Function Weibull

Weibull analysis Weibull distribution Weibull function Weibullmodulus... 

Dorko et al. [442] have used the Weibull distribution function for the consideration of reactions in which decomposition is accompanied by melting. Following a procedure described by Kao [446], they used a mixed Weibull function, written as a linear combination of separate functions, viz. 

It should be emphasized that models other than the Weibull function represented in Eq. (2) could also be proposed and tested. For these models, the possibility of over-parameterization should first be checked using the correlation matrix of the estimates. Of those tested, the best model can be... 

Three cases have been constructed by Weibull functions according to Eqs. (la) and (lb), as these best reflect systematic differences in the sequence. In all cases, a reference profile is defined by extent Eo l.O, scale parameter / = 2.0, and shape parameter a = 1.5. In each case, one parameter is altered to illustrate its influence. 

Alternative methods and algorithms may be used, such as the model-independent approach to compare similarity limits derived from multi-variate statistical differences (MSD) combined with a 90% confidence interval approach for test and reference batches (21). Model-dependent approaches such as the Weibull function use the comparison of parameters obtained after curve fitting of dissolution profiles. See Chapters 8 and 9 for further discussion of these methods. 

With the exception of the formulation obtained by direct compression in a rotary machine, the dissolution profiles were well fitted by the Weibull function. A high density in the centre of the tablets may explain the sigmoid dissolution profiles. A percentage of the drug remains imprisoned in the matrix after the dissolution test. It is possible to suppose that, in the densified central zone of the tablet, the diameters of the pores are smaller than the diameters of the drug particles covered by the inert polymers. 

As the AUC value is almost the same, may the P parameter of the Weibull function be used for evaluating the variability between tablets and between processes ... 

The analysis of covariance between a continuous variable (P is the curve shape parameter from the Weibull function) and a discrete variable (process) was determined using the general linear model (GLM) procedure from the Statistical Analysis System (SAS). The technique of the heterogeneity of slopes showed that there was no significant difference (Tables 5 and 6). 

The free induction decay following 90° pulse has a line shape which generally follows the Weibull functions (Eq. (22)). In the homogeneous sample the FID is described by a single Weibull function, usually exponential (Lorentzian) (p = 1) or Gaussian (p = 2). The FID of heterogeneous systems, such as highly viscous and crosslinked polydimethylsiloxanes (PDMS) 84), hardened unsaturated polyesters 8S), and compatible crosslinked epoxy-rubber systems 52) are actually a sum of three... 

A pseudo solid-like behavior of the T2 relaxation is also observed in i) high Mn fractionated linear polydimethylsiloxanes (PDMS), ii) crosslinked PDMS networks, with a single FID and the line shape follows the Weibull function (p = 1.5)88> and iii) in uncrosslinked c/.s-polyisoprenes with Mn > 30000, when the presence of entanglements produces a transient network structure. Irradiation crosslinking of polyisoprenes having smaller Mn leads to a similar effect91 . The non-Lorentzian free-induction decay can be a consequence of a) anisotropic molecular motion or b) residual dipolar interactions in the viscoelastic state. 

When carbon (S) spins are locked following a CP 90° pulse, the HB term becomes nonsecular and, after these oscillations vanish, the S spin-lock magnetization is fractionally reduced. Thus, the observed decay is generally a sum of two Weibull functions, usually exponentials (Equation 20 p = 1) the initial slope reflects a Tle dominated by a spin-lattice process and the final slope yields a Tt dominated... 

The general problem that we will focus on in this section is the escape of drug molecules1 from a cylindrical vessel. Initially, theoretical aspects are presented demonstrating that the Weibull function can describe drug release kinetics from cylinders, assuming that the drug molecules move inside the matrix by a Fickian... 

The above reasoning shows that the stretched exponential function (4.14), or Weibull function as it is known, may be considered as an approximate solution of the diffusion equation with a variable diffusion coefficient due to the presence of particle interactions. Of course, it can be used to model release results even when no interaction is present (since this is just a limiting case of particles that are weakly interacting). 

It is clear that it cannot be proven that the Weibull function is the best choice of approximating the release results. There are infinitely many choices of the form g (t) and some of them may be better than the Weibull equation. This reasoning merely indicates that the Weibull form will probably be a good choice. The simulation results below show that it is indeed a good choice. The above reasoning is quite important since it provides a physical model that justifies the use of the Weibull function in order to fit experimental release data. 

The parameters a and b are somehow connected to the geometry and size of the matrix that contains the particles. This connection was investigated by performing release simulations for several cylinder sizes and for several initial drug concentrations [82]. The Weibull function was fitted to the simulated data to obtain estimates for a and b. If one denotes by Aieak the number of leak sites and by ATtot the total number of sites, in the continuum limit the ratio /V eak / At.ot. 

The Weibull Function Describes Drug Release from Fractal Matrices... 

Figure 4.13 Plot of the number of particles (normalized) remaining in the percolation fractal as a function of time t for lattice sizes 100 x 100, 150 x 150, and 200 X 200. n (t) is the number of particles that remain in the lattice at time t and no is the initial number of particles. Simulation results are represented by points. The solid lines represent the results of nonlinear fitting with a Weibull function.

In Figure 4.13, n (t) /no is plotted as a function of time for different lattice sizes. The data were fitted with a Weibull function (4.14), where the parameter a ranges from 0.05 to 0.01 and the exponent b from 0.35 to 0.39. It has been shown [82] that (4.14) also holds in the case of release from Euclidean matrices. In that case the value of the exponent b was found to be b 0.70. 

Hn the phamaceutical literature the exponential in the Weibull function is written as exp (—At11) and therefore A has dimension time-In the version used herein (equation 5.11), the dimension of A is time-1. 

The successful use of the Weibull function in modeling the dissolution profiles raises a plausible query What is the rationale of its success The answer will be sought in the relevance of the Weibull distribution to the kinetics prevailing during the dissolution process. 

Most importantly, it was shown that the structure of the Weibull function captures the time-dependent character of the rate coefficient governing the dissolution process. These considerations agree with Elkoski s [120] analysis of the... 

This measure of heterogeneity generalizes the notion of heterogeneity as a departure from the classical first-order model initially introduced [121] for the specific case of the Weibull function. In addition, the above equation can also be used for comparison between two experimentally obtained dissolution profiles [131]. 

Kosmidis, K., Argyrakis, P., and Macheras, P., A reappraisal of drug release laws using Monte Carlo simulations The prevalence of the Weibull function, Pharmaceutical Research, Vol. 20, No. 7, 2003, pp. 988-995. 

Papadopoulou, V., Kosmidis, K., Vlachou, M., and Macheras, P., On the use of the Weibull function for the discernment of drug release mechanisms, International Journal of Pharmaceutics, In press. 

Pressure-time function (modified Weibull function) [39, 96-98] (Figure 16)... 

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