### Multistage model

A multistage model with *n* stages can be written as:

where *i* = 1,...,*n*-1, *x*
_{
j
}(*t*) is the frequency of individuals born at time 0 that are in stage *j* at time *t*, *u*
_{
j
}(*t*) is the rate at which individuals move from stage *j* to stage *j*+1, *d*
_{
j
}(*t*) is the death rate from other causes for individuals in stage *j*, and the dot is the derivative with respect to age. An individual dies when it arrives in the *n*th stage, thus *x*
_{
n
}(*t*) is the cumulative probability that an individual dies from the final stage of disease progression between ages 0 and *t*. The probability interpretation arises by letting *x*
_{0}(0) = 1, *x*
_{
k
}(0) = 0 for *k* = 1,...,*n*, and accumulating deaths by other causes in a class, with *D*(0) = 0.

### Age-specific incidence and acceleration

The cumulative probability of death from the final stage of a particular disease from age 0 to age *t* is *x*
_{
n
}(*t*). The rate of change in the cumulative probability of death at age *t* is , which is the probability of death per year at age *t*. (Here, *t* is continuous, and is the probability density at the point *t*.) Data on death are often presented as age-specific incidence, which is the probability of death per year at age *t* from a particular disease divided by the probability of surviving to age *t*. In the notation here, the age-specific incidence is , where survival is . The age-specific incidence has dimensions 1/*t*. Age-specific acceleration is the derivative of age-specific incidence, d*I*(*t*)/d*t*, which has dimensions 1/*t*
^{2}.

### Interpretation of log-log acceleration

Use of logarithms provides a scale-free measure of change. In other words, differences on a logarithmic scale summarize percentage change in a variable independently of the value of the variable. This can be seen by examining the derivative of the logarithm for a variable *x*, which is:

The right side is the change in *x* divided by *x*, which measures the fractional change in *x* independently of how large or small *x* is.

For example, if we wanted to measure the percentage increase in the age-specific incidence for a given percentage increase in age, then we need to measure in a scale-free way changes in both age-specific incidence and age. We obtain a scale-free measure by defining the log-log acceleration (LLA) at age *t* as:

The previous section showed that d*I*(*t*)/d*t* is the age-specific acceleration, so LLA is just a normalized (nondimensional) measure of age-specific acceleration.

### Log-log acceleration in the multistage model

The simplest models assume that the *u*
_{
j
}values do not change with age [8]. If nearly all members of a cohort at birth (age 0) concentrate in stage 0, then newborns mostly have *n* steps remaining. Later, as long as most of the probability among those alive remains concentrated in stage 0, the cumulative probability of death is *x*
_{
n
}(*t*) ≈ *U* *t*
^{n}, where *U* depends on the various constant transition rates, *u*
_{
j
}(see below).

The first derivative with respect to age is the age-specific mortality rate,

, and so *I*(*t*) = *Unt*
^{n-1}/*S*(*t*). Taking the logarithm of both sides yields log(*I*) ≈ log(*Un*) + (*n*-1)log(*t*) - log(*S*(*t*)). Thus, with constant transition rates, the log-log plot of age-specific incidence is approximately linear with age and has a slope of *n*-1. This can be seen from the log-log acceleration, which is d log (*I*)/d log(*t*) ≈ *n*-1, as long as the total number of surviving individuals, *S*(*t*), does not change too fast.

To sum up so far, with the simple model of constant transition rates between stages, the standard approximation for the multistage model that has been applied to cancer [8] yields a constant log-log acceleration of *n*-1 that is independent of age. The standard approximation is not a proper analysis – the point here is to show how one might conclude that a multistage model with *n* steps yields a log-log plot of incidence versus age that is linear with a slope of *n*-1.

I pointed out that as individuals age, they are no longer concentrated mostly in stage 0 [12]. If at age *t*, most individuals had passed to stage *a*, then the log-log acceleration at age *t* would be *n*-*a*-1. The actual acceleration at age *t* depends on the distribution of individuals over stages at that age, that is, the values of the *x*
_{
i
}(*t*).

As individuals grow older, they move through the early stages and become concentrated in the later stages. This causes acceleration to decline with age [12]. Among old individuals, most will have only one or few stages remaining, and the acceleration drops toward zero.

The analysis of acceleration in this section assumes that the transition rates, *u*
_{
j
}(*t*), do not change with time. Below, I consider the case in which the *u*
_{
j
}rise with time, causing acceleration to rise with age over the middle years of life [12].

### Solution of a multistage model with equal transition rates

If the transitions rates are constant and equal, *u*
_{
j
}= *u* for all *j*, and the nonspecific death rates are constant and equal, *d*
_{
j
}= *d* for all *j*, then we can obtain an explicit solution for the multistage model. This provides a special case that helps to interpret more complex assumptions that must be evaluated numerically. The solution is *x*
_{
i
}(*t*) = *e*
^{-(u+d)t}(*ut*)^{i}/*i*! for *i* = 0,...,*n*-1, with the initial condition that *x*
_{0}(0) = 1 and *x*
_{
i
}(0) = 0 for *i* > 0.

In the multistage model given above, the derivative of *x*
_{
n
}(*t*), which is the age-specific mortality rate, is . From the solution for *x*
_{
n-1}(*t*), we have

Age-specific incidence is:

Log-log acceleration is:

where

.

If transition rates between steps vary, all steps influence acceleration early in life, whereas the slowest steps dominate the number of remaining steps and the acceleration later in life.

If the transition rates, *u*, rise with age, then many older individuals will have passed through the early stages, causing a strong decline in acceleration later in life.

### Midlife rise in acceleration caused by increasing transition rates

An increase in transition rates, *u*
_{
j
}(*t*), with advancing age causes a midlife rise in acceleration. And, as mentioned in the preceding paragraph, faster transitions move more older individuals into later stages, causing a late-life decline in acceleration.

To provide a simple model of changes in transition rates, let *u*
_{
j
}(*t*) = *uf(t)*, where *f* is a function that describes changes in transition rates over different ages. We will usually want *f* to be a nondecreasing function that changes little in early life, rises in midlife, and perhaps levels off late in life. In numerical work, one commonly uses the cumulative distribution function (CDF) of the beta distribution to obtain various curve shapes that have these characteristics. Following this tradition, I use:

where *T* is maximum age so that *t*/*T* varies over the interval [0,1], and the parameters *a* and *b* control the shape of the curve.

We need *f* to vary over [1,*F*], where the lower bound arises when *f* has no effect, and *F* sets the upper bound. So, let *f*(*t*) = 1+(*F*-1)β(*t*). See Figure 3 for an example.

### The lifetable aging rate

Previous work on aging has studied acceleration in mortality by using the lifetable aging rate measure [2, 7, 15, 16]. The classical measure of mortality rate in the standard lifetable is *m*(*x*), the rate of death at age *x*. Although *m*(*x*) is usually calculated from data, it has roughly the same meaning as *I*(*t*) in the multistage analysis above.

The lifetable aging rate (LAR) is defined as the slope of the mortality rate at age *x* normalized by the mortality rate at that age [2, 7, 15, 16], that is:

This measure analyzes scale-free changes in mortality relative to scale-specific changes in age. For example, one may wish to know by what percentage mortality changes in a particular year of life. By contrast, log-log acceleration measures percentage change in mortality relative to percentage change in age.

Both measures provide a picture of mortality acceleration. If one does not have an underlying model of the processes that influence aging, then the LAR measure provides a reasonable approach because it arises from analysis of the statistics of life and death at particular ages. In the context of a multistage model of the aging process, the log-log acceleration gives a weighted measure of the number of stages remaining at a particular age. Extensions to the multistage model can also be incorporated – for example, changes in the transition rates, as mentioned above. Also, to the extent that aging rates change exponentially with age, logarithmic scaling of age may provide a more natural measure.