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March  2016, 21(2): 357-372. doi: 10.3934/dcdsb.2016.21.357

Nutrient limitations as an explanation of Gompertzian tumor growth

Ebraheem O. Alzahrani 1, and  Yang Kuang 2,

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

Received  January 2015 Revised  April 2015 Published  November 2015

An intuitive and influential two-compartment model of cancer cell growth proposed by Gyllenberg and Webb in 1989 [18] with transition rates between proliferating and quiescent cells reproduces some important features of the well known Gompertzian growth model. However, other plausible mechanisms may also be capable of producing similar dynamics. In this paper, we formulate a resource limited three-compartment model of avascular spherical solid tumor growth and study its dynamics. The resource, such as oxygen, is assumed to enter the tumor proportional to its surface area and the dead cells form the necrotic core inside the tumor. We show the tumor growth of our model mimics that of Gompertzian model, and the solutions of our model are naturally bounded. We also identify general and explicit expressions of the tumor final sizes and study the stability of the tumor at steady states. In contrast to the Gyllenberg-Webb model, our model confirms that tumor size at the positive steady state is strictly decreasing function of the dead cell removal rate. We also present two intriguing mathematical open questions.
Keywords: proliferation, resource limitation, Nonlinear tumor model, quiescence, stability..
Mathematics Subject Classification: Primary: 34K20, 92C50; Secondary: 92D2.
Citation: Ebraheem O. Alzahrani, Yang Kuang. Nutrient limitations as an explanation of Gompertzian tumor growth. Discrete & Continuous Dynamical Systems - B, 2016, 21 (2) : 357-372. doi: 10.3934/dcdsb.2016.21.357
References:
[1]

J. A. Adam and N. Bellomo (ed), A Survey of Models on Tumour Immune Systems Dynamics, Birkhüauser, 1997. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

L. von Bertalanffy, Quantitative laws in metabolism and growth, Quart. Rev. Biol., 32 (1957), 217-231. Google Scholar

[6]

A. C. Burton, Rate of growth of solid tumours as a problem of diffusion, Growth, 30 (1966), 157-176. Google Scholar

[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.  Google Scholar

[8]

L. Edelstein-Keshet, Mathematical Models in Biology, SIAM, 1988.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

M. Gyllenberg and G. F. Webb, Quiescence as an explanation of Gompertzian tumor growth, Growth Dev Aging, 53 (1989), 25-33. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

Y. Kuang, Delay Differential Equations: With Applications in Population Dynamics, Academic Press, 1993.  Google Scholar

[23]

A. K. Laird, Dynamics of tumor growth, Brit. J. Cancer, 18 (1964), 490-502. doi: 10.1038/bjc.1964.55.  Google Scholar

[24]

A. O. Martinez and R. J. Griego, Growth dynamics of multicell spheroids from three murine tumors, Growth, 44 (1980), 112-122. Google Scholar

[25]

M. Marusic and S. Vuk-Pavlovic, Prediction power of mathematical models for tumor growth, Journal of Biological Systems, 1 (1993), 69-78. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30]

D. Wallace and X. Guo, Properties of tumor spheroid growth exhibited by simple mathematical models, Frontiers in Oncology, 3 (2013), 1-9. Google Scholar

show all references

References:
[1]

J. A. Adam and N. Bellomo (ed), A Survey of Models on Tumour Immune Systems Dynamics, Birkhüauser, 1997. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

L. von Bertalanffy, Quantitative laws in metabolism and growth, Quart. Rev. Biol., 32 (1957), 217-231. Google Scholar

[6]

A. C. Burton, Rate of growth of solid tumours as a problem of diffusion, Growth, 30 (1966), 157-176. Google Scholar

[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.  Google Scholar

[8]

L. Edelstein-Keshet, Mathematical Models in Biology, SIAM, 1988.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

M. Gyllenberg and G. F. Webb, Quiescence as an explanation of Gompertzian tumor growth, Growth Dev Aging, 53 (1989), 25-33. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

Y. Kuang, Delay Differential Equations: With Applications in Population Dynamics, Academic Press, 1993.  Google Scholar

[23]

A. K. Laird, Dynamics of tumor growth, Brit. J. Cancer, 18 (1964), 490-502. doi: 10.1038/bjc.1964.55.  Google Scholar

[24]

A. O. Martinez and R. J. Griego, Growth dynamics of multicell spheroids from three murine tumors, Growth, 44 (1980), 112-122. Google Scholar

[25]

M. Marusic and S. Vuk-Pavlovic, Prediction power of mathematical models for tumor growth, Journal of Biological Systems, 1 (1993), 69-78. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30]

D. Wallace and X. Guo, Properties of tumor spheroid growth exhibited by simple mathematical models, Frontiers in Oncology, 3 (2013), 1-9. Google Scholar

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