\
ESTIMATION OF TRANSPORT RATES
BY RADIOISOTOPE STUDIES
OF NON-STEADY-STATE SYSTEMS
earl M. Metzler, Gennard Matrone and H. L. Lucas, Jr.
This investigation was made possible with the aid
of a fellowship under Public Health Service Training
Grant Number GM-618 from the Division of General
Medical Sciences and computing service provided
under Public Health Service Grant Number FR-OOOll
from the Division of Research Facilities and
Resources.
Institute of StatiS"tics M1meo Series No. 446
~pt~r~65
.A-l~
iv
TABLE OF CONTENTS
Page
LIST OF TABLES •
v
LIST OF FIGURES
vi
1.
INTRODUCTION
•
2•
REVIEW OF LITERATURE
1
2
2.1 Tracers in Biological Research
2
4
7
14
15
Compartment Analysis
2.3 Criticism of Compartment Analysis
2.4 Other Methods of Analysis •
2.5 Summary of Literature Review.
2.2
16
3.
A BIOLOGICAL PROBLEM
4.
MATHEMATICAL FORMULATION AND EXPERIMENTAL CONSIDERATION,
SYSTEM I: THE PLASMA-RUMEN-SODIUM TRANSPORT PROBLEM •
4.1
The Problem
4.2 Definitions and Assumptions
4.3 Derivation of the Estimates
4.4 Experimental Considerations
5.
EXTENSIONS OF THE METHOD
·
20
•
20
·
22
• 27
• 35
•
39
5.1 Two Compartment Models •
5.2 Three Compartment Systems •
5.3 Limitations to Application of the Method •
6.
44
53
61
AN APPLICATION AND COMPUTER SlMULATION
6.1 An Analysis of Slyter Experiments
6.2 Computer Simulation of System I .
7.
39
DISCUSSION.
7.1 Advantages of this Method •
7.2 Weaknesses of this Method •
7.3 Suggestions for Further Investigation.
•
61
67
82
82
83
84
8.
SUMMARY.
86
9.
LIST OF REFERENCES
88
APPENDICES.
93
10.
10.1 Solutions of Systems of Linear Equations
10.2 Data from Slyter Experiments
93
96
v
LIST 01" TABLES
Page
6.1
K+
r
6' .c
'Io';,:
·."C'
J(t) from data of Sl.yter
::.L'
{'i>U:;
J'
''''U~",:
6.4
Gl'<':
L,
,"::JCiiu.m transported from plasma to rumen
':Lth analys1s of variance
63
64
of' sheep and weight in kilograms
64
.:,:!i.l' transp,Jrted per kilogram of body weight
l::.u.':'< a1'Ce1' i.n,jection of tracer
66
69
],
(t) fry! Eim;u.Jated curves.
''r.
~_
75
10. j
97
10.2
98
10.3
Dn:.. '; '"
99
vi
LIST OF FIGURES
Page
",
Example of a transfer problem
0
3
1+ • .l
Sys tem I
27
1>.2
Example of a system not satisfyi.ng Assumption 4.8
34
System r(a):
a modification of System I
39
System I(b):
the two-compartment open model
42
1.::
,'.,
:,.., f,l::,
SysteDl I
....
T.
).4
System IIJ.:
)0>
System 1II(a):
~).
An N-compartment, one-way catenary system
(;
e. thrE'e cumpartment J modified catenary system
5.7 System III(b):
a model for the plasma-rumen-omasum system
a three-compartment catenary sYstem.
46
49
52
52
.
54
)·9 System IV(a)
57
5. 13 System IV
.~ig 10
-
/
:.~'
G
L
0.)
System IV(b)
.
59
An analog of System I for a simulation study
67
Simulated Curves 100, 120, 125 and 130; illustrating the
steady·~state situation and the effect of changes in B
O
70
Simulated Curves 130) 138 and 139; showing the effect of
changes in total rumen sodium
71
Simulated Curves 126, 127 and 136; illustrating the effect
of changes in B •
l
6.6
Simulated Curves 130, 136 and 137; illustrating the effect
of changes in B •
3
True values of B21 and average and s.d. of b
for
21
Series 120 and Series 130 •
True values of B21 for Curve 130 and b
values for three
21
error curves 01' Series 130
73
77
78
vii
LIS'r OJP FIGURES (continued)
'rrue values .• B91 , and average and s.d. of b
for
21
Series 160 ana: 161
79
True values, B , and average and s.d. of b
for
21
21
Series 137
80
TNTEom iCTIQTJ
V and bi.ocbero.5,stry
to~)l"
the rc,eiU'CfJ lnvesti gator.
r'JdiC'j2~)L)r'.';;
:.1re a major
The reason for this" parti.cularly in
and metabolism., is indIcated by the followi.ng
"'I'hr',)ughout pructic811y the whole history
bluchemical
r)f
.1 nveBtigatJon Bttem:pts have been made tD f:ind a method of
labeling 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 thIs
FIypO;38 arr;;
eli f'ficuJt t:] fi·t',d. part.i:~1)l
if
€~.rou.ps are '~~~:·.":~cl'u.j(~d 'vJbj .:::,.l':'. ~Xt·(·::
tel the t:is.sues :Jf thr'.h. .oJ.!!
u it~'pb.y;~:·
al ti
~')r f::-)Y'f>i,gn
: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
empirica1 methods w1:
1.
vid,,,:) in!.;") two classes:
(.1.) th2 intui.tive,.
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
its development from <h'fi n j 1:, ions and stated
biolog.ice-I ex.pedment:a Lion,
a~osUli1ption~l"
b~)th
sound in
and, useful i.n
2
2.
2.1
REVIEW OF
LITERA~
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 radioisotopes.
Even with these, in the
out in plant and animal biology.
19301~
many investigations were carried
One of the more important series of
experiments was that which showed the constant turnover of body constituents 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
and transfer or transport studies.
Examples of uptake
determinations,
~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
Ot~er
Hofer, 1934).
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.
Compartment I at a rate f
i
A substance K flows into
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
at the rate f
12
•
21
, and from Compartment II to Compartment I
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
1__m_f
21
is of more interest
i
I
Figure 2.1
II
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
in Compartment II, or has left
later than t
2
the system
l
in Compartment I,
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 analysis
I
which is based on some combination of the assumptions in the next paragraphs.)
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
matical approaches.
a~lysis,
but included other mathe-
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 compartments 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.
by
In this example Bergner shows
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
which it seemed to answer.
o~
the problems
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 compartmentswith 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 transport studies, it is of interest to examine the eight assumptions in some
detail, considering why they have been made, the mathematical consequences
c)f
each assumption and their biological validity.
The first assumption, that of identical behavior of tracer and substance, 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
U
well 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 equilibrium
has been overstressed in attempts to make precise determinations,
or to facilitate the development of mathematical descriptions
of the phenomena of hemeostasis, with corresponding neglect of
the equally important physiological phenomena of adaptation and
growth which only appear and are measurable when dynamic
equilibrium 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 the
specific: activity of the product at various times will not
give us the turnover rate. More information is required about
the peel size and the rate of change in specific activity of
both precursor and product. If these data are known, and this
requires more effort than rr.ost people are willing to do, then
we will have reasonable approXimations of the turnover rate
for 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 information 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.
Zier1er
(1964)
However Bergner
(1962)
and
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. has the implication that
0
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 interconnections 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
d' f;i,ve exponential ter.ms (Moore j 1962L others who have made
SliX"
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),
that th:is
i;5
1:1
It ls clear from the statistical literature
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
matics, and
(:i:1.1) in an att€':mpt
lllstlon from tb,e data.
(u) to permit
"';;0
get the
more tractable ffittthe·,
m8ximu.m
am.ount of 5.n1'o1"'"
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,
&. 19:59"
J
p. 344) o.n traCE.!.' methods
ftnS:inc(' even the simplest cell is an extn:::wely cc,m:p11cated
system 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 to
make a number of s:tmpl:i.fytng assumpt:loxliB 'before any of these
reactJons can be treated .in a s:i.mple mathemati,ca1 :rr..anner. The
solving of HLe mathemat:ical equations cb.osen to fit the system
under Investigation is therefore frequently a much easier matter
th(;,n dec.iding whether the equa.ti.ons are strictly applicable to
the 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 i t is possible to determine the distribution of the tracer
at the points of entry into and exit from the compartments.
14
g.4
Att~Ulpts
other Methods of
Ana1:Xs~
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 nonconstant 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
Thls
l'(~vje1;~ ~:yf
of tracer me:thods "
Review,
tbe literature has not included a complete account
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.
systems nut In
8.
stpbdy~state
rrhe attempts to analyze
condition have been relatively few.
)
BIOL()GICAI~
j\
0
PROBLt-:M
Ln animal n,!.l.tri,t.i.on has indi,cated trwt purif,ied
Re~;,.;;.:n':::l:,
•
\
r~
1
dit-;'t~, J
tLat
adequate
'"
i.mmature anima12 stop gI'CWll"'.,<
~".~'~
a~d!Cl:"3~;'J
i diet 6 :uakeE them
28 tisfactor;y
ij:'Kl
fo:: sheep c
;.,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
24
experllnents J'Ia"
int:: the
::'I"iilC~l
was injected into the bl.ood
of a sheep J and t11e radioactivi ty of the
'~esael,
p1::H3L8
or
'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. ·solids and liquid in the rumen.
o
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).
excreted from U:e body.
Also sodium is
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 exchanges 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.
reported in Section 6.1.
This analysis is
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
In Secti.cm
0
to systems ·wi'tJ::. ether
'5 the
c:ompaxtme:~lt,
The problem of
met-hud of th.Ls sectJ.cm 113 extende::i
ci:Jr.J'igm,'e.t,lons
0
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 1951
:
c
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 196
4L
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::ngeswi th 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,
these values.
The model may also indicate the functional form of
~~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
system; or j.t
cells,
ma~r
c;f
the c:Lrculatory
be a collection of" cells J such as the red blood
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).
total mass
ol~
Unless otherwise indicated; N. (t) is the
1
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 isotopes
1
of the element .i.n the compartment at time to
23
Assumption 40LE')};: amc:unt of t~he SUbGta:lce in a compartment
, such a marilier".st;
th .
changes 1.n
'I"" (. \ ..
!~,
l
t)
~J'i tb
Das
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
:)nl~i
me!1t changes
amou:~1t
of substance in a compart;=
graduall.y,1 and. if the removal or introduction of a
quanti.t;v of the eubstance causes N (t) to have a ju.mp~::Uscont.inuity
i
this will hapfer!. only a finite D\JlJfber of times
the study"
jiJ
r
the course erf
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 stud J 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
Compartment i
per unit timeo
Bt
The amDunt of tracer in
time t will be denoted by T (tL the units being counts
i
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 ratio
TIN,
The isotope ratio of any maES N of substance is
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 ure lativEo
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:
fG
(t) _
-
d
any function f(t) J
fClT
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
interval (tljt~).
...
-
j
in the t:tme
Note that B, .(t) is not defined for i • j •
lJ
The units of B
Ji
(t) are mass per unit t.ime.
Note that I(t ,t2 .,B )
ji
1
is the total mass of all particles of the substance that go from
Compartment i to Compartm.ent j in the time interval (t ,t )J the same
l 2
25
fun~tt(;nfcr
:ca!e
tile movement of substance into
transport rate function fer
C::Jll1.pa!"tment ,j
the movement of E'Jbst·e::1c;e ,:rom. Compartment j to unspecified compartm.ents,
and B ' (t) :is
O,1
substance
[Ie transport rate function for the total movement of
t
llcm
Ccmpar:tment jo
ITJDVement o:fr;race:c
t~)ereLs
Thus B/"o(t) ""
vJ
r;
Bi,(t)"
<1'.' l '
u J.~ ,c.
J
For the
a cDrresponding transport rate function"
j
is a continuous funct:ic)!l B'JIt,. (t)
f
such that for
greater tI18.n
which moves from CompartmenT, J
,~ I(tl.)
t:o
.J,.
-¥...
; .B . ,. ) i.s the amc;unt of trac er
,
JJ.
Compartm-ent j in the interval (t
l
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.n the li.terature c
i:(~
DeflYii ti.on 4·,5 has been vari.ously def.ined
rrerms which have been used for this or s:i.milar
concepts a.re fl1.L,<:.} flow.' transfer.' exchange rates, turnover, appearance
and dIsappearance"
state i or
sense of
U
Most
Jf
these have usually been reserved for isteady"
equilil,ri1lIll U co':lditions.
t,ran~)por1~
ra1;,e ftL'lction as defined in Defini.tion 405, and
St.eppard uses tranBpcrt rate (1962).
Definition
4)~·
Kamen (1957) uses transport in the
Tb.e i se,tope ratio defined in
: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) may
Jl
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,
i t is convenient to wrHe B., et)' as a function of t alone,
Jl
'iI"
B ..
Jl
(t) will
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
Com.partm.E.'Dt 1 1.s the
"Ihe. system is composed of two compartments;
plaBma.~]
Compartment 2 is the rumen fluid
0
f.rhe
"'4
substance is sodium and the tracer is the radioactive isotope NaC:"
A
''''4
quantity 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 B (t) for the
2l
movement of sodium from the plasma to the rumen y and a transport rate
.ji
function B (t) for the movement of tracer from plasma to rumen.
21
is no other flow of sodium into the rumen, !.~." B (t)
2I
*
B (t )
21
and
.jt-
=:
B (t ) ,
21
Assumption 4060
*0
= B21 (t),
'I'here
'Ihere are transport rate functi.ons B (t) and
02
.3 :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.
B (t)
21
Compartment I
plasma
K
........
B (t)
12
............. -
Figure 4.1
Compartment 2
'"r
rumen
'Ij
~B02(t)
-- - - -
E2 (t)
B
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 B
does not preclude transport by several pathways,
21
~.~.~
(t)
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 N (t) and
2
T (t)
2
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)
28
If
;2
not one of thepctnt!3 where N('t) or T( t) i.s not
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.
d
(4.3)
(. 4.
•
Q
...w( +-'
'-L
dt
* .
~ B
L:r)'.
(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).
1ciLJ
FrulIl the assumptio.':1s made
I
BUs the following ratios are well defined:
(4, )
NDte
(t)
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 B (t) gIves
02
(4.6)
Equating the right hand sides of equations (4,6) and (4,7) yields
29
R.;)(tj
] =
R),!. t.)
( 4,.8)
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,
obtai,ns an equatLm sjmilDr ::)
in which t:nere is
nG
(LL8) fGr
''k
.
a ,-,tead:y-state catenary system
11\cne of these
t"ecy<:,U.ng c1 'Cracer.
discuss the problem 'Jf' observing
'r v (t)"
Robertson (19yr)
develo1~ments
Dut riccording to the development
"
here.. if the Ri..j (, t) \ S are meani.ngful expre,;, i3 lens J and ,if they a1 (mg
NU (t) and
I'H
(t) can be determ.ined by experimental
Wi U,
observations.~ ct',en (4 )'))
makes it possible to obtain values of B (t) under very minImal biological
21
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~
If 'IJ::,e ,compartment is homogeneous, in
0
the sense that in t::.very element; :)["
.;.. 'J
has the same value, say J,1o:
•. /\....
vO.lu.'6.~
_,Lt; r.,, k,,{it, • \( t . ')
Ie)'
t "'. "C'
,
J"
the
'"
12c.LOl)e
,;~
,J' .
A
(+ .,
,h. \ .. I
0
ratio at time t
A* (t)
:J..L,_.
,") , ',4
J
' 'a 'nb e
'-
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
mixed cOffip&rtmenL
,1 [;
1'.
lY;)JDgeneous., well-
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 A (t) may be estimated by the jS~jt:.:)p',:; ratio of the sodium in
1
a sample of plasma at time to
The motili.ty 'Jf
rumen '\.wlls
DE
'-laE
of homogeneity as
1~he
rumen due
diEcussed in
WOE
UEEUmeQ
'that as ECldium and tracer
t~)
the frequ.ent contractions of the
Sec~;iun
) d')ec
Il',)t
imply the same kind
above for tbe pLEwma, but it does suggest
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 R* (t) to the /;lv-erage isotope ratio in that part of the
02
rumen near the rumen walls is next consideredu
Suppose that the rate of flow at time t) B (tL is divided into n
02
equal elements of rate of flow,
B(k).(t )
02
~
B ( t )/n, k=lj2,.o.,ll.
02
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.
(t) =
B
6t > 0 each
tracer
B(k)(t)
k=l 02
02
For
~
B;~)(t)6t
B~k)(t)6t.
Then
is a mass of sodium containing an amount of
Letting
the following equation is
true~
B* (k) (t)
=
02
or
n
...
E
k=l
This last equation says that
R~2(t)
n
is the average of the
R;~k)(t)
.
1j('(k)
Note that R
(t) can also be written
02
flS
'*(k)
I(t,t~t,.B02*(k))
(t) == 11m
02
.6t.... O .
.(k)
I (t , t+6t; B 02)
R
If b.t 1s small enough each
R~k) (t)
desjred by the isotope ratio
will be approxi.mated as closely as
~(k)(t)
of' the compartment which the sodium
of the sodlum in that volume
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
==
~
k=l
R*(k)(t)/n
02
*
It thus appears that R
02
Thus
~ ~ ~(k)(t)/n
k.l
(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 R* (t) to the i.sotope ratio of the
02
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
33
close
to the ayerage isotope ratio along the rumen wall.
evidence that this is the case.,
There is some
&'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
adjacent to the rumen wall,H
"rhus . RO*~(t)
is first approximated by the
.:::
j
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,
Assumption 4,,8,
.
iTt
*' )
It t 5 C+6;t.; B
02
;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 R* (t) and T( t) !N( t) is not that assumed
02
in Assumption 408"
34
Compar:ment 1
m,,~
.!cut;
+
- , aVel.
~" age
. C0nl.e,L",,·r·"·.lvr,
',.' ~ '~r ·,~t',·"~·,~,,1S
",,,,,rIll
c db'
, · , t~
u.,.e
J Hart.. ('19"'7)
. ), , WI,' t 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
compartments
0';:
many
Hart does not interltl. that average concentration be used as
0
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
Since T(t) -.:;-
~(t) are estimated by sampling the rumen fluido
~(t) N(t},
(409)
TO (t)
*
= ~(t)
N° (t) + N(t) A*' (t) 0
.
*
* and ~l
* (t) ~ ~* (t)
SUbstituting from (4,8) and Llsing R (t) ~ ~(t)
02
as impl:i.ed by Assumptions 4 7 and 408 y equation (408) can be written as
0
B
21
(t)
~
=
A;{t) N° (t) ~ ~'(t) N8 (t) - N(t) ~a (t)
. _ - - ~(t)
N{t) ~o (t)
Ar(t~(t)
=
~(t)
35
'rhe expressions Bij(t).. N (t)} A:(t) and Ti(t) represent true
values.
1
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
J.J
With this notation the
estimate of B (t) is
21
(4011)
Without knowing either (1) exactly what compartments exchange with
Compartment 1, or (ii) exactly what compartments the flow B (t) goes
02
to, it is not possible to compute an estimate of B (t)0
12
In the case
of rumen sodium problem B (t) :1s composed of B (t) plUS a component
12
02
to the omasum. Knowing the rate at which sodium goes from the rumen to
the omasum would make it possible to estimate B (t) •
12
This will be
discussed in detail in Section 5020
4.4
Experimental Considerations
According to (4.11) estimates of B (t) can be computed from the
21
values net),
*
a~{t),
a * (t)
observa~ions
or computed from observations.
2
and 8 *g(t).
2
These values maybe elther direct
Because of Assumption 407,
-It
(t) i.s the estimated isotope ratio of the sodium in a sample of plasma
1
taken at time to Possible sources of error in this observation are
8
(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
so
high~
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
in the rumE;U"
liquid volume of the rumen and the sodium concentration
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 b
2l
(t) .
Est:lmaLl.on of the sodium concentration of the rumen presents the
same difficulty as does estimation of
the
non~J:romogeneous
* .
~ (t),
namely that stemmi.ng from
character of the rumen contentso
sample of rumen fluid yields estimates of
*
~(t)
Since the same
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
the same
t j me
1.S
th.at two or more samples taken at
yield estim.ates of the varieb.Hay of the values of the
isotope ratic)s and sodtum concentratIons in the ramen
*"
TIle estimation of ~
0
(t) involves the difficulties considered above
1\
for ~ (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"
problem"
There is li.ttle theory
8Y81181:;le
rels-r::Lng to th,i,s
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 35 4 ) 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
'where the nuniber of data points is
expected to cover several peri.ods
large~
o1~
greater than 50,
car~
91:':).
any periodic fluctuatlons
0
l~, ,,~. :J
be
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 B 1. (t) will be subject to large di.stortion' of' the denom:i.nator
2
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 i t 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-
between ~ ("t) and
'If.-
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
B (t) is as
2l
systemo
follows~
'llie radioisotope is injected in"to the circulatory
After allowing an interval of time for the tracer to become
mixed in the plasma) a series of obseryations is made,
are of
(i) the isotope ratio of the
of the rumen fluid J
plasma~
The observatIons
(i1) the isotope ratio
(iii,) the concentration of sodium in the rumen}
(iv) the liquj.d volume of the rumen
0
and
Some or all of these observati.ons
are smoothed and equati.on (4o:11) is used to compute estimates of B (t),
21
The values of b (t) may also then be smoothed by a quadratic moving
2l
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
501
EX'TENSIONS OF 'lliE ME"rHOn
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
with the analagous system of compartrr,ent
The systems are
analysis,~
CC:':Tf,Bl'ed
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 wi th 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 experimental
1J
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
B~l(t)
\
,,;
Compartment 1
Compartment 2
~
~
.I.'~gure 501
"
B .
12
System I(a)~
(t)
a modification of System I
·40
estimat:es are
"
~
,ltD".
n2 {t) ''\:: I,t)
<-_=o>",.~_"",.-.~-..-...-..~_.~
b 12 ('t.) -~ b 21'it"',) ~ Jp~ ('· t,
2
)
If in addition It is known that B'Yl,{t) "'" B1;l(t)" a condition that.
"""
is true if and only j f N (t) "" C a cons ':am;. !hen for all t 3 H~ ( t) "" 0
2
and from (408) b
21
(t) can be 'writteE aie:
- ------_."
Of the two expressions fo.!' b
21
(t) given in (5,2), the eDITect one
to use would be determi.ned by the experimental conditionso
2(",)
sHuations i t i,8 possible to measure '1
In some
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 1ctcr;/ tc; measure the amount G of
i
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
Equation (50.?) is obtained ~:.'y dividing both sides of (502) by Co
itselfo
Since B
21
(t)
= B12 (t).,
this yields
A
~l (t)
b
21
(t)
= ---c-'
J
41
where
~l (t)
= IS.2 (t)
the Crate constants 'I
,
::>1
k"
:1,)
"_-cLs1 "
J
The quanti ty KJ.2 (t) is analogous to
ufr.:,en used in tracer studi,es (Gregg, 1963;
Solomon, 1960)., except that here K.12 (t) may be a non~constant function of
~l (t) is the rat:i.o (·f inflow to the amount in the compartment)
time,
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 B (t) as the
21
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"L, l.'
. ( L~1 i.6 not applicable" since Assumption 408 requ:.i:res that
all sodium PClTl;::.·::Les have the ,same chance of leaving the cells
case a model si)ch
be appropriate
DS
0
In thi.s
that proposed. by Shemin and Rittenberg (1946) might
0
Another e;ys't;em often analyzed is sho'''n in Figure 5020
is not in the fnlIllework of the general
Although i t
plasma~compartment systems ,
of i,nterest because it is the configuration of the
two~,compartme:nt
it 1s
open
system seen so (lften in the literature of tracer sb.ldies"
'"
~lE(t)
B {t)
21
Compartment 1
,~
~
~
"
\.
Compartm.ent 2
B (t)
12
B {-t}
E2
.~~
Figure 502
System I(b):
the two-compartment cpen model
Note that in System l(b), Bl1(t) = BlE(t) + B (t),9 and
l2
B (t) = B (t) + B (t)o
02
12
E2
It is assumed that the substance is excreted
out of the system at the rate B (t); thus i f tracer is injected into
E2
either compartment it will leave by way of Compartment 2) but no tracer
wHl enter the system other than the amount injectedo
describe the system,
Equations (5,4)
(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,
b 1,J (t) = b1
j J etc"
,0
•
0
Thus Ni(t) "" Ni "
but unless otherw:I.se stated all functions are still
to be considered as possibly no:n=constant functions of time,)
The
estimate of B is again given by (4011), and the other
2l
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
b
(5.6)
2l
..
b 12 =
n a
2 2
,
* a?*
al--
-l [
a2*
*
a
*u
a1 ~ 2
.*
a *l - a2
,if !
a
+ a*
l nl + nl l
] ,
44
[ n,..,
j
b
lE
=
'-'\
<::::
;~
8
'*1
a* n
'T'
2
l
b
E2
~,
[
r~
*u
a~
~
t,
a
8
5,2
]
'it
8
=:
*0
a
1 + !ll 1
It-
n
l
2
*0
~
'1
1 + '1
8
j
]
+ ~
1
*
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
(plasma)
~
,
./
,
ts
)
Compartment 3
(r,b.co)
13
B
21
"'"
~
Compartment 2
'"
,~B02
(rumen)
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
b
02 "" b 2l
b~,
",d.
~
!l2
,9
U
*
'"" n 3 a 3'* /(8 1
- 8 *3 )
Since Compartment, :2 and Compartment 3
1".,£:1'118
J
no direct excha:r.ge they
are in a se 1: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),
to two aspects of the system configuration,
The modification :is
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 B
ij
(t) as :in
System II would glve estimates of the Btj(t) for any number of compartments.
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
'",,,
d J 1962) shown in F:igu.re
cat enary sys t em lJ)neppar,
5" 4 J in wh:i.ch
so~called
Compartme~t
3 exche.:nges substance with the plasma only through the intermediate
Compartment 20
Compartment 1
:B.;<,r-
r~ _ _,":=_;i::~·_ ~_..,...;'~
I--._~-~---~
Figure 5,4
System III.~
Again B
E2
= B~)
Compartment .3
a three compartment., modified c:a tenary system
- £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
'0 3
"
-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 .8
02
and equating the twa gives
so that if
:l.Tl terms of the observable variables as
__
·.'I'r
!
a,.-,
b
21
=
......
.-~
Cot·
c.
.
·Ii
\8
.it"
·2
+
a*) n
~
3
a .*
a 1 - a2
2
~
s
~
3
:1
a *9
3 3
a*
1
.A comparison of equation (l~o11) with equation (':),11) shows that the
second term in tbe right hand side of C5,llL
ij
(5012)
(a* ,~ a*)
":_2 ..- L n)
8
*
2
=
,~
8
*u
n a
3 3
51
*
1
is the error that will be in the estimate of B
if b
i.s computed
21
2l
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':.
a~
8 -c
c.
1
Jl:V
wh.i.ch is a
Thus b
quan:.it.y j s:tnce
PQS:lti\i c
i.D.
b:;j.. 8 period of ti.me 9
.1.2 IC"i
3
The
as cGmpu ted
2l
Live,.
*"1
value 0:' A
That
3
}S
J
for- a given value of :IVA,'
./
*v
the larger is B32J the mere rapidly will T increase, so that ~5 will
3
'b'""
Ja"g'"
-
...
v
0
'G'_,,'{.';'~
.l:
,.
~
~.;'l·
..Yn..':>'d•
-
'181.. 'lP.
n_f K
+b
.. '~. ,.;,••cu.
"'m"'] . '1',-...
p"
,-.
~')'2 J
v_....
*.H
'''fer'''''-r'
A.:5
b~>
~T ,9'i.,·,~
·t·1.-,,,,}. .:..
-~.3
}'"~
S
will beo
Az t i.nCye83eS the denoml::l.!3.tor of the errcr t,erm
8
j
l<!'
2
a*
l
~>
decreases J
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"lo
c.
Another error that could occur would be estimating B
as though
31
Compartment 2 d,i.d not exist o
rile estimate in that case would be
b
=<
31
which differs f'r'Jill the rs'te of inflow of sc,bstance into Compartment
only in that. the denominator is
8
*
1
-,
8
*,
3
rather than
8
~
2 - 8 3 as
4
~)
(5,8)
v
If in this case the tracer is in,jected i.r..to Compartment Ijl then
early in the experiment
8
*>
1
8
ji-
2
so that the rate of inflOl." would be
On the other hand, if it is
underestimated,
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,
that case a *
2
>
*
al~
and the rate of inflow would be overestimated,
In
49
System l.Ilts)
the model for the
113 9
slight; mod.ification of System III which could be
I:laBma,~rU!l1e!l'-omasum system
referred to in Section 403.
System III(a) is shown in Figure 5050
E'?l
~
,
Compartment 1
,
(plasma)
Compartment 2
'(rumen)
1
B'1'
~c.
K
-,
I
.B
Figure 505
SJI"stem rIlta) z
03
(
,
-32
Compartment 3
(omasum)
a model for the plasma~rumen=omasum system
The estimates of System I are applicable to Compartment 2 and
Compartment ,3 of System III(a) y ,so the estimates of B
and B
are
03
32
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\f
2
dt
d
-B2i
'r2
dt
;~
- A B~
1 0::.1
Sol
+
,'it
~,
and, e:lue.ti.ng
,,1 U
(A '., AI)
,J.
rC
2
~~."~~~~;:-~-_.~
A
2
..
Hence y if
'1(.
Ac:. "
AI" thc;; '?c','.;ima te of B
1s g1.ven by
21
_ . _ - " " '......""""""
<.~._.
a
Using equb tion (.''-.,9) fer t
Compartment 2
[lGS
'*'*1
2
,~a
o
2
the estimate E18Y be written
::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 B
given :in equatio,n (5"11) is due to
21
Compartment 2 recel '/1 r.tg s'..:tostance from Compartments 1 and 3 .in Sys tem ILL
From the first equDtlon of (:;,,,14)
Bl~
:c:
rnav
"
be estimated as
51
b 1 :2
i
"' n,_,
+
c.
A nOi;ew:yctny pr5tcti.cal poi.nt J.s that (5,17) involves three numer:cal
differentiatiuus
9
chus b
12
would. have a high vari.ance unless the errora
in "the cb3ervnti.C'ns 'i!ere quite small
:B:y
·~OX18
0
'.i,a
only souree of 2uDEtance in'to Cowpartment l i s from Compartment j
substance with Compartment j
,9
J
then.,
the estimate
of B.\. is
1,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
the amount of
tt;~
:i.sotope ratios in all compartments J and by' observing
sut,s~!Jnce
in all compartments except Compartment 10
In the one-way catenary sys"tem i.t is assumed that none of the
B
lead direc'Lly to any
Oi
The fii1.al
s~{ste1':.
O!~
the Compartments 2 J
23
may
"be
0
J
No
'I'he quantities B21J B
32
estimated by the expressi.ons given in equations (508) and
~ 'Ll) f~~
o'~~~n'
. J......
U
.....l:..-: ..... e 1 1·1'
,J..()
( ./0,
,,L,'
0
for which estimates are derived .is another
modificat.l..on of System. nI J shown in F::.gure 5,7,
and B
0
>
B~.:
/.-
'"
I;~" ,
'----n--"-,~
"-'-="=:~/-Ok
'I,
~~
t., .. _"
¥
(;
o
o
J
If £JN
).l~ -1.
-J
lo-_,
1--_.
B....2.....l~_.~'')
fo----------~1Y1I""1
F ... g~~re
,
Compartrr,ent 2
,,' n
C'.
...
cr
3
}
<'=
~:: ~-
Reference L.c'
the method
1><',; made:!l Se(::ti.on 5, i
to one s itu.ation :in ',lhich
in Se':::tion 4 a~ld extendedln Section 5 18 not
j,:; 'lCc:
applicable;tt:u::, 13, to systems where Same of the substance is not free
to exchange,
Eybtem j or course., whi.ch does not satisfy the assumptions
made in Sect'i:.JD.
require
9
L;.
cmmot be anaLyzed by th.is method,; such a system may
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
(ii) there is one
compartment has ;:;l:Iy one COUTee of the sUbstance) or
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
or more rate fun:'LJ.o(lS was estimated,
J
the sum of two
'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
:he transpc;rt rate fU.nct:icms ,1 ·can be est·lmated,
r;Jf
If} however;,
c:
""v~
__.__
Ccmpart;.nent 2
,
Compartment 3
O!i!"..#---------......
.~'~-'RB.-?_...
"----L
-.,
F:gure 508 System IV
desc.::!:'~he
The equations which
d
I~·~·
T'
.:*.
.)0;
.',
/
:;
-eFCd N
2
dt .
d T
2
dt
:;
A B + ~. B
l 31
32
+ B2 "1
B;?1
_ ..L
.~
~
the system are
.,.
A.
.J..
I:i.
~21
't
~,
'*
~ B23
- B32
A)* B:23
~
A*.:J By
~~
)
-'
55
'f'.,n
'"'1at,",.'x
J.
u.
q"",~,.,~"""
:."'co.'"
. : .....
1j.,J''-,'-,r..;
....
..
',.",<,,-,:'-,,:;,t,',:.n.n
__/_'-' __ ,__
~~;
'-'',~
'0
., ')~
.-,. o"L
~")'a-t-~,~'n~
~\'-;1"", . . . . . ...L'-'~,,\.:;l
19\ ar"e,..l'1J:
I\;Q,..
II<=;.
~.;lO_~
;
...
R,,':'
'=- _ ,
where
",1
0
'*
g-
0
-A"
""
1
1
..
A,*
1..,
A ",
1,
'1t'
.J.
5
B '"'
0
"·1
0
",,A.,
3
,Tj
c
23
D
B
3l
""
""
3
N
2
T
2
"
.~...
j
N
£21
~
A,
;I'
A
I
,A-
*
In the r.18t','rix A", row 1;. "" A, row ,~ "' row') + A~1 row 3,
....
:I'hjs~ine9r
relation amo:cJ.t.S the ro·.rs impLies that the determi.nant of A is zero
By
0
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,
total. amount of suts ,::a[iC'e
lZl
Compartments
;i
ani
t)
together
That is J the
:L5
(N + N.j..J ,~ so
2
o
that the
ch::J1J.ge
In tt::.s ":ctal
given by (N + N )
2
3
1,S
U
a
3
,'"",
3 "" B21 + B;:;.,
Likewise,~ T + T) "" A (B
+ B31)'~ so cr.at
2
21
1
N) "" 'r2 +
thus the rows of the augmented matrix (A
t· N
(N
= B:21 + B , or
31
0
t
~'
linearly dependent 3 ard rank(A) and rank(A
1,£)
I Q)
a:re both less than four
are
v
will be s:'nown that these matrices both have rank three 3 so that by
Theorem 1002 three of
t~e
HI' :.::an be
,J...J
arb:ttrary value to the fC:.Lrth B..
l.J
four transport
rat~e
determi~'1ed
only b:v assi.gning some
ltwi.ll also be shown that anyone 0:::'
v
farlctions may be ass igned
aTI
artl trary value
0
Let A be the raatr:ix of the coeffic i.e::lte of B y B } and B
in the
1
23
31
32
first three::' equatio::1s of (5019)
Then
0
~l
A,
.L
""
1
'*3
'"A
1
1\.'*
0
1
'l+
~
=1
It
.:-i,
~
A-A.}
WtiC!l
ze::"o if' and only- if
:tiS
/
*
~
,:;
of
A_*y
,~
,1
*
,~ ~,
and t,he determ.in,snt of the coefficients of B :, B
and B
is
21
23
3l
'it
~
A, "
.l.
,w'
Thus 1f
after
Whi.C!l
A~'
1
the
f
I
,*
A_ ~ .!'\,3)
or A'* ~ A*
y
:'~.
l
,~>
""
Dr'
A and (A
'11'0[]
::t:;'ee are un:t quely determ,i.ned
01:1:':2':':'
I f!)
have rank
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 19) .
Q
'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
two very sim.ilar an:mals J such as identical'Cwins
0
would be using
'I'hese las'c two sug-
gestlons have the d.:ifficul ty 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
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_.L } or
2
whe~her
the outflow is not thus restricted,
0
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
,-
A
:II
0
0
-1
0
0
~A
1
*
3
o
~,*
T
D
,
=,.L
.!t,
",A
2
0
1
1
A*
l
A
*3
0
,
,
.''-
In this case the determinant of A is tAl "' A'''')'tAl';e'
*
A
1
1= A*' and
3
A"*
1
I: A2* ,
3
r:
3
*"
~)3
so that if
then by thE: Ccrollar'y of SectIon 1001 t.he system has
,:..;..
~.""
A,,,,
A.
3
'Ii.'f:
'It
A, (A_ ~, A: )
.1.
3
d
*
A
3
1.
A'l
.j
,~~~
A-
,~
'j«
A:)
,~
t:
~
~.
.~.
A,*
A, )
L
"
A'*
3
a
*
A
:r'.»
.-
3
.j..
"
n
'..)
B., ,.1
2
N
-'3
2
'T!
0
'Jt
)
...
~L5
"'2
.;t.
(AI
Here agalJl there
:N
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
two
'::omp.9rt.m,er~·t,.s
which
CBn
can only increase
0
and,
,rt'::Te might be
!::::ic<~ogj C'9.l
sJrstems
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.
Figure 5010). pI'ovides for an outflow
frC;!ll
System IV(b) J shown in
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
compartments
.~J:...d
tracer
i!l
0
The changes in amounts of tracer and substance in Compartments
and 4 of System IV(b) are described by
where
the
3J 2
,59
Compartment
.1
-:,
B
21
~31
~
B~2
~
~
'" ""!!Io
v
Compa...r t,'T,,".::n t
Compartment 2
II(.
,
~
,~~
...
B2 .3
B
, 42
,
Compartment 4
~if
B
04
System IV(t)
Figure 5010
a
1
*
'l.
=1
1
0
A*
3
0
=~
1
1
*
'~1
0
B ,·
2 .l.
N
2
*
=A-,
0
B_
'T'
0
0
B'l'j
N'l
!
c
-2
c,3
a
A ...
0
=1
0
-A*
3
0
0
*
.>,,_J,.,
B '*
D
"
'"
A*
~
0
0
B
IT'
0
0
0
1
-1
B
NJ
0
0
0
~
1
4:3
32
'J
'iii-
=A*4
42
'+
-r4
B·
04-
?
J
There is a linear relation among the rows of A J namely
row 2 • A*
l
(row
1 + row 31\
=
the determinant of A is zeroo
(*
A
l
=
*)( A*
~
4 row 5
=
row b~)/ (A*
4
*
A2.)
0
Thus
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"
""cl
is assigned
an assumed value y then (5021) uniquely determines the vallies of the other
five BijUSo
But in the case of System IV(b) the B
1j
are assigned must be one of B21~ B
B
or B u
2y 31
32
to which assumed values
Although Compartment 4
has only one source of sUbstance y determining values af BOLt and B
42
sufficient to uniquely determine the rest of the systelIL
is not
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",><
tat ion of tracer studies, developed i.ll this thesis
0
l'hey are l:imitations
only .in the sense that i t 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 8
21
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;'"
In experiments 4 and 5 Na
added,
system; in experi.men<t 6 Na
24
24 was injected into the circulatory
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 Sperber et al, (1963)
j
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 B the estimated isotope ratio, a *, and the esti2
21
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 _ ), y(ti=l), y(t: ), y(t
) and y(t +2 ) by least
i
i
i 2
itl
squares,
Then the smoothed value
fitted quadratic at t.,
1
~(t.)
1
was taken to be the value of the
The first two points in the experiment were
smoothed by letting ~(tl) and ~(t2) be the values at t
1
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
(4011L and these b
Estimates of B
21
were then obtained by the use of
values were then smoothed by the same procedure
21
0
Tne
estimates of the transport rate functions for the three experimentB are
reported in 2'able 6010
the tracer
0
The units of time are minutes since in,jection of
The units of b
negative values of b
2l
21
are milligrams of sodium per minute
0
'The
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
difference
* = Ait'
2J
~
*.a
0
Since the derivative A
2
changes sign at the point where
if these two values are nearly equal errors in the observations
may result in a
*a
2
being computed with the wrong signo
sheep, due either to large transport rate
fQ~ctions
In some of the
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
ferent diets the estimates of B
21
following the injection of tracero
dif<~
were i.ntegrated over the 8 hour period
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
Diet 11+
Hay Diet
b21 (t)
time
time
b?,(t)
15
10018
10006
10060
11037
6062
_..l.
Diet 11
time
b21 ( t;)
~--
Experiment 4
480.36
15
48,05
.55
78
52070
125
78075
122,92
180
1:53.86
270
104006
372
00-,,6091
482
~~.51. 86
600
788
-71077
=9,98
970
80
120
180
270
360
480
600
7E59
966
5+070
'=7" Lt·9
391t
,"j~4,36
600
,,,57019
1.3012
817
967
Experiment 5
1.5
7099
12041
45
16074
75
114
26079
46002
171
270
57.44
43008
351
480
15·95
6069
609
783
15059
52.. 03
960
15
45
75
123
180
270
363
480
606
795
979
902.3
9,97
11038
15,07
23055
33024
32,44
29039
44014
22088
-29018
15
45
Experiment 6
48055
15
52080
45
57034
75
120
66081
180
74 032
270
79003
65018
360
480
36072
600
33,54
780
35003
29,31
960
1440
-18,43
15
45
7'"
120
180
270
360
480
600
780
960
1440
5.17
8.99
13001
20.86
0
48
,/
.,
:c~7 .3~;
0
:52,05
27012
19·37
15089
19,62
19009
5041
15
45
75
126
180
270
13,53
15063
17055
20030
22060
270:31
.5h 38
o
.3.)0
75
120
180
270
360
514
609
801
980
15
45
75
120
180
270
360
480
600
780
966
14L.o
113016
75,:28
",46065
8003
8099
9085
10098
lle8.3
13.62
14090
13029
9·91
20.61
43.66
.:"
., ,C'i.
r; .,",, __ ..-'
2047
2032
.3067
5095
7097
7051
'7076
8,22
8003
6062
=303.3
64
Table 6.2
Total grams of sodium transported from plasma to
rumen 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
experiment 6
17·12
30.60
lL78
11.62
11.06
.5·97
2.89
29·98
11.67
4.43
mean
14.85
Analysis of variance
SS
df
MS
F
total
8
1361.14
experi.ments
2
96.87
48.43
1
diets
2
4
519·97
56.08
9·27
error
1059.91:.
224.33
pr(F2 , 4
Table 6.3
:s 9. 2 7). =:
.965
Identification of sheep and weight in kilograms
Hay Diet
Diet 11+
sheep
number
weight
sheep
number
experi.ment 4
5882
53.1
265
experiment 5
experiment 6
5882
54.0
5882
55·3
Diet 11
sheep
number
weight
47.2
260
37·6
312
49.5
309
.34.5
312
49·9
309
32.2
weight
time.
The period of 8 hours was choosen because of the increasing uncer-
tainty of the estimates after 8 hours.
The b were integrated by summing
2l
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 .. -(t _ +t )/2 for
i
i l i
1-4, ••• , N-2j t 1+-(t i +t i +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 B after four hours were considered unusable
2l
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.
these values and an analysis of variance.
Table 6.4 presents
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 body
weight in 8 hours after injection of tracer
Hay Diet
Diet 11+
Diet 11
4
.7955
.2582
,1179
experiment 5
•.3171
Q'~?;.82
,lD1
experiment 6
05529
02216
00879
,5551
02393
01269
experiment
mean
mean
02880
Analysis of variance
ss
df
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 b
21
~
s and for integrating
them were programed for the IBM 1620 digital computer; the programs, written 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 B
were computed from data obtained in
2l
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 b
the
2l
were taken.
In order to get some insight into the behavi.or of
estimates~~their bias
and
variance«,~a
int System I on a digital computer 0
small study was done by
simulat~,
The system shewn in Figure 6.1
used as an analog of System. I; Compartment 3 represerlts the
1'1813
extracell1l1~1r
sodium "rhieh exchanges rapidly with the sodium in the plasma.
B
Compartment 1
"B
"
~
~
12
..
,
,.,
(plasma)
~
31
B
Compartment 3
l3
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 ~ N and N..z. 1..rere
,1
./
assumed to be constant.
A constant value was assigned to B ,
13
rate of excretion from Compartment
BE3~
the
3, was assumed to be a constant percent,
p, of the amount of sodium in Compartment L
The transport rate B
1.8 a
2l
function of time, as is the rate of outflow from Compartment 2) B •
E2
Transport rates B
and B
'were adjusted to the necessary values to
12
31
tain Nand N constant.
1
.3
ma:in,~
The changes in am.ount of tracer 1.n the three
compartments are described by
68
d T
1
dt
d T
2
(601)
* B13 +
"" A3
'CIt ""
oro
B
21
A*
l
=
*
p.N ) A_.
(B
21 +
:c.
1
(B,z
poN
1"
1.
.L.)
(B21 + poNl + B ) i.i't2
E2
+ B21 )
*
~
)
)
simulate the system (601) was integrated numerically to give arrrcmnts
of tracer in each compartment following the i'l,jectl.;)!1
partment 1
0
"I'h,,::,
oftrtE~cl
1n1:-:)
Com~
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 B o. were varied until Isotope rati.o curves were obL.ained
l.J
for the plasma and for the rumen that resembled closely those of Table 1001.
It was assumed that the transport rate function B .l was the sine function
2
B (t) "" B + B sin(t!B
2l
2
l
O
»)
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 B J and by changes in the sod.:iu.m content of
2l
the rumen 0
These curves are listed in 'rable 605.
were used for all of the curves except Curve
100~
Tr:e' followil'.g values
A*.{O ) ::e5.jOOO~ N =4350.'1
1
1
N =4N ; B13",,1350; p=O.00075; rumen volume",,3200/2.2; QO""LO} ~=.<:>OO.,
l
3
~ =-.0015.
and Ql=O.Oo
Curve 100 differed in the following values
~
p=(LO.;
%::>().O.,
69
Table 605
Simulati.ous
.-.-f'
l...J<lI..
S~/s"t,I.:;m
I
i.n:.tl.sl
rumen flg',lre
sodium number
BO
B:J,.
:00
/·... !7
c,
0000
0
120
130
'7
4,25
4025
115
115
6400
6400
6400
602
6,2
602 y 603
605
nurriber
13
2
-~~-
27
4025
11'5
6400
602
27
9,00
65
6~tOO
6.lt
27
1.8.00
65
65
15
6J.t oo
6400
47
,~,25
126
127
136
137
138
4,25
4.25
4,25
27
27
27
27
139
11,5
11,5
4,25
6~
6i+OO
,4
f L'J-c.,..
h '1.
·jo
4800
bo ,5
6.3
,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 3
21
,
Thus the relative
magnitudes of the B
cannot be judged solely on the criterIa of how
2l
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
curves for a
steady,~sta'teJ
the assumption of a
Curve 130, are not markedly different from the
~o~ •••
closed system J represented by Curve 100,
steady,~state
system cannot be verified by
'rhus
cons:idera~,
tion of tracer curves alone,
Figures 604 and 605 indi,cate that
period of B
21
;i ifferences
in the am:pl:i tude ani
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
formance of equat ion (l~ .11)
5S
or ~~st:i.l'f~"'t<)r cd
~o
look at the per ..
frc:L
~1
()
",~1
+'
co
::l
.p
°rl
W
OJ
+J
tJ1
.p
(Q
u
;:».
ir~
lJ)
(Ii
P
i.:Q
Ii!
,
r:~
"~')
~
~.,
'.:-1
p
CV
~.
):J
0
cd
p
CJ
~ )
Ul
~l
,-·1
r-!
.,!' ,~
(J)
qn;)
l:l
,'d
(t)
t..i
l:l
.... i
tJ]
tJ]
OJ
+)
::l
~
0,.\
()
(\.1
S
(),~
0
"'"'
r-~i
'(\.1d
10.
OJ
r-n
()
(.x:~
~'J
'r'.~
0 '"
()j
,--j
,-~
'"
()
(I)
tV
~\
tJ1
,-~
,.h~
0
C·
rJ
W
<Ii
:>
t~
()
".".pill
~-
.
0
p
<:)
,Jj
tf-~
1-1
III
m ...'Iir'
,-I
:::$
~
t:'
S 'd
,,..,l
U:i
0
0
0
ll\
0
()
~
0
0
0
"'"'
lUTITpoS ma.rllHTpn .red
0
0
0
0
0
(>l
e~.nu1m
0
,--I
0
(~
lC\
0
(\J
0
\0
(()
.rad
s~uno;)
~j
t10
od
P'.
~
(1j
71
8-'
".'1
't)
0
CD
R
(JJ
~
~.
r--l
r.u
..,
.p
L.'
-.:'
5::.~
°rl
IQ
<f'
1:1\)
Q
,,1
11v.~
t:)
tH
,)
.j.>
(-"
tD
y._~
~~
0
p
",-j
()
ij~
tV
(jj
.r:l
.j.>
(l)
'r;)
Q
""..1
OJ
tJ
s::
~
'-'ri
~-
0
..c:
rn
°l-~
Vl
CD
OJ
+)
~
'rl
S
,<>,
0\
r<\
.-I
'Ea:l
CO
r<\
,-1
'"
()
r<\
,-j
tJl
Qj
~::.,
8
(.)
rrj
<11
.. p
r.u
.-l
§
0
0
0
0
:1\
0
0
0
•.:;t.
c>'
0
0
r<\
0
0
0
(\J
0
0
0
r!
0
0
l!\
.,...;
(I)
r<\
0
\{)
OJ
~i
iil
orl
"'"
;._j
rq
~
ur~
to
(1)
~
a.1
).~
()
I:t.~
0
()
()
W\
c
l1)
[....j
,+.~
(V
(])
~1
.p
~
l="!
0
",-I
'"",,)
r~)
(I)
"r-)
$. ~1
()
CO
,-1
"';
p
'U
Hp
((1
C,
,'!
t<-~
"~i!
',-1
t1J
0
\Ll
~~
.~
orl
r<\
,--I
(I)
'Ii
Ul
Q.I
+'
~1
s:l
o
~
F
~
,."
,~
a:l
t'-
~
'''',
'.0
~'J
(.0
OJ
:~¥'
~-l
:J
I:;)
't:'
Q)
·t;)
a:l
:-1
~
''1'4
CQ
.~t
0
0
0
0
If\
0
0
0
.~t·
UJl';~pos
0
0
0
0
"'"
me.rzlf.H T'Ul
()
0
(\j
.I ad
0
0
0
,....j
0
0
If\
\0
(l,l
fj
':iJ
'd
aq.mlT'iU .l'au s'iuno;:;
~J
(\J
Iq
l7::
...~ ~
(J)
(l)
1:lO
C(
iri
~~~l
(l
\~ .~
0
,
\
.~
tli
4-,
1H
(I}
QJ
"':;
p
~p
C")
.::t
(lJ
f-1
()
",ol
p
,e
()
1l
C'J
f:j
-..d
"
'd
~)
tU
H
~.
"(f)
;-,
r""
r'.-1
o,~
OJ
0
r
'n
1'<\
r1
1-1
')<'"1:1
c
•
lQ
ffl
(j)
!11
a1
,l,J
.;j
~
>,-1
\0
!'C\
,-j
S
'"
C)
1'0,
.- .~
(f)
m
;>
~-1
0
'el
OJ
p
a:I
,--1
§
,,-1
ro
In.
0
0
0
l("\
0
0
~'f
0
0
0
!'C\
c>
0
0
(\J
0
0
0
'·'1
0
0
ll\
0
\()
tV
~~
be
~f-~
r"l
74
oj(-
it.. .
~,
1;;\i
ere 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.
error.
The normal deviate:::: were obtai::led fr'cl:~
Tne
D2.X;Xl
aul Massey (:";;)7;
f:is'.,'t;li
as ·:r:.e ,31:.andHr::
CC';'l~
with the tru.e valc.€sQ, :B2.1'
the average of the 20 0'-;)'1 (t.. ) i S "\;illS COl.llJ".lted;
t;...J...
.1,
Figures 6,,6, 6.8 and 6,9,
of B
for three of the error curves i:n Series 130,
21
Series 120 of error curves was cased on CUTve 1.2C' ,,rith
int.;ervals betweeD observatiC!ls averagir.g '705; 1..5
a!.~d
5C
;;_i.)2pl::L~lg
at ::':,
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 b
21
at these
POil1tS
from values whi.ch were not sn:aothed w:!.th t.::1ese
was af !lecess::.t;.,' computed
pc~rJta
as t.r:e central poi.nts,
e
e
e
Table 6.6 Estimates of B21 (t) from simulated curves
time
Series 120
30
48
60
75
87
105
123
135
150
165
177
195
210
225
240
255
Series 160
30
36
45
54
60
66
75
84
B21 (t)
ave.
b21 (t)
s.d. of
b21 (t)
8.09
8.72
9·11
9·57
9·91
10.36
10·72
10·91
11.10
11.21
11.24
11.21
11.11
10·93
10.69
10.39
8.34
8.79
9·10
9.48
9.88
10.43
10.81
10.96
11.14
11.35
11.45
11,23
10.93
10.80
10.88
11018
1.086
.414
,415
.424
.427
.593
.574
.496
,613
.680
0561
.815
.812
0967
20202
4.155
28009
28.30
28062
28.92
29011
29·30
29·57
29.83
28069
28065
28.72
28091
29.34
30.02
29098
30010
4.099
2.243
10422
1.440
1.528
20390
2.334
2.109
time
Series 130
30
48
60
75
87
105
123
135
150
165
177
195
210
225
240
255
Series 161
15
45
75
102
135
165
195
225
B21 (t)
28.09
28.72
29.11
29.57
29·91
30.36
30.72
30·91
31.10
ave.
b21 (t)
s.d. of
b21 (t)
31.24
31.21
31.11
30·93
30.69
30039
28.73 2.633
29·17 1.059
29·43 1.195
29.63 1.209
30.07 1.180
30.80 1.622
31 .34 1.745
31.42 1.490
31·55 1.770
32.02 2.098
32.41 1.841
31·91 3.012
31.04 3.082
30·97 30443
31.85 9.1 65
34.03 18.908
27055
28.62
29·57
30029
30,91
3121
31.21
30.93
27090
29007
30.02
30.64
31.39
31.97
32.30
31.94
31~21
1.789
.814
1.134
1.240
1.139
10756
2.116
1.953
-.;]
VI
-
e
e
Table 6.6 (continued)
time
Series 160
90
93
99
105
114
123
129
135
Series 137
30
45
60
7(;;.
1/
90
105
120
135
150
165
180
195
210
225
240
255
B21 (t)
aveo
b21 (t,)
s odo of
b21 (t)
(~ontinued)
29·99
30.07
30.22
30.36
30·55
30·72
30082
30.91
30.33
31.10
31.69
31.32
30.38
30.24
30·77
31.76
30.86
27059
23.78
22·92
25081
29079
31.20
28.75
24.68
22.75
24.71
28.78
31.21
29.76
25·77
22.91
28.57
26066
25067
1~546
2503 4
1.0297
26oE31
lo3Cr2
29·00
29·82
28.49
26020
24.65
25026
27·59
29.5429024
26.81
22025
10318
lc835
1,705
1·559
10933
L813
1.744
2·527
2.1.30
6.032
140244
3a601
30782
2$736
2.865
20701
4.411
8.850
1:;1 035
0
time
B,",~
<:::J.
(t)
aveo
b21 (t)
sod of
b (t l
':l
21' ,
Series 161 (continued)
225
30039
3.139 2,080
31,14- 2.115
285
29.61
28.66
31.18 20816
."il..... '"
29.76
345
40979
27·59
26049
6.036
27047
375
24046 40588
25.42
405
200124
28.36
24.45
435
23.66
465
37014 50Q155
./
./
:·,553
~.g14
--...1
U\
e
e
e
I
I
38
, ....
34
Q)
+J
:;j
30
°8
H
~
.#
>:::
26
,-/
,
I
/
~
oM
........
_"...
__ "..
.....-.
-
---'
- ,.~ ""'
'
' -
odc
rage b 2'~
'" .... ....
\average - 2 Bodo
22
\
\
tIJ
0
,/.ave
S
~21
'0
0
r..;
average + 2
---,
.
Ser~es
130
/
'" / -
..••• "
, , , •••
~ ....
••
,,"
•
•
., • •
____
.
...;;;...;;..;;...;..-------=-:.:..:.~.~:..~.
.. "
•
Q)
PI
--
-
--
~
18
~
e.u
H
QO
oM
14
------ -----.. --
rl
rl
oM
S
10
...........
-
~,,--­
---
~
/ - - ---~
"../
6
o
20
Series 120
60
100
140
"
avez\.~
180
220
B
21
=
2 Bodo
"
260
minutes since i.njection
Figure 6.6
True values of B and average and sod. of b
for Series 120 and Series 130
21
21
-.:;
-.J
e
e
e
130~7
,,_ ...
;
I
I
I
,
ClJ
~
/~..
~
oM
S
~
. - .
~I
'\
32
\
ClJ
PI
0';", '0.
\
0' '
~,
.'
\
'\
\
-..-I
'8
....
I I .'"
..........
./
l1.l
"
lH
o
~
co
"of
.............
.../
/
-
c')
""
/........
\
§
/ - , ,,'" .
\:".
'/
.../"
'. .
•
\
..
~
\
\
......
....
.
..
I
"'''''
/
130=4
~-
II
\.;
\
I
I
\'G
~
~
E!
I
\\ .
\'"
\ .
.....--
iI
I
'\
\
/
B21
/
J..
.
... 130~..L
•
••
1
0
•••••••
oM
r-l
r-l
or-!
II
24'
22
o
40
80
120
160
200
240
minutes since injection
Figure 60'7
True values of B for Ourve 130 and b
values for three error curves of Series 130
21
21
.-:j
OJ
e
e
e
I
I
,
,,, average + 2 s.d.
,
f
46
42
38
Q)
~
s:!
'g
,
'-,
\
\
\
34
Q)
~
..-l
'8
u.l
30
-
26
o
u.l
S
eel
,/
22
H
bO
I
I
'
'
,-,
"
-'\
\
~
I
1i
-
----
-- --
----- -
ave. + 2 s ,d. (161)
b 21 (160)
-
_
I
\
-~
~
ave. b
(161)
21
- ------
----
B
2l
~6!j----
= 2_ s.d.
(
-ave.
- -
/ " ' \\
\
\
'r-!
r-!
rl
,
-
,-
.,/'
,,-,
CH
....... /
-
H
PI
,,I•
(160)
\
18
\
\ average
\
14
=
:2 s.d. (160)
.
\
\
l
\
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
to
0.!
(\1
·f
o
1..0
0.1
OJ
~
to
0
t....
~
___
N\
r4
~
to
fI.1
Q)
orl
~.
>~
(Li
rn
(\!
~f
()J
0
"H
t't~
()
r
c-'
"
-~
0
·,,·1
,\-1
0l)
r'\
"'J
r.r!
c:
I.V
,1.>
Qi.l
0
to
~1
"..-I
Tn
/
I
((1
C',
m
"'1--:)
'"d
I
it_,.·/
f.,
Q)
;:.
ro
'I1
ill
P
~_1
~
~
~
ro
~.
,,'1
(\.1
if!
':JJ
'u
I:'"~~'
m
t>
(!)
;::;
~1
H
()'\
\0
IV
o
. e.l
.
\0
r<"\
",;t·
N'
=+
1\J
1'1\
~
('0,,
I'i\
(0
\0
;J
(\1
I',\,~
('.1
(1.1
OJ
0
01
Lil:lD
'r!
rx.u
81
the rnagnitude of B
is incre8i3ed the d,1:ffe:xeLce b12twes::l
21
age of the b
2l
D
.B,_~
coL
a:r.J.d the
aver~
S 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 A .is also
2
partly responsible j that is, the errors BddEd to
tion
eq~al
Aa'*
have a s"tandar:j devia-
* ~~t'~e
root of
to
i,8 A...,
Co:
are the added errors.
t.he variabLLity of the estimates.
'I~hat
by analogy to linear regression pro'blel1K,
function, B , wIth more deftlrture from
21
this should be true is 8'l;.ggested
,H~iwever"
0
fJ
for a traw:,:,port rate
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,
d,
was, however j little effect on the standard
d""'''lB.ti~m
B'Yl
c_
C:ltrve.
Tn,ere
Df the estimates.
7.
7.1
The
ITJ.8in
DISCUSSION
Advar,tages of i;:,his f1ethod
value of (4.11) as an est,lmate of E (t)
21
to systems that are not in a steady=state can,Ht:ic;::"
be made about carls'tant volurnes j
concentratiar~s
this is an important advantage over other
there are Dlliny biological
By~tema
that
:i.3 i':;8
apl;li::abiljty
No assumptions nee1
!.:-r rate constants.
e8tLm.s.~e3
a~e ~Qt ~.11
1e
d.~1eto
'.I'hat
the fa:"c
a ateady=s'ta"te
1)"'i8t
~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
parameters of a fi tte<l
t~n.a:n
to the rate constants depends on the model 8ss'J.me,L
C'l.l.:'\re
'wb.ose
'I'hJ.s errors
re:lattcl~
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 B
21
(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 111 t in
v:eaknesses in the method
Thus the freedom from fitting a single curve to
0
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
tage 13 relative,
fJf
0
This disadvcm-
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).
quadratic moving average
It is emphasized that the use of a five point
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
data needs less than this or its equi.valent,
11<,.:
..
~:.:aQ)
of interpret:i.ng tracer
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 assumptions 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 b
21
in variotls situatio.ns.
analytically the distribution of '0
wri tten in terl11.B of the
soarces of
;:)1'
]
II
;: "••-.-..-.----.-~.
21
experimerrtal cib8ervables '!iLich are tbe
act~lal
t'
SOUl-U.n". umcsn.trstL::n
.1E :C~J.:[ner.,
rUrrIp.Y:,
1::
may be s\lggested by look::ng at (1~.,Ll)
21
thRt is
variation~
[V-~jl'illne
'l'he d.ifficulty of determining
J -=_ [eE:;tlnk~t."';
a
~
(:1 ]
d~;r~<·<i-t.rr.l Vf;
,~~=
'''-~'~-~'-~~~--"
'.
...--••.-~~.~'"<~~=. .~.~.~~~~-,~~".-.~-~--.<
'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 B
21
(t)
the effec't of different spar::ir,gs
J
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 derivati ves
shc~J.ld
Furtl1er comparison ShOl11d be made
also be .studied.
'~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~~:lOS n ,
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
ti.on of homogeneous com},(artments.
system~
It
W3S
and whIch modi.fied the assump-
.::.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,itted 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
lation.
simu~
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
and Sons J New York.
1959.
0
Meta"bolism in the Rumen.
John Wiley
BarlmM} Co F. and L,o Jo Roth. 19620 Water spaces of brain studied with
radioisotopic indicators. Proceedings of a Conference on the Use
of Rad:ioisotopes in Animal Biolbgy and the Medical Sciences 1~279­
294.
Barnett J A. J. Go and R. L. Reid. 19610
Edward Arnold (Publishers) .. London.
Reactions in the Rumen.
Berger J E. Y. 1963, Transfer :rates in two~'com:partment system not 1.11
dynamic equilibrium,. Annals of the l~ew York Acaden[,,/ of Sciences
108(1) ~217-229.
Bergner, Po-E. E. 1962. The sig:nifi.cance of certain tracer kinet.ical
methods especially with respect to the tracer dynamic definition
of metabolic turnover. Acta Rad:i.ologica ~ Supplementum 210 ~ 1-59 0
Bergner,~
P. -E. E. 1964. Kinetic theory ~ some aspects on the study of
metabolic processes) pp. 1-16. In R" Mo Knisley and W. N. Tauxe
(eds.), Dynamic Clinical Studies with Radioisotopes. U. S. Atomic
Energy Commission) Washington, D. Co
Berman, M. 1963. Formulation and testing of models.
New York Acaderrij of Sciences l08(1)~182-194.
Annals of the
Berman) M., E. Shahn and M. F. Weiss. 1962. The routine f:i tting of
kinetic data to models. A mathematical formulation for digital
computers
Biophysical Journal 2 ~275oo287
0
0
Bigeleisen~
Jo 1949. Validity of the use of tracers to follow chemical
reactions. Science 110~14=16.
Carter, M., G. Matrone and W. MendenhalL 1964. Estimation of the life
span of red bleod cells
Journal of General Physiology 47~851-8~xL
0
Copp~
D. H. 1962. The use of radioisotopes in physiology. Proceedings
of a Conference on the Use of Radioisotopes in Ani.mal Biology and
the Medical Sciences 1 ~23°· 300
CoreYJ K. RO j D. Weber~ M, Merleno E. Greenberg j Po Kenney and J. So
Laughteno 1964. Calcium turnover iYl man~ PPo '519-536. In R. M.
Knisely and W. No Tau;xe (eds. L Dynam,ic Clinical Studies with
Radioisotopes, Uo So Atomic Energy Commission, Washington, D. C.
j
W. Jo and Fo J. Masseyo 1957. Introduction to Statistical
Analysi.s.
McGraw~Hill Book Company, Inc. j New York.
Dixon~
Dukes.' JIo He
19'):;,
'I'he PhyBie;logy of
P'Llbl,:i,sb.ir~ A:5:3C,c:la~,es,~'
Dom';:a~:ic
k'limals v
Comstock
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 jn
biochemistry-co P.r'oee2di.c gs 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,
1
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 into
SC>'-l.r~lal of Gerleral Physiology 38 ~ 389-·484 0
19~J80
The digestion of
Gray., Fo \[0., Ao F" P.llgrirIl and IL 11.0 W~,.i.le:L
foodEd::u;ffs in the s't·omacb of t;'b:·,
and the passage of digezta
through Hs compartments" ErHi.at ,Jcmrnal of Nutrition 12 ~404~420.
F Ao 1961. An Intpodu.cti em to I.i.near Statistical Models J
Volume 10 McGraw·,H:i1l BDCK Compa:::.y) Inc" y New Yorko
Gra;>-'bill~
0
Gregg." E. Co 19630 1m analog computer fer the generali zed ffiulti=
compartment model. of transport l.n biologi.ca1 systems, Annals of the
New York Academy of Sci.erlces 108(1):128~·146,
Guest JI Po G. 19610 Nl.:.merical Me·thods of Curve Fitting"
University Press y London,
Cambridge
Harty Ho Eo 1955, Analysis of tracer experiments in non.,conservative,
steady-state systems
Bulletin of Mat~lemati.ca1 Biophysics 17~87=940
0
Hart JI E. E. 19570 Analysis of tracer experiments ~ II 0 Non~·conservati ve JI
non-steady-state systems" BU.lletin of Mathemat~.cal Biophysics 19~
61"'720
Hartley; Ho 0,
non~l:L"lear
:5 ~269-28o"
19610 The modified Ga'J.ss~Newton met;hod for fitting of
regression functions by .least squares
Technometrics
Hevesy" Go 19230 '.llie absorption and
Biochemical Journal 17~439-445,
0
trar~slocation
01' lead by plants.
Hevesy y G, 1962a. Historical progress of the isotope methodology and
.its influences ::m the biological sc.iences J yoL 2) ppo 997,,10220
In 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 Radioisotope
Research, Pergamon Press.l New York. O.r'ig:inally publi shed i.n Acta
Phyi3iologica Scandaniva 1~347(l941).
90
HevesY;j G. a~'1d Eo Ecfer v 193"+"
NaturE\)oE,'.ler'.)
L ~ Sr)"
lL.:CiJ..D.Btioc c)f 'water from tl:e buman cody.
Hildebrand) 1<'0 :B, 19560 I!lrrciuctim"l tc Kl.lliler:::al ALal;{sis
Hi11 Book CCJti:pa/,y, .1:'.1(.,: I'J·s(';lic.ck,
0
McGraw~
Jaffa;/j Hol96.?o Meas'J.ring turnover X3tcS L',.!~he (lcm",stead.y sta-ce"9
vol 1 I'po ;217,,22].0 1:'1 So Rcsl';c1:ild ::.d 5' Advances in 'rracer
v
0
Metbodolcgy
Kamens M. Dv
York"
)
P1.e!J.e'J.m PTes B) New' Y"'::Jl".k,
19.570
IsotGI-ic :Fracex's
~n
Biclog:y"
AcademiC Press} New
K.'1isely} R. f'iL IjDd 'N, :~L Tal.x." ed.:::> 0)
.. C::Uni.cal Sr~l..;.dies
',viti';.
6)t,. l'P"
l.c"",<l 3tcoitei.' AtcmLi.cEnergy COTIlXll.is3ion J
Was:hltl.J3tcns D'Q C·o
K.rriselys R. M. and W, 1\1", 'I'au:xev 196~b, FrE:face, ppo :ii::;>.. ivo In Ro M.
Krnsel.'! a;,d W. N, 'raux,:' (>:';d8"
"i. D~..(:narrllC~ Clinical Studies With Radio.
isotopes u U v S, A"t.:;)mic Ene.rgy Gomm:' s3ion,! WashingtO!lJ Do Co
~
~
~
Lipton} So 8..~d C, M:<;:.ncr..rist, 196,)0 MaxlJy,uxn LkelJ.hood esti.mators of
parameters in double expa:,:,£nt.1.91 ['egres,sion, Biometrics 19~144.-15L
Laushbaugh~ C, C,} Ao
Kretchmar and Wo Gibbs u
1964v
L.i.Yer fUDct:ion
measured by the blood clearance of Ro,3e Bengal~I13l~ a review and a
model based 0:" compartmental aEa:ly8is of changes in army blood and
l~lver radioactiVity} pp, 3}9""~151,
In Ro M, Kn:i.sely and W, No 'rauxe
(eda, L Dynam:Lc Clin:1.ca.l Stud.ie,s '''Hh RadIoisotopes
U S. Atomic
E.nergy Connnissio::l., WashiDgton: IL C"
0
0
Matrone} G,) lio Ao Ramsey a:'ld G. 110 Wise, 1959, Effect of volatile fatty
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19620 A ,:,ompariscD of HTOlr, plasma and expired water vapor,
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Moore y H,
,
Myhill; ,L" CL Po Wadswortr. and CL 1.0 BroIJl:r..elL 19650 I:nvest.igations of
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Bia.physical Ji)ur:nal 5~89-i07
0
Pochin) Eo E, 1964, .Liver cor;c.e;o.tX'ar~)'.):n of tl'.y:t·old metabolites) pp, 41.3=
432,. Irl R, NL Knisely and W, :::10 Tau.xe (eds 0) 5 Dynamic Clinical
Stud.ies wi til Radiolso,:opes, lL S AtC'JL]c: E~:ergy Co:mmissioT.l J
WashingtoD J Do C .
Price 'I Do c.. 19640 Iren tllrnover :in man} p:pv 5Y7~563,
In Ro M. Knisely
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Uo S. AtomIc Energy Commission, Wasr..ington) D, Co
j
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Un:l';'2rsity Press J Cambridge} M.assacrr;lsetts
0
Sega1 J So., M, Berman and Ao Bla1ro
196L
'rrr~e metabolism of vaY'i.o~lsly
C14~labeled glucose in man and an e,s-r.;lmatio:::. of the extent of gl·'.1cose
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0
Sbarney) IJu" 1.0 R, Wasserman,) La S':hwar·';·z d:r:fl D, '['",T,alel', :1.9630 MUlt:tple
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0
Shemin J Do and D, Rittenberg
19460 'I'he b.l.olog::Lcal utilization of
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Journal of Biological Chemistry 166 z621 ..,636 v
0
Sheppard y Co Wo 19620 Basic PrincIples of the Tracer Methodo
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,John
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0
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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
The estimates of the B,. in 'this thesis
IJ
of linear equahons in the B
Obtained as the solutions
: some cf the coeffi.c:Lents of these equations
IJ'
were values o,f the f\.:mc tions ~
Nk'~
"I'k.. ani
It is a convenience
uated at time t.
W6):"e
Equa~
+r
;.•·v
*
~.9
and their derivatives, eval=
use
mat~-:'ices
a.nd veeto:cs and the
equations.
B., .,
K.:"
is tt.e
A ::: (8.
E X 1
rs
) is the
a .lgmented matrix Dbtalr.ed by adjo:i Ding t.1':;,e ve :::tor Q totl:e matdx A as the
1
n+l-th column.
Tne system of
systems can be 'written
lir~ear
equations descr:i.bing the various
82
Tne folloWing well knol¥n and usefu,':. 'tteorems of linen.'1' algebra are
from Graybil.l. (1961, p. 10),
.L." :
ern 10.1.
of equal.ions
A£ .:
A r:-:"essary and s\.lff: c i.ent c('r.di,tion that the system
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
B:
ij
If :rank(A) _. ran.k.(A
i 12)
"" p" then nor ,.,f the unknovns
can be assigned any desired value and the remaining p .)f the B
ij
will
c(lefi'ic :i.ents of t.b.e rern8.j.ning p
If raI'Jc(A
If~)
AI' "" D
B
hay€; rank p.
unknc"r~lS
=:
n ::
ffi J
sher '2. is a unique vector B that
satisfies A:B .,. D.
SGlut~!..on
the
estilniite£7
~'n.e
of the B"
i.:';;
.i.n Secti 0:<.18 1" an:l
lJ
.t:s·/,~ ')1':1 que
'5
sol utic)l:';j j thus
¥
:letcrmi.DI\I": of A is A
1
AE "" D has a
where
1
0
'Ql
0
'~A
.*
,t
t\
A
1
0
2
0
l*
=A_
.1
*
~
1::(
IT'
J.,./
B
0
N
2
1 ·;,)
~2
,..L.oI-.
:::
=1
B
21
0
1
0
,'"
""
D
N
1
B
02
B
1I
.~
J
T1
95
(1\'I'
The determi,nant of A is
AB = D has a unique solution.
:::?,n be ;,lri tten AB
'2quations for System
= ~,
where
A
:::
0
..::,1
0
~~A3
*
0
1
0
A*
2
B
,
-1
1
.L
B
21
N
B:23
'1'3
3
D
~
()
BO::>
""
i
N""
C
(·r_~
*
Al
A*z
:J
-A2.*
-lE'
A~
G,
rJ. A-)(y
LB32
OJ'
The determinant of A. is (AI
I
- A2 )(A;
-l('
"
A
T
2
,'\
3
)
j
chat if Al*
"'"
*
f ~,
and
the system ha.s a uniquc.: solution.
The equations for System III(a) are (5.1)+) and can be written ~ = ]2,
where
A
=
°
0
1
0
0
A
2
~l
1
-A,.,*
A,*
*
N
3
-A*,3
B "
2 .1
'1'3
0
-~*
0
-)(,
3
The
12
·,1
The determinant of A 1.S (A
* f A*y
B
B
,~,
~
-1
- A*"J (A*
2
D
""
2
.u
::
i
B
32
N
2
,
By",
( ,
IT'
~2
A* ):; so
l
then the system has a uni,que solution.
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 b
21
(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 a * and a * are counts per minute per
2
l
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.
* were obtained by dividing the observed counts per
The values of al(t)
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 extraction method.
97
Table 10.1 Data from Sylter experiment 4
time
a*1(t)
a2* (t)
volume
sodium
conc.
260
15
48
80
120
180
270
360
480
600
789
966
8694
7444
6960
6330
6315
5905
5580
5467
5386
5039
5134
372
978
1546
2135
2851
4350
5445
5730
5900
6416
7217
2220
2204
2198
2185
2150
2120
2080
2030
1980
1900
1796
1.880
1.718
1.565
1.635
1.733
2.155
2.371
2.426
2.415
2·557
2.318
265
15
45
75
126
180
270
394
480
600
817
967
6256
5491
5179
4867
4503
4239
4148
4153
4106
3956
3869
186
337
480
905
1187
1713
2403
2558
3409
3652
4094
5645
5630
5600
5570
5503
5470
5400
5295
5250
5145
4999
2.481
2.374
2.415
2.243
2.201
2.243
2.353
2.489
2·571
2.514
15
55
78
125
180
270
372
482
600
788
970
2562
2096
1935
1773
1699
1676
1747
1741
1609
1558
1591
68
552
547
939
1171
1598
1544
1593
1641
2002
2023
5750
5695
5670
5600
5510
5390
5270
5115
4950
4700
4395
1·991
1.919
1.616
1·922
2.119
1.971
2.464
2.546
2.703
2.608
2.579
sheep
5882
~.464
98
Table 10.2
Data from Slyter experiment 5
time
a * (t)
l
8
309
15
45
75
120
,180
270
360
514
609
801
980
4592
4157
3970
3858
3690
3430
3315
3343
3363
3350
3235
243
342
536
855
1134
1616
1999
2411
2722
2677
3197
3255
3550
3240
3225
3200
3150
3120
3065
3035
2945
2880
1.854
1.765
1.801
1.721
1.739
1.801
1·522
1.901
1.714
1.869
1·771
15
45
75
123
180
270
363
480
606
795
979
3628
2944
2802
2723
2447
2532
2455
2354
2402
2318
48
276
308
440
809
1329
1720
1854
2088
2288
2387
4650
4585
4530
4450
4340
4180
4000
3750
3580
3220
2880
, 2.311
1.933
1.731
1·912
1.744
2.026
2.025
2.089
1.947
2.185
2.218
15
45
75
114
171
270
351
480
609
783
960
3228
2699
2566
2390
2327
2094
2215
1996
1968
1952
1834
16
51
195
283
567
1287
1395
1379
1,448
1503
1614
6170
6150
6140
6125
6095
6035
5980
5915
5850
5750
5657
2.113
2.300
2.145
2.113
2.171
2.296
2.760
2.875
2.645
2.540
2.776
312
5882
~073
*
2 (t)
sodium
cone.
sheep
volume
99
Table 10.3 Data from Slyter experiment 6
e
a2* (t)
23340
22277
22300
22436
21312
17977
17229
14840
13676
9823
10089
8530
volume
sodium
cone.
15
45
75
120
180
270
360
480
600
780
966
1440
a1* (t)
55
58
172
255
377
553
908
1250
1374
1710
1942
2463
3660
3635:5625
3600
:5565
3515
3475
3400
3350
3250
3196
2992
1.7610
1·7550
1·7730
1·7550
1·7790
1.9070
1.8460
1.8880
:1..9370
2.35 40
2.1930
2.1070
312
15
45
75
120
180
270
360
480
600
780
960
1440
159
523
861
1315
1490
1794
1946
2150
2317
2327
2562
2575
12739
13338
12776
11431
11025
7777
6997
6375
5632
4853
4255
3403
5800
5780
5765
5750
5700
5650
5585
5510
5450
5315
5213
4916
2.2080
2.1770
1.9780
1.9780
1.9010
1.9930
2.0240
2.0390
2.0390
2.0700
2.2390
2.5460
5882
15
45
75
120
180
270
360
480
600
780
960
1440
111
430
643
1076
1399
1672
1904
1956
2165
2689
2177
2403
17536
17173
15328
11379
9674
6008
4457
4414
3522
3300
3072
2723
7610
7560
7480
7370
7250
1040
6800
6475
6215
5740
5385
4295
2.1240
1.9210
1.8530
1.8130
1·7930
1.8940
2.1110
1. 7250
2.0970
2.1130
2.1280
2.0550
sheep
time
309