[Contents] [Section 1] [Section 2] [Section 3] [Section 4] [Section 5] [

**6.Corrections for quantitative
PET in 2D and 3D mode**

6.1 Introduction

6.2 Attenuation correction

6.3 Correction for random coincidences

6.4 Scatter correction

6.5 Detector normalization

6.6 Dead-time correction

PET offers the possibility of quantitative
measurements of tracer concentration *in vivo*. However, there
are several issues which must be addressed in order to realise this
potential. These issues are discussed in this section, and some of
the complicating factors associated with operation in 3D mode are
introduced.

In 2D PET, attenuation correction factors are
usually measured by illuminating the FOV with circular or rotating
rod sources with the subject in the field of view. Sources containing
quite large amounts of activity can be used to speed up the process,
and scatter can be minimised by a technique called "rod windowing",
whereby only LORs passing through the rod source are used for the
transmission measurement (Thompson
*et al* 1986). In certain cases it
may be possible dispense with the measurement by using a calculated
attenuation correction (Siegel
and Dahlbom 1992), or to improve it by
reconstructing the attenuation data and segmenting it into regions
with similar linear attenuation factors. This segmented image may
then be reprojected to obtain the attenuation correction factors for
each LOR (e.g. Xu
*et al* 1996).

With the septa retracted, the problem becomes more
complex. If highly active transmission sources are used without
septa, the detectors near the source will experience unacceptably
large levels of dead-time. The amount of scatter also rises
significantly, affecting the quantitative values of the attenuation
factors. For cameras with septa, the septa may be extended into the
FOV prior to the attenuation measurements being made. For septaless
cameras, this is not an option. Work is still proceeding on the issue
of 3D mode attenuation measurements. Promising avenues include the
method of "singles attenuation correction", where a collimated source
of photons of energy similar to annihilation photons is used in a
manner analogous to the acquisition of an X-ray CT scan
(Karp
*et al* 1995). Segmentation may be
used to reduce the errors due to scatter and the variation of linear
attenuation with photon energy.

**6.3 Correction for random
coincidences**

To obtain quantitative data in PET it is necessary to estimate and subtract the random coincidences from the measured data in each LOR to yield the sum of the true and scattered coincidences. As shown in section 2.5, the rate of random coincidences on a particular LOR is given by

* R*_{ij} =
2t *r*_{i}*
r*_{j} (14)

where *R*_{ij}
is the random coincidence rate on the LOR defined by channels i and
j, *r*_{i} is the singles rate on
channel i, *r*_{j} is the singles
rate on channel j and t is the coincidence resolving time. Therefore if *r*_{i} and
*r*_{j} can be measured and t is
known, *R*_{ij} can be calculated
for each line of response (e.g. Cooke
*et al* 1984). This method has the
advantage that in a particular acquisition the singles rates are
generally much higher than the coincidence rates, so that the
statistical quality of the estimate of
*R*_{ij} tends to be good. A more
commonly implemented method for estimating the randoms rate in a
particular LOR is the delayed coincidence channel method. Here timing
signals from one detector are delayed by a time significantly greater
that the coincidence resolving time of the circuitry. There will
therefore be no true coincidences in the delayed coincidence channel
(although it is possible for an event from one true coincidence to be
split from its partner and paired with an event from another), and
the number of coincidences found is a good estimate of the number of
random coincidences in the prompt signal. The estimate from the
delayed channel may be subtracted from the prompt signal on-line, or
stored as a separate sinogram for later processing. The advantage of
this method is that the delayed channel has identical dead-time
properties to the prompt channel. The disadvantage is that the
statistical quality of the randoms estimate is poorer, as
*R*_{ij} is a much smaller
quantity than *r*_{i }and
*r*_{j}. A method for improving
the noise characteristics of random coincidence estimates obtained in
this way was described by Casey
and Hoffman (1986), and characterised for
3D PET by Badawi
*et al* 1999b.

As stated in section
3.3, the sensitivity to scattered
coincidences is greater in 3D mode than in 2D mode. In 2D mode, many
workers ignore scatter altogether. However, in 3D mode the
amount of scatter in the signal can become extremely large
(Cherry
*et al* 1991, Badawi
*et al* 1996), and accurate scatter
correction methods are required. Many schemes have been proposed for
scatter correction in 3D mode. These include convolution-subtraction
techniques (e.g. Bailey
and Meikle 1994, Bentourkia
*et al* 1995), Monte-Carlo modelling
techniques (e.g. Levin
*et al* 1995), direct measurement
techniques (Cherry
*et al*, 1993) and multiple energy
window methods (e.g. Shao
*et al* 1994, Grootoonk
*et al* 1996). The methods in widest
use to date are the "Gaussian fit" technique (e.g. Stearns,
1995, Cherry
and Huang 1995), and model-based scatter
correction algorithms (Ollinger
1996, Watson
*et al* 1996).

The Gaussian fit method consists of fitting a Gaussian profile to the scatter tails found at the edge of each projection. This works well in brain scanning, where the activity and the scattering medium is fairly uniformly distributed and concentrated in the centre of the field of view, resulting in a simple slowly changing scatter distribution. It fails in the body, as the scatter tails available for fitting are very much shorter (because the body occupies a large portion of the field of view) and the scatter distribution contains more structure.

The model-based scatter correction algorithms use the attenuation map obtained from a transmission scan together with the emission data and a model of the scanner geometry and detector systems to calculate the percentage of photons falling on each detector, using the Klein-Nishina formula. The Klein-Nishina formula gives the differential scattering cross-section ds /dW as a function of scattering angle q as follows:

(1.15)

where *E* is the energy of the incident
photon,
*m*_{0}*c*^{2}
is the rest mass of the electron,
*r** _{0}* is the classical
electron radius and Z is the atomic number of the scattering atom.
Since the original emission data contains scatter, the correction
method must be applied iteratively. In the case where all the
activity is contained within the field of view, these methods are
highly accurate. However, where there is activity outside the field
of view these methods start to fail. In whole-body scanning, it may
sometimes be possible to obtain emission and attenuation data for
most of the regions contributing to scatter, thus improving the
accuracy of the scatter estimate, but in general this is not a
practical option.

Because of the limitations described, scatter correction in 3D PET remains an area of active research.

Fourier-based reconstruction techniques assume that all LORs have the same sensitivity. Unfortunately this is not the case for experimentally acquired data. For example, the sensitivity of a particular LOR is strongly affected by the angle that the LOR makes with the two detector faces at each end. This means that the sensitivity of the LOR relative to the mean is affected both by the geometry of the camera and the LOR position. Apart from such geometric effects, the block detectors themselves vary in efficiency, as the PMT gains are not all exactly the same (and may change with time), and the scintillation crystals are not all identical. The process of correcting for these effects is referred to as normalisation, and the individual correction factors for each LOR are referred to as normalisation coefficients (NCs).

The most straightforward way of obtaining a full
set of NCs is to perform a scan where every possible LOR is
illuminated by the same coincidence source. NCs are then proportional
to the inverse of the counts recorded in any given LOR. This
approach, known as* direct normalisation*, unfortunately has a
number of disadvantages. Scattered coincidences require a
different normalisation to trues (Ollinger
1995), and direct normalisation does not
yield these different factors. In 3D mode, small amounts of activity
must be used to reduce dead-time effects (Liow
and Strother 1995), and this means that
the amount of time required to obtain sufficient counts for
reasonable statistical accuracy in each LOR quite large (tens of
hours). Since NCs can change with time and should be measured as part
of routine quality control, this poses a significant practical
problem.

Both these problems can be overcome in
conventional PET cameras by using a component-based variance
reduction method (e.g. Hoffman
*et al* 1989). NCs are modelled as
the product of intrinsic crystal efficiencies and a small number of
geometric factors that account, for example, for the variation in
crystal efficiency with photon incidence angle. The NCs are not all
independent, as any given crystal efficiency is a factor for many
NCs, and, if the geometric factors are accurately known, the number
of unknowns is reduced from the number of LORs to the number of
crystals. There is a trade-off between systematic errors and
statistical accuracy that depends on the complexity of the model -
however, for an ECAT 951R operating in 3D mode there are about 1.25
million LORs and just 8192 detectors, so the potential for variance
reduction is very large (Badawi
*et al*1998).

While component-based normalisation is a promising
technique, it remains a developing field of study in 3D PET, and
several authors have reported the presence of residual artefacts in
images reconstructed from normalised acquisitions of uniform
cylindrical phantoms (e.g. Bailey
*et al* 1996, Oakes
*et al* 1998, Badawi
and Marsden, 1999a).

In both 2D and 3D mode, there will be losses due
to detector and system dead-time. To obtain quantitative results,
acquired data should be corrected for these losses. This is usually
done by modelling the dead-time losses as a combination of
paralyzable and non-paralyzable components and obtaining parameters
for the model by means of experiments involving repeated measurements
of a decaying source (e.g. Casey
*et al* 1995).

As discussed in section 5.6, a feature of block-detector systems is event mis-positioning at high count-rates due to pulse pile-up. In 2D mode, mis-positioning due to pulse pile-up has been shown to be unimportant except at very high activity concentrations (Germano and Hoffman, 1990). In 3D mode, pile-up can lead to high-frequency image artefacts and quantitative error if normalization measurements are carried out at significantly different count-rates to the emission measurements. A first-order correction scheme for this effect has been described by Badawi and Marsden (1999c).

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Revision date:12 Jan 1999