Mathematical and numerical analysis of a model for anti-angiogenic therapy in metastatic cancers
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 46 (2012) no. 2, p. 207-237

We introduce a phenomenological model for anti-angiogenic therapy in the treatment of metastatic cancers. It is a structured transport equation with a nonlocal boundary condition describing the evolution of the density of metastases that we analyze first at the continuous level. We present the numerical analysis of a lagrangian scheme based on the characteristics whose convergence establishes existence of solutions. Then we prove an error estimate and use the model to perform interesting simulations in view of clinical applications.

DOI : https://doi.org/10.1051/m2an/2011041
Classification:  35F16,  65M25,  92C50
Keywords: anticancer therapy modelling, angiogenesis, structured population dynamics, lagrangian scheme
@article{M2AN_2012__46_2_207_0,
     author = {Benzekry, S\'ebastien},
     title = {Mathematical and numerical analysis of a model for anti-angiogenic therapy in metastatic cancers},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {46},
     number = {2},
     year = {2012},
     pages = {207-237},
     doi = {10.1051/m2an/2011041},
     zbl = {1273.92025},
     mrnumber = {2855641},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2012__46_2_207_0}
}
Benzekry, Sébastien. Mathematical and numerical analysis of a model for anti-angiogenic therapy in metastatic cancers. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 46 (2012) no. 2, pp. 207-237. doi : 10.1051/m2an/2011041. http://www.numdam.org/item/M2AN_2012__46_2_207_0/

[1] O. Angulo and J.C. Lopez-Marcos, Numerical schemes for size-structured population equations. Math. Biosci. 157 (1999) 169-188. | MR 1686473

[2] D. Barbolosi, A. Benabdallah, F. Hubert and F. Verga, Mathematical and numerical analysis for a model of growing metastatic tumours. Math. Biosci. 218 (2009) 1-14. | MR 2501837 | Zbl 1157.92020

[3] D. Barbolosi, C. Faivre and S. Benzekry, Mathematical modeling of MTD and metronomic temozolomide, 2nd Workshop on Metronomic Anti-Angiogenic Chemotherapy in Paediatric Oncology (2010).

[4] C. Bardos, Problèmes aux limites pour les équations aux dérivées partielles du premier ordre à coefficients réels; théorèmes d'approximation; application à l'équation de transport. Ann. Sci. Éc. Norm. Supér. 3 (1970) 185-233. | Numdam | MR 274925 | Zbl 0202.36903

[5] R. Beals and V. Protopopescu, Abstract time-dependent transport equations. J. Math. Anal. Appl. 2 (1987) 370-405. | MR 872231 | Zbl 0657.45007

[6] D. Barbolosi and A. Iliadis, Optimizing drug regimens in cancer chemotherapy: a simulation study using a PK-PD model. Comput. Biol. Med. 31 (2001) 157-172.

[7] S. Benzekry, Mathematical analysis of a two-dimensional population model of metastatic growth including angiogenesis, J. Evol. Equ. 11 (2011) 187-213. | MR 2780578 | Zbl 1252.35126

[8] S. Benzekry, Passing to the limit 2D-1D in a model for metastatic growth, to appear in J. Biol. Dyn., doi:10.1080/17513758.2011.568071. | MR 2928368

[9] F. Billy, B. Ribba, O. Saut, H. Morre-Trouilhet, T. Colin, D. Bresch, J. Boissel, E. Grenier and J. Flandrois, A pharmacologically based multiscale mathematical model of angiogenesis and its use in investigating the efficacy of a new cancer treatment strategy. J. Theor. Biol. 260 (2009) 545-562. | MR 2973110

[10] F. Boyer, Trace theorems and spatial continuity properties for the solutions of the transport equation. Differential Integral Equations 18 (2005) 891-934. | MR 2150445 | Zbl 1212.35049

[11] A. Devys, T. Goudon and P. Laffitte, A model describing the growth and the size distribution of multiple metastatic tumours. Discret. Contin. Dyn. Syst. Ser. B 12 (2009) 731-767. | MR 2552072 | Zbl 1195.35054

[12] A. D'Onofrio and A. Gandolfi, Tumour eradication by antiangiogenic therapy: analysis and extensions of the model by Hahnfeldt et al. (1999). Math. Biosci. 191 (2004) 159-184. | MR 2090896 | Zbl 1050.92039

[13] A. D'Onofrio, U. Ledzewicz, H. Maurer and H. Schättler, On optimal delivery of combination therapy for tumours. Math. Biosci. 222 (2009) 13-26. | MR 2597086 | Zbl 1176.92033

[14] M. Doumic, Analysis of a population model structured by the cells molecular content. Math. Model. Nat. Phenom. 2 (2007) 121-152. | MR 2455395

[15] J.M.L Ebos, C.R. Lee, W. Cruz-Munoz, G.A. Bjarnason, J.G. Christensen and R.S. Kerbel, Accelerated metastasis after short-term treatment with a potent inhibitor of tumour angiogenesis. Cancer Cell 15 (2009) 232-239.

[16] J. Folkman, Antiangiogenesis: new concept for therapy of solid tumours. Ann. Surg. 175 (1972)

[17] P. Hahnfeldt, D. Panigraphy, J. Folkman and L. Hlatky, Tumour development under angiogenic signaling: a dynamical theory of tumour growth, treatment, response and postvascular dormancy. Cancer Res. 59 (1999) 4770-4775.

[18] P. Hahnfeldt, J. Folkman and L. Hlatky, Minimizing long-term tumour burden: the logic for metronomic chemotherapeutic dosing and its antiangiogenic basis. J. Theor. Biol. 220 (2003) 545-554.

[19] K. Iwata, K. Kawasaki and N. Shigesada, A dynamical model for the growth and size distribution of multiple metastatic tumours. J. Theor. Biol. 203 (2000) 177-186.

[20] R.K. Jain, Normalizing tumour vasculature with anti-angiogenic therapy: A new paradigm for combination therapy. Nature Med. 7 (2001) 987-989.

[21] F. Lignet, S. Benzekry, F. Billy, B. Cajavec Bernard, O. Saut, M. Tod, P. Girard, G. Freyer, E. Grenier, T. Colin and B. Ribba, Identifying optimal combinations of anti-angiogenesis drugs and chemotherapies using a theoretical model of vascular tumour growth (in preparation).

[22] M. Paez-Ribes, E. Allen, J. Hudock, T. Takeda, H. Okuyama, F. Vinals, M. Inoue, G. Bergers, D. Hanahan and O. Casanovas, Antiangiogenic therapy elicits malignant progression of tumours to increased local invasion and distant metastasis. Cancer Cell 15 (2009) 220-231.

[23] B. Perthame, Transport equations in biology. Frontiers in Mathematics, Birkhaüser Verlag, Basel (2007). | MR 2270822 | Zbl 1185.92006

[24] G.J. Riely et al., Randomized phase II study of pulse erlotinib before or after carboplatin and paclitaxel in current or former smokers with advanced non-small-cell lung cancer. J. Clin. Oncol. (2009) 264-270.

[25] G.W. Swan, Applications of optimal control theory in biomedicine. Math. Biosci. 101 (1990) 237-284. | Zbl 0702.92007

[26] S.L. Tucker and S.O. Zimmerman, A nonlinear model of population dynamics containing an arbitrary number of continuous structure variables. SIAM J. Appl. Math. 48 (1988) 549-591. | MR 941101 | Zbl 0657.92011

[27] B. You, C. Meille, D. Barbolosi, B. tranchand, J. Guitton, C. Rioufol, A. Iliadis and G. Freyer, A mechanistic model predicting hematopoiesis and tumour growth to optimize docetaxel + epirubicin (ET) administration in metastatic breast cancer (MBC): Phase I trial. J. Clin. Oncol.(Meeting abstracts) 25 (2007) 13013.