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 new implicit numerical method for the space and time fractional Bloch-Torrey equation

In recent years, it has been found that many phenomena in engineering, physics, chemistry and other sciences can be described very successfully by models using mathematical tools from fractional calculus. Recently, noted a new space and time fractional Bloch-Torrey equation (ST-FBTE) has been proposed (see Magin et al. (2008)), and successfully applied to analyse diffusion images of human brain tissues to provide new insights for further investigations of tissue structures. In this paper, we consider the 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. Firstly, we propose a new effective implicit numerical method (INM) for the STFBTE whereby we discretize the Riesz fractional derivative using a fractional centered difference. Secondly, we prove that the implicit numerical method for the ST-FBTE is unconditionally stable and convergent, and the order of convergence of the implicit numerical method is ( T2 - α + h2 x + h2 y + h2 z ). Finally, some numerical results are presented to support our theoretical analysis.

Analytical and numerical solutions of the space and time fractional bloch-torrey equation

Fractional order dynamics in physics, particularly when applied to diffusion, leads to an extension of the concept of Brown-ian motion through a generalization of the Gaussian probability function to what is termed anomalous diffusion. As MRI is applied with increasing temporal and spatial resolution, the spin dynamics are being examined more closely; such examinations extend our knowledge of biological materials through a detailed analysis of relaxation time distribution and water diffusion heterogeneity. Here the dynamic models become more complex as they attempt to correlate new data with a multiplicity of tissue compartments where processes are often anisotropic. Anomalous diffusion in the human brain using fractional order calculus has been investigated. Recently, a new diffusion model was proposed by solving the Bloch-Torrey equation using fractional order calculus with respect to time and space (see R.L. Magin et al., J. Magnetic Resonance, 190 (2008) 255-270). However effective numerical methods and supporting error analyses for the fractional Bloch-Torrey equation are still limited. In this paper, the space and time fractional Bloch-Torrey equation (ST-FBTE) is considered. The time and space derivatives in the ST-FBTE are replaced by the Caputo and the sequential Riesz fractional derivatives, respectively. Firstly, we derive an analytical solution for the ST-FBTE with initial and boundary conditions on a finite domain. Secondly, we propose an implicit numerical method (INM) for the ST-FBTE, and the stability and convergence of the INM are investigated. We prove that the implicit numerical method for the ST-FBTE is unconditionally stable and convergent. Finally, we present some numerical results that support our theoretical analysis.

Reply to William B. Grant

THE UVI working group acknowledges the contribution of Vitamin D to bone health as stated in our paper. However, we concluded that an optimal level of Vitamin D for humans has not yet been established with any certainty...

Platelet-derived endothelial cell growth factor expression and angiogenesis in cervical intraepithelial neoplasia and squamous cell carcinoma of the cervix

Growth and metastatic spread of invasive carcinoma depends on angiogenesis, the formation of new blood vessels. Platelet-derived endothelial cell growth factor (PD-ECGF) is an angiogenic growth factor for a number of solid tumors, including lung, bladder, colorectal, and renal cell cancer. Cervical intraepithelial neoplasia (CIN) is the precursor to squamous cell cervical carcinoma (SCC). Mean vessel density (MVD) increases from normal cervical tissue, through low- and high-grade CIN to SCC. We evaluated PD-ECGF immunoreactivity and correlated its expression with MVD in normal, premalignant, and malignant cervical tissue. PD-ECGF expression was assessed visually within the epithelial tissues and scored on the extent and intensity of staining. MVD was calculated by counting the number of vessels positive for von Willebrand factor per unit area subtending normal or CIN epithelium or within tumor hotspots for SCC. Cytoplasmic and/or nuclear PD-ECGF immunoreactivity was seen in normal epithelium. PD-ECGF expression significantly increased with histologic grade from normal, through low- and high-grade CIN, to SCC (P < .02). A progressive significant increase in the microvessel density was also seen, ranging from a mean of 28 vessels for normal tissue to 57 for SCC (P < .0005). No correlation was found between PD-ECGF expression and MVD (P = .45). We conclude that PD-ECGF expression and MVD increase as the cervix transforms from a normal to a malignant phenotype. PD-ECGF is thymidine phosphorylase, a key enzyme in the activation of fluoropyrimidines, including 5-fluorouracil. Evaluation of PD-ECGF thymidine phosphorylase expression may be important in designing future chemotherapeutic trials in cervical cancer. Copyright (C) 2000 by W.B. Saunders Company.

Angiogenesis as a biomarker and target in cancer chemoprevention

Angiogenesis, the formation of new blood vessels from existing vasculature, is essential to the late stages of carcinogenesis, allowing tumours to grow beyond 1-2 mm in diameter, invade surrounding tissue, and metastasise. However, more than two decades ago, angiogenesis that preceded neoplastic transformation was seen. Indeed, it can be detected in inflammatory and infectious diseases that increase the risk of developing cancer. Recent advances in fluorescence endoscopy and histological assessment suggest that, for certain cancers, the degree of new blood-vessel formation may differ between the early and late stages of carcinogenic progression. The association between angiogenesis and cancer occurrence, and ease of detection of this process in accessible tissues early in carcinogenesis, mean that angiogenesis fulfils the criteria for a biomarker of the effectiveness of chemopreventive intervention. There is also some evidence that biochemical assays of angiogenic growth factors may after similar potential as surrogate biomarkers. Many natural and synthetic chemopreventive agents in development or in clinical use inhibit new vessel formation in vivo. Validation of angiogenesis as a biomarker for the effectiveness of chemoprevention should further the advancement of some chemopreventive agents.

Cungellela by William ‘Dura Danje’ Leisha and Shem ‘Curan Danje’ Leisha

Research background: Cungelela is an intercultural music project undertaken in collaboration with William ‘Dura Danje’ Leisha and Shem ‘Curan Danje’ Leisha. The project contributes to cultural maintenance for Australian First Nations peoples, and is informed by prior work in this area by scholars including Peter Dunbar-Hall, Chris Gibson and Karl Neuenfeldt. These existing studies have discussed the complexities of intercultural collaboration, and the types of cultural politics that are involved when Indigenous and non-Indigenous musicians and scholars work together on projects of cultural significance. Critical race theory has also informed the creative work, as a means of interpreting the implicit and explicit discourses of race that arise through intercultural creative practice. The project asked the research question, in what ways can collaborative music making contribute to intercultural understanding and support cultural maintenance for Australian First Nations people affected by the Stolen Generations? Research contribution: This project has identified that collaborative production of recorded popular music can produce shared affective, embodied and transformative forms of knowledge about the impact of the Stolen Generations on Australian First Nations peoples. Research significance: The compact disc was presented by Aunty Anne Leisha as part of an invited presentation at the World Indigenous Nations Higher Education Consortium in New Mexico, 2013. The work also formed part of a refereed conference presentation at the 2013 conference of the International Association for the Study of Popular Music held at the University of Oviedo, Gijon, Spain.

Genome-wide meta-analysis identifies 56 bone mineral density loci and reveals 14 loci associated with risk of fracture

Bone mineral density (BMD) is the most widely used predictor of fracture risk. We performed the largest meta-analysis to date on lumbar spine and femoral neck BMD, including 17 genome-wide association studies and 32,961 individuals of European and east Asian ancestry. We tested the top BMD-associated markers for replication in 50,933 independent subjects and for association with risk of low-trauma fracture in 31,016 individuals with a history of fracture (cases) and 102,444 controls. We identified 56 loci (32 new) associated with BMD at genome-wide significance (P < 5 × 10−8). Several of these factors cluster within the RANK-RANKL-OPG, mesenchymal stem cell differentiation, endochondral ossification and Wnt signaling pathways. However, we also discovered loci that were localized to genes not known to have a role in bone biology. Fourteen BMD-associated loci were also associated with fracture risk (P < 5 × 10−4, Bonferroni corrected), of which six reached P < 5 × 10−8, including at 18p11.21 (FAM210A), 7q21.3 (SLC25A13), 11q13.2 (LRP5), 4q22.1 (MEPE), 2p16.2 (SPTBN1) and 10q21.1 (DKK1). These findings shed light on the genetic architecture and pathophysiological mechanisms underlying BMD variation and fracture susceptibility.

William James, Sciences of Mind, and Anti-imperial Discourse

In the past few decades, the humanities and social sciences have developed new methods of reorienting their conceptual frameworks in a “world without frontiers.” In this book, Bernadette M. Baker offers an innovative approach to rethinking sciences of mind as they formed at the turn of the twentieth century, via the concerns that have emerged at the turn of the twenty-first. The less-visited texts of Harvard philosopher and psychologist William James provide a window into contemporary debates over principles of toleration, anti-imperial discourse, and the nature of ethics. Baker revisits Jamesian approaches to the formation of scientific objects including the child mind, exceptional mental states, and the ghost to explore the possibilities and limits of social scientific thought dedicated to mind development and discipline formation around the construct of the West.