Section: New Results
Tumor Growth Modelling
Tumor growth modelling describes mathematically the dynamics of the tumor growth process. General models try to include as many details of the process as possible but they are difficult to personalize for a given patient due to the problem of observability and the lack of available observations.
Olivier Clatz has applied a macroscopic growth model, which is mathematically simpler and much more flexible, to the patient specific case. This model is based on the reaction-diffusion formalism and uses tissue and structural information of the brain obtained from MR and DT-MR images. Based on this model, we derived a formulation to help radiotherapists determine the irradiation region for a patient taking in account the growth dynamics of the tumor  . CT and MR images are not successful in visualizing all the tumor infiltration in the brain, especially regions where tumor cell density is low are not enhanced. Radiotherapists deal with these regions by irradiating them, and because of the lack of visualization they use a constant therapy margin (2cm) around the surrection region. This formulation tries to solve the problem of visualizing regions, where tumor cell density is low, in MR and CT images to help determining irradiation regions. It extrapolates invasion margins of a tumor from its visible part in the MR image (see Fig. 20 ).
In applying the aforementioned model to any patient, several further steps should be taken, like automatically determining correct patient specific parameters of the model from medical images taken at successive times, including effects of radiotherapy and chemotherapy in the model and including the uncertainty in the model, which is coming from many different sources of which the primary one is the variation of characteristics in different tumor cells.
This research work has been presented at the DIMACS Workshop on Computational Tumor Modeling  at Rutgers University, USA, and is now acknowledged at the Center for the Development of a Virtual Tumor(https://www.cvit.org/), supported by the NIH-National Cancer Institute.
Towards patient-specific models of the heart
This work is done in the context of the INRIA national action CardioSense3D(http://www-sop.inria.fr/CardioSense3D/ ), and in collaboration with the Division of Imaging Sciences, King's College London, United Kingdom.
The integration of knowledge from biology, physics and computer science makes it possible to combine in vivo observations, in vitro experiments and in silico simulations. From these points of view, knowledge of the heart function has greatly improved at the nanoscopic, microscopic and mesoscopic scales, along with an impressive development of the observations possibilities through medical imaging.
Our work aims at introducing computational models of the heart in clinical applications. Such models can be used to introduce prior knowledge in image analysis methods  . We showed that it makes it possible to regularize the segmentation process in a physiological way.
But such models can also be used to estimate hidden parameters of the cardiac function from clinical data, through data assimilation  . Such parameters give a much more interesting information for diagnosis, as they really represent the underlying physiology.
In order to build such models, it is necessary to be able to segment the cardiovascular system from medical images (see Fig. 21 ). More specifically, we work on heart segmentation from MRI, using active models, and on statistical classification of aorta's shape (see Fig. 22 ). The goal is to achieve the effective segmentation of the atria and proximal arteries in order to complete the current anatomical model where only ventricles are available. This augmented segmentation will participate in improving simulation of the electromechanical activity of the heart.
Recent developments of medical imaging and electrophysiological measures in cardiology also allow the realisation of realistic models of the heart electrophysiology which can be used to help diagnosis. Numerous models of electric propagation through the myocardium have been developed. However, very few calibration methods have been proposed. We proposed a method to adjust the parameters of an integrated 3D model of the left and right ventricles of the heart.
Many of the functional models of the heart are designed to reproduce in a realistic manner the cardiac activity (especially ionic gates and concentrations), often leading to high computational costs and the manual tuning of a very large set of parameters. In our approach, we rather selected a model involving a limited number of parameters, based on Aliev & Panfilov  . Thus allowing the identification of the model parameters from clinical measurements on a specific patient by solving the inverse problem.
It is well known that muscle fiber orientations vary across the myocardial wall. Fibres have an important impact on the behaviour of the depolarization wave as the action potential propagation is around 3 times faster in fiber direction than in radial directions. Therefore, a first step was to define an analytical linear fiber model to set the diffusion tensor on each vertex. To adjust a parameter, we browsed the parameter space solving the direct problem with a different parameter value each time, to match reference measures (action potential duration, APD, or depolarization speed) provided by activation times (ie: when the depolarization wave pass by the mesh vertices on the heart surface). The relation between the parameter and the measure is then approximated with a rational model and least square resolution. This method can be used for the entire mesh or one zone at a time (segmentation provided by American Heart Association) or on each vertex.
We obtained very promising results on simulated APD (see figure 23 ). But before getting the same quality on matching depolarization speed, several difficulties have to be tackled. The stability of the model with regard to its parameters, in particular, has to be studied precisely.
The collaboration with the Division of Imaging Sciences, King's College London(http://www.kcl.ac.uk/schools/medicine/research/imaging/), in Guy's Hospital provides unique data from their XMR facility which combines X-rays and MRI in the same room. Thus we can obtain anatomy, motion and electrophysiology from the same patient in the same spatial coordinates (see Fig 24 ).
One aim of this joint work is the development of computer tools allowing the cardiologists to plan the implantation of cardiac pacemakers on patients with arrhythmias. Then, we have to evaluate the heart motion from tagged MRI, this can be done with different image processing tools or using the phase information. Next, we have to estimate the parameters of the heart model in order to minimize the differences between the observed and the simulated motion. Finally, we will develop an optimization method to determine the position and the activation times of the cardiac pacemakers which give the best cardiac output.