Spatial mechanistic modeling for prediction of the growth of asymptomatic meningiomas

Mathematical modeling of tumor growth draws interest from the medical community as they have the potential to improve patients' care and the use of public health resources. The main objectives of this work are to model the growth of meningiomas - slow-growing benign tumors requiring extended imaging follow-up - and to predict tumor volume and shape at a later desired time using only two times examinations.

We develop two variants of a 3D partial differential system of equations (PDE) which yield after a spatial integration systems of ordinary differential equations (ODE) that relate tumor volume with time. Estimation of models parameters is a crucial step to obtain a personalized model for a patient that can be used for descriptive or predictive purposes. As PDE and ODE systems share the same parameters, they are both estimated by fitting the ODE systems to the tumor volumes obtained from MRI examinations acquired at different times. A population approach allows to compensate for sparse sampling times and measurement uncertainties by constraining the variability of the parameters in the population.

Description capabilities of the models are investigated in 39 patients with benign asymptomatic meningiomas who had had at least three surveillance MRI examinations. The two models can fit to the data accurately and more realistically than a naive linear regression. Prediction performances are validated for 33 patients using a population approach. Mean relative errors in volume predictions are less than 10% with ODE systems versus 12.5% with the naive linear model using only two times examinations. Concerning the shape, the mean Sørensen-Dice coefficients are 85% with the PDE systems in a subset of 10 representative patients.

Our strategy - based on personalization of mathematical model - provides a good insight on meningioma growth and may help decide whether to extend the follow-up or to treat the tumor.