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.

Is language a barrier for quality care in hospitals? A case series from an Emergency Department of a teaching hospital in Brisbane

Aim: The aim of this pilot study is to describe the use of an Emergency Department (ED) at a large metropolitan teaching hospital by patients who speak English or other languages at home. Methods: All data were retrieved from the Emergency Department Information System (EDIS) of this tertiary teaching hospital in Brisbane. Patients were divided into two groups based on the language spoken at home: patients who speak English only at home (SEO) and patients who do not speak English only or speak other language at home (NSEO). Modes of arrival, length of ED stay and the proportion of hospital admission were compared among the two groups of patients by using SPSS V18 software. Results: A total of 69,494 patients visited this hospital ED in 2009 with 67,727 (97.5%) being in the SEO group and 1,281 (1.80%) in the NSEO group. The proportion of ambulance utilisation in arrival mode was significantly higher among SEO 23,172 (34.2%) than NSEO 397 (31.0%), p <0.05. The NSEO patients had longer length of stay in the ED (M = 337.21, SD = 285.9) compared to SEO patients (M= 290.9, SD = 266.8), with 46.3 minutes (95%CI 62.1, 30.5, p <0.001) difference. The admission to the hospital among NSEO was 402 (31.9%) higher than SEO 17,652 (26.6%), p <0.001. Conclusion: The lower utilisation rates of ambulance services, longer length of ED stay and higher hospital admission rates in NSEO patients compared to SEO patients are consistent with other international studies and may be due to the language barriers.

What health problems overcrowd hospital emergency departments in Queensland and has this changed over time? A case study of two hospitals in Queensland 2001 - 2009

Aim: to describe what health problems patients attending emergency department with and whether this changed over time. Methods: Electronic data was retrieved from EDIS (Emergency Department Information System) and HBCIS (Hospital Based Clinical Information System) in two hospitals in Queensland in the period 2001-2009. The ICD-10 code of patient's diagnosis was then extrapolated and then group into ICD-10 chapters, such that the health problem can be presented. Results: Among the specific health problems, Chapter XIX 'Injury and poisoning' ranked number one consistently (ranging from 22.1% to 31.2% of the total presentations) in both the urban and remote hospitals in Queensland. The top ten specific presenting health problems in both the urban and remote hospital include Chapter XI 'Digestive system', Chapter XIV 'Genitourinary system', Chapter IX 'Circulatory system', and Chapter XIII 'Musculoskeletal system and connective tissue'. Chapter X 'Respiratory system' made the top ten presenting Chapters in both hospitals, but ranked much higher (number four consistently for the eight years, ranging from 6.8% to 8.3%) in the remote hospital. Chapter XV 'Pregnancy childbirth and puerperium' made to the top ten in the urban hospital only while Chapter XII 'Skin and subcutaneous tissue', Chapter I 'Infectious and parasitic diseases' made the top ten in the remote hospital only. Conclusion: The number one health problem presenting to both the urban and remote hospitals in Queensland is Chapter XIX 'Injury and poisoning', and it did not change in the period 211 - 2009.

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.

Mixed-dimensional consistent coupling by multi-point constraint equations for efficient multi-scale modeling

Finding an appropriate linking method to connect different dimensional element types in a single finite element model is a key issue in the multi-scale modeling. This paper presents a mixed dimensional coupling method using multi-point constraint equations derived by equating the work done on either side of interface connecting beam elements and shell elements for constructing a finite element multiscale model. A typical steel truss frame structure is selected as case example and the reduced scale specimen of this truss section is then studied in the laboratory to measure its dynamic and static behavior in global truss and local welded details while the different analytical models are developed for numerical simulation. Comparison of dynamic and static response of the calculated results among different numerical models as well as the good agreement with those from experimental results indicates that the proposed multi-scale model is efficient and accurate.