# Numerical Simulation of a Multiscale Cell Motility Model Based on the Kinetic Theory of Active Particles

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Description of the Model

- the term $\mathcal{H}$, modeling haptotaxis, is:$$\begin{array}{cc}\hfill \phantom{\rule{-17.07182pt}{0ex}}\mathcal{H}(f,Q)(t,x,v,y)& :=\phantom{\rule{-0.166667em}{0ex}}{\int}_{V}{\int}_{{\mathbb{S}}^{N-1}}{p}_{h}(t,x,{v}^{\prime},y)\psi (v;{v}^{\prime},\theta )f(t,x,{v}^{\prime},y)Q(t,x,\theta )d\theta \phantom{\rule{3.33333pt}{0ex}}d{v}^{\prime}\hfill \\ & -{p}_{h}(t,x,v,y)\phantom{\rule{4pt}{0ex}}f(t,x,v,y){\int}_{V}{\int}_{{\mathbb{S}}^{N-1}}\psi ({v}^{\prime};v,\theta )Q(t,x,\theta )d\theta d{v}^{\prime};\hfill \end{array}$$
- the turning operator $\mathcal{L}$ models random changes in velocity,$$\begin{array}{cc}\hfill \mathcal{L}(f)(t,x,v,y)& :={\int}_{V}{p}_{l}(t,x,{v}^{\prime},y){\alpha}_{1}(y)T(v,{v}^{\prime})f(t,x,{v}^{\prime},y)d{v}^{\prime}\hfill \\ & -\phantom{\rule{4pt}{0ex}}{p}_{l}(t,x,v,y){\alpha}_{1}(y)f(t,x,v,y){\int}_{V}T({v}^{\prime},v)d{v}^{\prime};\hfill \end{array}$$
- and the chemotactic term, $\mathcal{C}$, reads:$$\begin{array}{cc}\hfill \mathcal{C}(f,L)(t,x,v,y)& :={\int}_{V}{p}_{c}(t,x,{v}^{\prime},y){\alpha}_{2}(y)K[\nabla L](v,{v}^{\prime})f(t,x,{v}^{\prime},y)d{v}^{\prime}\hfill \\ & -\phantom{\rule{4pt}{0ex}}{p}_{c}(t,x,v,y){\alpha}_{2}(y)f(t,x,v,y){\int}_{V}K[\nabla L]({v}^{\prime},v)d{v}^{\prime}.\hfill \end{array}$$

#### The Hyperbolic Scaling

## 3. The Numerical Scheme

- First, we solved the transport term, related to the evolution of position and cell activity;
- Second, the integral operators were treated, thus giving the new velocities;
- Finally, we dealt with the equations for the ECM compounds, Q and L.

`SmoothN`, to avoid problems with subsequent steps; then, we calculated ${\Delta}_{x}L(t)$ using the implemented function

`Del2`, which computed the discrete Laplacian. Then, the dependence of the equation with respect to the spatial variable could be considered parametrical, so we solved the equation as an ODE in time: numerical computation of the integral terms, and used the explicit Euler to obtain the next iteration.

#### 3.1. Adaptation of Cells to the ECM: A Measure of Alignment

#### 3.2. Comparison between the Behavior of Activity $\rho W$ and Its Limiting Counterpart ${\rho}_{0}{W}_{0}$

#### 3.3. First Example: Fully-Oriented ECM

#### 3.4. Second Example: Radially-Oriented ECM

#### 3.5. Third Example: Random ECM

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Ananthakrishnan, R.; Ehrlicher, A. The Forces Behind Cell Movement. Int. J. Biol. Sci.
**2007**, 3, 303–317. [Google Scholar] [CrossRef] [PubMed] - Friedl, P.; Wolf, K. Tumour–cell invasion and migration: Diversity and escape mechanisms. Nat. Rev. Cancer
**2003**, 3, 362–374. [Google Scholar] [CrossRef] [PubMed] - Wolf, K.; Friedl, P. Molecular mechanisms of cancer cell invasion and plasticity. Br. J. Dermatol.
**2006**, 154 (Suppl. 1), 11–15. [Google Scholar] [CrossRef] [PubMed] - Keller, H.U.; Wissler, J.H.; Ploem, J. Chemotaxis is not a special case of haptotaxis. Experientia
**1979**, 35, 1669–1671. [Google Scholar] [CrossRef] [PubMed] - Elosegui-Artola, A.; Bazellières, E.; Allen, M.D.; Andreu, I.; Oria, R.; Sunyer, R.; Gomm, J.J.; Marshall, J.F.; Jones, J.L.; Trepat, X.; et al. Rigidity sensing and adaptation through regulation of integrin types. Nat. Mater.
**2014**, 13, 631–637. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Wehrle–Haller, B. Assembly and disassembly of cell matrix adhesions. Curr. Opin. Cell Biol.
**2012**, 24, 569–581. [Google Scholar] [CrossRef] - Guo, N.H.; Krutzsch, H.C.; Negre, E.; Zabrenetzky, V.S.; Roberts, D.D. Heparin-binding peptides from the Type I repeats of Thrombospondin. J. Biol. Chem.
**1992**, 267, 19349–19355. [Google Scholar] - Guo, N.H.; Zabrenetzky, V.S.; Chandrasekaran, L.; Sipes, J.M.; Lawler, J.; Krutzsch, H.C.; Roberts, D.D. Differential roles of protein Kinase C and Pertussis Toxin–sensitive G-binding proteins in modulation of melanoma cell proliferation and motility by Thrombospondin 1. Cancer Res.
**1998**, 58, 3154–3162. [Google Scholar] - Taraboletti, G.; Roberts, D.D.; Liotta, L.A. Thrombospondin–induced tumor cell migration: Haptotaxis and chemotaxis are meditated by different molecular domains. J. Cell Biol.
**1987**, 105, 2409–2415. [Google Scholar] [CrossRef] - Berry, H.; Larreta-Garde, V. Oscillatory behavior of a simple kinetic model for proteolysisis during cell invasion. Biophys. J.
**1999**, 77, 655–665. [Google Scholar] [CrossRef] - Erban, R.; Othmer, H. From signal transduction to spatial pattern formation in E. coli: A paradigm for multiscale modeling in biology. Multiscale Model. Simul.
**2005**, 3, 362–394. [Google Scholar] [CrossRef] - Patlak, C. Random walk with persistence and external bias. Bull. Math. Biophys.
**1953**, 15, 311–338. [Google Scholar] [CrossRef] - Keller, E.; Segel, L. Model for Chemotaxis. J. Theor. Biol.
**1971**, 30, 225–234. [Google Scholar] [CrossRef] - Hillen, T.; Painter, K.J. A user’s guide to PDE models for chemotaxis. J. Math. Biol.
**2009**, 58, 183–217. [Google Scholar] [CrossRef] [PubMed] - Chalub, F.A.; Markovich, P.; Perthame, B.; Schmeiser, C. Kinetic models for chemotaxis and their drift–diffusion limits. Monatsh. Math.
**2004**, 142, 123–141. [Google Scholar] [CrossRef] - Othmer, H.G.; Hillen, T. The diffusion limit of transport equations II: Chemotaxis equations. SIAM J. Appl. Math.
**2002**, 62, 1222–1250. [Google Scholar] [CrossRef] - Bellomo, N.; Bellouquid, A.; Nieto, J.; Soler, J. On the asymptotic theory from microscopic to macroscopic growing tissue models: An overview with perspectives. Math. Models Methods Appl. Sci.
**2012**, 22, 1130001. [Google Scholar] [CrossRef] - Bellomo, N.; Bellouquid, A.; Tao, Y.; Winkler, M. Toward a mathematical theory of Keller-Segel models of pattern formation in biological tissues. Math. Models Methods Appl. Sci.
**2015**, 25, 1663–1763. [Google Scholar] [CrossRef] - Arias, M.; Campos, J.; Soler, J. Cross-diffusion and traveling waves in porous-media flux-saturated Keller-Segel models. Math. Models Methods Appl. Sci.
**2018**, 28, 2103–2129. [Google Scholar] [CrossRef] - Bellouquid, A.; Nieto, J.; Urrutia, L. About the kinetic description of fractional diffusion equations modeling chemotaxis. Math. Models Methods Appl. Sci.
**2016**, 26, 249–268. [Google Scholar] [CrossRef] - Bellomo, N.; Bellouquid, A.; Chouhad, N. From a multiscale derivation of nonlinear cross-diffusion models to Keller-Segel models in a Navier-Stokes fluid. Math. Model. Methods Appl. Sci.
**2016**, 26, 2041–2069. [Google Scholar] [CrossRef] - Oster, G.; Murray, J.D.; Harris, A.K. Mechanical aspects of mesenchymal morphogenesis. J. Embryol. Exp. Morphol.
**1983**, 78, 83–125. [Google Scholar] - Mallet, D.G.; Pettet, G.J. A mathematical model of integrin–mediated haptotactic cell migration. Bull. Math. Biol.
**2006**, 68, 231–253. [Google Scholar] [CrossRef] - Painter, K. Modelling cell migration strategies in the extracellular matrix. J. Math. Biol.
**2009**, 58, 511–543. [Google Scholar] [CrossRef] - Chaplain, M.A.J.; Lolas, G. Mathematical modelling of cancer cell invasion of tissue: The role of the urokinase plasminogen activation system. Math. Models Methods Appl. Sci.
**2005**, 15, 1685–1734. [Google Scholar] [CrossRef] - Kim, Y.; Jeon, H.; Othmer, H. The Role of the Tumor Microenvironment in Glioblastoma: A Mathematical Model. IEEE Trans. Biomed. Eng.
**2017**, 64, 519–527. [Google Scholar] [CrossRef] - Zhigun, A.; Surulescu, C.; Uatay, A. Global existence for a degenerate haptotaxis model of cancer invasion. Z. Angew. Math. Phys.
**2016**, 67, 147. [Google Scholar] [CrossRef] - Stinner, C.; Surulescu, C.; Uatay, A. Global existence for a go-or-grow multiscale model for tumor invasion with therapy. Math. Models Methods Appl. Sci.
**2016**, 11, 2163–2201. [Google Scholar] [CrossRef] - Bellomo, N.; Knopoff, D.; Soler, J. On the difficult interplay between life, “complexity”, and mathematical sciences. Math. Models Methods Appl. Sci.
**2013**, 23, 1861–1913. [Google Scholar] [CrossRef] - Bellouquid, A.; De Angelis, E.; Knopoff, D. From the modeling of the immune hallmarks of cancer to a black swan in biology. Math. Models Methods Appl. Sci.
**2013**, 23, 949–978. [Google Scholar] [CrossRef] - Ajmone Marsan, G.; Bellomo, N.; Gibelli, L. Stochastic evolutionary differential games toward a systems theory of behavioral social dynamics. Math. Models Methods Appl. Sci.
**2016**, 26, 1051–1093. [Google Scholar] [CrossRef] - Dolfin, M.; Knopoff, D.; Leonida, L.; Patti, D. Escaping the trap of “blocking”: A kinetic model linking economic development and political competition. Kinet. Relat. Models
**2017**, 10, 423–443. [Google Scholar] [CrossRef] - Bellomo, N.; Bellouquid, A.; Knopoff, D. From the micro-scale to collective crowd dynamics. Multiscale Model. Simul.
**2013**, 11, 943–963. [Google Scholar] [CrossRef] - Bellomo, N.; Bellouquid, A.; Nieto, J.; Soler, J. On the multiscale modeling of vehicular traffic: From kinetic to hydrodynamics. Discret. Cont. Dyn. Syst. Ser. B
**2014**, 19, 1869–1888. [Google Scholar] [CrossRef] - Bellomo, N.; Gibelli, L. Toward a mathematical theory of behavioral-social dynamics for pedestrian crowds. Math. Mod. Methods Appl. Sci.
**2015**, 25, 2417–2437. [Google Scholar] [CrossRef] - Bellomo, N.; Gibelli, L.; Outada, N. On the interplay between behavioral dynamics and social interactions in human crowds. Kinet. Relat. Model.
**2019**, 12, 397–409. [Google Scholar] [CrossRef] [Green Version] - Bellomo, N.; Ha, S.-Y. A quest toward a mathematical theory of the dynamics of swarms. Math. Mod. Methods Appl. Sci.
**2017**, 27, 745–770. [Google Scholar] [CrossRef] - Dimarco, G.; Pareschi, L. Numerical methods for kinetic equations. Acta Numer.
**2014**, 23, 369–520. [Google Scholar] [CrossRef] [Green Version] - Outada, N.; Vauchelet, N.; Akrid, T.; Khaladi, M. From kinetic theory of multicellular systems to hyperbolic tissue equations: Asymptotic limits and computing. Math. Mod. Methods Appl. Sci.
**2016**, 26, 2709–2734. [Google Scholar] [CrossRef] [Green Version] - Burini, D.; Chouhad, N. Hilbert method toward a multiscale analysis from kinetic to macroscopic models for active particles. Math. Mod. Methods Appl. Sci.
**2017**, 27, 1327–1353. [Google Scholar] [CrossRef] - Banasiak, J.; Lachowicz, M. Methods of Small Parameter in Mathematical Biology; Series: Modeling and Simulation in Science, Engineering and Technology; Birkhäuser: Boston, MA, USA, 2014. [Google Scholar]
- Kelkel, J.; Surulescu, C. A multiscale approach to cell migration in tissue networks. Math. Mod. Methods Appl. Sci.
**2012**, 22, 1150017. [Google Scholar] [CrossRef] - Nieto, J.; Urrutia, L. A multiscale modeling of cell mobility: From kinetic to hydrodynamics. J. Math. Anal. Appl.
**2016**, 433, 1055–1071. [Google Scholar] [CrossRef] - Araujo, R.P.; McElwain, D.L.S. A history of the study of solid tumour growth: The contribution of mathematical modelling. Bull. Math. Biol.
**2004**, 66, 1039–1091. [Google Scholar] [CrossRef] [PubMed] - Holden, H.; Karlsen, K.; Lie, K.; Risebro, N. Splitting Methods for Partial Differential Equations with Rough Solutions: Analysis and MATLAB Programs; EMS Series of Lectures in Mathematics; European Mathematical Society: Zürich, Switzerland, 2010. [Google Scholar]
- Changede, R.; Xu, X.; Margadant, F.; Sheetz, M.P. Nascent integrin adhesions form on all matrix rigidities after integrin activation. Dev. Cell
**2015**, 35, 614–621. [Google Scholar] [CrossRef] - Welf, E.S.; Naik, U.P.; Ogunnaike, B.A. A spatial model for integrin clustering as a result of feedback between integrin activation and integrin binding. Biophys. J.
**2012**, 103, 1379–1389. [Google Scholar] [CrossRef] - Litvinov, B.A.; Mekler, A.; Shuman, H.; Bennett, J.S.; Barsegov, V.; Weisel, J.W. Resolving two-dimensional kinetics of the integrin αIIbβ3–fibrinogen interactions using binding-unbinding correlation spectroscopy. J. Biol. Chem.
**2012**, 287, 35275–35285. [Google Scholar] [CrossRef] [PubMed] - Saragosti, J.; Calvez, V.; Bournaveas, N.; Perthame, B.; Buguin, A.; Silberzan, P. Directional persistence of chemotactic bacteria in a traveling concentration wave. Proc. Natl. Acad. Sci. USA
**2011**, 108, 16235–16240. [Google Scholar] [CrossRef] [Green Version]

**Figure 1.**First example: initial conditions for the ECM fibers (oriented in the direction (0,1)), cell population, and the alignment functions ${C}_{(0,1)}$ (dotted) and ${F}_{(0,1)}$ (dashed).

**Figure 2.**Graph representing the evolution of the ECM (left), cell population (center), and the alignment functions ${C}_{(0,1)}$ (dotted) and ${F}_{(0,1)}$ (dashed) for the first example.

**Figure 3.**Second example: initial conditions for the ECM fibers, cell population, and the alignment functions $CR$ (dotted) and $FR$ (dashed).

**Figure 4.**Graph representing the evolution of the ECM (left), cell population (center), and the alignment functions $C{R}_{(0,1)}$ (dotted) and $F{R}_{(0,1)}$ (dashed) for the second example.

**Figure 5.**Third example: initial conditions for the ECM fibers (oriented in the direction (0,1)), cell population, and the alignment functions ${C}_{(0,1)}$ (dotted) and ${F}_{(0,1)}$ (dashed).

**Figure 6.**Graph representing the evolution of the ECM (left), cell population (center), and the alignment functions ${C}_{(0,1)}$ (dotted) and ${F}_{(0,1)}$ (dashed) for the third example.

Symbol | Description of the Constant | Value | Reference |
---|---|---|---|

${R}_{0}$ | Maximum concentration of integrins on the cell membrane | 1000 integrins/$\mathsf{\mu}$m${}^{2}$ | [10,46] |

${k}_{i}$ | Binding rate of the integrins with the chemical compounds | 1.5–0.34 s${}^{-1}$, 1.4–2.3$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}\phantom{\rule{3.33333pt}{0ex}}\mathsf{\mu}$m${}^{2}$/s | [10,47,48] |

${k}_{-i}$ | Unbinding rate of the integrins with the chemical compounds | 0.1–3.4 s${}^{-1}$, 2.42–0.6 s${}^{-1}$ | [10,47,48] |

${\alpha}_{1}$ | Relative influence of the random turning in the cellular movement | $1/27$ | [9] |

${\alpha}_{2}$ | Relative influence of the chemotaxis in the cellular movement | $26/27$ | [9] |

${R}_{C}$ | Typical size of cells | 20–200 $\mathsf{\mu}$m | [2] |

s | Typical migration speed for the cells | 0.1–1.0 $\mathsf{\mu}$m/min | [2] |

${r}_{L}$ | Degradation rate of the chemoattractant | $5\times {10}^{-3}$ mol/s | [49] |

${D}_{L}$ | Diffusion coefficient of the chemical L | $8\times {10}^{-6}{\mathrm{cm}}^{2}/\mathrm{s}$ | [49] |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Knopoff, D.A.; Nieto, J.; Urrutia, L.
Numerical Simulation of a Multiscale Cell Motility Model Based on the Kinetic Theory of Active Particles. *Symmetry* **2019**, *11*, 1003.
https://doi.org/10.3390/sym11081003

**AMA Style**

Knopoff DA, Nieto J, Urrutia L.
Numerical Simulation of a Multiscale Cell Motility Model Based on the Kinetic Theory of Active Particles. *Symmetry*. 2019; 11(8):1003.
https://doi.org/10.3390/sym11081003

**Chicago/Turabian Style**

Knopoff, Damián A., Juanjo Nieto, and Luis Urrutia.
2019. "Numerical Simulation of a Multiscale Cell Motility Model Based on the Kinetic Theory of Active Particles" *Symmetry* 11, no. 8: 1003.
https://doi.org/10.3390/sym11081003