An Open-Source Mesh Generation Platform for Biophysical Modeling Using Realistic Cellular Geometries

Advances in imaging methods such as electron microscopy, tomography, and other modalities are enabling high-resolution reconstructions of cellular and organelle geometries. Such advances pave the way for using these geometries for biophysical and mathematical modeling once these data can be represented as a geometric mesh, which, when carefully conditioned, enables the discretization and solution of partial differential equations. In this work, we outline the steps for a naïve user to approach the Geometry-preserving Adaptive MeshER software version 2, a mesh generation code written in C++ designed to convert structural data sets to realistic geometric meshes while preserving the underlying shapes. We present two example cases: 1) mesh generation at the subcellular scale as informed by electron tomography and 2) meshing a protein with a structure from x-ray crystallography. We further demonstrate that the meshes generated by the Geometry-preserving Adaptive MeshER software are suitable for use with numerical methods. Together, this collection of libraries and tools simplifies the process of constructing realistic geometric meshes from structural biology data.     

Excess electrical noise during current flow through porous membranes separating ionic solutions.

Spectral analysis of electrical noise from various artificial membrane systems suggests that excess noise of an f-n spectral form, where n is approximately unity, is not primarily a bulk phenomenon simply dependent on the number of charge carriers. Measurements from aqueous and nonaqueous electrolytic resistors, comprised of several different ionic species, show only flat power density spectra under applied currents, even at extreme dilutions. Excess noise of f-n form is observed under applied d-c current in single pore membranes, as previously reported, but is also seen in multipore and polymer mesh membranes. Calculations based on single pore membrane noise data are in significant variance with the bulk charge carrier model proposed by Hooge. These observations suggest that such excess noise occurs in conjunction with anisotropic constraints to ion flow.     

Anisotropic adaptive finite element method for modelling blood flow

In this study, we present an adaptive anisotropic finite element method (FEM) and demonstrate how computational efficiency can be increased when applying the method to the simulation of blood flow in the cardiovascular system. We use the SUPG formulation for the transient 3D incompressible Navier-Stokes equations which are discretised by linear finite elements for both the pressure and the velocity field. Given the pulsatile nature of the flow in blood vessels we have pursued adaptivity based on the average flow over a cardiac cycle. Error indicators are derived to define an anisotropic mesh metric field. Mesh modification algorithms are used to anisotropically adapt the mesh according to the desired size field. We demonstrate the efficiency of the method by first applying it to pulsatile flow in a straight cylindrical vessel and then to a porcine aorta with a stenosis bypassed by a graft. We demonstrate that the use of an anisotropic adaptive FEM can result in an order of magnitude reduction in computing time with no loss of accuracy compared to analyses obtained with uniform meshes.     

Anisotropic adaptive finite element method for modelling blood flow

In this study, we present an adaptive anisotropic finite element method (FEM) and demonstrate how computational efficiency can be increased when applying the method to the simulation of blood flow in the cardiovascular system. We use the SUPG formulation for the transient 3D incompressible Navier–Stokes equations which are discretised by linear finite elements for both the pressure and the velocity field.

Given the pulsatile nature of the flow in blood vessels we have pursued adaptivity based on the average flow over a cardiac cycle. Error indicators are derived to define an anisotropic mesh metric field. Mesh modification algorithms are used to anisotropically adapt the mesh according to the desired size field. We demonstrate the efficiency of the method by first applying it to pulsatile flow in a straight cylindrical vessel and then to a porcine aorta with a stenosis bypassed by a graft. We demonstrate that the use of an anisotropic adaptive FEM can result in an order of magnitude reduction in computing time with no loss of accuracy compared to analyses obtained with uniform meshes.     

SPHARA - A Generalized Spatial Fourier Analysis for Multi-Sensor Systems with Non-Uniformly Arranged Sensors: Application to EEG

Important requirements for the analysis of multichannel EEG data are efficient techniques for signal enhancement, signal decomposition, feature extraction, and dimensionality reduction. We propose a new approach for spatial harmonic analysis (SPHARA) that extends the classical spatial Fourier analysis to EEG sensors positioned non-uniformly on the surface of the head. The proposed method is based on the eigenanalysis of the discrete Laplace-Beltrami operator defined on a triangular mesh. We present several ways to discretize the continuous Laplace-Beltrami operator and compare the properties of the resulting basis functions computed using these discretization methods. We apply SPHARA to somatosensory evoked potential data from eleven volunteers and demonstrate the ability of the method for spatial data decomposition, dimensionality reduction and noise suppression. When employing SPHARA for dimensionality reduction, a significantly more compact representation can be achieved using the FEM approach, compared to the other discretization methods. Using FEM, to recover 95% and 99% of the total energy of the EEG data, on average only 35% and 58% of the coefficients are necessary. The capability of SPHARA for noise suppression is shown using artificial data. We conclude that SPHARA can be used for spatial harmonic analysis of multi-sensor data at arbitrary positions and can be utilized in a variety of other applications.     

Adaptive generation of multimaterial grids from imaging data for biomedical Lagrangian fluid–structure simulations

Spatial discretization of complex imaging- derived fluid–solid geometries, such as the cardiac environment, is a critical but often overlooked challenge in biomechanical computations. This is particularly true in problems with Lagrangian interfaces, where the fluid and solid phases share a common interface geometrically. For simplicity and better accuracy, it is also highly desirable for the two phases to have a matching surface mesh at the interface between them. We outline a method for solving this problem, and illustrate the approach with a 3D fluid–solid mesh of the mouse heart. An MRI dataset of a perfusion-fixed mouse heart with 50μm isotropic resolution was semi-automatically segmented using a customized multimaterial connected-threshold approach that divided the volume into non-overlapping regions of blood, tissue, and background. Subsequently a multimaterial marching cubes algorithm was applied to the segmented data to produce two detailed, compatible isosurfaces, one for blood and one for tissue. Both isosurfaces were simultaneously smoothed with a multimaterial smoothing algorithm that exactly conserves the volume for each phase. Using these two isosurfaces, we developed and applied novel automated meshing algorithms to generate anisotropic hybrid meshes on arbitrary biological geometries with the number of layers and the desired element anisotropy for each phase as the only input parameters. Since our meshes adapt to the local feature sizes and include boundary layer prisms, they are more efficient and accurate than non-adaptive, isotropic meshes, and the fluid–structure interaction computations will tend to have relative error equilibrated over the whole mesh.     

Nature and distribution of vascular resistance in hypoxic pig lungs

We used the vascular occlusion technique in pig lungs isolated in situ to describe the effects of hypoxia on the distribution of vascular resistance and to determine whether the resistive elements defined by this technique behaved as ohmic or Starling resistors during changes in flow at constant outflow pressure, changes in outflow pressure at constant flow, and reversal of flow. During normoxia, the largest pressure gradient occurred across the middle compliant region of the vasculature (delta Pm). The major effect of hypoxia was to increase delta Pm and the gradient across the relatively noncompliant arterial region (delta Pa). The gradient across the noncompliant venous region (delta Pv) changed only slightly, if at all. Both delta Pa and delta Pv increased with flow but delta Pm decreased. The pressure at the arterial end of the middle region was independent of flow and, when outflow pressure was increased, did not increase until the outflow pressure of the middle region exceeded 8.9 Torr during normoxia and 18.8 Torr during hypoxia. Backward perfusion increased the total pressure gradient across the lung, mainly because of an increase in delta Pm. These results can be explained by a model in which the arterial and venous regions are represented by ohmic resistors and the middle region is represented by a Starling resistor in series and proximal to an ohmic resistor. In terms of this model, hypoxia exerted its major effects by increasing the critical pressure provided by the Starling resistor of the middle region and the ohmic resistance of the arterial region.     

Three-dimensional structure determination of capsid of Aedes albopicus C6/36 cell densovirus

The three-dimensional structure of capsid of Aedes albopictus C6/36 densovirus was determined to 14-(A) resolution by electron cryomicroscopy and computer reconstruction. The triangulation number of the capsid is 1. There are 12 holes in each triangular face and a spike on each 5-fold vertex. The validity of the capsid and nucleic acid densities in the reconstructions was discussed.     

Three-dimensional structure determination of capsid of Aedes albopictus C6/36 cell densovirus

Apreviously-unknownviruswasfoundchroni-callyinfectingC6/36cellsthatwereusedtoculturedenguevirus.Afterisolationandpurification,RT,PCRandsequencingwereperformedwithrandomprimers.TheresultsshowthattheviralgenomeisssDNAandis4096ntinlength(GenBankAccessionNo.AY095351).Thenucleicacidsequenceofthispreviously-undescribedvirussharesabout90%iden-titywiththatofAedesaegyptidensonucleosisvirusandtheaminoacidsequencesofthreeproteinsen-codedbythenucleicacidshareabout89%—93%identitieswiththoseofAedesa…     

Geometry of gene expression dynamics

MOTIVATION: A gene expression trajectory moves through a high dimensional space where each axis represents the mRNA abundance of a different gene. Genome wide gene expression has a dynamic structure, especially in studies of development and temporal response. Both visualization and analyses of such data require an explicit attention to the temporal structure. RESULTS: Using three cell cycle trajectories from Saccharomyces cerevisiae to illustrate, we present several techniques which reveal the geometry of the data. We import phase-delay time plots from chaotic systems theory as a dynamic data visualization device and show how these plots capture important aspects of the trajectories. We construct an objective function to find an optimal two-dimensional projection of the cell cycle, demonstrate that the system returns to this plane after three different initial perturbations, and explore the conditions under which this geometric approach outperforms standard approaches such as singular value decomposition and Fourier analysis. Finally, we show how a geometric analysis can isolate distinct parts of the trajectories, in this case the initial perturbation versus the cell cycle. CONTACT: junhyong.kim@yale.edu     

Mice recognize the center of an enclosure

In recent years, it has been shown that animals can localize the geometric center of an area by reference to the shape of the environment. We trained a group of mice (experimental group) to search for a pellet hidden under sand in the center of a square-shaped dry maze. Three weeks later, they were tested in a triangular enclosure half the size of the training area and a circular enclosure double the size of the training area to see transfer to these enclosures. We compared their searching behavior with that of subjects that had received no training. The results show that the experimental group searched the geometric center of each enclosure in both transfer tests, while the untrained control group walked along the walls. This indicates that the experimental group localized the center not by reference to the absolute distance from the corners but by equal distances from all walls (geometric center).     

Merging of intersecting triangulations for finite element modeling

Surface mesh generation over intersecting triangulations is a problem common to many branches of biomechanics. A new strategy for merging intersecting triangulations is described. The basis of the method is that object surfaces are represented as the zero-level iso-surface of the distance-to-surface function defined on a background grid. Thus, the triangulation of intersecting objects reduces to the extraction of an iso-surface from an unstructured grid. In a first step, a regular background mesh is constructed. For each point of the background grid, the closest distance to the surface of each object is computed. Background points are then classified as external or internal by checking the direction of the surface normal at the closest location and assigned a positive or negative distance, respectively. Finally, the zero-level iso-surface is constructed. This is the final triangulation of the intersecting objects. The overall accuracy is enhanced by adaptive refinement of the background grid elements. The resulting surface models are used as support surfaces to generate three-dimensional grids for finite element analysis. The algorithms are demonstrated by merging arterial branches independently reconstructed from contrast-enhanced magnetic resonance images and by adding extra features such as vascular stents. Although the methodology is presented in the context of finite element analysis of blood flow, the algorithms are general and can be applied in other areas as well.     

Three-dimensional geometric modeling of membrane-bound organelles in ventricular myocytes: bridging the gap between microscopic imaging and mathematical simulation

A general framework of image-based geometric processing is presented to bridge the gap between three-dimensional (3D) imaging that provides structural details of a biological system and mathematical simulation where high-quality surface or volumetric meshes are required. A 3D density map is processed in the order of image pre-processing (contrast enhancement and anisotropic filtering), feature extraction (boundary segmentation and skeletonization), and high-quality and realistic surface (triangular) and volumetric (tetrahedral) mesh generation. While the tool-chain described is applicable to general types of 3D imaging data, the performance is demonstrated specifically on membrane-bound organelles in ventricular myocytes that are imaged and reconstructed with electron microscopic (EM) tomography and two-photon microscopy (T-PM). Of particular interest in this study are two types of membrane-bound Ca²⁺-handling organelles, namely, transverse tubules (T-tubules) and junctional sarcoplasmic reticulum (jSR), both of which play an important role in regulating the excitation–contraction (E–C) coupling through dynamic Ca²⁺ mobilization in cardiomyocytes.     

Fitting low-resolution cryo-EM maps of proteins using constrained geometric simulations

Recent experimental advances in producing density maps from cryo-electron microscopy (cryo-EM) have challenged theorists to develop improved techniques to provide structural models that are consistent with the data and that preserve all the local stereochemistry associated with the biomolecule. We develop a new technique that maintains the local geometry and chemistry at each stage of the fitting procedure. A geometric simulation is used to drive the structure from some appropriate starting point (a nearby experimental structure or a modeled structure) toward the experimental density, via a set of small incremental motions. Structural motifs such as α-helices can be held rigid during the fitting procedure as the starting structure is brought into alignment with the experimental density. After validating this procedure on simulated data for adenylate kinase and lactoferrin, we show how cryo-EM data for two different GroEL structures can be fit using a starting x-ray crystal structure. We show that by incorporating the correct local stereochemistry in the modeling, structures can be obtained with effective resolution that is significantly higher than might be expected from the nominal cryo-EM resolution.     

Linear network representation of multistate models of transport.

By introducing external driving forces in rate-theory models of transport we show how the Eyring rate equations can be transformed into Ohm's law with potentials that obey Kirchhoff's second law. From such a formalism the state diagram of a multioccupancy multicomponent system can be directly converted into linear network with resistors connecting nodal (branch) points and with capacitances connecting each nodal point with a reference point. The external forces appear as emf or current generators in the network. This theory allows the algebraic methods of linear network theory to be used in solving the flux equations for multistate models and is particularly useful for making proper simplifying approximation in models of complex membrane structure. Some general properties of linear network representation are also deduced. It is shown, for instance, that Maxwell's reciprocity relationships of linear networks lead directly to Onsager's relationships in the near equilibrium region. Finally, as an example of the procedure, the equivalent circuit method is used to solve the equations for a few transport models.     

Three-dimensional structure determination of capsid of<Emphasis Type="Italic">Aedes albopictus</Emphasis> C6/36 cell densovirus

The three-dimensional structure of capsid ofAedes albopictus C6/36 densovirus was determined to 14-Å resolution by electron cryomicroscopy and computer reconstruction. The triangulation number of the capsid is 1. There are 12 holes in each triangular face and a spike on each 5-fold vertex. The validity of the capsid and nucleic acid densities in the reconstructions was discussed.     

Subdivision meshes for organizing spatial biomedical data

As biomedical images and volumes are being collected at an increasing speed, there is a growing demand for efficient means to organize spatial information for comparative analysis. In many scenarios, such as determining gene expression patterns by in situ hybridization, the images are collected from multiple subjects over a common anatomical region, such as the brain. A fundamental challenge in comparing spatial data from different images is how to account for the shape variations among subjects, which make direct image-to-image comparisons meaningless. In this paper, we describe subdivision meshes as a geometric means to efficiently organize 2D images and 3D volumes collected from different subjects for comparison. The key advantages of a subdivision mesh for this purpose are its light-weight geometric structure and its explicit modeling of anatomical boundaries, which enable efficient and accurate registration. The multi-resolution structure of a subdivision mesh also allows development of fast comparison algorithms among registered images and volumes.     

Pressure-flow effects on endurance of inspiratory muscles

We examined the effects of varying inspiratory pressures and flows on inspiratory muscle endurance. Four normal subjects performed voluntary forced breathing with various assigned inspiratory tasks. Duty cycle, tidal volume, and mean lung volume were the same in all tasks. Mean esophageal pressure, analogous to a pressure-time integral (PTes), was varied over a wide range. In each task the subject maintained an assigned PTes while breathing on one of a range of inspiratory resistors, and this gave a range of inspiratory flows at any given PTes. Inspiratory muscle endurance for each task was assessed by the length of time the task could be maintained (Tlim). For a given resistor, Tlim increased as PTes decreased. At a given PTes, Tlim increased as the external resistance increased and therefore as mean inspiratory flow rate (VI) decreased. Furthermore, for a given Tlim, PTes and VI were linearly related with a negative slope. We conclude that inspiratory flow, probably because of its relationship to the velocity of muscle shortening, is an independent variable importantly influencing endurance of the inspiratory muscles.     

Noise Properties in the Ideal Kirchhoff-Law-Johnson-Noise Secure Communication System

In this paper we determine the noise properties needed for unconditional security for the ideal Kirchhoff-Law-Johnson-Noise (KLJN) secure key distribution system using simple statistical analysis. It has already been shown using physical laws that resistors and Johnson-like noise sources provide unconditional security. However real implementations use artificial noise generators, therefore it is a question if other kind of noise sources and resistor values could be used as well. We answer this question and in the same time we provide a theoretical basis to analyze real systems as well.