Wave-Driven Assembly of Quasiperiodic Patterns of Particles.

We theoretically show that a superposition of plane waves causes small (compared to the wavelength) particles dispersed in a fluid to assemble in quasiperiodic two or three-dimensional patterns. We experimentally demonstrate this theory by using ultrasound waves to assemble quasiperiodic patterns of carbon nanoparticles in water using an octagonal arrangement of ultrasound transducers, and we document good agreement between theory and experiments. The theory also applies to obtaining quasiperiodic patterns in other situations where particles move with linear waves, such as optical lattices.

Internal resonances and relaxation memory kernels in composites.

In heterogeneous composite materials, the behaviour of the medium on larger scales is determined by the microgeometry and properties of the constituents on finer scales. To model the influence of the microlevel processes in composite materials, they are described as materials with memory in which the constitutive relations between stress and strain are given as time-domain convolutions with some relaxation kernel. The paper reveals the relationship between the viscoelastic relaxation kernel and the spectral measure in the Stieltjes integral representation of the effective properties of composites. This spectral measure contains all information about the microgeometry of the material, thus providing a link between the relaxation kernel and the microstructure of the composite. We show that the internal resonances of the microstructure determine the characteristic relaxation times of the fading memory kernel and can be used to introduce a set of internal variables that captures dissipation at the microscale. This article is part of the theme issue 'Modelling of dynamic phenomena and localization in structured media (part 2)'.

Anderson Transition for Classical Transport in Composite Materials.

The Anderson transition in solids and optics is a wave phenomenon where disorder induces localization of the wave functions. We find here that the hallmarks of the Anderson transition are exhibited by classical transport at a percolation threshold-without wave interference or scattering effects. As long range order or connectedness develops, the eigenvalue statistics of a key random matrix governing transport cross over toward universal statistics of the Gaussian orthogonal ensemble, and the field eigenvectors delocalize. The transition is examined in resistor networks, human bone, and sea ice structures.

Analytical approach to recovering bone porosity from effective complex shear modulus.

This work deals with the study of the analytical relations between porosity of cancellous bone and its mechanical properties. The Stieltjes representation of the effective shear complex modulus of cancellous bone is exploited to recover porosity. The microstructural information is contained in the spectral measure in this analytical representation. The spectral function can be recovered from the effective measurements over a range of frequencies. The problem of reconstruction of the spectral measure is very ill-posed. Regularized algorithm is derived to ensure stability of the results. The proposed method does not use any specific assumptions about the microgeometry of bone. The approach does not rely on correlation analysis, it uses analytical relationships. For validation purposes, complex shear modulus over a range of frequencies was calculated by the finite element method using micro-computed tomography (micro-CT) images of human cancellous bone. The calculated values were used in numerical algorithm to recover bone porosity. At the microlevel, bone was modeled as a heterogeneous medium composed of trabeculae tissue and bone marrow treated as transversely isotropic elastic and isotropic viscoelastic materials, respectively. Recovered porosity values are in excellent agreement with true porosity found from the corresponding micro-CT images.

Modeling the Kinetics of Lipid-Nanoparticle- Mediated Delivery of Multiple siRNAs to Evaluate the Effect on Competition for Ago2.

Drug combinations can improve the control of diseases involving redundant and highly regulated pathways. Validating a multi-target therapy early in drug development remains difficult. Small interfering RNAs (siRNAs) are routinely used to selectively silence a target of interest. Owing to the ease of design and synthesis, siRNAs hold promise for combination therapies. Combining siRNAs against multiple targets remains an attractive approach to interrogating highly regulated pathways. Currently, questions remain regarding how broadly such an approach can be applied, since siRNAs have been shown to compete with one another for binding to Argonaute2 (Ago2), the protein responsible for initiating siRNA-mediated mRNA degradation. Mathematical modeling, coupled with in vitro and in vivo experiments, led us to conclude that endosomal escape kinetics had the highest impact on Ago2 depletion by competing lipid-nanoparticle (LNP)-formulated siRNAs. This, in turn, affected the level of competition observed between them. A future application of this model would be to optimize delivery of desired siRNA combinations in vitro to attenuate competition and maximize the combined therapeutic effect.

Mathematical Modeling: A Tool for Optimization of Lipid Nanoparticle-Mediated Delivery of siRNA.

Lipid nanoparticles (LNPs) have been used to successfully deliver small interfering RNAs (siRNAs) to target cells in both preclinical and clinical studies and currently are the leading systems for in vivo delivery. Here, we propose the use of an ordinary differential equation (ODE)-based model as a tool for optimizing LNP-mediated delivery of siRNAs. As a first step, we have used a combination of experimental and computational approaches to develop and validate a mathematical model that captures the critical features for efficient siRNA-LNP delivery in vitro. This model accurately predicts mRNA knockdown resulting from novel combinations of siRNAs and LNPs in vitro. As demonstrated, this model can be effectively used as a screening tool to select the most efficacious LNPs, which can then further be evaluated in vivo. The model serves as a starting point for the future development of next generation models capable of capturing the additional complexity of in vivo delivery.

Characterization of structure and properties of bone by spectral measure method.

Novel mathematical method called spectral measure method (SMM) is developed for characterization of bone structure and indirect estimation of bone properties. The spectral measure method is based on an inverse homogenization technique which allows to derive information about the structure of composite material from measured effective electric or viscoelastic properties. The mechanical properties and ability to withstand fracture depend on the structural organization of bone as a hierarchical composite. Information about the bone structural parameters is contained in the spectral measure in the Stieltjes integral representation of the effective properties. The method is based on constructing the spectral measure either by calculating it directly from micro-CT images or using measurements of electric or viscoelastic properties over a frequency range. In the present paper, we generalize the Stieltjes representation to the viscoelastic case and show how bone microstructure, in particular, bone volume or porosity, can be characterized by the spectral function calculated using measurements of complex permittivity or viscoelastic modulus. For validation purposes, we numerically simulated measured data using micro-CT images of cancellous bone. Recovered values of bone porosity are in excellent agreement with true porosity estimated from the micro-CT images. We also discuss another application of this method, which allows to estimate properties difficult to measure directly. The spectral measure method based on the derived Stieltjes representation for viscoelastic composites, has a potential for non-invasive characterization of bone structure using electric or mechanical measurements. The method is applicable to sea ice, porous rock, and other composite materials.

Spectral analysis and connectivity of porous microstructures in bone.

Cancellous bone is a porous composite of calcified tissue interspersed with soft marrow. Sea ice is also a porous composite, consisting of pure ice with brine, air, and salt inclusions. Interestingly, the microstructures of bone and sea ice exhibit notable similarities. In recent years, we have developed mathematical and experimental techniques for imaging and characterizing the brine microstructure of sea ice, such as its volume fraction and connectivity, as well as a range of theoretical approaches for studying fluid, thermal, and electromagnetic transport in sea ice. Here we explore the application of our sea ice techniques to investigate trabecular bone. For example, percolation theory that quantifies brine connectivity and its thermal evolution can also help assess the impact of osteoporosis on trabecular structure. Central to our approach is the spectral measure of a composite material, which contains detailed information about the mixture geometry, and can be used in powerful integral representations to compute the effective properties. The spectral measure is obtained from the eigenvalues and eigenvectors of a self-adjoint operator determined exclusively by the composite microgeometry. Here we compute the spectral measures for discretizations of images of healthy and osteoporotic bone. The measures are used to compute the effective electromagnetic properties of the bone specimens. These data are then inverted to reconstruct the porosity of the original specimens, with excellent agreement.