Probabilistic vs. Retention-Time Models

We look now for the evaluation of the state probability p (t) that the molecule is in the compartment at time t in the case of a one-compartment model. To this end, consider the partition 0 = ai a2 a i an = t and the n— 1 mutually exclusive events that the molecule leaves the compartment between its age instants aj i and aj. The state probability p(t) equals the probability of the complement of the above n — 1 mutually exclusive events, 

Therefore, the survival function S (a) plays the same role as the state probability p (t). But the former independent variable a is defined as the endogenous 

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 

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. 

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