A mechanobiological model to study upstream cell migration guided by tensotaxis

Abstract

Cell migration is a process of crucial importance for the human body. It is responsible for important processes such as wound healing and tumor metastasis. Migration may occur in response to stimuli of chemical, physical and mechanical nature occurring in the cellular microenvironment. The interstitial flow (IF) can generate mechanical stimuli in cells that influence the cell behavior and interactions of the cells with the extracellular matrix (ECM). One of the phenomena is upstream migration, which is observed in some tumors. In this work, we present a new approach to study the adherent cell migration in a porous medium using a mechanobiological model, attempting to understand if upstream migration can be generated exclusively by mechanical factors. The influence of IF on the behavior of cells and the extracellular matrix was considered. The model is based on a system of coupled nonlinear differential equations solved by the finite element method. Several simulations were performed to study the upstream cell migration and evaluate the effects of pressure, permeability, ECM stiffness and cellular concentration variations on the cell velocity. The results indicated that upstream migration can occur in the presence of mechanical stimuli generated by IF and that the tested parameters have a direct influence on the cellular velocity, especially the pressure and the permeability.

Appendix: Weak formulation of the finite element method

In the finite element discretization method, the application of weak formulation in the differential equations is required, which allows developing the systems of integrable equations. For the application of this method, test functions of the variable unknowns $\left( {\mathbf {u}}, n, P\right)$ are considered. After the application of the weak formulation, first Green identity was used to reduce the order of derivatives in differential equations. The next step for developing the system of integrable equations was temporal and spatial discretization. For temporal discretization, the backward Euler method was used. For spatial discretization, the interpolation of the model variables was performed through the shape functions associated with ${\mathbf {u}}$, n and P. For each discretized element ${\Omega ^e}$, the interpolation can be written as follows:

$$\begin{aligned} n^h|_{\varOmega ^e} = {\mathbf {N}}_n{\mathbf {n}}^e P^h|_{\varOmega ^e} = {\mathbf {N}}_P{\mathbf {P}}^e {\mathbf {u}}^h|_{\varOmega ^e} = \mathbf {N_{u}}{\mathbf {u}}^e \end{aligned}$$

(11)

where ${\mathbf {n}}^e$, ${\mathbf {P}}^e$ and ${\mathbf {u}}^e$ are the nodal values vectors and ${\mathbf {N}}_n$, ${\mathbf {N}}_P$ and $\mathbf {N_{u}}$ the matrices of shape functions. Both are evaluated for each discretized element ${\Omega ^e}$ and each integration point h.

Nonlinear differential equations are solved through the Newton–Raphson method. The application of this method is based on an iterative process that involves solving of linear system of equations developed by weak formulation of equations and their discretization. Basically, the Newton–Raphson method is based on the assembly of the residual vector, denoted by ${\mathbf {R}}^a$, and a Jacobian matrix, also called tangent stiffness matrix, denoted by ${\mathbf {K}}_{T}\left( {\mathbf {X}}^a\right)$, where a is the ath increment and ${\mathbf {X}}^a$ represents the vector of model variables. The residual vector is a result of equilibrium of internal and external forces of the model. This equilibrium is expressed in the form:

$$\begin{aligned} {\mathbf {f}}_{i, k+1}^e = {\mathbf {f}}_{i, k+1}^{\mathrm{{int}},e} - {\mathbf {f}}_{i, k+1}^{\mathrm{{ext}},e},\quad i = {\mathbf {u}}, n, P \end{aligned}$$

(12)

The residual vector is represented as follows:

$$\begin{aligned} {\mathbf {R}}_{k+1}^e = \left[ {\mathbf {f}}_{{\mathbf {u}}, k+1}^e, {\mathbf {f}}_{n, k+1}^e, {\mathbf {f}}_{P, k+1}^e\right] ^{\mathrm{T}} \end{aligned}$$

(13)

and the contribution of the each component is shown below:

$$\begin{aligned} {\mathbf {f}}_{{\mathbf {u}},k+1}^e= & {} \int _{{\Omega }^{e}}{\mathbf {B}}_{{\mathbf {u}}}^{\mathrm{T}}\left[ {\mathbf {D}}_{\mathrm{ecm}}{\mathbf {H}}{\mathbf {u}}^{h}+{\mathbf {D}}_{\mathrm{cell}} {\mathbf {H}}{\mathbf {u}}^{h}+\frac{K_{\mathrm{cell}}n^h}{1+\lambda n^h}{\mathbf {I}}\right] _{k+1}{\mathrm{d}}{\Omega }\nonumber \\&+\,\int _{\Omega ^{e}}{\mathbf {N}}_{{\mathbf {u}}}^{\mathrm{T}} \left[ \nabla P^{h} + c_{a}\frac{\kappa }{\phi \eta }n^{h}\nabla P^{h}\right] _{k+1}{\mathrm{d}}{\Omega } \end{aligned}$$

(14)

$$\begin{aligned} {\mathbf {f}}_{n,k+1}^e= & {} \int _{{\Omega }^{e}}{\mathbf {N}}_{n}^{\mathrm{T}}\frac{n_{k+1}^h - n_{k}^h}{\Delta t}{\mathrm{d}}{\Omega } + \int _{{\Omega }^{e}}\nabla {\mathbf {N}}_{n}^{\mathrm{T}}\left[ D\nabla n^h \right. \nonumber \\&-\, n^h\frac{{\mathbf {u}}^h - {\mathbf {u}}_{k}^h}{\Delta t} - MK_{\mathrm{cell}}\frac{n^h}{1+\lambda n^h}\nabla \theta ^h\nonumber \\&\left. -\, \frac{Mp_{cell}\theta ^h}{\left( 1+\lambda n^h\right) ^2}\nabla n^h \right] _{k+1}{\mathrm{d}}{\Omega } \end{aligned}$$

(15)

$$\begin{aligned} {\mathbf {f}}_{P,k+1}^e= & {} \int _{{\Omega }^{e}}{\mathbf {N}}_{P}^{\mathrm{T}}\gamma \frac{P_{k+1}^h - P_{k}^h}{\Delta t}{\mathrm{d}}{\Omega } + \int _{{\Omega }^{e}} \nabla {\mathbf {N}}_{P}^{\mathrm{T}}\frac{\kappa }{\eta }\nabla P_{k+1}^h {\mathrm{d}}{\Omega } \nonumber \\&- \int _{{\Omega }^{e}}\nabla {\mathbf {N}}_{P}^{\mathrm{T}} \frac{{\mathbf {u}}_{k+1}^h - {\mathbf {u}}_{k}^h}{\Delta t}{\mathrm{d}}{\Omega } \end{aligned}$$

(16)

The tangent stiffness matrix, or Jacobian matrix, is obtained by linearizing the external and internal forces of the system of equations resulting from the application of the weak formulation by the finite element method. The relationship between the components associated with internal and external stiffness matrix can be expressed as follows:

$$\begin{aligned} {\mathbf {k}}^e|_{k+1} = {\mathbf {k}}^{\mathrm{{int}},e}|_{k+1} + {\mathbf {k}}^{\mathrm{{ext}},e}|_{k+1} \end{aligned}$$

(17)

In the matrix form, the tangent stiffness matrix can be expressed in the form:

$$\begin{aligned} {\mathbf {K}}_{T}^e|_{k+1} = \begin{bmatrix} {\mathbf {k}}_{\mathbf {uu}}^e &{}&{} {\mathbf {k}}_{{\mathbf {u}}n}^e &{}&{} {\mathbf {k}}_{{\mathbf {u}}P}^e \\ {\mathbf {k}}_{n{\mathbf {u}}}^e &{}&{} {\mathbf {k}}_{nn}^e &{}&{} {\mathbf {0}} \\ {\mathbf {k}}_{P{\mathbf {u}}}^e &{}&{} {\mathbf {0}} &{}&{} {\mathbf {k}}_{PP}^e \end{bmatrix}_{k+1} {\mathrm{where}} \quad {\mathbf {k}}_{ij}^e = \frac{\partial {\mathbf {f}}_{i}^e}{\partial j} \end{aligned}$$

(18)

and the contribution of each component in tangent stiffness matrix is:

$$\begin{aligned} {\mathbf {k}}_{\mathbf {uu},k+1}^e= & {} \int _{{\Omega }^{e}}{\mathbf {B}}_{{\mathbf {u}}}^{\mathrm{T}}{\mathbf {D}}_{\mathrm{ecm}}{\mathbf {B}}_{\mathbf {u}}{\mathrm{d}}{\Omega }+ \int _{\Omega ^{e}}{\mathbf {B}}_{{\mathbf {u}}}^{\mathrm{T}}{\mathbf {D}}_{\mathrm{cell}}{\mathbf {B}}_{\mathbf {u}}{\mathrm{d}}{\Omega } \end{aligned}$$

(19)

$$\begin{aligned} {\mathbf {k}}_{{\mathbf {u}}n,k+1}^e= & {} \int _{{\Omega }^{e}}{\mathbf {B}}_{{\mathbf {u}}}^{\mathrm{T}}\left[ {\mathbf {D}}_{\mathrm{cell}}^*{\mathbf {H}}{\mathbf {u}}^h + \frac{K_{\mathrm{cell}}^*}{\left( 1+\lambda n^h\right) ^2}{\mathbf {I}}\right] {\mathrm{d}}\Omega \nonumber \\&+ \int _{{\Omega }^{e}}{\mathbf {N}}_{{\mathbf {u}}}^{\mathrm{T}}c_{a} \frac{\kappa }{\eta }\nabla P_{k+1}^h{\mathbf {N}}_n{\mathrm{d}}\Omega \end{aligned}$$

(20)

$$\begin{aligned} {\mathbf {k}}_{{\mathbf {u}}p,k+1}^e= & {} \int _{{\Omega }^{e}}{\mathbf {N}}_{{\mathbf {u}}}^{\mathrm{T}}\nabla {\mathrm{N}}_P{\mathrm{d}}\Omega \nonumber \\&+ \int _{{\Omega }^{e}}{\mathbf {N}} _{{\mathbf {u}}}^{\mathrm{T}}c_a\frac{\kappa }{\phi \eta }n_{k+1}^h\nabla {\mathrm{N}}_P{\mathrm{d}}\Omega \end{aligned}$$

(21)

$$\begin{aligned} {\mathbf {k}}_{nn,k+1}^e= & {} \frac{1}{\Delta t}\int _{{\Omega }^{e}}{\mathbf {N}}_{\mathrm{n}}^{\mathrm{T}}{\mathbf {N}}_{\mathrm{n}}{\mathrm{d}}\Omega + \int _{{\Omega }^{e}} \nabla {\mathbf {N}}_{\mathrm{n}}^{\mathrm{T}}\left[ D \right. \nonumber \\&\left. - \frac{Mp_{\mathrm{cell}}\theta ^h}{\left( 1 + \lambda n^h\right) ^2}\right] _{k+1}\nabla {\mathbf {N}}_P {\mathrm{d}}\Omega \nonumber \\&+ \int _{{\Omega }^{e}}\nabla {\mathbf {N}}_{\mathrm{n}}^{\mathrm{T}}\left[ \frac{2\lambda Mp_{\mathrm{cell}}\theta ^h}{\left( 1+\lambda n^h\right) ^3}\nabla n^h \right. \nonumber \\&\left. - \frac{{\mathbf {u}}_{k+1}^h - {\mathbf {u}}_{k}^h}{\Delta t} - \frac{MK_{\mathrm{cell}}\nabla \theta ^h}{\left( 1 + \lambda n^h\right) }\right] _{k+1}{\mathbf {N}} _n{\mathrm{d}}\Omega \end{aligned}$$

(22)

$$\begin{aligned} {\mathbf {k}}_{n{\mathbf {u}},k+1}^e= & {} -\frac{1}{\Delta t}\int _{{\Omega }^{e}}\nabla {\mathbf {N}}_{n}^{\mathrm{T}}n_{k+1}^h{\mathbf {N}}_{{\mathbf {u}}}{\mathrm{d}}\Omega \nonumber \\&- \int _{{\Omega }^{e}}\nabla {\mathbf {N}}_{n}^{\mathrm{T}}\frac{MK_{\mathrm{cell}}n_{k+1}^h}{1 + \lambda n_{k+1}^h}\mathrm{Hess}\mathbf {N_u}{\mathrm{d}}\Omega \nonumber \\&- \int _{{\Omega }^{e}}\nabla {\mathbf {N}}_{n}^{\mathrm{T}}\frac{MK_{\mathrm{cell}}}{\left( 1 + \lambda n_{k+1}^h\right) ^2}\nabla n_{k+1}^h\nabla ^{*}\mathbf {N_u}{\mathrm{d}} \Omega \end{aligned}$$

(23)

$$\begin{aligned} {\mathbf {k}}_{PP,k+1}^e= & {} \frac{1}{\Delta t}\int _{{\Omega }^{e}}{\mathbf {N}}_{P}^{\mathrm{T}}\gamma {\mathbf {N}}_{P}{\mathrm{d}}\Omega + \int _{{\Omega }^{e}}\nabla {\mathbf {N}} _{P}^{\mathrm{T}}\frac{\kappa }{\eta }\nabla {\mathbf {N}}_{P}{\mathrm{d}}\Omega \end{aligned}$$

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$$\begin{aligned} {\mathbf {k}}_{P{\mathbf {u}},k+1}^e= & {} -\frac{1}{\Delta t}\int _{{\Omega }^{e}}\nabla {\mathbf {N}}_{P}^{\mathrm{T}}\gamma {\mathbf{N}}_{P}{\mathrm{d}}\Omega \end{aligned}$$

(25)

Using the local residual vector (Eq. 13) and local tangent stiffness matrix (Eq. 18), it is possible to assemble the global residual vector and the global tangent stiffness matrix required to solve the equation by the Newton–Raphson method.