Bayesian parameter identification in Cahn-Hilliard models for biological growth
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In this paper we consider the identification of internal parameters in a diffuse interface model for tumour growth. The tumour is modelled by a phase field that acts as a smooth indicator function to distinguish the healthy cells and the tumour cells inside the tissue. Our model is a variant of the model proposed in [Garcke et al., Math. Models Methods Appl. Sci., 26 (2016), pp. 1095--1148] and contains three phenomenological parameters: the proliferation rate for the tumour growth, the consumption rate for the nutrition consumption by the tumour, and a parameter to control the influence of chemotaxis, i.e., the growing of the tumour in the direction of the gradient of the nutrition concentration. These parameters need to be adapted to particular observations, and we apply concepts from Bayesian inversion for this task. After a brief summary of the analytical properties of the model we discuss the well-posedness of the posterior measure for general prior measures. We approximate the posterior measure by the sequential Monte Carlo approach with tempering, and present test results for two numerical examples. In the first example we simulate a tumour with given parameters and add artificial noise to obtain synthetic data. In the second example we use real-world measurements of the volume of spherical tumours and estimate the corresponding internal variables.
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