A TRACER KINETICS THAT WAS
by
Per-
1 THE CLIQUE
After World War II the rapid development of radioisotope techniques caused the so-called tracer procedures to grow and assume structure. With the new isotope methods available investigators started to inject radioisotopes here and there, and then looked for the radioactivity to appear in other places. Although biology and medicine dominated these activities, the methodology found applications far outside these fields. For instance, the radioisotope procedures were soon adopted in chemical engineering for, say, the study of mixing in large reactors, detection of bottle necks, and more generally to see what was going on in complex industrial plants.
Some of the early findings and concepts became consolidated, and unquestionably, as we can see it today, there are features of these activities that remind more of religion than of science. Yes, one could say that, after some time, an international clique crystallized where there were a few “truths” that were not to be questioned but, as a consequence of the historic process, today some of them (the truths) appear a bit startling. However, before I discuss them it is necessary with a slightly more formal formulation.
2 TRACER AND MOTHER SUBSTANCE
Let R stand for
a “complicated” system like a living organism (R for rabbit). Let also there be a
constant input into R (e.g.
grams per hour) of some substance. Under general conditions it is assumed that
there is steady state in the sense that the amount of the considered
substance is constant everywhere in R. We are now speaking of a specified and naturally present substance in
R and it is called mother substance; to fix ideas we shall assume that the
mother substance defined by one kind of reasonable molecule. We further assume that it has been
possible to replace one of the atoms in the molecule with a corresponding
radioactive isotope. That is, we got a radioactive version of the mother
substance that, when introduced into R, can serve as a so-called tracer, and basically the procedure was at the time entertained
to observe the tracer amount as time function
in different parts of R (after a
single “injection”). The basic idea is that the tracer behaves in R identically with mother substance. It
is then assumed that there are no “isotope effects”: the tracer’s behavior is
identical to that of the mother substance (today it is known that this
assumption is far from trivial).
The very object of the tracer method was to describe the dynamics of mother substance, namely amount of that substance in various parts of the system and flows of that substance between those parts. Again, always and everywhere present has been the assumption that R is in steady state with respect to the mother substance: the amount and flow of mother substance is time independent through out R.
3 EXAMPLE 1
The amount of
potassium in man is occasionally considered as a clinically important quantity,
and various methods has been employed to estimate
that parameter, and over the years quite common such a method has been
the so-called dilution method, using for instance a salt of the radioisotope (e.g.
the nitrate) as tracer. The method is based on the
belief that if one injects into the blood a tiny dose of this tracer (a
non-disturbing amount) it will eventually be distributed in the body in the
same way as is the mother substance, the naturally present potassium that is.
Yes, this was self-evident: R treats tracer and mother substance equivalently and, hence, the two
substances must become equally distributed (common sense does not question the
self evident).
4 EXAMPLE 2
The thyroid gland manufactures the clinically important hormone thyroxin and releases it into the circulating blood. The hormone molecule contains iodine atoms and, hence, if inorganic iodine-131 is injected into the blood it is picked up by the gland and the radioactivity will appear in the gland and in the blood hormone. This can be measured as time functions and Fig. 1 illustrates the possible look of the curves data will scatter around.
Fig. 1 Thought curves with the purpose to illustrate what the radioactivity might look like in unit amount of tissue of the thyroid gland and in the hormone thyroxin in blood, following a single injection into the blood stream of inorganic iodine-131. This inorganic iodine is picked up by the gland and incorporated into the thyroxin molecule that the gland manufactures and releases into the blood.
It was soon recognized that such data in general (not only from radioactive iodine) can often be represented by sum of exponentials:
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If
time increases the first term in the sum (the term with the smallest -value)
will dominate; that is, the last part of the curves in Fig. 1 is mono
exponential. Of course, this is self-evident but had nevertheless consequences
for a reasoning that today is difficult for us to understand. The following
deliberation is an attempt of mine to describe that reasoning.
Fig. 2 The basics of the notion
of well-stirred and open system: the vessel should be of modest size (no more than 500
ml, say (cf. Appendix 2)). Well-stirred means here that should a small amount
of table salt (NaCl) be added to the water of the container, the ions become immediately
and uniformly distributed
throughout the volume). The disappearance from the container of the ions in
terms of, say, mg/min will be mono-exponential (a so-called first order kinetics); this is a result of the vessel being well-stirred so that the ion
distributions over the volume remains uniform all the time. It can be
(mathematically) demonstrated: if R consists of n “compartments”, between which the
tracer can move according to first order kinetics (the linearityfirst
order kinetics
is
warranted by the relative smallness of the amount of tracer everywhere in the
system), the recorded time-series data will be of the form depicted by (1). For
instance, if
in (1) R can be represented by a compartment
model according Fig 3. The investigator’s problem consists then of two steps:
firstly, to estimate from data the values of the
and
parameters in (1), and secondly, to calculate from those estimates the
values of the rate constants
in Fig 3.
Using the
principle of a small and well-stirred vessel (Fig. 2) as an analogue the clique
coined the notion of compartment: a physico-chemical
domain of R that has the “well stirred properties”
indicated by Fig. 2 (see Appendix 2). The reason for doing this was that it
could be (mathematically) demonstrated that if for instance R consists of three such entities, as
depicted by Fig. 3, then data can be represented by (1) with (the alpha and beta parameters become
functions of the rate constants
). A data analysis in such terms became known
as compartment analysis and was often applied to the
thyroid data discussed above for estimation of the hormone production. If it
appears complicated it was
both experimentally and theoretically
and a lot of ingenuity was invested (the
physical and the clinical gain was nil).
5 FAULT I
In 1962 one could read in a text book on tracer methodology: “As tracers are allowed to equilibrate… …all specific activities will tend to equality throughout the system”. Here the term “specific activity” can be interpreted simply as a quantity proportional to the ratio between the amount of tracer and the amount of mother substance in a volume element; hence, when the specific activities are equal the two substances are equally distributed (and the converse).
The man who wrote this had an excellent physico-mathematical background and, therefore, the statement demonstrates how the reasoning behind the dilution method was very much a matter of faith rather than of knowledge. In short, to make a pedestrian formulation, in an open complex system the distribution of tracer (after a single “injection”) does hardly ever equal that of mother substance; we are in fact dealing with two different distributions: the tracer distribution is a limit distribution whereas the mother substance distribution is a steady state distribution. But, thus, the clique stated otherwise: when the curves in Fig. 1 are mono exponential the tracer and the mother substance are equally distributed (cf. the mono-exponentially behaviour in the system depicted by Fig. 2).
Appendix 1 suggests another way to look at it. Thus, to justify the dilution method one used a macroscopic property (time data become gradually mono exponential) for deduction of a microscopic property (equality in particle distributions) and that obviously contradicts the mm-rule.
Fig. 3 The
kind of compartment models that dominated tracer studies of medicine and
biology during the later half of the 20th centaury. The system (e.g. a rabbit)
is imagined to consist of a few compartments (here 3, which has been a common
figure in the past) between which the tracer moves according to first order
kinetics (e.g. if is a positive constant and
is the total amount of substance in compartment
3, then
is the flow of substance into compartment 1
from compartment 3). A compartment is a domain that is well stirred (Fig. 2).
Today it is difficult to understand how it was possible that (in the mid of
last century) clinicians were sitting, with contracted brows, looking at this
kind of pictures as representing fundamental properties of patients.
6 FAULT 2
The “definition” of compartment is not of much help to the experimenter. And this is obvious from the literature where investigators regularly have mixed up function and morphology. For instance, in medicine an organ may be considered to be a compartment, not because of its function but because it is anatomically easy to recognize it as a well-defined subsystem. And numerous are those investigations where the blood-circulatory system is looked upon as a compartment, in spite of the fact that blood is physico-chemically about the most heterogeneous material known to man.
But instead
of going into details, let us consider the following scenario:
An investigator K observes
the amount of tracer (as function of time) in a part of R, and finds that (1) works fine for .
So here we have a three-compartment system! But K makes similar recordings also in another part of R and finds that also in this case (1) functions
fine for
.
Hence the very thinking does work! Well, there is a small detail, namely that
the two data sets produce different
-values. That is, K is actually facing a six-compartment system. So now K records a third part of R in order to get a third data set… Yes,
the number of compartments (and, hence, the complexity) grows fast and, in
fact, this kind of analysis has been compared with the epicycle approach of
Ptolemy in astronomy (Bergner, 1967), and it still seems that the parallel is quite
a valid one: complexity is one of the prices one has to pay for artificially
simple notions.
There are in fact a number of fundamental matters that go against this kind of compartment analysis, and to avoid unworthy space-spending let me just refer to Appendix 1: the procedure violates not only the mm-rule but also the time-scale rule. That the compartment concept has, in spite of all this, been so popular in various contexts (cf. Jacquez, 1985) is something that should stimulate investigators of the history of science. It is a psychological riddle.
Appendix 1
The notion of time process is dominating here, and the question of how to describe such a process is a major matter, so it should indeed be justified to consider the meaning and properties of the word “time”. However, as everybody knows, time is quite knotty a concept and the literature is crowded with writings on the topic; yes, it is easy to get lost in a variety of facets, so here I shall limit the issue to a practical aspect, namely to the influence of change of time scale.
In spite of being practical, the theme of time scale is quite a tricky one, and I shall try to deal with it by introducing time unit as a primary concept. As a primary concept it cannot be defined so let us simply try to use it illustratively. For instance, if one observes a time process every tenth minute it is natural to consider ten minutes as a time unit. Should we instead make one observation every half an hour, the time interval half-an-hour becomes a natural time unit. This thus means that we should observe the same time process in two different time scales, and Fig. 4 is supposed to depict a possible outcome: please do note that it is the same process that is observed, the difference between the two pictures being a dissimilarity in what commonly is called time scale (more precisely, a difference in the time unit that determines the frequency of observation).
Instead of going further into the notion of time scale let us now consider the concept of “difference in time scale”: the two pictures in Fig. 4 depict the same time process in different time scales (the scales are made up of different units). It is instinctive to say that the lower scale is larger than is the upper scale simply because the time unit of the former is larger than the one of the latter: a unit of the “b-scale” is three times the unit of the “a-scale”, when the two scales are considered in the same unit (e.g. minute). For instance, it is common practice to say that the daily time scale of man is much larger than is that of molecule.
The difference between the man universe and the molecule
universe has been used in a kind of self-evident fashion in statistical mechanics;
and this difference could be expressed as a difference in time unit. In the
context of ergodicity, an implication of this scale difference is that the
expression “ ” in the “world of atoms” has habitually been
looked upon as an instant for the human investigator: an instantaneous
macroscopic observation is interpreted as a time-average for the underlying
molecular events. Or otherwise
expressed: the “natural” time unit for molecule is immensely much smaller than
that for man.
Fig. 4 One and the same stochastic time-process observed in two different time scales: the scale of “a” being smaller than the scale of “b” (the latter has larger time unit). A thing made clear by the figure is the triviality that details of the process (e.g. the presence of inflexion points) disappear when the time scale is increased. From the Supplement it follows that at comparison of time processes such time-scale dependent properties cannot be used, and one reason is simply that there is no absolute time scale. That is, comparison can only be made in terms of quantities with time-scale independent significance.
A glance at Fig. 4 reveals a triviality, namely that increase in time scale implies that details in the time process disappear. To at all mention this kind of thing might appear as pettiness but, as it happens, there are a few practically significant consequences involved: the problem is to truly accept this triviality. For instance, assume that a small amount (i.e. a non-disturbing amount) of some substance is instantaneously injected into a man and into a rat; and assume further that the venous concentration of the substance is observed as a time-function in the two subjects. The outcome could quite possibly be something according to the lines depicted by Fig. 4, where (a) corresponds to man and (b) to rat. The difference in appearance is likely due to, in part, the simple fact that rat does things much faster than does man. In other words, although the observations are made in a time scales that to the investigator are the same for the two objects they are “virtually” made in a larger time scale for rat than for man. In other words, a comparison of the two curves is not without problems. The difficulty is much due to the simple fact that there is no absolute time scale.
Time-Scale Rule: If a description of a time process is to be used for comparison, the significance of the description must be time-scale independent.
For instance, a property such as existence-nonexistence of inflection point is obviously time-scale dependent whereas the maximum of the curves has a time-scale independent meaning.
To proceed from here is somewhat problematic and I must
appeal to the reader’s intuition and indulgence. Thus, let us first note that
it is an old tradition to use the two terms macroscopic and microscopic, where
the first term means that something is large enough for being observable
whereas the latter label means that this something is too small. Classic an
example would be that the pressure of one litre of a gas is a macroscopic
quantity whereas the underlying causethe
molecular collisions
is
a microscopic matter (well, this is a subject of experimental set up). If one
recalls what I have just stated above about observations and differences in
time scale it is no difficult distance to the next rule
MM Rule: Usually it is not possible to determine from macroscopic data the underlying microscopic structure.
where “mm” stands for microscopic-macroscopic.
Still in the earlier part of the last century it was
commonly believed that macroscopic data did reveal deductible properties about
the microscopic structure, but since the middle of that century the mm-rule has
been, as it seems, rather generally accepted among physicists. However, outside
the discipline of physics the rule has seemingly been less honoured, and an
example is the compartment analysis. Another example is the so-called “target
theory” that in biomedical work has been as popular as compartment thinking
during the later half of the last century. A simplified description of this
target theory would be that investigators have tried to estimate from
(macroscopic) dose-response data e.g. radioactivity dose vs. red-blood-cell
count
the
number of “hits” by particles (e.g. photons) a target (e.g. nerve synapses)
must be subjected to for observable response to occur (this is analogous to how
one has tried, in compartment analysis, to estimate from kinetic data the
number of compartments the subject consists of).
In summary, at comparison of time-series data there are
two things to be emphasized. The first one of these things is the importance of
time-scale independence; more precisely, the parameter’s physical significance
must be time scale independent. And the second matter can be expressed as a
rule of thumb: usually it is not possible to go (quantitatively or
qualitatively) from macro cosmos to micro cosmos; to contradict this rule has
occasionally been used for amusement purposes (I am thinking of Gamow’s classic
books on the adventures of Mr. even among physicists
that the “communication of model” is one-way,
namely from micro cosmos to macro cosmos (the mm rule); e.g. one can deduce
macroscopic consequences of microscopic models, but one cannot formally
deduce microscopic models from
macroscopic data.
Appendix 2 Homogeneity and rapid mixing
Hen and egg go necessarily together in spite of the two things being physically quite different, and similar holds for the two notions homogeneity and rapid mixing (Fig. 2). Although formal definitions of these two concepts can be given it suffices presently to note that the former means that the particles are uniformly distributed over the vessel’s volume whereas the latter indicates that, should some substance be added to a small volume element in the vessel, the homogeneity distribution (over the whole volume) is achieved instantaneously.
We may of course consider particle distribution over abstract spaces and the term volume might be replaced by something like, say, “fully connected domain”. But the very word “volume” is habitual and familiar to such an extent that, to me, there is no harm in using it as long as one keeps in mind what it actually stands for. However, personally I usually use the term of volume with its classic and more restricted geometrical significance, and use other terms for other size measures but, of course, I am not totally rigid in that regard.
The togetherness between the two notions of homogeneity and rapid mixing is so strong that often only one of the terms is used and the other being considered self-evident or just silently understood. Other terms for this “concept pair” appear in the literature, like efficient stirring or “the domain is well-stirred”, which are terms that actually denote a single set of two physically quite different notions (like hen and egg are different).
It is important that the vessel’s volume is quite modest (no more than 500ml, say) for a good “mixing” to occur. The reason is that with larger volumes there are often problems with the stirring procedure. It might be that the mixing process takes too long a time, simply because if the stirring is increased the energy input into the vessel will be too large and result in increased temperature of the water (Fig. 2). It is very much a matter of the ratio (surface area) / (volume), and it constitutes a general problem in chemical industry, namely the risk involved in, say, transforming a synthesis from laboratory scale to full industrial scale (where for instance the volumes of the containers are usually impressively large): insufficient mixing can result in decreased yield, with economic consequences that might be disastrous.
The “problem of absolute size” is rather common, and in a way well recognized though frequently ignored. A close-of-being-trivial example is that a mouse can fall to the hard ground from the third floor without much of a problem, whereas a human is likely to get seriously hurt by a similar fall. It is simply a matter of the ratio “surface area” (= air resistance) to “volume” (= weight): at increase in absolute size, volume increases faster than does surface area, which means that mouse has higher air resistance per unit weight than does man. Yes, quite a small and self-evident matter of some profound consequences (a classic example is that organisms must not be all that large in order to necessarily be multicellular).
Appendix 3 The epicycle language of Ptolemy
According to Ptolemy
When the amount of data increased (there were qualitative improvements as well) it became necessary gradually to superimpose epicycles upon epicycles and the model grew terribly complex (each heavenly body had its own main circle and set of epicycles, with its own set of angular velocities and radii). This was in direct violation of Rule 2 and it is amazing that the polyparametric system generated by the model could at all be handled, centuries before the computer. However, the model did fit data and, in spite of its huge complexity, it proved practically useful (e.g. in navigation at sea). Among the “great astronomers” Tycho Brahe seems to have been the last one to use (to believe in?) the epicycle model.
The persistence of the Ptolemy theory (several centuries)
was very much a result of the church’s activity (to some extent against
Ptolemy’s own thinking). Firstly, the church dictated
that the Earth is the centre of the world. Secondly, the church dictated that the heavenly bodies must move in circles:
the motions were divine and must therefore be perfect, and the perfect motion
is circular. Today’s multi-compartment models in, say, medicine and ecology
(cf. Jacquez, 1985) are a modern version of the epicycle model (Bergner,
1967b). In the compartment analysis it is dictated
that motions of a cohort of particles follow a decent number of coupled first
order ordinary and linear differential equations with constant coefficients
and, hence, the “metabolic” systems consist of a decent number of homogeneous
(well-stirred) subsystems so called compartments.
This compartment concept is an almost perfect analog to the notion of epicycle:
as with epicycles, successively more compartments have to be introduced in the
models (when the amount of data increases) with increasing complexity as a
consequence; moreover, like the notion of epicycle the compartment concept is a
purely mental construct
with
no physico-biologic counterpart. One might indeed ask oneself why this kind of
analysis has become so common when it can be easily shown that it does not work
(cf. Bergner, 1962: play it on the computer) and, moreover, this happens at a
time when medieval sciences are so downgraded for, as a common reasoning goes,
being non-critical (authoritarian) and conceptually close to magic. Well, our
today’s notions of probability and stochastic processes might in the future be
looked upon in a similar manner.
REFERENCES
Bergner, P-E. E. (1962): The Significance of Certain Tracer Kinetical Methods. Acta Radiol. Suppl.210
(1967b): Panel Discussion. In Compartments,
Pools and Spaces. Bergner and Lushbaugh (Eds.) CONF-661010. Clearing
House for Federal and Technical Information,
Jacquez, J.A. (1985): Compartment
Analysis in Biology and Medicine.