A simple rule of thumb based on resolution is not adequate to identify the best treatment of atomic displacements in macromolecular structural models. The choice to use isotropic B factors, anisotropic B factors, TLS models or some combination of the three should be validated through statistical analysis of the model refinement.
In choosing and refining any crystallographic structural model, there is tension between the desire to extract the most detailed information possible and the necessity to describe no more than what is justified by the observed data. A more complex model is not necessarily a better model. Thus, it is important to validate the choice of parameters as well as validating their refined values. One recurring task is to choose the best model for describing the displacement of each atom about its mean position. At atomic resolution one has the option of devoting six model parameters (a ‘thermal ellipsoid’) to describe the displacement of each atom. At medium resolution one typically devotes at most one model parameter per atom to describe the same thing (a ‘B factor’). At very low resolution one cannot justify the use of even one parameter per atom. Furthermore, this aspect of the structure may be described better by an explicit model of bulk displacements, the most common of which is the translation/libration/screw (TLS) formalism, rather than by assigning some number of parameters to each atom individually. One can sidestep this choice between atomic displacement parameters and TLS descriptions by including both treatments in the same model, but this is not always statistically justifiable. The choice of which treatment is best for a particular structure refinement at a particular resolution can be guided by general considerations of the ratio of model parameters to the number of observations and by specific statistics such as the Hamilton R-factor ratio test.
atomic displacements; B factors; TLS models; model parameters
The benefit of computer-assisted navigation depends on the registration process, at which patient features are correlated to some preoperative imagery. The operator-induced uncertainty in localizing patient features – the User Localization Error (ULE) - is unknown and most likely dominating the application accuracy. This initial feasibility study aims at providing first data for ULE with a research navigation system.
Active optical navigation was done in CT-images of a plastic skull, an anatomic specimen (both with implanted fiducials) and a volunteer with anatomical landmarks exclusively. Each object was registered ten times with 3, 5, 7, and 9 registration points. Measurements were taken at 10 (anatomic specimen and volunteer) and 11 targets (plastic skull). The active NDI Polaris system was used under ideal working conditions (tracking accuracy 0.23 mm root mean square, RMS; probe tip calibration was 0.18 mm RMS. Variances of tracking along the principal directions were measured as 0.18 mm2, 0.32 mm2, and 0.42 mm2. ULE was calculated from predicted application accuracy with isotropic and anisotropic models and from experimental variances, respectively.
The ULE was determined from the variances as 0.45 mm (plastic skull), 0.60 mm (anatomic specimen), and 4.96 mm (volunteer). The predicted application accuracy did not yield consistent values for the ULE.
Quantitative data of application accuracy could be tested against prediction models with iso- and anisotropic noise models and revealed some discrepancies. This could potentially be due to the facts that navigation and one prediction model wrongly assume isotropic noise (tracking is anisotropic), while the anisotropic noise prediction model assumes an anisotropic registration strategy (registration is isotropic in typical navigation systems). The ULE data are presumably the first quantitative values for the precision of localizing anatomical landmarks and implanted fiducials. Submillimetric localization is possible for implanted screws; anatomic landmarks are not suitable for high-precision clinical navigation.
application accuracy; navigation; human localization error; registration
The general principles behind the macromolecular crystal structure refinement program REFMAC5 are described.
This paper describes various components of the macromolecular crystallographic refinement program REFMAC5, which is distributed as part of the CCP4 suite. REFMAC5 utilizes different likelihood functions depending on the diffraction data employed (amplitudes or intensities), the presence of twinning and the availability of SAD/SIRAS experimental diffraction data. To ensure chemical and structural integrity of the refined model, REFMAC5 offers several classes of restraints and choices of model parameterization. Reliable models at resolutions at least as low as 4 Å can be achieved thanks to low-resolution refinement tools such as secondary-structure restraints, restraints to known homologous structures, automatic global and local NCS restraints, ‘jelly-body’ restraints and the use of novel long-range restraints on atomic displacement parameters (ADPs) based on the Kullback–Leibler divergence. REFMAC5 additionally offers TLS parameterization and, when high-resolution data are available, fast refinement of anisotropic ADPs. Refinement in the presence of twinning is performed in a fully automated fashion. REFMAC5 is a flexible and highly optimized refinement package that is ideally suited for refinement across the entire resolution spectrum encountered in macromolecular crystallography.
Biomolecular X-ray structures typically provide a static, time- and ensemble-averaged view of molecular ensembles in crystals. In the absence of rigid-body motions and lattice defects, B-factors are thought to accurately reflect the structural heterogeneity of such ensembles. In order to study the effects of averaging on B-factors, we employ molecular dynamics simulations to controllably manipulate microscopic heterogeneity of a crystal containing 216 copies of villin headpiece. Using average structure factors derived from simulation, we analyse how well this heterogeneity is captured by high-resolution molecular-replacement-based model refinement. We find that both isotropic and anisotropic refined B-factors often significantly deviate from their actual values known from simulation: even at high 1.0 Å resolution and Rfree of 5.9%, B-factors of some well-resolved atoms underestimate their actual values even sixfold. Our results suggest that conformational averaging and inadequate treatment of correlated motion considerably influence estimation of microscopic heterogeneity via B-factors, and invite caution in their interpretation.
The structural heterogeneity of a biomolecular crystal structure is typically captured using atomic B-factors, determined during structure refinement. Here, the authors use molecular dynamics to show that this strategy is flawed, and that crystallographic B-factors underestimate structural heterogeneity.
The local dilation of the infrarenal abdominal aorta, termed an abdominal aortic aneurysm (AAA), is often times asymptomatic and may eventually result in rupture —an event associated with a significant mortality rate. The estimation of in-vivo stresses within AAAs has been proposed as a useful tool to predict the likelihood of rupture. For the current work, a previously-derived anisotropic relation for the AAA wall was implemented into patient-specific finite element simulations of AAA. There were 35 AAAs simulated in the current work which were broken up into three groups: elective repairs (n = 21), non-ruptured repairs (n = 5), and ruptured repairs (n = 9). Peak stresses and strains were compared using the anisotropic and isotropic constitutive relations. There were significant increases in peak stress when using the anisotropic relationship (p<0.001), even in the absence of the ILT (p = 0.014). Rutpured AAAs resulted in elevated peak stresses as compared to non-ruptured AAAs when using both the isotropic and anisotropic simulations, however these comparisons did not reach significance (pani = 0.55, piso = 0.73). While neither the isotropic or anisotropic simulations were able to significantly discriminate ruptured vs. non-ruptured AAAs, the lower p-value when using the anisotropic model suggests including it into patient-specific AAAs may help better identify AAAs at high risk.
Anisotropy; Biaxial testing; Aneurysm; AAA; Stress; Finite element method
When movement outcome differs consistently from the intended movement, errors are used to correct subsequent movements (e.g., adaptation to displacing prisms or force fields) by updating an internal model of motor and/or sensory systems. Here, we examine changes to an internal model of the motor system under changes in the variance structure of movement errors lacking an overall bias. We introduced a horizontal visuomotor perturbation to change the statistical distribution of movement errors anisotropically, while monetary gains/losses were awarded based on movement outcomes. We derive predictions for simulated movement planners, each differing in its internal model of the motor system. We find that humans optimally respond to the overall change in error magnitude, but ignore the anisotropy of the error distribution. Through comparison with simulated movement planners, we found that aimpoints corresponded quantitatively to an ideal movement planner that updates a strictly isotropic (circular) internal model of the error distribution. Aimpoints were planned in a manner that ignored the direction-dependence of error magnitudes, despite the continuous availability of unambiguous information regarding the anisotropic distribution of actual motor errors.
To plan effective movements of the limbs, the human motor system must keep track of certain parameters: Obvious examples are the lengths and masses of to-be-controlled limb segments. In addition, the nervous system tracks its own motor outcome noise, which is important for selecting among movement plans where there are substantial costs associated with movement inaccuracies (e.g., reaching past a glass of red wine on a cluttered dinner table). Here, we introduce a change in motor noise in a reaching task: reaches were perturbed unpredictably by activating a reflex that introduced unplanned horizontal arm motion at the ends of reaches. We show that the motor system updates an internal model of the overall increase in motor noise induced by this reflex perturbation, but fails to represent the anisotropic component of the noise. This result is consistent with current theories of motor planning and control in which reach magnitudes and directions are represented independently, because a system that updates only a circular representation of recent motor errors is equivalent to a system that monitors only the magnitudes of recent errors, and ignores their directions.
The synthesis and crystallographic characterization of a new family of M(μ-CN)Ln complexes are reported. Two structural series have been prepared by reacting in water rare earth nitrates (LnIII = La, Pr, Nd, Sm, Eu, Gd, Dy, Ho) with K3[M(CN)6] (MIII = Fe, Co) in the presence of hexamethylenetetramine (hmt). The first series consists of six isomorphous heterobinuclear complexes, [(CN)5M-CNLn(H2O)8]·2hmt ([FeLa] 1, [FePr] 2, [FeNd] 3, [FeSm] 4, [FeEu] 5, [FeGd] 6), while the second series consists of four isostructural ionic complexes, [Ln(H2O)8][M(CN)6]·hmt ([FeDy] 7, [FeHo] 8, [CoEu] 9, [CoGd] 10). The hexamethylenetetramine molecules contribute to the stabilization of the crystals by participating in an extended network of hydrogen bond interactions. In both series the aqua ligands are hydrogen bonded to the nitrogen atoms from both the terminal CN groups and the hmt molecules. The [FeGd] complex has been analyzed with 57Fe Mössbauer spectroscopy, EPR, and magnetic susceptibility measurements. We have also analyzed the [FeLa] complex, in which the paramagnetic GdIII is replaced by diamagnetic LaIII, to obtain information about the low-spin FeIII site that is not accessible in the presence of a paramagnetic ion at the complementary site. For the same reason, the [CoGd] complex, containing diamagnetic CoIII, was studied with EPR and magnetic susceptibility measurements, which confirmed the S = 7/2 spin of GdIII. Prior knowledge about the paramagnetic sites in [FeGd] allows a detailed analysis of the exchange interactions between them. In particular, the question of whether the exchange interaction in [FeGd] is isotropic or anisotropic has been addressed. Standard variable-temperature magnetic susceptibility measurements provide only the value for a linear combination of Jx, Jy, and Jz but contain no information about the values of the individual exchange parameters Jx, Jy, and Jz. In contrast, the spin-Hamiltonian analysis of the variable-field, variable-temperature Mössbauer spectra reveals an exquisite sensitivity on the anisotropic exchange parameters. Analysis of these dependencies in conjunction with adopting the g-values obtained for [FeLa], yielded the values Jx = Ŝ1 · J · Ŝ2 +0.11 cm−1, Jy = +0.33 cm−1, and Jz = +1.20 cm−1 (convention). The consistency of these results with magnetic susceptibility data is analyzed. The exchange anisotropy is rooted in the spatial anisotropy of the low-spin FeIII ion. The condition for anisotropic exchange is the presence of low-lying orbital excited states at the ferric site that (i) effectively interact through spin-orbit coupling with the orbital ground state and (ii) have an exchange parameter with the Gd site with a value different from that for the ground state. DFT calculations, without spin-orbit coupling, reveal that the unpaired electron of the t2g5 ground configuration of the FeIII ion occupies the xy orbital, i.e. the orbital along the plane perpendicular to the Fe⋯Gd vector. The exchange-coupling constants for this orbital, jxy, and the other t2g orbitals, jyz and jxz, have been determined using a theoretical model that relates them to the anisotropic exchange parameters and the g-values of FeIII. The resulting values, jyz = −5.7 cm−1, jxz = −4.9 cm−1, and jxy = +0.3 cm−1 are quite different. The origin of the difference is briefly discussed.
3d-4f interaction; Mössbauer; EPR; magnetic susceptibility; anisotropic exchange coupling
An extension is proposed to the rigid-bond description of atomic thermal motion in crystals.
The rigid-bond model [Hirshfeld (1976 ▶). Acta Cryst. A32, 239–244] states that the mean-square displacements of two atoms are equal in the direction of the bond joining them. This criterion is widely used for verification (as intended by Hirshfeld) and also as a restraint in structure refinement as suggested by Rollett [Crystallographic Computing (1970 ▶), edited by F. R. Ahmed et al., pp. 167–181. Copenhagen: Munksgaard]. By reformulating this condition, so that the relative motion of the two atoms is required to be perpendicular to the bond, the number of restraints that can be applied per anisotropic atom is increased from about one to about three. Application of this condition to 1,3-distances in addition to the 1,2-distances means that on average just over six restraints can be applied to the six anisotropic displacement parameters of each atom. This concept is tested against very high resolution data of a small peptide and employed as a restraint for protein refinement at more modest resolution (e.g. 1.7 Å).
rigid-bond test; refinement restraints; anisotropic displacement parameters
The vocal folds are known to be mechanically anisotropic due to the microstructural arrangement of fibrous proteins such as collagen and elastin in the lamina propria. Even though this has been known for many years, the biomechanical anisotropic properties have rarely been experimentally studied. We propose that an indentation procedure can be used with uniaxial tension in order to obtain an estimate of the biomechanical anisotropy within a single specimen. Experiments were performed on the lamina propria of three male and three female human vocal folds dissected from excised larynges. Two experiments were conducted: each specimen was subjected to cyclic uniaxial tensile loading in the longitudinal (i.e. anterior-posterior) direction, and then to cyclic indentation loading in the transverse (i.e. medial-lateral) direction. The indentation experiment was modeled as contact on a transversely isotropic half-space using the Barnett-Lothe tensors. The longitudinal elastic modulus EL was computed from the tensile test, and the transverse elastic modulus ET and longitudinal shear modulus GL were obtained by inverse analysis of the indentation force-displacement response. It was discovered that the average of EL/ET was 14 for the vocal ligament and 39 for the vocal fold cover specimens. Also, the average of EL/GL, a parameter important for models of phonation, was 28 for the vocal ligament and 54 for the vocal fold cover specimens. These measurements of anisotropy could contribute to more accurate models of fundamental frequency regulation and provide potentially better insights into the mechanics of vocal fold vibration.
Indentation; Tensile deformation; Anisotropy; Larynx; Biomechanics
Due to the avascular nature of articular cartilage, solute transport through its extracellular matrix is critical for the maintenance and the functioning of the tissue. What’s more, diffusion of macromolecules may be affected by the microstructure of the extracellular matrix in both undeformed and deformed cartilage and experiments demonstrate diffusion anisotropy in the case of large solute. However, these phenomena have not received sufficient theoretical attention to date.
We hypothesize here that the diffusion anisotropy of macromolecules is brought about by the particular microstructure of the cartilage network. Based on this hypothesis, we then propose a mathematical model that correlates the diffusion coefficient tensor with the structural orientation tensor of the network. This model is shown to be successful in describing anisotropic diffusion of macromolecules in undeformed tissue and is capable of clarifying the effects of network reorientation as the tissue deforms under mechanical load. Additionally, our model explains the anomaly that at large strain, in a cylindrical plug under unconfined compression, solute diffusion in the radial direction increases with strain.
Our results indicate that in cartilage the degree of diffusion anisotropy is site specific but depends also on the size of the diffusing molecule. Mechanical loading initiates and/or further exacerbates this anisotropy. At small deformation, solute diffusion is near isotropic in a tissue that is isotropic in its unstressed state, becoming anisotropic as loading progresses. Mechanical loading leads to an attenuation of solute diffusion in all directions when deformation is small. However, loading, if it is high enough, enhances solute transport in the direction perpendicular to the load line, instead of inhibiting it.
diffusion - convection; anisotropy; soft tissue; deformation; cartilage
In the framework of a recently proposed method for in vivo lung morphometry, acinar lung airways are considered as a set of randomly oriented cylinders covered by alveolar sleeves. Diffusion of 3He in each airway is anisotropic and can be described by distinct longitudinal and transverse diffusion coefficients. This macroscopically isotropic but microscopically anisotropic model allows estimation of these diffusion coefficients from multi b-value MR experiments despite the airways being too small to be resolved by direct imaging. Herein a Bayesian approach is used for analyzing the uncertainties in the model parameter estimates. The approach allows evaluation of relative errors of the parameter estimates as functions of the “true” values of the parameters, the signal-to noise ratio, the maximum b-value and the total number of b-values used in the experiment. For a given set of the “true” diffusion parameters, the uncertainty in the estimated diffusion coefficients has a minimum as a function of maximum b-value and total number of data points. Choosing the MR pulse sequence parameters corresponding to this minimum optimizes the diffusion MR experiment and gives the best possible estimates of the diffusion coefficients. The mathematical approach presented can be generalized for models containing arbitrary numbers of estimated parameters.
diffusion MRI; hyperpolarized gas; lung airways; Bayesian analysis
The structure of human carbonic anhydrase II has been solved with a sulfonamide inhibitor at 0.9 Å resolution. Structural variation and flexibility is seen on the surface of the protein and is consistent with the anisotropic ADPs obtained from refinement. Comparison with 13 other atomic resolution carbonic anhydrase structures shows that surface variation exists even in these highly ordered isomorphous crystals.
Carbonic anhydrase has been well studied structurally and functionally owing to its importance in respiration. A large number of X-ray crystallographic structures of carbonic anhydrase and its inhibitor complexes have been determined, some at atomic resolution. Structure determination of a sulfonamide-containing inhibitor complex has been carried out and the structure was refined at 0.9 Å resolution with anisotropic atomic displacement parameters to an R value of 0.141. The structure is similar to those of other carbonic anhydrase complexes, with the inhibitor providing a fourth nonprotein ligand to the active-site zinc. Comparison of this structure with 13 other atomic resolution (higher than 1.25 Å) isomorphous carbonic anhydrase structures provides a view of the structural similarity and variability in a series of crystal structures. At the center of the protein the structures superpose very well. The metal complexes superpose (with only two exceptions) with standard deviations of 0.01 Å in some zinc–protein and zinc–ligand bond lengths. In contrast, regions of structural variability are found on the protein surface, possibly owing to flexibility and disorder in the individual structures, differences in the chemical and crystalline environments or the different approaches used by different investigators to model weak or complicated electron-density maps. These findings suggest that care must be taken in interpreting structural details on protein surfaces on the basis of individual X-ray structures, even if atomic resolution data are available.
carbonic anhydrase; structure comparison; metalloproteins; atomic resolution
The application of a new normal-mode-based X-ray crystallographic refinement method to a total of eight structures of moderate resolution is illustrated.
The structural refinement of large complexes at the lower resolution limit is often difficult and inefficient owing to the limited number of reflections and the frequently high-level structural flexibility. A new normal-mode-based X-ray crystallographic refinement method has recently been developed that enables anisotropic B-factor refinement using a drastically smaller number of thermal parameters than even isotropic refinement. Here, the method has been systematically tested on a total of eight systems in the resolution range 3.0–3.9 Å. This series of tests established the most applicable scenarios for the method, the detailed procedures for its application and the degree of structural improvement. The results demonstrated substantial model improvement at the lower resolution limit, especially in cases in which other methods such as the translation–libration–screw (TLS) model were not applicable owing to the poorly converged isotropic B-factor distribution. It is expected that this normal-mode-based method will be a useful tool for structural refinement, in particular at the lower resolution limit, in the field of X-ray crystallography.
conformational flexibility; anisotropic thermal parameters; moderate resolution
To measure spatial variations in mechanical properties of biological materials, prior studies have typically performed mechanical tests on excised specimens of tissue. Less invasive measurements, however, are preferable in many applications, such as patient-specific modeling, disease diagnosis, and tracking of age- or damage-related degradation of mechanical properties. Elasticity imaging (elastography) is a nondestructive imaging method in which the distribution of elastic properties throughout a specimen can be reconstructed from measured strain or displacement fields. To date, most work in elasticity imaging has concerned incompressible, isotropic materials. This study presents an extension of elasticity imaging to three-dimensional, compressible, transversely isotropic materials. The formulation and solution of an inverse problem for an anisotropic tissue subjected to a combination of quasi-static loads is described, and an optimization and regularization strategy that indirectly obtains the solution to the inverse problem is presented. Several applications of transversely isotropic elasticity imaging to cancellous bone from the human vertebra are then considered. The feasibility of using isotropic elasticity imaging to obtain meaningful reconstructions of the distribution of material properties for vertebral cancellous bone from experiment is established. However, using simulation, it is shown that an isotropic reconstruction is not appropriate for anisotropic materials. It is further shown that the transversely isotropic method identifies a solution that predicts the measured displacements, reveals regions of low stiffness, and recovers all five elastic parameters with approximately 10% error. The recovery of a given elastic parameter is found to require the presence of its corresponding strain (e.g., a deformation that generates ε12 is necessary to reconstruct C1212), and the application of regularization is shown to improve accuracy. Finally, the effects of noise on reconstruction quality is demonstrated and a signal-to-noise ratio (SNR) of 40 dB is identified as a reasonable threshold for obtaining accurate reconstructions from experimental data. This study demonstrates that given an appropriate set of displacement fields, level of regularization, and signal strength, the transversely isotropic method can recover the relative magnitudes of all five elastic parameters without an independent measurement of stress. The quality of the reconstructions improves with increasing contrast, magnitude of deformation, and asymmetry in the distributions of material properties, indicating that elasticity imaging of cancellous bone could be a useful tool in laboratory studies to monitor the progression of damage and disease in this tissue.
elastography; anisotropy; nondestructive imaging; modulus; trabecular bone; inverse problems
This report describes complete practical guidelines and insights for the crystalline sponge method, which have been derived through the first use of synchrotron radiation on these systems, and includes a procedure for faster synthesis of the sponges. These guidelines will be applicable to crystal sponge data collected at synchrotrons or in-house facilities, and will allow researchers to obtain reliable high-quality data and construct chemically and physically sensible models for guest structural determination.
A detailed set of synthetic and crystallographic guidelines for the crystalline sponge method based upon the analysis of expediently synthesized crystal sponges using third-generation synchrotron radiation are reported. The procedure for the synthesis of the zinc-based metal–organic framework used in initial crystal sponge reports has been modified to yield competent crystals in 3 days instead of 2 weeks. These crystal sponges were tested on some small molecules, with two being unexpectedly difficult cases for analysis with in-house diffractometers in regard to data quality and proper space-group determination. These issues were easily resolved by the use of synchrotron radiation using data-collection times of less than an hour. One of these guests induced a single-crystal-to-single-crystal transformation to create a larger unit cell with over 500 non-H atoms in the asymmetric unit. This led to a non-trivial refinement scenario that afforded the best Flack x absolute stereochemical determination parameter to date for these systems. The structures did not require the use of PLATON/SQUEEZE or other solvent-masking programs, and are the highest-quality crystalline sponge systems reported to date where the results are strongly supported by the data. A set of guidelines for the entire crystallographic process were developed through these studies. In particular, the refinement guidelines include strategies to refine the host framework, locate guests and determine occupancies, discussion of the proper use of geometric and anisotropic displacement parameter restraints and constraints, and whether to perform solvent squeezing/masking. The single-crystal-to-single-crystal transformation process for the crystal sponges is also discussed. The presented general guidelines will be invaluable for researchers interested in using the crystalline sponge method at in-house diffraction or synchrotron facilities, will facilitate the collection and analysis of reliable high-quality data, and will allow construction of chemically and physically sensible models for guest structural determination.
X-ray crystallography; crystalline sponge method; metal–organic framework; single-crystal-to-single-crystal transformation; synchrotron radiation
Single crystals of warwickite, trimagnesium titanium(IV) dioxide bis(borate), Mg3TiO2(BO3)2, were prepared by slow cooling of the melt. The title compound is isotypic with Co3TiO2(BO3)2. In contrast to the previous refinement of warwickite [Moore & Araki (1974 ▶). Am. Mineral.
59, 985–1004], that reported only isotropic atomic displacement parameters for all atoms, anisotropic displacement parameters of all atoms were refined during the current redetermination. All atoms are situated on special positions (site symmetry .m.). One of the two Mg sites is statistically disordered with Ti atoms (ratio 1:1), while the other is fully occupied by Mg atoms. The occupancy ratio of the Mg and Ti atoms is similar to that reported in the previous study. Metal atoms (M) at the Ti/Mg and Mg sites are coordinated by six O atoms in form of distorted octahedra. Four edge-sharing MO6 octahedra form M
4O18 units, which are connected by common corners into layers parallel to (010). Adjacent layers are linked along  into a framework structure by sharing common edges. The B atoms are located in the triangular prismatic tunnels of the framework.
Spatial models of functional magnetic resonance imaging (fMRI) data allow one to estimate the spatial smoothness of general linear model (GLM) parameters and eschew pre-process smoothing of data entailed by conventional mass-univariate analyses. Recently diffusion-based spatial priors (Harrison et al., 2008) were proposed, which provide a way to formulate an adaptive spatial basis, where the diffusion kernel of a weighted graph-Laplacian (WGL) is used as the prior covariance matrix over GLM parameters. An advantage of these is that they can be used to relax the assumption of isotropy and stationarity implicit in smoothing data with a fixed Gaussian kernel. The limitation of diffusion-based models is purely computational, due to the large number of voxels in a brain volume. One solution is to partition a brain volume into slices, using a spatial model for each slice. This reduces computational burden by approximating the full WGL with a block diagonal form, where each block can be analysed separately. While fMRI data are collected in slices, the functional structures exhibiting spatial coherence and continuity are generally three-dimensional, calling for a more informed partition. We address this using the graph-Laplacian to divide a brain volume into sub-graphs, whose shape can be arbitrary. Their shape depends crucially on edge weights of the graph, which can be based on the Euclidean distance between voxels (isotropic) or on GLM parameters (anisotropic) encoding functional responses. The result is an approximation the full WGL that retains its 3D form and also has potential for parallelism. We applied the method to high-resolution (1mm3) fMRI data and compared models where a volume was divided into either slices or graph-partitions. Models were optimized using Expectation-Maximization and the approximate log-evidence computed to compare these different ways to partition a spatial prior. The real high-resolution fMRI data presented here had greatest evidence for the graph partitioned anisotropic model, which was best able to preserve fine functional detail.
High-resolution functional magnetic resonance images; graph-Laplacian; diffusion-based spatial priors; graph partitioning; Expectation-Maximization; model comparison
Accurate representations and measurements of skull electrical conductivity are essential in developing appropriate forward models for applications such as inverse EEG or Electrical Impedance Tomography of the head. Because of its layered structure, it is often assumed that skull is anisotropic, with an anisotropy ratio around 10. However, no detailed investigation of skull anisotropy has been performed. In this paper we investigate four-electrode measurements of conductivities and their relation to tissue anisotropy ratio (ratio of tangential to radial conductivity) in layered or anisotropic biological samples similar to bone. It is shown here that typical values for the thicknesses and radial conductivities of individual skull layers produce tissue with much smaller anisotropy ratios than 10. Moreover, we show that there are very significant differences between the field patterns formed in a three-layered isotropic structure plausible for bone, and those formed assuming that bone is homogeneous and anisotropic.We performed a measurement of conductivity using an electrode configuration sensitive to the distinction between three-layered and homogeneous anisotropic composition and found results consistent with the sample being three-layered. We recommend that the skull be more appropriately represented as three isotropic layers than as homogeneous and anisotropic.
skull conductivity; anisotropy; finite element model; head model
This paper describes the concepts and implementation of an MRI method, Multiple Echo Diffusion Tensor Acquisition Technique (MEDITATE), which is capable of acquiring apparent diffusion tensor maps in two scans on a 3T clinical scanner. In each MEDITATE scan, a set of RF-pulses generates multiple echoes whose amplitudes are diffusion-weighted in both magnitude and direction by a pattern of diffusion gradients. As a result, two scans acquired with different diffusion weighting strengths suffice for accurate estimation of diffusion tensor imaging (DTI)-parameters. The MEDITATE variation presented here expands previous MEDITATE approaches to adapt to the clinical scanner platform, such as exploiting longitudinal magnetization storage to reduce T2-weighting. Fully segmented multi-shot Cartesian encoding is used for image encoding. MEDITATE was tested on isotropic (agar gel), anisotropic diffusion phantoms (asparagus), and in vivo skeletal muscle in healthy volunteers with cardiac-gating. Comparisons of accuracy were performed with standard twice-refocused spin echo (TRSE) DTI in each case and good quantitative agreement was found between diffusion eigenvalues, mean diffusivity, and fractional anisotropy derived from TRSE-DTI and from the MEDITATE sequence. Orientation patterns were correctly reproduced in both isotropic and anisotropic phantoms, and approximately so for in vivo imaging. This illustrates that the MEDITATE method of compressed diffusion encoding is feasible on the clinical scanner platform. With future development and employment of appropriate view-sharing image encoding this technique may be used in clinical applications requiring time-sensitive acquisition of DTI parameters such as dynamical DTI in muscle.
Diffusion; DTI; Multiple Modulation Multiple Echo; Fast Acquisition
Evaluate the performance of a new 3T high-resolution trabecular bone (TB) imaging technique at two resolution regimes in terms of serial reproducibility and sensitivity.
Materials and Methods
The left distal tibial metaphysis of seven healthy volunteers was imaged at three time-points using a FLASE (fast large-angle spin-echo) pulse sequence at 137×137×410 μm3 and (160 μm)3 voxel sizes. Image artifacts, motion degradation, and serial image volume misalignments were controlled to maximize reproducibility of image-derived measures of scale, topology, orientation in terms of structural anisotropy, and finite-element derived Young’s and shear moduli. Coefficients of variation (CV) and intraclass correlation coefficients (ICC) for structural and mechanical parameters were evaluated as measures of reproducibility and reliability. The ability of structural and mechanical parameters to distinguish between subjects was tested by analysis of variance (ANOVA).
Reproducibility was generally higher in the anisotropic data (CVs 1–5% versus 1–9% for isotropic images). Anisotropic voxel size yielded greater measurement reliability (ICC 0.75–0.99, average 0.92 versus 0.62–0.99, average 0.86, at isotropic resolution) and better discrimination of the seven subjects (75% versus 50% of the possible comparisons were significantly different (p<0.05)) except for measures of structural anisotropy and topology. Isotropic resolution improved detection of structural orientation and permitted visualization of small perforations in longitudinal trabecular plates not detected at anisotropic resolution.
Improved image acquisition and processing techniques enhance reproducibility of structural and mechanical parameters derived from high-resolution MRI of metaphyseal bone in the distal tibia.
trabecular bone; micro-MRI; osteoporosis; resolution isotropy; quantitative imaging
The structural organization of biological tissues and cells often produces anisotropic transport properties. These tissues may also undergo large deformations under normal function, potentially inducing further anisotropy. A general framework for formulating constitutive relations for anisotropic transport properties under finite deformation is lacking in the literature. This study presents an approach based on representation theorems for symmetric tensor-valued functions and provides conditions to enforce positive semi-definiteness of the permeability or diffusivity tensor. Formulations are presented which describe materials that are orthotropic, transversely isotropic, or isotropic in the reference state, and where large strains induce greater anisotropy. Strain-induced anisotropy of the permeability of a solid-fluid mixture is illustrated for finite torsion of a cylinder subjected to axial permeation. It is shown that, in general, torsion can produce a helical flow pattern, rather than the rectilinear pattern observed when adopting a more specialized, unconditionally isotropic spatial permeability tensor commonly used in biomechanics. The general formulation presented in this study can produce both affine and non-affine reorientation of the preferred directions of material symmetry with strain, depending on the choice of material functions. This study addresses a need in the biomechanics literature by providing guidelines and formulations for anisotropic strain-dependent transport properties in porous-deformable media undergoing large deformations.
Advances in magnetic resonance imaging (MRI) have contributed greatly to the study of neurodegenerative processes, psychiatric disorders, and normal human development, but the effect of such improvements on the reliability of downstream morphometric measures has not been extensively studied. We examined how MRI-derived neurostructural measures are affected by three technological advancements: parallel acceleration, increased spatial resolution, and the use of a high bandwidth multiecho sequence. Test-retest data were collected from 11 healthy participants during 2 imaging sessions occurring approximately 2 weeks apart. We acquired 4 T1-weighted MP-RAGE sequences during each session: a non-accelerated anisotropic sequence (MPR), a non-accelerated isotropic sequence (ISO), an accelerated isotropic sequence (ISH), and an accelerated isotropic high bandwidth multiecho sequence (MEM). Cortical thickness and volumetric measures were computed for each sequence to assess test-retest reliability and measurement bias. Reliability was extremely high for most measures and similar across imaging parameters. Significant measurement bias was observed, however, between MPR and all isotropic sequences for all cortical regions and some subcortical structures. These results suggest that these improvements in MRI acquisition technology do not compromise data reproducibility, but that consistency should be maintained in choosing imaging parameters for structural MRI studies.
We employ a generalized van der Waals-Onsager perturbation theory to construct a free energy functional capable of describing the thermodynamic properties and orientational order of the isotropic and nematic phases of attractive disc particles. The model mesogen is a hard (purely repulsive) cylindrical disc particle decorated with an anisotropic square-well attractive potential placed at the centre of mass. Even for isotropic attractive interactions, the resulting overall inter-particle potential is anisotropic, due to the orientation-dependent excluded volume of the underlying hard core. An algebraic equation of state for attractive disc particles is developed by adopting the Onsager trial function to characterize the orientational order in the nematic phase. The theory is then used to represent the fluid-phase behaviour (vapour-liquid, isotropic-nematic, and nematic-nematic) of the oblate attractive particles for varying values of the molecular aspect ratio and parameters of the attractive potential. When compared to the phase diagram of their athermal analogues, it is seen that the addition of an attractive interaction facilitates the formation of orientationally-ordered phases. Most interestingly, for certain aspect ratios, a coexistence between two anisotropic nematic phases is exhibited by the attractive disc-like fluids.
equation of state; discotics; attractive cylindrical disc; nematic-nematic equilibria; anisotropic square well; phase diagrams
The development of ultrasound for use in dental tissues is hampered by the complex, multilayered nature of the teeth. The purpose of this preliminary study was to obtain the phase and group velocities associated with several directions of ultrasonic wave propagation in relation to the tooth structure, which would then lead to the determination of the elastic constants in dental hard tissue. Knowledge of these elastic constants can be used to feed back into numerical models (such as finite element) in order to simulate/predict ultrasonic wave propagation and behavior in the teeth. This will help to optimize ultrasonic protocols as potential noninvasive therapeutic tools for novel dental regenerative therapies.
An extracted human second molar was used to determine time-of-flight information from A-scan signatures obtained at various angles of inclination and rotation using a scanning acoustic microscope at 10 MHz. Phase and group velocities and associated slowness curves were calculated in order to determine the independent elastic constants in the human teeth.
Results show that as the tooth was inclined at three azimuthal angles (Θin = 0°, 15°, and 30°) and rotated from Φin = 0° to 360° in order to cover the whole perimeter of the tooth, slowness curves constructed from the computed phase and group velocities versus angle of rotation confirm the inhomogeneous and anisotropic nature of the tooth as indicated by the nonuniform appearance of uneven circular shape patterns of the measurements when compared to those produced in a control isotropic fused quartz sample.
This study demonstrates that phase and group velocities of ultrasound as determined by acoustic microscopy change and are dependent on the direction of the tooth structure. Thus, these results confirm that the tooth is indeed a multilayered anisotropic structure underscoring that there is no single elastic constant sufficient to represent the complex structure of the tooth. The findings underline the importance to take into account these crucial characteristics in order to develop and optimize therapeutic as well as diagnostic applications of ultrasound in dental tissue repair, and further studies are warranted to analyze ultrasound transmission at various frequencies and intensities in different teeth to develop appropriate models for ultrasound biophysical behavior in dental tissues.
Dental ultrasound; Therapeutic ultrasound; Tooth repair; Phase velocities; Slowness curves
The crystal structure of kovdorskite, ideally Mg2PO4(OH)·3H2O (dimagnesium phosphate hydroxide trihydrate), was reported previously with isotropic displacement paramaters only and without H-atom positions [Ovchinnikov et al. (1980 ▶). Dokl. Akad. Nauk SSSR.
255, 351–354]. In this study, the kovdorskite structure is redetermined based on single-crystal X-ray diffraction data from a sample from the type locality, the Kovdor massif, Kola Peninsula, Russia, with anisotropic displacement parameters for all non-H atoms, with all H-atom located and with higher precision. Moreover, inconsistencies of the previously published structural data with respect to reported and calculated X-ray powder patterns are also discussed. The structure of kovdorskite contains a set of four edge-sharing MgO6 octahedra interconnected by PO4 tetrahedra and O—H⋯O hydrogen bonds, forming columns and channels parallel to . The hydrogen-bonding system in kovdorskite is formed through the water molecules, with the OH− ions contributing little, if any, to the system, as indicated by the long H⋯A distances (>2.50 Å) to the nearest O atoms. The hydrogen-bond lengths determined from the structure refinement agree well with Raman spectroscopic data.