Hierarchical Complexity and Aging - Towards a Physics of Aging

In this paper we extend the previous work of Witten and her team on defining a classical physics driven model of survival in aging populations (Eakin, 1994; Eakin and Witten, 1995a; Eakin and Witten, 1995b; Witten and Eakin, 1997) by revisiting the concept of a force of aging and introducing the concepts of a momentum of aging, a kinetic energy and a potential energy of an aging. As an example of the use of these constructs, we then explore the implications of these concepts with respect to the (Yu et al., 1982) diet restriction experiments. 1 HISTORY OF RELIABILITY The history of the demographics of aging is tightly bound to the field of survival analysis (Witten, 1981; Elandt-Johnson and Johnson, 1999). Survival analysis, however, emerged from the earlier discipline of reliability theory (Abdel-Hameed et al., 1984; Ansell and Phillips, 1994). The constructs of reliability theory emerged from the 1950’s gedankt experiments of the computer scientist John Von Neumann. His interest (Neumann, 1956) was in how one would go about building a reliable biological organism out of unreliable parts. Until the thought experiments of von Neumann, the concept of reliability had not been well-defined. Von Neumann’s argument proceeded as follows. He began by defining the concept of the conditional instantaneous failure rate, denoted by λ(t). We interpret this as follows. The condition is that the failure has not occurred at time t given that the organism has survived until time t. With this in mind, we may then define the reliability R(t) of an organism as the probability of no failure of the organism before time t. If we let f (t) be the time to (first) failure (this is the same as the failure density function), then the reliability R(t) is given by R(t) = 1−F(t) where F(t) = ∫ t 0 f (τ)dτ ((Abdel-Hameed et al., 1984; Deshpande and Purohit, 2005; Elandt-Johnson and Johnson, 1999; Kalbfleish and Prentice, 2002; Lawless, 2003)). How do we actually obtain an equation for the reliability R(t)? We do this as follows. Suppose we ask what is the reliability R(t +∆t) where ∆t is a small time increment. In other words, suppose that we know the reliability of the organism at time t and we want to know the organism’s reliability at a small time increment ∆t later than time t. In order for the organism to be operational at time t+∆t, the organism must have been operational until at least time t and then not have failed in the time interval (t, t +∆t). We can express this mathematically as follows. The reliability R(t +∆t) is given by R(t +∆t) = R(t)−λ(t)R(t)∆t (1) Reading equation [1], we see that to be functional (operational) at time t+∆t, the organisms had to be functional at time t (denoted by the reliability term R(t) on the right hand side of the equation). Next, we have to subtract out all of the items that failed in the time interval (t, t +∆t) (given by the second term on the right hand side of equation [1]). What remains after this subtraction is all of the organisms or items that remain functional at time t +∆t. A bit of algebraic rearrangement and we have R(t +∆t)−R(t) ∆t =−λ(t)R(t) (2) It follows that letting ∆t → 0 (remembering our calculus), Equation [1] becomes the simple differential equation given by