



INTRODUCTION AND OVERVIEW 
CONTENTS  
Photons in an xray beam and those emitted from a radioactive source are random
events with respect to both time and location. The random emission of
photons with respect to time makes it somewhat difficult to get a precise
measurement of radioactivity. The random distribution of photons
with respect to location within an image area is the major source of
visual noise in both xray and radionuclide imaging. This is because the
random distribution produces a
fluctuation in the number of photons emitted from one time interval to
another or from one area in an image to another.. If the photons from a
radioactive sample or in a small area of an image are counted for several
consecutive time intervals, the number of counts recorded will be
different for each, as illustrated below. This natural variation
with time introduces an error in the measurement of activity that is generally
referred to as the statistical counting error. The variation or random
distribution of photons from one area of an image to another is visible as
image noise.
In this chapter we first consider the nature of the random variation,
or fluctuation, in photons from a radioactive source, and then show how
this knowledge can be used to increase the precision of activity
measurements (counting) and the quality of nuclear images. The same
general statistics applies to photons in an xray beam.
We must first know something about the extent of the fluctuation that is the
source of error and image noise, and
if it is related to a factor over which we have control. One approach
is to imagine we are conducting an experiment for the purpose of
studying the random nature of photon emissions. Let us assume we have a
source of radiation in a scintillation well counter that is being
counted again and again for 1minute intervals. We quickly notice that
the number of counts, or photons, varies from one interval to another.
Some of the values we observe might be 87, 102, 118, 96, 124, 92, 108,
73, 115, 97, 105, and 82. Although these data show that there is a
fluctuation, they do not readily show the range of fluctuations. The
amount of fluctuation will become more apparent if we arrange our data
in the form of a graph, as shown in the figure below. In this graph, we
plotted the number of times we measured a specific number of counts
versus the actual number of counts observed. Obviously, we would have
to count the number of photons from our source many many times to
obtain the data to plot this type of graph.
Graph
Showing the Relative Number of Times Different Count Values (Number of
Counts) Are Obtained
When the data are presented in this manner, it is apparent that some
count values occurred more frequently than others. In our experiment we
observed 100 counts more frequently than any other value. Also, most of
the count values fell within the range of 70 to 130 counts. Within this
range, the number of times we observed specific count values is
distributed in the Gaussian, or normal, distribution pattern. (This is
actually a special type of Gaussian distribution known as the Poisson
distribution and is discussed later.)
At this point we need to raise a very significant question: Of
all of these values, which one is the "true" count rate that best
represents the activity of our sample? In our example, the average, or
mean, of all of the values is 100 counts. This is considered to be the
value that best represents the true activity of the sample. 

COUNTING ERROR 
CONTENTS 
In our experiment, we took one sample of radioactive material and
counted it many times to determine the fluctuation in the number of
counts. or photons, from the sample. However, in the clinical
laboratory we normally count a sample only one time. Any time we make a
single count on a source we are faced with a question: How close is our
measured count value to the true count value for that particular
sample? As illustrated below, the difference between these two values
is the error in our observed count value. At this point we have a
problem: Since we do not know what the true count value is, we have no
way of knowing what the error associated with a single measurement is.
The Amount of Error Is the Difference between the Measured and True Count Values



Error Ranges 

From our
earlier experiment, we know that the value of any individual count falls
within a certain range around the true count value. In our experiment, we
observed that all counts fell within 30 counts (plus or minus) of the true
value (100 counts). Based on this observation, we could predict the
maximum error that could occur when we make a single count. In our case.
the maximum error would be ± 30 counts ( ± 30%). We also observed that
very few count values approached the maximum error. In fact, a large
proportion of the count values are clustered relatively close to the true
value. In other words, the error associated with many individual counts is
obviously much less than the maximum error. To assume that the error in a
single count is always the maximum possible error is overstating the
problem. Although it is necessary to recognize that a certain maximum
error is possible, we must be more realistic in assigning values to the
error itself because it is usually much less than the maximum possible
error. If, at this
point, we were to go back to the data from our earlier experiment and
analyze it from the standpoint of how the individual count values are
clustered around the true value (mean), we would get results similar to
those illustrated in the following figure. For the purpose of this
analysis, we established three error ranges around the true value. In our
particular case, the error ranges are in tencount increments. The first
range is ± 10 counts (10% error), the second range is ± 20 counts (20%
error), and the largest range is ± 30 counts (30% error). At this point we
are interested in how often the value of a single measurement fell within
the various error ranges. Upon careful analysis of our data, we find that
68% of the time the count values are within the first error range ( ±
10%), 95% of all count values are within the next error range (± 20%), and
essentially all values (theoretically 99.7%) are within the largest error
range ( ± 30%).
With this information as background, let us now see what we can say about the error of an individual measurement. Note that in the case of a single measurement, there is no way to determine the actual error because the true value is unknown. Therefore, we must think in terms of the probability of being within certain error ranges. With this in mind, we can now make several statements concerning the error of an individual measurement in our earlier experiment:
While we are
still not able to predict what the actual error is, we can make a
statement as to the probability that the error is within certain
stated limits.
It might appear that the error ranges used above were chosen because
they were in simple increments of ten counts. Actually, they were
chosen because they represent "standard" error ranges used for values
distributed in a Gaussian manner. Error ranges can be expressed in
units, or increments, of standard deviations (s). In our example, one
standard deviation (s) is equivalent to ten counts. However, one
standard deviation is not always equivalent to ten counts.
The general situation is illustrated in the figure which follows the
table below. For values distributed in a Gaussian manner, the
relationship between the probability of a value falling within a
specific error range remains constant when the error range is expressed
in terms of standard deviations. For the general case, we have the
following relationship between error limits and the probability of a
value falling within the specific limits: Error
Limits
Probability
It might be helpful to draw an analogy between the error limits and a
bull'seye target, as shown below. The small bull'seye in the center
represents the true count value for a specific sample. If we make one
measurement, we can expect the count value to "hit" somewhere within
the overall target area. Although there is no way to predict where the
value of a single measurement will fall, we do know something about the
probability, or chance, of it falling within certain areas. For
example, there is a 68% chance that our count value will fall within
the smallest circle, which represents an error range of one standard
deviation. There is a 95% probability that the value will fall within
the next largest circle, which represents an error range of two
standard deviations. Essentially all of the values (99.7%) will fall
within the largest circle, which represents an error range of three
standard deviations.
Error Range
Confidence Level
A clear distinction between an error range and a confidence level is
necessary. Error range describes how far a single measurement value
might deviate, or miss, the true count value of a sample. Confidence
level expresses the probability, or chance, that a single measurement
will fall within a specific error range. Notice that as we increase the
size of our error range our confidence level also increases. In terms
of our target, this simply means we are more confident that our shot
will hit within a larger circle than within a smaller circle.
The relationship between confidence level and error range expressed in
standard deviations does not change for measurement values distributed
in a Gaussian manner. What does change, however, is the relationship
between an error range expressed in standard deviations and an error
range expressed in actual number of counts or percentages. In our
earlier example, one standard deviation was equal to ten counts, or
10%. We will now find that for other measurements one standard
deviation can be a different number of counts. Radiation events (such as count values), unlike the value of many other nonradiation variables, are distributed in a very special way. The value of the standard deviation, expressed in number of counts, is related to the actual number of photons counted during the measurement. Theoretically, the value of the standard deviation is the square root of the mean of a large number of measurements. In actual practice, we never know what the true count value of a sample is. In most cases, our measurement value will be sufficiently close to the true value so that we can use it to estimate the value of the standard deviation as follows:
______________

The error range, expressed as a percentage of the measured value,
decreases as the number of counts in an individual measurement is
increased. The real significance of this is that the precision of a
radiation measurement is determined by the actual number of counts
recorded during the measurement. The error limits for different count
values and levels of confidence are shown in Table 2.
We can use the information in Table 2 to plan a radiation measurement
that has a specific precision. For example, if we want our measurement
to be within a 2% error range at the 95% confidence level, it will be
necessary to record at least 10,000 counts. Most radiation counters can
be set to record counts either for a specific time interval or until a
specific number of counts are accumulated. In either case, the count
rate of the sample (relative activity) is determined by dividing the
number of counts recorded by the amount of time. Presetting the number
of counts and then measuring the time required for that number of
counts to accumulate allows the user to obtain a specific precision in
the measurement. 
Combined Errors when Subtracting and Adding 

In some applications, it is necessary to add or subtract counts in order to obtain a desired parameter. One example is the subtraction of background from a sample measurement. Another example is the subtraction of one xray image from another as in digitalsubtraction angiography. Here we will consider the
process of measuring radioactivity. The first step is to count the
sample. (Counts from background radiation are included in this
measurement.) The next step is to remove the radioactive sample from
the counting system, and then measure only the background radiation.
The background count rate is then subtracted from the
sampleplusbackground count rate to obtain a measurement of the
relative sample activity. The question now arises: What is the error in
the sample count rate that was obtained by subtracting one count value
from another?
Let us use the example shown below to investigate the error in the
difference between two count values. Assume that we have two radiation
sources. For the same counting time, one has a true value of 3,600
counts and the other a true value of 6,400 counts. The true difference
between the two is 2,800 counts. If we now measure the two samples, we
expect the measured values to fall somewhere within the error ranges
indicated in the figure below. With respect to the measured difference
we now have two errors to contend with, one for each of the sample
measurements. The question is now: What will be the error range for the
difference between the two measured values? In some instances, the
error in the measurement of one sample might be in a direction that
compensates for the error in the measurement of the other sample, and
the net error in the difference would be relatively small. There is
also the possibility that the two errors are in opposite directions, in
which case the error in the difference would be relatively large.
When making measurements on two samples, we have no way of knowing
either the amount or direction of the individual errors. Therefore, we
must consider the range of errors possible in the difference between
the two measured count values. Because of the possibility of errors
compounding (by being in opposite directions), it should be obvious
that the error range for the difference (s_{d}) will be larger than the error range associated with the individual measurement
(s_{1} and s_{2}). When the error ranges are expressed in terms of standard deviations, the relationship becomes
________ Let us now examine the actual values in the figure above and see what the error range will be for the difference between the two count values. The first sample measurement has a true value of 3,600 counts. By taking the square root of this number, we find that the standard deviation is 60 counts, or 1.67%. The second sample has a count total of 6,400 counts with a standard deviation of 80 counts, or 1.25%. If we now determine the standard deviation for the difference by using the relationship given above, we see that
__________ A standard deviation of 100 counts is 3.6% of the difference between the two count values, 2,800 counts.
Any time we add or subtract count values, the error range (standard
deviation) of the sum or difference will be larger than the error range
of the individual measurements. When two count values are added, the
standard deviation of the sum (s_{s}) is related to the standard deviation of the individual measurements
(s_{1} and s_{2}) by
________
Notice that this is the same relationship as that for the difference
between two count values. A common mistake is to assume that the sign
between the two standard deviation values is different for addition and
subtraction. It does not change; it is positive in both cases.
Let us now determine the error range of the sum of the two count values
in the above figure. As we have just seen, the standard deviation for
the sum is the same as the standard deviation for the difference. That
is, in this case, 100 counts, but since the sum of the two count values
is 10,000 counts this now represents an error range of only 1 %. This
is the same error range as we would find on a single measurement of
10,000 counts. When expressed as a percentage, the error range increases when we take the difference between two measurements, but it decreases when we add the results of the two measurements. When applied to imaging:
This is
discussed in much more detail in the section on Image Noise.



IMAGE NOISE 

The statistical nature and fluctuation of photons is the predominant source of visual noise in both xray and radionuclide imaging. The general concept is illustrated here.
It is the random distribution of photons as they interact with the image receptor (radiographic, fluoroscopic, gamma camera, etc) that produces the noise that is visible in images. The amount of noise is determined by the quantity or concentration of photons interacting with the receptor. For xray imaging this is strongly related to patient exposure. The noise can be decreased by changing several of the imaging parameters but that results in increased exposure to the patient. The relationship between image noise and patient exposure is one of the major factors that must be considered in the process of optimizing all forms of xray imaging, including CT. This is considered in detail in the Chapter on Image Noise. 


CLINICAL APPLICATIONS 

