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Special issue dedicated to the memory of Paul Waltman
Nutrient limitations as an explanation of Gompertzian tumor growth
1. | Department of Mathematics, Faculty of Science, King Abdulaziz University, P. O. Box 80203, Jeddah 21589, Saudi Arabia |
2. | School of Mathematics and Statistical Sciences, Arizona State University, Tempe, AZ 85281 |
References:
[1] |
J. A. Adam and N. Bellomo (ed), A Survey of Models on Tumour Immune Systems Dynamics, Birkhüauser, 1997. |
[2] |
E. O. Alzahrani, A. Asiri, M. M. El-Dessoky and Y. Kuang, Quiescence as an explanation of Gompertzian tumor growth revisited, Mathematical Biosciences, 254 (2014), 76-82.
doi: 10.1016/j.mbs.2014.06.009. |
[3] |
R. P. Araujo and D. L. S. McElwain, A history of the study of solid tumour growth: The contribution of mathematical modelling, Bull. Math. Biol., 66 (2004), 1039-1091.
doi: 10.1016/j.bulm.2003.11.002. |
[4] |
E. Beretta and Y. Kuang, Modeling and analysis of a marine bacteriophage infection with latency period, Nonlinear Analysis: Real World Applications, 2 (2001), 35-74.
doi: 10.1016/S0362-546X(99)00285-0. |
[5] |
L. von Bertalanffy, Quantitative laws in metabolism and growth, Quart. Rev. Biol., 32 (1957), 217-231. |
[6] |
A. C. Burton, Rate of growth of solid tumours as a problem of diffusion, Growth, 30 (1966), 157-176. |
[7] |
H. M. Byrne, J. R. King, D. L. S. McElwain and L. Preziosi, A two-phase model of solid tumour growth, Applied Mathematics Letters, 16 (2003), 567-573.
doi: 10.1016/S0893-9659(03)00038-7. |
[8] |
L. Edelstein-Keshet, Mathematical Models in Biology, SIAM, 1988. |
[9] |
S. E. Eikenberry, J. D. Nagy and Y. Kuang, The evolutionary impact of androgen levels on prostate cancer in a multi-scale mathematical model, Biology Direct, 5 (2010), p24.
doi: 10.1186/1745-6150-5-24. |
[10] |
S. E. Eikenberry, T. Sanker, M. C. Preul, E. J. Kostelich, C. Thalhauser and Y. Kuang, The virtual glioblastoma: Growth, migration, and treatment in a three-dimensional mathematical model, Cell Proliferation, 42 (2009), 511-528. (pdf 987k).
doi: 10.1111/j.1365-2184.2009.00613.x. |
[11] |
S. E. 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, 18pp.
doi: 10.1371/journal.pcbi.1000362. |
[12] |
R. A. Everett, Y. Zhao, K. B. Flores and Y. Kuang, Data and implication based comparison of two chronic myeloid leukemia models, Math. Biosc. Eng., 10 (2013), 1501-1518.
doi: 10.3934/mbe.2013.10.1501. |
[13] |
C. L. Frenzen and J. D. Murray, A cell kinetics justification for Gompertz equation, SIAM J. Appl. Math., 46 (1986), 614-629.
doi: 10.1137/0146042. |
[14] |
S. Geritz and M. Gyllenberg, A mechanistic derivation of the DeAngelis-Beddington functional response, Journal of Theoretical Biology, 314 (2012), 106-108.
doi: 10.1016/j.jtbi.2012.08.030. |
[15] |
B. Gompertz, On the nature of the function expressive of the law of human mortality, and on a new mode of determining the value of life contingencies, Phil. Trans. Royal Soc. London, 115 (1825), 513-583. |
[16] |
S. A. Gourley and Y. Kuang, A stage structured predator-prey model and its dependence on maturation delay and death rate, J. of Math. Biol., 49 (2004), 188-200.
doi: 10.1007/s00285-004-0278-2. |
[17] |
H. P. Greenspan, Models for the growth of a solid tumor by diffusion, Stud. Appl. Math., 51 (1972), 317-340.
doi: 10.1002/sapm1972514317. |
[18] |
M. Gyllenberg and G. F. Webb, Quiescence as an explanation of Gompertzian tumor growth, Growth Dev Aging, 53 (1989), 25-33. |
[19] |
M. Gyllenberg and G. F. Webb, A nonlinear structured population model of tumor growth with quiescence, J. Math. Biol., 28 (1990), 671-694.
doi: 10.1007/BF00160231. |
[20] |
F. Kozusko and Z. Bajzer, Combining Gompertzian growth and cell population dynamics, Math. Biosc., 185 (2003), 153-167.
doi: 10.1016/S0025-5564(03)00094-4. |
[21] |
Y. Kuang, Global stability for a class of nonlinear nonautonomous delay equations, Nonlinear Analysis: TMA, 17 (1991), 627-634.
doi: 10.1016/0362-546X(91)90110-M. |
[22] |
Y. Kuang, Delay Differential Equations: With Applications in Population Dynamics, Academic Press, 1993. |
[23] |
A. K. Laird, Dynamics of tumor growth, Brit. J. Cancer, 18 (1964), 490-502.
doi: 10.1038/bjc.1964.55. |
[24] |
A. O. Martinez and R. J. Griego, Growth dynamics of multicell spheroids from three murine tumors, Growth, 44 (1980), 112-122. |
[25] |
M. Marusic and S. Vuk-Pavlovic, Prediction power of mathematical models for tumor growth, Journal of Biological Systems, 1 (1993), 69-78. |
[26] |
L. Norton, R. Simon, H. D. Brereton and A. E. Bogden, Predicting the course of Gompertzian growth, Nature, 264 (1976), 542-545.
doi: 10.1038/264542a0. |
[27] |
T. Portz, Y. Kuang and J. D. Nagy, A clinical data validated mathematical model of prostate cancer growth under intermittent androgen suppression therapy, AIP Advances, 2 (2012), 011002.
doi: 10.1063/1.3697848. |
[28] |
J. A. Sherratt and M. J. Chaplain, A new mathematical model for avascular tumor growth, J. Math. Biol., 43 (2001), 291-312.
doi: 10.1007/s002850100088. |
[29] |
C. J. Thalhauser, T. Sankar, M. C. Preul and Y. Kuang, Explicit separation of growth and motility in a new tumor cord model, Bulletin of Math. Biol., 71 (2009), 585-601.
doi: 10.1007/s11538-008-9372-8. |
[30] |
D. Wallace and X. Guo, Properties of tumor spheroid growth exhibited by simple mathematical models, Frontiers in Oncology, 3 (2013), 1-9. |
show all references
References:
[1] |
J. A. Adam and N. Bellomo (ed), A Survey of Models on Tumour Immune Systems Dynamics, Birkhüauser, 1997. |
[2] |
E. O. Alzahrani, A. Asiri, M. M. El-Dessoky and Y. Kuang, Quiescence as an explanation of Gompertzian tumor growth revisited, Mathematical Biosciences, 254 (2014), 76-82.
doi: 10.1016/j.mbs.2014.06.009. |
[3] |
R. P. Araujo and D. L. S. McElwain, A history of the study of solid tumour growth: The contribution of mathematical modelling, Bull. Math. Biol., 66 (2004), 1039-1091.
doi: 10.1016/j.bulm.2003.11.002. |
[4] |
E. Beretta and Y. Kuang, Modeling and analysis of a marine bacteriophage infection with latency period, Nonlinear Analysis: Real World Applications, 2 (2001), 35-74.
doi: 10.1016/S0362-546X(99)00285-0. |
[5] |
L. von Bertalanffy, Quantitative laws in metabolism and growth, Quart. Rev. Biol., 32 (1957), 217-231. |
[6] |
A. C. Burton, Rate of growth of solid tumours as a problem of diffusion, Growth, 30 (1966), 157-176. |
[7] |
H. M. Byrne, J. R. King, D. L. S. McElwain and L. Preziosi, A two-phase model of solid tumour growth, Applied Mathematics Letters, 16 (2003), 567-573.
doi: 10.1016/S0893-9659(03)00038-7. |
[8] |
L. Edelstein-Keshet, Mathematical Models in Biology, SIAM, 1988. |
[9] |
S. E. Eikenberry, J. D. Nagy and Y. Kuang, The evolutionary impact of androgen levels on prostate cancer in a multi-scale mathematical model, Biology Direct, 5 (2010), p24.
doi: 10.1186/1745-6150-5-24. |
[10] |
S. E. Eikenberry, T. Sanker, M. C. Preul, E. J. Kostelich, C. Thalhauser and Y. Kuang, The virtual glioblastoma: Growth, migration, and treatment in a three-dimensional mathematical model, Cell Proliferation, 42 (2009), 511-528. (pdf 987k).
doi: 10.1111/j.1365-2184.2009.00613.x. |
[11] |
S. E. 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, 18pp.
doi: 10.1371/journal.pcbi.1000362. |
[12] |
R. A. Everett, Y. Zhao, K. B. Flores and Y. Kuang, Data and implication based comparison of two chronic myeloid leukemia models, Math. Biosc. Eng., 10 (2013), 1501-1518.
doi: 10.3934/mbe.2013.10.1501. |
[13] |
C. L. Frenzen and J. D. Murray, A cell kinetics justification for Gompertz equation, SIAM J. Appl. Math., 46 (1986), 614-629.
doi: 10.1137/0146042. |
[14] |
S. Geritz and M. Gyllenberg, A mechanistic derivation of the DeAngelis-Beddington functional response, Journal of Theoretical Biology, 314 (2012), 106-108.
doi: 10.1016/j.jtbi.2012.08.030. |
[15] |
B. Gompertz, On the nature of the function expressive of the law of human mortality, and on a new mode of determining the value of life contingencies, Phil. Trans. Royal Soc. London, 115 (1825), 513-583. |
[16] |
S. A. Gourley and Y. Kuang, A stage structured predator-prey model and its dependence on maturation delay and death rate, J. of Math. Biol., 49 (2004), 188-200.
doi: 10.1007/s00285-004-0278-2. |
[17] |
H. P. Greenspan, Models for the growth of a solid tumor by diffusion, Stud. Appl. Math., 51 (1972), 317-340.
doi: 10.1002/sapm1972514317. |
[18] |
M. Gyllenberg and G. F. Webb, Quiescence as an explanation of Gompertzian tumor growth, Growth Dev Aging, 53 (1989), 25-33. |
[19] |
M. Gyllenberg and G. F. Webb, A nonlinear structured population model of tumor growth with quiescence, J. Math. Biol., 28 (1990), 671-694.
doi: 10.1007/BF00160231. |
[20] |
F. Kozusko and Z. Bajzer, Combining Gompertzian growth and cell population dynamics, Math. Biosc., 185 (2003), 153-167.
doi: 10.1016/S0025-5564(03)00094-4. |
[21] |
Y. Kuang, Global stability for a class of nonlinear nonautonomous delay equations, Nonlinear Analysis: TMA, 17 (1991), 627-634.
doi: 10.1016/0362-546X(91)90110-M. |
[22] |
Y. Kuang, Delay Differential Equations: With Applications in Population Dynamics, Academic Press, 1993. |
[23] |
A. K. Laird, Dynamics of tumor growth, Brit. J. Cancer, 18 (1964), 490-502.
doi: 10.1038/bjc.1964.55. |
[24] |
A. O. Martinez and R. J. Griego, Growth dynamics of multicell spheroids from three murine tumors, Growth, 44 (1980), 112-122. |
[25] |
M. Marusic and S. Vuk-Pavlovic, Prediction power of mathematical models for tumor growth, Journal of Biological Systems, 1 (1993), 69-78. |
[26] |
L. Norton, R. Simon, H. D. Brereton and A. E. Bogden, Predicting the course of Gompertzian growth, Nature, 264 (1976), 542-545.
doi: 10.1038/264542a0. |
[27] |
T. Portz, Y. Kuang and J. D. Nagy, A clinical data validated mathematical model of prostate cancer growth under intermittent androgen suppression therapy, AIP Advances, 2 (2012), 011002.
doi: 10.1063/1.3697848. |
[28] |
J. A. Sherratt and M. J. Chaplain, A new mathematical model for avascular tumor growth, J. Math. Biol., 43 (2001), 291-312.
doi: 10.1007/s002850100088. |
[29] |
C. J. Thalhauser, T. Sankar, M. C. Preul and Y. Kuang, Explicit separation of growth and motility in a new tumor cord model, Bulletin of Math. Biol., 71 (2009), 585-601.
doi: 10.1007/s11538-008-9372-8. |
[30] |
D. Wallace and X. Guo, Properties of tumor spheroid growth exhibited by simple mathematical models, Frontiers in Oncology, 3 (2013), 1-9. |
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