An enzyme kinetics model of tumor dormancy, regulation of secondary metastases

Previous Article
A computational study of avian influenza
DCDS-S Home
This Issue
Next Article
The dynamics of zeroth-order ultrasensitivity: A critical phenomenon in cell biology

December 2011, 4(6): 1465-1498. doi: 10.3934/dcdss.2011.4.1465

An enzyme kinetics model of tumor dormancy, regulation of secondary metastases

Yangjin Kim ^1, and Khalid Boushaba ^2,

1.	Department of Mathematics & Statistics, University of Michigan, Dearborn, MI 48128, United States
2.	Iowa State University, Department of Mathematics, 482 Carver Hall Ames, IA 50011

Received March 2009 Revised November 2009 Published December 2010

Abstract
Related Papers

In this paper we study 1 dimensional (1D) and 2D extended version of a two compartment model for tumor dormancy suggested by Boushaba et al. [3]. The model is based on the idea that the vascularization of a secondary tumor can be suppressed by inhibitor originating from a larger primary tumor. It has been observed emergence of a polypoid melanoma at a site remote from a primary polypoid melanoma after excision of the latter. The authors observed no recurrence of the melanoma at the primary site, but did observe secondary tumors at secondary sites five to seven centimeters from the primary site within a period of one month after the excision of the primary site. 1D and 2D simulations show that when the tumors are sufficiently remote, the primary tumor will not influence the secondary tumors while, if they are too close together, the primary tumor can effectively prevent the growth of the secondary tumors, even after it is removed. The sensitivity analysis was carried out for the 1D model. It has been long observed that surgery should be followed by other treatment options such as chemotherapy. 2D simulation suggests a possible treatment options with different dosage schedule after a surgery in order to achieve better clinical outcome.

Keywords: Tumor dormancy, Enzyme kinetics, metastases..

Mathematics Subject Classification: Primary: 92C45, 92C50; Secondary: 92B0.

Citation: Yangjin Kim, Khalid Boushaba. An enzyme kinetics model of tumor dormancy, regulation of secondary metastases. Discrete and Continuous Dynamical Systems - S, 2011, 4 (6) : 1465-1498. doi: 10.3934/dcdss.2011.4.1465

References:

[1]	S. G. Anderson, R. H. Buckingham and C. G. Kurland, Does codon composition influence ribosome function?, EMBO, 3 (1983), 91-94.
[2]	L. S. Beck, W. P. L. DeGuzman, Y. X. Lee, M. W. Siegel and E. P. Amento, One systemic administration of transforming growth factor-beta 1 reverses age- or glucocorticoid-impaired wound healing, J. Clin. Invest., 92 (1993), 2841-2849. doi: 10.1172/JCI116904.
[3]	K. Boushaba, H. A. Levine and M. Nilsen-Hamilton, A mathematical model for the regulation of tumor dormancy based on enzyme kinetics, Bull Math. Biol., 68 (2006), 1495-1526. doi: 10.1007/s11538-005-9042-z.
[4]	P. G. Braunschweiger, L. M. Schiffer and S. Betancourt, Tumor cell proliferation and sequential chemotherapy after partial tumor resection in C3H/HeJ mammary tumors, Breast Cancer Research and Treatment, 2 (1982), 323-329. doi: 10.1007/BF01805873.
[5]	F. Brivio, P. Lissoni, G. Alderi, S. Barni, F. Lavorato and L. Fumagalli, Preoperative interleukin-2 subcutaneous immunotherapy may prolong the survival time in advanced colorectal cancer patients, Oncology, 53 (1996), 263-268. doi: 10.1159/000227571.
[6]	F. Brivio, P. Lissoni, M. S. Perego, A. Lissoni and L. Fumagalli, Abrogation of surgery-induced IL-6 hypersecretion by presurgical immunotherapy with IL-2 and its importance in the prevention of postoperative complications, J. Biol. Regul. Homeost Agents, 15 (2001), 370-374.
[7]	Y. Cao, M. S. O'Reilly, B. Marshall, E. Flynn, R. W. Ji and J. Folkman, Expression of angiostatin cDNA in a murine fibrosarcoma suppresses primary tumor growth and produces long-term dormancy of metastases, J. Clin. Invest., 101 (1998), 1055-1063. doi: 10.1172/JCI1558.
[8]	R. Castello, A. Estelles, C. Vazquez, C. Falco, F. Espana, S. M. Almenar, C. Fuster and J. Aznar, Quantitative real-time reverse transcription-pcr assay for urokinase plasminogen activator, plasminogen activator inhibitor type 1, and tissue metalloproteinase inhibitor type 1 gene expressions in primary breast cancer, Clin. Chem., 48 (2002), 1288-1295.
[9]	W. L. Chandler, M. C. Alessi, M. F. Aillaud, P. Vague and I. Juhan-Vague, Formation, inhibition and clearance of plasmin in vivo, Haemostasis, 30 (2002), 204-218.
[10]	J. Cheng and L. Weiner, Tumors and their microenvironments: tilling the soil commentary re: A. m. scott et al., a phase i dose-escalation study of sibrotuzumab in patients with advanced or metastatic fibroblast activation protein-positive cancer, Clin Cancer Res, 9 (2003), 1590-1595.
[11]	J. C. Coffey, M. Doyle, L. O'Mahony et al., Probiotics confer protection against perioperative metastatic tumour growth, Annals of Surg. Oncol., 85 (2001), 273-278.
[12]	J. C. Coffey, J. H. Wang, T. G. Cotter and H. P. Redmond, Cytoreductive surgery enhances tumorogenicityby downregulating mitochondrial apoptosis, Ann Surg Oncol, 10 (2003), S24.
[13]	J. C. Coffey, J. H. Wang, M. J. Smith, D. Bouchier-Hayes, T. G. Cotter and H. P. Redmond, Excisional surgery for cancer cure: Therapy at a cost, Lancet Oncol., 4 (2003), 760-768. doi: 10.1016/S1470-2045(03)01282-8.
[14]	M. L. Costa and H. P. Redmond and D. J. Bouchier-Hayes, Taurolidine improves survival by abrogating the accelerated development and proliferation of solid tumors and development of organ metastases from circulating tumor cells released, J. Surg. Res., 101 (2001), 111-119. doi: 10.1006/jsre.2001.6250.
[15]	B. Davis, Reinforced random walks, Probal. Theory Related Fields, 84 (1990), 203-229. doi: 10.1007/BF01197845.
[16]	G. De Crescenzo, S. Grothe, J. Zwangstra, M. Tsang and M. D.O'Connor-McCourt, Real-time monitoring of the interactions of transforming growth factor- (TGF- ) isoforms with latency-associated protein and the ectodomains of the TGF- type II and III receptors reveals different kinetic models and stoichiometries of binding, J. Biol. Chem., 276 (2001), 29632-29643. doi: 10.1074/jbc.M009765200.
[17]	G. De Crescenzo, P. L. Pham, Y. Durocher and M. D.O'Connor-McCourt, Transforming growth factor-beta(tgf-$\beta$ binding to the extracellular domain of the type ii (tgf-$\beta$ receptor: Receptor capture on a biosensor surface using a new coiled-coil capture system demonstrates that avidity contributes significantly to high affinity binding, J. Mol. Biol., 328 (2003), 1173-1183. doi: 10.1016/S0022-2836(03)00360-7.
[18]	V. De Giorgi, D. Massai, G. Gerlini, F. Mannone, E. Quercioli and P. Carli, Immediate local and regional recurrence after the excision of a polypoid melanoma: tumor dormancy or tumor activation?, Derm. Surgery, 29 (2003), 664-667. doi: 10.1046/j.1524-4725.2003.29163.x.
[19]	R. Demicheli, P. Valagussa and G. Bonadonna, Does surgery modify growth kinetics of breast cancer micrometastases?, Br. J. Cancer, 85 (2001), 490-492. doi: 10.1054/bjoc.2001.1969.
[20]	R. Demicheli, Tumour dormancy: findings and hypotheses from clinical research on breast cancer, Semin. Cancer Biol., 11 (2001), 297-306. doi: 10.1006/scbi.2001.0385.
[21]	E. R. Edelman, N. M. A. and M. J. Karnovsky, Perivascular and intravenous administration of basic fibroblast growth factor: Vascular and solid organ deposition, Proc. Natl. Acad. Sci., 90 (1993), 1513-1517. doi: 10.1073/pnas.90.4.1513.
[22]	S. Eikenberry, C. Thalhauser and Y. Kuang, Tumor-immune interaction, surgical treatment, and cancer recurrence in a mathematical model of melanoma, PLoS Comput. Biol., 5 (2009) e1000362. doi: 10.1371/journal.pcbi.1000362.
[23]	V. Ellis, N. Behrendt and K. Dano, Plasminogen activation by receptor-bound urokinase. a kinetic study with both cell-associated and isolated receptor, J. Biol. Chem., 266 (1991), 12752-12758.
[24]	R. J. Filion and A. S. Popel, A reaction-diffusion model of basic fibroblast growth factor integrations with cell surface receptors, Ann. Biochem. Eng., 32 (2004), 645-663. doi: 10.1023/B:ABME.0000030231.88326.78.
[25]	B. Fisher, E. Saffer, C. Rudock, J. Coyle and N.Gunduz, Effect of local or systemic treatment prior to primary tumor removal on the production and response to a serum growth-stimulating factor in mice, Cancer Res., 49 (1989), 2002-2004.
[26]	B. Fisher, N. Gunduz, J. Coyle, C. Rudock and E. Saffer, Presence of a growth-stimulating factor in serum following primary tumor removal in mice, Cancer Res., 49 (1989), 1996-2001.
[27]	J. Glotzman, M. Mikula, E. Andreas, R. Schulte-Hermann, R. Foisner, H. Beug and W. Mikulits, Molecular aspects of epithelial cell plasticity;implications for local tumor invasion and metastasis, Mutation Res., 566 (2004), 9-20. doi: 10.1016/S1383-5742(03)00033-4.
[28]	M. Guba, G. Cernaianu, G. Koehl, E. K. Geissier, K. Jauch, M. Anthuber, W. Falk and M. Steinbauer, A primary tumor promotes dormancy of solitary tumor cells before inhibiting angiogenesis, Cancer Res., 61 (2001), 5375-5379.
[29]	N. Gunduz, B. Fisher and E. A. Saffer, Effect of surgical removal on the growth and kinetics of residual tumor, Cancer Res, 39 (1979), 3861-3865.
[30]	L. He and B. Niemeyer, A novel correlation for protein diffusion coefficients based on molecular weight and radius of gyration, Biotechnol. Prog., 19 (2003), 544-548. doi: 10.1021/bp0256059.
[31]	L. Holmgren, M. S. O'Reilly and J. Folkman, Dormancy of micrometastases: balanced proliferation and apoptosis in the presence of angiogenesis suppression, Nat Med, 1 (1995), 149-153. doi: 10.1038/nm0295-149.
[32]	S. P. Jung, B. Siegrist, C. A. Hornick, Y.-Z. Wang, M. Wade, C. T. Anthony and E. A. Woltering, Effect of humen recombinant endostatin^® protein on human angiogenesis, Angiogenesis, 5 (2002), 111-118. doi: 10.1023/A:1021540328613.
[33]	Y. Kim, M. Stolarska and H. G. Othmer, A hybrid model for tumor spheroid growth in vitro I: Theoretical development and early results, Math. Models Methods in Appl. Scis., 17 (2007), 1773-1798.
[34]	Y. Kim, A. Friedman, J. Wallace, F. Li and M. Ostrowski, Transformed epithelial cells and fibroblasts/myofibroblasts interaction in breast tumor: A mathematical model and experiments, J. Math. Biol., 61 (2010), 401-421. doi: 10.1007/s00285-009-0307-2.
[35]	Y. Kim and A. Friedman, Interaction of tumor with its microenvironment: A mathematical model, Bull Math. Biol., 72 (2010), 1029-1068. doi: 10.1007/s11538-009-9481-z.
[36]	M. Kirach, G. Schakert and P. M. Black, Angiogenesis, metastasis and endogenous inhibition, J. Neurooncol., 50 (2000), 173-180. doi: 10.1023/A:1006453428013.
[37]	D. F. Lazarous, M. Shou, J. A. Stiber, D. M. Dadhania, V. Thirumurti, E. Hodge and E. F. Unger, Pharmacodynamics of basic fibroblast growth factor: Route of administration determines myocardial and systemic distribution, Cardiovasc Res., 36 (1997), 78-85. doi: 10.1016/S0008-6363(97)00142-9.
[38]	C. Leaf, Why we're losing the war on cancer (and how to win it), Fortune, 149 (2004), 76-97.
[39]	H. A. Levine, S. Pamuk, B. D. Sleeman and M. Nilsen-Hamilton, Mathematical modeling of capillary formation and development in tumor angiogenesis: Penetration into the stroma, Bull. Math. Biol., 63 (2001), 801-863. doi: 10.1006/bulm.2001.0240.
[40]	H. A. Levine, B. D. Sleeman and M. Nilsen-Hamilton, Mathematical modeling of the onset of capillary formation initiating angiogenesis, J. Math. Biol., 42 (2001), 195-238. doi: 10.1007/s002850000037.
[41]	H. A. Levine, B. D. Sleeman and M. Nilsen-Hamilton, A mathematical model for the roles of pericytes and macrophages in the initiation of angiogenesis: I. The role of protease inhibitors in preventing angiogenesis, Mathematical Biosciences, 168 (2000), 77-115. doi: 10.1016/S0025-5564(00)00034-1.
[42]	H. A. Levine, A. L. Tucker and N.-H. M., A mathematical model for the role of cell signaling and transduction in the initiation of angiogenesis, Growth Factors, 20 (2002), 155-175. doi: 10.1080/0897719031000084355.
[43]	H. R. Lijnen, P. Carmeliet, A. Bouche, L. Moons, V. A. Ploplis, E. Plow and D. Collen, Restoration of thrombolytic potential in plasminogen-deficient mice by bolus administration of plasminogen, Blood, 88 (1996), 870-876.
[44]	R. M. Lyons, L. E. Gentry, A. F. Purchio and H. L. Mosesl, Mechanism of activation of latent recombinant transforming growth factor $\beta$1 by plasmin, J. Cell Biol., 110 (1990), 1361-1367. doi: 10.1083/jcb.110.4.1361.
[45]	R. M. Lyons, J. Keski-Oja and H. L. Mosesl, Proteyolytic activation of latent transforming growth factor-$\beta$ from fibroblast conditioned medium, J. Cell Biol., 106 (1988), 1659-1665. doi: 10.1083/jcb.106.5.1659.
[46]	G. D. MacLean and B. M. Longenecker, New possibilities for cancer therapy with advances in cancer immunology, Can J Oncol, 4 (1994), 249-254.
[47]	O. Mandelboim, M. Feldman and L. Eisenbach, H-2K double transfectants of tumor cells as antimetastatic cellular vaccines in heterozygous recipients. Implications for the T cell repertoire, J. Immunol., 148 (1992), 3666-3673.
[48]	Y. Maniwa, M. Kanki and Y. Okita, Importance of the control of lung recurrence soon after surgery of pulmonary metastases, Am. J. Surg., 179 (2000), 122-125. doi: 10.1016/S0002-9610(00)00244-0.
[49]	J. N. Mansbridge, K. Liu, R. E. Pinney, R. Patch, A. Ratcliffe and G. K. Naugnton, Growth factors secreted by fibroblasts: Role in healing diabetic foot ulcers, diabetes, Obesity and Metabolism, 1 (1999), 265-279. doi: 10.1046/j.1463-1326.1999.00032.x.
[50]	S. Marino, I. B. Hogue, C. J. Ray and D. E. Kirschner, A methodology for performing global uncertainty and sensitivity analysis in systems biology, J. Theor. Biol., 254 (2008), 178-196. doi: 10.1016/j.jtbi.2008.04.011.
[51]	T. Mitsudomi, K. Nishioka, R. Maruyama, G. Saitoh, M. Hamatake, Y. Fukuyama, H. Yaita, T. Ishida and K. Sugimachi, Kinetic analysis of recurrence and survival after potentially curative resection of nonsmall cell lung cancer, J Surg Oncol, 63 (1996), 159-165. doi: 10.1002/(SICI)1096-9098(199611)63:3<159::AID-JSO5>3.0.CO;2-C.
[52]	J. Murray, "Mathematical Biology, Biomathematics Texts," Springer-Verlag, 1989.
[53]	M. A. Nugent and E. R. Edelman, Kinetics of basic fibroblast growth factor binding to its receptor and heparan sulfate proteoglycan: A mechanism for cooperativity, Biochemistry, 31 (1992), 8876-8883. doi: 10.1021/bi00152a026.
[54]	R. T. Oliver, Does surgery disseminate or accelerate cancer?, Lancet, 346 (1995), 1506-1507. doi: 10.1016/S0140-6736(95)92046-3.
[55]	M. S. O'Reilly, L. Holmgren, Y. Shing, C. Chen, R. A. Rosenthal, M. Moses, W. S. Lane, Y. Cao, E. H. Sage and J. Folkman, Angiostatin: a novel angiogenesis inhibitor that mediates the suppression of metastases by a Lewis lung carcinoma, Cell, 79 (1994), 315-328.
[56]	M. S. O'Reilly, T. Boehm, Y. Shing, N. Fukai, G. Vasios, W. S. Lane, E. Flynn, J. R. Birkhead, B. R. Olsen and J. Folkman, Endostatin: An endogenous inhibitor of angiogenesis and tumor growth, Cell, 88 (1997), 277-285.
[57]	H. G. Othmer and A. Stevens, Aggregation, blow up and collapse: The abc's of taxis and reinforced random walks, SIAM J. Appl. Math., 57 (1997), 1044-1081. doi: 10.1137/S0036139995288976.
[58]	M. Y. pavlov and M. Ehernberg, Rate of translation of natural mrnas in an optimized in vitro system, Arch. Biochem. Biophys., 328 (1996), 9-16.
[59]	V. A. Ploplis, P. Carmeliet, S. Vazirzadeh, I. Van Vlaenderen, L. Moons, E. F. Plow and D. Collen, Effects of disruption of the plasminogen gene on thrombosis, growth, and health in mice, Circulation, 92 (1995), 2585-2593.
[60]	A. Sadlonova, Z. Novak, M. Johnson, D. Bowe, S. Gault, G. Page, J. Thottassery, D. Welch and A. Frost, Breast fibroblasts modulate epithelial cell proliferation in three-dimensional in vitro co-culture, Breast Cancer Res., 7 (2005), R46-R59. doi: 10.1186/bcr949.
[61]	M. Samoszuk, J. Tan and G. Chorn, Clonogenic growth of human breast cancer cells co-cultured in direct contact with serum-activated fibroblasts, Breast Cancer Res., 7 (2005), R274-R283. doi: 10.1186/bcr995.
[62]	Y. Sato and D. B. Rifkin, Inhibition of endothelial cell movement by perycytes and smooth muscle cells: Activation of latent tgf-$beta 1$ like molecule by plasma during co-culture, J. Cell Biol., 109 (1989), 309-315. doi: 10.1083/jcb.109.1.309.
[63]	Y. Sato, R. Tsuboi, R. Lyons, H. Moses and D. B. Rifkin, Characterization of the activation of latent tgf-$\beta$ by co-cultures of endothelial cells and pericytes of smooth muscle cells: A self-regulating system, J. Cell Biol., 111 (1990), 757-764. doi: 10.1083/jcb.111.2.757.
[64]	M. B. Sporn, The war on cancer, Lancet, 347 (1996), 1377-1381. doi: 10.1016/S0140-6736(96)91015-6.
[65]	J. Stebbing, E. Copson and S. O'Reilly, Herceptin (trastuzamab) in advanced breast cancer, Cancer Treat Rev, 26 (2000), 287-190. doi: 10.1053/ctrv.2000.0182.
[66]	M. Stolarska, Y. Kim and H. G. Othmer, Multi-scale models of cell and tissue dynamics, Phil. Trans. Roy. Soc. A, 367 (2009), 3525-3553. doi: 10.1098/rsta.2009.0095.
[67]	Y. Takeda and M. Nakabayashi, Physicochemical and biological properties of human and canine plasmins, J. Clin. Invest., 53 (1974), 154-162. doi: 10.1172/JCI107533.
[68]	Y. Tsuzuki, D. Fukumura, B. Oosthuyse, C. Koike, P. Carmeliet and R. K. Jain, Vascular endothelial growth factor (vegf) modulation by targeting hypoxia-inducible factor-1alpha$\rightarrow$ hypoxia response element$\rightarrow$ vegf cascade differentially regulates vascular response and growth rate in tumors, Cancer Res., 60 (2000), 6248-6252.
[69]	D. Voet and J. Voet, "Biochemistry," 2nd edition, John Wiley and Sons, 1995.
[70]	L. M. Wakefield, T. S. Winokur, R. S. Hollands, K. Christopherson, A. D. Levinson and M. B. Sporn, ecombinant latent transforming growth factor beta 1 has a longer plasma half-life in rats than active transforming growth factor beta 1, and a different tissue distribution, J Clin Invest, 86 (1990), 1976-7684. doi: 10.1172/JCI114932.
[71]	F. T. Wall, "Chemical Thermodynamics," Freeman, 1958.
[72]	G. F. Whalen, Y. Shing and J. Folkman, The fate of intravenously administered bfgf and the effect of heparin, Growth Factors, 1 (1989), 157-164. doi: 10.3109/08977198909029125.
[73]	M. Yashiro, K. Ikeda, M. Tendo, T. Ishikawa and K. Hirakawa, Effect of organ-specific fibroblasts on proliferation and differentiation of breast cancer cells, Breast Cancer Res Treat, 90 (2005), 307-313. doi: 10.1007/s10549-004-5364-z.
[74]	S. S. Yoon, H. Eto, C. Lin, H. Nakamura, T. M. Pawlik, S. U.Song and K. K. Tanabe, Mouse endostatin inhibits the formation of lung and liver metastases, Cancer Res., 99 (1999), 6251-6236.
[75]	E. D. Yorke, L. Fuks, L. Norton, W. Whitemore and C. C. Ling, Modeling the development of metastases from primary and locally recurrent tumors: comparison with a clinical database for prostatic cancer, Cancer Res., 53 (1993), 2987-2993.
[76]	B. R. Zetter, Angiogenesis and tumor metastasis, review, Ann. Rev. Me, 49 (1998), 407-22. doi: 10.1146/annurev.med.49.1.407.
[77]	S. A. Zioncheck, T. F.and Chen, L. Richardson, M. Mora-Worms, C. Lucas, D. Lewis, J. D. Green and J. Mordenti, Pharmacokinetics and tissue distribution of recombinant human transforming growth factor beta 1 after topical and intravenous administration in male rats, Pharm. Res., 11 (1994), 213-220. doi: 10.1023/A:1018995005775.

show all references

References:

[1]	S. G. Anderson, R. H. Buckingham and C. G. Kurland, Does codon composition influence ribosome function?, EMBO, 3 (1983), 91-94.
[2]	L. S. Beck, W. P. L. DeGuzman, Y. X. Lee, M. W. Siegel and E. P. Amento, One systemic administration of transforming growth factor-beta 1 reverses age- or glucocorticoid-impaired wound healing, J. Clin. Invest., 92 (1993), 2841-2849. doi: 10.1172/JCI116904.
[3]	K. Boushaba, H. A. Levine and M. Nilsen-Hamilton, A mathematical model for the regulation of tumor dormancy based on enzyme kinetics, Bull Math. Biol., 68 (2006), 1495-1526. doi: 10.1007/s11538-005-9042-z.
[4]	P. G. Braunschweiger, L. M. Schiffer and S. Betancourt, Tumor cell proliferation and sequential chemotherapy after partial tumor resection in C3H/HeJ mammary tumors, Breast Cancer Research and Treatment, 2 (1982), 323-329. doi: 10.1007/BF01805873.
[5]	F. Brivio, P. Lissoni, G. Alderi, S. Barni, F. Lavorato and L. Fumagalli, Preoperative interleukin-2 subcutaneous immunotherapy may prolong the survival time in advanced colorectal cancer patients, Oncology, 53 (1996), 263-268. doi: 10.1159/000227571.
[6]	F. Brivio, P. Lissoni, M. S. Perego, A. Lissoni and L. Fumagalli, Abrogation of surgery-induced IL-6 hypersecretion by presurgical immunotherapy with IL-2 and its importance in the prevention of postoperative complications, J. Biol. Regul. Homeost Agents, 15 (2001), 370-374.
[7]	Y. Cao, M. S. O'Reilly, B. Marshall, E. Flynn, R. W. Ji and J. Folkman, Expression of angiostatin cDNA in a murine fibrosarcoma suppresses primary tumor growth and produces long-term dormancy of metastases, J. Clin. Invest., 101 (1998), 1055-1063. doi: 10.1172/JCI1558.
[8]	R. Castello, A. Estelles, C. Vazquez, C. Falco, F. Espana, S. M. Almenar, C. Fuster and J. Aznar, Quantitative real-time reverse transcription-pcr assay for urokinase plasminogen activator, plasminogen activator inhibitor type 1, and tissue metalloproteinase inhibitor type 1 gene expressions in primary breast cancer, Clin. Chem., 48 (2002), 1288-1295.
[9]	W. L. Chandler, M. C. Alessi, M. F. Aillaud, P. Vague and I. Juhan-Vague, Formation, inhibition and clearance of plasmin in vivo, Haemostasis, 30 (2002), 204-218.
[10]	J. Cheng and L. Weiner, Tumors and their microenvironments: tilling the soil commentary re: A. m. scott et al., a phase i dose-escalation study of sibrotuzumab in patients with advanced or metastatic fibroblast activation protein-positive cancer, Clin Cancer Res, 9 (2003), 1590-1595.
[11]	J. C. Coffey, M. Doyle, L. O'Mahony et al., Probiotics confer protection against perioperative metastatic tumour growth, Annals of Surg. Oncol., 85 (2001), 273-278.
[12]	J. C. Coffey, J. H. Wang, T. G. Cotter and H. P. Redmond, Cytoreductive surgery enhances tumorogenicityby downregulating mitochondrial apoptosis, Ann Surg Oncol, 10 (2003), S24.
[13]	J. C. Coffey, J. H. Wang, M. J. Smith, D. Bouchier-Hayes, T. G. Cotter and H. P. Redmond, Excisional surgery for cancer cure: Therapy at a cost, Lancet Oncol., 4 (2003), 760-768. doi: 10.1016/S1470-2045(03)01282-8.
[14]	M. L. Costa and H. P. Redmond and D. J. Bouchier-Hayes, Taurolidine improves survival by abrogating the accelerated development and proliferation of solid tumors and development of organ metastases from circulating tumor cells released, J. Surg. Res., 101 (2001), 111-119. doi: 10.1006/jsre.2001.6250.
[15]	B. Davis, Reinforced random walks, Probal. Theory Related Fields, 84 (1990), 203-229. doi: 10.1007/BF01197845.
[16]	G. De Crescenzo, S. Grothe, J. Zwangstra, M. Tsang and M. D.O'Connor-McCourt, Real-time monitoring of the interactions of transforming growth factor- (TGF- ) isoforms with latency-associated protein and the ectodomains of the TGF- type II and III receptors reveals different kinetic models and stoichiometries of binding, J. Biol. Chem., 276 (2001), 29632-29643. doi: 10.1074/jbc.M009765200.
[17]	G. De Crescenzo, P. L. Pham, Y. Durocher and M. D.O'Connor-McCourt, Transforming growth factor-beta(tgf-$\beta$ binding to the extracellular domain of the type ii (tgf-$\beta$ receptor: Receptor capture on a biosensor surface using a new coiled-coil capture system demonstrates that avidity contributes significantly to high affinity binding, J. Mol. Biol., 328 (2003), 1173-1183. doi: 10.1016/S0022-2836(03)00360-7.
[18]	V. De Giorgi, D. Massai, G. Gerlini, F. Mannone, E. Quercioli and P. Carli, Immediate local and regional recurrence after the excision of a polypoid melanoma: tumor dormancy or tumor activation?, Derm. Surgery, 29 (2003), 664-667. doi: 10.1046/j.1524-4725.2003.29163.x.
[19]	R. Demicheli, P. Valagussa and G. Bonadonna, Does surgery modify growth kinetics of breast cancer micrometastases?, Br. J. Cancer, 85 (2001), 490-492. doi: 10.1054/bjoc.2001.1969.
[20]	R. Demicheli, Tumour dormancy: findings and hypotheses from clinical research on breast cancer, Semin. Cancer Biol., 11 (2001), 297-306. doi: 10.1006/scbi.2001.0385.
[21]	E. R. Edelman, N. M. A. and M. J. Karnovsky, Perivascular and intravenous administration of basic fibroblast growth factor: Vascular and solid organ deposition, Proc. Natl. Acad. Sci., 90 (1993), 1513-1517. doi: 10.1073/pnas.90.4.1513.
[22]	S. Eikenberry, C. Thalhauser and Y. Kuang, Tumor-immune interaction, surgical treatment, and cancer recurrence in a mathematical model of melanoma, PLoS Comput. Biol., 5 (2009) e1000362. doi: 10.1371/journal.pcbi.1000362.
[23]	V. Ellis, N. Behrendt and K. Dano, Plasminogen activation by receptor-bound urokinase. a kinetic study with both cell-associated and isolated receptor, J. Biol. Chem., 266 (1991), 12752-12758.
[24]	R. J. Filion and A. S. Popel, A reaction-diffusion model of basic fibroblast growth factor integrations with cell surface receptors, Ann. Biochem. Eng., 32 (2004), 645-663. doi: 10.1023/B:ABME.0000030231.88326.78.
[25]	B. Fisher, E. Saffer, C. Rudock, J. Coyle and N.Gunduz, Effect of local or systemic treatment prior to primary tumor removal on the production and response to a serum growth-stimulating factor in mice, Cancer Res., 49 (1989), 2002-2004.
[26]	B. Fisher, N. Gunduz, J. Coyle, C. Rudock and E. Saffer, Presence of a growth-stimulating factor in serum following primary tumor removal in mice, Cancer Res., 49 (1989), 1996-2001.
[27]	J. Glotzman, M. Mikula, E. Andreas, R. Schulte-Hermann, R. Foisner, H. Beug and W. Mikulits, Molecular aspects of epithelial cell plasticity;implications for local tumor invasion and metastasis, Mutation Res., 566 (2004), 9-20. doi: 10.1016/S1383-5742(03)00033-4.
[28]	M. Guba, G. Cernaianu, G. Koehl, E. K. Geissier, K. Jauch, M. Anthuber, W. Falk and M. Steinbauer, A primary tumor promotes dormancy of solitary tumor cells before inhibiting angiogenesis, Cancer Res., 61 (2001), 5375-5379.
[29]	N. Gunduz, B. Fisher and E. A. Saffer, Effect of surgical removal on the growth and kinetics of residual tumor, Cancer Res, 39 (1979), 3861-3865.
[30]	L. He and B. Niemeyer, A novel correlation for protein diffusion coefficients based on molecular weight and radius of gyration, Biotechnol. Prog., 19 (2003), 544-548. doi: 10.1021/bp0256059.
[31]	L. Holmgren, M. S. O'Reilly and J. Folkman, Dormancy of micrometastases: balanced proliferation and apoptosis in the presence of angiogenesis suppression, Nat Med, 1 (1995), 149-153. doi: 10.1038/nm0295-149.
[32]	S. P. Jung, B. Siegrist, C. A. Hornick, Y.-Z. Wang, M. Wade, C. T. Anthony and E. A. Woltering, Effect of humen recombinant endostatin^® protein on human angiogenesis, Angiogenesis, 5 (2002), 111-118. doi: 10.1023/A:1021540328613.
[33]	Y. Kim, M. Stolarska and H. G. Othmer, A hybrid model for tumor spheroid growth in vitro I: Theoretical development and early results, Math. Models Methods in Appl. Scis., 17 (2007), 1773-1798.
[34]	Y. Kim, A. Friedman, J. Wallace, F. Li and M. Ostrowski, Transformed epithelial cells and fibroblasts/myofibroblasts interaction in breast tumor: A mathematical model and experiments, J. Math. Biol., 61 (2010), 401-421. doi: 10.1007/s00285-009-0307-2.
[35]	Y. Kim and A. Friedman, Interaction of tumor with its microenvironment: A mathematical model, Bull Math. Biol., 72 (2010), 1029-1068. doi: 10.1007/s11538-009-9481-z.
[36]	M. Kirach, G. Schakert and P. M. Black, Angiogenesis, metastasis and endogenous inhibition, J. Neurooncol., 50 (2000), 173-180. doi: 10.1023/A:1006453428013.
[37]	D. F. Lazarous, M. Shou, J. A. Stiber, D. M. Dadhania, V. Thirumurti, E. Hodge and E. F. Unger, Pharmacodynamics of basic fibroblast growth factor: Route of administration determines myocardial and systemic distribution, Cardiovasc Res., 36 (1997), 78-85. doi: 10.1016/S0008-6363(97)00142-9.
[38]	C. Leaf, Why we're losing the war on cancer (and how to win it), Fortune, 149 (2004), 76-97.
[39]	H. A. Levine, S. Pamuk, B. D. Sleeman and M. Nilsen-Hamilton, Mathematical modeling of capillary formation and development in tumor angiogenesis: Penetration into the stroma, Bull. Math. Biol., 63 (2001), 801-863. doi: 10.1006/bulm.2001.0240.
[40]	H. A. Levine, B. D. Sleeman and M. Nilsen-Hamilton, Mathematical modeling of the onset of capillary formation initiating angiogenesis, J. Math. Biol., 42 (2001), 195-238. doi: 10.1007/s002850000037.
[41]	H. A. Levine, B. D. Sleeman and M. Nilsen-Hamilton, A mathematical model for the roles of pericytes and macrophages in the initiation of angiogenesis: I. The role of protease inhibitors in preventing angiogenesis, Mathematical Biosciences, 168 (2000), 77-115. doi: 10.1016/S0025-5564(00)00034-1.
[42]	H. A. Levine, A. L. Tucker and N.-H. M., A mathematical model for the role of cell signaling and transduction in the initiation of angiogenesis, Growth Factors, 20 (2002), 155-175. doi: 10.1080/0897719031000084355.
[43]	H. R. Lijnen, P. Carmeliet, A. Bouche, L. Moons, V. A. Ploplis, E. Plow and D. Collen, Restoration of thrombolytic potential in plasminogen-deficient mice by bolus administration of plasminogen, Blood, 88 (1996), 870-876.
[44]	R. M. Lyons, L. E. Gentry, A. F. Purchio and H. L. Mosesl, Mechanism of activation of latent recombinant transforming growth factor $\beta$1 by plasmin, J. Cell Biol., 110 (1990), 1361-1367. doi: 10.1083/jcb.110.4.1361.
[45]	R. M. Lyons, J. Keski-Oja and H. L. Mosesl, Proteyolytic activation of latent transforming growth factor-$\beta$ from fibroblast conditioned medium, J. Cell Biol., 106 (1988), 1659-1665. doi: 10.1083/jcb.106.5.1659.
[46]	G. D. MacLean and B. M. Longenecker, New possibilities for cancer therapy with advances in cancer immunology, Can J Oncol, 4 (1994), 249-254.
[47]	O. Mandelboim, M. Feldman and L. Eisenbach, H-2K double transfectants of tumor cells as antimetastatic cellular vaccines in heterozygous recipients. Implications for the T cell repertoire, J. Immunol., 148 (1992), 3666-3673.
[48]	Y. Maniwa, M. Kanki and Y. Okita, Importance of the control of lung recurrence soon after surgery of pulmonary metastases, Am. J. Surg., 179 (2000), 122-125. doi: 10.1016/S0002-9610(00)00244-0.
[49]	J. N. Mansbridge, K. Liu, R. E. Pinney, R. Patch, A. Ratcliffe and G. K. Naugnton, Growth factors secreted by fibroblasts: Role in healing diabetic foot ulcers, diabetes, Obesity and Metabolism, 1 (1999), 265-279. doi: 10.1046/j.1463-1326.1999.00032.x.
[50]	S. Marino, I. B. Hogue, C. J. Ray and D. E. Kirschner, A methodology for performing global uncertainty and sensitivity analysis in systems biology, J. Theor. Biol., 254 (2008), 178-196. doi: 10.1016/j.jtbi.2008.04.011.
[51]	T. Mitsudomi, K. Nishioka, R. Maruyama, G. Saitoh, M. Hamatake, Y. Fukuyama, H. Yaita, T. Ishida and K. Sugimachi, Kinetic analysis of recurrence and survival after potentially curative resection of nonsmall cell lung cancer, J Surg Oncol, 63 (1996), 159-165. doi: 10.1002/(SICI)1096-9098(199611)63:3<159::AID-JSO5>3.0.CO;2-C.
[52]	J. Murray, "Mathematical Biology, Biomathematics Texts," Springer-Verlag, 1989.
[53]	M. A. Nugent and E. R. Edelman, Kinetics of basic fibroblast growth factor binding to its receptor and heparan sulfate proteoglycan: A mechanism for cooperativity, Biochemistry, 31 (1992), 8876-8883. doi: 10.1021/bi00152a026.
[54]	R. T. Oliver, Does surgery disseminate or accelerate cancer?, Lancet, 346 (1995), 1506-1507. doi: 10.1016/S0140-6736(95)92046-3.
[55]	M. S. O'Reilly, L. Holmgren, Y. Shing, C. Chen, R. A. Rosenthal, M. Moses, W. S. Lane, Y. Cao, E. H. Sage and J. Folkman, Angiostatin: a novel angiogenesis inhibitor that mediates the suppression of metastases by a Lewis lung carcinoma, Cell, 79 (1994), 315-328.
[56]	M. S. O'Reilly, T. Boehm, Y. Shing, N. Fukai, G. Vasios, W. S. Lane, E. Flynn, J. R. Birkhead, B. R. Olsen and J. Folkman, Endostatin: An endogenous inhibitor of angiogenesis and tumor growth, Cell, 88 (1997), 277-285.
[57]	H. G. Othmer and A. Stevens, Aggregation, blow up and collapse: The abc's of taxis and reinforced random walks, SIAM J. Appl. Math., 57 (1997), 1044-1081. doi: 10.1137/S0036139995288976.
[58]	M. Y. pavlov and M. Ehernberg, Rate of translation of natural mrnas in an optimized in vitro system, Arch. Biochem. Biophys., 328 (1996), 9-16.
[59]	V. A. Ploplis, P. Carmeliet, S. Vazirzadeh, I. Van Vlaenderen, L. Moons, E. F. Plow and D. Collen, Effects of disruption of the plasminogen gene on thrombosis, growth, and health in mice, Circulation, 92 (1995), 2585-2593.
[60]	A. Sadlonova, Z. Novak, M. Johnson, D. Bowe, S. Gault, G. Page, J. Thottassery, D. Welch and A. Frost, Breast fibroblasts modulate epithelial cell proliferation in three-dimensional in vitro co-culture, Breast Cancer Res., 7 (2005), R46-R59. doi: 10.1186/bcr949.
[61]	M. Samoszuk, J. Tan and G. Chorn, Clonogenic growth of human breast cancer cells co-cultured in direct contact with serum-activated fibroblasts, Breast Cancer Res., 7 (2005), R274-R283. doi: 10.1186/bcr995.
[62]	Y. Sato and D. B. Rifkin, Inhibition of endothelial cell movement by perycytes and smooth muscle cells: Activation of latent tgf-$beta 1$ like molecule by plasma during co-culture, J. Cell Biol., 109 (1989), 309-315. doi: 10.1083/jcb.109.1.309.
[63]	Y. Sato, R. Tsuboi, R. Lyons, H. Moses and D. B. Rifkin, Characterization of the activation of latent tgf-$\beta$ by co-cultures of endothelial cells and pericytes of smooth muscle cells: A self-regulating system, J. Cell Biol., 111 (1990), 757-764. doi: 10.1083/jcb.111.2.757.
[64]	M. B. Sporn, The war on cancer, Lancet, 347 (1996), 1377-1381. doi: 10.1016/S0140-6736(96)91015-6.
[65]	J. Stebbing, E. Copson and S. O'Reilly, Herceptin (trastuzamab) in advanced breast cancer, Cancer Treat Rev, 26 (2000), 287-190. doi: 10.1053/ctrv.2000.0182.
[66]	M. Stolarska, Y. Kim and H. G. Othmer, Multi-scale models of cell and tissue dynamics, Phil. Trans. Roy. Soc. A, 367 (2009), 3525-3553. doi: 10.1098/rsta.2009.0095.
[67]	Y. Takeda and M. Nakabayashi, Physicochemical and biological properties of human and canine plasmins, J. Clin. Invest., 53 (1974), 154-162. doi: 10.1172/JCI107533.
[68]	Y. Tsuzuki, D. Fukumura, B. Oosthuyse, C. Koike, P. Carmeliet and R. K. Jain, Vascular endothelial growth factor (vegf) modulation by targeting hypoxia-inducible factor-1alpha$\rightarrow$ hypoxia response element$\rightarrow$ vegf cascade differentially regulates vascular response and growth rate in tumors, Cancer Res., 60 (2000), 6248-6252.
[69]	D. Voet and J. Voet, "Biochemistry," 2nd edition, John Wiley and Sons, 1995.
[70]	L. M. Wakefield, T. S. Winokur, R. S. Hollands, K. Christopherson, A. D. Levinson and M. B. Sporn, ecombinant latent transforming growth factor beta 1 has a longer plasma half-life in rats than active transforming growth factor beta 1, and a different tissue distribution, J Clin Invest, 86 (1990), 1976-7684. doi: 10.1172/JCI114932.
[71]	F. T. Wall, "Chemical Thermodynamics," Freeman, 1958.
[72]	G. F. Whalen, Y. Shing and J. Folkman, The fate of intravenously administered bfgf and the effect of heparin, Growth Factors, 1 (1989), 157-164. doi: 10.3109/08977198909029125.
[73]	M. Yashiro, K. Ikeda, M. Tendo, T. Ishikawa and K. Hirakawa, Effect of organ-specific fibroblasts on proliferation and differentiation of breast cancer cells, Breast Cancer Res Treat, 90 (2005), 307-313. doi: 10.1007/s10549-004-5364-z.
[74]	S. S. Yoon, H. Eto, C. Lin, H. Nakamura, T. M. Pawlik, S. U.Song and K. K. Tanabe, Mouse endostatin inhibits the formation of lung and liver metastases, Cancer Res., 99 (1999), 6251-6236.
[75]	E. D. Yorke, L. Fuks, L. Norton, W. Whitemore and C. C. Ling, Modeling the development of metastases from primary and locally recurrent tumors: comparison with a clinical database for prostatic cancer, Cancer Res., 53 (1993), 2987-2993.
[76]	B. R. Zetter, Angiogenesis and tumor metastasis, review, Ann. Rev. Me, 49 (1998), 407-22. doi: 10.1146/annurev.med.49.1.407.
[77]	S. A. Zioncheck, T. F.and Chen, L. Richardson, M. Mora-Worms, C. Lucas, D. Lewis, J. D. Green and J. Mordenti, Pharmacokinetics and tissue distribution of recombinant human transforming growth factor beta 1 after topical and intravenous administration in male rats, Pharm. Res., 11 (1994), 213-220. doi: 10.1023/A:1018995005775.

[1]	Z. Jackiewicz, B. Zubik-Kowal, B. Basse. Finite-difference and pseudo-spectral methods for the numerical simulations of in vitro human tumor cell population kinetics. Mathematical Biosciences & Engineering, 2009, 6 (3) : 561-572. doi: 10.3934/mbe.2009.6.561
[2]	Ben Sheller, Domenico D'Alessandro. Analysis of a cancer dormancy model and control of immuno-therapy. Mathematical Biosciences & Engineering, 2015, 12 (5) : 1037-1053. doi: 10.3934/mbe.2015.12.1037
[3]	Jean-Pierre Françoise, Hongjun Ji. The stability analysis of brain lactate kinetics. Discrete and Continuous Dynamical Systems - S, 2020, 13 (8) : 2135-2143. doi: 10.3934/dcdss.2020182
[4]	Guy Katriel. Existence of periodic solutions for enzyme-catalysed reactions with periodic substrate input. Conference Publications, 2007, 2007 (Special) : 551-557. doi: 10.3934/proc.2007.2007.551
[5]	Juan Su, Bing Xu, Lan Zou. Bifurcation analysis of an enzyme-catalyzed reaction system with branched sink. Discrete and Continuous Dynamical Systems - B, 2019, 24 (12) : 6783-6815. doi: 10.3934/dcdsb.2019167
[6]	Justin P. Peters, Khalid Boushaba, Marit Nilsen-Hamilton. A Mathematical Model for Fibroblast Growth Factor Competition Based on Enzyme. Mathematical Biosciences & Engineering, 2005, 2 (4) : 789-810. doi: 10.3934/mbe.2005.2.789
[7]	Hyeuknam Kwon, Yoon Mo Jung, Jaeseok Park, Jin Keun Seo. A new computer-aided method for detecting brain metastases on contrast-enhanced MR images. Inverse Problems and Imaging, 2014, 8 (2) : 491-505. doi: 10.3934/ipi.2014.8.491
[8]	Carole Guillevin, Rémy Guillevin, Alain Miranville, Angélique Perrillat-Mercerot. Analysis of a mathematical model for brain lactate kinetics. Mathematical Biosciences & Engineering, 2018, 15 (5) : 1225-1242. doi: 10.3934/mbe.2018056
[9]	Robert P. Gilbert, Philippe Guyenne, Ying Liu. Modeling of the kinetics of vitamin D$_3$ in osteoblastic cells. Mathematical Biosciences & Engineering, 2013, 10 (2) : 319-344. doi: 10.3934/mbe.2013.10.319
[10]	Dan Stanescu, Benito Chen-Charpentier. Random coefficient differential equation models for Monod kinetics. Conference Publications, 2009, 2009 (Special) : 719-728. doi: 10.3934/proc.2009.2009.719
[11]	Sarthok Sircar, Anthony Roberts. Ion mediated crosslink driven mucous swelling kinetics. Discrete and Continuous Dynamical Systems - B, 2016, 21 (6) : 1937-1951. doi: 10.3934/dcdsb.2016030
[12]	Boris Baeumer, Lipika Chatterjee, Peter Hinow, Thomas Rades, Ami Radunskaya, Ian Tucker. Predicting the drug release kinetics of matrix tablets. Discrete and Continuous Dynamical Systems - B, 2009, 12 (2) : 261-277. doi: 10.3934/dcdsb.2009.12.261
[13]	Qiuyan Zhang, Lingling Liu, Weinian Zhang. Bogdanov-Takens bifurcations in the enzyme-catalyzed reaction comprising a branched network. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1499-1514. doi: 10.3934/mbe.2017078
[14]	Amy H. Lin. A model of tumor and lymphocyte interactions. Discrete and Continuous Dynamical Systems - B, 2004, 4 (1) : 241-266. doi: 10.3934/dcdsb.2004.4.241
[15]	Elena Izquierdo-Kulich, José Manuel Nieto-Villar. Mesoscopic model for tumor growth. Mathematical Biosciences & Engineering, 2007, 4 (4) : 687-698. doi: 10.3934/mbe.2007.4.687
[16]	Elena Izquierdo-Kulich, José Manuel Nieto-Villar. Morphogenesis of the tumor patterns. Mathematical Biosciences & Engineering, 2008, 5 (2) : 299-313. doi: 10.3934/mbe.2008.5.299
[17]	Roy Malka, Vered Rom-Kedar. Bacteria--phagocyte dynamics, axiomatic modelling and mass-action kinetics. Mathematical Biosciences & Engineering, 2011, 8 (2) : 475-502. doi: 10.3934/mbe.2011.8.475
[18]	Annegret Glitzky. Energy estimates for electro-reaction-diffusion systems with partly fast kinetics. Discrete and Continuous Dynamical Systems, 2009, 25 (1) : 159-174. doi: 10.3934/dcds.2009.25.159
[19]	Evans K. Afenya, Calixto P. Calderón. Growth kinetics of cancer cells prior to detection and treatment: An alternative view. Discrete and Continuous Dynamical Systems - B, 2004, 4 (1) : 25-28. doi: 10.3934/dcdsb.2004.4.25
[20]	Pei Zhang, Siyan Liu, Dan Lu, Ramanan Sankaran, Guannan Zhang. An out-of-distribution-aware autoencoder model for reduced chemical kinetics. Discrete and Continuous Dynamical Systems - S, 2022, 15 (4) : 913-930. doi: 10.3934/dcdss.2021138

2021 Impact Factor: 1.865

Tools

Metrics

Other articles
by authors

on AIMS
Yangjin Kim Khalid Boushaba
on Google Scholar
Yangjin Kim Khalid Boushaba

[Back to Top]