Linearity related substances

In general, therefore, the surface flux of each substance is linearly related Co all the concentration gradients in the adjacent bulk phase. The coefficients in this linear relation depend on the bulk phase concentrations,... 

It is not a trivial point that 0fj vs. E curves are practically linear. In a reversible system the electrode potential can be linked to the activities (concentrations) of the potential-determining substances. In the system being discussed, this substance is atomic hydrogen. According to the Nemst equation we have E = const - (RTIF) X In Cjj. It follows that the degree of coverage, 0, is linearly related to the logarithm of concentration c in the solution ... 

According to Stevens law the logarithm of the perceived intensity is linearly related to the logarithm of the odour intensity. In the figure this relationship is given for two substances, one with a slope of 1.00 and one with a slope of. 67. As can be seen from the figure, this means that an odour concentration of 100 odour units/m3 is related to very different perceived odour intensities for the two substances. This means that odour concentrations computed in odour units/m3 should not be used as an indication of perceived odour intensity, but can only be used in relative measurements where the effects of measures taken to reduce odour pollution are compared, or in studies where dispersion models are used to find the distance to the source at which threshold is reached. 

If a linearity curve (Figure 3.1) is constructed for both the related substance and the drug substance by plotting the response versus the concentration, the relative response factor can also be determined by... 

Sample Concentration (Method Sensitivity). To maintain linearity at the concentration range of the drug substance, scientists may try to lower the sample concentration to improve peak shape for the drug substance. However, if the sample concentration is too low, it will affect the method sensitivity, and the ability to detect low levels of related substances may not be adequate. 

Reduced Linear Range. Unlike the area percent and high-low methods, which use the response of the drug substance in sample injections for calculation, an external standard method uses a standard curve. Typically, the concentration range of the calibration curve is similar to that of related substances in the sample (e.g., 1 to 5% of the nominal sample concentration). Therefore, this method requires a small linear range. 

Case 1. Linearity demonstrated from 50% of the ICH reporting limit to a nominal concentration of drug substance in the sample solution. In addition, no significant v-in(creep is observed (Figure 3.6). In this case, area percent calculation is suitable because the linearity range covers the responses of related substances and that of the drug substance in the sample solution. Therefore, these responses can be used directly to calculate the area percentage of each related substance. 

Concentration (% related substance) Figure 3.6. Linearity case 1. 

Case 3. Linearity demonstrated from 50% of the ICH reporting limit to 150% of the shelf life specification of a related substance, and a significant y-intercept is observed (Figure 3.8). Due to the significant -intercept, a single-point calibration (e.g., high-low or one-point external standard calibration) is not suitable. In this case, multiple-point external standard calibration is the most appropriate. See Section 3.3.3 for more discussion of the significant y-intercept. 

Range. Ideally, linearity should be established from 50% of the ICH reporting limit to the nominal concentration of drug substance in the sample solution (for area percent method). If the linearity does not support such a wide range of concentration, determine the linearity from 50% of the ICH reporting level to 150% of the proposed shelf life specifications of the related substance (for the high-low and external standard methods) as a minimum. This will ensure a linear response for related substances at all concentration levels to be detected during stability. 

Experimental Requirements. Solutions of known concentrations are used to determine the linearity. A plot of peak area versus concentration (in percent related substance) is used to demonstrate the linearity. Authentic samples of related substances with known purity are used to prepare these solutions. In most cases, for the linearity of a drug product, spiking the related substance authentic sample into excipients is not necessary, as the matrix effect should be investigated in method accuracy. 

Different Approaches for Linearity Determination. The first approach is to weigh different amounts of authentic sample directly to prepare linearity solutions of different concentrations. Since solutions of different concentration are prepared separately from different weights, if the related substances reach their solubility limit, they will not be completely dissolved and will be shown as a nonlinear response in the plot. However, this is not suitable to prepare solutions of very low concentration, as the weighing error will be relatively high at such a low concentration. In general, this approach will be affected significantly by weighing error in the preparation. 

Another approach is to prepare a stock solution of high concentration, then perform serial dilution from the stock solution to obtain solutions of lower concentrations for linearity determination. This is a more popular approach, as serial dilution can be used to prepare solutions of very low concentrations. Since the low concentrations are prepared by serial dilution, this approach does not need to weigh a very small quantity of related substance. In addition, since all solutions are diluted from the same stock solution, weighing error in preparing the stock solution will not affect the linearity determination. 

Intrinsic Accuracy. Intrinsic accuracy indicates the bias caused by sample matrix and sample preparation. In this approach, a stock solution is prepared by using known quantities of related substance and drug substance. The stock solution is further diluted to obtained solutions of lower concentrations. These solutions are used to generate linearity results. In addition, these linearity solutions of different concentrations are spiked into placebo. The spiked solutions are prepared according to the procedure for sample analysis. The resulting solutions, prepared from the spiked solution, are then analyzed. If the same stock solution is used for both linearity and accuracy and all of these solutions are analyzed on the same HPLC run, the response of linearity (without spike into matrix) and accuracy (with spike into matrix) can be compared directly. Any differences in response indicate the bias caused by matrix interference or sample preparation. To determine the intrinsic accuracy at each concentration level, one can compare the peak area of accuracy (with matrix) with that of linearity (without matrix) at the same concentration (Figure 3.11). This is the simplest approach, and one would expect close to 100% accuracy at all concentration levels. 

Typically, linearity and accuracy determination covers a wide concentration range (e.g., 50% of the ICH reporting limit to 150% of specification). However, the concentration range for precision will be limited by the availability of sample of different related substance levels. Therefore, to ensure an appropriate method validation range with respect to precision, it is critical to use samples of low and high levels of related substance in precision experiments (e.g., fresh and stressed samples). 

At first sight, there seems to be a basic difference between the two regimes with respect to the influence of Kia/Vl. In the water-phase-controlled regime, the overall exchange velocity, via/w, is independent of Kia/v/, whereas in the air-phase controlled regime v(a/w is linearly related to Ga/w. Yet, this asymmetry is just a consequence of our decision to relate all concentrations to the water phase. In fact, for substances with small Kia/v/ values, the aqueous phase is not the ideal reference system to describe air-water exchange. This can be best demonstrated for the case of exchange of water itself (Kia/V1 = 2.3 x 10 5 at 25°C), that is, for the evaporation of water. 

One essential assumption is that all substances experience the same film thickness. Therefore, the model predicts that for given conditions the exchange velocities of different compounds, i and j, should be linearly related to their molecular diffusivities ... 

At a high solubility of the solid in the liquid phase, account must be taken of a variation of the volume of their solution with passing time, due to the transition of the dissolving substance into the bulk of the liquid. For most systems, an exact dependence of the volume of the solution upon its concentration is not known. In such cases, use is made of various approximate methods of determining this dependence. Usually, satisfactory results may be obtained assuming a linear relation between the solution volume and the concentration of the dissolved substance... 

If this linear relation between the chemical potential and the logarithm of the molar fraction of i holds valid in the whole concentration range extending from x, = 0 to xf = 1, the unitary part of the chemical potential (r,p) is identical with the chemical potential Li (T,p) of the pure substance i. The linear relation of Eq. 5.22, however, is not necessarily valid over the whole range of concentrations but in the range of dilute concentrations only. In such a case the unitary part of pi (T,p) is usually set at the value estimated by extrapolation from the dilute concentration range to the mole fraction of xt - 1. 

Two cases then arise with respect to the ideality of mixtures One is the case in which the mixture is ideal for all values of x and for all constituent substances. This type of mixture is thermodynamically called the perfect mixture, for which the Raoult s law (a linear relation between pt and In x. in the whole range of concentrations) holds valid and in which the unitary chemical potential pi (T,p) of i equals the chemical potential pi, T, p) of pure substance i for all the substances in the system as shown in Eq. 5.24 ... 

In this case the unitary value of the chemical potential of solute substance i can be estimated, as mentioned above, by extrapolating the chemical potential of dilute constituent i to xt = 1 from the dilute concentration range in which the linear relation of Eq. 5.22 holds. 

Sutoo et al. developed a high sensitivity and high linearity fluorescence microphotometry system for distribution analysis of neurotransmitter and related substances in the small brain regions [42]. The method makes use of the fluorescence intensity of not more than 10,000 points in animal brain slices, which are immunohistochemically and histochemically stained. 

The external forces may be optical tweezers, microneedles, or the viscous load of the substance that is carried. These generalized forces create motion, characterized by an average velocity v, and average rate of ATP consumption JT. Molecular motors mostly operate far from equilibrium, and the velocity and rate of ATP consumptions are not linear functions of the forces. However, in the vicinity of the linear region, where A kBT, linear relations hold... 

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