64
DETAILED UPTAKE OF ACTIVITY IN FORMING HAVERSIAN
SYSTEMS
ance, 7k, values of which are available for the
earth elements in man and many animals.
Continuous or Multiple Injection
Single Injection
Several experiments have verified that the deposition
Tetracycline labeling in dogs and cats’
of alkaline earth activity in forming haversian systems
occurs at the specific activity of the blood plasma at the
time of that formation.
Therefore, a knowledge of the average speeifie activity of the plasma during the period of tracer intake
yields the specific activity of the new bone formed
during the same period. An observed ratio has been used
for this estimate heretofore, but the observed ratio
method does not take into account the dilution of the
blood plasma with unlabeled calcium transferring from
bone to blood for a considerable period of time after
the start of the continuous tracer intake. Furthermore,
the observed ratio method is not intimately connected
with the mechanisms of calcium transfer to and from
the blood plasma, since It is expressed as a product of
discrimination factors for urine, feces, and bone—
mechanisms which in fact are tn parallel, not in series.
A more powerful method for this particular calculation follows from the excretion postulate (basic postulate 1{d) above, and postulate I of Reference 5). If
you Know the shape of the retention curve R for a
radioisotope in the bodyfollowing a single injection,
:
.
together with the rate of excretory plasma clearance,
following a single injection, then you can derive that
for: plasma specific activity following the start of continuous tracer intake simply by taking the time integral
of the single injection curve. This procedure vields the
simple result that the specific activity of the plasma
under continuous tracer intake is
So
=
(G/nk) (1
—_ R;),
(3)
where
¢
= the rate of tracer introduction into the blood
nk = the rate of excretory plasma clearance in
grams of calcium per day (equivalent)
S,. = the plasma specific activity at any time {
after the start of the continuous tracer intake
ae
Ei
he
a
be
oF ee E
a
ees
eae
and the specifie activity of bone formed at this
time
R, = the whole-body retention of the tracer at the
same time ¢ after a single injection.
This useful result depends only upon the excretion
postulate and upon the assumption of a steady state
over a relatively short period of time. It is well verified
by experiment. This equilibrium value of S, for a given
g depends only upontherate of excretory plasmaclear-
(7-9)
a
labeling in a dog’” have shownthat haversian
in the process of their formation lay down b:
linear apposition rate that decreases in prop«
the current size of the haversian canal. To a :
proximation this maybe expressed
dr/dt = — Br,
where 7 is the radius of the canal as a function
and @ is the fractional rate of closure in unitsof(
Marshall found 8 = 0.03 + 0.01 day™ in anac
Lee found @ values of 0.04, 0.03, and 0.026 «
dogs of age 3 months, 1 year, and >1 year.
tively. Manson and Waters in experiments «
and cats found that the data on osteon grow
well represented by
‘y= kro ;
where 7; is the radius of the first label and 72 1s th
of the second label.
Expression (5) is consistent with expressi
because the solution of (4) yields
then the curve of the plasma specific activity versus
time is given by the time derivative of this function
nk. If you knowthe curve of plasma specific activity
:
To
=
reo,
where ty» is the time interval between the la:
follows that
k= eae,
Manson and Waters found 8 values of 0.044 an
dayin cats of age 9 months and 2.5 years, respe
and 8 values of 0.055 and 0.045 day~ in two adu!
These expressions imply that the individual
blast lays down bone more rapidly in the early
of osteon formation when the canal is large |
does in later stages as the canal closes. In ad
there is the fact that the canal surface and, th«
the number of osteoblasts decrease in direet pro]
to canal radius so that the mass of bone being lak
per unit of time decreases as the square of the
The amount of ealeium being laid down in a fi
haversian system per unit length as a function «
is then
_
d(Area)
ff,
a")
g = Pa p (zr
’
where
g = gramsof calcium per day per cm length 01
r
p
radius of canal as a function of time
*
3
grams calcium per em’ of newbone.