Extrapolating glioma invasion margin in brain magnetic resonance images: Suggesting new irradiation margins

Abstract

Radiotherapy for brain glioma treatment relies on magnetic resonance (MR) and computed tomography (CT) images. These images provide information on the spatial extent of the tumor, but can only visualize parts of the tumor where cancerous cells are dense enough, masking the low density infiltration. In radiotherapy, a 2 cm constant margin around the tumor is taken to account for this uncertainty. This approach however, does not consider the growth dynamics of gliomas, particularly the differential motility of tumor cells in the white and in the gray matter. In this article, we propose a novel method for estimating the full extent of the tumor infiltration starting from its visible mass in the patients’ MR images. This estimation problem is a time independent problem where we do not have information about the temporal evolution of the pathology nor its initial conditions. Based on the reaction–diffusion models widely used in the literature, we derive a method to solve this extrapolation problem. Later, we use this formulation to tailor new tumor specific variable irradiation margins. We perform geometrical comparisons between the conventional constant and the proposed variable margins through determining the amount of targeted tumor cells and healthy tissue in the case of synthetic tumors. Results of these experiments suggest that the variable margin could be more effective at targeting cancerous cells and preserving healthy tissue.

Introduction

Brain gliomas form the major class of tumors in the central nervous system (Tovi, 1993, Price et al., 2003). Among them, high grade ones remain incurable despite the state-of-the-art therapy, and patients have an average life expectancy of 1 year (Swanson et al., 2002). For diagnosis and therapy of gliomas, clinicians rely on medical images, such as magnetic resonance (MR) and computed tomography (CT). These images show the dense part of the tumor. However, with the current technology they are not able to expose the low density infiltration as different studies show the presence of tumor cells beyond the radiological signal (Tracqui et al., 1995, Swanson et al., 2004, Burger et al., 1983, Watanabe et al., 1992). This poses a problem for the experts in outlining the whole tumor and in understanding its extent. Fig. 1 illustrates this observability problem. In radiotherapy the invisible low density infiltration is addressed by outlining the tumor volume (more precisely the clinical target volume CTV2) and assuming the whole tumor infiltration would be contained within a 2 cm constant margin around that volume (Seither et al., 1995, Kantor et al., 2001). The irradiation region is constructed accordingly. This approach however, does not take into account the infiltration dynamics of gliomas, particularly the higher motility of tumor cells in the white matter compared to the gray matter (Giese et al., 1996). As a result, the irradiation region ignoring these dynamics would not reach the full extent of the tumor infiltration in the white matter and irradiate gray matter where probability of finding tumor cells would be very low. Mathematical tumor growth models can offer solutions to this problem by integrating clinical information and theoretical knowledge about tumor cell dynamics (Swanson et al., 2002, Swanson et al., 2008, Stamatakos et al., 2006a, Stamatakos et al., 2006b). In this respect, here we describe a formulation based on medical images, which aims to solve the problem of estimating tumor cell density distribution beyond the visible part in the image (low density infiltration) for gliomas. It uses the anatomical MR images and the diffusion tensor imaging (DTI) to suggest irradiation margins taking into account the inhomogeneity of the tumor invasion.

Over the last decade researchers proposed a variety of different tumor growth models. These different approaches can be coarsely classified into two groups, microscopic and macroscopic models. Microscopic models try to describe the progression of the tumor at the cellular level in terms of interactions between individual cells and their surrounding (Cristini et al., 2003, Patel et al., 2001, Drasdo and Höhme, 2005, Kansal et al., 2000, Hogea et al., 2006, Araujo and McElwain, 2004, Frieboes et al., 2007, Sanga et al., 2007, Maini et al., 2004, Byrne and Preziosi, 2003, Byrne et al., 2006, Breward et al., 2003, Athale and Deisboeck, 2006). These models are detailed, discrete and aim to capture the stochastic nature of tumor growth. However, they often include microscopic parameters that are not observable from medical images. This makes them harder to adapt to specific patient cases. Macroscopic models on the other hand, describe the average behavior of tumor cells and model the evolution of local tumor cell densities rather than individual cells (Swanson et al., 2000, Clatz et al., 2005, Jbabdi et al., 2005, Cristini et al., 2003, Hogea et al., 2007, Tracqui et al., 1995, Cruywagen et al., 1995, Frieboes et al., 2007, Mohamed and Davatzikos, 2005, Stein et al., 2007). Therefore, it is harder for them to capture the stochastic characteristic of tumor growth. On the other hand, they usually have fewer equations and parameters which can be identified from medical images making macroscopic models easier to adapt to specific patient data.

Most of the macroscopic models, Swanson et al., 2000, Clatz et al., 2005, Jbabdi et al., 2005, Hogea et al., 2007, Stein et al., 2007, are based on the reaction–diffusion formalism Murray introduced in the early 1990 and later formulated as a conservation equation (Murray, 2002, Swanson et al., 2002). This formalism uses the general class of partial differential equations (PDEs) called the reaction–diffusion and reaction–diffusion–advection type. They describe the temporal change of tumor cell densities at different locations in the brain through diffusion, migration and mitosis of tumor cells. The complexity of these models vary depending on the number of factors they take into account. In this work we derive our method starting from the reaction–diffusion equations. Since we focus on clinically available data (i.e. medical images), we choose to base our method on simpler reaction–diffusion models as given in (Swanson et al., 2000, Clatz et al., 2005, Jbabdi et al., 2005). The number of parameters for these models are few and can be directly related to the available clinical data such as the anatomical MRI and the DTI.

The literature on predicting irradiation margins on medical images using automatic methods is rather limited. Kaspari et al. (1997) used artificial neural networks to model statistically the way the radiotherapist constructs the irradiation margin. In their work they do not include the growth dynamics of gliomas. Zizzari et al. started from the same framework and included mathematical growth models in their prediction of the irradiation volume (Zizzari et al., 2004). They use their model to predict further growth of the tumor and then use this prediction to construct the irradiation margins through artificial neural networks. However, they do not focus on the spatial distribution of tumor cells at a given time and they do not include the differential motility of glioma cells in different tissues. Swanson et al. (2002) also focus on the problem of limited visualization of gliomas. For virtual tumors grown by reaction–diffusion models, they compare the visible part of the tumor with the extent of the invisible infiltration. This work is important in demonstrating the visualization problem for brain gliomas. However, their work does not provide concrete methods to solve this problem for patient images.

In this article, we propose a formulation to extrapolate the tumor cell density distribution of diffusive gliomas beyond their visible mass in MR images taking into account the growth tendencies of the tumor. In deriving the formulation we start from the reaction–diffusion type growth models as given in (Clatz et al., 2005, Jbabdi et al., 2005, Swanson et al., 2002). Directly applying these models to solve the mentioned problem poses several difficulties. First, reaction–diffusion models require the knowledge of tumor cell densities at every point in the brain while in reality only CTV1 and/or CTV2 contours are observable in the images. In addition, reaction–diffusion models describe the temporal evolution of tumor cell density distribution, however, the problem we are tackling is static, dealing with the distribution of tumor cell density at a single time instance. We use asymptotic approximations to overcome these difficulties and derive a static formulation that uses the information in the images. The proposed method starts from the delineation of the tumor (manual delineation or automatic segmentation) and constructs an approximation for the low density infiltration taking into account the underlying tissue characteristics by using anatomical and diffusion tensor images. With such a formulation, we aim to construct irradiation margins that would be more efficient in targeting tumor cells and reducing the irradiation of healthy brain tissues.

In Section 2, we explain the reaction–diffusion type models in detail and derive our formulation for extrapolating low density infiltration using a single image. In Section 3 we assess the quality of the approximation constructed by the proposed formulation using virtually grown tumors. Following this, we use our formulation to construct a variable irradiation margin. We compare these margins to the conventionally used constant ones in terms of number of tumor cells and volume of healthy tissue targeted in the case of synthetic tumors. In Section 4 we conclude by summarizing the work with our results and provide future directions.