RADIONUCLIDE IMAGING MII 3073

LECTURE 2

Nuclear Counting Statistics

and

Radionuclide Production

LESSON OUTCOMES

• At the end of lesson, student should be able to:-

• describe

Introduction

• All nuclear medicine procedures are based on radiation counting measurements.

• All measurements are subject to measurement error.

• This includes physical measurements, such as radiation counting measurements used in nuclear medicine procedures, as well as in biologic and clinical studies, such as evaluation of the effectiveness of an imaging technique.

Types of Measurement Error

• Measurement errors are of three general types: blunders, systematic errors, and random errors.

Blunders • Are errors that are adequately described by their name. • Usually they produce grossly inaccurate results and their

occurrence is easily detected. • Examples in radiation measurements include the use of

incorrect instrument settings, incorrect labeling of sample containers, and injecting the wrong radiopharmaceutical into the patient.

• There is no way to "analyze" errors of this type, only to avoid them by careful work.

Systematic Errors • It is caused by the use of mis-calibrated instruments,

results with either under/overestimate measurement.

• The term ‘accuracy’ is used to describe errors. Random Errors • caused by some uncontrollable factors. • The term ‘precision’ is used to describe errors. • Because random errors are always present in

radiation measurements, it is necessary to be able to analyze them and to obtain estimates of their magnitude.

• This is done using methods of statistical analysis

Random Errors in Radiation Counting Measurements

• The methods of analysis can be divided into:-

1. The Poisson Distribution

2. The Standard Deviation

3. The Gaussian Distribution

Poisson Distribution

• If a long-lived radioactive sample is counted repeatedly under properly counting system, there will be variant numbers of counts recorded. (i.e N1, N2, N3,…etc.)

• The “true value” of measurements will be questioned. To overcome this, it is possible to make a large number of measurements and use average (N) to estimate the “true value”.

• Unfortunately, multiple measurements are impractical in routine practice. One must usually be satisfied with taking only one measurement.

• So, how good is the result of a single measurement as an estimate of the "true value” ??

• The answer to this depends on the frequency distribution of the measurement results.

• This graph shows a typical frequency distribution curve for a series of radiation counting measurements.

• It is a graph showing the different possible results, (i.e., number of counts recorded) versus the probability of getting each result.

• The curve is peaked at a mean value m, which is the "true value" for the measurement.

• Thus, if a large number of measurements were made and their results averaged, one would obtain:-

• The probability of getting a certain result N when the true value is m is

• where e (= 2.718 . . .) is the base of natural logarithms and N! (N factorial) is the product of all integers up to N (i.e., 1 X 2 X 3 X . . . X N). From Graph Image it is apparent that the probability of getting the exact result N = m is rather small; however, one could hope that the result would at least be "close to" m.

• The probability that a measurement result will be "close to" m depends on the relative width, or dispersion, of the frequency distribution curve.

• This is related to a parameter called the variance, σ2, of the distribution.

• The variance is a number such that 68.3 percent (~2/3) of the measurement results fall within ±σ (i.e. square root of the variance) of the true value m. For the Poisson distribution, the variance is given by

• Thus one expects to find approximately 2/3 of the counting measurement results within the range ±√m of the true value m.

• Given only the result of a single measurement, N, one does not know the exact value of m or of σ; however, one can reasonably assume that N ≈ m, and thus that σ = √N.

• One can therefore say that if the result of the measurement is N, there is a 68.3 percent chance that the true value of the measurement m is within the range N ± √N.

• This is called the "68.3 percent confidence interval" for m; i.e., one is 68.3 percent confident that m is in the range N ± √N

Standard Deviation

• The variance σ2 is related to a statistical index called the standard deviation (SD).

• The standard deviation is a number that is calculated for a series of measurements.

• If n counting measurements are made, with results N1, N2, N3,. . ., Nn and a mean value N for those results is found, the standard deviation is

• The standard deviation is a measure of the dispersion of measurement results about the mean and is in fact an estimate of σ, the square root of the variance.

• For radiation counting measurements one should therefore obtain

• This can be used as a test to determine if the random error observed in series of counting measurements is consistent with that predicted from random variations in source decay rate or if there are conditional random errors present e.g. from faulty instrument performance.

Gaussian Distribution

• When the mean value m is "large" (m ≥ 20) the Poisson distribution can be approximated by the Gaussian distribution (Normal Distribution).

• The equation describing the Gaussian distribution is

where m and σ are again the mean and variance.

• It describes a symmetric "bell-shaped" curve similar to Pearson Graph.

• The Gaussian distribution with σ = m describes the results of radiation counting measurements when the only random error present is that due to random variations in source decay rate.

• When additional sources of random error are present--e.g., a random error or uncertainty of ΔN counts due to variations in sample preparation technique, counting system variations, etc.-the results are described by the Gaussian distribution with variance given by

Type and Energy of Emission

• The radiation given off should be sufficiently penetrating to allow it to be detected externally even though it may need to pass through several centimetres of tissue. This limits the choice to gamma rays or X-rays.

• The energy of the radiation will also affect its ability to penetrate tissue: the higher the energy the better it will be. However, the higher the energy the more difficult it will be to stop the gamma ray in the detector of the imaging device.

• In practice gamma rays with energies between 50 keV and 300 keV are preferred, about 150 keV being ideal.

Methods of Radionuclides Production

• Radionuclides can be produced from three sources: – nuclear reactor

– cyclotron

– generator

• Nuclear reactor consists of fuel/fissile rods that contain large atoms (typically Uranium-235, Uranium-238 or Plutonium-239) that inherently unstable.

• These atoms undergo nuclear fission where 2 or 3 neutrons and heat are emitted.

• These neutrons leave the nucleus with moderately high kinetic energy: fast neutrons.

• The neutrons are slowed with a moderator such as graphite, water or heavy water.

• However, these slow neutrons sometimes can still undergo nuclear fission and create a chain reaction.

• If this chain reaction is not controlled, it may end with an explosion. Therefore, cadmium control rods are inserted to absorb the neutrons in the reactor.

• Medical nuclides are made in reactors by two processes:

1. Nuclear fission

2. Neutron capture

Nuclear Reactor

A schematic diagram of nuclear reactor

Chain reaction of 235U atoms

Fission of a 235U nucleus

235U

Neutrons

Heat

Fission fragment

Fission fragment

235Uranium (neutron, fission) daughter nuclide eg:235U(n,f)131I and 235U(n,f)99Mo

Neutron Capture

• In neutron capture, the target atom captures a neutron.

• The new atom is radioactive and emits gamma photons or charged particles to produce the daughter nuclide.

• This reaction is written as: – Target (neutron, gamma)

daughter nuclide eg:98Mo(n,γ)99Mo