Stochastic Considerations

The dissolution process can be interpreted stochastically since the profile of the accumulated fraction of amount dissolved from a solid dosage form gives the probability of the residence times of drug molecules in the dissolution medium. In fact, the accumulated fraction of the drug in solution, q (t) /goo, has a statistical sense since it represents the cumulative distribution function of the random variable dissolution time T, which is the time up to dissolution for an individual drug fraction from the dosage form. Hence, q (t) /q can be defined statistically as the probability that a molecule will leave the formulation prior to t, i.e., that the particular dissolution time T is smaller than t  

Since g (t) /goo is a distribution function, it can be characterized by its statistical moments. The first moment is defined as the mean dissolution time (MDT) and corresponds to the expectation of the time up to dissolution for an individual drug fraction from the dosage form  

Since the fundamental rate equation of the diffusion layer model has the typical form of a first-order rate process (5.1), using (5.4) and (5.14), the MDT is found equal to the reciprocal of the rate constant k  

As a matter of fact, all dissolution studies, which invariably rely on (5.1) and do not make dose considerations, utilize (5.15) for the calculation of the MDT. However, the previous equation applies only when the entire available amount 

In fact, it will be shown below that MDT is dependent on the dose-solubility ratio if one takes into account the dose qo actually utilized [90]. Also, it will be shown that the widely used (5.15) applies only to a special limiting case. Multiplying both parts of (5.1) by V/qo (volume of the dissolution medium/actual dose), one gets the same equation in terms of the fraction of the actual dose of drug dissolved, p(t) = q(t) /qo  

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Stochastic Considerations on Chromatographic Dispersion, J. C. Giddings, J. Chem. Phys., 26, 169 (1957). 

For both cases, the retention-time distribution functions fev (t) and fiv (t) are similar to the input functions vev (t) and Viv (t), respectively, defined for the deterministic models. The only difference is that in the stochastic consideration, the drug amounts are not included is these input functions. 

The reaction of the proton with its emitter—when both are confined in a cavity—should not, a priori, be treated by classical kinetic formalism, but according to stochastic considerations. The short observation time practically isolates the measured site from the bulk, thus the number of the reactants in the observed space is an integer and the dynamics should be treated as a probability function. 

Stochastic considerations are usually not of primary concern in reactor work because of the fact that the neutron number is large in all power reactors. Taking Njl as the rate of loss of neutrons and also the rate of production of neutrons in a critical reactor, leads to Njvl as the rate of fissions. Using v = 2.5 and the fact that about 3 x fissions per second give one watt of power, it follows that... 

It is very important to make classification of dynamic models and choose an appropriate one to provide similarity between model behavior and real characteristics of the material. The following general classification of the models is proposed for consideration deterministic, stochastic or their combination, linear, nonlinear, stationary or non-stationary, ergodic or non-ergodic. 

Hamiltonian, but in practice one often begins with a phenomenological set of equations. The set of macrovariables are chosen to include the order parameter and all otlier slow variables to which it couples. Such slow variables are typically obtained from the consideration of the conservation laws and broken synnnetries of the system. The remaining degrees of freedom are assumed to vary on a much faster timescale and enter the phenomenological description as random themial noise. The resulting coupled nonlinear stochastic differential equations for such a chosen relevant set of macrovariables are collectively referred to as the Langevin field theory description. 

The long term behavior of any system (3) is described by so-called invariant measures a probability measure /r is invariant, iff fi f B)) = ft(B) for all measurable subsets B C F. The associated invariant sets are defined by the property that B = f B). Throughout the paper we will restrict our attention to so-called SBR-measures (cf [16]), which are robust with respect to stochastic perturbations. Such measures are the only ones of physical interest. In view of the above considerations about modelling in terms of probabilities, the following interpretation will be crucial given an invariant measure n and a measurable set B C F, the value /r(B) may be understood as the probability of finding the system within B. 

The Separation Stage. A fundamental quantity, a, exists in all stochastic separation processes, and is an index of the steady-state separation that can be attained in an element of the process equipment. The numerical value of a is developed for each process under consideration in the subsequent sections. The separation stage, which in a continuous separation process is called the transfer unit or equivalent theoretical plate, may be considered as a device separating a feed stream, or streams, into two product streams, often called heads and tails, or product and waste, such that the concentrations of the components in the two effluent streams are related by the quantity, d. For the case of the separation of a binary mixture this relationship is... 

Special considerations are required in estimating paraimeters from experimental measurements when the relationship between output responses, input variables and paraimeters is given by a Monte Carlo simulation. These considerations, discussed in our first paper 1), relate to the stochastic nature of the solution and to the fact that the Monte Carlo solution is numerical rather than functional. The motivation for using Monte Carlo methods to model polymer systems stems from the fact that often the solution... 

The quantities AUMC and AUSC can be regarded as the first and second statistical moments of the plasma concentration curve. These two moments have an equivalent in descriptive statistics, where they define the mean and variance, respectively, in the case of a stochastic distribution of frequencies (Section 3.2). From the above considerations it appears that the statistical moment method strongly depends on numerical integration of the plasma concentration curve Cp(r) and its product with t and (r-MRT). Multiplication by t and (r-MRT) tends to amplify the errors in the plasma concentration Cp(r) at larger values of t. As a consequence, the estimation of the statistical moments critically depends on the precision of the measurement process that is used in the determination of the plasma concentration values. This contrasts with compartmental analysis, where the parameters of the model are estimated by means of least squares regression. 

A general theory of the equilibrium polycondensation of an arbitrary mixture of monomers, described by the FSSE model, has been developed [75]. Proceeding from rigorous thermodynamic considerations a branching process has been indicated which describes the chemical structure of condensation polymers and expressions have been derived which relate the probability parameters of this stochastic process to the thermodynamic parameters of the FSSE model. 

The new edition of Principles of Electrochemistry has been considerably extended by a number of new sections, particularly dealing with electrochemical material science (ion and electron conducting polymers, chemically modified electrodes), photoelectrochemistry, stochastic processes, new aspects of ion transfer across biological membranes, biosensors, etc. In view of this extension of the book we asked Dr Ladislav Kavan (the author of the section on non-electrochemical methods in the first edition) to contribute as a co-author discussing many of these topics. On the other hand it has been necessary to become less concerned with some of the classical topics the details of which are of limited importance for the reader. 

In the stochastic approach, the Markovian random process is usually used for the description of the solvent, and it is assumed that the velocity relaxation is much faster than the coordinate relaxation.74 Such a description is applicable at long time intervals which considerably exceed the characteristic times of the electron... 

Equation (7.24) can be compared with the corresponding prescribed-diffusion equation, namely d(N)/dt = (l/2)pi(t)(N)((N) - 1) (Clifford et al., 1982a). These two equations would be equivalent if (N2) = (N)2—that is, the variance of N would be zero. This implies that all spurs would have exactly the same number of radicals at a given time. Since stochasticity denies this, a considerable difference is expected between the results of these two methods however, this difference tends to decrease with the spur size Ng (Clifford et al, 1982a Pimblott and Green, 1995). 

The stochastic tools used here differ considerably from those used in other fields of application, e.g., the investigation of measurements of physical data. For example, in this article normal distributions do not appear. On the other hand random sums, invented in actuary theory, are important. In the first theoretical part we start with random demand and end with conditional random service which is the basic quantity that should be used to decide how much of a product one should produce in a given period of time. 

The WATS model is formulated in deterministic terms. However, an extension to include simple Monte-Carlo stochastic simulation is possible, taking into consideration a measured variability of the process parameters. 

As pointed out by Mikhail, both functional and stochastic models must be considered together at all times, as there may be several possible combinations, each representing a possible mathematical model. The functional model describes the physical events using an intelligible system, suitable for analysis. It is linked to physical realities by measurements that are themselves physical operations. In simpler situations, measurements refer directly to at least some elements of the functional model. However, it is not necessary, and often not practical, that all the elements of the model be observable. That is, from practical considerations, direct access to the system may not be possible or in some cases may be very poor, making the selection of the measurements of capital importance. 

The full characterization of the stochastic properties of a surface requires consideration of higher order correlations of the height function. However, it can be difficult to construct surfaces in this manner without experimental input. As an approximation, it may be reasonable to neglect the higher order terms. 

The logic and rigor of statistics also reside at the heart of modern science. Statistical evaluation is the first step in considering whether a body of measurements allows one to discriminate among rival models for a biochemical process. Beyond their value in experimental sciences, probabilistic considerations also help us to formulate theories about the behavior of molecules and particles and to conceptualize stochastic and chaotic events. 

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