## Attenuation Correction

One of the most important data correction techniques for PET (and also SPECT) studies is the correction for attenuation. Although the basic principles of image reconstruction in emission computed tomography (PET and SPECT) are the same as transmission tomography (X-ray CT), there is a distinct difference in these two modalities on the data to be reconstructed. In X-ray CT, image reconstruction gives attenuation coefficient distribution of a known source while scattering is usually ignored. In PET (and SPECT), image reconstruction provides the number of photon emissions from unknown sources at unknown positions, and the photons have gone through attenuation by unknown matter (tissue) before they are externally detected. Therefore, attenuation correction factors must be estimated accurately to recover the original signals.

Attenuation occurs when high-energy photons emitted by the radiopharmaceutical in the patient are scattered and/or absorbed by the matter (tissue) between the detector and the emission site of the photon(s). The fraction of photon absorbed depends on a number of factors, including density and thickness of the intervening tissue, and photon energy. Typically, the attenuation coefficients (at 511 keV) for bone, soft tissue, and lungs are 0.151 cm-1, 0.095 cm-1, and 0.031 cm-1, respectively.

Mathematically, the fraction of photons that will pass through a matter with linear attenuation coefficient ¡x is:

where x is the thickness of the matter. If the matter is made up of different materials, then the total fraction of photons that passes through the matter would be the sum of the attenuation coefficients for each material multiplied by the thickness of the material that the photons pass through:

where ¡i is the attenuation coefficient of the ith material and xi is the thickness of the ith material that the photons pass through. Accordingly, if a detector measures Na counts per unit time from a source without attenuation (for example, in air, where the attenuation coefficient is close to zero), the attenuated counts, N, after placing a matter with varying linear attenuation coefficient in between, is:

where jx(x) is a distance-dependent attenuation coefficient function which

d d accounts for the varying attenuation within the matter, and d is the distance between the source and the detector (in cm). Therefore, in PET and SPECT, attenuation artifacts can cause a significant reduction in measured counts, particularly for deep structures. For example, attenuation artifacts can resemble hypoperfusion in the septal and inferior-posterior parts of the myocardium in cardiac PET or SPECT study. Failure to correct for attenuation can cause severe error in interpretation and quantitation. As the attenuation coefficient varies with different tissue types, the extent of photon attenuation/absorption will also vary even though the distance between the emission site of the photons and the detector remains unchanged. Therefore, spatial distribution of attenuation coefficients, i.e. an attenuation map, is required for each individual patient in order to correct for photon attenuation accurately.

Consider the attenuation in an object whose total thickness is D, measured along the LoR, and the attenuation coefficient is f, as shown in Fig. 2.7. If the annihilation event occurs at position x, measured along the LoR, then the probabilities for the two gamma rays to reach the opposing detectors are e-f(D-x) and e-fx, respectively The probability of registering the coincidence event is the product of the probabilities of detection of the gamma rays by the opposing detectors, i.e. e-f(D-x) ■ = e-fD, which is independent of the source position, x. This remains true when the attenuation coefficient is not uniform within the cross-section of the body. Thus, the attenuation is always the same even if the source position is outside the object.

The measured projection data will differ from the unattenuated projection data in the same fashion. Suppose /x(x, y) denotes the attenuation coefficient

Object Figure 2.7: Attenuation of the gamma rays in an object for a given line of response.

map of the object, the general equation for the attenuated projection data can be described by the attenuated Radon transform r ^ / r l(x,y) \

pm(r 0) = J f (x, y) exp i - J i(x', y~)ds\ dlr,e (2.18)

where pm(r, 0) is the measured projection data, l(x, y) is the distance from the detector to a point (x, y) in the object, while and r have the same definitions as in equations (2.6) and (2.7). It should be noted that unlike the unattenuated Radon transform as in equation (2.6), there is no analytical inversion formula available for equation (2.18).

The attenuation correction in PET is simpler and easier as compared to SPECT due to the difference in the photon detection schemes. In SPECT, the attenuation depends not only on the source position, but also on the total path length that the photon travels through the object. It is not straightforward to correct for attenuation or find an inversion of equation (2.18) for image reconstruction. On the contrary, the attenuation in PET is independent of the source position because both gamma rays must escape from the body for external detection and the LoR can be determined. Therefore, the exponential term in equation (2.18) can be separated from the outer integral. The unattenuated projection data and the measured projection data can then be related as follows:

where p(r, 0) is the unattenuated projection data, and pi(r, 0) = exp - ^ i(x, y) dlr^ (2.20)

is the projection data of the attenuation map. Therefore, if the attenuation coefficient map i(x, y) or its projection data p^(r, 0) is known, then the unattenuated projection data p(r, 0) of the object can be calculated as:

and f(x, y) can then be reconstructed without attenuation artifacts.

Since the attenuation is always the same regardless of the source position inside the FOV, it is possible to use an external (transmission) positron-emitting source that comprises a fixed ring or rotating rod sources, to measure the attenuation correction factors through two extra scans: blank scan and transmission scan. A blank scan is acquired with nothing inside the FOV, and a transmission

Blank scan Transmission scan

Blank scan Transmission scan rotating rod source

Figure 2.8: Attenuation correction in PET using a rotating rod source of 68Ge. Blank and transmission scans are generally acquired before tracer administration.

rotating rod source

Figure 2.8: Attenuation correction in PET using a rotating rod source of 68Ge. Blank and transmission scans are generally acquired before tracer administration.

scan is acquired to measure the coincidence rate when the patient being imaged is in the FOV but has not been given an injection of positron emitter. Figure 2.8 shows a schematic for measured attenuation correction using a rotating rod source of positron emitter 68Ge. Attenuation correction factors are then determined by calculating the pixelwise ratio of the measured projection data obtained from the blank scan and the transmission scan. The major drawback of this approach is that statistical noise in the transmission data would propagate into the emission images [46,47]. Therefore, transmission scans of sufficiently long duration have to be acquired to limit the effect of noise propagation. Depending on the radioactivity present in the external radiation source and on the dimension and composition of the body, transmission scans of 15-30 min are performed to minimize the propagation of noise into the emission data through attenuation correction, at the expense of patient throughput. Further, lengthened scan duration increases the likelihood of patient movement, which can cause significant artifacts in the attenuation factors for particular LoRs.

Application of analytical, so-called calculated attenuation correction eliminates the need for a transmission scan, thus making this method attractive in many clinical PET centers. This method assumes uniform skull thickness and constant attenuation in the brain and skull. However, such assumptions do not hold for sections that pass through sinuses and regions where the adjacent bone is much thicker. Alternatively, the transmission scan may be performed after tracer administration, referred to as postinjection transmission (PIT) scanning , which utilizes strong rotating rod (or point) sources for the transmission source. A small fraction of "transmission" coincidences contains in the sinogram data can be distinguished from emission coincidences that originate from the administered radiopharmaceuticals by knowing the positions of the orbiting sources. Another approach is to integrate measured and calculated attenuation that makes use of the advantages of each approach. A transmission scan is still required and the attenuation coefficient images derived from the transmission and blank scans are reconstructed and then segmented into a small number of tissue types, which are assigned with a priori known attenuation coefficients [49-51]. These processes greatly reduce noise propagation from the transmission data into the reconstructed emission images.