Recently, mathematical modeling of cancer started drawing great interest from the medical community. Indeed, developing models able to describe accurately tumor growth may help monitoring the disease evolution or even predicting the efficacy of different therapeutic strategies. Such applications are made possible through the routine monitoring of patients with imaging devices. This offers a consistent amount of valuable data to elaborate and validate the mathematical models. The aim of my talk is to present a quick overview of the strategies that we have recently developed in our team at Bordeaux.

In a first part, I will quickly present a simple tumor growth model based on a mechanistic description of the healthy and tumor cell densities evolution over time. This model is valid for example for meningiomas or for some lung metastases, i.e. when there is no treatment (only the growth is considered) and when the shape is approximatively the same over time. This model presents both the interest to be parametrizable - using the tumor volumes at 2 times and the initial shape tumor - for each considered patient and to produce reliable predictions and 3D extension simulation within a reasonable computing timescale. The parameters estimation strategy based on a reduced 0D-model and on stochastic methods (Monte-Carlo-like methods) will be presented.

When we consider treatments (tumour decay) or/and time-evolving shapes, more complex models - with different tumour cell densities - have to be written and more information - issued from medical imaging - is necessary to parametrize them. In a second part of my talk, I will introduce these complex models and I will present the information that can be extracted from medical imaging such as textures or shapes of the lesions. I will show why the parameter estimation approach used for the simple model is not anymore available and I will propose a new strategy based on data assimilation strategy. I will present a Luenberger - also called nudging - state observer coupled with a parameter Kalman-based observer to perform a joint state and parameter estimation. I will illustrate my results with synthetic and real data.