Response analysis of rigid structures rocking on viscoelastic foundation

In this paper the rocking response of slender/rigid structures stepping on a viscoelastic foundation is revisited. The study examines in depth the motion of the system with a non-linear analysis that complements the linear analysis presented in the past by other investigators. The non-linear formulation combines the fully non-linear equations of motion together with the impulse-momentum equations during impacts. The study shows that the response of the rocking block depends on the size, shape and slenderness of the block, the stiffness and damping of the foundation and the energy loss during impact. The effect of the stiffness and damping of the foundation system along with the influence of the coefficient of restitution during impact is presented in rocking spectra in which the peak values of the response are compared with those of the rigid block rocking on a monolithic base. Various trends of the response are identified. For instance, less slender and smaller blocks have a tendency to separate easier, whereas the smaller the angle of slenderness, the less sensitive the response to the flexibility, damping and coefficient of restitution of the foundation

fMRI-guided tractography method: group analysis.

<p>Steps A–F described in the text (Materials and Methods section) are here graphically explained (A: Registration, B: Masking, C: Dilation, D: Individual fMRI-guided tractography, E: fMRI-guided probabilistic map, F: fMRI-guided <i>ensemble</i> tract). The final result is the <i>ensemble</i> tract, i.e. a tractographic template of the activated tracts in the considered population.</p

fMRI activation maps and fMRI-guided <i>ensemble</i> tracts for a) HC, b) aMCI and c) AD.

<p>From left to right, <u>PANEL 1</u>: second level fMRI group activation map resulting from one sample t-tests performed separately for each group, verbal fluency task, axial views; <u>PANELS 2/4</u>: anterior (A), left lateral (L) and right lateral (R) views of the <i>ensemble</i> tract based on the activation of the first panel. The yellow circles highlight the left CB (anterior view) and the left AF (left view), observable with different portions in all the three groups.</p

Demographic data of the study groups.

<p>Chi square was used for gender comparison. One-way ANOVA test with Bonferroni correction for multiple comparisons was used for age and MMSE score comparisons (significance level: pcorr<0.05).</p><p>Abbreviations: HC  =  healthy controls; aMCI  =  amnestic mild cognitive impairment; AD =  Alzheimer's disease; M/F =  males/females; CDR =  clinical dementia rating scale; MMSE =  mini-mental state examination; SD =  standard deviation.</p><p>*All differences resulted significant between the groups.</p

The main effect of verbal fluency task in HC (n = 14), subjects with aMCI (n = 15) and AD (n = 14), who completed the fMRI assessments.

<p>One sample t-test with cluster correction (Z>2.3), significance threshold of p = 0.05. X, Y, Z coordinates expressed in millimiters.</p><p>Abbreviations: Sup  =  superior; Inf  =  inferior; Mid  =  middle; R  =  right; L  =  left; Gy  =  gyrus; SMA  =  supplementary motor area; BA  =  Brodmann area; HC  =  healthy controls; aMCI  =  amnestic mild cognitive impairment; AD  =  Alzheimer's disease.</p

Measure of integrity and percentages of involvement in the activation of WM bundles for the three groups.

<p>The percentages over 5% in at least one group are reported. For each group, the percentages of involvement were computed superimposing the <i>ensemble</i> tract to the atlases of the WM fiber bundles and represent the number of overlapping voxels between the two, with respect to the number of voxels of the atlas. Abbreviations: WM  =  white matter; HC  =  healthy controls; aMCI  =  amnestic mild cognitive impairment; AD  =  Alzheimer's disease; FA  =  fractional anisotropy; SD  =  standard deviation; CB  =  cingulum bundle; AF  =  arcuate fasciculus; CC  =  corpus callosum; sup  =  superior; par  =  parietal; post  =  posterior; temp  =  temporal.</p><p>(*)The average FA was found to be significantly different between AD and HC (p<0.05 corrected for multiple comparisons).</p><p>(#)The average FA was found to be significantly different between aMCI and AD (p<0.05 corrected for multiple comparisons).</p

Relationship of clinical measures to gross thalamic volume in CD patients.

<p>The relationship between individual gross thalamic volume (manually segmented, in mm³) and clinical measures suggests that reduced gross thalamic volume is a risk factor for dystonia, and is not a secondary effect of dystonia symptoms. Data are shown for all variables, even if excluded from regression models, so raw data for CD and SD can be viewed and compared. Both CD patients and controls (A) showed declining volume with age. Patient:control gross thalamic volume showed a qualitative divergence with age between CD and controls, but the divergence was not statistically significant. (B) There was no relationship between volume and age in the SD cohort (for either SD or controls), presumably reflecting the smaller age range in this cohort. Likewise, there was no divergence of slopes with age between SD and controls. Age at CD onset (C) appeared to correlate with gross thalamic volume, but this was likely driven by the relationship between age at scan and age of onset; age at SD onset (D) was not correlated with volume. Gross thalamic volume did not correlate significantly with duration for either CD (E); this relationship was not evaluated statistically for SD due to collinearity with other variables, but the positive slope suggests no indication of a decline in volume with increasing duration (F). Thalamic volume also did not correlate with severity of dystonia for either CD or SD in the multiple regression model, as measured by the Tsui scale for CD (G) or the voice-related quality of life score for SD (H), although the SD relationship to severity showed a trend toward significance (V-RQOL, p = 0.056), and appeared significant when evaluated post hoc as a single variable (p = 0.012). The asymmetry of muscles affected with cervical dystonia (as gauged by laterality of units of botulinum toxin injected) did not correlate with asymmetries in thalamic volume (I, p = 0.89). Note that for (C) and (D), thalamic volumes for control subjects are plotted vs. the age at onset for the matched patient, as control subjects do not have an age at onset. Volume for control subjects is included in C and D as a reference only (designated by Ω), to illustrate that patient:control differences persist (and in fact are more robust) when demographics (including age) are matched: with the exception of a single CD/control dyad, every patient showed lower volume than his/her matched control.</p

Regional automated and manual gross volume measures.

<p>A reduction in thalamic volume, not seen in other regions involved in the control of movement, was seen in both cervical dystonia and spasmodic dysphonia. Total volume (i.e., number of voxels in left plus right hemispheres) is shown for each region of interest (mean ± standard error of the mean). Given the large differences in volume between brain regions, the axis has been adjusted to focus on each cluster of values. Breaks in the y-axis are indicated by hashed horizontal bars. All p-values corrected for multiple comparisons (Bonferroni corrected significance threshold, p = 0.00625); * p≤0.0060; ** p = 0.00020. Abbreviation: BA6 = Brodmann Area 6; Thal auto = automated thalamic segmentation.</p

Voxel based morphometry in cervical dystonia.

<p>Voxel based morphometry demonstrated reduced gray matter local tissue volume in the posterior cingulate (A and B, blue voxels, shown at two significance thresholds, presented as family-wise uncorrected p-values), but no differences in the thalamus, in cervical dystonia (family-wise error corrected p = 0.9996). When the analysis was restricted to only those voxels in a thalamic mask (to minimize the loss of statistical power by multiple-comparisons correction; C, green voxels), no significant differences in local tissue volume were noted (p = 0.34). Significant voxels (A, B) and thalamic mask (C) overlie the mean gray matter structural image. Note that identical structural scans were used in VBM analyses and segmentation analyses (i.e., scans used in this figure were the same as those used for data in Figs <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0155302#pone.0155302.g002" target="_blank">2</a>, <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0155302#pone.0155302.g003" target="_blank">3</a> and <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0155302#pone.0155302.g005" target="_blank">5</a>). VBM results were corrected using threshold-free cluster enhancement (TFCE). All axial and coronal views are from a single plane, indicated in MNI Talairach coordinates. Color bars at bottom indicate TFCE-corrected p-values for the images above. Abbreviations: pat = patients; ctrl = controls. R = Right hemisphere; L = Left hemisphere.</p

Demographic information and gross volumetric negative-control contrasts.

<p>Demographic and volumetric negative-control measures for each experimental group demonstrate high group similarity. Within each experimental group (cervical dystonia–CD; spasmodic dysphonia–SD), controls were matched to patients for gender, handedness, and age +/- five years (A). Each volumetric negative-control measure is expressed as the percentage of mean control volume (<i>e</i>.<i>g</i>., estimated total intracranial volume (eTIV) for CD patients is 2% larger for patients than for matched controls). No large-scale volumetric measures differed between patients and controls (A, p-values uncorrected). The reduction of thalamic volume in patients maintained significance following normalization for all volumetric negative-control measures (B).</p