Towards the Personalized Treatment of Glioblastoma: Integrating Patient-Specific Clinical Data in a Continuous Mechanical Model

Abstract

Glioblastoma multiforme (GBM) is the most aggressive and malignant among brain tumors. In addition to uncontrolled proliferation and genetic instability, GBM is characterized by a diffuse infiltration, developing long protrusions that penetrate deeply along the fibers of the white matter. These features, combined with the underestimation of the invading GBM area by available imaging techniques, make a definitive treatment of GBM particularly difficult. A multidisciplinary approach combining mathematical, clinical and radiological data has the potential to foster our understanding of GBM evolution in every single patient throughout his/her oncological history, in order to target therapeutic weapons in a patient-specific manner. In this work, we propose a continuous mechanical model and we perform numerical simulations of GBM invasion combining the main mechano-biological characteristics of GBM with the micro-structural information extracted from radiological images, i.e. by elaborating patient-specific Diffusion Tensor Imaging (DTI) data. The numerical simulations highlight the influence of the different biological parameters on tumor progression and they demonstrate the fundamental importance of including anisotropic and heterogeneous patient-specific DTI data in order to obtain a more accurate prediction of GBM evolution. The results of the proposed mathematical model have the potential to provide a relevant benefit for clinicians involved in the treatment of this particularly aggressive disease and, more importantly, they might drive progress towards improving tumor control and patient's prognosis.

Fig 1. Excesses stress Σ exerted by the cells in the case of homogeneous tissue, i.e. ∇ϕ _c = 0.

For physical and biological consistency, when ϕ _c < ϕ _e cells experience an adhesive force (f(ϕ _c) < 0), whereas for ϕ _c > ϕ _e, as cells are very close, a repulsive force acts among them and f(ϕ _c) > 0. The threshold value ϕ _e is called state of mechanical equilibrium. The repulsive force becomes infinite in the limit that cells fill the whole volume.

Fig 2. Post-contrast T1-MR of a patient affected by GBM and corresponding segmented slices.

(A) Axial, sagittal and coronal slices of post-contrast T1-MR in a patient with right parietal GBM (white arrow), used for image segmentation. (B) In the segmented brain image, the white region represents the white matter, the grey areas indicate the grey matter, while the cerebrospinal fluid is labeled by the blue color.

Fig 3. Patient-specific medical and numerical DTI data, depicted on a slice cut along the plane xy.

(A) A single component of the tensor D, obtained from the DTI medical images, is represented for each image: the intensity of the voxels is related to the diffusion coefficient along the relative direction (see the gray-scale at the bottom). (B) Numerical patient-specific components of the diffusion tensor D depicted on the same slice of the medical images: the diffusion coefficient is higher in the region occupied by the cerebrospinal fluid (red colored areas), where the diffusion is unconstrained. (C) Corresponding patient-specific components of the tensor of preferential directions T: in isotropic region, e.g. the cerebrospinal fluid and the grey matter, T _xx ≈ T _yy ≈ T _zz ≈ 1 and T _xy ≈ T _xz ≈ T _yz ≈ 0, while in the white matter, instead, 0 < T _ii < 3 with i = (x, y, z) and 0 < T _ij < 1 with i, j = (x, y, z) and i ≠ j, denoting an anisotropic region.

Fig 4. Tumor size parameters.

The anisotropic growth of an initially spherical tumor is evaluated measuring the ratio between the major semi-axis and the minor ones of the grown tumor ellipsoid.

Fig 5. Sensitivity analysis of the parameters k_n and M.

The influence of the parameters k _n and M on the cells volume fraction distribution at time t = 6 days is studied. The resulting tumor are characterized in terms of: the ratio between the maximum volume fraction at the final time, ϕ ^M, and maximum initial volume fraction, ${\varphi }_0^M$; the ratio between the final and the initial volume; the ratio between the major semi-axis, Δx, and the two minor semi-axes, Δy and Δz, defined as in Fig 4.

Fig 6. Sensitivity analysis of the parameters S_n and δ_n.

The influence of the parameters S _n and δ _n on the cell volume fraction and on the dimensionless nutrient concentration is reported at time t = 9 days.

Fig 7. Diagonal components of the tensor T over the brain mesh cut along each plane.

The components T _ii are represented over the brain mesh cut along the xy, xz and yz planes. The initial location of the virtual tumor, that corresponds to the corpus callosum, is indicated by a white cross.

Fig 8. Comparison between anisotropic and isotropic growth.

For both the anisotropic and the isotropic simulations, we report the tumor volume fraction distribution, the dimensionless nutrient concentration and the tumor contour plot at t = 5 day, t = 15 day and t = 25 day over the computational mesh cut along the xy-plane. In the anisotropic simulation the tensor D and T are the one reported in Fig 3(B)–3(C), respectively, whereas in the isotropic simulation we set D = D _n I and T = I.

Fig 9. Influence of brain fibers’ alignment on tumor growth.

(A) Tumor concentration plotted over the T _xx component (in transparency), at times t = 5 day, t = 15 day, t = 25 day: the cellular fraction shows an anisotropic distribution that follows the preferential direction determined by the T _xx component. (B) Tumor volume at t = 25 day overlapped to the maps of T _xx over the brain mesh cut along xy and xz planes and to the map of T _zz over the brain mesh cut along xz-plane: the glioblastoma assumes an elongated shape along the x direction, whereas it has a flat top in the z-direction, as T _zz is almost null there.