Bright solitary waves in malignant gliomas

Víctor M. Pérez-García, Gabriel F. Calvo, Juan Belmonte-Beitia, David Diego, and Luis Pérez-Romasanta

Phys. Rev. E 84, 021921 – Published 19 August 2011

Abstract

We put forward a nonlinear wave model describing the fundamental dynamical features of an aggressive type of brain tumors. Our model accounts for the invasion of normal tissue by a proliferating and propagating rim of active glioma cancer cells in the tumor boundary and the subsequent formation of a necrotic core. By resorting to numerical simulations, phase space analysis, and exact solutions we prove that bright solitary tumor waves develop in such systems. Possible implications of our model as a tool to extract relevant patient specific tumor parameters in combination with standard preoperative clinical imaging are also discussed.

Received 8 March 2011

DOI:https://doi.org/10.1103/PhysRevE.84.021921

Authors & Affiliations

Víctor M. Pérez-García¹, Gabriel F. Calvo², Juan Belmonte-Beitia¹, David Diego¹, and Luis Pérez-Romasanta³

¹Departamento de Matemáticas, E. T. S. I. Industriales and Instituto de Matemática Aplicada a la Ciencia y la Ingeniería, Universidad de Castilla-La Mancha, E-13071 Ciudad Real, Spain
²Departamento de Matemáticas, E. T. S. I. Caminos, Canales y Puertos and Instituto de Matemática Aplicada a la Ciencia y la Ingeniería, Universidad de Castilla-La Mancha, E-13071 Ciudad Real, Spain
³Radiation Oncology Service, Hospital General Universitario de Ciudad Real, c/ Obispo Torija, s/n, E-13005 Ciudad Real, Spain

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Vol. 84, Iss. 2 — August 2011

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Images

Figure 1
(a) Sagittal magnetic resonance image of a patient with a GBM in the occipital (posterior) lobe of the brain. The contrast enhanced region (marked by arrows) contains the active tumor cells and the darker inner part of the rim is the necrotic tissue. (b) Schematic representation of the different areas. Arrows indicate the tumor and necrotic core growth directions.Reuse & Permissions
Figure 2
Simulations of Eqs. (1a) in two spatial dimensions leading to the formation of the glioma rim, Parameter values are $D = 0.02$ mm $^{2}$ /day, $ρ = 0.5$ day $^{- 1}$ , $α = 1 / 30$ day $^{- 1}$ , and $γ = 0.2$ day $^{- 1}$ . Shown are pseudocolor plots of the tumor cell density $u (x, y, t)$ (upper row) and the necrotic core $w (x, y, t)$ (lower row) for times $t = 25, 75$ , and 150 days. The spatial region shown is approximately the square $[- 4, 4] \times [- 4, 4]$ with $x, y$ measured in centimeters.Reuse & Permissions
Figure 3
Simulations of Eqs. (1a) in one spatial dimension leading to the formation of a bright soliton. Parameter values are $D = 0.02$ mm $^{2}$ /day, $ρ = 0.5$ day $^{- 1}$ , $α = 1 / 30$ day $^{- 1}$ , and $γ = 0.25$ day $^{- 1}$ . Shown are the profiles of the density of the different densities: (a) tumor $u (x, t)$ , (b) normal tissue $v (x, t)$ , and (c) necrotic core $w (x, t)$ for times $t = 15, 60, 120$ , and 180 days. All of the densities are measured in units of $u_{*}$ .Reuse & Permissions
Figure 4
Simulations of Eqs. (1a) in one spatial dimension leading to the formation of a bright soliton. Parameter values are as in Fig. 3. Evolution of the total cell number (a), center (b), and speed (c) of the soliton traveling to the left, defined as $N (t) = \int_{- \infty}^{0} u (x, t) d x$ , $X (t) = \int_{- \infty}^{0} x u (x, t) d x / N (t)$ , and $d X / d t$ , respectively.Reuse & Permissions
Figure 5
Bright and antikink solitons for $β = 0.2$ , $φ_{0} = 0.4$ , and $c = 3$ . (a) Profiles from Eq. (5) (solid curve) and the explicit solution given by Eq. (8) (dashed curve). (b) Profiles from Eq. (9) by substituting the solution of Eq. (5) (solid curve) and the explicit solution given by Eq. (10) (dashed curve) for $v_{*} / u_{*} = 0.3$ .Reuse & Permissions
Figure 6
(a) T2-weighted axial MRI of a low-grade astrocytoma in the right fronto-parietal lobe (marked with an arrow). Notice that the tumor limits are fuzzy. (b) T1-weighted axial MRI with intravenous contrast. The tumor (marked with an arrow) remains almost undetectable (there is no contrast uptake by the tumor). (c) T1-weighted axial MRI with intravenous contrast of the same patient 30 months later. The tumor (transformed into a secondary GBM) uptakes contrast heterogeneously. The rim and the necrotic core are clearly visible. (d) Schematic of the characteristic widths $δ x$ of the tumor rims when the cell density is above the T1 detection limit $u_{T 1}$ .Reuse & Permissions

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