A least squares method for estimation of Bezier curves and surfaces and its applicability to multivariate analysis

A new least squares estimation method for Bezier polynomial curves and surfaces is described and illustrated. The Cartesian coordinates of the vertices of polygons defining the curves separated well natural populations of Anodonta cygnea L. and human sagittal profiles of different sexes and age classes. Multivariate comparisons of the coordinates of Bezier polygon vertices and Euclidean distance measures showed that the polygon coordinates revealed shape differences better than distances between homologous points on the curves. Further, polygon coordinates separated groups for more than two variables as well as or better than the equivalent number distances. In addition, polygon coordinates permit construction of mean shapes and their variances. Possible applications in trend surface analyses and for illustration in computer-aided identification programs are suggested.     

3D articulated growth model of the fetus skeleton,envelope and soft tissues

Fetal dosimetry studies require the development of accurate 3D models of the fetus. This paper proposes a 3D articulated fetal growth model including skeleton, body envelope, brain and lungs based on medical images of ten different fetuses acquired in clinical routine. The structures of interest were semi-manually segmented from the images and surface meshes were generated. A generic mesh of each structure has been deformed towards the segmented ones. By interpolating linearly between the subjects of the database, each structure can be estimated at any age and in any position. This process results in an automated model, the operator being only required to specify the age and position of the desired estimated fetus.     

Nanostructure design using protein building blocks enhanced by conformationally constrained synthetic residues

Increasing efforts are being invested in the construction of nanostructures with desired shapes and physical and chemical properties. Our strategy involves nanostructure design using naturally occurring protein building blocks. Inspection of the protein structural database (PDB) reveals the richness of the conformations, shapes, and chemistries of proteins and their building blocks. To increase the population of the native fold in the selected building block, we mutate natural residues by engineered, constrained residues that restrict the conformational freedom at the targeted site and have favorable interactions, geometry, and size. Here, as a model system, we construct nanotubes using building blocks from left-handed beta-helices which are commonly occurring repeat protein architectures. We pick two-turn beta-helical segments, duplicate and stack them, and using all-atom molecular dynamics simulations (MD) with explicit solvent probe the structural stability of these nanotubular structures as indicated by their capacity to retain the initial organization and their conformational dynamics. Comparison of the results for the wild-type and mutated sequences shows that the introduction of the conformationally restricted 1-aminocyclopropanecarboxylic acid (Ac3c) residue in loop regions greatly enhances the stability of beta-helix nanotubes. The Ac3c geometrical confinement effect is sequence-specific and position-specific. The achievement of high stability of nanotubular structures originates not only from the reduction of mobility at the mutation site induced by Ac3c but also from stabilizing association forces between building blocks such as hydrogen bonds and hydrophobic contacts. For the selected synthetic residue, similar size, hydrophobicity, and backbone conformational tendencies are desirable as in the Ac3c.     

Analyzing ion distributions around DNA

We present a new method for analyzing ion, or molecule, distributions around helical nucleic acids and illustrate the approach by analyzing data derived from molecular dynamics simulations. The analysis is based on the use of curvilinear helicoidal coordinates and leads to highly localized ion densities compared to those obtained by simply superposing molecular dynamics snapshots in Cartesian space. The results identify highly populated and sequence-dependent regions where ions strongly interact with the nucleic and are coupled to its conformational fluctuations. The data from this approach is presented as ion populations or ion densities (in units of molarity) and can be analyzed in radial, angular and longitudinal coordinates using 1D or 2D graphics. It is also possible to regenerate 3D densities in Cartesian space. This approach makes it easy to understand and compare ion distributions and also allows the calculation of average ion populations in any desired zone surrounding a nucleic acid without requiring references to its constituent atoms. The method is illustrated using microsecond molecular dynamics simulations for two different DNA oligomers in the presence of 0.15 M potassium chloride. We discuss the results in terms of convergence, sequence-specific ion binding and coupling with DNA conformation.     

Advances in Geometric Morphometrics

Geometric morphometrics is the statistical analysis of form based on Cartesian landmark coordinates. After separating shape from overall size, position, and orientation of the landmark configurations, the resulting Procrustes shape coordinates can be used for statistical analysis. Kendall shape space, the mathematical space induced by the shape coordinates, is a metric space that can be approximated locally by a Euclidean tangent space. Thus, notions of distance (similarity) between shapes or of the length and direction of developmental and evolutionary trajectories can be meaningfully assessed in this space. Results of statistical techniques that preserve these convenient properties—such as principal component analysis, multivariate regression, or partial least squares analysis—can be visualized as actual shapes or shape deformations. The Procrustes distance between a shape and its relabeled reflection is a measure of bilateral asymmetry. Shape space can be extended to form space by augmenting the shape coordinates with the natural logarithm of Centroid Size, a measure of size in geometric morphometrics that is uncorrelated with shape for small isotropic landmark variation. The thin-plate spline interpolation function is the standard tool to compute deformation grids and 3D visualizations. It is also central to the estimation of missing landmarks and to the semilandmark algorithm, which permits to include outlines and surfaces in geometric morphometric analysis. The powerful visualization tools of geometric morphometrics and the typically large amount of shape variables give rise to a specific exploratory style of analysis, allowing the identification and quantification of previously unknown shape features.     

Motor primitives in vertebrates and invertebrates

In recent years different lines of evidence have led to the idea that motor actions and movements in both vertebrates and invertebrates are composed of elementary building blocks. The entire motor repertoire can be spanned by applying a well-defined set of operations and transformations to these primitives and by combining them in many different ways according to well-defined syntactic rules. Motor and movement primitives and modules might exist at the neural, dynamic and kinematic levels with complicated mapping among the elementary building blocks subserving these different levels of representation. Hence, while considerable progress has been made in recent years in unravelling the nature of these primitives, new experimental, computational and conceptual approaches are needed to further advance our understanding of motor compositionality.     

Reversible network reconnection model for simulating large deformation in dynamic tissue morphogenesis

Morphogenesis of tissues in organ development is accompanied by large three-dimensional (3D) deformations, in which mechanical interactions among multiple cells are spatiotemporally regulated. To reveal the deformation mechanisms, in this study, we developed the reversible network reconnection (RNR) model. The model is developed on the basis of 3D vertex model, which expresses a multicellular aggregate as a network composed of vertices. 3D vertex models have successfully simulated morphogenetic dynamics by expressing cellular rearrangements as network reconnections. However, the network reconnections in 3D vertex models can cause geometrical irreversibility, energetic inconsistency, and topological irreversibility, therefore inducing unphysical results and failures in simulating large deformations. To resolve these problems, we introduced (1) a new definition of the shapes of polygonal faces between cellular polyhedrons, (2) an improved condition for network reconnections, (3) a new condition for potential energy functions, and (4) a new constraint condition for the shapes of polygonal faces that represent cell–cell boundaries. Mathematical and computational analyses demonstrated that geometrical irreversibility, energetic inconsistency, and topological irreversibility were resolved by suppressing the geometrical gaps in the network and avoiding the generation of irreversible network patterns in reconnections. Lastly, to demonstrate the applicability of the RNR model, we simulated tissue deformation of growing cell sheets and showed that our model can simulate large tissue deformations, in which large changes occur in the local curvatures and layer formations of tissues. Thus, the RNR model enables in silico recapitulation of complex tissue morphogenesis.     

Parametric-equation representation of biconcave erythrocytes

The representation of the shape of a biconcave erythrocyte by a set of three parametric equations was achieved by using the expressions that transform the curvilinear coordinates from the disc-cyclide coordinate system [denoted J2R; Moon and Spencer (1988), Field Theory Handbook, Springer-Verlag, Berlin] to Cartesian coordinates. The equations are products of elliptic functions, so the challenge was to relate the three major ’shape-defining’ measurements of the human erythrocyte in Cartesian coordinates to three parameters in the new curvilinear coordinates, to give a realistic representation of the shape of the membrane-surface. The relationships between the coefficients of the Cartesian degree-4 surface that describes the discocyte and the coordinate transformation equations were derived with the aid of Mathematica; and the membrane-surface of the cell was drawn using the ParametricPlot3D function in this ‘package’. By having the erythrocyte shape expressed in its new form it is readily amenable to further transformations that might be used to model those changes in shape that are seen when the cells are immersed in media of various osmolalities, or when they change metabolic ’states’. On the other hand, the relationship between the coefficients of the Cartesian expression for the disc-cyclide surface is relevant to image analysis of erythrocytes, as determined by physical methods that rely on Cartesian imaging ’slices’. These methods include confocal microscopy and various nuclear magnetic resonance microimaging procedures.     

Principles of nanostructure design with protein building blocks

Currently there is increasing interest in nanostructures and their design. Nanostructure design involves the ability to predictably manipulate the properties of the self-assembly of autonomous units. Autonomous units have preferred conformational states. The units can be synthetic material science-based or derived from functional biological macromolecules. Autonomous biological building blocks with available structures provide an extremely rich and useful resource for design. For proteins, the structural databases contain large libraries of protein molecules and their building blocks with a range of shapes, surfaces, and chemical properties. The introduction of engineered synthetic residues or short peptides into these can expand the available chemical space and enhance the desired properties. Here we focus on the principles of nanostructure design with protein building blocks.     

Structure by design: from single proteins and their building blocks to nanostructures

Nanotechnology realizes the advantages of naturally occurring biological macromolecules and their building-block nature for design. Frequently, assembly starts with the choice of a "good" molecule that is synthetically optimized towards the desired shape. By contrast, we propose starting with a pre-specified nanostructure shape, selecting candidate protein building blocks from a library and mapping them onto the shape and, finally, testing the stability of the construct. Such a shape-based, part-assembly strategy is conceptually similar to protein design through the combinatorial assembly of building blocks. If the conformational preferences of the building blocks are retained and their interactions are favorable, the nanostructure will be stable. The richness of the conformations, shapes and chemistries of the protein building blocks suggests a broad range of potential applications; at the same time, it also highlights their complexity. In this Opinion article, we focus on the first step: validating such a strategy against experimental data.     

Design and experimental evaluation of adjustable bone plates for mandibular fracture fixation

Conventional bone plates are commonly used for surgical mandibular fracture fixation. Improper alignment between bone segments, however, can result in malocclusion. Current methods of fixation require a surgeon to visually align segments of bone and affix a metal plate using bone screws, after which little can be done to adjust alignment. A method of adjusting fracture alignment after plate placement, without screw removal, presents an improvement over costly and risky revision surgery. A modified bone plate has been designed with a deformable section to give surgeons the ability to reduce misalignments at the fracture site. The mechanics of deformation for various adjustment mechanisms was explored analytically, numerically, and experimentally to ensure that the adjustable plate is comparable to conventional bone plates. A static force of 358.8 N is required to deform the adjustable bone plate, compared with predicted values of 351 N using numerical simulation and 362 N using a simple beam theory. Dynamic testing was performed to simulate in vivo loading conditions and evaluate load-capacity in both deformed and un-deformed bone plates. Results indicate that bending stiffness of a rectangular bone plate is 709 N/mm, compared with 174 N/mm for an octagonal plate and 176 N/mm for standard plates. Once deformed, the rectangular and octagonal plates had a stiffness of 323 N/mm and 228 N/mm, respectively. Un-deformed and deformed adjustable bone plates have efficacy in bone segment fixation and healing.     

Use of RNA structure flexibility data in nanostructure modeling

In the emerging field of RNA-based nanotechnology there is a need for automation of the structure design process. Our goal is to develop computer methods for aiding in this process. Towards that end, we created the RNA junction database, which is a repository of RNA junctions, i.e. internal, multi-branch and kissing loops with emanating stem stubs, extracted from the larger RNA structures stored in the PDB database. These junctions can be used as building blocks for nanostructures. Two programs developed in our laboratory, NanoTiler and RNA2D3D, can combine such building blocks with idealized fragments of A-form helices to produce desired 3D nanostructures. Initially, the building blocks are treated as rigid objects and the resulting geometry is tested against the design objectives. Experimental data, however, shows that RNA accommodates its shape to the constraints of larger structural contexts. Therefore we are adding analysis of the flexibility of our building blocks to the full design process. Here we present an example of RNA-based nanostructure design, putting emphasis on the need to characterize the structural flexibility of the building blocks to induce ring closure in the automated exploration. We focus on the use of kissing loops (KL) in nanostructure design, since they have been shown to play an important role in RNA self-assembly. By using an experimentally proven system, the RNA tectosquare, we show that considering the flexibility of the KLs as well as distortions of helical regions may be necessary to achieve a realistic design.     

On the shapes of leaves

The properties of a function G(x, y) of 2-dimensional Cartesian coordinates x, y can be used to obtain complex shapes through a nonlinear iterative procedure. It is conjectured that at least some biological shapes are determined by this procedure, and those of leaves are considered as a test of the hypothesis. It is shown that this procedure will generate the shapes of even very complicated leaves such as those of the holly tree to a considerable degree of accuracy. Certain issues of evolutionary biology appear in our view in a clearer light according to this procedure, and these are discussed towards the end of the paper. © 1994 Wiley-Liss, Inc.     

Primitive Fitting Based on the Efficient multiBaySAC Algorithm

Although RANSAC is proven to be robust, the original RANSAC algorithm selects hypothesis sets at random, generating numerous iterations and high computational costs because many hypothesis sets are contaminated with outliers. This paper presents a conditional sampling method, multiBaySAC (Bayes SAmple Consensus), that fuses the BaySAC algorithm with candidate model parameters statistical testing for unorganized 3D point clouds to fit multiple primitives. This paper first presents a statistical testing algorithm for a candidate model parameter histogram to detect potential primitives. As the detected initial primitives were optimized using a parallel strategy rather than a sequential one, every data point in the multiBaySAC algorithm was assigned to multiple prior inlier probabilities for initial multiple primitives. Each prior inlier probability determined the probability that a point belongs to the corresponding primitive. We then implemented in parallel a conditional sampling method: BaySAC. With each iteration of the hypothesis testing process, hypothesis sets with the highest inlier probabilities were selected and verified for the existence of multiple primitives, revealing the fitting for multiple primitives. Moreover, the updated version of the initial probability was implemented based on a memorable form of Bayes’ Theorem, which describes the relationship between prior and posterior probabilities of a data point by determining whether the hypothesis set to which a data point belongs is correct. The proposed approach was tested using real and synthetic point clouds. The results show that the proposed multiBaySAC algorithm can achieve a high computational efficiency (averaging 34% higher than the efficiency of the sequential RANSAC method) and fitting accuracy (exhibiting good performance in the intersection of two primitives), whereas the sequential RANSAC framework clearly suffers from over- and under-segmentation problems. Future work will aim at further optimizing this strategy through its application to other problems such as multiple point cloud co-registration and multiple image matching.     

Contrast sensitivity functions to stimuli defined in Cartesian, polar and hyperbolic coordinates

Recent electrophysiological studies indicate that cells in the LGN, V1, V2, and V4 areas in monkeys are specifically sensitive to Cartesian, polar and hyperbolic stimuli. We have characterized the contrast sensitivity functions (CSF) to stimuli defined in these coordinates with the two-alternatives forced-choice paradigm. CSFs to Cartesian, concentric, and hyperbolic stimuli have had similar shapes, with peak sensitivity at approximately 3 c/deg. However, the Cartesian CSF peak sensitivity has been at least 0.1 log units higher than that to stimuli in any other coordinate system. The concentric-Bessel CSF has a low-pass shape, peaking at 1.5 c/deg or below. The radial CSF has a bell shape with maximum sensitivity at 8 c/360 degrees. Only the concentric-Bessel CSF could be explained in terms of the components of maximum amplitude of the Fourier transform. Neural models, which in previous studies predicted the responses to Cartesian and polar Glass patterns, failed to account for the full CSFs data.     

Dynamic Lung Modeling and Tumor Tracking Using Deformable Image Registration and Geometric Smoothing

A greyscale-based fully automatic deformable image registration algorithm, based on an optical flow method together with geometric smoothing, is developed for dynamic lung modeling and tumor tracking. In our computational processing pipeline, the input data is a set of 4D CT images with 10 phases. The triangle mesh of the lung model is directly extracted from the more stable exhale phase (Phase 5). In addition, we represent the lung surface model in 3D volumetric format by applying a signed distance function and then generate tetrahedral meshes. Our registration algorithm works for both triangle and tetrahedral meshes. In CT images, the intensity value reflects the local tissue density. For each grid point, we calculate the displacement from the static image (Phase 5) to match with the moving image (other phases) by using merely intensity values of the CT images. The optical flow computation is followed by a regularization of the deformation field using geometric smoothing. Lung volume change and the maximum lung tissue movement are used to evaluate the accuracy of the application. Our testing results suggest that the application of deformable registration algorithm is an effective way for delineating and tracking tumor motion in image-guided radiotherapy.     

Development and validation of a generic 3D model of the distal femur

The development and validation of a virtual generic 3D model of the distal femur using computer graphical methods is presented. The synthesis of the generic model requires the following steps: acquisition of bony 3D morphology using standard computed tomography (CT) imaging; alignment of 3D models reconstructed from CT images with a common coordinate system; computer graphical sectioning of the models; extraction of bone contours from the image sections; combining and averaging of extracted contours; and 3D reconstruction of the averaged contours. The generic models reconstructed from the averaged contours of six cadaver femora were validated by comparing their surface geometry on a point to point basis with that of the CT reconstructed reference models. The mean errors ranged from 0.99 to 2.5 mm and were in agreement with the qualitative assessment of the models.     

Model-based inverse estimation for active contraction stresses of tongue muscles using 3D surface shape in speech production

This paper presents a novel inverse estimation approach for the active contraction stresses of tongue muscles during speech. The proposed method is based on variational data assimilation using a mechanical tongue model and 3D tongue surface shapes for speech production. The mechanical tongue model considers nonlinear hyperelasticity, finite deformation, actual geometry from computed tomography (CT) images, and anisotropic active contraction by muscle fibers, the orientations of which are ideally determined using anatomical drawings. The tongue deformation is obtained by solving a stationary force-equilibrium equation using a finite element method. An inverse problem is established to find the combination of muscle contraction stresses that minimizes the Euclidean distance of the tongue surfaces between the mechanical analysis and CT results of speech production, where a signed-distance function represents the tongue surface. Our approach is validated through an ideal numerical example and extended to the real-world case of two Japanese vowels, /ʉ/ and /ɯ/. The results capture the target shape completely and provide an excellent estimation of the active contraction stresses in the ideal case, and exhibit similar tendencies as in previous observations and simulations for the actual vowel cases. The present approach can reveal the relative relationship among the muscle contraction stresses in similar utterances with different tongue shapes, and enables the investigation of the coordination of tongue muscles during speech using only the deformed tongue shape obtained from medical images. This will enhance our understanding of speech motor control.     

An anatomically based patient-specific finite element model of patella articulation: towards a diagnostic tool

A 3D anatomically based patient-specific finite element (FE) model of patello-femoral (PF) articulation is presented to analyse the main features of patella biomechanics, namely, patella tracking (kinematics), quadriceps extensor forces, surface contact and internal patella stresses. The generic geometries are a subset from the model database of the International Union of Physiological Sciences (IUPS) (http://www.physiome.org.nz) Physiome Project with soft tissue derived from the widely used visible human dataset, and the bones digitised from an anatomically accurate physical model with muscle attachment information. The models are customised to patient magnetic resonance images using a variant of free-form deformation, called 'host-mesh' fitting. The continuum was solved using the governing equation of finite elasticity, with the multibody problem coupled through contact mechanics. Additional constraints such as tissue incompressibility are also imposed. Passive material properties are taken from the literature and implemented for deformable tissue with a non-linear micro-structurally based constitutive law. Bone and cartilage are implemented using a 'St-Venant Kirchoff' model suitable for rigid body rotations. The surface fibre directions have been estimated from anatomy images of cadaver muscle dissections and active muscle contraction was based on a steady-state calcium-tension relation. The 3D continuum model of muscle, tendon and bone is compared with experimental results from the literature, and surgical simulations performed to illustrate its clinical assessment capabilities (a Maquet procedure for reducing patella stresses and a vastus lateralis release for a bipartite patella). Finally, the model limitations, issues and future improvements are discussed.     

A biomechanical model of mammographic compressions

A number of biomechanical models have been proposed to improve nonrigid registration techniques for multimodal breast image alignment. A deformable breast model may also be useful for overcoming difficulties in interpreting 2D X-ray projections (mammograms) of 3D volumes (breast tissues). If a deformable model could accurately predict the shape changes that breasts undergo during mammography, then the model could serve to localize suspicious masses (visible in mammograms) in the unloaded state, or in any other deformed state required for further investigations (such as biopsy or other medical imaging modalities). In this paper, we present a validation study that was conducted in order to develop a biomechanical model based on the well-established theory of continuum mechanics (finite elasticity theory with contact mechanics) and demonstrate its use for this application. Experimental studies using gel phantoms were conducted to test the accuracy in predicting mammographic-like deformations. The material properties of the gel phantom were estimated using a nonlinear optimization process, which minimized the errors between the experimental and the model-predicted surface data by adjusting the parameter associated with the neo-Hookean constitutive relation. Two compressions (the equivalent of cranio-caudal and medio-lateral mammograms) were performed on the phantom, and the corresponding deformations were recorded using a MRI scanner. Finite element simulations were performed to mimic the experiments using the estimated material properties with appropriate boundary conditions. The simulation results matched the experimental recordings of the deformed phantom, with a sub-millimeter root-mean-square error for each compression state. Having now validated our finite element model of breast compression, the next stage is to apply the model to clinical images.