Nonlocal adhesion models for two cancer cell phenotypes in a multidimensional bounded domain

Abstract

Cell–cell adhesion is an inherently nonlocal phenomenon. Numerous partial differential equation models with nonlocal term have been recently presented to describe this phenomenon, yet the mathematical properties of nonlocal adhesion model are not well understood. Here we consider a model with two kinds of nonlocal cell–cell adhesion, satisfying no-flux conditions in a multidimensional bounded domain. We show global-in-time well-posedness of the solution to this model and obtain the uniform boundedness of solution.

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Notes

  1. 1.

    There is the other steady state, \(u^{*}=-\frac{\mu }{\lambda }v^{*}, v^{*}=\frac{k(1-\frac{m}{\lambda })}{1-\frac{\mu }{\lambda }}\), which are of different signs, hence unrealistic.

  2. 2.

    For case II, we could change the shrinking rate and shape of E(x) as x approaches the boundary so that the regularity of adhesion terms is possibly worse.

  3. 3.

    Eckardt et al. considered averaging nonlocal operators \(\mathcal A_r\) and \(\mathring{\nabla _r}\). Following notations in this paper, we have

    $$\begin{aligned} \mathcal A_Ru(x) = \frac{1}{R}\frac{1}{|B_R(0)|} \int \limits _{E(x)} u(x+y) \frac{y}{|y|} w(|y|) dy= \frac{1}{M_{11}R}\frac{1}{|B_R(0)|}\mathcal K [u,0] \end{aligned}$$

    by choosing \({\omega }(x)=\frac{y}{|y|} w(|y|)\) in (1.2).

  4. 4.

    We follow a definition of \(C^1\)-functions on a submanifold \(\partial {\Omega }\times \mathbb {R}\) embedded in \(\mathbb {R}^n \times \mathbb {R}\). The open set E(V)is called a tubular neighborhood of \(\partial {\Omega }\).

  5. 5.

    \(A_{\beta }\) is a \(W^{s,p}\)- realization of A if \( A= A_{\beta }\) in D(A) and the range of \(A_{\beta }\) is in \(W^{s,p}\).

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Acknowledgements

Jaewook Ahn’s work is supported by NRF-2018R1D1A1B07047465. Jihoon Lee’s work is supported by SSTF-BA1701-05. Myeongju Chae’s work is supported by NRF-2018R1A1A3A04079376.

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Appendix

Appendix

Let \({\Omega }, {\mathcal {K}}, {\mathcal {S}}\) and \({\omega }\) be given as in Section 3. We define the operator \( F: W^{2,p}_B \times W^{2,p}_B \rightarrow L^2\times L^2\) by

$$\begin{aligned} F(u,v)&= (F_1(u,v), F_2(u,v)),\\ F_1(u,v)&= \Delta u-\nabla \cdot (u \,\mathcal {K}[u,v] )+(1-m)u +\displaystyle \frac{\lambda }{k} u(k-(u+v)) \\ F_2(u,v)&= \Delta v-\nabla \cdot (v \,\mathcal {S}[u,v] )+mu+v +\displaystyle \frac{\mu }{k} v(k-(u+v)). \end{aligned}$$

We denote the Gâteaux derivative of F at \(U=(u,v)\) by \(T_{U}\);

$$\begin{aligned} T_{U}(W) = \lim _{t\rightarrow 0} \frac{F(U+tW)- F(U)}{t} = (\delta _W F_1(U), \delta _W F_2(U)) \end{aligned}$$

where \( W= (w, z) \in W^{2,p}_B \times W^{2,p}_B\). By computation, we have

$$\begin{aligned} \delta _WF_1(0,k)&= \Delta w-mw \\ \delta _WF_2(0,k)&= \Delta z - \nabla \cdot ( k{\mathcal {S}}[w,0]+kS[w,z]+z{\mathcal {S}}[0,k]) + (m-\mu )w - \mu z\\ \delta _WF_1(0,0)&= \Delta w - mw\\ \delta _WF_2(0,0)&= \Delta z + mw + \mu z. \end{aligned}$$

We consider the two linearized equations at (0, k) and (0, 0), respectively, with initial data \((w_0, z_0)\in W^{2,p}_B \times W^{2,p}_B\);

$$\begin{aligned} \begin{aligned} \partial _t w&= \Delta w-mw\\ \partial _t z&= \Delta z - \nabla \cdot ( k{\mathcal {S}}[w,0]+kS[w,z]+z{\mathcal {S}}[0,k]) + (m-\mu )w - \mu z, \end{aligned} \end{aligned}$$
(3.21)

and

$$\begin{aligned} \begin{aligned} \partial _t w&= \Delta w-mw \\ \partial _t z&= \Delta z + mw + \mu z. \end{aligned} \end{aligned}$$
(3.22)

Equations (3.21), (3.22) are decoupled, and it is immediate that

$$\begin{aligned} \Vert w\Vert _{W^{1,p}({\Omega })} \le e^{-mt}\Vert w_0\Vert _{W^{1,p}}, \quad p\ge 1 \end{aligned}$$
(3.23)

from

$$\begin{aligned} \partial _t(e^{m t} w ) = \Delta (e^{m t} w ). \end{aligned}$$

Let \( \tilde{z}\) denote \(e^{\mu t} z\). Multiplying to the z- equation of (3.21) by \({e^{\mu t}}\), we have

$$\begin{aligned} \partial _t \tilde{z}- \Delta \tilde{z}= -\nabla \cdot ( 2k{\mathcal {S}}[e^{\mu t}w,0]+kS[0,\tilde{z}]+ \tilde{z}{\mathcal {S}}[0,k]) + (m-\mu ) e^{\mu t}w. \end{aligned}$$
(3.24)

It holds that

$$\begin{aligned} \frac{d}{dt}\int \limits _{{\Omega }}|\tilde{z}|\le |m-\mu | e^{\mu t} \int \limits |w| \le |m-\mu | e^{-(m-\mu )t} \Vert w_0\Vert _{L^1({\Omega })}, \end{aligned}$$

which implies

$$\begin{aligned} \int \limits _{{\Omega }}|\tilde{z}|&\le \int \limits _{{\Omega }}|z_0| - \frac{|m-\mu |}{m-\mu }(e^{-(m-\mu )t}-1)\int \limits _{{\Omega }}|w_0| \end{aligned}$$

and

$$\begin{aligned} \int \limits _{{\Omega }}|z |&\le e^{-\mu t} \int \limits _{{\Omega }}|z_0| + (e^{-mt}- e^{-\mu t})\int \limits _{{\Omega }}|w_0| , \end{aligned}$$
(3.25)

where we abuse the notation by \( | m-\mu |/(m-\mu ) =0\) if \( \mu =m\). When \(m > \mu \), it holds that

$$\begin{aligned} \int \limits | \tilde{z}| \le \Vert z_0\Vert _{L^1({\Omega })} + \Vert w_0\Vert _{L^1({\Omega })}. \end{aligned}$$
(3.26)

In what follows, we find that different signs of \({\mp } \mu z\) in (3.21) and (3.22) imply that (0, k) is linearly stable and (0, 0) is linearly unstable as expected.

Proposition 1

Each of the linearized equations (3.21), (3.22) have a unique global solution (wz) for each in

$$\begin{aligned} C([0,t); W^{1,p}(\Omega ))\cap W^{1,p}(0,t; L^{p}(\Omega ))\cap L^{p}(0,t;W^{2,p}(\Omega )) \end{aligned}$$

for any \(t>0\). When \(m> \mu \), the solution (wz) for (3.21) is asymptotically stable such that

$$\begin{aligned} \Vert z\Vert _{L^p({\Omega }) }\le e^{-\mu t}\Vert z_0\Vert _{L^p({\Omega })} \quad \hbox { for } p\ge 1. \end{aligned}$$
(3.27)

The solution (wz) for (3.22) grows exponentially in its \(L^1\)-norm if the initial data is non-negative;

$$\begin{aligned} \int \limits _{\Omega }|z| \ge e^{\mu t}\int \limits _{\Omega }|z_0|. \end{aligned}$$
(3.28)

Proof

Due to the a priori estimates (3.23) and (3.25), the global well-posedness part for (3.21) follows from the same argument in Sect. 2.3 or Sect. 3.2. Repeating the argument of Lemma 6 to (3.21) it holds that

$$\begin{aligned} \Vert z\Vert _{L^{\infty }({\Omega })}\le C(\Vert w_0\Vert _{L^1\cap L^{\infty }({\Omega })}, \Vert z_0\Vert _{L^1\cap L^{\infty }({\Omega })}). \end{aligned}$$
(3.29)

For details, see (3.31)–(3.33) for \(\tilde{z}\), where the similar estimates are given. By Lemma 9, it also holds that

$$\begin{aligned} \Vert z\Vert _{W^{1,p}} \le C(\Vert w_0\Vert _{L^1\cap L^{\infty }({\Omega })}, \Vert z_0\Vert _{L^1\cap L^{\infty }({\Omega })} ) \end{aligned}$$
(3.30)

for any \(p\ge 1\). Let us prove (3.28) first. The solution (wz) remains non-negative and we have

$$\begin{aligned} \int \limits _{{\Omega }}w&= e^{-mt}\int \limits _{{\Omega }}w_0,\\ \frac{d}{dt}(e^{-\mu t} \int \limits _{{\Omega }}z)&= m e^{-(\mu + m)t} \int \limits _{{\Omega }}w_0. \end{aligned}$$

Integrating the second equation, we have (3.28).

For (3.27), we proceed as in Lemma 6. Multiplying \(|\tilde{z}|^{p-2}\tilde{z}\) into (3.24) for \(p \ge 2\), we have

$$\begin{aligned} \frac{1}{p} \frac{d}{dt}\int \limits _{{\Omega }}|\tilde{z}|^{p}+ \frac{4(p-1)}{p^2}\int \limits _{{\Omega }}|\nabla \tilde{z}^{\frac{p}{2}}|^2 =&\int \limits _{{\Omega }}|\tilde{z}|^{p-2}\tilde{z}\nabla \cdot ( 2k{\mathcal {S}}[e^{\mu t}w,0]+kS[0,\tilde{z}]+ \tilde{z}{\mathcal {S}}[0,k]) \end{aligned}$$
(3.31)
$$\begin{aligned}&+ (m-\mu ) \int \limits _{{\Omega }}e^{\mu t}w |\tilde{z}|^{p-2}\tilde{z}. \end{aligned}$$
(3.32)

By Lemma 7, (3.23), (3.30) and using \(m >\mu \), we have

$$\begin{aligned}&\Vert \nabla \cdot {\mathcal {S}}[ e^{\mu t}w, 0]\Vert _{L^{\infty }({\Omega })}\le C\Vert e^{\mu t}w\Vert _{W^{1,q}({\Omega })} \le C \Vert w_0\Vert _{W^{1,q}({\Omega })} (q >n)\\&\Vert {\mathcal {S}}[ 0, \tilde{z}] \Vert _{L^{\infty }({\Omega })}\le C\Vert \tilde{z}\Vert _{L^1({\Omega })} \le C(\Vert w_0\Vert _{L^1}, \Vert z_0\Vert _{L^1})\\&\Vert \nabla {\mathcal {S}}[0, k]\Vert _{L^{\infty }({\Omega })}\le C \end{aligned}$$

and estimate the right hand side of (3.31) as follows

$$\begin{aligned}&\int \limits _{{\Omega }}|\tilde{z}|^{p-2}\tilde{z}\nabla \cdot ( 2k{\mathcal {S}}[e^{\mu t}w,0]) + (m-\mu ) \int \limits _{{\Omega }}e^{\mu t}w |\tilde{z}|^{p-2}\tilde{z}\le C\int \limits _{{\Omega }}|\tilde{z}|^{p-1}\\&\int \limits _{{\Omega }}|\tilde{z}|^{p-2}\tilde{z}\nabla \cdot kS[0,\tilde{z}] \le \frac{p-1}{p^2} \int \limits _{{\Omega }}| \nabla \tilde{z}^{\frac{p}{2}|}|^2 + C(p-1) \int \limits \tilde{z}^{p-2} \Vert {\mathcal {S}}[0, \tilde{z}\Vert _{{L^{\infty }({\Omega })}}^2 \\&\le \frac{p-1}{p^2} \int \limits _{{\Omega }}| \nabla \tilde{z}^{\frac{p}{2}}|^2 + C \frac{p-1}{p} \left( |{\Omega }| +(p-2)\int \limits _{{\Omega }}|\tilde{z}|^{p}\right) ,\\&\int \limits _{{\Omega }}|\tilde{z}|^{p-2}\tilde{z}\nabla \cdot ( \tilde{z}{\mathcal {S}}[0,k]) \le \frac{1}{p^2} \int \limits _{{\Omega }}| \nabla \tilde{z}^{\frac{p}{2}}|^2 + C\int \limits _{{\Omega }}|\tilde{z}|^{p}\Vert \nabla {\mathcal {S}}[0, k]\Vert _{{L^{\infty }({\Omega })}}. \end{aligned}$$

Summing up, we have

$$\begin{aligned} \frac{1}{p} \frac{d}{dt}\int \limits _{{\Omega }}|\tilde{z}|^{p}+ \frac{3(p-1)}{p^2}\int \limits _{{\Omega }}|\nabla \tilde{z}^{\frac{p}{2}}|^2 \le C+ C\int \limits _{{\Omega }}|\tilde{z}|^{p}, \quad p\ge 2, \end{aligned}$$

where C is a uniform constant depending on \(\Vert w_0\Vert _{L^1({\Omega })}\), \(\Vert z_0\Vert _{L^1({\Omega })}\), and given constants \(\mu , m, k\) etc.. As was derived from (2.30) for v in Lemma 6, it holds that

$$\begin{aligned} \sup _ {0<t \le T} \Vert \tilde{z}\Vert _{L^{p_k}({\Omega })} \le C(\Vert z_0\Vert _{L^1({\Omega })}, \Vert z_0\Vert _{{L^{\infty }({\Omega })}}) \sup _ {0 <t \le T}\left( \Vert \tilde{z}\Vert _{L^1({\Omega })} +C \right) \quad p_k= 2^k, k=0, 1\dots . \end{aligned}$$

and

$$\begin{aligned} \sup _ {0 <t \le T} \Vert \tilde{z}\Vert _{L^{\infty }({\Omega })}\le C(\Vert w_0\Vert _{L^1\cap L^{\infty }({\Omega })}, \Vert z_0\Vert _{L^1\cap L^{\infty }({\Omega })}). \end{aligned}$$
(3.33)

That implies (3.27). \(\square \)

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Ahn, J., Chae, M. & Lee, J. Nonlocal adhesion models for two cancer cell phenotypes in a multidimensional bounded domain. Z. Angew. Math. Phys. 72, 48 (2021). https://doi.org/10.1007/s00033-021-01485-y

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Keywords

  • Cell–cell adhesion
  • Non-local models
  • No-flux boundary conditions
  • Global existence
  • Semigroups

Mathematics Subject Classification

  • 92C17
  • 35Q92
  • 35K51