Realistic simulation of the 3-D growth of brain tumors in MR images coupling diffusion with biomechanical deformation

Abstract

We propose a new model to simulate the three-dimensional (3-D) growth of glioblastomas multiforma (GBMs), the most aggressive glial tumors. The GBM speed of growth depends on the invaded tissue: faster in white than in gray matter, it is stopped by the dura or the ventricles. These different structures are introduced into the model using an atlas matching technique. The atlas includes both the segmentations of anatomical structures and diffusion information in white matter fibers. We use the finite element method (FEM) to simulate the invasion of the GBM in the brain parenchyma and its mechanical interaction with the invaded structures (mass effect). Depending on the considered tissue, the former effect is modeled with a reaction-diffusion or a Gompertz equation, while the latter is based on a linear elastic brain constitutive equation. In addition, we propose a new coupling equation taking into account the mechanical influence of the tumor cells on the invaded tissues. The tumor growth simulation is assessed by comparing the in-silico GBM growth with the real growth observed on two magnetic resonance images (MRIs) of a patient acquired with 6 mo difference. Results show the feasibility of this new conceptual approach and justifies its further evaluation.

Fig. 1

MR images of a patient (A) T2; (B) GTV1 (dark red) and GTV2 (light blue) segmentations overlaid on the T2 MRI.

Fig. 2

Flowchart of the proposed approach

Fig. 3

Close up view of the tumor region on the first (#1), and second (#2) MRI series (T1 − T1 + gadolinium injection − T2), acquired respectively in March and September 2001.

Fig. 4

Visualization of the brain surface mesh and the different structures included in the model: (1) the skull, (2) gray matter, (3) white matter, (4) ventricles, (5) falx cerebri.

Fig. 5

Behavior of the reaction-diffusion equation on test cases. (Left) isotropic. (Right) anisotropic. The tumor cell density is fixed to C_max for the central part of the tumor (initialized as a random shape, central blue part in the cube).

Fig. 6

Diffusion model and boundary conditions summary.

Fig. 7

Tumor initialization in the finite element model. (A) GTV1. (B) GTV2.

Fig. 8

(Left) constitutive equation proposed by Miller and linear approximation; (right) stress error made with the linear approximation.

Fig. 9

Mechanical model and boundary conditions summary.

Fig. 10

Displacement of the tissues induced by the tumor mass effect

Fig. 11

Visualization of the mass effect. 1. T1 MRI 03/2001, 2. T1 MRI 09/2001, 3. T1 MRI 03/2001 deformed with the simulated displacement field.

Fig. 12

Result of the GBM growth simulation on slices # 1 → 5 of the brain. (A) MR T2 image of the patient in March 2003. (B) MR T2 image in March 2003 + superimposed simulation initialization contours. (C) MR T2 image of the patient in September 2003 (corresponding slice after rigid registration). (D) Contours of the 6 months tumor growth simulation above 8000 cells mm⁻³ superimposed on the MR T2 image in September 2003.