40 60 Age, years
Fig. 1. Typical patterns of intersection between male and female overall cancer incidence rates: Denmark (1988-1992), India (Bombay, 1988-1992), USA (Connecticut, 1960-1962 and 1988-1992), and Japan (Miyagi prefecture, 1962-1964 and 1988-1992). 'M' - males, 'F' - females; data source: -.
Common sense suggests that male and female age patterns of overall cancer incidence rate must differ because of biologically-based differences in specific cancer sites (such as breast, ovarian cancers for females and prostate cancer for males). However, the differing age-pattern, and its relative stability over time and place, cannot be predicted from such a consideration.
The differences in mechanisms involved in cancer initiation and development for males and females would be better understood if one could explain forces shaping the age-trajectories of cancer incidence rates, evaluate the role
Japan (Miyagi) USA (New York State)
Japan (Miyagi) USA (New York State)
- Fig. 2. Female and male "cohort" cancer incidence rates in Japan (Miyagi prefecture), 1929 "cohort" and in USA (New York State), 1920 "cohort". Data source: -.
of gender in this process, as well as factors responsible for observed timetrends of these rates. Below, we describe the approach, which has the capability to explain the relative stability of the age pattern of cancer incidence and mortality rates for males and females, as well as their change over time. The approach explores the possibility to represent cancer incidence rate in terms of age-related processes. This involves a new mathematical model of carcinogenesis . This model represents cancer incidence rate as a sum of two components reflecting basic types of age-related changes in an organism (see ). We show that in contrast to traditional models of carcinogenesis, the new model, which we call the ontogenetic model, captures main features of the age pattern and time-trend of cancer incidence rates. It also explains the relative stability of the intersection pattern of male and female cancer incidence rates. We illustrate this model by the application to data on overall cancer incidence rates in Japan (Miyagi prefecture) (data source: -).
We apply our model to data on female and male cancer incidence rates in Japan (Miyagi prefecture). The International Agency for Research on Cancer (IARC) provides the data on cancer incidence in different countries, in seven volumes (-). Each volume covers a time period of several years (usually 3-5 years) for each country (or province and/or ethnic group). The periods vary for different countries. In each volume, female and male average annual cancer incidence per 100000 over the corresponding time period are given for the specific country (province and/or ethnic group), in five-year age groups up to age 85+ (for some countries the first group 0-4 is separated into 0 and 1-4). The data are given for separate sites and for all sites combined. Not all countries are presented in each volume. The longest time series are available for Japan (Miyagi prefecture). Each of the seven volumes contains the data on cancer incidence in this territory over different time periods. This data set is the foremost one to analyze time trends in cancer incidence rates over time, and is used in this study.
3 Three Components of the Individual Aging Process
Ukraintseva and Yashin  suggested studying individual aging by analyzing three internal biological processes that have different age-related dynamics. These include basal, ontogenetic, and exposure-related components. These processes also affect the shape of cancer incidence rate. We assume that any observed age pattern of this rate is the result of the combined influence of these three age-related processes.
The main characteristic of the basal component is the age-related decline in the individual rate of living (i.e., in the metabolic and information processing rates). This component is responsible for the deceleration of change in many physiological parameters of an organism with advanced age. It can be responsible for the leveling-off of the morbidity rate at old ages, observed for many chronic diseases (see ). This component may also contribute towards the acceleration in rates of onset of acute health disorders leading to death (due to deceleration in the potency to recover, and hence due to the progressive decline in individual stress resistance at old ages).
The term ontogenetic refers to the developmental history of an organism. The ontogenetic component of aging represents effects of metabolic switches accompanying changes in stages of ontogenesis during life (e.g., in infancy, in the reproductive period and at the climacteric). This component of individual aging can be responsible for non-monotonic change in vulnerability of an organism to stress and diseases due to a variation in hormonal balance in an organism. The exposure-related component is responsible for long-term accumulation of specific lesions in an organism, which contribute to an increase in the morbidity rate.
A properly balanced combination of all these components may be used for an explanation of age-specific morbidity and mortality patterns in human populations, including cancer morbidity. The obvious advantage of such an approach is that by dividing individual aging into the processes with different age-related dynamics, one has an opportunity to use information from different studies focused on specific aspects of individual aging. For example, the age pattern of ontogenetic vulnerability used in the respective component of cancer incidence rate in our study was obtained from asthma studies (see ). A similar pattern is also produced in the studies of other chronic diseases, as shown in . The limitations of this approach are associated with the large amounts of data required for identification of model parameters.
4 The Incorporated Ontogenetic Model of Cancer
To capture the age pattern, time-trends, as well as the intersection of age-specific incidence rates for males and females, we incorporate the three-component model of individual aging  into the tumor latency model of carcinogenesis . We specify patterns of age-dependence for different components in the oldest cohort, and set a rule of changing these components from one cohort to the next to construct the corresponding period rates. Following this idea we define cancer incidence rates as where i = 1 ...n stands for a cohort, hi (x) is an age-specific intensity of unrepaired lesion formation in ith cohort, and F(t) is a cumulative probability distribution function of progression times. We suppose that progression times are gamma distributed with fixed shape and scale parameters k and A and the functions F(t) are the same for all cohorts.
We also assume that the age-specific intensity of unrepaired lesion formation hi(x) is a result of the combined influence of age-related processes in an organism which are represented by the basal, ontogenetic and exposure-related components described above.
The part of the hazard rate, associated with the basal component, should be increasing with the declining rate with age. Respectively, the part of the hazard rate, associated with the exposure-related component, should exhibit accelerated increase with age by definition of this component. For the sake of simplicity, we combine the exposure-related and basal effects and specify one general pattern of hazard rate for these components (referred to as time-component ). We denote this general component hfme (x), where index i is associated with the birth year of the cohort, and x is an individual's age. Thus, the exposure-related lesions in an organism accumulate with age, on the grounds of a basal deceleration in the individual rate of living (e.g., due to general deceleration in information processing) in an organism.
The ontogenetic component has a wave-like shape for both males and females, with peaks at early ages and around ages of climacterics for females, and between ages 55 and 65 for males. The peaks correspond to the ages of hormonal imbalance where this component largely influences risks of morbidity and mortality. A similar pattern of morbidity is observed for many human chronic diseases (see , , , ). In principle, one can use these patterns to model the ontogenetic component. However, these rates, in essence, reflect not only the ontogenetic changes, but also the other factors responsible for the manifestation of the disease. Thus, to model the ontogenetic changes at advanced ages influencing unrepaired lesion formation, we use the function with a pronounced peak around some specific age, and zero otherwise. The peak is around the age of menopause for females, and the pattern is shifted x x