Characteristic functionals in imaging and image-quality assessment: tutorial.

Characteristic functionals are one of the main analytical tools used to quantify the statistical properties of random fields and generalized random fields. The viewpoint taken here is that a random field is the correct model for the ensemble of objects being imaged by a given imaging system. In modern digital imaging systems, random fields are not used to model the reconstructed images themselves since these are necessarily finite dimensional. After a brief introduction to the general theory of characteristic functionals, many examples relevant to imaging applications are presented. The propagation of characteristic functionals through both a binned and list-mode imaging system is also discussed. Methods for using characteristic functionals and image data to estimate population parameters and classify populations of objects are given. These methods are based on maximum likelihood and maximum a posteriori techniques in spaces generated by sampling the relevant characteristic functionals through the imaging operator. It is also shown how to calculate a Fisher information matrix in this space. These estimators and classifiers, and the Fisher information matrix, can then be used for image quality assessment of imaging systems.

Radiance and photon noise: imaging in geometrical optics, physical optics, quantum optics and radiology.

The statistics of detector outputs produced by an imaging system are derived from basic radiometric concepts and definitions. We show that a fundamental way of describing a photon-limited imaging system is in terms of a Poisson random process in spatial, angular, and wavelength variables. We begin the paper by recalling the concept of radiance in geometrical optics, radiology, physical optics, and quantum optics. The propagation and conservation laws for radiance in each of these domains are reviewed. Building upon these concepts, we distinguish four categories of imaging detectors that all respond in some way to the incident radiance, including the new category of photon-processing detectors (capable of measuring radiance on a photon-by-photon basis). This allows us to rigorously show how the concept of radiance is related to the statistical properties of detector outputs and to the information content of a single detected photon. A Monte-Carlo technique, which is derived from the Boltzmann transport equation, is presented as a way to estimate probability density functions to be used in reconstruction from photon-processing data.

Impact of the Fano Factor on Position and Energy Estimation in Scintillation Detectors.

The Fano factor for an integer-valued random variable is defined as the ratio of its variance to its mean. Light from various scintillation crystals have been reported to have Fano factors from sub-Poisson (Fano factor < 1) to super-Poisson (Fano factor > 1). For a given mean, a smaller Fano factor implies a smaller variance and thus less noise. We investigated if lower noise in the scintillation light will result in better spatial and energy resolutions. The impact of Fano factor on the estimation of position of interaction and energy deposited in simple gamma-camera geometries is estimated by two methods - calculating the Cramér-Rao bound and estimating the variance of a maximum likelihood estimator. The methods are consistent with each other and indicate that when estimating the position of interaction and energy deposited by a gamma-ray photon, the Fano factor of a scintillator does not affect the spatial resolution. A smaller Fano factor results in a better energy resolution.

High-Resolution Anamorphic SPECT Imaging.

We have developed a gamma-ray imaging system that combines a high-resolution silicon detector with two sets of movable, half-keel-edged copper-tungsten blades configured as crossed slits. These apertures can be positioned independently between the object and detector, producing an anamorphic image in which the axial and transaxial magnifications are not constrained to be equal. The detector is a 60 mm × 60 mm, one-millimeter-thick, one-megapixel silicon double-sided strip detector with a strip pitch of 59 µm. The flexible nature of this system allows the application of adaptive imaging techniques. We present system details; calibration, acquisition, and reconstruction methods; and imaging results.

Fisher Information Analysis of Depth-of-Interaction Estimation in Double-Sided Strip Detectors.

In very-high-spatial-resolution gamma-ray imaging applications, such as preclinical PET and SPECT, estimation of 3D interaction location inside the detector crystal can be used to minimize parallax error in the imaging system. In this work, we investigate the effect of bias voltage setting on depth-of-interaction (DOI) estimates for a semiconductor detector with a double-sided strip geometry. We first examine the statistical properties of the signals and develop expressions for likelihoods for given gamma-ray interaction positions. We use Fisher Information to quantify how well (in terms of variance) the measured signals can be used for DOI estimation with different bias-voltage settings. We performed measurements of detector response versus 3D position as a function of applied bias voltage by scanning with highly collimated synchrotron radiation at the Advanced Photon Source at Argonne National Laboratory. Experimental and theoretical results show that the optimum bias setting depends on whether or not the estimated event position will include the depth of interaction. We also found that for this detector geometry, the z-resolution changes with depth.

Synthetic phase-shifting for optical testing: point-diffraction interferometry without null optics or phase shifters.

An innovative iterative search method called the synthetic phase-shifting (SPS) algorithm is proposed. This search algorithm is used for maximum-likelihood (ML) estimation of a wavefront that is described by a finite set of Zernike Fringe polynomials. In this paper, we estimate the coefficient, or parameter, values of the wavefront using a single interferogram obtained from a point-diffraction interferometer (PDI). In order to find the estimates, we first calculate the squared-difference between the measured and simulated interferograms. Under certain assumptions, this squared-difference image can be treated as an interferogram showing the phase difference between the true wavefront deviation and simulated wavefront deviation. The wavefront deviation is the difference between the reference and the test wavefronts. We calculate the phase difference using a traditional phase-shifting technique without physical phase-shifters. We present a detailed forward model for the PDI interferogram, including the effect of the finite size of a detector pixel. The algorithm was validated with computational studies and its performance and constraints are discussed. A prototype PDI was built and the algorithm was also experimentally validated. A large wavefront deviation was successfully estimated without using null optics or physical phase-shifters. The experimental result shows that the proposed algorithm has great potential to provide an accurate tool for non-null testing.

Skeletal muscle satellite cell migration to injured tissue measured with 111In-oxine and high-resolution SPECT imaging.

The delivery of adult skeletal muscle stem cells, called satellite cells, to several injured muscles via the circulation would be useful, however, an improved understanding of cell fate and biodistribution following their delivery is important for this goal to be achieved. The objective of this study was to evaluate the ability of systemically delivered satellite cells to home to injured skeletal muscle using single-photon emission computed tomography (SPECT) imaging of (111)In-labeled satellite cells. Satellite cells labeled with (111)In-oxine and green fluorescent protein (GFP) were injected intravenously after bupivicaine-induced injury to the tibialis anterior muscle. Animals were imaged with a high-resolution SPECT system called FastSPECT II for up to 7 days after transplantation. In vivo FastSPECT II imaging demonstrated a three to five-fold greater number of transplanted satellite cells in bupivicaine-injured muscle as compared to un-injured muscle after transplantation; a finding that was verified through autoradiograph analysis and quantification of GFP expression. Satellite cells also accumulated in other organs including the lung, liver, and spleen, as determined by biodistribution measurements. These data support the ability of satellite cells to home to injured muscle and support the use of SPECT and autoradiograph imaging techniques to track systemically transplanted (111)In labeled satellite cells in vivo, and suggest their homing may be improved by reducing their entrapment in filter organs.

Maximum-likelihood estimation of parameterized wavefronts from multifocal data.

A method for determining the pupil phase distribution of an optical system is demonstrated. Coefficients in a wavefront expansion were estimated using likelihood methods, where the data consisted of multiple irradiance patterns near focus. Proof-of-principle results were obtained in both simulation and experiment. Large-aberration wavefronts were handled in the numerical study. Experimentally, we discuss the handling of nuisance parameters. Fisher information matrices, Cramér-Rao bounds, and likelihood surfaces are examined. ML estimates were obtained by simulated annealing to deal with numerous local extrema in the likelihood function. Rapid processing techniques were employed to reduce the computational time.

Mesenchymal Stem Cell-mediated delivery of the sodium iodide symporter supports radionuclide imaging and treatment of breast cancer.

Mesenchymal Stem Cells (MSCs) migrate specifically to tumors in vivo, and coupled with their capacity to bypass immune surveillance, are attractive vehicles for tumor-targeted delivery of therapeutic agents. This study aimed to introduce MSC-mediated expression of the sodium iodide symporter (NIS) for imaging and therapy of breast cancer. Tumor bearing animals received an intravenous or intratumoral injection of NIS expressing MSCs (MSC-NIS), followed by (99m) Technetium pertechnetate imaging 3-14 days later using a BazookaSPECT γ-camera. Tissue was harvested for analysis of human NIS (hNIS) expression by relative quantitative-polymerase chain reaction. Therapy animals received an i.p. injection of (131) I or saline 14 days after injection of MSC-NIS, and tumor volume was monitored for 8 weeks. After injection of MSC-NIS, BazookaSPECT imaging revealed an image of animal intestines and chest area at day 3, along with a visible weak tumor image. By day 14, the tumor was visible with a significant reduction in radionuclide accumulation in nontarget tissue observed. hNIS gene expression was detected in the intestines, heart, lungs, and tumors at early time points but later depleted in nontarget tissues and persisted at the tumor site. Based on imaging/biodistribution data, animals received a therapeutic dose of (131) I 14 days after MSC-NIS injection. This resulted in a significant reduction in tumor growth (mean ± SEM, 236 ± 62 mm(3) vs. 665 ± 204 mm(3) in controls). The ability to track MSC migration and transgene expression noninvasively in real time before therapy is a major advantage to this strategy. This promising data supports the feasibility of this approach as a novel therapy for breast cancer.

Objective assessment of image quality. V. Photon-counting detectors and list-mode data.

A theoretical framework for detection or discrimination tasks with list-mode data is developed. The object and imaging system are rigorously modeled via three random mechanisms: randomness of the object being imaged, randomness in the attribute vectors, and, finally, randomness in the attribute vector estimates due to noise in the detector outputs. By considering the list-mode data themselves, the theory developed in this paper yields a manageable expression for the likelihood of the list-mode data given the object being imaged. This, in turn, leads to an expression for the optimal Bayesian discriminant. Figures of merit for detection tasks via the ideal and optimal linear observers are derived. A concrete example discusses detection performance of the optimal linear observer for the case of a known signal buried in a random lumpy background.

Simulating photon-transport in uniform media using the radiative transport equation: a study using the Neumann-series approach.

We present the implementation, validation, and performance of a Neumann-series approach for simulating light propagation at optical wavelengths in uniform media using the radiative transport equation (RTE). The RTE is solved for an anisotropic-scattering medium in a spherical harmonic basis for a diffuse-optical-imaging setup. The main objectives of this paper are threefold: to present the theory behind the Neumann-series form for the RTE, to design and develop the mathematical methods and the software to implement the Neumann series for a diffuse-optical-imaging setup, and, finally, to perform an exhaustive study of the accuracy, practical limitations, and computational efficiency of the Neumann-series method. Through our results, we demonstrate that the Neumann-series approach can be used to model light propagation in uniform media with small geometries at optical wavelengths.

Three-dimensional Neumann-series approach to model light transport in nonuniform media.

We present the implementation, validation, and performance of a three-dimensional (3D) Neumann-series approach to model photon propagation in nonuniform media using the radiative transport equation (RTE). The RTE is implemented for nonuniform scattering media in a spherical harmonic basis for a diffuse-optical-imaging setup. The method is parallelizable and implemented on a computing system consisting of NVIDIA Tesla C2050 graphics processing units (GPUs). The GPU implementation provides a speedup of up to two orders of magnitude over non-GPU implementation, which leads to good computational efficiency for the Neumann-series method. The results using the method are compared with the results obtained using the Monte Carlo simulations for various small-geometry phantoms, and good agreement is observed. We observe that the Neumann-series approach gives accurate results in many cases where the diffusion approximation is not accurate.

A System Calibration and Fast Iterative Reconstruction Method for Next-Generation SPECT Imagers.

Recently, high-resolution gamma cameras have been developed with detectors containing > 105-106 elements. Single-photon emission computed tomography (SPECT) imagers based on these detectors usually also have a large number of voxel bins and therefore face memory storage issues for the system matrix when performing fast tomographic reconstructions using iterative algorithms. To address these issues, we have developed a method that parameterizes the detector response to a point source and generates the system matrix on the fly during MLEM or OSEM on graphics hardware. The calibration method, interpolation of coefficient data, and reconstruction results are presented in the context of a recently commissioned small-animal SPECT imager, called FastSPECT III.

Multiple-hit parameter estimation in monolithic detectors.

We examine a maximum-a-priori (MAP) method for estimating the primary interaction position of gamma rays with multiple-interaction sites (hits) in a monolithic detector. In assessing the performance of a multiple-hit estimator over that of a conventional one-hit estimator, we consider a few different detector and readout configurations of a 50-mm-wide square LSO block. For this study, we use simulated data from SCOUT, a Monte-Carlo tool for photon tracking and modeling scintillation-camera output. With this tool, we determine estimate bias and variance for a multiple-hit estimator and compare these with similar metrics for a conventional ML estimator, which assumes full energy deposition in one hit. We also examine the effect of event filtering on these metrics; for this purpose, we use a likelihood threshold to reject signals that are not likely to have been produced under the assumed likelihood model.Depending on detector design, we observe a 1-12% improvement of intrinsic resolution for a 1-or-2-hit estimator as compared with a 1-hit estimator. We also observe improved differentiation of photopeak events using a 1-or-2-hit estimator as compared with the 1-hit estimator; more than 6% of photopeak events that were rejected by likelihood filtering for the 1-hit estimator were accurately identified as photo peak events and positioned without loss of resolution by a 1-or-2-hit estimator.

Stochastic models for objects and images in oncology and virology: application to PI3K-Akt-mTOR signaling and COVID-19 disease.

Purpose: The goal of this research is to develop innovative methods of acquiring simultaneous multidimensional molecular images of several different physiological random processes (PRPs) that might all be active in a particular disease such as COVID-19. Approach: Our study is part of an ongoing effort at the University of Arizona to derive biologically accurate yet mathematically tractable models of the objects of interest in molecular imaging and of the images they produce. In both cases, the models are fully stochastic, in the sense that they provide ways to estimate any estimable property of the object or image. The mathematical tool we use for images is the characteristic function, which can be calculated if the multivariate probability density function for the image data is known. For objects, which are functions of continuous variables rather than discrete pixels or voxels, the characteristic function becomes infinite dimensional, and we refer to it as the characteristic functional. Results: Several innovative mathematical results are derived, in particular for simultaneous imaging of multiple PRPs. Then the application of these methods to cancers that disrupt the mammalian target of rapamycin signaling pathway and to COVID-19 are discussed qualitatively. One reason for choosing these two problems is that they both involve lipid rafts. Conclusions: We found that it was necessary to employ a new algorithm for energy estimation to do simultaneous single-photon emission computerized tomography imaging of a large number of different tracers. With this caveat, however, we expect to be able to acquire and analyze an unprecedented amount of molecular imaging data for an individual COVID patient.

Erratum: Publisher's Note: Stochastic models for objects and images in oncology and virology: application to PI3K-Akt-mTOR signaling and COVID-19 disease.

[This corrects the article DOI: 10.1117/1.JMI.8.S1.S16001.].

Singular-value decomposition of a tomosynthesis system.

Tomosynthesis is an emerging technique with potential to replace mammography, since it gives 3D information at a relatively small increase in dose and cost. We present an analytical singular-value decomposition of a tomosynthesis system, which provides the measurement component of any given object. The method is demonstrated on an example object. The measurement component can be used as a reconstruction of the object, and can also be utilized in future observer studies of tomosynthesis image quality.

Kinetic characterization of a novel cationic (99m)Tc(I)-tricarbonyl complex, (99m)Tc-15C5-PNP, for myocardial perfusion imaging.

BACKGROUND: Intense liver uptake of (99m)Tc-sestamibi (MIBI) often interferes with visualization of myocardial perfusion in the inferior wall of the left ventricle. To develop improved myocardial perfusion agents, crown ether-containing dithiocarbamates and bisphosphines have been introduced in recent years. This study was designed to investigate the myocardial imaging properties and in vivo kinetics of a cationic (99m)Tc(I)-tricarbonyl complex, (99m)Tc-15C5-PNP, in comparison with MIBI. METHODS: Dynamic cardiac images were acquired for 60 minutes after intravenous tracer injection using a small-animal SPECT system in healthy control rats and rats with myocardial infarcts. Myocardial and liver time-activity curves were generated for radiopharmaceutical kinetic analysis. RESULTS: Good visualization of the left ventricular wall and perfusion defects could be achieved 20 minutes after (99m)Tc-15C5-PNP administration. (99m)Tc-15C5-PNP images in all hearts with infarcts showed perfusion defects, which were comparable to MIBI images. The kinetic curves plotted from 1 to 60 minutes demonstrated that (99m)Tc-15C5-PNP has a shorter washout half-life (6.4 ± 3.2 vs 124 ± 30.5 minutes, P < .01) in the liver, lower residual liver activity (14.5 ± 10.2% vs 36.5 ± 28.9%, P < .01), and higher heart/liver ratio than MIBI. CONCLUSIONS: (99m)Tc-15C5-PNP has potential for rapid myocardial perfusion imaging with low liver uptake.

Maximum-Likelihood Estimation With a Contracting-Grid Search Algorithm.

A fast search algorithm capable of operating in multi-dimensional spaces is introduced. As a sample application, we demonstrate its utility in the 2D and 3D maximum-likelihood position-estimation problem that arises in the processing of PMT signals to derive interaction locations in compact gamma cameras. We demonstrate that the algorithm can be parallelized in pipelines, and thereby efficiently implemented in specialized hardware, such as field-programmable gate arrays (FPGAs). A 2D implementation of the algorithm is achieved in Cell/BE processors, resulting in processing speeds above one million events per second, which is a 20× increase in speed over a conventional desktop machine. Graphics processing units (GPUs) are used for a 3D application of the algorithm, resulting in processing speeds of nearly 250,000 events per second which is a 250× increase in speed over a conventional desktop machine. These implementations indicate the viability of the algorithm for use in real-time imaging applications.

Is there a role for image science in the brave new world of artificial intelligence?

Knowledge of the principles of image science is essential to the successful application of artificial intelligence in medical imaging.