Basic Failure Models

Reliability of the system (or its component) refers to its ability to operate properly according to a specified standard (Crowder et al., 1991). Reliability is described by the reliability function S(x), which is the probability that a system (or component) will carry out its mission through time x (Rigdon and Basu, 2000). The reliability function (also called the survival function) evaluated at time x is just the probability P, that the failure time X, is beyond time x. Thus, the reliability function is defined in the following way:

S(x) = P(X > x) = 1 — P(X < x) = 1 — F(x)

where F(x) is a standard cumulative distribution function in the probability theory (Feller, 1968). The best illustration for the reliability function S(x) is a survival curve describing the proportion of those still alive by time x (the lx column in life tables).

Failure rate, ^(x), or instantaneous risk of failure, also called the hazard rate, h(x), or mortality force is defined as the relative rate for reliability function decline:

In those cases when the failure rate is constant (does not increase with age), we have nonaging system (component) that does not deteriorate (does not fail more often) with age:

The reliability function of nonaging systems (components) is described by the exponential distribution:

This failure law describes ''lifespan'' distribution of atoms of radioactive elements, and, therefore, it is often called an exponential decay law. Interestingly, this failure law is observed in many wild populations with high extrinsic mortality (Finch, 1990; Gavrilov and Gavrilova, 1991). This kind of distribution is observed if failure (death) occurs entirely by chance, and it is also called a ''one-hit model,'' or a ''first order kinetics.'' The nonaging behavior of a system can be detected graphically when the logarithm of the survival function decreases with age in a linear fashion.

Recent studies found that at least some cells in the aging organism might demonstrate nonaging behavior. Specifically, the rate of neuronal death does not increase with age in a broad spectrum of aging-related neurodegenerative conditions (Heintz, 2000). These include 12 different models of photoreceptor degeneration, ''excitotoxic'' cell death in vitro, loss of cerebellar granule cells in a mouse model, and Parkinson's and Huntington's diseases (Clarke et al., 2000). In this range of diseases, five different neuronal types are affected. In each of these cases, the rate of cell death is best fit by an exponential decay law with constant risk of death independent of age (death by chance only), arguing against models of progressive cell deterioration and aging (Clarke et al., 2000, 2001a). An apparent lack of cell aging is also observed in the case of amyotrophic lateral sclerosis (Clarke et al., 2001a), retinitis pigmentosa (Burns, 2002; Clarke et al., 2000, 2001a; Massof, 1990) and idiopathic Parkinsonism (Calne, 1994; Clarke et al., 2001b; Schulzer et al., 1994). These observations correspond well with another observation that ''an impressive range of cell functions in most organs remain unimpaired throughout the life span'' (Finch, 1990, p. 425), and these unimpaired functions might reflect the ''no-aging'' property known as ''old as good as new'' property in survival analysis (Klein and Moeschberger, 1997, p. 38). Thus we come to the following fundamental question about the origin of aging: How can we explain the aging of a system built of nonaging elements? This question invites us to think about the possible systemic nature of aging and to wonder whether aging may be a property of the system as a whole. We would like to emphasize the importance of looking at the bigger picture of the aging phenomenon in addition to its tiny details, and we will suggest a possible answer to the posed question in this chapter.

If failure rate increases with age, we have an aging system (component) that deteriorates (fails more often) with age. There are many failure laws for aging systems, and the most famous one in biology is the Gompertz law with exponential increase of the failure rates with age (Finch, 1990; Gavrilov and Gavrilova, 1991; Gompertz, 1825; Makeham, 1860; Strehler, 1978):

An exponential (Gompertzian) increase in death rates with age is observed for many biological species including fruit flies Drosophila melanogaster (Gavrilov and Gavrilova, 1991), nematodes (Brooks et al., 1994; Johnson, 1987, 1990), mosquitoes (Gavrilov, 1980), human lice, Pediculus humanus (Gavrilov and Gavrilova, 1991), flour beetles, Tribolium confusum (Gavrilov and Gavrilova, 1991), mice (Kunstyr and Leuenberger, 1975; Sacher, 1977), rats (Gavrilov and Gavrilova, 1991), dogs (Sacher, 1977), horses (Strehler, 1978), mountain sheep (Gavrilov, 1980), baboons (Bronikowski et al., 2002) and, perhaps most important, humans (Finch, 1990; Gavrilov and Gavrilova, 1991; Gompertz, 1825; Makeham, 1860; Strehler, 1978). According to the Gompertz law, the logarithm of failure rates increases linearly with age. This is often used in order to illustrate graphically the validity of the Gompertz law—the data are plotted in the semilog scale (known as the Gompertz plot) to check whether the logarithm of the failure rate is indeed increasing with age in a linear fashion.

For technical systems one of the most popular models for failure rate of aging systems is the Weibull model, the power-function increase in failure rates with age (Weibull, 1939):

This law was suggested by Swedish engineer and mathematician W. Weibull in 1939 to describe the strength of materials (Weibull, 1939). It is widely used to describe aging and failure of technical devices (Barlow and Proschan, 1975; Rigdon and Basu, 2000; Weibull, 1951), and occasionally it was also applied to a limited number of biological species (Eakin et al., 1995; Hirsch and Peretz, 1984; Hirsch et al., 1994; Janse et al., 1988; Ricklefs and Scheuerlein, 2002; Vanfleteren et al., 1998). According to the Weibull law, the logarithm of failure rate increases linearly with the logarithm of age with a slope coefficient equal to parameter p. This is often used in order to illustrate graphically the validity of the Weibull law—the data are plotted in the log-log scale (known as the Weibull plot) to check whether the logarithm of the failure rate is indeed increasing with the logarithm of age in a linear fashion.

We will show later that both the Gompertz and the Weibull failure laws have a fundamental explanation rooted in reliability theory. Therefore it may be interesting and useful to compare these two failure laws and their behavior.

Figure 5.1a presents the dependence of the logarithm of the failure rate on age (Gompertz plot) for the Gompertz and the Weibull functions.

Note that in Figure 5.1a this dependence is strictly linear for the Gompertz function (as expected), and concave-down for the Weibull function. So the Weibull function looks like decelerating with age if compared to the Gompertz function.

Figure 5.1b presents the dependence of the logarithm of the failure rate on the logarithm of age (Weibull plot) for the Gompertz and the Weibull functions. Note that this dependence is strictly linear for the Weibull function (as expected), and concave-up for the Gompertz function. So the Gompertz function looks like the accelerating one with the logarithm of age if compared to the Weibull function.

There are two fundamental differences between the Weibull and the Gompertz functions.

First, the Weibull function states that the system is immortal at starting age—when the age X is equal to zero, the failure rate is equal to zero too, according to the Weibull formula. This means that the system should be initially ideal (immortal) in order for the Weibull law to be applicable to it. On the contrary, the Gompertz function states that the system is already vulnerable to failure at starting age—when the age X is equal to zero, the failure rate is already above zero, equal to parameter R in the Gompertz formula. This means that the partially damaged systems having some initial damage load are more likely to follow the Gompertz failure law, while the initially perfect systems are more likely to follow the Weibull law.

Second, there is a fundamental difference between the Gompertz and the Weibull functions regarding their response to misspecification of the starting age (''age zero''). This is an important issue, because in biology there is an ambiguity regarding the choice of a ''true'' age, when aging starts. Legally, it is the moment of birth, which serves as a starting moment for age calculation. However, from a biological perspective there are reasons to consider a starting age as a date either well before the birth date (the moment of conception in genetics, or a critical month of pregnancy in embryology), or long after the birth date (the moment of maturity, when the formation of a body is finally completed). This uncertainty in starting age has very different implications for data analysis with the Gompertz or the Weibull functions.

Figure 5.1. Plots of Gompertz and Weibull functions in different coordinates. (a) semilog (Gompertz) coordinates, (b) log-log (Weibull) coordinates. Source: Gavrilov and Gavrilova, 2005.

For the Gompertz function a misspecification of a starting age is not as important, because the shift in the age scale will still produce the same Gompertz function with the same slope parameter a. The data generated by the Gompertz function with different age shifts will all be linear and parallel to each other in the Gompertz plot. The situation is very different for the Weibull function—it is linear in the Weibull plot for only one particular starting age, and any shifts in a starting age produce a different function. Specifically, if a "true" starting age is larger than assumed, then the resulting function will be a nonlinear concave-up curve in the Weibull plot indicating model misspecification and leading to a bias in estimated parameters. Thus, researchers choosing the Weibull function for data analysis have first to resolve an uneasy biological problem—at what age does aging start?

An alternative graceful mathematical solution of this problem would be to move from a standard two-parameter Weibull function to a more general three-parameter Weibull function, which has an additional "location parameter'' y (Clark, 1975):

^(x) = a(x — y)^, x > y, and equal to zero otherwise

Parameters of this formula, including the location parameter y, could be estimated from the data through standard fitting procedures, thus providing a computational answer to a question "when does aging start?'' However, this computational answer might be shocking to researchers, unless they are familiar with the concept of initial damage load, which is discussed elsewhere (Gavrilov and Gavrilova, 1991; 2001; 2004b; 2005).

In addition to the Gompertz and the standard two-parameter Weibull laws, a more general failure law was also suggested and theoretically justified using the systems reliability theory. This law is known as the binomial failure law (Gavrilov and Gavrilova, 1991; 2001; 2005), and it represents a special case of the three-parameter Weibull function with a negative location parameter:

The parameter x0 in this formula is called the initial virtual age of the system, IVAS (Gavrilov and Gavrilova, 1991; 2001; 2005). This parameter has the dimension of time and corresponds to the age by which an initially ideal system would have accumulated as many defects as a real system already has at the starting age (at x = 0). In particular, when the system is initially undamaged, the initial virtual age of the system is zero and the failure rate grows as a power function of age (the Weibull law). However, as the initial damage load is increasing, the failure kinetics starts to deviate from the Weibull law, and eventually it evolves to the Gompertz failure law at high levels of initial damage load. This is illustrated in Figure 5.2, which represents the Gompertz plot for the data generated by the binomial failure law with different levels of initial damage load (expressed in the units of initial virtual age).

Note that as the initial damage load increases the failure kinetics evolves from the concave-down curves typical to the Weibull function, to an almost linear dependence between the logarithm of failure rate and age (the Gompertz function). Thus, biological species dying according to the Gompertz law may have a high initial damage load, presumably because of developmental noise, and a clonal expansion of mutations occurred in the early development (Gavrilov and Gavrilova, 1991; 2001; 2003a; 2004b).

Figure 5.2. Failure kinetics of systems with different levels of initial damage. Dependence 1 is for initially ideal system (with no damage load). Dependence 2 is for system with initial damage load equivalent to damage accumulated by 20-year-old system. Dependencies 3 and 4 are for systems with initial damage load equivalent to damage accumulated respectively by 50-year-old and 100-year-old system. Source: Gavrilov and Gavrilova, 2005.

Figure 5.2. Failure kinetics of systems with different levels of initial damage. Dependence 1 is for initially ideal system (with no damage load). Dependence 2 is for system with initial damage load equivalent to damage accumulated by 20-year-old system. Dependencies 3 and 4 are for systems with initial damage load equivalent to damage accumulated respectively by 50-year-old and 100-year-old system. Source: Gavrilov and Gavrilova, 2005.

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