A statistical analogy between collapse of solids and death of living organisms: Proposal for a ‘law of life’
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 , a function of the position vector , as , 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):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 [1]; this can be deduced by setting (=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
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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