Compactness properties

Let y be a normed space and V be its dual. A sequence of elements Un gV is called weakly converging to an element u if for every fixed u G V  

We can consider a dual space V with respect to V. For an arbitrary fixed element u G V the functional u — u u) can be defined, where u G V. This functional is linear and continuous on V and therefore is an element of the space V. For every u gV the functional u G V can be pointed out such that 

Thus we obtain the imbedding of the space V into the second dual one y. The imbedding operator is denoted by tt. If Try = V, the space y is called reflexive (Kantorovich, Akilov, 1984). The simplest example of reflexive spaces are LP fl) for 1 p 00 since 

At the same time the spaces T (fi) and T°°(f2) are nonreflexive a space dual of T (fi) coincides with T°°(f2), but a dual one of T°°(f2) is wider as compared with The notion of -weak convergence will be used for 

The compactness properties are closely connected with the reflexivity of spaces. On that score we formulate two theorems widely used in this book (Vainberg, 1972). 

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Materials that do not compress well produce soft tablets. In addition, brittle crystalline materials will yield brittle tablets. Hiestand was the first pharmaceutical scientist to quantify rationally the compaction properties of pharmaceutical powders [23-28], The results... 

N. Kaneniwa, K. Imagaw, and J.-I. Ichikawa, The effect of particle size on the compaction properties and compaction mechanism of sulfadimethoxine and sulfaphenazole, Chem. Pharm. Bull, 36, 2531 (1988). 

The family F is called a finite cover if F contains only a finite number of F, subsets. If every open cover of a subset A of a topological space X contains a finite subcover, then the subset A of the topological space X is compact. The compactness property is a generalization of the elementary properties of closed and bounded intervals. 

Water uptake causes a host of problems in drug products and the inactive and active ingredients contained in them. Moisture uptake has been shown to be an important factor in the decomposition of drug substances [1-8]. Moisture has also been shown to change surface properties of solids [9,10], alter flow characteristics of powders [11,12], and affect the compaction properties of solids [13]. This chapter discusses various mathematical models that can be used to describe moisture uptake by deliquescent materials. 

As with particle size, particle shape can influence the compaction properties of solids. While work in this area is limited, some work has been reported. Ridgway and Scotton [17], for example, investigated the effect of particle shape on die-fill weight and found that more angular materials had a greater weight variation. Also, Rupp [18] investigated the effects of particle shape on tablet... 

Tetrazene (C2H8N10O) is a pale yellow crystalline explosive generally used in ignition caps, where a small amount is added to the explosive composition to improve its sensitivity to percussion and friction. Tetrazene is not suitable for filling detonators because its compaction properties make the transition from burning to detonation very difficult. This primary explosive is stable in ambient temperatures. Its ignition temperature is lower and it is slightly more sensitive to impact than mercury fulminate. 

Optimal compression or compaction properties of powders are critical for a robust solid dosage form. Although prediction of compaction properties of powders is not fully... 

Commercial products vary from 25 to 45% w/w solids contents and addition levels required to produce concrete of almost self-compacting properties would be 1.0-3.0% by weight of cement. 

The typical processing conditions and powder characteristics are shown in Table 8.1.6 and Table 8.1.7, respectively. The compaction property of this powder was... 

To solve the preceding set of equations, Equation 5.62 is plugged into Equation 5.60. By separately determining the compaction properties of the fiber bed [32] an evolution equation for the pressure can be obtained. Because this is a moving boundary problem the derivative in the thickness direction can be rewritten [32] in terms of an instantaneous thickness. The pressure field can then be solved for by finite difference or finite element techniques. Once the pressure is obtained and the velocity computed, the energy and cured species conservation equations can be solved using the methodology outlined in Section 5.4.1. 

Successful scale-up of the tableting process also requires control of the raw materials used in compaction. Typically, pharmaceutical excipients vary in their physicochemical properties, which result in batch-to-batch variations. The tableting process, especially direct compression processes where there is limited raw material alteration before compaction, is susceptible to raw material variation, which may be magnified upon scale-up. Compaction science affords the ability to fingerprint raw materials, including the drug substance, to determine if the same compaction properties will be observed from batch to batch. This also allows for a rational approach for determining alternate vendor sources of the same materials. 

Compare compaction properties of different salt forms, polymorphs, or hydrates. 

Coprocessed excipients have been mainly used in DC tableting because of their better flow ability and compressibility, and the excipient formed is a filler-binder. The compressibility of several coprocessed excipients such as Cellactose (45), SMCC (42,44), and Ludipress (BASF AG, Ludwigshafen, Germany) (46) have been reported to be superior to the physical mixtures of their constituent excipients. While comparing the compressibility profile of SMCC with MCC in the presence of high compression forces, the former was found to retain the compaction properties,... 

Compaction properties Compaction simulator or instrumented tablet press ... 

Figure 2 Determination of compaction properties of Mannogem EZ. Data points are mean of duplicate trials. Curve is best-fit quadratic equation.

Compaction properties of each material were determined with a standardized test performed on a custom-built hydraulic compaction simulator using 8 mm (0.3150 in.) round flat-faced punches. A linear saw-tooth upper punch position profile was selected with a punch velocity of 300 mm/sec for both punch extension and retraction. The lower punch position was at a fixed position within the die during the compaction event. The powder weight loaded into the die for each compression was calculated from the equation below so as to form a cylindrical tablet having a thickness-to-diameter ratio of 0.30 at a theoretical SF of 1.0. These dimensions are typical of commercially elegant tablets. 

Compaction properties Compression stress at 0.85 SF (MPa) Maximum compression stress (MPa)... 

Williams RO, Sriwongjanya M, Barron MK. Compaction properties of microcrystalline cellulose using tableting indices. Drag Dev Ind Pharma 1997 23 695-704. 

Fell JT. The flow and compaction properties of lactose. Pharmaceutisch Weekblad 1976 111 (28) 681 685. 

Masaki H, Akinobu O, Fukuji H. Effect of particle size of lactose on the compaction properties of powder and tensile strength of tablets. Yakuzaigaku 1986 46(l) 50-57. 

Tomohiro Y, Forbes RT, Peter Y, Yoshiaki K. The improved compaction properties of mannitol after a moisture-induced polymorphic transition. Int J Pharma 2003 258(1-2) 121 131. 

In a demonstration of the pharmaceutical advantage that can be realized through the use of a cocrystal form of a substance, it was shown that the 1 1 cocrystal of caffeine and methyl gallate exhibited significantly improved powder compaction properties [64], The compression characteristics of the cocrystal were reported to be excellent over the entire pressure range studied, with the tablet tensile strength of the cocrystal being twice that of caffeine at pressures less than 200 MPa. The superior compaction properties of the cocrystal product were attributed to the presence of slip planes in crystal structure. 

To improve compaction properties of the powder mix Granulation is known to improve the compaction properties of powders that otherwise do not readily lead to compactable mixes. This is linked to the use of some excipients (i.e., binders) whose distribution within the granules lead to easier compaction. Most binders facilitate compression by softening and deforming plastically under load. 

For a two-level factorial design, only two excipients can be selected for each factor. However, for the filler-binder, a combination of brittle and plastic materials is preferred for optimum compaction properties. Therefore, different combinations such as lactose with MCC or mannitol with starch can count as a single factor. Experimental responses can be powder blend flowability, compactibility, blend uniformity, uniformity of dose unit, dissolution, disintegration, and stability under stressed storage conditions. The major advantage of using a DOE to screen prototype formulations is that it allows evaluation of all potential factors simultaneously, systematically, and efficiently. It helps the scientist understand the effect of each formulation factor on each response, as well as potential interaction between factors. It also helps the scientist identify the critical factors based on statistical analysis. DOE results can define a prototype formulation that will meet the predefined requirements for product performance stability and manufacturing. 

Compact Property Symbol Equation Incorporating Effect of Porosity on Compact Property... 

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