Mathematical Analysis of a Non-Local Mixed ODE-PDE Model for Tumor Invasion and Chemotherapy
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Mathematical Analysis of a Non-Local Mixed ODE-PDE Model for Tumor Invasion and Chemotherapy Anderson L.A. de Araujo1 Luís F. Salvino1
· Artur C. Fassoni2 ·
Received: 11 September 2019 / Accepted: 16 June 2020 © Springer Nature B.V. 2020
Abstract We present a new mathematical model for acid-mediated tumor invasion encompassing chemotherapy treatment. The model consists of a mixed ODE-PDE system with four differential equations, describing the spatio-temporal dynamics of normal cells, tumor cells, lactic acid concentration, and chemotherapy drug concentration. The model assumes non-local diffusion coefficients for tumor cells. We provide an analysis on the existence and uniqueness of model solutions. We also provide numerical simulations illustrating the model behavior, showing the invasion and the treatment phases, and comparing the model solutions with the case of constant diffusion coefficients instead of the non-local terms. Keywords Nonlinear system · Existence of solutions · Tumor growth · Acid-mediated tumor invasion · Chemotherapy Mathematics Subject Classification Primary 35K45 · 35K57 · Secondary 92C50 · 92C37
1 Introduction In this work, we present a rigorous mathematical analysis and numerical simulations of a non-local mixed ODE-PDE system of nonlinear differential equations describing the growth and treatment of a tumor. The model is described as follows. Let ⊂ R2 , be an open and bounded set; let also 0 < T < ∞ be a final time and denote t the time within [0, T ], the Anderson L.A. de Araujo was partially supported by FAPEMIG FORTIS-10254/2014.
B A.L.A. de Araujo
[email protected] A.C. Fassoni [email protected] L.F. Salvino [email protected]
1
Departamento de Matemática, Universidade Federal de Viçosa, Viçosa, MG, Brazil
2
Instituto de Matemática e Computação, Universidade Federal de Itajubá, Itajubá, MG, Brazil
A.L.A. de Araujo et al.
space-time cylinder Q = × (0, T ), and the space-time boundary = ∂ × (0, T ). ‘The model is given by: ⎧ ∂N ⎪ = rN − μN N − β1 N A − αH γH N H − αN γN N D, ⎪ ⎪ ⎪ ∂t ⎪ ⎪ ⎪ ⎪ ∂A A ⎪ ⎪ A (N, t) A + rA A 1 − = ξ − μA A − β3 N A − αA γA AD, ⎪ ⎪ ⎪ ∂t kA ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ∂H = ξ H + ν A − τ H − γ N H, H H H H ∂t ⎪ ⎪ ⎪ ∂D ⎪ ⎪ = ξD D + νD χω − τD D − γA AD − γN N D, ⎪ ⎪ ∂t ⎪ ⎪ ⎪ ⎪ ∂H ∂D ∂A ⎪ ⎪ (·) = (·) = (·) = 0, ⎪ ⎪ ⎪ ∂η ∂η ∂η ⎪ ⎪ ⎩ N (·, 0) = N0 (·), A(·, 0) = A0 (·), H (·, 0) = H0 (·), D(·, 0) = D0 (·), where
N (x, t)dx ξA (N, t) := ξA 1 − (1 − 1 ) ||rN /μN
in
Q,
in
Q,
in
Q,
in
Q,
(1.1)
on , in
,
(1.2)
with 0 < 1 ≤ 1. In the model, N (x, t) represents the normal cells within the tissue, A(x, t) represents the tumor cells, H (x, t) represents the excess of normal tissue acid concentration, which is produced by tumor cells, and D(x, t) represents the chemotherapy drug concentration. The model assumptions are the following. Normal cells are produced by the tissue with a constant rate rN and die with rate μN . A constant influx rN is considered instead of a density-dependent growth because in many tissues, such as the s
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