Kinetic space

Electrochemical anodization of multi-layer AI/Ta/Al thin film compositions was developed to fabrieate regular nanostructures of tantalum oxide (TaiOs). Anodization kinetics, space characteristics of TaaOs nanopillars and electrical properties of Al/TaiOs/Al structures have been studied. Al/Ta/Al thin film compositions were shown to permit formation of regular nanostructured layers appropriate for photonic crystal and nanoelectronic applications. 

With the noncompartmental model, there are two kinetic spaces of interest, the sampling or central kinetic space and the peripheral kinetic space. The average length of time that all drug molecules spend in a specific kinetic space is the mean residence time of that kinetic space. 

The mean times of these kinetic spaces can be calculated as follows when the drug is input into the central kinetic space and the drug leaving the system (body) from the peripheral kinetic space is negligible. 

A kinetic space defines fhe presence of a molecular moiety in one or more states. The states referred to are chemical (e.g., drug and metabolites) or physical states (e.g., distribution kinetic spaces) or both. 

The transit time is the time from when a molecule enters a kinetic space to when it subsequently leaves the kinetic space. 

The MTT of drug molecules in a kinetic space is the average time taken by drug molecules from entering the kinetic space to subsequently leaving the kinetic space. 

If all molecules exit irreversibly from a kinetic space, then MRT = MTT. If all the molecules enter the kinetic space at the same time (t = 0) and the drug leaves irreversibly from the kinetic space, then MTT and MRT are simply given by... 

Theorem 16.9. MRT Partitioning Consider a kinetic space partitioned in an arbitrary way into any number of mutually exclusive kinetic spaces. The MRT of the kinetic space is equal to the sum of the MRTs of the mutually exclusive kinetic spaces. Applying the MRT partition theorem to bioavailable drug molecules gives ... 

FIGURE 16.4 Arbitrary partitioning of kinetic space into mutnally exclnsive space. 

The partition theorem can also be applied to the disintegration and absorption kinetics of an oral tablet to determine the MRT of the drug in the various kinetic spaces in the drug delivery sequence (Figure 16.5A). The residence of the drug can be partitioned into four kinetic spaces (Figure 16.5) ... 

FIGURE 16.5 Partitioning of kinetic spaces in a subject receiving an oral tablet, a solution, and an IV bolus adminisUation. GI = gastrointestinal. 

Because the transfer of the drug between the kinetic spaces (1 to 3 in Figure 16.5A) is largely irreversible, the MTTs through these spaces are very similar to the MRTs. From intrasubject administrations of a drug eliminated from the disposition space at a rate proportional to the measured drug concentration, it then follows that... 

The MAT of drug molecules entering a kinetic space is the average time it takes the molecules to arrive in the kinetic space. The MAT for the absorption. 

K inCO denotes the rate of input of drug into a kinetic space and/ouj(t) denotes the rate of irreversible elimination from that space, then the MRT of the drug in the kinetic space is... 

The residence time fnnction RT(t) for a kinetic space describes the probabilitiy that a molecnle that enters the kinetic space at the arbitrary time T is present in the kinetic space at time T + t. This definition allows the drug to reenter the kinetic space any nnmber of times after the iifitial entry. The inclnsion of the arbitrary entry time T is consistent with a time-invariant system. 

Theorem 16.10 The MRT in a kinetic space is equal to the total integral of the residence time function of the kinetic space ... 

The residence time function of a kinetic space of a linear system is the UIR of the kinetic space expressed in terms of the dose-normalized amount vs. time function with respect to drug input into the kinetic space. For example, let the kinetic space be the unchanged drug molecules in the general systemic blood circulation, and let c(f) denote the systemic drug concentration following an IV bolus injection then. 

Theorem 16.11. The MTT of a kinetic space is related to the initial derivative of the residence time function. 

The mean residence number (MRN) of a kinetic space is the average number of times the drug molecules enter a kinetic space. 

Traditionally, distribution kinetics has been described in terms of volume of distribution parameters (V and K,s) and structure-specific distribution parameters, e.g., the k 2, 21 parameters of the two-compartment model. LSA offers a less-structured alternative that considers the net effect of the distribution kinetics based on the disposition decomposition analysis. For example, the partition/distribution properties of a drag may be expressed in terms of the affinity of the drag molecules to a kinetic space expressed as the MRT in the kinetic space. Accordingly, it is meaningful to consider ratios of MRTs for two kinetic spaces as a metric for the relative affinity. Thus, residence time coefficients (RTC) similar to a partition coefficient may be defined Residence Time Coefficient The residence time coefficient RTC for the distribution of a drug in two kinetic spaces is the ratio of the MRT of the drug in the two kinetic spaces. It is readily shown thaC ... 

Electronegativity (EL) and chemical hardness (HD) comprise an orthogonal 2D chemical space in which chemical reactivity can be described analogous to physical phenomena in the kinetic space of velocity and acceleration... 

Because of slow kinetics, space velocities must allow residence times of 20 to 120 min and are therefore around two to dtree volumes per volume per hour, in two or three in-line columns, two to three meters high. This set up allows bed permutation and maximum saturation in the first column (bed specific gravity of 0.42 to 0.46). 

As we have pointed out at several instances the present equations are essentially analogous to the development of suitable master equations in statistical mechanics [4-7], where the wavefunction here plays the role of suitable probability distributions. Note for instance the similarity between the reduced resolvent, based on J-[ (z), and the collision operator of the Prigogine subdynamics. The eigenvalues of the latter define the spectral contributions corresponding to the projector that defines the map of an arbitrary initial distribution onto a kinetic space obeying semigroup evolution laws, for more details we refer to Ref. [6] and the following section. 

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