Estimating fiber orientation in ultrasound imaging through correlation and multivariate Granger causality.

Neuromonitoring-derived indices play an important role in implementing personalised medicine for traumatic brain injury patients. A well-established example is the pressure reactivity index (PRx), calculated from spontaneous fluctuations of arterial blood pressure (ABP) and intracranial pressure (ICP). PRx assumes causal relationship between ABP and ICP but lacks the check for this assumption. Granger causality (GC) — a method of assessing causal interactions between time series data — is gaining popularity in neurosciences. In our work, we used ABP and ICP data recorded at the frequency of 100 Hz or higher from 235 traumatic brain injury patients. We focused on time domain GC. Analysis was first performed directly on high-resolution data, which included pulse waves. We showed that due to the measurement delay in high-resolution ABP data, GC analysis may erroneously indicate strong ICP→ABP causal relation. Subsequently, the data were downsampled to 0.1 Hz, effectively removing pulse and respiratory waves. We aimed to investigate how different ways of calculating GC influence results and which way should be recommended for ABP-ICP recordings. We considered aspects like selecting autoregressive model order and dealing with data non-stationarity. In addition, we generated simulated signals to investigate the influence of gaps and different procedures of missing data imputation on GC estimation. We showed that unlike methods which interpolate missing data, replacing missing data by white Gaussian noise did not increase the rate of false GC detection. Python source code used in this study is available at: https://github.com/m-m-placek/python-icmplus-granger-causality. Statement of significanceAssessing causality between time series data is of particular interest when neuromonitoring indices are derived from those time series and causal interaction between them is assumed. Causality assessment can improve reliability of such indices and open pathways for their safe clinical implementation. Granger Causality (GC) has recently been investigated in data collected from traumatic brain injury patients. However, there are two main issues related to applications suggested in these studies. Firstly, they considered GC for entire multi-day data recordings or for 24-h long episodes. There is interest in considering causal relationships in finer granularity, also in terms of their potential real-time applications at the bedside. Secondly, GC calculation requires selecting some parameters and there is no unique nor standardised way of doing that. Many papers often provide very brief description of data pre-processing and GC calculation. For this reason, it can be harder to reproduce and compare results derived from GC application. Different ways of obtaining GC may potentially lead to inconsistent results. Here, we attempted to explore possibility of time-varying GC of finer granularity and to provide general guidelines for application of GC to neurocritical care time series affected by periods of missing values.

Dictionary-based fiber orientation estimation with improved spatial consistency.

Granger causality (GC) is one of the most popular measures to reveal causality influence of time series and has been widely applied in economics and neuroscience. Especially, its counterpart in frequency domain, spectral GC, as well as other Granger-like causality measures have recently been applied to study causal interactions between brain areas in different frequency ranges during cognitive and perceptual tasks. In this paper, we show that: 1) GC in time domain cannot correctly determine how strongly one time series influences the other when there is directional causality between two time series, and 2) spectral GC and other Granger-like causality measures have inherent shortcomings and/or limitations because of the use of the transfer function (or its inverse matrix) and partial information of the linear regression model. On the other hand, we propose two novel causality measures (in time and frequency domains) for the linear regression model, called new causality and new spectral causality, respectively, which are more reasonable and understandable than GC or Granger-like measures. Especially, from one simple example, we point out that, in time domain, both new causality and GC adopt the concept of proportion, but they are defined on two different equations where one equation (for GC) is only part of the other (for new causality), thus the new causality is a natural extension of GC and has a sound conceptual/theoretical basis, and GC is not the desired causal influence at all. By several examples, we confirm that new causality measures have distinct advantages over GC or Granger-like measures. Finally, we conduct event-related potential causality analysis for a subject with intracranial depth electrodes undergoing evaluation for epilepsy surgery, and show that, in the frequency domain, all measures reveal significant directional event-related causality, but the result from new spectral causality is consistent with event-related time-frequency power spectrum activity. The spectral GC as well as other Granger-like measures are shown to generate misleading results. The proposed new causality measures may have wide potential applications in economics and neuroscience.

Comparison of diffusion MRI and CLARITY fiber orientation estimates in both gray and white matter regions of human and primate brain

Wiener and Granger have introduced an intuitive concept of causality (Granger causality) between two variables which is based on the idea that an effect never occurs before its cause. Later, Geweke generalized this concept to a multivariate Granger causality, i.e. n variables Granger-cause another variable. Although Granger causality is not "effective causality" in the Aristothelic sense, this concept is useful to infer directionality and information flow in observational data. Granger causality is usually identified by using VAR (Vector Autoregressive) models due to their simplicity. In the last few years, several VAR-based models were presented in order to model gene regulatory networks. Here, we generalize the multivariate Granger causality concept in order to identify Granger causalities between sets of gene expressions, i.e. whether a set of n genes Granger-causes another set of m genes, aiming at identifying the flow of information between gene networks (or pathways). The concept of Granger causality for sets of variables is presented. Moreover, a method for its identification with a bootstrap test is proposed. This method is applied in simulated and also in actual biological gene expression data in order to model regulatory networks. This concept may be useful for the understanding of the complete information flow from one network or pathway to the other, mainly in regulatory networks. Linking this concept to graph theory, sink and source can be generalized to node sets. Moreover, hub and centrality for sets of genes can be defined based on total information flow. Another application is in annotation, when the functionality of a set of genes is unknown, but this set is Granger-caused by another set of genes which is well studied. Therefore, this information may be useful to infer or construct some hypothesis about the unknown set of genes.

Discovery of gene regulatory network (GRN) from gene expression data gives an insight into tumor developments and underlying structures. Granger causality (GC) is considered as a powerful tool to detect the interactions between elements of a network. Among the various methods suggested for GC, we use Pairwise GC (PGC), Kernel GC (KGC) and Correntropy method. Also GC is defined in two domains. In time domain, GC cannot correctly determine how strongly one time series influences the other when there is directional causality between them, this limitation necessitates an alternative method. In this regard, GC in frequency domain is being applied as a solution. In this paper, first, we conduct a frequency domain analysis on these methods theoretically. We then evaluate the performance of PGC in both domains by applying real HeLa dataset with three experiments and compare it with previous work. Finally, we apply all methods to both synthetic data and a 94-gene HeLa data to illustrate the discovered networks. We show that frequency domain has better performance in discovery of relations at all experiments.

Granger causality (GC) has been widely applied in economics and neuroscience to reveal causality influence of time series. In our previous paper (Hu etal., in IEEE Trans on Neural Netw, 22(6), pp. 829-844, 2011), we proposed new causalities in time and frequency domains and particularly focused on new causality in frequency domain by pointing out the shortcomings/limitations of GC or Granger-alike causality metrics and the advantages of new causality. In this paper we continue our previous discussions and focus on new causality and GC or Granger-alike causality metrics in time domain. Although one strong motivation was introduced in our previous paper (Hu etal., in IEEE Trans on Neural Netw, 22(6), pp. 829-844, 2011) we here present additional motivation for the proposed new causality metric and restate the previous motivation for completeness. We point out one property of conditional GC in time domain and the shortcomings/limitations of conditional GC which cannot reveal the real strength of the directional causality among three time series. We also show the shortcomings/limitations of directed causality (DC) or normalize DC for multivariate time series and demonstrate it cannot reveal real causality at all. By calculating GC and new causality values for an example we demonstrate the influence of one of the time series on the other is linearly increased as the coupling strength is linearly increased. This fact further supports reasonability of new causality metric. We point out that larger instantaneous correlation does not necessarily mean larger true causality (e.g., GC and new causality), or vice versa. Finally we conduct analysis of statistical test for significance and asymptotic distribution property of new causality metric by illustrative examples.

Gradient nonlinearities in magnetic resonance imaging (MRI) cause spatially varying mismatches between the imposed and the effective gradients and can cause significant biases in rotationally invariant diffusion MRI measures derived from, for example, diffusion tensor imaging. The estimation of the orientational organization of fibrous tissue, which is nowadays frequently performed with spherical deconvolution techniques ideally using higher diffusion weightings, can likewise be biased by gradient nonlinearities. We explore the sensitivity of two established spherical deconvolution approaches to gradient nonlinearities, namely constrained spherical deconvolution (CSD) and damped Richardson‐Lucy (dRL). Additionally, we propose an extension of dRL to take into account gradient imperfections, without the need of data interpolation. Simulations show that using the effective b‐matrix can improve dRL fiber orientation estimation and reduces angular deviations, while CSD can be more robust to gradient nonlinearity depending on the implementation. Angular errors depend on a complex interplay of many factors, including the direction and magnitude of gradient deviations, underlying microstructure, SNR, anisotropy of the effective response function, and diffusion weighting. Notably, angular deviations can also be observed at lower b‐values in contrast to the perhaps common assumption that only high b‐value data are affected. In in vivo Human Connectome Project data and acquisitions from an ultrastrong gradient (300 mT/m) scanner, angular differences are observed between applying and not applying the effective gradients in dRL estimation. As even small angular differences can lead to error propagation during tractography and as such impact connectivity analyses, incorporating gradient deviations into the estimation of fiber orientations should make such analyses more reliable.

Estimation of fiber orientations using neighborhood information.

Diffusion tensor imaging is widely used for studying neural fiber trajectories in white matter and for quantifying changes in tissue using diffusion properties at each voxel in the brain. To better model the nature of crossing fibers within complex architectures, rather than using a simplified tensor model that assumes only a single fiber direction at each image voxel, a model mixing multiple diffusion tensors is used to profile diffusion signals from high angular resolution diffusion imaging (HARDI) data. Based on the HARDI signal and a multiple tensors model, spherical deconvolution methods have been developed to overcome the limitations of the diffusion tensor model when resolving crossing fibers. The Richardson-Lucy algorithm is a popular spherical deconvolution method used in previous work. However, it is based on a Gaussian distribution, while HARDI data are always very noisy, and the distribution of HARDI data follows a Rician distribution. This current work aims to present a novel solution to address these issues. By simultaneously considering both the Rician bias and neighbor correlation in HARDI data, the authors propose a localized Richardson-Lucy (LRL) algorithm to estimate fiber orientations for HARDI data. The proposed method can simultaneously reduce noise and correct the Rician bias. Mean angular error (MAE) between the estimated Fiber orientation distribution (FOD) field and the reference FOD field was computed to examine whether the proposed LRL algorithm offered any advantage over the conventional RL algorithm at various levels of noise. Normalized mean squared error (NMSE) was also computed to measure the similarity between the true FOD field and the estimated FOD filed. For MAE comparisons, the proposed LRL approach obtained the best results in most of the cases at different levels of SNR and b-values. For NMSE comparisons, the proposed LRL approach obtained the best results in most of the cases at b-value = 3000 s/mm(2), which is the recommended schema for HARDI data acquisition. In addition, the FOD fields estimated by the proposed LRL approach in regions of fiber crossing regions using real data sets also showed similar fiber structures which agreed with common acknowledge in these regions. The novel spherical deconvolution method for improved accuracy in investigating crossing fibers can simultaneously reduce noise and correct Rician bias. With the noise smoothed and bias corrected, this algorithm is especially suitable for estimation of fiber orientations in HARDI data. Experimental results using both synthetic and real imaging data demonstrated the success and effectiveness of the proposed LRL algorithm.

Fibre orientation in short fibre reinforced thermoplastics depends on injection moulding parameters. There are a lot of different parameters that must be established and controlled to achieve proper injection moulding of a plastic part. These parameters fall within four major areas: pressure, temperature, time, and distance. The aim of this article is estimation of fibre orientation in injection moulding plastics parts and comparison of these results with numerical simulated ones. Stereological metallography was used for estimation of experimental orientation of fibres. The orientation of simple fibre may be defined by the two angles θ and Φ. In a Short Fibre Reinforced Thermoplastic (SFRT) component there are frequently millions of fibres, therefore each individual fibre orientation specifying is very impractical. The fibres orientation in space can be described by the probability distribution function (PDF), Ψ(θ, Φ). Numerical modelling of fibre orientation was realised using MOLDEX3D software. Moldex3D is the CAE product for the plastics injection moulding industry. This software allows to view results of fibre orientation as an orientation of the X direction, Y direction, Z direction, the total orientation and orientation at surface. These first three orientations are relevant for the establishment of second-order orientation tensor. They belong to tensor ́s values a11, a22 and a33. Utilization of stereological metallography for short fibre orientation in plastic matrix is very similar to its utilization for estimation of grain boundaries orientation in polycrystalline alloys cased by plastic deformation. In the case of short glass fibres reinforced thermoplastics it’s structure consist of thermoplastic matrix and reinforcing fibres, which has some preferred orientation in most of cases – the structure is anisotropy. The way of scalar measurement of structure anisotropy is determination of degree of orientation. The anisotropic microstructure is decomposed into isotropic, planar or linear oriented components using stereology methods.

The detection of causality in gene regulatory networks from experimental data, such as gene expression measurements, is a challenging problem. Granger causality, based on a vector autoregressive model, is one of the most popular methods for uncovering the temporal dependencies between time series, and so it can be used for estimating the causal relationships between the genes in the network. The application of multivariate Granger causality to the networks with a big number of variables (genes) requires a variable selection procedure. For fighting with lack of informative data, the so called regularization procedures are applied. Lasso method is a well known example of such a procedure and the multivariate Granger causality method with the Lasso is called Graphical Lasso Granger method. It is widely accepted that the Graphical Lasso Granger method with an inappropriate parameter setting tends to select too many causal relationships, which leads to spurious results. In our previous work, we proposed a thresholding strategy for Graphical Lasso Granger method, called two-level-thresholding and demonstrated how the variable over-selection of the Graphical Lasso Granger method can be overcome. Thus, an appropriate thresholding, i.e. an appropriate choice of the thresholding parameter, is crucial for the accuracy of the Graphical Lasso Granger method. In this paper, we compare the performance of the Graphical Lasso Granger method with an appropriate thresholding to two other Lasso Granger methods (the regular Lasso Granger method and Copula Granger method) as well as to the method combining ordinary differential equations with dynamic Bayesian Networks. The comparison of the methods is done on the gene expression data of the human cancer cell line for a regulatory network of nineteen selected genes. We test the causal detection ability of these methods with respect to the selected benchmark network and compare the performance of the mentioned methods on various statistical measures. The discussed methods apply a dynamic decision making. They are scalable and can be easily extended to networks with a higher number of genes. In our tests, the best method with respect to the precision and computational cost turns out to be the Graphical Lasso Granger method with two-level-thresholding. Although the discussed algorithms were motivated by problems coming from genetics, they can be also applied to other real-world problems dealing with interactions in a multi-agent system.

Granger causality analysis is a popular method for inference on directed interactions in complex systems of many variables. A shortcoming of the standard framework for Granger causality is that it only allows for examination of interactions between single (univariate) variables within a system, perhaps conditioned on other variables. However, interactions do not necessarily take place between single variables but may occur among groups or "ensembles" of variables. In this study we establish a principled framework for Granger causality in the context of causal interactions among two or more multivariate sets of variables. Building on Geweke's seminal 1982 work, we offer additional justifications for one particular form of multivariate Granger causality based on the generalized variances of residual errors. Taken together, our results support a comprehensive and theoretically consistent extension of Granger causality to the multivariate case. Treated individually, they highlight several specific advantages of the generalized variance measure, which we illustrate using applications in neuroscience as an example. We further show how the measure can be used to define "partial" Granger causality in the multivariate context and we also motivate reformulations of "causal density" and "Granger autonomy." Our results are directly applicable to experimental data and promise to reveal new types of functional relations in complex systems, neural and otherwise.

Diffusion magnetic resonance imaging (dMRI) enables in vivo investigation of white matter tracts, where the estimation of fiber orientations (FOs) is a crucial step. Dictionary-based methods have been developed to compute FOs with a lower number of dMRI acquisitions. To reduce the effect of noise that is inherent in dMRI acquisitions, spatial consistency of FOs between neighbor voxels has been incorporated into dictionary-based methods. Because many fiber tracts are tube- or sheet-shaped, voxels belonging to the same tract could share similar FO configurations even when they are not adjacent to each other. Therefore, it is possible to use nonlocal information to improve the performance of FO estimation. In this work, we propose an FO estimation algorithm, Fiber Orientation Reconstruction using Nonlocal and Local Information (FORNLI), which adds nonlocal information to guide FO computation. The diffusion signals are represented by a set of fixed prolate tensors. For each voxel, we compare its patch-based diffusion profile with those of the voxels in a search range, and its nonlocal reference voxels are determined as the k nearest neighbors in terms of diffusion profiles. Then, FOs are estimated by iteratively solving weighted \(\ell _{1}\)-norm regularized least squares problems, where the weights are determined using local neighbor voxels and nonlocal reference voxels. These weights encourage FOs that are consistent with the local and nonlocal information. FORNLI was performed on simulated and real brain dMRI, which demonstrates the benefit of incorporating nonlocal information for FO estimation.

Diffusion tensor imaging (DTI) is a magnetic resonance imaging technique that allows for the visualization and characterization of the white matter tracts of the brain in vivo. DTI does not assess white matter directly. Rather, it capitalizes on the fact that diffusion is isotropic (equal in all directions) in cerebral spinal fluid and cell bodies but anisotropic (greater in one direction than the other directions) in axons that comprise white matter. It provides quantitative information about the degree and direction of water diffusion within individual units of volume within the magnetic resonance image, and by inference, about the integrity of white matter. Measures from DTI can be applied throughout the brain or to regions of interest. Fiber tract reconstruction, or tractography, creates continuous 3-dimensional tracts by sequentially piecing together estimates of fiber orientation from the direction of diffusion within individual volume units. DTI has increased our understanding of white matter structure and function. DTI shows nonlinear growth of white matter tracts from childhood to adulthood. Delayed maturation of the white matter in the frontal lobes may explain the continued growth of cognitive control into adulthood. Relative to good readers, adults and children who are poor readers have evidence of white matter differences in a specific region of the temporo-parietal lobe, implicating differences in connections among brain regions as a factor in reading disorder. Measures from DTI changed in poor readers who improved their reading skills after intense remediation. DTI documents injury to white matter tracts after prematurity. Measures indicative of white matter injury are associated with motor and cognitive impairment in children born prematurely. Further research on DTI is necessary before it can become a routine clinical procedure.