Nonlinear waves in a simple model of high-grade glioma

The application of tools from nonlinear dynamical systems to describe in a quantitative manner the underlying mechanisms encountered in a number of biologically- and/or medically-oriented scenarios has attracted an increasing interest in recent years [2, 11, 12, 13, 19, 32, 44, 45, 56]. As multidisciplinary collaboration between mathematical modelers with experimental biologists and/or clinicians is becoming a more common practice, it is possible to tackle complex problems and develop new armamentaria that could potentially display a great impact in public health, with cancer being a quite prominent example [38].

Here, we analyze in detail a mathematical model that was proposed in [39] to describe the salient physiopathological hallmarks of a very aggressive type of primary brain tumor: glioblastoma (GBM), the most frequent high-grade glioma in adults. At present, GBM carries a dismal prognosis; the median overall survival from diagnosis is 14.6 months with the current standard therapy [46], which includes neurosurgery followed by radiotherapy in combination with the alkylating agent temozolomide, and various salvage therapies (e.g., antiangiogenic) once tumor progression recurs [51]. Large-scale research endeavors are ongoing to provide a comprehensive understanding of all the genetic and epigenetic alterations that underlie GBM [48]. Despite all of these efforts, GBM remains incurable (the five-year overall survival rate is less than 10%), although it should be mentioned that, albeit still very small, a slow but steadily increase in the number of long-term surviving patients (more than five years) has been reported [47].

Various mathematical models have been proposed to describe specific aspects of GBM in vivo [3, 4, 5, 6, 7, 18, 23, 28, 40, 49, 50, 54]. Most of these models assume that the tumor evolves according to a simple reaction-diffusion equation (or a combination of): the Fisher-Kolmogorov (FK) equation [33], which epitomizes the simplest form of a nonlinear reaction-diffusion process

∂u∂t=∇⋅D∇u+ρ1−uu∗u,$$\begin{array}{} \displaystyle \frac{\partial u}{\partial t} = \nabla\cdot\left(D\nabla u\right) + \rho\left(1-\frac{u}{u_{*}}\right)u, \end{array}$$(1)

where D, ρ and u_∗ are positive parameters accounting, in the case of GBM, for the diffusion constant (cell migration), the proliferation rate and the maximum cell density (carrying capacity). This equation was first proposed by Fisher [17] in a completely different scenario; as a deterministic version of a stochastic model for the spatial spread of a favoured gene in a population. It is also the natural extension of the logistic growth population model when the population disperses via linear diffusion [10, 34]. The same equation appears in flame propagation [55], neurophysiology [53], and Brownian motion processes [8] among other situations. This equation and its variants have attracted the attention of many mathematicians, which have calculated different exact solutions under various conditions (see for instance [41, 42] and references therein).

In one-dimensional scenarios the FK equation has solitary wave solutions of kink-type [1, 29], which account for the progression of the tumor front edge. In dimensions higher than one, the FK equation does not display any kind of travelling wave solution [9], thus its analysis must resort to numerical methods. However, some essential features of brain tumor lesions in their transit to high-grade gliomas (i.e. from low-grade astrocytoma to GBM) are neglected by the FK model, such as the formation of a necrotic core responsible for the increase of the interstitial pressure and the subsequent intracranial deformation that may result in patient death. Furthermore, they do not incorporate the interaction of the tumor with the adjacent normal tissue (brain parenchyma).

Consequently, in order to understand glioma progression from low to high grade and thinking of connections with clinical practice there is a need for: (i) models accounting for the key features of the tumor microenviroment dynamics without involving too many details on any of the intervening specific processes, and (ii) models that are simple enough to allow for some quantitative understanding, e.g. using the tools of nonlinear dynamical systems and/or nonlinear waves well developed in other scenarios [45].

In this paper we present in detail one of those models allowing both for a description of features seen in real GBM and for a theoretical analysis. Interestingly, this model leads to the existence of multicomponent solitary waves encompassing bright solitary waves in the glioma compartment coupled with kink and antikink waves for the normal tissue and necrotic core respectively. We will study the model’s phenomenology using a combination of numerical simulation methods, dynamical systems theory and qualitative analysis. In the course of the study we wish to bring out a number of distinguishing features with respect to the FK model that are also relevant when comparing their predictions of how GBMs evolve.

Our plan is as follows, first in Sec. 2 we present the model in detail. Next in Sec. 3 we analyze the existence of traveling wave solutions of the system of equations using qualitative analysis and geometric perturbation theory. Then in Sec. 4 we calculate bright soliton solutions and determine numerically the minimum speed of these solitary waves solutions. The linear stability analysis of small initial data is discussed in Sec. 5 together with its application to the computation of the relapse time after surgery. Finally Sec. 6 summarizes our conclusions.

The model to be discussed in detail in this paper for describing the progression of GBM was proposed in [39]. To account for the spatial structure occupied by the malignant glioma cells and their environment, let Ω be a domain of ℝⁿ, for n = 1, 2, 3, which is open, bounded, connected and measurable with a given size (length, area or volume) μ(Ω). The model incorporates three relevant real functions u, v and w, defined on the closure of Ω × ℝ⁺, which are nonnegative, twice differentiable in the interior of their domain and satisfy the initial-value problem

∂u∂t(x,t)=∇⋅D(x)∇u(x,t)+ρ(x,t)1−u(x,t)+v(x,t)+w(x,t)u∗(x)u(x,t)−α(x,t)u(x,t),$$\begin{array}{} \displaystyle \frac{\partial u}{\partial t}(x,t) = \nabla\cdot\left(D(x) \nabla u(x,t)\right) + \rho(x,t)\!\left(1-\frac{u(x,t)+v(x,t)+w(x,t)}{u_{*}(x)}\right)\!u(x,t) - \alpha(x,t) u(x,t), \end{array}$$(2a)

∂v∂t(x,t)=−Fx,t,u(x,t),v(x,t),w(x,t),$$\begin{array}{} \displaystyle \frac{\partial v}{\partial t}(x,t) = -\mathscr{F}\left(x,t,u(x,t),v(x,t),w(x,t)\right), \end{array}$$(2b)

∂w∂t(x,t)=Fx,t,u(x,t),v(x,t),w(x,t)+α(x,t)u(x,t),$$\begin{array}{} \displaystyle \frac{\partial w}{\partial t}(x,t) = \mathscr{F}\left(x,t,u(x,t),v(x,t),w(x,t)\right) + \alpha(x,t) u(x,t), \end{array}$$(2c)

∀x ∈ Ω and ∀t ∈ ℝ⁺

u(x,0)=u0(x),∀x∈Ω¯,$$\begin{array}{} \displaystyle u(x,0) = u_{0}(x), \quad \forall x\in \overline{\Omega}, \end{array}$$(3a)

v(x,0)=v0(x),∀x∈Ω¯,$$\begin{array}{} \displaystyle v(x,0) = v_{0}(x), \quad \forall x\in \overline{\Omega}, \end{array}$$(3b)

w(x,0)=w0(x),∀x∈Ω¯,$$\begin{array}{} \displaystyle w(x,0) = w_{0}(x), \quad \forall x\in \overline{\Omega}, \end{array}$$(3c)

for some nonnegative continuous functions u₀, v₀, w₀ : Ω̄ → ℝ, together with the homogeneous Neumann boundary conditions

∂u∂n(x,t)=0,∀x∈∂Ω,∀t∈R+∪{0},$$\begin{array}{} \displaystyle \frac{\partial u}{\partial{\bf n}}(x,t) = 0, \quad \forall x\in\partial\Omega, \forall t\in \mathbb{R}^{+}\cup\{0\}, \end{array}$$(4a)

∂v∂n(x,t)=0,∀x∈∂Ω,∀t∈R+∪{0},$$\begin{array}{} \displaystyle \frac{\partial v}{\partial{\bf n}}(x,t) = 0, \quad \forall x\in\partial\Omega, \forall t\in \mathbb{R}^{+}\cup\{0\}, \end{array}$$(4b)

∂w∂n(x,t)=0,∀x∈∂Ω,∀t∈R+∪{0}.$$\begin{array}{} \displaystyle \frac{\partial w}{\partial{\bf n}}(x,t) = 0, \quad \forall x\in\partial\Omega, \forall t\in \mathbb{R}^{+}\cup\{0\}. \end{array}$$(4c)

The biological meaning of all the above quantities is the following: u denotes the tumor cell density, v the adjacent normal cell density (brain parenchyma) and w the density of dead cells (necrotic core). The nonnegative and continuous functions D(x), ρ(x, t), u_∗(x) and α(x, t) represent the diffusion coefficient (or more generally, a tensor), the proliferation rate, the carrying capacity (or maximum cell density) and the tumor cell death rate, respectively.

Eqs. (2) add a number of quantities with respect to the Fisher-Kolmogorov Eq. (1). The standard FK equation is recovered by setting α = v = w = 0. However, in order to properly account for the observed physiopathology, we include both the population of the normal tissue and the developing necrotic core, the last one being a distinctive clinical feature of GBM observable in magnetic resonance images as well as in histopathological examination of the tumor samples from patients.

In Eqs. (2), tumor cell spreading is taken into consideration by using a standard Fickian diffusion term in (2a). This is the simplest transport mechanism and the one employed in most of the continuous mathematical models on cell motility. More realistic and complicated diffusion terms in gliomas should probably be governed by fractional (anomalous) diffusion [14, 16, 35] or other more elaborate approaches [15, 27, 52] to account for the high infiltration observed in this type of tumors [21, 36] and the fact that cells do not behave like purely random walkers and may actually remain immobile for a significant amount of time before compelled to migrate to a more favorable place in terms of resources. Other possibilities are to build in (at least) two cell phenotypes observed in malignant gliomas [26, 43]: one of invasive type migrating through the brain parenchyma and the other, more proliferative, dependent on angiogenesis and migrating due to the effect caused by the destruction of the anomalous tumor vasculature. A number of mathematical models have been proposed to incorporate such a tumor heterogeneity [24, 30, 31], however, here we will maintain a single tumor cell phenotype that encompasses many of the features observed in GBM progression.

The nonlinear term in (2a) corresponds to proliferation and is mediated by a competition for space that is occupied by the two cell types and the necrotic core, which comprises apoptotic cells and cell debris. There is a maximum cell density u_∗ that may depend on space without loss of generality but, here, we will assume u_∗ to be a constant. We also explicitly add a tumor cell death term to include the fact that tumor cells, although generally lacking the apoptotic (programmed cell death) mechanisms [22], may succumb by means of a number of mechanisms that include the interaction with the immune system (e.g. microglia) in the normal tissue, hypoxia and acidosis generated by the anomalous metabolism of the glioma cells in the high density tumor areas, and deficiency of nutrients and physical support in the necrotic core. This tumor cell death term assumes that, on average, the characteristic tumor cell life time is 1/α. The fact that α may depend on time would correspond to the administration of radiotherapy and/or chemotherapy.

As to the normal cell dynamics, via Eq. (2b), due to their differentiation state, the proliferation of normal brain tissue is almost negligible within the characteristic time scales of GBM progression, thus we will assume that the normal tissue will not be able to regenerate. This is why in Eq. (2b) we have represented the cell loss due to the interaction with the tumor by means of an arbitrary form 𝓕(x, t, u, v, w) depending on all the densities and/or space and/or time. The details of the interaction may be very complicated and remain unclear. Finally, the space occupied by the necrotic core is the same space occupied by the original cells and grows at the expense of the other two compartments, as described by Eq. (2c). More elaborate models could include a reduction coefficient to account for the shrinkage of the cells and/or the destruction of their cytoplasm and the release of the cellular content to the necrotic area. Notice that the imposed boundary conditions (4) reflect the fact that, in GBM, the tumor cells are confined within the brain during progression since metastasis from the brain to other ectopic organs and tissues is a very rare event.

To simplify the analysis and to focus on the main dynamical features, we will restrict ourselves to a one spatial dimension and assume that D(x), ρ(x, t), u_∗(x) and α(x, t) are constants (do not depend on space and/or time), although we emphasize that, on a qualitative basis, many of the phenomena persist in higher dimensions.

It is convenient to introduce the new functions

U(x,t)=u(x,t)/u∗,V(x,t)=(v(x,t)+w(x,t))/u∗,$$\begin{array}{} \displaystyle U(x,t) = u(x,t)/u_*, \\ V(x,t) = (v(x,t) + w(x,t))/u_*, \end{array}$$

and the rescaled variables

s=xρ/D,τ=ρt.$$\begin{array}{} \displaystyle s = x\sqrt{\rho/D}, \\ \displaystyle\tau = \rho t. \end{array}$$

Notice that V represents the normal cell and the necrotic tissue compartments altogether (henceforth referred to as the normal-necrotic compartment); it accounts for the effect of the peritumoral environment [39]. Then, Eqs. (2) in one dimension can be cast in the form

∂U∂τ=∂2U∂s2+U(1−β−U−V),$$\begin{array}{} \displaystyle \frac{\partial{U}}{\partial \tau} = \frac{\partial^{2}U}{\partial s^{2}} + U (1-\beta-U-V), \end{array}$$(5a)

∂V∂τ=βU,$$\begin{array}{} \displaystyle \frac{\partial V}{\partial \tau} = \beta U, \end{array}$$(5b)

with 0 ≤ β = α/ρ < 1.

Travelling waves with constant speed are solutions of Eq. (5) depending on z = s – cτ (′ = d/dz)

U″+cU′+U(1−β−U−V)=0,$$\begin{array}{} \displaystyle U''+c U'+ U(1-\beta-U-V) =0, \end{array}$$(6a)

cV′+βU=0.$$\begin{array}{} \displaystyle cV' + \beta U =0. \end{array}$$(6b)

Using Eq. (6b) to get U and substituting into Eq. (6b), we obtain a third order nonlinear differential equation

V‴+cV″+(1−β)V′+cβV′2−VV′=0.$$\begin{array}{} \displaystyle V''' +cV''+(1-\beta)V'+\frac{c}{\beta}\left(V'\right)^{2}-VV'=0. \end{array}$$(7)

3.1

Fast solitons

Let us first consider the case c ≫ 1. To do so we define the new variable ξ = z/c = εz, and the new function 𝓥(ξ) = V(z). Then, Eq. (7) becomes

ε2Vξξξ+Vξξ+(1−β)Vξ+Vξ2β−VVξ=0.$$\begin{array}{} \displaystyle \varepsilon^{2}\mathscr{V}_{\xi\xi \xi}+\mathscr{V}_{\xi\xi}+(1-\beta)\mathscr{V}_{\xi}+\frac{\left(\mathscr{V}_{\xi}\right)^{2}}{\beta}-\mathscr{V}\mathscr{V}_{\xi}=0. \end{array}$$(8)

In the limit c ≫ 1, ε ≪ 1 and Eq. (8) is equivalent to

Vξξ+(1−β)Vξ+Vξ2β−VVξ=0.$$\begin{array}{} \displaystyle \mathscr{V}_{\xi\xi}+(1-\beta)\mathscr{V}_{\xi}+\frac{\left(\mathscr{V}_{\xi}\right)^{2}}{\beta}-\mathscr{V}\mathscr{V}_{\xi}=0. \end{array}$$(9)

Defining 𝓥_ξ = 𝓦, Eq. (9) can be written as the autonomous system

Vξ=W,$$\begin{array}{} \displaystyle \mathscr{V}_{\xi}= \mathscr{W}, \end{array}$$(10a)

Wξ=−(1−β)W−1βW2+WV.$$\begin{array}{} \displaystyle \mathscr{W}_{\xi} =-(1-\beta)\mathscr{W} -\frac{1}{\beta}\mathscr{W}^2+\mathscr{WV}. \end{array}$$(10b)

The equilibria of Eqs. (10) are of the form (𝓥, 𝓦) = (𝓥_∗, 0), with 𝓥_∗ being an arbitrary real number. We are interested in values of 𝓥_∗ ≥ 0. Thus, linearizing Eq. (10) around (𝓥_∗, 0), we obtain that the eigenvalues of the Jacobian matrix are given by

λ1=0,λ2=V∗−(1−β),$$\begin{array}{} \displaystyle \lambda_{1}=0,\quad \lambda_{2}=\mathscr{V}_*-(1-\beta), \end{array}$$(11)

and its corresponding eigenvectors are (1, 0) and (1, 𝓥_∗ + β – 1), respectively (we are assuming that λ₂ ≠ 0). These points are nonhyperbolic points. If 𝓥_∗ > 1 – β, then, the fixed point (𝓥_∗,0) possesses a local unstable manifold and a local center manifold. Otherwise, (𝓥_∗, 0) has a local stable manifold and a local center manifold. Thus, to get a heteroclinic orbit joining two points, say (𝓥_–, 0) and (𝓥₊, 0), with 𝓥_– > 𝓥₊, it is a necessary condition that 𝓥₊ < 1 – β and 𝓥_– > 1 – β (we are further interested in 𝓥_– < 1). These two conditions can be summarized in the physical constraint imposing that the normal-necrotic compartment density is always strictly larger than the normal compartment density; the difference originating in the fraction of tumor cell loss.

Bright solitons, when existing, correspond to positive U(z) solutions, thus to negative 𝓦 ∼ V′ heteroclinic orbits connecting two positive equilibria for 𝓥. The equation of the orbits can be obtained from Eqs. (10) and reads as

dWdV=V−W/β−(1−β).$$\begin{array}{} \displaystyle \frac{d\mathscr{W}}{d\mathscr{V}}=\mathscr{V}-\mathscr{W}/\beta-(1-\beta). \end{array}$$(12)

Its solution is

W(V)=Ce−V/β+β(V−1),$$\begin{array}{} \displaystyle \mathscr{W}(\mathscr{V})= C e^{-\mathscr{V}/\beta}+\beta(\mathscr{V}-1), \end{array}$$(13)

for any arbitrary C. Our solutions, when existing, must satisfy 𝓥(± ∞) = 𝓥_± with 𝓥_∗ = 𝓥_– > 𝓥₊ (cf. Eq. (6b)) and 𝓦(± ∞) = 0, thus C must be chosen to allow for the equation

Ce−V/β+β(V−1)=0,$$\begin{array}{} \displaystyle C e^{-\mathscr{V}/\beta}+\beta(\mathscr{V}-1) = 0 , \end{array}$$(14)

to have two positive roots 𝓥_– > 𝓥₊ > 0. Thus, for the initial condition 𝓦(𝓥₊) = 0, Eq. (13) becomes

W(V)=β(V−1)+β(1−V+)e(V+−V)/β.$$\begin{array}{} \displaystyle \mathscr{W}(\mathscr{V})=\beta(\mathscr{V}-1)+\beta(1-\mathscr{V}_{+})e^{(\mathscr{V}_{+}-\mathscr{V})/\beta}. \end{array}$$(15)

It is straightforward to prove that, given the initial condition 𝓦(𝓥₊) = 0, with 0 < 𝓥₊ < 1 – β, there exists a value, say 𝓥_–, with 𝓥₊ < 1 – β < 𝓥_– < 1, such that there exists a heteroclinic orbit joining both values, 𝓥_– and 𝓥₊; the function 𝓦(𝓥) has a single minimum at

Vm=V+−βlogβ1−V+,$$\begin{array}{} \displaystyle \mathscr{V}_{m}=\mathscr{V}_{+}-\beta\log\left(\frac{\beta}{1-\mathscr{V}_{+}}\right), \end{array}$$(16)

which is larger than 𝓥₊ for 𝓥₊ < 1 – β. As 𝓦(𝓥₊) = 0 and the function 𝓦(𝓥) is decreasing in the interval (𝓥₊, 𝓥_m), it is clear that

W(Vm)<0.$$\begin{array}{} \displaystyle \mathscr{W}(\mathscr{V}_{m}) \lt 0. \end{array}$$(17)

Therefore, as 𝓦(1) > 0, the existence of a value 𝓥_– in the interval (𝓥_m, 1), such that 𝓦(𝓥_–) = 0, follows from a direct application of Bolzano’s Intermediate Value Theorem.

Moreover, it is straightforward to verify that no orbits can cross the triangle T, given by

0≤V≤1,β(V−1)≤W≤0,$$\begin{array}{} \displaystyle 0\leq \mathscr{V}\leq 1,\quad \beta(\mathscr{V}-1) \leq\mathscr{W}\leq 0, \end{array}$$(18)

from the outside. Therefore, T is a negative invariant region. Thus, all orbits inside T must emanate from equilibrium points 𝓥_– such that 1 – β < 𝓥_– < 1, since these points have an unstable manifold inside T. However, the points 𝓥₊, which satisfy 𝓥₊ < 1 – β, have a stable manifold inside T. Now, let a point 𝓥₊ ∈ T such that 0 ≤ 𝓥₊ < 1 – β. As this point has a stable manifold and belongs to T, its orbit is in T. Thus, 𝓥₊ is a ω-limit point. Hence, there must exist a point 𝓥_– ∈ T, with 1 – β < 𝓥_– < 1, such that there is always an orbit connecting 𝓥_– to 𝓥₊ inside T.

Thus, we have shown the following result: there exists a heteroclinic orbit that connects 𝓥_– and 𝓥₊ and, therefore, 𝓥 > 0, (i.e. to remain entirely in the fourth quadrant).

In general, these fast moving solitons are small ones as Fig. 1(c) shows. Let us notice that

Fig. 1

Vξ=β(V−1)+β(1−V+)e(V+−V)/β,$$\begin{array}{} \displaystyle \mathscr{V}_{\xi} =\beta(\mathscr{V}-1)+\beta(1-\mathscr{V}_{+})e^{(\mathscr{V}_{+}-\mathscr{V})/\beta}, \end{array}$$(19)

that allows for a solution for 𝓥 in the form of quadratures, once initial data is specified,

ξ−ξ0=1β∫V+Vds(1−V+)e−(s−V+)/β+s−1.$$\begin{array}{} \displaystyle \xi-\xi_{0}=\frac{1}{\beta}\int_{\mathscr{V}_{+}}^{\mathscr{V}}\frac{ds}{(1-\mathscr{V}_{+})e^{-(s-\mathscr{V}_{+})/\beta}+s-1}\, . \end{array}$$(20)

Finally, in the limit ξ → +∞, the asymptotic behavior of 𝓥 is 𝓥 ∼ 𝓥₊ + e^{–(1–β–𝓥_∗–𝓥₊)(ξ–ξ₀)}. In a similar way, for ξ → –∞, 𝓥 ∼ 𝓥_– + e^{–(1–β–𝓥_∗–𝓥_–)(ξ–ξ₀)}.

Instead of considering kink solutions, via Eq. (7), we can work in the variable U that represents a localized unimodal function of tumor cell density (corresponding to homoclinic paths in the phase plane).

From Eqs. (5) and defining S = V − 1+β we have ∂U∂τ=∂2U∂s2−(U+S)U,$$\begin{array}{} \displaystyle \frac{\partial U}{\partial\tau} \, =\,\frac{\partial^{2}U}{\partial s^{2}} - (U+S)U\, , \end{array} $$(41a)∂S∂τ=βU.$$\begin{array}{} \displaystyle \frac{\partial S}{\partial\tau} \,=\, \beta U\, . \end{array} $$(41b)

Combining Eqs. (41) into a single (and more general) equation for the tumor density we get 1UUτ−Uss+U2τ=−βU,$$\begin{array}{} \displaystyle \left[ \frac{1}{U}\left( U_{\tau} - U_{ss} + U^{2}\right)\right]_{\tau} = -\beta U\, , \end{array} $$(42)

where subscripts denote partial differentiation. Notice that Eq. (42) admits (trivial) exact solutions of the form U(s,τ)=U0(s)e−βτ,$$\begin{array}{} U(s,\tau)=U_{0}(s)e^{-\beta \tau}, \end{array} $$(43)

where U₀(s) is a sufficiently regular arbitrary function of the spatial coordinate. Also, if β = 0, (42) possesses the following exact solution in terms of the rational function U(s,τ)=6C12+12(2+6)C2+120(12+56)τ+12(4+6)C1s+12(4+6)s2C2+10(3+6)τ+C1s+s22,$$\begin{array}{} \displaystyle U(s,\tau) = \frac{6C_{1}^{2} + 12(2+\sqrt{6})C_{2} +120(12+5\sqrt{6})\tau + 12(4+\sqrt{6})C_{1}s + 12(4+\sqrt{6})s^{2}}{\left( C_{2} +10(3+\sqrt{6})\tau + C_{1}s + s^{2}\right)^{2}}\, , \end{array} $$(44)

where C₁ and C₂ are arbitrary constants. (44) remains localized if 4C2+40(3+6)τ>C12,∀τ≥0.$\begin{array}{} 4C_{2} + 40(3+\sqrt{6})\tau \gt C_{1}^{2}, \forall \tau \geq 0. \end{array} $ However, just as is occurs with (43), it decays to zero for τ → ∞. We will be interested in searching for solutions which remain finite (but nonzero) for arbitrary large times.

4.1

Solitary wave solutions

We will look for bright solitary solutions of Eq. (42) in the form U(s − cτ) = ϕ(η), where c is their velocity (positive or negative for right- or left-moving fronts, respectively). These solutions will be required to satisfy ϕ < 1 − β, together with ϕ → 0 and ϕ_η → 0 as |η| → ∞. Upon substitution in Eq. (42) and defining ξ = η/c, we arrive at a third order nonlinear autonomous differential equation βϕ3−ϕξ2−ϕ2ϕξ+ϕϕξξ−1c2(ϕξϕξξ−ϕϕξξξ)=0.$$\begin{array}{} \displaystyle \beta\phi^{3} -\phi_{\xi}^{2} - \phi^{2}\phi_{\xi} + \phi\phi_{\xi\xi} - \frac{1}{c^{2}}(\phi_{\xi}\phi_{\xi\xi} -\phi\phi_{\xi\xi\xi})= 0\, . \end{array} $$(45)

Just as we did with the kink-type solutions of Eq. (7) representing the normal-necrotic compartment, within the fast limit c ≫ 1 and assuming ϕ < 1 − β, it can be shown [39] that Eq. (45) admits the approximate explicit form ϕ(ξ)=ϕ0sech2ϕ0β2(ξ−ξ0),$$\begin{array}{} \displaystyle \phi(\xi) = \phi_{0}\,\textrm{sech}^{2}\!\left[ \sqrt{\frac{\phi_{0}\beta}{2}}(\xi - \xi_{0})\right] \! , \end{array} $$(46)

with constants ϕ₀ and ξ₀, representing the amplitude and a shift, respectively, which, without loss of generality, can be set ξ₀ = 0. Figure 2 compares the exact numerical profile from Eq. (45) to the explicit form Eq. (46) and their corresponding homoclinic orbits.

Fig. 2

4.2

Minimum speed of positive solutions

We now wish to determine the minimum speed c₀ above which the solutions of Eq. (45) are positive for all ξ, since biologically feasible solutions for the tumor density must be positive for all ξ. Let χ = ϕ_ξ/ϕ, then Eq. (45) becomes β−χ+1+3χc2χχϕ+ϕχc2χϕ2+χχϕϕ=0.$$\begin{array}{} \displaystyle \beta - \chi + \left( 1 + \frac{3\chi}{c^{2}}\right)\chi\chi_{\phi} + \frac{\phi\chi}{c^{2}}\left( \chi_{\phi}^{2} + \chi\chi_{\phi\phi}\right) =0 . \end{array} $$(47)

By probing the parameter space for 0 < ϕ < 1 − β we have verified numerically that the following approximation 3χ2χϕ+ϕχ(χϕ2+χχϕϕ)≃2χ2χϕ$\begin{array}{} 3\chi^{2}\chi_{\phi} + \phi\chi( \chi_{\phi}^{2} + \chi\chi_{\phi\phi}) \simeq 2\chi^{2}\chi_{\phi} \end{array} $ holds. Hence, Eq. (47) reduces to β−χ+1+2χc2χχϕ=0.$$\begin{array}{} \displaystyle \beta - \chi + \left( 1 + \frac{2\chi}{c^{2}}\right)\chi\chi_{\phi} =0 . \end{array} $$(48)

This autonomous differential equation possesses an exact explicit quadrature. Assuming that χ < β, together with the conditions ϕ = ϕ₀ and χ = 0 at ξ = 0, we obtain 1+2βc2χ+βlog1−χβ+χ2c2=ϕ−ϕ0.$$\begin{array}{} \displaystyle \left( 1 + \frac{2\beta}{c^{2}}\right)\!\left( \chi + \beta\log\left\vert 1 - \frac{\chi}{\beta}\right\vert\right) + \frac{\chi^{2}}{c^{2}} = \phi - \phi_{0} . \end{array} $$(49)

Positive solutions of Eq. (45) require that χ remains finite for all ξ (if c < c₀, χ exhibits discontinuities and ϕ becomes negative in nonzero measure sets for ξ > 0). As ξ → ∞ (ξ → −∞), χ → χ₊ < 0 (χ → χ_– > 0). These asymptotic values can be found from Eq. (49) by means of the ancillary function G=χ−β+βexp−1βχ+c2ϕ0+χ2c2+2β.$$\begin{array}{} \displaystyle G = \chi - \beta + \beta\exp\!\left[-\frac{1}{\beta}\left( \chi + \frac{c^{2}\phi_{0} + \chi^{2}}{c^{2}+2\beta}\right)\right] \! . \end{array} $$(50)

The nonlinear equation G(χ, c, β, ϕ₀) = 0 has two real roots χ₊ < 0 and χ_– > 0 if c ≥ c₀, whereas for 0 < c < c₀ the only root of G is χ_– > 0. At c = c₀, χ₊ satisfies G(χ=χ+,c=c0)=0,∂G∂χχ=χ+,c=c0=0,∂2G∂χ2χ=χ+,c=c0<0,$$\begin{array}{} \displaystyle G(\chi = \chi_{+}, c=c_{0}) = 0, \quad \left.\frac{\partial G}{\partial\chi}\right\vert_{\chi = \chi_{+}, c=c_{0}} = 0, \quad \left. \frac{\partial^{2} G}{\partial\chi^{2}}\right\vert_{\chi = \chi_{+}, c=c_{0}} \lt 0 , \end{array} $$(51)

that is, χ₊ becomes a double root of G.

By imposing the first two conditions of Eq. (51) to Eq. (50) we arrive at the two simultaneous transcendetal equations χ+−β+βexp−1βχ++c02ϕ0+χ+2c02+2β=0,$$\begin{array}{} \displaystyle \chi_{+} - \beta +\beta\exp\!\left[-\frac{1}{\beta}\left( \chi_{+} + \frac{c_{0}^{2}\phi_{0} + \chi_{+}^{2}}{c_{0}^{2}+2\beta}\right)\right] \,=\,0, \end{array} $$(52a)1−1+2χ+c2+2βexp−1βχ++c02ϕ0+χ+2c02+2β=0.$$\begin{array}{} \displaystyle 1 - \left(1 + \frac{2\chi_{+}}{c^{2}+2\beta}\right) \exp\!\left[-\frac{1}{\beta}\left( \chi_{+} + \frac{c_{0}^{2}\phi_{0} + \chi_{+}^{2}}{c_{0}^{2}+2\beta}\right)\right] \,=\, 0 . \end{array} $$(52b)

Eqs. (52) give rise to two solutions for χ₊, χ₊ = 0 and χ₊ = − c022$\begin{array}{} \frac{c_{0}^{2}}{2} \end{array} $, the second one being the only biologically meaningful one (χ₊ = 0 leads to c₀ = 0). Upon substitution of χ₊ = − c022$\begin{array}{} \frac{c_{0}^{2}}{2} \end{array} $ into Eq. (50), we derive an implicit expression for the speed c₀ in terms of β and ϕ₀1+c022β−expc02+4β−4ϕ0c024c02+2ββ=0.$$\begin{array}{} \displaystyle 1 + \frac{c_{0}^{2}}{2\beta} - \exp\!\left[\frac{\left(c_{0}^{2}+4\beta -4\phi_{0}\right)c_{0}^{2}}{4\left(c_{0}^{2}+2\beta\right)\beta}\right] =0 . \end{array} $$(53)

The dependence of c₀ on β and ϕ₀ is depicted in Fig. 3.

Fig. 3

It is worth comparing the values of the minimum velocity that arise in the present model with those of the FK equation. It is well known [33] that the only solution of the FK equation evolving from positive compactly supported initial data that remains bounded and propagates as an antikink solitary wave is the one with the constant minimal speed cmin(FK)=2Dρ.$\begin{array}{} c_\textrm{min}^{(\textrm{FK})}=2\sqrt{D\rho}. \end{array} $ In view of the profiles displayed in Fig. 3, we observe that the minimal speed cmin=c0D(ρ−α),$\begin{array}{} c_\textrm{min}=c_{0}\sqrt{D(\rho-\alpha)}, \end{array} $ where c₀ > 0, can actually be lower than the limit set by the FK equation. The consequence of this is that for the same values of D and ρ, the predicted velocities of tumor expansion by the FK equation are generally larger than the ones resulting from the present model.

In this section we explore the possibility of calculating the time of tumor recurrence after the macroscopic tumor volume (detectable in MRI) has been surgically removed. To study this scenario from a mathematical point of view we examine the linear stability of the solutions

U(s)=0,V(s)=V∗,$$\begin{array}{} \displaystyle U(s)=0,\quad V(s)=V_{*}, \end{array} $$(54)

for the system (5), where V_* is a positive constant. In fact, we are interested in the stability of the tumoral density U. Since the brain is bounded, we can consider a characteristic length of the brain, 2L say, for our problem.

Now consider a small perturbation on (U, V) of the form

Up(s,τ)=U(s)+εu(s,τ),$$\begin{array}{} \displaystyle U_{p}(s,\tau)=U(s)+\varepsilon u(s,\tau), \end{array} $$(55a)

Vp(s,τ)=V(s)+εv(s,τ),$$\begin{array}{} \displaystyle V_{p}(s,\tau)=V(s)+\varepsilon v(s,\tau), \end{array} $$(55b)

with 0 < ε ≪ 1. The solution U is stable to perturbations u(s, τ) if

limτ→∞u(s,τ)=0orlimτ→∞u(s,τ)=dU(s)dτ.$$\begin{array}{} \displaystyle \lim_{\tau\to\infty}u(s,\tau)=0\quad \text{or}\quad \lim_{\tau\to\infty}u(s,\tau)=\frac{dU(s)}{d\tau}. \end{array} $$(56)

Otherwise, U is unstable. In a similar way, V is stable to perturbations v(s, τ) if an analogous relation as (56) for v(s, τ) is satisfied. Otherwise, V is unstable.

The fact that U(s) is stable if the second of these holds is because u(s, τ) then represents a small translation of the wave along the s−axis since

U(s+δs)≈U(s)+δsdUds.$$\begin{array}{} \displaystyle U(s+\delta s)\approx U(s)+\delta s\frac{dU}{ds}. \end{array} $$(57)

Substituting Eqs. (55a) into Eq. (5) and keeping only the first-order terms in ε we get the equations governing (u(s, τ),v(s, τ)) as

uτ=uss+(1−β−V∗)u,$$\begin{array}{} \displaystyle u_{\tau}=u_{ss}+(1-\beta-V_{*})u, \end{array} $$(58a)

vτ=βu.$$\begin{array}{} \displaystyle v_{\tau}=\beta u. \end{array} $$(58b)

We now look for solutions to the linear Eqs. (58a) by setting

u(s,τ)=g1(s)e−λ1τ,v(s,τ)=g2(s)e−λ2τ,$$\begin{array}{} \displaystyle u(s,\tau)=g_{1}(s)e^{-\lambda_{1}\tau},\quad v(s,\tau)=g_{2}(s)e^{-\lambda_{2}\tau}, \end{array} $$(59)

which, upon substituting into (58a) and (58b), gives

g1″+(λ1−{β+V∗−1})g1=0,$$\begin{array}{} \displaystyle g_{1}''+\left(\lambda_{1}-\left\{\beta+V_{*}-1\right\}\right)g_{1}=0, \end{array} $$(60a)

λ2g2e−λ2τ+βg1e−λ1τ=0.$$\begin{array}{} \displaystyle \lambda_{2}g_{2}e^{-\lambda_{2}\tau}+\beta g_{1}e^{-\lambda_{1}\tau}=0. \end{array} $$(60b)

From Eq. (60b) we obtain that

λ1=λ2=λ,λg2(s)+βg1(s)=0.$$\begin{array}{} \displaystyle \lambda_{1}=\lambda_{2}=\lambda,\quad \lambda g_{2}(s)+\beta g_{1}(s)=0 . \end{array} $$(61)

Therefore, one has the following condition for the instability of the solution U = 0

λ<0.$$\begin{array}{} \displaystyle \lambda \lt 0. \end{array} $$(62)

Thus, if λ < 0, from (59), both u(s, τ) and v(s, τ) tend to ∞ as τ → ∞. Now we use the fact that u(s, τ) and v(s, τ) are nonzero only in a finite domain which, from (59), means that the boundary conditions

g1(±Ls)=0,g2(±Ls)=0,$$\begin{array}{} \displaystyle g_{1}(\pm L_{s})=0,\quad g_{2}(\pm L_{s})=0, \end{array} $$(63)

with L_s being a dimensionless characteristic length, Ls=Lρ/D$\begin{array}{} L_{s}=L\sqrt{\rho/D} \end{array} $ and where g₁ and g₂ are the respective solutions to Eqs. (60a) and (60b). Notice that Eq. (60a), together with boundary conditions (63), is a regular Sturm-Liouville problem and has two infinite families of solutions for

λ<β+V∗−1,$$\begin{array}{} \displaystyle \lambda \lt \beta+V_{*}-1, \end{array} $$(64)

and for

β+V∗−1<λ.$$\begin{array}{} \displaystyle \beta+V_{*}-1 \lt \lambda. \end{array} $$(65)

For the first case, the only solution of Eq. (60a) satisfying the boundary conditions (63) is the trivial solution. Thus, we focus on the second case: β+V_* –1 < λ < 0. For this case, the eigenvalues of the Sturm-Liouville problem (60a), (63) are given by

λn=(β+V∗−1)+((2n−1)π2Ls)2,n=1,2,3…$$\begin{array}{} \displaystyle \lambda_{n}=(\beta+V_{*}-1)+\left(\frac{(2n-1)\pi}{2L_{s}}\right)^{2}, \quad n=1,2,3\dots \end{array} $$(66)

Hence, the most unstable eigenvalue is achieved for n = 1, i.e., for the lowest frequency. From (66), it is straightforward to prove that there exists a value for n, say n₁, such that λ_n1+1 > 0 and λ_n1 < 0. For this value of n₁, L_s must verify

|Ls|>(2n1−1)π2(1−β−V∗).$$\begin{array}{} \displaystyle |L_{s}| \gt \frac{(2n_{1}-1)\pi}{2\sqrt{(1-\beta-V_{*})}} \, . \end{array} $$(67)

Therefore, the eigenfunctions are given by

g1(s)=cos⁡((2n−1)πs2Ls),n=1,2,3…$$\begin{array}{} \displaystyle g_{1}(s)=\cos\left(\frac{(2n-1)\pi s}{2L_{s}}\right),\quad n=1,2,3\dots \end{array} $$(68)

and, from (61), we obtain the function g₂

g2(s)=−λnβsin⁡((2n−1)πs2Ls),n=1,2,3…$$\begin{array}{} \displaystyle g_{2}(s)=-\frac{\lambda_{n}}{\beta}\sin\left(\frac{(2n-1)\pi s}{2L_{s}}\right),\quad n=1,2,3\dots \end{array} $$(69)

Equivalently, another way to get the same result is by the following way: If we multiplying both sides of Eq. (58a) by exp(−(1−β −V_*)τ) and rearranging the terms, we have

∂(ue−(1−β−V∗)τ)∂τ=∂2(ue−(1−β−V∗)τ)∂s2.$$\begin{array}{} \displaystyle \frac{\partial \left(ue^{-\left(1-\beta-V_{*}\right)\tau}\right)}{\partial \tau}=\frac{\partial^{2}\left(ue^{-\left(1-\beta-V_{*}\right)\tau}\right)}{\partial s^{2}} . \end{array} $$(70)

This equation has the same form as the heat equation. The boundary conditions for such equation are

u(±Ls,τ)=0,u(s,0)=f(s),$$\begin{array}{} \displaystyle u(\pm L_{s},\tau)=0,\quad u(s,0)=f(s), \end{array} $$(71)

where

f(s)={bea(s+s0),s<−s0,0,s∈[−s0,s0],be−a(s−s0),s>s0,$$\begin{array}{} \displaystyle f(s)= \begin{cases} be^{a(s+s_{0})}, & {s \lt -s_{0},}\\ 0, & {s\in[-s_{0},s_{0}],}\\ be^{-a(s-s_{0})}, & {s \gt s_{0},} \end{cases} \end{array} $$(72)

is the profile of the initial condition, where a, b, s₀ > 0.

The solution of Eq. (70) is given by

u(s,τ)=∑n=1∞Bncos⁡((2n−1)πs2Ls)e−λnτ,$$\begin{array}{} \displaystyle u(s,\tau)=\sum_{n=1}^{\infty}B_{n}\cos\left(\frac{(2n-1)\pi s}{2L_{s}}\right)e^{-\lambda_{n}\tau}, \end{array} $$(73)

where λ_n is provided by (66) and B_n is

Bn=2bLs(a2+[2n−12πLs]2)[(2n−1)π2Ls((−1)n−1e−a(Ls−s0)−sin⁡((2n−1)πs02Ls))+acos⁡((2n−1)πs02Ls)].$$\begin{array}{} \displaystyle B_{n}=\frac{2b}{L_{s}\bigg(a^{2}+\left[\frac{2n-1}{2}\frac{\pi}{L_{s}}\right]^{2}\bigg)} \Bigg[\frac{(2n-1)\pi}{2L_{s}}\bigg((-1)^{n-1}e^{-a(L_{s}-s_{0})}-\sin\left(\frac{(2n-1)\pi s_{0}}{2L_{s}}\right)\bigg)\\\displaystyle \quad\,\,\,\,+\,a\cos\left(\frac{(2n-1)\pi s_{0}}{2L_{s}}\right)\Bigg]. \end{array} $$(74)

Now, we wish to calculate the time of tumor recurrence. We assume that the amplitude of the perturbation is appreciable for a value M, for s = s₀. It is clear that the most unstable modes occur when λ_n < 0. Thus, from Eq. (66), it is straightforward to find the first value of n for which λ_n > 0, namely

n=⌊12+Lsπ(1−β−V∗)⌋,$$\begin{array}{} \displaystyle n=\left\lfloor\frac{1}{2}+\frac{L_{s}}{\pi}\sqrt{(1-\beta-V_{*})}\right\rfloor , \end{array} $$(75)

where the symbol ⌊⋅⌋ denotes the integer part of a number. In the biological literature, there is a vast range of values for the diffusion and proliferation coefficients. To carry out the estimations, we resort to the following value for the proliferation ρ = 0.2 day⁻¹, which is in the range [0.01–0.5] day⁻¹, taken from [28, 54] and D = 0.05 mm²/day (which is in the range [0.0004–0.1] mm²/day) [39].

Finally, we take α = 1/10 day⁻¹, L = 85 mm, x₀ = 10 mm, c=cmin=2(1−β),$\begin{array}{} c=c_{\text{min}}=2\sqrt{(1-\beta)}, \end{array} $M = 0.3, b = 0.005, a=(c−c2−4(1−β−V∗))/2,$\begin{array}{} \displaystyle a=\left(c-\sqrt{c^2-4(1-\beta-V_{*}})\right)/2, \end{array} $ and V_* = 0.3 in units of u_*. Therefore, the recurrence time is t ≅ 6 months, which is in the range clinically observed for GBM. In Figs. 4 (a) and (b), the relationship between the depth of resection b in the surgery and the recurrence time t is depicted for different values of ρ.

Fig. 4

In this paper we have analyzed a simple model of glioma progression incorporating the normal tissue, tumor cells and the necrotic core. In comparison with the Fisher-Kolmogorov equation, widely used to model GBM, the model studied here only adds a single extra effective parameter accounting for the finite tumor cell lifetime in the tumor microenvironment due to the effect of vascular degeneration, competition for space and resources, hypoxia, acidosis and interaction with the immune system. Remarkably, with only this simple addition the model displays many of the signatures of GBM embodied as bright solitary wave solutions behaving as attractors of the dynamics of the tumor rim. We have studied the formation and propagation speed of this tumor front, we have proven numerically (and analytically in certain limits) that it takes the form of a solitary wave, indeed a vector soliton composed of a bright soliton in the malignant cell population coupled to a kink in the normal cell population and an antikink in a necrotic core compartment. We have also computed what is the relapse time of the tumor after surgical extirpation of the tumor even when it is possible to achieve total resection of the visible part of the tumor. It is worth mentioning that the outcomes of this model, such as the relation between the width of the bright solitons with the biological constants (diffusion, proliferation and death reate constants), have been used to predict novel imaging biomarkers for GBM patients [37]. In particular, the average size of the tumor rim, as observed in contrast enhanced T1-weighted magnetic resonance images, is a statistically significant metric for predicting the overall survival of GBM patients.