10.1 Leading Causes of Death

Figure 10.1 illustrates mortality patterns for non-Hispanic white females in the United States for the years 1999 and 2000. The top row of panels shows the age-specific death rate per 100,000 individuals on a log-log scale. The columns plot all causes of death, death by heart disease, and death by cancer.

Figure 10.1

Age-specific mortality patterns by cause of death. Data averaged for the years 1999 and 2000 for non-Hispanic white females in the United States from statistics distributed by the National Center for Health Statistics, http://www.cdc.gov/nchs/, Worktable (more...)

The curves for death rate in the top row have different shapes. To study quantitative characteristics of death rates, it is useful to present the data in a different way. The second row of panels shows the same data, but plots the age-specific acceleration of death instead of the age-specific rate of death. The log-log acceleration (LLA) is simply the slope of the rate curve in the top panel at each age. Plots of acceleration emphasize how changes in the rate of mortality vary with age (Horiuchi and Wilmoth 1997, 1998; Frank 2004a).

The bottom row of panels shows one final plotting transformation to aid in visual inspection of mortality patterns. The bottom row takes the plots in the row above, transforms the age axis to a linear scale to spread the ages more evenly, and applies a mild smoothing algorithm that retains the same shape but smooths the jagged curves. I use the transformations in Figure 10.1 to plot mortality patterns for the leading causes of death in Figure 10.2, using the style of plot in the bottom row of Figure 10.1.

Figure 10.2

Age-specific acceleration of mortality by cause of death. Data averaged for the years 1999 and 2000 for non-Hispanic white males in the United States from statistics distributed by the National Center for Health Statistics http://www.cdc.gov/nchs/, Worktable (more...)

Figure 10.2 illustrates the mortality patterns for non-Hispanic white males in the United States for the years 1999 and 2000. Each plot shows a different cause of death and the percentage of deaths associated with that cause.

The panels in the left column of Figure 10.2 show causes that account for about one-half of all deaths. Each of those causes shares two attributes of age-specific acceleration. From early life until about age 80, the acceleration in mortality increases in an approximately linear way. After age 80, acceleration declines sharply and linearly for the remainder of life. Some of the causes of death also have a lower peak between 30 and 40 years.

The panels in the upper-right column of Figure 10.2 show causes that account for about one-third of all deaths. These causes follow steep, linear rises in mortality acceleration up to 40–50 years, and then steep, nearly linear declines in acceleration for the remainder of life. The bottom-right column of panels shows two minor causes of mortality that are intermediate between the left and upper-right columns.

What can we conclude from these mortality curves? The patterns by themselves do not reveal the underlying processes. However, the patterns do constrain the possible explanations for changes in age-specific mortality. For example, any plausible explanation must satisfy the constraint of generating an early-life rise in acceleration and a late-life decline in acceleration, with the rise and fall being nearly linear in most cases. A refined explanation would also account for the minor peak in acceleration before age 40 for certain causes.

10.2 Multistage Hypotheses

The mortality curves show a rise in acceleration to a mid- or late-life peak, followed by a steep and nearly linear decline at later ages.

In earlier chapters, I provided an extensive analysis of multistage models. Within the multistage framework, many alternative assumptions can often be fit to the same age-incidence pattern. Thus, fits to the data can only be regarded as a way to generate specific hypotheses. With that caveat in mind, I describe some multistage assumptions that fit the mortality curves and thus provide one line for the development of particular hypotheses (Frank 2004a).

Several alternative models may cause a rise in acceleration through the first part of life. Perhaps the simplest alternative focuses on the transition rates between stages of progression. If transition rates increase slowly with age, then acceleration will rise with age (Figures 6.8, 6.9).

With regard to the late-life decline in acceleration, all multistage models produce a force that pushes acceleration down at later ages. That downward force comes from the progression of individuals, as they grow older, through the early stages of disease (Figures 6.1, 6.2).

If, for example, n stages remain before death, then the predicted slope of the log-log plot (acceleration) is n − 1. As individuals age, they tend to progress through the early stages. If there are n stages remaining at birth, then later in life the typical individual will have progressed through some of the early stages, say a of those stages. Then, at that later age, there are n − a stages remaining and the slope of the log-log plot (acceleration) is n − a − 1. As time continues, a rises and the acceleration declines (Frank 2004a, 2004b).

10.3 Reliability Models

For cancer, I have been using various stepwise multistage models. Those stepwise models were originally developed for cancer in the 1950s (see Chapter 4) based on the idea of a sequence of changes to cells or tissues, for example, a sequence of somatic mutations in a cell lineage. Later empirical research has supported stepwise progression, based on both genetic and morphological stages in tumorigenesis.

Cancer researchers sometimes argue about what kinds of changes to cells and tissues determine stages in progression, the order of such changes, the number of different pathways of progression for a given type tissue and tumor, and how many rate-limiting changes must be passed for carcinogenesis. But those arguments take place within the multistage framework, which provides the only broad theoretical structure for studies of cancer. The multistage framework developed internally within the history of cancer research, with relatively little outside influence. For those reasons, I have presented the multistage theory with reference only to cancer.

By contrast, studies of heart disease and other causes of mortality face different biological problems and have a different theoretical tradition. On the biological side, most diseases do not have widely accepted stages of progression or widely accepted processes, such as somatic mutation, that drive transitions between stages. Certainly, some multistage progression ideas exist for noncancerous diseases (Peto 1977), and some theories about somatic mutation have been posed (e.g., Andreassi et al. 2000; Vijg and Dolle 2002; Kirkwood 2005; Wallace 2005; Bahar et al. 2006). But those ideas and theories do not form a cohesive framework in current studies of mortality.

Several theories of age-specific mortality have been based on multiple stages or multiple states of progression. Specific models almost always derive from reliability theory—the engineering field that evaluates time to failure for manufactured devices (Gavrilov and Gavrilova 2001).

In engineering, components of a device that protect against failure may be arranged in various pathways. Serial protection means that system failure follows a pathway in which first one component fails, followed by a second component, and so on; the probability of failure of later components in the sequence occurs conditionally on the failure of earlier components in the sequence. Parallel protection describes functional redundancy, in which any single functioning component keeps the system going; failure occurs only after all redundant components fail; and component failures occur independently. Various combinations of serial and parallel pathways may be designed.

Reliability theory calculates time to failure (mortality) based on assumptions about component failure rates and pathways by which components are related. Obviously, the multistage theory I developed earlier forms a branch of reliability theory. However, the reliability theory found in texts focuses on engineering problems, and those problems rarely match the particular biological scenarios for cancer progression. So, although the principles exist in reliability texts, many of the specific results in my theory chapters are new.

Gavrilov and Gavrilova (2001) provided a nice review of reliability theory applied to human mortality. They note that when system failure depends on the simultaneous failure of several components, the acceleration of age-specific mortality declines later in life. I have already discussed the idea several times. If system failure requires failure of n components, then log-log acceleration (LLA) is n − 1. As systems age and components fail, say a have failed, then LLA tends to drop toward n − a − 1. Details vary, but the idea holds widely. Vaupel (2003) gives a good, intuitive description of how multicomponent reliability may explain the late-life mortality plateau.

In light of reliability theory, we can state more generally an explanation for the late-life decline in the acceleration of mortality (Frank 2004a). Suppose a measurable disease outcome, such as death, occurs only after several different rate-limiting events have occurred. Each event has at least some aspect of its time course that is independent of other events. If so, then the dynamics of onset will not follow the course for a single event model, and will instead be the outcome of a multi-event model. The events do not have to follow one after another or be arranged in any particular pattern. The key is at least partial independence in the time course of progression for each event, and final measured outcome (mortality) only occurring after multiple events have occurred.

Similarly, a condition for a midlife rise in acceleration is a slow increase in the rate at which individual components fail (Frank 2004a).

10.4 Conclusions

I have included a discussion of mortality in a book otherwise devoted to cancer for two reasons. First, from the vantage point of the general reliability problem, one can more easily see what is necessary to explain patterns of cancer incidence. Second, the extensive development of multistage theory I presented in earlier chapters provides just the sort of quantitative background needed to use reliability theory fruitfully in the general study of mortality.

One might now ask: If reliability theory applies to everything, then does it have any explanatory power? This question seems reasonable, but I think it is the wrong question. The reliability framework provides tools to help us formulate testable hypotheses. That framework by itself is not a hypothesis.

For cancer, I have shown how multistage theory leads to many useful hypotheses. For example, I have used the theory to predict how age-incidence curves change in response to genetic perturbations (inherited mutations) and environmental perturbations (mutagens and mitogens). Reliability theory will develop into a useful tool for studies of mortality and aging to the extent that one can develop useful hypotheses about how age-incidence curves change in response to measurable perturbations.

10.5 Summary

This chapter finishes my three empirical analyses of disease dynamics in light of multistage progression models. The three empirical analyses covered genetics, chemical carcinogenesis, and aging. The next section of the book turns to evolutionary processes: What factors shape the population frequencies of predisposing genetic variants? How does tissue architecture affect the somatic evolution of cancer?