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 & 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.  doi: 10.1016/S0377-0427(00)00470-2.  Google Scholar

[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.  doi: 10.1016/j.mbs.2008.11.008.  Google Scholar

[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.  doi: 10.1007/s10269-011-2028-6.  Google Scholar

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

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S. Benzekry, Modélisation et Analyse Mathématique de Thérapies Anti-cancéreuses Pour Les Cancers Métastatiques,, Ph.D thesis, (2011).   Google Scholar

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

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

[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.  doi: 10.1090/S0025-5718-1982-0658219-8.  Google Scholar

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

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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.  doi: 10.3934/dcdsb.2009.12.731.  Google Scholar

[11]

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

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

[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.   Google Scholar

[14]

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

[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).  doi: 10.1158/0008-5472.CAN-14-0721.  Google Scholar

[16]

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

[17]

M. Ianelli, Mathematical Theory of Age-Structured Population Dynamics,, Applied Mathematical Monographs, (1995).   Google Scholar

[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.  doi: 10.1006/jtbi.2000.1075.  Google Scholar

[19]

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

[20]

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

[21]

B. Perthame, Transport Equations in Biology,, Frontiers in Mathematics, (2007).   Google Scholar

[22]

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

[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,, , ().   Google Scholar

[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, (2011).   Google Scholar

[25]

F. Verga, Modélisation Mathématique de Processus Métastatiques,, Ph.D thesis, (2010).   Google Scholar

[26]

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

[27]

T. E. Wheldon, Mathematical Models in Cancer Reseach,, Medical Science Series, (1988).   Google Scholar

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.  doi: 10.1016/S0377-0427(00)00470-2.  Google Scholar

[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.  doi: 10.1016/j.mbs.2008.11.008.  Google Scholar

[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.  doi: 10.1007/s10269-011-2028-6.  Google Scholar

[4]

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

[5]

S. Benzekry, Modélisation et Analyse Mathématique de Thérapies Anti-cancéreuses Pour Les Cancers Métastatiques,, Ph.D thesis, (2011).   Google Scholar

[6]

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

[7]

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

[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.  doi: 10.1090/S0025-5718-1982-0658219-8.  Google Scholar

[9]

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

[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.  doi: 10.3934/dcdsb.2009.12.731.  Google Scholar

[11]

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

[12]

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

[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.   Google Scholar

[14]

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

[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).  doi: 10.1158/0008-5472.CAN-14-0721.  Google Scholar

[16]

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

[17]

M. Ianelli, Mathematical Theory of Age-Structured Population Dynamics,, Applied Mathematical Monographs, (1995).   Google Scholar

[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.  doi: 10.1006/jtbi.2000.1075.  Google Scholar

[19]

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

[20]

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

[21]

B. Perthame, Transport Equations in Biology,, Frontiers in Mathematics, (2007).   Google Scholar

[22]

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

[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,, , ().   Google Scholar

[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, (2011).   Google Scholar

[25]

F. Verga, Modélisation Mathématique de Processus Métastatiques,, Ph.D thesis, (2010).   Google Scholar

[26]

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

[27]

T. E. Wheldon, Mathematical Models in Cancer Reseach,, Medical Science Series, (1988).   Google Scholar

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