Physical stability calculation

Equation (8.29) provides no guarantee of stability. It is a necessary condition for stability that is imposed by the discretization scheme. Practical experience indicates that it is usually a sufficient condition as well, but exceptions exist when reaction rates (or heat-generation rates) become very high, as in regions near thermal runaway. There is a second, physical stability criterion that prevents excessively large changes in concentration or temperature. For example. An, the calculated change in the concentration of a component that is consumed by the reaction, must be smaller than a itself Thus, there are two stability conditions imposed on Az numerical stability and physical stability. Violations of either stability criterion are usually easy to detect. The calculation blows up. Example 8.8 shows what happens when the numerical stability limit is violated. 

Bindschaedler and Gurny [12] published an adaptation of the simplex technique to a TI-59 calculator and applied it successfully to a direct compression tablet of acetaminophen (paracetamol). Janeczek [13] applied the approach to a liquid system (a pharmaceutical solution) and was able to optimize physical stability. In a later article, again related to analytical techniques, Deming points out that when complete knowledge of the response is not initially available, the simplex method is probably the most appropriate type [14]. Although not presented here, there are sets of rules for the selection of the sequential vertices in the procedure, and the reader planning to carry out this type of procedure should consult appropriate references. 

Since the aim of this study was to evaluate the use of the change in the tablet properties during storage as a parameter to express the physical stability of tablet formulations, the Storage to Initial Ratio of the crushing strength (SIR(S)) and of the disintegration time (SIR(D)) were calculated and used as response values for equation (2). The calculations of SIR(S) and SIR(D) were performed as in equation (1). The mean of the measurements directly after preparation were used as initial value. 

With the aid of multiple linear regression, model coefficients were calculated, which describe the effect of the variables pn the physical stability of the tablets. Since two levels of each variable were studied it was possible to calculate the linear contribution of the variables. The general form of the model which describes the effect of the variables is given by the following formula ... 

The magnitudes of the solubility of forms I and II of this drug varied signiLcantly in water, decyl alcohol, and dodecyl alcohol. However, their data showed that the solubility ratio was independent of solvent, but dependent on temperature. Figure 19.3 shows the data for these polymorphs in water The difference in slopes (indicating a difference in enthalpies of fusion) for the two polymorphs can be used to calculate a transition temperature, where both forms have the same physical stability The identity of the metastable form and the degree of solubility enhancement both depend on the temperature chosen for comparison. 

In general, it has been shown that the applicability of the simple calculated GFA and liquid GS parameters in the field of small organic molecules is limited and that conclusions drawn from the inorganic sciences do not hold true for pharmaceuticals. As the physical stability is of great importance in the pharmaceutical field, other methods of attempting to predict the stability are required to circumvent time- and material-consuming experiments. 

In small-scale smdies with a limited number of drugs (i. e., two to three), the calculated relaxation time showed some degree of correlation with stability of the amorphous form however, when the sample set was increased, this correlation could not be supported (Graeser et al. 2009b). The predictive potential of the KWW equation in terms of physical stability is small. 

A straightforward way to assess the relaxadon time from easily obtainable DSC data is outlined in the literature (Mao et al. 2006a). Despite the improvement of the AG equation over the KWW equation, relaxation times calculated from the AG equation also did not show the capability of predicting the physical stability for a set of drugs. 

Mujica A and R J Needs 1993. First-principles Calculations of the Structural Properties, Stability, aind Band Structure of Complex Tetrahedral Phases of Germanium ST12 and BC8. Physical Review B48 17010-17017. 

The described method can generate a first-order backward or a first-order forward difference scheme depending whether 0 = 0 or 0 = 1 is used. For 9 = 0.5, the method yields a second order accurate central difference scheme, however, other considerations such as the stability of numerical calculations should be taken into account. Stability analysis for this class of time stepping methods can only be carried out for simple cases where the coefficient matrix in Equation (2.106) is symmetric and positive-definite (i.e. self-adjoint problems Zienkiewicz and Taylor, 1994). Obviously, this will not be the case in most types of engineering flow problems. In practice, therefore, selection of appropriate values of 6 and time increment At is usually based on trial and error. Factors such as the nature of non-linearity of physical parameters and the type of elements used in the spatial discretization usually influence the selection of the values of 0 and At in a problem. 

A one-dimensional mesh through time (temporal mesh) is constructed as the calculation proceeds. The new time step is calculated from the solution at the end of the old time step. The size of the time step is governed by both accuracy and stability. Imprecisely speaking, the time step in an explicit code must be smaller than the minimum time it takes for a disturbance to travel across any element in the calculation by physical processes, such as shock propagation, material motion, or radiation transport [18], [19]. Additional limits based on accuracy may be added. For example, many codes limit the volume change of an element to prevent over-compressions or over-expansions. 

Many of the properties oj -hydroxypyridines are typical of phenols. It was long assumed that they existed exclusively in the hydroxy form, and early physical measurements seemed to confirm this. For example, the ultraviolet spectrum of a methanolic solution of 3-hydroxypyridine is very similar to that of the 3-methoxy analog, and the value of the dipole moment of 3-hydroxypyridine obtained in dioxane indicates little, if any, zwitterion formation. However, it has now become clear that the hydroxy form is greatly predominant only in solvents of low dielectric constant. Comparison of the pK values of 3-hydroxypyridine with those of the alternative methylated forms indicated that the two tautomeric forms are of comparable stability in aqueous solution (Table II), and this was confirmed using ultraviolet spectroscopy. The ratios calculated from the ultraviolet spectral data are in good agreement with those de-... 

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