February  2017, 14(1): 45-66. doi: 10.3934/mbe.2017004

On the mathematical modelling of tumor-induced angiogenesis

1. 

G. Millán Institute, Fluid Dynamics, Nanoscience and Industrial Mathematics, Universidad Carlos Ⅲ de Madrid, 28911 Leganés, Spain

2. 

ADAMSS, Universitá degli Studi di Milano, 20133 MILANO, Italy

* Corresponding author: Vincenzo Capasso

Received  November 23, 2015 Accepted  April 13, 2016 Published  October 2016

An angiogenic system is taken as an example of extremely complex ones in the field of Life Sciences, from both analytical and computational points of view, due to the strong coupling between the kinetic parameters of the relevant branching -growth -anastomosis stochastic processes of the capillary network, at the microscale, and the family of interacting underlying biochemical fields, at the macroscale. To reduce this complexity, for a conceptual stochastic model we have explored how to take advantage of the system intrinsic multiscale structure: one might describe the stochastic dynamics of the cells at the vessel tip at their natural microscale, whereas the dynamics of the underlying fields is given by a deterministic mean field approximation obtained by an averaging at a suitable mesoscale. But the outcomes of relevant numerical simulations show that the proposed model, in presence of anastomosis, is not self-averaging, so that the "propagation of chaos" assumption cannot be applied to obtain a deterministic mean field approximation. On the other hand we have shown that ensemble averages over many realizations of the stochastic system may better correspond to a deterministic reaction-diffusion system.

Citation: Luis L. Bonilla, Vincenzo Capasso, Mariano Alvaro, Manuel Carretero, Filippo Terragni. On the mathematical modelling of tumor-induced angiogenesis. Mathematical Biosciences & Engineering, 2017, 14 (1) : 45-66. doi: 10.3934/mbe.2017004
References:
[1]

L. AmbrosioV. Capasso and E. Villa, On the approximation of mean densities of random closed sets, Bernoulli, 15 (2009), 1222-1242. doi: 10.3150/09-BEJ186.

[2]

A. R. A. Anderson and M. A. J. Chaplain, Continuous and discrete mathematical models of tumour-induced angiogenesis, Bull. Math. Biol., 60 (1998), 857-900.

[3]

C. BirdwellA. Brasier and L. Taylor, Two-dimensional peptide mapping of fibronectin from bovine aortic endothelial cells and bovine plasma, Biochem. Biophys. Res. Commun., 97 (1980), 574-581. doi: 10.1016/0006-291X(80)90302-2.

[4]

L. L. BonillaV. CapassoM. Alvaro and M. Carretero, Hybrid modelling of tumor-induced angiogenesis, Physical Review E, 90 (2014), 062716.

[5] P. Bremaud, Point Processes and Queues. Martingale Dynamics, Springer-Verlag, New-York, 1981.
[6]

M. BurgerV. Capasso and D. Morale, On an aggregation model with long and short range interactions, Nonlinear Anal. Real World Appl., 8 (2007), 939-958. doi: 10.1016/j.nonrwa.2006.04.002.

[7]

M. BurgerV. Capasso and L. Pizzocchero, Mesoscale averaging of nucleation and growth models, Multiscale Modeling and Simulation, 5 (2006), 564-592. doi: 10.1137/050626120.

[8]

V. Capasso, Randomness and Geometric Structures in Biology in Pattern Formation in Morphogenesis. Problems and Mathematical Issues (eds. V. Capasso, M. Gromov, A. Harel-Bellan, N. Morozova and L. L. Pritchard), Springer, Heidelberg, 2013.

[9] V. Capasso and D. Bakstein, An Introduction to Continuous-time Stochastic Processes, 3 edition, Birkhäuser, Boston, 2015. doi: 10.1007/978-1-4939-2757-9.
[10]

V. Capasso and D. Morale, Stochastic modelling of tumour-induced angiogenesis, J. Math. Biol., 58 (2009), 219-233. doi: 10.1007/s00285-008-0193-z.

[11]

V. CapassoD. Morale and G. Facchetti, The Role of Stochasticity for a Model of Retinal Angiogenesis, IMA J. Appl. Math., 77 (2012), 729-747. doi: 10.1093/imamat/hxs050.

[12]

V. Capasso and E. Villa, On the geometric densities of random closed sets, Stoch. Anal. Appl., 26 (2008), 784-808. doi: 10.1080/07362990802128396.

[13]

P. F. Carmeliet, Angiogenesis in life, disease and medicine, Nature, 438 (2005), 932-936. doi: 10.1038/nature04478.

[14]

P. Carmeliet and R. K. Jain, Molecular mechanisms and clinical applications of angiogenesis, Nature, 473 (2011), 298-307. doi: 10.1038/nature10144.

[15]

P. Carmeliet and M. Tessier-Lavigne, Common mechanisms of nerve and blood vessel wiring, Nature, 436 (2005), 193-200. doi: 10.1038/nature03875.

[16]

A. Carpio and G. Duro, Well posedness of a kinetic model for angiogenesis, Nonlinear Analysis; Real World Applications, 30 (2016), 184-212. doi: 10.1016/j.nonrwa.2016.01.002.

[17]

N. Champagnat and S. Méléard, Invasion and adaptive evolution for individual-based spatially structured populations, J. Math. Biol., textbf55 (2007), 147-188. doi: 10.1007/s00285-007-0072-z.

[18]

M. Chaplain and A. Stuart, A model mechanism for the chemotactic response of endothelial cells to tumour angiogenesis factor, IMA J. Math. Appl. Med. Biol., 10 (1993), 149-168. doi: 10.1093/imammb/10.3.149.

[19]

M. CoradaL. ZanettaF. OrsenigoF. BreviarioM. G. LampugnaniS. BernasconiF. LiaoD. J. HicklinP. Bohlen and E. Dejana, A monoclonal antibody to vascular endothelialcadherin inhibits tumor angiogenesis without side effects on endothelial permeability, Blood, 100 (2002), 905-911. doi: 10.1182/blood.V100.3.905.

[20]

S. L. Cotter, V. Klika, L. Kimpton, S. Collins and A. E. P. Heazell, A stochastic model for early placental development J. R. Soc. Interface, 11 (2014), 20140149, Available from: http://rsif.royalsocietypublishing.org/content/11/97/20140149. doi: 10.1098/rsif.2014.0149.

[21]

R. F. Gariano and T. W. Gardner, Retinal angiogenesis in development and disease, Nature, 438 (2004), 960-966. doi: 10.1038/nature04482.

[22]

J. Folkman, Tumour angiogenesis, Adv. Cancer Res., 19 (1974), 331-358.

[23]

J. W. Gibbs, Elementary Principles of Statistical Mechanics Yale Bicentennial Publications, Scribner and Sons, New York, 1902. doi: 10.1017/CBO9780511686948.

[24]

H. A. HarringtonM. MaierL. NaidooN. Whitaker and P. G. Kevrekidis, A hybrid model for tumor-induced angiogenesis in the cornea in the presence of inhibitors, Mathematical and Computer Modelling, 46 (2007), 513-524. doi: 10.1016/j.mcm.2006.11.034.

[25]

R. K. Jain and P. F. Carmeliet, Vessels of death or life, Sci. Am., 285 (2001), 38-45. doi: 10.1038/scientificamerican1201-38.

[26] S. Karlin and H. M. Taylor, A Second Course in Stochastic Processes, Academic Press, New York, 1981.
[27]

N. V. MantzarisS. Webb and H. G. Othmer, Mathematical modeling of tumor-induced angiogenesis, J. Math. Biol., 49 (2004), 111-187. doi: 10.1007/s00285-003-0262-2.

[28]

D. MoraleV. Capasso and K. Ölschlaeger, An interacting particle system modelling aggregation behavior: From individuals to populations, J. Math. Biol., 50 (2005), 49-66. doi: 10.1007/s00285-004-0279-1.

[29]

K. Oelschläger, On the derivation of reaction-diffusion equations as limit dynamics of systems of moderately interacting stochastic processes, Probab. Theor. Relat. Fields, 82 (1989), 565-586. doi: 10.1007/BF00341284.

[30]

M. J. Plank and B. D. Sleeman, Lattice and non-lattice models of tumour angiogenesis, Bull. Math. Biol., 66 (2004), 1785-1819. doi: 10.1016/j.bulm.2004.04.001.

[31]

M. HubbardP. F. Jones and B. D. Sleeman, The foundations of a unified approach to mathematical modelling of angiogenesis, Int. J. Adv. Eng. Sci. and Appl. Math., 1 (2009), 43-52. doi: 10.1007/s12572-009-0004-9.

[32] P. E. Protter, Stochastic Integration and Differential Equations, Second Edition, Springer-Verlag, Heidelberg, 2004.
[33] G. G. Roussas, A Course in Mathematical Statistics, 2 edition, Academic Press, San Diego, CA, 1997.
[34]

M. SciannaL. Munaron and L. Preziosi, A multiscale hybrid approach for vasculogenesis and related potential blocking therapies, Prog. Biophys. Mol. Biol., 106 (2011), 450-462. doi: 10.1016/j.pbiomolbio.2011.01.004.

[35]

A. StéphanouS. R. McDougallA. R. A. Anderson and M. A. J. Chaplain, Mathematical modelling of the influence of blood rheological properties upon adaptative tumour-induced angiogenesis, Math. Comput. Modelling, 44 (2006), 96-123. doi: 10.1016/j.mcm.2004.07.021.

[36]

C. L. Stokes and D. A. Lauffenburger, Analysis of the roles of microvessel endothelial cell random motility and chemotaxis in angiogenesis, J. Theor. Biol., 152 (1991), 377-403. doi: 10.1016/S0022-5193(05)80201-2.

[37]

S. SunM. F. WheelerM. Obeyesekere and C. W. Patrick Jr., A deterministic model of growth factor-induced angiogenesis, Bull. Math. Biol., 67 (2005), 313-337. doi: 10.1016/j.bulm.2004.07.004.

[38]

S. SunM. F. WheelerM. Obeyesekere and C. W. Patrick Jr., A multiscale angiogenesis modeling using mixed finite element methods, Multiscale Model. Simul., 4 (2005), 1137-1167. doi: 10.1137/050624443.

[39]

K. R. SwansonR. C. RockneJ. ClaridgeM. A. ChaplainE. C. Alvord Jr and A. R. A. Anderson, Quantifying the role of angiogenesis in malignant progression of gliomas: In silico modeling integrates imaging and histology, Cancer Res., 71 (2011), 7366-7375. doi: 10.1158/0008-5472.CAN-11-1399.

[40]

A. S. Sznitman, Topics in propagation of chaos, in École d'Été de Probabilités de Saint-Flour XIX-1989 , Lecture Notes in Math., Springer, Berlin, 1464 (1991), 165-251. doi: 10.1007/BFb0085169.

[41]

F. TerragniM. CarreteroV. Capasso and L. L. Bonilla, Stochastic model of tumor-induced angiogenesis: Ensemble averages and deterministic equations, Physical Review E, 93 (2016), 022413. doi: 10.1103/PhysRevE.93.022413.

[42]

S. Tong and F. Yuan, Numerical simulations of angiogenesis in the cornea, Microvascular Research, 61 (2001), 14-27. doi: 10.1006/mvre.2000.2282.

show all references

References:
[1]

L. AmbrosioV. Capasso and E. Villa, On the approximation of mean densities of random closed sets, Bernoulli, 15 (2009), 1222-1242. doi: 10.3150/09-BEJ186.

[2]

A. R. A. Anderson and M. A. J. Chaplain, Continuous and discrete mathematical models of tumour-induced angiogenesis, Bull. Math. Biol., 60 (1998), 857-900.

[3]

C. BirdwellA. Brasier and L. Taylor, Two-dimensional peptide mapping of fibronectin from bovine aortic endothelial cells and bovine plasma, Biochem. Biophys. Res. Commun., 97 (1980), 574-581. doi: 10.1016/0006-291X(80)90302-2.

[4]

L. L. BonillaV. CapassoM. Alvaro and M. Carretero, Hybrid modelling of tumor-induced angiogenesis, Physical Review E, 90 (2014), 062716.

[5] P. Bremaud, Point Processes and Queues. Martingale Dynamics, Springer-Verlag, New-York, 1981.
[6]

M. BurgerV. Capasso and D. Morale, On an aggregation model with long and short range interactions, Nonlinear Anal. Real World Appl., 8 (2007), 939-958. doi: 10.1016/j.nonrwa.2006.04.002.

[7]

M. BurgerV. Capasso and L. Pizzocchero, Mesoscale averaging of nucleation and growth models, Multiscale Modeling and Simulation, 5 (2006), 564-592. doi: 10.1137/050626120.

[8]

V. Capasso, Randomness and Geometric Structures in Biology in Pattern Formation in Morphogenesis. Problems and Mathematical Issues (eds. V. Capasso, M. Gromov, A. Harel-Bellan, N. Morozova and L. L. Pritchard), Springer, Heidelberg, 2013.

[9] V. Capasso and D. Bakstein, An Introduction to Continuous-time Stochastic Processes, 3 edition, Birkhäuser, Boston, 2015. doi: 10.1007/978-1-4939-2757-9.
[10]

V. Capasso and D. Morale, Stochastic modelling of tumour-induced angiogenesis, J. Math. Biol., 58 (2009), 219-233. doi: 10.1007/s00285-008-0193-z.

[11]

V. CapassoD. Morale and G. Facchetti, The Role of Stochasticity for a Model of Retinal Angiogenesis, IMA J. Appl. Math., 77 (2012), 729-747. doi: 10.1093/imamat/hxs050.

[12]

V. Capasso and E. Villa, On the geometric densities of random closed sets, Stoch. Anal. Appl., 26 (2008), 784-808. doi: 10.1080/07362990802128396.

[13]

P. F. Carmeliet, Angiogenesis in life, disease and medicine, Nature, 438 (2005), 932-936. doi: 10.1038/nature04478.

[14]

P. Carmeliet and R. K. Jain, Molecular mechanisms and clinical applications of angiogenesis, Nature, 473 (2011), 298-307. doi: 10.1038/nature10144.

[15]

P. Carmeliet and M. Tessier-Lavigne, Common mechanisms of nerve and blood vessel wiring, Nature, 436 (2005), 193-200. doi: 10.1038/nature03875.

[16]

A. Carpio and G. Duro, Well posedness of a kinetic model for angiogenesis, Nonlinear Analysis; Real World Applications, 30 (2016), 184-212. doi: 10.1016/j.nonrwa.2016.01.002.

[17]

N. Champagnat and S. Méléard, Invasion and adaptive evolution for individual-based spatially structured populations, J. Math. Biol., textbf55 (2007), 147-188. doi: 10.1007/s00285-007-0072-z.

[18]

M. Chaplain and A. Stuart, A model mechanism for the chemotactic response of endothelial cells to tumour angiogenesis factor, IMA J. Math. Appl. Med. Biol., 10 (1993), 149-168. doi: 10.1093/imammb/10.3.149.

[19]

M. CoradaL. ZanettaF. OrsenigoF. BreviarioM. G. LampugnaniS. BernasconiF. LiaoD. J. HicklinP. Bohlen and E. Dejana, A monoclonal antibody to vascular endothelialcadherin inhibits tumor angiogenesis without side effects on endothelial permeability, Blood, 100 (2002), 905-911. doi: 10.1182/blood.V100.3.905.

[20]

S. L. Cotter, V. Klika, L. Kimpton, S. Collins and A. E. P. Heazell, A stochastic model for early placental development J. R. Soc. Interface, 11 (2014), 20140149, Available from: http://rsif.royalsocietypublishing.org/content/11/97/20140149. doi: 10.1098/rsif.2014.0149.

[21]

R. F. Gariano and T. W. Gardner, Retinal angiogenesis in development and disease, Nature, 438 (2004), 960-966. doi: 10.1038/nature04482.

[22]

J. Folkman, Tumour angiogenesis, Adv. Cancer Res., 19 (1974), 331-358.

[23]

J. W. Gibbs, Elementary Principles of Statistical Mechanics Yale Bicentennial Publications, Scribner and Sons, New York, 1902. doi: 10.1017/CBO9780511686948.

[24]

H. A. HarringtonM. MaierL. NaidooN. Whitaker and P. G. Kevrekidis, A hybrid model for tumor-induced angiogenesis in the cornea in the presence of inhibitors, Mathematical and Computer Modelling, 46 (2007), 513-524. doi: 10.1016/j.mcm.2006.11.034.

[25]

R. K. Jain and P. F. Carmeliet, Vessels of death or life, Sci. Am., 285 (2001), 38-45. doi: 10.1038/scientificamerican1201-38.

[26] S. Karlin and H. M. Taylor, A Second Course in Stochastic Processes, Academic Press, New York, 1981.
[27]

N. V. MantzarisS. Webb and H. G. Othmer, Mathematical modeling of tumor-induced angiogenesis, J. Math. Biol., 49 (2004), 111-187. doi: 10.1007/s00285-003-0262-2.

[28]

D. MoraleV. Capasso and K. Ölschlaeger, An interacting particle system modelling aggregation behavior: From individuals to populations, J. Math. Biol., 50 (2005), 49-66. doi: 10.1007/s00285-004-0279-1.

[29]

K. Oelschläger, On the derivation of reaction-diffusion equations as limit dynamics of systems of moderately interacting stochastic processes, Probab. Theor. Relat. Fields, 82 (1989), 565-586. doi: 10.1007/BF00341284.

[30]

M. J. Plank and B. D. Sleeman, Lattice and non-lattice models of tumour angiogenesis, Bull. Math. Biol., 66 (2004), 1785-1819. doi: 10.1016/j.bulm.2004.04.001.

[31]

M. HubbardP. F. Jones and B. D. Sleeman, The foundations of a unified approach to mathematical modelling of angiogenesis, Int. J. Adv. Eng. Sci. and Appl. Math., 1 (2009), 43-52. doi: 10.1007/s12572-009-0004-9.

[32] P. E. Protter, Stochastic Integration and Differential Equations, Second Edition, Springer-Verlag, Heidelberg, 2004.
[33] G. G. Roussas, A Course in Mathematical Statistics, 2 edition, Academic Press, San Diego, CA, 1997.
[34]

M. SciannaL. Munaron and L. Preziosi, A multiscale hybrid approach for vasculogenesis and related potential blocking therapies, Prog. Biophys. Mol. Biol., 106 (2011), 450-462. doi: 10.1016/j.pbiomolbio.2011.01.004.

[35]

A. StéphanouS. R. McDougallA. R. A. Anderson and M. A. J. Chaplain, Mathematical modelling of the influence of blood rheological properties upon adaptative tumour-induced angiogenesis, Math. Comput. Modelling, 44 (2006), 96-123. doi: 10.1016/j.mcm.2004.07.021.

[36]

C. L. Stokes and D. A. Lauffenburger, Analysis of the roles of microvessel endothelial cell random motility and chemotaxis in angiogenesis, J. Theor. Biol., 152 (1991), 377-403. doi: 10.1016/S0022-5193(05)80201-2.

[37]

S. SunM. F. WheelerM. Obeyesekere and C. W. Patrick Jr., A deterministic model of growth factor-induced angiogenesis, Bull. Math. Biol., 67 (2005), 313-337. doi: 10.1016/j.bulm.2004.07.004.

[38]

S. SunM. F. WheelerM. Obeyesekere and C. W. Patrick Jr., A multiscale angiogenesis modeling using mixed finite element methods, Multiscale Model. Simul., 4 (2005), 1137-1167. doi: 10.1137/050624443.

[39]

K. R. SwansonR. C. RockneJ. ClaridgeM. A. ChaplainE. C. Alvord Jr and A. R. A. Anderson, Quantifying the role of angiogenesis in malignant progression of gliomas: In silico modeling integrates imaging and histology, Cancer Res., 71 (2011), 7366-7375. doi: 10.1158/0008-5472.CAN-11-1399.

[40]

A. S. Sznitman, Topics in propagation of chaos, in École d'Été de Probabilités de Saint-Flour XIX-1989 , Lecture Notes in Math., Springer, Berlin, 1464 (1991), 165-251. doi: 10.1007/BFb0085169.

[41]

F. TerragniM. CarreteroV. Capasso and L. L. Bonilla, Stochastic model of tumor-induced angiogenesis: Ensemble averages and deterministic equations, Physical Review E, 93 (2016), 022413. doi: 10.1103/PhysRevE.93.022413.

[42]

S. Tong and F. Yuan, Numerical simulations of angiogenesis in the cornea, Microvascular Research, 61 (2001), 14-27. doi: 10.1006/mvre.2000.2282.

Figure 1.  Upper panels: Snapshots of the vessel network inside a central square of side $L$ at three different times. The level curves of the TAF density $C(t,\mathbf{x})$ are also depicted. Middle panels: Density plots of the TAF density at the same times as in the upper panels showing its consumption as the vessel tips advance. Lower panels: Surface plots of the $x$-component of the chemotactic force at the same times showing how it pushes the tips toward the tumor
Figure 2.  Density plots of the marginal tip density $\tilde{p}(t,x,y)$ calculated from (36) with $\mathcal{N}=50$ replicas for the initial time and the same times as in Figure 1. The panels show how tips are created at $x=0$ and march toward the tumor at $x=L$
Figure 3.  Density plots of the marginal tip density calculated from the deterministic description for the same times as in Figure 2
Figure 4.  Comparison of the marginal tip density at the $x$ axis, $\tilde{p}(t,x,y=0)$, as calculated from the deterministic description (upper panels) and from ensemble averages over 50 replicas of the stochastic process (lower panels)
Table 1.  Dimensionless parameters. $\tilde{v}_0=40 \mu$m/h is a typical cell velocity
$\delta$ $\beta$ $A$ $\Gamma$ $\Gamma_1$ $\kappa$ $\chi$
$\frac{d_1C_R}{\tilde{v}_0^2}$ $\frac{kL}{\tilde{v}_0}$ $\frac{\alpha_1L}{\tilde{v}_0^3}$ $\frac{\gamma}{\tilde{v}_0^2}$ $\gamma_1C_R$ $\frac{d_2}{\tilde{v}_0 L}$ $\frac{\eta}{L}$
1.5 5.88 22.42 0.145 1 0.0045 0.002
$\delta$ $\beta$ $A$ $\Gamma$ $\Gamma_1$ $\kappa$ $\chi$
$\frac{d_1C_R}{\tilde{v}_0^2}$ $\frac{kL}{\tilde{v}_0}$ $\frac{\alpha_1L}{\tilde{v}_0^3}$ $\frac{\gamma}{\tilde{v}_0^2}$ $\gamma_1C_R$ $\frac{d_2}{\tilde{v}_0 L}$ $\frac{\eta}{L}$
1.5 5.88 22.42 0.145 1 0.0045 0.002
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