Retention-Time Distribution Models

A stochastic model may also be defined on the basis of its retention-time distributions. In some ways, this conceptualization of the inherent chance mechanism is more satisfactory since it relies on a continuous-time probability distribution rather than on a conditional transfer probability in discretized units of size At. 

One first needs the basic notions associated with a continuous probability distribution. Consider the age or the retention time of a molecule in the compartment as a random variable, A. Let  

From the above relations, the hazard function h (a) is defined as 

Also from this definition, the simple relationship 

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The link between the probabilistic transfer model and retention-time distribution model may be explicitly demonstrated by deriving the conditional probability implied in the one-compartment probabilistic transfer model. We look for the probability, S (a + A a), that a particle survives to age (a + A a). Clearly, the necessary events are that the particle survives to age a, associated with the state probability S (a) AND that it remains in the compartment during the interval from a to (a + A a), associated with the conditional probability [1 — hAa, where h is the probabilistic hazard rate. Therefore, the probability of the desired joint event may be written as... 

Then, the probabilistic hazard rate h is the particular hazard function value h (a) evaluated at a specified age a. For the retention-time distribution models, h (a) A a gives the conditional probability that a molecule that has remained in the compartment for age a leaves by a + Aa. In other words, the probabilistic hazard rate is the instantaneous speed of transfer. 

Noteworthy is that only for the exponential distribution is the hazard rate h a) = f (a) /S (a) = k not a function of the age a, i.e., the molecule has no memory and this is the main characteristic of Markovian processes. In other words, the assumption of an exponential retention time is equivalent to the assumption of an age-independent hazard rate. One practical restriction of this model is that the transfer mechanism must not discriminate on the basis of the accrued age of a molecule in the compartment. In summary, it is clear that the formulations in the probabilistic transfer model and in the retention-time distribution model are equivalent. In the probabilistic transfer model we assume an age-independent hazard rate and derive the exponential distribution, whereas in the retention-time distribution model we assume an exponential distribution and derive an age-independent hazard rate. 

Reaction engineering texts provide two simple single-parameter models that represent two extremes of flow. These can be used to obtain clues to which flow regimes are occurring in the vessel. These two equations also can be used to predict the retention-time distribution of this vessel, within the limiting assumptions. 

Here it was assumed that n=3. From bqs. 14 and IS, we calculate that o2(r0)=0.333. and K/vl. 0 115. Be tween the two predictions is a composite curve. This is an interpolation predicting what the retention time distribution should be for the constant-level skim tank based on the two models. According to this curve, the peak concentration is predicted to occur at t-O. Stp, and the time span at one half the peak concentration is t. The peak concentration should be about 0.9 (/ ). The variance. aJ, of this vessel, should he 0.333 the mean residence time is 0.78fp. The actual retention time distribution for this vessel is plotted in Fig. 6. It can he seen that the peak concentration actually occurs at /=O.35r0. From Eq. 8. T=54.9 minutes (1.0lro) for this distribution and we can calculate that 02( d)sO.3O3. Thus, in this vessel the actual constants are rt 3.29 and A/vT.=0.106, using Eqs. 14 and IS. 

The retention-time distributions based on these constants are plotted in Fig. 6. Comparing Figs. 5 and 6 shows how similar the actual performance was to the ideal performance. The variance of the actual retention-time distribution was only slightly less than the variance predicted by the models. T his is an indication that slightly less mixing and turbulence was occurring than was expected. 

For multicompartment models, in addition to the retention-time distributions within each compartment, we require the specification of the transition probabilities LJij of transfer among compartments. These ujtJ is, assumed age-invariant, give the probabilities of transfer from a donor compartment i to each possible recipient compartment j. From (9.1), it follows that uiij = hij/ha is the probability that a particle in i will transfer to j on the next departure. 

Consider now a multicompartment structure aiming not only to describe the observed data but also to provide a rough mechanistic description of how the data were generated. This mechanistic system of compartments is envisaged with the drug flowing between the compartments. The stochastic elements describing these flows are the transition probabilities as previously defined. In addition, with each compartment in this mechanistic structure, one can associate a retention-time distribution (a). The so-obtained multicompartment model is referred to as the semi-Markov formulation. The semi-Markov model has two properties, namely that ... 

For instance, in the simple one-compartment model associated with a gamma retention-time distribution A Gam(A,/r),... 

This section proposes the use of a semi-Markov model with Erlang- and phase-type retention-time distributions as a generic model for the kinetics of systems with inhomogeneous, poorly stirred compartments. These distributions are justified heuristically on the basis of their shape characteristics. The overall objective is to find nonexponential retention-time distributions that adequately describe the flow within a compartment (or pool). These distributions are then combined into a more mechanistic (or physiologically based) model that describes the pattern of drug distribution between compartments. The new semi-Markov model provides a generalized compartmental analysis that can be applied to compartments that are not well stirred. 

The Erlang distributions used as retention-time distributions fi (a) have interesting mathematical properties considerably simplifying the modeling. For the Erlang distribution, it is well known that if v independent random variables Z, are distributed according to the exponential distribution... 

The phase-type distribution has an interpretation in terms of the compartmental model. Indeed, if the phenomenological compartment in the model, which is associated with a nonexponential retention-time distribution, is considered as consisting of a number of pseudocompartments (phases) with movement... 

The phase-type distributions are designed to serve as retention-time distributions in semi-Markov models. To obtain the equations of the model for a phenomenological compartmental configuration, one has to follow the following procedure ... 

Express the retention-time distribution for each phenomenological compartment by using phase-type distributions. However, the phase-type distributions for these sites are determined empirically. There is no assurance of finding the best phase-type distribution. This step leads to the expanded model involving pseudocompartments generating the desired phase-type distribution. 

Erlang- and phase-type distributions provide a versatile class of distributions, and are shown to fit naturally into a Markovian compartmental system, where particles move between a series of compartments, so that phase-type compartmental retention-time distributions can be incorporated simply by increasing the size of the system. This class of distributions is sufficiently rich to allow for a wide range of behaviors, and at the same time offers computational convenience for data analysis. Such distributions have been used extensively in theoretical studies (e.g., [366]), because of their range of behavior, as well as in experimental work (e.g., [367]). Especially for compartmental models, the phase-type distributions were used by Faddy [364] and Matis [301,306] as examples of long-tailed distributions with high coefficients of variation. 

This model is a special case of the model studied by Matis and Wehrly [369] in which Ai Erl(Ai, u ) and A2 Ev A2.V2) retention-time distributions are associated with the first and second compartments, respectively. The analysis of the characteristic polynomial of this model implies that there are at least two complex eigenvalues, except for the case v = 2 with parameters satisfying the condition... 

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. 

It is important to realize that many important processes, such as retention times in a given chromatographic column, are not just a simple aspect of a molecule. These are actually statistical averages of all possible interactions of that molecule and another. These sorts of processes can only be modeled on a molecular level by obtaining many results and then using a statistical distribution of those results. In some cases, group additivities or QSPR methods may be substituted. 

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