WO2000072038A1 - Determination d'une distribution empirique statistique du tenseur de diffusion dans un procede irm - Google Patents

Determination d'une distribution empirique statistique du tenseur de diffusion dans un procede irm Download PDF

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WO2000072038A1
WO2000072038A1 PCT/US2000/014290 US0014290W WO0072038A1 WO 2000072038 A1 WO2000072038 A1 WO 2000072038A1 US 0014290 W US0014290 W US 0014290W WO 0072038 A1 WO0072038 A1 WO 0072038A1
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diffusion
weighted signals
interest
tensor
time point
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PCT/US2000/014290
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WO2000072038A9 (fr
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Peter J. Basser
Sinisa Pajevic
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The Government Of The United States, As Represented By The Secretary Of The Dept. Of Health And Human Services
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Priority to JP2000620373A priority Critical patent/JP4271873B2/ja
Priority to CA002373536A priority patent/CA2373536C/fr
Priority to US09/979,013 priority patent/US6845342B1/en
Priority to AU51603/00A priority patent/AU780940B2/en
Priority to EP00936262A priority patent/EP1204880A1/fr
Publication of WO2000072038A1 publication Critical patent/WO2000072038A1/fr
Publication of WO2000072038A9 publication Critical patent/WO2000072038A9/fr

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    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01R—MEASURING ELECTRIC VARIABLES; MEASURING MAGNETIC VARIABLES
    • G01R33/00—Arrangements or instruments for measuring magnetic variables
    • G01R33/20—Arrangements or instruments for measuring magnetic variables involving magnetic resonance
    • G01R33/44—Arrangements or instruments for measuring magnetic variables involving magnetic resonance using nuclear magnetic resonance [NMR]
    • G01R33/48—NMR imaging systems
    • G01R33/54—Signal processing systems, e.g. using pulse sequences ; Generation or control of pulse sequences; Operator console
    • G01R33/56—Image enhancement or correction, e.g. subtraction or averaging techniques, e.g. improvement of signal-to-noise ratio and resolution
    • G01R33/563—Image enhancement or correction, e.g. subtraction or averaging techniques, e.g. improvement of signal-to-noise ratio and resolution of moving material, e.g. flow contrast angiography
    • G01R33/56341—Diffusion imaging

Definitions

  • the invention relates to diffusion tensor magnetic resonance imaging (DT-MRI) and. more particularly, to anah zing DT-MR signals using statistical properties of the diffusion tensor
  • the first article describing the calculation of the diffusion weighting factors needed to calculate the diffusion tensor from the measured diffusion wei ⁇ hted images is (3) J. Mattiello. P. J. Basser. D. LeBihan. Analytical expression for the b matrix in NMR diffusion imaging and spectroscopy. J. Magn. Reson. A 108, 131 -141 ( 1994).
  • the first article describing the estimation of the diffusion tensor from the diffusion weighed signals or echoes are:
  • a demonstration of DT-MRI of the brain of a human subject is provided in: (20) C. Pierpaoli. P. Jezzard, P. J. Basser. High-resolution diffusion tensor imaging of the human brain. SMR/ESMRMB Joint Meeting. Nice. 1995. p. 899. (21 ) P. Jezzard. C. Pierpaoli. Dual-Echo Navigator Approach to Minimization of Eddy Current and Motion Artifacts in Echo-Planar Diffusion Imaging. Proceedings of the ISMRM. New York. 1996, p. 189.
  • the first paper describing DT-MRI is: (44) P. J. Basser. J. Mattiello. D. LeBihan. MR diffusion tensor spectroscopy and imaging. Biophys f 66, 1. 259-267 (1994).
  • NMR noise A description of nuclear magnetic resonance (NMR) noise is provided in:
  • D The effective diffusion tensor of water.
  • D (or D eff ). measured via diffusion tensor magnetic resonance imaging (DT-MRI). is sensitive to tissue structure and its changes in disease, development, aging, etc. However, these D measurements are noisy.
  • D is described by a multivariate Gaussian (or normal) distribution, and have developed quantitative hypothesis tests to determine the statistical significance of observed changes in D (and quantities derived from it), as well as a technique for reporting these data using commonly understood and well established statistical methods.
  • the statistical distribution of D is now known from the invention, it can be determined whether the DT-MRI data is corrupted by systematic artifacts that would cause the distribution to deviate from normality.
  • This invention makes DT-MR imaging and spectroscopy data more valuable since one can now assess its reliability.
  • this new information can be used to diagnose stroke (both acute and chronic) and identify the ischemic ⁇ "penumbra. *"
  • the invention can advantageously be used in evaluating soft tissue and muscle structure in orthopedic and cardiology applications.
  • the invention can advantageously be used in neonatal screening and diagnostic applications.
  • the invention can advantageously be used to segment, cluster, and classify different tissue types based on their diffusion properties.
  • this information can advantageously improve studies to screen for the efficacy of newly discovered drugs.
  • the invention can reduce cost of and the time required to perform clinical trials by reducing patient number and identifying unsuitable subjects.
  • the invention can advantageously be used to provide quantitative support for claims of safety and efficacy necessary to obtain Food and Drug Administration (FDA) approval for drugs and medical devices and tissue engineered products.
  • FDA Food and Drug Administration
  • these improved imaging parameters obtained with the invention can help determine phenotypic alterations in tissue structure and architecture for a given genotypic modification.
  • This information obtained with the invention can advantageously help materials engineers develop and test the quality of new and processed materials (e.g., polymers, films, gels, plastics, coatings, etc.).
  • the invention includes a novel use of the non-parametric Bootstrap method, or variant, for generating estimates of second and higher moments of the diffusion tensor, and functions of it in each voxel.
  • Non-parametric methods as used in this invention, provide a technique for generating a probability distribution of the diffusion tensor data using a set of empirical diffusion weighted images (e.g.. acquired during a single patient scanning session), from which the moments and other features of a probability distribution can be determined. This aspect of the invention enables testing the integrity of experimental diffusion tensor data in each voxel, because it is shown, in the absence of systematic artifacts, that the distribution of the diffusion tensor is statistically normal.
  • voxels can also be grouped together into regions of interest (ROIs).
  • ROIs regions of interest
  • Statistical hypothesis tests are then applied, for example, in a disease diagnosis application, to determine whether there are significant differences between tissue in different ROIs. or. for example, in a longitudinal study to determine whether there are any differences between tissue within the same ROI observed at different time points.
  • the invention can be embodied as a method, an apparatus, and as an article of manufacture.
  • the method of the invention includes analyzing diffusion tensor magnetic resonance signals, comprising the steps of: acquiring a plurality of diffusion weighted signals, each signal having a plurality of voxels; sampling the plurality of diffusion weighted signals to obtain at least one set of resampled diffusion weighted signals: determining a diffusion tensor for each voxel from each set of the resampled diffusion weighted signals: and determining an empirical statistical distribution for a quantity associated with the diffusion tensor from the diffusion tensors determined from the at least one set of the resampled diffusion weighted signals. Thereafter, the empirical statistical distribution can be used to diagnose the subject used to generate the diffusion weighted signals.
  • the method of the invention includes analyzing diffusion tensor magnetic resonance signals, comprising the steps of: acquiring a plurality of diffusion weighted signals, each signal having a plurality of voxels: identifying at least two regions of interest in the signals, each region of interest comprising at least one voxel: determining a diffusion tensor for each voxel in each region of interest from the diffusion weighted signals; and determining an empirical statistical distribution for a quantity associated with the diffusion tensor from the diffusion tensors for each voxel in each region of interest. Thereafter, the empirical statistical distribution can be used to diagnose the subject used to generate the diffusion weighted signals.
  • the method of the invention includes analyzing diffusion tensor magnetic resonance signals, comprising the steps of: acquiring a plurality of diffusion weighted signals for a first time point, each signal having a plurality of voxels: identifying a region of interest in the diffusion weighted signals for the first time point, the region of interest comprising at least one voxel: determining a diffusion tensor for each voxel in the region of interest for the first time point from the diffusion weighted signals for the first time point; determining an empirical statistical distribution for a quantity associated with the diffusion tensor from the diffusion tensors for each voxel in the region of interest for the first time point: acquiring a plurality of diffusion weighted signals for a second time point, each signal having a plurality of voxels; identifying a region of interest in the diffusion weighted signals for the second time point, the region of interest comprising at least one voxel; determining a diffusion tensor for each
  • the method of the invention includes analyzing diffusion tensor magnetic resonance signals, comprising the steps of: acquiring a plurality of diffusion weighted signals, each signal having a plurality of voxels; determining a quantity associated with a diffusion tensor for each voxel from the diffusion weighted signals; and determining an empirical statistical distribution for the quantity associated with the diffusion tensor from the quantity associated with the diffusion tensor. Thereafter, the empirical statistical distribution can be used to diagnose the subject used to generate the diffusion weighted signals.
  • the apparatus for the invention includes a computer programmed with software to operate the computer in accordance with the method of the invention.
  • the article of manufacture for the invention comprises a computer-readable medium embodying software to control a computer to perform the method of the invention.
  • a "computer” refers to any apparatus that is capable of accepting a structured input, processing the structured input according to prescribed rules, and producing results of the processing as output.
  • Examples of a computer include: a computer; a general pu ⁇ ose computer: a supercomputer: a mainframe; a super mini-computer: a mini-computer; a workstation; a micro-computer: a server; an interactive television: and a hybrid combination of a computer and an interactive television.
  • a computer also refers to two or more computers connected together via a network for transmitting or receiving information between the computers.
  • An example of such a computer includes a distributed computer system for processing information via computers linked by a network.
  • a "computer-readable medium” refers to any storage device used for storing data accessible by a computer Examples of a computer-readable medium include a magnetic hard disk, a floppy disk, an optical disk, such as a CD-ROM. a magnetic tape, a memory chip, and a carrier wave used to carry computer-readable electronic data, such as those used in transmitting and receiving e-mail or in accessing a network
  • a “computer system” refers to a system having a computer, where the computer includes a computer-readable medium embodying software to operate the computer
  • a “network” refers to a number of computers and associated de ⁇ ⁇ ces that are connected by communication facilities
  • a network involves permanent connections such as cables or temporary connections such as those made through telephone or other communication links Examples of a network include an internet, such as the Internet, an intranet, a local area network (LAN), a wide area network (WAN), and a combination of networks, such as an internet and an intranet
  • Figure 1 illustrates a flow diagram for analyzing DT-MR signals
  • Figure 2 illustrates a flow diagram for analyzing DT-MR signals for at least two regions of interest
  • Figure 3 illustrates a flow diagram for analyzing DT-MR signals for first and second time points
  • Figure 4 illustrates a flow diagram for analyzing DT-MR signals without determining the diffusion tensor
  • Figure 5 illustrates probability distributions of the six independent components of the diffusion tensor obtained from Monte Carlo simulations
  • FIG. 17- Figure 6 illustrates a probability distribution of the trace within a region of interest (ROI).
  • Figure 7 illustrates a probability distribution obtained within a single voxel.
  • Figure 8 illustrates a comparison between a "true " probability distribution tensor component and estimates obtained from an original set.
  • Figure 9A illustrates data generated using a diffusion phantom.
  • Figure 9B illustrates an image that is the reciprocal of an angle subtending the smallest cone of uncertainty for three eigenvectors obtained from the bootstrap method.
  • Figure 10 illustrates selecting individual voxels.
  • Figure 1 1 A illustrates empirical bootstrap data obtained from a single voxel in Figure
  • Figure 1 1 B illustrates empirical bootstrap data obtained from a single voxel in Figure 10 containing white matter.
  • Figure 1 1 C illustrates empirical bootstrap data obtained from a single voxel in Figure 10 containing CSF.
  • Figure 12 illustrates an hypothesis test of whether the difference in an ROI-averaged Trace measurement is statistically significant at a fixed statistical power.
  • Figure 13 illustrates the detection of artifacts in a diffusion weighted images.
  • Figure 1 illustrates a flow diagram for analyzing DT-MR signals. Further details of the flow diagram in Figure 1 are discussed below in the following sections.
  • diffusion weighted signals are acquired, and each signal contains several voxels. Preferably, each signal contains at least one voxel.
  • the diffusion weighted signals are acquired by a measuring technique.
  • the diffusion weighted signals are based on measuring a subject, and the subject is one of a living creature and a non-living object. Examples of a living creature are a human being and an animal. Examples of a non-living object are a phantom, a dead human being, a dead animal, a polymer, a film, a gel. a plastic, and a coating.
  • the diffusion weighted signals are obtained from a simulation using a computer and software. Alternatively, the diffusion weighted signals are stored on and read from a computer-readable medium.
  • the diffusion weighted signals are sampled to obtain at least one set of resampled diffusion weighted signals.
  • the resampled diffusion weighted signals are used in a bootstrap simulation, which is discussed further below.
  • the diffusion weighted signals are diffusion weighted images.
  • a diffusion tensor is determined for each voxel from each set of resampled diffusion weighted signals. Techniques for determining the diffusion tensor are well known.
  • the resampling is performed a plurality of times.
  • the diffusion tensor can be determined for each voxel in a subset of the voxels of the resampled diffusion weighted signals.
  • the subsets can be referred to as a region of interest (ROI).
  • the number of voxels in the ROI can be as many as the number of voxels in the resampled diffusion weighted signals and as few as one voxel in the resampled diffusion weighted signals.
  • the diffusion tensor can be determined for each voxel in at least one subset of the voxels of the resampled diffusion weighted signals.
  • the subsets can be referred to as a regions of interest (ROIs).
  • the number of voxels in each ROI can be as many as the number of voxels in the resampled diffusion weighted signals and as few as one voxel in the resampled diffusion weighted signals.
  • an empirical statistical distribution is determined for a quantity associated with the diffusion tensor.
  • the empirical statistical distribution is determined using a non-parametric technique, for example, the bootstrap method discussed further below. Other non-parametric techniques, such as an empirical resampling scheme, can also be used.
  • the empirical statistical distribution is at least one of the following: a probability density function, a cumulative density function, and a histogram.
  • the quantity associated with the diffusion tensor is at least one of the following: an element of the diffusion tensor, a linear combination of the elements of the diffusion tensor, and a function of the diffusion tensor.
  • the linear combination of the elements of the diffusion tensor is the Trace of the diffusion tensor.
  • Other empirical distributions and associated quantities can also be determined.
  • a subject is diagnosed based on the empirical statistical distribution of the quantity.
  • the subject is diagnosed based on features of the empirical statistical distribution for the quantity.
  • the empirical statistical distribution is one of a probability density function and a cumulative density function, and the features of the empirical statistical distribution are the first and higher moments of the empirical statistical distribution.
  • the subject can be diagnosed by segmenting, clustering, or classifying a portion of the subject based on features of the empirical statistical distribution of the quantity.
  • systematic artifacts in the diffusion weighted signals of the subject can be identified based on features of the empirical statistical distribution for the quantity.
  • the empirical statistical distribution for the quantity is compared to a parametric statistical distribution for the quantity.
  • the parametric distribution can be a multivariate Gaussian distribution where the quantity is the diffusion tensor.
  • the parametric distribution can be a univariate Gaussian distribution where the quantity is the Trace of the diffusion tensor.
  • a diffusion tensor is determined for each voxel in the diffusion weighted signals, or for each voxel of a subset of voxels in the diffusion weighted signals. This subset can be referred to as a region of interest. As an option, the diffusion tensor can be determined for two or more subsets of voxels in the diffusion weighted signals, where the subsets are referred to as regions of interest.
  • Figure 2 illustrates this concept as a flow diagram for analyzing DT-MR signals for at least two regions of interest. Further details of the flow diagram in Figure 2 are discussed below in the following sections.
  • diffusion weighted signals are acquired, and each signal contains voxels.
  • each signal contains at least one voxel.
  • Block 20 is similar to block 10.
  • Block 21 the diffusion weighted signals are sampled to obtain at least one set of resampled diffusion weighted signals.
  • Block 21 is similar to block 1 1.
  • At least two regions of interest (ROIs) in the resampled diffusion weighted singals are identified, where each ROI comprises at least one voxel.
  • a diffusion tensor is determined for each voxel in each ROI from the resampled diffusion weighted signals.
  • Block 23 is similar to block 1 . except the diffusion tensor is determined within at least two ROIs.
  • an empirical statistical distribution is determined for a quantity associated with the diffusion tensor from the diffusion tensors for each voxel in each ROI.
  • Block 24 is similar to block 12. except the diffusion tensor is determined for at least two ROIs.
  • the empirical statistical distributions for the quantity are compared across the ROIs to obtain differences in features of the empirical statistical distributions for the quantity between the ROIs.
  • the differences in the features of the empirical statistical distributions for the quantity are determined by performing hypothesis tests on the differences in the features of the empirical statistical distributions for the quantity between the regions of interest.
  • similar analyses as those described for block 14 can be performed for the ROIs in block 25.
  • the flow diagram is for a single time point.
  • the flow diagram of Figure 1 can be performed for two or more time points for a subject.
  • the flow diagram of Figure 1 can be performed shortly after the stroke occurs as a first time point and can be performed a month or so thereafter as a second time point.
  • the results for the first time point and the second time point can thereafter be compared to analyze differences in the human being over time. Additional time points can be obtained for further comparison and analysis.
  • Figure 3 illustrates this concept as a flow diagram for analyzing DT-MR signals for first and second time points. The technique can be modified to include additional time points. Further details of the flow diagram in Figure 3 are discussed below in the following sections.
  • Block 30 diffusion weighted signals for a first time point are acquired, and each signal contains voxels. Preferably, each signal contains at least one voxel.
  • Block 30 is similar to block 10. except for the first time point.
  • Block 31 the diffusion weighted signals for the first time point are sampled to obtain at least one set of resampled diffusion weighted signals for the first time point.
  • Block 31 is similar to block 1 1. except for the first time point.
  • an ROI of the resampled diffusion weighted signals for the first time point is identified, where the ROI comprises at least one voxel.
  • the number of voxels in the ROI can be as many as the number of voxels in the resampled diffusion weighted signals and as few as one voxel.
  • Block 33 a diffusion tensor is determined for each voxel in the ROI from the resampled diffusion weighted signals for the first time point.
  • Block 33 is similar to block 12, except for the first time point.
  • Block 34 an empirical statistical distribution is determined for a quantity associated with the diffusion tensor for the first time point.
  • Block 34 is similar to block 13. except for the first time point.
  • each signal contains at least one voxel. Preferably, each signal contains at least one voxel.
  • Block 35 is similar to block 30. except for the second time point.
  • the diffusion weighted signals for the second time point are sampled to obtain at least one set of resampled diffusion weighted signals for the second time point.
  • Block 36 is similar to block 31. except for the second time point.
  • an ROI of the resampled diffusion weighted signals for the second time point is identified, where the ROI comprises at least one voxel.
  • the number of voxels in the ROI can be as many as the number of voxels in the resampled diffusion weighted signals and as few as one voxel.
  • approximately the same ROIs are identified as in blocks 32 and 37.
  • a diffusion tensor is determined for each voxel in the ROI from the resampled diffusion weighted signals for the second time point.
  • Block 38 is similar to block 33, except for the second time point.
  • an empirical statistical distribution is determined for a quantity associated with the diffusion tensor for the second time point.
  • Block 39 is similar to block 34. except for the second time point.
  • features of the empirical statistical distributions are compared for the quantity for the at least one ROI between the first and second time points. By comparing the empirical statistical distributions for the two different time points, changes in time can be monitored and analyzed. In addition, similar analyses as those described for block 14 can be performed for the two time points in block 40. After first determining the diffusion tensor in block 12 of Figure 1. an empirical statistical distribution of a quantity associated with the diffusion tensor is determined in block 13 of Figure 1.
  • the diffusion tensor may not need to be determined first, as in block 21.
  • Figure 4 illustrates this concept as a flow diagram for analyzing DT-MR signals without determining the diffusion tensor. Further details of the flow diagram in Figure 4 are discussed below in the following sections.
  • each signal contains voxels.
  • each signal contains at least one voxel.
  • Block 45 is similar to block 10.
  • a quantity associated with the diffusion tensor of the diffusion weighted signals is determined from the diffusion weighted signals.
  • the quantity is an approximation of the Trace of the diffusion tensor of the diffusion weighted signals.
  • the Trace(D) can be approximated using a total of five. four. two. or one diffusion weighted image(s).
  • Block 47 an empirical statistical distribution is determined for the quantity associated with the diffusion tensor. Block 47 is similar to block 13.
  • Block 48 a subject is diagnosed based on the empirical statistical distribution of the quantity. Block 48 is similar to block 14.
  • one or more ROIs can be analyzed using the technique of Figure 4 combined with the technique of Figure 2.
  • two or more time points can be analyzed using the technique of Figure 4 combined with the technique of Figure 3.
  • the background noise properties of magnitude MRI images are known to conform to a Ricean distribution (46), (47).
  • DT-MRI diffusion tensor magnetic resonance imaging
  • the components of the diffusion tensor are estimated from noisy diffusion weighted (DW) magnitude images using regression analysis, usually linear regression of the log-linearized DW magnitude signals ( 17).
  • DW diffusion weighted
  • 17 log-linearized DW magnitude signals
  • Diffusion tensor data was generated using Monte Carlo simulations by assuming mat the noise in the magnitude images is described by a Ricean distribution. Different types of diffusion processes were modeled. Both isotropic and anisotropic diffusion tensors were used, covering typical values obtained in a living human brain. Also, a wide range of S N values ( 1-100) and imaging parameters were explored (e.g.. the number of images acquired for regression ni was varied from 7 to 70).
  • KS Kolmogorov-Smirnov
  • ⁇ and ⁇ are straightforward. Estimates of the sample covariance matrix were obtained empirically, using bootstrap analysis or Monte Carlo methods, and analytically, using linear regression models ( 17). The linear model estimates for log-linearized data are biased for low S/N. particularly the variances of the diagonal elements of the tensor. Also, the prediction of ⁇ degrades for high diffusion attenuation (i.e.. large Trace(b)) when the signal approaches the background noise level. Based on experiments for isotropic diffusion of water, the diffusion tensor components follow a Gaussian distribution.
  • Figure 5 illustrates probability distributions of the six independent components of the effective diffusion tensor, namely the anisotropic D eff components, obtained from Monte Carlo simulations.
  • the solid line indicates fit to a Gaussian distribution.
  • Diffusion Tensor Magnetic Resonance Imaging have grown significantly in recent years. From the measured diffusion tensor, D tff , one can calculate the principal diffusivities and other rotationally invariant measures, such as the Trace of the diffusion tensor. Trace(D eff ), the relative anisotropy, and other histological stains. To date, only mean values, or first order moments of these quantities, could be estimated in each voxel. Knowing the uncertainties or higher moments of these quantities, and their probability distributions, could help design DT-MRI experiments more efficiently, and improve the ability to analyze DT-MRI data.
  • the invention uses a bootstrap method (48), which is an empirical, non-parametric re-sampling technique to obtain estimates of standard errors, confidence intervals, probability distributions, and other measures of uncertainty.
  • bootstrap samples are used to infer the properties of a sample. The applicability of the bootstrap method to DT-MRI is demonstrated, and the limitations of the bootstrap method are assessed using experimental DT-MRI data and data synthesized using Monte Carlo simulations. 3.2.
  • the effective diffusion tensor. D efr . is estimated by applying magnetic field gradients in at least six oblique directions specified by the symmetric matrix, b. From b and the measured T 2 -weighted signals. D eff is estimated using weighted multivariate linear regression (17), where A(0). the T2-weighted signal in the absence of diffusion weighting, is also estimated. Monte Carlo simulations of the DT-MRI experiment were implemented (40). Using the Monte Carlo simulation, a set of n noisy signals per oblique direction and also n signals using a zero/small magnetic field gradient were obtained and yielded a total of 7n signals, which are referred to as the original set.
  • a bootstrap set is created by drawing n random samples (with replacement) from each of the seven sets to obtain a "'new " set of In signals. By repeating this step many times. N ⁇ , many estimates of each of D eff are obtained, along with other quantities of interest, e.g., diffusion tensor. Trace(D eff ), eigenvalues, eigenvectors, etc. The ""true” distribution of these quantities can also be predicted using Monte Carlo simulations to validate the bootstrap results.
  • a bootstrap standard error is the non-parametric maximum likelihood estimate of the true SE. It was found that the bootstrap method is particularly suitable for DT-MRI measurements since reliable estimates of errors can be obtained for a single voxel within inhomogeneous regions. This information cannot be obtained using conventional approaches, since only n samples can be used for estimating the SE (n is commonly less than 5). If one uses voxels within an ROI to estimate errors, the SE is largely exaggerated due to inhomogeneities within the ROI.
  • Figure 6 illustrates a probability distribution of D within the ROI obtained using bootstrap analysis which reveals the existence of different modes.
  • the ROI is drawn in gray matter but close to CSF hence partial volume artifacts can be seen.
  • Figure 7 illustrates the probability distribution of D xx obtained within a single voxel.
  • the probability distribution of D xx is obtained within a single voxel in gray matter using the bootstrap analysis.
  • the validity of the bootstrap method has been confirmed using both real data and Monte Carlo simulations.
  • the bootstrap method facilitates the use of many new statistical approaches to design and analyze DT-MRIs.
  • the inventors discovered that estimates of the uncertainty and the probability distribution of measured and calculated variables in DT-MRI can be obtained using bootstrap analysis, provided a sufficient number of images are used in the bootstrap resampling scheme. Also, estimates of bias, outliers, and confidence levels can be obtained using the bootstrap method, and hypothesis tests can be performed using the bootstrap method.
  • mean values of diffusion tensor components are estimated using statistical methods, such as regression. Using these measured tensor components, one can calculate the principal diffusivities and other rotationally invariant measures, such as the Trace of the diffusion tensor. Trace(D cff ). the relative anisotropy. etc. To date, only mean values, or first (1 st ) order moments have been estimated in each voxel. However, knowing the uncertainties or higher moments of these quantities and their probability distributions could help design DT-MRI experiments more efficiently and also improve analysis of DT- MRI data.
  • the invention uses a novel application of the bootstrap method (48). which is an empirical, non-parametric resampling technique to obtain estimates of standard errors, confidence intervals, probability distributions, and other measures of parameter uncertainty.
  • the bootstrap method is applied to DT-MRI. and the limitations of this approach are assessed using experimental DT-MRI data and data synthesized using Monte Carlo simulations.
  • the effective diffusion tensor, D eff is estimated by applying magnetic field gradients in at least six oblique directions specified by a symmetric matrix b. From b and the measured T 2 -weighted signals. D eff is estimated using weighted multivariate linear regression (17), where A(0) is also estimated. Monte Carlo simulations of the DT-MRI experiment were implemented (40). Using the Monte Carlo simulation technique, a set of /? noisy signals per oblique direction and also n signals using a zero/small magnetic field gradient were obtained and yielded a total of In signals, which are referred to as the original set.
  • a bootstrap set is also created by drawing n random samples (with replacement) from each of the seven signals in the original sets to obtain a "new" set of 7n signals.
  • N ⁇ many estimates of each of D ef were obtained, along with other associated quantities of interest, e.g., diffusion tensor deviatoric. eigenvectors, etc.
  • computation time scales with the number N ⁇ , on a parallel computer the bootstrap method can be implemented more efficiently.
  • the "true " distribution can also be predicted using Monte Carlo simulations to validate the bootstrap results.
  • Figure 8 illustrates a comparison between a "true " probability distribution of D x tensor component (thick solid line) and three bootstrap estimates corresponding to three different original sets (thin solid lines), and the estimates obtained from an original set (vertical lines).
  • the thick dashed line represents the true distribution, and the solid lines represent the bootstrap estimates.
  • the vertical dashed lines are estimates of the original sets used with the bootstrap estimates. Since the probability with which bootstrap samples are drawn is uniform, the estimate of the mean does not change while the standard error (SE) is the nonparametric maximum likelihood estimate of the true SE. With the smaller the number of images n used, the greater are the artifacts that the bootstrap distributions produce.
  • SE standard error
  • Figure 9A illustrates data generated using a diffusion phantom.
  • Figure 9B illustrates an image that is the reciprocal of the angle subtending the smallest cone of uncertainty for the three eigenvectors obtained from the bootstrap method.
  • the intensity of each voxel is inversely proportional to the value of the smallest of the three cone angles.
  • Trace(D) Since the first measurement of Trace(D) ( 1 ), where D is the effective diffusion tensor of water, and the first demonstration of Trace MR images (44). it has been widely reported that Trace(D) is uniform in normal brain parenchyma, drops substantially in ischemic regions during acute stroke, and increases significantly in chronic stroke. Still, distinguishing changes in Trace(D) caused by alterations in the tissue ' s physiologic state, its physical properties, or its microstructure. from those caused by background noise, is still problematic.
  • hypotheses can be formulated about the moments of Trace(D) within ROIs, and tested quantitatively (e.g., see (51 )) for the first time using the invention.
  • the relevant hypotheses such as whether the Trace(D) is uniform in normal brain parenchyma, whether the Trace(D) drops significantly in ischemic areas, or whether the Trace(D) is elevated in diseased areas of brain parenchyma, can be tested clinically or biologically.
  • the hypothesis that the variance of the Trace in different ROIs can be tested. Prior to the invention the variance of the Trace has never been examined, and now. with the invention, may provide valuable physiological information.
  • the variance of the Trace(D) can be examined using F-tests adapted for this pu ⁇ ose.
  • F-tests adapted for this pu ⁇ ose.
  • a Kolmogorov-Smirnov test or one of its variants can be used. These tests indicate whether Trace data is corrupted by systematic artifacts (e.g., by motion).
  • FIG 10 illustrates selecting individual voxels as regions of interest. Voxels containing gray matter (GM). white matter (WM). and CSF are indicated as being selected. Each region of interest has a single voxel.
  • GM gray matter
  • WM white matter
  • CSF CSF
  • Figure 1 I B illustrates empirical bootstrap data obtained from a single voxel in Figure 10 containing white matter, and data in Figure 1 I B was also fitted using a Gaussian distribution.
  • Figure 1 1 C illustrates empirical bootstrap data obtained from a single voxel in Figure 10 containing CSF. and data in Figure 1 I C was also fitted using a Gaussian distribution.
  • the empirical statistical distributions for Trace(D) closely track the parametric statistical distributions for Trace(D), and it can be inferred that almost no artifacts exist in the diffusion weighted images.
  • Quantitative hypothesis tests can be constructed for differences or changes in Trace(D) between individual voxels either by using non-parametric methods to generate many replicates of a DT-MRI experiment from a single study, or by pooling data from a single study to construct a distribution of D within an ROI.
  • useful features can be determined about Trace(D) in a single voxel, within an ROI. and between different ROIs.
  • quantitative hypothesis tests for Trace(D) can be constructed for the following:
  • Trace measurement is statistically significant at a fixed statistical power.
  • ⁇ is plotted against SNR for an ROI containing 50 voxels and a statistical power of 0.95.
  • the different curves in Figure 12 correspond to different p-levels. and the higher the confidence level desired, the larger ⁇ is at each SNR.
  • the left axis of Figure 12 displays the minimum detectable change in Trace(D) ( ⁇ r/sec). Also, the nearly hyperbolic relationship between ⁇ and SNR indicates the tradeoff between ⁇ and SNR. Similar results are obtained for the minimal detectable significant difference in the variance of Trace(D).
  • Trace(D) is unbiased for SNR above approximately 5 (62) and is shown here to be Gaussian distributed.
  • This new methodology of the invention should significantly improve the quality, clinical value, and biological utility of DT-MRI Trace data because the statistical analysis of the invention can now be repeated for any DWI acquisition.
  • the invention provides the first quantitative techniques for testing hypotheses about Trace(D), and its changes in health, development, degeneration, and disease. For the first time. Trace MR imaging experiments can be designed, inte ⁇ reted. and analyzed (including the detection of statistical artifacts) using rigorous statistical tests rather than ad hoc statistical methods. Confidence intervals can be assigned to observe changes in Trace(D). This approach can find applications to longitudinal, multi-site, drug discovery, and therapeutic efficacy studies, as well as to diagnostic assessment of. for example, stroke, tumors, and other diseases and conditions.
  • the invention can be used to improve the ability to detect subtle changes of diffusion properties, such as required in grading tumors or in identifying the ischemic penumbra. Further, the invention can be used to detect systematic artifacts (e.g., motion) in DWIs on a voxel-by-voxel basis. Finally, this statistical methodology can also be applied to segmenting tissues by their diffusion properties. This framework can also be extended to treat variability of other DT-MRI "stains.”
  • Trace of an effective diffusion tensor is described by a univariate Gaussian distribution whose mean and variance can be estimated directly from DT-MRI data.
  • Quantitative statistical hypotheses test can be designed to discriminate between biologically meaningful changes in Trace(D) (e.g., those occurring in health, development, degeneration, and disease) and MR background noise present in all diffusion weighted images. Such tests include determining for any particular signal-to-noise ratio (SNR), the minimum detectable change in Trace(D). whether Trace(D) is uniform in normal brain parenchyma, or whether there is a significant difference between Trace(D) in one brain region and another.
  • SNR signal-to-noise ratio
  • Trace(D) data can be assessed in both longitudinal and multi-site studies.
  • the significance of differences in the variance of Trace(D) can be measured between different voxels, different ROIs, and different subjects.
  • Non-parametric (e.g., bootstrap) methods can be used to determine second and higher moments of the distribution of Trace(D) within each voxel.
  • the departure from normality of the distribution of Trace(D) can be used as a sensitive measure of systematic artifacts, for example, caused by subject motion, in diffusion weighted image (DWI) acquisition.
  • DWI diffusion weighted image
  • knowing the distribution of Trace(D) permits the use of statistically based image segmentation schemes, such as k-means. for distinguishing between different tissues compartments on the basis of their diffusion properties.
  • the decrease of Trace(D) in ischemic areas has been reported to be as large as 50% (54); in chronic stroke, the increase can be of the same magnitude (54).
  • the concomitant changes in tissue microstructure, architecture, or physical properties may be small.
  • changes in Trace(D) could be difficult to distinguish from signal variations due to background radio frequency (RF) noise. Examples include: identifying and grading tumors (55); measuring small differences in mean diffusivitv between white and gray matter in acute stroke (56); identifying MS lesions (57). (58); and following diffuse changes in neurodegenerative disorders such as Pelizaeus-Merzbacher disease (36).
  • Trace(D) Wallerian degeneration (25), and Parkinson ' s and Alzheimer ' s diseases (59). If the spatial uniformity of Trace(D) in a normal brain can be established quantitatively, a significant deviation from Trace(D) could be a useful and sensitive clinical marker for abnormal development or disease, and may also advance methods for their early detection or screening.
  • the new statistical testing framework can be used in reverse to design MRI scanning protocols that are optimal (e.g., that require the fewest DWIs, take the shortest time, etc.). More generally, despite the usefulness of Trace(D) in clinical diagnosis, there is presently no quantitatively rigorous methodology available to compare its values in different voxels or regions of interest (ROI) for the same subject during a single scan (e.g.. to compare normal and abnormal tissue), to compare its values in the same ROI of the same subject scanned at different time points (e.g., in a longitudinal study), or to compare its values among ROIs in different subjects (e.g.. in a multi-center study).
  • ROI regions of interest
  • diffusion weighted images can be contaminated by systematic artifacts, for example, subject motion, eddy currents, susceptibility differences, etc. (60).
  • artifacts for example, subject motion, eddy currents, susceptibility differences, etc.
  • This section further provides a prescription for detecting systematic artifacts in each voxel of a DWI by assessing the departure of the distribution of Trace(D) from its known distribution.
  • Trace(D) is distributed according to: p( Tracei D)) ) (6) where ⁇ is the population mean and ⁇ ⁇ " is the population variance.
  • the population mean of Trace(D) is the sum of the population means of the three diagonal elements of D.
  • the population variance of Trace(D) is the sum of the variances of the diagonal elements of the diffusion tensor, D x , Dyy, and Dzz, (e.g., Var(D )), and of their covariances (e.g., Cov(D , Dv ).
  • the variances are all positive, the covariances can be either sign, and can contribute to the total variance.
  • ⁇ of Trace(D) positively or negatively.
  • the population values are ⁇ , ⁇ x", and ⁇ are unknown a priori.
  • the population values must be estimated from experimental data, which can be performed in two manners.
  • D is estimated in each voxel using multivariate linear regression on a complete set of DWIs. from which sample estimates of ⁇ . ⁇ and ⁇ (17) are obtained, or these estimates are obtained.
  • D is estimated in each voxel using the bootstrap method, in which a subset of the original data set is repeatedly sampled with replacement, as described in previous sections, and the moments of the diffusion tensor elements and functions of them, such as Trace(D). are empirically determined.
  • the hypotheses about its moments within individual voxels or within ROIs can be formulated and tested quantitatively (e.g., see (51)) for the first time with the invention.
  • clinical or biological relevant hypotheses such as whether the Trace(D) is uniform in normal brain parenchyma, or whether the Trace(D) drops or is elevated in diseased areas of brain parenchyma, can be tested.
  • Trace(D) is uniform within an ROI. For example, the following tests can be performed: whether the distribution of Trace(D) is described by a distribution with a single mean for the given level of background noise; whether the distribution of Trace(D) is described by a distribution with a single variance for the given level of background noise; whether the distribution of Trace(D) is described by a normal distribution for a given level of background noise: using a t-test for testing null hypothesis of a single mean of Trace(D); using an F-test for testing null hypothesis of a single variance of Trace(D); and using a Kolmogorov-Smirnov test for testing normal distribution of Trace(D).
  • Trace(D) is normally distributed within each voxel, and if Trace(D) is not normally distributed, the data has been corrupted by a systematic artifact.
  • Figure 13 illustrates the detection of artifacts in diffusion weighted images.
  • the distribution of Trace(D) obtained via the bootstrap method is not described by a Gaussian probability density function.
  • the test of Trace(D) can be preformed on a voxel-by- voxel basis.
  • the test is accomplished by using the bootstrap method, which is a resampling technique, where an empirical distribution of diffusion tensor values is obtained in each voxel from a single set of diffusion weighted images. This method, and its applications to DT-MRI. are described in the previous sections.
  • Trace(D) is uniform, (i.e., Trace(D) is consistent with being drawn from a Gaussian distribution with the same mean and variance).
  • the Trace(D) and other quantities are determined based on a determination of the diffusion tensor from, preferably, seven or more diffusion weighted images. As an option, these quantities, such as Trace(D). can be approximated using fewer diffusion weighted images, such as five. four, two or one diffusion weighted images.
  • estimates of Trace(D) can be obtained by summing four tetrahedrally encoded diffusion weighted images, each of which is normalized by one non-diffusion weighted image. This approach was proposed in (63) and (64). An estimate of Trace(D) can also be obtained by summing three apparent diffusion coefficients obtained from three diffusion weighted images whose diffusion gradients are applied along three orthogonal directions. An example of this approach is given in (53).
  • a water phantom is used in this case containing 0.5% polvacrvlic acid (500.000 MW) by volume.
  • the long chain polymer present in such a low concentration does not affect water diffusivitv of the bulk solution but does suppress a number of potential experimental artifacts that could be mistaken for water diffusion, such as mechanical vibrations (shear and bulk modes) and convection, owing to thermal gradients that may arise in the phantom.
  • a thermal blanket was also used to ensure temperature uniformity throughout the phantom to within 0.1 ° C. This further reduces the possibility of Rayleigh convection arising in this sample, and limits the thermal variation in the diffusion coefficient in the sample, which is approximately 1.5% per 1 °C (61 ).
  • the MRI system was precalibrated to ensure that correct gradient scale factors were applied when obtaining DWIs. using a method similar to one outlined previously (18). Other artifacts, such as eddy currents effects were also mitigated by using well established approaches (see (21 )). Susceptibility artifacts are only present near the walls of the container, so the regions of interest are always chosen far from them.
  • the b-matrix for each DWI acquisition is numerically calculated (13), ( 12), (3) using software and a computer. Diffusion tensors are estimated using the method outlined in (17) using the measured intensities of the DWIs and their corresponding calculated b-matrices using multivariate linear regression.
  • the described embodiments, as well as the examples discussed herein, are non- limiting examples.

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Abstract

Dans cette invention, des signaux de résonance magnétique de tenseur de diffusion sont analysés. Des signaux de diffusion pondérés sont acquis, chaque signal présentant un groupe de plusieurs voxels (10). Les signaux de diffusion pondérés sont échantillonnés de manière à obtenir au moins un ensemble de signaux (11) de diffusion pondérés reéchantillonnés. Un tenseur de diffusion pour chaque voxel est déterminé à partir de chaque ensemble des signaux (12) de diffusion pondérés reéchantillonnés. Une distribution statistique empirique est déterminée pour une quantité associée au tenseur de diffusion à partir des tenseurs de diffusion déterminés à partir de l'ensemble de signaux (13) de diffusion pondérés reéchantillonnés.
PCT/US2000/014290 1999-05-21 2000-05-19 Determination d'une distribution empirique statistique du tenseur de diffusion dans un procede irm WO2000072038A1 (fr)

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WO2002086478A2 (fr) * 2001-04-23 2002-10-31 Metabometrix Limited Procedes d'analyse de donnees spectrales et leurs applications
WO2002086478A3 (fr) * 2001-04-23 2003-04-10 Metabometrix Ltd Procedes d'analyse de donnees spectrales et leurs applications
US7901873B2 (en) 2001-04-23 2011-03-08 Tcp Innovations Limited Methods for the diagnosis and treatment of bone disorders
CN104634805A (zh) * 2013-11-13 2015-05-20 中国科学院苏州纳米技术与纳米仿生研究所 仿水黾腿结构液下接触角的测量方法
CN104634805B (zh) * 2013-11-13 2017-05-03 中国科学院苏州纳米技术与纳米仿生研究所 仿水黾腿结构液下接触角的测量方法
CN110288587A (zh) * 2019-06-28 2019-09-27 重庆同仁至诚智慧医疗科技股份有限公司 一种缺血性脑卒中磁共振影像的病灶识别方法

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