Random Hazard-Rate Models

After bolus administration and keeping the CL constant, Weiss [245] obtained the simple time-concentration profile 

We consider now a class of models that introduce particle heterogeneity through random rate coefficients. In this conceptualization, the particles are assumed different due to variability in such characteristics as age, size, molecular conformation, or chemical composition. The hazard rates h are now considered to be random variables that vary influenced by extraneous sources of fluctuation 

We now consider models that combine the sources of stochastic variability identified previously [375]. The experimental context reproducing the randomness of h can be conceived as follows  

The variance expression is composed of two terms m0ps (t) generalizes the variance of a standard binomial distribution and is attributable to the stochastic transfer mechanism (structural heterogeneity) and rn pp (f) reflects the random nature of h (functional heterogeneity). 

The random hazard rate model is easily obtained from the above by considering a single unit, mo = 1, and no particles initially administered into the system. The first two moments are obtained by summing n0 independent and identically distributed experiments  

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Probabilistic Models with Random Hazard Rates... 

The solution of the probabilistic transfer equations leads to the exponential model (9.3). The presence of negative exponentials in the model may simplify somewhat the choice of distribution associated with the random hazard rate. In fact, the elements p(t,h) of the state probability matrix exp(Ht) in (9.3) are exponentials, and integrating (9.27) over the random variable h, we obtain... 

For the one-compartment model with n0 initial conditions, the distribution of the random hazard rate h can be simply mixed with the state probability p t,h) = exp (—/it), and relations (9.27) become... 

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. 

Each molecule has its own hazard rate, and if we assume a constant volume of distribution V, each molecule will have its own clearance defined as CL = Vh. Then CL becomes a random variable, and there follows the distribution of h with expectation E[CL] = VE h = V/i/X. Regardless of the molecule s clearance, the systemic clearance may be obtained on the basis of the expected profile E [IV (f)] using either the plateau evaluation during a long-term infusion or the total area under the curve. Both evaluations give CL = V (ji 1)/A. Note that for p, = 1, the systemic clearance cannot be defined albeit individual molecular clearances exist. The discrepancy between E CL and CL is due to the randomness of the model parameter h. 

The hazard rate is suitable to describe the failure behaviour (early, random, wearout) of a product. The reliability characteristics mentioned above can be transformed directly into each other. If one of these is known all other characteristics can be determined easily. Due to using warranty data empirical reliability characteristics are applied in the following model. They are defined analogously to the theoretical reliability characteristics. 

Random hardware failures assessment contains all the items suggested in Appendix 2 of this book. Include reliability model, CCF model, justification of choice of failure rate data, coverage of all the hazardous failure modes ... 

The purpose of this paper is to present the quantification of the risk rates, that is the probability per unit of time (hour) for each and every of the 63 hazards. The paper is organised as follows Section 2 outlines the modelling of the arrival of occupational accidents as a Poisson random process and briefly describes the procedure followed for identifying the number of accidents and the exposure of the Dutch working population to the occupational hazards during a given period of time. With these two sets of data point estimates of the risk rates are obtained. Section 3 presents the assessment of the uncertainties associated with this estimation following a Bayesian approach. Finally Section 4 discusses the obtained results. 

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