Hostname: page-component-848d4c4894-4hhp2 Total loading time: 0 Render date: 2024-04-30T18:21:48.740Z Has data issue: false hasContentIssue false
  • English
  • Français

Mathematical and numerical analysis of a model for anti-angiogenic therapy in metastatic cancers

Published online by Cambridge University Press:  05 October 2011

Sébastien Benzekry
Sébastien Benzekry*
Affiliation:
LATP, UMR 6632. Université de Provence, Technopole Château-Gombert, 39 rue F. Joliot Curie, 13453 Marseille Cedex 13, France. Laboratoire de Toxicocinétique et Pharmacocinétique UMR-MD3, 27 boulevard Jean Moulin, 13005 Marseille, France. benzekry@phare.normalesup.org
Get access

Abstract

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.

Keywords

Anticancer therapy modellingangiogenesisstructured population dynamicslagrangian scheme
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 46 , Issue 2 , March 2012 , pp. 207 - 237
Copyright
© EDP Sciences, SMAI, 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Angulo, O. and Lopez-Marcos, J.C., Numerical schemes for size-structured population equations. Math. Biosci. 157 (1999) 169–188. CrossRef
Barbolosi, D., Benabdallah, A., Hubert, F. and Verga, F., Mathematical and numerical analysis for a model of growing metastatic tumours. Math. Biosci. 218 (2009) 1–14. CrossRef
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).
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.
Beals, R. and Protopopescu, V., Abstract time-dependent transport equations. J. Math. Anal. Appl. 2 (1987) 370-405. CrossRef
Barbolosi, D. and Iliadis, A., Optimizing drug regimens in cancer chemotherapy: a simulation study using a PK–PD model. Comput. Biol. Med. 31 (2001) 157–172. CrossRef
S. Benzekry, Mathematical analysis of a two-dimensional population model of metastatic growth including angiogenesis, J. Evol. Equ. 11 (2011) 187–213. CrossRef
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. CrossRef
Billy, F., Ribba, B., Saut, O., Morre-Trouilhet, H., Colin, T., Bresch, D., Boissel, J., Grenier, E. and Flandrois, J., 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. CrossRef
Boyer, F., Trace theorems and spatial continuity properties for the solutions of the transport equation. Differential Integral Equations 18 (2005) 891–934.
Devys, A., Goudon, T. and Laffitte, P., A model describing the growth and the size distribution of multiple metastatic tumours. Discret. Contin. Dyn. Syst. Ser. B 12 (2009) 731–767. CrossRef
d'Onofrio, A. and Gandolfi, A., Tumour eradication by antiangiogenic therapy: analysis and extensions of the model by Hahnfeldt et al. (1999). Math. Biosci. 191 (2004) 159–184. CrossRef
d'Onofrio, A., Ledzewicz, U., Maurer, H. and Schättler, H., On optimal delivery of combination therapy for tumours. Math. Biosci. 222 (2009) 13–26. CrossRef
Doumic, M., Analysis of a population model structured by the cells molecular content. Math. Model. Nat. Phenom. 2 (2007) 121–152. CrossRef
Ebos, J.M.L, Lee, C.R., Cruz-Munoz, W., Bjarnason, G.A., Christensen, J.G. and Kerbel, R.S., Accelerated metastasis after short-term treatment with a potent inhibitor of tumour angiogenesis. Cancer Cell 15 (2009) 232–239. CrossRef
J. Folkman, Antiangiogenesis: new concept for therapy of solid tumours. Ann. Surg. 175 (1972)
Hahnfeldt, P., Panigraphy, D., Folkman, J. and Hlatky, L., Tumour development under angiogenic signaling: a dynamical theory of tumour growth, treatment, response and postvascular dormancy. Cancer Res. 59 (1999) 4770–4775.
Hahnfeldt, P., Folkman, J. and Hlatky, L., Minimizing long-term tumour burden: the logic for metronomic chemotherapeutic dosing and its antiangiogenic basis. J. Theor. Biol. 220 (2003) 545–554. CrossRef
Iwata, K., Kawasaki, K. and Shigesada, N., A dynamical model for the growth and size distribution of multiple metastatic tumours. J. Theor. Biol. 203 (2000) 177–186. CrossRef
Jain, R.K., Normalizing tumour vasculature with anti-angiogenic therapy: A new paradigm for combination therapy. Nature Med. 7 (2001) 987–989. CrossRef
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).
Paez-Ribes, M., Allen, E., Hudock, J., Takeda, T., Okuyama, H., Vinals, F., Inoue, M., Bergers, G., Hanahan, D. and Casanovas, O., Antiangiogenic therapy elicits malignant progression of tumours to increased local invasion and distant metastasis. Cancer Cell 15 (2009) 220–231. CrossRef
B. Perthame, Transport equations in biology. Frontiers in Mathematics, Birkhaüser Verlag, Basel (2007).
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.
Swan, G.W., Applications of optimal control theory in biomedicine. Math. Biosci. 101 (1990) 237–284. CrossRef
Tucker, S.L. and Zimmerman, S.O., A nonlinear model of population dynamics containing an arbitrary number of continuous structure variables. SIAM J. Appl. Math. 48 (1988) 549–591. CrossRef
You, B., Meille, C., Barbolosi, D., 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.