130 and b21

values for threeerror curves 01' Series 130 78

LIS'r OJP FIGURES (continued)

'rrue values .• B91 , and average and s.d. of b21

forSeries 160 ana: 161

True values, B21

, and average and s.d. of b21

forSeries 137

vii

79

80

TNTEomiCTIQTJ

V and bi.ocbero.5,stry r'JdiC'j2~)L)r'.';; :.1re a major

to~)l" the rc,eiU'CfJ lnvesti gator. The reason for this" parti.cularly in

and metabolism., is indIcated by the followi.ng

"'I'hr',)ughout pructic811y the whole history r)f bluchemical.1 nveBtigatJon Bttem:pts have been made tD f:ind a method oflabeling organic compcu.nds which 'vlould enable these com­pounds to be traced in their passage through, and excre­t.i.on from, the animal body. La'bels Euitable for thIsFIypO;38 arr;; eli f'ficuJt t:] fi·t',d. part.i:~1)l if€~.rou.ps are '~~~:·.":~cl'u.j(~d 'vJbj .:::,.l':'. ~Xt·(·:: uit~'pb.y;~:· al ti ~')r f::-)Y'f>i,gntel the t:is.sues :Jf thr'.h. .oJ.!!

:y::came o'Iailable the

met.be-ds nec8s,sary fOJ' thf:.:i.r u::;'" in biological experim':cntswere devel:::Jped.,

The mat:nemat.i.cs used L)t' tJJ·' analysis of data collected in these experi-'

ments can be rc)!].gt?l~y 1. vid,,,:) in!.;") two classes: (.1.) th2 intui.tive,.

empirica1 methods w1: L: were' c1::>se1y tJed to the experi.ment) and, often

:had nei.ther a ,soUd 7nf.1tl'Jemat"ical foundation nor sufficient precision to

permit valid conclu;::j:~':GC;.• and (U) sophisticated mat,hemat.-ical formalisms,.

Dl0St

10gicaJl~,.. uD.leasoni3hLf." asmm·pt1.on:s to ,permit their applicaU':ln tC) msn,';

biological problems,

The purpose of LbiJ thesis .is tC) deve10p a mathenlatical method for

the analysis of tracer data from transport studies whicfJ is b~)th sound in

its development from <h'fi n j 1:, ions and stated a~osUli1ption~l" and, useful i.n

biolog.ice-I ex.pedment:a Lion,

2

2. REVIEW OF LITERA~

2.1 Tracers in Biolosical Research

Although radioisotopes have been used in biological studies for

45 years, it is only in the last 20 years that their use has become

widespread and routine in biological ~aboratories (Copp, 1962). In

1923 Hevesy reported the first Qiological experiment using radioisotopes

as tracers in plants. He reported the uptake of RaD labeled lead by

bean Seedlings and the subsequent release by the plant when placed in a

~olution containing non-radioactive +ead (Hevesy, 1923). The first

studies with radioisotopes were restricted to naturally occuring radio­

isotopes. Even with these, in the 19301~ many investigations were carried

out in plant and animal biology. One of the more important series of

experiments was that which showed the constant turnover of body con­

stituents by the metabolic processes of synthesis and degradation

(Schoenheimer, 19~).

The production of artificial radioisotopes considerably widened

the scope of biological investigations, but even with the development of

the cyclotron the amounts of the various i~otopes were limited. With the

availability of the products of the neutron piles at the end of tpe

Second World War this situation was changed, and radioisotopes of many

elements became widely available.

The applications of radioisotopes in biology can roughly be

divided into four areas: uptake, metabolism, volume determinations,

and transfer or transport studies. Examples of uptake ~tudies are

Hevesy's pioneer work with beans, studies of uptake of iodine by the

3

thyroid (Pachin, 1964), and uptake of calcium by the skeleton (Corey,

et al., 1964). Examples of metabolism studies are Schoenheimer's work,--the absorption and excretion of irop by the body (Price, 1964), and

glucose metabolism (Segal, et ~., 1961). One of the first studies of

the volume of a body compartment by the use of r~dioisotopes was the

determination of the volume of water present in the body (Hevesy and

Hofer, 1934). Ot~er examples are the volume of plasma (Zierler, 1964)

and water spaces of the brain (Barlow and Roth, 1962).

'+he following simple example illustrates transfer and also the

necessity for radioisotopes as tracers in many biological studie:;;.

Suppose there are two compartments, Corp.partment I and Compartment II,

separated by a membrane m, as in Figure 2.1. A substance K flows into

Compartment I at a rate f i and out of Compartment II at a rate of f o '

The substance K also flows through the membrane m from Compartment I to

Compartment II at the rate f 21 , and from Compartment II to Compartment I

at the rate f 12 • By measuring the inflow f. and the outflow f , as well~ 0

as the volumes of the compartments, it would be possible to determine

the net flow of the substance K from Compartment I to Compartment II.

But in many biological problems the one-way flow f 21 is of more interest

III

1_i _m_f

Figure 2.1 Example of a transfer problem

4

than the :net flow. To measure the one-way flow the particles of K must

be labeled so that it is possible to tell whether a particle which is

in Compartment I at time t l is at time t 2 later than t l in Compartment I,

in Compartment II, or has left the system As another example,

consider what happens when a person drinks a glass of water; where do the

molecules of this water go in the body? To answer this question there

must be some way of distinguishing the molecules of water formerly in

the body from those molecules which have just been drunk. Rariioisoto'pe~

provide a means of labeling at least some of the particles of K or of

the water. This thesis is concerned only with problems of transport,

and examples will be given in Section 4.1. Extensive reviews and

bibliographies of tracer studies may be found in Hevesy (1962a), Copp

(1962), and 9heppard (1962). Reports of recent, but specialized symposia

on the uses of tracers were edited by Whipple and Hart (1963) and by

Knisely and Tauxe (1964a).

2.2 Compartment Analysis

The mere presence of radioactive particles of the substance K in

Compartment II after the introduction of a radioisotope of K in Compartment I

is evidence of uptake of K by Compartment II from Compartment I. As it is

usually applied the dilution technique for estimating volumes involves

little more mathematics than the manipulation of proportions. But when

investigators attack.ed the problem of estimating from tracer studies the

quanti.tative rate at which a substance moves from one place to another

then the mathematics became more difficult. In the case of studies of

transfer or transport of substances in organisms, the problems of

5

observation in the various regions ot the organism, of interpreting the

data of observations of radioactivity, and the desire to estimate as

many transfer rates as possible led to the development of a complex

mathematical formalism which came to be known as compartment analysis.

(Compartment analysis could mean the analysis of any system in terms of

compartments into which the system is divided. In this thesis, however,

compartment analysis will be used in the narrower sense to mean analysisI

which is based on some combination of the assumptions in the next para-

graphs.) An important early paper which did much to stimulate this

development was by Sheppard and Householder (1951). The intensive research

effort that was expended on problems of transfer in the 1950's is indicated

by the review and extensive bibliography of Robertson (1957), which was

largely concerned with compartment a~lysis, but included other mathe-

matical approaches. Compartment analysis seemed to offer large rewards

for experimental efforts, apd by 1962 a large amount of effort had been

spent on the development of this mathematical approach to the interpretation

of tracer data. The report of a conference held in 1962 suggests that the

mathematics had perhaps outreached the biological considerations (Robertson,

1963). Indeed, in 1964 compartment analysis was being referred to as a

subspecialty of mathematics (Knisely and Tauxe, 1964b).

In the literature the mathematics used in the interpretation of

tracer experiments by compartment analysis is based on some combination

of the following assumptions (Wrenshall and Hetenyi, 1963).

Assumption 1. The tracer and the substance of interest have the same

chemical and biological behavior; the biological system being studied

cannot distinguish them in any way.

6

Assumption 2. A dynamic steady-state condition exists for the

substance of interest in the system.

Assumption 2 has been interpreted or alternately stated as

ASsUDwtions 3 and 4 together.

Assumption 3. The rates of transfer of the substance between

compartments of the system are constant.

Assumption 4. The volumes of the compartments are constant.

AssumptIon 5. The substance has uni.form concentration in every

compartment. Wi.th Assumption 4 this implies that the amount of the sub­

stance remains constant in each compartment. Closely associated with

Assumption 5 and sometimes used interchangeably, although not equivalent,

is Assumption 6.

Assumption 6. As the substance, with or without tracer, enters a

compartment it is instantly mixed with the substa:pce and tracer already

present in the compartment.

Assumption 7. The number of compartments in the system and their

connections are known or can be assumed. This provides knowledge of the

recycling of tracer.

Assumption 8. Introduction of tracer into the system does not

change the system's behavior.

Other implicit assumptions are that the system can be divided into

compartments and the corresponding mathematical and biological compart-

ments can be identified.

~lese assumptions of compartment analysis permit the movement of

tracer to be described by a system of first order linear differential

equations with constant coefficients; the solution of the system is a set

7

of functions which are sums of exponentials. These sums of exponentials

describe the radioactivity in each compartment, but the parameters of

the functions are related to the rate constants. Thus compartment

analysis ultimately involves the fitting of sums of exponentials to tracer

data.

2.3 Criticism of Compartment Analysis

The invalidity of any of the assumptions on which the analysis is

based invalidates the conclusions reached from the analysis. Compartment

analysis has been criticised recently on the basis of the validity of

the assumptions. ~ergner (1962) examines the fitting of exponential

curves to radioactivity measurements and gives an example to show that

thl.s can lead to erroneous conclusions. In this example Bergner shows

by means of a system simulated on a digital computer that the observations

of the change of specific actiVity in one compartment is not sufficient

to determine, even in a relatively simple case, the number of compartments

in the system. Wrenshall and Hetenyi (1963), in work with hydrodynamic

models, have shown that compartment analysis is very insensitive to large

changes in the outflow of inaccessible compartments, and to large changes

in the contents of such compartments. This estimation of flow into and

from inaccessible compartments is one of the problems which motivated

the development of cqmpartment analysis, and is one o~ the problems

which it seemed to answer. Zierler (1964) points out that many experiments

which have been analysed by compartment analysis have extended over many

h()urs and even days, and that it is not likely that the volumes and rate

constants have held constant over such intervals. Zierler also points out

8

that some applications of compartment analysis ignore delay, dispersion,

threshold, acti.ve transport, and saturation, all of which are found in

many biological systems and cannot be interpreted by fitting exponential

curves to the data. In a review article Wilde (1955) discusses the

difficulty of i.nterpretation and identification of mathematical com­

partmentswith biological compartments, and the failures of compartment

analysis due to non=constant rates, concentration gradients, 'lumping'

of two or m.ore cQrllpartments, and the non=homogeniety of compartments.

Since the purpose of this thesis is to develop an alternate method

to compartment analysis for the interpretation of tracer data in trans­

port studies, it is of interest to examine the eight assumptions in some

detail, considering why they have been made, the mathematical conse­

quences c)f each assumption and their biological validity.

The first assumption, that of identical behavior of tracer and sub­

stance, is often referred to as the absence of isotope effect. Since

isotopes of a given chemical element have the same atomic number but

different masses, it might be expected that they would behave differently.

For hydrogen isotopes in particu~ar this is true. There is

both a chemical and biological difference in the behaviors of water,

deuterium and triterium, and this difference has been used to advantage

in radioisotope experiments (Glascock, 1962). But for elements of greater

atomic number the ratio of the masses of the isotopes approaches unity,

and Bigeleisen (1949), in a study of isotope effects, concludes that

those tracers which are isotopes of carbon, or elements of a greater

atomic number, are 'faithful tracers', that is, any difference in

9

behavior due to isotope effect is negligible, This is likely true for

transport studies of biological systems, although better experimental

techniques may require a consideration of the isotope effect, as is now

done in studying chemical reactions, In the biological systems discussed

in this thesis the isotope effect will be considered negligible,

The term isteady-state U or idynamic equilibrium' has usually meant

that transfer rates and pool sizes are constant (Berger, 1963), That

thi.s assum.ption is not valid in the case of growth, disease, or other

streeE, is cl,~ar~ but even in bi.ological systems that are m.ature and oft.en

cons idered to be in a i steady=state i ccmdition volumes J rates of transfer

and concentrations can oscillate over a period of hours, Some of the

attempts to imply two-compartment systems from tracer activity curves

that are Uwell fit i by a curve whic1~ :i.B the sum of two exponentials might

better be explained on the basis of changing Urate constants'. This

unrealistic assumption of steady-state conditions seems to have been

made in order to simplify the mathematical analysis, Wrenshall and Lax

(1953, p. 19) say

" It appears probable that the concept of dynamic equilibriumhas been overstressed in attempts to make precise determinations,or to facilitate the development of mathematical descriptionsof the phenomena of hemeostasis, with corresponding neglect ofthe equally important physiological phenomena of adaptation andgrowth which only appear and are measurable when dynamicequilibrium does not exist."

In a similar remark Jaffay (1963)j after discussing various

simplifications which have been attempted so that turnover rates can be

calculated in non-steady conditions, concludes

10

" ••• in each case we find that a simple measurement of thespecific: activity of the product at various times will notgive us the turnover rate. More information is required aboutthe peel size and the rate of change in specific activity ofboth precursor and product. If these data are known, and thisrequires more effort than rr.ost people are willing to do, thenwe will have reasonable approXimations of the turnover ratefor the conditions under study.1i

Thus the assumptions of steady~state seem to be ones of convenience,

and although some authors claim that tracer transfer rates yield infor-

mation on substance transfer rates only under steady=state condi.tions

(Bergner, 1964), certainly many non=steadyo,state situations are

Important and of interest, and as the authors above i.:r.!dicate methods

are needed to analyze tracer data in such situations.

Assumption 3 and Assumption 4..1 constant rate functi.ons and constant

volumes, seem to have been made for the reasons above, particularly to

avoid difi'erenti.al equations with non~":;on8tant coefficients, and also

in some cases 1.n order that volumes WQuld not haye to be measured. It

is also often implicitly assumed that in different systems of the same

class, ~.~., a group of similar animals, the rate constants will be the

same. At least this would seem to be the assn.mpti.on that permits the

averaging of data from several experiments before the rate constants

are estimated.

For compartment analysis the number of compartments and their

connections must be known in order to derive the system of differential

equations describing the system. Since the assumptions of compartment

analysis imply that the specific activity curves are sums of exponentials,

it has "been suggested (Berman, 1963) that the number of exponential terms

needed to give the 'best fit' to the data can be used to indicate the

11

number of compartments in the system. However Bergner (1962) and

Zier1er (1964) have shown that the specific activity curves can resemble

sum s ')f exponentials with fewer terms than the number of compartments

in the system. Especially is this true if some of the assumptions do

not hold, and these assumptions cannot usually be verifi,ed by consideration

of the t.racer data alone.

Assumfltion 5, that of uniform concentration, :l.s DDt alwa;ys made

expl:i.cit but is necessary for compartment analysis. The basic r,;:qu:irem,ent

in the use of t.racers to study transport phenomena is tithe quick

presentati.on to the cell surface of a known steady isotope ratio which

is to travel :into the cells." (Wj,lde~ 1955,? p. :n) This statement, made

in the context of' transport through merribranes clearly shows a prablem

of tracer studies: knowing the distribution of the tracer at the point

or points where the substance and the tracer are enteri.ng and leaVing

the compartment. In Figure, 2 .1 this would mean kno'vli!lg the distribution

of the tracer over the surface of the membrane m. AssunU.ng uniform

concentrat:bn and instant m:Lxing.~ ASSUIn:ptLon 6.0 has the implication that

the d:istrHJt.ltion over the area of the ml(~mbrane is uniform and can be

determined by the concentration of the tracer :i,n the compartment. Thus

one sample from the compartment gives the concentration of the tracer at

the membrane at the time the sample :l,s taken. Sheppard (1962) discusses

some of the (;ff'ects on estimation of rate constants if this assumption is

not val:i.d. Un:i.form concentration is also needed for the determination of

pool sizes by dilution methods.

12

Constant '\lOlumes are assumed in order to a\roid non=constant

coeffic ients :in the different:i.al equat:i.ons j) and because often there is

no convenient way to measure the volumes of com;partments. Also it is

possible to compute Uturnover rates a without knowing pool sizes if constant

volumes and constant concentrations are ass'~edo

Recycling of tracer 1.S determined.;> of coursE'j by the assumed

connections between compartments ~ but often no recycling :i.iS assumed

so that the emnputati.o!J. of rate constants from ·the parameters of the

fitted sum.s of ex.ponent-ialswlll t,e s:inrpl1fled (Robe:rtscmj) 1957) 0 In any

case assl..un:pt:ions about rec:rcli:ng imp.l.:y ass1.mrp't:i.o:ns about the inter­

connections of the entire systemo

Assumption 8 has been thought neC€Eisary i:::1 cJrder that the con~

clusi.OllS ,:)f the tracer study apply to t,r,.e system of :interest ~ and not to

the system as perturbed by the :inject:tcn c,f tracer. Those ,;,rho have used

thi.s aSB1.l..Ulpti.on r.laye atgued that in the systems at udied, and with the very

small amou.nt of substance introduced i.nto the system with the tracer, any

disturbance caused by the introduction of the tracer disappears in a

very short time. But this is not really suff:i.cient for compartment

analysis, since the curves are fit from the time of injection.

Another practical diffi.culty in compartment analysis which many

biologists ignore, at least in their p1.ibl.ished results)' is the difficulty

of estimating from experimental data the parameters of a function which is

the sum of exponentialso When the data are subject to large error, as

in the case With much biological data, the problem becomes especially

severe, While some investigators report the fitting of a curve which

13

is the SliX" d' f;i,ve exponential ter.ms (Moore j 1962L others who have made

detailed ti:,udies of the estimation problem and the resultant errors,

report IfHge uncertainties in estimat:l.ng the parameters i.n sums of two

or three I;":xponentials, and give analyses for given size errors (Myhill,

.::! ~o ,j 196)3 Gregg, 1963), It ls clear from the statistical literature

that th:is i;5 1:1 far from resolved problem (Lipton and McGilchrist, 1963).

It thus appears that theseassUffipti.onswere fnvoked in compartment

anal;ysis (.:',.) as 8. stfbstitute for ex.per:i.mental:::,rBf';tvations that were

diffi,cult or impossible to make.9 (u) to permit more tractable ffittthe·,

matics, and (:i:1.1) in an att€':mpt "';;0 get the m8ximu.m am.ount of 5.n1'o1"'"

lllstlon from tb,e data. It also a.ppears that the development of a mathE>~

.mat.leal ;formalism was at tImes easi.er than appl.lcat:io,n. of the 1I.i.8thematlc3.

As one text (Francis ~ et, &. J 19:59" p. 344) o.n traCE.!.' methods

ftnS:inc(' even the simplest cell is an extn:::wely cc,m:p11catedsystem contaLni.ng a wide var:iety o.f substances ~lndergo:ing i:r..ter­reaction tn a highly organlzed m..a:nner:J itLs u.suall.y necessary tomake a number of s:tmpl:i.fytng assumpt:loxliB 'before any of thesereactJons can be treated .in a s:i.mple mathemati,ca1 :rr..anner. Thesolving of HLe mathemat:ical equations cb.osen to fit the systemunder Investigation is therefore frequently a much easier matterth(;,n dec.iding whether the equa.ti.ons are strictly applicable tothe by-stem, and to what extent the assunlptions are justif:ied. 16

:Il1Js dIscussion, and the rest of the thesis, will indicate that there

are two basic assumptions that need to be Dlade to validate tracer studies

of transfer Dr transport rates between compartments ,; one J that the tracer

and substance of interest behave the same chemically and biologically,

and two~ that it is possible to determine the distribution of the tracer

at the points of entry into and exit from the compartments.

14

g.4 other Methods of Ana1:Xs~

Att~Ulpts have been made to analyze tracer experiments without

invold:ng 8'11 of the assumptions used in compartment analysis. Berger

(196,3);, tn a study of' sodium transport between plasma and the intestinal

lumen, relaxes the steady-state assumption to the extent of allowing the

rate constants in a two-compartment closed system to be unequal; thus

compartment volumes change. This means that i,n a finite t:1me one of

the compartments would be empty. Wrenshall (1955) considers systems in

which some of' the compartments do not have rapid mJ.x.:tng of their constants,

but all other assu.mpt:ions hold. His method requi.res extrapolation of the

curves back to the time of injection, which has the disadvantage of

increasIng the error of estimation, the method also requires the use of

i average absolute spec:ific activities u. Wrenshall and Lax (195,3) use

hydrodynamic models to study the behavJ.or of'tracers in non~steady-state

conditions.

Sheppard (1962) discusses the use of numerical solutions to two-

and three-compartment models where the volumes and rates may be non­

constant functions of time. Hart (1955, 1957) discusses non-steady-state,

non-conservative systems, but his results are formal mathematical ones not

suitable for application to experimental data. In general, for a study

of an n-compartment system Hart assumes the use of. n tracers and access

to all n compartments, as well as 'smal.l u errors. He hes shown that

the preaence of concentration grad:1.ents in the compartments may give rise

to spec:tf':lc activlty curves that are sums of exponential terms.

ModelS have been proposed for interpreting tracer data which do not

fHHlmnel,hat all :plitrt;1,(~les of the substan<;e have the same chance of

15

leaving the compartlnent. 'l'he model of Shemin and Rittenberg (1946),

which was extended by Carter, et a1., (1964), for the transport of

glycine into red blood cells is an example of a mathematical approach to

the interpretation of tracer data in a situation where there is a

mechanism which selectively determines the particles of the substance

that leave the compartment.

S:,d.~narL2£.Literature Review,

Thls l'(~vje1;~ ~:yf tbe literature has not included a complete account

of tracer me:thods " Wlr h88 it included the great number of papers .1hich

have appeared In lJirJl'-:igi.cal, wed leal and b:iochemical journals, and which

have applLed compal'trnent ana.lys:1.s uncritically to data collected in

tracer studies .in Ii and medi.cine. It has intended to show that

there arE many bi.oL.1g:Lcal ;3ystems that can be studied in terms of the

C()Olpartments vhi.ch together make up the system, but that the usual

compartment ar,1BlyE;1.s 1d not appropri.ate for studying these systems

"becau.se GfU..":: :tE.;,s tlJcLlve aSEumptions made. rrhe attempts to analyze

systems nut In 8. stpbdy~state condition have been relatively few.

) 0 j\ BIOL()GICAI~ PROBLt-:M

Re~;,.;;.:n':::l:, Ln animal n,!.l.tri,t.i.on has indi,cated trwt purif,ied dit-;'t~, J tLat

• \ r~ 1 adequate

'"

i.mmature anima12 stop gI'CWll"'.,< ij:'Kl

~".~'~ i diet 6 :uakeE them 28 tisfactor;y fo:: sheep c

a~d!Cl:"3~;'J ;.,ric purif::.ed. d:ietE spent much less time rWni~la".~ing; this

.i, cd.'.: e,s of' experiments was conducted to test this" as "o'ie2-l. as otter J

24experllnents J'Ia" was injected into the bl.ood '~esael, or

int:: the ::'I"iilC~l of a sheep J and t11e radioactivi ty of the p1::H3L8 'J"Kl. c:f

Tr:.e specific 8ctiV':.ty curve3 ob'c.ained for sheep on di£'fere,'l+: ,j,i,ets

C)!, '.:. the plasma cO'':'::.d be cons i.dered a c:01T..p8.r''.:!iJeYl~ .. ; 8nduie

17

anb.lysis seemed .to be indicated. An attempt was made to fit the specific

e.ct5.vity data to sum of exponential curves. This attempt was made on

the North Carolina State University IBM 1620 computer using Hartley's

modHied non~linear least squares procedure (Hartley, 1961), and at the

Natio!lal Institutes of Health using the program of Dr. Mones Berman of

the Office of Mathematical Research (Berman, et ~., 1962). Both of

these attempts were unsatisfactory in that it was not clear whether a

twc),-compartment closed model, a two=compartment open model, or a three=

compartment model should be used. The esti.mates of the parameters for

the models considered had very large uncertainties; the estimates of the

standard devlatiDns of the estimates were often larger than the estimates

of the parameters. A more critical consideration of the data, and of the

biology of the problem, suggested that most of the assumptions needed

for compartment analysis were probably not valid in this system.

Both the volume and composition of the rumen change markedly in a

period of less than 24 hours (Gray, et a1., 1958), so that during a

peri.od of 16 to 24 hours the liquid volume, sodium concentration, and

total sodium in the rumen, as well as the concentration of solids, can

be expected to vary appreciably (Dukes, 1955). The very noticeable

lack of smoothne[,s in the rumen specific actiVity curves was greater than

would be expected from the variation in the chemical analysis procedure

or from the Poisson distribution of the radioactivity counting rates;

st~gesting that the concentration of the sodium and of the tracer in the

rumen was not uniform. The nonhomogeneous nature of the rumen contents

has been noted in non-tracer studies of the rumen (Barnett and Reid,

18

This nonhomogeniety seems to be due to the mixture

()f solids) s2ml. o·solids and liquid in the rumen. These studies indicate

that the rnrnen contents are kept well churned by the frequent contractions

of the rumen waLls; these contractions normally occurring several times

ebch minute. TrillS the variation in observations of sodium concentration

in the rumen may be due to sampling solids as well as the liquid phase,

which is the phase of interest.

These factors suggest that it i.s unlikely that the rate of flow of

sodi.lml from plasma via saliva and through the rumen wall to the rumen

is constant, but rathe:c i.t may be a function of the following factors,

'Wtlich are not necessarily independent, nor exhaustive:

o. time sInce feeding,

b. amount and t.>rpe of diet,

c. acidity of rumen contents,

d. concentration and/or total amount of sodium in the rumen,

e. liquid volume and concentration of solids,

f.. ,iH'ference in sodium concentration between plasma and rumen,

g. aUDunt of cheWing and salivation,

h. rate of flow of blood to capillaries in the rumen wall.

Even with all the above factors the same rate of flow may vary from

animal to animal or from week to week in the same animal.

It is Imown that the sodium in the plasma exchanges with other

sodium 'compartments' .in the body; in particular exchange with the

extracellular fluids is quite rapid (Hevesy, 1962b). Also sodium is

excreted from U:e body. These considerations suggest that two

19

compartments are not adequate to describe the system, and the data are

probably not capable of' resolving more than two compartments in the

usual compartment analysis.

Thus a model or method is needed which makes it possible to get

estimates of the rate of flow of sodium from the plasma to the rumen

without making assumptions that are made for convenience only and which

are contrary to or not based on the biological evidence. ·The assumptions

that are used sbould be based as much as poss:ible on preVious knOWledge

or experience, should be formulated clearly, only introduced as needed,

and in a manner that indicates how they m:ight be tested for valid:ity.

Possible errors introduced with the assumptions should be recognized and

an attempt made to evaluate these errors.

The problem is orle of transport, and the model should be formulated

with the view of its addaptability to some of the many other biological

problems of transport. In particular those problems were, as in this

problem, the plasma is the agent of transport; a compartment which ex­

changes with one or more other compartments (Bergner, 1962). The model

should be such that compartments that exchange with the plasma but not

with the compartments of interest can be ignored.

The data obtained from nine of the experiments reported by Slyter

were analyzed by the method developed in this thesis. This analysis is

reported in Section 6.1.

20

MATFJ!:MA'l'J:CAL FORM!.:Lt\'I'ION AND Edlil'ERIME:.~,rAL CONSIDERATION)S~{S':rEM I ~ 'IfHE PIA8MA,Rl1MIN SODJDJ:M 'l'RANSPORr PROBI~

4'• u,J..

In Section 4 the 1118c;':",,,mati.ca1 fornmla'tio:a is presented for the tracer

study of the movement of Bod ,tum. .from the plasma to the rumen :in s:teep;

this formulati,on being aI=pl,iccftl1e to other systems with the same cOJl=

figu.rat'ion and mee

is also consIdered 0 In Secti.cm '5 the met-hud of th.Ls sectJ.cm 113 extende::i

to systems ·wi'tJ::. ether c:ompaxtme:~lt, ci:Jr.J'igm,'e.t,lons 0

The problem of of "particles ":.:y the circulator)' system is

i.mportantin that many biological a:nd medical problems 9.re answered

directly in term,s of "t,his transport) and the answers to many other

problems depend on it :indirectl.y (Wrenshall, 1955; Sheppard y 1962;

Sheppard and Householder J 1951c: Kamen) 1957) 0 Specific examples of

this are the :..ransport of plasma album.1.n (Bergner y 1964)" iodine

metabolism (Sheppard) 1962 L 11 'reI' function (Lushbaugh., ~ !l)" J 1964L

kidney function (Greggs 1963)s sodium movement between the plasma and

the intestinal lumen .in dogs (Berger .,)1963)" and iron metabo1:i.sm

(Sharney, ~ &< J 1963) Q :Most inve,stigato:cs have considered the circu~

lation as a mamilla:ry system." that :iSy one in which a central compartment y

the plasma" excr.a::ngeswith sEyeral .peripheral compartments 0 This term

is not used here s however y as :i.t is considered that the pheriperal

compartments may exchange w:lth otb.er compartment or excrete particles

out of the systemo This thes:ls cons:i.der:s substances wh:i.ch are transported

21

frOi::'. c::\(.';·,<J:.' U :.l~r; to compartment, and does not, consider substances wh:i.ch

undergcJ c'X:-,'0!-l '_'81 chazlge or are formed from or broken down into other

substances v

TbE: aim 18 to formulate the problem with a mj.nimum of biological

assumptions.; "i'1hen such assumptions are introduced the mathematical,

biological or experimental consideration motivating their introduction

will l,e discussed. The emphasis on minimal assumptions indicates the

suitability of the model for exploratory- and introdllC't(Jr~{ studi.es where

little is known abOllt the system being studied> as well as for systems

that have mm=constant parameters, The goal is to avoid the many bio··

logical assumptions of compartment analysis and to evolve a model which

will allow analysis that will indicate the non-constancies in the system

and Vlhi.ch will est:imate the amount by which volumes, concentrations and

rates are changing, The model may also indicate the functional form of

these values. ~~is ir~ormation could then be incorporated into a more

complex model '""hich would be used i.n further investigation.

As Zi1versmit (1963) has pointed out, tracer studies have suffered

from a variety of definitions and inadequate termInology. Some reports~

for exam.pIe, use a theoretical definj,tiol1 and an operational definition

for such concepts as the ratio of the tracer to the substance being

traced, There are no standard definitions J and the definiti.ons in thi.s

thesis, while attempting to be precise and the basis for a logical

development, are formulated with some of the reali.ties of experimental.

biology and chemistry in mind,

22

4.2 Definitions and ABaumptio~

Tne definitions are stated in a manner that will make them

applicable to tt.e systems in the next section~ as well as to the system

of this section,

Definiti.on 4,1. A compartment is a region with physical boundaries

in the biologi.cal system or organism being studied,

Thus a compartment may be an organ and its conte!'lts J such as the

r~~cm.en; it. may be a certai.n fluId .. such as the plasma c;f the c:Lrculatory

system; or j.t ma~r be a collection of" cells J such as the red blood

cells, For convenience of discussion 'when considering systems with more

than two compartments the compartments will be labeled 1 J 2 J , •• , N.

Unless otherwise stated the plasma will always be Compartment 1.

Definition 4,20 The substance of interest, .substance for short J

is the naturally' occurring chemical element or compound that is of

interest and whose transport between compartments is being studied,

With the restriction to problems of transport only, if' the substance

is a compound .it must be one which is not involved in any chemical

reactions ,ihich would destroy it or form it in the compartments being

studied 0 In the biological system that is being considered in thi.s

section the sl,)~bstance of interest is sodium, The am')unt of the substance

in Compartment i at time t is the mass of the substance in .compartment i,

and will be denoted by Ni(t). Unless otherwise indicated; N. (t) is the1

total mass ol~ the substance i.n Compartment i, both that whi.ch is naturally

there and any whi.ch may have been introduced as a tracer. In the case

that the substance is a chemical element N.{t) is the mass of all isotopes1

of the element .i.n the compartment at time to

23

Assumption 40LE')};: amc:unt of t~he SUbGta:lce in a compartment

, t h .. 'I"" (. \ ..changes 1.n such a marilier".st; !~, t) Dasl

~J'i tb respect to

t at all but at m:)st a finite number of points of time Ln the time

interval in which the system is being studied"

This assumpt.icn means that the amou:~1t of substance in a compart;=

me!1t changes :)nl~i graduall.y,1 and. if the removal or introduction of a

quanti.t;v of the eubstance causes Ni(t) to have a ju.mp~::Uscont.inuity

this will hapfer!. only a finite D\JlJfber of times jiJ r the course erf

the study" This assumptIon is needed to eDEure "that some of the later

rnathemati.cal operaticns are poss:I.ble» 8J:ld seemJ:, natural. b.L:;log:i.ca.l.ly

since i.t. is almost equivalent to sayingt.!l8t. The 9!Jl(.)ijl1t of the E1..:."bstfHlCJ:'

is H::out,inuo1,1S functi.orJ at almost all Im:lt:anta of timeo

Defin:i.tion 4.)0 The .traceJ;:, :l.S a quantity of the iSubsta!lce \{hieh is

l.abeled by its radioacti.vltysnd has beE::l introduced i:c.to the system in

orderh:J studJ the behavior of the substanceo

A,sS'.)l'jpticm 4,2, The system cannot di.sting:;ish tracer from Eubstance;

the tracer aEd substance have identical biolC'g:ieal and chemical behavic)l'

in the system,

The ta"t~ljl amount of the tracer in a compsrtment will be some m.ass J

but sincE the tracer and substance cannot be disti:nguished chemically

there is no easy way to determi.ne this mass" 'Jl.'.15 the amo"L:nt of tracer

is measured in terms of its radioactivity as recorded by the counting

procedure of the particular experimento The amDunt of tracer in

Compartment i Bt

per unit timeo

time t will be denoted by Ti(tL the units being counts

It is also assumed that T,(t) Is a differentiable f~lnction).

at all but a finite number of points of time 0

24

ASSUillpt:ion 1+ 0.'5. 'rhe pbysics and geomeny of the counting procedure

of samples taken from the system remain constar.t throughout the experi"'

ment, so that the counts per unit time per unit mass of tracer, when

corrected for radioactive decay, is constant throughout the experim.ent,

Definition L+.40 The isotope ratio of any maES N of substance is

the ratio TIN, where 'II is the amount of tracer 1.n the mass N of the

substance; the units of an i,sotope ratio are counts per uni.t time per

unit mass.

The concept just defined as :l.sotope rati.o is defined as urelativEo

specific actiVityG by Sheppard and H::mseholder (1951),

For con-'lenience and clarity inwri.t:i.ng equations and definitions

the fol1ow:ll1g notation will be used: fClT any function f(t) J

fG (t) _ d Llll- dt "

and the Riemann integral

Definition 4.5. The transport~ funcgon for the movement of

substance from Compartment i to Compartment j 1s a continuous function

BJi(t) such that for t 2 greater than tl~ I{tl~t2;Bji) is the mass of

substance which moves from Compartment i to Compartment j in the t:tme

interval (tljt~). Note that B, .(t) is not defined for i • j •... - lJ

The units of BJi (t) are mass per unit t.ime. Note that I(t1,t2 .,Bji

)

is the total mass of all particles of the substance that go from

Compartment i to Compartm.ent j in the time interval (tl ,t2 )J the same

25

C::Jll1.pa!"tment ,j

:ca!e fun~tt(;nfcr tile movement of substance into

transport rate function fer

the movement of E'Jbst·e::1c;e ,:rom. Compartment j to unspecified compartm.ents,

and BO

' (t) :is t [Ie transport rate function for the total movement of,1

substance llcm Ccmpar:tment jo Thus B/"o(t) "" r; Bi,(t)" For thevJ <1'.' l' J

u J.~ ,c.

ITJDVement o:fr;race:c t~)ereLs a cDrresponding transport rate function"

such that for greater tI18.n ,~ I(tl.)

j is a continuous funct:ic)!l B'JIt,. (t)f .J,.

-¥...; .B . ,.. ) i.s the amc;unt of trac er, JJ.

which moves from CompartmenT, J t:o Compartm-ent j in the interval (tl

J t 2 ) 0

*The units of B" are crounts .'OE:;Y unit time per unit time •. , (+) ",] 1. -.J.

The transpm:t rate functi');n for the substance is assumed to be

positive and the trarl8}Jort rate function for the tracer 1.S assumed to be

non-negat.i.Ye. ~rhis assumptio::l is need.ed to obtain equation (4.5)0

The concept dcflrl'i:::d i:(~ DeflYiiti.on 4·,5 has been vari.ously def.ined

i.n the li.terature c rrerms which have been used for this or s:i.milar

concepts a.re fl1.L,<:.} flow.' transfer.' exchange rates, turnover, appearance

and dIsappearance" Most Jf these have usually been reserved for isteady"

state i or Uequilil,ri1lIll U co':lditions. Kamen (1957) uses transport in the

sense of t,ran~)por1~ ra1;,e ftL'lction as defined in Defini.tion 405, and

St.eppard uses tranBpcrt rate (1962). Tb.e i se,tope ratio defined in

Definition 4)~· :is eften called specific a,:;tivity~ although spec:i.:fic

activity is uften defined as the ratio of two masses but reported as the

rati.o of counts to mass.

26

Ax:. was dlscm3i3ed in Sectlon "5 the transport rate functi.on B .. (t) mayJl

be a non··constant function of time and of a vector of parameters which

are deterrn:Lned by the state of the system. Since at th:is point it is

only values of the transport rate functi.on which are being estimated,

it is convenient to wrHe B., et)' as a function of t alone,Jl

'iI"B .. (t) willJl

be a function of the above parameters as well as of the time which has

elapsed since the tracer 'W'as Lntroduced into the system.

Wi.th the,se dE;f::Lnit.i.o;ns srll asslJ,mpti(ms as a general. foundation"

assumptions are no'w stated which will formulate the plasma to rumen

sodIum transport proble.mo "Ihe. system is composed of two compartments;

Com.partm.E.'Dt 1 1.s the plaBma.~] Compartment 2 is the rumen fluid 0 f.rhe

"'4substance is sodium and the tracer is the radioactive isotope NaC:" A

''''4quantity of nac,· is injected into the plasma; the amount being so small

relative to the mass of sodium naturally in the plasma that the physical

and biological properties of the system are not disturbed, or at least

the disturbances are minor and of short duration.

ASStunption 4040 The rumen fluid and the plasma are compartments as

defined in Defi.nition 4010

Assumption 4050 There is a transport rate function B2l

(t) for the

movement of sodium from the plasma to the rumen y and a transport rate.ji

function B21

(t) for the movement of tracer from plasma to rumen. 'I'here

is no other flow of sodium into the rumen, !.~." B2I(t) = B21(t), and

* .jt-B

21( t ) =: B

21( t ) ,

Assumption 4060 'Ihere are transport rate functi.ons B02 (t) and

*.30:2(t) for the movement of sodium and tracer from the rumen.

27

A system satisfying Assumptions 4,4 through 406 is called System I

and is represented schematically in Figure 4.1.

Compartment IB21

(t) Compartment 2'"

plasmar

rumen

K........ B12 (t) ~B02(t) BE2 (t)'Ij

............. - -- ---Figure 4.1 System I

The plasma compartment is open at the top to suggest the other

compartmentswHh which the plasma exchanges sodium; the dotted line

indIcates that some sodium flows from the rumen directly back into the

plasma through the rumen walls and some may get there more indi.rectly,

~o~., through the intestine. The directness of the arrow labeled B21

(t)

does not preclude transport by several pathways, ~.~.~ through capillary

walls and by way of the salivary glands.

For most of this section only the contents of one compartment,

the rumen flUid, need to be considered, so the subscripts of N2

(t) and

T2

(t) will be dropped; in Sections 4.3 and 404 N(t) will be the amount of

sodium in the rumen, and T(t) will be the amount of tracer in the rumen.

Note again that the mass of any tracer in the rumen at time t is included

in N(t).

4.3 Derivation of the Estimates

The changes in the amounts of total sodium and of tracer (marked

sodium) in the rumen are

(4.1)

If ; 2 not one of thepctnt!3 where N('t) or T( t) i.s not

28

differentiable a:ad [.:it is a pesibive num'c,er,9 and if each of these equati():1s

is divided by 6t and the limit Liken as 6t approaches zero, the fol1m"Tlng

equations are obtained.

(4.3) d w( +-'... '-Ldt

(. 4. L: '.• Q r)

* .~ B (t)­;;:'1' (t: )

It seems letui ti'rely evident thst if aLytLLng i.8 tel 1:e J.earned E3.b::iiJ.t

tb';; movement of sc.cUum from tracer experlmeY,;:.,s.1 then tbel'e must be SOIT·'"

.*relati.oD 'between B, , (t) endE,. (t). FrulIl the assumptio.':1s

1ciLJ

BUs the following ratios are well defined:

made

I(4, ) (t)

NDte that these ay" not isotope rattans as defined by Definittan 404 .. b~lt

rather the Jjlnits as 6t app,rc,acbes zero,)f the iSGtope ratios

Substituting the expressions (4.5) into equation (4.4) and solving

equations (4.4) and (4.3) for B02 (t) gIves

(4.6)

Equating the right hand sides of equations (4,6) and (4,7) yields

R.;)(tj] =

29

( 4,.8) R),!. t.)c;..",....

Having 01.: tai ned B... (t ') .<.:'.1 ~ '" (\:) can be nb':E\ l:lcd. by ucLa..s; either equ8U,'::n

( I~ 06) or

c:t.ne'.c JrJ.v'es

der.~.ved s Lmilar equatiD!1S under t.he

aSEumpt.ion t.hat

ment (if a

the entire system ie Iii a 2 t",ady·,s+.;ate cD":dltion, Robertson (19yr)

obtai,ns an equatLm sjmilDr ::) (LL8) fGr a ,-,tead:y-state catenary system

in which t:nere is nG t"ecy<:,U.ng c1 'Cracer. 11\cne of these develo1~ments

discuss the problem 'Jf' observing 'rv(t)" Dut riccording to the development

''k . "here.. if the Ri..j (, t) \ S are meani.ngful expre,;, i3 lens J and ,if they a1 (mg Wi U,

NU (t) and I'H (t) can be determ.ined by experimental observations.~ ct',en (4 )'))

makes it possible to obtain values of B21

(t) under very minImal biological

assumptions" The problems raised in this last se~tence will now be

c:ons.1dered,

With ASSillnpticn 4,2, thatLhe system cannot d:i.stinguish between

.it' .

naturai sodium and the rWUuuct.i ve Eod-Tum, the ratio R.. (t) of tracer tr',.' lJ

sodium leaVing a compartmelltwll.1 reflect the isotope ratlos in those

parts of the cornpdrtrne~1t unrnediately adJacent tG the poInts where the

30

sodium leaves the u.:llnpartment~0 If 'IJ::,e ,compartment is homogeneous, in

the sense that in t::.very element; :)[" vO.lu.'6.~ the 12c.LOl)e ratio at time t

has the same value, say J,1o: / .;.. 'J t "'. "C' , ,,{it ( . ') A ,;~ (+ ., ,") , ',4 A* (t) '''b•. \.... , _,Lt; r., k , • \ t '" ,h. \ .. I 0 :J..L,_. '- a n eJ" Ie)' ,J' . J

estimated by taki.ng a sample from t,he ce,mpartment at time t and measuring

*the isotope ratio of the i30dlum inche sample, Which ratio 1.,rill be A. (t),J

Concerning the plasma ccmlp':lrLment, illO,lj' 1:",V2,stig8.tcrs have Hsswned

emil ,1 [; 1'. lY;)JDgeneous., well-

mixed cOffip&rtmenL Two author:::: who actlwl.l.y state this assumption are

Bergner (1964) and Gregg (1963) ID:ly,)ne using compLrtrnent analysis to

interpret data obtained from the plasma is making this assumption,

although many do not explicitly c:taLe it. ThEre is little in the

literature conce.rning vel'ificat;j(j:l of this as sumption J but Annison and

Lewis (1959) discuss the dif.ficult~y of verifying thiE aSEumption as

regards the pleEmu wh:i.ch circulates to the rUme!L

Asswnption 4070 Within a short time period after the injection of

tracer into the circulatory eye tern the plnem£.l i2 e. "well-mixed compartment,

in the sense that in all elemente of volume the isotope ratior,: of the

sodium i.n those volume!:? iE the some and equals the isotope ratio of the

total maSE of sodium in the plasmao

Assumptions 4,7 and 4,2 1m.ply thut n;l(t) "" A~(t) = Tl(t)/N1(t),

*and that A1(t) may be estimated by the jS~jt:.:)p',:; ratio of the sodium in

a sample of plasma at time to

The motili.ty 'Jf 1~he rumen due t~) the frequ.ent contractions of the

rumen '\.wlls DE '-laE diEcussed in Sec~;iun ) d')ec Il',)t imply the same kind

of homogeneity as WOE UEEUmeQ above for tbe pLEwma, but it does suggest

'that as ECldium and tracer enl:,l'l' the 1 t.Ullen fJu:td they are mixed in such

31

a manner that the Gaverage isotope ratio G of the sodium in the rumen at

time t is approximately the same as the 'average isotope ratio' of the

sodium leaving the rumen 1n the small time interval (t~ t+~t). The

*relation of R02 (t) to the /;lv-erage isotope ratio in that part of the

rumen near the rumen walls is next consideredu

Suppose that the rate of flow at time t) B02 (tL is divided into n

equal elements of rate of flow,

(k). ) ( )/B02 (t ~ B02 t n, k=lj2,.o.,ll.

This division may be visualized as a division of the rumen wall into

elements of area (possibly unequal) such that equal amounts of sodium

leave the rumen through each of the elements of area. Then

B (t) = ~ B(k)(t)02 k=l 02

For 6t > 0 each B;~)(t)6t is a mass of sodium containing an amount of

tracer B~k)(t)6t. Letting

the following equation is true~

B* (k) (t)02 =

or

n... E

k=l n

This last equation says that R~2(t) is the average of the R;~k)(t) .

1j('(k)Note that R02 (t) can also be written flS

'*(k) I(t,t~t,.B02*(k))R (t) == 11m

02 .6t....O . .(k)I (t , t+6t; B 02)

If b.t 1s small enough each R~k) (t) will be approxi.mated as closely as

desjred by the isotope ratio ~(k)(t) of the sodlum in that volume

of' the compartment which the sodium B~~)(t)l:I,t was in at time t.o And

for small .6t this volume will be close to the rumen 1<lall y or close to

the other points where the sodium leaves the rumenu Thus

== ~ R*(k)(t)/n ~ ~ ~(k)(t)/nk=l 02 k.l

*It thus appears that R02 (t) can be closely approximated as by the

average isotope ratio of sodium at the rumen wall, and that it is not

necessary that th.e sodium or tracer be uniformly di.strn;.uted over the

points where the sodium leaves the rumen. It should be noted that this

average isotope ratio is the average over elements of volume containing

the same amount of sodium, and is not an average of the isotope ratios

over equal elements of volume; in the case of a homogeneous compartment

equal elements of volume would contain equal amounts of the substance.

Determining the average isotope ratio near the rumen wall may not

*be possible so the; relationshi.p of R02 (t) to the i.sotope ratio of the

total rumen is next considered. If there do not exist concentration

gradients which uni.formly increase or decrease with distance from the

rumen wall" then the average isotope ratio over the ,,'hole rum.en may be

adjacent to the rumen wall,H

Assumption 4,,8,

33

close to the ayerage isotope ratio along the rumen wall. There is some

evidence that this is the case., &'1nison and Lewis (1959 j po 139) suggest

that, I!'lhe mixing of contents due to contracti.ons minimizes the differences

in metabolic concentrations in the midst of the rumen fluid, and in areas

"rhus j RO*~(t) is first approximated by the. .:::

average isotope ratio near the rumen wall;; which isotope ratio is

approximately the isotope ratio of the whole rumen)' and Assumption 4.8 is

motivated,. *' )It t 5 C+6;t.; B

02iTt ;t.y6t i"B

o2)

is closely approximated by ;;(t) "" T(t)!N(t)"

This assumption is possibly the most restrictive biological

assumption that has been made in this development, It is not obvious how

it would be tested experimentally, If a means were devised for obtaini~

samples from specified parts of the rumen J then a comparison of the

distribution of the isotope ratios of samples from near the rumen walls

with the distribution of isotope ratios from samples from other parts

of the rumen would give some indication of the validity of the assumption.

It is easily seen that in some compartments Assumption 4.8 would

not be valid, As an example consider the system shown in Figure 4.2.

Compartment 2 .is a cyH.nder in which the substance enters at one end,

moves through the cylinder in a uniform flow J and leaves at the other

*end. Thus the relati.on between R02 (t) and T(t)!N( t) is not that assumed

in Assumption 408"

34

Compar:ment 1

m,,~ +-, ~" . ',.' ~ '~r ·,~t',·"~·,~,, c db' H . ('19"'7) 't,·,t.!cut; ",,,,,rIll aVel. age C0nl.e,L",,·r·"·.lvr, 1S u.,.e J art. . ), , WI, not, ~

being well defined,. for the purpose of showing that in compartments

whi.ch are :not homogeneous the speci.fic activity curves may resemble sums

of' exponentials as though the compartment were composed 0';: many

compartments 0 Hart does not interltl. that average concentration be used as

an experimental method for determing rate constants; he remarks that the

average concentration of' a compartment would be very difficult to

determine,

In some sItuations it might be possible to determine 'T( t) by

countIng oyer the whole compartme:nt J or by other means, but in the rumen

sodium problem N(t) and ~(t) are estimated by sampling the rumen fluido

Since T(t) -.:;- ~(t) N(t},

* *'(409) TO (t) =~(t) N° (t) + N(t) A (t) 0

. * * * *SUbstituting from (4,8) and Llsing R02 (t) ~ ~(t) and ~l(t) ~ ~ (t)

as impl:i.ed by Assumptions 4 0 7 and 408 y equation (408) can be written as

B (t) ~21

=

A;{t) N° (t) ~ ~'(t) N8 (t) - N(t) ~a (t)

._-- ~(t) = ~(t)

N{t) ~o (t)

Ar(t~(t)

35

'rhe expressions Bij(t).. N1(t)} A:(t) and Ti(t) represent true

values. To distingu:lsh true val.ues from estimates obtained from experi. ..,

mental observations J estimates of these and other quantities 'will be

represented by lower case letters. Thus bij(t) is the estimate of

B, .(t), net) is the estimate of N(t)j etco With this notation theJ.J

estimate of B21

(t) is

(4011)

Without knowing either (1) exactly what compartments exchange with

Compartment 1, or (ii) exactly what compartments the flow B02 (t) goes

to, it is not possible to compute an estimate of B12

(t)0 In the case

of rumen sodium problem B02(t) :1s composed of B12

(t) plUS a component

to the omasum. Knowing the rate at which sodium goes from the rumen to

the omasum would make it possible to estimate B12

(t) • This will be

discussed in detail in Section 5020

4.4 Experimental Considerations

According to (4.11) estimates of B21

(t) can be computed from the

* * *gvalues net), a~{t), a2 (t) and 8 2 (t). These values maybe elther direct

observa~ions or computed from observations. Because of Assumption 407,-It

81(t) i.s the estimated isotope ratio of the sodium in a sample of plasma

taken at time to Possible sources of error in this observation are

(1) the Poisson character of the counting procedure, (11) error in the

chemical analysis of the concentration of sodium in the plasma, and

(iii) error in determi,ning the volume of the sample of plasmao Normally

36

c:crc:rs (it) ',:nd (iii) should be quite small J and .if the tracer is

Injected i!ltothe blood J the actIvity of the plasma should be high~ so

that error (J) is a small per cent of the observation,

N( t) ,s the amount of sodlum in the rumen, can be computed as a

funct1,on of U~e liquid volume of the rumen and the sodium concentration

in the rumE;U" The dilution of a given quantity of a suitable tracer

in,lec ted .i.nto the rumen gives a measure of the liquid volume, Sperber}

et. .~lo J (1953) have investigated methods of determini.ng the liquid volume

of the rumen 1.n phys iological and nutri tional studies,~ and have reported

that polyethylene glycol is a suitable reference substance in that it

does not pass through the rumen wall nor is it destroyed by the

di.gestive processes of the rumen 0 Although concluding that polyethylene

glycol gave a valid measure of rumen volumes, they did not present any

indication of the errors which might be encountered. In the experiments

reported by Sl;j'ter (1963) ,the volume of the rumen was determined by the

polyethylene glycol method at the beginning and at the end of each

experiment} and a linear relationship was assumed for the liqUid volume

during the experiment. It would be possible to determine the rumen

volume at mDre frequent intervals and to assume some other relationship

to est:imate the rumen liquid volume at the times needed to compute b2l(t) .

Est:lmaLl.on of the sodium concentration of the rumen presents the

* .same difficulty as does estimation of ~ (t), namely that stemmi.ng from

the non~J:romogeneous character of the rumen contentso Since the same

*sample of rumen fluid yields estimates of ~(t) and of sodium concentration

the same assumptions are made about both estimates, One is that at time

t the estimates of the .isotope ratio and of the sodium concentration

.:.'-

frcm a samrJi:: ~C"'~ gCCJd estimbtes of ~ (t) a:od. cf 'the sodium eC!lcentrat:lOD

of the eat ire ,'wnen J and the other 1.S th.at two or more samples taken at

the same tj me yield estim.ates of the varieb.Hay of the values of the

isotope ratic)s and sodtum concentratIons in the ramen 0

for

*"TIle estimation of ~ (t) involves the difficulties considered above

1\~ (t) and the addHional difficu1ty of estimating the derivative

of a funct:to::l from isolated values ()f t.he function '.-Ihen the functional

this difficulty" There is li.ttle theory 8Y81181:;le rels-r::Lng to th,i,s

problem" The Literature of Eumerlcal 8.na1;Y218 g:lves mU.ch at.tention to

formulas for evenly spaced abSC1.S"8S ,1 l:mt :in gene.!"81 do not consIder

errors in the ordi.nate:3" The advi'::e is given t;o aV'oid numerical

differentiati.on 1.f at all possible} and if the data are empirical and

subject to considera1::J.1.e error they should first be smoothed in some way

(Hildebrand,! 1956) Q Guest (1961 3 po 354) says, '~No general rule can be

giyen, and the choi.ce of the amount of smoothing desirable is largel:r

a matter of personal judgment," StatisticBl literature discusses the

problem of smoothir.g almost entirely in the context of. time series:i l~, ,,~. :J

'where the nuniber of data points is large~ greater than 50, 91:':). car~ be

expected to cover several peri.ods o1~ any periodic fluctuatlons 0 In the

absence of any theory a lim:ited study was made of the effect of a five"

point quadratic moVing average, The details of this smoothing procedure

and the results of the sampling study are given i.n Section 60

Since the observati.cns are subject to error J ::l.t 1.13 clear that the

estimates of B21. (t) will be subject to large di.stortion' of' the denom:i.nator

38

of the right hand s ide of (4011) beccmes smalL Thus observations should

be taken when this difference is large compared to error sizeo If it Is

desired to carry out observations over long periods of time then repeated

injections of tracer should be used to maintain a large difference

'If- 'If.-between ~ ("t) and A2 (t) , It should be not.ed that the estimate given by

(4011) is not :l.n any way dependent on the number of injections of tracer J

nor on the "time since the tracer was injected} except as this affects

the difference in values just discussed,

In summary} the suggested experimental procedure for estimating

B2l(t) is as follows~ 'llie radioisotope is injected in"to the circulatory

systemo After allowing an interval of time for the tracer to become

mixed in the plasma) a series of obseryations is made, The observatIons

are of (i) the isotope ratio of the plasma~ (i1) the isotope ratio

of the rumen fluid J (iii,) the concentration of sodium in the rumen} and

(iv) the liquj.d volume of the rumen 0 Some or all of these observati.ons

are smoothed and equati.on (4o:11) is used to compute estimates of B21(t),

The values of b2l (t) may also then be smoothed by a quadratic moving

average y or some other smoothing procedure, It:i,s suggested that two or

more samples of rumen fluid be taken si.multaneously so as to get not

only the average isotope ratio and average sodIum concentration y but also

an indicati.on of how variable these values are,

39

50 EX'TENSIONS OF 'lliE ME"rHOn

501 Two Compartment Models

In Section 5 the method is extended 't,o various modifications of

System I,~ Figure 401 J and to other systems which have been analyzed ir.

the literature by compartment ana,lysis 0 The systems are CC:':Tf,Bl'ed

with the analagous system of compartrr,ent analysis,~ and some of the

differences between :i,nterpretation by compartment analysi sand In:er,q

pretation by the method deve}:)I"ed here are considered 0 ODe of the

differences of importance is the ope:n-,ended cr.aracter of the plasma,

compartment as it is consi,dered here} that is, it is not necessary to

completely specify all compartments wh,ich exchange substance with the

plasma compartment in order to get estimates of the transport rate

functions for some of the compartments 0 For the most part only the

estimates of the B.. (t) are given, the assumptions and experimental1J

considerations are in general the same as those discussed in Section 4

for System 10

A slight modification of System I 1.s shown schematically in Figure

501 In this system the only flow of substance from Compartment 2 is

back to Compartment 10 In this case

12

B~l(t) \,;

Compartment 1,

Compartment 2~

~

" B . (t)

.I.'~gure 501 System I(a)~ a modification of System I

·40

estimat:es are " ,ltD".n2 {t) ''\:: I,t)

~ <-_=o>",.~_"",.-.~-..-...-..~_.~

b ('t.) ~ b it ) p~ (' , )12 - 21' "', ~ J2 ·t

If in addition It is known that B'Yl,{t) "'" B1;l(t)" a condition that.""" -

is true if and only jf N2

(t) "" C a cons ':am;. !hen for all t 3 H~ ( t) "" 0

and from (408) b21

(t) can be 'writteE aie:

- ------_."

Of the two expressions fo.!' b21 (t) given in (5,2), the eDITect one

to use would be determi.ned by the experimental conditionso In some

sHuations it i,8 possible to measure '12(",) d:i.rectly, as with a whol.e-

organ counter; in other situations ,9 as when an organ is surrounded by

b Load vessels J it would be more sa',:;] ef i1ctcr;/ tc; measure the amount G of

substance in the compartment 0

There may be some situations:; such as when maki:J,g compar:isons from

animal to animal .• when the :rstio of the imler\..;' of slibstance to the amO:lnt

of substance in the compartment :1 s more meaningful than the rate of inflow

itselfo Equation (50.?) is obtained ~:.'y dividing both sides of (502) by Co

Since B21

(t) = B12

(t)., this yields

A b21

(t)~l (t) =---c-' J

41

where ~l (t) = IS.2 (t) ::>1 "_-cLs1 " The quanti ty KJ.2 (t) is analogous to

the Crate constants 'I , k" J ufr.:,en used in tracer studi,es (Gregg, 1963;:1,)

Solomon, 1960)., except that here K.12 (t) may be a non~constant function of

time, ~l(t) is the rat:i.o (·f inflow to the amount in the compartment)

whereas lS.2 (t) is the ratio of outflow to amount. ,in the compartment" Thls

situation is a semi-stead:y"etate condition ,in that the amount of substance

in the compartment remains constant" but the rates of flow may be

changing,

1m example of a biologIcal system sueh as SY'stem I(a) is the plasma

and the red blood cells; the red blood cells being suspended in and

surrounded by the plasma can only exchange substance with the plasma,

Sheppard and Gold (1955) and Gold and Solomon (1955) investigated the

transport of potassium and sodium, respectively, into the red blood cel18

from the plasma, Both investigations indicated that the correct model

for the system was not a two·,compartment model» the evidence cited beillg

the lack of fit of a single-exponential curve to the observations of

specific activity, and difference in specific activities of the plasma

and of the cells 15 hours after the tracer was injected into the plasma,

Gold and Solomon correctly point out that there are other explanations

for the inequality of the specific activities in addition to the

possibility of more than two compartments, If there are simply two

regions of the erythrocy"te j one of which has a rapid transport rate

function relative to the other y then (4011) would estimate B21

(t) as the

sum of the two rate functionso If on the other hand, there is some

mechanism in the red blood cell which ties up some of the sodium so that

it cannot exchange with the sodium in the plasma J then this method of

42

esti.mating B" .. ( L1 i.6 not applicable" since Assumption 408 requ:.i:res thatL, l.' ~

all sodium PClTl;::.·::Les have the ,same chance of leaving the cells 0 In thi.s

case a model si)ch DS that proposed. by Shemin and Rittenberg (1946) might

be appropriate 0

Another e;ys't;em often analyzed is sho'''n in Figure 5020 Although it

is not in the fnlIllework of the general plasma~compartment systems , it 1s

of i,nterest because it is the configuration of the two~,compartme:nt open

system seen so (lften in the literature of tracer sb.ldies"

'"~lE(t)

B21

{t)\.~

Compartment 1 ~

, Compartm.ent 2~

" B12

(t)

BE2

{-t}

.~~

Figure 502 System I(b): the two-compartment cpen model

Note that in System l(b), Bl1(t) = BlE(t) + Bl2 (t),9 and

B02 (t) = B12 (t) + BE2 (t)o It is assumed that the substance is excreted

out of the system at the rate BE2

(t); thus if tracer is injected into

either compartment it will leave by way of Compartment 2) but no tracer

wHl enter the system other than the amount injectedo Equations (5,4)

describe the system, (To emphasize their possible non-constant character

all functions of ti,me have been written in the functional notation, f(t),

In the remainder Df this thesis, for conciseness and convenfence j such

functions will be written without the argument t, Thus Ni(t) "" Ni "

b 0 • (t) = b 0 j J etc" but unless otherw:I.se stated all functions are still1,J 1,-

to be considered as possibly no:n=constant functions of time,)

The estimate of B2l is again given by (4011), and the other

estimates, obtained from equations (504),9 are given by

In terms of the experimental observables, namely the isotope ratios

and. the compartment contents, these estimates are given by

*u

b2l ..n2a2* *

,al-- a?

* *u

(5.6) -l [ a1 ~a2 * ,if ! ]b12 =

* * .* + al nl + nla

l ,a2 al - a2

44

[ '*1 * *0 ]j n,.., 82 'T' a

ln

1 + !ll a1blE = '-'\ ;~ <::::

'it8 2

*u It- ~ *0 j

[ r~ a~ t, al

n1 + '1 8

1 ]bE2=: ~, ~ '1 + ~

*82

5,2 :Three Compart~!IL,§lSterns

Subsystems slJ.ch as System I and System r( a) CBn be put together tc

form a s;.rstem. wl.th more coxnpart;ments J and the equations derIved i.n

Section 403 and 501 can again be used, For example, if the movement of

sodium between the plasma and the red blood cells was of interest, as

well as the movement of sodium from the plasma to the rumen, the system

shown in Flgure 5.3 would be appropriate.

B, '31 ~

Compartment 1 ) Compartment 3,(plasma) ./ (r,b.co),

ts13

B21

"'" ~

Compartment 2'"

(rumen) ,~B02

Figure 503 System II~ a three compartment system

The estimates of the B . are obtained from equations (4,6).9 (4,11)'J

and (5,1) by substitution of the proper sUbscripts.~ and are given by

b02 "" b2l

~ !l2 ,9

'* U * *b~, '"" n3

a3

/(81 - 8

3) J

",d.

Since Compartment, :2 and Compartment 3 1".,£:1'118 no direct excha:r.ge they

are in a se1:Lse 1:nd,epe:r1':l.e:n.t,< and are included together in one system on·l.y

as a convenience l':j that 'informatIon abcmt both may be obtained from

one experimenL Any number of' systems such as System I and System 1(a)

may be put together to fo:rm a system which is a modifIcation of what is

often called a mamm:illary system (Sheppard J 1962), The modification :is

to two aspects of the system configuration, One J the peripheral

compartments of' t,he usual mammillary system exchange substance only 'with

the central compartment.~ but in the systems described here the peripheral

compartments J while receiving substance only from the central compartment

(plasma) y may have outflow to other compartments not being consi.dered,

The second modifl.cati.on i.s a concept which has alrea.dy been di.scussed; it

is not necessary to speci.fy completely the compartments whi.ch may have

exchange with the plasma.

An extension of the development of es'ti.mates of Bij (t) as :in

System II would glve estimates of the Btj(t) for any number of compart­

ments. But a practical limitation might be :imposed :i.n many cases by the

necessity for makirJg observations i.n every compartment for which est:l.mates

of transport rate function are desiredo

46

'I'Le rn(~tt:od C6.n also be applied to the rnodification of the so~called

t t '",,, dca enary sys em lJ)neppar, J 1962) shown in F:igu.re 5" 4 J in wh:i.ch Compartme~t

3 exche.:nges substance with the plasma only through the intermediate

Compartment 20

Compartment 1

:B.;<,r-r~__,":=_;i::~·_~_..,...;'~

I--._~-~---~Compartment .3

Figure 5,4 System III.~ a three compartment., modified c:a tenary system

Again BE2 = B~) - £32' The estimates of System I(a) apply to

Compartments 2 and 3 of System Ill .. so thal; with a change of subscripts j

equation (501) yields.1 for

'03"-c.. ,I

The changes ir. the ;)mounts of substance and tracer in Compartment 2

are described by

47

Solv:LGg Each equation of {5,,9) for .802

and equating the twa gives

so that if

:l.Tl terms of the observable variables as

·.'I'r ! .it" * s *9a,.-, \8 ~ a3

) n ~ :13

a3b21

c. ·2 3= .-~......__.

+Cot· ·Ii .* a*a - a2

a2~

1 1

.A comparison of equation (l~o11) with equation (':),11) shows that the

second term in tbe right hand side of C5,llL

(a* ,~ a*)ij *u

(5012) n) ,~ n3a3":_2 ..-L 51

* *82

= 81

is the error that will be in the estimate of B21

if b2l i.s computed

from the equations 0:1' System .1 when System III i.s the true situationo

This error mi.ght occur becaU;:ie Compartment 3 is not known or recognized ..

or because observations in Compartment 3 are difficult or impossi.ble to

make, If the tracer is injected into Compartment 1 and the observations

** *are taken while ~ .i.s much larger than ~., then ~ ~ IJ. < 0 J and

If N~ i.s a constant, then the error term (5012) reduces to

48

.",v

:r,:<" 8:z,_,~_L

.1':. *8-c a~

1 c.Jl:V

wh.i.ch is a PQS:lti\i c quan:.it.y j s:tnce i.D. b:;j.. 8 period of ti.me 93

.1.2 IC"i Live,.

Thus b2l

as cGmpu ted The

*"1value 0:' A

3That }S J for- a given value of :IVA,'

./

*vthe larger is B32J the mere rapidly will T3 increase, so that ~5 will

*.H'b'" Ja"g'" 0 'G'_,,'{.';'~ ~.;'l·..Y.':>'d- '181.. 'lP. n_f K +b.. '~. "'m"'] '1' p" }' S ~T ·t·1.-,,,,} '''fer'''''-r' A

" - ... v .l: ,. ~ • n. - ,- -. ~')'2 J v_.... ,.;, ••cu.. ,-... "~ -~.3 ,9'i.,·,~ . .:.. b~> .:5

will beo

l<!' *Az t i.nCye83eS the denoml::l.!3.tor of the errcr t,erm j 82

~> al

J decreases J

)li,

thus tendH\g Te, i,ncrease the error term) but at the same time 8) will

tend to fla":;t~en out5 :n.aking a;v smaller 0 It i.e clear that (5012) could

introduce an apprec fable error into the es·timate of B"loc.

Another error that could occur would be estimating B31

as though

Compartment 2 d,i.d not exist o rile estimate in that case would be

b31

=<

which differs f'r'Jill the rs'te of inflow of sc,bstance into Compartment ~)

* * ~ 4only in that. the denominator is 81

-, 83

, rather than 8 2 - 8 3as (5,8) v

If in this case the tracer is in,jected i.r..to Compartment Ijl then

* ji-early in the experiment 81

> 8 2 so that the rate of inflOl." would be

underestimated, On the other hand, if it is knOW!1 that tracer goes from

Compartment 3 to Compartment 1 (in this case through the unrecognized.

Compartment 2)J then the tracer might be injected i.nto Compartment 3, In

* *that case a2 > al~ and the rate of inflow would be overestimated,

49

System l.Ilts) 113 9 slight; mod.ification of System III which could be

the model for the I:laBma,~rU!l1e!l'-omasumsystem referred to in Section 403.

System III(a) is shown in Figure 5050

E'?l ~

Compartment 1 Compartment 2,,

(plasma) '(rumen)1

B'1'~c.

K-32-,

I Compartment 3.B03 ( ,

(omasum)

Figure 505 SJI"stem rIlta) z a model for the plasma~rumen=omasumsystem

The estimates of System I are applicable to Compartment 2 and

Compartment ,3 of System III(a) y ,so the estimates of B32

and B03

are

given by

Although the c01n.figuratton of System I applies here~ the assU!nption

of rapid and complete miXing which was assumed in regard to the plasma

is not as reasonabLe an assumption i,n regard to the rumen; thus a further

approximat,ion and source of error have been introduced 0

50

Ii'1 Compartment 2 t.he c.hanges In "total. s:1bstance and 1:(. total tracer

are given by

d l\f2

dt -B2i

d 'r2

dt

;~

- A B~1 0::.1

Sol

,'it ~,

(A '., AI)rC

~~."~~~~;:-~-_.~

A2

+ and, e:lue.ti.ng

U,,1,J.2

.. '1(.

Hence y if Ac:. " AI" thc;; '?c','.;ima te of B21 1s g1.ven by

<.~._._._-""'......""""""

'*'*a ,~a

2 1

oUsing equb tion (.''-.,9) fer t

2the estimate E18Y be written

Compartment 2 [lGS ::mly Com.partmem; .... :13 .its source of substance, the:!. ;:<'-21

is es t:imated by trie "arne expresslon in the twa cases 0 The complexity

of the expression fo,l' estimating B21

given :in equatio,n (5"11) is due to

Compartment 2 recel '/1 r.tg s'..:tostance from Compartments 1 and 3 .in Sys tem ILL

From the first equDtlon of (:;,,,14) Bl~ rnav be estimated as:c: "

51

b1:2

+i"' n,_,

c.

A nOi;ew:yctny pr5tcti.cal poi.nt J.s that (5,17) involves three numer:cal

differentiatiuus 9 chus b12

would. have a high vari.ance unless the errora

in "the cb3ervnti.C'ns 'i!ere quite small 0

:B:y ·~OX18 '.i,a

only souree of 2uDEtance in'to Cowpartment lis from Compartment j J then.,

substance with Compartment j ,9 the estimate

of B.\. is1,J

The same ercor e:Y!f3ideraticns apply as in the discu.ssi.on wh.i.ch follow.s

equation (:50 'L'hu5 all the transport rate functions in a one-,way

catenary, 2m b';:CI1 as shown :i.n Figure 506 can be estimated by making

observatiom;; ;If tt;~ :i.sotope ratios in all compartments J and by' observing

the amount of sut,s~!Jnce in all compartments except Compartment 10

In the one-way catenary sys"tem i.t is assumed that none of the

BOi

lead direc'Lly to any O!~ the Compartments 2 J 0 0 0 J No

The fii1.al s~{ste1':. for which estimates are derived .is another

modificat.l..on of System. nI J shown in F::.gure 5,7, 'I'he quantities B21J B32and B

23may "be estimated by the expressi.ons given in equations (508) and

( ~ 'Ll) f~~ o'~~~n' 1·1'./0, ,,L,' .J...... U .....l:..-: ..... e 1 > ,J..()

B~.:/.-

'"

I;~" , "-'-="=:~/-Ok'----n--"-,-~ 'I,

~~ t., .. _"

¥(;oo

JIf £JN).l~ -1.

,lo-_, -J

1--_. B....2.....l~_.~'')fo----------~1Y1I""1 Compartrr,ent 2

F ...g~~re

cr

,,' n3 }

C'. <'=

~::... ~-

Reference L.c' 1><',; made:!l Se(::ti.on 5, i to one s itu.ation :in ',lhich

the method j,:; 'lCc: in Se':::tion 4 a~ld extendedln Section 5 18 not

applicable;tt:u::, 13, to systems where Same of the substance is not free

to exchange, Eybtemj or course., whi.ch does not satisfy the assumptions

made in Sect'i:.JD. L;. cmmot be anaLyzed by th.is method,; such a system may

require 9 chang<:::n':.he method or a different method l.r the transport

rate functicn.s 8t:;LC be estimated,

In all the syetem considered in Sections 4 and 5 either (i) each

compartment has ;:;l:Iy one COUTee of the sUbstance) or (ii) there is one

compartment in tnc 2yBtem which has only one source of the substance and

the estImates cf tl:c tr'3~lS1=Crt rate fUi:1.ctionsinto and from this compart-

ment were used tn u')tainLng estimates of the remainir.g transport rate

functions i:n. th';:2yc3 tem, Alternately, as in System I J the sum of two

or more rate fun:'LJ.o(lS was estimated,

'y+

There al':::

because of the.i:1: ~ c:!':erc:onnectL:n:s) the t,ra::lsport rate

functions cannot be eat1fficlted. 'wi 7":'h the use of c:;'"ly one tracer in a

single experiment, Syste!1.\ .in Figure 508 is an example of such a systeDL

tU'ee of the trans.por+; rat.e fu.nctions) cr three

linear combinations r;Jf :he transpc;rt rate fU.nct:icms ,1 ·can be est·lmated,

If} however;,

Compartment 3,

__.__ ""v~

c: I~·~·Ccmpart;.nent 2

O!i!"..#---------.......~'~-'RB.-?_..."----L -'

-.,

F:gure 508 System IV

The equations which desc.::!:'~he the system are

d T' .)0; .:*. '*.', :; Al B31 + ~. B32~,

~ B23/ )

-eFC-

d N2

dt . :; B;?1 + B2 "1 - B32_ ..L.~

d T2 * A*

.,.~ A. I:i. 't A) B:23

~ Bydt .J.. ~21 .:J~~

55

'f'.,n '"'1at,",.'x ',.",<,,-,:'-,,:;,t,',:.n.n "'co.'" q"",~,.,~""" '0 ., ')~ ~")'a-t-~,~'n~ II<=;. 19\ ar"e I\;Q,.. = R,,':'J. u. __/_'-' __ ,__ :...: ..... ~~; 1j.,J''-,'-,r..; .... '-'',~ .. .-,. o"L ~\'-;1"",..... ...L'-'~,,\.:;l ~.;lO_~ ; ... ,..l'1J: '-_,

where

0 ",1 ,A-I

£21 N3

g- '* '1t'0 -A" A, ~ 5

23,Tj

;I' .J.c

3A "" B '"' D ""

1 1 0 "·1 B3l

N2

* ..""A, A 0 ",,A.,

" T21, ", .~...1.., j

* ~In the r.18t','rix A", row 1;. "" A, row ,~ "' row') + A1 row 3, :I'hjs~ine9r....

relation amo:cJ.t.S the ro·.rs impLies that the determi.nant of A is zero 0 By

considering the Bum of the contents of Compartments :2 and 3 it can be seen

that the same .:Linea: relationshi.p holds for the row's of D, That is J the

are

t· N3 "" B21 + B;:;.,a 0 ~'

(N3 t N) "" 'r2 +

total. amount of suts ,::a[iC'e lZl Compartments t) ani ;i together :L5 (N2

+ N...J ~ soj ,

o

that the ch::J1J.ge In tt::.s ":ctal 1,S given by (N2 + N3

) = B:21 + B31

, or

U ,'"",

Likewise,~ T2 + T) "" A1 (B21 + B31)'~ so cr.at

thus the rows of the augmented matrix (A I Q)

linearly dependent 3 ard rank(A) and rank(A 1,£) a:re both less than four v It

will be s:'nown that these matrices both have rank three 3 so that by

Theorem 1002 three of t~e HI' :.::an be determi~'1ed only b:v assi.gning some,J...J

arb:ttrary value to the fC:.Lrth B.. v ltwi.ll also be shown that anyone 0:::'l.J

four transport rat~e farlctions may be ass igned aTI artl trary value 0

Let A1

be the raatr:ix of the coeffic i.e::lte of B23

y B31

} and B32

in the

first three::' equatio::1s of (5019) 0 Then

~l 1 1

'* '* 'l+A, "" '"A3 1\. ~.L

1 0 =1

.:-i,~ A-A.}

/

*WtiC!l :tiS ze::"o if' and only- if ~,:;

*,~ A_ y,1

*,~ ~,

'it~ A, "

.l.

of

and t,he determ.in,snt of the coefficients of B21

:, B23

and B3l

is

,w'

"",~> * '* *Thus 1f A~'

I A_ ~,

Al ~ Ay '11'0[] A and (A I f!) have rankf Dr' .!'\,3) or

1 :'~.

after Whi.C!l the 01:1:':2':':' ::t:;'ee are un:t quely determ,i.ned Q

All of 'tne :ra;:sp':;rt ra::'e functions of S;Y3tem IV could be est::imatedLf

two tracers wereli3e~0 ':'t;,e injection of :9 seccnd trseer L1tO a iifferent

compartment 1Mould make • "- possible to obtain an eqCl8tion which would be

lndependent of any of' the equations of (5 Q 19) . 'l:nere are other ways In

which a fOUT'th independent eq':l,ati.or.i might be Gbtained without using a second.

tracer 0 If there were re8Bcn t.() believe that':he '1alues of the rate

functions would remain unchanged t.hrough time one tracer could be used in

two successive experiments; the tra;:erwculd be injected into different

compartments int'he::w(, exper~ments, A!lother p():3sib~,.lity would be using

two very sim.ilar an:mals J such as identical'Cwins 0 'I'hese las'c two sug-

gestlons have the d.:ifficulty of knoWing that the system is really uncha:n,ged

in the two sltuatic)!L3 i:rvolved1 !o~, J d:i.:t'ferent times and different animals 0

A modification sf Sys-cem IV 1,13 sho'wn in Figure 5090 Tr.:.e estimates are

the same whether the ol;;tflo'H J BO'.2 J from Compartment 2 is directly back to

Compartment 1 9 B_2

} or whe~her the outflow is not thus restricted,, .L

57

Compartment; 1

,

Figure 509 S'y"',stel..ll IV! a)..' ,

SyS1.'';;J]] IV(a) can be dE:::scribed l\y the matr:,K eguat1Jl'::.

(5020)

where,-

0 0 -1 1

* *0 0 ~A~,3

A :II

,1 1 0=,.L

.!t,* *",A Al A

30

2

oT3

D r:

,, .''-

In this case the determinant of A is tAl

* *' "* *A1

1= A3

and A1

I: A2

, then by thE: Ccrollar'y of

'''')' ';e' *""' A3 tAl ~)3 so that if

SectIon 1001 t.he system has

~.""A,,,, A.3 1.

'j« '*A ~ A3

N'l -'3~.

* *a

A, " A3

:r'.».j.. .-

"n :N2'..)

0 'T!

"'2

,:..;..

A:).j

.~.

,~ A, )L

,~~~

,~ A­t:

*A3

'Ii.'f: 'ItA, (A_ ~, A

3: )

.1. d

B2, ,. .1

'Jt(AI...

.;t.

)

Here agalJl there ~L5 a com.pa,rtmenc, J

source of substance.

It might be said that S:rstem IV 1S ncy:: reaI16+iJ;::'z: that: thE::'e .15 n()

outflow from the corrib:i.ned Compartments 2 and,

two '::omp.9rt.m,er~·t,.s can only increase 0 ,rt'::Te might be !::::ic<~ogjC'9.l sJrstems

which CBn be assumed to correspond to System II! for a peri.ad of time» but

sooner or later outfl.ow would have to b,,,, ass7..uned. System IV(b) J shown in

Figure 5010). pI'ovides for an outflow frC;!ll Cowpa.c''':mE'.nt <'::J but as in

System IV) the values of the transport rat·s functlcms eannot be estimated

by observBtiGns of the changes in amounts of substance .~J:...d tracer i!l the

compartments 0

The changes in amounts of tracer and substance in Compartments 3J 2

and 4 of System IV(b) are described by

where

,59

Compartment .1

B21

-:,~31

~ B~2~ ~

'"""!!Iov

Compartment 2 Compa...r t,'T,,".::nt ~,~~

II(....

B2 .3B42, ,,

Compartment 4

B04~if

Figure 5010 System IV(t)

a

1 1 0 =1 '~1 0 B2

,· N2.l.

* * * * !

'l. A3

0 =~ =A-, 0 B_3

'T'c c, -2

a

0 =1 1 1 0 0 B'l'j N'lA ... B '*

.>,,_J,.,

D '" "

* A* *0 -A ~ 0 0 B32IT'

3 1 4:3'J

0 0 0 0 1 -1 B42 NJ'+

'iii- *0 0 0 0 ~ =A4 B· -r4? 04- J

There is a linear relation among the rows of AJ namely

* ( \ (* *)( * ~)/ *row 2 • Al row 1 + row 31 = Al = ~ A4 row 5 = row b (A4*A2.) 0 Thus

the determinant of A is zeroo As before a consideration of the total

amounts of sodium and tracer in Compartments :2 and 3 together shows that

in D the aame linear dependence holds amoriS the rows as ir.. n:atr:ix Ao ~rrrr(,la

60

the determinant of (A I~) is also zerou As ,,'as done for S;/stem IV;, 1t can

be shown that A and (A I~) t.aire ran..~ fLve J e:li i.f o::'1e oftt',e B" is assigned""cl

an assumed value y then (5021) uniquely determines the vallies of the other

five BijUSo But in the case of System IV(b) the B1j

to which assumed values

are assigned must be one of B21~ B2y B31

or B32

u Although Compartment 4

has only one source of sUbstance y determining values af BOLt and B42 is not

sufficient to uniquely determine the rest of the systelIL Here again y the

equation would make it possible to e:stimate all of the:'ransport rate

function valueso

These examples illustrate a limitation of the method of interpr",><

tation of tracer studies, developed i.ll this thesis 0 l'hey are l:imitations

only .in the sense that it is not possible to estImate all the transport

rate functions of some systems 'with the use of a s:1.ngle tracer i.n a

single experi.ment,

61

6, Pili APPLICA'l'ION AND COMPU'I'ER SIMULA:I'ION

6,1 An Analysis of Slyter EXEe£iments

used to compute est imates of the transport rate functions by the eq:16tLons

developed in Section 4, Although these experiments were conducted before

the formulation of the method given here J the observaticms required fol'

estimating 821

by equation (4,11) "fere made :l.n three of' the eX'pe.ri.T.er~teo: v

In each of experiments 4, 5 and 6 tracer studies 1,Jere made of three sheep,;:

each sheep in an experiment was on a di.fferent diet, One sheep\-Jas on a

hay diet; one sheep was on Diet 11, a purified diet; and one steep was on

Diet lIt, the same purified diet but 'with potass:Lum and 8C;dllrrc b:csrb':m11;'"

added, In experiments 4 and 5 Na24 was injected into the circulatory

24system; in experi.men<t 6 Na was introduced into the rumen, In every case

the sheep were force fed prior to the injection of the tracer, The volumes

of the rumina were measured by the method of Sperberj et al, (1963) 0 The

data of these experiments by Slyter are recorded in Tables 10.1, 10,2 and

10.3, The complete details of the experlments are reported by Slyter (1963),

*Before estimating B21

the estimated isotope ratio, a2 , and the esti-

mated total sodium in the rumen, n, were smoothed by a five point quadratic

moving average, By a five point quadratic moving average H is meant thaT.;

a value y( t,) was smoothed by fitting a quadratic curve through the five~

successive points y(t i _2 ), y(ti=l), y(t:i), y(titl ) and y(t i +2 ) by least

squares, Then the smoothed value ~(t.) was taken to be the value of the1

fitted quadratic at t., The first two points in the experiment were1

smoothed by letting ~(tl) and ~(t2) be the values at t1

and t 2 of the

62

quadratic fitted to the first five points of the experimenL Likewide the

last two points were smoothed by the quadrati;:: curve which was fitted to

the last five pointso Estimates of B21

were then obtained by the use of

(4011L and these b21

values were then smoothed by the same procedure 0 Tne

estimates of the transport rate functions for the three experimentB are

reported in 2'able 6010 The units of time are minutes since in,jection of

the tracer 0 The units of b21 are milligrams of sodium per minute 0 'The

negative values of b2l

in Table 601 are the resul.t of us1ng data collected

at a tIme when the isotope ratios in the plasma and in the rumen were of

nearly equal valueo AB has been pointed out before» when these 1sotope

ratios are of nearly the same magnitude J error in the observ8Lions can

cause the estimated difference to have a different sign than the true*.a

difference 0 Since the derivative A2

changes sign at the point where

* it'~ = A2J if these two values are nearly equal errors in the observations

*amay result in a2 being computed with the wrong signo In some of the

sheep, due either to large transport rate fQ~ctions or due to small amounts

of sodium in the rumen, these isotope ratios were nearly equal in less than

eight hours after injection of tracer, Even where negative estimates were

not obtained estimates of the transport rate functions flucuated widely in

these sheepo

To make comparisons between the flow of sodium in sheep on the dif<~

ferent diets the estimates of B21

were i.ntegrated over the 8 hour period

following the injection of tracero It was considered that the total flow

of sodium in some interval after the force feeding was of biological i.nter=

est in that it indicated how much sodium had come into the rumen in this

63

Table 6,1 Est.imates of B21(t) from data of Slyter

Hay Diet Diet 11+ Diet 11

time b21(t) time b?,(t) time ( )b21 t;_..l.

~--

Experiment 4

15 480.36 15 10018 15 13,53.55 48,05 48 10006 45 1506378 52070 80 10060 75 17055

125 78075 120 11037 126 20030180 122,92 180 6062 180 22060270 1:53.86 270 ., 270 270:31372 104006 360 5+070 391t .5h o 38482 00-,,6091 480 '=7" Lt·9 .3.)0600 ~~.51. 0 86 600 ,"j~4,36 600 113016788 -71077 7E59 ,,,57019 817 75,:28970 =9,98 966 1.3012 967 ",46065

Experiment 51.5 7099 15 902.3 15 800345 12041 45 9,97 45 809975 16074 75 11038 75 9085

114 26079 123 15,07 120 10098171 46002 180 23055 180 lle8.3270 57.44 270 33024 270 13.62351 43008 363 32,44 360 14090480 15·95 480 29039 514 13029609 6069 606 44014 609 9·91783 15059 795 22088 801 20.61960 52.. 03 979 -29018 980 43.66

Experiment 6

15 48055 15 5.17 15 .:- ., ,C'i." r; .,",, __ ..-'

45 52080 45 8.99 45 204775 57034 7'" 13001 75 2032,/

120 66081 120 20.86 120 .3067180 74032 180 :c~7 0 .3~; 180 5095270 79003 270 :52,05 270 7097360 65018 360 27012 360 7051480 36072 480 19·37 480 '7076600 33,54 600 15089 600 8,22780 35003 780 19,62 780 8003960 29,31 960 19009 966 6062

1440 -18,43 1440 5041 14L.o =303.3

Table 6.2 Total grams of sodium transported from plasma torumen in 8 hours; with analysis of variance

Hay Diet Diet 11+ Diet 11 mean

expe:c:iment 4 42.22 12.18 4.44 19.61

experiment 5 17·12 lL78 .5·97 11.62

experiment 6 30.60 11.06 2.89 14.85

mean 29·98 11.67 4.43

Analysis of variance

df SS MS F

total 8 1361.14

experi.ments 2 96.87 48.43 1

diets 2 1059.91:. 519·97 9·27error 4 224.33 56.08

pr(F2 4 :s 9.27). =: .965,

Table 6.3 Identification of sheep and weight in kilograms

Hay Diet Diet 11+ Diet 11

sheep sheep sheepnumber weight number weight number weight

experi.ment 4 5882 53.1 265 47.2 260 37·6experiment 5 5882 54.0 312 49.5 309 .34.5experiment 6 5882 55·3 312 49·9 309 32.2

64

time. The period of 8 hours was choosen because of the increasing uncer­

tainty of the estimates after 8 hours. The b2l were integrated by summing

the integrals I(ti~.,ti+;ql) from i-, to i-N-2, where qi is the quadratic

curve Which was fit in the smoothing of the b2l j t i .. -(ti _l +ti )/2 for

1-4, ••• , N-2j t 1+-(t i +ti +l )/2 for 1-3, ••• , N-'; t, ..-o and t N.•2+-480.

Here N is the number of pointe at which observations were made in the

experiment.

The results of this integration and an analysis of variance of the

totals are reported in Table 6.2. In the case of the sheep on Diet 11 in

experiment 4 the estimates of B2l after four hours were considered unusable

because of the rapidity With which the two isotope ratios approached

equality. The total for eight hours was obtained by extrapolating the

values of the first four estimates of ~l'

Before any conclusions about differences in sodium transport are

drawn from Table 6.2 it should be noted that the sheep involved differed

considerably in weight. In Tabl.e 6.3 the sheep used in these experiments

are identified and their weights on the da,}" of the experiment are giwn.

Rather than comparing total sodium transported in eight hours, 8 C{)l».­

parison of grams of sodium transported per kUogram of blldy wei,ght in 8

hours might be more rel.evant to the biological problem. Table 6.4 presents

these values and an analysis of variance. Tables 6.2 and 6.4 indicate

that there is a significant difference between the rates at which sodium

is transported from the plasma to the rumen in sheep on purified diets and

on hay diets. There are two ccmslderat!cma that qUAlify this conclusion.

One, the estima'tes of' some of' the B21 flucuet,ed widely 'because of the

66

relatively small difference between the plasma and rumen isctope ratios J

and two, the valIdity of the necessar;y- assumptIons for the usual analysis

of variance has not been verified,

Table 6,4 Grams of sodium transported per kilogram of bodyweight in 8 hours after injection of tracer

experiment 4

experiment 5

experiment 6

mean

Hay Diet Diet 11+ Diet 11

.7955 .2582 ,1179

•.3171 Q'~?;.82 ,lD1

05529 02216 00879

,5551 02393 01269

Analysis of variance

mean

02880

df ss M8 F

total 8 04144

experiments 2 0034.3 .01715 00812

diets 2 ,29.57 014785 70007

error 4 ,0844 002110

pr(F2 4 ::: 70007) > "9.5,

The computations necessary for obtaini.:r-tg the b21 ~ s and for integrating

them were programed for the IBM 1620 digital computer; the programs, writ-

ten in PDQ Fortran, are available,

Consideration of differences between animals wlthin diets In the

amount of sodium transported suggests that the transport rate functions

may be influenced by the following factors: differences in plasma and rumen

sodi.um concentrations, rumen volume, and the acidity of the rumen,

~2 _..90mputer Sim\~lasion of Sys:tem I

In Section 601 estimates of B2l

were computed from data obtained in

biological experiments. It was stated that some or the estimates were

highly variable due to the time at which t.he observations used to compute

those b2l

were taken. In order to get some insight into the behavi.or of

the estimates~~theirbias and variance«,~a small study was done by simulat~,

int System I on a digital computer 0 The system shewn in Figure 6.1 1'1813

used as an analog of System. I; Compartment 3 represerlts the extracell1l1~1r

sodium "rhieh exchanges rapidly with the sodium in the plasma.

B31 ..Compartment 1 ,

Compartment 3(plasma) ,.,

"" Bl3~ ~

B12 ~ B2l

Compartment 2

(rumen)

Figure 6.1 An analog of System I for a si.lT!ulat,ion study

Assuming that the plasma and extracellular sodium. are maintained at

nearly constant amounts over periods of 12 hours or less ~ N1

and N..z. 1..rere, ./

assumed to be constant. A constant value was assigned to B13

, BE3~ the

rate of excretion from Compartment 3, was assumed to be a constant percent,

p, of the amount of sodium in Compartment L The transport rate B2l

1.8 a

function of time, as is the rate of outflow from Compartment 2) BE2 •

Transport rates B31

and B12

'were adjusted to the necessary values to ma:in,~

tain Nand N3

constant. The changes in am.ount of tracer 1.n the three1 .

compartments are described by

68

d T1 * * *"" A3 B13 + (B

21 + p.N1

) A_. (B,z 1" poN + B21

) ~ )dt :c. .L.) 1.

d T2 * i.i't(601) 'CIt "" B

21Al

= (B21 + poNl + B

E2) )2

oro simulate the system (601) was integrated numerically to give arrrcmnts

of tracer in each compartment following the i'l,jectl.;)!1 oftrtE~cl 1n1:-:) Com~

partment 1 0 "I'h,,::, values for the amount:; of i3c:di UXrcl!1 the rumen and pla;sm.8.j

and the volumes of the plasma and rU:T:.en1,Jere tahm tot."'~ ahr:mtthe same as

those of sheep 265 in experiment 4 of the Slyter study.~) Table 1001.. The

values for the Bo. were varied until Isotope rati.o curves were obL.ainedl.J

for the plasma and for the rumen that resembled closely those of Table 1001.

It was assumed that the transport rate function B2

.l was the sine function

B2l(t) "" BO

+ Blsin(t!B2 »)

and B:E2 was assumed to be the exponential function

A number of curves were simulated on the digital computer to observe

the effect on the isotope ratios of the rumen and plasma caused by changes

in the magnitude and period of B2l

J and by changes in the sod.:iu.m content of

the rumen 0 These curves are listed in 'rable 605. Tr:e' followil'.g values

*. )were used for all of the curves except Curve 100~ A1

{O ::e5.jOOO~ N1

=4350.'1

N3

=4Nl ; B13",,1350; p=O.00075; rumen volume",,3200/2.2; QO""LO} ~=.<:>OO.,

~=-.0015. Curve 100 differed in the following values ~ p=(LO.; %::>().O.,and Ql=O.Oo

69

Table 605 Simulati.ous .-.-f' S~/s"t,I.:;m Il...J<lI..

i.n:.tl.slrumen flg',lre

nurriber BO B:J,. 132

sodium number-~~-

:00 /·... !7 0000 0 6400 602c,

120 '7 4,25 115 6400 6,2130 27 4025 115 6400 602 y 603

605,~,25 47 4025 11'5 6400 602126 27 9,00 65 6~tOO 6.lt

127 27 1.8.00 65 6i+OO 6~ ,4136 27 4,25 65 6J.too f L h '1

·jo 'J-c.,....

137 27 4.25 15 6400 bo ,5138 27 4,25 11,5 4800 6.3139 27 4,25 11,5 ,320,) 6.3

A comparison of Figures 6.2 and 6,3 shows that d1.f'ferer:ces :in the

amount of sodium in the rumen can have .ss great an effect on the isotope

ratio curves as do differences in the magni,tude of 321

, Thus the relative

magnitudes of the B2l cannot be judged solely on the criterIa of how

rapidly the rumen and plasma isotope ratio curves approach equality,

Figure 6.2 also indicates that the isotope ;:"at.io curves for some non·"

steady-state systems J ~o~ ••• Curve 130, are not markedly different from the

curves for a steady,~sta'teJ closed systemJ represented by Curve 100, 'rhus

the assumption of a steady,~state system cannot be verified by cons:idera~,

tion of tracer curves alone,

Figures 604 and 605 indi,cate that ;i ifferences in the am:pl:i tude ani

period of B21 must be qui'te large before they are reflected in noticeable

changes in the isotope ratio curveso

Some of these simulated curves were also used ~o look at the per ..

formance of equat ion (l~ .11) 5S or ~~st:i.l'f~"'t<)r cd frc:L

()(\.1,-~

00 0 0 0 0 00 () 0 0 0 (~

0 ~0 0 0 lC\

ll\ "'"' (>l ,--I

lUTITpoS ma.rllHTpn .red e~.nu1m .rad s~uno;)

~1()",~1

+'co::l.p°rlW

OJ+JtJ1.p(Q

u;:».ir~

lJ)(Ii

Pi.:Q

Ii!r:~,"~')

~~.,

'.:-1pCV~.

):J~ )

Ul0 ~l

cd ,-·1p r-!CJ .,!' ,~

(J)qn;)

l:l (),~

,'d 0

"'"'(t) r-~i

t..il:l ''d.... itJ] (\.1

10.()

tJ] (.x:~OJ OJ+) r-n ~'J::l 'r'.~

~ '"0,.\ 0 (I)S ()j tV

,--j~\

'" tJ1() ,-~

0,.h~

rJC·

W~- ..

<Ii0

:> p

t~ <:),Jj

() tf-~

"." 1-1III

ill.p

'Iim r',-I ... ~:::$

t:'

S 'd,,..,l~U:i (1j

(\J0

\0

(()

~jt10odP'.

0

0 0 c 0 0 0>'0 0 0 0 0 00 0 0 0 0 l!\:1\ •.:;t. r<\ (\J r!

71

8-'

".'1't)0CD

R(JJ

~~.

r--lr.u.p..,L.'

-.:'5::.~

°rl

IQ<f'1:1\)Q,,111v.~

t:)

tH,)

.j.>(-"tD

y._~

ij~

~~ tV0",-j (jjp .r:l() .j.>(l)'r;)

~Q""..1 '-'ri

~-OJ 0tJ ..c:s:: rn°l-~

Vl ,<>,

0\CD r<\OJ .-I+)

~ 'E'rl a:lS

COr<\,-1

'"()r<\,-j

tJlQj~::.,

8(.)

rrj<11

..pr.u.-l

§.,...;(I)

r<\0

\{)

OJ

~iiilorl

"'"

o~

()CO,-1

()()W\

;._j

rq

~ur~

to(1)

~a.1).~()

I:t.~

0

cl1)

[....j

,+.~

(V

(])~1.p

~"';

l="! p0 'U

",-I H-'"",,) pr~) ((1(I) C,"r-) ,'!$. ~1 t<-~

"~i! ',-1

t1J0 \Ll~~ r<\.~

orl ,--I(I)

'IiUl

,.",~

Q.I a:l+' t'-~1s:l

~F~'''',

'.0

~'J(.0OJ:~¥'

~-l

:JI:;)

't:'Q)

·t;)a:l:-1

~''1'4CQ

.~t

\0

(l,l

fj':iJ'd~J

0

0 0 0 0 0 00 0 0 () 0 00 0 0 0 0 If\If\ .~t· "'" (\j ,....j

UJl';~pos me.rzlf.H T'Ul .Iad aq.mlT'iU .l'au s'iuno;:;

(\JIq

l7::...~ ~

(J)(l)

1:lOC(iri~~~l(l

\~ .~

0

,.~ \

tli4-,1H(I}

QJ

"':;p

~p"'d

f-1 ~)

() tU",ol Hp ~.

() "(f)

1l ;-,C'J

r""f:j r'.-1-..d o,~

OJ ')<'"1:1

0 rc•

1-1 1'<\'n r1lQ

ffl !11(j) a1,l,J

.;j \0~ !'C\

>,-1 ,-j

S'"C)

1'0,.- .~

(f)

m;>

~-10

'elOJpa:I,--1

§,,-1

ro

In.\()

tV

~~be~f-~

r"l

C")

.::t(lJ

00 0 0 c> 0 00 0 0 0 0 00 ~'f 0 0 0 ll\l("\ !'C\ (\J '·'1

,e

74

oj(-

it.... 1;;\iere asslll.D£;d t;o have errore i..:r:::,:,ch 'Wel"'i2~,

distributed &B POiSSSD

random v8:;:·.iables ~ which could be appr::)xim.ated by nc,rn:.al var:tates. Tne CC';'l~

error. The normal deviate:::: were obtai::led fr'cl:~ D2.X;Xl aul Massey (:";;)7;

with the tru.e valc.€sQ :B, 2.1'

the average of the 20 0'-;)'1 (t .. ) i S "\;illS COl.llJ".lted; f:is'.,'t;li as ·:r:.e ,31:.andHr::t;...J... .1,

Figures 6,,6, 6.8 and 6,9,

of B21

for three of the error curves i:n Series 130,

Series 120 of error curves was cased on CUTve 1.2C' ,,rith ;;_i.)2pl::L~lg at ::':,

int.;ervals betweeD observatiC!ls averagir.g '705; 1..5 a!.~d 5C ITci.m..t;:;6

Seriea 1.37 'W'as based on Curve 137 wlth sampling at

A ::consideration c.f: .Figures 6.6., 6.8 and 6.9 can

for conduc1;ing biological experiments and suggest l.ic.es of 1nve"'+,igatlO?.l

for a larger sampl:i;ng study of (4,11) 8t;: a~'l estimat~~ of IL.~l_

I:asxal:ll.:n1ng

these figures .the f'irst two points a;-Jd the last two p.:;ints sho·,J:!.d bE: ccn-

sidered sepE.rately, since b21

at these POil1tS was af !lecess::.t;.,' computed

from values whi.ch were not sn:aothed w:!.th t.::1ese pc~rJta as t.r:e central poi.nts,

e e e

Table 6.6 Estimates of B21(t) from simulated curves

ave. s.d. of ave. s.d. oftime B21(t) b21(t) b21(t) time B21(t) b21(t) b21(t)

Series 120 Series 130

30 8.09 8.34 1.086 30 28.09 28.73 2.63348 8.72 8.79 .414 48 28.72 29·17 1.05960 9·11 9·10 ,415 60 29.11 29·43 1.19575 9·57 9.48 .424 75 29.57 29.63 1.20987 9·91 9.88 .427 87 29·91 30.07 1.180

105 10.36 10.43 .593 105 30.36 30.80 1.622123 10·72 10.81 .574 123 30.72 31.34 1.745135 10·91 10.96 .496 135 30·91 31.42 1.490150 11.10 11.14 ,613 150 31.10 31·55 1.770165 11.21 11.35 .680 165 31~21 32.02 2.098177 11.24 11.45 0561 177 31.24 32.41 1.841195 11.21 11,23 .815 195 31.21 31·91 3.012210 11.11 10.93 .812 210 31.11 31.04 3.082225 10·93 10.80 0967 225 30·93 30·97 30443240 10.69 10.88 20202 240 30.69 31.85 9.165255 10.39 11018 4.155 255 30039 34.03 18.908

Series 160 Series 16130 28009 28069 4.099 15 27055 27090 1.78936 28.30 28065 2.243 45 28.62 29007 .81445 28062 28.72 10422 75 29·57 30.02 1.13454 28.92 28091 1.440 102 30029 30.64 1.24060 29011 29.34 1.528 135 30,91 31.39 1.13966 29·30 30.02 20390 165 3121 31.97 1075675 29·57 29098 2.334 195 31.21 32.30 2.11684 29.83 30010 2.109 225 30.93 31.94 1.953

-.;]VI

e - e

Table 6.6 (continued)

aveo s odo of aveo sod ':l oftime B21(t) ( ) b21(t) time B,",~ (t) b21(t) b (t lb21 t,

<:::J. 21' ,

Series 160 (~ontinued) Series 161 (continued)

90 29·99 30.33 3a601 225 30039 3.139 2,08093 30.07 31.10 30782 285 29.61 31,14- 2.11599 30.22 31.69 2$736 "il '" 28.66 31.18 20816../ ..... ./

105 30.36 31.32 2.865 345 27·59 29.76 40979114 30·55 30.38 20701 375 26049 27047 6.036123 30·72 30.24 4.411 405 25.42 24046 40588129 30082 30·77 8.850 435 24.45 28.36 200124135 30.91 31.76 1:;1 0 035 465 23.66 37014 50Q155

Series 137

30 30.86 28.57 :·,55345 27059 26066 1~546

60 23.78 25067 ~.g147(;;. 22·92 25034 1.02971/

90 25081 26oE31 lo3Cr2105 29079 29·00 10318120 31.20 29·82 lc835135 28.75 28.49 1,705150 24.68 26020 1·559165 22.75 24.65 10933180 24.71 25026 L813195 28.78 27·59 1.744210 31.21 29.54- 2·527225 29.76 29024 2.1.30240 25·77 26.81 6.032255 22.91 22025 140244

--...1U\

e e e

II

B21

"avez\.~ = 2 Bodo

"

\average - 2 Bodo\\

--------- -----.. --

Series 120

••• " ,/.ave, , , ••• ~ .... •• ,," • • ., • • ' rage b____...;;;...;;..;;...;..-------=-:.:..:.~.~:..~. "" 2'. - ,.~ ~-..---,--

/- -''" --~__ ".. average + 2 S odc

-- ........ _"... .....-.

....

/

~ ---~/--

.. ".# ~ , I • - ~/ ---, 21

/ 'S . 'er~es 130 - '" .... ....

........... ~,,--­-

,

"../

38

34

Q) 30+J:;j>:::°8H 26Q)PI

~ 22oM'00tIJ

r..; 180

~e.uH 14QOoMrlrloMS 10

6

o 20 60 100 140 180 220 260

minutes since i.njection

Figure 6.6 True values of B21 and average and sod. of b21 for Series 120 and Series 130-.:;-.J

e e e

,_ ... 130~7,

B21J..

/ 130=4

//

;III,

I/~.. i;", ". I I0' '0. /-, ,,' \:". I

' . - . '. ~-0' "of c') '\ I.' "" \ • I~, .

/........ '/ \ .. I

....~I \ \'"I " " \. II .' .../ \. I

.....-- \'G- \ \.;\ ~ .

\ ...... I •• 1

.... . "''''' 0 130~..L.. ... ..• •••••••

./

" /............. .../

'\\\

\\

'\\

..........

24'

32

lHo

ClJ

~~

oMS~ClJPI

~co~~

oMr-lr-lor-!E!

§-..-I

'8l1.l

22

o 40 80 120 160 200 240

minutes since injection

Figure 60'7 True values of B21

for Ourve 130 and b21

values for three error curves of Series 130.-:jOJ

e e e

B2l

-- -- ~6!j----2 s.d. (ave. = _

ave. + 2 s ,d. (161) ~-- -~--

- ----

\\\ average = :2 s.d. (160)\ .

\\l\

II,f,,

, average + 2 s.d. (160),•I,,

'-, ,....... /

,- ----------

- b21 (160) _ ave. b21

(161)- - --------

,\\\

-.,/' -' /"'\,,-, ,-, \ I \

' " \ I \/ ~ \

' \,I

I22

46

30

42

14

26

34

38

18

CHo

HQ)PI

~..-l

'8u.l

Q)

~s:!'g

u.lSeelHbO'r-!r-!rl

1i

30 60 100 140 180 220 260 300

minutes since injection

Figure 6.8 True values J BI')1 J and average and s ,d, of b~:)1 for Series 160 and 161~~ -~

-,J\0

OJ

0.!

·f

OJ~to~

~___ to

~.

I/I

to

(\1

o1..00.1

(\!()J

r

o. e.l

0

t....N\r4

fI.1Q)orl>~(Lirn~f0

"H

t't~

()

c-'" -~

0 it_,.·/

·,,·1,\-1 ((1C',m 0l)

"'1--:) c:r'\ r.r!"'J

'"dI.V

,1.> Qi.l0 to~1 f.,

"..-I Q)Tn ;:.

ro'I1ill

~P~_1 ro~

~~.

,,'1(\.1

if!

':JJ'u'"'I: ~~mt>(!);::;~1H

()'\

\0

IV

Lil:lD'r!rx.u

.. =+ ~

\0 ",;t· 1\J('0,, (0 \0 ;J (1.1 0

r<"\ N' 1'1\ I'i\ (\1 I',\,~ ('.1 OJ 01

81

the rnagnitude of B21

is incre8i3ed the d,1:ffe:xeLce b12twes::l .B,_~ a:r.J.d the aver~coL

age of the b2l

DS increases, 6E does the standard deviation of' "th~ sample 0

A part of this increase might be attribu.talile to the decreasi':Jg\lalue of

the denominator in (4.11).oli'

'!'he naV.l.re of t:i::e err02:S added to A2

.is also

'*partly responsible j that is, the errors BddEd to Aa have a s"tandar:j devia-

tion eq~al to root of *i,8 A..., ~~t'~eCo:

are the added errors.

t.he variabLLity of the estimates. 'I~hat this should be true is 8'l;.ggested

by analogy to linear regression pro'blel1K, 0 ,H~iwever" for a traw:,:,port rate

function, B21

, wIth more deftlrture from fJ qu.adratLc curve over U,e pertod

of the experiment, too long an interval between observat::'.O!l8 7J),uy result in

the estimation pro.::edure hiding the true nature ~)f the fur.etion. Th::Ls i.B

illustrated in Figure 6.9, where thE,; quadratic smcothlng p:::"C>c,:edure result.ed

in an average b,...1 curve with smal1rc'(' ampl:.i.tt~de tbm tte, B'Yl C:ltrve. Tn,ered, c_

was, however j little effect on the standard d""'''lB.ti~m Df the estimates.

7. DISCUSSION

7.1 Advar,tages of i;:,his f1ethod

The ITJ.8in value of (4.11) as an est,lmate of E21

(t) :i.3 i':;8 apl;li::abiljty

to systems that are not in a steady=state can,Ht:ic;::" No assumptions nee1

be made about carls'tant volurnes j concentratiar~s !.:-r rate constants. '.I'hat

this is an important advantage over other e8tLm.s.~e3 1e d.~1eto the fa:"c 1)"'i8t

there are Dlliny biological By~tema that a~e ~Qt ~.11 a ateady=s'ta"te ~o~di'~

as this phrase is usually defined.

g1.ven here does DDt depend on fitting a ,:uy've t::: specifie ac;:-,:lvity :tn a

compartment from Lime of injection of" -t.:;:be tracer. ':i.1"J."15 tra;;lsient dLst:lrb,

ances d'.J.e to injection of tbe tracer or to a m:ixing time can be avoided v

Also j inj ections can be repeated to restore the amount of i.n.formation .in

the system (Sheppard, 1962)/ Th:is method estimates sc).cceSSi"\te values of

the rate f'urlction rather t~n.a:n parameters of a fitte<l C'l.l.:'\re 'wb.ose re:lattcl~

to the rate constants depends on the model 8ss'J.me,L 'I'hJ.s errors WI~icb

result from fitting the wrong curve to the speci.fic act,ivi't,y da1:e are

avoided. The estimates computed from (4.11) when pl::Jtted against '!,lme

give an :indication of the form of B21(t).

The estimate (4.11) does not depend on lwCiw1edge of'':he entire -biolc.g~

ical system. Thus it can be used when.:,l~· D. r.£.:'":: of the sy;; rem 13 of

immedi.ate interest J or when l,t is impossible er di.ffi.cul:, to ocserve changes

in tLe cLtire system.

83

1:.2 Weaknesses of this Meth()d

Some of the features of the method i-lhicb. result in the advantages

dlscussed in previous sect:Lar.. are ,"he same features that :ce S 111t in

v:eaknesses in the method 0 Thus the freedom from fitting a single curve to

all the data presents tte problem of smoothing the observations, and the

associated problem of estimatir~ the derivative of a function from the

values of the function (plus errors) at discrete poinc;s 0 This disadvcm-

tage 13 relative, fJf coun:e, since the pr6blem of fitting some curves in

the presence of noise in the data presents difficulties of the same

magnitude, This is especially true of curves which are sums of exponentials

(:MYhill) et a1, 1965). It is emphasized that the use of a five point

quadratic moving average L~J smooth the data is an entirely empirical

procedure.

This m.<:::th0d requires observations of the isotope ratios in all af the

compartments of interest, and estimates of the volumes and concentrations

of the substance (or total amounts of the substance an.dtraeer) in most of

the compartments 0 It is n~:Yt clear that any 11<,.: .. of interpret:i.ng tracer

data needs less than this or its equi.valent, ~:.:aQ) knOWing that volumes

are constant.

This method requires knOWledge of the distribution of the tracer in

each compartment j or lackii'lg that knowledge} requires that some assump-

tions must be made about hJrnageneity !'t{IJ/or average isotope ratios 0 As

was discussed in Section 5,3 there are some systems in which estimates of

the transport rate functions cannot be made with the use of a single tracer.

• 84

,1.3 Suggestio£:s for PUl:'tLer Inv~~igat:iun.

A large :scale sampling study should be done to deterrrd:Gc t.he ti as and

variance of b21

in variotls situatio.ns. 'l'he d.ifficulty of determining

analytically the distribution of '021

may be s\lggested by look::ng at (1~.,Ll)

written in terl11.B of the act~lal experimerrtal cib8ervables '!iLich are tbe

soarces of variation~ thRt is

[V-~jl'illne ;:)1' ] II t' SOUl-U.n". umcsn.trstL::nJ_ [eE:;tlnk~t."'; (:1 ]rUrrIp.Y:, .1E :C~J.:[ner., -= a d~;r~<·<i-t.rr.lVf;

~ ,~~= '.

1:: ;: "••-.-..-.----.-~. '''-~'~-~'-~~~--" ...--••.-~~.~'"<~~=. .~.~.~~~~-,~~".-.~-~--.<21

'I't.,';.ce are varic,us factors which could be studied in such a S8ffi.I;:1ing

studY'. Some of them are: the performance of (4.11) as an estimate for

different functional forms of B21

(t) J the effec't of different spar::ir,gs

of 'the times of sampling) and the effect of varic)U8 types and sizes of

errors. 'l'ne effect af the smoothIng proCedtlrei3 and other Vlays of

;3moothing and estimatir,g derivatives shc~J.ld also be .studied.

Furtl1er comparison ShOl11d be made '~ri t11 ot~tler methods or llaodels for

interpreting tracer data, although s'llch a comparison would be d.:tff.ic'J.lt

because some of." the computati.onal details of se-me of trLe other methods

have not t"en Bpecif.:ed, ~'~.'.1 estimot:1ng expcnenti.al parameters. Criter:ia

fGr comparing different models would also have to be developed.

A biolcJgi:;al question related to Hie appllcation in Section 6.1 and

vih!"ch needs further investigation is the variati.on of sodium and t:.::'acer

concentration in the rumen. The distribution of subaten,::€; and tracer in

to :L:1cllJ.de t,his kno'¥'ledge in some 'Way o·ther ttan by US:l.llg ;laverags: .isotcr::~

ra~~:lOSn ,

T.r.. is method mi.ght also 'be extended to pro1.:1em.6 ct::er 'c,han txatlsfc,r-:;.i

86

8, SUMMA..11Y

In this thesis the analysis of tracer data frc'ill f.:xper:Lme:nts using

radioisoto,r,es tC) study trar.sport phenomena htis becniisc:us,:;ed j arid a ne,,;

method. has been deve2.oped to est,irIlate ";,. port. r2:p"e rune tiC'ilS Wit,hO:lt

applied to many tiological s;ysterns indicated the fle,::d f'Cl:3C{1K-o other method 0

In Section 3 the biological problem c.f trendj);:)r+, of sodii)JL frerei tte

. plasma to the rumen fluid in rum.inantsW&8 described J and the faIlure of

the standard methods of compartment analysis to prcwide ir:f'ormation about

this system was discussed, This failure ,L"·,tJi",,ti::'r..he development in

Section 4 of a new method for interpreting tracer datawh:ich was not based

on the assumptions of a steady-state system~ and whIch modi.fied the assump-

ti.on of homogeneous com},(artments. It W3S .::.hc·i,;nt118t some ,systems cannot be

completely studi.ed with only one tracer in a sirJ€le exper.imf;nL

In Sec tiOD 6,1 some :..,f the eX'pc:,ime::-.ts itll:1i eh had beer.. ,;ondueted earl.ier

in the stu.dy of digest1.on i.n sbeepwere analyzed by this new method v It

was seen that even though some of the data 'Were unsatisfactory for these

estimating equBtbns because of the nearly equal values of the isotope

ratios in the rumen and in the plasms J estimates were obtained which perrr,it­

ted conclusi.ons which iNcre in accord with the biological theory (Sl;y-t-:er j

1963) and other experimental findings.

In Section 6.2 a limited stud.y o~ s:imi..llation of the plaSmi21-rWUen

system on a digital c:>".p'ILer i.ndicated that the estimates obtained had

relatively small biases and va:ri,ances with the errors assumed in the simu~

lation. This study also indicated the need for a large scale sampling stu.dy

to determine the characteristics of the estimates under a wider variety of

conditions.

88

Annison} Eo F. and D. Lewis 0 1959. Meta"bolism in the Rumen. John Wileyand Sons J New York.

BarlmM} Co F. and L,o Jo Roth. 19620 Water spaces of brain studied withradioisotopic indicators. Proceedings of a Conference on the Useof Rad:ioisotopes in Animal Biolbgy and the Medical Sciences 1~279­

294.

Barnett J A. J. Go and R. L. Reid. 19610 Reactions in the Rumen.Edward Arnold (Publishers) .. London.

Berger J E. Y. 1963, Transfer :rates in two~'com:partment system not 1.11dynamic equilibrium,. Annals of the l~ew York Acaden[,,/ of Sciences108(1) ~217-229.

Bergner, Po-E. E. 1962. The sig:nifi.cance of certain tracer kinet.icalmethods especially with respect to the tracer dynamic definitionof metabolic turnover. Acta Rad:i.ologica ~ Supplementum 210 ~ 1-59 0

Bergner,~ P. -E. E. 1964. Kinetic theory ~

metabolic processes) pp. 1-16. In R"(eds.), Dynamic Clinical Studies withEnergy Commission) Washington, D. Co

some aspects on the study ofMo Knisley and W. N. TauxeRadioisotopes. U. S. Atomic

Berman, M. 1963. Formulation and testing of models. Annals of theNew York Acaderrij of Sciences l08(1)~182-194.

Berman) M., E. Shahn and M. F. Weiss. 1962. The routine f:i tting ofkinetic data to models. A mathematical formulation for digitalcomputers 0 Biophysical Journal 2 ~275oo287 0

Bigeleisen~ Joreactions.

1949. Validity of the use of tracers to follow chemicalScience 110~14=16.

Carter, M., G. Matrone and W. MendenhalL 1964. Estimation of the lifespan of red bleod cells 0 Journal of General Physiology 47~851-8~xL

Copp~ D. H. 1962. The use of radioisotopes in physiology. Proceedingsof a Conference on the Use of Radioisotopes in Ani.mal Biology andthe Medical Sciences 1 ~23°·300

CoreYJ K. RO j D. Weber~ M, Merleno j E. Greenberg j Po Kenney and J. SoLaughteno 1964. Calcium turnover iYl man~ PPo '519-536. In R. M.Knisely and W. No Tau;xe (eds. L Dynam,ic Clinical Studies withRadioisotopes, Uo So Atomic Energy Commission, Washington, D. C.

Dixon~ W. Jo and Fo J. Masseyo 1957. Introduction to StatisticalAnalysi.s. McGraw~Hill Book Company, Inc. j New York.

Dukes.' JIo He 19'):;, 'I'he PhyBie;logy of Dom';:a~:ic k'limals v ComstockP'Llbl,:i,sb.ir~ A:5:3C,c:la~,es,~' It~haca;: l\{el..J' ~iG::':k'J

Francf.s) Go EO J ~L M~lllgan and Ao Tl'lc>nr.!El~Li..o 1.959" Isotopic Tracers,University of' Lo:n:icn.9 I'he Athlone Press.9 London,

Glascock~ Ro l!', 1962 v Sen;e eX9.n:ples of t:ne use of radioisotopes jnbiochemistry-co P.r'oee2di.c1gs of a Co;:..I",·rer.ce 0[[ "the Use of Radio~"

isotope,s "J,n An.:~mal B~ology aId the Med1.2al Sc:i.encE:s 1~)+9~67,

Goldy CL L. and. A, K. Sclomo:r:"11uman eT'y"tb.r·'ocytee- tn 'tr:t'\rc c'

195':;' " 'I'he transport of sodium intoSC>'-l.r~lal of Gerleral Physiology 38 ~ 389-·484 0

Gray., Fo \[0., Ao F" P.llgrirIl and IL 11.0 W~,.i.le:L 19~J80 The digestion offoodEd::u;ffs in the s't·omacb of t;'b:·, and the passage of digeztathrough Hs compartments" ErHi.at ,Jcmrnal of Nutrition 12 ~404~420.

Gra;>-'bill~ F 0 AoVolume 10

1961. An Intpodu.cti em to I.i.near Statistical Models J

McGraw·,H:i1l BDCK Compa:::.y) Inc" y New Yorko

Gregg." E. Co 19630 1m analog computer fer the generali zedcompartment model. of transport l.n biologi.ca1 systems,New York Academy of Sci.erlces 108(1):128~·146,

ffiulti=Annals of the

Guest JI Po G. 19610 Nl.:.merical Me·thods of Curve Fitting" CambridgeUniversity Press y London,

Harty Ho Eo 1955, Analysis of tracer experiments in non.,conservative,steady-state systems 0 Bulletin of Mat~lemati.ca1 Biophysics 17~87=940

Hart JI E. E. 19570 Analysis of tracer experiments ~ II 0 Non~·conservative JI

non-steady-state systems" BU.lletin of Mathemat~.cal Biophysics 19~

61"'720

Hartley; Ho 0, 19610 The modified Ga'J.ss~Newton met;hod for fitting ofnon~l:L"lear regression functions by .least squares 0 Technometrics:5 ~269-28o"

Hevesy" Go 19230 '.llie absorption and trar~slocation 01' lead by plants.Biochemical Journal 17~439-445,

Hevesy y G, 1962a. Historical progress of the isotope methodology and.its influences ::m the biological sc.iences J yoL 2) ppo 997,,10220In Go HevesYJ Adventures.in Radio:1so+'.opc Resear~h, Pergamon Press,New York, Originally published .in Minerva Nucleare 1~182(1957)o

Hevesy J Go 1962t 0 Ra te of penetration of 10;'lS through the c~pillary

wall.> 1!OL 1, pp. 423",,436. In Go HevesYJ Adventures ::Ln RadioisotopeResearch, Pergamon Press.l New York. O.r'ig:inally publi shed i.n ActaPhyi3iologica Scandaniva 1~347(l941).

90

HevesY;j G. a~'1d Eo Ecfer v 193"+" lL.:CiJ..D.Btioc c)f 'water from tl:e buman cody.NaturE\)oE,'.ler'.) L~ Sr)"

Hildebrand) 1<'0 :B, 19560 I!lrrciuctim"l tc Kl.lliler:::al ALal;{sis 0 McGraw~

Hi11 Book CCJti:pa/,y, .1:'.1(.,: I'J·s(';lic.ck,

Jaffa;/j Hol96.?o Meas'J.ring turnover X3tcSvol 0 1 I'po ;217,,22].0 1:'1 So Rcsl';c1:ildMetbodolcgy P1.e!J.e'J.m PTes B) New' Y"'::Jl".k,

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93

:1.0. APPENDICES

_1_C'_,_1_,__S_()lE~t.,;,:i~o;..:r'_.s_o.;.,f__S",-;Y5tem.6 of:".Llne3.r Equa~

The estimates of the B,. in 'this thesis W6):"e Obtained as the solutionsIJ

of linear equahons in the B : some cf the coeffi.c:Lents of these equationsIJ'

*were values o,f the f\.:mc tions ~ Nk'~ "I'k.. ani ~.9 and their derivatives, eval=

uated at time t. It is a convenience +r;.•·v use mat~-:'ices a.nd veeto:cs and the

equations.

B., .,K.:"

is tt.e E X 1 A ::: (8. ) is thers

a 1.lgmented matrix Dbtalr.ed by adjo:i Ding t.1':;,e ve :::tor Q totl:e matdx A as the

n+l-th column. Tne system of lir~ear equations descr:i.bing the various

systems can be 'written 82

Tne folloWing well knol¥n and usefu,':. 'tteorems of linen.'1' algebra are

from Graybil.l. (1961, p. 10),

.L." : ern 10.1. A r:-:"essary and s\.lff: c i.ent c('r.di,tion that the system

of equal.ions A£ .: D be CODs:tstent (:tI8.·;re at least CJne vector ~ satisfying

it:) is that the rank of the c ;Jeffic :i.ent matrix A be equal to the rank of

the augmented matrix (A IQ).

Tneorem 10.2 If :rank(A) _. ran.k.(A i 12) "" p" then nor ,.,f the unknovns

B:ij

can be assigned any desired value and the remaining p .)f the Bij

will

c(lefi'ic :i.ents of t.b.e rern8.j.ning p unknc"r~lS hay€; rank p.

If raI'Jc(A If~) =: n :: ffi J sher'2. is a unique vector B that

satisfies A:B .,. D.

AI' "" D i.:';; BSGlut~!..on

the estilniite£7 of the B" .i.n Secti 0:<.18 1" an:l '5 .t:s·/,~ ')1':1 que solutic)l:';j j thuslJ

¥~'n.e :letcrmi.DI\I": of A is A

1

AE "" D has a

where

1 0 'Ql 0 B21N2

,t .*t\ 0 '~A 0 1::( IT'

1 2 J.,./ 1 ·;,) ~2,..L.oI-.

A ::: B "" D .~

=1 0 0 1 B02

N1

l * *=A_ ~ 0 '" B1I T1,

.1 J

95'I'

The determi,nant of A is (1\AB = D has a unique solution.

'2quations for System

where

:::?,n be ;,lri tten AB = ~,

0 ..::,1 0 1 B21

N3

* *0 ~~A3 0 A2 B:23 '1'3

A ::: B ~ D "" i,1 -1 () BO::> N"".L

(·r_~ C

* * *I

Al Az -A OJ' LB32T2:J 2.

-lE' -l(' " ,'\ * f *The determinant of A. is (AI - A2 )(A; A3

) j " chat if Al ~, and"'

A~ J. -)(G, r Ay the system ha.s a uniquc.: solution.

The equations for System III(a) are (5.1)+) and can be written ~ = ]2,

where

° 0 1 -1 B12

N3

* *0 0 A2 -A B2 " '1'3,3 .1

A = B "" D :: i

~l 1 ·,1 0 B32

N2

* * *-A,., A, -~ 0 By", , IT',~, ( , ~2

-)(, *" * *The determinant of A 1.S (A3

- A2

J (A2

.u Al

):; so

* *~ f Ay then the system has a uni,que solution.

The non~equalities above that are necessary ani ~ufficient for the

existence of uni,que solutior..s are the non-equaliti es that it was necessary

to use in toe first derivation of the estimates in Sections 4 and 5.

96

10.2 Data from Slyter Experiments

Tables 10.1 through 10.3 present the observations and estimates that

were used to compute the b21(t) values of Table 6.1, and from which the

values in Tables 6.2 and 6.4 were computed. Table 10.1, experiment 4, was

compiled from Appendix Table 6 and Appendix Table 8 of Slyter's thesis

(1963, pp. 89-91, 93). Table 10.2, experiment 5, was compiled from

Appendix Table 10 and Appendix Table 11 (Slyter, 1963, pp. 97-100), and

Table 10.3, experiment 6, was compiled from Appendix Tables 12 and 16

(Slyter, 1963, pp. 101-103, 107).

In Tables 10.1 through 10.3 the units of tim~ are minutes since

* *injection of tracer; the units of al

and a2

are counts per minute per

milligram of sodium. The rumen volume is recorded in mi11iters and the

concentration of sodium in the rumen fluid is reported as milligrams of

sodium per milliter of rumen fluid.

*The values of al(t) were obtained by dividing the observed counts per

milliter of plasma by the average of the observed concentration of sodium

in the plasma, since it was assumed that the true concentration of sodium

in the plasma was constant. The values of sodium concentration in the

rumen in Tables 10.1 through 10.3 are those obtained by the water extrac-

tion method.

97

Table 10.1 Data from Sylter experiment 4

* *sheep time a1(t) a2(t) volume sodiumconc.

260 15 8694 372 2220 1.88048 7444 978 2204 1.71880 6960 1546 2198 1.565

120 6330 2135 2185 1.635180 6315 2851 2150 1.733270 5905 4350 2120 2.155360 5580 5445 2080 2.371480 5467 5730 2030 2.426600 5386 5900 1980 2.415789 5039 6416 1900 2·557966 5134 7217 1796 2.318

265 15 6256 186 5645 2.48145 5491 337 5630 2.37475 5179 480 5600 2.415

126 4867 905 5570 2.243180 4503 1187 5503 2.201270 4239 1713 5470 2.243394 4148 2403 5400 ~.464

480 4153 2558 5295 2.353600 4106 3409 5250 2.489817 3956 3652 5145 2·571967 3869 4094 4999 2.514

5882 15 2562 68 5750 1·99155 2096 552 5695 1.91978 1935 547 5670 1.616

125 1773 939 5600 1·922180 1699 1171 5510 2.119270 1676 1598 5390 1.971372 1747 1544 5270 2.464482 1741 1593 5115 2.546600 1609 1641 4950 2.703788 1558 2002 4700 2.608970 1591 2023 4395 2.579

98

Table 10.2 Data from Slyter experiment 5

* *sodium

sheep time al (t) 8 2(t) volume cone.

309 15 4592 243 3255 1.85445 4157 342 3550 1.76575 3970 536 3240 1.801

120 3858 855 3225 1.721,180 3690 1134 3200 1.739270 3430 1616 3150 1.801360 3315 1999 3120 1·522514 3343 2411 3065 1.901609 3363 2722 3035 1.714801 3350 2677 2945 1.869980 3235 3197 2880 1·771

312 15 3628 48 4650 , 2.31145 ~073 276 4585 1.93375 2944 308 4530 1.731

123 2802 440 4450 1·912180 2723 809 4340 1.744270 2447 1329 4180 2.026363 2532 1720 4000 2.025480 2455 1854 3750 2.089606 2354 2088 3580 1.947795 2402 2288 3220 2.185979 2318 2387 2880 2.218

5882 15 3228 16 6170 2.11345 2699 51 6150 2.30075 2566 195 6140 2.145

114 2390 283 6125 2.113171 2327 567 6095 2.171270 2094 1287 6035 2.296351 2215 1395 5980 2.760480 1996 1379 5915 2.875609 1968 1,448 5850 2.645783 1952 1503 5750 2.540960 1834 1614 5657 2.776

99

Table 10.3 Data from Slyter experiment 6

* *sodium

sheep time a1(t) a2(t) volume cone.

309 15 55 23340 3660 1.761045 58 22277 3635- 1·755075 172 22300 :5625 1·7730

120 255 22436 3600 1·7550180 377 21312 :5565 1·7790270 553 17977 3515 1.9070360 908 17229 3475 1.8460480 1250 14840 3400 1.8880600 1374 13676 3350 :1..9370780 1710 9823 3250 2.3540966 1942 10089 3196 2.1930

1440 2463 8530 2992 2.1070

312 15 159 12739 5800 2.208045 523 13338 5780 2.177075 861 12776 5765 1.9780

120 1315 11431 5750 1.9780180 1490 11025 5700 1.9010270 1794 7777 5650 1.9930360 1946 6997 5585 2.0240480 2150 6375 5510 2.0390600 2317 5632 5450 2.0390780 2327 4853 5315 2.0700960 2562 4255 5213 2.2390

1440 2575 3403 4916 2.5460

5882 15 111 17536 7610 2.124045 430 17173 7560 1.921075 643 15328 7480 1.8530

120 1076 11379 7370 1.8130180 1399 9674 7250 1·7930270 1672 6008 1040 1.8940360 1904 4457 6800 2.1110480 1956 4414 6475 1. 7250600 2165 3522 6215 2.0970780 2689 3300 5740 2.1130960 2177 3072 5385 2.1280

1440 2403 2723 4295 2.0550

e