## Accumulation Of Defects With Constant Rate Of Damage Flow

A wide variety of concepts about the destruction of the organism can lead to the model where the rate of damage flow, numerically equal to the mean number of ''hits'' per unit of time, is practically independent of the state of the organism and is on average constant in time. In the simplest case, the model corresponds to a situation in which the organism is affected by a random flow of traumatic loads with an on average constant rate independent of the state of the organism (exogenous environmental damage like cosmic radiation, viruses, etc.)

However, there is also the possibility of other mechanisms of destruction leading to this particular model of the accumulation of defects. In particular, this model can be obtained after a critical reinterpretation of the assumptions underlying the previously described models. In fact, these models contain an assumption that the death of the organism occurs only when all the elements in a block fail. It is possible that this hypothesis may be justified for some of the organism's systems (stem cell populations, for example). However, in the majority of cases this hypothesis seems contentious. For example, it is hard to imagine that a single surviving liver cell (hepatocyte) can assume the functions of an entire destroyed liver. Significantly more realistic is the hypothesis that the system initially contains an enormous number of elements that greatly exceeds the critical number of defects leading to the death of the organism. In this case we arrive at a schema for the accumulation of damage in which the rate of damage flow (equal to the product of the number of elements and their failure rate) turns out to be practically constant in view of the incommensurability of the high initial numbers of elements, and the much smaller permitted number of defects (Gavrilov and Gavrilova, 1991).

Another advantage of this model is that it allows us to take into account the influence of living conditions on the value for the critical number of defects incompatible with the survival of the organism. The key to the solution of this problem is the replacement of the parallel connection hypothesis (assumed in previously described models) with the more realistic assumption that there exists a critical number of defects incompatible with the survival of the organism. In this case, it is natural to expect that under harsher conditions the critical number of defects leading to death might be less than under more comfortable living conditions. In particular, in the wild, when an animal is deprived of care and forced to acquire its own food, as well as to defend itself against predators, the first serious damage to the organism can lead to death. It is therefore not surprising that the mortality of many animals (in particular, birds) is practically independent of age in the wild. This follows directly from the singlestage destruction of the organism model. On the other hand, the greater the number of defects the organism can accumulate while remaining alive, the greater its lifespan will be.

If the rate of the damage flow equals k, and an organism dies after the accumulation of n defects, the density of the survival distribution is identical to the density of the gamma function (see Barlow and Proschan, 1965; 1975). At the initial moment in time, this distribution corresponds to a power (Weibull) law of mortality with an exponent equal to (n — 1).

A fundamentally different result is obtained when the initial damage of organisms is taken into account (Gavrilov and Gavrilova, 1991). If at the initial moment in time the average number of random defects in the population equals 1, the probability of encountering a living organism, Pt, with i defects may be approximated by the Poisson law (see Gavrilov and Gavrilova, 1991, pp. 272-276, for more detail).

Since the death of an organism with i defects occurs after n — i additional hits, the density of the lifespan distribution for such organisms is given by f(x) =

where

The density of the survival distribution for the whole population, which is a mixture of organisms with i = 0, 1, 2,... n — 1 initial defects, equals f (x) = £ Pifi(x) = Cke i=0

It is not difficult to see that at the initial moment in time this model leads to the binomial law of mortality, with an initial virtual age of the population equal to 1/ k. A more detailed analysis of the model is formally similar to the analysis of the other models described in previous sections. We merely note that during the initial time period when x ^ 1/k, the model leads to an exponential growth of failure rate with age (the Gompertz law) with an exponent, a, of k(n —1)/1 and a pre-exponential factor, R, of CkXn—1/(n — 1)!. It is

easy to see that an inverse relationship between these Gompertz parameters (the compensation effect of mortality) can arise both as a result of variation in parameter l (the degree to which the organisms are initially damaged) and of variation in parameter n (the critical number of defects, dependent on the harshness of living conditions).

Thus the basic mortality phenomena can equally be explained within the framework of the model of accumulation of defects with the constant rate of damage flow, as long as the organisms initially contain a significant number of defects.

Summarizing this brief review of reliability models, we note the striking similarity between the formulas and conclusions of the considered models. It must, however, be noted that we are dealing only with a superficial similarity in behavior between fundamentally different and competing models. The existence of a multitude of competing models is therefore compatible with the reliable and meaningful interpretation of a number of mortality phenomena, since pluralism of models does not preclude their agreement on a number of issues. All these models predict a mortality deceleration, no matter what assumptions are made regarding initial population heterogeneity, or its complete initial homogeneity. Moreover, these reliability models of aging produce mortality plateaus as the inevitable outcome for any values of considered parameters (Gavrilov and Gavrilova, 1991). The only constraint is that the elementary steps of the multistage destruction process of a system should occur only by chance, independent of age. The models also predict that an initially homogeneous population will become highly heterogeneous for risk of death over time (acquired heterogeneity).

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