Bracketing

The factors that may be bracketed in a stability study are outlined in ICF1Q1A and described in further detail below. The types of drug products and the types of submissions to which bracketing design can be applied are also discussed. 

Bracketing design is applicable to most types of drug products, including immediate- and modified-release oral solids, liquids, semisolids, and injectables. Certain types of drug products, such as MDIs, DPIs, and transdermal delivery systems, may not be amenable to, or may need additional justification for, bracketing design. 

If the formulation is significantly different among the different strengths (e.g., addition or deletion of an excipient, except colorant or flavoring), bracketing is generally not applicable. 

Because of the complexity in product formulation, applicants are advised to consult the appropriate chemistry review team in advance when questions arise in the above situations or where justification is needed for bracketing design. In the case in which the strength and the container or fill size of a drug product both vary, bracketing design may be applicable if justified. 

The stability data obtained under a bracketing protocol should be subjected to the same type of statistical analysis described in Section VII.E. The same principle and procedure on poolability should be applied (i.e., testing data from different batches for similarity before combining them into one overall estimate). If the statistical assessments of the extremes are found to be dissimilar, the intermediate sizes or strengths should be considered to be no more stable than the least stable extreme. 

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For condensable components, we use the symmetric normaliza-L as x - 1 therefore, the quantity in brackets is the fugacity of pure liquid i at system temperature and pressure. 

Sum given by first term in brackets in Equation (A-49). Sum given by bracketed term in Equation (A-44). 

Bracketed figures are for the most commonly available isotope in the case of artificial elements. 

Cranking Simulator), by a pumpability temperature limit measured by a rotating mini viscometer, and by the minimum kinematic viscosity at 100°C. The five summer grades are defined by bracketing kinematic viscosities at 100°C. 

Under this programme 25% of the stiffeners and adjacent plate material in bulk carriers will be examined with thickness measurements being taken at upper, lower and mid points in both face and web plates of the stiffeners and associated end brackets. The extent of thickness measurement applied to transverse bulkheads has also been increased. In addition. Classification has revised the scantling requirements for new buildings to nominate thicker material in the bottom structure and at the transverse watertight bulkheads of the No 1 cargo holds. 

This can be illustrated by showing the net work involved in various adiabatic paths by which one mole of helium gas (4.00 g) is brought from an initial state in whichp = 1.000 atm, V= 24.62 1 [T= 300.0 K], to a final state in whichp = 1.200 atm, V= 30.7791 [T= 450.0 K]. Ideal-gas behaviour is assumed (actual experimental measurements on a slightly non-ideal real gas would be slightly different). Infomiation shown in brackets could be measured or calculated, but is not essential to the experimental verification of the first law. 

The factor in wavy brackets is obviously an exact differential because the coefficient of d9 is a fiinction only of 9 and the coefficient of dVis a fiinction only of V. (The cross-derivatives vanish.) Manifestly then... 

For the quantum mechanical case, p and Ware operators (or matrices in appropriate representation) and the Poisson bracket is replaced by the connnutator [W, p] If the distribution is stationary, as for the systems in equilibrium, then Bp/dt = 0, which implies... 

A stationary ensemble density distribution is constrained to be a functional of the constants of motion (globally conserved quantities). In particular, a simple choice is pip, q) = p (W (p, q)), where p (W) is some fiinctional (fiinction of a fiinction) of W. Any such fiinctional has a vanishing Poisson bracket (or a connnutator) with Wand is thus a stationary distribution. Its dependence on (p, q) through Hip, q) = E is expected to be reasonably smooth. Quanttun mechanically, p (W) is die density operator which has some fiinctional dependence on the Hamiltonian Wdepending on the ensemble. It is also nonnalized Trp = 1. The density matrix is the matrix representation of the density operator in some chosen representation of a complete orthononnal set of states. If the complete orthononnal set of eigenstates of die Hamiltonian is known ... 

The CS pressures are close to the machine calculations in the fluid phase, and are bracketed by the pressures from the virial and compressibility equations using the PY approximation. Computer simulations show a fluid-solid phase transition tiiat is not reproduced by any of these equations of state. The theory has been extended to mixtures of hard spheres with additive diameters by Lebowitz [35], Lebowitz and Rowlinson [35], and Baxter [36]. 

The brackets symbolize fiinction of, not multiplication.) Smce there are only two parameters, and a, in this expression, the homogeneity assumption means that all four exponents a, p, y and S must be fiinctions of these two hence the inequalities in section A2.5.4.5(e) must be equalities. Equations for the various other thennodynamic quantities, in particular the singidar part of the heat capacity Cy and the isothemial compressibility Kp may be derived from this equation for p. The behaviour of these quantities as tire critical point is approached can be satisfied only if... 

The linear response of a system is detemiined by the lowest order effect of a perturbation on a dynamical system. Fomially, this effect can be computed either classically or quantum mechanically in essentially the same way. The connection is made by converting quantum mechanical conmuitators into classical Poisson brackets, or vice versa. Suppose tliat the system is described by Hamiltonian where denotes an... 

An identical expression holds classically [32] if- //) times the conmuitator is replaced by the classical Poisson bracket. 

We note that the expression in brackets is just the b c tensor element of the electronic polarizability in the ground electronic state,, (ttj)- Thus... 

In an earlier section, measurements were described in which the equilibrium constant, K, for bimolecular reactions involving gas-phase ions and neutral molecules were detennined. Another method for detemiining the proton or other affinity of a molecule is the bracketing method [ ]. The principle of this approach is quite straightforward. Let us again take the case of a proton affinity detemiination as an example. In a reaction... 

The two factors on the right are both positive, real numbers less than one. If the magnitudes of U(h and U h ) are both close to one, therefore, the magnitude of the difference between the temis within the brackets on the left (complex numbers in general) must be small. 

Here we have adopted a Dirac bracket notation which should be distmguished from the ensemble... 

These methods try to bracket the transition state from both the reactant and the product side [72, 73]. For example, in the method of Dewar etal [73], two stmctiires, one in the reactant valley and one hi the product valley, are optimized simultaneously. The lower-energy stmcture is moved to reduce the distance separating the two stmctures by a small amount, e.g. by 10%, and its stmcture is reoptimized under the constraint that the distance is fixed. This process is repeated until the distance between the two stmctures is sufficiently small. 

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