Weibull equation

Here (simulated) in vitro release profiles that differ by at least 10% are shown (panels a and b), as well as the (simulated) resulting plasma concentration-time profiles for a drug with a 1-hr half-life (panel c) and 6-hr half-life (panel d). The simulated-release profiles are described by the following Weibull equation ... 

Figure 3 Observed in vitro dissolution data for three ER formulations (panel a) fast ( target 80%=12hr), medium (o target 80%=lbhr), and slow ( target 80% = 20hr). Also shown are the predicted lines corresponding to fitting the data to the double Weibull equation (fitted parameter values are listed in Table 2). The associated rate plot for the three formulations is shown in panel b (fast,--------- medium,--------- slow,---).

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. 

Any model can be applied to in vitro dissolution data and fitted by linear or non-linear regression, as appropriate. Sometimes a first-order model [A(t) = A - Ae kt where A(t) is the amount dissolved after time t, A is the initial amount and k is the first-order dissolution rate constant] or even a zero-order model (A(t) = A-Akt) is sufficiently sophisticated to determine a dissolution rate that is representative for the whole process. However, a more general equation that is commonly applied to dissolution data is the Weibull equation (Langenbucher 1976) ... 

Figure 29. Degradation of ascorbic acid in the presence of mannitol ([ascorbic acid] [mannitol] = 1 9) plotted according to the Weibull equation, (water content 2%). (Reproduced from Ref. 292 with permission.)...

Figure 187. Arrhenius plots for discoloration of parenteral formulations of ascorbic acid (1), reserpine (2), thiamine hydrochloride (3), and ATP (4). The parameters k and m represent the constants obtained using the Weibull equation (time unit day). (Reproduced from Ref. 730 with permission.)...

The kinetics of solid-state degradation of peptide and protein pharmaceuticals is difficult to describe for many of the same reasons that it is difficult to describe solid-state inactivation of small molecules. Apparent inactivation of digestive enzymes such as lipase was analyzed empirically using the Weibull equation (Eq. 2.69), as shown in Fig. 209,880-881 whereas apparent inactivation of dry horse serum cholinesterase was adequately described by a first-order equation even for inactivation in the solid state.882... 

Using the Weibull equation, it is easily shown that... 

In summary, it must be stated that the strength distribution is a direct function of the size distribution of the flaws which act as fracture origins. The analytical Weibull equation [ Eq. (12)] is a special case of a more general distribution which often - but... 

The fibers taken from the same bundle showed a wide range of diameters, which is a typical drawback of natural fibers, justifying the need for the use of a more accurate statistical distribution function. In fact, all the fibers showed wide dispersion of strength with respect to diameter data, indicated by the dimensionless shape parameter of the Weibull equation. By contrast, the advanced statistical approach based on neural network algorithms (PDF estimation technique) mentioned above resulted in asymmetric curves of diameter distributions of four lignocellulosic fibers mentioned above (Fig. 8.7). 

This is the general form of the Weibull equation for arbitrarily loaded components. 

If the Weibull modulus m and the reference stress [Pg.241]

It is not necessary to use the volume-dependent Weibull equation (7.6) because we determine the parameters for the specimen volume Vb-... 

Equation (8.42) shows that the relation between the fine powder ratio and external force P can be expressed by the Weibull equation. The broken curve (the relationship curve between the fine powder ratio and external force) is a Weibull curve. This is closely related to the agreement between SPCS and the Weibull distribution. Experimental results in the literature show that the parameter M of the WSCS has a value very close to the Weibull modulus in the SPCS data. Both relationships are reflections of the distribution of defects in the material. In the measurement of WSCS, the disabled probability of [E(P)] is the fine powder ratio (dm/m). 

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