# American Institute of Mathematical Sciences

December  2017, 22(10): 3771-3782. doi: 10.3934/dcdsb.2017189

## Tumor growth dynamics with nutrient limitation and cell proliferation time delay

 1 Department of Mathematics, University of Surrey, Guildford, Surrey GU2 7XH, United Kingdom 2 Department of Mathematics, Faculty of Science, King Abdulaziz University, P. O. Box 80203, Jeddah 21589, Saudi Arabia 3 Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt 4 School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85287, USA

Received  October 2016 Revised  May 2017 Published  July 2017

It is known that avascular spherical solid tumors grow monotonically, often tends to a limiting final size. This is repeatedly confirmed by various mathematical models consisting of mostly ordinary differential equations. However, cell growth is limited by nutrient and its proliferation incurs a time delay. In this paper, we formulate a nutrient limited compartmental model of avascular spherical solid tumor growth with cell proliferation time delay and study its limiting dynamics. The nutrient is assumed to enter the tumor proportional to its surface area. This model is a modification of a recent model which is built on a two-compartment model of cancer cell growth with transitions between proliferating and quiescent cells. Due to the limitation of resources, it is imperative that the population values or densities of a population model be nonnegative and bounded without any technical conditions. We confirm that our model meets this basic requirement. From an explicit expression of the tumor final size we show that the ratio of proliferating cells to the total tumor cells tends to zero as the death rate of quiescent cells tends to zero. We also study the stability of the tumor at steady states even though there is no Jacobian at the trivial steady state. The characteristic equation at the positive steady state is complicated so we made an initial effort to study some special cases in details. We find that delay may not destabilize the positive steady state in a very extreme situation. However, in a more general case, we show that sufficiently long cell proliferation delay can produce oscillatory solutions.

Citation: Ahuod Alsheri, Ebraheem O. Alzahrani, Asim Asiri, Mohamed M. El-Dessoky, Yang Kuang. Tumor growth dynamics with nutrient limitation and cell proliferation time delay. Discrete and Continuous Dynamical Systems - B, 2017, 22 (10) : 3771-3782. doi: 10.3934/dcdsb.2017189
##### References:
 [1] J. A. Adam and B. 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] E. O. Alzahrani and Y. Kuang, Nutrient limitations as an explanation of Gompertzian tumor growth, Discrete Cont. Dyn. Syst.-B., 21 (2016), 357-372.  doi: 10.3934/dcdsb.2016.21.357. [4] 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. [5] L. von Bertalanffy, Quantitative laws in metabolism and growth, Quart. Rev. Biol., 32 (1957), 217-231.  doi: 10.1086/401873. [6] 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. [7] 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. [8] 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. [9] 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. [10] 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. [11] 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. [12] 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. [13] M. Gyllenberg and G. F. Webb, Quiescence as an explanation of Gompertzian tumor growth, Growth Dev Aging, 53 (1989), 25-33. [14] 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. [15] 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. [16] Y. Kuang, Delay Differential Equations: With Applications in Population Dynamics, Academic Press, 1993. [17] Y. Kuang, J. D. Nagy and S. E. Eikenberry Introduction to Mathematical Oncology, CRC Press, 2016. [18] A. O. Martinez and R. J. Griego, Growth dynamics of multicell spheroids from three murine tumors, Growth, 44 (1980), 112-122. [19] M. Marusic and S. Vuk-Pavlovic, Prediction power of mathematical models for tumor growth, Journal of Biological Systems, 1 (1993), 69-78. [20] 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. [21] 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. [22] 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. [23] 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. [24] D. Wallace and X. Guo, Properties of tumor spheroid growth exhibited by simple mathematical models, Frontiers in Oncology, 3 (2013), 1-9.  doi: 10.3389/fonc.2013.00051. [25] R. Yafia, Dynamics analysis and limit cycle in a delayed model for tumor growth with quiescence, Nonlinear Analysis: Modelling and Control, 11 (2006), 95-110.

show all references

##### References:
 [1] J. A. Adam and B. 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] E. O. Alzahrani and Y. Kuang, Nutrient limitations as an explanation of Gompertzian tumor growth, Discrete Cont. Dyn. Syst.-B., 21 (2016), 357-372.  doi: 10.3934/dcdsb.2016.21.357. [4] 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. [5] L. von Bertalanffy, Quantitative laws in metabolism and growth, Quart. Rev. Biol., 32 (1957), 217-231.  doi: 10.1086/401873. [6] 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. [7] 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. [8] 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. [9] 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. [10] 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. [11] 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. [12] 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. [13] M. Gyllenberg and G. F. Webb, Quiescence as an explanation of Gompertzian tumor growth, Growth Dev Aging, 53 (1989), 25-33. [14] 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. [15] 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. [16] Y. Kuang, Delay Differential Equations: With Applications in Population Dynamics, Academic Press, 1993. [17] Y. Kuang, J. D. Nagy and S. E. Eikenberry Introduction to Mathematical Oncology, CRC Press, 2016. [18] A. O. Martinez and R. J. Griego, Growth dynamics of multicell spheroids from three murine tumors, Growth, 44 (1980), 112-122. [19] M. Marusic and S. Vuk-Pavlovic, Prediction power of mathematical models for tumor growth, Journal of Biological Systems, 1 (1993), 69-78. [20] 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. [21] 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. [22] 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. [23] 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. [24] D. Wallace and X. Guo, Properties of tumor spheroid growth exhibited by simple mathematical models, Frontiers in Oncology, 3 (2013), 1-9.  doi: 10.3389/fonc.2013.00051. [25] R. Yafia, Dynamics analysis and limit cycle in a delayed model for tumor growth with quiescence, Nonlinear Analysis: Modelling and Control, 11 (2006), 95-110.
Bifurcation diagrams of model (2.4) with $f(r)= \frac{kr}{ar+1}$ and $g(r)= \frac{c}{r+m}$ using the cell proliferation time delay $\tau$ as the bifurcation parameter. The parameter values are $k=2, a=1, m=2, \mu=0.1, c=1, \theta=2/3.$ The positive steady state appears to be globally attractive for short time delay but lost its stability for larger values of $\tau$. As cell proliferation time delay increases, tumor size oscillates more noticeably and at a lower lever. Notice that the percentage of the average amount of proliferating cells decreases as $\tau$ increases
 [1] Mostafa Adimy, Fabien Crauste, Laurent Pujo-Menjouet. On the stability of a nonlinear maturity structured model of cellular proliferation. Discrete and Continuous Dynamical Systems, 2005, 12 (3) : 501-522. doi: 10.3934/dcds.2005.12.501 [2] Junde Wu, Shihe Xu. Asymptotic behavior of a nonlinear necrotic tumor model with a periodic external nutrient supply. Discrete and Continuous Dynamical Systems - B, 2020, 25 (7) : 2453-2460. doi: 10.3934/dcdsb.2020018 [3] Zejia Wang, Haihua Zhou, Huijuan Song. The impact of time delay and angiogenesis in a tumor model. Discrete and Continuous Dynamical Systems - B, 2022, 27 (7) : 4097-4119. doi: 10.3934/dcdsb.2021219 [4] Lifeng Han, Changhan He, Yang Kuang. Dynamics of a model of tumor-immune interaction with time delay and noise. Discrete and Continuous Dynamical Systems - S, 2020, 13 (9) : 2347-2363. doi: 10.3934/dcdss.2020140 [5] Mary Ballyk, Ross Staffeldt, Ibrahim Jawarneh. A nutrient-prey-predator model: Stability and bifurcations. Discrete and Continuous Dynamical Systems - S, 2020, 13 (11) : 2975-3004. doi: 10.3934/dcdss.2020192 [6] Kazuhiko Yamamoto, Kiyoshi Hosono, Hiroko Nakayama, Akio Ito, Yuichi Yanagi. Experimental data for solid tumor cells: Proliferation curves and time-changes of heat shock proteins. Discrete and Continuous Dynamical Systems - S, 2012, 5 (1) : 235-244. doi: 10.3934/dcdss.2012.5.235 [7] Min Yu, Gang Huang, Yueping Dong, Yasuhiro Takeuchi. Complicated dynamics of tumor-immune system interaction model with distributed time delay. Discrete and Continuous Dynamical Systems - B, 2020, 25 (7) : 2391-2406. doi: 10.3934/dcdsb.2020015 [8] Juping Ji, Lin Wang. Bifurcation and stability analysis for a nutrient-phytoplankton model with toxic effects. Discrete and Continuous Dynamical Systems - S, 2020, 13 (11) : 3073-3081. doi: 10.3934/dcdss.2020135 [9] Zhiqi Lu. Global stability for a chemostat-type model with delayed nutrient recycling. Discrete and Continuous Dynamical Systems - B, 2004, 4 (3) : 663-670. doi: 10.3934/dcdsb.2004.4.663 [10] C. Connell McCluskey. Global stability of an $SIR$ epidemic model with delay and general nonlinear incidence. Mathematical Biosciences & Engineering, 2010, 7 (4) : 837-850. doi: 10.3934/mbe.2010.7.837 [11] Youshan Tao, J. Ignacio Tello. Nonlinear stability of a heterogeneous state in a PDE-ODE model for acid-mediated tumor invasion. Mathematical Biosciences & Engineering, 2016, 13 (1) : 193-207. doi: 10.3934/mbe.2016.13.193 [12] Avner Friedman, Yangjin Kim. Tumor cells proliferation and migration under the influence of their microenvironment. Mathematical Biosciences & Engineering, 2011, 8 (2) : 371-383. doi: 10.3934/mbe.2011.8.371 [13] Ebraheem O. Alzahrani, Yang Kuang. Nutrient limitations as an explanation of Gompertzian tumor growth. Discrete and Continuous Dynamical Systems - B, 2016, 21 (2) : 357-372. doi: 10.3934/dcdsb.2016.21.357 [14] Pham Huu Anh Ngoc. Stability of nonlinear differential systems with delay. Evolution Equations and Control Theory, 2015, 4 (4) : 493-505. doi: 10.3934/eect.2015.4.493 [15] David Schley, S.A. Gourley. Linear and nonlinear stability in a diffusional ecotoxicological model with time delays. Discrete and Continuous Dynamical Systems - B, 2002, 2 (4) : 575-590. doi: 10.3934/dcdsb.2002.2.575 [16] Jianquan Li, Xin Xie, Dian Zhang, Jia Li, Xiaolin Lin. Qualitative analysis of a simple tumor-immune system with time delay of tumor action. Discrete and Continuous Dynamical Systems - B, 2021, 26 (10) : 5227-5249. doi: 10.3934/dcdsb.2020341 [17] Hongying Shu, Lin Wang. Global stability and backward bifurcation of a general viral infection model with virus-driven proliferation of target cells. Discrete and Continuous Dynamical Systems - B, 2014, 19 (6) : 1749-1768. doi: 10.3934/dcdsb.2014.19.1749 [18] Xin-Guang Yang, Jing Zhang, Shu Wang. Stability and dynamics of a weak viscoelastic system with memory and nonlinear time-varying delay. Discrete and Continuous Dynamical Systems, 2020, 40 (3) : 1493-1515. doi: 10.3934/dcds.2020084 [19] Nabil T. Fadai, Michael J. Ward, Juncheng Wei. A time-delay in the activator kinetics enhances the stability of a spike solution to the gierer-meinhardt model. Discrete and Continuous Dynamical Systems - B, 2018, 23 (4) : 1431-1458. doi: 10.3934/dcdsb.2018158 [20] Simone Göttlich, Elisa Iacomini, Thomas Jung. Properties of the LWR model with time delay. Networks and Heterogeneous Media, 2021, 16 (1) : 31-47. doi: 10.3934/nhm.2020032

2021 Impact Factor: 1.497