217742a0Nature2175130196802247427430028-0836196810.1038/217742a0ukNatureNatureNATUREnatureNature is a weekly international journal publishing the finest peer-reviewed research in all fields of science and technology on the basis of its originality, importance, interdisciplinary interest, timeliness, accessibility, elegance and surprising conclusions. Nature also provides rapid, authoritative, insightful and arresting news and interpretation of topical and coming trends affecting science, scientists and the wider public./nature/journal/v217/n5130issueJournal homeArchiveCurrent issueAdvance online publicationPrivacy policySubscribeNature Publishing GroupCurrent issue217742a0Radial Diffusion and Power Function Retention of Alkaline Earth Radio-isotopes in Adult Bone
AU  - MARSHALL, JOHN H.
AU  - ONKELINX, CLAUDERadiological Physics Division, Argonne National Laboratory, Argonne, Illinois.THE retention of alkaline earth radioisotopes in skeletal systems can be approximated by the power function1 where b is a fraction between 0.0 and 0.5, t is the time after intravenous injection and [epsi] is a short time which for adult dog and man is the order of 1 day. A physiological explanation for this form of retention function is now presented which may be applicable to normal adult bone.Suppose that at times which are long compared with e the principal effect on tracer retention is the transfer of tracer ions between the blood and existing volumes of bone, the diffuse component2, rather than the formation or resorption of bone. This supposition seems to hold for adult dogs3'4 and rabbits5. The diffuse component has also been observed in man, but whether it has a dominant effect on retention has riot been established.
To reach the volume of existing bone the tracer must pass through canaliculi; longitudinal flow of tracer along canaliculi occurs in the liquid phase and so is relatively rapid. According to this new explanation, there is, then, a diffusion radially outwards from each canalicuhis; this is very slow because it takes place in the essentially solid, calcined matrix. Retention of tracer in this process is long because there is a large mass of mineral with which to exchange. Furthermore, retention is enhanced by the geometry of the cylinder, because once an ion has left the cylinder axis it is improbable that it will return by a random path. The retention problem is therefore similar to that for the diffusion of an instantaneous source of tracer from the axis of a cylinder.
Consider a long cylinder with a very small canal running along the cylinder axis. If a diffusible tracer is distributed uniformly along the canal at time zero, the amount of tracer remaining in the canal as a function of time can be calculated by solving the diffusion equation in cylindrical co-ordinates
BO ? _ n[ _ l cl Ht  La^ + rlrJ (2)
where C is the concentration of tracer as a function of time t at any radius r and D is the coefficient of diffusion6. For the initial conditions given here, the solution of equation (2) for the concentration of tracer at r=0 is
C = 4nDt (3)
where q' is the initial quantity of tracer per unit length deposited along the axis at t = 0 (ref. 6). The quantity of activity remaining in the canal depends exactly on the power function t?1. If the canal is assumed to have a nonzero diameter, C remains finite at time zero. The difference between t?1 and the functions t?1'2 to t?1'5 observed for blood specific activities1 at times that are long compared with e can be explained as the effect of excretion.
The effect of excretion on equation (2) can be taken into account by introducing a small drain on the axial canal after instantaneous labelling. This analysis was performed on the 'PACE' analogue computer using an eighteen compartment series model. The differential equations for the model reduced to equation (2) as the number of compartments was made large. Furthermore, when the drain was set equal to zero, the analogue computer yielded a t?1 dependence for axial activity as in expression (3). This provided a check on the model. When the drain was increased, a series of curves for axial activity against time was produced which could be well approximated by a set of power function expressions varying continuously from t?1.0 to t?1.5. We assume that the specific activities of blood and canalicular fluid are equal at times which are long compared with e. Because the specific activity of blood is proportional to the time derivative of retention1, the corresponding retention curve can be approximated by t?b with b between 0.0 and 0.5. When the drain was adjusted to a value consistent with the endogenous excretion rate for radium-226 in dogs, the model gave a good fit to the data of Van Dilla et al.7 over three decades of time as shown in Fig. 1.
Fig. 1. Data for radiuni-226 in the plasma (O) and in the body (?) of beagles injected at 1.3 years of age7. The curves (????) show the corresponding analogue computer results for the specific activity of canalicular fluid and of the body as a whole, considered as a large number of canaliculi together with their surrounding territories. C = 135 g calcium; 10 mg/100 ml.
Fig. 2. Kadial diffusion of an atom of calcium or other alkaline earth from a canaliculus into the surrounding territory of calcined matrix. The dimensions are from Baud's8 electron microscope measurements of mouse bone.
We picture this radial diffusion as involving a penetration of alkaline earth ions through a complex material consisting of collagen, ground substance, amorphous component and embedded bone crystals (Fig. 2). We have taken the diameter (0.15?.) and average spacing (l.7?) of canaliculi from electron microscope measurements of mouse bone8. The data for calcium-45, strontium-90 and radium-226 in adult dogs1 can be fitted with an effective coefficient of diffusion of the order of 10?16 cm2/s. At this rate, it would take almost a year for an ion to diffuse from a canaliculus to the limit of canalicular territory. The theory therefore represents a marked departure from the common hypothesis that the typical bone crystal is bathed by extracellular fluid which is in fairly rapid exchange with the blood.
It may be possible to test this theory of slow radial diffusion by carefully timed electron microscope autc-radiography.
This work was carried out under the auspices of the US Atomic Energy Commission.Marshall, , J. H., J. Theoret. Biol., 6, 386 (1964).ISIChemPortMarshall, , J. H., Rowland, , R. E., and Jowsey, , J., Radiat. Res., 10, 258 (1959).PubMedISIChemPortRowland, , R. E., Radiat. Res., 15, 126 (1961).PubMedISIChemPortJowsey, , J., Lafferty, , W., and Rabinowitz, , J., J. Bone and Joint Surgery, 47A, 359 (1965).ChemPortLloyd, , E., Calcified Tissues, 381 (University of Liege, 1965).Barrer, , R. M., Diffusion In and Through Solids, 32, 46 (Cambridge University Press, New York, 1951).Van Dilla, , M. A., Stover, , B. J., Floyd, , R. L., Atherton, , D. R., and Taysum, , D. H., Radiat. Res., 8, 417 (1958).PubMedChemPortBaud, , C. A., Acta Anat., 51, 209 (1962).PubMedISIChemPort