Time-dependent linear distribution

Figure 11. Time-dependent linear distribution coefficients for sorption of by a-Al203, silica gel 40, and silica gel 100. Symbols with a dot at the center represent CMBR systems initiated at an aqueous-phase phenanthrene concentration of 50 ig/L open symbols represent CMBR systems initiated at 900 iig/L (Adapted from ref 51).

Colburn, W.A. A time- dependent Volume of distribution term used to describe linear concentration-time profiles. J. Pharmacoki net. Bi opha rm 11(4) 389-400, 1983. 

In both calculations, the boundary conditions are linear with respect to 0 and its first-order derivatives. The solution of the Fourier equation, with respect to the space variables, may be developed in a series of orthogonal functions, winch are exponential with respect to the time variable [for the solution of similar problems, see (45)]- The time-dependance of the temperature distribution along a single space variable r, resulting from a unit pulse, is therefore given by... 

Mahaney, J. E., Thomas, D. D. and Froehlich, J. P. The time-dependent distribution of phosphorylated intermediates in native sarcoplasmic reticulum Ca2+-ATPase from skeletal muscle is not compatible with a linear kinetic model. Biochemistry 43 4400-4416, 2004. 

The first term in Eq. (68) describes the steady-state properties of the system, as exploited by flux balance analysis to constrain the stoichiometrically feasible flux distributions. Since we consider infinitesimal perturbations only, quadratic terms in the expansion are neglected. In this case, the time-dependent behavior of an infinitesimal perturbation AS(t) = S — S° in the vicinity of S° is described by a linear differential equation... 

After the administration of a drug, its concentration in plasma rises, reaches a peak, and then declines gradually to the starting level, due to the processes of distribution and elimination (p. 46). Plasma concentration at a given point in time depends on the dose administered. Many drugs exhibit a linear relationship between plasma concentration and dose within the therapeutic range (dose-linear kinetics (A) note different scales on ordinate). However, the same does not apply to drugs whose elimination processes are already sufficiently activated at therapeutic plasma levels so as to preclude further proportional increases in the rate of elimination when the concentration is increased further. Under these conditions, a smaller proportion of the dose administered is eliminated per unit of time. 

Using time-resolved crystallographic experiments, molecular structure is eventually linked to kinetics in an elegant fashion. The experiments are of the pump-probe type. Preferentially, the reaction is initiated by an intense laser flash impinging on the crystal and the structure is probed a time delay. At, later by the x-ray pulse. Time-dependent data sets need to be measured at increasing time delays to probe the entire reaction. A time series of structure factor amplitudes, IF, , is obtained, where the measured amplitudes correspond to a vectorial sum of structure factors of all intermediate states, with time-dependent fractional occupancies of these states as coefficients in the summation. Difference electron densities are typically obtained from the time series of structure factor amplitudes using the difference Fourier approximation (Henderson and Moffatt 1971). Difference maps are correct representations of the electron density distribution. The linear relation to concentration of states is restored in these maps. To calculate difference maps, a data set is also collected in the dark as a reference. Structure factor amplitudes from the dark data set, IFqI, are subtracted from those of the time-dependent data sets, IF,I, to get difference structure factor amplitudes, AF,. Using phases from the known, precise reference model (i.e., the structure in the absence of the photoreaction, which may be determined from... 

Thus we have found the distribution of the fluctuations around the macroscopic value. They have been computed to order Q 1/2 relative to the macroscopic value n, which will be called the linear noise approximation. In this order of Q the noise is Gaussian even in time-dependent states far from equilibrium. Higher corrections are computed in X.6 and they modify the Gaussian character. However, they are of order 2 1 relative to n and therefore of the order of a single molecule. 

Figure 6.2. Relations between shear stress, deformation rate, and viscosity of several classes of fluids, (a) Distribution of velocities of a fluid between two layers of areas A which are moving relatively to each other at a distance x wider influence of a force F. In the simplest case, F/A = fi(du/dx) with ju constant, (b) Linear plot of shear stress against deformation, (c) Logarithmic plot of shear stress against deformation rate, (d) Viscosity as a function of shear stress, (e) Time-dependent viscosity behavior of a rheopectic fluid (thixotropic behavior is shown by the dashed line). (1) Hysteresis loops of time-dependent fluids (arrows show the chronology of imposed shear stress).

Linear response theory is reviewed in Section II in order to establish contact between experiment and time-correlation functions. In Section III the memory function equation is derived and applied in Section IV to the calculation of time-correlation functions. Section V shows how time-correlation functions can be used to guess time-dependent distribution functions and similar methods are then applied in Section VI to the determination of time-correlation functions. In Section VII a succinct review is given of other exact and experimental calculations of time-correlation functions. 

This latter expression has been used to simplify KD(t)- Note that the time dependences of the linear and angular momentum autocorrelation functions depend only on interactions between a molecule and its surroundings. In the absence of torques and forces these functions are unity for all time and their memories are zero. There is some justification then for viewing these particular memory functions as representing a molecule s temporal memory of its interactions. However, in the case of the dipolar correlation function, this interpretation is not so readily apparent. That is, both the dipolar autocorrelation function and its memory will decay in the absence of external torques. This decay is only due to the fact that there is a distribution of rotational frequencies, co, for each molecule in the gas phase. In... 

The buildup of the H2 concentration, for any given depth x, starts with all its time derivatives zero at t = 0, increases gradually, and after a depth-dependent induction time becomes linear in t. The unbounded growth can be truncated by allowing the molecules either to dissociate or to diffuse. Dissociation will of course modify the development of the H° distribution molecular diffusion will not. As regards dissociation, there are to date no time-dependent solutions for this problem available presumably if the molecules are immobile, they would show an approach to a flat thermal-equilibrium distribution, which would extend to deeper depths at longer times. The case of diffusion without dissociation will be taken up in the paragraphs to follow. 

Many variables used and phenomena described by fracture mechanics concepts depend on the history of loading (its rate, form and/or duration) and on the (physical and chemical) environment. Especially time-sensitive are the level of stored and dissipated energy, also in the region away from the crack tip (far held), the stress distribution in a cracked visco-elastic body, the development of a sub-critical defect into a stress-concentrating crack and the assessment of the effective size of it, especially in the presence of microyield. The role of time in the execution and analysis of impact and fatigue experiments as well as in dynamic fracture is rather evident. To take care of the specihcities of time-dependent, non-linearly deforming materials and of the evident effects of sample plasticity different criteria for crack instability and/or toughness characterization have been developed and appropriate corrections introduced into Eq. 3, which will be discussed in most contributions of this special Double Volume (Vol. 187 and 188). 

Tissue uptake clearance is a useful parameter to characterize the in vivo distribution properties of a drag since it is independent of the drag concentration in plasma. However, when the tissue uptake process depends on the drag concentration in plasma and follows non-linear kinetics, the calculated CZapp i represents an average value of its time-dependent clearance for the overall experimental period (Nishikawa et al., 1992). 

Figures 16.38 and 16.39 demonstrate that the form of the particle size distributions is once again almost constant during the process time, and consequently the pneumatic recycled dust is not used for seed production. Dust is deposited on the particles because the nozzle position is close to the dust recycle tube (uniform wetted dust), and this leads to an enlarged particle growth. The measured time-dependent gas outlet temperature and the measured time-dependent conversion corresponds with simulations (Fig 16.40). The bed mass growth is linear at constant liquid injection rates (Fig. 16.41). The change in particle size distribution value and of the Sauter diameter is, again, declining.

Model uncertainty is principally based upon (1) modelling errors (i.e. non-consideration of parameters) and (2) relation (dependency) errors (i.e. drawing incorrect conclusions from correlations). For example, the mathematical formula of a model describing the concentration-time curve of a substance emitted from, for example, glue should consider an algorithm that describes the time-dependent increase of the substance in a room, its distribution and the decrease of the concentration. Body weight correlates non-linearly with... 

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