In the previous sections, we examined a situation in which series-connected blocks have varying degrees of redundancy within each organism, while the organisms themselves were considered to be initially identical to each other and to have the same risk of death. This latter assumption can be justified in some special cases (see Gavrilov and Gavrilova, 1991) and also when the focus is on the analysis of the individual risks of failure. In a more general case the population heterogeneity needs to be taken into account, because there is a large variation in the numbers of cells for the organisms of the same species (Finch and Kirkwood, 2000).
In this section, we demonstrate that taking into account the heterogeneity of the population provides an explanation for all the basic laws of mortality. This model of heterogeneous redundant systems was proposed by Gavrilov and Gavrilova in 1991 (pp. 264-272).
The model considers the simplest case when the organism consists of a single vital block with n elements connected in parallel with q being the probability that an element is initially functional. Then the probability of encountering an organism with i initially functional elements out of a total number n of elements is given by the binomial distribution law.
The final formula for failure rate in heterogeneous population, ^p(x), is (see Gavrilov and Gavrilova, 1991 for more details):
F0(x) _ nqy,e-lxx(1 - qe-llx)n-1 1 - F(x) = 1 - (1 - qe-^x)n
^ Cnq^(1 — q + q^x)n 1 for x ^ 1/^ ^ ^ for x ^ 1 /^
where C is a normalizing factor.
Thus the hazard rate of a heterogeneous population at first grows with age according to the binomial law n1
of mortality, then asymptotically approaches an upper limit ß:
where x0 = (1 — q)/qM, a parameter which we propose to call the initial virtual age of the population. This parameter has the dimension of time, and corresponds to the age by which an initially homogeneous population would have accumulated as many damaged organisms as a real population actually possesses at the initial moment in time (at x = 0). In particular, when q = 1, i.e., when all the elements in each organism are functional at the outset, the initial virtual age of the population is zero and the hazard rate of population grows as a power function of age (the Weibull law), this being the case described in causes of failure rate increase with age. However when the population is not initially homogeneous (q < 1), we arrive at the already mentioned binomial law of mortality. Thus, the heterogeneous population model proposed here can also provide a theoretical justification for the binomial law of mortality.
If a population is initially heterogeneous (q < 1), the hazard rate in the initial period of time grows exponentially with age (according to the Gompertz law).
The heterogeneous population model not only provides an explanation for the exponential growth in the failure rate with age, but also the compensation law of mortality (Gavrilov and Gavrilova, 1991). The compensation effect of mortality is observed whenever differences in mortality are brought about by interpopulation differences in the number of elements in the organism (n), while the other parameters, including the rate of aging (the rate of irreversible elements failure m), are similar for all compared populations of a particular species (presumably because of homeostasis in physiological parameters). It is not difficult to see the similarity between this explanation for the compensation effect of mortality and the explanations which emerge from the models of individual system described in preceding sections.
Figure 5.9 presents the age kinetics of failure rate in heterogeneous population where redundancy is distributed by the Poisson law (a special case of binomial distribution) with different mean number of functional elements (2 = 1, 5, 10, 15 and 20).
Note that the logarithm of the failure rate is increasing with age in almost a linear fashion, indicating a reasonable applicability of the Gompertz law in this case. Also note that the slope of the lines is increasing with higher mean redundancy levels (2), and the lines
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