Random hazard rate

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 ... 

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. 

We present the one-compartment case in which the drug amount no is given over a period T by a constant-rate infusion. Assuming a random hazard rate h over the molecules, the state probability that a molecule associated with a hazard rate h is in the compartment at time t is... 

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... 

Assume that too independent units were introduced initially into the system with a transfer mechanism whose hazard rate h applies to all units in the experiment. The random movement of individual units in the heterogeneous process will result in a state probability p (t, h) depending on the specific h of all units in that experiment. Using the binomial distribution, the conditional expectation and variance are... 

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. 

In survival analysis, X can be viewed as the lifetime of some subject under study and 9 is the associated hazard rate. The subject may continue to be in the study program rmtil some death occurs or it drops off the program according to some random censoring variable Y. 

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. 

The hazard rate h(t) is another important reliability characteristic which can be used to describe the field failure behaviour when using a Weibull distribution. All three elements of the well known bath-tub-curve can be displayed by this theoretical distribution function. Thus it is possible to describe early failure, random failure and wearout failure. 

This is one of the simplest continuous random variable distribution frequently used in the industry, particularly in performing reliability studies because many engineering items exhibit a constant hazard rate during their useful life [14]. In addition, it is relatively easy to handle in performing reliability analysis-related studies. 

During the useful-life period, the hazard rate remains constant. Some of the reasons for the occurrence of failures in this region are higher random stress than expected, low safety factors, undetectable defects, human errors, abuse, and natural failures. 

The rheological behaviour of thermotropic polymers is complex and not yet well understood. It is undoubtedly complicated in some cases by smectic phase formation and by variation in crystallinity arising from differences in thermal history. Such variations in crystallinity may be associated either with the rates of the physical processes of formation or destruction of crystallites, or with chemical redistribution of repeating units to produce non-random sequences. Since both shear history and thermal history affect the measured values of viscosity, and frequently neither is adequately defined, comparison of results between workers and between polymers is at present hazardous. 

---