2016, 13(6): 1185-1206. doi: 10.3934/mbe.2016038

Limiting the development of anti-cancer drug resistance in a spatial model of micrometastases

1. 

Department of Biology, The College of New Jersey, Ewing, NJ, United States

2. 

Integrated Mathematical Oncology Department and Center of Excellence in Cancer Imaging and Technology, H. Lee Mott Cancer Center and Research Institute, Department of Oncologic Sciences, University of South Florida, Tampa, FL, United States

3. 

Department of Mathematics and Statistics, The College of New Jersey, Ewing, NJ, United States

Received  October 2015 Revised  May 2016 Published  August 2016

While chemoresistance in primary tumors is well-studied, much less is known about the influence of systemic chemotherapy on the development of drug resistance at metastatic sites. In this work, we use a hybrid spatial model of tumor response to a DNA damaging drug to study how the development of chemoresistance in micrometastases depends on the drug dosing schedule. We separately consider cell populations that harbor pre-existing resistance to the drug, and those that acquire resistance during the course of treatment. For each of these independent scenarios, we consider one hypothetical cell line that is responsive to metronomic chemotherapy, and another that with high probability cannot be eradicated by a metronomic protocol. Motivated by experimental work on ovarian cancer xenografts, we consider all possible combinations of a one week treatment protocol, repeated for three weeks, and constrained by the total weekly drug dose. Simulations reveal a small number of fractionated-dose protocols that are at least as effective as metronomic therapy in eradicating micrometastases with acquired resistance (weak or strong), while also being at least as effective on those that harbor weakly pre-existing resistant cells. Given the responsiveness of very different theoretical cell lines to these few fractionated-dose protocols, these may represent more effective ways to schedule chemotherapy with the goal of limiting metastatic tumor progression.
Citation: Ami B. Shah, Katarzyna A. Rejniak, Jana L. Gevertz. Limiting the development of anti-cancer drug resistance in a spatial model of micrometastases. Mathematical Biosciences & Engineering, 2016, 13 (6) : 1185-1206. doi: 10.3934/mbe.2016038
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I. Bozic, J. Reiter, B. Allen, T. Antal, K. Chatterjee, P. Shah, Y. S. Moon, A. Yaqubie, N. Kelly, D. Le, E. Lipson, P. Chapman, L. Diaz, B. Vogelstein and M. Nowak, Evolutionary dynamics of cancer in response to targeted combination therapy,, eLife, 2 (2013). doi: 10.7554/eLife.00747.

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L. Buffoni, D. Dongiovanni, C. Barone, C. Fissore, D. Ottaviani, V. Dongiovanni, R. Grillo, A. Salvadori, N. Birocco, M. Schena and O. Bertetto, Fractionated dose of cisplatin (CDDP) and vinorelbine (VNB) chemotherapy for elderly patients with advanced non-small cell lung cancer: Phase II trial,, Lung Cancer, 54 (2006), 353. doi: 10.1016/j.lungcan.2006.08.013.

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J. Casciari, S. Sotirchos and R. Sutherland, Variations in tumor cell growth rates and metabolism with oxygen concentration, glucose concentration and extracellular pH,, J. Cellular Physiol., 151 (1992), 386. doi: 10.1002/jcp.1041510220.

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A. Coldman and J. Goldie, A stochastic model for the origin and treatment of tumors containing drug-resistant cells,, Bull. Math. Biol., 48 (1986), 279. doi: 10.1007/BF02459682.

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J. Cunningham, J. Brown, T. L. Vincent3 and R. A. Gatenby, Divergent and convergent evolution in metastases suggest treatment strategies based on specific metastatic sites,, Evolution, 2015 (2015), 76. doi: 10.1093/emph/eov006.

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J. Gevertz, Z. Aminzare, K.-A. Norton, J. Perez-Velazquez, A. Volkening and K. Rejniak, Emergence of anti-cancer drug resistance: Exploring the importance of the microenvironmental niche via a spatial model,, in Applications of Dynamical Systems in Biology and Medicine (eds. T. Jackson and A. Radunskaya), (2015), 1. doi: 10.1007/978-1-4939-2782-1_1.

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J. Greene, O. Lavi, M. Gottesman and D. Levy, The impact of cell density and mutations in a model of multidrug resistance in solid tumors,, Bull. Math. Biol., 76 (2014), 627. doi: 10.1007/s11538-014-9936-8.

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M. Hadjiandreou and G. Mitsis, Mathematical modeling of tumor growth, drug- resistance, toxicity, and optimal therapy design,, IEEE Trans. Biomed. Eng., 61 (2013), 415.

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D. Hanahan, G. Bergers and E. Bergsland, Less is more, regularly: Metronomic dosing of cytotoxic drugs can target tumor angiogenesis in mice,, The Journal of Clinical Investigations, 105 (2000), 1045. doi: 10.1172/JCI9872.

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T. Jackson and H. Byrne, A mathematical model to study the effects of drug resistance and vasculature on the response of solid tumors to chemotherapy,, Math. Biosci., 164 (2000), 17. doi: 10.1016/S0025-5564(99)00062-0.

[24]

J. Kim and I. Tannock, Repopulation of cancer cells during therapy: An important cause of treatment failure,, Nat. Rev. Cancer, 5 (2005), 516. doi: 10.1038/nrc1650.

[25]

N. Komarova and D. Wodarz, Drug resistance in cancer: Principles of emergence and prevention,, Proc. Natl. Acad. Sci., 102 (2005), 9714. doi: 10.1073/pnas.0501870102.

[26]

N. Komarova and D. Wodarz, Stochastic modeling of cellular colonies with quiescence: An application to drug resistance in cancer,, Theor. Popul. Biol., 72 (2007), 523. doi: 10.1016/j.tpb.2007.08.003.

[27]

O. Lavi, M. Gottesman and D. Levy, The dynamics of drug resistance: A mathematical perspective,, Drug Resist. Update, 15 (2012), 90. doi: 10.1016/j.drup.2012.01.003.

[28]

O. Lavi, J. Greene, D. Levy and M. Gottesman, The role of cell density and intratumoral heterogeneity in multidrug resistance,, Cancer Res., 73 (2013), 7168. doi: 10.1158/0008-5472.CAN-13-1768.

[29]

U. Ledzewicz, B. Amni and H. Schattler, Dynamics and control of a mathematical model for metronomic chemotherapy,, Mathematical Biosciences and Engineering, 12 (2015), 1257. doi: 10.3934/mbe.2015.12.1257.

[30]

U. Ledzewicz and H. Schattler, Drug resistance in cancer chemotherapy as an optimal control problem,, Discret. Contin. Dyn-B, 6 (2006), 129.

[31]

U. Ledzewicz, H. Schattler, M. Gahrooi and S. Dehkordi, On the MTD paradigm and optimal control for multi-drug cancer chemotherapy,, Math. Biosci. Eng., 10 (2013), 803. doi: 10.3934/mbe.2013.10.803.

[32]

A. Lorz, T. Lorenzi, J. Clairambault, A. Escargueil and B. Perthame, Modeling effects of space structure and combination therapies on phenotypic heterogeneity and drug resistance in solid tumors,, Bull. Math. Biol., 77 (2015), 1. doi: 10.1007/s11538-014-0046-4.

[33]

A. Lorz, T. Lorenzi, M. Hochberg, J. Clairambault and B. Perthame, Populational adaptive evolution, chemotherapeutic resistance and multiple anti-cancer therapies,, ESAIM: Math. Model. Num. Anal., 47 (2013), 377. doi: 10.1051/m2an/2012031.

[34]

V. Malik P.S.and Raina and N. Andre, Metronomics as maintenance treatment in oncology: Time for chemo-switch,, Front. Oncol., 4 (2014).

[35]

F. Meineke, C. Potten and M. Loeffler, Cell migration and organization in the intestinal crypt using a lattice-free model,, Cell Prolif., 34 (2001), 253. doi: 10.1046/j.0960-7722.2001.00216.x.

[36]

S. Menchon, The effect of intrinsic and acquired resistances on chemotherapy effectiveness,, Acta Biother., 63 (2015), 113. doi: 10.1007/s10441-015-9248-x.

[37]

K. Mross and S. Steinbild, Metronomic anti-cancer therapy - an ongoing treatment option for advanced cancer patients,, Journal of Cancer Therapeutics & Research, 1 (2012), 1. doi: 10.7243/2049-7962-1-32.

[38]

S. Mumenthaler, J. Foo, K. Leder, N. Choi, D. Agus, W. Pao, P. Mallick and F. Michor, Evolutionary modeling of combination treatment strategies to overcome resistance to tyrosine kinase inhibitors in non-small cell lung cancer,, Mol. Pharm., 8 (2011), 2069. doi: 10.1021/mp200270v.

[39]

O. G. Scharovsky, L. Mainetti and V. Rozados, Metronomic chemotherapy: Changing the paradigm that more is better,, Current Oncology, 16 (2012), 7.

[40]

P. Orlando, R. Gatenby and J. Brown, Cancer treatment as a game: Integrating evolutionary game theory into the optimal control of chemotherapy,, Phys. Biol., 9 (2012).

[41]

K. Pietras and D. Hanahan, A multitargeted, metronomic, and maximum-tolerated dose "chemo-switch" regimen is antiangiogenic, producing objective responses and survival benefit in a mouse model of cancer,, J. Clinical Oncology, 23 (2005), 939. doi: 10.1200/JCO.2005.07.093.

[42]

A. Pisco and S. Huang, Non-genetic cancer cell plasticity and therapy-induced stemness in tumour relapse: 'What does not kill me strengthens me',, Br. J. Cancer, 112 (2015), 1725. doi: 10.1038/bjc.2015.146.

[43]

G. Powathil, M. Chaplain and M. Swat, Investigating the development of chemotherapeutic drug resistance in cancer: A multiscale computational study,, , ().

[44]

S. Saxena and G. Christofori, Rebuilding cancer metastasis in the mouse,, Molecular Oncology, 7 (2013), 283. doi: 10.1016/j.molonc.2013.02.009.

[45]

A. Silva and R. Gatenby, A theoretical quantitative model for evolution of cancer chemotherapy resistance,, Biol. Direct., 5 (2010). doi: 10.1186/1745-6150-5-25.

[46]

O. Trédan, C. Galmarini, K. Patel and I. Tannock, Drug resistance and the solid tumor microenvironment,, J. Natl. Cancer Inst., 99 (2007), 1441.

[47]

R. Turner and S. J. Charlton, Assessing the minimum number of data points required for accurate IC50 determination,, Assay Drug Dev Technol., 3 (2005), 525.

[48]

M. Vives, M. Ginesta, K. Gracova, M. Graupera, O. Casanovas, G. Capella, T. Serrano, B. Laquente and F. Vinals, Motronomic chemotherapy following the maximum tolerated dose is an effective anti-tumour therapy affecting angiogenesis, tumour dissemination and cancer stem cells,, International Journal of Cancer, 133 (2013), 2464.

[49]

B. Waclaw, I. Bozic, M. Pittman, R. Hruban, B. Vogelstein and M. Nowak, A spatial model predicts that dispersal and cell turnover limit intratumor heterogeneity,, Nature, 525 (2015), 261.

[50]

A. Wu, K. Loutherback, G. Lambert, L. Estévez-Salmeron, T. Tlsty, R. Austin and J. Sturma, Cell motility and drug gradients in the emergence of resistance to chemotherapy,, Proc. Natl. Acad. Sci., 110 (2013), 16103. doi: 10.1073/pnas.1314385110.

[51]

H. Zahreddine and K. Borden, Mechanisms and insights into drug resistance in cancer,, Front. Pharmacol., 4 (2013). doi: 10.3389/fphar.2013.00028.

show all references

References:
[1]

B. Baguley, Multiple drug resistance mechanisms in cancer,, Mol. Biotechnol., 46 (2010), 308. doi: 10.1007/s12033-010-9321-2.

[2]

S. Benzekry and P. Hahnfeldt, Maximum tolerated dose versus metronomic scheduling in the treatment of metastatic cancers,, Journal of Theoretical Biology, 335 (2013), 235. doi: 10.1016/j.jtbi.2013.06.036.

[3]

I. Bozic, J. Reiter, B. Allen, T. Antal, K. Chatterjee, P. Shah, Y. S. Moon, A. Yaqubie, N. Kelly, D. Le, E. Lipson, P. Chapman, L. Diaz, B. Vogelstein and M. Nowak, Evolutionary dynamics of cancer in response to targeted combination therapy,, eLife, 2 (2013). doi: 10.7554/eLife.00747.

[4]

T. Brocato, P. Dogra, E. J. Koay, A. Day, Y.-L. Chuang, Z. Wang and V. Cristini, Understanding drug resistance in breast cancer with mathematical oncology,, Curr. Breast Cancer Rep., 6 (2014), 110. doi: 10.1007/s12609-014-0143-2.

[5]

L. Buffoni, D. Dongiovanni, C. Barone, C. Fissore, D. Ottaviani, V. Dongiovanni, R. Grillo, A. Salvadori, N. Birocco, M. Schena and O. Bertetto, Fractionated dose of cisplatin (CDDP) and vinorelbine (VNB) chemotherapy for elderly patients with advanced non-small cell lung cancer: Phase II trial,, Lung Cancer, 54 (2006), 353. doi: 10.1016/j.lungcan.2006.08.013.

[6]

J. Casciari, S. Sotirchos and R. Sutherland, Variations in tumor cell growth rates and metabolism with oxygen concentration, glucose concentration and extracellular pH,, J. Cellular Physiol., 151 (1992), 386. doi: 10.1002/jcp.1041510220.

[7]

A. Coldman and J. Goldie, A stochastic model for the origin and treatment of tumors containing drug-resistant cells,, Bull. Math. Biol., 48 (1986), 279. doi: 10.1007/BF02459682.

[8]

J. Cunningham, J. Brown, T. L. Vincent3 and R. A. Gatenby, Divergent and convergent evolution in metastases suggest treatment strategies based on specific metastatic sites,, Evolution, 2015 (2015), 76. doi: 10.1093/emph/eov006.

[9]

J. Cunningham, R. Gatenby and J. Brown, Evolutionary dynamics in cancer therapy,, Mol. Pharm., 8 (2011), 2094. doi: 10.1021/mp2002279.

[10]

R. De Souza, P. Zahedi, R. Badame, C. Allen and M. Piquette-Miller, Chemotherapy dosing schedule influences drug resistance development in ovarian cancer,, Mol. Cancer Ther., 10 (2011), 1289. doi: 10.1158/1535-7163.MCT-11-0058.

[11]

U. Emmenegger, G. Francia, A. Chow, Y. Shaked, A. Kouri, S. Man and R. Kerbel, Tumor that acquire resistance to low-dose metronomic cyclophosphamide retain sensitivity to maximum tolerated dose cyclophosphamide,, Neoplasia, 13 (2011), 40. doi: 10.1593/neo.101174.

[12]

J. Foo, K. Leder and S. Mumenthaler, Cancer as a moving target: Understanding the composition and rebound growth kinetics of recurrent tumors,, Evol. Appl., 6 (2013), 54. doi: 10.1111/eva.12019.

[13]

J. Foo and F. Michor, Evolution of resistance to targeted anti-cancer therapies during continuous and pulsed administration strategies,, PLoS Comput. Biol., 5 (2009). doi: 10.1371/journal.pcbi.1000557.

[14]

J. Foo and F. Michor, Evolution of resistance to anti-cancer therapy during general dosing schedules,, J. Theor. Biol., 263 (2010), 179. doi: 10.1016/j.jtbi.2009.11.022.

[15]

J. Foo and F. Michor, Evolution of acquired resistance to anti-cancer therapy,, J. Theor. Biol., 355 (2014), 10. doi: 10.1016/j.jtbi.2014.02.025.

[16]

J. Freyer and R. Sutherland, A reduction in the in situ rates of oxygen and glucose consumption of cells in EMT6/Ro spheroids during growth,, J. Cellular Physiol., 124 (1985), 516. doi: 10.1002/jcp.1041240323.

[17]

F. Fu, M. Nowak and S. Bonhoeffer, Spatial heterogeneity in drug concentrations can facilitate the emergence of resistance to cancer therapy,, PLoS Comput. Biol., 11 (2015). doi: 10.1371/journal.pcbi.1004142.

[18]

R. Gatenby, A. Silva, R. Gillies and B. Frieden, Adaptive therapy,, Cancer Res., 69 (2009), 4894. doi: 10.1158/0008-5472.CAN-08-3658.

[19]

J. Gevertz, Z. Aminzare, K.-A. Norton, J. Perez-Velazquez, A. Volkening and K. Rejniak, Emergence of anti-cancer drug resistance: Exploring the importance of the microenvironmental niche via a spatial model,, in Applications of Dynamical Systems in Biology and Medicine (eds. T. Jackson and A. Radunskaya), (2015), 1. doi: 10.1007/978-1-4939-2782-1_1.

[20]

J. Greene, O. Lavi, M. Gottesman and D. Levy, The impact of cell density and mutations in a model of multidrug resistance in solid tumors,, Bull. Math. Biol., 76 (2014), 627. doi: 10.1007/s11538-014-9936-8.

[21]

M. Hadjiandreou and G. Mitsis, Mathematical modeling of tumor growth, drug- resistance, toxicity, and optimal therapy design,, IEEE Trans. Biomed. Eng., 61 (2013), 415.

[22]

D. Hanahan, G. Bergers and E. Bergsland, Less is more, regularly: Metronomic dosing of cytotoxic drugs can target tumor angiogenesis in mice,, The Journal of Clinical Investigations, 105 (2000), 1045. doi: 10.1172/JCI9872.

[23]

T. Jackson and H. Byrne, A mathematical model to study the effects of drug resistance and vasculature on the response of solid tumors to chemotherapy,, Math. Biosci., 164 (2000), 17. doi: 10.1016/S0025-5564(99)00062-0.

[24]

J. Kim and I. Tannock, Repopulation of cancer cells during therapy: An important cause of treatment failure,, Nat. Rev. Cancer, 5 (2005), 516. doi: 10.1038/nrc1650.

[25]

N. Komarova and D. Wodarz, Drug resistance in cancer: Principles of emergence and prevention,, Proc. Natl. Acad. Sci., 102 (2005), 9714. doi: 10.1073/pnas.0501870102.

[26]

N. Komarova and D. Wodarz, Stochastic modeling of cellular colonies with quiescence: An application to drug resistance in cancer,, Theor. Popul. Biol., 72 (2007), 523. doi: 10.1016/j.tpb.2007.08.003.

[27]

O. Lavi, M. Gottesman and D. Levy, The dynamics of drug resistance: A mathematical perspective,, Drug Resist. Update, 15 (2012), 90. doi: 10.1016/j.drup.2012.01.003.

[28]

O. Lavi, J. Greene, D. Levy and M. Gottesman, The role of cell density and intratumoral heterogeneity in multidrug resistance,, Cancer Res., 73 (2013), 7168. doi: 10.1158/0008-5472.CAN-13-1768.

[29]

U. Ledzewicz, B. Amni and H. Schattler, Dynamics and control of a mathematical model for metronomic chemotherapy,, Mathematical Biosciences and Engineering, 12 (2015), 1257. doi: 10.3934/mbe.2015.12.1257.

[30]

U. Ledzewicz and H. Schattler, Drug resistance in cancer chemotherapy as an optimal control problem,, Discret. Contin. Dyn-B, 6 (2006), 129.

[31]

U. Ledzewicz, H. Schattler, M. Gahrooi and S. Dehkordi, On the MTD paradigm and optimal control for multi-drug cancer chemotherapy,, Math. Biosci. Eng., 10 (2013), 803. doi: 10.3934/mbe.2013.10.803.

[32]

A. Lorz, T. Lorenzi, J. Clairambault, A. Escargueil and B. Perthame, Modeling effects of space structure and combination therapies on phenotypic heterogeneity and drug resistance in solid tumors,, Bull. Math. Biol., 77 (2015), 1. doi: 10.1007/s11538-014-0046-4.

[33]

A. Lorz, T. Lorenzi, M. Hochberg, J. Clairambault and B. Perthame, Populational adaptive evolution, chemotherapeutic resistance and multiple anti-cancer therapies,, ESAIM: Math. Model. Num. Anal., 47 (2013), 377. doi: 10.1051/m2an/2012031.

[34]

V. Malik P.S.and Raina and N. Andre, Metronomics as maintenance treatment in oncology: Time for chemo-switch,, Front. Oncol., 4 (2014).

[35]

F. Meineke, C. Potten and M. Loeffler, Cell migration and organization in the intestinal crypt using a lattice-free model,, Cell Prolif., 34 (2001), 253. doi: 10.1046/j.0960-7722.2001.00216.x.

[36]

S. Menchon, The effect of intrinsic and acquired resistances on chemotherapy effectiveness,, Acta Biother., 63 (2015), 113. doi: 10.1007/s10441-015-9248-x.

[37]

K. Mross and S. Steinbild, Metronomic anti-cancer therapy - an ongoing treatment option for advanced cancer patients,, Journal of Cancer Therapeutics & Research, 1 (2012), 1. doi: 10.7243/2049-7962-1-32.

[38]

S. Mumenthaler, J. Foo, K. Leder, N. Choi, D. Agus, W. Pao, P. Mallick and F. Michor, Evolutionary modeling of combination treatment strategies to overcome resistance to tyrosine kinase inhibitors in non-small cell lung cancer,, Mol. Pharm., 8 (2011), 2069. doi: 10.1021/mp200270v.

[39]

O. G. Scharovsky, L. Mainetti and V. Rozados, Metronomic chemotherapy: Changing the paradigm that more is better,, Current Oncology, 16 (2012), 7.

[40]

P. Orlando, R. Gatenby and J. Brown, Cancer treatment as a game: Integrating evolutionary game theory into the optimal control of chemotherapy,, Phys. Biol., 9 (2012).

[41]

K. Pietras and D. Hanahan, A multitargeted, metronomic, and maximum-tolerated dose "chemo-switch" regimen is antiangiogenic, producing objective responses and survival benefit in a mouse model of cancer,, J. Clinical Oncology, 23 (2005), 939. doi: 10.1200/JCO.2005.07.093.

[42]

A. Pisco and S. Huang, Non-genetic cancer cell plasticity and therapy-induced stemness in tumour relapse: 'What does not kill me strengthens me',, Br. J. Cancer, 112 (2015), 1725. doi: 10.1038/bjc.2015.146.

[43]

G. Powathil, M. Chaplain and M. Swat, Investigating the development of chemotherapeutic drug resistance in cancer: A multiscale computational study,, , ().

[44]

S. Saxena and G. Christofori, Rebuilding cancer metastasis in the mouse,, Molecular Oncology, 7 (2013), 283. doi: 10.1016/j.molonc.2013.02.009.

[45]

A. Silva and R. Gatenby, A theoretical quantitative model for evolution of cancer chemotherapy resistance,, Biol. Direct., 5 (2010). doi: 10.1186/1745-6150-5-25.

[46]

O. Trédan, C. Galmarini, K. Patel and I. Tannock, Drug resistance and the solid tumor microenvironment,, J. Natl. Cancer Inst., 99 (2007), 1441.

[47]

R. Turner and S. J. Charlton, Assessing the minimum number of data points required for accurate IC50 determination,, Assay Drug Dev Technol., 3 (2005), 525.

[48]

M. Vives, M. Ginesta, K. Gracova, M. Graupera, O. Casanovas, G. Capella, T. Serrano, B. Laquente and F. Vinals, Motronomic chemotherapy following the maximum tolerated dose is an effective anti-tumour therapy affecting angiogenesis, tumour dissemination and cancer stem cells,, International Journal of Cancer, 133 (2013), 2464.

[49]

B. Waclaw, I. Bozic, M. Pittman, R. Hruban, B. Vogelstein and M. Nowak, A spatial model predicts that dispersal and cell turnover limit intratumor heterogeneity,, Nature, 525 (2015), 261.

[50]

A. Wu, K. Loutherback, G. Lambert, L. Estévez-Salmeron, T. Tlsty, R. Austin and J. Sturma, Cell motility and drug gradients in the emergence of resistance to chemotherapy,, Proc. Natl. Acad. Sci., 110 (2013), 16103. doi: 10.1073/pnas.1314385110.

[51]

H. Zahreddine and K. Borden, Mechanisms and insights into drug resistance in cancer,, Front. Pharmacol., 4 (2013). doi: 10.3389/fphar.2013.00028.

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