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 .
Here, we analyze in detail a mathematical model that was proposed in  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 , which includes neurosurgery followed by radiotherapy in combination with the alkylating agent temozolomide, and various salvage therapies (e.g., antiangiogenic) once tumor progression recurs . Large-scale research endeavors are ongoing to provide a comprehensive understanding of all the genetic and epigenetic alterations that underlie GBM . 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 .
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 , which epitomizes the simplest form of a nonlinear reaction-diffusion process
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  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 , neurophysiology , and Brownian motion processes  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 , 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 .
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.
2 A simple extension of the Fisher equations
The model to be discussed in detail in this paper for describing the progression of GBM was proposed in . 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
∀x ∈ Ω and ∀t ∈ ℝ+
for some nonnegative continuous functions u0, v0, w0 : Ω̄ → ℝ, together with the homogeneous Neumann boundary conditions
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 , 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
and the rescaled variables
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 . Then, Eqs. (2) in one dimension can be cast in the form
with 0 ≤ β = α/ρ < 1.
3 Analysis of travelling waves
Travelling waves with constant speed are solutions of Eq. (5) depending on z = s – cτ (′ = d/dz)
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
In the limit c ≫ 1, ε ≪ 1 and Eq. (8) is equivalent to
Defining 𝓥ξ = 𝓦, Eq. (9) can be written as the autonomous system
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
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
Its solution is
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
to have two positive roots 𝓥– > 𝓥+ > 0. Thus, for the initial condition 𝓦(𝓥+) = 0, Eq. (13) becomes
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
which is larger than 𝓥+ for 𝓥+ < 1 – β. As 𝓦(𝓥+) = 0 and the function 𝓦(𝓥) is decreasing in the interval (𝓥+, 𝓥m), it is clear that
Therefore, as 𝓦(1) > 0, the existence of a value 𝓥– in the interval (𝓥m, 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
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
that allows for a solution for 𝓥 in the form of quadratures, once initial data is specified,
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).
3.2 Finite speed solutions
Our previous analysis proves the existence of the heteroclinic orbit in the limit c ≫ 1, when ε → 0. If ε ≪ 1 (but not zero), we can recast Eq. (8) as a system
which has the critical points (𝓥∗, 0, 0), for 𝓥∗ ∈ ℝ.
On the other hand, writing ζ = ξ/ε2, system (21) becomes
which is a two-dimensional submanifold of ℝ3. We choose any compact subset of such manifold and designate this compact subset as M0. Geometric perturbation theory uses both the above systems: (22) provides us with an invariant manifold Mε close to M0 and we study the flow of (21) restricted to this manifold.
Let us introduce the following definition, which is useful for our purposess:
The manifold M0 is said to be normally hyperbolic if the linearisation of (22) at each point in M0, restricted to M0, has exactly dim M0 eigenvalues on the imaginary axis.
Now, we use the Fenichel’s invariant manifold theory [20, 25] to prove that for ε sufficiently small, there exists a two-dimensional submanifold Mε of ℝ3 which is within distance ε of M0 and which is invariant for the flow (22):
Ifε > 0, but sufficiently small, there exists a manifoldMεthat lies within 𝓞(ε) ofM0and is diffeomorphic toM0. Moreover, it is locally invariant under the flow of(22), andCrfor anyr < +∞.
Thus, from the previous definition, such a perturbed invariant manifold Mε will exist if M0 is normally hyperbolic.
The linearisation of the fast system (22), restricted to M0 has the matrix
and the eigenvalues of this are λ = 0 (double) and λ = –1 (simple). Thus, M0 is normally hyperbolic and, by Fenichel’s theory, for sufficiently small ε, there exists a perturbed manifold Mε with the properties listed above.
We would like to determine the dynamics on Mε. Since (𝓥, 𝓦) are the slow variables and 𝓤 is the fast variable, the manifold Mε is given by
where hε(𝓥, 𝓦) ∈ Cr for any r < +∞ satisfying h0(𝓥, 𝓦) = 0, and the equations on Mε are
We Taylor expand hε in the variable ε around ε = 0,
Equating powers of ε, we get the following hierarchy of equations for εi, i = 0, 1, 2 …
and the equations on Mε become
These equations determine the dynamics on the manifold Mε.
When ε = 0, (33) reduces to system (10). Existence of a heteroclinic orbit joining the points (𝓥–,0) and (𝓥+,0), 𝓥– > 𝓥+ ≥ 0, of (10) was proved in the previous subsection. Note that when ε > 0, the system (33) still has the same critical points as system (10). Let (𝓥0,𝓦0) be the solution of (33) for ε = 0, we are going to use the Fredholm alternative to show that when ε > 0 is sufficiently small a heteroclinic connection exists between the critical points (𝓥–,0) and (𝓥+,0).
Thus, as ξ → ∞, any solution of (40) is a sum of constant and exponential functions. Thus, the only solution in L2 for the adjoint problem is therefore the zero solution s(ξ) = 0 and consequently the Fredholm ortogonality conditionally holds. We have proved the existence of the searched connection in the manifold Mε. Thus, we have the following theorem
There exists δ > 0 sufficiently small such that, for every ε ∈ (0, δ], Eq. (8)admits a travelling front solution 𝓥(ξ) satisfying 𝓥(−∞) = 𝓥–and 𝓥(∞) = 𝓥+.
Since the relationship between 𝓥(ξ) and 𝓤(ξ) is given by (6b), we have the following corollary about the existence of bright solitons
Let a travelling wave front solution 𝓥(ξ) of Eq. (8) whose existence is given by Theorem 2. Then 𝓤(ξ) is a homoclinic solution or bright soliton solution of the system (6), with ξ = ε z and z = s − cτ.
4 Bright solitons
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).
where C1 and C2 are arbitrary constants. (44) remains localized if
4.1 Solitary wave solutions
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.
4.2 Minimum speed of positive solutions
that is, χ+ becomes a double root of G.
The dependence of c0 on β and ϕ0 is depicted in 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  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
5 Linear Stability
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
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
with 0 < ε ≪ 1. The solution U is stable to perturbations u(s, τ) if
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
We now look for solutions to the linear Eqs. (58a) by setting
From Eq. (60b) we obtain that
Therefore, one has the following condition for the instability of the solution U = 0
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
with Ls being a dimensionless characteristic length,
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
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 λn1+1 > 0 and λn1 < 0. For this value of n1, Ls must verify
Therefore, the eigenfunctions are given by
and, from (61), we obtain the function g2
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
This equation has the same form as the heat equation. The boundary conditions for such equation are
is the profile of the initial condition, where a, b, s0 > 0.
The solution of Eq. (70) is given by
where λn is provided by (66) and Bn is
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 λn < 0. Thus, from Eq. (66), it is straightforward to find the first value of n for which λn > 0, namely
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) .
Finally, we take α = 1/10 day−1, L = 85 mm, x0 = 10 mm,
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 . 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.
We wish to thank Víctor M. Pérez-García and David Diego for discussions. This work has been supported by Ministerio de Economía y Competitividad/FEDER, Spain [grant numbers: MTM2012-31073 and MTM2015-71200-R], Consejería de Educación Cultura y Deporte from Junta de Comunidades de Castilla-La Mancha (Spain) [grant number PEII-2014-031-P] and James S. Mc. Donnell Foundation (USA) 21st Century Science Initiative in Mathematical and Complex Systems Approaches for Brain Cancer (Special Initiative Collaborative-Planning Grant 220020420 and Collaborative award 220020450).
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