Ebook: A Stochastic Model for Immunological Feedback in Carcinogenesis: Analysis and Approximations
Author: Neil Dubin (auth.)
- Tags: Mathematics general
- Series: Lecture Notes in Biomathematics 9
- Year: 1976
- Publisher: Springer-Verlag Berlin Heidelberg
- Edition: 1
- Language: English
- pdf
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.
Content:
Front Matter....Pages N2-XIII
Introduction....Pages 1-2
Background of Statistical Studies of Carcinogenesis....Pages 3-13
Immunological Response as a Factor in Carcinogenesis....Pages 14-15
The Mathematical Model....Pages 16-32
Some Unsuccessful Approaches to the Approximation Problem....Pages 33-44
Stochastic Linearization....Pages 45-48
Van Kampen’s Method....Pages 49-59
Method of Linearized Transition Probabilities....Pages 60-106
The Quadratic Death Process....Pages 107-133
The Collective Model....Pages 134-139
Further Implications of the Immunological Feedback Model....Pages 140-145
Conclusion....Pages 146-148
Back Matter....Pages 149-163
Content:
Front Matter....Pages N2-XIII
Introduction....Pages 1-2
Background of Statistical Studies of Carcinogenesis....Pages 3-13
Immunological Response as a Factor in Carcinogenesis....Pages 14-15
The Mathematical Model....Pages 16-32
Some Unsuccessful Approaches to the Approximation Problem....Pages 33-44
Stochastic Linearization....Pages 45-48
Van Kampen’s Method....Pages 49-59
Method of Linearized Transition Probabilities....Pages 60-106
The Quadratic Death Process....Pages 107-133
The Collective Model....Pages 134-139
Further Implications of the Immunological Feedback Model....Pages 140-145
Conclusion....Pages 146-148
Back Matter....Pages 149-163
....