Alternating direction implicit numerical method for the space and time fractional Bloch-Torrey equation in 3-D

Recently, some authors have considered a new diffusion model–space and time fractional Bloch-Torrey equation (ST-FBTE). Magin et al. (2008) have derived analytical solutions with fractional order dynamics in space (i.e., _ = 1, β an arbitrary real number, 1 < β ≤ 2) and time (i.e., 0 < α < 1, and β = 2), respectively. Yu et al. (2011) have derived an analytical solution and an effective implicit numerical method for solving ST-FBTEs, and also discussed the stability and convergence of the implicit numerical method. However, due to the computational overheads necessary to perform the simulations for nuclear magnetic resonance (NMR) in three dimensions, they present a study based on a two-dimensional example to confirm their theoretical analysis. Alternating direction implicit (ADI) schemes have been proposed for the numerical simulations of classic differential equations. The ADI schemes will reduce a multidimensional problem to a series of independent one-dimensional problems and are thus computationally efficient. In this paper, we consider the numerical solution of a ST-FBTE on a finite domain. The time and space derivatives in the ST-FBTE are replaced by the Caputo and the sequential Riesz fractional derivatives, respectively. A fractional alternating direction implicit scheme (FADIS) for the ST-FBTE in 3-D is proposed. Stability and convergence properties of the FADIS are discussed. Finally, some numerical results for ST-FBTE are given.

A computationally effective alternating direction method for the space and time fractional Bloch–Torrey equation in 3-D

The space and time fractional Bloch–Torrey equation (ST-FBTE) has been used to study anomalous diffusion in the human brain. Numerical methods for solving ST-FBTE in three-dimensions are computationally demanding. In this paper, we propose a computationally effective fractional alternating direction method (FADM) to overcome this problem. We consider ST-FBTE on a finite domain where the time and space derivatives are replaced by the Caputo–Djrbashian and the sequential Riesz fractional derivatives, respectively. The stability and convergence properties of the FADM are discussed. Finally, some numerical results for ST-FBTE are given to confirm our theoretical findings.

Ex vivo investigation of novel wound healing therapies and development of a 3-D human skin equivalent wound model

It has previously been found that complexes comprised of vitronectin and growth factors (VN:GF) enhance keratinocyte protein synthesis and migration. More specifically, these complexes have been shown to significantly enhance the migration of dermal keratinocytes derived from human skin. In view of this, it was thought that these complexes may hold potential as a novel therapy for healing chronic wounds. However, there was no evidence indicating that the VN:GF complexes would retain their effect on keratinocytes in the presence of chronic wound fluid. The studies in this thesis demonstrate for the first time that the VN:GF complexes not only stimulate proliferation and migration of keratinocytes, but also these effects are maintained in the presence of chronic wound fluid in a 2-dimensional (2-D) cell culture model. Whilst the 2-D culture system provided insights into how the cells might respond to the VN:GF complexes, this investigative approach is not ideal as skin is a 3-dimensional (3-D) tissue. In view of this, a 3-D human skin equivalent (HSE) model, which reflects more closely the in vivo environment, was used to test the VN:GF complexes on epidermopoiesis. These studies revealed that the VN:GF complexes enable keratinocytes to migrate, proliferate and differentiate on a de-epidermalised dermis (DED), ultimately forming a fully stratified epidermis. In addition, fibroblasts were seeded on DED and shown to migrate into the DED in the presence of the VN:GF complexes and hyaluronic acid, another important biological factor in the wound healing cascade. This HSE model was then further developed to enable studies examining the potential of the VN:GF complexes in epidermal wound healing. Specifically, a reproducible partial-thickness HSE wound model was created in fully-defined media and monitored as it healed. In this situation, the VN:GF complexes were shown to significantly enhance keratinocyte migration and proliferation, as well as differentiation. This model was also subsequently utilized to assess the wound healing potential of a synthetic fibrin-like gel that had previously been demonstrated to bind growth factors. Of note, keratinocyte re-epitheliasation was shown to be markedly improved in the presence of this 3-D matrix, highlighting its future potential for use as a delivery vehicle for the VN:GF complexes. Furthermore, this synthetic fibrin-like gel was injected into a 4 mm diameter full-thickness wound created in the HSE, both keratinocytes and fibroblasts were shown to migrate into this gel, as revealed by immunofluorescence. Interestingly, keratinocyte migration into this matrix was found to be dependent upon the presence of the fibroblasts. Taken together, these data indicate that reproducible wounds, as created in the HSEs, provide a relevant ex vivo tool to assess potential wound healing therapies. Moreover, the models will decrease our reliance on animals for scientific experimentation. Additionally, it is clear that these models will significantly assist in the development of novel treatments, such as the VN:GF complexes and the synthetic fibrin-like gel described herein, ultimately facilitating their clinical trial in the treatment of chronic wounds.

3-D reconstruction of an ancient Egyptian mummy using X-ray computer tomography

Computer tomography has been used to image and reconstruct in 3-D an Egyptian mummy from the collection of the British Museum. This study of Tjentmutengebtiu, a priestess from the 22nd dynasty (945-715 BC) revealed invaluable information of a scientific, Egyptological and palaeopathological nature without mutilation and destruction of the painted cartonnage case or linen wrappings. Precise details on the removal of the brain through the nasal cavity and the viscera from the abdominal cavity were obtained. The nature and composition of the false eyes were investigated. The detailed analysis of the teeth provided a much closer approximation of age at death. The identification of materials used for the various amulets including that of the figures placed in the viscera was graphically demonstrated using this technique.

Towards automated and in-situ, near-real time 3-D reconstruction of coral reef environments

Coral reefs are biologically complex ecosystems that support a wide variety of marine organisms. These are fragile communities under enormous threat from natural and human-based influences. Properly assessing and measuring the growth and health of reefs is essential to understanding impacts of ocean acidification, coastal urbanisation and global warming. In this paper, we present an innovative 3-D reconstruction technique based on visual imagery as a non-intrusive, repeatable, in situ method for estimating physical parameters, such as surface area and volume for efficient assessment of long-term variability. The reconstruction algorithms are presented, and benchmarked using an existing data set. We validate the technique underwater, utilising a commercial-off-the-shelf camera and a piece of staghorn coral, Acropora cervicornis. The resulting reconstruction is compared with a laser scan of the coral piece for assessment and validation. The comparison shows that 77% of the pixels in the reconstruction are within 0.3 mm of the ground truth laser scan. Reconstruction results from an unknown video camera are also presented as a segue to future applications of this research.

Collective dynamics and control of a 3-D small-world network with time delays

The paper presents a detailed analysis on the collective dynamics and delayed state feedback control of a three-dimensional delayed small-world network. The trivial equilibrium of the model is first investigated, showing that the uncontrolled model exhibits complicated unbounded behavior. Then three control strategies, namely a position feedback control, a velocity feedback control, and a hybrid control combined velocity with acceleration feedback, are then introduced to stabilize this unstable system. It is shown in these three control schemes that only the hybrid control can easily stabilize the 3-D network system. And with properly chosen delay and gain in the delayed feedback path, the hybrid controlled model may have stable equilibrium, or periodic solutions resulting from the Hopf bifurcation, or complex stranger attractor from the period-doubling bifurcation. Moreover, the direction of Hopf bifurcation and stability of the bifurcation periodic solutions are analyzed. The results are further extended to any "d" dimensional network. It shows that to stabilize a "d" dimensional delayed small-world network, at least a "d – 1" order completed differential feedback is needed. This work provides a constructive suggestion for the high dimensional delayed systems.

An in-vitro 3-D cell culture model for studying pathomechanisms in AMD

Purpose To develop a novel 3-D cell culture model with the view to studying the pathomechanisms underlying the development of age-related macular degeneration (AMD). Our central hypothesis is that the silk structural protein fibroin used in conjunction with cultured human cells can be used to mimic the structural relationships between the RPE and choriocapillaris in health and disease. Methods Co-cultures of human RPE cells (ARPE-19 cells grown in Miller’s medium) and microvascular endothelial cells (HMEC-1 cells grown in endothelial culture medium) were established on opposing sides of a synthetic Bruch’s membrane (3 microns thick) constructed from B mori silk fibroin. Cell attachment was facilitated by pre-coating the fibroin membrane with vitronectin (for ARPE-19 cells) and gelatin (for HMEC-1 cells) respectively. The effects of tropoelastin on attachment of ARPE-19 cells was also examined. Barrier function was examined by measurement of trans-epithelial resistance (TER) using a voltohmmeter (EVOM-2). The phagocytic activity of the synthetic RPE was tested using vitronectin-coated microspheres (2 micron diameter FluoSpheres). In some cultures, membrane defects were created by puncturing within a 24 G needle. The architecture of the synthetic tissue before and after wounding was examined by confocal microscopy after staining for ZO-1 and F-actin. Results The RPE layer of the 3D model developed a cobblestoned morphology (validated by staining for ZO-1 and F-actin), displayed barrier function (validated by measurement of TER) and demonstrated cytoplasmic uptake of vitronectin-coated microspheres. Attachment of ARPE-19 cells to fibroin was unaffected by tropoelastin. Microvascular endothelial cells attached well to the gelatin-coated surface of the fibroin membrane and remained physically separated from the overlaying RPE layer. The fibroin membranes were amenable to puncturing without collapse thus providing the opportunity to study transmembrane migration of the endothelial cells. Conclusions Synthetic Bruch’s membranes constructed from silk fibroin, vitronectin and gelatin, support the co-cultivation of RPE cells and microvascular endothelial cells. The resulting RPE layer displays functions similar to that of native RPE and the entire tri-layered structure displays potential to be used as an in vitro model of choroidal neovascularization.

Improved estimates of percolation and anisotropic permeability from 3-D X-ray microtomography using stochastic analyses and visualization

X-ray microtomography (micro-CT) with micron resolution enables new ways of characterizing microstructures and opens pathways for forward calculations of multiscale rock properties. A quantitative characterization of the microstructure is the first step in this challenge. We developed a new approach to extract scale-dependent characteristics of porosity, percolation, and anisotropic permeability from 3-D microstructural models of rocks. The Hoshen-Kopelman algorithm of percolation theory is employed for a standard percolation analysis. The anisotropy of permeability is calculated by means of the star volume distribution approach. The local porosity distribution and local percolation probability are obtained by using the local porosity theory. Additionally, the local anisotropy distribution is defined and analyzed through two empirical probability density functions, the isotropy index and the elongation index. For such a high-resolution data set, the typical data sizes of the CT images are on the order of gigabytes to tens of gigabytes; thus an extremely large number of calculations are required. To resolve this large memory problem parallelization in OpenMP was used to optimally harness the shared memory infrastructure on cache coherent Non-Uniform Memory Access architecture machines such as the iVEC SGI Altix 3700Bx2 Supercomputer. We see adequate visualization of the results as an important element in this first pioneering study.

A simple method for fabricating 3-D multilayered composite scaffolds

One limitation of electrospinning stems from the charge build-up that occurs during processing, preventing further fibre deposition and limiting the scaffold overall thickness and hence their end-use in tissue engineering applications targeting large tissue defect repair. To overcome this, we have developed a technique in which thermally induced phase separation (TIPS) and electrospinning are combined. Thick three-dimensional, multilayered composite scaffolds were produced by simply stacking individual polycaprolactone (PCL) microfibrous electrospun discs into a cylindrical holder that was filled with a 3% poly(lactic-co-glycolic acid) (PLGA) solution in dimethylsulfoxide (a good solvent for PLGA but a poor one for PCL). The construct was quenched in liquid nitrogen and the solvent removed by leaching out in cold water. This technique enables the fabrication of scaffolds composed principally of electrospun membranes that have no limit to their thickness. The mechanical properties of these scaffolds were assessed under both quasi-static and dynamic conditions. The multilayered composite scaffolds had similar compressive properties to 5% PCL scaffolds fabricated solely by the TIPS methodology. However, tensile tests demonstrated that the multilayered construct outperformed a scaffold made purely by TIPS, highlighting the contribution of the electrospun component of the composite scaffold to enhancing the overall mechanical property slate. Cell studies revealed cell infiltration principally from the scaffold edges under static seeding conditions. This fabrication methodology permits the rapid construction of thick, strong scaffolds from a range of biodegradable polymers often used in tissue engineering, and will be particularly useful when large dimension electrospun scaffolds are required.

The role of vascularization in 3-D tissue engineering

The role of vascularization in 3-D tissue engineering was studied. Mouse fat, angiogenic growth factors, adult human stem cells and fat tissue have been inserted and subsequent tissue growth was monitored. Human fat grafts or human lipoaspirates in SCID mouse chambers induced mouse fat generation at 6 weeks. Tissue engineering models utilizing intrinsic vascularization have major advantages including rapid and appropriate vascularization of new tissues.