Cumulant analysis

The basic elements and considerations for assay development, validation, and specification assignment are reviewed briefly. Assay development produces a method that requires validation for the analysis and release of materials (bulk or formulated finished product) for use in clinical development. The cumulative analysis of materials and stability considerations is then used to established specifications for internal and regulatory submission. 

Equation (2.5) is a stochastic differential equation. Some required characteristics of stochastic process may be obtained even from this equation either by cumulant analysis technique [43] or by other methods, presented in detail in Ref. 15. But the most powerful methods of obtaining the required characteristics of stochastic processes are associated with the use of the Fokker-Planck equation for the transition probability density. 

A. N. Malakhov, Cumulant Analysis of Random Non-Gaussian Processes and Its Transformations, Sovetskoe Radio, Moscow, 1978, in Russian. 

The logic of the above form of gi(sttd) and additional details are available in advanced books on DLS, and the above description is meant only to illustrate the basic ideas and one data-analysis approach. The cumulant analysis is often used as a first step before more advanced analytical procedures (each of which has its own advantages and disadvantages) are attempted. Most DLS instruments come with computer programs for the analysis of the size distribution, but we should bear in mind that each analysis technique has specific, and often restrictive, assumptions and none is exact. As a consequence, the results of size distributions from DLS are best interpreted as semiquantitative indicators of polydispersity rather than a true representation of the distribution. 

Structure of aggregates probed by small-angle scattering Effective diameter of an enzyme from DLS Cumulant analysis of DLS data... 

A statistical analysis of light-scattering data can compensate for polydispersity. In cumulant analysis, lng(z) is expanded in a power series and coefficients of the different terms are evaluated against the experimentally obtained t, in search of the closest-fitting average selected by the smallness of the standard deviation. In a histogram method, the experimental t is... 

Since the PS reference sample is almost monodisperse, a cumulant analysis of that material would yield a very small Q, say Q < 0.03. That is, all the correction terms are negligible and Eqs. (17) collapse to Eqs. (12). But cumulant analysis is a useful way to handle practical samples such as pigments, inks, microemulsions, swollen micelles, globular proteins, and spherical virus particles, where there is a size distribution but one that is not very broad (say Q < 0.3). This analysis should be made for the milk data using a non-linem teast-squares fitting of Eq. (17a), neglecting /1.3 and all higher order terms. Report the F, D, and R values as well as the second cumulant /t2 aiid the polydispersity index Q. 

Software for cumulant analysis from Brookhaven Instruments was used to fit the measured correlation function C(f) to equation 9 by using a weighted second-order polynomial nonlinear regression. The measured base line B of the correlation function C t) was determined from the average of four delay channels (1029-1032) multiplied by the sample time Af. The calculated and measured base lines were within 0.1% for all runs used in the analysis. 

Figure 30. Effect of extreme static disorder in EXAFS analysis. Left Gaussian distributions of equal area but with much different widths. Both the sum of the two functions and the single narrow distribution function are shown. Right Simulated EXAFS functions for the functions at left. It is seen that there is no detectable difference in the EXAFS except at low k-values. This difference would overlap with XANES and be extremely difficult to analyze. Hence physical distributions with a broad tail will have re duced coordination numbers via standard EXAFS analysis, as well as an artificially produced distance contraction . For cases not as severe as this, cumulate analysis can quantify the degree of static disorder and allow more correct results. After Kortright et al. (1983).

Correlate the raw data to Z average mean size using a cumulative analysis by the Zetasizer 3000HS software package. 

Fig. lO.a The inset shows the postulated variation of the solubility parameter 8 caused by deuterium labeling (symbols and V correspond to labeled and nonlabeled copolymers, respectively) and due to the change in ethyl ethylene fraction x. The cumulative analysis, described in text, yields the absolute 8 value for deuterated dx (A) and protonated hx (V) copolymers as a function of x at a reference temperature Tref=100 °C determined interaction parameters (as in Fig. 9) allow us to determine two sets of differences AS adjusted here to fit independent PVT data [140,141] measured at 83 °C ( ) and at 121 °C (O). b The interaction parameter, yE/EE, arising from the microstructural difference contribution to the overall effective interaction parameter (hxj/dxpej) in Eq. (19) as a function of the average blend composition (xi+Xj)/2 at a reference temperature of 100 °C.%E/ee values are calculated (see text) from coexistence data ( points correspond to [91,143] and O symbols to [136]) for blend pairs, structurally identical but with swapped labeled component. X marks %e/ee yielded directly [134] for a blend with both components protonated. Solid line is the best fit to data... 

Microemulsions were characterized by Dynamic Light Scattering (DLS) to determine the size distribution of the water pools. Measurements were performed on a Malvern Zetasizer. Interactions existing between aqueous droplets were neglected which is a prerequisite to use the hard sphere model. The linear mode of the apparatus was chosen this corresponds to a cumulative analysis. 

For a narrow particle size distribution the cumulant analysis is usually satisfactory. The cumulant method is based on the assumption that, for monodisperse suspensions gi(r) is monoexponential. Hence, the log of gi(r) versus t yields a straight line with a slope equal to F,... 

Unfortunately, in a typical NSE experiment the number of points obtained as a representation of this function is in most cases too small and the error of the individual points is too high to allow for a fit with more than two or three adjustable parameters or for an analysis using Laplace transformation - and maximum entropy methods [56]. Without information from additional experiments it is only possible to compute an effective diffusion coefficient from the data by using a first- or second-order cumulant analysis [57]. 

The cumulative analysis of experimental data allows estimation of the quality of prepared compositions and various mixing duration and intensity. The experimental data on modeling dependences make it possible to optimize the specified... 

Polydispersity of simple bile salt micelles can only be assessed by modem QLS techniques employing the 2nd cumulant analysis of the time decay of the autocorrelation function [146,161]. These studies have shown, in the cases of the 4 taurine conjugates in 10 g/dl concentrations in both 0.15 M and 0.6 M NaCl, that the distribution in the polydispersity index (V) varies from 20% for small n values to 50% for large n values [6,146]. Others [112] have foimd much smaller V values (2-10%) for the unconjugated bile salts in 5% (w/v) solutions. Recently, the significance of QLS-derived polydispersities have been questioned on the basis of the rapid fluctuation in n of micellar assemblies hence V may not actually represent a micellar size distribution [167-169]. This argument is specious, since a micellar size distribution and fast fluctuations in aggregation number are identical quantities on the QLS time scale (jusec-msec) [94]. 

Particle-size distribution for powder (a) differential analysis (b) cumulative analysis. 

Calculations of average particle size, specific surface area, or particle population of a mixture may be based on either a differential or a cumulative analysis. In principle, methods based on the cumulative analysis are more precise than those based on the differential analysis, since when the cumulative analysis is used, the assumption that all particles in a single fraction are equal in size is not needed. The accuracy of particle-size measurements, however, is rarely great enough to warrant the use of the cumulative analysis, and calculations are nearly always based on the differential analysis. 

In view of McWhirter and Pike s work, noise will limit the number of cumulants that we can actually measure. In most cases, only (E) and p.2 t)c determined. Cumulants are used for monomodal size distributions that have a PI not larger than 0.3. In bimodals or other more complex distributions, cumulant analysis will be meaningless and will only give some type of rough screening of the data. Even more, T>app( )> which is what is usually determined, is also dependent on the type of correlator, whether linear or nonlinear correlator, and the number of channels used [55]. 

This would mean that the particles on the 170 mesh screen lie between 88 p,m and 125 p,m. This representation of the data is referred to as a differential analysis. In a cumulative analysis, the particles retained on each screen are summed sequentially starting from either the receiver or the screen with the largest aperture. 

TEM observation confirmed the formation of spherical micelles through the irradiation. The TEM image of the micelles is shown in Figure 2-4. The diameter of the micelles was estimated to average 40.6 nm based on the TEM. Compared to the micellar size determined by the cumulant analysis, the TEM exhibited a smaller diameter of the micelles than the dynamic light scattering. The estimation of the micelles as the smaller size can be accoimted for by the fact that the micelles in the solution contracted when isolated in air. 

Meta-analysis is a detailed, cumulative analysis of data from several similar studies, which is usually based on scientific publications. One of its many advantages is that the assessment or testing of any hypothesis can be carried out on a much larger sample set than in any single tests. It is widely used both in chnical and epidemiological studies. 

The uninitiated reader may wish to consult books on polydispersity analysis (30,31). In practice, one can use a fast but limited cumulants analysis to obtain the average line width and relative width /u-2/ of G(F) (20), wherein [G Hq, t) - A]/A is expanded as... 

By the cumulant analysis the intensity weighted hydrodynamic diameter (z-average) and a polydispersity index (PDI, a parameter for the width of the size distribution) are obtained. These values give, however, just an "overview" over the whole sample. For samples with a very broad size distribution (PDI > 0.5), the z-average does not reliably represent the size of the nanoparticles and interpretation is often difficult or even meaningless. SLN dispersions normally possess relatively broad size distributions with PDI values between 0.15-0.30. By complex calculations (e.g. using the Contin algorithm), the size distribution (by intensity, volume or number) can be estimated. For calculations of the size distribution, the optical properties of the particles and the dispersant (refractive indices, absorption) have to be known or estimated. 

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