Optimisation of time-scheduled regimen for anti-cancer drug infusion
ESAIM: Modélisation mathématique et analyse numérique, Volume 39 (2005) no. 6, pp. 1069-1086.

The chronotherapy concept takes advantage of the circadian rhythm of cells physiology in maximising a treatment efficacy on its target while minimising its toxicity on healthy organs. The object of the present paper is to investigate mathematically and numerically optimal strategies in cancer chronotherapy. To this end a mathematical model describing the time evolution of efficiency and toxicity of an oxaliplatin anti-tumour treatment has been derived. We then applied an optimal control technique to search for the best drug infusion laws. The mathematical model is a set of six coupled differential equations governing the time evolution of both the tumour cell population (cells of Glasgow osteosarcoma, a mouse tumour) and the mature jejunal enterocyte population, to be shielded from unwanted side effects during a treatment by oxaliplatin. Starting from known tumour and villi populations, and a time dependent free platinum Pt (the active drug) infusion law being given, the mathematical model allows to compute the time evolution of both tumour and villi populations. The tumour population growth is based on Gompertz law and the Pt anti-tumour efficacy takes into account the circadian rhythm. Similarly the enterocyte population is subject to a circadian toxicity rhythm. The model has been derived using, as far as possible, experimental data. We examine two different optimisation problems. The eradication problem consists in finding the drug infusion law able to minimise the number of tumour cells while preserving a minimal level for the villi population. On the other hand, the containment problem searches for a quasi periodic treatment able to maintain the tumour population at the lowest possible level, while preserving the villi cells. The originality of these approaches is that the objective and constraint functions we use are L criteria. We are able to derive their gradients with respect to the infusion rate and then to implement efficient optimisation algorithms.

DOI: 10.1051/m2an:2005052
Classification: 49M29, 92B05, 92C50
Mots-clés : dynamical systems, optimisation, circadian rhythms, drugs, therapeutics, cancer
@article{M2AN_2005__39_6_1069_0,
     author = {Basdevant, Claude and Clairambault, Jean and L\'evi, Francis},
     title = {Optimisation of time-scheduled regimen for anti-cancer drug infusion},
     journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
     pages = {1069--1086},
     publisher = {EDP-Sciences},
     volume = {39},
     number = {6},
     year = {2005},
     doi = {10.1051/m2an:2005052},
     mrnumber = {2195905},
     zbl = {1078.92027},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/m2an:2005052/}
}
TY  - JOUR
AU  - Basdevant, Claude
AU  - Clairambault, Jean
AU  - Lévi, Francis
TI  - Optimisation of time-scheduled regimen for anti-cancer drug infusion
JO  - ESAIM: Modélisation mathématique et analyse numérique
PY  - 2005
SP  - 1069
EP  - 1086
VL  - 39
IS  - 6
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/m2an:2005052/
DO  - 10.1051/m2an:2005052
LA  - en
ID  - M2AN_2005__39_6_1069_0
ER  - 
%0 Journal Article
%A Basdevant, Claude
%A Clairambault, Jean
%A Lévi, Francis
%T Optimisation of time-scheduled regimen for anti-cancer drug infusion
%J ESAIM: Modélisation mathématique et analyse numérique
%D 2005
%P 1069-1086
%V 39
%N 6
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/m2an:2005052/
%R 10.1051/m2an:2005052
%G en
%F M2AN_2005__39_6_1069_0
Basdevant, Claude; Clairambault, Jean; Lévi, Francis. Optimisation of time-scheduled regimen for anti-cancer drug infusion. ESAIM: Modélisation mathématique et analyse numérique, Volume 39 (2005) no. 6, pp. 1069-1086. doi : 10.1051/m2an:2005052. http://www.numdam.org/articles/10.1051/m2an:2005052/

[1] Z. Agur, R. Arnon and B. Schechter, Effect of the dosing interval on myelotoxicity and survival in mice treated by cytarabine. Eur. J. Cancer 28A (1992) 1085-1090.

[2] L.K. Andersen and M.C. Mackey, Resonance in periodic chemotherapy: a case study of acute myelogenous leukemia. J. Theor. Biol. 209 (2001) 113-130.

[3] J.F. Bonnans, J.C. Gilbert, C. Lemarechal and C.A. Sagastizabal, Numerical Optimization: Theoretical and Practical Aspects. Springer Universitext (2003). | MR | Zbl

[4] N.A. Boughattas, F. Lévi, et al., Circadian Rhythm in Toxicities and Tissue Uptake of 1,2-diamminocyclohexane(trans-1)oxaloplatinum(II) in Mice. Cancer Research 49 (1989) 3362-3368.

[5] N.A. Boughattas, B. Hecquet, C. Fournier, B. Bruguerolle, A. Trabelsi, K. Bouzouita, B. Omrane and F. Lévi, Comparative pharmacokinetics of oxaliplatin (L-OHP) and carboplatin (CBDCA) in mice with reference to circadian dosing time. Biopharmaceutics and drug disposition 15 (1994) 761-773.

[6] N.F. Britton, N.A. Wright and J.D. Murray, A mathematical model for cell population kinetics in the intestine. J. Theor. Biol. 98 (1982) 531-541.

[7] L. Canaple, T. Kazikawa and V. Laudet, The days and nights of cancer cells. Cancer Research 63 (2003) 7545-7552.

[8] J. Clairambault, D. Claude, E. Filipski, T. Granda and F. Lévi, Toxicité et efficacité antitumorale de l'oxaliplatine sur l'ostéosarcome de Glasgow induit chez la souris : un modèle mathématique. Pathologie-Biologie 51 (2003) 212-215.

[9] L. Cojocaru and Z.A. Agur, Theoretical analysis of interval drug dosing for cell-cycle-phase-specific drugs. Math. Biosci. 109 (1992) 85-97. | Zbl

[10] B.F. Dibrov, M.A. Zhabotinski, Yu.A. Neyfakh, M.P. Orlova and L.I. Churikova, Mathematical model of cancer chemotherapy. Periodic schedules of of phase-specific cytotoxic agent administration increasing the selectivity of therapy. Math. Biosci. 73 (1985) 1-34. | Zbl

[11] B.F. Dibrov, Resonance effect in self-renewing tissues. J. Theor. Biol. 192 (1998) 15-33.

[12] L. Edelstein-Keshet, Mathematical Models in Biology. NY: McGraw-Hill (1988) 210-270. | Zbl

[13] A.W. El-Kareh and T.W. Secomb, A mathematical model for cisplatin cellular pharmacodynamics. Neoplasia 5 (2004) 161-169.

[14] S. Faivre, D. Chan, R. Salinas, B. Woynarowska and J.M. Woynarowski, DNA Single Strand Breaks and apoptosis induced by oxaliplatin in cancer cells. Biochemical Pharmacology 66 (2003) 225-237.

[15] K.R. Fister and J.C. Panetta, Optimal control applied to cell-cycle-specific cancer chemotherapy. SIAM J. Appl. Math. 60 (2000) 1059-1072. | Zbl

[16] L. Fu, H. Pellicano, J. Liu, P. Huang and C.C. Lee, The Circadian Gene Period2 Plays an Important Role in Tumor Suppression and DNA Damage Response In Vivo. Cell 111 (2002) 41-50.

[17] L. Fu and C.C. Lee, The circadian clock: pacemaker and tumour suppressor. Nature Reviews 3 (2003) 351-361.

[18] T.G. Granda, R.-M. D'Attino, E. Filipski, et al., Circadian optimisation of irinotecan and oxaliplatin efficacy in mice with Glasgow osteosarcoma. Brit. J. Cancer 86 (2002) 999-1005.

[19] T.G. Granda, X.H. Liu, R. Smaaland, N. Cermakian, E. Filipski, P. Sassone-Corsi and F. Levi, Circadian regulation of cell cycle and apoptosis proteins in mouse bone marrow and tumor. FASEB J. 19 (2005) 304.

[20] M. Gyllenberg and G.F. Webb, Quiescence as an explanation of gompertzian tumor growth. Growth, Development and Aging 53 (1989) 25-33.

[21] M. Gyllenberg and G.F. Webb, A nonlinear structured population model of tumor growth with quiescence. J. Math. Biol. 28 (1990) 671-694. | Zbl

[22] M.H. Hastings, A.B. Reddy and E.S. Maywood, A clockwork web: circadian timing in brain and periphery, in health and disease. Nat. Rev. Neurosci. 4 (2003) 649-661.

[23] A. Iliadis and D. Barbolosi, Optimising drug regimens in cancer chemotherapy by an efficacy-toxicity mathematical model. Computers Biomed. Res. 33 (2000) 211-226.

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

[25] M. Kimmel and A. Swierniak, Using control theory to make cancer chemotherapy beneficial from phase dependence and resistant to drug resistance. Technical report #7, Ohio State University, Nov. 2003, available on line at http://mbi.osu.edu/publications/techreport7.pdf (2003). | MR

[26] F. Lévi, G. Metzger, C. Massari and G. Milano, Oxaliplatin: Pharmacokinetics and Chronopharmacological Aspects. Clin. Pharmacokinet. 38 (2000) 1-21.

[27] F. Lévi (Ed.), Cancer Chronotherapeutics. Special issue of Chronobiology International 19 #1 (2002).

[28] T. Matsuo, S. Yamaguchi, S. Mitsui, A. Emi, F. Shimoda and H. Okamura, Control mechanism of the circadian clock for timing of cell division in vivo. Science 302 (5643) (2003) 255-259.

[29] M.C. Mckeage, T. Hsu, G. Haddad and B.C. Baguley, Nucleolar damage correlates with neurotoxicity induced by different platinum drugs. Br. J. Cancer 85 (2001) 1219-1225.

[30] M. Mishima, G. Samimi, A. Kondo, X. Lin and S.B. Howell, The cellular pharmacology of oxaliplatin resistance. Eur. J. Cancer 38 (2002) 1405-1412.

[31] C.S. Potten and M. Loeffler, Stem cells: attributes, cycles, spirals, pitfalls and uncertainties. Lessons for and from the crypt. Development 110 (1990) 1001-1020.

[32] U. Schibler, Circadian rhythms. Liver regeneration clocks on. Science 302 (5643) (2003) 234-235.

[33] G. Swan, Role of optimal control theory in cancer chemotherapy. Math. Biosci. 101 (1990) 237-284. | Zbl

[34] G.F. Webb, Resonance phenomena in cell population chemotherapy models. Rocky Mountain J. Math. 20 (1990) 1195-1216. | Zbl

Cited by Sources: