March  2015, 20(2): 445-467. doi: 10.3934/dcdsb.2015.20.445

Efficient resolution of metastatic tumor growth models by reformulation into integral equations

1. 

Aix Marseille Université, CNRS, Centrale Marseille, I2M, UMR 7373, 13453 Marseille, France

Received  January 2014 Revised  October 2014 Published  January 2015

The McKendrick/Von Foerster equation is a transport equation with a non-local boundary condition that appears frequently in structured population models. A variant of this equation with a size structure has been proposed as a metastatic growth model by Iwata et al.
    Here we will show how a family of metastatic models with 1D or 2D structuring variables, based on the Iwata model, can be reformulated into an integral equation counterpart, a Volterra equation of convolution type, for which a rich numerical and analytical theory exists. Furthermore, we will point out the potential of this reformulation by addressing questions coming up in the modelling of metastatic tumour growth. We will show how this approach permits to reduce the computational cost of the numerical resolution and to prove structural identifiability.
Citation: Niklas Hartung. Efficient resolution of metastatic tumor growth models by reformulation into integral equations. Discrete and Continuous Dynamical Systems - B, 2015, 20 (2) : 445-467. doi: 10.3934/dcdsb.2015.20.445
References:
[1]

C. H. T. Baker, A perspective on the numerical treatment of Volterra equations, J Comput Appl Math, 125 (2000), 217-249. doi: 10.1016/S0377-0427(00)00470-2.

[2]

D. Barbolosi, F. Verga, A. Benabdallah and F. Hubert, Mathematical and numerical analysis for a model of growing metastatic tumors, Math Biosci, 218 (2009), 1-14. doi: 10.1016/j.mbs.2008.11.008.

[3]

D. Barbolosi, F. Verga, B. You, A. Benabdallah, F. Hubert, C. Mercier, J. Ciccolini and C. Faivre, Modélisation du risque d'évolution métastatique chez les patients supposés avoir une maladie localisée, Oncologie, 13 (2011), 528-533. doi: 10.1007/s10269-011-2028-6.

[4]

S. Benzekry, Mathematical analysis of a two-dimensional population model of metastatic growth including angiogenesis, J Evol Equ, 11 (2011), 187-213. doi: 10.1007/s00028-010-0088-5.

[5]

S. Benzekry, Modélisation et Analyse Mathématique de Thérapies Anti-cancéreuses Pour Les Cancers Métastatiques, Ph.D thesis, Aix-Marseille Université, 2011.

[6]

S. Benzekry, Mathematical and numerical analysis of a model for anti-angiogenic therapy in metastatic cancers, ESAIM-Math Model Num, 46 (2012), 207-237. doi: 10.1051/m2an/2011041.

[7]

S. Benzekry, A. Gandolfi and P. Hahnfeldt, Global Dormancy of Metastases due to Systemic Inhibition of Angiogenesis, PLoS One, 9 (2014), e84249. doi: 10.1371/journal.pone.0084249.

[8]

H. Brunner, E. Hairer and S. P. Nøorsett, Runge-Kutta theory for Volterra integral equations of the second kind, Math Comp, 39 (1982), 147-163. doi: 10.1090/S0025-5718-1982-0658219-8.

[9]

L. C. Chaffer and R. A. Weinberg, A perspective on cancer cell metastasis, Science, 331 (2011), 1559-1564. doi: 10.1126/science.1203543.

[10]

A. Devys, T. Goudon and P. Lafitte, A model describing the growth and the size distribution of multiple metastatic tumors, Discret Contin Dyn S, 12 (2009), 731-767. doi: 10.3934/dcdsb.2009.12.731.

[11]

G. P. Gupta and J. Massagué, Cancer metastasis: Building a framework, Cell, 127 (2006), 679-695. doi: 10.1016/j.cell.2006.11.001.

[12]

M. Gyllenberg and G. F. Webb, Quiescence as an explanation of Gompertzian tumor growth, Growth Develop Aging, 53 (1989), 25-33.

[13]

P. Hahnfeldt, D. Panigrahy, J. Folkman and L. Hlatky, Tumor development under angiogenic signaling: A dynamical theory of tumor growth, response and postvascular dormancy, Cancer Res, 59 (1999), 4770-4775.

[14]

E. Hairer, C. Lubich and M. Schlichte, Fast numerical solution of nonlinear Volterra convolution equations, J Sci Stat Comp, 5 (1985), 532-541. doi: 10.1137/0906037.

[15]

N. Hartung, S. Mollard, D. Barbolosi, A. Benabdallah, G. Chapuisat, J. Ciccolini, C. Faivre, S. Giacometti, G. Henry, A. Iliadis and F. Hubert, Mathematical modeling of tumor growth and metastatic spreading: Validation in tumor-bearing mice, Cancer Res, 15 (2014), p6397. doi: 10.1158/0008-5472.CAN-14-0721.

[16]

V. Haustein and U. Schumacher, A dynamic model for tumour growth and metastasis formation, J Clin Bioinforma, 2 (2012), p11. doi: 10.1186/2043-9113-2-11.

[17]

M. Ianelli, Mathematical Theory of Age-Structured Population Dynamics, Applied Mathematical Monographs, C.N.R.I., Giardini Editori e Stampatori, Pisa, 1995.

[18]

K. Iwata, K. Kawasaki and N. Shigesada, A Dynamical Model for the Growth and Size Distribution of Multiple Metastatic Tumors, J Theor Biol, 203 (2000), 177-186. doi: 10.1006/jtbi.2000.1075.

[19]

T. Lalescu, Introduction À la Théorie Des Équations Intégrales. Avec Une Préface de É. Picard, A. Hermann et Fils, Paris, 1912.

[20]

A. G. McKendrick, Applications of mathematics to medical problems, Proc Edinburgh Math Soc, 44 (1926), 98-130.

[21]

B. Perthame, Transport Equations in Biology, Frontiers in Mathematics, Birkhäuser, Basel, 2007.

[22]

P. Pouzet, Étude en vue de leur traitement numérique des équations intégrales de type Volterra, Rev Franç Traitement Information Chiffres, 6 (1963), 79-112.

[23]

J. G. Scott, P. Gerlee, D. Basanta, A. G. Fletcher, P. K. Maini and A. R. A. Anderson, Mathematical modelling of the metastatic process, preprint, arXiv:1305.4622.

[24]

A. Stein, D. DeWoskin, M. Higley, K. Lemoi, B. Owens, A. Rahman, H. Rotstein, D. Rumschitzki, S. Swaminathan, M. Tanzy, O. Varfolomiyev, T. Witelski and V.Zubekov, Dynamic Models of Metastatic Tumor Growth, Final Report of the 27th Annual Workshop on Mathematical Problems in Industry, New Jersey Institute of Technology, 2011.

[25]

F. Verga, Modélisation Mathématique de Processus Métastatiques, Ph.D thesis, Aix-Marseille Université, 2010.

[26]

H. Von Foerster, Some remarks on changing populations, in The Kinetics of Cell Proliferation, (1959), 382-407.

[27]

T. E. Wheldon, Mathematical Models in Cancer Reseach, Medical Science Series, Adam Hilger, Bristol and Philadelphia, 1988.

show all references

References:
[1]

C. H. T. Baker, A perspective on the numerical treatment of Volterra equations, J Comput Appl Math, 125 (2000), 217-249. doi: 10.1016/S0377-0427(00)00470-2.

[2]

D. Barbolosi, F. Verga, A. Benabdallah and F. Hubert, Mathematical and numerical analysis for a model of growing metastatic tumors, Math Biosci, 218 (2009), 1-14. doi: 10.1016/j.mbs.2008.11.008.

[3]

D. Barbolosi, F. Verga, B. You, A. Benabdallah, F. Hubert, C. Mercier, J. Ciccolini and C. Faivre, Modélisation du risque d'évolution métastatique chez les patients supposés avoir une maladie localisée, Oncologie, 13 (2011), 528-533. doi: 10.1007/s10269-011-2028-6.

[4]

S. Benzekry, Mathematical analysis of a two-dimensional population model of metastatic growth including angiogenesis, J Evol Equ, 11 (2011), 187-213. doi: 10.1007/s00028-010-0088-5.

[5]

S. Benzekry, Modélisation et Analyse Mathématique de Thérapies Anti-cancéreuses Pour Les Cancers Métastatiques, Ph.D thesis, Aix-Marseille Université, 2011.

[6]

S. Benzekry, Mathematical and numerical analysis of a model for anti-angiogenic therapy in metastatic cancers, ESAIM-Math Model Num, 46 (2012), 207-237. doi: 10.1051/m2an/2011041.

[7]

S. Benzekry, A. Gandolfi and P. Hahnfeldt, Global Dormancy of Metastases due to Systemic Inhibition of Angiogenesis, PLoS One, 9 (2014), e84249. doi: 10.1371/journal.pone.0084249.

[8]

H. Brunner, E. Hairer and S. P. Nøorsett, Runge-Kutta theory for Volterra integral equations of the second kind, Math Comp, 39 (1982), 147-163. doi: 10.1090/S0025-5718-1982-0658219-8.

[9]

L. C. Chaffer and R. A. Weinberg, A perspective on cancer cell metastasis, Science, 331 (2011), 1559-1564. doi: 10.1126/science.1203543.

[10]

A. Devys, T. Goudon and P. Lafitte, A model describing the growth and the size distribution of multiple metastatic tumors, Discret Contin Dyn S, 12 (2009), 731-767. doi: 10.3934/dcdsb.2009.12.731.

[11]

G. P. Gupta and J. Massagué, Cancer metastasis: Building a framework, Cell, 127 (2006), 679-695. doi: 10.1016/j.cell.2006.11.001.

[12]

M. Gyllenberg and G. F. Webb, Quiescence as an explanation of Gompertzian tumor growth, Growth Develop Aging, 53 (1989), 25-33.

[13]

P. Hahnfeldt, D. Panigrahy, J. Folkman and L. Hlatky, Tumor development under angiogenic signaling: A dynamical theory of tumor growth, response and postvascular dormancy, Cancer Res, 59 (1999), 4770-4775.

[14]

E. Hairer, C. Lubich and M. Schlichte, Fast numerical solution of nonlinear Volterra convolution equations, J Sci Stat Comp, 5 (1985), 532-541. doi: 10.1137/0906037.

[15]

N. Hartung, S. Mollard, D. Barbolosi, A. Benabdallah, G. Chapuisat, J. Ciccolini, C. Faivre, S. Giacometti, G. Henry, A. Iliadis and F. Hubert, Mathematical modeling of tumor growth and metastatic spreading: Validation in tumor-bearing mice, Cancer Res, 15 (2014), p6397. doi: 10.1158/0008-5472.CAN-14-0721.

[16]

V. Haustein and U. Schumacher, A dynamic model for tumour growth and metastasis formation, J Clin Bioinforma, 2 (2012), p11. doi: 10.1186/2043-9113-2-11.

[17]

M. Ianelli, Mathematical Theory of Age-Structured Population Dynamics, Applied Mathematical Monographs, C.N.R.I., Giardini Editori e Stampatori, Pisa, 1995.

[18]

K. Iwata, K. Kawasaki and N. Shigesada, A Dynamical Model for the Growth and Size Distribution of Multiple Metastatic Tumors, J Theor Biol, 203 (2000), 177-186. doi: 10.1006/jtbi.2000.1075.

[19]

T. Lalescu, Introduction À la Théorie Des Équations Intégrales. Avec Une Préface de É. Picard, A. Hermann et Fils, Paris, 1912.

[20]

A. G. McKendrick, Applications of mathematics to medical problems, Proc Edinburgh Math Soc, 44 (1926), 98-130.

[21]

B. Perthame, Transport Equations in Biology, Frontiers in Mathematics, Birkhäuser, Basel, 2007.

[22]

P. Pouzet, Étude en vue de leur traitement numérique des équations intégrales de type Volterra, Rev Franç Traitement Information Chiffres, 6 (1963), 79-112.

[23]

J. G. Scott, P. Gerlee, D. Basanta, A. G. Fletcher, P. K. Maini and A. R. A. Anderson, Mathematical modelling of the metastatic process, preprint, arXiv:1305.4622.

[24]

A. Stein, D. DeWoskin, M. Higley, K. Lemoi, B. Owens, A. Rahman, H. Rotstein, D. Rumschitzki, S. Swaminathan, M. Tanzy, O. Varfolomiyev, T. Witelski and V.Zubekov, Dynamic Models of Metastatic Tumor Growth, Final Report of the 27th Annual Workshop on Mathematical Problems in Industry, New Jersey Institute of Technology, 2011.

[25]

F. Verga, Modélisation Mathématique de Processus Métastatiques, Ph.D thesis, Aix-Marseille Université, 2010.

[26]

H. Von Foerster, Some remarks on changing populations, in The Kinetics of Cell Proliferation, (1959), 382-407.

[27]

T. E. Wheldon, Mathematical Models in Cancer Reseach, Medical Science Series, Adam Hilger, Bristol and Philadelphia, 1988.

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