Analysis And Simulation On Chemotaxis-Haptotaxis Models Of Tumor Invasion

Recently,there is an increasing biological and mathematical interest in mathematical models of cancer invasion(see[7-8,16,18,33,37,40-43,47]).In[7]Chaplain and Lolas (2005) developed a mathematical model of the urokinase plasminogen activation(uPA) system and its role in tissue invasion.Cancer invasion is associated with the degradation of the extracellular matrix(ECM),which is degraded by matrix degrading enzymes(MDEs) secreted by tumor cells.In addition to random motion,the migration of invasive cells is directed either by a mechanism termed chemotaxis(cellular locomotion directed in response to a concentration gradient of the diffusible MDE) or by a mechanism termed haptotaxis(cellular locomotion directed in response to a concentration gradient of the non-diffusible adhesive molecules within extracellular matrix).Chaplain and Lolas[7,8] proposed a chemotaxis-haptotaxis model describing interactions between tumor cells, matrix degrading enzymes and the host tissue(ECM).Actually,Chaplain and Lolas' model can be regarded as an extension of the classical chemotaxis model which may be first proposed in 1970 by Keller and Segel(see[29]).This paper mostly studies the properties of the solutions of Chaplain and Lolas' model.First,in Chapter 2,this paper studies a parabolic-elliptic chemotaxis-haptotaxis model. This chapter deals with a 3×3 chemotaxis-haptotaxis system modeling cancer invasion. The model consists of a parabolic chemotaxis-haptotaxis partial differential equation(PDE) describing the evolution of tumor cell density,an elliptic PDE governing the evolution of proteolytic enzyme concentration and an ordinary differential equation(ODE) modeling the proteolysis of extracellular matrix.In three space dimensions,the existence,uniqueness and uniform-in-time boundedness of global classical solutions to above system is proved for largeμ＞0 by raising the a priori estimate of a solution from L~1(Ω) to L~2(Ω),and then to L~4(Ω);in two space dimensions,the existence,uniqueness and boundedness is proved for anyμ＞0 by raising the a priori estimate of a solution in the following way: L~1(Ω)→L~3(Q_T)→L~2(Ω)→L~4(Q_T)→L~3(Ω).The above-mentionedμis the logistic growth rate of cancer cells,Ω(?)R~d(d=2 or 3) is a bounded domain,and Q_T=Ω×(0,T).The central point of this paper is to develop new L~p -estimate techniques for a 3×3 chemotaxis-haptotaxis system.Second,in Chapter 3,this paper studies a parabolic-parabolic chemotaxis-haptotaxis model.This chapter deals with a mathematical model of tumor invasion of tissue recently proposed by Chaplain and Lolas.The model consists of a reaction-diffusion-taxis PDE describing the evolution of tumor cell density,a reaction-diffusion PDE governing the evolution of the proteolytic enzyme concentration,and an ODE modelling the proteolysis of extracellular matrix(ECM).In one space dimension,global existence and uniqueness of a classical solution to this combined chemotactic-haptotactic model is proved for any chemotactic coefficientχ＞0 andμ＞0.In three space dimensions,the global existence is proved for largeμ＞0(whereμis the logistic growth rate of the tumor cells).In two space dimensions,the global existence is proved for anyμ＞0 in paper[42].The fundamental point of proof is to raise the regularity of a solution from L~1 to L~p(p＞3). Here,we should point out:there are two main differences between above-mentioned 3×3 parabolic-ODE-elliptic chemotaxis-haptotaxis system under consideration in Chapter 2 and the 3×3 parabolic-ODE-parabolic chemotaxis-haptotaxis system studied in Chapter 3 and paper[42].The solution of the former has a weaker t-direction regularity than that of the latter since the chemoattractant concentration solves an elliptic equation in the former.Hence,in the proof of the local existence of classical solutions,the parabolic-ODE-elliptic chemotaxis-haptotaxis system will need carefully choosing the mapping space X_M and the mapping F which differ from those of the parabolic-ODE-parabolic chemotaxis-haptotaxis system in this paper.On the other hand, the fundamental idea for proving the global existence of classical solutions in paper[42] and present paper is to raise the regularity estimate of a solution.However,the parabolic-ODE-elliptic chemotaxis-haptotaxis system in this paper raises the regularity estimate of a solution from L~1(Ω) to L~3(Ω),whereas paper[42]raises the regularity estimate of a solution from L~1(Ω) to L~3(Ω×(0,T)).The L~p estimate strategies in paper [42]and present paper are different.In addition to the global existence of solutions under afore-mentioned assumptions,the solutions of the parabolic-ODE-elliptic chemotaxis-haptotaxis system in this paper are shown to be bounded under the same assumptions as those for global existence.We should note that the possible boundedness of solutions under some assumptions in paper[42]and the parabolic-ODE-parabolic chemotaxis-haptotaxis system of this paper is still an open problem up to our knowledge.Finally,we do some simulation on the simplified chemotaxis-haptotaxis model.The simulation shows that the solution probably blows-up in a finite time.