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August  2022, 15(8): 2053-2086. doi: 10.3934/dcdss.2022044

A coupled 3D-1D multiscale Keller-Segel model of chemotaxis and its application to cancer invasion

1. 

Laboratoire Jacques-Louis Lions UMR7598, Sorbonne Universite, CNRS, Université de Paris, Inria, F-75005 Paris, France

2. 

MOX Laboratory, Dipartimento di Matematica, Politecnico di Milano, Milan, Italy

* Corresponding author

Received  September 2021 Revised  January 2022 Published  August 2022 Early access  March 2022

Many problems arising in biology display a complex system dynamics at different scales of space and time. For this reason, multiscale mathematical models have attracted a great attention as they enable to take into account phenomena evolving at several characteristic lengths. However, they require advanced model reduction techniques to reduce the computational cost of solving all the scales.

In this work, we present a novel version of the Keller-Segel model of chemotaxis on embedded multiscale geometries, i.e., one-dimensional networks embedded in three-dimensional bulk domains. Applying a model reduction technique based on spatial averaging for geometrical order reduction, we reduce a fully three-dimensional Keller-Segel system to a coupled 3D-1D multiscale model. In the reduced model, the dynamics of the cellular population evolves on a one-dimensional network and its migration is influenced by a three-dimensional chemical signal evolving in the bulk domain. We propose the multiscale version of the Keller-Segel model as a realistic approach to describe the invasion of malignant cancer cells along the collagen fibers that constitute the extracellular matrix. Performing several numerical simulations, we investigate how the invasive abilities of the cells are affected by the topology of the network (i.e., matrix fibers orientation and alignment) as well as by three-dimensional spatial effects. We discuss these results in light of biological evidences.

Citation: Federica Bubba, Benoit Perthame, Daniele Cerroni, Pasquale Ciarletta, Paolo Zunino. A coupled 3D-1D multiscale Keller-Segel model of chemotaxis and its application to cancer invasion. Discrete and Continuous Dynamical Systems - S, 2022, 15 (8) : 2053-2086. doi: 10.3934/dcdss.2022044
References:
[1]

L. AlmeidaF. BubbaB. Perthame and C. Pouchol, Energy and implicit discretization of the Fokker-Planck and Keller-Segel type equations, Netw. Heterog. Media, 14 (2019), 23-41.  doi: 10.3934/nhm.2019002.

[2]

A. Anderson, A hybrid mathematical model of solid tumour invasion: The importance of cell adhesion, Math. Med. Biol., 22 (2005), 163-186. 

[3]

A. AndersonM. ChaplainE. NewmanR. Steele and A. Thompson, Mathematical modelling of tumour invasion and metastasis, Journal of Theoretical Medicine, 2 (2000), 129-154. 

[4]

N. BellomoA. BellouquidY. Tao and M. Winkler, Towards a mathematical theory of Keller-Segel models of pattern formation in biological tissues, Math. Models Methods Appl. Sci., 25 (2015), 1663-1763.  doi: 10.1142/S021820251550044X.

[5]

V. BitsouniM. A. J. Chaplain and R. Eftimie, Mathematical modelling of cancer invasion: The multiple roles of TGF-$\beta$ pathway on tumour proliferation and cell adhesion, Math. Models Methods Appl. Sci., 27 (2017), 1929-1962.  doi: 10.1142/S021820251750035X.

[6]

R. BorscheS. GöttlichA. Klar and P. Schillen, The scalar Keller-Segel model on networks, Math. Models Methods Appl. Sci., 24 (2014), 221-247.  doi: 10.1142/S0218202513400071.

[7]

G. Bretti and R. Natalini, On modeling maze solving ability of slime mold via a hyperbolic model of chemotaxis, Journal of Computational Methods in Sciences and Engineering, 18 (2018), 85-115. 

[8]

G. BrettiR. Natalini and M. Ribot, A hyperbolic model of chemotaxis on a network: A numerical study, ESAIM Math. Model. Numer. Anal., 48 (2014), 231-258.  doi: 10.1051/m2an/2013098.

[9]

A. ButtenschönT. HillenA. Gerisch and K. Painter, A space-jump derivation for non-local models of cell-cell adhesion and non-local chemotaxis, J. Math. Biol., 76 (2018), 429-456.  doi: 10.1007/s00285-017-1144-3.

[10]

L. Cattaneo and P. Zunino, A computational model of drug delivery through microcirculation to compare different tumor treatments, Int. J. Numer. Methods Biomed. Eng., 30 (2014), 1347-1371.  doi: 10.1002/cnm.2661.

[11]

M. A. J. ChaplainM. LachowiczZ. Szymanska and D. Wrzosek, Mathematical modelling of cancer invasion: The importance of cell-cell adhesion and cell-matrix adhesion, Math. Models Methods Appl. Sci., 21 (2011), 719-743.  doi: 10.1142/S0218202511005192.

[12]

M. A. J. Chaplain and G. Lolas, Mathematical modelling of cancer cell invasion of tissue: The role of the urokinase plasminogen activation system, Math. Models Methods Appl. Sci., 15 (2005), 1685-1734.  doi: 10.1142/S0218202505000947.

[13]

M. A. J. Chaplain and G. Lolas, Mathematical modelling of cancer cell invasion of tissue: Dynamic heterogeneity, Netw. Heterog. Media, 1 (2006), 399-439.  doi: 10.3934/nhm.2006.1.399.

[14]

A. Chauviére and L. Preziosi, Mathematical framework to model migration of cell population in extracellular matrix, Cell Mechanics: From Single Scale-Based Models to Multiscale Modeling, 2010.

[15]

A. Chauviére, L. Preziosi and T. Hillen, Modeling the motion of a cell population in the extracellular matrix, Discrete Contin. Dyn. Syst., (2007), 250–259.

[16]

C. D'Angelo, Multiscale modelling of metabolism and transport phenomena in living tissues, PhD Thesis, EPFL Lausanne, 2007.

[17]

C. D'Angelo, Finite element approximation of elliptic problems with Dirac measure terms in weighted spaces: Applications to one- and three-dimensional coupled problems, SIAM J. Numer. Anal., 50 (2012), 194-215.  doi: 10.1137/100813853.

[18]

C. D'Angelo and A. Quarteroni, On the coupling of 1d and 3d diffusion-reaction equations. Application to tissue perfusion problems, Math. Models Methods Appl. Sci., 18 (2008), 1481-1504.  doi: 10.1142/S0218202508003108.

[19]

D. Drasdo and S. Höhme, A single-cell-based model of tumor growth in vitro: Monolayers and spheroids, Physical Biology, 2 (2005), 133-147. 

[20]

J. Eble and S. Niland, The extracellular matrix in tumor progression and metastasis, Clinical & Experimental Metastasis, 36 (2019), 171-198. 

[21]

M. EgebladM. Rasch and V. Weaver, Dynamic interplay between the collagen scaffold and tumor evolution, Current Opinion in Cell Biology, 22 (2010), 697-706. 

[22]

Y. Epshteyn, Discontinuous Galerkin methods for the chemotaxis and haptotaxis models, J. Comput. Appl. Math., 224 (2009), 168-181.  doi: 10.1016/j.cam.2008.04.030.

[23]

C. FrantzK. Stewart and V. Weaver, The extracellular matrix at a glance, Journal of Cell Science, 123 (2010), 4195-4200. 

[24]

T. Hillen and H. G. Othmer, The diffusion limit of transport equations derived from velocity-jump processes, SIAM J. Appl. Math., 61 (2000), 751-775.  doi: 10.1137/S0036139999358167.

[25]

R. KayP. LangridgeD. Traynor and O. Hoeller, Changing directions in the study of chemotaxis, Nature Reviews Molecular Cell Biology, 9 (2008), 455. 

[26]

E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol., 26 (1970), 399-415.  doi: 10.1016/0022-5193(70)90092-5.

[27]

E. Keller and L. Segel, Model for chemotaxis, Journal of Theoretical Biology, 30 (1971), 225-234. 

[28]

F. Laurino and P. Zunino, Derivation and analysis of coupled PDEs on manifolds with high dimensionality gap arising from topological model reduction, ESAIM Math. Model. Numer. Anal., 53 (2019), 2047-2080.  doi: 10.1051/m2an/2019042.

[29]

X. H. LiC.-W. Shu and Y. Y. Yang, Local discontinuous Galerkin method for the Keller-Segel chemotaxis model, J. Sci. Comput., 73 (2017), 943-967.  doi: 10.1007/s10915-016-0354-y.

[30]

N. Loy and L. Preziosi, Kinetic models with non-local sensing determining cell polarization and speed according to independent cues, J. Math. Biol., 80 (2020), 373-421.  doi: 10.1007/s00285-019-01411-x.

[31]

G. MacDonaldJ. A. MackenzieM. Nolan and R. H. Insall, A computational method for the coupled solution of reaction-diffusion equations on evolving domains and manifolds: Application to a model of cell migration and chemotaxis, J. Comput. Phys., 309 (2016), 207-226.  doi: 10.1016/j.jcp.2015.12.038.

[32]

J. Mackenzie, M. Nolan and R. Insall, Local modulation of chemoattractant concentrations by single cells: Dissection using a bulk-surface computational model, Interface Focus, 6 (2016).

[33]

J. D. Murray, Mathematical Biology, vol. Ⅰ: An introduction, Springer, 2002.

[34]

H. G. Othmer and T. Hillen, The diffusion limit of transport equations Ⅱ: Chemotaxis equations, SIAM J. Appl. Math., 62 (2002), 1222-1250.  doi: 10.1137/S0036139900382772.

[35]

H. G. Othmer and C. Xue, The mathematical analysis of biological aggregation and dispersal: Progress, problems and perspectives, Dispersal, Individual Movement And Spatial Ecology, Lecture Notes in Math., 2071 (2013), 79–127. doi: 10.1007/978-3-642-35497-7_4.

[36]

K. J. Painter, Modelling cell migration strategies in the extracellular matrix, J. Math. Biol., 58 (2009), 511-543.  doi: 10.1007/s00285-008-0217-8.

[37]

K. J. Painter and T. Hillen, Volume-filling and quorum-sensing in models for chemosensitive movement, Can. Appl. Math. Q., 10 (2002), 501-543. 

[38]

L. Possenti, S. di Gregorio, F. M. Gerosa, et al., A computational model for microcirculation including Fahraeus-Lindqvist effect, plasma skimming and fluid exchange with the tissue interstitium, Int. J. Numer. Methods Biomed. Eng., 35 (2019), e3165, 27 pp. doi: 10.1002/cnm.3165.

[39]

A. Poulain, Scalar auxiliary variable finite element scheme for the parabolic-parabolic Keller-Segel model: Positivity preserving and energy stability, in Preparation, 2020.

[40]

L. Preziosi and M. Scianna, Mathematical models of the interaction of cells and cell aggregates with the extracellular matrix, Mathematical Models and Methods for Living Systems. Lecture Notes Mathematics, 2167 (2016), 131-210. 

[41]

A. QuarteroniA. Veneziani and C. Vergara, Geometric multiscale modeling of the cardiovascular system, between theory and practice, Computer Methods in Applied Mechanics and Engineering, 302 (2016), 193-252. 

[42]

I. Ramis-CondeM. A. J. Chaplain and A. R. A. Anderson, Mathematical modelling of cancer cell invasion of tissue, Math. Comput. Modelling, 47 (2008), 533-545.  doi: 10.1016/j.mcm.2007.02.034.

[43]

Y. Renard and K. Poulios, GetFEM: Automated FE modeling of multiphysics problems based on a generic weak form language, CM Trans. Math. Software, 47 (2021), Art. 4, 31 pp. doi: 10.1145/3412849.

[44]

H. L. RochaR. C. AlmeidaE. A. B. F. LimaA. C. M. ResendeJ. T. Oden and T. E. Yankeelov, A hybrid three-scale model of tumor growth, Math. Models Methods Appl. Sci., 28 (2018), 61-93.  doi: 10.1142/S0218202518500021.

[45]

E. RoussosJ. Condeelis and A. Patsialou, Chemotaxis in cancer, Nature Reviews Cancer, 11 (2011), 573. 

[46]

D. SchlüterI. Ramis-Conde and M. Chaplain, Computational modeling of single-cell migration: The leading role of extracellular matrix fibers, Biophysical Journal, 103 (2012), 1141-1151. 

[47]

N. SfakianakisA. Madzvamuse and A. Chaplain, A hybrid multiscale model for cancer invasion of the extracellular matrix, Multiscale Modeling & Simulation, 18 (2020), 824-850. 

[48]

P. Souplet and M. Winkler, Blow-up profiles for the parabolic-elliptic Keller-Segel system in dimensions $n\ge3$, Comm. Math. Phys., 367 (2019), 665-681.  doi: 10.1007/s00220-018-3238-1.

[49]

D. TrucuP. DomschkeA. Gerisch and M. Chaplain, Multiscale computational modelling and analysis of cancer invasion, Mathematical Models and Methods for Living Systems. Lecture Notes in Mathematics, 2167 (2016), 275-321. 

[50]

G. Wadhams and J. Armitage, Making sense of it all: Bacterial chemotaxis, Nature Reviews Molecular Cell Biology, 5 (2004).

[51]

E. Yekaterina and K. Alexander, New interior penalty discontinuous Galerkin methods for the Keller-Segel Chemotaxis model, SIAM Journal on Numerical Analysis, 47 (2008), 386-408. 

show all references

References:
[1]

L. AlmeidaF. BubbaB. Perthame and C. Pouchol, Energy and implicit discretization of the Fokker-Planck and Keller-Segel type equations, Netw. Heterog. Media, 14 (2019), 23-41.  doi: 10.3934/nhm.2019002.

[2]

A. Anderson, A hybrid mathematical model of solid tumour invasion: The importance of cell adhesion, Math. Med. Biol., 22 (2005), 163-186. 

[3]

A. AndersonM. ChaplainE. NewmanR. Steele and A. Thompson, Mathematical modelling of tumour invasion and metastasis, Journal of Theoretical Medicine, 2 (2000), 129-154. 

[4]

N. BellomoA. BellouquidY. Tao and M. Winkler, Towards a mathematical theory of Keller-Segel models of pattern formation in biological tissues, Math. Models Methods Appl. Sci., 25 (2015), 1663-1763.  doi: 10.1142/S021820251550044X.

[5]

V. BitsouniM. A. J. Chaplain and R. Eftimie, Mathematical modelling of cancer invasion: The multiple roles of TGF-$\beta$ pathway on tumour proliferation and cell adhesion, Math. Models Methods Appl. Sci., 27 (2017), 1929-1962.  doi: 10.1142/S021820251750035X.

[6]

R. BorscheS. GöttlichA. Klar and P. Schillen, The scalar Keller-Segel model on networks, Math. Models Methods Appl. Sci., 24 (2014), 221-247.  doi: 10.1142/S0218202513400071.

[7]

G. Bretti and R. Natalini, On modeling maze solving ability of slime mold via a hyperbolic model of chemotaxis, Journal of Computational Methods in Sciences and Engineering, 18 (2018), 85-115. 

[8]

G. BrettiR. Natalini and M. Ribot, A hyperbolic model of chemotaxis on a network: A numerical study, ESAIM Math. Model. Numer. Anal., 48 (2014), 231-258.  doi: 10.1051/m2an/2013098.

[9]

A. ButtenschönT. HillenA. Gerisch and K. Painter, A space-jump derivation for non-local models of cell-cell adhesion and non-local chemotaxis, J. Math. Biol., 76 (2018), 429-456.  doi: 10.1007/s00285-017-1144-3.

[10]

L. Cattaneo and P. Zunino, A computational model of drug delivery through microcirculation to compare different tumor treatments, Int. J. Numer. Methods Biomed. Eng., 30 (2014), 1347-1371.  doi: 10.1002/cnm.2661.

[11]

M. A. J. ChaplainM. LachowiczZ. Szymanska and D. Wrzosek, Mathematical modelling of cancer invasion: The importance of cell-cell adhesion and cell-matrix adhesion, Math. Models Methods Appl. Sci., 21 (2011), 719-743.  doi: 10.1142/S0218202511005192.

[12]

M. A. J. Chaplain and G. Lolas, Mathematical modelling of cancer cell invasion of tissue: The role of the urokinase plasminogen activation system, Math. Models Methods Appl. Sci., 15 (2005), 1685-1734.  doi: 10.1142/S0218202505000947.

[13]

M. A. J. Chaplain and G. Lolas, Mathematical modelling of cancer cell invasion of tissue: Dynamic heterogeneity, Netw. Heterog. Media, 1 (2006), 399-439.  doi: 10.3934/nhm.2006.1.399.

[14]

A. Chauviére and L. Preziosi, Mathematical framework to model migration of cell population in extracellular matrix, Cell Mechanics: From Single Scale-Based Models to Multiscale Modeling, 2010.

[15]

A. Chauviére, L. Preziosi and T. Hillen, Modeling the motion of a cell population in the extracellular matrix, Discrete Contin. Dyn. Syst., (2007), 250–259.

[16]

C. D'Angelo, Multiscale modelling of metabolism and transport phenomena in living tissues, PhD Thesis, EPFL Lausanne, 2007.

[17]

C. D'Angelo, Finite element approximation of elliptic problems with Dirac measure terms in weighted spaces: Applications to one- and three-dimensional coupled problems, SIAM J. Numer. Anal., 50 (2012), 194-215.  doi: 10.1137/100813853.

[18]

C. D'Angelo and A. Quarteroni, On the coupling of 1d and 3d diffusion-reaction equations. Application to tissue perfusion problems, Math. Models Methods Appl. Sci., 18 (2008), 1481-1504.  doi: 10.1142/S0218202508003108.

[19]

D. Drasdo and S. Höhme, A single-cell-based model of tumor growth in vitro: Monolayers and spheroids, Physical Biology, 2 (2005), 133-147. 

[20]

J. Eble and S. Niland, The extracellular matrix in tumor progression and metastasis, Clinical & Experimental Metastasis, 36 (2019), 171-198. 

[21]

M. EgebladM. Rasch and V. Weaver, Dynamic interplay between the collagen scaffold and tumor evolution, Current Opinion in Cell Biology, 22 (2010), 697-706. 

[22]

Y. Epshteyn, Discontinuous Galerkin methods for the chemotaxis and haptotaxis models, J. Comput. Appl. Math., 224 (2009), 168-181.  doi: 10.1016/j.cam.2008.04.030.

[23]

C. FrantzK. Stewart and V. Weaver, The extracellular matrix at a glance, Journal of Cell Science, 123 (2010), 4195-4200. 

[24]

T. Hillen and H. G. Othmer, The diffusion limit of transport equations derived from velocity-jump processes, SIAM J. Appl. Math., 61 (2000), 751-775.  doi: 10.1137/S0036139999358167.

[25]

R. KayP. LangridgeD. Traynor and O. Hoeller, Changing directions in the study of chemotaxis, Nature Reviews Molecular Cell Biology, 9 (2008), 455. 

[26]

E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol., 26 (1970), 399-415.  doi: 10.1016/0022-5193(70)90092-5.

[27]

E. Keller and L. Segel, Model for chemotaxis, Journal of Theoretical Biology, 30 (1971), 225-234. 

[28]

F. Laurino and P. Zunino, Derivation and analysis of coupled PDEs on manifolds with high dimensionality gap arising from topological model reduction, ESAIM Math. Model. Numer. Anal., 53 (2019), 2047-2080.  doi: 10.1051/m2an/2019042.

[29]

X. H. LiC.-W. Shu and Y. Y. Yang, Local discontinuous Galerkin method for the Keller-Segel chemotaxis model, J. Sci. Comput., 73 (2017), 943-967.  doi: 10.1007/s10915-016-0354-y.

[30]

N. Loy and L. Preziosi, Kinetic models with non-local sensing determining cell polarization and speed according to independent cues, J. Math. Biol., 80 (2020), 373-421.  doi: 10.1007/s00285-019-01411-x.

[31]

G. MacDonaldJ. A. MackenzieM. Nolan and R. H. Insall, A computational method for the coupled solution of reaction-diffusion equations on evolving domains and manifolds: Application to a model of cell migration and chemotaxis, J. Comput. Phys., 309 (2016), 207-226.  doi: 10.1016/j.jcp.2015.12.038.

[32]

J. Mackenzie, M. Nolan and R. Insall, Local modulation of chemoattractant concentrations by single cells: Dissection using a bulk-surface computational model, Interface Focus, 6 (2016).

[33]

J. D. Murray, Mathematical Biology, vol. Ⅰ: An introduction, Springer, 2002.

[34]

H. G. Othmer and T. Hillen, The diffusion limit of transport equations Ⅱ: Chemotaxis equations, SIAM J. Appl. Math., 62 (2002), 1222-1250.  doi: 10.1137/S0036139900382772.

[35]

H. G. Othmer and C. Xue, The mathematical analysis of biological aggregation and dispersal: Progress, problems and perspectives, Dispersal, Individual Movement And Spatial Ecology, Lecture Notes in Math., 2071 (2013), 79–127. doi: 10.1007/978-3-642-35497-7_4.

[36]

K. J. Painter, Modelling cell migration strategies in the extracellular matrix, J. Math. Biol., 58 (2009), 511-543.  doi: 10.1007/s00285-008-0217-8.

[37]

K. J. Painter and T. Hillen, Volume-filling and quorum-sensing in models for chemosensitive movement, Can. Appl. Math. Q., 10 (2002), 501-543. 

[38]

L. Possenti, S. di Gregorio, F. M. Gerosa, et al., A computational model for microcirculation including Fahraeus-Lindqvist effect, plasma skimming and fluid exchange with the tissue interstitium, Int. J. Numer. Methods Biomed. Eng., 35 (2019), e3165, 27 pp. doi: 10.1002/cnm.3165.

[39]

A. Poulain, Scalar auxiliary variable finite element scheme for the parabolic-parabolic Keller-Segel model: Positivity preserving and energy stability, in Preparation, 2020.

[40]

L. Preziosi and M. Scianna, Mathematical models of the interaction of cells and cell aggregates with the extracellular matrix, Mathematical Models and Methods for Living Systems. Lecture Notes Mathematics, 2167 (2016), 131-210. 

[41]

A. QuarteroniA. Veneziani and C. Vergara, Geometric multiscale modeling of the cardiovascular system, between theory and practice, Computer Methods in Applied Mechanics and Engineering, 302 (2016), 193-252. 

[42]

I. Ramis-CondeM. A. J. Chaplain and A. R. A. Anderson, Mathematical modelling of cancer cell invasion of tissue, Math. Comput. Modelling, 47 (2008), 533-545.  doi: 10.1016/j.mcm.2007.02.034.

[43]

Y. Renard and K. Poulios, GetFEM: Automated FE modeling of multiphysics problems based on a generic weak form language, CM Trans. Math. Software, 47 (2021), Art. 4, 31 pp. doi: 10.1145/3412849.

[44]

H. L. RochaR. C. AlmeidaE. A. B. F. LimaA. C. M. ResendeJ. T. Oden and T. E. Yankeelov, A hybrid three-scale model of tumor growth, Math. Models Methods Appl. Sci., 28 (2018), 61-93.  doi: 10.1142/S0218202518500021.

[45]

E. RoussosJ. Condeelis and A. Patsialou, Chemotaxis in cancer, Nature Reviews Cancer, 11 (2011), 573. 

[46]

D. SchlüterI. Ramis-Conde and M. Chaplain, Computational modeling of single-cell migration: The leading role of extracellular matrix fibers, Biophysical Journal, 103 (2012), 1141-1151. 

[47]

N. SfakianakisA. Madzvamuse and A. Chaplain, A hybrid multiscale model for cancer invasion of the extracellular matrix, Multiscale Modeling & Simulation, 18 (2020), 824-850. 

[48]

P. Souplet and M. Winkler, Blow-up profiles for the parabolic-elliptic Keller-Segel system in dimensions $n\ge3$, Comm. Math. Phys., 367 (2019), 665-681.  doi: 10.1007/s00220-018-3238-1.

[49]

D. TrucuP. DomschkeA. Gerisch and M. Chaplain, Multiscale computational modelling and analysis of cancer invasion, Mathematical Models and Methods for Living Systems. Lecture Notes in Mathematics, 2167 (2016), 275-321. 

[50]

G. Wadhams and J. Armitage, Making sense of it all: Bacterial chemotaxis, Nature Reviews Molecular Cell Biology, 5 (2004).

[51]

E. Yekaterina and K. Alexander, New interior penalty discontinuous Galerkin methods for the Keller-Segel Chemotaxis model, SIAM Journal on Numerical Analysis, 47 (2008), 386-408. 

Figure 1.  Outline of the unit cubic domain $ \Omega $ (in black) and the embedded single-branch network $ \Lambda $ (red line)
Figure 2.  Evolution of the cellular density $ U $ on a single-branch network and of the corresponding three-dimensional chemoattractant concentration $ c $ at four different times, $ t = 0.0, \, 0.08,\, 0.26 {\rm{\;and\;}} 1.2 $.
Figure 3.  Initial conditions and subsequent states of the evolution in time of the cellular density $ U $ and of the concentration of the chemoattractant $ c $ along the single-branch network, thus for $ s \in [0,0.8] $ (or, equivalently, for $ x \in [0.1,0.9] $).
Figure 4.  Evolution in time of the solution of the $ 3 $D-$ 1 $D Keller-Segel model, on a one-dimensional bifurcated network embedded in a unit three-dimensional domain, at four different times of the simulation: $ t = 0, \, 0.04, \, 0.16, \, 1 $
Figure 5.  (a)(c) Time evolution of three initial aggregates of cells on a symmetric network and the corresponding chemical concentration at three different times, $ t = 0, \, 0.04, {\rm{\;and\;}} 0.3 $. (d) Internal visualization of the chemoattractant concentration (on a different scale of values)
Figure 6.  Evolution of the cellular density and of the chemical concentration, solutions to the multiscale Keller-Segel model, at six different times
Figure 7.  Schematic representation of the metastatic cascade, adapted from [20]. Malignant tumor cells escape from the primary tumor site by breaching the extracellular barriers (such as the basement membrane) (1), take advantage of the collagen fibrils and networks to invade the tissue (2) and enter the blood circulation (3)
Figure 8.  Time evolution of the cellular density $ U $ (blue line) and the corresponding ECM density $ m $ (red line) on the one-dimensional single-branch network, for $ x \in [0.1,0.9] $ and at four different times $ t = 0, \, 0.12, \, 0.24,\, 1.2 $
Figure 9.  Solutions of the multiscale Keller-Segel-type model (20) in the single-branch test case at three different times. The evolution of the cellular density $ U $ on the one-dimensional segment is displayed together with the density of the extracellular matrix $ m $ (left panels) and with the MDEs concentration $ c $ (right panels)
Figure 10.  The time evolution of the cellular density $ U $ and the density of the extracellular matrix $ m $ for two different networks. The invasion is slowed down ((a)(b)) or sped up ((c)(d)) compared to the simulation in Figure 9
Figure 11.  Initial density $ m_0 $ of the extracellular matrix for $ x\in[0,1] $
Figure 12.  Evolution in time of the cellular density $ U $ and the corresponding density of the ECM $ m $, solutions of the multiscale Keller-Segel-type model (20), at six subsequent times of the simulations
Figure 13.  Evolution in time of the cellular density $ U $ and the corresponding density of the ECM $ m $, solutions of the multiscale Keller-Segel-type model 20. The invasion is more effective without the presence of transversal branches, compared to the solutions displayed in Figures 12
Figure 14.  Comparison of the evolution in time of the cellular density $ U $ for the network $ \mathcal{N}_1 $ (left panels) and $ \mathcal{N}_2 $ (right panels) at three subsequent times of the simulations
Figure 15.  Comparison of the evolution in time of the cellular density $ U $ on the three parallel branches $ \Lambda_1 $ (red line), $ \Lambda_2 $ (green line) and $ \Lambda_3 $ (blue line) for the networks $ \mathcal{N}_1 $ (left panels) and $ \mathcal{N}_2 $ (right panels)
[1]

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