An analysis of a mathematical model describing the growth of a tumor treated with chemotherapy

Anderson L. A. de Araujo 1 , Artur C. Fassoni 2 , and Luís F. Salvino 3
  • 1 Departamento de Matemática, Universidade Federal de Viçosa, Viçosa, Brazil
  • 2 Instituto de Matemática e Computação, Universidade Federal de Itajubá, Itajubá, Brazil
  • 3 Instituto de Ciências Exatas, Universidade Federal de Viçosa - Campus Florestal, Florestal, Brazil
Anderson L. A. de Araujo, Artur C. Fassoni and Luís F. Salvino
  • Instituto de Ciências Exatas, Universidade Federal de Viçosa - Campus Florestal, Florestal, MG, Brazil
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Abstract

We present a mathematical analysis of a mixed ODE-PDE model describing the spatial distribution and temporal evolution of tumor and normal cells within a tissue subject to the effects of a chemotherapeutic drug. The model assumes that the influx of chemotherapy is restricted to a limited region of the tissue, mimicking a blood vessel passing transversely. We provide results on the existence and uniqueness of the model solution and numerical simulations illustrating different model behaviors.

1 Introduction

The main objective of this work is to perform a rigorous mathematical analysis of a system of nonlinear partial differential equations corresponding to a generalization of a mathematical model describing the growth of a tumor proposed in [6].

To describe the model, let Ω ⊂ IR2, be an open and bounded set; let also 0 < T < ∞ be a given final time of interest and denote t the times between [0, T] and Q = Ω × (0, T), the space-time cylinder and Γ¯=Ω×(0,T) , the space-time boundary. Then, the system of equations we are considering is the following:
{Nt=rNμNNβ1NAαNγNDN,inQ,At=rAA(1AkA)(μA+εA)AαAγADA,inQ,Dt=σΔD+μχωγADAγNDNτD,inQ,Dη=0,onΓ,N(,0)=N0(),A(,0)=A0(),D(,0)=D0(),inΩ.
In [6], Fassoni studied an ODE system corresponding to system (1.1) in a spatially homogeneous setting. Such model describes the growth of a tumor and its effect on the normal tissue, the tissue response to the tumor and the application of chemotherapeutic treatments, without spatial heterogeneity. The aim of the authors was to understand the phenomena of cancer onset and treatment as transitions between different basins of attraction of the underlying ODE system. The equations of the model that were studied in [6] are
{dNdt=rNμNNβ1NAαNγNDN,dAdt=rAA(1AkA)(μA+εA)Aβ3NAαAγADA,dDdt=μγADAγNDNτD,
where N represents the number of normal cells in a given tissue of the human body, A represents the number of tumor cells in the tissue and D represents the concentration of a chemotherapeutic drug used to treat such a tumor.

Parameter rN represents a constant influx of new normal cells produced by the tissue stem cells and μN presents the natural mortality of normal cells. A constant influx is considered because the imperative dynamics within a formed tissue is the maintenance of a homeostatic state through the natural replenishment of old and dead cells, see [14].

On the other hand, tumor cells maintain their own growth program [7]. Thus, a density dependent growth is considered for tumor cells. The logistic growth is chosen due to its simplicity. Parameter μA represents the natural mortality of tumor cells, and εA represents an extra mortality rate due to apoptosis [4].

Parameters β1 and β3 encompass the many negative interactions exerted by tumor cells on normal cells and vice-versa, such as competition for nutrients and oxygen. Besides competition, parameter β3 encompasses also the effects on normal cells of anti-growth and death signals released by normal cells. In the same way, the parameter β1 encompasses also mechanisms developed by tumor cells that damage normal tissue, such as increased local acidity, growth suppression, and release of death signals [9].

The third equation of (1.2) describes the dynamics of chemotherapeutic drug concentration according the following assumptions. The drug has a constant infusion rate μ and a clearance rate τ. Such constant infusion rate mimics a metronomic dosage, i.e., a near continuous and long-term administration of the drug. The absorption and deactivation of the drug by normal and cancerous cells are described in terms of the law of mass action with rates γN and γA. Following the log-linear hypothesis [3], it is assumed that the amounts of drug absorbed by normal (γNND) and cancerous cells (γAAD) kill such cells with rates αN and αA, respectively. Although many models of cancer treatment do not consider drug absorption explicitly, in [6], the authors believe that it is an important fact to be considered, since, this phenomenon contributes to decrease the concentration of drug as time passes.

System (1.2) is similar to the classical Lotka-Volterra competition model, frequently used in models for tumor growth and population dynamics. The fundamental difference here is the use of a constant flux for normal cells instead of a logistic growth. Such constant flux, also used in other well-known models of cancer [5], removes the symmetry observed in the Lotka-Volterra equations, so that there is no steady state with N = 0. Thus, it is impossible to observe the extinction of one of the populations (the normal cells in this case), as opposed to the Lotka-Volterra models. The authors of [6] claim that this is a realistic result since, roughly speaking, cancer “does not win” by killing all the cells in the tissue, but by reaching a dangerous size that disrupts the proper functioning of the tissue and threatens the health of the individual.

In this work, we are not interested in analyzing the dynamics (stability, asymptotic behavior) of the model, as such study has already been made in [6]. Our objective is to study the existence and uniqueness of the solution of system (1.1). Such system extends the ODE model (1.2) to a more realistic situation by considering spatial variation of normal and cancer cells and the diffusion of the chemotherapeutic drug through the tissue, with diffusion coefficient σ [2]. Further, it is also assumed that the drug influx is restricted to a limited region of the tissue, corresponding to a blood vessel passing transversely in such region. This is mathematically described in the model by the expression μχω, where χω is the characteristic function of the subset ω ⊂ Ω. Finally, due to mathematical necessity to simplify the model, we set β3 = 0. This corresponds to a situation where normal cells do not exert negative effects on tumor cells, and is a plausible biological assumption, since there are many tumors that develop resistance to the normal tissue’ mechanisms which suppress tumor growth [9].

The paper is organized as follows. In Section 2 we present the technical hypothesis and state our main result. In Section 3 we study an auxiliary problem. Using its solution, we prove our main result in Section 4. In Section 5 we present numerical simulations illustrating model behavior.

2 Technical hypotheses and main result

Let Ω ⊂ IR2 be a domain with boundary Ω, 0 ≤ T < ∞, and denote Q = Ω × (0, T) and Γ = Ω × (0, T). We will use standard notations for Sobolev spaces, i.e., given 1 ≤ p ≤ +∞ and k ∈ ℕ, we denote
Wpk(Ω)={fLp(Ω):DαfLp(Ω),|α|k};
when p = 2, as usual we denote W2k(Ω)=Hk(Ω) ; properties of these spaces can be found for instance in Adams [1, Theorem 5.4, p. 97]. Problem (1.1) will be studied in the standard functional spaces denoted by
Wq2,1(Q)={fLq(Q):DαfLq(Q),1|α|2,ftLq(Q)},W={fL(Q):ftL(Q)}
and
Lp(0,T;B)={f:(0,T)B:f(t)Lp(0,T;B)<+},
where B is suitable Banach space, and the norm is given by ||f(t)||Lp (0, T; B) = ||||f(t)||B ||Lp ((0, T)). We remark that Lp(Q) = Lp((0, T);Lp(Ω)). Results concerning these spaces can be found for instance in Ladyzhenskaya [10] and Mikhaylov [15].

Next, we state some hypotheses that will be assumed throughout this article.

2.1 Technical Hypotheses

  1. (i)Ω ⊂ ℝ2 is a bounded C2-domain;
  2. (ii)0 < T < ∞, and Q = Ω × (0, T);
  3. (iii)N0, A0L (Ω) and D0W432(Ω) , satisfying D0η()=0 , on Ω;
  4. (iv)0D0μτ and N0, A0 ≥ 0 a.e. on Ω.
Remark 2.1.

The constraints imposed in(iv)on the initial conditions are natural biological requirements.

2.2 Main result

Theorem 2.2
Assume that the Technical Hypotheses 2.1 hold; then, there exists a unique nonnegative solution(N,A,D)W×W×W42,1(Q)of Problem (1.1). Moreover, N, A and D are functions satisfying
N||N0||L(Q)+rNT,ACλ||A0||L(Ω)a.e.inQ
and
||N||W+||A||W+||D||W42,1(Q)C,
where C is a constant depending on rN, μN, β1, αN, γN, Cλ, rA, kA, μA, εA, αA, γA, μ, τ, T, ω, ‖N0L (Ω), ‖A0L (Ω)and||D0||W432(Ω) .
Remark 2.3

The explicit knowledge on how the constant C appearing in the above estimates depends on the given data is important for applications in related control problems.

2.3 Known technical results

To ease the references, we also state some technical results to be used in this paper. The first one is sometimes called the Lions-Peetre embedding theorem (see Lions [11], pp.15); it is also a particular case of Lemma 3.3, pp.80, in Ladyzhenskaya [10]: (obtained by taking l = 1 and r = s = 0).

Lemma 2.4
Let Ω be a domain of IRn with boundary ∂Ω satisfying the cone property. Then, the functional spaceWp2,1(Q)is continuously embedded in uLq(Q) for q satisfying:(i)1qp(n+2)n+22p , ifp<n+22 ; (ii) 1 ≤ q < ∞, ifp=n+22and(iii)q = ∞, ifp>n+22 . In particular, for such q and any functionuWp2,1(Q)we have that
uLq(Q)CuWp2,1(Q),
with a constant C depending only on Ω, T , p, q, n.

In the cases(ii), (iii)or in(i)when1q<p(n+2)n+22p , the referred embedding is compact.

Next, we consider the following simple parabolic initial-boundary value problem:
{uti,j=1naij(x,t)u2xixj+j=1nai(x,t)uxj+a(x,t)u=finQ,i=1nbi(x,t)uxi+b(x,t)u=0onΓ,u(,0)=u0()inΩ.

Existence and uniqueness of solutions for this problem is a particular case of Theorem 9.1, pp.341, in Ladyzenskaya [10] for the case of Neumann boundary condition, according to the remarks at the end Chapter IV, section 9, p. 351 in [10]. In the following, we state this particular result, stressing the dependencies certain norms of the coefficients, that will be important in our future arguments.

Proposition 2.5

Let Ω be a bounded domain inn, with a C2boundary ∂Ω, aij be bounded continuous functions in Q, and p > 1. Assume that

  1. aijC(Q¯) , i, j = 1,..., n; [aij]n×nis a real positive matrix such that for some positive constant β we have i,j=1naij(x,t)ξiξjβ|ξ|2for all (x, t) ∈ Q and all ξ ∈ Rn,;
  2. fLp(Q);
  3. aiLr(Q) with either r = max p, n + 2 if pn + 2 or r = n + 2 + ε, for any ε > 0, if p = n + 2;
  4. aLs(Q) with either s = max (p, (n + 2)/2) if p ≠ (n + 2)/2 or s = (n + 2)/2 + ε, for any ε > 0, if p = (n + 2)/2.
  5. bi,bC2(Γ¯) , i = 1,..., n, and the coefficients bi(x, t) satisfy the condition|i=1nbi(x,t)ηi(x)|δ>0for a.e. in ∂Ω × (0, T), where ηi(x) is the ith-component of the unitary outer normal vector to ∂Ω in x Ω;
  6. u0Wp22p(Ω)with p ≠ 3 and satisfying the compatibility conditioni=1nbiu0xi+bu0=0on ∂ Ω when p > 3.

Then, there exists a unique solutionuWp2,1(Q)of Problem (2.1); moreover, there is a positive constant Cp such that the solution satisfies
uWp2,1(Q)Cp(fLp(Q)+u0Wp22p(Ω)).

Such constant Cp depends only on Ω, T, p, r, s, β, δ and on the normsbiC2(Γ¯) , bC2(Γ¯) , aijC(Q¯) , ||ai||Lr (Q)and ||ai||Ls (Q). Moreover, we may assume that the dependencies of Cp on stated the norms are non decreasing.

Remark 2.6

The result set out in Proposition 2.5 can be formulated for the parabolic problem with Dirichlet conditions (see Ladyzenskaya [10, Theorem 9.1, pp.341]). In the problem with Dirichlet condition the compatibility condition in Proposition 2.5-(6) can be replaced by u0 = 0 on ∂Ω when p > 3/2. This way, all the results in this paper holds if we replaced the Neumann conditions by Dirichlet conditions.

3 An auxiliary problem

In this section we will prove an auxiliary result to be used in the proof of Theorem 2.2. To cope with difficulties with the signs of certain terms during the derivation of the estimates, we firstly have to consider the following modified problem:
{N^t=rNμNN^β1N^A^αNγN|D^|N^,inQ,A^t=rAA^(1A^kA)(μA+εA)A^αAγA|D^|A^,inQ,D^t=σΔD^+μχωγD^A^γND^N^τD^,inQ,D^η()=0,onΓ,N^(,0)=N0(),A^(,0)=A0(),D^(,0)=D0(),inΩ.
Now we observe that, since the equation for N^ in this last problem is, for each x ∈ Ω, an ordinary differential equation which is linear in N^ , we can find an explicit expression for it in terms of  and |D^| . However,  is, for each x ∈ Ω, a nonlinear differential equation inÂ, and we can determine its explicit expression in terms of |D^| using Bernoulli’s method. Using these observations and setting λ = rA − (μA + εA), we introduce operators Λ : L (Q) → L (Q) and Θ : L (Q) → L (Q), defined respectively by
Λ(ϕ)(x,t)=A0(x)kAeλteαAγA0t|ϕ(ξ,x)|dξkA+A0(x)rA0teλseαAγA0s|ϕ(ξ,x)|dξds
and
Θ(ϕ)(x,t)=N0(x)+rN0teμNseαNγN0s|ϕ(x,ξ)|dξeβ10sΛ(ϕ)(x,ξ)dξdseμNteαNγN0t|ϕ(x,ξ)|dξeβ10tΛ(ϕ)(x,ξ)dξ,
where 0 ≤ stT.
Remark 3.1
Thus, ( N^ , Â, D^ ) is a solution of (3.1) if, and only if,N^=Θ(D^) , A^=Λ(D^)andD^satisfies the following integro-differential system:
{D^t=σΔD^+μχωγD^Λ(D^)γND^Θ(D^)τD^,inQ,D^η()=0,onΓ,D^(,0)=D0(),inΩ.
Remark 3.2

Notice that, to guarantee that (N, A, D), withD=D^ , N=Θ(D^)andA=Λ(D^)is also a solution of system (1.1), it is enough to prove that the solutionD^of Problem (3.4) is nonnegative.

For the Problem 3.4, we have the following existence result:

Proposition 3.3
Assuming that the Technical Hypotheses 2.1 hold, there exists at least one nonnegative solutionD^W42,1(Q)of Problem(3.4). Moreover,D^μτa.e. in Q and
||D^||W42,1(Q)C,
where C is a constant depending on μ, T , ω and||D0||W432(Ω) .
Lemma 3.4

Let f : (0, T) → ℝ differentiable such that f (t) > 0 and f′(t) ≥ 0. Ifg(t)=0tf(x)dxf(t) , then g(t) ≤ T, for all t ∈ (0, T).

Proof
Since f is continuous in (0, T), it follows that
g(t)=f(t)2f(t)0tf(x)dxf(t)2=1f(t)f(t)g(t).

As f (t) > 0 we have g(t) ≥ 0 and using the fact that f′(t) ≥ 0 we obtain f(t)f(t)g(t)0 . Therefore, g′(t) ≤ 1, which suggests g(t) ≤ t, for all t ∈ (0, T). Thus, g(t) ≤ T, as intended.

Since in the proof of existence of solutions of (3.4) the expression of Λ and Θ will play important roles, we state some of their properties in the following:

Lemma 3.5

If N0, A0L (Ω) and Cλ = max {1, eλT}, then for any ϕ, ϕ1, ϕ2L (Q) and for almost every (x, t) ∈ Q, there holds

  1. (i)0 ≤ Θ(ϕ)(x, t) ≤ ||N0||L (Ω) + rNT;
  2. (ii)0 ≤ Λ(ϕ)(x, t) ≤ Cλ ||A0||L (Ω);
  3. (iii)||Λ(ϕ1) − Λ(ϕ2)||L (Q)C1||ϕ1ϕ2||L (Q), where C1is a constant depending on rA, kA, αA, γA, Cλ, T and ||A0||L (Ω);
  4. (iv)||Θ(ϕ1) − Θ(ϕ2)||L (Q)C2||ϕ1 >− ϕ2||L (Q), where C2is a constant depending on rN, μN, β1, αN, γN, Cλ , C1, T, ||ϕ1||L (Q), ||ϕ2||L (Q), ||N0||L (Ω)and ||A0||L (Ω).

Proof (i) and (ii)
By the expressions (3.2) and (3.3) it is immediate that Λ(ϕ)(x, t), Θ(ϕ)(x, t) ≥ 0. To prove that Θ(ϕ)(x, t) ≤ ||N0||L (Ω) + rNT , we observe that
Θ(ϕ)(x,t)=N0(x)+rN0teμNseαNγN0s|ϕ(x,ξ)|dξeβ10sΛ(ϕ)(x,ξ)dξdseμNteαNγN0t|ϕ(x,ξ)|dξeβ10tΛ(ϕ)(x,ξ)dξN0(x)+rN0teμNseαNγN0s|ϕ(x,ξ)|dξeβ10sΛ(ϕ)(x,ξ)dξdseμNteαNγN0t|ϕ(x,ξ)|dξeβ10tΛ(ϕ)(x,ξ)dξ.
Fixed x ∈ Ω, we define
g(x,t)=0teμNseαNγN0s|ϕ(x,ξ)|dξeβ10sΛ(ϕ)(x,ξ)dξdseμNteαNγN0t|ϕ(x,ξ)|dξeβ10tΛ(ϕ)(x,ξ)dξ,
and using the Lemma 3.4 with f(x,t)=eμNteαNγN0t|ϕ(x,ξ)|dξeβ10tΛ(ϕ)(x,ξ)dξ , it follows that
Θ(ϕ,φ)(x,t)N0(x)+rNT||N0||L(Ω)+rNT.
To prove that Λ(ϕ)(x, t) ≤ Cλ ||A0||L (Ω), note that
Λ(ϕ)(x,t)=A0(x)kAeλteαAγA0t|ϕ(ξ,x)|dξkA+A0(x)rA0teλseαAγA0s|ϕ(ξ,x)|dξds1kAA0(x)kAeλteαAγA0t|ϕ(x,ξ)|dξA0(x)eλtCλA0(x)Cλ||A0||L(Ω).
Proof (iii)
We firstly need to observe that, due to the mean value inequality, given any z1, z2 ∈ IR, there is θ = θ (z1, z2) such that ez2ez1 = e(1−θ)z1 + θz2(z2z1); in particular, for any z1, z2 ≤ 0 we also have (1 − θ)z1 + θz2 ≤ 0 and thus
|ez2ez1||z2z1|,z1,z20.
Secondly, we note that by the inequality (3.5) and by ϕiL (Q), i = 1, 2, we obtain
|eαAγA0t|ϕ1(x,ξ)|dξeαAγA0t|ϕ2(x,ξ)|dξ||αAγA0t(|ϕ1(x,ξ)||ϕ2(x,ξ)|)dξ|αAγAT||ϕ1ϕ2||L(Q).
Thirdly, we observe that
|eαAγA0t|ϕ1(x,ξ)|dξ0teλseαAγA0s|ϕ2(x,ξ)|dξdseαAγA0t|ϕ2(x,ξ)|dξ0teλseαAγA0s|ϕ2(x,ξ)|dξds||eαAγA0t|ϕ1(x,ξ)|dξeαAγA0t|ϕ2(x,ξ)|dξ|0teλseαAγA0s|ϕ2(x,ξ)|dξds+eαAγA0t|ϕ2(x,ξ)|dξ0teλs|eαAγA0s|ϕ1(x,ξ)|dξeαAγA0s|ϕ2(x,ξ)|dξ|ds.
How eλTCλ and eαAγA||ϕ2||L (Q) ≤ 1, and using study analogous to that done in (3.6), we obtain that
|eαAγA0t|ϕ1(x,ξ)|dξ0teλseαAγA0s|ϕ2(x,ξ)|dξdseαAγA0t|ϕ2(x,ξ)|dξ0teλseαAγA0s|ϕ1(x,ξ)|dξds|2αAγACλT2||ϕ1ϕ2||L(Q).
Finally, the expression in (3.3) suggests
|Λ(ϕ1)(x,t)Λ(ϕ2)(x,t)|A0(x)eλt|eαAγA0t|ϕ1(x,ξ)|dξeαAγA0t|ϕ2(x,ξ)|dξ|+1kAA0(x)2rAeλt|eαAγA0t|ϕ1(x,ξ)|dξ0teλseαAγA0s|ϕ2(x,ξ)|dξdseαAγA0t|ϕ2(x,ξ)|dξ0teλseαAγA0s|ϕ1(x,ξ)|dξds|,
and using the estimates obtained in (3.6) and (3.7) and making the possible simplifications, we obtain
|Λ(ϕ1)(x,t)Λ(ϕ2)(x,t)|||A0||L(Ω)CλαAγAT||ϕ1ϕ2||L(Q)+2kA||A0||L(Ω)2rACλ2αAγAT2||ϕ1ϕ2||L(Q),
for almost everything (x, t) ∈ Q, i.e.,
||Λ(ϕ1)Λ(ϕ2)||L(Q)C1||ϕ1ϕ2||L(Q).
Proof (iv)
First, note that
|eαNγN0t|ϕ2(x,ξ)|dξeβ10tΛ(ϕ2)(x,ξ)dξeαNγN0t|ϕ1(x,ξ)|dξeβ10tΛ(ϕ1)(x,ξ)dξ|eαNγN0t|ϕ2(x,ξ)|dξ|eβ10tΛ(ϕ2)(x,ξ)dξeβ10tΛ(ϕ1)(x,ξ)dξ|+eβ10tΛ(ϕ1)(x,ξ)dξ|eαNγN0t|ϕ2(x,ξ)|dξeαNγN0t|ϕ1(x,ξ)|dξ|,
and by the inequality (3.5) and by Λ(ϕi), ϕiL (Q), i = 1, 2, we obtain
|eαNγN0t|ϕ2(x,ξ)|dξeβ10tΛ(ϕ2)(x,ξ)dξeαNγN0t|ϕ1(x,ξ)|dξeβ10tΛ(ϕ1)(x,ξ)dξ|eαNγN0t|ϕ2(x,ξ)|dξβ1T||Λ(ϕ1)Λ(ϕ2)||L(Q)+eβ10tΛ(ϕ1)(x,ξ)dξαNγNT||ϕ1ϕ2||L(Q).
Since
|eαNγN0t|ϕ2(x,ξ)|dξeβ10tΛ(ϕ2)(x,ξ)dξ0teμNseαNγN0s|ϕ1(x,ξ)|dξeβ10sΛ(ϕ1)(x,ξ)dξdseαNγN0t|ϕ1(x,ξ)|dξeβ10tΛ(ϕ1)(x,ξ)dξ0teμNseαNγN0s|ϕ2(x,ξ)|dξeβ10sΛ(ϕ2)(x,ξ)dξds|eαNγN0t|ϕ2(x,ξ)|dξeβ10tΛ(ϕ2)(x,ξ)dξ×0teμNs|eαNγN0s|ϕ1(x,ξ)|dξeβ10sΛ(ϕ1)(x,ξ)dξeαNγN0s|ϕ2(x,ξ)|dξeβ10sΛ(ϕ2)(x,ξ)dξ|ds+|eαNγN0s|ϕ2(x,ξ)|dξeβ10sΛ(ϕ2)(x,ξ)dξeαNγN0s|ϕ1(x,ξ)|dξeβ10sΛ(ϕ1)(x,ξ)dξ|×0teμNseαNγN0s|ϕ2(x,ξ)|dξeβ10sΛ(ϕ2(x,ξ)dξ,
doing ||ϕ||L (Q) = max {||ϕ1||L (Q), ||ϕ2||L (Q)} and study analogous to that done in (3.9), guarantees us
|eαNγN0t|ϕ2(x,ξ)|dξeβ10tΛ(ϕ2)(x,ξ)dξ0teμNseαNγN0s|ϕ1(x,ξ)|dξeβ10sΛ(ϕ1)(x,ξ)dξdseαNγN0t|ϕ1(x,ξ)|dξeβ10tΛ(ϕ1)(x,ξ)dξ0teμNseαNγN0s|ϕ2(x,ξ)|dξeβ10sΛ(ϕ2)(x,ξ)dξds|eαNγN0t|ϕ2(x,ξ)|dξeβ10tΛ(ϕ2)(x,ξ)dξeμTeαNγNT||ϕ||L(Q)β1T2×||Λ(ϕ1)Λ(ϕ2)||L(Q)+eαNγN0t|ϕ2(x,ξ)|dξeβ10tΛ(ϕ2)(x,t)dξeμTeβ1TCλ||A0||L(Ω)αNγNT2×||ϕ1ϕ2||L(Q)+eαNγN0t|ϕ2(x,ξ)|dξβ1T2||Λ(ϕ1)Λ(ϕ2)||L(Q)×eμNTeαNγNT||ϕ||L(Q)eβ1TCλ||A0||L(Ω)+eβ10tΛ(ϕ1)(x,ξ)dξαNγNT2||ϕ1ϕ2||L(Q)×eμNTeαNγNT||ϕ||L(Q)eβ1TCλ||A0||L(Ω).
Finally, the expression in (3.3) suggests
|Θ(ϕ1)(x,t)Θ(ϕ2)(x,t)|1eαNγN0t|ϕ1(x,ξ)|dξeβ10tΛ(ϕ1)(x,ξ)dξeαNγN0t|ϕ2(x,ξ)|dξeβ10tΛ(ϕ2)(x,ξ)dξ×(N0(x)|eαNγN0t|ϕ2(x,ξ)|dξeβ10tΛ(ϕ2)(x,ξ)dξeαNγN0t|ϕ1(x,ξ)|dξeβ10tΛ(ϕ1)(x,ξ)dξ|+rN|eαNγN0t|ϕ2(x,ξ)|dξeβ10tΛ(ϕ2)(x,ξ)dξ0teμNseαNγN0s|ϕ1(x,ξ)|dξeβ10sΛ(ϕ1)(x,ξ)dξdseαNγN0t|ϕ1(x,ξ)|dξeβ10tΛ(ϕ1)(x,ξ)dξ0teμNseαNγN0s|ϕ2(x,ξ)|dξeβ10sΛ(ϕ2)(x,ξ)dξ|).
and using the estimates obtained in (3.8), (3.9) and (3.10) and making the possible simplifications, we obtain
|Θ(ϕ1)(x,t)Θ(ϕ2)(x,t)|||N0||L(Ω)eαNγNT||ϕ||L(Q)β1T2C1||ϕ1ϕ2||L(Q)+||N0||L(Ω)eβ1TCλ||A0||L(Ω)αNγNT2||ϕ1ϕ2||L(Q)+rNeμNTeαNγNT||ϕ||L(Q)eβ1TCλ||A0||L(Ω)β1T2C1||ϕ1ϕ2||L(Q)+rNeμNTeαNγNT||ϕ||L(Q)eβ1TCλ||A0||L(Ω)αNγNT2||ϕ1ϕ2||L(Q)
for almost everything (x, t) ∈ Q, i.e.,
||Θ(ϕ1)Θ(ϕ2)||L(Q)C2||ϕ1ϕ2||L(Q).

3.1 Proof of Proposition 3.3

To not overburden the notation, in this subsection we denote D as a generic solution of the equations that follows.

To get a solution of problem (3.4), we will apply the Leray-Schauder fixed point theorem to the mapping Ψ defined as follows:
Ψ:[0,1]×L(Q)L(Q)(l,ϕ)D,
where D is the unique solution of
{Dt=σΔD+μχωlγDΛ(ϕ)lγNDΘ(ϕ)τD,inQ,Dη()=0,onΓ,D(,0)=D0(),inΩ,
with Λ(ϕ) and Θ(ϕ) given by (3.2) and (3.3), respectively.

To apply such theorem we present next a sequence of lemmas:

Lemma 3.6

Suppose N0, A0L (Ω) andD0W432(Ω) . Then the mapping Ψ : [0, 1] × L (Q) → L (Q) is well defined.

Proof
We affirm that the coefficients of the Problem 3.12 satisfy the hypotheses of the Proposition 2.5. For example, it is immediate that −Λ(ϕ) −NΘ(ϕ) − τL4(Q), because by Lemma 3.5, Λ(ϕ), Θ(ϕ) ∈ L (Q). Thus, we conclude that there is a unique solution DW42,1(Q) of problem 3.12. Moreover, D satisfies the following estimate:
||D||W42,1(Q)Cp(||μχω||L4(Q)+||D0||W432(Ω))Cp(μ|ω|14T14+||D0||W432(Ω)).

Finally, from Lemma 2.4, we have W42,1(Q)L(Q) , and we conclude that the operator Ψ in well defined.

Lemma 3.7

Suppose D is a solution of (3.12) and0D0μτa.e. in Ω, then0Dμτa.e. in Q.

Proof
Multiplying the first equation in (3.12) by D and integrating into Ω, we get
12ddtΩ(D)2dx=σΩ|D|2dxμωDdxlγΩΛ(ϕ)(D)2dxlγNΩΘ(ϕ)(D)2dxτΩ(D)2dx.
Thus,
ddtΩ(D)2dx0,
and using Gronwall’s inequality and the fact that D0 ≥ 0 a.e. in Ω, we obtain
Ω(D)2dxΩ(D0)2dx=0,
that is, ||D (·, t)||L2 (Ω) = 0 for all t ∈ (0, T), where we conclude that D = 0 a.e. in Q and therefore D ≥ 0 a.e. in Q.
Now, we observe that the first equation in (3.12) can be rewritten as
t(Dμτ)=σΔ(Dμτ)lγΛ(ϕ)DlγNΘ(ϕ)Dτ(Dμχωτ).
Multiplying by (Dμτ)+ and integrating in Ω, we obtain
12ddtΩ((Dμτ)+)2dx=σΩ|(Dμτ)+|2dxlγΩΛ(ϕ)D(Dμτ)+dxlγNΩΘ(ϕ)D(Dμτ)+dxτω((Dμτ)+)2dxτΩ\ωD(Dμτ)+dx,
that is,
ddtΩ((Dμτ)+)2dx0.
Thus, using Gronwall’s inequality and the fact that D0μτ a.e. in Ω, it follows that
Ω((Dμτ)+)2dxΩ((D0μτ)+)2dx=0,
that is, ||(D(,t)μτ)+||L2(Ω)=0 for all t ∈ (0, T), and therefore (Dμτ)+=0 a.e. in Q, and we conclude that Dμτ a.e. in Q.
Lemma 3.8

For each fixed l ∈ [0, 1], the mapping Ψ(l, ·) : L (Q) → L (Q) is compact, i.e., it is continuous and maps bounded sets into relatively compacts sets.

Proof
The functions Ψ(l, ϕ1) = D1 and Ψ(l, ϕ2) = D2 satisfy the system
{Dit=σΔDi+μχωlγDiΛ(ϕ1)lγNDiΘ(ϕi)τDi,inQ,Diη()=0,onΓ,Di(,0)=D0(),inΩ,
with i = 1, 2; letting D˜=D1D2 , we have
{D˜tσΔD˜+lγD˜Λ(ϕ2)+lγND˜Θ(ϕ2)+τD˜=lγD1(Λ(ϕ1)Λ(ϕ2))lγND1(Θ(ϕ1)Θ(ϕ2)),inQ,D˜η()=0,onΓ,D˜(,0)=D˜0()=0,inΩ.
Using the Proposition 2.5 and the fact that L (Q) ↪ L4(Q) and D1μτ , we get
||D˜||W42,1(Q)Cp||lγD1(Λ(ϕ1)Λ(ϕ2))lγND1(Θ(D1)Θ(D2))||L4(Q)C¯p||lγD1(Λ(ϕ1)Λ(ϕ2))lγND1(Θ(ϕ1)Θ(ϕ2))||L(Q)C¯pγμτ||Λ(ϕ1)Λ(ϕ2)||L(Q)+C¯pγNμτ||Θ(ϕ1)Θ(ϕ2)||L(Q).
Then, by Lemmas 3.5 and 2.4, we finally have
||Ψ(l,ϕ1)Ψ(l,ϕ2)||L(Q)C||ϕ1ϕ2||L(Q),
where C depends on C¯p , C1, C2, γ, γN, μ, τ and the immersion constant.

To show that Ψ(l, ·) is compact, we use the fact that the immersion W42,1(Q)L(Q) is compact and that Ψ(l, ·) is the composition between the inclusion operator and the solution operator, i.e., Ψ(l,):L(Q)W42,1(Q)L(Q) .

Lemma 3.9

Given a bounded subset BL (Q), for each ϕB, the mapping Ψ(·, ϕ) : [0, 1] → L (Q) is uniformly continuous with respect to B.

Proof
Since BL (Q) is bounded, there is rB ≥ 0 such that, for any ϕB, we have ||ϕ||L (Q)rB. Now, let us fix ϕL (Q) and consider l1, l2 ∈ [0, 1] and denote Ψ(l1, ϕ) = D1, Ψ(l2, ϕ) = D2 and D˜=D1D2 . Then, D˜ satisfies
{D˜tσΔD˜+γl2Λ(ϕ)D˜+γNl2Θ(ϕ)D˜+τD˜=γΛ(ϕ)D1(l1l2)γNΘ(ϕ)D1(l1l2),inQ,D˜η=0,onΓ,D˜(,0)=D˜0()=0,inΩ.
Using the Proposition 2.5 and the fact that L (Q) ↪ L4(Q), D1μτ , we get
||D˜||W42,1(Q)Cp||γΛ(ϕ)D1(l1l2)γNΘ(ϕ)D1(l1l2)||L4(Q)C¯pγμτ|l1l2|||Λ(ϕ)||L(Q)+C¯pγNμτ|l1l2|||Θ(ϕ)||L(Q).
Then, by Lemmas 3.5 and 2.4, we finally have
||Ψ(l1,ϕ)Ψ(l2,ϕ)||L(Q)C|l1l2|,
where C depends on C¯p , γ, γN, μ, τ, rN, T, Cλ, ||N0||L (Ω), ||A0||L (Ω) and the immersion constant.
Lemma 3.10

SupposeD0μτa.e. in Ω, then there exists a number ρ > 0 such that, for any l ∈ [0, 1] and any possible fixed point DL (Q) of Ψ(l, ·), there holds ||D||L (Q) < ρ.

Proof

Let DL (Q) such that Ψ(l, D) = D. The analogous demonstration made in Proposition 3.7 guarantees us ||D||L(Q)μτ . Therefore, just take ρ=μτ+1 .

Lemma 3.11

The mapping Ψ(0, ·) : L (Q) → L (Q) has a unique fixed point.

Proof
Indeed, letting l = 0 in 3.12, D is a fixed point of Ψ(0, ·) if, and only if, D is the unique solution to the problem
{Dt=σΔD+μχωτD,inQ,Dη()=0,onΓ,D(,0)=D0(),inΩ.

But Proposition 2.5 guarantees the existence of a unique solution DW42,1(Q)L(Q) of this last problem; therefore Ψ(0, ·) has a unique fixed point in L (Q).

Proposition 3.12

There is a nonnegative solutionD^W42,1(Q)of the problem (3.4).

Proof

From Lemmas 3.6, 3.8, 3.9, 3.10 and 3.11, we conclude that the mapping Ψ : [0, 1] × L (Q) → L (Q) satisfies the hypotheses of the Leray-Schauder’s fixed point theorem (see Friedman [8, pp. 189, Theorem 3]). Thus, there exists D^L(Q) such that Ψ(1,D^)=D^ . Moreover, by Lemmas 3.6 and 3.7, D^W42,1(Q) is nonnegative and D^ is the required solution of (3.4).

4 Proof of Theorem 2.2

Proposition 4.1

There is a nonnegative solution(N^,A^,D^)L(Q)×L(Q)×W42,1(Q)of the modified problem (3.1).

Proof

Just combine the Proposition 3.12, the Remark 3.1 and the Lemma 3.5.

Remark 4.2
We affirm thatN^,A^W . Indeed, by Lemma 3.5 we know thatN^=Θ(D^),A^=Λ(D^)L(Q) . Moreover, returning to the first equation of (3.1), using the Lemmas 3.5 and 3.7, it follows that:
|N^t|rN+μN(||N0||L(Ω)+rNT)+β1(||N0||L(Ω)+rNT)Cλ||A0||L(Ω)+αNγNμτ(||N0||L(Ω)+rNT),
a.e. in Q, i.e.,N^tL(Q) .
Moreover, returning to the second equation of (3.1) and using, again, the Lemmas 3.5 and 3.7, we get:
|A^t|rACλ||A0||L(Ω)+rAkA(Cλ||A0||L(Ω))2+(μA+εA)Cλ||A0||L(Ω)+αAγAμτCλ||A0||L(Ω),
a.e. in Q, i.e.,ÂtL (Q).
Proposition 4.3

There is a nonnegative solution(N,A,D)W×W×W42,1(Q)of problem (1.1).

Proof

Just combine the Proposition 4.1 and the Remarks 3.2 and 4.2.

Proposition 4.4

The solution (N, A, D) of the problem (1.1) is unique.

Proof
Let (N1, A1, D1) and (N2, A2, D2) be solutions to the problem (1.1); if Ñ = N1N2, Ã = A1A2 and D˜=D1D2 , then Ñ, Ã and H˜ satisfy the following problems, respectively:
{N˜t=μNN˜β1A1N˜β1N2A˜αNγNN1D˜αNγND2N˜,inQ,N˜(,0)=N˜0()=0,inΩ,
{A˜t=rAA˜rAkA(A1+A2)A˜(μA+εA)A˜αAγAA1D˜αAγAD2A˜,inQ,A˜(,0)=A˜0()=0,inΩ,
{D˜t=σΔD˜γA1D˜γD2A˜γNN1D˜γND2N˜τD˜,inQ,D˜η()=0,onΓ,D˜(,0)=D˜0()=0,inΩ.
Multiplying the first equation of (4.3) byÑ, integrating into Ω, using the fact that N1 ≤ ||N0||L (Ω) + rNT and the inequality of Young, we have
12ddtΩN˜2dx=μNΩN˜2dxβ1ΩN2A˜N˜dxαNγNΩN1D˜N˜dxαNγNΩD2N˜2dx(||N0||L(Ω)+rNT)(β1Ω|A˜||N˜|dx+αNγNΩ|D˜||N˜|dx)CΩ(A˜2+N˜2+H˜2)dx,
where C depends on β1, αN, γN, rN, T and ||N0||L (Ω).
Now, multiplying the first equation of (4.4) byÃ, integrating into Ω, using the fact that A1Cλ ||A0||L (Ω) and the inequality of Young, we obtain
12ddtΩA˜2dx=rAΩA˜2dxrAkAΩ(A1+A2)A˜2dx(μA+εA)ΩA˜2dxαAγAΩA1D˜A˜dxαAγAΩD2A˜2dxrAΩ|A˜|2dx+αAγACλ||A0||L(Ω)Ω|D˜||A˜|dxCΩ(A˜2+N˜2+H˜2)dx,
where C depends on rA, αA, γA Cλ and ||A0||L (Ω).
Lastly, multiplying the first equation of (4.5) by H˜ , integrating into Ω, using the fact that Dμτ and the inequality of Young, we obtain
12ddtΩD˜2dx=σΩ|D˜|2dxγΩA1D˜2dxγΩD2A˜D˜dxγNΩN1D˜2dxγNΩD2N˜D˜dxτΩD˜2dxγμτΩ|A˜||D˜|dx+γNμτΩ|N˜||D˜|dxCΩ(A˜2+N˜2+H˜2)dx,
where C depends on γ, γN, μ and τ.
Thus,
ddt(Ω(|N˜|2+|A˜|2+|D˜|2)dx)CΩ(|N˜|2+|A˜|2+|D˜|2)dx,
and using the Gronwall’s inequality, we finally
Ω(|N˜|2+|A˜|2+|D˜|2)dxeCTΩ(|N˜0|2+|A˜0|2+|D˜0|2)dx=0,
that is, ||N˜(,t)||L2(Ω)2+||A˜(,t)||L2(Ω)2+||D˜(,t)||L2(Ω)2=0 , for all t ∈ (0, T). Where we concludeÑ =Ã = D˜ = 0 a.e. in Q and therefore N1 = N2, A1 = A2 and D1 = D2 a.e. in Q.

5 Numerical simulations

In this section, we provide numerical simulations illustrating different model behaviors. The settings and methods used to implement the simulations are the following. We consider the spatial domain as a square Ω = [0, L] × [0, L], with L = 1, discretized with n = 50 steps Δx = Δy = L/n = 0.02. The Laplacian ΔD is approximated by second order centered finite differences and the coupled ODE system arising from such discretization is solved with the method of lines in the software Mathematica. The simulations run from time t = 0 until t = 25 (which is enough to achieve stationary behavior in all simulations).

The initial conditions for numerical simulations are N(x, 0) = N2, A(x, 0) = A2, D(x, 0) = 0, where (N2, A2, 0) is a globally asymptotically stable equilibrium point for the ODE system (1.2) without treatment (ν = 0). The expressions for N2 and A2 are:
N2=rNμN+β1A2,2=rAμAεArAKA.

Such equilibrium is allways globally asymptotically stable in system (1.2) (see details in [6]). From the biological point of view, these initial conditions correspond to the start of chemotherapy application when a tumor is already a formed, where the normal cells were not able to control tumor growth, and no chemotherapy was applied until the tumor reached a stationary state.

To avoid large numbers and numerical instabilities, we re-scale the populations with respect to their possible maximum values, setting NN/(rN/μN) and AA/KA. Therefore, the population sizes range from 0 to 1. The re-scaled parameter values used in the model simulations were fixed to
rN=1,μN=1,A=1,A=1,β1=1.5,μA=0.05,εA=0.05,τH=0.9,γN=0.1,αN=1,γA=1.

These values were chosen to describe: normal cells that reach the equilibrium N = rN/μN = 1 at absence of tumor cells; a tumor with the same carrying capacity of normal cells (KA = rN/μN = 1) and a greater absorption of the chemotherapeutic drug by tumor cells in comparison with normal cells (γA > γN), due to the drug specificity.

In order to illustrate different biological outcomes in the model simulations, we allowed the following parameters to assume different values: the chemotherapeutic drug cytotoxicity against cancer cells αA, the diffusion coefficient of the chemotherapeutic drug σ and the chemotherapy infusion rate μ. We will show that these properties of the drug and the infusion rate are crucial for determining an effective treatment. We also simulated different positions for the subset ω, which is a mathematical description of a blood vessel crossing the tissue, from where the chemotherapy enters the tissue. The values for parameters αA, σ, μ and the position of σ used in each simulation are indicated in Table 1. We present the following results.

Table 1

Set-up of different simulations an their biological outcomes. Each row indicates the numerical values used for the chemotherapeutic parameters αA (cytotoxicity), σ (diffusion coefficient), μ (infusion rate), and the position of ω ⊂ Ω ⊂ ℝ2. Simulation 1 was performed in a two-dimensional domain Ω = [0, 1] × [0, 1], while simulations 2–5 were performed in a one-dimensional domain Ω = [0, 1].

SimulationFigureOutcomeαAμσω
11tumor persistence530.1[0.45, 0.55] × [0.45, 0.55]
22tumor persistence1030.1[0, 0.1]
33tumor extinction1060.1[0, 0.1]
44tumor extinction1030.2[0, 0.1]
55tumor extinction2030.1[0, 0.1]

In the first simulation of system (1.1), we confirm that our model and numerical methods are able to reproduce the expected biological behavior (Figure 1). The blood vessel crosses the tissue at its center, i.e., ω = [0.45, 0.55] × [0.45, 0.55]. We use the following parameter values: αA = 5, μ = 3, and σ = 0.1. With such values, the chemotherapy is not able to lead to tumor extinction. We observe that tumor cells that are near the blood vessel are eliminated but not extinct by the chemotherapeutic effect, and those which are distant from the blood vessel persist (Figure 1).

Fig. 1
Fig. 1

Results of Simulation 1, for model (1.1) within a two-dimensional domain Ω = [0, L] × [0, L] = [0, 1] × [0, 1]. Plots of model solutions N(x, y, t) (normal cells, blue, top row), A(x, y, t) (cancer cells, red, middle row) and D(x, y, t) (chemotherapeutic drug concentration, green, bottom row) at time points t = 0, 1, 15 (columns 1,2 and 3, respectively). See Table 1 for parameter values used here. At time t = 0, the tumor is spread trough the tissue, and as chemotherapy is applied (t > 0), the tumor cells are reduced in the vicinity of the blood vessel, while the distant tumor cells persist along time (the shape of the solution at time t = 15 is stationary). Within the vicinity of the blood vessel, the removal of tumor cells allows the normal tissue to recover and grow.

Citation: Applied Mathematics and Nonlinear Sciences 5, 2; 10.2478/amns.2020.2.00037

In order to make easier to illustrate the model dynamics, we present the results of next simulations in a one-dimensional domain Ω = [0, 1]. In Simulation 2, we use the same parameters values used in Simulation 1 (see Table 1), but increase the chemotherapy toxicity αA and move the blood vessel to the left side of the tissue, ω = [0, 0.1]. Although the tumor cells in the vicinity of the blood vessel are extinct, the chemotherapy is still not able to eliminate the distant tumor cells (Figure 2). Thus, we observe tumor persistence in the long-term. In Simulation 3, we keep the parameters as in Simulation 2, but increase the chemotherapy infusion rate μ (mimicking a higher dose). We observe that the tumor cells are extinct in the entire tissue (Figure 3). In Simulation 4, we illustrate other mechanism to achieve tumor extinction: instead of increasing drug dose, we adopt the parameter values of Simulation 2, but increase the drug diffusion σ, so that it is capable to spread over the entire tissue and effectively eliminate all tumor cells (Figure 4). Finally, in Simulation 5, we also adopt the parameter values of Simulation 2, but increase the chemotherapy toxicity against tumor cells αA. This also leads to tumor extinction (Figure 5). An advantage of the strategies adopted in Simulations 4 and 5, in comparison with Simulation 3 (increasing dose), is that the former lead to less side effects. Simulation 3 describes the use of a drug which spreads faster, while Simulation 5 illustrates the use of a more potent and specific drug, which targets more tumor cells but not more normal cells (αN was not changed). Taken together, these simulations and the different outcomes observed for different parameter values confirm the ability of the model to consistently describe tumor chemotherapy and illustrate the potential of mathematical models to provide testable hypothesis that could be studied together with clinicians in order to achieve better results in the treatment of cancer.

Fig. 2
Fig. 2

Results of Simulation 2, with Ω = [0, L] = [0, 1]. Plots of model solutions A(x, t) (cancer cells, red), N(x, t) (normal cells, blue) and D(x, t) (chemotherapeutic drug concentration, green) at time points t = 0, 3, 6, 9, 12, 15. See Table 1 for parameter values used here. At time t = 0, the tumor is spread trough the tissue, and as chemotherapy is applied (t > 0), the tumor cells are reduced and extinct within a given distance from the blood vessel (x < 0.6), but not in the entire tissue (x > 0.6). Within the region of tumor extinction, the removal of tumor cells release the normal tissue to recover and grow.

Citation: Applied Mathematics and Nonlinear Sciences 5, 2; 10.2478/amns.2020.2.00037

Fig. 3
Fig. 3

Results of Simulation 3, with Ω = [0, L] = [0, 1]. Plots of model solutions A(x, t) (cancer cells, red), N(x, t) (normal cells, blue) and D(x, t) (chemotherapeutic drug concentration, green) at time points t = 0, 3, 6, 9, 12, 15. See Table 1 for parameter values used here. At time t = 0, the tumor is spread trough the tissue, and as chemotherapy is applied (t > 0), the tumor cells are reduced and in the entire tissue. In comparison with Simulation 2, the tumor extinction is reached because the drug infusion rate μ is increased here. Within the entire tissue, the removal of tumor cells release the normal tissue to recover and grow.

Citation: Applied Mathematics and Nonlinear Sciences 5, 2; 10.2478/amns.2020.2.00037

Fig. 4
Fig. 4

Results of Simulation 4, with Ω = [0, L] = [0, 1]. Plots of model solutions A(x, t) (cancer cells, red), N(x, t) (normal cells, blue) and D(x, t) (chemotherapeutic drug concentration, green) at time points t = 0, 3, 6, 9, 12, 25. See Table 1 for parameter values used here. At time t = 0, the tumor is spread trough the tissue, and as chemotherapy is applied (t > 0), the tumor cells are reduced and in the entire tissue. In comparison with Simulation 2, the tumor extinction is reached because the drug diffusion coefficient σ is increased here.

Citation: Applied Mathematics and Nonlinear Sciences 5, 2; 10.2478/amns.2020.2.00037

Fig. 5
Fig. 5

Results of Simulation 5, with Ω = [0, L] = [0, 1]. Plots of model solutions A(x, t) (cancer cells, red), N(x, t) (normal cells, blue) and D(x, t) (chemotherapeutic drug concentration, green) at time points t = 0, 3, 6, 9, 12, 15. See Table 1 for parameter values used here. At time t = 0, the tumor is spread trough the tissue, and as chemotherapy is applied (t > 0), the tumor cells are reduced and in the entire tissue. In comparison with Simulation 2, the tumor extinction is reached because the chemotherapy toxicity against tumor cells, αA, was increased.

Citation: Applied Mathematics and Nonlinear Sciences 5, 2; 10.2478/amns.2020.2.00037

Acknowledgement

The first author was partially sponsored by FAPEMIG FORTIS-10254/2014.

References

  • [1]

    Adams, R. A., Sobolev Spaces. New York: Academic Press, 1975.

  • [2]

    Anderson, A.R.A., A hybrid mathematical model of solid tumour invasion: the importance of cell adhesion. Math. Med. Biol. 22, (2005) 163–186.

    • Crossref
    • PubMed
    • Export Citation
  • [3]

    Benzerky, S.; Pasquier, E.; Barbolosi, D.; Lacarelle, B.; Barlesi, F.; Andre, N.; Ciccolini, J., Metronomic reloaded: theoretical models bringing chemotherapy into the era of precision medicine. In: ELSEVIER. Seminars in Cancer Biology. [S.l.], 2015. v. 35, p. 53–61.

    • Crossref
    • PubMed
    • Export Citation
  • [4]

    Daniel, N.N.; Korsmeyer, S.J., Cell death: critical control points. Cell 116 (2), (2004) 205–219.

    • Crossref
    • PubMed
    • Export Citation
  • [5]

    Eftimie, R.; Bramson, J.L.; Earn, D.J.D., Interactions between the immune system and cancer: a brief review of non-spatial mathematical models. Bull. Math. Biol. 73 (1), (2011) 2–32.

    • Crossref
    • Export Citation
  • [6]

    Fassoni, A. C., Mathematical modeling in cancer addressing the early stage and treatment of avascular tumors. PhD thesis, University of Campinas, 2016.

  • [7]

    Fedi, P.; Tronick, S.R.; Aaronson, S.A., Growth factors. Cancer Med. 4, (1997) 1–64.

  • [8]

    Friedman, A., Partial Differential Equations of Parabolic Type. New York: Mineola, Dover Publications, 2008

  • [9]

    Hanahan, D.; Weinberg, R.A., Hallmarks of cancer: the next generation. Cell 144(5), (2011), 646–674.

    • Crossref
    • PubMed
    • Export Citation
  • [10]

    Ladyzhenskaya, O.; Solonikov, V.; Uraltseva, N., Linear and Quasilinear Equations of Parabolic Type. Amer. Math. Soc., 1968

  • [11]

    Lions, Jacques-Louis., Contrôle des Systèmes Distribués Singuliers. Méthodes Mathématiques de L’informatique, Gautier-Villars, 1983.

  • [12]

    Mcgillen, J.B.; Gaffney, E.A.; Martin, N.K.; Maini, P.K., A general reaction-diffusion model of acidity in cancer invasion. J. Math. Biol. 68 (5), (2014) 1199–1224.

    • Crossref
    • PubMed
    • Export Citation
  • [13]

    Sarapata, E.A.; Pillis, L.G. de., A comparison and catalog of intrinsic tumor growth models. Bull. Math. Biol. 76 (8), (2014) 2010–2024.

    • Crossref
    • PubMed
    • Export Citation
  • [14]

    Simons, B.D.; Clevers, H., Strategies for homeostatic stem cell self-renewal in adult tissues. Cell 145 (6), (2011) 851–862.

    • Crossref
    • PubMed
    • Export Citation
  • [15]

    Mikhaylov, V. P., Partial Differential Equations, Mir Publishers, Moscow, 1978.

If the inline PDF is not rendering correctly, you can download the PDF file here.

  • [1]

    Adams, R. A., Sobolev Spaces. New York: Academic Press, 1975.

  • [2]

    Anderson, A.R.A., A hybrid mathematical model of solid tumour invasion: the importance of cell adhesion. Math. Med. Biol. 22, (2005) 163–186.

    • Crossref
    • PubMed
    • Export Citation
  • [3]

    Benzerky, S.; Pasquier, E.; Barbolosi, D.; Lacarelle, B.; Barlesi, F.; Andre, N.; Ciccolini, J., Metronomic reloaded: theoretical models bringing chemotherapy into the era of precision medicine. In: ELSEVIER. Seminars in Cancer Biology. [S.l.], 2015. v. 35, p. 53–61.

    • Crossref
    • PubMed
    • Export Citation
  • [4]

    Daniel, N.N.; Korsmeyer, S.J., Cell death: critical control points. Cell 116 (2), (2004) 205–219.

    • Crossref
    • PubMed
    • Export Citation
  • [5]

    Eftimie, R.; Bramson, J.L.; Earn, D.J.D., Interactions between the immune system and cancer: a brief review of non-spatial mathematical models. Bull. Math. Biol. 73 (1), (2011) 2–32.

    • Crossref
    • Export Citation
  • [6]

    Fassoni, A. C., Mathematical modeling in cancer addressing the early stage and treatment of avascular tumors. PhD thesis, University of Campinas, 2016.

  • [7]

    Fedi, P.; Tronick, S.R.; Aaronson, S.A., Growth factors. Cancer Med. 4, (1997) 1–64.

  • [8]

    Friedman, A., Partial Differential Equations of Parabolic Type. New York: Mineola, Dover Publications, 2008

  • [9]

    Hanahan, D.; Weinberg, R.A., Hallmarks of cancer: the next generation. Cell 144(5), (2011), 646–674.

    • Crossref
    • PubMed
    • Export Citation
  • [10]

    Ladyzhenskaya, O.; Solonikov, V.; Uraltseva, N., Linear and Quasilinear Equations of Parabolic Type. Amer. Math. Soc., 1968

  • [11]

    Lions, Jacques-Louis., Contrôle des Systèmes Distribués Singuliers. Méthodes Mathématiques de L’informatique, Gautier-Villars, 1983.

  • [12]

    Mcgillen, J.B.; Gaffney, E.A.; Martin, N.K.; Maini, P.K., A general reaction-diffusion model of acidity in cancer invasion. J. Math. Biol. 68 (5), (2014) 1199–1224.

    • Crossref
    • PubMed
    • Export Citation
  • [13]

    Sarapata, E.A.; Pillis, L.G. de., A comparison and catalog of intrinsic tumor growth models. Bull. Math. Biol. 76 (8), (2014) 2010–2024.

    • Crossref
    • PubMed
    • Export Citation
  • [14]

    Simons, B.D.; Clevers, H., Strategies for homeostatic stem cell self-renewal in adult tissues. Cell 145 (6), (2011) 851–862.

    • Crossref
    • PubMed
    • Export Citation
  • [15]

    Mikhaylov, V. P., Partial Differential Equations, Mir Publishers, Moscow, 1978.

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  • View in gallery

    Results of Simulation 1, for model (1.1) within a two-dimensional domain Ω = [0, L] × [0, L] = [0, 1] × [0, 1]. Plots of model solutions N(x, y, t) (normal cells, blue, top row), A(x, y, t) (cancer cells, red, middle row) and D(x, y, t) (chemotherapeutic drug concentration, green, bottom row) at time points t = 0, 1, 15 (columns 1,2 and 3, respectively). See Table 1 for parameter values used here. At time t = 0, the tumor is spread trough the tissue, and as chemotherapy is applied (t > 0), the tumor cells are reduced in the vicinity of the blood vessel, while the distant tumor cells persist along time (the shape of the solution at time t = 15 is stationary). Within the vicinity of the blood vessel, the removal of tumor cells allows the normal tissue to recover and grow.

  • View in gallery

    Results of Simulation 2, with Ω = [0, L] = [0, 1]. Plots of model solutions A(x, t) (cancer cells, red), N(x, t) (normal cells, blue) and D(x, t) (chemotherapeutic drug concentration, green) at time points t = 0, 3, 6, 9, 12, 15. See Table 1 for parameter values used here. At time t = 0, the tumor is spread trough the tissue, and as chemotherapy is applied (t > 0), the tumor cells are reduced and extinct within a given distance from the blood vessel (x < 0.6), but not in the entire tissue (x > 0.6). Within the region of tumor extinction, the removal of tumor cells release the normal tissue to recover and grow.

  • View in gallery

    Results of Simulation 3, with Ω = [0, L] = [0, 1]. Plots of model solutions A(x, t) (cancer cells, red), N(x, t) (normal cells, blue) and D(x, t) (chemotherapeutic drug concentration, green) at time points t = 0, 3, 6, 9, 12, 15. See Table 1 for parameter values used here. At time t = 0, the tumor is spread trough the tissue, and as chemotherapy is applied (t > 0), the tumor cells are reduced and in the entire tissue. In comparison with Simulation 2, the tumor extinction is reached because the drug infusion rate μ is increased here. Within the entire tissue, the removal of tumor cells release the normal tissue to recover and grow.

  • View in gallery

    Results of Simulation 4, with Ω = [0, L] = [0, 1]. Plots of model solutions A(x, t) (cancer cells, red), N(x, t) (normal cells, blue) and D(x, t) (chemotherapeutic drug concentration, green) at time points t = 0, 3, 6, 9, 12, 25. See Table 1 for parameter values used here. At time t = 0, the tumor is spread trough the tissue, and as chemotherapy is applied (t > 0), the tumor cells are reduced and in the entire tissue. In comparison with Simulation 2, the tumor extinction is reached because the drug diffusion coefficient σ is increased here.

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    Results of Simulation 5, with Ω = [0, L] = [0, 1]. Plots of model solutions A(x, t) (cancer cells, red), N(x, t) (normal cells, blue) and D(x, t) (chemotherapeutic drug concentration, green) at time points t = 0, 3, 6, 9, 12, 15. See Table 1 for parameter values used here. At time t = 0, the tumor is spread trough the tissue, and as chemotherapy is applied (t > 0), the tumor cells are reduced and in the entire tissue. In comparison with Simulation 2, the tumor extinction is reached because the chemotherapy toxicity against tumor cells, αA, was increased.