Retention-Time Models with Random Hazard Rates

2 Retention-Time Models with Random Hazard Rates 

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. 

This approach is presented for the two-compartment model of Section 9.2.7. At the second level in (9.16), we assume that A is a gamma-distributed random variable, A Gam(A2,p2)- The Laplace transform of the state probability is 

Actually, the inverse problem should be solved, i.e., given the data n(t) containing errors, obtain a plausible candidate / (h) associated with a known function p(t,h). This function, termed kernel, is assumed to be a retentiontime distribution other than an exponential one otherwise, the problem has a tractable solution by means of the moment generating functions as presented earlier. This part aims to supply some indications on how to select the density of h. For a given probability density function f (h), one has to mix the kernel with / (h)  

Equation (9.32) is a linear Fredholm integral equation of the first kind. It is also known as an unfolding or deconvolution equation. One can preanalyze the data and try to solve this first-kind integral equation. Besides the complexity of this equation, there is a paucity of numerical methods for determining the unknown function / (h) [208,379] with special emphasis on methods based on the principle of maximum entropy [207,380]. The so-obtained density function may be approximated by several models, gamma, Weibull, Erlang, etc., or by phase-type distributions. 

---