Bayesian Parameter Identification in Cahn--Hilliard Models for Biological Growth

Abstract

We consider the inverse problem of parameter estimation in a diffuse interface model for tumour growth. The model consists of a fourth-order Cahn-Hilliard system and contains three phenomenological parameters: the tumour proliferation rate, the nutrient consumption rate, and the chemotactic sensitivity. We study the inverse problem within the Bayesian framework and construct the likelihood and noise for two typical observation settings. One setting involves an infinite-dimensional data space where we observe the full tumour. In the second setting we observe only the tumour volume, hence the data space is finite-dimensional. We show the well-posedness of the posterior measure for both settings, building upon and improving the analytical results in [C. Kahle and K.F. Lam, Appl. Math. Optim. (2018)]. A numerical example involving synthetic data is presented in which the posterior measure is numerically approximated by the sequential Monte Carlo approach with tempering. are able to ignore apoptosis (programmed cell death) signals, remain elusive to attacks from the immune system, and, most dangerously, have the ability to induce the growth of new blood vessels towards itself (angiogenesis). This leads to the spreading of cancer to other parts of the body, and the formation of secondary tumours (metastasis).The study of tumour growth can be roughly divided according to the physical and chemical phenomena occuring at three scales [47]: the tissue scale which is commonly observed in experiments involving movement of cells (such as metastasis and growth into the extracellular matrix) and nutrient diffusion; the cellular scale consisting of activities and interactions between individual cells such as mitosis and the activation of receptors; and sub-cellular scale where genetic mutations and DNA degradation occur. We focus on the tissue-scaled phenomena, as they are the first to be detected in a routine diagnosis, and can be described fairly well with help of continuum models consisting of differential equations.Since the seminal work in [11] and [27] where simple mathematical models for tumour growth are employed, there has been an explosion in the number of models proposed for modelling the multiscale nature of cancer, see for instance [19,21,47] and the references cited therein. The diversity of model variants reflects the difficulties when we try to identify key biological phenomena that are responsible for experimental observations.As metastasis is an important hallmark of cancer, we restrict our attention to continuum models that can capture such events. Continuum models often rely on a mathematical description to distinguish tumour tissue from healthy host tissues. To be able to capture metastasis the models have to allow for some form of topological change of the separation layers between the tumour and the host tissues. The classical description represents the separation layers as idealised hypersurfaces, known also as the sharp interface approach. In this case complicated boundary conditions have to be imposed t...

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“…Several authors have discussed, what we choose to call, (Lipschitz, Hellinger) well-posedness for a variety of Bayesian inverse problems. For example, elliptic partial differential equations [8,22], level-set inversion [23], Helmholtz source identification with Dirac sources [13], a Cahn-Hilliard model for tumour growth [24], hierarchical prior measures [25], stable priors in quasi-Banach spaces [36,37], convex and heavy-tailed priors [20,21]. Moreover, to show well-posedness, Stuart [34] has proposed a set of sufficient but not necessary assumptions.…”

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“…For example, the simulated tempering method (Marinari and Parisi 1992) uses a single chain and varies the temperature within this chain. In addition, tempering forms the basis of efficient particle filtering methods for stationary model parameters in Sequential Monte Carlo settings (Beskos et al 2016(Beskos et al , 2015Kahle et al 2019;Kantas et al 2014;Latz et al 2018) and Ensemble Kalman Inversion (Schillings and Stuart 2017).…”