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to the right for males (see Fig. 3). We ignore the peak at early ages for the sake of simplicity. This component is the same for all cohorts.

Denote hont (x) the value of the ontogenetic component of hazard rate at age x and let hoime (x) be the value of the general component (combined from exposure-related and basal effects) at age x for the oldest cohort. We suppose that the last component may change for different cohorts due to an increasing influence of harmful factors on an organism. The dynamics of this component for ith cohort, i = 1. ..n is described as where parameter d characterizes the growth rate of the hazard rate over time. The introduced values hont (x) and hfme (x) are used to define the age-specific intensity of unrepaired lesion formation for ith cohort, i = 1. ..n, as a sum of these two components,

5 Application of the Ontogenetic Model to Data on Cancer Incidence Rate by Sex

We apply the model to data on cancer incidence in Japan (Miyagi prefecture) (data source: [3]—[9]). The parameters of the model are fixed at d = 0.2, k = 25, and A =1. The patterns of the ontogenetic component (hont (x)) and the time-dependent component in the oldest cohort (hoime (x)), for both males and females, are shown in Fig. 3. The trajectories of hoime (x) were assumed piecewise constant and were estimated using Matlab's least-square routine.

The observed and estimated male and female incidence rates are shown in Fig. 3.2. Table 1 illustrates the fit of the model. Note that for the sake of simplicity, we used a rather straightforward pattern of htlme (x) and a number of fixed parameters of the model. A more elaborated specification of htlme (x) and an estimation of all the parameters would likely provide a better fit to the data. However, the message here is that this model captures all the features of the observed cancer incidence rates mentioned above. It describes an increase of the rates over time, the deceleration and decline of the rates at the oldest old ages, and the intersection of male and female incidence rate curves near the age of female climacteric. Increasing htlme (x) in cohorts gives an increase of the period incidence rates over time. The specification of a cumulative probability distribution function of progression times and difference in ontogenetic component for males and females produces a decline of the rates at oldest old ages and the intersection of the male and female rates.

Fig. 3. The ontogenetic model of cancer applied to data on overall cancer incidence rates in Japan (Miyagi prefecture): curves of the combined exposure-related and basal component ("time component") in the oldest cohort and the ontogenetic component for males and females. Data source: [3]-[9].

Fig. 3. The ontogenetic model of cancer applied to data on overall cancer incidence rates in Japan (Miyagi prefecture): curves of the combined exposure-related and basal component ("time component") in the oldest cohort and the ontogenetic component for males and females. Data source: [3]-[9].

Table 1. The ontogenetic model of cancer applied to data on overall female and male cancer incidence rates in Japan (Miyagi prefecture): norm of differences (columns 'Norm') and correlation (columns 'Corr') between modeled and observed incidence rates. Data source: [3]-[9].

Period Norm (Females) Corr (Females) Norm (Males) Corr (Males)

Table 1. The ontogenetic model of cancer applied to data on overall female and male cancer incidence rates in Japan (Miyagi prefecture): norm of differences (columns 'Norm') and correlation (columns 'Corr') between modeled and observed incidence rates. Data source: [3]-[9].

Period Norm (Females) Corr (Females) Norm (Males) Corr (Males)

1959-

1960 436.105

0.972

384.487

0.990

1962-

-1964 285.894

0.982

614.039

0.973

1968-

1971 211.032

0.993

198.371

0.998

1973-

1977 200.417

0.995

502.704

0.994

1978-

1981 233.626

0.998

452.061

0.999

1983-

1987 94.588

0.999

196.311

0.999

1988-

1992 165.258

0.999

201.993

males (observed)

g 1500

males (observed)

g 1500

30 40 50 60 70 80

30 40 50 60 70 80

males (observed)

males (observed)

30 40 50 60 70 80

0 10 20 30 40 50 60 70 80

30 40 50 60 70 80

1988-92

1988-92

Fig. 4. The ontogenetic model of cancer applied to data on overall cancer incidence rates in Japan (Miyagi prefecture): male and female observed and modeled rates for different time periods. Data source: [3]-[9].

6 Conclusion

The analysis of epidemiological data on cancer shows that cancer became the leading cause of death in most productive ages of human life. The range of ages where cancer maintains its leading role tends to increase with years. In many developed countries the overall cancer incidence rate still tends to increase. Many factors associated with the economic progress could be mainly responsible for the increase in cancer incidence rate. Among those are the improved cancer diagnostics, elevated exposure to external carcinogens such as car exhaust pollution, and factors associated with a Western-like life style (such as dietary habits, new medicines and home-use chemicals). This increase is not likely to be explained by the improvement in cancer diagnostics alone. The survival of cancer patients differs in different countries, despite continuing efforts in sharing medical information on efficiency of cancer treatment procedures and respective drugs.

Different models of carcinogenesis can explain some of the observed phenomena of human cancer incidence rates. The literature on cancer modeling is extensive. The list of classical models includes the multistage model of cancer by Armitage-Doll (AD model), the two-event model by Moolgavkar-Venzon-Knudson (MVK model), and the tumor latency model by Yakovlev and Tsodikov. These models describe biological mechanisms involved in cancer initiation and development, and derive mathematical representation for cancer incidence rate. This representation can then be used in the statistical estimation procedures to test hypotheses about regularities of respective mechanisms and the validity of basic assumptions. The multi-stage model of carcinogenesis [2] explains the increase of the rates over age, but does not describe the entire age-trajectory of cancer incidence rate and does not explain the intersection of male and female incidence rates. The two-mutation model [10], as well as the tumor latency model (see [16], [17]), is capable of describing the entire age-trajectory of cancer incidence rate. However, they cannot explain the stable intersection pattern of male and female cancer incidence rates.

It is clear that the overall cancer incidence rates for males and females do have different age patterns. This conclusion stems from the basic biological knowledge about the difference between male and female organisms. This difference is responsible for the different susceptibility to cancer of certain sites (e.g., breast cancer). The exposure to hazardous materials can also be different for males and females because of their difference in social and economic life. There is, however, neither a theory nor a mathematical model that predicts how age-trajectories of cancer incidence rates will behave, and to what extent these trajectories are affected by environmental and living conditions experienced by populations in different countries.

In this paper we show that the relative difference in age patterns of male and female cancer incidence rates may be explained by the difference in onto-genetic curves of age-dependent susceptibility to cancer for males and females.

This is because the peak of hormonal imbalance in females is between ages 45 and 55, when the reproductive system ultimately stops functioning. In males this peak is shifted to the right (between 55 and 65). The age pattern of cancer incidence rate reflects the contribution of the ontogenetic component of age-related processes in an organism. The heterogeneity in individual frailty may also have a substantial contribution. The ontogenetic model is capable of describing the time trends and the stable pattern of intersection in the male and female incidence rates. In our recent paper [1], we pointed out that the universal pattern of male/female cancer incidence rates might also be a result of different strategies of resource allocation between "fighting" against external stresses and "fighting" against physiological aging used by male and female organisms. This effect needs further explanation, from both biological and mathematical perspectives. The availability of molecular-biological and epidemiological data on stress resistance (e.g., cellular sensitivity to oxidative stress) would allow for the development of more sophisticated mathematical models of such mechanisms. New models are also needed to explain age pattern and time-trends in male/female cancer mortality rates. These models should include information on cancer incidence rates as well as on survival of cancer patients.

Acknowledgements

The authors wish to thank Prof. James W. Vaupel for the opportunity to complete this work at the Max Planck Institute for Demographic Research, Germany.

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