Bifurcations and transition to chaos in generalized fractional maps of the orders 0 < α < 1.

In this paper, we investigate the generalized fractional maps of the orders 0<α<1. Commonly used in publications, fractional and fractional difference maps of the orders 0<α<1 belong to this class of maps. As an example, we numerically solve the equations, which define asymptotically periodic points to draw the bifurcation diagrams for the fractional difference logistic map with α=0.5. For periods more than four (T>4), these bifurcation diagrams are significantly different from the bifurcation diagrams obtained after 105 iterations on individual trajectories. We present examples of transition to chaos on individual trajectories with positive and zero Lyapunov exponents. We derive the algebraic equations, which allow the calculation of bifurcation points of generalized fractional maps. We use these equations to calculate the bifurcation points for the fractional and fractional difference logistic maps with α=0.5. The results of our numerical simulations allow us to make a conjecture that the cascade of bifurcations scenarios of transition to chaos in generalized fractional maps and regular maps are similar, and the value of the generalized fractional Feigenbaum constant δf is the same as the value of the regular Feigenbaum constant δ=4.669….

On stability of fixed points and chaos in fractional systems.

In this paper, we propose a method to calculate asymptotically period two sinks and define the range of stability of fixed points for a variety of discrete fractional systems of the order 0<α<2. The method is tested on various forms of fractional generalizations of the standard and logistic maps. Based on our analysis, we make a conjecture that chaos is impossible in the corresponding continuous fractional systems.

On the fractional Eulerian numbers and equivalence of maps with long term power-law memory (integral Volterra equations of the second kind) to Grünvald-Letnikov fractional difference (differential) equations.

In this paper, we consider a simple general form of a deterministic system with power-law memory whose state can be described by one variable and evolution by a generating function. A new value of the system's variable is a total (a convolution) of the generating functions of all previous values of the variable with weights, which are powers of the time passed. In discrete cases, these systems can be described by difference equations in which a fractional difference on the left hand side is equal to a total (also a convolution) of the generating functions of all previous values of the system's variable with the fractional Eulerian number weights on the right hand side. In the continuous limit, the considered systems can be described by the Grünvald-Letnikov fractional differential equations, which are equivalent to the Volterra integral equations of the second kind. New properties of the fractional Eulerian numbers and possible applications of the results are discussed.

Percutaneous vertebroplasty: a review for the primary care physician.

The purpose of this article is to help primary care physicians who are often challenged with the management of vertebral compression fracture (VCF) by presenting clinical background and identifying candidates for percutaneous vertebroplasty, a minimally invasive procedure for treatment of VCF.

Phase II clinical trial of the epothilone B analog, ixabepilone, in patients with non small-cell lung cancer whose tumors have failed first-line platinum-based chemotherapy.

PURPOSE: Ixabepilone is the first in a new class of antineoplastic agents, the epothilones and their analogs. This international, randomized, phase II trial assessed two administration schedules of ixabepilone as second-line therapy in patients with non-small-cell lung cancer (NSCLC). PATIENTS AND METHODS: Patients had experienced disease progression after one prior cisplatin- or carboplatin-based chemotherapy regimen. Ixabepilone was administered as a single 32 mg/m(2) 3-hour infusion (77 patients; arm A) or a 6 mg/m(2) 1-hour infusion daily for 5 consecutive days (69 patients; arm B) in a 3-week cycle. RESULTS: The intent-to-treat objective response rate was 14.3% in arm A and 11.6% in arm B. Median duration of response was 8.7 months (95% CI, 5.3 to 9.5 months) in arm A and 9.6 months (95% CI, 6.1 to 19.7 months) in arm B. Median time to progression was 2.1 months (95% CI, 1.4 to 2.8 months) for arm A and 1.5 months (95% CI, 1.4 to 2.8 months) for arm B. Median survival was 8.3 months (95% CI, 5.8 to 11.5 months) for arm A, and 7.3 months (95% CI, 5.7 to 11.7 months) for arm B; the 1-year survival rate (both cohorts) was 38%. Responses occurred in patients with taxane-pretreated and platinum-refractory tumors. Both regimens had an acceptable toxicity profile. Myelosuppression was manageable, manifesting primarily as neutropenia and leukopenia. Neuropathy was primarily sensory, generally mild to moderate in severity, and mostly reversible (both regimens). CONCLUSION: Single-agent ixabepilone had clinically relevant activity and an acceptable safety profile in patients with advanced NSCLC whose tumors had failed one prior platinum-based chemotherapy regimen.

MR angiography of aortoiliac occlusive disease: a phase III study of the safety and effectiveness of the blood-pool contrast agent MS-325.

PURPOSE: To evaluate prospectively the safety and effectiveness of aortoiliac magnetic resonance (MR) angiography enhanced with MS-325 (gadofosveset trisodium) at a dose of 0.03 mmol/kg; effectiveness was defined as accuracy relative to the reference standard, conventional angiography. MATERIALS AND METHODS: Study was approved by institutional review boards of participating institutions, and required national approvals were obtained. Study protocol conformed to Good Clinical Practice guidelines, and informed patient consent was obtained. Patients with known or suspected peripheral vascular disease received 0.03 mmol/kg MS-325 for aortoiliac MR angiography. They were also examined with conventional angiography. MS-325-enhanced MR was evaluated for safety and effectiveness. Along with unenhanced two-dimensional time-of-flight MR angiography, it was compared with conventional angiography for presence of vascular stenosis. Student t tests were used to identify significant improvement in diagnostic sensitivity, specificity, and accuracy, as well as quantitative characterization of stenoses by three blinded readers. Correlations between readers of conventional angiograms were calculated and compared with MR results. RESULTS: In 174 patients, MS-325-enhanced MR angiography showed significant improvement (P < or = .001) in sensitivity, specificity, and accuracy for diagnosis of clinically significant (> or =50%) stenosis, compared with unenhanced MR. For all readers, areas under the receiver operating characteristic curve for both quantitative and qualitative measures of significant disease increased (P < .001) for MS-325-enhanced MR compared with time-of-flight MR. All readers also expressed higher confidence in diagnosis (P < .001) and found fewer images uninterpretable with MS-325 enhancement. All measures of interpretation accuracy approached corresponding measures of correlation between readers of conventional angiograms. Incidence of severe and serious adverse events with MS-325 was low. No patients were withdrawn from study due to adverse events or abnormalities in laboratory results. There were no clinically important trends in findings at hematology, blood chemistry, urinalysis, electrocardiography, or physical examination. CONCLUSION: MR angiography with MS-325 provides significant improvement in effectiveness over unenhanced MR (and minimal and transient side effects) at a dose of 0.03 mmol/kg and was safe and effective for MR evaluation of patients with aortoiliac occlusive disease.

MR angiography with gadofosveset trisodium for peripheral vascular disease: phase II trial.

PURPOSE: To evaluate the dose response and safety of gadofosveset trisodium-enhanced magnetic resonance (MR) angiography compared with nonenhanced two-dimensional time-of-flight MR angiography and with x-ray angiography as the standard. MATERIALS AND METHODS: In this randomized, 20-center, double-blind study, 238 men and women who had peripheral vascular disease or were suspected of having it received intravenous injection of placebo or gadofosveset (0.005, 0.01, 0.03, 0.05, or 0.07 mmol per kilogram of body weight). MR angiographic images were evaluated by three blinded readers, and x-ray angiographic images were evaluated by two readers. Hypothesis testing for the presence of a dose response was based on a linear test for trend for increase in area under the receiver operating characteristic curve as a function of dose for each reader of MR angiographic images independently. RESULTS: Gadofosveset administration resulted in a dose-dependent increase in diagnostic accuracy for detection of aortoiliac occlusive disease as reflected in the area under the receiver operating characteristic curve for each reader (P <.001). The plateau in effectiveness improvement began at the 0.03 mmol/kg dose. At doses of 0.03 mmol/kg and higher, gadofosveset-enhanced MR angiography provided an approximate 20% increase in accuracy over nonenhanced MR angiography for diagnosis of clinically significant aortoiliac occlusive disease. Gadofosveset exhibited a good safety profile in all dose groups. Three serious adverse events were possibly or probably related to gadofosveset administration. There were no dose-related trends in severe or serious adverse events in patients receiving gadofosveset. CONCLUSION: A dose of 0.03 mmol/kg of gadofosveset was safe and effective for evaluation of aortoiliac occlusive disease with MR angiography.