Sparse Solution of Fiber Orientation Distribution Function by Diffusion Decomposition

Fiber orientation is the key information in diffusion tractography. Several deconvolution methods have been proposed to obtain fiber orientations by estimating a fiber orientation distribution function (ODF). However, the L ₂ regularization used in deconvolution often leads to false fibers that compromise the specificity of the results. To address this problem, we propose a method called diffusion decomposition, which obtains a sparse solution of fiber ODF by decomposing the diffusion ODF obtained from q-ball imaging (QBI), diffusion spectrum imaging (DSI), or generalized q-sampling imaging (GQI). A simulation study, a phantom study, and an in-vivo study were conducted to examine the performance of diffusion decomposition. The simulation study showed that diffusion decomposition was more accurate than both constrained spherical deconvolution and ball-and-sticks model. The phantom study showed that the angular error of diffusion decomposition was significantly lower than those of constrained spherical deconvolution at 30° crossing and ball-and-sticks model at 60° crossing. The in-vivo study showed that diffusion decomposition can be applied to QBI, DSI, or GQI, and the resolved fiber orientations were consistent regardless of the diffusion sampling schemes and diffusion reconstruction methods. The performance of diffusion decomposition was further demonstrated by resolving crossing fibers on a 30-direction QBI dataset and a 40-direction DSI dataset. In conclusion, diffusion decomposition can improve angular resolution and resolve crossing fibers in datasets with low SNR and substantially reduced number of diffusion encoding directions. These advantages may be valuable for human connectome studies and clinical research.     

Q-ball imaging of macaque white matter architecture

Diffusion-weighted magnetic resonance imaging holds substantial promise as a technique for non-invasive imaging of white matter (WM) axonal projections. For diffusion imaging to be capable of providing new insight into the connectional neuroanatomy of the human brain, it will be necessary to histologically validate the technique against established tracer methods such as horseradish peroxidase and biocytin histochemistry. The macaque monkey provides an ideal model for histological validation of the diffusion imaging method due to the phylogenetic proximity between humans and macaques, the gyrencephalic structure of the macaque cortex, the large body of knowledge on the neuroanatomic connectivity of the macaque brain and the ability to use comparable magnetic resonance acquisition protocols in both species. Recently, it has been shown that high angular resolution diffusion imaging (HARDI) can resolve multiple axon orientations within an individual imaging voxel in human WM. This capability promises to boost the accuracy of tract reconstructions from diffusion imaging. If the macaque is to serve as a model for histological validation of the diffusion tractography method, it will be necessary to show that HARDI can also resolve intravoxel architecture in macaque WM. The present study therefore sought to test whether the technique can resolve intravoxel structure in macaque WM. Using a HARDI method called q-ball imaging (QBI) it was possible to resolve composite intravoxel architecture in a number of anatomic regions. QBI resolved intravoxel structure in, for example, the dorsolateral convexity, the pontine decussation, the pulvinar and temporal subcortical WM. The paper concludes by reviewing remaining challenges for the diffusion tractography project.     

Diffusion MRI of complex neural architecture

While functional brain imaging methods can locate the cortical regions subserving particular cognitive functions, the connectivity between the functional areas of the human brain remains poorly understood. Recently, investigators have proposed a method to image neural connectivity noninvasively using a magnetic resonance imaging method called diffusion tensor imaging (DTI). DTI measures the molecular diffusion of water along neural pathways. Accurate reconstruction of neural connectivity patterns from DTI has been hindered, however, by the inability of DTI to resolve more than a single axon direction within each imaging voxel. Here, we present a novel magnetic resonance imaging technique that can resolve multiple axon directions within a single voxel. The technique, called q-ball imaging, can resolve intravoxel white matter fiber crossing as well as white matter insertions into cortex. The ability of q-ball imaging to resolve complex intravoxel fiber architecture eliminates a key obstacle to mapping neural connectivity in the human brain noninvasively.     

Angular smoothing and radial regularization of ODF fields: Application on deterministic crossing fiber tractography

The advent of high angular resolution diffusion imaging (HARDI) has opened up new perspectives for the delineation of crossing and branching fiber pathways. However, image acquisition under clinical conditions with limited measurement time faces the problem of poor spatial and angular resolution and the technique’s high susceptibility to noise. In this paper we present a straightforward spatial filter for ODF fields that uses the data-inherent structural information around a voxel as part of a directionally selective method for angular smoothing and radial regularization (ASRR). Especially in regions where fibers cross (multimodal voxels), the method allows us to reduce noise, improve the accuracy of ODF diffusion peaks, and strengthen signals of non-dominant fibers. Moreover, we propose a dynamic scheme in which regularization is applied only to ODFs classified as multimodal. The approach is quantitatively evaluated on synthetic datasets of various configurations. With an in vivo dataset of a human subject, measured under clinical imaging conditions, we demonstrate the method’s ability to improve tractography of non-dominant transcallosal fiber pathways and the long fibers of the superior longitudinal fasciculus.     

Assessment of the Characteristics of Orientation Distribution Functions in HARDI Using Morphological Metrics

Orientation distribution functions (ODFs) are widely used to resolve fiber crossing problems in high angular resolution diffusion imaging (HARDI). The characteristics of the ODFs are often assessed using a visual criterion, although the use of objective criteria is also reported, which are directly borrowed from classic signal and image processing theory because they are intuitive and simple to compute. However, they are not always pertinent for the characterization of ODFs. We propose a more general paradigm for assessing the characteristics of ODFs. The idea consists in regarding an ODF as a three-dimensional (3D) point cloud, projecting the 3D point cloud onto an angle-distance map, constructing an angle-distance matrix, and calculating metrics such as length ratio, separability, and uncertainty. The results from both simulated and real data show that the proposed metrics allow for the assessment of the characteristics of ODFs in a quantitative and relatively complete manner.     

Mapping complex myoarchitecture in the bovine tongue with diffusion-spectrum magnetic resonance imaging

The ability to resolve complex fiber populations in muscular tissues is important for relating tissue structure with mechanical function. To address this issue in the case of tongue, we employed diffusion spectrum imaging (DSI), an MRI method for determining three-dimensional myoarchitecture where myofiber populations are variably aligned. By specifically varying gradient field strength, molecular displacement in a tissue can be determined by Fourier-transforming the echo intensity against gradient strength at fixed gradient pulse spacing. The displacement profiles are visualized by graphing three-dimensional isocontour icons for each voxel, with the isocontour shape and size representing the magnitude and direction of the constituting fiber populations. To validate this method, we simulated a DSI experiment within the constraints of arbitrary crossing fibers, and determined that DSI accurately depicts the angular relationships between these fibers. Considering the fiber relationships in the whole bovine tongue, we compared the images obtained by DSI with those obtained by diffusion tensor imaging in an anterior slice of the lingual core, a region known to possess extensive fiber crossing. In contrast to diffusion tensor imaging, which depicts the anterior core solely as a region with low anisotropy due to the presence of mixed-orientation fiber populations, DSI shows two distinct fiber populations, with an explicit orthogonal relationship to each other. In imaging the whole lingual tissue, we discerned arrays of crossing and noncrossing fibers involving the intrinsic and extrinsic muscles, which merged at regions of interface. We conclude that DSI has the capacity to determine three-dimensional fiber orientation in structurally complex muscular tissues.     

Beyond Cytoarchitectonics: The Internal and External Connectivity Structure of the Caudate Nucleus

While there is ample evidence on the functional and connectional differentiation of the caudate nucleus (CN), less is known about its potential microstructural subdivisions. However, this latter aspect is critical to the local information processing capabilities of the tissue. We applied diffusion MRI, a non-invasive in vivo method that has great potential for the exploration of the brain structure-behavior relationship, in order to characterize the local fiber structure in gray matter of the CN. We report novel evidence of a functionally meaningful structural tri-partition along the anterior-posterior axis of this region. The connectivity of the CN subregions is in line with connectivity evidence from earlier invasive studies in animal models. In addition, histological validation using polarized light imaging (PLI) confirms these results, corroborating the notion that cortico-subcortico-cortical loops involve microstructurally differentiated regions in the caudate nucleus. Methodologically speaking, the comparison with advanced analysis of diffusion MRI shows that diffusion tensor imaging (DTI) yields a simplified view of the CN fiber architecture which is refined by advanced high angular resolution imaging methods.     

Segmentation of parotid glands from registered CT and MR images

PurposeTo develop an automatic multimodal method for segmentation of parotid glands (PGs) from pre-registered computed tomography (CT) and magnetic resonance (MR) images and compare its results to the results of an existing state-of-the-art algorithm that segments PGs from CT images only.MethodsMagnetic resonance images of head and neck were registered to the accompanying CT images using two different state-of-the-art registration procedures. The reference domains of registered image pairs were divided on the complementary PG regions and backgrounds according to the manual delineation of PGs on CT images, provided by a physician. Patches of intensity values from both image modalities, centered around randomly sampled voxels from the reference domain, served as positive or negative samples in the training of the convolutional neural network (CNN) classifier. The trained CNN accepted a previously unseen (registered) image pair and classified its voxels according to the resemblance of its patches to the patches used for training. The final segmentation was refined using a graph-cut algorithm, followed by the dilate-erode operations.ResultsUsing the same image dataset, segmentation of PGs was performed using the proposed multimodal algorithm and an existing monomodal algorithm, which segments PGs from CT images only. The mean value of the achieved Dice overlapping coefficient for the proposed algorithm was 78.8%, while the corresponding mean value for the monomodal algorithm was 76.5%.ConclusionsAutomatic PG segmentation on the planning CT image can be augmented with the MR image modality, leading to an improved RT planning of head and neck cancer.     

Microstructure of temporo-parietal white matter as a basis for reading ability: evidence from diffusion tensor magnetic resonance imaging

Diffusion tensor magnetic resonance imaging (MRI) was used to study the microstructural integrity of white matter in adults with poor or normal reading ability. Subjects with reading difficulty exhibited decreased diffusion anisotropy bilaterally in temporoparietal white matter. Axons in these regions were predominantly anterior-posterior in direction. No differences in T1-weighted MRI signal were found between poor readers and control subjects, demonstrating specificity of the group difference to the microstructural characteristics measured by diffusion tensor imaging (DTI). White matter diffusion anisotropy in the temporo-parietal region of the left hemisphere was significantly correlated with reading scores within the reading-impaired adults and within the control group. The anisotropy reflects microstructure of white matter tracts, which may contribute to reading ability by determining the strength of communication between cortical areas involved in visual, auditory, and language processing.     

A model for diffusion in white matter in the brain

Diffusion of molecules in brain and other tissues is important in a wide range of biological processes and measurements ranging from the delivery of drugs to diffusion-weighted magnetic resonance imaging. Diffusion tensor imaging is a powerful noninvasive method to characterize neuronal tissue in the human brain in vivo. As a first step toward understanding the relationship between the measured macroscopic apparent diffusion tensor and underlying microscopic compartmental geometry and physical properties, we treat a white matter fascicle as an array of identical thick-walled cylindrical tubes arranged periodically in a regular lattice and immersed in an outer medium. Both square and hexagonal arrays are considered. The diffusing molecules may have different diffusion coefficients and concentrations (or densities) in different domains, namely within the tubes' inner core, membrane, myelin sheath, and within the outer medium. Analytical results are used to explore the effects of a large range of microstructural and compositional parameters on the apparent diffusion tensor and the degree of diffusion anisotropy, allowing the characterization of diffusion in normal physiological conditions as well as changes occurring in development, disease, and aging. Implications for diffusion tensor imaging and for the possible in situ estimation of microstructural parameters from diffusion-weighted MR data are discussed in the context of this modeling framework.     

Characterizing Brain Structures and Remodeling after TBI Based on Information Content,Diffusion Entropy

Background

To overcome the limitations of conventional diffusion tensor magnetic resonance imaging resulting from the assumption of a Gaussian diffusion model for characterizing voxels containing multiple axonal orientations, Shannon''s entropy was employed to evaluate white matter structure in human brain and in brain remodeling after traumatic brain injury (TBI) in a rat.

Methods

Thirteen healthy subjects were investigated using a Q-ball based DTI data sampling scheme. FA and entropy values were measured in white matter bundles, white matter fiber crossing areas, different gray matter (GM) regions and cerebrospinal fluid (CSF). Axonal densities'' from the same regions of interest (ROIs) were evaluated in Bielschowsky and Luxol fast blue stained autopsy (n = 30) brain sections by light microscopy. As a case demonstration, a Wistar rat subjected to TBI and treated with bone marrow stromal cells (MSC) 1 week after TBI was employed to illustrate the superior ability of entropy over FA in detecting reorganized crossing axonal bundles as confirmed by histological analysis with Bielschowsky and Luxol fast blue staining.

Results

Unlike FA, entropy was less affected by axonal orientation and more affected by axonal density. A significant agreement (r = 0.91) was detected between entropy values from in vivo human brain and histologically measured axonal density from post mortum from the same brain structures. The MSC treated TBI rat demonstrated that the entropy approach is superior to FA in detecting axonal remodeling after injury. Compared with FA, entropy detected new axonal remodeling regions with crossing axons, confirmed with immunohistological staining.

Conclusions

Entropy measurement is more effective in distinguishing axonal remodeling after injury, when compared with FA. Entropy is also more sensitive to axonal density than axonal orientation, and thus may provide a more accurate reflection of axonal changes that occur in neurological injury and disease.     

Differential development of human brain white matter tracts

Neuroscience is increasingly focusing on developmental factors related to human structural and functional connectivity. Unfortunately, to date, diffusion-based imaging approaches have only contributed modestly to these broad objectives, despite the promise of diffusion-based tractography. Here, we report a novel data-driven approach to detect similarities and differences among white matter tracts with respect to their developmental trajectories, using 64-direction diffusion tensor imaging. Specifically, using a cross-sectional sample comprising 144 healthy individuals (7 to 48 years old), we applied k-means cluster analysis to separate white matter voxels based on their age-related trajectories of fractional anisotropy. Optimal solutions included 5-, 9- and 14-clusters. Our results recapitulate well-established tracts (e.g., internal and external capsule, optic radiations, corpus callosum, cingulum bundle, cerebral peduncles) and subdivisions within tracts (e.g., corpus callosum, internal capsule). For all but one tract identified, age-related trajectories were curvilinear (i.e., inverted 'U-shape'), with age-related increases during childhood and adolescence followed by decreases in middle adulthood. Identification of peaks in the trajectories suggests that age-related losses in fractional anisotropy occur as early as 23 years of age, with mean onset at 30 years of age. Our findings demonstrate that data-driven analytic techniques may be fruitfully applied to extant diffusion tensor imaging datasets in normative and neuropsychiatric samples.     

Optimization of in vivo high-resolution DTI of non-human primates on a 3T human scanner

Magnetic resonance (MR) diffusion tensor imaging (DTI) has emerged as a unique technique to reveal small anatomical structures of brain by characterizing the diffusion process of water molecules within an image voxel. Combined with fiber tractography techniques, DTI can be further used to reveal white matter fibers and connectivity in the brain non-invasively. The non-human primate brain study provides important supplemental means for human brain exploration since the two species share close anatomical and functional similarities. There is therefore increasing interest in in vivo non-human primate DTI studies. However, several technical challenges need to be addressed to perform non-human primate brain DTI and fiber tractography. We have established an imaging protocol together with a post-acquisition procedure for high-resolution in vivo non-human primate DTI studies using a 3T human clinical scanner. Data acquired with this procedure is appropriate for accurate diffusion tensor quantification and fiber tractography, and is accessible within an acceptable scan time. We investigated in detail the effects of spatial resolution and SNR on diffusion tensor-derived quantities and fiber tractography. Our results should be of general utility for implementation of in vivo non-human primate DTI studies.     

Integrating Genetic,Neuropsychological and Neuroimaging Data to Model Early-Onset Obsessive Compulsive Disorder Severity

We propose an integrative approach that combines structural magnetic resonance imaging data (MRI), diffusion tensor imaging data (DTI), neuropsychological data, and genetic data to predict early-onset obsessive compulsive disorder (OCD) severity. From a cohort of 87 patients, 56 with complete information were used in the present analysis. First, we performed a multivariate genetic association analysis of OCD severity with 266 genetic polymorphisms. This association analysis was used to select and prioritize the SNPs that would be included in the model. Second, we split the sample into a training set (N = 38) and a validation set (N = 18). Third, entropy-based measures of information gain were used for feature selection with the training subset. Fourth, the selected features were fed into two supervised methods of class prediction based on machine learning, using the leave-one-out procedure with the training set. Finally, the resulting model was validated with the validation set. Nine variables were used for the creation of the OCD severity predictor, including six genetic polymorphisms and three variables from the neuropsychological data. The developed model classified child and adolescent patients with OCD by disease severity with an accuracy of 0.90 in the testing set and 0.70 in the validation sample. Above its clinical applicability, the combination of particular neuropsychological, neuroimaging, and genetic characteristics could enhance our understanding of the neurobiological basis of the disorder.     

Correlation of diffusion and metabolic alterations in different clinical forms of multiple sclerosis

Diffusion tensor imaging (DTI) and MR spectroscopic imaging (MRSI) provide greater sensitivity than conventional MRI to detect diffuse alterations in normal appearing white matter (NAWM) of Multiple Sclerosis (MS) patients with different clinical forms. Therefore, the goal of this study is to combine DTI and MRSI measurements to analyze the relation between diffusion and metabolic markers, T2-weighted lesion load (T2-LL) and the patients clinical status. The sensitivity and specificity of both methods were then compared in terms of MS clinical forms differentiation. MR examination was performed on 71 MS patients (27 relapsing remitting (RR), 26 secondary progressive (SP) and 18 primary progressive (PP)) and 24 control subjects. DTI and MRSI measurements were obtained from two identical regions of interest selected in left and right centrum semioval (CSO) WM. DTI metrics and metabolic contents were significantly altered in MS patients with the exception of N-acetyl-aspartate (NAA) and NAA/Choline (Cho) ratio in RR patients. Significant correlations were observed between diffusion and metabolic measures to various degrees in every MS patients group. Most DTI metrics were significantly correlated with the T2-LL while only NAA/Cr ratio was correlated in RR patients. A comparison analysis of MR methods efficiency demonstrated a better sensitivity/specificity of DTI over MRSI. Nevertheless, NAA/Cr ratio could distinguish all MS and SP patients groups from controls, while NAA/Cho ratio differentiated PP patients from controls. This study demonstrated that diffusivity changes related to microstructural alterations were correlated with metabolic changes and provided a better sensitivity to detect early changes, particularly in RR patients who are more subject to inflammatory processes. In contrast, the better specificity of metabolic ratios to detect axonal damage and demyelination may provide a better index for identification of PP patients.     

Multi-Fiber Tractography Visualizations for Diffusion MRI Data

In recent years, several new diffusion MRI approaches have been proposed to explore microstructural properties of the white matter, such as Q-ball imaging and spherical deconvolution-based techniques to estimate the orientation distribution function. These methods can describe the estimated diffusion profile with a higher accuracy than the more conventional second-rank diffusion tensor imaging technique. Despite many important advances, there are still inconsistent findings between different models that investigate the “crossing fibers” issue. Due to the high information content and the complex nature of the data, it becomes virtually impossible to interpret and compare results in a consistent manner. In this work, we present novel fiber tractography visualization approaches that provide a more complete picture of the microstructural architecture of fiber pathways: multi-fiber hyperstreamlines and streamribbons. By visualizing, for instance, the estimated fiber orientation distribution along the reconstructed tract in a continuous way, information of the local fiber architecture is combined with the global anatomical information derived from tractography. Facilitating the interpretation of diffusion MRI data, this approach can be useful for comparing different diffusion reconstruction techniques and may improve our understanding of the intricate white matter network.     

Multi-Shell Hybrid Diffusion Imaging (HYDI) at 7 Tesla in TgF344-AD Transgenic Alzheimer Rats

Diffusion weighted imaging (DWI) is widely used to study microstructural characteristics of the brain. Diffusion tensor imaging (DTI) and high-angular resolution imaging (HARDI) are frequently used in radiology and neuroscience research but can be limited in describing the signal behavior in composite nerve fiber structures. Here, we developed and assessed the benefit of a comprehensive diffusion encoding scheme, known as hybrid diffusion imaging (HYDI), composed of 300 DWI volumes acquired at 7-Tesla with diffusion weightings at b = 1000, 3000, 4000, 8000 and 12000 s/mm² and applied it in transgenic Alzheimer rats (line TgF344-AD) that model the full clinico-pathological spectrum of the human disease. We studied and visualized the effects of the multiple concentric “shells” when computing three distinct anisotropy maps–fractional anisotropy (FA), generalized fractional anisotropy (GFA) and normalized quantitative anisotropy (NQA). We tested the added value of the multi-shell q-space sampling scheme, when reconstructing neural pathways using mathematical frameworks from DTI and q-ball imaging (QBI). We show a range of properties of HYDI, including lower apparent anisotropy when using high b-value shells in DTI-based reconstructions, and increases in apparent anisotropy in QBI-based reconstructions. Regardless of the reconstruction scheme, HYDI improves FA-, GFA- and NQA-aided tractography. HYDI may be valuable in human connectome projects and clinical research, as well as magnetic resonance research in experimental animals.     

Inter Subject Variability and Reproducibility of Diffusion Tensor Imaging within and between Different Imaging Sessions

The aim of these studies was to provide reference data on intersubject variability and reproducibility of diffusion tensor imaging. Healthy volunteers underwent imaging on two occasions using the same 3T Siemens Verio magnetic resonance scanner. At each session two identical diffusion tensor sequences were obtained along with standard structural imaging. Fractional anisotropy, apparent diffusion coefficient, axial and radial diffusivity maps were created and regions of interest applied in normalised space. The baseline data from all 26 volunteers were used to calculate the intersubject variability, while within session and between session reproducibility were calculated from all the available data. The reproducibility of measurements were used to calculate the overall and within session 95% prediction interval for zero change. The within and between session reproducibility data were lower than the values for intersubject variability, and were different across the brain. The regional mean (range) coefficient of variation figures for within session reproducibility were 2.1 (0.9–5.5%), 1.2 (0.4–3.9%), 1.2 (0.4–3.8%) and 1.8 (0.4–4.3%) for fractional anisotropy, apparent diffusion coefficient, axial and radial diffusivity, and were lower than between session reproducibility measurements (2.4 (1.1–5.9%), 1.9 (0.7–5.7%), 1.7 (0.7–4.7%) and 2.4 (0.9–5.8%); p<0.001). The calculated overall and within session 95% prediction intervals for zero change were similar. This study provides additional reference data concerning intersubject variability and reproducibility of diffusion tensor imaging conducted within the same imaging session and different imaging sessions. These data can be utilised in interventional studies to quantify change within a single imaging session, or to assess the significance of change in longitudinal studies of brain injury and disease.     

Diffusion tensor imaging differences relate to memory deficits in diffuse traumatic brain injury

Background  

Memory is one of the most impaired functions after traumatic brain injury (TBI). We used diffusion tensor imaging (DTI) to determine the structural basis of memory deficit. We correlated fractional anisotropy (FA) of the fasciculi connecting the main cerebral regions that are involved in declarative and working memory functions.     

Quantitative analysis of three-dimensional-resolved fiber architecture in heterogeneous skeletal muscle tissue using nmr and optical imaging methods

The determination of principal fiber directions in structurally heterogeneous biological tissue substantially contributes to an understanding of its mechanical function in vivo. In this study we have depicted structural heterogeneity through the model of the mammalian tongue, a tissue comprised of a network of highly interwoven fibers responsible for producing numerous variations of shape and position. In order to characterize the three-dimensional-resolved microscopic myoarchitecture of the intrinsic musculature of the tongue, we viewed its fiber orientation at microscopic and macroscopic length scales using NMR (diffusion tensor MRI) and optical (two-photon microscopy) imaging methods. Diffusion tensor imaging (DTI) of the excised core region of the porcine tongue resulted in an array of 3D diffusion tensors, in which the leading eigenvector corresponded to the principal fiber orientation at each location in the tissue. Excised axially oriented lingual core tissues (fresh or paraffin-embedded) were also imaged with a mode-locked Ti-Sapphire laser, (76 MHz repetition rate, 150 femtosecond pulse width), allowing for the visualization of individual myofibers at in situ orientation. Fiber orientation was assessed by computing the 3D autocorrelation of discrete image volumes, and deriving the minimal eigenvector of the center voxel Hessian matrix. DTI of the fibers, comprising the intrinsic core of the tongue, demonstrated directional heterogeneity, with two distinct populations of fibers oriented orthogonal to each other and in-plane to the axial perspective. Microscopic analysis defined this structural heterogeneity as discrete regions of in-plane parallel fibers, with an angular separation of ~80 degrees, thereby recapitulating the macroscopic angular relationship. This analysis, conceived at two different length scales, demonstrates that the lingual core is a spatially complex tissue, composed of repeating orthogonally oriented and in-plane fiber patches, which are capable of jointly producing hydrostatic elongation and displacement.