Ulteriori informazioni
Stochastic processes often pose the difficulty that, as soon as a model devi ates from the simplest kinds of assumptions, the differential equations obtained for the density and the generating functions become mathematically formidable. Worse still, one is very often led to equations which have no known solution and don't yield to standard analytical methods for differential equations. In the model considered here, one for tumor growth with an immunological re sponse from the normal tissue, a nonlinear term in the transition probability for the death of a tumor cell leads to the above-mentioned complications. Despite the mathematical disadvantages of this nonlinearity, we are able to consider a more sophisticated model biologically. Ultimately, in order to achieve a more realistic representation of a complicated phenomenon, it is necessary to examine mechanisms which allow the model to deviate from the more mathematically tractable linear format. Thus far, stochastic models for tumor growth have almost exclusively considered linear transition probabilities.
Sommario
1. Introduction.- 2. Background of Statistical Studies of Carcinogenesis.- 3. Immunological Response as a Factor in Carcinogenesis.- 4. The Mathematical Model.- 5. Some Unsuccessful Approaches to the Approximation Problem.- 6. Stochastic Linearization.- 7. van Kampen's Method.- 8. Method of Linearized Transition Probabilities.- 9. The Quadratic Death Process.- 10. The Collective Model.- 11. Further Implications of the Immunological Feedback Model.- 12. Conclusion.- Appendix I.- Evaluation of the Integral I(t) from van Kampen's Method.- Appendix II.- Derivation of the Probability Density Function Obtained by the Method of Linearized Transition Probabilities.- Appendix III.- Fortran Program for the Computer Simulation.- Appendix IV.- Mathematical Induction for the Probability Density of the General Death Process.- Appendix V.- Iterative Routine in Fortran for the Quadratic Death Process.
Riassunto
Stochastic processes often pose the difficulty that, as soon as a model devi ates from the simplest kinds of assumptions, the differential equations obtained for the density and the generating functions become mathematically formidable. Worse still, one is very often led to equations which have no known solution and don't yield to standard analytical methods for differential equations. In the model considered here, one for tumor growth with an immunological re sponse from the normal tissue, a nonlinear term in the transition probability for the death of a tumor cell leads to the above-mentioned complications. Despite the mathematical disadvantages of this nonlinearity, we are able to consider a more sophisticated model biologically. Ultimately, in order to achieve a more realistic representation of a complicated phenomenon, it is necessary to examine mechanisms which allow the model to deviate from the more mathematically tractable linear format. Thus far, stochastic models for tumor growth have almost exclusively considered linear transition probabilities.
