Skip to main content

Advertisement

SpringerLink
Log in

A Mathematical Model for the Regulation of Tumor Dormancy Based on Enzyme Kinetics

  • Original Article
  • Published:
Bulletin of Mathematical Biology Aims and scope Submit manuscript

Abstract

In this paper we present a two-compartment model for tumor dormancy based on an idea of Zetter [1998, Ann. Rev. Med. 49, 407–422] to wit: The vascularization of a secondary (daughter) tumor can be suppressed by an inhibitor originating from a larger primary (mother) tumor. We apply this idea at the avascular level to develop a model for the remote suppression of secondary avascular tumors via the secretion of primary avascular tumor inhibitors. The model gives good agreement with the observations of [De Giorgi et al., 2003, Derm. Surgery 29, 664–667]. These authors reported on the 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 5–7 cm from the primary site within a period of 1 month after the excision of the primary site. We attempt to provide a reasonable biochemical/cell biological model for this phenomenon. We show that when the tumors are sufficiently remote, the primary tumor will not influence the secondary tumor while, if they are too close together, the primary tumor can effectively prevent the growth of the secondary tumor, even after it is removed. It should be possible to use the model as the basis for a testable hypothesis.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Anderson, S.G., Buckingham, R.H., Kurland, C.G., 1983. Does codon composition influence ribosome function? EMBO 3, 91–94.

    Google Scholar 

  • Beck, L.S., DeGuzman, W.P.L., Lee, Y.X., Siegel, M.W., Amento, E.P., 1993. One systemic administration of transforming growth factor-beta 1 reverses age- or glucocorticoid-impaired wound healing. J. Clin. Invest. 92, 2841–2849.

    Article  PubMed  CAS  Google Scholar 

  • Castello, R., Estelles, A., Vazquez, C., Falco, C., Espana, F., Almenar, S.M., 2002. 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, 1288–1295.

    PubMed  CAS  Google Scholar 

  • Chandler, W.L., Alessi, M.C., Aillaud, M.F., Vague, P., Juhan-Vague, I., 2002. Formation, inhibition and clearance of plasmin in vivo. Haemostasis 30, 204–218.

    Article  Google Scholar 

  • Davis, B., 1990. Reinforced random walks. Probal. Theory Relat. Fields 84, 203–229.

    Article  MATH  ADS  Google Scholar 

  • De Crescenzo, G., Grothe, S., Zwangstra, J., Tsang, M., O'Connor-McCourt, M.D., Real time monitoring of the interactions of transforming growth factor-β (tgf-βisoforms with latency associated protein and the ectodomains of the (tgf-β). J. Biol. Chem. 276, 29632–29643.

  • De Crescenzo, G., Pham, P.L., Durocher, Y., O'Connor-McCourt, M.D., 2003. Transforming growth factor-beta(tgf-β binding to the extracellular domain of the type ii (tgf-β 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, 1173-1183.

    Article  PubMed  CAS  Google Scholar 

  • De Giorgi, V., Massai, D., Gerlini, G., Mannone, F., Quercioli, E., Carli, P., 2003. Immediate local and regional recurrence after the excision of a polypoid melanoma: Tumor dormancy or tumor activation? Dermatol. Surg. 29, 664–667.

    Article  PubMed  Google Scholar 

  • Edelman, E.R., Nugent, M.A., Karnovsky, M.J., 1993. Perivascular and intravenous administration of basic fibroblast growth factor: Vascular and solid organ deposition. Proc. Natl. Acad. Sci. 90, 1513–1517.

    Article  PubMed  CAS  ADS  Google Scholar 

  • Edelman, E.R., Nugent, M.A., Karnovsky, M.U., 1993. Perivascular and intravenous administration of basic fibroblast gowth factor: Vascular and solid organ deposition. Proc. Natl. Acad. Sci. 90, 1513–1517.

    Article  PubMed  CAS  ADS  Google Scholar 

  • Ellis, V., Behrendt, N., Dano, K., 1991. Plasminogen activation by receptor-bound urokinase. a kinetic study with both cell-associated and isolated receptor. J. Biol. Chem. 266, 12752-12758.

    PubMed  CAS  Google Scholar 

  • Filion, R.J., Popel, A.S., 2004. A reaction-diffusion model of basic fibroblast growth factor integrations with cell surface receptors. Ann. Biochem. Eng. 32, 645–663.

    Article  Google Scholar 

  • Glotzman, J., Mikula, M., Andreas, E., Schulte-Hermann, R., Foisner, R., Beug, H., Mikulits, W., 2004. Molecular aspects of epithelial cell plasticity;implications for local tumor invasion and metastasis. Mutat. Res. 566, 9–20.

    Article  CAS  Google Scholar 

  • Guba, M., Cernaianu, G., Koehl, G., Geissier, E.K., Jauch, K., Anthuber, M., Falk, W., Steinbauer, M., 2001. A primary tumor promotes dormancy of solitary tumor cells before inhibiting angiogenesis. Cancer Res. 61, 5375–5379.

    Google Scholar 

  • He, L., Niemeyer, B., 2003. A novel correlation for protein diffusion coefficients based on molecular weight and radius of gyration. Biotechnol. Prog. 19, 544–548.

    Article  PubMed  CAS  Google Scholar 

  • Jung, S.P., Siegrist, B., Hornick, C.A., Wang, Y.-Z., Wade, M., Anthony, C.T., Woltering, E.A., 2002. Effect of humen recombinant endostatin∘ledR protein on human angiogenesis. Angiogenesis 5, 111–118.

    Article  PubMed  CAS  Google Scholar 

  • Kirach, M., Schakert, G., Black, P.M., 2000. Angiogenesis, metastasis and endogenous inhibition. J. Neurooncol. 50, 173–180.

    Article  Google Scholar 

  • Lazarous, D.F., Shou, M., Stiber, J.A., Dadhania, D.M., Thirumurti, V., Hodge, E., Unger, E.F., 1997. Pharmacodynamics of basic fibroblast growth factor: Route of administration determines myocardial and systemic distribution. Cardiovasc. Res. 36, 78–85.

    Article  PubMed  CAS  Google Scholar 

  • Leaf, C., 2004. Why we're losing the war on cancer (and how to win it). Fortune 149, 76–97.

    PubMed  Google Scholar 

  • Levine, H.A., Pamuk, S., Sleeman, B.D., Nilsen-Hamilton, M., 2001a. Mathematical modeling of capillary formation and development in tumor angiogenesis: Penetration into the stroma. Bull. Math. Biol. 801–863.

  • Levine, H.A., Sleeman, B.D., Nilsen-Hamilton, M., 2001b. Mathematical modeling of the onset of capillary formation initiating angiogenesis. J. Math. Biol. 195–238.

  • Levine, H.A., Sleeman, B.D., Nilsen-Hamilton, M., 2002a. A mathematical model for the roles of pericytes and macrophages in the initiation of angiogenesis I: The role of protease inhibitors in preventing angiogenesis. Math. Biosci. 77–115.

  • Levine, H.A., Tucker, A.L., Nilsen-Hamilton, M., 2002b. A mathematical model for the role of cell signaling and transduction in the initiation of angiogenesis. Growth Factors 155–176.

  • Lijnen, H.R., Carmeliet, P., Bouche, A., Moons, L., Ploplis, V.A., Plow, E., Collen, D., 1996. Restoration of thrombolytic potential in plasminogen-deficient mice by bolus administration of plasminogen. Blood 88, 870–876.

    PubMed  CAS  Google Scholar 

  • Lyons, R.M., Gentry, L.E., Purchio, A.F., Mosesl, H.L., 1990. Mechanism of activation of latent recombinant transforming growth factor β1 by plasmin. J. Cell Biol. 110, 1361–1367.

    Article  PubMed  CAS  Google Scholar 

  • Lyons, R.M., Keski-Oja, J., Mosesl, H.L., 1988. Proteyolytic activation of latent transforming growth factor-β from fibroblast conditioned medium. J. Cell Biol. 106, 1659–1665.

    Article  PubMed  CAS  Google Scholar 

  • Mansbridge, J.N., Liu, K., Pinney, R.E., Patch, R., Ratcliffe, A., Naugnton, G.K., 1999. Growth factors secreted by fibroblasts: role in healing diabetic foot ulcers. Diabetes, Obes. Metab. 1, 265–279.

    Article  CAS  Google Scholar 

  • Murray, J., 1989. Mathematical Biology, Biomathematics Texts, Springer-Verlag, Berlin Heidelberg New York.

    Google Scholar 

  • Nugent, M.A., Edelman, E.R., 1992. Kinetics of basic fibroblast growth factor binding to its receptor and heparan sulfate proteoglycan: A mechanism for cooperativity. Biochemistry 31, 8876–8883.

    Article  PubMed  CAS  Google Scholar 

  • Othmer, H.G., Stevens, A., 1997. Aggregation, blow up and collapse: The abc's of taxis and reinforced random walks. SIAM J. Appl. Math. 1044–1081.

  • Pavlov, M.Y., Ehernberg, M., 1996. Rate of translation of natural mrnas in an optimized in vitro system. Arch. Biochem. Biophys. 328, 9–16.

    Article  PubMed  CAS  Google Scholar 

  • Ploplis, V.A., Carmeliet, P., Vazirzadeh, S., Van Vlaenderen, I., Moons, L., Plow, E.F., Collen, D., 1995. Effects of disruption of the plasminogen gene on thrombosis, growth, and health in mice. Circulation 92, 2585–2593.

    PubMed  CAS  Google Scholar 

  • Sato, Y., Rifkin, D.B., 1989. 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, 309–315.

    Article  PubMed  CAS  Google Scholar 

  • Sato, Y., Tsuboi, R., Lyons, R., Moses, H., Rifkin, D.B., 1990. Characterization of the activation of latent tgf-β by co-cultures of endothelial cells and pericytes of smooth muscle cells: a self-regulating system. J. Cell Biol. 111, 757–764.

    Article  PubMed  CAS  Google Scholar 

  • Sporn, M.B., 1996. The war on cancer. Lancet 347, 1377–1381.

    Article  PubMed  CAS  Google Scholar 

  • Takeda, Y., Nakabayashi, M., 1974. Physicochemical and biological properties of human and canine plasmins. J. Clin. Invest. 53, 154–162.

    PubMed  CAS  Google Scholar 

  • Tsuzuki, Y., Fukumura, D., Oosthuyse, B., Koike, C., Carmeliet, P., Jain, R.K., 2000. Vascular endothelial growth factor (vegf) modulation by targeting hypoxia-inducible factor-1alpha→ hypoxia response element→ vegf cascade differentially regulates vascular response and growth rate in tumors. Cancer Res. 6248–6252.

  • Voet, D., Voet, J., 1995. Biochemistry, 2nd edn. Wiley, New York.

  • Wakefield, L.M., Winokur, T.S., Hollands, R.S., Christopherson, K., Levinson, A.D., Sporn, M.B., 1990. 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, 1976–7684.

    PubMed  CAS  Google Scholar 

  • Wall, F.T., 1958. Chemical Thermodynamics. Freeman, San Francisco.

    Google Scholar 

  • Whalen, G.F., Shing, Y., Folkman, J., 1989. The fate of intravenously administered bfgf and the effect of heparin. Growth Factors 1, 157–164.

    PubMed  CAS  Google Scholar 

  • Yoon, S.S., Eto, H., Lin, C., Nakamura, H., Pawlik, T.M., Song, S.U., Tanabe, K.K., 1999. Mouse endostatin inhibits the formation of lung and liver metastases. Cancer Res. 99, 6251–6236.

    Google Scholar 

  • Yorke, E.D., Fuks, L., Norton, L., Whitemore, W., Ling, C.C., 1993. Modeling the development of metastases from primary and locally recurrent tumors: comparison with a clinical database for prostatic cancer. Cancer Res. 53, 2987–2993.

    PubMed  CAS  Google Scholar 

  • Zetter, B.R., 1998. Angiogenesis and tumor metastasis, review. Ann. Rev. Med. 49, 407–422.

    Article  PubMed  CAS  Google Scholar 

  • Zioncheck, S.A., Chen, T.F., Richardson, L., Mora-Worms, M., Lucas, C., Lewis, D., Green, J.D., Mordenti, J., 1994. Pharmacokinetics and tissue distribution of recombinant human transforming growth factor beta 1 after topical and intravenous administration in male rats. Pharm. Res. 11, 213–220.

    Article  PubMed  CAS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Iowa State University, Ames, Iowa, 50011, USA

    Khalid Boushaba & Howard A. Levinea

  2. Department of Biochemistry, Biophysics and Molecular Biology, Iowa State University, Ames, Iowa, 50011, USA

    Marit Nilsen-Hamiltonb

Authors
  1. Khalid Boushaba
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Howard A. Levinea
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Marit Nilsen-Hamiltonb
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Khalid Boushaba.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Boushaba, K., Levinea, H.A. & Nilsen-Hamiltonb, M. A Mathematical Model for the Regulation of Tumor Dormancy Based on Enzyme Kinetics. Bull. Math. Biol. 68, 1495–1526 (2006). https://doi.org/10.1007/s11538-005-9042-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11538-005-9042-z

Keywords

Search

Navigation