Manfred R. Trummer

Slow-rotation dynamic SPECT with a temporal second derivative constraint.

PURPOSE Dynamic tracer behavior in the human body arises as a result of continuous physiological processes. Hence, the change in tracer concentration within a region of interest (ROI) should follow a smooth curve. The authors propose a modification to an existing slow-rotation dynamic SPECT reconstruction algorithm (dSPECT) with the goal of improving the smoothness of time activity curves (TACs) and other properties of the reconstructed image. METHODS The new method, denoted d2EM, imposes a constraint on the second derivative (concavity) of the TAC in every voxel of the reconstructed image, allowing it to change sign at most once. Further constraints are enforced to prevent other nonphysical behaviors from arising. The new method is compared with dSPECT using digital phantom simulations and experimental dynamic 99mTc -DTPA renal SPECT data, to assess any improvement in image quality. RESULTS In both phantom simulations and healthy volunteer experiments, the d2EM method provides smoother TACs than dSPECT, with more consistent shapes in regions with dynamic behavior. Magnitudes of TACs within an ROI still vary noticeably in both dSPECT and d2EM images, but also in images produced using an OSEM approach that reconstructs each time frame individually, based on much more complete projection data. TACs produced by averaging over a region are similar using either method, even for small ROIs. Results for experimental renal data show expected behavior in images produced by both methods, with d2EM providing somewhat smoother mean TACs and more consistent TAC shapes. CONCLUSIONS The d2EM method is successful in improving the smoothness of time activity curves obtained from the reconstruction, as well as improving consistency of TAC shapes within ROIs.

An APD-based iterative reconstruction method for simultaneous technetium-99m/iodine-123 SPECT imaging

In this paper, we present an iterative reconstruction method for Dual-Isotope SPECT (DI-SPECT) imaging. The method includes accurately analytically computed cross-talk and scatter corrections provided by the APD (Analytical Photon Distribution) algorithm. Simulation and experimental results demonstrate significant improvement in image quality over uncorrected images.

Segmentation-based regularization of dynamic SPECT reconstruction

Dynamic SPECT reconstruction using a single slow camera rotation is a highly underdetermined problem, which requires the use of regularization techniques to obtain useful results. The dSPECT algorithm (Farncombe et al. 1999) provides temporal but not spatial regularization, resulting in poor contrast and low activity levels in organs of interest, due mostly to blurring. In this paper we incorporate a user-assisted segmentation algorithm (Saad et al. 2008) into the reconstruction process to improve the results. Following an initial reconstruction using the existing dSPECT technique, a user places seeds in the image to indicate regions of interest (ROIs). A random-walk based automatic segmentation algorithm then assigns every voxel in the image to one of the ROIs, based on its proximity to the seeds as well as the similarity between time activity curves (TACs). The user is then able to visualize the segmentation and improve it if necessary. Average TACs are extracted from each ROI and assigned to every voxel in the ROI, giving an image with a spatially uniform TAC in each ROI. This image is then used as initial input to a second run of dSPECT, in order to adjust the dynamic image to better fit the projection data. We test this approach with a digital phantom simulating the kinetics of Tc99m-DTPA in the renal system, including healthy and unhealthy behaviour. Summed TACs for each kidney and the bladder were calculated for the spatially regularized and non-regularized reconstructions, and compared to the true values. The TACs for the two kidneys were noticeably improved in every case, while TACs for the smaller bladder region were unchanged. Furthermore, in two cases where the segmentation was intentionally done incorrectly, the spatially regularized reconstructions were still as good as the non-regularized ones. In general, the segmentation-based regularization improves TAC quality within ROIs, as well as image contrast.

Spectral methods in computing invariant tori

We present two new spectral implementations for computing invariant tori. The underlying nonlinear partial differential equation (Dieci et al., 1991), although hyperbolic by nature, has periodic boundary conditions in both space and time. Our first approach uses a spatial spectral discretization, and finds the solution via a shooting method. The second one employs a full two-dimensional Fourier spectral discretization, and uses Newtons method. This leads to very large, sparse, unsymmetric systems, although with highly structured matrices. A modified conjugate gradient type iterative solver was found to perform best when the dimensions get too large for direct solvers. The two methods are implemented for the van der Pol oscillator, and compared to previous algorithms.

Segmentation-Based Regularization of Dynamic SPECT Reconstructions

Dynamic SPECT reconstruction using a single slow camera rotation is a highly underdetermined problem, which requires the use of regularization techniques to obtain useful results. The dSPECT algorithm (Farncombe et al. 1999) provides temporal but not spatial regularization, resulting in poor contrast and low activity levels in organs of interest, due mostly to blurring. In this paper we incorporate a user-assisted segmentation algorithm (Saad et al. 2008) into the reconstruction process to improve the results. Following an initial reconstruction using the existing dSPECT technique, a user places seeds in the image to indicate regions of interest (ROIs). A random-walk based automatic segmentation algorithm then assigns every voxel in the image to one of the ROIs, based on its proximity to the seeds as well as the similarity between time activity curves (TACs). The user is then able to visualize the segmentation and improve it if necessary. Average TACs are extracted from each ROI and assigned to every voxel in the ROI, giving an image with a spatially uniform TAC in each ROI. This image is then used as initial input to a second run of dSPECT, in order to adjust the dynamic image to better fit the projection data. We test this approach with a digital phantom simulating the kinetics of Tc99m-DTPA in the renal system, including healthy and unhealthy behaviour. Summed TACs for each kidney and the bladder were calculated for the spatially regularized and non-regularized reconstructions, and compared to the true values. The TACs for the two kidneys were noticeably improved in every case, while TACs for the smaller bladder region were unchanged. Furthermore, in two cases where the segmentation was intentionally done incorrectly, the spatially regularized reconstructions were still as good as the non-regularized ones. In general, the segmentation-based regularization improves TAC quality within ROIs, as well as image contrast.

An APD-based Iterative Reconstruction Method for Simultaneous Technetium-99m/lodine-123 SPECT Imaging

In this paper, we present an iterative reconstruction method for Dual-Isotope SPECT (DI-SPECT) imaging. The method includes accurately analytically computed cross-talk and scatter corrections provided by the APD (Analytical Photon Distribution) algorithm. Simulation and experimental results demonstrate significant improvement in image quality over uncorrected images.

Preconditioning for a Class of Spectral Differentiation Matrices

We propose an efficient preconditioning technique for the numerical solution of first-order partial differential equations (PDEs). This study has been motivated by the computation of an invariant torus of a system of ordinary differential equations. We find the torus by discretizing a nonlinear first-order PDE with a full two-dimensional Fourier spectral method and by applying Newton’s method. This leads to large nonsymmetric linear algebraic systems. The sparsity pattern of these systems makes the use of direct solvers prohibitively expensive. Commonly used iterative methods, e.g., GMRes, BiCGStab and CGNR (Conjugate Gradient applied to the normal equations), are quite slow to converge. Our preconditioner is derived from the solution of a PDE with constant coefficients; it has a fast implementation based on the Fast Fourier Transform (FFT). It effectively increases the clustering of the spectrum, and speeds up convergence significantly. We demonstrate the performance of the preconditioner in a number of linear PDEs and the nonlinear PDE arising from the Van der Pol oscillator

A finite difference scheme for computing inertial manifolds

A fully discrete method is presented for computing inertial manifolds of dissipative partial differential equations. In particular, only an approximate spectral decomposition of the dominant differential operator needs to be known. The first few of the smallest eigenvalues and eigenvectors of the discretized operator are approximated using the Lanczos algorithm. Numerical experiments are performed for an equation in one space dimension by discretizing the space variable on a sufficiently fine grid. The basic ideas and techniques are exemplified for selected bifurcation diagrams of an integrated form of the Kuramoto-Sivashinsky equation.

Radial basis functions for solving differential equations

High-order numerical methods for solving differential equations are, in general, fairly sensitive to perturbations in their data. A previously proposed radial basis function (RBF) method, namely an integrated multiquadric scheme (IMQ), is applied to two-point boundary value problems whose solutions exhibit thin boundary layers. As frequently observed among RBF methods, the matrices arising are ill-conditioned, in this paper to the point of numerical singularity. The sensitivity of the method to perturbations and round-off error is investigated, and evidence is provided that perturbations are not nearly as strongly amplified as suggested by the large condition numbers of the matrices used in the computation.

Preconditioning of spectral methods via Birkhoff interpolation

High-order differentiation matrices as calculated in spectral collocation methods usually include a large round-off error and have a large condition number (Baltensperger and Berrut Computers and Mathematics with Applications 37(1), 41–48 1999; Baltensperger and Trummer SIAM J. Sci. Comput. 24(5), 1465–1487 2003; Costa and Don Appl. Numer. Math. 33(1), 151–159 2000). Wang et al. (Wang et al. SIAM J. Sci. Comput. 36(3), A907–A929 2014) present a method to precondition these matrices using Birkhoff interpolation. We generalize this method for all orders and boundary conditions and allowing arbitrary rows of the system matrix to be replaced by the boundary conditions. The preconditioner is an exact inverse of the highest-order differentiation matrix in the equation; thus, its product with that matrix can be replaced by the identity matrix. We show the benefits of the method for high-order differential equations. These include improved condition number and, more importantly, higher accuracy of solutions compared to other methods.