广东工业大学学报  2024, Vol. 41Issue (1): 11-18.  DOI: 10.12052/gdutxb.230111. 0

### 引用本文

Wang Zhen-you, Huang Ya-ting. Qualitative Analysis and Numerical Simulation of Generative Model of Tumor Lymphatic Vessels Under ECM Remodeling[J]. JOURNAL OF GUANGDONG UNIVERSITY OF TECHNOLOGY, 2024, 41(1): 11-18. DOI: 10.12052/gdutxb.230111.

### 文章历史

ECM重塑下肿瘤淋巴管生成模型的定性分析与数值模拟

Qualitative Analysis and Numerical Simulation of Generative Model of Tumor Lymphatic Vessels Under ECM Remodeling
Wang Zhen-you, Huang Ya-ting
School of Mathematics and Statistics, Guangdong University of Technology, Guangzhou 510520, China
Abstract: Tumor metastasis is an important link in the process of tumor development, and it is also one of the main reasons for cancer deterioration and treatment failure. Taking tumor metastasis as the background, a study is conducted on the generative model of tumor lymphatics based on the interaction between tumor and extracellular matrix (ECM). First, mathematical language is used to sort out the biological principles of tumor lymphangiogenesis, and then assumptions made and mathematical models established and qualitative analysis carried out. The proof of the uniqueness of the existence of local solutions of the model is mainly carried out by means of approximation methods, the qualitative theory of partial differential equations and Banach's immovable point theorem, as well as the uniqueness of the existence of the overall solution of the model with the help of the regularity estimate of the local solution and the embedding inequality. Finally, the difference numerical method is used to carry out numerical simulation to illustrate the reliability and accuracy of the model. This research is of great significance for in-depth understanding the mechanism of tumor metastasis, guiding cancer treatment, and promoting related research.
Key words: tumor lymphangiogenesis    extracellular matrix(ECM)    reaction diffusion    existence    uniqueness

1 数学模型及方程

 图 1 肿瘤淋巴管生成模型示意图 Figure 1 Schematic diagram of generative model of tumor lymphatic vessels

 ${\varOmega _T} = \left\{ {\left( {x,t} \right) \left| {0 < x < L,0 < t < T} \right.} \right\},L > 0,T > 0$

② 定义$W_p^{2,1}\left( {{\varOmega _T}} \right)$

 ${ W_p^{2,1}\left( {{\varOmega _T}} \right) = \left\{ {u,v \in {L^p}\left( {{\varOmega _T}} \right) \left| {{u_t},{v_t},\nabla u,} \right.\nabla v,{\nabla ^2}u,{\nabla ^2}v \in {L^p}\left( {{\varOmega _T}} \right) } \right\}}$

$1 \leqslant p < \infty$，记空间$W_p^{2,1}\left( {{\varOmega _T}} \right)$中的函数$u\left( {x,t} \right)$的范数${\left\| u \right\|_{W_p^{2,1}\left( {{\varOmega _T}} \right) }}$如下：${\left\| u \right\|_{W_p^{2,1}\left( {{\varOmega _T}} \right) }} = $${\left\| {{D_x}u} \right\|_{{L^p}\left( {{\varOmega _T}} \right) }} + {\left\| {D_x^2u} \right\|_{{L^p}\left( {{\varOmega _T}} \right) }} + {\left\| {{D_t}u} \right\|_{{L^p}\left( {{\varOmega _T}} \right) }} ③ 对于p > \dfrac{5}{2}，记{D_P}\left( {0,1} \right)$$t = 0$$W_p^{2,1}\left( {{\varOmega _T}} \right) 的迹空间，即\varphi \in {D_P}\left( {0,1} \right) 当且仅当\exists u \in W_p^{2,1}\left( {{\varOmega _T}} \right) 使得 u\left(\cdot,0\right) =\phi 。定义{D_P}\left( {0,1} \right) 空间中的函数\varphi \left( {x,t} \right) 的范数{\left\| \varphi \right\|_{{D_P}\left( {0,1} \right) }}  {\left\| \varphi \right\|_{{D_P}\left( {0,1} \right) }}= \{{T}^{-\tfrac{1}{P}}{\Vert u\Vert }_{{W}_{p}^{2,1}\left({\varOmega }_{T}\right) }|u\in {W}_{p}^{2,1}\left({\varOmega }_{T}\right) ,u\left(\cdot,0\right) =\varphi \} 由于p > \dfrac{5}{2}时，W_p^{2,1}\left( {{\varOmega _T}} \right) 连续嵌入到C\left( {{\varOmega _T}} \right) ，因此上面的定义是有意义的。而且，显然如果\varphi \in {W^{2,p}}\left( {0,1} \right) ，则\varphi \in {D_P}\left( {0,1} \right)$${\left\| \varphi \right\|_{{D_P}\left( {0,1} \right) }} \leqslant {\left\| \varphi \right\|_{{W^{2,p}}\left( {0,1} \right) }}$

④ 记$C_{x,t}^{k + \alpha ,\beta }\left( {{\varOmega _T}} \right)$，整数$k \geqslant 0$$\alpha ,\beta 满足0 < \alpha < 1, 0 < \beta < 1$$C_{x,t}^{k + \alpha ,\beta }\left( {{\varOmega _T}} \right)$空间中的函数$u\left( {x,t} \right)$有如下范数：

 ${\left\| u \right\|_{C_{x,t}^{k + \alpha ,\beta }\left( {{\varOmega _T}} \right) }} = \sum\limits_{\left| l \right| = 0}^k {\left[ {\mathop {\sup }\limits_{{\varOmega _T}} \left| {D_x^lu} \right| + \left\langle {D_x^lu} \right\rangle _{x,{\varOmega _T}}^\alpha + \left\langle {D_x^lu} \right\rangle _{t,{\varOmega _T}}^\beta } \right]}$

 $\begin{split} &\left\langle u \right\rangle _{x,{\varOmega _T}}^\alpha = \mathop {\sup }\limits_{\left( {x,t} \right) ,\left( {y,t} \right) \in {\varOmega _T}} \dfrac{{\left| {u\left( {x,t} \right) - u\left( {y,t} \right) } \right|}}{{{{\left| {x - y} \right|}^\alpha }}}\\ &\left\langle u \right\rangle _{t,{\varOmega _T}}^\beta = \mathop {\sup }\limits_{\left( {x,t} \right) ,\left( {x,\tau } \right) \in {\varOmega _T}} \dfrac{{\left| {u\left( {x,t} \right) - u\left( {x,\tau } \right) } \right|}}{{{{\left| {t - \tau } \right|}^\beta }}} \end{split}$

 $\left\{ \begin{array}{l} \dfrac{{\partial u}}{{\partial t}} = D\dfrac{{{\partial ^2}u}}{{\partial {x^2}}} + a\left( {x,t} \right) \dfrac{{\partial u}}{{\partial x}} + b\left( {x,t} \right) u + f\left( {x,t} \right) \\ \left\{ {\left. {\left( {x,t} \right) } \right|0 < x < L,0 < t < T} \right\} \\ x = 0,L:Bu = \varphi ,0 < t < T \\ u\left( {x,0} \right) = {u_0}\left( x \right) ,0 \leqslant x \leqslant L \end{array} \right.$

 $\begin{gathered} {\left\| u \right\|_{W_p^{2,1}\left( {{\varOmega _T}} \right) }} \leqslant \\ {C_P}\left( T \right) \left( {{{\left\| {{u_0}\left( x \right) } \right\|}_{{D_P}\left( {0,L} \right) }} + {{\left\| {\varphi \left( {x,t} \right) } \right\|}_{{W^{2,P}}\left( {0,T} \right) }} + {{\left\| f \right\|}_P}} \right) \\ \end{gathered}$

(2) 当$\alpha {{ = 1,}}\;\beta \geqslant 0$时，有${\left\| u \right\|_{{C^{2 + \alpha ,1 + {\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. } 2}}}\left( {{{\overline \varOmega }_T}} \right) }} \leqslant $${\left\| {{u_0}} \right\|_{{C^{2 + \alpha }}\left[ {0,L} \right]}} + {C_\alpha }\left( T \right) \left( {{{\left\| \varphi \right\|}_{{C^{1 + {\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. } 2}}}\left[ {0,T} \right]}} + {{\left\| f \right\|}_{{C^{\alpha ,{\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. } 2}}}\left( {{{\overline \varOmega }_T}} \right) }}} \right) 这里的 {C_\alpha }\left( T \right) 是仅依赖于D,T,$${\left\| {a\left( {x,t} \right) } \right\|_{{C^{\alpha ,{\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. } 2}}}\left( {{{\overline \varOmega }_T}} \right) }}, {\left\| {b\left( {x,t} \right) } \right\|_{{C^{\alpha ,{\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. } 2}}}\left( {{{\overline \varOmega }_T}} \right) }}$的常数。

 ${\left\| {u\left( {x,t} \right) - u\left( {x,0} \right) } \right\|_{{C^{1 + \alpha ,{{\left( {1 + \alpha } \right) } \mathord{\left/ {\vphantom {{\left( {1 + \alpha } \right) } 2}} \right. } 2}}}\left( {{{\overline \varOmega }_T}} \right) }} \leqslant C\eta \left( T \right) {\left\| u \right\|_{{C^{2 + \alpha ,1 + {\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. } 2}}}\left( {{{\overline \varOmega }_T}} \right) }}$

3 局部解的存在唯一性

$\forall T > 0$和一个正常数$M$,引入度量空间$\left( {{X_M},d} \right)$，对任意的$m\left( {x,t} \right) ,n\left( {x,t} \right) ,c\left( {x,t} \right) ,$$s\left( {x,t} \right) ,q\left( {x,t} \right) \in {X_M}\left( {0 < x < L,0 < t < T} \right) 满足条件：m\left( {x,t} \right) ,n\left( {x,t} \right) , c\left( {x,t} \right) , s\left( {x,t} \right) ,q\left( {x,t} \right)$$ \in C_{x,t}^{1 + \alpha ,{{\left( {1{\text{ + }}\alpha } \right) } \mathord{\left/ {\vphantom {{\left( {1{\text{ + }}\alpha } \right) } 2}} \right. } 2}}\left( {{{\overline \varOmega }_T}} \right)$，并满足问题(1)~(5)且

 $\begin{split} & {\left\| {m\left( {x,t} \right) } \right\|_{C_{x,t}^{1 + \alpha ,{{\left( {1{\text{ + }}\alpha } \right) } \mathord{\left/ {\vphantom {{\left( {1{\text{ + }}\alpha } \right) } 2}} \right. } 2}}\left( {{\varOmega _T}} \right) }} \leqslant M,{\left\| {n\left( {x,t} \right) } \right\|_{C_{x,t}^{1 + \alpha ,{{\left( {1{\text{ + }}\alpha } \right) } \mathord{\left/ {\vphantom {{\left( {1{\text{ + }}\alpha } \right) } 2}} \right. } 2}}\left( {{\varOmega _T}} \right) }} \leqslant M \\ & {\left\| {c\left( {x,t} \right) } \right\|_{C_{x,t}^{1 + \alpha ,{{\left( {1{\text{ + }}\alpha } \right) } \mathord{\left/ {\vphantom {{\left( {1{\text{ + }}\alpha } \right) } 2}} \right. } 2}}\left( {{\varOmega _T}} \right) }} \leqslant M,{\left\| {s\left( {x,t} \right) } \right\|_{C_{x,t}^{1 + \alpha ,{{\left( {1{\text{ + }}\alpha } \right) } \mathord{\left/ {\vphantom {{\left( {1{\text{ + }}\alpha } \right) } 2}} \right. } 2}}\left( {{\varOmega _T}} \right) }} \leqslant M \\ &{\left\| {q\left( {x,t} \right) } \right\|_{C_{x,t}^{1 + \alpha ,{{\left( {1{\text{ + }}\alpha } \right) } \mathord{\left/ {\vphantom {{\left( {1{\text{ + }}\alpha } \right) } 2}} \right. } 2}}\left( {{\varOmega _T}} \right) }} \leqslant M \end{split}$

$u = \left( {m,n,c,s,q} \right) ,\widetilde u = \left( {\widetilde m,\widetilde n,\widetilde c,\widetilde s,\widetilde q} \right)$，定义${X_M}$中的度量为$d( {u,\widetilde u} ) = {\| {u - \widetilde u} \|_{{C^{1 + \alpha ,{{\left( {1{\text{ + }}\alpha } \right) } \mathord{\left/ {\vphantom {{\left( {1{\text{ + }}\alpha } \right) } 2}} \right. } 2}}}\left( {{\varOmega _T}} \right) }}$

 $\frac{{\partial \widetilde m}}{{\partial t}} = - \sigma q\widetilde m$ (6)
 $\frac{{\partial \widetilde n}}{{\partial t}} = {D_n}{\nabla ^2}\widetilde n + \lambda {n_0}\left( {1 - \frac{{\widetilde n}}{{{n_M}}}} \right) - {\mu _n}\widetilde n$ (7)
 $\frac{{\partial \widetilde c}}{{\partial t}} = {D_c}{\nabla ^2}\widetilde c - \nabla \left( {\frac{{\theta \rho }}{{1 + \rho }} \widetilde c\nabla s} \right) - \nabla \left( {\widetilde c{\chi _c}\nabla m} \right) - {\mu _c}\widetilde c$ (8)
 $\frac{{\partial \widetilde s}}{{\partial t}} = {D_s}{\nabla ^2}\widetilde s + {\alpha _s}n + {\beta _s}m - \varphi c\widetilde s$ (9)
 $\frac{{\partial \widetilde q}}{{\partial t}} = {D_q}{\nabla ^2}\widetilde q - \nabla \left( {\widetilde q{\chi _q}\nabla m} \right) + {\alpha _q}n + {\gamma _q}c$ (10)

(1) 下面先证明$F$映射到${X_M}$自身。

(a) 由引理2可知，问题(1) 存在唯一解$\widetilde m\left( {x,t} \right) \in C_{x,t}^{2 + \alpha ,1 + {\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. } 2}}\left( {{\varOmega _T}} \right)$且满足

 ${\| {\widetilde m} \|_{{C^{2 + \alpha ,1 + {\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. } 2}}}\left( {{{\overline \varOmega }_T}} \right) }} \leqslant {\| {\widetilde m} \|_{{C^{2 + \alpha }}\left[ {0,L} \right]}} = {C_1}\left( {T,M} \right)$

 $\begin{split} {\| {\widetilde m} \|_{C_{x,t}^{2 + \alpha ,1 + {\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. } 2}}\left( {{{\overline \varOmega }_T}} \right) }} \leqslant &\; {\left\| {{m_0}} \right\|_{{C^{1 + \alpha }}\left[ {0,L} \right]}} + {\| {\widetilde m - {m_0}} \|_{C_{x,t}^{2 + \alpha ,{{\left( {1 + \alpha } \right) } \mathord{\left/ {\vphantom {{\left( {1 + \alpha } \right) } 2}} \right. } 2}}\left( {{{\overline \varOmega }_T}} \right) }} \leqslant \\ &{\left\| {{m_0}} \right\|_{{C^{1 + \alpha }}\left[ {0,L} \right]}} + C\eta \left( T \right) {\| {\widetilde m} \|_{C_{x,t}^{2 + \alpha ,1 + {\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. } 2}}\left( {{{\overline \varOmega }_T}} \right) }} \leqslant \\ & {\left\| {{m_0}} \right\|_{{C^{1 + \alpha }}\left[ {0,L} \right]}}{\text{ + }}C\eta \left( T \right) {C_1}\left( {T,M} \right) \end{split}$

 $\begin{split} & {\| {\widetilde n} \|_{C_{x,t}^{2 + \alpha ,1 + {\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. } 2}}\left( {{{\overline \varOmega }_T}} \right) }} \leqslant {\left\| {{n_0}} \right\|_{{C^{1 + \alpha }}\left[ {0,L} \right]}}{\text{ + }}C\eta \left( T \right) {C_2}\left( {T,M} \right) \\ & {\| {\widetilde s} \|_{C_{x,t}^{2 + \alpha ,1 + {\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. } 2}}\left( {{{\overline \varOmega }_T}} \right) }} \leqslant {\left\| {{s_0}} \right\|_{{C^{1 + \alpha }}\left[ {0,L} \right]}}{\text{ + }}C\eta \left( T \right) {C_4}\left( {T,M} \right) \end{split}$

(b) 考虑问题(3) ，为方便记

 $\begin{split} & {a_c}\left( {x,t} \right) = - \frac{{\theta \rho }}{{1 + \rho }}\nabla s - {\chi _c}\nabla m \\ & {b_c}\left( {x,t} \right) = - \frac{{\theta \rho }}{{1 + \rho }}{\nabla ^2}s - {\chi _c}{\nabla ^2}m \\ & {h_c}\left( {x,t} \right) = - {\mu _c}c \end{split}$

 ${\left\| {{a_c}\left( {x,t} \right) } \right\|_{C_{x,t}^{\alpha ,{\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. } 2}}\left( {{{\overline \varOmega }_T}} \right) }} + {\left\| {{b_c}\left( {x,t} \right) } \right\|_{C_{x,t}^{\alpha ,{\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. } 2}}\left( {{{\overline \varOmega }_T}} \right) }} \leqslant C\left( T \right) M$

 $\frac{{\partial \widetilde c}}{{\partial t}} = {D_c}{\nabla ^2}\widetilde c + {a_c}\left( {x,t} \right) \nabla \widetilde c + {b_c}\left( {x,t} \right) \widetilde c + {h_c}\left( {x,t} \right)$

(2) 下面证明映射$F$是压缩映射。对任意的${u_1}, {u_2} \in {X_M}$，假设${\widetilde u_1} = F{u_1},{\widetilde u_2} = F{u_2}, $${\widetilde u^*} = {\widetilde u_1} - {\widetilde u_2},\delta = \left\| {u_1} - {u_2} \right\|_{{C^{1 + \alpha ,{{\left( {1 + \alpha } \right) } \mathord{\left/ {\vphantom {{\left( {1 + \alpha } \right) } 2}} \right. } 2}}}\left( {{{\overline \varOmega }_T}} \right) } {\widetilde m^*} = {\widetilde m_1}^* - {\widetilde m_2}^* ，则有  \frac{{\partial {{\widetilde m}^*}}}{{\partial t}} = - \sigma q{\widetilde m^*} + {h_m}\left( {x,t} \right) 式中： {h_m}\left( {x,t} \right) = - \sigma \left( {{q_1} - {q_2}} \right) {\widetilde m_2} ，则由引理2得  \begin{split} & {\| {{{\widetilde m}_1} - {{\widetilde m}_2}} \|_{{C^{2 + \alpha ,1 + {\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. } 2}}}\left( {{{\overline \varOmega }_T}} \right) }} \leqslant\\ & {C_\alpha }\left( T \right) {\left\| {{h_m}} \right\|_{{C^{\alpha ,{\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. } 2}}}\left( {{{\overline \varOmega }_T}} \right) }} \leqslant\\ & {C_\alpha }\left( T \right) ( {\sigma {{\left\| {{q_1} - {q_2}} \right\|}_{{C^{\alpha ,{\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. } 2}}}\left( {{{\overline \varOmega }_T}} \right) }}{{\| {{{\widetilde m}_2}} \|}_{{C^{\alpha ,{\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. } 2}}}\left( {{{\overline \varOmega }_T}} \right) }}} ) \leqslant\\ & C\left( T \right) M \end{split} (11) {\widetilde n^*} = {\widetilde n_1}^* - {\widetilde n_2}^* ，则有  \frac{{\partial {{\widetilde n}^*}}}{{\partial t}} = {D_n}{\nabla ^2}{\widetilde n^*} - \left( {\frac{{\lambda {n_0}}}{{{n_M}}} + {\mu _n}} \right) {\widetilde n^*} + {h_n}\left( {x,t} \right) 式中： {h_n}\left( {x,t} \right) = - \dfrac{{\lambda {n_0}}}{{{n_M}}}{\widetilde n_2} - {\mu _n}{\widetilde n_2} ，则由引理2得  \begin{split} & {\| {{{\widetilde n}_1} - {{\widetilde n}_2}} \|_{{C^{2 + \alpha ,1 + {\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. } 2}}}\left( {{{\overline \varOmega }_T}} \right) }} \leqslant\\ & {C_\alpha }\left( T \right) {\left\| {{h_n}} \right\|_{{C^{\alpha ,{\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. } 2}}}\left( {{{\overline \varOmega }_T}} \right) }} \leqslant\\ & {C_\alpha }\left( T \right) \left( {\frac{{\lambda {n_0}}}{{{n_M}}}{{\| {{{\widetilde n}_2}} \|}_{{C^{\alpha ,{\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. } 2}}}\left( {{{\overline \varOmega }_T}} \right) }} + {\mu _n}{{\| {{{\widetilde n}_2}} \|}_{{C^{\alpha ,{\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. } 2}}}\left( {{{\overline \varOmega }_T}} \right) }}} \right) \leqslant\\ & C\left( T \right) M \end{split} (12) {\widetilde c^*} = {\widetilde c_1}^* - {\widetilde c_2}^* ，则有  \frac{{\partial {{\widetilde c}^*}}}{{\partial t}} = {D_c}{\nabla ^2}{\widetilde c^*} + {a_c}\left( {x,t} \right) \nabla {\widetilde c^*} + {b_c}\left( {x,t} \right) {\widetilde c^*} + {h_c^\prime} \left( {x,t} \right) 式中：  \begin{split} & {a_c}\left( {x,t} \right) = - \frac{{\theta \rho }}{{1 + \rho }}\nabla s - {\chi _c}\nabla m \\ & {b_c}\left( {x,t} \right) = - \frac{{\theta \rho }}{{1 + \rho }}{\nabla ^2}s - {\chi _c}{\nabla ^2}m \\ & {h_c^\prime} \left( {x,t} \right) = - \frac{{\theta \rho }}{{1 + \rho }}\nabla \left( {{s_1} - {s_2}} \right) - {\chi _c}\nabla \left( {{m_1} - {m_2}} \right) - \\ &\qquad\qquad \frac{{\theta \rho }}{{1 + \rho }}{\nabla ^2}\left( {{s_1} - {s_2}} \right) - {\chi _c}{\nabla ^2}\left( {{m_1} - {m_2}} \right) - {\mu _c}{{\widetilde c}_2} \end{split} 则由引理2得  \begin{split} {\left\| {{{\widetilde c}_1} - {{\widetilde c}_2}} \right\|_{{C^{2 + \alpha ,1 + {\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. } 2}}}\left( {{{\overline \varOmega }_T}} \right) }} \leqslant &\;{C_\alpha }\left( T \right) {\left\| {{h_c^\prime }} \right\|_{{C^{\alpha ,{\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. } 2}}}\left( {{{\overline \varOmega }_T}} \right) }} \leqslant \\ & C\left( T \right) M \end{split} (13) {\widetilde s^*} = {\widetilde s_1}^* - {\widetilde s_2}^* ，则有  \frac{{\partial {{\widetilde s}^*}}}{{\partial t}} = {D_s}{\nabla ^2}{\widetilde s^*} - \varphi c{\widetilde s^*} + {h_s^\prime} \left( {x,t} \right) 式中： {h_s^\prime} \left( {x,t} \right) = - \varphi \left( {{c_1} - {c_2}} \right) {\widetilde s_2} + {\alpha _s}\left( {{n_1} - {n_2}} \right)$$ + {\beta _s}\left( {{m_1} - {m_2}} \right)$。由引理2得

 $\begin{split} & {\| {{{\widetilde s}_1} - {{\widetilde s}_2}} \|_{{C^{2 + \alpha ,1 + {\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. } 2}}}\left( {{{\overline \varOmega }_T}} \right) }} \leqslant\\ & {C_\alpha }\left( T \right) {\left\| {{h_s^\prime}} \right\|_{{C^{\alpha ,{\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. } 2}}}\left( {{{\overline \varOmega }_T}} \right) }} \leqslant\\ & {C_\alpha }\left( T \right) (\varphi {\left\| {{c_1} - {c_2}} \right\|_{{C^{\alpha ,{\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. } 2}}}\left( {{{\overline \varOmega }_T}} \right) }}{\| {{{\widetilde s}_2}} \|_{{C^{\alpha ,{\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. } 2}}}\left( {{{\overline \varOmega }_T}} \right) }} - \\ & {\alpha _s}{\left\| {{n_1} - {n_2}} \right\|_{{C^{\alpha ,{\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. } 2}}}\left( {{{\overline \varOmega }_T}} \right) }} - {\beta _s}{\left\| {{m_1} - {m_2}} \right\|_{{C^{\alpha ,{\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. } 2}}}\left( {{{\overline \varOmega }_T}} \right) }}) \leqslant\\ & C\left( T \right) M \end{split}$ (14)

${\widetilde q^*} = {\widetilde q_1}^* - {\widetilde q_2}^*$，则有

 $\frac{{\partial {{\widetilde q}^*}}}{{\partial t}} = {D_q}{\nabla ^2}{\widetilde q^*} + {a_q}\left( {x,t} \right) \nabla {\widetilde q^*} + {b_q}\left( {x,t} \right) {\widetilde q^*} + {h_q^\prime} \left( {x,t} \right)$

 $\begin{split} & {a_q}\left( {x,t} \right) = - {\chi _q}\nabla m,{b_q}\left( {x,t} \right) = - {\chi _q}{\nabla ^2}m, \\ & {h_q^\prime} \left( {x,t} \right) = - {\chi _q}\nabla \left( {{m_1} - {m_2}} \right) - {\chi _q}{\nabla ^2}\left( {{m_1} - {m_2}} \right) + \\ &\qquad\qquad {\alpha _q}\left( {{n_1} - {n_2}} \right) + {\gamma _q}\left( {{c_1} - {c_2}} \right) \end{split}$

 $\begin{split} {\| {{{\widetilde q}_1} - {{\widetilde q}_2}} \|_{{C^{2 + \alpha ,1 + {\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. } 2}}}\left( {{{\overline \varOmega }_T}} \right) }} \leqslant &\;{C_\alpha }\left( T \right) {\| {{h_q^\prime }} \|_{{C^{\alpha ,{\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. } 2}}}\left( {{{\overline \varOmega }_T}} \right) }} \leqslant\\ & C\left( T \right) M \end{split}$ (15)

 $\begin{split} {\| {{{\widetilde m}_1} - {{\widetilde m}_2}} \|_{{C^{2 + \alpha ,{{\left( {1{\text{ + }}\alpha } \right) } \mathord{\left/ {\vphantom {{\left( {1{\text{ + }}\alpha } \right) } 2}} \right. } 2}}}\left( {{{\overline \varOmega }_T}} \right) }} \leqslant &\;C\eta (T) {\| {{{\widetilde m}_1} - {{\widetilde m}_2}} \|_{{C^{2 + \alpha ,{{1 + \alpha } \mathord{\left/ {\vphantom {{1 + \alpha } 2}} \right. } 2}}}\left( {{{\overline \varOmega }_T}} \right) }} \leqslant\\ & \eta \left( T \right) C\left( T \right) \delta \end{split}$

 $\begin{split} & {\| {{{\widetilde n}_1} - {{\widetilde n}_2}} \|_{{C^{2 + \alpha ,{{\left( {1{\text{ + }}\alpha } \right) } \mathord{\left/ {\vphantom {{\left( {1{\text{ + }}\alpha } \right) } 2}} \right. } 2}}}\left( {{{\overline \varOmega }_T}} \right) }} \leqslant \eta \left( T \right) C\left( T \right) \delta \\ & {\| {{{\widetilde c}_1} - {{\widetilde c}_2}} \|_{{C^{2 + \alpha ,{{\left( {1{\text{ + }}\alpha } \right) } \mathord{\left/ {\vphantom {{\left( {1{\text{ + }}\alpha } \right) } 2}} \right. } 2}}}\left( {{{\overline \varOmega }_T}} \right) }} \leqslant \eta \left( T \right) C\left( T \right) \delta \\ & {\| {{{\widetilde s}_1} - {{\widetilde s}_2}} \|_{{C^{2 + \alpha ,{{\left( {1{\text{ + }}\alpha } \right) } \mathord{\left/ {\vphantom {{\left( {1{\text{ + }}\alpha } \right) } 2}} \right. } 2}}}\left( {{{\overline \varOmega }_T}} \right) }} \leqslant \eta \left( T \right) C\left( T \right) \delta \\ & {\| {{{\widetilde q}_1} - {{\widetilde q}_2}} \|_{{C^{2 + \alpha ,{{\left( {1{\text{ + }}\alpha } \right) } \mathord{\left/ {\vphantom {{\left( {1{\text{ + }}\alpha } \right) } 2}} \right. } 2}}}\left( {{{\overline \varOmega }_T}} \right) }} \leqslant \eta \left( T \right) C\left( T \right) \delta \end{split}$

$T > 0$充分小，使得$0 < \eta \left( T \right) C\left( T \right) < 1$，此时$F$${X_M}上的压缩映射。由Banach不动点定理可知，当T > 0充分小时，F存在唯一的不动点 \left( {m,n,c,s,q} \right) ，是问题(6)~(10)在区域{\varOmega _T}中的唯一解。由证明过程可知T依赖于初值m\left( {x,0} \right) ,n\left( {x,0} \right) ,c\left( {x,0} \right) ,s\left( {x,0} \right) ,q\left( {x,0} \right) 在空间{C^{2 + \alpha }}\left( {0,L} \right) 中的范数的上确界。上述结果可以整理为定理1。 定理1 存在T > 0，对所有t \in \left[ {0,T} \right]，原问题(1)~(5)的逼近问题(6)~(10)在区域{\varOmega _T}内存在唯一的解 ，其中T依赖于m\left( {x,0} \right) ,n\left( {x,0} \right) ,c\left( {x,0} \right) ,s\left( {x,0} \right) ,q\left( {x,0} \right)$${C^{2 + \alpha }}\left( {0,L} \right)$中的范数的上确界。

4 整体解的存在唯一性

 $0 \leqslant n \leqslant C\sup \left( {{n_0}} \right) \equiv {n_M}$

 $0 \leqslant X \leqslant C\sup \left( X \right) \equiv {X_M}$

 $\begin{split} &{\Vert m\Vert }_{{L}^{k}\left({\varOmega }_{T}\right) }\leqslant {C}_{k}\left(T\right) ,{\Vert n\Vert }_{{L}^{k}\left({\varOmega }_{T}\right) }\leqslant {C}_{k}\left(T\right) \\ &{\Vert c\Vert }_{{L}^{k}\left({\varOmega }_{T}\right) }\leqslant {C}_{k}\left(T\right) ,{\Vert s\Vert }_{{L}^{k}\left({\varOmega }_{T}\right) }\leqslant {C}_{k}\left(T\right) \\ &{\Vert q\Vert }_{{L}^{k}\left({\varOmega }_{T}\right) }\leqslant {C}_{k}\left(T\right) \end{split}$

 ${\left\| m \right\|_{{L^k}\left( {{\varOmega _T}} \right) }} \leqslant {C_k}\left( T \right) ,{\left\| n \right\|_{{L^k}\left( {{\varOmega _T}} \right) }} \leqslant {C_k}\left( T \right)$ (16)

 $\begin{split} & \frac{1}{{k + 1}}\int_0^L {\frac{\partial }{{\partial t}}} \int_0^t {{s^{k + 1}}} {\rm{d}}x{\rm{d}}t + k{D_s}\int_0^t {\int_0^L {{{\left| {\nabla s} \right|}^2}} } {s^{k - 1}}{\rm{d}}x{\rm{d}}t \leqslant\\ & {\alpha _s}\int_0^t {\int_0^L n } {s^k}{\rm{d}}x{\rm{d}}t + {\beta _s}\int_0^t {\int_0^L m } {s^k}{\rm{d}}x{\rm{d}}t \leqslant\\ & {\alpha _s}{\left( {\int_0^t {\int_0^L {{s^{k + 1}}} } {\rm{d}}x{\rm{d}}t} \right) ^{\tfrac{k}{{k + 1}}}}{\left( {\int_0^t {\int_0^L {{n^{k + 1}}} } {\rm{d}}x{\rm{d}}t} \right) ^{\tfrac{1}{{k + 1}}}} + \\ & {\beta _s}{\left( {\int_0^t {\int_0^L {{s^{k + 1}}} } {\rm{d}}x{\rm{d}}t} \right) ^{\tfrac{k}{{k + 1}}}}{\left( {\int_0^t {\int_0^L {{m^{k + 1}}} } {\rm{d}}x{\rm{d}}t} \right) ^{\tfrac{1}{{k + 1}}}} \end{split}$ (17)

${\phi _s}\left( t \right) = \displaystyle\int_0^t {\displaystyle\int_0^L {{s^{k + 1}}} {\rm{d}}x{\rm{d}}t}$代入式(17)，由式(16)可得$\dfrac{{{\rm{d}}{\phi _s}}}{{{\rm{d}}t}} \leqslant C{\phi _s}^{\tfrac{k}{{k + 1}}} \leqslant {C_k}\left( {{\phi _s} + 1} \right)$

 ${\phi _s} \leqslant {C_k}(T)$ (18)

 $\begin{split} & \frac{1}{{k + 1}}\int_0^L {\frac{\partial }{{\partial t}}} \int_0^t {{{\left( {c + q} \right) }^{k + 1}}} {\rm{d}}x{\rm{d}}t + \\ & k{D_{cq}}\int_0^t {\int_0^L {{{\left| {\nabla \left( {c + q} \right) } \right|}^2}} } {\left( {c + q} \right) ^{k - 1}}{\rm{d}}x{\rm{d}}t \leqslant \\ & {\alpha _q}\int_0^t {\int_0^L n } {\left( {c + q} \right) ^k}{\rm{d}}x{\rm{d}}t \leqslant \\ &{\alpha _q}{\left( {\int_0^t {\int_0^L {{{\left( {c + q} \right) }^{k + 1}}} } {\rm{d}}x{\rm{d}}t} \right) ^{\frac{k}{{k + 1}}}}{\left( {\int_0^t {\int_0^L {{n^{k + 1}}} } {\rm{d}}x{\rm{d}}t} \right) ^{\tfrac{1}{{k + 1}}}} \end{split}$ (19)

 ${\phi _{cq}}\left( t \right) = \int_0^t {\int_0^L {{{\left( {c + q} \right) }^{k + 1}}} {\rm{d}}x{\rm{d}}t}$

 $\frac{{{\rm{d}}{\phi _{cq}}}}{{{\rm{d}}t}} \leqslant C{\phi _{cq}}^{\tfrac{k}{{k + 1}}} \leqslant {C_k}\left( {{\phi _{cq}} + 1} \right)$

${\left\| {\left( {c + q} \right) } \right\|_{{L^{k + 1}}\left( {{\varOmega _T}} \right) }} \leqslant {C_k}\left( T \right)$，又$c \geqslant 0,q \geqslant 0$，则

 ${\left\| c \right\|_{{L^{k + 1}}\left( {{\varOmega _T}} \right) }} \leqslant {C_k}\left( T \right) ,{\left\| q \right\|_{{L^{k + 1}}\left( {{\varOmega _T}} \right) }} \leqslant {C_k}\left( T \right)$ (20)

 $\begin{split} &{\Vert m\Vert }_{{W}_{p}^{2,1}\left({\varOmega }_{T}\right) }\leqslant {C}_{p}\left(T\right) ,{\Vert n\Vert }_{{W}_{p}^{2,1}\left({\varOmega }_{T}\right) }\leqslant {C}_{p}\left(T\right) \\ &{\Vert c\Vert }_{{W}_{p}^{2,1}\left({\varOmega }_{T}\right) }\leqslant {C}_{p}\left(T\right) ,{\Vert s\Vert }_{{W}_{p}^{2,1}\left({\varOmega }_{T}\right) }\leqslant {C}_{p}\left(T\right) \\ &{\Vert q\Vert }_{{W}_{p}^{2,1}\left({\varOmega }_{T}\right) }\leqslant {C}_{p}\left(T\right) \end{split}$

 ${\left\| m \right\|_{W_p^{2,1}\left( {{\varOmega _T}} \right) }} \leqslant {C_p}\left( T \right) ,{\left\| n \right\|_{W_p^{2,1}\left( {{\varOmega _T}} \right) }} \leqslant {C_p}\left( T \right) ,{\left\| s \right\|_{W_p^{2,1}\left( {{\varOmega _T}} \right) }} \leqslant {C_p}\left( T \right) ,$

 $W_p^{2,1}\left( {{\varOmega _T}} \right) \subset \subset C_{x,t}^{\alpha ,{\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. } 2}}( {{{\overline \varOmega }_T}} ) ,\alpha = 2 - \frac{5}{p}\left( {p > \frac{5}{2}} \right)$

 $\begin{split} & {\left\| m \right\|_{C_{x,t}^{\alpha ,{\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. } 2}}\left( {{{\overline \varOmega }_T}} \right) }} \leqslant {C_p}\left( T \right) ,{\left\| n \right\|_{C_{x,t}^{\alpha ,{\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. } 2}}\left( {{{\overline \varOmega }_T}} \right) }} \leqslant {C_p}\left( T \right) \\ & {\left\| s \right\|_{C_{x,t}^{\alpha ,{\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. } 2}}\left( {{{\overline \varOmega }_T}} \right) }} \leqslant {C_p}\left( T \right) \end{split}$

 $\frac{{\partial c}}{{\partial t}} = {D_c}{\nabla ^2}c + {a_c}\left( {x,t} \right) \nabla c + {b_c}\left( {x,t} \right) c + {h_c}\left( {x,t} \right)$

 $\begin{split} & {a_c}\left( {x,t} \right) = - \frac{{\theta \rho }}{{1 + \rho }}\nabla s - {\chi _c}\nabla m \\ & {b_c}\left( {x,t} \right) = - \frac{{\theta \rho }}{{1 + \rho }}{\nabla ^2}s - {\chi _c}{\nabla ^2}m \\ & {h_c}\left( {x,t} \right) = - {\mu _c}c \end{split}$

 $\frac{{\partial q}}{{\partial t}} = {D_q}{\nabla ^2}q + {a_q}\left( {x,t} \right) \nabla q + {b_q}\left( {x,t} \right) q + {h_q}\left( {x,t} \right)$

 $\begin{split} & {a_q}\left( {x,t} \right) = - {\chi _q}\nabla m,{b_q}\left( {x,t} \right) = - {\chi _q}{\nabla ^2}m, \\ & {h_q}\left( {x,t} \right) = {\alpha _q}n + {\gamma _q}c \end{split}$

 ${b_c}(x,t) + {h_c}(x,t) ,{b_q}(x,t) + {h_q}(x,t) \in {L^p}\left( {{\varOmega _T}} \right)$

${a_c}(x,t) ,{a_q}(x,t)$是连续有界函数，由引理1可得${\left\| c \right\|_{W_p^{2,1}\left( {{\varOmega _T}} \right) }} \leqslant {C_p}\left( T \right) ,{\left\| q \right\|_{W_p^{2,1}\left( {{\varOmega _T}} \right) }} \leqslant {C_p}\left( T \right)$，引理6得证。

 $W_p^{2,1}\left( {{\varOmega _T}} \right) \subset \subset C_{x,t}^{\alpha ,{\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. } 2}}\left( {{{\overline \varOmega }_T}} \right) ,\alpha = 2 - \frac{5}{p}\left( {p > \frac{5}{2}} \right)$

5 数值模拟

 图 3 从$t = 0.1$到$t = 0.5$各成分的浓度变化图 Figure 3 Concentration variation diagram of each component from $t = 0.1$ to $t = 0.5$

6 结论与展望

 [1] DILLEKS H, ROGERS M S, STRAUME O. Are 90% of deaths from cancer caused by metastases?[J]. Cancer Medicine, 2019, 8(12): 5574-5576. DOI: 10.1002/cam4.2474. [2] LIU P, DING P, SUN C, et al. Lymphangiogenesis in gastric cancer: function and mechanism[J]. European Journal of Medical Research, 2023, 28(1): 405. [3] ALEJANDRA G L, TATIANA V, PETROVA. Development and aging of the lymphatic vascular system[J]. Advanced Drug Delivery Reviews, 2021, 169: 63-78. DOI: 10.1016/j.addr.2020.12.005. [4] YANG Y L, CAO Y H. The impact of VEGF on cancer metastasis and systemic disease[J]. Seminars in Cancer Biology, 2022, 86(3): 251-261. [5] KAI F B, DRAIN A P, WEAVER V M. The extracellular matrix modulates the metastatic Journey[J]. Developmental Cell, 2019, 49(3): 332-346. DOI: 10.1016/j.devcel.2019.03.026. [6] LE X N, NILSSON M, GOLDMAN J, et al. Dual EGFR-VEGF pathway inhibition: a promising strategy for patients with EGFR-mutant NSCLC[J]. Journal of Thoracic Oncology, 2021, 16(2): 205-215. DOI: 10.1016/j.jtho.2020.10.006. [7] LIU Y, CAO X T. Characteristics and significance of the pre-metastatic niche[J]. Cancer Cell, 2016, 30(5): 668-681. DOI: 10.1016/j.ccell.2016.09.011. [8] QUINTERO-FABIÁN S, RODRIGO A, ECERRIL-VILLANUEVA, et al. Role of matrix metalloproteinases in angiogenesis and cancer[J]. Frontiers in Oncology, 2019, 9: 1307. DOI: 10.3389/fonc.2019.01307. [9] 周云, 卫雪梅. 一个具有Robin自由边界的双曲肿瘤生长模型解的定性分析[J]. 广东工业大学学报, 2021, 38(2): 60-65. ZHOU Y, WEI X M. A qualitative analysis of a hyperbolic tumor growth model with robin free boundary[J]. Journal of Guangdong University of Technology, 2021, 38(2): 60-65. DOI: 10.12052/gdutxb.200109. [10] 梁小珍, 卫雪梅. 结肠癌细胞代谢模型解的存在性[J]. 广东工业大学学报, 2019, 36(5): 38-42. LIANG X Z, WEI X M. Existence of the solution to the metabolic model of colon cancer cells[J]. Journal of Guangdong University of Technology, 2019, 36(5): 38-42. DOI: 10.12052/gdutxb.180177. [11] CUI S B. Analysis of a free boundary problem modeling tumor growth[J]. Acta Mathematica Sinica, 2005, 21(5): 1071-1082. DOI: 10.1007/s10114-004-0483-3. [12] LADYZHENSKAYA O A, SOLONNIKOV V A, URAL'TSEVA N N. Linear and quasi-linear equations of parabolic type[M]. Translations of Mathematical Monographs. USA: Am Math Soc, 1968: 23. [13] FRIEDMAN A, LOLAS G. Analysis of a mathematical model of tumor lymphangiogenesis[J]. Mathematical Models & Methods in Applied Sciences, 2005, 15(1): 95-107. [14] WEI X, CUI S. Existence and uniqueness of global solutions for a mathematical model of antiangiogenesis in tumor growth[J]. Nonlinear Analysis:Real World Applications, 2008, 9(5): 1827-1836. DOI: 10.1016/j.nonrwa.2007.05.013. [15] WEI X, GUO C. Global existence for a mathematical model of the immune response to cancer[J]. Nonlinear Analysis Real World Applications, 2010, 11(5): 3903-3911. DOI: 10.1016/j.nonrwa.2010.02.017. [16] 王术. Sobolev空间与偏微分方程引论[M]. 北京: 科学出版社, 2009. [17] LAI X, FRIEDMAN A. Combination therapy for melanoma with braf/mek inhibitor and immune checkpoint inhibitor: a mathematical model[J]. BMC Systems Biology, 2017, 11(1): 70. DOI: 10.1186/s12918-017-0446-9.