It was demonstrated in the previous section that the failure rate of a simple parallel system grows with age according to the Weibull law. This model analyzed initially ideal structures in which all the elements are functional from the outset. This standard assumption may be justified for technical devices manufactured from pretested components, but it is not justified for living organisms, replete with initial defects (see Gavrilov and Gavrilova, 1991; 2001; 2004b; 2005).
Following the tradition of the reliability theory, we start our analysis with reliability of an individual system (or homogeneous population). This model of series-parallel structure with distributed redundancy was suggested by Gavrilov and Gavrilova in 1991 and described in more detail in 2001.
Consider first a series-parallel model in which initially functional elements occur very rarely with low probability q, so that the distribution of the organism's subsystems (blocks) according to the initial functioning elements they contain is described by the Poisson law with parameter l = nq. Parameter l corresponds to the mean number of initially functional elements in a block.
As has already been noted, the failure rate of a system constructed out of m blocks connected in series is equal to the sum of the failure rates of these blocks, Tb (Barlow et al., 1965):
where Pi is the probability of a block to have i initially functioning elements. Parameter C is a normalizing factor that ensures the sum of the probabilities of all possible outcomes being equal to unity (see Gavrilov, Gavrilova, 1991; 2001). For sufficiently high values of n and l, the normalizing factor turns out to be hardly greater than unity.
Using the formula for failure rate of a block of elements connected in parallel (see the simplest reliability model of aging section), we obtain the final expression for the series-parallel system with distributed redundancy:
Ts = ixImCe-1 ^ ^T^T ^ R^™ - £(x)) ^ Reax where R = CmX^e-1, a = It fi(x) is close to zero for large n and small x (initial period of life; see Gavrilov, Gavrilova, 1991, 2001 for more detail).
In the early-life period (when x ^ 1 /t) the mortality kinetics of this system follows the exponential Gompertzian law.
In the late-life period (when x ^ 1/t), the failure rate levels off and the mortality plateau is observed:
If the age-independent mortality (A) also exists in addition to the Gompertz function, we obtain the well-known Gompertz-Makeham law described earlier. At advanced ages the rate of mortality decelerates and approaches asymptotically an upper limit equal to m^.
The model explains not only the exponential increase in mortality rate with age and the subsequent leveling-off, but also the compensation law of mortality:
ln(R) = ln(Cma) - a = ln(M) - Ba where M = Cma, B = 1/t.
According to this model, the compensation law is inevitable whenever differences in mortality arise from differences in the parameter l (the mean number of initially functional elements in the block), while the ''true aging rate'' (rate of elements' loss, t) is similar in different populations of a given species (presumably because of homeostasis). In this case, the species-specific lifespan estimated from the compensation law as an expected age at mortality convergence (95 years for humans, see Gavrilov and Gavrilova, 1991) characterizes the mean lifetime of the elements (1 /t).
The model also predicts certain deviations from the exact mortality convergence in a specific direction because the parameter M proved to be a function of the parameter a according to this model (see earlier). This prediction could be tested in future studies.
It also follows from this model that even small progress in optimizing the processes of ontogenesis and increasing the numbers of initially functional elements (l) can potentially result in a remarkable fall in mortality and a significant improvement in lifespan.
The model assumes that most of the elements in the system are initially nonfunctional. This interpretation of the assumption can be relaxed, however, because most nonfunctional elements (e.g., cells) may have already died and been eliminated by the time the adult organism is formed. In fact, the model is based on the hypothesis that the number of functional elements in the blocks is described by the Poisson distribution, and the fate of defective elements and their death in no way affects the conclusions of the model. Therefore, the model may be reformulated in such a way that stochastic events in early development determine later-life aging and survival through variation in initial redundancy of organs and tissues (see, for example, Finch and Kirkwood, 2000). Note that this model does not require an assumption of initial population heterogeneity in failure risks. Instead the model is focused on distributed redundancy of physiological systems within a given organism, or a group of initially identical organisms.
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