ee ee te HES ee te oN. 61 i ae f ndependently verified by observing the overlapping of | ayversian systems in bone of known age and remodeling a id for thnttion of bone generations shown in [igure 51. (See ng i the valuefeection on the generation distribution of remodeli LL ute. A truly random process would produce a distribu- semuewelin # bone in’ 4 TT TTT TT Toa TTT | q 4 4 — Lil (1) It follows from postulates 1(b), 1(c); and 1(d) hat the area under the curveof the specific activity of some ex the body between the time of injection and infinite svstem vbtime|is equal to the area under the curve of specific 7 mentation (b) This area is simply q/nk, where q is the activitye at bone. cae and nk is the rate of excretory plasma clearanc n the last xpressed as grams of calcium per day(liters per day . mentationtimes the calcium content per liter). go. Ther ToT 2 » Derived Relationships onelucive activity of the blood plasma versus time. ; 0 L on rate ibne.) dent Por s JF > [_ 5 4 = - > r~ uJ = [— L. - a _ 4 o Lo oO z o 4 4 -j L re =o i Ol fe 4 I = [7 4 (c) This requirement of area together with the early jon ¢ateftetention. curve determines the time constant of thefinal onds uporbesponential which is characteristic of age-invariant at region fsystems. If the system is age-invariant, then a tracer "following a single injection must eventually reach tranmds upon sient. equilibrium: a state in which the tracer concentra- nentatioWein differ in different compartments, but all decrease erimentee proportion to the same exponential function of ed by the!itime.' 4. Vew Associations (1), (a) Observation and theory show that the removal of tracer activity from bone by diminution (diffusion) ~ d 5 days ean be represented quite closely by the power function. pposition’ At short and intermediate times after tracer intake, given in power function retention in bone existingat the timeof in years.pintake can result from diffusion of tracer in cylindrical dl year,’ geometry around canaliculi.“ In bone formed shortly tion. For; aftcr intake and in existing bone at long times after ~alizationt intake, power function retention can result from the nly the’ decrease of the coefficient of diffusion with local bone ne. This, @xe. > rate off (b) On the other hand, in an age-invariant system -diminu-f the removal of tracer activity by resorption of bone bone. Inf could not produce power function or multi-exponential pears tof retention unless there were an extremely wide distribu- nsfer be-g tion of the turnover times of bonein different parts of J the skeleton. Bone turnover times that differed by four Porfive orders of magnitude in different locations would be required to produce the sort of power function or multi-exponential function that is observed for skeletal of boneg Tetention, The ratio between the fastest and the slowest re of thet turnover by apposition-resorption is probably not ‘ould bef ®reuter than four or six. Therefore, the effect. of resorp- tion -hould be associated only with the final one or found a poxsibly two exponential terms. The earlier terms (or the power funetion part of retention) should be associ- Oo, +L 0 | 05 ! Lo } plo Et 1.5 2 25 TIME AFTER ADOLESCENCE — AT Fia. 51—The distribution of generations within a region of bone that has been remodeling at the rate A in locations governed entirely by chance (osteons) for a period of time T since the original formation of the skeleton (assumed to occur at the age of adolescence). Figures from Table 28. These predictions, if verified, could be used to verify the applicability of the assumption of randomnessin local regions of remodeling. ated with diminution. In viewof the little data on this point, we assume that the diminution of activity from hotspots and from the diffuse component is the same so that the specific activities of individual hotspots and the diffuse component decrease in parallel with the power function part of the whole-body retention curve. 4, Bone Model (a) Let us assume that there are twoclasses of bone, cortical bone and trabecular bone, each with its own rate of turnover by apposition-resorption. Let the rate of cortical turnover be (in units of time~'). Then let the rate of trabecular turnover be oA. ¢ probablyis of the order of 4 to 6. Let the fraction of the skeleton that is trabecular be 7, and the fraction that is cortical be 1 — 7 (fraction by calcium content). + is probably about 0.2. (b) Then it is easy to show that when an injected tracer has finally equilibrated within this skeletal system, practically no activity remains in the trabeculae. The skeletal activity is almost wholly in the slowlyturning-over cortex so that it is the turnover time of dad