Calculating tumor incidence rates in stochastic models of carcinogenesis. (English) Zbl 0859.92013
Summary: Multistage models of carcinogenesis are increasingly used in the estimation of risks from exposure to environmental agents. The two-stage model of carcinogenesis is routinely used because it agrees with much of the existing tumor incidence data, parallels the biological two-stage model, and has much of its mathematical details derived. However, recent findings on the mechanisms of carcinogenesis has led researchers to believe that there are a greater number of stages and a more complex structure to these models than a single pathway.
In this paper, a method for readily computing tumor incidence rates for arbitrarily complex multistage models is derived. The formulas for the two-stage model with time-varying rates are given explicitly. Simple rules for more complicated models are given, and computer codes able to implement these formulas are provided.
In this paper, a method for readily computing tumor incidence rates for arbitrarily complex multistage models is derived. The formulas for the two-stage model with time-varying rates are given explicitly. Simple rules for more complicated models are given, and computer codes able to implement these formulas are provided.
MSC:
| 92C50 | Medical applications (general) |
| 60J85 | Applications of branching processes |
| 92-08 | Computational methods for problems pertaining to biology |
| 92-04 | Software, source code, etc. for problems pertaining to biology |
Keywords:
Kolmogorov backward equations; carcinogenesis; tumor incidence rates; arbitrarily complex multistage models; two-stage model; time-varying ratesReferences:
| [1] | Muller, H. J., Radiation damage to the genetic material, Sci. Prog., 7, 93-493 (1951) |
| [2] | Nordling, C. O., A new theory on the cancer inducing mechanism, Br. J. Cancer, 8, 68-72 (1953) |
| [3] | Stocks, P., A study of the age curve for cancer of the stomach in connection with the theory of the cancer producing mechanism, Br. J. Cancer, 7, 407-417 (1953) |
| [4] | Slaga, T. J., Mechanisms involved in multistage skin tumor carcinogenesis, (Huberman, E.; Barr, S., Carcinogenesis, Vol. 10 (1985), Raven Press: Raven Press New York), 189-199 |
| [5] | Whittemore, A.; Keller, J., Quantitative theories of carcinogenesis, SIAM Rev., 20, 1-30 (1978) |
| [6] | Armitage, P.; Doll, R., The age distribution of cancer and a multistage theory of carcinogenesis, Br. J. Cancer, 8, 1-12 (1954) |
| [7] | Armitage, P.; Doll, R., A two-stage theory of carcinogenesis in relation to the age distribution of human cancer, Br. J. Cancer, 11, 161-169 (1957) |
| [8] | Kendall, D. G., Birth and death processes and the theory of carcinogenesis, Biometrika, 47, 316-330 (1960) |
| [9] | Neyman, J.; Scott, E., Statistical aspects of the problem of carcinogenesis, (Fifth Berkeley Symposium on Mathematical Statistics and Probability (1967), Univ. California Press: Univ. California Press Berkeley, CA), 745-776 |
| [10] | Moolgavkar, S.; Venzon, D., Two-event models for carcinogenesis: incidence curves for childhood and adult tumors, Math. Biosci., 47, 55-77 (1979) · Zbl 0422.92006 |
| [11] | Moolgavkar, S.; Knudson, A., Mutation and cancer: a model for human carcinogenesis, JNCI, 66, 1037-1052 (1981) |
| [12] | Greenfield, R. E.; Ellwein, L. B.; Cohen, S. M., A general probabilistic model of carcinogenesis: analysis of experimental urinary bladder cancer, Carcinogenesis, 5, 437 (1984) |
| [13] | Portier, C.; Kopp-Schneider, A., A multistage model of carcinogenesis incorporating DNA damage and repair, Risk Anal., 11, 535-543 (1991) |
| [14] | Kopp-Schneider, A.; Portier, C. J., A stem cell model for carcinogenesis, Math. Biosci., 120, 211-232 (1994) · Zbl 0795.92011 |
| [15] | Moolgavkar, S., Model for human carcinogenesis: actions of environmental agents, Environ. Health Perspect., 50, 285-291 (1983) |
| [16] | Portier, C.; Bailer, A., Two-stage models of carcinogenesis for historical control animals from the National Toxicology Program, J. Toxicol. Environ. Health, 27, 21-45 (1989) |
| [17] | Moolgavkar, S.; Luebeck, E. G., Multistage carcinogenesis: a population perspective on colon cancer, J. Nat. Cancer Inst., 84, 8, 610-618 (1992) |
| [18] | Moolgavkar, S. H.; Luebeck, E. G.; de Gunst, M.; Port, R. E.; Schwarz, M., Quantitative analysis of enzyme-altered foci in rat hepatocarcinogenesis experiments. I: Single agent regimen, Carcinogenesis, 11, 1271-1278 (1990) |
| [19] | Luebeck, E. G.; Moolgavkar, S. H.; Buchmann, A.; Schwarz, M., Effects of polychlorinated biphenyls in rat liver: quantitative analysis of enzyme-altered foci, Toxicol. Appl. Pharmacol., 111, 469-484 (1991) |
| [20] | Portier, C.; Kohn, M.; Sherman, C.; Lucier, G., Modeling the number and size of hepatic focal lesions following exposure to 2,3,7,8-TCDD, Organohalogen Comp., 21, 393-397 (1994) |
| [21] | Kopp-Schneider, A.; Portier, C. J., Birth and death/differentiation rates of papillomas in mouse skin, Carcinogenesis, 13, 973-978 (1992) |
| [22] | Tan, W., Stochastic Models of Carcinogenesis (1991), Marcel Dekker: Marcel Dekker New York |
| [23] | Barrett, J. C.; Wiseman, R. W., Cellular and molecular mechanisms of multistep carcinogenesis: relevance to carcinogen risk assessment, Environ. Health Perspect., 76, 65-70 (1987) |
| [24] | Vogelstein, B.; Fearon, E.; Hamilton, S.; Kern, S.; Preisinger, A.; Leppert, M.; Nakamura, Y.; White, R.; Smits, A.; Bos, J., Genetic alterations during colorectal-tumor development, N. Engl. J. Med., 319, 525-532 (1988) |
| [25] | Portier, C., Statistical properties of a two-stage model of carcinogenesis, Environ. Health Perspect., 76, 125-132 (1987) |
| [26] | Kopp-Schneider, A.; Portier, C. J., Carinoma formation in mouse skin painting studies is a process with greater than two stages, Carcinogenesis, 16, 53-59 (1995) |
| [27] | Sherman, C. D.; Portier, C. J.; Kopp-Schneider, A., Multistage models of carcinogenesis: an approximation for the size and number distribution of late-stage clones, Risk Anal., 14, 6, 1039-1048 (1994) |
| [28] | Sherman, C. D.; Portier, C. J., Quantitative analysis of multiple phenotype enzyme-altered foci in rat hepatocarcinogenesis experiments, Carcinogenesis, 16, 2499-2506 (1995) |
| [29] | Degunst, M.; Luebeck, E., Quantitative analysis of two-dimensional observations of premalignant clones in the presence or absence of malignant tumors, Math. Biosci., 119, 5-34 (1994) · Zbl 0828.92018 |
| [30] | Harris, T., The Theory of Branching Processes (1963), Springer-Verlag: Springer-Verlag Berlin · Zbl 0117.13002 |
| [31] | Parzen, E., Stochastic Processes (1962), Holden-Day: Holden-Day San Francisco · Zbl 0107.12301 |
| [32] | Kimmel, M., An equivalence result for integral equations with application to branching processes, Bull. Math. Biol., 44, 1, 1-15 (1982) · Zbl 0472.60069 |
| [33] | Karlin, S.; Taylor, H., A Second Course in Stochastic Processes (1981), Academic: Academic New York · Zbl 0469.60001 |
| [34] | Portier, C.; Sherman, C., The potential effects of chemical mixtures on the carcinogenic process within the context of the mathematical multistage model, (Yang, R., Risk Assessment of Chemical Mixtures: Biological and Toxicological Issues (1994), Academic: Academic New York), 665-686 |
| [35] | Tritshcer, A.; Goldstein, J.; Portier, C.; McCoy, Z.; Clark, G.; Lucier, G., Dose-response relationships for chronic exposure to 2,3,7,8-tetrachlorodibenzo-\(p\)-dioxin in a rat tumor promotion model: quantification and immunolocalization of CYP1A1 and CYP1A2 in the liver, Cancer Res., 52, 3436-3442 (1992) |
| [36] | Kohn, M.; Lucier, G.; Clark, G.; Sewall, C.; Tritscher, A.; Portier, C., A mechanistic model of the effects of dioxin on gene expression in the rat liver, Toxicol. Appl. Pharmacol., 120, 138-154 (1993) |
| [37] | Kopp-Schneider, A.; Portier, C. J.; Sherman, C. D., The exact formula for tumor incidence in the two-stage model, Risk Anal., 14, 6, 1079-1080 (1994) |
| [38] | Zheng, Q., On the exact hazard and survival functions of the MVK stochastic carcinogenesis model, Risk Anal., 14, 6, 1081-1084 (1994) |
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