On some descriptive and predictive methods for the dynamics of cancer growth

Iulian T. Vlad, Jorge Mateu, Elvira Romano


Cancer is a widely spread disease that affects a large proportion of the human population, and many research teams are developing algorithms to help medics to understand this disease. In particular, tumor growth has been studied from different viewpoints and several mathematical models have been proposed. In this paper, we review a set of comprehensive and modern tools that are useful for prediction of cancer growth in space and time. We comment on three alternative approaches. We first consider spatio-temporal stochastic processes within a Bayesian framework to model spatial heterogeneity, temporal dependence and spatio-temporal interactions amongst the pixels, providing a general modeling framework for such dynamics. We then consider predictions based on geometric properties of plane curves and vectors, and propose two methods of geometric prediction. Finally we focus on functional data analysis to statistically compare tumor contour evolutions. We also analyze real data on brain tumor.


Geometric methods; Prediction methods; Space-time modeling; Tumor growth

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A. G. BELYAEV, E. V. ANOSHKINA, S. YOSHIZAWA, M. YANO (1999). Polygonal curve evolutions for planar shape modeling and analysis. International Journal of Shape Modeling 5: 195-217.

M. BRAMSON, D. GRIFFEATH (1981). On the Williams-Bjerknes tumour growth model. The Annals of Probability 9: 173-185.

N. CRESSIE (1991) Modelling growth with random sets. Spatial Statistics and Imaging, A. Possolo and C.A. Hayward, eds, IMS Lecture Notes Monogr. Ser. 20. IMS, Hayward, CA.

I. EPIFANIO, N. VENTURA-CAMPOS (2011) Functional data analysis in shape analysis. Journal Computational Statistics and Data Analysis 55: 2758-2773.

J. B. ILLIAN, S. H. SORBYE, H. RUE (2012). A toolbox for fitting complex spatial point processes models using integreted nested Laplace approximations (INLA). The Annals of Applied Statistics 6: 1499-1530.

INLA (2012) R-INLA project, 2012.

A. R. KANSAL, S. TORQUATO, G. R. HARSH, E. A. CHIOCCA, T. S. DEISBOECK (2000). Simulated brain tumor growth dynamics using a three-dimensional cellular automaton. J. Theor. Biol 203: 367-382.

T. LEE, R. COWAN (1994). A stochastic tessellation of digital space. In Mathematical Morphology and Its Applications to Image Processing (J. Serra, ed.) 217-224. Dordrecht: Kluwer.

F. LINDGREN, H. RUE, J. LINDSTROM (2011). An explicit link between Gaussian fields and Gaussian Markov random fields: The SPDE approach (with discussion). Journal of the Royal Statistical Society, Series B 73: 423-498.

F. LINDGREN, H. RUE (2015). Bayesian spatial modelling with R-INLA. Journal of Statistical Software 63.

MATLAB (2014). Image processing toolbox. Matlab user guide. Available at URL www.mathworks.co.uk/help/pdf doc/images/images tb.pdf.

A. S. QI, X. ZHENG, C. Y. DU, B. S. AN (1993). A cellular automaton model of cancerous growth. Journal of Theoretical Biology 161: 1-12.

J. O. RAMSAY, B. W. SILVERMAN (2005). Functional Data Analysis. Second Edition, Springer-Verlag, NY.

E. ROMANO, I. T. VLAD, J. MATEU (2012). Automatic Contour Detection and Functional Prediction of Brain Tumour Boundary. Analysis and Modeling of Complex Data in Behavioural and Social Sciences. PADOVA: CLEUP, ISBN: 978-88-6129-916-0, Anacapri, Italy, 3-4 Settembre 2012.

E. ROMANO, J. MATEU, I. T. VLAD (2014a) Principal differential analysis for modeling dynamic contour evolution. A distance-based approach for the analysis of Gioblastoma Multiform. Submitted.

E. ROMANO, J. MATEU, I. T. VLAD (2014b).A functional predictive model for monitoring variation in shapes of brain tumours. Technical Report.

H. RUE, L. HELD (2005) Gaussian Markov Random Fields. Theory and Applications. Chapman & Hall/CRC: Boca Raton.

H. RUE, S. MARTINO (2006) Approximate Bayesian inference for hierarchical Gaussian Markov random fields models. Journal of Statistical Planning and Inference 137: 3177-3192.

H. RUE, S. MARTINO, N. CHOPIN (2007). Approximate bayesian inference for latent gaussian models using integrated nested laplace approximations. Statistics Report No. 1, Department of Mathematical Sciences, Norwegian University of Science and Technology, Trondheim, Norway.

H. RUE, S. MARTINO, N. CHOPIN (2009). Approximate Bayesian Inference for Latent Gaussian Models using Integrated Nested Laplace Approximations (with discussion). Journal of the Royal Statistical Society B 71: 319-392.

I. T. VLAD, P. JUAN, J. MATEU (2015a) Bayesian spatio-temporal prediction of cancer dynamics. Computers and Mathematics with Applications 70: 857-868.

I. T. VLAD, J. MATEU (2015) A geometric approach to cancer growth prediction based on Cox processes. Journal of Statistics: Advances in Theory and Applications 13: 1-32.

I. T. VLAD, J. MATEU, X. GUAL-ARNAU (2015b). Two handy geometric prediction methods of cancer growth. Current Medical Imaging Reviews 11: 254-261.

DOI: 10.6092/issn.1973-2201/6096