Parameter non-identifiability of the Gyllenberg–Webb ODE model
An ODE model introduced by Gyllenberg and Webb (Growth Develop Aging 53:25–33, 1989) describes tumour growth in terms of the dynamics between proliferating and quiescent cell states. The passage from one state to another and vice versa is modelled by two functions \(r_o\) and \(r_i\) depending on the total tumour size. As these functions do not represent any observable quantities, they have to be identified from the observations. In this paper we show that there is an infinite number of pairs (\(r_o, r_i\)) corresponding to the same solution of the ODE system and the functions (\(r_o, r_i\)) will be classified in terms of this equivalence. Surprisingly, the technique used for this classification permits a uniqueness proof of the solution of the ODE model in a non-Lipschitz case. The reasoning can be widened to a more general setting including an extension of the Gyllenberg–Webb model with a nonlinear birth rate. The relevance of this result is discussed in a preclinical application scenario.
KeywordsTumour Growth Quiescence Parameter identifiability
Mathematics Subject Classification (2000)92C50 34A12 34A55
The author would like to thank Dr. T. Pourcher for providing the SPECT images and for his helpful comments on the usage of SPECT imaging in his laboratory. In addition to that, the remarks and propositions of Florence Hubert and Guillemette Chapuisat were of great value. Finally, the referees’ feedback permitted to improve the quality and readability of this manuscript substantially. The author was partially supported by the Agence Nationale de la Recherche under grant ANR-09-BLAN-0217-01 and by the Cancéropôle PACA.
- WHO cancer fact sheet. http://www.who.int/mediacentre/factsheets/fs297/en/
- Amann H, Escher J (2006) Analysis II. Grundstudium Mathematik, BirkhäuserGoogle Scholar
- Gyllenberg M, Webb G (1989) Quiescence as an explanation of Gompertzian tumor growth. Growth Develop Aging 53:25–33Google Scholar
- Hahnfeldt P, Panigrahy D, Folkman J, Hlatky L (1999) Tumor development under angiogenic signaling: a dynamical theory of tumor growth, response and postvascular dormancy. Cancer Res 59:4770–4775Google Scholar
- Hartung N (2011) Une étude de l’applicabilité du modèle de Gyllenberg et Webb. http://www.cmi.univ-mrs.fr/~hartung/memoire_master2_niklas_hartung.pdf
- Richard Fiardo P, Franken P, Lamit A, Marsault R, Guglielmi J, Cambien B, Graslin F, Lindenthal S, Darcourt J, Pourcher T, Vassaux G (2012) Normalisation to blood activity is a requirement for the accurate quantification of Na/I symporter ectopic expression by SPECT/CT. PLoS One, accepted.Google Scholar