PROCESOS PUNTUALES, DENSIDADES PRODUCTO Y BIOLOGÍA CELULAR

En este artículo se explica cómo los procesos puntuales han sido usados con éxito para modelar problemas de poblaciones celulares. En particular, se muestra la utilidad de la técnica de densidades producto para tratar un problema de células sujetas a radiación. Se proponen dos tipos de modelos estocásticos: el primero trata sobre la reparación de células basada en cinética enzimática y el segundo sobre la paridad de células en el crecimiento de un tumor. Mediante la técnica de densidades producto hallamos varias características estadísticas claves de estos dos modelos.

In this paper an attempt is made to explain how point processes have been successfully used in modeling cell biology problems. In particular we demonstrate the utility of the product density techniques in dealing with cells subject to radiation. In this paper, we propose two types of stochastic models: the first one is concerned with cell repair based on enzyme kinetics and the second one with parity of cells in tumor growth. Using product density techniques, we device several key statistical characteristics of the models.

Procesos puntuales, densidades producto y biología celular VISWANATHAN ARUNACHALAM¹

¹Departamento de Matemáticas, Universidad de los Andes, Sede Bogotá. E-mail: aviswana@uniandes.edu.co., Departamento de Estadística, Universidad Nacional de Colombia, Sede Bogotá.

---

Resumen

En este artículo se explica cómo los procesos puntuales han sido usados con éxito para modelar problemas de poblaciones celulares. En particular, se muestra la utilidad de la técnica de densidades producto para tratar un problema de células sujetas a radiación. Se proponen dos tipos de modelos estocásticos: el primero trata sobre la reparación de células basada en cinética enzimática y el segundo sobre la paridad de células en el crecimiento de un tumor. Mediante la técnica de densidades producto hallamos varias características estadísticas claves de estos dos modelos.

Palabras Clave: Modelos estocásticos, procesos puntuales, reparación de células, paridad de células.

---

Abstract

In this paper an attempt is made to explain how point processes have been successfully used in modeling cell biology problems. In particular we demonstrate the utility of the product density techniques in dealing with cells sub- ject to radiation. In this paper, we propose two types of stochastic models: the first one is concerned with cell repair based on enzyme kinetics and the second one with parity of cells in tumor growth. Using product density techniques, we device several key statistical characteristics of the models.

Keywords: Stochastic models, point processes, cell repair, parity of cells.

---

Texto completo disponible en PDF

---

Referencias

1. Albright, N. W. (1989), "A markov formulation of the repair-misrepair model of cell survival", Radiation Research 118, 1–20.

2. Bertuzzi, A., Gandolfi, A. & Giovenco, M. A. (1981), "Mathematical models of the cell cycle with a view to tumor studies", Mathematical Biosciences 53, 159-188.

3. Braby, L. A. & Nelson, J. M. (1991), Linear-quadratic dose kinetics or dose dependent repair/misrepair, in "Workshop on Biological Modelling of Radiation Effects", Padue - Italy, pp. 331–334.

4. Cox, D. R. & Isham, V. (1980), Point Processes, Chapman and Hall, New York.

5. Curtis, S. B. (1986), "Lethal and potentially lethal lesions induced by radiation: a unified repair model", Radiation Research 106, 252–270.

6. Daley, D. J. & Vere-Jones, D. (2003), An Introduction to Theory of Point Processes: Elementary Theory and Methods, Vol. Volume I, 2 edn, Springer, New York.

7. Gani, J. & Saunders, I. W. (1976), "On the parity of individuals in a branching process", Journal of Applied Probability 13, 219–230.

8. Goodhead, D. T. (1985), "Saturable repair models of radiation action in mammalian cells", Radiation Research 104, S58–S67.

9. Janssen, I. (1987), "A stochastic repair-misrepair model for effects of radiation on cells", Journal of Mathematical Biology 24, 681–689.

10. Ramakrishnan, A. (1950), "Stochastic processes relating to particles distributed in a continuous infinity of states", Proceedings Cambridge Philosophical Society 46, 595–602.

11. Rangan, A. & Arunachalam, V. (1998), "A stochastic model for cell repair based on enzyme kinetics", Journal of Biological Systems 5, 139–150.

12. Rangan, A. & Arunachalam, V. (1999), "On the parity of cells in tumor growth", Stochastic Processes and Applications pp. 61–72. Narosa, New Delhi.

13. Sontag, W. (1987), "A cell survival model with saturable repair after irradiation", Radiation Environmental Biophysics 26, 63–79.

14. Srinivasan, S. K. (1974), Stochastic Point Processes and their Applications, Griffin, London.

15. Srinivasan, S. K. & Ranganathan, C. R. (1982), "On the parity of individuals in birth and death processes", Advances in Applied probability 14, 484–501.

16. Tobias, C. A. (1985), "The repair-misrepair model in radiobiology: Comparison to other models", Radiation Research 104, S77–S95.