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Nonlinear Bayesian estimation of BOLD signal under non-Gaussian noise. COMPUTATIONAL AND MATHEMATICAL METHODS IN MEDICINE 2015; 2015:389875. [PMID: 25691911 PMCID: PMC4321086 DOI: 10.1155/2015/389875] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 08/02/2014] [Revised: 10/13/2014] [Accepted: 10/20/2014] [Indexed: 11/18/2022]
Abstract
Modeling the blood oxygenation level dependent (BOLD) signal has been a subject of study for over a decade in the neuroimaging community. Inspired from fluid dynamics, the hemodynamic model provides a plausible yet convincing interpretation of the BOLD signal by amalgamating effects of dynamic physiological changes in blood oxygenation, cerebral blood flow and volume. The nonautonomous, nonlinear set of differential equations of the hemodynamic model constitutes the process model while the weighted nonlinear sum of the physiological variables forms the measurement model. Plagued by various noise sources, the time series fMRI measurement data is mostly assumed to be affected by additive Gaussian noise. Though more feasible, the assumption may cause the designed filter to perform poorly if made to work under non-Gaussian environment. In this paper, we present a data assimilation scheme that assumes additive non-Gaussian noise, namely, the e-mixture noise, affecting the measurements. The proposed filter MAGSF and the celebrated EKF are put to test by performing joint optimal Bayesian filtering to estimate both the states and parameters governing the hemodynamic model under non-Gaussian environment. Analyses using both the synthetic and real data reveal superior performance of the MAGSF as compared to EKF.
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Adrian DW, Maitra R, Rowe DB. Ricean over Gaussian modelling in magnitude fMRI Analysis-Added Complexity with Negligible Practical Benefits. Stat (Int Stat Inst) 2013; 2:303-316. [PMID: 29326483 PMCID: PMC5759793 DOI: 10.1002/sta4.34] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
Abstract
It is well-known that Gaussian modeling of functional Magnetic Resonance Imaging (fMRI) magnitude time-course data, which are truly Rice-distributed, constitutes an approximation, especially at low signal-to-noise ratios (SNRs). Based on this fact, previous work has argued that Rice-based activation tests show superior performance over their Gaussian-based counterparts at low SNRs and should be preferred in spite of the attendant additional computational and estimation burden. Here, we revisit these past studies and after identifying and removing their underlying limiting assumptions and approximations, provide a more comprehensive comparison. Our experimental evaluations using ROC curve methodology show that tests derived using Ricean modeling are substantially superior over the Gaussian-based activation tests only for SNRs below 0.6, i.e SNR values far lower than those encountered in fMRI as currently practiced.
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Affiliation(s)
- Daniel W. Adrian
- Research and Development Division, National Agricultural Statistics Service, Fairfax, VA, USA
| | - Ranjan Maitra
- Department of Statistics and Statistical Laboratory, Iowa State University, Ames, Iowa, USA
| | - Daniel B. Rowe
- Department of Mathematics, Statistics and Computer Science at Marquette University, Milwaukee, Wisconsin, USA
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Walker-Samuel S, Orton M, McPhail LD, Boult JKR, Box G, Eccles SA, Robinson SP. Bayesian estimation of changes in transverse relaxation rates. Magn Reson Med 2011; 64:914-21. [PMID: 20806382 DOI: 10.1002/mrm.22478] [Citation(s) in RCA: 28] [Impact Index Per Article: 2.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/06/2022]
Abstract
Although the biasing of R(2)* estimates by assuming magnitude MR data to be normally distributed has been described, the effect on changes in R(2)* (DeltaR(2)*), such as induced by a paramagnetic contrast agent, has not been reported. In this study, two versions of a novel Bayesian maximum a posteriori approach for estimating DeltaR(2)* are described and evaluated: one that assumes normally distributed data and the other, Rice-distributed data. The approach enables the robust, voxelwise determination of the uncertainty in DeltaR(2)* estimates and provides a useful statistical framework for quantifying the probability that a pixel has been significantly enhanced. This technique was evaluated in vivo, using ultrasmall superparamagnetic iron oxide particles in orthotopic murine prostate tumors. It is shown that assuming magnitude data to be normally distributed causes DeltaR(2)* to be underestimated when signal-to-noise ratio is modest. However, the biasing effect is less than is found in R(2)* estimates, implying that the simplifying assumption of normally distributed noise is more justifiable when evaluating DeltaR(2)* compared with when evaluating precontrast R(2)* values.
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Affiliation(s)
- Simon Walker-Samuel
- Cancer Research UK & EPSRC Cancer Imaging Centre, The Institute of Cancer Research, Sutton, Surrey, United Kingdom.
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Domingo Köhler S, Spitzbarth M, Diederichs K, Exner TE, Drescher M. A short note on the analysis of distance measurements by electron paramagnetic resonance. JOURNAL OF MAGNETIC RESONANCE (SAN DIEGO, CALIF. : 1997) 2011; 208:167-70. [PMID: 21044853 DOI: 10.1016/j.jmr.2010.10.005] [Citation(s) in RCA: 11] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/01/2010] [Revised: 09/13/2010] [Accepted: 10/04/2010] [Indexed: 05/24/2023]
Abstract
In electron paramagnetic resonance (EPR) distance distributions between site-directedly attached spin labels in soft matter are obtained by measuring their dipole-dipole interaction. The analysis of these distance distributions can be misleading particularly for broad distributions, because the most probable distance deviates from the distance between the most probable label positions. The current manuscript studies this effect using numerically generated spin label positions, molecular dynamics simulations, and experimental data of a model system. An approach involving Rice distributions is proposed to overcome this problem.
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Zhu H, Li Y, Ibrahim JG, Shi X, An H, Chen Y, Gao W, Lin W, Rowe DB, Peterson BS. Regression Models for Identifying Noise Sources in Magnetic Resonance Images. J Am Stat Assoc 2009; 104:623-637. [PMID: 19890478 DOI: 10.1198/jasa.2009.0029] [Citation(s) in RCA: 38] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/21/2022]
Abstract
Stochastic noise, susceptibility artifacts, magnetic field and radiofrequency inhomogeneities, and other noise components in magnetic resonance images (MRIs) can introduce serious bias into any measurements made with those images. We formally introduce three regression models including a Rician regression model and two associated normal models to characterize stochastic noise in various magnetic resonance imaging modalities, including diffusion-weighted imaging (DWI) and functional MRI (fMRI). Estimation algorithms are introduced to maximize the likelihood function of the three regression models. We also develop a diagnostic procedure for systematically exploring MR images to identify noise components other than simple stochastic noise, and to detect discrepancies between the fitted regression models and MRI data. The diagnostic procedure includes goodness-of-fit statistics, measures of influence, and tools for graphical display. The goodness-of-fit statistics can assess the key assumptions of the three regression models, whereas measures of influence can isolate outliers caused by certain noise components, including motion artifacts. The tools for graphical display permit graphical visualization of the values for the goodness-of-fit statistic and influence measures. Finally, we conduct simulation studies to evaluate performance of these methods, and we analyze a real dataset to illustrate how our diagnostic procedure localizes subtle image artifacts by detecting intravoxel variability that is not captured by the regression models.
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Affiliation(s)
- Hongtu Zhu
- Hongtu Zhu is Associate Professor, Department of Biostatistics and Biomedical Research Imaging Center and Department of Radiology, University of North Carolina at Chapel Hill, NC 27599 ( ). Yimei Li is a Ph.D. student, Department of Biostatistics and Biomedical Research Imaging Center and Department of Radiology, University of North Carolina at Chapel Hill, NC 27599 ( ). Joshep G. Ibrahim is Alumni Distinguished Professor, Department of Biostatistics and Biomedical Research Imaging Center and Department of Radiology, University of North Carolina at Chapel Hill, NC 27599 ( ). Xiaoyan Shi is a Ph.D. student, Department of Biostatistics and Biomedical Research Imaging Center and Department of Radiology, University of North Carolina at Chapel Hill, NC 27599 ( . Hongyu An is Research Assistant Professor, Department of Radiology, University of North Carolina at Chapel Hill, NC 27599 ( ). Yashen Chen is Research Fellow, Department of Radiology, University of North Carolina at Chapel Hill, NC 27599 ( ). Wei Gao is a Ph.D. student, Department of Radiology, University of North Carolina at Chapel Hill, NC 27599 ( ). Weili Lin is Professor, Department of Radiology, University of North Carolina at Chapel Hill, NC 27599 ( ). Daniel B. Rowe is Associate Professor, Department of Biophysics, Medical College of Wisconsin, Milwaudee, WI 53226 ( ). Bradley S. Peterson is Professor, Department of Psychiatry, Columbia Medical Center and the New York State Psychiatric Institiute, New York, NY 10032 ( )
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BOLD Noise Assumptions in fMRI. Int J Biomed Imaging 2006; 2006:12014. [PMID: 23165015 PMCID: PMC2324015 DOI: 10.1155/ijbi/2006/12014] [Citation(s) in RCA: 30] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/30/2006] [Revised: 06/13/2006] [Accepted: 06/17/2006] [Indexed: 11/17/2022] Open
Abstract
This paper discusses the assumption of Gaussian noise in the
blood-oxygenation-dependent (BOLD) contrast for functional MRI
(fMRI). In principle, magnitudes in MRI images follow a Rice
distribution. We start by reviewing differences between Rician and
Gaussian noise. An analytic expression is derived for the null
(resting-state) distribution of the difference between two Rician
distributed images. This distribution is shown to be symmetric,
and an exact expression for its standard deviation is derived.
This distribution can be well approximated by a Gaussian, with
very high precision for high SNR, and high precision for lower
SNR. Tests on simulated and real MR images show that subtracting
the time-series mean in fMRI yields asymmetrically distributed
temporal noise. Subtracting a resting-state time series from the
first results in symmetric and nearly Gaussian noise. This has
important consequences for fMRI analyses using standard
statistical tests.
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