The role of Mittag-Leffler functions in anomalous relaxation

The inertia-corrected Debye model of rotational Brownian motion of polar molecules was generalized by Coffey et al. [Phys. Rev. E, 65, 32 102 (2002)] to describe fractional dynamics and anomalous rotational diffusion. The linear- response theory of the normalized complex susceptibility was given in terms of a Laplace transform and as a function of frequency. The angular-velocity correlation function was parametrized via fractal Mittag-Leffler functions. Here we apply the latter method and complex-contour integral- representation methods to determine the original time-dependent amplitude as an inverse Laplace transform using both analytical and numerical approaches, as appropriate. (C) 2004 Elsevier B.V. All rights reserved.

Green function for the diffusion limit of one-dimensional continuous time random walks

It is shown how the fractional probability density diffusion equation for the diffusion limit of one-dimensional continuous time random walks may be derived from a generalized Markovian Chapman-Kolmogorov equation. The non-Markovian behaviour is incorporated into the Markovian Chapman-Kolmogorov equation by postulating a Levy like distribution of waiting times as a kernel. The Chapman-Kolmogorov equation so generalised then takes on the form of a convolution integral. The dependence on the initial conditions typical of a non-Markovian process is treated by adding a time dependent term involving the survival probability to the convolution integral. In the diffusion limit these two assumptions about the past history of the process are sufficient to reproduce anomalous diffusion and relaxation behaviour of the Cole-Cole type. The Green function in the diffusion limit is calculated using the fact that the characteristic function is the Mittag-Leffler function. Fourier inversion of the characteristic function yields the Green function in terms of a Wright function. The moments of the distribution function are evaluated from the Mittag-Leffler function using the properties of characteristic functions and a relation between the powers of the second moment and higher order even moments is derived. (C) 2004 Elsevier B.V. All rights reserved.

Phase I study of GSK461364, a specific and competitive Polo-like Kinase 1 (PLK1) inhibitor in patients with advanced solid malignancies.

Purpose: GSK461364 is an ATP-competitive inhibitor of polo-like kinase 1 (Plk1). A phase I study of two schedules of intravenous GSK461364 was conducted. Experimental Design: GSK461364 was administered in escalating doses to patients with solid malignancies by two schedules, either on days 1, 8, and 15 of 28-day cycles (schedule A) or on days 1, 2, 8, 9, 15, and 16 of 28-day cycles (schedule B). Assessments included pharmacokinetic and pharmacodynamic profiles, as well as marker expression studies in pretreatment tumor biopsies. Results: Forty patients received GSK461364: 23 patients in schedule A and 17 in schedule B. Dose-limiting toxicities (DLT) in schedule A at 300 mg (2 of 7 patients) and 225 mg (1 of 8 patients) cohorts included grade 4 neutropenia and/or grade 3–4 thrombocytopenia. In schedule B, DLTs of grade 4 pulmonary emboli and grade 4 neutropenia occurred at 7 or more days at 100 mg dose level. Venous thrombotic emboli (VTE) and myelosuppression were the most common grade 3–4, drug-related events. Pharmacokinetic data indicated that AUC (area under the curve) and C max (maximum concentration) were proportional across doses, with a half-life of 9 to 13 hours. Pharmacodynamic studies in circulating tumor cells revealed an increase in phosphorylated histone H3 (pHH3) following drug administration. A best response of prolonged stable disease of more than 16 weeks occurred in 6 (15%) patients, including 4 esophageal cancer patients. Those with prolonged stable disease had greater expression of Ki-67, pHH3, and Plk1 in archived tumor biopsies. Conclusions: The final recommended phase II dose for GSK461364 was 225 mg administered intravenously in schedule A. Because of the high incidence (20%) of VTE, for further clinical evaluation, GSK461364 should involve coadministration of prophylactic anticoagulation.