zbMATH — the first resource for mathematics

A mathematical model to study the effects of drug resistance and vasculature on the response of solid tumors to chemotherapy. (English) Zbl 0947.92013
Summary: A mathematical model is developed that describes the reduction in volume of a vascular tumor in response to specific chemotherapeutic administration strategies. The model consists of a system of partial differential equations governing intratumoral drug concentration and cancer cell density. In the model the tumor is treated as a continuum of two types of cells which differ in their proliferation rates and their responses to the chemotherapeutic agent. The balance between cell proliferation and death within the tumor generates a velocity field which drives expansion or regression of the spheroid. Insight into the tumor’s response to therapy is gained by applying a combination of analytical and numerical techniques to the model equations.

92C50 Medical applications (general)
35Q92 PDEs in connection with biology, chemistry and other natural sciences
65C20 Probabilistic models, generic numerical methods in probability and statistics
Full Text: DOI
[1] Adam, J.A., A mathematical model of tumor growth. ii: effects of geometry and spatial uniformity on stability, Math. biosci., 86, 183, (1987) · Zbl 0634.92002
[2] Adam, J.A., A mathematical model of tumor growth iii: comparison with experiment, Math. biosci., 86, 213, (1987) · Zbl 0634.92003
[3] Adam, J.A.; Maggelakis, S.A., Diffusion regulated growth characterisitcs of a spherical prevascular carcinoma, Bull. math. biol., 52, 549, (1990) · Zbl 0712.92010
[4] Byrne, H.M.; Chaplain, M.A.J., Growth of non-necrotic tumors in the presence and absence of inhibitiors, Math. biosci., 130, 151, (1995) · Zbl 0836.92011
[5] Byrne, H.M.; Chaplain, M.A.J., Growth of necrotic tumors in the presence and absence of inhibitors, Math. biosci., 135, 187, (1996) · Zbl 0856.92010
[6] Greenspan, H.P., Models for the growth of a solid tumor by diffusion, Stud. appl. math., 52, 317, (1972) · Zbl 0257.92001
[7] Greenspan, H.P., On the growth and stability of cell cultures and solid tumors, J. theor. biol., 56, 229, (1976)
[8] Maggelakis, S.A.; Adam, J.A., Mathematical model of prevascular growth of a spherical carcinoma, Math. comput. modelling, 13, 23, (1990) · Zbl 0706.92010
[9] Ward, J.P.; King, J.R., Mathematical modeling of avascular-tumor growth, IMA J. math. appl. med. biol., 14, 39, (1997) · Zbl 0866.92011
[10] Folkman, J.; Hochberg, M., Self-regulation of growth in three-dimensions, J. exp. med., 138, 743, (1973)
[11] Kunz-Schughart, L.A.; Kreutz, M.; Kneuchel, R., Multicellular spheroids: a three- dimensional in vitro culture system to study tumor biology, Int. J. eng. path., 79, 1, (1979)
[12] Sutherland, R.M.; Durand, R.E., Growth and cellular characteristics of multicell spheroids, Recent results cancer res., 95, 510, (1994)
[13] Marusic, M.; Bajzer, Z.; Vuk-Pavlovic, S.; Freyer, J.P., Tumor growth in vivo and as multicellular spheroids compared by mathematical models, Bull. math. biol., 56, 617, (1994) · Zbl 0800.92117
[14] P.D. Senter, J.D. Murray, Development and validation of a mathematical model to describe anti-cancer prodrug activation by antibody – enzyme conjugates, J. Theor. Med., in press · Zbl 0965.92024
[15] Jackson, T.L.; Lubkin, S.R.; Murray, J.D., Theoretical analysis of conjugate localization in two-step cancer chemotherapy, J. math. bio., 39, 353, (1999) · Zbl 0948.92011
[16] Gerlowski, L.E.; Jain, R.K., Microvascular permeability of normal and neoplastic tissues, Microvas. res., 31, 288, (1986)
[17] Jain, R.K., Barriers to drug delivery in solid tumors, Sci. am., 271, 58, (1994)
[18] Siemers, N.O.; Kerr, D.E.; Yarnold, S.; Stebbins, M.; Vrudgyka, V.M.; Hellstrom, I.; Hellstrom, K.E.; Senter, P.D., L49-sfv-β-lactamase a single chain anti-p97 antibody fusion protein, Bioconjugate chem., 8, 510, (1997)
[19] Crank, J., Free and moving boundary problems, Oup, (1988)
[20] Birkhead, B.G.; Rankin, E.M.; Gallivan, S.; Dones, L.; Rubens, R.D., A mathematical model of the development of drug resistance to cancer chemotherapy, Eur. J. cancer clin. oncol., 23, 1421, (1987)
[21] Goldie, J.H.; Coldman, A.J., A mathematical model for relating the drug sensitivity of tumors to their spontaneous mutation rate, Cancer treat. rep., 63, 1727, (1971)
[22] Crowther, D., A rational approach to the chemotherapy of human malignant disease – ii, Rit. med. J., 4, 216, (1974)
[23] Bagshawe, K.D.; Begent, R.H.J., First clinical experience with ADEPT, Adv. drug delivery rev., 22, 365, (1996)
[24] Kerr, D.E.; Schreiber, G.J.; Vrudhula, V.M.; Svensson, H.P.; Hellstrom, K.E.; Senter, P.D., Regressions and cures of melanoma xenografts following treatment with monoclonal antibody β-lactamase conjugates in combination with anticancer prodrugs, Cancer res., 55, 3558, (1995)
[25] Agur, Z.; Arnon, R.; Schechter, B., Reduction of cytotoxicity to normal tissues by new regimens of cell cycle phase specific drugs, Math. biosci., 92, 1, (1988) · Zbl 0655.92004
[26] Byrne, H.M.; Chaplain, M.A.J., Necrosis and apoptosis: distinct cell loss mechanisms in a mathematical model of avascular tumor growth, J. theor. med., 1, 223, (1998) · Zbl 0917.92013
[27] Folkman, J., Fighting cancer by attacking its blood supply, Sci. am., 150, (1996)
[28] Hanahan, D.; Folkman, J., Patterns and emerging mechanisms of the angiogenic switch during tumorigenesis, Cell, 86, 353, (1996)
[29] O’Reilly, M.S.; Boehm, T.; Shing, Y.; Fukai, N.; Vasios, G.; Lane, W.S.; Flynn, E.; Birkhead, J.R.; Folkman, J., Endostatin: an endogenous inhibitor of angiogenesis and tumor growth, Cell, 88, 277, (1997)
[30] Byrne, H.M.; Gourley, S.A., The role of growth factors in avascular tumour growth, Math. comput. modeling, 26, 83, (1997) · Zbl 0898.92018
[31] Byrne, H.M.; Chaplain, M.A.J., Free boundary value problems associated with the growth and development of multicellular spheroids, Eur. J. appl. math., 8, 639, (1997) · Zbl 0906.92016
[32] Byrne, H.M., A weakly nonlinear analysis of a model of avascular solid tumour growth, J. math. bio., 39, 59, (1997) · Zbl 0981.92011
[33] Baxter, L.T.; Yuan, F.; Jain, R.K., Pharamacokinetic analysis of the perivascular distribution of bifunctional antiboies and haptens: comparison with experimental data, Cancer res., 52, 5838, (1992)
[34] Nugent, L.J.; Jain, R.K., Extravascular diffusion in normal and neoplastic tissues, Cancer res., 44, 238, (1984)
[35] Dordal, M.S.; Winter, J.N.; Atkinson, A.J., Kinetic analysis of p-glycoprotein-mediated doxorubicin efflux, J. pharmacol. exp. ther., 263, 762, (1992)
[36] Greenblatt, D.J.; Koch-Weser, J., Drug therapy: clinical pharmacokinetics, NE J. med., 293, 702, (1975)
[37] Jain, R.K., Transport of macromolecules across tumor vasculature, Cancer metastasis rev., 6, 559, (1987)
[38] Robert, J.; Illiadia, A.; Hoerniand, B.; Cano, J.; Durand, M.; Lagarde, C., Pharamacokinetics of adrianmycin in patients with breast cancer: correlation between pharmacokinetic parameters and clinical short-term respons, Eur. J. cancer clin. oncol., 18, 739, (1982)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.