Advances in Pure Mathematics
Vol.06 No.12(2016), Article ID:72061,16 pages
10.4236/apm.2016.612066
Nonlinear Evolution Equations and Its Application to a Tumour Invasion Model
Akisato Kubo1, Yuto Miyata2, Hidetoshi Kobayashi3, Hiroki Hoshino1, Naoki Hayashi4
1Department of Mathematics, School of Health Sciences, Fujita Health University, Toyoake, Japan
2Medical Radiation Sciences, Graduate School of Health Sciences, Fujita Health University, Toyoake, Japan
3Department of Radiation Oncology, School of Medicine, Fujita Health University, Toyoake, Japan
4Faculty of Radiological Technology, School of Health Sciences, Fujita Health University, Toyoake, Japan
Copyright © 2016 by authors and Scientific Research Publishing Inc.
This work is licensed under the Creative Commons Attribution International License (CC BY 4.0).
http://creativecommons.org/licenses/by/4.0/
Received: September 12, 2016; Accepted: November 14, 2016; Published: November 17, 2016
ABSTRACT
We consider nonlinear evolution equations with logistic term satisfying initial Neumann-boundary condition and show global existence in time of solutions to the problem in arbitrary space dimension by using the method of energy. Applying the result to a mathematical model of tumour invasion, we discuss the property of the rigorous solution to the model. Finally we will show the time depending relationship and interaction between tumour cells, the surrounding tissue and matrix degradation enzymes in the model by computer simulations. It is seen that our mathematical result of the existence and asymptotic behaviour of solutions verifies our simulations, which also confirm the mathematical result visibly.
Keywords:
Nonlinear Evolution Equation, Mathematical Analysis, Tumour Invasion, Proliferation, Re-Establishment
1. Introduction
In this paper we consider the initial Neumann-boundary value problem of nonlinear evolution equations with logistic term, arising from tumour invasion models with proliferation and re-establishment: (NE)
where 
for
, D and μ are positive constants, Ω is a bounded domain in 
and 
is a smooth boundary of Ω and 
is the outer unit normal vector.
Let us introduce function spaces used in below. First, 
denotes the usual Sobolev space 
of order l on Ω. For functions 
and 
defined in
, we denote
where 
is a multi-index for
.
The eigenvalues of 
with the homogeneous Neumann boundary conditions are denoted by 
satisfying 
and 
indicates the 
normalized eigenfunction corresponding to
. For a non-negative integer l, we set 
as a closure of 
in the function space
. It is noticed that we have 
for
, which enables us to use Poincare’s Inequality.
Putting 
in (1) it follows from (1) that
where 
and 
are positive parameters. (NE) is rewritten by the following problem.
where 
We will show the global existence in time of solutions of (RP), which gives our desired result of (NE).
Applying the above result to the following mathematical model of tumour invasion proposed by Chaplain and Lolas [1] , we have a rigorous mathematical understanding to tumour invasion for the key variables
, 
and
.
where 
is the density of tumour cells, 
is matrix degradation enzymes (MDEs) concentration and 
is extracellular matrix (ECM) density in 
and
, 
, 
, 
, 
, 
, 
and 
are positive constants. In the right hand side of (4) the second, third and fourth terms mean chemotaxis, haptotaxis and proliferation of tumour cell respectively. Also the second term of the right hand side of (5) describes the re-establishment of ECM. We consider an initial boundary value problem for (C-L) satisfying
,
where 
are initial data of
.
Chaplain and Anderson [2] , corresponding to the case of 
in (C-L), proposed a mathematical model on generic solid tumour growth at the avascular stage. While most tumours are asymptomatic at this stage, it is still possible for cells to escape and migrate to the lymph nodes and for more aggressive tumours to invade.
In our previous papers [3] [4] , we consider only the case of 
and 
for our convenience. In this paper we may consider the case where 
is sufficiently small positive constant. Actually compared with the effect of haptotaxis, the one of chemotaxis is very small. Furthermore we study the case of
, which describes tumour invasion phenomena with tumour cell proliferation and re-establishment of ECM respectively.
In the final section by computer simulations of our model (by Mathematica 8) we can easily observe time-dependent interaction and the relationship between the above components in complicated procedure of tumour invasion and a comparison to our rigorous mathematical result. Comparing our mathematical result with computer simulations we will gain a better understanding of the mechanism of tumour invasion.
2. Existence Theorem of (NE)
By deriving the energy estimate of (RP) (see [3] - [9] ) and considering the iteration scheme we obtain existence of solutions to (RP) by the standard argument to show the convergence of solutions of the iteration scheme.
In the same way as used in [3] [4] [5] [6] [7] we have the following estimates of (RP). We begin with 
in order to obtain a basic estimate of (RP). Then we have for
,
by the integration by parts
(7)
It is noticed that the following estimate is obtained in [4] [5] [6] [7] for
,
(8)
Then for the nonlinear term we have by using (8)
(9)
where we used Dionne [10] for the estimate of nonlinear terms and 
is sufficiently small positive constant. Therefore we have by integrating the both sides of (7) over 
and using (9)
(10)
where we denote 
Since the last term of the right hand side of (10) is negligible for sufficiently small
, we have by integration by parts with respect to t
(11)
Taking 
sufficiently large for the second term of the right hand side of (11) is negligible. Since 
if
, the third term in the left hand side of (11) is positive. Then we obtain a basic energy estimate
(12)
Replacing
, 
instead of 
in the above procedure, we obtain the following estimate of higher order.
Lemma 1 (Energy estimate of (RP)) Assume that 
for 
satisfies appropriate smooth regularity condition. Supposed that 
we have the energy estimate of (RP) for 
(13)
where we denote for any non-negative integer
, 
We consider the iteration scheme of (RP):
where
(13) guarantees the uniformly bounded estimate of 
for
. We determine 
by the solution of the following ordinary equations with initial data.
The local existence in time of 
is shown by the theory of ordinary differential equations. Therefore, by the energy estimates, the global existence in time of the solution 
satisfying the regularity required for Lemma 1 and justification of the limiting process are assured by the standard method. The energy estimate enables us to get the solution of (RP) in
by considering 
and standard argument of convergence for 
(see [3] - [9] ).
Then we obtain the following result of (NE) by using the above result of (RP).
Theorem 2 Assume that 
for 
satisfies appropriate smooth regularity condition, initial data 
are sufficiently smooth for
, 
and that 
is small enough. For sufficiently large
, there is a solution for 
to (NE) such that it satisfies the following asymptotic behavior
3. Application to a Tumour Invasion Model
In the last several decades, a number of mathematical models describing the procedure of tumour growth have been the remarkable subject of research (cf. [1] [2] [11] - [19] , further references therein). Especially our main concern in this section is mathematical models of avascular tumour growth proposed by Chaplain et al. (see [1] [2] ). They are considered mainly by three components in the process of tumour invasion, tumour cells, ECM (extracellular matrix) and MDEs (matrix degradation enzymes) without the effect of proliferation of tumour cell. Anderson and Chaplain [2] has been developed by Chaplain and Lolas [1] additionally considering into chemotaxis, proliferation of tumour cells and re-establishment of ECM.
Their mathematical approach to above models mainly depends on numerical analysis. In this paper first we show the rigorous mathematical result of (C-L) and then computer simulations, of which the validity is guaranteed by our mathematical result.
On the other hand, there are many mathematical models which can be found in the literature describing tumour angiogenesis. In [20] Levine and Sleeman applied the mathematical model of Othmer and Stevens [21] for the understanding of tumour angiogenesis, which arises in the theory of reinforced random walk. Anderson and Chaplain [12] proposed a model of tumour angiogenesis taking account of endothelial tip-cell migration. The model describes cell migration governed by three factors: diffusion, chemotaxis and haptotaxis.
Rigorous Mathematical approaches to tumour growth models have been known (see [3] - [9] [20] [21] [22] [23] [24] ). Levine and Sleeman [20] and Yang, Chen and Liu [24] studied the global existence in time of solutions and blow up ones to a simplified Othmer and Stevens model. Kubo et al. [3] - [9] show the time global solution and asymptotic behavior of the solution to the mathematical models proposed by [2] [12] [20] [21] [23] .
3.1. Reduced Problem
Following to Levine and Sleeman [20] we reduce our problem to a simpler system (see [3] - [9] [20] ). It is easily seen in (5) that 
is written by
(14)
Integrating (14) over 
for 
Put 
and 
for a constant
. Then we have
denoting 
and 
(15)
Substituting 
by the right hand side of (14), from (4) and (6) it follows that for 
(16)
and
(17)
The nonlinear evolution Equation (1) involves (16) and so we can apply Theorem 2 to (16).
3.2. Existence Theorem of (C-L) with 
The Equations (16) and (17) are essentially regarded as the same type of equation as (1). Hence the energy estimates of 
and 
are obtained and combining these estimates we obtain the desired estimate (cf. [3] [4] [5] [6] [7] ).
Lemma 3 (Energy estimate of (C-L)) We obtain the energy inequality of the reduced problem (16) and (17) with zero-Neumann boundary condition for 
and sufficiently large 
(18)
where 
as
.
Then applying the same argument as used for Theorem 2 to the above mathematical model, we obtain global existence in time and asymptotic behaviour of the solutions to our mathematical model.
Our main result for 
is as follows.
Theorem 4 For smooth initial data 
assume that 
- norms of
, 
and 
are sufficiently small, then there are classical solutions of (C-L) for
: 
such that they satisfy the following asymptotic behavior for 
Also we can deal with the case of 
as follows.
3.3. Existence Theorem of (C-L) with 
In 
for
, especially we may consider 
-inner product composed of the chemotaxis term and 
in the same way as in Section 2
where we can take 
small enough for sufficiently small
. It is noticed that compared with the effect of haptotaxis, the one of chemotaxis may be quite small. Hence since we can have the same estimate as Lemma 3 for sufficiently small
, the same result as in Theorem 4 holds for this case too.
Theorem 5 Under the same assumption as in Theorem 4 we further assume that 
is sufficiently small, then we obtain the same result as stated in Theorem 4.
4. Computer Simulations
In Kolev and Zubik-Kowal [16] the same type model of (C-L) for 
was considered and they obtained numerical solutions. In this paper our advantage is to show global existence in time of rigorous mathematical smooth solutions of the model and based on it carry out computer simulations.
In the following Figures 1-7 illustrated below we show the graphs along the time by the computer simulations of the model at 
and 1.35 by Mathematica 8. We use the parameter values of
, 
, 
, 
, 
, 
, 
, 
and 
specified below.
We will observe the time dependent relationship and interaction between tumour cells (Red line), the surrounding tissue (Black line) and degradation enzymes (Green line). In the graphs below a coordinate axis of the horizontal direction indicates the spatial position and vertical direction indicates the density or concentration of each component of the model.
Figure 1. Interactions between the tumour and the surrounding tissue without proliferation of tumour cell, migration, and ECM re-establishment: The parameter values dn = 0.001, dm = 0.001, γ = 0.02, η = 10, μ1 = 0, μ2 = 0, α = 0 and β = 0.1, xn = 0. We can observe that MDEs degradates the surrounding tissue, and makes space into which tumour cells move. Further tumour cells form a small peak and keep going forward inside ECM, preserving the shape.
Figure 2. Tumour cell proliferation, migration, and interactions between the tumour and the surrounding tissue without ECM re-establishment: The parameter values dn = 0.001, dm = 0.001, γ = 0.02, η = 10, μ1 = 5, μ2 = 0, α = 0.1 and β = 0.1, xn = 0. It is seen that tumour cell density becomes much higher than in Figure 1 inside ECM and maintains the upper bound of the logistic curve constantly.
Figure 3. Tumour cell proliferation, migration, without ECM re-establishment, and interactions between the tumour and the surrounding tissue: The parameter values dn = 0.001, dm = 0.001, γ = 0.02, η = 10, μ1 = 10, μ2 = 0, α = 0.1 and β = 0.1, xn = 0. Increasing μ1 more, it is observed that tumour cell density inside ECM become higher than in Figure 2.
Figure 4. Tumour cell proliferation, migration, ECM re-establishment, and interactions between the tumour and the surrounding tissue: The parameter values dn = 0.001, dm = 0.001, γ = 0.02, η = 10, μ1 = 5, μ2 = 0.001, α = 0.1 and β = 0.1, xn = 0. We take μ2 = 0.001 only and other parameters are same as in Figure 2. Then the tumour cell density is almost same as in Figure 2.
Figure 5. Tumour cell proliferation, migration, ECM re-establishment, and interactions between the tumour and the surrounding tissue: The parameter values dn = 0.001, dm = 0.001, γ = 0.02, η = 10, μ1 = 5, μ2 = 1, α = 0.1 and β = 0.1, xn = 0. When taking μ2 = 0.001, compared with Figure 2 and Figure 4, it is clear that the invasive area of tumour cells inside ECM is much more limited.
Figure 6. Tumour cell proliferation, migration, ECM re-establishment, and interactions between the tumour and the surrounding tissue: The parameter values dn = 0.001, dm = 0.001, γ = 0.02, η = 10, μ1 = 5, μ2 = 5, α = 0.1 and β = 0.1, xn = 0. Compared with Figure 4 and Figure 5, the reachable range of tumour cells is observed to be more suppressed by taking μ2 = 5.
Figure 7. Tumour cell proliferation, chemotaxis, migration, ECM re-establishment, and interactions between the tumour and the surrounding tissue: The parameter values dn = 0.001, dm = 0.001, γ = 0.02, η = 10, μ1 = 5, μ2 = 0, α = 0.1 and β = 0.1, xn = 0.01. By the effect of chemotaxis (xn = 0.01), tumour cells are attracted by MDEs, the density is beyond 1 at t = 0.25 ~ 0.5, eventually it converges to 1 and after that keeps it constantly.
5. Conclusions
In order to obtain the global existence in time and asymptotic profile of solutions of a mathematical model of tumour invasion proposed by Chaplain and Lolas, we investigate nonlinear evolution equations with logistic term related to our mathematical models as an initial Neumann-boundary value problem. We could show the global existence in time of rigorous mathematical solutions to the initial boundary value problem for the model in arbitrary space dimension by using the energy inequalities. Applying the result to our model we show global existence in time of mathematical solutions of the model.
By Figures 1-7, it is recognized that our rigorous mathematical result of the existence and asymptotic behaviour of smooth solutions verifies our computer simulations and confirms the pattern form of each component of the model in the graphs respectively. Then we can gain the understanding of the process of tumour invasion more in details.
Acknowledgements
This work was supported in part by the Grants-in-Aid for Scientific Research (C) 19540200, 22540208, 25400148 and 16K05214 from Japan Society for the Promotion of Science.
Cite this paper
Kubo, A., Miyata, Y., Kobayashi, H., Hoshino, H. and Hayashi, N. (2016) Nonlinear Evolution Equations and Its Application to a Tumour Invasion Model. Advances in Pure Mathematics, 6, 878-893. http://dx.doi.org/10.4236/apm.2016.612066
References
- 1. Chaplain, M.A.J. and Lolas, G. (2006) Mathematical Modeling of Cancer of Tissue: Dynamic Heterogeneity. Networks and Heterogeneous Media, 1, 399-439.
http://dx.doi.org/10.3934/nhm.2006.1.399 - 2. Anderson, A.R.A. and Chaplain, M.A.J. (2003) Mathematical Modelling of Tissue Invasion, In: Preziosi, L., Ed., Cancer Modelling and Simulation, Chapman Hall/CRC, 269-297.
- 3. Kubo, A. and Kimura, K. (2014) Mathematical Analysis of Tumour Invasion with Proliferation Model and Simulations. WSEAS Transaction on Biology and Biomedicine, 11, 165-173.
- 4. Kubo, A. and Hoshino, H. (2015) Nonlinear Evolution Equation with Strong Dissipation and Proliferation, Current Trends in Analysis and Its Applications. Springer, Birkhauser, 233-241.
- 5. Kubo, A. and Suzuki, T. (2004) Asymptotic Behavior of the Solution to a Parabolic ODE System Modeling Tumour Growth. Differential and Integral Equations, 17, 721-736.
- 6. Kubo, A., Suzuki, T. and Hoshino, H. (2005) Asymptotic Behavior of the Solution to a Parabolic ODE System. Mathematical Sciences and Applications, 22, 121-135.
- 7. Kubo, A. and Suzuki, T. (2007) Mathematical Models of Tumour Angiogenesis. Journal of Computational and Applied Mathematics, 204, 48-55.
http://dx.doi.org/10.1016/j.cam.2006.04.027 - 8. Kubo, A., Saito, N., Suzuki, T. and Hoshino, H. (2006) Mathematical Models of Tumour Angiogenesis and Simulations, Theory of Bio-Mathematics and Its Application. RIMS Kokyuroku, 1499, 135-146.
- 9. Kubo, A. (2011) Nonlinear Evolution Equations Associated with Mathematical Models, Discrete and Continuous Dynamical Systems, Supplement, 881-890.
- 10. Dionne, P. (1962) Sur les problèmes de Cauchy hyperboliques bien posés. Journal d'Analyse Mathematique, 10, 1-90.
http://dx.doi.org/10.1007/BF02790303 - 11. Andasari, V., Roper, R.T., Swat, M.H. and Chaplain, M.A.J. (2011) Integrating Intracellular Dynamics Using CompuCell3D and Bionetsolver: Applications to Multiscale Modelling of Cancer Cell Growth and Invasion. PLoS ONE, 7, e33726.
http://dx.doi.org/10.1371/journal.pone.0033726 - 12. Anderson, A.R.A. and Chaplain, M.A.J. (1998) Continuous and Discrete Mathematical Models of Tumour-Induced Angiogenesis. Bulletin of Mathematical Biology, 60, 857-899.
http://dx.doi.org/10.1006/bulm.1998.0042 - 13. Deakin, N.E. and Chaplain, M.A.J. (2013) Mathematical Modeling of Cancer Invasion: The Role of Membrane-Bound Matrix Metalloproteinases. Frontiers in Oncology, 3, 70.
http://dx.doi.org/10.3389/fonc.2013.00070 - 14. Hatami, F. and Ghaemi, M.B. (2013) Numerical Solution of Model of Cancer Invasion with Tissue. Applied Mathematics, 4, 1050-1058.
http://dx.doi.org/10.4236/am.2013.47143 - 15. Kim, Y. and Othmer, H.G. (2013) A Hybrid Model of Tumor-Stromal Interactions in Breast Cancer. Bulletin of Mathematical Biology, 75, 1304-1350.
http://dx.doi.org/10.1007/s11538-012-9787-0 - 16. Kolev, M. and Zubik-Kowal, B. (2011) Numerical Solutions for a Model of Tissue Invasion and Migration of Tumourcells. Computational and Mathematical Methods in Medicine, 2011, Article ID: 452320.
http://dx.doi.org/10.1155/2011/452320 - 17. Mahiddin, N. and Hashim, A. (2014) Approximate Analytical Solutions for Mathematical Model of Tumourinvasion and Metastasis Using Modified Adomian Decomposition and Homotopy Perturbation Methods. Journal of Applied Mathematics, 2014, Article ID: 654978.
http://dx.doi.org/10.1155/2014/654978 - 18. Märkl, C., Meral, G. and Surulescu, C. (2013) Mathematical Analysis and Numerical Simulations for a System Modeling Acid-Mediated Tumor Cell Invasion. International Journal of Analysis, 2013, Article ID: 878051.
- 19. Orlando, P.A., Gatenby, R.A. and Brown, J.S. (2013) Tumor Evolution in Space: The Effects of Competition Colonization Tradeoffs on Tumor Invasion Dynamics. Frontiers in Oncology, 3, 45.
http://dx.doi.org/10.3389/fonc.2013.00045 - 20. Levine, H.A. and Sleeman, B.D. (1997) A System of Reaction and Diffusion Equations Arising in the Theory of Reinforced Random Walks. SIAM Journal on Applied Mathematics, 57, 683-730.
http://dx.doi.org/10.1137/S0036139995291106 - 21. Othmer, H.G. and Stevens, A. (1997) Aggregation, Blowup, and Collapse: The ABCs of Taxis in Reinforced Random Walks. SIAM Journal on Applied Mathematics, 57, 1044-1081.
http://dx.doi.org/10.1137/S0036139995288976 - 22. Chaplain, M.A.J., Lachowicz, M., Szymanska, Z. and Wrzosek, D. (2011) Mathematical Modeling of Cancer Invasion: The Importance of Cell-Cell Adhesion and Cell-Matrix Adhesion. Mathematical Models & Methods in Applied Sciences, 21, 719-743.
http://dx.doi.org/10.1142/S0218202511005192 - 23. Sleeman, B.D. and Levine, H.A. (2001) Partial Differential Equations Chemotaxis and Angiogenesis. Mathematical Methods in the Applied Sciences, 24, 405-426.
http://dx.doi.org/10.1002/mma.212 - 24. Yang, Y. Chen, H. and Liu, W. (1997) On Existence and Non-Existence of Global Solutions to a System of Reaction-Diffusion Equations Modeling Chemotaxis. SIAM Journal on Applied Mathematics, 33, 763-785.
http://dx.doi.org/10.1137/S0036141000337796