A Stochastic Model for the Inheritance of the Cancer Proneness Phenotype

Summary

A continuous parameter, stationary Gaussian process is introduced as a first approach to the probabilistic representation of the phenotype inheritance process. With some specific assumptions about the components of the covariance function, it may describe the temporal behaviour of the “cancer-proneness phenotype” (CPF) as a quantitative continuous trait. Upcrossing a fixed level (“threshold”) u and reaching level zero are the extremes of the Gaussian process considered; it is assumed that they might be interpreted as the transformation of CPF into a “neoplastic disease phenotype” or as the non-proneness to cancer, respectively.

Ein stetiger stationärer Gauss-Prozess wird als erster Ansatz einer probabilistischen Darstellung des phänotypischen Vererbungsprozesses vorgestellt. Unter Annahme einiger spezifischer Komponenten der Kovarianzfunktion kann er das zeitliche Verhalten des krebsempfänglichen Phänotyps als quantitatives, stetiges Charakteristikum beschreiben. Das Überschreiten eines bestimmten Schwellenwertes und das Erreichen eines Null-Niveaus sind die Extreme des betrachteten Gauss-Prozesses: es wird angenommen, daß sie. als Transformation des krebsempfänglichen Phänotyps in einen “neoplastischen Krankheitsphänotyp” oder als Nichtemp-fänglichkeit für Krebs interpretiert werden könnten.

Key-Words:

Cancer Proneness Phenotype - Familial Cancer - Gaussian Process

Schlüssel-Wörter:

Krebsempfänglicher Phänotyp - familiärer Krebs - Gauss-Prozess

• REFERENCES

• 1 Bcaty T. H. Discriminating among single locus models using small pedigrees. Amer. J. med. Genet 6 1980; 229-240.
• 2 Belyaev Yu. K. On the number of intersections of a level by Gaussian stochastic processes I, II. Theor. Prob. Appl 11 1966; 106-113. 12 (1967) 392-404
• 3 Berman S. M. Asymptotic independence of the numbers of high and low level crossings of stationary Gaussian processes. Ann. Math. Statist 42 1971; 927-945.
• 4 Brémaud P. Point Processes and Queues. Martingale Dynamics. New York - Hei-delberg - Berlin: Springer; 1981
• 5 Burns T. L, Moll P. P. Schork, M.A.: Comparisons of different sampling designs for the determination of genetic transmission mechanisms in quantitative traits. Amer. J. hum. Genet 36 1984; 1060-1074.
• 6 Butler W. H, Driver H. E. Clinical significance of preneoplasia. In: Stoll B. A. (Ed.): Risk Factors and Multiple Cancers Chichester: Wiley; 1984: 135-154.
• 7 Cavalli-Sforza L. L, Feldman M. W. Evolution of continuous variation, III. Joint transmission of genotype, phenotype and environment. Genetics 90 1978; 391-425.
• 8 Cavalli-Sforza L. L, Feldman M. W. Dynamics and statistics of traits under the influence of cultural transmission. In: Morton N. E, Chung C. S. (Eds): Genetic Epidemiology New York: Academic Press; 1978: 133-143.
• 9 Chaganti R. S. K. Genetic indicators of cancer predisposition. In: Chaganti R. S, Genan J. (Eds): Genetics in Clinical Oncology New York - Oxford: Oxford University Press; 1985: 149-158.
• 10 Chatterji S. D, Mandrekar V. Equivalence and singularity of Gaussian measures and applications. In: Bharucha-Reid A. T. (Ed.): Probabilistic Analysis and Related Topics, Vol. 1 New York: Academic Press; 1978: 169-197.
• 11 Dobrushin R. L. Properties of sample functions of a stationary Gaussian process. Theor. Prob. Appl 5 1960; 120-122.
• 12 Doob J. L. Stochastic Processes. New York: Wiley; 1953
• 13 Emery M. Convariance des semimartin-gales gaussiennes. C. R. Acad. Sci. Paris 295 1982; 703-706.
• 14 Falconer D. S. The inheritance of liability to certain diseases, estimated from the incidence in relatives. Ann. hum. Genet 29 1965; 51-76.
• 15 Feldman M. W, Cavalli-Sforza L. L. On hereditary transmission in diseases of com-plex etiology. in: Sing D. V, Skolnick M. (Eds): Genetic Analysis of Common Diseases: Applications to Predictive Factors in Coronary Disease New York: A. R. Liss; 1979: 203-227.
• 16 Fieger W. Die Anzahl der γ-Niveau-Kreuzungspunkte von stochastischen Pro-zessen. Z. Wahrscheinlichkeitstheorie verw. Geb 18 1971; 227-260.
• 17 Gal'chuk L. I. Gaussian semimartingales. In: Krylov N. V, Lipster R. Sh, Novi-kov A. A. (Eds): Statistics and Contrai of Stochastic Processes New York: Optimization Software; 1985: 102-121.
• 18 Itô K. The expected number of zeros of continuous stationary Gaussian processes. J. Math. Kyoto Univ 3 1964; 207-216.
• 19 Jain N. C, Monrad D. Gaussian quasi-martingales. Z. Wahrsch.theor. verw. Geb 59 1982; 139-159.
• 20 Karlin S. Approaches in modeling mode of inheritance with distributed traits. In: Sing C. F, Skolnick M. (Eds): Genetic-Analysis of Common Diseases: Applications to Predictive Factors in Coronary Disease New York: A. R. Liss; 1979: 229-257.
• 21 Karlin S. Models of multifactorial inheritance. Theor. Popul. Biol 15 1979; 308-355. 356-393; 394-423; 424-438 17 (1980) 255-275
• 22 Karlin S, Cameron E. C, Williams P. T. Structured exploratory data analysis applied to mode of inheritance. In: Krish-naiah P. R. (Ed.): Developments in Statistics, Vol.4 New York: Academic Press; 1983: 185-277.
• 23 Kopp P. E. Martingales and Stochastic Integrals. Cambridge: Cambridge Univ. Press; 1984
• 24 Leadbetter M. R, Lindgren G, Root-zén H. Extremes and Related Properties of Random Sequences and Processes. New York - Heidelberg – Berlin: Springer-Verlag; 1983
• 25 Lewontin R. C. Models of natural selection. Lecture Notes in Biomathematics, Vol. 39. Berlin - Heidelberg -New York: Springer-Verlag; 1980: 15-27.
• 26 Lindgren G. Extreme values of stationary normal processes. Z. Wahrscheinlichkeits-theorie verw. Geb 77 1971; 39-47.
• 27 Lindgren G. Model processes in nonlinear prediction with application to detection and alarm. Ann. Probability 8 1980; 775-792.
• 28 Lipkin M. The development of a risk profile for colorectal cancer by utilizing the proliferative and antigenic characteristics of colonic epithelial cells. In: Rozen P, Winawer S. J. (Eds): Secondary Prevention of Colorectal Cancer Basel: Karges; 1986: 257-269.
• 29 Liptser R. S, Shiryayev A. N. Statistics of conditionally Gaussian random sequen-ces. Proc. 6th Berkeley Symp. Math. Statist. Probability, Vol II. Berkeley: Univ. California Press; 1972: 389-422.
• 30 Lynch H. T. Genetics, etiology and human cancer. Prevent. Med 9 1980; 231-243.
• 31 Marcus M. B. Level crossings of a stochastic process with absolutely continuous sample paths. Ann. Probability 5 1977; 52-71.
• 32 Martin R. D. Schwartz, S. C: On mixture, quasi-mixture and nearly normal pro-cesses. Ann. Math. Statist 43 1972; 948-967.
• 33 Morton N. E, Rao D. C, Lalouel J.-M. Methods in Genetic Epidemiology. Basel: Karger; 1983
• 34 Motulsky A. G. Ecogenetics: Genetic variation in susceptibility to environmental agents. Proc. 5th Intern. Congress of Human Genet Amsterdam: Excerpta Medica; 1977: 375-385.
• 35 Neveu J. Discrete Parameter Martingales. Amsterdam: North Holland; 1975
• 36 Orey S. F-Processes. 5th Berkeley Symp. Math. Statist. Probabil., Vol.11. Berkeley: Univ. Calif Press; 1965: 301-313.
• 37 Schneider N. R, Williams W. R, Chaganti R. S. K. Genetic epidemiology of familial aggregation of cancer. Adv. Cancer Res 47 1985; 1-36.
• 38 Tautu P, Wagner G. An approach to oncological genetics. J. Cancer Res. Clin. Oncol 707 1984; 141-148.
• 39 Thompson J. N. Quantitative variation and gene number. Nature 255 1975; 665-668.
• 40 Veale A. M. O. Investigation of a genetic model. In: Lynch P. M, Lynch H. . (Eds): Colon Cancer Genetics New York: Van Nostrand-Reinhold; 1985: 17-29.
• 41 Vogel F, Motulsky A. G. Human Genetics. Problems and Approaches. 2nd edit. Berlin - Heidelberg - New York – Tokyo: Springer; 1986
• 42 Wagener D. K. Preferential mating: Nonrandom mating of a continuous phenotype. Theor. Popul. Biol 10 1976; 185-204.
• 43 Williams D. Diffusions, Markov Processes and Martingales. Chichester: Wiley; 1979
• 44 Ylvisaker N. D. The expected number of zeros of a stationary Gaussian process. Ann. Math. Statist 36 1965; 1043-1046.