An Expectation Maximization Method for Joint Estimation of Emission Activity Distribution and Photon Attenuation Map in PET

The SNR of time-of-flight positron emission tomography data for joint reconstruction of the activity and attenuation images

Objective.Measurement of the time-of-flight (TOF) difference of each coincident pair of photons increases the effective sensitivity of positron emission tomography (PET). Many authors have analyzed the benefit of TOF for quantification and hot spot detection in the reconstructed activity images. However, TOF not only improves the effective sensitivity, it also enables the joint reconstruction of the tracer concentration and attenuation images. This can be used to correct for errors in CT- or MR-derived attenuation maps, or to apply attenuation correction without the help of a second modality. This paper presents an analysis of the effect of TOF on the variance of the jointly reconstructed attenuation and (attenuation corrected) tracer concentration images.Approach.The analysis is performed for PET systems that have a distribution of possibly non-Gaussian TOF-kernels, and includes the conventional Gaussian TOF-kernel as a special case. Non-Gaussian TOF-kernels are often observed in novel detector designs, which make use of two (or more) different mechanisms to convert the incoming 511 keV photon to optical photons. The analytical result is validated with a simple 2D simulation.Main results.We show that if two different TOF-kernels are equivalent for image reconstruction with known attenuation, then they are also equivalent for joint reconstruction of the activity and the attenuation images. The variance increase in the activity, caused by also jointly reconstructing the attenuation image, vanishes when the TOF-resolution approaches perfection.Significance.These results are of interest for PET detector development and for the development of stand-alone PET systems.

Algorithms for joint activity-attenuation estimation from positron emission tomography scatter

Background

Attenuation correction in positron emission tomography remains challenging in the absence of measured transmission data. Scattered emission data may contribute missing information, but quantitative scatter-to-attenuation (S2A) reconstruction needs to input the reconstructed activity image. Here, we study S2A reconstruction as a building block for joint estimation of activity and attenuation.

Methods

We study two S2A reconstruction algorithms, maximum-likelihood expectation maximization (MLEM) with one-step-late attenuation (MLEM-OSL) and a maximum-likelihood gradient ascent (MLGA). We study theoretical properties of these algorithms with a focus on convergence and convergence speed and compare convergence speeds and the impact of object size in simulations using different spatial scale factors. Then, we propose joint estimation of activity and attenuation from scattered and nonscattered (true) emission data, combining MLEM-OSL or MLGA with scatter-MLEM as well as trues-MLEM and the maximum-likelihood transmission (MLTR) algorithm.

Results

Shortcomings of MLEM-OSL inhibit convergence to the true solution with high attenuation; these shortcomings are related to the linearization of a nonlinear measurement equation and can be linked to a new numerical criterion allowing geometrical interpretations in terms of low and high attenuation. Comparisons using simulated data confirm that while MLGA converges largely independent of the attenuation scale, MLEM-OSL converges if low-attenuation data dominate, but not with high attenuation. Convergence of MLEM-OSL can be improved by isolating data satisfying the aforementioned low-attenuation criterion. In joint estimation of activity and attenuation, scattered data helps avoid local minima that nonscattered data alone cannot. Combining MLEM-OSL with trues-MLEM may be sufficient for low-attenuation objects, while MLGA, scatter-MLEM, and MLTR may additionally be needed with higher attenuation.

Conclusions

The performance of S2A algorithms depends on spatial scales. MLGA provides lower computational complexity and convergence in more diverse setups than MLEM-OSL. Finally, scattered data may provide additional information to joint estimation of activity and attenuation through S2A reconstruction.

Technical Note: Emission expectation maximization look-alike algorithms for x-ray CT and other applications

PURPOSE

In emission tomography, the expectation maximization (EM) algorithm is easy to use with only one parameter to adjust - the number of iterations. On the other hand, the EM algorithms for transmission tomography are not so user-friendly and have many problems. This paper develops a new transmission algorithm similar to the emission EM algorithm.

METHODS

This paper develops a family of emission-EM-look-alike algorithms by expressing the emission EM algorithm in the additive form and changing the weighting factor. One of the family members can be applied to transmission tomography such as the x-ray computed tomography (CT).

RESULTS

Computer simulations are performed and compared with a similar algorithm by a different group using the transmission CT noise model. Our algorithm has the same convergence rate as theirs, and our algorithm provides better contrast-to-noise ratio for lesion detection.

CONCLUSIONS

For any noise variance function, an emission-EM-look-alike algorithm can be derived. This algorithm preserves many properties of the emission EM algorithm such as multiplicative update, non-negativity, faster convergence rate for the bright objects, and ease of implementation.

Application of the 4-D XCAT Phantoms in Biomedical Imaging and Beyond

The four-dimensional (4-D) eXtended CArdiac-Torso (XCAT) series of phantoms was developed to provide accurate computerized models of the human anatomy and physiology. The XCAT series encompasses a vast population of phantoms of varying ages from newborn to adult, each including parameterized models for the cardiac and respiratory motions. With great flexibility in the XCAT's design, any number of body sizes, different anatomies, cardiac or respiratory motions or patterns, patient positions and orientations, and spatial resolutions can be simulated. As such, the XCAT phantoms are gaining a wide use in biomedical imaging research. There they can provide a virtual patient base from which to quantitatively evaluate and improve imaging instrumentation, data acquisition, techniques, and image reconstruction and processing methods which can lead to improved image quality and more accurate clinical diagnoses. The phantoms have also found great use in radiation dosimetry, radiation therapy, medical device design, and even the security and defense industry. This review paper highlights some specific areas in which the XCAT phantoms have found use within biomedical imaging and other fields. From these examples, we illustrate the increasingly important role that computerized phantoms and computer simulation are playing in the research community.

Joint estimation of activity and attenuation for PET using pragmatic MR-based prior: application to clinical TOF PET/MR whole-body data for FDG and non-FDG tracers

Accurate and robust attenuation correction remains challenging in hybrid PET/MR particularly for torsos because it is difficult to segment bones, lungs and internal air in MR images. Additionally, MR suffers from susceptibility artifacts when a metallic implant is present. Recently, joint estimation (JE) of activity and attenuation based on PET data, also known as maximum likelihood reconstruction of activity and attenuation, has gained considerable interest because of (1) its promise to address the challenges in MR-based attenuation correction (MRAC), and (2) recent advances in time-of-flight (TOF) technology, which is known to be the key to the success of JE. In this paper, we implement a JE algorithm using an MR-based prior and evaluate the algorithm using whole-body PET/MR patient data, for both FDG and non-FDG tracers, acquired from GE SIGNA PET/MR scanners with TOF capability. The weight of the MR-based prior is spatially modulated, based on MR signal strength, to control the balance between MRAC and JE. Large prior weights are used in strong MR signal regions such as soft tissue and fat (i.e. MR tissue classification with a high degree of certainty) and small weights are used in low MR signal regions (i.e. MR tissue classification with a low degree of certainty). The MR-based prior is pragmatic in the sense that it is convex and does not require training or population statistics while exploiting synergies between MRAC and JE. We demonstrate the JE algorithm has the potential to improve the robustness and accuracy of MRAC by recovering the attenuation of metallic implants, internal air and some bones and by better delineating lung boundaries, not only for FDG but also for more specific non-FDG tracers such as 68Ga-DOTATOC and 18F-Fluoride.