Stability, Convergence, and Sensitivity Analysis of the FBLM and the Corresponding FEM

Abstract

We study in this paper the filament-based lamellipodium model (FBLM) and the corresponding finite element method (FEM) used to solve it. We investigate fundamental numerical properties of the FEM and justify its further use with the FBLM. We show that the FEM satisfies a time step stability condition that is consistent with the nature of the problem and propose a particular strategy to automatically adapt the time step of the method. We show that the FEM converges with respect to the (two-dimensional) space discretization in a series of characteristic and representative chemotaxis and haptotaxis experiments. We embed and couple the FBLM with a complex and adaptive extracellular environment comprised of chemical and adhesion components that are described by their macroscopic density and study their combined time evolution. With this combination, we study the sensitivity of the FBLM on several of its controlling parameters and discuss their influence in the dynamics of the model and its future evolution. We finally perform a number of numerical experiments that reproduce biological cases and compare the results with the ones reported in the literature.

Appendices

Appendix A: The FEM for the FBLM

We numerically solve the FBLM (6) with a problem-specific FEM that was first presented in Manhart et al. (2016). Here we present some of its components.

The maximal filament length varies around the lamellipodium, and in effect the computational domain

$$\begin{aligned} B(t) = \left\{ (\alpha ,s):\, 0\le \alpha<2\pi ,\, -L(\alpha ,t)\le s < 0\right\} \end{aligned}$$

is non-rectangular. For consistency and stability reasons we recover the orthogonality of the domain B(t), using the coordinate transformation

$$\begin{aligned} (\alpha ,s,t) \rightarrow \left( \alpha , L(\alpha ,t)s, t\right) , \end{aligned}$$

and replace it by

$$\begin{aligned} B_0 := [0,2\pi )\times [-\,1,0)\ni (\alpha ,s). \end{aligned}$$

(41)

See also Fig. 17.

Fig. 17

(Color figure online) Graphical representation of the $\mathbf F^\pm$ that map the domain of dependence $B_0$ (41) to the lamellipodium. The $s=0$ boundary of $B_0$ is mapped to the membrane of the cell and the $s=-\,1$ to the minus ends of the filaments inside the cell. The filaments and the other functions of $\alpha$ are periodic with respect to $\alpha$. The “filaments” plotted in the lamellipodium correspond to the interfaces of the discretization cells of $B_0$ along the $\alpha$ direction

Accordingly, the weak formulation of (6), recasts into

$$\begin{aligned} 0 =&\int _{B_0}\eta \left( \mu ^B \partial ^2_s\mathbf {F} \cdot \partial _s^2\mathbf {G} + L^4\mu ^A \widetilde{D_t} \mathbf {F} \cdot \mathbf {G} + L^2\lambda _{\text {inext}} \partial _s\mathbf {F} \cdot \partial _s\mathbf {G} \right) \mathrm{d}(\alpha ,s) \nonumber \\&+\int _{B_0} \eta \eta ^*\left( L^4\widehat{\mu ^{S}} \left( \widetilde{D_t}\mathbf {F} - \widetilde{D_t^*}\mathbf {F}^* \right) \cdot \mathbf {G} \mp L^2\widehat{\mu ^{T}}(\phi -\phi _0)\partial _s\mathbf {F}^{\perp } \cdot \partial _s \mathbf {G} \right) \ \mathrm{d}(\alpha ,s) \nonumber \\&- \int _{B_0} p(\varrho ) \left( L^3\partial _{\alpha } \mathbf {F}^{\perp }\cdot \partial _s \mathbf {G} - \frac{1}{L}\partial _s \mathbf {F}^{\perp }\cdot \partial _{\alpha } (L^4\mathbf {G})\right) \mathrm{d}(\alpha ,s) \nonumber \\&+ \int _0^{2\pi } \eta \left( L^2 f_{\text {tan}}\partial _s\mathbf {F} + L^3 f_{\text {inn}}\mathbf {V} \right) \cdot \mathbf {G} \Bigm |_{s=-1}\mathrm{d}\alpha \mp \int _0^{2\pi } L^3\lambda _{\text {tether}}\nu \cdot \mathbf {G} \Bigm |_{s=0} \mathrm{d}\alpha , \end{aligned}$$

(42)

with $\mathbf {F}, \mathbf {G}\in H^1_{\alpha }\left( (0,2\pi );\,H^2_s(-\,1,0)\right)$. In a similar manner the modified material derivative and inextensibility conditions read

$$\begin{aligned} \widetilde{D_t} = \partial _t - \left( \frac{v}{L} + \frac{s \partial _t L}{L}\right) \partial _s \end{aligned}$$

and

$$\begin{aligned} \left| \partial _s {\mathbf {F}}(\alpha ,s,t)\right| = L(\alpha ,t). \end{aligned}$$

We decompose $B_0$ into disjoined rectangular computational cells as follows:

$$\begin{aligned} B_0 = \bigcup _{i=1}^{N_a} \bigcup _{j=1}^{N_s -1} C_{i,j},\quad \text{ where }\quad C_{i,j}=[\alpha _i,\alpha _{i+1})\times [s_j,s_{j+1}), \end{aligned}$$

(43)

for $\alpha _i = (i-1)\Delta \alpha , \Delta \alpha = \frac{2\pi }{N_{\alpha }}, i = 1,\ldots ,N_{\alpha }+1$, and $s_j = -\,1 + (j-1)\Delta s, \Delta s = \frac{1}{N_s - 1}, j = 1,\ldots ,N_s$. The resolution of the grid along the $\alpha$ and s directions is denoted by $\alpha _{\text {nodes}}, s_{\text {nodes}}$. The $\alpha$-periodicity assumption suggests that $\alpha _{N_{\alpha }+1} = 2\pi$ is identified with $\alpha _1 = 0$.

We follow Manhart et al. (2016) and set the conforming FE space

$$\begin{aligned} {\mathcal {V}}&:= \Bigl \{ \mathbf {F}\in C_{\alpha }\left( [0,2\pi ];\, C^1_s([-\,1,0])\right) ^2 \text { such that } \mathbf {F}\bigm |_{C_{i,j}}(\cdot ,s) \in \mathbb {P}^1_{\alpha }, \nonumber \\&\quad \mathbf {F}\bigm |_{C_{i,j}}(\alpha ,\cdot ) \in \mathbb {P}^3_s\quad \text{ for } i=1,\ldots ,N_{\alpha }\,;\, j = 1,\ldots ,N_s-1\Bigr \}, \end{aligned}$$

(44)

of continuous functions that are continuously differentiable with respect to s, and such that on each computational cell they coincide with a first-order polynomial in $\alpha$, and a third-order polynomial in s.

In particular, we consider for $i = 1,\ldots ,N_{\alpha }+1, i = j,\ldots ,N_s$, and $(\alpha ,s)\in C_{i,j}$, that

$$\begin{aligned} \left\{ \begin{array}{lcl} H_1^{i,j}(\alpha ,s)=L_1^{i,j}(\alpha ) G_1^{i,j}(s),&{}~&{} H_5^{i,j}(\alpha ,s)=L_2^{i,j}(\alpha ) G_1^{i,j}(s)\\ H_2^{i,j}(\alpha ,s)=L_1^{i,j}(\alpha ) G_2^{i,j}(s),&{}&{} H_6^{i,j}(\alpha ,s)=L_2^{i,j}(\alpha ) G_2^{i,j}(s)\\ H_3^{i,j}(\alpha ,s)=L_1^{i,j}(\alpha ) G_3^{i,j}(s),&{}&{} H_7^{i,j}(\alpha ,s)=L_2^{i,j}(\alpha ) G_3^{i,j}(s)\\ H_4^{i,j}(\alpha ,s)=L_1^{i,j}(\alpha ) G_4^{i,j}(s),&{}&{} H_8^{i,j}(\alpha ,s)=L_2^{i,j}(\alpha ) G_4^{i,j}(s) \end{array}\right. \end{aligned}$$

(45)

with

$$\begin{aligned} \left\{ \begin{array}{lcl} L_1^{i,j}(\alpha ) =\frac{\alpha _{i+1}-\alpha }{\Delta \alpha }, &{}&{}\quad G_1^{i,j}(s)=1-\frac{3(s-s_j)^2}{\Delta s^2}+\frac{2(s-s_j)^3}{\Delta s^3} \\ L_2^{i,j}(\alpha )=1-L_1^{i,j}(\alpha ), &{}&{}\quad G_2^{i,j}(s)=s-s_j-\frac{2(s-s_j)^2}{\Delta s}+\frac{(s-s_j)^3}{\Delta s^2}\\ &{}&{}\quad G_3^{i,j}(s)=1-G_1^{i,j}(s)\\ &{}&{}\quad G_4^{i,j}(s)=-G_2^{i,j}(s_j+s_{j+1}-s) \end{array}\right. \end{aligned}$$

(46)

and that $H_k^{i,j}(\alpha ,s)=0, k=1,\ldots ,8$, whenever $(\alpha ,s)\not \in C_{i,j}$. The basis functions are then defined as:

$$\begin{aligned} \left\{ \begin{array}{r} \Phi _{i,j} := H_7^{i-1,j-1}+ H_5^{i-1,j} + H_3^{i,j-1} + H_1^{i,j} \\ \Psi _{i,j} :=H_8^{i-1,j-1}+ H_6^{i-1,j} + H_4^{i,j-1} + H_2^{i,j} \end{array}\right. \end{aligned}$$

(47)

for $i=1,\ldots ,N_{\alpha },\, j=1,\ldots ,N_s$, and the element $\mathbf {F}\in {\mathcal {V}}$ can be represented in terms of the point values $\mathbf {F}_{i,j}$ and the s-derivatives $\partial _s \mathbf {F}_{i,j}$ at the discretization nodes, as:

$$\begin{aligned} \mathbf {F}(\alpha ,s)=\sum _{i=1}^{N_{\alpha }} \sum _{j=1}^{N_s} \big ( \mathbf {F}_{i,j} \Phi _{i,j}(\alpha ,s) + \partial _s\mathbf {F}_{i,j} \Psi _{i,j}(\alpha ,s) \big ) . \end{aligned}$$

(48)

The FE formulation of the lamellipodium problem on the time interval [0, T] is to find $\mathbf {F}\in C^1\big ([0,T];\,{\mathcal {V}}\big )$, such that (42) holds for all $\mathbf {G}\in C\big ([0,T];\,{\mathcal {V}}\big )$.

We only mention here that the enforcement of the inextensibility condition (3), and in effect the computation of $\lambda _{\text {inext}}$ in (42) is done via and augmented Lagrangian method (Hestenes 1969; Powell 1969).

For the implementation details of the augmented Lagrangian and the rest of the terms of (42), we refer to Manhart et al. (2016).

Appendix B: The FV Method the Environment

We solve the (27) using a FV method that was previously developed in Kolbe et al. (2016) and Sfakianakis et al. (2017) where we refer for details. Here we provide some information.

We consider the advection–reaction–diffusion (ARD) system

$$\begin{aligned} \mathbf w_t = A(\mathbf w) + R(\mathbf w) + D(\mathbf w), \end{aligned}$$

(49)

where $\mathbf w$ represents the solution vector, and A, R, and D the advection, reaction, and diffusion operators, respectively.

Table 5 Butcher tableaux for the explicit (upper) and the implicit (lower) parts of the third-order IMEX scheme (52), see also Kennedy and Carpenter (2003)

We denote by $\mathbf w_h(t)$ the corresponding (semi-)discrete numerical approximation, indexed by the maximal diameter of the spatial grid h, that satisfies the system of ODEs

$$\begin{aligned} \partial _t \mathbf w_h = {\mathcal {A}}(\mathbf w_h) + {\mathcal {R}}(\mathbf w_h) + {\mathcal {D}}(\mathbf w_h), \end{aligned}$$

(50)

where the numerical operators ${\mathcal {A}}, {\mathcal {R}}$, and ${\mathcal {D}}$ are discrete approximations of the operators A, R, and D in (49), respectively.

We split (50) in an explicit and an implicit part as

$$\begin{aligned} \partial _t \mathbf w_h = {\mathcal {I}}(\mathbf w_h) + {\mathcal {E}}(\mathbf w_h). \end{aligned}$$

(51)

The details of the splitting depend on the particular problem in hand but in a typical case, the advection terms ${\mathcal {A}}$ are explicit in time, the diffusion terms ${\mathcal {D}}$ implicit, and the reaction terms ${\mathcal {R}}$ partly explicit and partly implicit, according to the reaction rates.

More precisely, we employ a diagonally implicit RK method for the implicit part and an explicit RK for the explicit part

$$\begin{aligned} {\left\{ \begin{array}{ll} \mathbf W_i^*= \mathbf w_h^n + \tau _n \sum _{j=1}^{i-2}{{\bar{a}}}_{i,j}\mathbf E_j + \tau _n {{\bar{a}}}_{i,i-1}\mathbf E_{i-1},&{}\quad i=1\ldots s\\ \mathbf W_i = \mathbf W_i^*+ \tau _n \sum _{j=1}^{i-1} a_{i,j}\mathbf I_j + \tau _n a_{i,i}\mathbf I_i,&{}\quad i=1\ldots s\\ \mathbf w_h^{n+1} = \mathbf w_h^n + \tau _n \sum _{i=1}^s{\bar{b}}_i \mathbf E_i + \tau _n \sum _{i=1}^sb_i\mathbf I_i \end{array}\right. }, \end{aligned}$$

(52)

where $s=4$ are the stages of the IMEX method, $\mathbf E_i=\mathcal E(\mathbf W_i), I_i={\mathcal {I}}(\mathbf W_i), i=1\ldots s, \{{\bar{b}},\, {\bar{A}}\}, \{b,\, A\}$ are, respectively, the coefficients for the explicit and the implicit part of the scheme, given in the Butcher Tableau in Table 5 (Kennedy and Carpenter 2003). The linear systems in (52) are solved using the iterative biconjugate gradient stabilized Krylov subspace method (Krylov 1931; Vorst 1992).