J. theor Biol. (1977) 65, 743-754
Theoretical Apects of the Distribution and Retention of Radionuclides in Biological Systems M. C. TIPORNE Medical Research Council, Radiobiology Unit, Hauwell, Didcot, Oxfordshire OX11 ORD, U.K. (Received 29 March 1976, and in revisedform
30 September 1976)
A general model of radionuclide metabolism in biological systems is developed. This model is used to investigate the effects of the internal dynamics of such systems upon the clearance of radionuclides from them. Both tracer and non-tracer amounts of radionuclides are considered.
1. Introduction General models for the behaviour of radionuclides in biological systems have been discussed by several authors (Stephenson, 1960; Bergner, 1964; Hart, 1966; Jacquez, 1972; Riihimaki, 1974). However, these authors have been primarily concerned with the transfer of the radionuclide between a finite number of compartments, e.g. Hart (1966), or its flux through a finite number of system or sub-system boundaries (Stephenson, 1960). In this paper an alternative model of radionuclide metabolism is developed which emphasizes the effects of the internal dynamics of a continuous system upon radionuclide retention. In this model the only assumption made is that the metabolic behaviour of a radioactive isotope of a particular element is indistinguishable from the metabolic behaviour of stabIe isotopes of that element. The model has been formulated with sufficient generality to be applicable not only to systems in which the radionuclide is introduced as a tracer but also to systems in which the radionuclide, or radionuclide plus carrier, represents a sizeable perturbation in the total amount of the element present in the system. Indeed, this extends to cases in which the introduced radionuclide constitutes all of the element present in the system. The formulation of such a general model, of necessity lacking in specific biological content, emphasizes some aspects of the analysis of radionuclide metabolism that are not always considered. There is, for example, a tendency to neglect the effects of the following. 743
744
M.
C.
THORNE
(i) Changes in the metabolic function of the system with time. This includes the effects of aging and the effects of entry into, and recovery from, abnormal metabolic states. (ii) Variation of level of intake of the tracee between otherwise identical systems. (iii) Variation of level of intake of the tracee with time in a particular system. In studies of biological systems where these factors have been taken into account they have often been shown to be of considerable significance. For example, Smiley, Dahl, Spraragen & Silver (1961) found that the half-time of retention of sodium in man varied from 5 to 335 days depending upon the level of stable sodium in the diet, and it has also been shown that stable iodine administered before, or soon after, a dose of radioactive iodine has a considerable effect on the uptake of the radionuclide by the thyroid (Ramsden, Passant, Peabody & Speight, 1967). Similarly, uptake from the gastrointestinal tract to blood is often much affected by the level of stable element present in the diet. This has, for example, proved to be the case for manganese (Howes, 1970) and cobalt (Smith, Edmonds & Barnaby, 1972). 2. The Generalized Model The generalized model developed in this paper is based on the conservation of the total number of atoms of the element under consideration, radioactive decay being regarded as loss from the system into an external “pool”. This conservation law leads to the equations (l)-(4) set out below, the various symbols used being defined in Table 1.
$ MI,
0 = $ dd&4h
42, M4z, 9--44z, 41, Mh
Ol+
+u#Q~ C-P(4,, Mh % dh,O ;
0.
= ! W~C~(h, 42, hd&, +-4h A, Oqd4i, Ol+ 0 +~RR(db0--l-~(419~>+4&R(41~0
QXO= i d4A4,v 0 - i dhJ'(+,> Oqd4,, 6 0
0
0
(2)
(3)
0
;i4:QdO = i d$J,(4,, O- 7 d4,[P(h O+h&dh
(1)
0.
(4)
These equations are equivalent to those for a non-linear multi-compartmental system (Jacquez, 1972) except that the compartmental indices have been
RADIONUCLIDES
IN BIOLOGICAL TABLE
SYSTEMS
745
1
Symbols used in equations Definition A distribution variable uniquely identifying each infinitesimal element of the system Time, defined from some arbitrary time origin The ouantitv of the radionuclide ner unit interval of d at time t The &anti~y of stable isotopes of the element per unit interval of 4 at time t The total quantity of stable isotopes of the element present in the system at time t The total quantity of the radionuclide present in the system at time r The rate constant for transfer of the radionuclide or stable isotope of the element from the region of the system designated by 4= to the region designated by il. Thus, a( II, dz, t) qs( 42, t) d d,d# 2 is the rate of flow of stable isotopes of the element from the region 42, 5 4< 4Z +d/, to the
region hI
4~ 4I+d4,
The rate of intake of stable isotopes of the element into the system per unit interval of 4 The rate of intake of radionuclide into the system per unit interval of 4 The rate constant for transfer of the radionuclide or stable isotopes of the element out of the system per uuit interval of 4 The radioactive decay constant of the radionuclide under consideration
replaced by the distribution variable 4. In this sense the above equations may be said to be the generalization of a non-linear multi-compartment system to an infinite number of infinitesimal compartments. It should be noted that the point 41 = C& must be excluded from the integrations in equations (l)-(4), since the transfer of material from a point in the system to itself has no physical interpretation. The exclusion of this point is most easily achieved by defining a(+,, &, t) to be identically equal to zero for +I = &. The only other constraint imposed upon the functions qA9,th qtt(A 0, 44, h, t), L(4, 0, 444, 0 and P(A t> is that they must all necessarily be non-negative for all values of t and 4. 3. Application to Simple Models (A> INTRODUCTIOI’4 OF RADIOACTIVE
TRACER
In this discussion the amount of radionuclide introduced into the system will be assumed to constitute an infinitesimal perturbation to the total amount of the element in the system. It will also be assumed that, prior to the introduction of the radionuclide, the system was in dynamic equilibrium. Then, because the perturbation to the system due to the introduction of the
746
M.
C.
THORME
radionuclide is infinitesimal, the dynamic equilibrium is not altered by its introduction and, at any time t;
of the stable isotope
And thus from equation (3):
This, of course, merely states that, in dynamic equilibrium, the rate of intake of stable isotopes is equal to their rate of loss from the system. Consider now the behaviour of the radionuchde. It will be assumed that this radionuclide is introduced continuously at a uniform rate into the system between a time To and a time Tl. The time interval (Tl - To) is assumed to be sufficiently large that by time Tl the radionuclide is uniformly mixed such that its distribution is identical to that of the stable isotopes. That such a uniform distribution is the most stable of the system may be deduced from thermodynamic arguments, or by time dependent perturbation theory, as discussed in the Appendix. It must be emphasized that uniform mixing is only necessarily approached in open systems under the administration conditions described above. Matthews (1971) has shown that following the instantaneous introduction of radioactive tracer into such a system uniform mixing may occur only transiently or not at all and similar conchrsions may be reached for other non-uniform regimes of administration. Uniform mixing implies that any aliquot taken from the system at a particular time will contain the radioactive isotope and the stable element in the same proportions. However, because of dilution due to the continuous intake of stable isotopes this proportion wih, in general, be a function of time. Formally this can be represented by saying that uniform mixing implies the relation : By integrating
d41, 0 = WMA, 0, equation (7) over 41 we obtain:
QdO = WQ,,
t>
t> Tl.
Ti.
(7) (8)
Since the quantity of stable isotopes (Qs) in the system must necessarily be constant if equation (5) is satisfied. Now, at time t > T,, no intakes of radionuclides from outside the system will be occurring, thus equation (4) may be rewritten : (9)
RADIONUCLIDES
IN
BIOLOGICAL
SYSTEMS
Equations (6), (7) and (8) may now be used in conjunction to yield:
; QR(O= - E d&1,(&, t)] $f
747
with this expression
- &QR(t).
00)
It will be noticed that, although the total amount of stable isotopes in the system has been constrained to be constant, the rate of intake of these isotopes may vary. Thus, if the total rate of intake of stable isotopes into the system at time t is $(t), equation (10) reduces to:
$ Qdt) = - 7 s Q&)-&Q&). Finally, if S,(t) is time independent reduces to : ’ dt
Qdt) =
we may write it S, and equation (11)
- [2 + &]
Q&h
This will be recognized as the differential formula for clearance from a single compartment of a model of the simple compartmental type, since the rate of loss of the radionuclide is proportional to the amount present and to the rate of dilution of the pool by stable isotopes of the element. It will be of interest to list the assumptions that have been made in deriving this result. (a) The total amount of the element in the system was assumed constant. (b) The rate of intake of the element into the system was assumed constant. (c) The radionuclide was assumed to have reached the stable equilibrium condition of uniform mixing. Although in many studies these assumptions will be reasonably well justified, in some they will necessarily be invalid. For example, an intake of one of the alkaline earth elements, calcium, strontium, barium or radium, takes many years to become uniformly distributed throughout mineral bone (ICRP Publication 20,1973) and thus equilibrium mixing will not be achieved in most studies with radioactive isotopes of these elements. Further, in shortterm studies neither the rate of intake of an element nor the total system content can generally be regarded as constant. If, for example, the system under consideration is man, or any other animal, then the pattern of intake will normally be regulated by the eating pattern of the individual. Thus, periods of finite intake will alternate with periods of zero intake and radionuclide metabolism may be influenced by the time of administration with respect to this pattern.
748
M. (B) INTRODUCTION
C.
THORNE
OF A NON-NORMAL
CONSTITUENT
OF THE SYSTEM
In this discussion it will be assumed that the radionuclide in use constitutes essentially all of the element present in the system. In this case only equations (2) and (4) are relevant. If S,(t) is defined as the total rate of intake into the system at time t, then equation (4) can be rewritten:
Now, if all parts of the system are homogeneous with respect to clearance, P(4, t) will be independent of $ and may be written P(t). Thus, the above expression reduces to : 04)
Further, if P(t) is made independent of time, this reduces to an equation describing clearance from a single simple compartment. However, very strong constraints have had to be placed on P(4, t) in this case and it is unlikely that such constraints will be generally appropriate in biological systems. Let us, for example, assume that there is a continuous range of binding sites available to the radionuclide in question and that these sites are filled in order of their binding strength. This is not an unrealistic assumption, since we are considering a radioisotope of an element that has no specific biochemical role to play. It will, therefore, tend to be retained at sites where it has some physico-chemical similarity to the normal occupant of those sites. However, unless it is a very close analogue of an element essential to metabolism it is likely to find a wide range of sites suited to it to different degrees. Further, if there is rapid exchange between these sites, or if the sphere of influence of the more strongly binding sites is larger than that of the weaker, then those sites which bind the radionuclide most strongly will tend to be occupied first. A simple model for this is to make P(4, t) decrease monotonically with increasing C$and to set qR(4, t) = 1 for 4 Q q&,,(t) and qR(+, t) = 0 for 9 > c&,,,,(t). Note that no loss of generality is involved in imposing these constraints on qR(cj, t) since if q’R(@, t) and P’(c$‘, t) are general functions of 4’ and t, it will always be possible to find a transformation 4 = T(@) such that qR(c$, t) satisfies the above constraints, i.e. it is always possible to map all the (b dependence into P(c$, t). Since the radionuclide is supposed to be the only isotope of the element present in the system, the above constraints on qR(4, t) allow us to write &(t) = &&t) and thus equation (13)
RADIONUCLIDES
IN
BIOLOGICAL
SYSTEMS
749
becomes :
or:
In order to make this example concrete, let us take P($, t) to have the form: (17) where a, b and m are arbitrary positive constants and a > b. This function is entirely arbitrary and has been chosen such that there is maximum binding [minimum P(4, t)] at 4 = 0 and minimum binding [maximum P(+, t)] at large values of (b. Substituting from equation (17) into equation (16) we obtain: (18)
If (b/a)Q,(t)‘”
< 1 for all t, this expression reduces to:
(1% If the radioactive decay of the radionuclide can be neglected compared with its rate of clearance from the system, this reduces to:
; QdO = S,(O - &
CQdW+ 'a
Assuming a quantity Q, of the radionuclide is introduced into the system at time t = 0, equation (20) reduces to:
$ Q&> = &
CQR(W+~ for t # 0
(20)
instantaneously
(21)
or: QR(T) dQ&) s Qo
=-1i&.
CQR(OI”+ ’ (m+ l>a o
(22)
Where QR(T) is the total quantity of the raclionuclide remaining in the system at time T. Equation (22) may be integrated to yield: (23)
750
M.
C.
THORNE
Which for T S- (m + l)a/mQ’;; reduces to :
This is a rather interesting result since it demonstrates that plausible assumptions about the distribution of bond strengths in a biological system lead to a retention function very different from the sum of exponentials which is often assumed. It will be readily apparent that other equally plausible assumptions could have been made and that few of the retention functions thus derived would have been adequately described by a few exponential components. Thus, for radionuclides introduced as non-normal constituents into biological systems, analyses based upon first-order kinetics must be regarded with some suspicion unless experimentally or theoretically justified for the system in question. 4. Discussion In this paper a general model of radionuclide distribution and retention has been presented and used to demonstrate some of the assumptions implicit in the use of multi-compartmental models in which intercompartmental exchange is governed by first-order kinetics. For systems into which a radioactive tracer had been introduced, three assumptions were found to be sufficient. These were that : (i) the total amount of the element in the system could be assumed constant ; (ii) the rate of intake of the element into the system could be assumed constant ; (iii) the radionuclide could be assumed to have reached the stable equilibrium condition of uniform mixing. As discussed in the text, none of these assumptions is necessarily valid in all circumstances. The total amount of the element in the system may change with time as is the case with calcium in man (Papworth & Vennart, 1973) or the rate of intake may exhibit seasonal variation as is the case with hydrogen (Morghissi, Carter & Bretthauer, 1972). The lack of equilibrium mixing may need to be taken account of in short-term blood flow studies (Brookes, 1971), in long-term models of alkaline earth metabolism (ICIZP, 1973) and in any experiment involving non-uniform administration of the radioactive tracer. Further, as is shown in the Appendix, the approach to uniform mixing is very dependent upon the internal dynamics of the system.
RADIONUCLIDES
IN BIOLOGICAL
SYSTElllS
751
Therefore, unless a considerable amount of knowledge is available concerning these internal dynamics, corrections for the lack of equilibrium mixing cannot be made with any confidence. In the case of systems in which the radionuclide constitutes the whole, or the greater part, of the element present much more restrictive assumptions have to be made. In such systems first-order kinetics cannot be assumed without some experimental justification. Such justification would be that the metabolic behaviour of the radionuclide did not change appreciably with increasing age of the system over the period of the study and that changes in the amount of the radionuclide introduced into the system did not affect its metabolic behaviour. Throughout this paper it has been assumed that, except for radioactive decay, stable and radioactive isotopes of an element are indistinguishable. However, this is not always the case since isotope mass may to some extent influence metabolism. This is particularly true in the case of isotopes of hydrogen (Feinendegen, 1967). In these circumstances it may be necessary to treat the radionuclide as a non-normal constituent of the system, its metabolism being modified by, but not entirely determined by, the metabolism of other isotopes of the element present in the system. Although a general model of radionuclide metabolism has been set out in the paper no attention has been paid to the metabolism of specific elements, in particular biological systems. This the author intends to rectify in future by publishing analyses based on the formalism discussed above. However, he hopes that this general formalism and the attendant discussion will, in itself, be of use to investigators concerned with the metabolism of radionuclides in many types of biological system. I would like to thank Dr J. Vennart and Dr R. H. Mole for valuable discussions during the preparation of this paper.
REFERENCES STEPHENSON, J. L. (1960).Bull. Math BERGNER, P. E. (1964).J. theor. Biol.
Biophys. 22, 113. 6, 137. HART,H. E. (1966).Bull. Math. Biophys. 28, 261. JACQUEZ, J. A. (1972). Compartmental Analysis in Biology and Medicine, Amsterdam:
ElsevierPublishingCompany. R~IHIMXKI, E. (1974).Dynamic Studies with Radioisotopes in Medicine, Vol. 1, p. 3, IAEASM-185/100. SMILEY, M. G., DAHL,L. K., SPRARAGEN, S.C. & SILVER, L. (1961).J. Lab. Clin. Med. S&60. RAWDEN,D., PASSANT, F. H., PEABODY, C. 0. & SPEIOHT, R. G. (1967).UealthPhys. 13, 633. How&s,A. D. (1969). Effect of Dietary Anomalies on Manganese Metabolism, Washington StateUniversity Thesis[NSA, 24: 25386(197011. SMITH,T., EDMONDS, C. J. & BARNABY, C. F. (1972).Health Pizys. 22, 359.
752
M.
C.
THORNE
MATTHEWS, C. M. E. (1971).In Radioisotopes in Medical Diagnosis (E. H. Belcher&
H. Vetter, eds).Ch. 11, pp. 236-257.London: Butterworths.
ICRP publication20 (1973).Alkaline Earth Metabolism in Adult Man, Oxford: Pergamon PRSS. FEINENDEGEN,L. E. (1967).Tritium LabelIedMoZecules in Biology and Medicine, New York:
AcademicPress. Mo~mssr,A. A., CARTER, M. W. &B RETIXAUJSR,E. W. (1972).Health Phys. 23, 805. BROOKES, M. (1971).The Blood Supply of Bone, London:Buttenvorths. PAPWORTH, D. G. & VENNART, J. (1973).Phys. Med. Biol. 18, 169.
APPENDIX A
Approach to Stable Equilibrium following Introduction Radioactive Tracer Following introduction
of a
section 3(A) to represent the proposed equilibrium of a radioactive tracer, it is possible to write:
To represent a perturbation this equation by:
following
et@, 0 = W&4, 0. (Al) at a particular value of 4, qb, say, we replace
cdh 0 = WE1 + AW(qb, - 4)lrlXA 0.
(A’4
Where 6(x) is a function of the arbitrary variable x defined by: 6(x) = 0
x#O
-Tm5(x) dx = 1.
And A (t) is the time dependent perturbation. into equation (2) in the main text yields:
Substituting
(A3) equation
(A2)
; CWW + A(OKh - ddlt~s~~,01 = r d4,C4bh W4OCl+AOM4, - 441 x “x ~(42, 0 --4h 4, OWD + AW(4, -~&IX&J, Ol- EfW, 0 + 12,lWLl+ A(W(4, - 4Md, 0.
(A4) Where it has been assumed that no intakes of radionuclide into the system from outside are occurring. This expression may be evaluated at C$= & :
$ CWEl +~(Ol~skb, 01
(A5)
RADIONUCLIDES
And at4
IN
BIOLOGICAL
SYSTEMS
753
# 41:
$ CWhddh Ql = ~dM4~,
42, OWCl +AtO&4, -h)l~s(~~, O-
- 49~ d~,OWh,tc4 91- CJY4,0 + W’4Oe(4, 0.
(-46)
Letting 4 approach & in equation (A6) and subtracting from equation (A5), i.e. assuming continuity in the various functions at the point 4 = +1 we obtain:
(A7)
754
M.
C.
THORNE
Thus, since
cannot be negative and since a(c),, (61, t) is defined to be identically zero, this expression implies that A(t) will always tend to zero with increasing time. This demonstrates that qR($, t) = k(t)q,(& t), for all 4, is a stable equilibrium state of the system. Note that this approach to stable equilibrium is governed by a general function of time which depends explicitly on the internal dynamics of the system. For this reason it is, in general, impossible to predict the time necessary to obtain essentially uniform mixing of the radionuclide throughout the system.
Theoretical Apects of the Distribution and Retention of Radionuclides in Biological Systems M. C. TIPORNE Medical Research Council, Radiobiology Unit, Hauwell, Didcot, Oxfordshire OX11 ORD, U.K. (Received 29 March 1976, and in revisedform
30 September 1976)
A general model of radionuclide metabolism in biological systems is developed. This model is used to investigate the effects of the internal dynamics of such systems upon the clearance of radionuclides from them. Both tracer and non-tracer amounts of radionuclides are considered.
1. Introduction General models for the behaviour of radionuclides in biological systems have been discussed by several authors (Stephenson, 1960; Bergner, 1964; Hart, 1966; Jacquez, 1972; Riihimaki, 1974). However, these authors have been primarily concerned with the transfer of the radionuclide between a finite number of compartments, e.g. Hart (1966), or its flux through a finite number of system or sub-system boundaries (Stephenson, 1960). In this paper an alternative model of radionuclide metabolism is developed which emphasizes the effects of the internal dynamics of a continuous system upon radionuclide retention. In this model the only assumption made is that the metabolic behaviour of a radioactive isotope of a particular element is indistinguishable from the metabolic behaviour of stabIe isotopes of that element. The model has been formulated with sufficient generality to be applicable not only to systems in which the radionuclide is introduced as a tracer but also to systems in which the radionuclide, or radionuclide plus carrier, represents a sizeable perturbation in the total amount of the element present in the system. Indeed, this extends to cases in which the introduced radionuclide constitutes all of the element present in the system. The formulation of such a general model, of necessity lacking in specific biological content, emphasizes some aspects of the analysis of radionuclide metabolism that are not always considered. There is, for example, a tendency to neglect the effects of the following. 743
744
M.
C.
THORNE
(i) Changes in the metabolic function of the system with time. This includes the effects of aging and the effects of entry into, and recovery from, abnormal metabolic states. (ii) Variation of level of intake of the tracee between otherwise identical systems. (iii) Variation of level of intake of the tracee with time in a particular system. In studies of biological systems where these factors have been taken into account they have often been shown to be of considerable significance. For example, Smiley, Dahl, Spraragen & Silver (1961) found that the half-time of retention of sodium in man varied from 5 to 335 days depending upon the level of stable sodium in the diet, and it has also been shown that stable iodine administered before, or soon after, a dose of radioactive iodine has a considerable effect on the uptake of the radionuclide by the thyroid (Ramsden, Passant, Peabody & Speight, 1967). Similarly, uptake from the gastrointestinal tract to blood is often much affected by the level of stable element present in the diet. This has, for example, proved to be the case for manganese (Howes, 1970) and cobalt (Smith, Edmonds & Barnaby, 1972). 2. The Generalized Model The generalized model developed in this paper is based on the conservation of the total number of atoms of the element under consideration, radioactive decay being regarded as loss from the system into an external “pool”. This conservation law leads to the equations (l)-(4) set out below, the various symbols used being defined in Table 1.
$ MI,
0 = $ dd&4h
42, M4z, 9--44z, 41, Mh
Ol+
+u#Q~ C-P(4,, Mh % dh,O ;
0.
= ! W~C~(h, 42, hd&, +-4h A, Oqd4i, Ol+ 0 +~RR(db0--l-~(419~>+4&R(41~0
QXO= i d4A4,v 0 - i dhJ'(+,> Oqd4,, 6 0
0
0
(2)
(3)
0
;i4:QdO = i d$J,(4,, O- 7 d4,[P(h O+h&dh
(1)
0.
(4)
These equations are equivalent to those for a non-linear multi-compartmental system (Jacquez, 1972) except that the compartmental indices have been
RADIONUCLIDES
IN BIOLOGICAL TABLE
SYSTEMS
745
1
Symbols used in equations Definition A distribution variable uniquely identifying each infinitesimal element of the system Time, defined from some arbitrary time origin The ouantitv of the radionuclide ner unit interval of d at time t The &anti~y of stable isotopes of the element per unit interval of 4 at time t The total quantity of stable isotopes of the element present in the system at time t The total quantity of the radionuclide present in the system at time r The rate constant for transfer of the radionuclide or stable isotope of the element from the region of the system designated by 4= to the region designated by il. Thus, a( II, dz, t) qs( 42, t) d d,d# 2 is the rate of flow of stable isotopes of the element from the region 42, 5 4< 4Z +d/, to the
region hI
4~ 4I+d4,
The rate of intake of stable isotopes of the element into the system per unit interval of 4 The rate of intake of radionuclide into the system per unit interval of 4 The rate constant for transfer of the radionuclide or stable isotopes of the element out of the system per uuit interval of 4 The radioactive decay constant of the radionuclide under consideration
replaced by the distribution variable 4. In this sense the above equations may be said to be the generalization of a non-linear multi-compartment system to an infinite number of infinitesimal compartments. It should be noted that the point 41 = C& must be excluded from the integrations in equations (l)-(4), since the transfer of material from a point in the system to itself has no physical interpretation. The exclusion of this point is most easily achieved by defining a(+,, &, t) to be identically equal to zero for +I = &. The only other constraint imposed upon the functions qA9,th qtt(A 0, 44, h, t), L(4, 0, 444, 0 and P(A t> is that they must all necessarily be non-negative for all values of t and 4. 3. Application to Simple Models (A> INTRODUCTIOI’4 OF RADIOACTIVE
TRACER
In this discussion the amount of radionuclide introduced into the system will be assumed to constitute an infinitesimal perturbation to the total amount of the element in the system. It will also be assumed that, prior to the introduction of the radionuclide, the system was in dynamic equilibrium. Then, because the perturbation to the system due to the introduction of the
746
M.
C.
THORME
radionuclide is infinitesimal, the dynamic equilibrium is not altered by its introduction and, at any time t;
of the stable isotope
And thus from equation (3):
This, of course, merely states that, in dynamic equilibrium, the rate of intake of stable isotopes is equal to their rate of loss from the system. Consider now the behaviour of the radionuchde. It will be assumed that this radionuclide is introduced continuously at a uniform rate into the system between a time To and a time Tl. The time interval (Tl - To) is assumed to be sufficiently large that by time Tl the radionuclide is uniformly mixed such that its distribution is identical to that of the stable isotopes. That such a uniform distribution is the most stable of the system may be deduced from thermodynamic arguments, or by time dependent perturbation theory, as discussed in the Appendix. It must be emphasized that uniform mixing is only necessarily approached in open systems under the administration conditions described above. Matthews (1971) has shown that following the instantaneous introduction of radioactive tracer into such a system uniform mixing may occur only transiently or not at all and similar conchrsions may be reached for other non-uniform regimes of administration. Uniform mixing implies that any aliquot taken from the system at a particular time will contain the radioactive isotope and the stable element in the same proportions. However, because of dilution due to the continuous intake of stable isotopes this proportion wih, in general, be a function of time. Formally this can be represented by saying that uniform mixing implies the relation : By integrating
d41, 0 = WMA, 0, equation (7) over 41 we obtain:
QdO = WQ,,
t>
t> Tl.
Ti.
(7) (8)
Since the quantity of stable isotopes (Qs) in the system must necessarily be constant if equation (5) is satisfied. Now, at time t > T,, no intakes of radionuclides from outside the system will be occurring, thus equation (4) may be rewritten : (9)
RADIONUCLIDES
IN
BIOLOGICAL
SYSTEMS
Equations (6), (7) and (8) may now be used in conjunction to yield:
; QR(O= - E d&1,(&, t)] $f
747
with this expression
- &QR(t).
00)
It will be noticed that, although the total amount of stable isotopes in the system has been constrained to be constant, the rate of intake of these isotopes may vary. Thus, if the total rate of intake of stable isotopes into the system at time t is $(t), equation (10) reduces to:
$ Qdt) = - 7 s Q&)-&Q&). Finally, if S,(t) is time independent reduces to : ’ dt
Qdt) =
we may write it S, and equation (11)
- [2 + &]
Q&h
This will be recognized as the differential formula for clearance from a single compartment of a model of the simple compartmental type, since the rate of loss of the radionuclide is proportional to the amount present and to the rate of dilution of the pool by stable isotopes of the element. It will be of interest to list the assumptions that have been made in deriving this result. (a) The total amount of the element in the system was assumed constant. (b) The rate of intake of the element into the system was assumed constant. (c) The radionuclide was assumed to have reached the stable equilibrium condition of uniform mixing. Although in many studies these assumptions will be reasonably well justified, in some they will necessarily be invalid. For example, an intake of one of the alkaline earth elements, calcium, strontium, barium or radium, takes many years to become uniformly distributed throughout mineral bone (ICRP Publication 20,1973) and thus equilibrium mixing will not be achieved in most studies with radioactive isotopes of these elements. Further, in shortterm studies neither the rate of intake of an element nor the total system content can generally be regarded as constant. If, for example, the system under consideration is man, or any other animal, then the pattern of intake will normally be regulated by the eating pattern of the individual. Thus, periods of finite intake will alternate with periods of zero intake and radionuclide metabolism may be influenced by the time of administration with respect to this pattern.
748
M. (B) INTRODUCTION
C.
THORNE
OF A NON-NORMAL
CONSTITUENT
OF THE SYSTEM
In this discussion it will be assumed that the radionuclide in use constitutes essentially all of the element present in the system. In this case only equations (2) and (4) are relevant. If S,(t) is defined as the total rate of intake into the system at time t, then equation (4) can be rewritten:
Now, if all parts of the system are homogeneous with respect to clearance, P(4, t) will be independent of $ and may be written P(t). Thus, the above expression reduces to : 04)
Further, if P(t) is made independent of time, this reduces to an equation describing clearance from a single simple compartment. However, very strong constraints have had to be placed on P(4, t) in this case and it is unlikely that such constraints will be generally appropriate in biological systems. Let us, for example, assume that there is a continuous range of binding sites available to the radionuclide in question and that these sites are filled in order of their binding strength. This is not an unrealistic assumption, since we are considering a radioisotope of an element that has no specific biochemical role to play. It will, therefore, tend to be retained at sites where it has some physico-chemical similarity to the normal occupant of those sites. However, unless it is a very close analogue of an element essential to metabolism it is likely to find a wide range of sites suited to it to different degrees. Further, if there is rapid exchange between these sites, or if the sphere of influence of the more strongly binding sites is larger than that of the weaker, then those sites which bind the radionuclide most strongly will tend to be occupied first. A simple model for this is to make P(4, t) decrease monotonically with increasing C$and to set qR(4, t) = 1 for 4 Q q&,,(t) and qR(+, t) = 0 for 9 > c&,,,,(t). Note that no loss of generality is involved in imposing these constraints on qR(cj, t) since if q’R(@, t) and P’(c$‘, t) are general functions of 4’ and t, it will always be possible to find a transformation 4 = T(@) such that qR(c$, t) satisfies the above constraints, i.e. it is always possible to map all the (b dependence into P(c$, t). Since the radionuclide is supposed to be the only isotope of the element present in the system, the above constraints on qR(4, t) allow us to write &(t) = &&t) and thus equation (13)
RADIONUCLIDES
IN
BIOLOGICAL
SYSTEMS
749
becomes :
or:
In order to make this example concrete, let us take P($, t) to have the form: (17) where a, b and m are arbitrary positive constants and a > b. This function is entirely arbitrary and has been chosen such that there is maximum binding [minimum P(4, t)] at 4 = 0 and minimum binding [maximum P(+, t)] at large values of (b. Substituting from equation (17) into equation (16) we obtain: (18)
If (b/a)Q,(t)‘”
< 1 for all t, this expression reduces to:
(1% If the radioactive decay of the radionuclide can be neglected compared with its rate of clearance from the system, this reduces to:
; QdO = S,(O - &
CQdW+ 'a
Assuming a quantity Q, of the radionuclide is introduced into the system at time t = 0, equation (20) reduces to:
$ Q&> = &
CQR(W+~ for t # 0
(20)
instantaneously
(21)
or: QR(T) dQ&) s Qo
=-1i&.
CQR(OI”+ ’ (m+ l>a o
(22)
Where QR(T) is the total quantity of the raclionuclide remaining in the system at time T. Equation (22) may be integrated to yield: (23)
750
M.
C.
THORNE
Which for T S- (m + l)a/mQ’;; reduces to :
This is a rather interesting result since it demonstrates that plausible assumptions about the distribution of bond strengths in a biological system lead to a retention function very different from the sum of exponentials which is often assumed. It will be readily apparent that other equally plausible assumptions could have been made and that few of the retention functions thus derived would have been adequately described by a few exponential components. Thus, for radionuclides introduced as non-normal constituents into biological systems, analyses based upon first-order kinetics must be regarded with some suspicion unless experimentally or theoretically justified for the system in question. 4. Discussion In this paper a general model of radionuclide distribution and retention has been presented and used to demonstrate some of the assumptions implicit in the use of multi-compartmental models in which intercompartmental exchange is governed by first-order kinetics. For systems into which a radioactive tracer had been introduced, three assumptions were found to be sufficient. These were that : (i) the total amount of the element in the system could be assumed constant ; (ii) the rate of intake of the element into the system could be assumed constant ; (iii) the radionuclide could be assumed to have reached the stable equilibrium condition of uniform mixing. As discussed in the text, none of these assumptions is necessarily valid in all circumstances. The total amount of the element in the system may change with time as is the case with calcium in man (Papworth & Vennart, 1973) or the rate of intake may exhibit seasonal variation as is the case with hydrogen (Morghissi, Carter & Bretthauer, 1972). The lack of equilibrium mixing may need to be taken account of in short-term blood flow studies (Brookes, 1971), in long-term models of alkaline earth metabolism (ICIZP, 1973) and in any experiment involving non-uniform administration of the radioactive tracer. Further, as is shown in the Appendix, the approach to uniform mixing is very dependent upon the internal dynamics of the system.
RADIONUCLIDES
IN BIOLOGICAL
SYSTElllS
751
Therefore, unless a considerable amount of knowledge is available concerning these internal dynamics, corrections for the lack of equilibrium mixing cannot be made with any confidence. In the case of systems in which the radionuclide constitutes the whole, or the greater part, of the element present much more restrictive assumptions have to be made. In such systems first-order kinetics cannot be assumed without some experimental justification. Such justification would be that the metabolic behaviour of the radionuclide did not change appreciably with increasing age of the system over the period of the study and that changes in the amount of the radionuclide introduced into the system did not affect its metabolic behaviour. Throughout this paper it has been assumed that, except for radioactive decay, stable and radioactive isotopes of an element are indistinguishable. However, this is not always the case since isotope mass may to some extent influence metabolism. This is particularly true in the case of isotopes of hydrogen (Feinendegen, 1967). In these circumstances it may be necessary to treat the radionuclide as a non-normal constituent of the system, its metabolism being modified by, but not entirely determined by, the metabolism of other isotopes of the element present in the system. Although a general model of radionuclide metabolism has been set out in the paper no attention has been paid to the metabolism of specific elements, in particular biological systems. This the author intends to rectify in future by publishing analyses based on the formalism discussed above. However, he hopes that this general formalism and the attendant discussion will, in itself, be of use to investigators concerned with the metabolism of radionuclides in many types of biological system. I would like to thank Dr J. Vennart and Dr R. H. Mole for valuable discussions during the preparation of this paper.
REFERENCES STEPHENSON, J. L. (1960).Bull. Math BERGNER, P. E. (1964).J. theor. Biol.
Biophys. 22, 113. 6, 137. HART,H. E. (1966).Bull. Math. Biophys. 28, 261. JACQUEZ, J. A. (1972). Compartmental Analysis in Biology and Medicine, Amsterdam:
ElsevierPublishingCompany. R~IHIMXKI, E. (1974).Dynamic Studies with Radioisotopes in Medicine, Vol. 1, p. 3, IAEASM-185/100. SMILEY, M. G., DAHL,L. K., SPRARAGEN, S.C. & SILVER, L. (1961).J. Lab. Clin. Med. S&60. RAWDEN,D., PASSANT, F. H., PEABODY, C. 0. & SPEIOHT, R. G. (1967).UealthPhys. 13, 633. How&s,A. D. (1969). Effect of Dietary Anomalies on Manganese Metabolism, Washington StateUniversity Thesis[NSA, 24: 25386(197011. SMITH,T., EDMONDS, C. J. & BARNABY, C. F. (1972).Health Pizys. 22, 359.
752
M.
C.
THORNE
MATTHEWS, C. M. E. (1971).In Radioisotopes in Medical Diagnosis (E. H. Belcher&
H. Vetter, eds).Ch. 11, pp. 236-257.London: Butterworths.
ICRP publication20 (1973).Alkaline Earth Metabolism in Adult Man, Oxford: Pergamon PRSS. FEINENDEGEN,L. E. (1967).Tritium LabelIedMoZecules in Biology and Medicine, New York:
AcademicPress. Mo~mssr,A. A., CARTER, M. W. &B RETIXAUJSR,E. W. (1972).Health Phys. 23, 805. BROOKES, M. (1971).The Blood Supply of Bone, London:Buttenvorths. PAPWORTH, D. G. & VENNART, J. (1973).Phys. Med. Biol. 18, 169.
APPENDIX A
Approach to Stable Equilibrium following Introduction Radioactive Tracer Following introduction
of a
section 3(A) to represent the proposed equilibrium of a radioactive tracer, it is possible to write:
To represent a perturbation this equation by:
following
et@, 0 = W&4, 0. (Al) at a particular value of 4, qb, say, we replace
cdh 0 = WE1 + AW(qb, - 4)lrlXA 0.
(A’4
Where 6(x) is a function of the arbitrary variable x defined by: 6(x) = 0
x#O
-Tm5(x) dx = 1.
And A (t) is the time dependent perturbation. into equation (2) in the main text yields:
Substituting
(A3) equation
(A2)
; CWW + A(OKh - ddlt~s~~,01 = r d4,C4bh W4OCl+AOM4, - 441 x “x ~(42, 0 --4h 4, OWD + AW(4, -~&IX&J, Ol- EfW, 0 + 12,lWLl+ A(W(4, - 4Md, 0.
(A4) Where it has been assumed that no intakes of radionuclide into the system from outside are occurring. This expression may be evaluated at C$= & :
$ CWEl +~(Ol~skb, 01
(A5)
RADIONUCLIDES
And at4
IN
BIOLOGICAL
SYSTEMS
753
# 41:
$ CWhddh Ql = ~dM4~,
42, OWCl +AtO&4, -h)l~s(~~, O-
- 49~ d~,OWh,tc4 91- CJY4,0 + W’4Oe(4, 0.
(-46)
Letting 4 approach & in equation (A6) and subtracting from equation (A5), i.e. assuming continuity in the various functions at the point 4 = +1 we obtain:
(A7)
754
M.
C.
THORNE
Thus, since
cannot be negative and since a(c),, (61, t) is defined to be identically zero, this expression implies that A(t) will always tend to zero with increasing time. This demonstrates that qR($, t) = k(t)q,(& t), for all 4, is a stable equilibrium state of the system. Note that this approach to stable equilibrium is governed by a general function of time which depends explicitly on the internal dynamics of the system. For this reason it is, in general, impossible to predict the time necessary to obtain essentially uniform mixing of the radionuclide throughout the system.
