isuriace, D, where
Rowland!
of ¢ for the’
hr. Kinetic:
D= do |
V dt
“6
wed that ¢
time of aly
= 0.5 g¢,/em> = the ealeium content per
unit volume of bone.
he alkalingbursan bone for the different alkaline earths, then
= (10° Bo./em” Yq/nk
df = re™dt.
(22)
rface poole)...
iat part off gin!
oft tissue,Faverage dose to bone in pCi-days/ghone
(25)
4, Then
he alkaling m:y be compared to the average dose to bone as a
se, Where whole calculated from the time integral of body specific
han Row, activity B, because the latter is also given by fo Bdt =
-term cal;
i
0
between t and ¢ + dé is
f= | redt = —e™“ |g = 1
hange. | f. the microcurie-days per cm’ of bone surface. This
(23)
is the total amount of bone in the region. .. The nor-
malization is correct.
5. If at adolescence all the bone is essentially new
bone, and if the time since adolescence is 7’, then the
age distribution of bone as a function of T is
df _ re’ + [e*? with age 7],
= (0.25 gcoa/Sbone)g/nk-
Rowland’s,
I
t
2. Then the amount of new bone formed in time dt
2 to associgfor "Cu in rabbits and dogs applies equally well to
y for
he time o
I
J
3. After a time ¢, e“ of this new bone will remain
unremodeled, so the fraction of the bone, df, with age
this point, "1herefore, if we assume that the 2000 A figure measured
26 dav f
!
[
is \ dt.
* maximum ec:aivalent depth of the pool
‘uptakeon;
to
to
I
| at
—'!
—
{
Lo
~
(21)
= (2x 107° em)(0.5 Zoa/em)q/ nk,
©
5
orrespond. where A = 2000A = 2 X 10-°cm = the measured
ts, suggests,
ry possibly,
{
tod
~
very well,‘hn order to convert expression (22) to the average dose
dt
(0<tsT)
(26)
ths in this to soft tissue within 40 u of the bone surface (applicable
onjecture? to alpha particles) :
where ¢ is bone age from 0 to 7.
6. Then
‘Ae in theSort tissue surface dose
s-/ Medt +e = —e™ |g +e = 1. (27)
veek after
ferentially’
°
2000 A is
-~
"Therefore,
T
_.
10 Bea/!em'(q/nk)
(24)
*, Normalization is again correct.
.
4
7. From the data given by Marshall® one can
.
express augmentation rates as a function of bone age:
, then wel Exc‘hangeable surface dose to soft tissue
sulate the
Average bone dose to hard tissue
exchange. .
Augs = at”,
=
0.25
(28)
where a = 30
0.00125
1ation bel
qg
8 = 0.70
Aug; isin %/year, and
= 0.005,
luction off
ould soon’ Which was obtained by dividing expression (24) by
ivities of expression (23). Therefore, this initial surface dose due
e pool ati to the rapidly exchanging calcium on bone surfacesis
rtion, the, Very small (0.5% for typical alpha particles) compared
tis the age of the bone in years.
8. Then the augmentation rate of the tissue as a
whole is found by summing the local augmentation
rates weighted by the fraction of bone in each age group:
3 and thet to the lifetime doses from long-lived isotopes to the
ection tof Skeleton as a whole. This intense short-term uptake on
bone surfaces may be of importance, however, in the
easc of “*Ra, which has a half-life of only 3.64 days
(20)2 and hence can never accumulate the long-term dose
to theskeleton.
n express
k AGE
x
'
z
e surface”
DISTRIBUTION
OF AUGMENTATION
RATES
pee ree
. Assume that a region of bone has a random
- of bone ae~orption-apposition rate of \ (time!)
9,
Augs
= f Augs(t) df.
for whole region
{29)
Augs = ar [ ie dt tale”.
(30)
10. From the calculations of the incomplete gamma
function by Rowland and Leuer,"” one can pick out
values for the integral in step 9. Table 23 gives the
values for the integral fre*"e™ dt and Table 24 the
values for T7°"e™*7.