ed by the
ar. Com‘der of 14
n and soil
from the
; made by
chosen by maximizing the percentage variation
for which the model could account. The percentage variation accounted for is defined by
> (M,—ayXnO.9, — 03gn)”
V=1—"
;
=
[10]
' missing
‘monthly
where 4, isthe observed milk level in the n
month and Jf is the average observed milk level
for 7 months. The parameter values for the
optimal model are shown in table 3. Note that
the «’s differ from region to region, reflecting
the differing relative emphasis on the soil level
and deposition rate factors.
The percentage variations accounted for are
smaller in stations wherethereis less variability
in the milk levels. From inspecting the varia-
ie overall
. constant
*e plotted
example,
is missing
g overall
le. Then,
suries per
e for the
rates will
959, The
uld occur
rlons are
ted rates.
on those
er greatly
h they get
alf-times.
ation was
bility of the milk levels during the barn-fed
period when one would expect them to be essentially constant, we can see that one could hardly
hope to account for more than 50 percent of the
variation in some of the stations. The results
from the best model for each station are shown
graphically in figure 4, where solid and dotted
lines represent observed and expected (calcuJated) values respectively. The average predicted values are based on the assumption that
the deposition rates will be about what. they
were in 1959.
Discussion
Percent
Variation
wecounted
for
The sharp peak which
sometimes appears in October, November, or
December, and is not: fitted well by the theoreti-
cal line might be a result of using feed from an
>, (A4,—M)?
n=l
1 of the
sampling variation.
It is interesting to note that the estimates for
the parameters in areas of similar environmental conditions are close to one another. This
is particularly reflected in the pasture halfresidence time 71,2. The areas of low rainfall,
Sacramento and Salt Lake City, have a longer
residence time than New York City and Cincinati, probably because of a much slower
washout. Also, the «’s are very similar in the
similar areas.
Since the deposition rates contain a large element of uncertainty, some of the minor peaks
are missed occasionally. In periods of low dep-
osition, one would not expect to predict fluctua-
tions of minor magnitude. The predicted line
remains essentially constant during the months
of barn feeding so that the deviations of the observed values during this period measure the
Vol. 77, No. 12, December 1962
early spring harvest. This factor might well
account for the fall peak sometimes observed
and should be investigated. It seems reason-
able on the basis of figure 4, however, to assume
that the model is capable of predicting large
fluctuations if the correct deposition input is
known.
Obviously, we have considered quite a range
of deposition inputs. The future milk values
will probably be a little less than the predicted
line shown as “expected.” Estimates in May
1962 place the expected deposition from 0 to 20
percent less than in 1959. The milk levels of
the five stations were averaged to arrive at national average observed and expected milk
levels. Figure 5 showsthe observed milk levels
along with the levels calculated by Knapp and
the model presented here. He predicts an average of 15 micromicrocuries of strontium 90 per
liter for 1962, whereas our model predicts an
average of 11 micromicrocuries of strontium
90 per liter. However, if the fallout deposition
should be twice what is expected at every station, our model predicts an average value of
15 micromicrocuries per liter in the U.S. milk
supply for 1962. Then, on the basis of fallout
deposition predictions and transport assumptions, it becomes possible to predict milk levels
for any given future time period in segments
of the country where the required measurements
are available. This leads to a muchbetter esti-
mate of the average yearly bone marrow dose
in any region.
Summary
A mathematical model has been developed
for the purpose of predicting strontium 90 concentrations in milk. It is the successor of several other models designed to predict the activity of radionuclides in milk. A relatively
large yield of strontium 90 is deposited on the
surface of the earth as a result of nuclear weapons tests. This radionuclide then finds its way
through the food chain to human beings. Milk
is one of the most important contributors to the
total dietary intake.
1061