Hongbin Guo

Clustering huge data sets for parametric PET imaging.

A new preprocessing clustering technique for quantification of kinetic PET data is presented. A two-stage clustering process, which combines a precluster and a classic hierarchical cluster analysis, provides data which are clustered according to a distance measure between time activity curves (TACs). The resulting clustered mean TACs can be used directly for estimation of kinetic parameters at the cluster level, or to span a vector space that is used for subsequent estimation of voxel level kinetics. The introduction of preclustering significantly reduces the overall time for clustering of multiframe kinetic data. The efficiency and superiority of the preclustering scheme combined with thresholding is validated by comparison of the results for clustering both with and without preclustering for FDG-PET brain data of 13 healthy subjects.

Characterization of the image-derived carotid artery input function using independent component analysis for the quantitation of [18F] fluorodeoxyglucose positron emission tomography images*

We previously developed a noninvasive technique for the quantification of fluorodeoxyglucose (FDG) positron emission tomography (PET) images using an image-derived input function obtained from a manually drawn carotid artery region. Here, we investigate the use of independent component analysis (ICA) for more objective identification of the carotid artery and surrounding tissue regions. Using FDG PET data from 22 subjects, ICA was applied to an easily defined cubical region including the carotid artery and neighboring tissue. Carotid artery and tissue time activity curves and three venous samples were used to generate spillover and partial volume-corrected input functions and to calculate the parametric images of the cerebral metabolic rate for glucose (CMRgl). Different from a blood-sampling-free ICA approach, the results from our ICA approach are numerically well matched to those based on the arterial blood sampled input function. In fact, the ICA-derived input functions and CMRgl measurements were not only highly correlated (correlation coefficients >0.99) to, but also highly comparable (regression slopes between 0.92 and 1.09), with those generated using arterial blood sampling. Moreover, the reliability of the ICA-derived input function remained high despite variations in the location and size of the cubical region. The ICA procedure makes it possible to quantify FDG PET images in an objective and reproducible manner.

A Regularized Total Least Squares Algorithm

Error-contaminated systems A x ≈ b, for which A is ill-conditioned, are considered. Such systems may be solved using Tikhonov-like regularized total least squares (R-TLS) methods. Golub et al, 1999, presented a direct algorithm for the solution of the Lagrange multiplier formulation for the R-TLS problem. Here we present a parameter independent algorithm for the approximate R-TLS solution. The algorithm, which utilizes the shifted inverse power method, relies only on a prescribed estimate for the regularization constraint condition and does not require the specification of other regularization parameters. An extension of the algorithm for nonsmooth solutions is also presented.

FDG-PET Parametric Imaging by Total Variation Minimization

Parametric imaging of the cerebral metabolic rate for glucose (CMRGlc) using [(18)F]-fluoro deoxyglucose positron emission tomography is considered. Traditional imaging is hindered due to low signal-to-noise ratios at individual voxels. We propose to minimize the total variation of the tracer uptake rates while requiring good fit of traditional Patlak equations. This minimization guarantees spatial homogeneity within brain regions and good distinction between brain regions. Brain phantom simulations demonstrate significant improvement in quality of images by the proposed method as compared to Patlak images with post-filtering using Gaussian or median filters.

Reducing modeling error of graphical methods for estimating volume of distribution measurements in PIB-PET study

Graphical analysis methods are widely used in positron emission tomography quantification because of their simplicity and model independence. But they may, particularly for reversible kinetics, lead to bias in the estimated parameters. The source of the bias is commonly attributed to noise in the data. Assuming a two-tissue compartmental model, we investigate the bias that originates from modeling error. This bias is an intrinsic property of the simplified linear models used for limited scan durations, and it is exaggerated by random noise and numerical quadrature error. Conditions are derived under which Logans graphical method either over-or under-estimates the distribution volume in the noise-free case. The bias caused by modeling error is quantified analytically. The presented analysis shows that the bias of graphical methods is inversely proportional to the dissociation rate. Furthermore, visual examination of the linearity of the Logan plot is not sufficient for guaranteeing that equilibrium has been reached. A new model which retains the elegant properties of graphical analysis methods is presented, along with a numerical algorithm for its solution. We perform simulations with the fibrillar amyloid beta radioligand [11C] benzothiazole-aniline using published data from the University of Pittsburgh and Rotterdam groups. The results show that the proposed method significantly reduces the bias due to modeling error. Moreover, the results for data acquired over a 70min scan duration are at least as good as those obtained using existing methods for data acquired over a 90min scan duration.

Estimation of uTƒ(A)v for large‐scale unsymmetric matrices

Fast algorithms, based on the unsymmetric look-ahead Lanczos and the Arnoldi process, are developed for the estimation of the functional Φ(ƒ)=uTƒ(A) v for fixed u, v and A, where A∈ℜn×n is a large-scale unsymmetric matrix. Numerical results are presented which validate the comparable accuracy of both approaches. Although the Arnoldi process reaches convergence more quickly in some cases, it has greater memory requirements, and may not be suitable for especially large applications. Copyright

Multisplitting for Regularized Least Squares with Krylov Subspace Recycling

SUMMARY The method of multisplitting (MS), implemented as a restricted additive Schwarz type algorithm, is extended for the solution of regularized least squares problems. The presented non-stationary version of the algorithm uses dynamic updating of the weights applied to the subdomains in reconstituting the global solution. Standard convergence results follow from extensive prior literature on linear MS schemes. Additional convergence results on nonstationary iterations yield convergence conditions for the presented nonstationary MS algorithm. The global iteration uses repeated solves of local problems with changing right hand sides but a fixed system matrix. These problems are solved inexactly using a conjugate gradient least squares algorithm which provides a seed Krylov subspace. Recycling of the seed system Krylov subspace to obtain the solutions of subsequent nearby systems of equations improves the overall efficiency of the MS algorithm, and is apparently novel in this context. The obtained projected solution is not always of sufficient accuracy to satisfy a reasonable inner convergence condition on the local solution. Improvements to accuracy may be achieved by reseeding the solution space either every few steps, or when the successive right hand sides are sufficiently close as measured by a provided tolerance. Restarting and augmenting the solution space are also discussed. Any time a new space is generated it is used for subsequent steps. Numerical simulations validate the use of the recycling algorithm. These numerical experiments use the standard reconstruction of the two dimensional Shepp–Logan phantom, as well as a two dimensional problem from seismic tomography. Copyright

Revisiting stopping rules for iterative methods used in emission tomography.

The expectation maximization algorithm is commonly used to reconstruct images obtained from positron emission tomography sinograms. For images with acceptable signal to noise ratios, iterations are terminated prior to convergence. A new quantitative and reproducible stopping rule is designed and validated on simulations using a Monte-Carlo generated transition matrix with a Poisson noise distribution on the sinogram data. Iterations are terminated at the solution which yields the most probable estimate of the emission densities while matching the sinogram data. It is more computationally efficient and more accurate than the standard stopping rule based on the Pearsons χ(2) test.

Reducing the noise effects in Logan graphic analysis for PET receptor measurements

Logans graphical analysis (LGA) is a widely-used approach for quantification of biochemical and physiological processes from Positron emission tomography (PET) image data. A well-noted problem associated with the LGA method is the bias in the estimated parameters. We recently systematically evaluated the bias associated with the linear model approximation and developed an alternative to minimize the bias due to model error. In this study, we examined the noise structure in the equations defining linear quantification methods, including LGA. The noise structure conflicts with the conditions given by the Gauss-Markov theorem for the least squares (LS) solution to generate the best linear unbiased estimator. By carefully taking care of the data error structure, we propose to use structured total least squares (STLS) to obtain the solution using a one-dimensional optimization problem. Simulations of PET data for [11C] benzothiazole-aniline (Pittsburgh Compound-B [PIB]) show that the proposed method significantly reduces the bias. We conclude that the bias associated with noise is primarily due to the unusual structure of he correlated noise and it can be reduced with the proposed STLS method.

Parallel variable distribution for total least squares

A novel parallel method for determining an approximate to- tal least squares (TLS) solution is introduced. Based on domain dis- tribution, the global TLS problem is partitioned into several dependent TLS subproblems. A convergent algorithm using the parallel variable distribution technique (Ferris and Mangasarian, 1994) is presented. Nu- merical results support the development and analysis of the algorithms.