Probabilistic Transfer Models

The development of probabilistic transfer models is based on two probabilities, a conditional probability and a marginal one, commonly stated as transfer and state probabilities, respectively. 

Rearranging, taking the limit At > 0 in the above difference equations, and neglecting the higher-order terms of At, one obtains m2 differential equations, namely the probabilistic transfer model... 

In what follows, we will rather restrict ourselves mainly to the standard Markov process in the probabilistic transfer model with time-independent hazard rates. This is equivalent to assuming that the transfer probabilities do not depend on either the time the particle has been in the compartment or the previous history of the process, and... 

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... 

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. 

The initial idea is to use the differential equations of a probabilistic transfer model with hazard rates varying with the age of the molecules, i.e., to enlarge the limiting hypothesis (9.2). The objective is to find nonexponential families of survival distributions that are mathematically tractable and yet sufficiently flexible to fit the observed data. In the simplest case, the differential equation (9.7) links hazard rates and survival distributions. Nevertheless, this relation was at the origin of an erroneous use of the hazard function. In fact, substituting in this relation the age a by the exogenous time t, we obtain... 

Let us examine now the conditions for which a probabilistic transfer model is equivalent to a retention-time model, both using the same hazard functions. More precisely, for the irreversible multicompartment structures, the study can be reduced to the analysis of an irreversible two-compartment model, where the compartment n°l embodies all compartments before the compartment n°2. One has to compare two situations ... 

The probabilistic transfer model whose differential form is... 

Probabilistic transfer model. The model is a special case of the two-compartment model presented in Figure 9.1, where compartment 1 is associated with the infusion balloon and compartment 2 is associated with the central compartment. The links between compartments are specified as h 2 = hiv (t), /121 = 0, h Q = 0, and /120 = h. The state probabilities associated with compartment 1 are pn (t) = piv (t) and p21 (t) = 0. The probabilistic transfer equation for the central compartment 2 is obtained directly from (9.4) ... 

If the hazard rate of any single particle out of a compartment depends on the state of the system, the equations of the probabilistic transfer model are still linear, but we have nonlinear rate laws for the transfer processes involved and such systems are the stochastic analogues of nonlinear compartmental systems. For such systems, the solutions for the deterministic model are not the same as the solutions for the mean values of the stochastic model. 

In the following, we show how to apply probabilistic transfer models with random hazard rates associated with the administration and elimination processes in a single-compartment configuration. 

We report the one-compartment probabilistic transfer model receiving the drug particles by an absorption process. In this model, the elimination rate h was fixed and the absorption constant hev was random. For the stochastic context, the difference hev — h = w is assumed to follow the gamma distribution, i.e., W Gam(A, //.) with density / (w, A, //.) and E [W] =... 

Like the previous ones, these models are two-level models. Now, the retentiontime model substitutes the probabilistic transfer model in the first level, and in the second level, parameters of this model are assumed to be random and they are associated with a given distribution. Consider, for instance, the one-compartment model with Erlang retention times where the parameter A is a random variable expressing the heterogeneity of the molecules. Nevertheless, even for the simplest one-compartment case, the model may reach extreme complexity. In these cases, analytical solutions do not exist and numerical procedures have to be used to evaluate the state probability profiles. 

Two-Compartment Model First, we develop full probabilistic transfer modeling. Consider the number of particles in the first and second compartments being rq and 712, respectively, at time t + At, where At is some small time interval. There are a number of mutually exclusive ways in which this event could have come about, starting from time t. Specifically, they are ... 

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