For want of a nail the shoe was lost, For want of a shoe the horse was lost, For want of a horse the rider was lost, For want of a rider the battle was lost, For want of a battle the kingdom was lost, And all for the want of a horseshoe nail. — English nursery rhyme (ca. 1390)

The models described in previous sections assumed that the failures of elements in the organism occur independently of each other. This assumption may be acceptable as the first approximation. In real biological systems many aging phenomena may be represented as a ''cascade of dependent failures'' which occurs when one of the organism's systems randomly fails (Gavrilov, 1978; Gavrilov et al., 1978). The idea that an avalanche-like mechanism is involved in the destruction of an organism during natural aging is worth further consideration. In fact, it is well-known that defects in an organism have a tendency to multiply following an avalanche-like mechanism. For example, if there are n cancer cells in the organism, each of which is capable of division, the rate at which the organism is transformed into a state with n + 1 cancer cells increases with the growth of the number of cancer cells (n) already accumulated. Infections of the organism follow similar regularities. The positive feedback between the degree and the rate of an organism's destruction also follows from the fact that when parts of the structure fail, the load on the remaining structures increases, accelerating the wearing-out. It seems that aging may be caused by similar cascades of dependent failures developing over long periods in a hidden, preclinical form. Therefore mathematical models of the avalanche-like destruction of the organism are of particular interest.

Consider the simplest model of the avalanche-like destruction of the organism (Gavrilov and Gavrilova, 1991). Let S0, Si,..., Sn denote the states of an organism with 0,1,2,..., n defects. Let 10 be the background rate at which defects accumulate being independent on the stage of destruction, which the organism has reached. Correspondingly, let u0 be the age-independent mortality (the Makeham term). In the simplest case, both of these quantities arise from random harmful effects of the external environment. In tandem, there is also an induced rate of deterioration (parameter l) and an induced failure rate (parameter u) which grow as the number of defects increases. At a first approximation, it can be assumed that both the induced rate of deterioration and the induced failure rate are proportional to the number of defects, so that for an organism with n defects the induced rate

1q 1q + 1 1q + 21 1q + 31 , 1q + nl iS0 - S-, —S2 _-" S3 *■ ■ Sn I

Figure 5.10. Avalanche-like mechanism of organism's destruction with age. In the initial state (So) organism has no defects. Then, as a result of random damage, it enters states Si, S2, ■■■ Sn, where n corresponds to the number of defects. Rate of new defects emergence has avalanche-like growth with the number of already accumulated defects (horizontal arrows). Hazard rate (vertical arrows directed down) also has an avalanche-like growth with the number of defects.

of deterioration is equal to nl, and the induced failure rate is nj.

With these assumptions, we can present the avalanchelike destruction of the organism by the schema presented in Figure 5.10.

This schema corresponds to the following system of differential equations:

dSn/dx = [I0 + (n — 1)l]Sn—1 — [I0 + U0 + n(1 + u)]Sn

A similar system of equations (not taking into account the age-independent mortality) was obtained and solved in a mathematical model linking the survival of organisms with chromosome damage (Le Bras, 1976). However, this ''chromosomal'' interpretation of the avalanche model could be applicable only to unicellular organisms, while for multicellular organisms including humans, where chromosomes are compartmentalized in separate cells, this model needs to be revised, or provided with a different ''nonchromosomal'' interpretation (as it is suggested in this section).

In the particular case when the rate at which defects multiply, the parameter l is significantly greater than the induced failure rate, parameter u,(2 ^ u), the hazard rate of an organism in the initial stage (with low values of x) grows according to the Gompertz-Makeham law:

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