A statistical analogy between collapse of solids and death of living organisms: Proposal for a ‘law of life’

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Summary

In this paper we present a statistical analogy between the collapse of solids and living organisms; in particular we deduce a statistical law governing their probability of death. We have derived such a law coupling the widely used Weibull Statistics, developed for describing the distribution of the strength of solids, with a general model for ontogenetic growth recently proposed in literature. The main idea presented in this paper is that cracks can propagate in solids and cause their failure as sick cells in living organisms can cause their death. Making a rough analogy, living organisms are found to behave as “growing” mechanical components under cyclic, i.e., fatigue, loadings and composed by a dynamic evolutionary material that, as an ineluctable fate, deteriorates. The implications on biological scaling laws are discussed. As an example, we apply such a Dynamic Weibull Statistics to large data collections on human deaths due to cancer of various types recorded in Italy: a significant agreement is observed.

Section snippets

Introduction: Weibull and West–Brown–Enquist approaches

Weibull Statistics [1] has been derived on the hypothesis of the weakest link theory and describes the statistical distribution for the strength of solids. Weibull derives the cumulative probability of failure Pf for a solid of volume V and subjected to a uniaxial local stress σ(x¯), a function of the position vector x¯, as Pf=1-exp-Vσ(x¯)σ0Vmdx3, where σ0V and m are, respectively, Weibull’s scale and shape parameters, governing the mean value and the standard deviation of the distribution.

Towards a statistical law of the life

Coupling the Weibull and West–Brown–Enquist approaches, we formulate a statistical law of the life giving the probability of death PD, or of survival PS, as suggested by Eqs. (1), (2a), (2b):PD=1-PS=1-exp-kMγ(t)tt0m(t)where M(t) is given by Eq. (2b), m(t) is an evolutionary modulus, k is a constant with anomalous physical dimension and t/t0 is a dimensionless time, defined by arbitrarily fixing t0, e.g., t0 = 1 year (note that mathematically PD  1 only for t  ∞, but practically PD  1 for realistic

Scaling laws in physics and biology

Let us consider solids composed by a given material but having different size-scales. It is well-known that Eq. (1) implies a scaling for their (nominal) failure stresses according to σfmVconst [1]; this can be deduced by setting Pf=const (=0.63, it being the value of the probability defining the nominal stress). The size-scale effect σf  V−1/m suggests that smaller is stronger as nowadays well-known in Physics [18], [19]. This represents the reason why nanostructures, such as nanotubes or

Deterioration as ineluctable fate

Let us consider cancer data. In particular we refer to the tables of mortality due to cancers of various types recorded in Italy [24]. The age of the individuals deaths are divided into time-intervals, 1–4, 5–9,  , i-(i+4) ,  , 75–79 years. For each time-interval i the number Ni of the observed deceases, for a specified year and in Italy, is reported. We consider the deaths related to the time-interval i as arising at its mean value ti and we calculate the cumulative probability of death as PD(ti)=j=

Cancer data analysis

We choose to treat other data on cancer deaths [24] to further test the statistics of Eq. (5a). In Fig. 3 we still refer to the year 1990 but considering separately males and females. Females (k−1  643.7 and tm  46.1 years) are found to be slightly stronger (larger k−1, tm) than males (k−1  513.8 and tm  43.5 years) against cancer. In Fig. 4 a comparison between the years 1974 (k−1  246.3 and tm  48.8 years) and 1984 (k−1  192.9 and tm  54.9 years) is reported. The influence of the time, related to the

Conclusions

Summarizing, the proposed statistical law of the life could represent an interesting tool for classifying and deducing statistical predictions on the natural deaths of living organisms, as here demonstrated for cancer in human individuals. Further investigations may reveal the necessity of considering the more general Eq. (3a) rather then its simplified version of Eq. (5a), or a different statistics. As the Weibull Statistics can be applied for predicting the probability of failure of a given

Acknowledgement

The author would like to thank P.P. Delsanto and F. Pugno for discussion, as well as Diane Dijak for the English grammar supervision.

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