Water mobility in heterogeneous emulsions determined by a new combination of confocal laser scanning microscopy, image analysis, nuclear magnetic resonance diffusometry, and finite element method simulation

Diffusion MRI simulation of realistic neurons with SpinDoctor and the Neuron Module

The diffusion MRI signal arising from neurons can be numerically simulated by solving the Bloch-Torrey partial differential equation. In this paper we present the Neuron Module that we implemented within the Matlab-based diffusion MRI simulation toolbox SpinDoctor. SpinDoctor uses finite element discretization and adaptive time integration to solve the Bloch-Torrey partial differential equation for general diffusion-encoding sequences, at multiple b-values and in multiple diffusion directions. In order to facilitate the diffusion MRI simulation of realistic neurons by the research community, we constructed finite element meshes for a group of 36 pyramidal neurons and a group of 29 spindle neurons whose morphological descriptions were found in the publicly available neuron repository NeuroMorpho.Org. These finite elements meshes range from having 15,163 nodes to 622,553 nodes. We also broke the neurons into the soma and dendrite branches and created finite elements meshes for these cell components. Through the Neuron Module, these neuron and cell components finite element meshes can be seamlessly coupled with the functionalities of SpinDoctor to provide the diffusion MRI signal attributable to spins inside neurons. We make these meshes and the source code of the Neuron Module available to the public as an open-source package. To illustrate some potential uses of the Neuron Module, we show numerical examples of the simulated diffusion MRI signals in multiple diffusion directions from whole neurons as well as from the soma and dendrite branches, and include a comparison of the high b-value behavior between dendrite branches and whole neurons. In addition, we demonstrate that the neuron meshes can be used to perform Monte-Carlo diffusion MRI simulations as well. We show that at equivalent accuracy, if only one gradient direction needs to be simulated, SpinDoctor is faster than a GPU implementation of Monte-Carlo, but if many gradient directions need to be simulated, there is a break-even point when the GPU implementation of Monte-Carlo becomes faster than SpinDoctor. Furthermore, we numerically compute the eigenfunctions and the eigenvalues of the Bloch-Torrey and the Laplace operators on the neuron geometries using a finite elements discretization, in order to give guidance in the choice of the space and time discretization parameters for both finite elements and Monte-Carlo approaches. Finally, we perform a statistical study on the set of 65 neurons to test some candidate biomakers that can potentially indicate the soma size. This preliminary study exemplifies the possible research that can be conducted using the Neuron Module.

Practical computation of the diffusion MRI signal of realistic neurons based on Laplace eigenfunctions

The complex transverse water proton magnetization subject to diffusion-encoding magnetic field gradient pulses in a heterogeneous medium such as brain tissue can be modeled by the Bloch-Torrey partial differential equation. The spatial integral of the solution of this equation in realistic geometry provides a gold-standard reference model for the diffusion MRI signal arising from different tissue micro-structures of interest. A closed form representation of this reference diffusion MRI signal called matrix formalism, which makes explicit the link between the Laplace eigenvalues and eigenfunctions of the biological cell and its diffusion MRI signal, was derived 20 years ago. In addition, once the Laplace eigendecomposition has been computed and saved, the diffusion MRI signal can be calculated for arbitrary diffusion-encoding sequences and b-values at negligible additional cost. Up to now, this representation, though mathematically elegant, has not been often used as a practical model of the diffusion MRI signal, due to the difficulties of calculating the Laplace eigendecomposition in complicated geometries. In this paper, we present a simulation framework that we have implemented inside the MATLAB-based diffusion MRI simulator SpinDoctor that efficiently computes the matrix formalism representation for realistic neurons using the finite element method. We show that the matrix formalism representation requires a few hundred eigenmodes to match the reference signal computed by solving the Bloch-Torrey equation when the cell geometry originates from realistic neurons. As expected, the number of eigenmodes required to match the reference signal increases with smaller diffusion time and higher b-values. We also convert the eigenvalues to a length scale and illustrate the link between the length scale and the oscillation frequency of the eigenmode in the cell geometry. We give the transformation that links the Laplace eigenfunctions to the eigenfunctions of the Bloch-Torrey operator and compute the Bloch-Torrey eigenfunctions and eigenvalues. This work is another step in bringing advanced mathematical tools and numerical method development to the simulation and modeling of diffusion MRI.

Interconnectivity imaged in three dimensions: Nano-particulate silica-hydrogel structure revealed using electron tomography

We have used Electron Tomography (ET) to reveal the detailed three-dimensional structure of particulate hydrogels, a material category common in e.g. controlled release, food science, battery and biomedical applications. A full understanding of the transport properties of these gels requires knowledge about the pore structure and in particular the interconnectivity in three dimensions, since the transport takes the path of lowest resistance. The image series for ET were recorded using High-Angle Annular Dark Field Scanning Transmission Electron Microscopy (HAADF-STEM). We have studied three different particulate silica hydrogels based on primary particles with sizes ranging from 3.6nm to 22nm and with pore-size averages from 18nm to 310nm. Here, we highlight the nanostructure of the particle network and the interpenetrating pore network in two and three dimensions. The interconnectivity and distribution of width of the porous channels were obtained from the three-dimensional tomography studies while they cannot unambiguously be obtained from the two-dimensional data. Using ET, we compared the interconnectivity and accessible pore volume fraction as a function of pore size, based on direct images on the nanoscale of three different hydrogels. From this comparison, it was clear that the finest of the gels differentiated from the other two. Despite the almost identical flow properties of the two finer gels, they showed large differences concerning the accessible pore volume fraction for probes corresponding to their (two-dimensional) mean pore size. Using 2D pore size data, the finest gel provided an accessible pore volume fraction of over 90%, but for the other two gels the equivalent was only 10-20%. However, all the gels provided an accessible pore volume fraction of 30-40% when taking the third dimension into account.

Numerical analysis of NMR diffusion measurements in the short gradient pulse limit

Pulsed gradient spin-echo (PGSE) NMR diffusion measurements provide a powerful technique for probing porous media. The derivation of analytical mathematical models for analysing such experiments is only straightforward for ideal restricting geometries and rapidly becomes intractable as the geometrical complexity increases. Consequently, in general, numerical methods must be employed. Here, a highly flexible method for calculating the results of PGSE NMR experiments in porous systems in the short gradient pulse limit based on the finite element method is presented. The efficiency and accuracy of the method is verified by comparison with the known solutions to simple pore geometries (parallel planes, a cylindrical pore, and a spherical pore) and also to Monte Carlo simulations. The approach is then applied to modelling the more complicated cases of parallel semipermeable planes and a pore hopping model. Finally, the results of a PGSE measurement on a toroidal pore, a geometry for which there is presently no current analytical solution, are presented. This study shows that this approach has great potential for modelling the results of PGSE experiments on real (3D) porous systems. Importantly, the FEM approach provides far greater accuracy in simulating PGSE diffraction data.

Determination of local diffusion properties in heterogeneous biomaterials

The coupling between structure and diffusion properties is essential for the functionality of heterogeneous biomaterials. Structural heterogeneity is defined and its implications for time-dependent diffusion are discussed in detail. The effect of structural heterogeneity in biomaterials on diffusion and the relevance of length scales are exemplified with regard to different biomaterials such as gels, emulsions, phase separated biopolymer mixtures and chocolate. Different diffusion measurement techniques for determination of diffusion properties at different length and time scales are presented. The interplay between local and global diffusion is discussed. New measurement techniques have emerged that enable simultaneous determination of both structure and local diffusion properties. Special emphasis is given to fluorescence recovery after photobleaching (FRAP). The possibilities of FRAP at a conceptual level is presented. The method of FRAP is briefly reviewed and its use in heterogeneous biomaterials, at barriers and during dynamic changes of the structure is discussed.

Brownian dynamics simulations in hydrogels using an adaptive time-stepping algorithm

The adaptive time-stepping algorithm for Brownian simulation of solute diffusion in three-dimensional complex geometries previously developed by the authors of this paper was applied to heterogeneous three-dimensional polymer hydrogel structures. The simulations were performed on reconstructed three-dimensional hydrogels. The obstruction effect from the gel strands on water and diffusion of dendrimers with different sizes were determined by simulations and compared with experimental nuclear magnetic resonance diffusometry data obtained from the same material. It was concluded that obstruction alone cannot explain the observed diffusion rates, but an interaction between the dendrimers and the gel strands should be included in the simulations. The effect of a sticky-wall interaction potential with geometrically distributed residence times on the diffusion rate has been studied. It was found that sticky-wall interaction is a possible explanation for the discrepancy between simulated and experimental diffusion data for dendrimers of different sizes diffusing in hydrogels.

Identification of the three-dimensional gel microstructure from transmission electron micrographs

Mass transport in gels depends crucially on local properties of the gel network. We propose a method for identifying the three-dimensional (3D) gel microstructure from statistical information in transmission electron micrographs. The gel strand network is modelled as a random graph with nodes and edges (branches). The distribution of edge length, the number of edges at nodes and the angles between edges at a node are estimated from transmission electron micrographs by image analysis methods. The 3D network is simulated by Markov chain Monte Carlo, with a probability function based on the statistical information found from the micrographs. The micrographs are projections of stained gel strands in slices, and we derive a formula for estimating the thickness of the stained gel slice based on the total projected gel strand length and the number of times that gel strands enter or exit the slice.

Influence of kappa-carrageenan gel structures on the diffusion of probe molecules determined by transmission electron microscopy and NMR diffusometry

The influence of the microstructures of different kappa-carrageenan gels on the self-diffusion behavior of poly(ethylene glycol) (PEG) has been determined by nuclear magnetic resonance (NMR) diffusometry and transmission electron microscopy (TEM). It was found that the diffusion behavior was determined mainly by the void size, which in turn was defined by the state of aggregation of the kappa-carrageenan. The kappa-carrageenan concentration was held constant at 1 w/w%, and the aggregation was controlled by the amount of potassium and/or sodium chloride and, for samples containing potassium, also by the cooling rate. Gels containing potassium formed microstructures where kappa-carrageenan strands are rather evenly distributed over the image size, while sodium gels formed dense biopolymer clusters interspersed with large openings. In a gel with small void sizes, relatively slow diffusion was found for all PEG sizes investigated. Extended studies of the self-diffusion behavior of the 634 g mol(-)(1) PEG showed that there is a strong time dependence in the measured PEG diffusion. An asymptotic lower time limit of the diffusion coefficient was found in all gels when the diffusion observation time was increased. According to the ratio, D/D(0), where D(0) is the diffusion coefficient in D(2)O and D is the diffusion coefficient in the gels, the gels could be divided into three classes: small, medium, and large voids. For quenched kappa-carrageenan solutions with salt concentrations of 20 mM K(+), 100 mM K(+), or 20 mM K(+)/200 mM Na(+) as well as slowly cooled solutions with only 20 mM K(+), D/D(0) ratios between 0.18 and 0.29 were obtained. By quenching a kappa-carrageenan solution with 100 mM K(+), the D/D(0) was 0.5, while D/D(0) ratios between 0.9 and 1 were obtained in a quenched solution with 250 mM Na(+) and slowly cooled samples with 20 mM K(+)/200 mM Na(+) or 250 mM Na(+).