M. Tamal

High accuracy multiple scatter modelling for 3D whole body PET

A new technique for modelling multiple-order Compton scatter which uses the absolute probabilities relating the image space to the projection space in 3D whole body PET is presented. The details considered in this work give a valuable insight into the scatter problem, particularly for multiple scatter. Such modelling is advantageous for large attenuating media where scatter is a dominant component of the measured data, and where multiple scatter may dominate the total scatter depending on the energy threshold and object size. The model offers distinct features setting it apart from previous research: (1) specification of the scatter distribution for each voxel based on the transmission data, the physics of Compton scattering and the specification of a given PET system; (2) independence from the true activity distribution; (3) in principle no scaling or iterative process is required to find the distribution; (4) explicit multiple scatter modelling; (5) no scatter subtraction or addition to the forward model when included in the system matrix used with statistical image reconstruction methods; (6) adaptability to many different scatter compensation methods from simple and fast to more sophisticated and therefore slower methods; (7) accuracy equivalent to that of a Monte Carlo model. The scatter model has been validated using Monte Carlo simulation (SimSET).

Noise Properties of Four Strategies for Incorporation of Scatter and Attenuation Information in PET Reconstruction Using the EM-ML Algorithm

Conventional methods for dealing with attenuation and scatter in PET can limit the reconstructed image quality, particularly if the attenuating medium is large (as in whole body 3D PET). In such cases, often a substantial scatter subtraction is performed followed by amplification of the remaining data (to correct for attenuation) resulting in noisy reconstructions. More recent iterative reconstruction methods include the attenuation in the system model in conjunction with either pre-scatter subtraction or a separate addition of the scatter component after each application of the forward model. This work compares these more conventional approaches of including attenuation and scatter to the case where attenuation and scatter information are both included within the system matrix used by the expectation maximization maximum likelihood (EM-ML) algorithm. For this case all acquired data are used and regarded as a source of information by the reconstruction algorithm. Multiple realisations of simulated data sets have been used to compare the performance of the unified attenuation and scatter model with other methods. For a large attenuating medium and low counts there are notable differences between the four main ways of including attenuation and scatter within the reconstruction-with full pre-correction of the data being inferior compared to all the other methods, and the method which models scatter and attenuation within the system matrix showing some advantages. This work suggests that if regularisation of the EM algorithm is carried out by early termination of the iterative process, the subtraction method is the better approach among the techniques considered. In contrast, if a post-reconstruction smoothing approach to regularisation is to be used (whereby highly iterated, noisy image estimates are smoothed), the full modeling method for attenuation and scatter yields the better results, albeit at the computational cost of many more iterations being required

An advanced analytic method incorporating the geometrical properties of scanner and radiation emissions into the system model for the true component of 3D PET data

An approach to an analytic system model for the true component of 3D PET data offering the accuracy of a Monte Carlo model run for an infinite time is presented. The system model, which is used in the expectation maximisation (EM) algorithm, accounts for the 3D nature of the emission process in which the geometric sensitivity to a point source varies along and across the tube of each line of response (LOR). Therefore the system model cannot be based just on line or tube integrals as it is for the model of transmission scanning. By accounting for the varying sensitivity within each LOR in the reconstruction process unbiased and quantitatively exact images can be achieved without using calibration or scaling factors after the reconstruction process. Two approaches to the model are investigated, i.e., LOR- and voxel-driven which are validated using high statistics Monte Carlo simulation (SimSET).

Impact of Scatter Modeling Error on 3D Maximum Likelihood Reconstruction in PET

In statistical image reconstruction for PET, the reconstructed image quality depends on the system matrix as well as the scatter correction method used, especially for the case of a large attenuating medium where the measurement process is dominated by photon attenuation and scatter. Accurate system and scatter modeling can improve image quality, but whatever the method employed systematic and/or random errors will always exist in the system model, inevitably impacting final reconstructed image quality. Theoretical expressions have been derived to study the error propagation from the scatter response function to the reconstructed images for the case of maximum likelihood (ML) reconstruction. The effect of system and scatter modeling errors for three different scatter correction methods are considered: a) scatter subtraction, b) adding scatter as a constant term to the forward model and c) a unified model where the scatter is completely modeled within the system matrix itself. First order approximations are used to derive the theoretical expressions for the error propagation, which account for errors in both the system matrix and the scatter estimates (when used outside the system matrix). These expressions are validated using simulated data. A close agreement is found between the measured and theoretically derived error images, with the unified system model being least sensitive to the errors. The theoretical expressions are useful to determine the required accuracy for the system matrix and scatter estimation.

Noise properties of four strategies for incorporation of scatter and attenuation information in PET reconstruction

Conventional methods for dealing with attenuation and scatter can degrade the reconstructed image quality, particularly if the attenuating medium is large (as in whole body 3D PET). In such cases, a substantial scatter subtraction is performed followed by amplification of the remaining data (to correct for attenuation), which results in noisy reconstructions. More recent methods used with iterative reconstruction include the attenuation in the system model in conjunction with either pre-scatter subtraction or a separate addition of the scatter component after each application of the forward model. This work compares these more conventional approaches to including attenuation and scatter within the EM algorithm with a fully unified scatter and attenuation model - whereby all attenuation and scattering effects are included within the system matrix. For this case all acquired data are used and regarded as information by the reconstruction algorithm. This work indicates that for a large attenuating medium there are notable differences between the four ways of including attenuation and scatter within the reconstruction - fully pre-correction of the data is inferior compared to all the other methods. For the case of simple shift-invariant Gaussian model of scatter - subtraction, additive and unified model methods shows similar variance-bias characteristics. Multiple realisations of simulated data sets have been used to compare the performance of this model with other methods.

High Accuracy Multiple Order Scatter Model for 3D Whole Body PET

A technique for modelling multiple Compton scatter as the absolute probabilities which relate the image space to the projection space in 3D whole body PET system is presented. Such modelling is advantageous for large attenuating media where scatter is a dominant component of the measured data, and where multiple scatter may constitute up to about half of the total scatter depending on the energy settings and attenuating object. The model developed in this work goes beyond the limitations of previous methods in three distinct ways: (1) specification of the scatter distribution for every voxel based on calculation of actual probabilities, using the transmission data, the physics of Compton scattering and the specification of a given PET system (independence from a true activity estimate); (2) in principle no scaling or iterative process is required to find the scatter distribution; (3) explicit multiple scatter modelling; (4) easily adaptable to time-of-flight PET scatter estimation. The model can be included in a system matrix for statistical image reconstruction methods (avoiding the additive or subtractive approaches to scatter compensation), and of course the method can be adapted to a whole host of differing scatter compensation methods of varying speed and complexity. The proposed, high accuracy model has been validated using Monte Carlo simulation (SimSET).

Towards an accurate voxel-based analytic unified scatter and attenuation system model for 3D PET

A voxel-based approach to 3D PET analytic system modeling which projects the unit sphere of Klein-Nishina probabilities onto the cylindrical detection sphere, is proposed and under development. The aim is to offer the accuracy of a Monte Carlo model but without the noise and computational limitations. The model, incorporating both photon attenuation and scattering, defines PET system matrix elements based only on knowledge of a transmission scan and the specifications of a given PET system. No knowledge of the true activity is required to define the model. Reconstruction with such a unified analytic model makes full use of all the information contained in the measured data (both true and scatter events). No scatter subtraction, or scatter addition to the forward model output, are required. For 3D PET with large attenuating media, full use of the acquired information results in lower noise levels in the final image. The core of the technique models object scatter and attenuation via the Klein-Nishina cross-section and the transmission scan. However, the model must also include component-based normalization which accounts for scanner geometry as well as the crystal type and their arrangement.