Bayesian Inference of the Stochastic Gompertz Growth Model for Tumor Growth

Paek, Jayeong;Choi, Ilsu

  • Received : 2014.08.30
  • Accepted : 2014.11.02
  • Published : 2014.11.30


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.


Stochastic diffusion;Gompertz growth model;tumor growth;Bayesian;Markov data structure;sparse data structure


  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.
  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,
  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.
  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.
  8. Lo, C. F. (2010). A modified stochastic Gompertz model for tumour cell growth, Computational and Mathematical Methods in Medicine, 11, 3-11.
  9. Lv, Q. and Pitchford, J. W. (2007). Stochastic von Bertalanffy models with applications to fish recruitment, Journal of Theoretical Biology, 244, 640-655.
  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.
  11. Paap, R. (2002). What are the advantages of MCMC based inference in latent variable models?, Statistica Neerlandica, 56, 2-22.
  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.


Supported by : Chonnam National University