ne alkalnggubstituting (4) into (8)
g = p(2n@r") = 2apBroe™,
(9)
The long-term activity deposited in growing osteons
is given by
H = gf,
iyvhere ro = the radius of the resorption cavity or cement
and Segjine of the osteon. When the osteon canal has closed to
n system it- inal radius, ry , growth stops so that
pone‘at
:
dortion
ry = 790,
(10)
_ good ap where & = the time necessary for the formation of a
haversian system. Therefore,
1
4
i, = B loge (10/Ty).
(11)
‘time For values 79 = 100 microns, ry; = 10 microns, 8 =
e
0.0; day”, the time necessary for the formation of an
idult dog:
dav7! Soxtcon is t = 11 weeks, and the average apposition
ay
itt dr/dt, 18 1.2 microns/day.
N= nth ,
(12)
(5; where 2 = the numberof osteons which start forming
. per unit area per unit time. Then the fraction of the
he radi Pbone volume which turns over per unit time is \ (the
sume \ used above for the rate of cortical turnover),
sion (4)
where
d= nx(ra — v7).
(6)
(13)
J From these relations one can calculate the number of
abels. It forming osteons per unit area of bone for a given bone
(7)
turnover rate.
The distribution of tracer uptake in these forming
fosteons for a single injection then follows from the
nd 0.030
ectivelys
ult dogs!
iL osteoe
consideration that the uptake in a given osteon is
proportional to the rate at which calcium is being laid
down in that osteon [g from expression (9)]. Furthermore, if we assume that the turnover rate \ has been
y stagesy constant for at least the osteon formation time & , then
than i the number of osteons with a growth rate between g
ddition, and g + dg will be proportional to the length of time
lereforeg dt ~pent in this interval of growth rate by a single
dove oxtcon. Differentiating expression (9) with respect to
id down ' time,
radius
forming
of time® o}
» 2 -a8e
= —289 dt
dg = —2mpB'roe
dg
di = —~~.,
(8)5
(14)
15
289
(15)
Then the numberof osteons per unit area with growth
F tates between g and g + dg is
dN
ndé
n
dg
dg
289
——| =,
16
(16)
0
(17)
in an osteon with growth rate g and ¢ is the time after
injection (diminution of this activity produces a peak
value of H at a relatively short time after injection.
Furthermore, / reaches its final value for relatively
short times ¢. Therefore, the growth rate g does not
have time to change very much during the major part
of the tracer uptake. The value of ¢ used here is not
critical—it should be the order of a week.
Then the distribution of hotspot intensities due to
the process of osteon formation is given by
dN
n
dH 28H 2BHx(? — 73)’
N
OB
t
[ = / S dt
and where H is the activity per unit length deposited
r, respec’...
.;
.
0 ied {he numberof forming osteons per unit area of bone,
n dog?
,. is then
wth we
where
where
(18)
Hmax = Qmaxl = 2rpGrol
Aunin = Jminl = Qaperil.
The normalization of this distribution is correct be-
cause one may integrate (18) over the range from
Hain to Hax and obtain expression (12) above. This
distribution of hotspot intensities following a single
injection of tracer is hyperbolic in that it depends upon
H™. The basis for this calculation is well established
experimentally because both expression (4) and (17)
have been verified by direct experiments in dogs and
cats, the latter for both “Ca and “*Ra. However, it
would be most interesting to obtain direct verification
of expression (18) by measuring the dose distribution
from “Ca in a dog old enoughforan equilibrium popula-
tion of growing osteons to have been established.
Note that a convenient expression for the time integral of the plasma specific activity S after a single
injection is given by the excretion postulate:
r= [ Sat= (a/R),
where gq is the amount of activity injected,
(19)
nk is the excretory plasma clearance,
Tis the time integral of S from injection until
time ¢, and
R is the whole-body retention of the isotope at
the same time .
The curve of whole-body retention can, therefore, be
used to indicate howclosely J has approached its final
value for infinite time. (Note the similarity between
expressions (19) and (3), both derived from the exere-
tion postulate and well verified by experiment.)