Principal eigenvector field segmentation for reproducible diffusion tensor tractography of white matter structures

The study was aimed to test the feasibility of utilizing an algorithmically determinable stable fiber mass (SFM) map obtained by an unsupervised principal eigenvector field segmentation (PEVFS) for automatic delineation of 18 white matter (WM) tracts: (1) corpus callosum (CC), (2) tapetum (TP), (3) inferior longitudinal fasciculus (ILF), (4) uncinate fasciculus (UNC), (5) inferior fronto-occipital fasciculus (IFO), (6) optic pathways (OP), (7) superior longitudinal fasciculus (SLF), (8) arcuate fasciculus (AF), (9) fornix (FX), (10) cingulum (CG), (11) anterior thalamic radiation (ATR), (12) superior thalamic radiation (STR), (13) posterior thalamic radiation (PTR), (14) corticospinal/corticopontine tract (CST/CPT), (15) medial lemniscus (ML), (16) superior cerebellar peduncle (SCP), (17) middle cerebellar peduncle (MCP) and (18) inferior cerebellar peduncle (ICP). Diffusion tensor imaging (DTI)-derived fractional anisotropy (FA) and the principal eigenvector field have been used to create the SFM consisting of a collection of linear voxel structures which are grouped together by color-coding them into seven natural classes to provide PEVFS signature segments which greatly facilitate the selection of regions of interest (ROIs) for fiber tractography using just a single mouse click, as compared with a manual drawing of ROIs in the classical approach. All the 18 fiber bundles have been successfully reconstructed, in all the subjects, using the single ROIs provided by the SFM approach, with their reproducibility characterized by the fact that the ROI selection is user independent. The essentially automatic PEVFS method is robust, efficient and compares favorably with the classical ROI methods for diffusion tensor tractography (DTT).     

Existence of contralateral abnormalities revealed by texture analysis in unilateral intractable hippocampal epilepsy

We selected 23 patients with unilateral temporal lobe epilepsy characterized by ipsilateral hippocampal sclerosis and an apparently normal contralateral hippocampus on MR imaging. Images were acquired on a 0.28 T MR scanner using a conventional Carr-Purcell Meiboom Gill sequence in all patients and in 9 healthy subjects. Texture analysis was applied to axial MR images of the first and tenth echoes. Texture analysis detects macroscopic lesions and microscopic abnormalities that can not be observed visually. The presence of texture differences in the between normal (controls) and sclerotic hippocampi was ascertained by statistical discriminant analysis. The apparently normal contralateral hippocampi can be classified into three categories in terms of texture: 4 apparently healthy, 8 similar to sclerosis, and 11 different from either healthy or sclerosis. These findings are related to a certain degree of hippocampal alteration, which further investigation might help better characterize.     

Alterations in diffusion properties of white matter in Williams syndrome

Diffusion tensor imaging (DTI) was used to investigate the involvement of brain white matter in Williams syndrome (WS), a genetic neurodevelopmental disorder. Whole-brain DTIs were obtained from 16 young adults with WS and 16 normal controls. A voxel-based analysis was performed to compare fractional anisotropy (FA) values between the two groups. A tract-based analysis was also performed to compare FA values between the two groups along two major white matter tracts that pass through the external capsule: the uncinate and inferior fronto-occipital fasciculi. Several regions of both increased and decreased FA were found within major white matter tracts that connect functional regions that have previously been implicated in the cognitive and neurological symptoms of the syndrome. The tract-based analysis provided additional insight into the involvement of specific white matter tracts implicated in the voxel-based analysis within the external capsule. The results from this study support previously reported changes in white matter diffusion properties in WS and demonstrate the potential usefulness for tract-based analysis in future studies of the disorder.     

Hippocampal MRI volumetrics and temporal lobe substrates in medial temporal lobe epilepsy

Forty-nine consecutive patients undergoing anteromedial temporal lobe resection for medically intractable temporal lobe seizures, and averaging 2 yr (range 6 mo to 4 yr) postoperative follow-up, were selected for a retrospective study. This study correlated magnetic resonance imaging (MRI) derived hippocampal volumetrics, preoperative demographics, postoperative seizure control, and tissue analysis, including hippocampal CA (cornu ammonis) field neuronal, and glial cell counts, and immunohistochemistry (IHC) evidence for dentate sprouting and reorganization. These measures were compared in hippocampi with or without an adjacent presumptive epileptogenic temporal lobe mass. Mesial temporal sclerosis (MTS) was defined as >50% neuronal cell loss averaged across all CA fields with NPY (neuropeptide-y) and somatostatin reorganization. These patients may or may not include granule cell sprouting as determined by dynorphin staining. Patients were divided into two groups based on CA field neuronal cell counts, one averaging >50% cell loss and one averaging <50% cell loss. For the MTS group (N = 38), 89% had significant volumetric atrophy of the ipsilateral hippocampus, 74% had dentate reorganization, and complete seizure control was seen in 76% of these patients. In one subgroup of the <50% cell loss group, patients with medial temporal lobe epilepsy caused by a mass in the medial temporal lobe (mass group) (N = 6), 33% demonstrated significant volumetric atrophy of the hippocampus ipsilateral to the mass, 0% had dentate sprouting, and seizures were completely controlled in 67%. For the second subgroup of the <50% cell loss group, patients without mass lesions (N = 5) who were classified as the paradoxical medial temporal lobe epilepsy group (paradoxical group), 20% had ipsilateral hippocampal atrophy, 0% had dentate reorganization, and complete seizure control was seen in 60% of these patients. In conclusion, for the MTS group, hippocampal atrophy proven by MRI volumetrics was highly predictive of significant neuronal cell loss and an excellent indicator of success. However, in patients who had a foreign mass, hippocampal atrophy was not necessarily indicative of significant neuronal cell loss and MRI volumetrics was not a factor in the determination of a successful outcome. Furthermore, patients without mass lesions who have normal volumetrics but demonstrate hippocampal disease through invasive electrode monitoring, are likely to have paradoxical medial temporal lobe epilepsy, seizures beginning at a later age, and a lower, but not insignificant, success rate than the classical mesial temporal sclerosis group.     

Neurofibrillary tangles and plaques are not accompanied by white matter pathology in aged triple transgenic-Alzheimer disease mice

Alzheimer's disease (AD) is a progressive neurodegenerative disorder that is the most common cause of dementia in aging populations. Although senile plaques and neurofibrillary tangles are well-established hallmarks of AD, changes in cerebral white matter correlate with cognitive decline and may increase the risk of the development of dementia. We used the triple transgenic (3xTg)-AD mouse model of AD, previously used to show that white matter changes precede plaque formation, to test the hypothesis that MRI detectable changes occur in the corpus callosum, external capsule and the fornix. T₂-weighted and diffusion tensor magnetic resonance imaging and histological stains were employed to assess white matter in older (11–17 months) 3xTg-AD mice and controls. We found no statistically significant changes in white matter between 3xTg-AD mice and controls, despite well-developed neurofibrillary tangles and beta amyloid immunoreactive plaques. Myelin staining was normal in affected mice. These data suggest that the 3xTg-AD mouse model does not develop MRI detectable white matter changes at the ages we examined.     

Evaluation of diffusion models of fiber tracts using diffusion tensor magnetic resonance imaging

Modeling of water diffusion in white matter is useful for revealing microstructure of the brain tissue and hence diagnosis and evaluation of white matter diseases. Researchers have modeled diffusion in white matter using mathematical and mechanical analysis at the cellular level. However, less work has been devoted to evaluate these models using macroscopic real data such as diffusion tensor magnetic resonance imaging (DTMRI) data. DTMRI is a noninvasive tool for evaluating white matter microstructure by measuring random motion of water molecules referred to as diffusion. It reflects directional information of microscopic structures such as fibers. Thus, it is applicable for evaluation and modification of mathematical models of white matter. Nevertheless, a realistic relation between a fiber model and imaging data does not exist. This work opens a promising avenue for relating DTMRI data to microstructural parameters of white matter. First, we propose a strategy for relating DTMRI and fiber model parameters to evaluate mathematical models in light of real data. The proposed strategy is then applied to evaluate and extend an existing model of white matter based on clinically available DTMRI data. Next, the proposed strategy is used to estimate microstructural characteristics of fiber tracts. We illustrate this approach through its application to approximation of myelin sheath thickness and fraction of volume occupied by fibers. Using sufficiently small imaging voxels, the proposed approach is capable of estimating model parameters with desirable precision.     

Morphometry in temporal lobe epilepsy

We demonstrate a method for quantitating changes in volume and morphology of the temporal lobe in epilepsy. The temporal lobes of 10 neurologically normal subjects and six subjects with well defined left-sided mesial temporal epilepsy were studied. From high resolution T₁-weighted magnetic resonance images, the grey and white matter were manually segmented over a predetermined extent. The volumes of the grey and white matter were determined. Using the segmented images, the grey matter/CSF surface and the white matter/grey matter surface were reconstructed, allowing estimates of the surface area and calculation of indices of curvature for the two surfaces. The index of curvature was calculated for each vertex of a polygonal mesh that was fitted to the surfaces. An index of grey matter thickness (grey matter volume/white matter surface area) was also calculated. There was a significant bilateral decrease in the total volume (p < .01), grey matter volume (p < .001) and grey matter thickness index (p < .05) in epileptic subjects. In addition, there was a bilateral decrease in white matter surface area (p < .05) and a small left-sided decrease in white matter volume (p < .05) in epileptic subjects. The average distributions of indices of curvature for both surfaces differed significantly (p < .05) between normal and epileptic subjects. In the grey matter/CSF surface of normal subjects, a large peak corresponding to surface concavity was present. The amplitude of this peak was significantly lower in epileptic subjects (p < .05 for the right hemisphere; p < .001 for the left hemisphere).     

Altered hemispheric symmetry found in left-sided mesial temporal lobe epilepsy with hippocampal sclerosis (MTLE/HS) but not found in right-sided MTLE/HS

Purpose

The purpose was to investigate the altered hemispheric asymmetry in patients with mesial temporal lobe epilepsy with unilateral hippocampus sclerosis (MTLE/HS).

Materials and methods

This study examined the hemispheric asymmetry of regional gray matter (GM) and white matter (WM) volume among a group of 13 patients with left-sided MTLE/HS, a group of 10 patients with right-sided MTLE/HS and a group of 21 age- and gender- matched healthy controls by optimized voxel-based morphometry (VBM) based on magnetic resonance imaging.

Results

Compared to healthy controls, abnormal asymmetries were detected in the left-sided MTLE/HS patients. The left-sided MTLE/HS patients had more GM asymmetries (L<R) in the temporal lobes, including the inferior temporal gyrus, middle temporal gyrus and parahippocampal gyrus. There was significant asymmetry (L<R) in subcortical WM of the mesial temporal lobe in left-sided MTLE/HS patients. However, no significant difference was detected in terms of GM and WM asymmetry between the group with right-sided MTLE/HS and normal controls.

Conclusion

We should approach hemispheric asymmetry in left- and right-sided MTLE/HS patients differently. The study also demonstrates potential future use of VBM in detecting hemispheric asymmetries and lateralization of brain functions.     

Multiple sclerosis: Benefits of q-space imaging in evaluation of normal-appearing and periplaque white matter

Introduction

Diffusion tensor imaging (DTI) reveals white matter pathology in patients with multiple sclerosis (MS). A recent non-Gaussian diffusion imaging technique, q-space imaging (QSI), may provide several advantages over conventional MRI techniques in regard to in vivo evaluation of the disease process in patients with MS. The purpose of this study is to investigate the use of root mean square displacement (RMSD) derived from QSI data to characterize plaques, periplaque white matter (PWM), and normal-appearing white matter (NAWM) in patients with MS.

Methods

We generated apparent diffusion coefficient (ADC) and fractional anisotropy (FA) maps by using conventional DTI data from 21 MS patients; we generated RMSD maps by using QSI data from these patients. We used the Steel–Dwass test to compare the diffusion metrics of regions of interest in plaques, PWM, and NAWM.

Results

ADC differed (P < 0.05) between plaques and PWM and between plaques and NAWM. FA differed (P < 0.05) between plaques and NAWM. RMSD differed (P < 0.05) between plaques and PWM, plaques and NAWM, and PWM and NAWM.

Conclusion

RMSD values from QSI may reflect microstructural changes and white-matter damage in patients with MS with higher sensitivity than do conventional ADC and FA values.     

A combined DTI and structural MRI study in medicated-naïve chronic schizophrenia

Disconnection in white matter (WM) pathway and alterations in gray matter (GM) structure have been hypothesized as pathogenesis in schizophrenia. However, the relationship between the abnormal WM integrity and the alteration of GM in anatomically connected areas remains uncertain. Moreover, the potential influence of antipsychotic medication on WM anisotropy and cortical morphology was not excluded in previous studies. In this study, a total number of 34 subjects were enrolled, including 17 medicated-naïve chronic schizophrenia patients and 17 healthy controls. Tract-based spatial statistics (TBSS) were applied to investigate the level of WM integrity. The FreeSurfer surface-based analysis was used to determine GM volume, cortical thickness and the surface area of GM regions which corresponded to abnormal WM fiber tracts. We observed that patients possessed lower fractional anisotropy (FA) values in the left inferior fronto-occipital fasciculus (IFOF) and left inferior longitudinal fasciculus (ILF), along with smaller GM volume and cortical thinning in temporal lobe than the healthy controls, which reflected the underlying WM and GM disruption that contributed to the disease. In the patient population, the lower connectivity of ILF and IFOF was positively associated with cortical thickness in left lateral orbitofrontal cortex, superior temporal gyrus and lingual gyrus in males, and positively correlated with GM volume in left lateral orbitofrontal cortex in females. On the other hand, it was negatively correlated with cortical area of middle temporal gyrus in males and temporal pole in females respectively, but not when genders were combined. These findings suggested that abnormal WM integrity and anatomical correspondence of GM alterations in schizophrenia were interdependent on gender-separated analysis in patients with schizophrenia. Moreover, combining TBSS and FreeSurfer might be a useful method to provide significant insight into interacting processes related to WM fiber tracts and GM changes in schizophrenia.     

Quantitative mapping of diffusion characteristics under the cortical surface

Recent studies have demonstrated regional segregations on several peripheral white matter (WM) regions, which may imply different anatomical or functional characteristics [Cereb Cortex 17(4) 2007 816–25; Neuroimage 37(2) 2007 599–610; J Cogn Neurosci 16(7) 2004 1227–33]. Nonetheless, little is known about overall patterns of peripheral WM across the regions. In this study, diffusion tensor imaging with 2-mm isovoxel resolution and cortical surface mapping were combined to determine peripheral WM structure. Fractional anisotropy (FA) mapping showed consistent regional patterns across the young normal subjects while significant high or low FA values were shown in the motor-somatosensory cortex, prefrontal cortex, temporal, and medial occipital cortex. By adopting both region of interest and connectivity analysis, results were then discussed with structural network properties as well as WM maturation process.     

Correlation of CSF neuroinflammatory molecules with leptomeningeal cortical subcortical white matter fractional anisotropy in neonatal meningitis

It has been previously hypothesized that the high fractional anisotropy (FA) values in leptomeningeal cortical subcortical white matter (LCSWM) regions of neonatal brain with bacterial meningitis is due to the presence of adhesion molecules in the subarachnoid space, which are responsible for adherence of inflammatory cells over the subarachnoid membrane. The aim of this study was to look for any relationship between FA values in LCSWM regions and various neuroinflammatory molecules (NMs) in cerebrospinal fluid (CSF) measured in neonates with bacterial meningitis. Diffusion tensor imaging was performed on 18 term neonates (median age, 10.5 days) having bacterial meningitis and 10 age-/sex-matched healthy controls. CSF enzyme-linked immunosorbent assay was performed to quantify NMs [soluble intracellular adhesion molecules (sICAM), tumor necrosis factor-alpha (TNF-alpha) and interleukin-1beta (IL-1beta)]. Significantly increased FA values were observed in LCSWM regions of the patients compared to controls. A significant positive correlation was observed between FA values in LCSWM regions and NMs [sICAM (r=0.67, P=.006), TNF-alpha (r=0.69, P=.005) and IL-1beta (r=0.82, P=.000)] in CSF of these patients. No difference in FA values (P=.99) in LCSWM regions was observed between patients with sterile (0.12+/-0.02) and culture-positive CSF study (0.12+/-0.02). FA may be used as noninvasive surrogate marker of NMs in neonatal meningitis in assessing therapeutic response in future.     

Moderating registration misalignment in voxelwise comparisons of DTI data: a performance evaluation of skeleton projection

An evaluative methodology and five accompanying performance measures were developed to quantitatively assess the performance of the skeleton projection algorithm constituting the heart of tract-based spatial statistics (TBSS). The performance measures were designed to quantify the accuracy of skeleton projection in its indented task of alleviating any residual misalignment that may remain after image registration. A ground truth fractional anisotropy (FA) image was slightly warped using a realistic warp field that served to model post-registration residual misalignment of varying magnitudes. Skeleton projection was then used to register the warped FA image to the ground truth. Performing skeleton projection was found to yield up to 50% better correspondence between the values of FA compared to smoothing, despite the fact that less than 10% of post-registration misalignment was corrected. The align-max-with-max strategy underlying TBSS was posited as a potential explanation for this high correspondence in the values of FA, at the expense of lesser alignment between anatomically concordant voxels.     

SPatial REgression Analysis of Diffusion tensor imaging (SPREAD) for longitudinal progression of neurodegenerative disease in individual subjects

Objectives

To develop a novel statistical method for analysis of longitudinal DTI data in individual subjects.

Materials and Methods

The proposed SPatial REgression Analysis of Diffusion tensor imaging (SPREAD) method incorporates a spatial regression fitting of DTI data among neighboring voxels and a resampling method among data at different times. Both numerical simulations and real DTI data from healthy volunteers and multiple sclerosis (MS) patients were used in the study to evaluate this method.

Results

Statistical inference based on SPREAD was shown to perform well through both group comparisons among simulated DTI data of individuals (especially when the group size is smaller than 5) and longitudinal comparisons of human DTI data within the same individual.

Conclusions

When pathological changes of neurodegenerative diseases are heterogeneous in a population, SPREAD provides a unique way to assess abnormality during disease progression at the individual level. Consequently, it has the potential to shed light on how the brain has changed as a result of disease or injury.     

White matter hyperintensities and dynamics of postural control

Background

White matter hyperintensities (WMHs) on MRI have been associated with age, cardiovascular risk factors and falls in the elderly. This study evaluated the relationship between WMHs and dynamics of postural control in older adults without history of falls.

Methods

We studied 76 community-living subjects without history of falls (age 64.5±7.3 years). Brain and WMH volume calculations and clinical rating were done on fluid-attenuation inversion recovery (FLAIR) and MP-RAGE MR images on 3 T. Balance was assessed from the center of pressure displacement using the force platform during 3 min of quiet standing using traditional and dynamic measures (using stabilogram-diffusion analysis). Gait speed was measured from 12-min walk.

Results

Age-adjusted periventricular and focal WMHs were associated with changes in certain dynamic balance measures, including reduced range of postural sway in anteroposterior direction (fronto-temporal WMHs, P=.045; parieto-occipital WMHs, P=.009) and more irregular long-term mediolateral fluctuations (P=.046). Normal walking speed was not affected by WMHs.

Conclusions

Periventricular and focal WMHs affect long-term dynamics of postural control, which requires engagement of feedback mechanisms, and may contribute to mobility decline in the elderly.     

Aging effects on cerebral asymmetry: a voxel-based morphometry and diffusion tensor imaging study

The hemispheres of the human brain are functionally and structurally asymmetric. The purpose of this study was to evaluate the effects of aging on gray and white matter asymmetry. Two hundred twenty-six right-handed normal volunteers aged 21–71 years were included in this study. The effects of aging on gray matter volume asymmetry and white matter fractional anisotropy asymmetry were evaluated with use of voxel-based morphometry and voxel-based analysis of fractional anisotropy maps derived from diffusion tensor imaging (DTI), respectively. The voxel-based morphometry showed no significant correlation between age and gray matter volume asymmetry. The voxel-based analysis of DTI also showed no significant correlation between age and white matter fractional anisotropy asymmetry. Our results showed no significant effects of aging on either gray matter volume asymmetry or white matter fractional anisotropy asymmetry.     

An investigation of a genomewide supported psychosis variant in ZNF804A and white matter integrity in the human brain

ZNF804A, a genomewide supported susceptibility gene for schizophrenia and bipolar disorder, has been associated with task-independent functional connectivity between the left and right dorsolateral prefrontal cortices. Several lines of evidence have converged on the hypothesis that this effect may be mediated by structural connectivity. We tested this hypothesis using diffusion tensor magnetic resonance imaging in three samples: one German sample of 50 healthy individuals, one Scottish sample of 83 healthy individuals and one Scottish sample of 84 unaffected relatives of bipolar patients. Voxel-based analysis and tract-based spatial statistics did not detect any fractional anisotropy (FA) differences between minor allele carriers and individuals homozygous for the major allele at rs1344706. Similarly, region-of-interest analyses and quantitative tractography of the genu of the corpus callosum revealed no significant FA differences between the genotype groups. Examination of effect sizes and confidence intervals indicated that this negative finding is very unlikely to be due to a lack of statistical power. In summary, despite using various analysis techniques in three different samples, our results were strikingly and consistently negative. These data therefore suggest that it is unlikely that the effects of genetic variation at rs1344706 on functional connectivity are mediated by structural integrity differences in large, long-range white matter fiber connections.     

Application of high field spectroscopic imaging in the evaluation of temporal lobe epilepsy

Previous spectroscopic imaging studies of temporal lobe epilepsy have used comparisons of metabolite content or ratios to lateralize the seizure focus. Although highly successful, these studies have shown significant variations within each of the groups of healthy subjects and patients. This variation may arise from the natural differences seen in metabolite concentration in gray and white matter, the complex anatomy seen about the hippocampus, and the large voxels typically employed at 1.5 T. Using a 4.1 T whole body system, we have acquired spectroscopic images with 0.5 cc nominal voxels (1 cc after filtering) to evaluate the regional variation in metabolite content of the hippocampus, temporal gray and white matter, midbrain, and cerebellar vermis. Using a threshold value of 0.90 for CR/NAA, a value 90% of all normal hippocampal voxels lay below, we have correctly identified the presence of epileptogenic tissue in patients with unilateral as well as bilateral seizures. By using comparisons to healthy values of the CR/NAA ratio, this method enables the visualization of bilateral disease and provides information on the extent of gray matter involvement.     

Correlations between microstructural alterations and severity of cognitive deficiency in Alzheimer's disease and mild cognitive impairment: a diffusional kurtosis imaging study

Object

Diffusional kurtosis imaging (DKI), a natural extension of diffusion tensor imaging (DTI), can characterize non-Gaussian diffusion in the brain. We investigated the capability of DKI parameters for detecting microstructural changes in both gray matter (GM) and white matter (WM) in patients with mild cognitive impairment (MCI) and Alzheimer's disease (AD) and sought to determine whether these DKI parameters could serve as imaging biomarkers to indicate the severity of cognitive deficiency.

Materials and Methods

DKI was performed on 18 AD patients and 12 MCI patients. Fractional anisotropy, kurtosis and diffusivity parameters in the temporal, parietal, frontal and occipital lobes were compared between the two groups using Mann–Whitney U test. The correlations between regional DKI parameters and mini-mental state examination (MMSE) score were tested using Pearson's correlation.

Results

In ADs, significantly increased diffusivity and decreased kurtosis parameters were observed in both the GM and WM of the parietal and occipital lobes as compared to MCIs. Significantly decreased fractional anisotropy was also observed in the WM of these lobes in ADs. With the exception of fractional anisotropy and radial kurtosis, all the five other DKI parameters exhibited significant correlations with MMSE score in both GM and WM.

Conclusion

Bearing additional information, the DKI model can provide sensitive imaging biomarkers for assessing the severity of cognitive deficiency in reference to MMSE score and potentially improve early detection and progression monitoring of AD based on characterizing microstructures in both the WM and especially the GM.     

Noise and nonlinear estimation with optimal schemes in DTI

In general, the estimation of the diffusion properties for diffusion tensor experiments (DTI) is accomplished via least squares estimation (LSE). The technique requires applying the logarithm to the measurements, which causes bad propagation of errors. Moreover, the way noise is injected to the equations invalidates the least squares estimate as the best linear unbiased estimate. Nonlinear estimation (NE), despite its longer computation time, does not possess any of these problems. However, all of the conditions and optimization methods developed in the past are based on the coefficient matrix obtained in a LSE setup. In this article, NE for DTI is analyzed to demonstrate that any result obtained relatively easily in a linear algebra setup about the coefficient matrix can be applied to the more complicated NE framework. The data, obtained using non-optimal and optimized diffusion gradient schemes, are processed with NE. In comparison with LSE, the results show significant improvements, especially for the optimization criterion. However, NE does not resolve the existing conflicts and ambiguities displayed with LSE methods.