Communications for Statistical Applications and Methods

The Korean Statistical Society (한국통계학회)
권호 표지
DOI QR코드

DOI QR Code

Bayesian Inference of the Stochastic Gompertz Growth Model for Tumor Growth

  • Paek, Jayeong (Department of Statistics, Chonnam National University) ;
  • Choi, Ilsu (Department of Statistics, Chonnam National University)
  • Received : 2014.08.30
  • Accepted : 2014.11.02
  • Published : 2014.11.30
Download PDF
⟨ Previous Next ⟩

Abstract

A stochastic Gompertz diffusion model for tumor growth is a topic of active interest as cancer is a leading cause of death in Korea. The direct maximum likelihood estimation of stochastic differential equations would be possible based on the continuous path likelihood on condition that a continuous sample path of the process is recorded over the interval. This likelihood is useful in providing a basis for the so-called continuous record or infill likelihood function and infill asymptotic. In practice, we do not have fully continuous data except a few special cases. As a result, the exact ML method is not applicable. In this paper we proposed a method of parameter estimation of stochastic Gompertz differential equation via Markov chain Monte Carlo methods that is applicable for several data structures. We compared a Markov transition data structure with a data structure that have an initial point.

Keywords

References

  1. Alili, L., Patie, P. and Pedersen, J. L. (2005). Representations of the first hitting time density of an Ornstein-Uhlenbeck process, Stochastic Models, 21, 967-980. https://doi.org/10.1080/15326340500294702
  2. Benzekry, S., Lamont, C., Beheshti, A., Tracz, A., Ebos, J. M. L., Hlatky, L. and Hahnfeldt P. (2014). Classical mathematical models for description and prediction of experimental tumor growth, arXiv preprint, arXiv, 1406-1446, http://arxiv.org. https://doi.org/10.1371/journal.pcbi.1003800
  3. Bonate, P. L., Peter, L. and Suttle, B. (2013). Effect of censoring due to progressive disease on tumor size kinetic parameter estimates, The American Association of Pharmaceutical Scientists Journal, 15, 832-839.
  4. Gutierrez-Jaimez, R., Roman, P., Romero, D., Serrano, J. and Torres, F. (2007). A new Gompertz-type diffusion process with application to random growth, Mathematical Biosciences, 208, 147-165. https://doi.org/10.1016/j.mbs.2006.09.020
  5. Karatzas, I. (1991). Brownian motion and stochastic calculus, Springer.
  6. Linetsky, V. (2004). Computing hitting time densities for CIR and OU diffusions: Applications to mean-reverting models, Journal of Computational Finance, 7, 1-22.
  7. Lo, C. F. (2007). Stochastic Gompertz model of tumour cell growth, Journal of Theoretical Biology, 248, 317-21. https://doi.org/10.1016/j.jtbi.2007.04.024
  8. Lo, C. F. (2010). A modified stochastic Gompertz model for tumour cell growth, Computational and Mathematical Methods in Medicine, 11, 3-11. https://doi.org/10.1080/17486700802545543
  9. Lv, Q. and Pitchford, J. W. (2007). Stochastic von Bertalanffy models with applications to fish recruitment, Journal of Theoretical Biology, 244, 640-655. https://doi.org/10.1016/j.jtbi.2006.09.009
  10. Nobile, A. G., Ricciardi, L. M. and Sacerdote, L. (1985). Exponential trends of Ornstein-Uhlenbeck first-passage-time densities, Journal of Applied Probability, 22, 360-369. https://doi.org/10.2307/3213779
  11. Paap, R. (2002). What are the advantages of MCMC based inference in latent variable models?, Statistica Neerlandica, 56, 2-22. https://doi.org/10.1111/1467-9574.00060
  12. Phillips, P. C. and Yu, J. (2009). Maximum likelihood and Gaussian estimation of continuous time models in finance, In Handbook of financial time series, 497-530, Springer, Berlin Heidelberg.
  13. Schuster, R. and Schuster, H. (1995). Reconstruction models for the Ehrlich Ascites Tumor for the mouse, Mathematical Population Dynamics, 2, 335-348.