Local existence and uniqueness of regular solutions in a model of tissue invasion by solid tumours

Local existence and uniqueness of regular solutions in a model of tissue invasion by solid tumours Cristian Morales-Rodrigo Institute of Applied Mathematics and Mechanics, Faculty of Informatics, Mathematics and Mechanics, Warsaw University, ul. Banacha 2, 02-097 Warsaw, Poland [email protected] Abstract In this paper we consider a nonlinear system of differential equations arising in tumour invasion which has been proposed in [1]. The system consists of two PDEs describing the evolution of tumour cells and proteases and an ODE which models the concentration of the extracellular matrix. We prove local existence and uniqueness of solutions in the class of H¨older spaces. The proof of local existence is done by Schauder’s fixed point theorem and for the uniqueness we use an idea from [2]. Keywords: Haptotaxis; Tumour invasion of tissue; Reaction-diffusion equations; Uniqueness. AMS Subject Classification: 35K45, 35K57, 92C17 1 Introduction The most dangerous feature of malignant tumour and the main cause of cancer deceases is the ability to metastasize. Metastasis is the formation of a secondary tumour foci at a site discontinuous from the primary tumour. Two main processes have to be taken into account during the metastasis. The first one is called angiogenesis. Tumour cells response to hypoxia by secreting tumour angiogenic factors (TAFs) which induce to the endothelial cells in a nearby vessel to proliferate and migrate chemotactically towards the tumour. The other important process occurring during metastasis is the invasion. Tumour cells on contact with extracellular matrix (ECM) induce the production of some proteolytic enzymes, such as metallo-proteases (MMPs) and serine-proteases. MMPs digest the ECM and this enables the cancer cells to migrate through the tissue. In order to understand better the mechanisms leading to angiogenesis and invasion, several models were proposed. For the area related to angiogenesis we just refer to the recent review paper [3] and the references therein. Concerning tumour invasion modelling we briefly recapitulate some papers. 2C. Morales-Rodrigo In [4] the authors proposed a model of invasion. In this model the diffusion of the tumour cells was neglected. They provided a travelling wave analysis for this model, finding a singular barrier which just can be crossed by the slowest member of the family of travelling waves connecting the steady-states. Later, in [5] the same system is studied but, by contrast with [4] where just regular travelling waves were founded, the authors showed travelling shock waves which jump over the singular barrier. In [6] basing on experimental data, the authors validate a model of invasion for the fibrosarcoma cell line HT1080. They showed that collagen concentration influences the proliferation of HT1080 in a biphasic manner. Recently, in [7] the author examined the role of the urokinase plasminogen system in cancer invasion, showing how this system influences the migratory properties of the cancer cells. In this paper we will consider a model of tissue invasion that has been proposed by Chaplain and Anderson in the recent review book about cancer modelling [1]. They considered the following variables and facts. Cancer Cells, n(x,t): The movement of cancer cells is supposed to be by a random motility and haptotaxis i.e. up to the spatial gradients in the extracellular matrix. Extracellular Matrix, f(x,t): The matrix is just degraded by the proteases produced by the tumour. Proteases, m(x,t): Factors influencing the protease concentration are assumed to be diffusion, production and natural decay. As a result, the model reads as ∂n ∂t = random motility z}|{ dn∆n haptotaxis z }| { −γ∇·(n∇f) in Ω ×(0, T ), ∂f ∂t = degradation z}|{ −ηmf in Ω ×(0, T), ∂m ∂t = random motility z }| { dm∆m decay z}|{ −αm production z}|{ +βn in Ω ×(0, T ), (1.1) where dn, dm, α, β, γ and ηare positive constants. Finally, denoting by νthe unit exterior vector to ∂Ω, the model is supplemented with no-flux boundary conditions on ∂Ω      ∂n ∂ν −n∂f ∂ν = 0 on ∂Ω×(0, T ), ∂m ∂ν = 0 on ∂Ω×(0, T ), (1.2) and the initial conditions            n(x, 0) = n0(x) in Ω, f(x, 0) = f0(x) in Ω, m(x, 0) = m0(x) in Ω. (1.3) Local existence and uniqueness in a model of tissue invasion 3 In what follows and in order to simplify the formulas we will suppose dn=dm=η=α= β=γ= 1. Let us point at that our calculations can be repeated without any problem for general positive constants. This paper is organized as follows. In section 2 we define the space in which is our solution. In section 3 we prove the existence and uniqueness of local-in-time solution in such space. 2 Notations In this paper Ω ⊂IRNis an open, connected set with regular boundary. QT= Ω ×(0, T ) is a cylinder of IRN+1. We consider the Banach space of H¨older continuous functions Hk+α,(k+α)/2(QT) where k≥0 is an integer and α∈(0,1). The associate norm to this space is given by |u|k+α QT:= huik+α x,QT+hui(k+α)/2 t,QT+ k X j=0huij QT, where huij QT:= X 2r+s=k max QT|Dr tDs xu|QT, huik+α x,QT:= X 2r+s=khDr tDs xuiα x,QT, huik+α t,QT:= X 0<α+k−2r−s<2hDr tDs sui(α+k−2r−s)/2 t,QT, and huiα x,QT:= sup (x,t),(x0,t)∈QT |x−x0|≤ρ0 |u(x, t)−u(x0, t)| |x−x0|α,0< α < 1, huit,QT:= sup (x,t),(x,t0)∈QT |t−t0|≤ρ0 |u(x, t)−u(x, t0)| |t−t0|α,0< α < 1. The norm in the space Lp(Ω), 1 ≤p≤ ∞ is denoted by k ·kp. The norm associated to the classical Sobolev spaces W1,p(Ω) will be denoted by k·k1,p. Finally, the norm in the space L∞(QT) is denoted by k·k∞,QT. 3 Local existence and uniqueness of regular solutions First of all we define a new variable q=e−fn, then our system is transformed into            ∂q ∂t −∆q−∇q·∇f=−qft=qmf in Ω ×(0, T ), ∂f ∂t =−mf in Ω ×(0, T), ∂m ∂t = ∆m−m+qefin Ω ×(0, T ), (3.1) 4C. Morales-Rodrigo with a new boundary      ∂q ∂ν = 0 on ∂Ω×(0, T ), ∂m ∂ν = 0 on ∂Ω×(0, T ). (3.2) Actually, this change of variable has been proposed in another papers before as [8] and [9]. The main advantage of this change is that the first equation of the system is in divergence form. In our proof, based on a fixed point argument, the following lemma will be required. Lemma 3.1. Let N≤3. Given fsuch that f∈Hα,α/2(QT),fx∈Hα,α/2(QT)then the problem      ∂q ∂t −∆q−∇q·∇f=qmf in Ω×(0, T), ∂m ∂t = ∆m−m+efqin Ω×(0, T ), (3.3) with Neumann boundary conditions and regular initial data admits a unique regular solution (m, q)∈(H2+α,1+α/2(QT))2. Moreover, if q0,m0≥0then q(x, t), m(x, t)≥0for all (x, t)∈QT. Proof. Consider the space of functions X=C([0, T]; L2(Ω)). We define the operator F:X→Xsuch that F(q) = mwhere mis the unique solution to the linear equation            ∂m ∂t = ∆m−m+efqin Ω ×0, T), ∂m ∂ν = 0 on ∂Ω×(0, T ), m(x, 0) = m0in Ω. (3.4) On multiplying (3.4) by mand integrating in QTwe obtain ∂ 2∂t ZT 0kmk2 2+ZT 0k∇mk2 2+ZT 0kmk2 2=ZT 0ZΩ efqm (3.5) Applying H¨older’s inequality and Young’s inequality to the right-hand-side of (3.5) ∂ 2∂t ZT 0kmk2 2+ZT 0k∇mk2 2+µ1−1 2αkefk∞,QT¶ZT 0kmk2 2≤α 2kefk∞,QTZT 0kqk2 2.(3.6) Choosing α > 0 large enough in (3.6) and integrating on the time interval [0, T] we get km(T)k2 2≤ km0k2 2+Tαkefk∞,QTZT 0kqk2 2.(3.7) Local existence and uniqueness in a model of tissue invasion 5 Now, we define the linear operator G:X→Xsuch that for each z G(z) is the unique solution to            ∂q ∂t −∆q−∇q·∇f=qzf in Ω ×0, T), ∂q ∂ν = 0 on ∂Ω×(0, T ), q(x, 0) = q0in Ω. (3.8) It is easy to see that q∈Xis a fixed point of H=G◦Fthen is a weak solution to (3.3). Taking z=F(q) = mand multiplying (3.8) by qwe obtain, after integrating in space. d 2dtkqk2 2+k∇qk2 2=ZΩ q2mf +ZΩ q∇q·∇f. (3.9) From the Sobolev inequality kqk3≤Ckqk1/2 1,2kqk1/2 2, (N≤3), H¨older’s inequality and Young’s inequality we infer d 2dtkqk2 2+k∇qk2 2≤ kfk∞,QTkqk3kqk6kmk2+k∇fk∞,QTµα 2kqk2 2+1 2αk∇qk2 2¶≤ ≤Ckfk∞,QTkqk3/2 1,2kqk1/2 2kmk2+k∇fk∞,QTµα 2kqk2 2+1 2αk∇qk2 2¶≤ ≤ kfk∞,QT(α0kqk2 1,2+Cα0kqk2 2kmk4 2) + k∇fk∞,QTµα 2kqk2 2+1 2αk∇qk2 2¶. Choosing α > 0 large enough and α0>0 small enough then d dtkqk2 2≤(2α0kfk∞,QT+ 2Cα0kmk4 2+αk∇fk∞,QT)kqk2 2:= β(t)kqk2 2. If we choose qsuch that kqkC([0,T];L2(Ω)) <kq0k2+ 1 = Rthen, thanks to the estimate (3.7) β(t)≤M∀t∈(0, T), for that kq(t)k2 2≤ kq0k2 2exp(tM),∀t∈(0, T ). Clearly, choosing Tsmall enough follows that kH(q) = qkC([0,T ];L2(Ω)) ∈BR. For that, H:BR→BR. Now, we are going to prove that His a contractive operator. Given q1, q2∈BR, then F(q1)−F(q2) = m1−m2satisfies the equation            zt−∆z+z=ef(q1−q2) in Ω ×(0, T ), ∂z ∂ν = 0 on ∂Ω×(0, T ), z(x, 0) = 0 in Ω. (3.10) On multiplying (3.10) by m1−m2and integrating in Ω we obtain, ∂ ∂tkm1−m2k2 2≤Ckq1−q2k2 2, 6C. Morales-Rodrigo for that, km1−m2kC([0,T];L2(Ω)) ≤C√Tkq1−q2kC([0,T ];L2(Ω)).(3.11) We have that H(q1)−H(q2) = q1−q2solves the equation            zt−∆z−∇z·∇f=fm1z+fq2(m1−m2) in Ω ×(0, T), ∂z ∂ν = 0 on ∂Ω×(0, T ), z(x, 0) = 0 in Ω. (3.12) Multiplying (3.12) by q1−q2and integrating in Ω gives us ∂ 2∂tkq1−q2k2 2+k∇(q1−q2)k2 2= =ZΩ (q1−q2)∇f·∇(q1−q2) + ZΩ fm1(q1−q2)2+ZΩ fq2(m1−m2)(q1−q2)≤ ≤ k∇fk∞,QT(²k∇(q1−q2)k2 2+C²kq1−q2k2 2)+ +kfk∞,QT(α0k∇(q1−q2)k2 2+ (α0+Cα0km1k4 2)kq1−q2k2 2)+ kfk∞,QT(²0(kq1−q2k2 2+k∇(q1−q2)k2 2) + C²0kq2k2 3km1−m2k2 2). Choosing α0, ², ²0positive and small enough we infer ∂ ∂tkq1−q2k2 2≤α(t)km1−m2k2 2+βkq1−q2k2 2, where βis a positive constant and α(t) = C²0kfk∞,QTkq2k2 3. Then, ∂ ∂t ³e−βtkq1−q2k2 2´≤α(t)e−βtkm1−m2k2 2.(3.13) From the Sobolev’s inequality kq2k2 3≤Ckq2k1/2 1,2kq2k1/2 2and taking in account that q2∈ C([0, T]; L2(Ω)) ∩L2(0, T ;H1(Ω)) we get ZT 0 α(s)ds ≤M. (3.14) Finally thanks to (3.14) and (3.11) we obtain kH(q1)−H(q2)kC([0,T];L2(Ω)) ≤CpMeβT pTkq1−q2kC([0,T];L2(Ω)). Choosing T≤Tsmall enough, H:BR→BRis contractive and from Banach’s fixed point theorem we infer that problem (3.3) have a unique solution in the space C([0, T]; L2(Ω)) ∩L2(0, T ;H1(Ω)). Local existence and uniqueness in a model of tissue invasion 7 Since N≤3 then the function q∈L2(0, T;L6(Ω)). We begin an iterative argument that will provided us regularization of our solution. Let p= 6, multiplying by pmp−1the second equation of (3.3) and integrating by parts, gives us ∂ ∂tkmkp p+4(p−1) pk∇(mp/2)k2 2+pkmkp p=pZΩ efqmp−1≤ ≤pkefk∞,QTkqkpkmp/2k2kmk p−2 2 p≤pkefk∞,QT(Cαkqk2 pkmkp−2 p+αkmp/2k2 2). Choosing α > 0 small enough the following estimate follows ∂ ∂tkmkp p≤pMkqk2 pkmkp−2 p. From this differential inequality we infer km(t)k2 p≤ km0k2 p+Zt 0 p2 2Mkq(s)k2 pds, (3.15) for all t∈[0, T]. Since q∈L2(0, T;Lp(Ω)) then the integral term on the right-hand-side of (3.15) is finite. Therefore m∈L∞(0, T;L6(Ω)). On multiplying the first equation (3.3) by pqp−1, then after integrating by parts, we obtain ∂ ∂tkqkp p+4(p−1) pk∇(qp/2)k2 2=pZΩ qp−1∇u·∇f+pZΩ qpmf. (3.16) Now, we are going to find the proper bounds of the two integrals on the right-hand-side. pZΩ qp−1∇q·∇f≤ k∇fk∞,QT(C²kqp/2k2 2+²p2kqp 2−1∇qk2 2) =k∇fk∞,QT(C²kqkp p+ 4²k∇(qp/2)k2 2). (3.17) Taking in account that m∈L∞(0, T;L6(Ω)) we get for p= 6 that pZΩ qpmf ≤pkfk∞,QTkqp/2k2kqp/2k3kmk6 ≤pkfk∞,QTkqp/2k3/2 2kqp/2k1/2 1,2kmk6 ≤pkfk∞,QTkqp/2k2kqp/2k1,2kmk6 ≤ kfk∞,QT(²0kqp/2k2 1,2+p2C²0kqp/2k2 2kmk2 6) ≤ kfk∞,QT(²0kqp/2k2 1,2+p2MC²0kqp/2k2 2). (3.18) Choosing ², ²0small enough and putting the estimates (3.17), (3.18) in (3.16) we obtain ∂ ∂tkqkp p+k∇(qp/2)k2 2≤Cp2kqkp p. Easily after integrating on [s, t]⊂[0, T], implies kq(t)kp p+Zt sk∇(qp/2)k2 2≤exp(Cp2(t−s))kq(s)kp p.(3.19) 8C. Morales-Rodrigo From (3.19) and following the same argument as in [10, p. 1197] we can prove that q∈L∞(QT). However, for completeness we present it here. Consider any t∈(0, T ]. For simplicity t=T, although this argument remains true for every t∈(0, T]. Take t0∈(T−1, T), σ= 3. Define pm= 6σmand δm= (T−t0)σ−2m−1. Observe that p2 mδm=c. Now, consider the intervals Im= [T−σδm, T −δm]. We define the sequence Nm= supτ∈[tm,T]kq(τ)kpmwhere tm∈Imwill be determined later. If we apply (3.19) with s=tm+1 and τ∈[tm+1, T ] then Nm+1 = sup τ∈[tm+1,T]kq(τ)kpm+1 ≤(exp(Cp2 m+1σδm+1))1/pm+1 kq(tm+1)kpm+1 (3.20) We have to determine kq(tm+1)kpm+1 . Thanks to the Sobolev’s embedding, kq(tm+1)kpm pm+1 =kqpm/2(tm+1)k2 6≤M(kq(tm+1)kpm pm+k∇(qpm/2(tm+1))k2 2) (3.21) We are going to determine k∇(qpm/2(tm+1))k2 2. Applying (3.19) for s=tmand t= T−δm+1 (so Im+1 ⊂[s, t]) we get inf τ∈Im+1 k∇(qpm/2(τ))k2 2≤ |Im+1|−1exp(Cp2 m(T−δm+1 −tm))kq(tm)kpm pm. Choosing tm+1 =τwe obtain the estimate we were looking for. Since |Im+1|<1 then with a similar argument we can estimate kq(tm+1)kpm pm. Putting this estimate in (3.21) we get kq(tm+1)kpm pm+1 ≤2σ2(σ3−1)−1δ−1 mexp(Cp2 m(1 −σ−2)δm)kq(tm)kpm pm ≤Cδ−1 mNpm m (3.22) Thanks to (3.22) and taking in account that p2 mδm=C1, we obtain from (3.20) Nm+1 ≤(exp(M)1/σCδ−1 m)1/pmNm ≤Ãm Y i=0 (C2σ2i)1/pi!N0=zmN0 Clearly zmis finite for all mbecause ln zm=Pm i=0 1 6σi(ln C2+2iln σ) where σ > 1. Finally, kq(T)k∞≤sup m≥1 Nm≤C3N0≤C3sup τ∈[0,T]kq(τ)k6<∞. Repeating the same argument for mwe get the same regularity. Now, the regularity can be improved thanks to [11, Chapter 3, Theorem 10.1] and [11, Chapter 3, Theorem 12.1] , the first one gives us q, m ∈Hα,α/2(QT) and then we can apply the second one obtaining q, m ∈H2+α,1+α/2(QT). Since fm is bounded in L∞(QT) then, from maximum principle for parabolic equations we get the positivity of q. Now, from the positivity of qwe can infer, thanks to the maximum principle, the positivity of m. Theorem 3.2. If the initial condition (1.3) are regular then the problem given by (1.1) with the boundary condition (1.2) and initial condition (1.3) respectively, has a unique local solution in the space (H2+α,1+α/2(QT))3. Local existence and uniqueness in a model of tissue invasion 9 Proof. We define the following ball in H={f:f∈Hα,α/2(QT)∧fx∈Hα,α/2(QT)} Bδ(f0) = {u:|u−f0|α QT< δ ∧|(u−f0)x|α QT< δ}. Now, we define the operator K:Bδ(f0)→H.K(f) is the unique solution to the ordinary differential equation ∂f ∂t =−mf, f(x, 0) = f0(x), where mis given as the solution to the second equation in (3.3) with f=f. For simplify the calculus, we consider f0= 1, the same calculus can be done with a general f0. We have, f(x, t) = 1 + Zt 0−m(x, s)f(x, s)ds = 1 + Zt 0−m(x, s)e−Rs 0m(x,θ)dθ By definition |f−1|α QT=hf−1iα x,QT+hf−1iα/2 t,QT+ maxQT|f−1|. hf−1iα x,QT:= sup (x,t),(x0,t)∈QT |x−x0|≤ρ0 |Rt 0−m(x, s)e−Rs 0m(x,θ)dθ +Rt 0−m(x0, s)e−Rs 0m(x0,θ)dθ| |x−x0|α≤ ≤sup (x,t),(x0,t)∈QT |x−x0|≤ρ0 kmk∞,QT³Rt 0ds Rs 0|m(x0, θ)−m(x, θ)|dθ´+Rt 0|m(x, s)−m(x0, s)|ds |x−x0|α≤ ≤T2kmk∞,QThmiα x,QT+Thmiα x,QT hf−1iα/2 t,QT:= sup (x,t),(x,t0)∈QT |t−t0|≤ρ0 |Rt 0−m(x, s)e−Rs 0m(x,θ)dθ +Rt0 0−m(x, s)e−Rs 0m(x,θ)dθ| |t−t0|α/2≤ ≤sup (x,t),(x0,t)∈QT |t−t0|≤ρ0 |Rt0 tm(x, s)e−Rs 0m(x,θ)dθ| |t−t0|α/2≤ |t−t0|1−α/2kmk∞,QT max QT|f−1| ≤ Tkmk∞,QT Also by definition |(f−1)x|α QT=hfxiα x,QT+hfxiα/2 t,QT+ maxQT|fx| (f−1)x=Zt 0 e−Rs 0m(x,θ)dθ µ−mx(x, s) + m(x, s)Zs 0 mx(x, θ)dθ¶ Let denote ax=e−Rs 0m(x,θ)dθ,bx=m(x, s), cx=Rs 0mx(x, θ)dθ and dx=mx(x, s) then h(f−1)xiα x,QT:= sup (x,t),(x0,t)∈QT |x−x0|≤ρ0 |Rt 0ax(−dx+bxcx)−ax0(−dx0+bx0cx0)ds| |x−x0|α≤