August 2018, 23(6): 2393-2414. doi: 10.3934/dcdsb.2018060

Qualitative analysis of kinetic-based models for tumor-immune system interaction

1. 

BCAM -Basque Center for Applied Mathematics, azarredo, 14, E-48009 Bilbao, Basque Country -Spain

2. 

Dipartimento di Scienze Matematiche, Fisiche e Informatiche, Università di Parma, Parco Area delle Scienze 53/A, 43124 Parma, Italy

* Corresponding author: Maria Groppi

Received  April 2017 Revised  September 2017 Published  February 2018

A mathematical model, based on a mesoscopic approach, describing the competition between tumor cells and immune system in terms of kinetic integro-differential equations is presented. Four interacting components are considered, representing, respectively, tumors cells, cells of the host environment, cells of the immune system, and interleukins, which are capable to modify the tumor-immune system interaction and to contribute to destroy tumor cells. The internal state variable (activity) measures the capability of a cell of prevailing in a binary interaction. Under suitable assumptions, a closed set of autonomous ordinary differential equations is then derived by a moment procedure and two three-dimensional reduced systems are obtained in some partial quasi-steady state approximations. Their qualitative analysis is finally performed, with particular attention to equilibria and their stability, bifurcations, and their meaning. Results are obtained on asymptotically autonomous dynamical systems, and also on the occurrence of a particular backward bifurcation.

Citation: Martina Conte, Maria Groppi, Giampiero Spiga. Qualitative analysis of kinetic-based models for tumor-immune system interaction. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2393-2414. doi: 10.3934/dcdsb.2018060
References:
[1]

J. Adam and N. Bellomo, A Survey of Models on Tumor Immune System Dynamics, Birkäuser, Boston, 1996. doi: 10.1007/978-0-8176-8119-7.

[2]

L. Arlotti and M. Lachowicz, Qualitative analysis of a non-linear integro-differential equation modelling tumor-host dynamics, Math. Comput. Model., 23 (1996), 11-29. doi: 10.1016/0895-7177(96)00017-9.

[3]

N. Bellomo and M. Delitala, From the mathematical kinetic, and stochastic game theory to modelling mutations, onset, progression and immune competition of cancer cells, Physics of Life Reviews, 5 (2008), 183-206. doi: 10.1016/j.plrev.2008.07.001.

[4]

N. Bellomo and G. Forni, Dynamics of tumor interaction with the host immune system, Math. Comput. Model., 20 (1994), 107-122. doi: 10.1016/0895-7177(94)90223-2.

[5]

A. Bellouquid and E. de Angelis, From kinetic models of multicellular growing systems to macroscopic biological tissue models, Nonlinear Anal. Real World Appl., 12 (2011), 1111-1122. doi: 10.1016/j.nonrwa.2010.09.005.

[6]

V. C. BoffiV. Protopopescu and G. Spiga, On the equivalence between the probabilistic, kinetic, and scattering kernel formulations of the Boltzmann equation, Physica A, 164 (1990), 400-410. doi: 10.1016/0378-4371(90)90203-5.

[7]

B. Buonomo and D. Lacitignola, On the backward bifurcation of a vaccination model with nonlinear incidence, Nonlinear Anal. Model. Control, 16 (2011), 30-46.

[8]

C. Castillo-Chavez and B. Song, Dynamical models of tubercolosis and their applications, Math. Biosci. Eng., 1 (2004), 361-404. doi: 10.3934/mbe.2004.1.361.

[9]

C. Castillo-Chavez and H. R. Thieme, Asymptotically autonomous epidemic models, in Mathematical Population Dynamics: Analysis of Heterogeneity 1 (ed. O. Arino), Wuerz, (1995), 33–50.

[10]

C. Cercignani, The Boltzmann Equation and Its Applications, Springer, New York, 1988.

[11]

E. de Angelis and B. Lods, On the kinetic theory for active particles: A model for tumor-immune system competition, Math. Comput. Model., 47 (2008), 196-209. doi: 10.1016/j.mcm.2007.02.016.

[12]

L. G. de Pillis and A. Radunskaya, A mathematical model of immune response to tumor invasion, in Computational Fluid and Solid Mechanics 2003. Proceedings of the second M. I. T. conference on computational fluid dynamics and solid mechanics (ed. K. J. Bathe), Elsevier Science, (2003), 1661-1668.

[13]

L. G. de PillisA. E. Radunskaya and C. I. Wiseman, A validated mathematical model of cell-mediated immune response to tumor growth, Cancer Res., 65 (2005), 7950-7958.

[14]

A. d'Onofrio, Metamodeling tumor-immune system interaction, tumor evasion and immunotherapy, Math. Comput. Model., 47 (2008), 614-637. doi: 10.1016/j.mcm.2007.02.032.

[15]

J. Duderstadt and W. Martin, Transport Theory, Wiley, New York, 1979.

[16]

R. EftimieJ. L. Bramson and D. J. D. Earn, Interactions between the immune system and cancer: A brief review of non-spatial mathematical models, Bull. Math. Biol., 73 (2011), 2-32. doi: 10.1007/s11538-010-9526-3.

[17]

R. EftimieJ. J. Gillard and D. A. Cantrell, Mathematical models for immunology: Current state of the art and future research directions, Bull. Math. Biol., 78 (2016), 2091-2134. doi: 10.1007/s11538-016-0214-9.

[18]

J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer, Berlin, 1983.

[19]

K. P. Hadeler and C. Castillo-Chavez, A core group model for disease transmission, Math. Biosci., 128 (1995), 41-55. doi: 10.1016/0025-5564(94)00066-9.

[20]

M. IoriG. Nespi and G. Spiga, Analysis of a kinetic cellular model for tumor-immune system interaction, Math. Comput. Model., 29 (1999), 117-129. doi: 10.1016/S0895-7177(99)00075-8.

[21]

E. Jäger and L. A. Segel, On the distribution of dominance in population of social organisms, SIAM J. Appl. Math., 52 (1992), 1442-1468. doi: 10.1137/0152083.

[22]

D. Kirschner and J. C. Panetta, Modeling immunotherapy of the tumor-immune interaction, J. Math. Biol., 37 (1998), 235-252. doi: 10.1007/s002850050127.

[23]

M. Kolev, Mathematical modelling of the competition between tumors and immune system considering the role of the antibodies, Math. Comput. Model., 37 (2003), 1143-1152. doi: 10.1016/S0895-7177(03)80018-3.

[24]

M. Kolev, A mathematical model of cellular immune response to leukemia, Math. Comput. Model., 41 (2005), 1071-1081. doi: 10.1016/j.mcm.2005.05.003.

[25]

Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, Applied Mathematical Sciences, Vol. 112, Springer Science + Business Media, New York, 1998.

[26]

K. L. LiaoX. F. Bai and A. Friedman, Mathematical modeling of Interleukin-35 promoting tumor growth and angiogenesis, PLoS One, 9 (2014), e110126. doi: 10.1371/journal.pone.0110126.

[27]

R. H. Martin Jr, Nonlinear Operators and Differential Equations in Banach Space, Wiley, New York, 1976.

[28]

A. MerolaC. Cosentino and F. Amato, An insight into tumor dormancy equilibrium via the analysis of its domain of attraction, Biomed. Signal Process. Control, 3 (2008), 212-219. doi: 10.1016/j.bspc.2008.02.001.

[29]

L. Preziosi, From population dynamics to modelling the competition between tumors and immune system, Math. Comput. Model., 23 (1996), 135-152. doi: 10.1016/0895-7177(96)00023-4.

[30]

H. R. Thieme, Asymptotically autonomous differential equations in the plane, Rocky Mt. J. Math., 24 (1993), 351-380.

show all references

References:
[1]

J. Adam and N. Bellomo, A Survey of Models on Tumor Immune System Dynamics, Birkäuser, Boston, 1996. doi: 10.1007/978-0-8176-8119-7.

[2]

L. Arlotti and M. Lachowicz, Qualitative analysis of a non-linear integro-differential equation modelling tumor-host dynamics, Math. Comput. Model., 23 (1996), 11-29. doi: 10.1016/0895-7177(96)00017-9.

[3]

N. Bellomo and M. Delitala, From the mathematical kinetic, and stochastic game theory to modelling mutations, onset, progression and immune competition of cancer cells, Physics of Life Reviews, 5 (2008), 183-206. doi: 10.1016/j.plrev.2008.07.001.

[4]

N. Bellomo and G. Forni, Dynamics of tumor interaction with the host immune system, Math. Comput. Model., 20 (1994), 107-122. doi: 10.1016/0895-7177(94)90223-2.

[5]

A. Bellouquid and E. de Angelis, From kinetic models of multicellular growing systems to macroscopic biological tissue models, Nonlinear Anal. Real World Appl., 12 (2011), 1111-1122. doi: 10.1016/j.nonrwa.2010.09.005.

[6]

V. C. BoffiV. Protopopescu and G. Spiga, On the equivalence between the probabilistic, kinetic, and scattering kernel formulations of the Boltzmann equation, Physica A, 164 (1990), 400-410. doi: 10.1016/0378-4371(90)90203-5.

[7]

B. Buonomo and D. Lacitignola, On the backward bifurcation of a vaccination model with nonlinear incidence, Nonlinear Anal. Model. Control, 16 (2011), 30-46.

[8]

C. Castillo-Chavez and B. Song, Dynamical models of tubercolosis and their applications, Math. Biosci. Eng., 1 (2004), 361-404. doi: 10.3934/mbe.2004.1.361.

[9]

C. Castillo-Chavez and H. R. Thieme, Asymptotically autonomous epidemic models, in Mathematical Population Dynamics: Analysis of Heterogeneity 1 (ed. O. Arino), Wuerz, (1995), 33–50.

[10]

C. Cercignani, The Boltzmann Equation and Its Applications, Springer, New York, 1988.

[11]

E. de Angelis and B. Lods, On the kinetic theory for active particles: A model for tumor-immune system competition, Math. Comput. Model., 47 (2008), 196-209. doi: 10.1016/j.mcm.2007.02.016.

[12]

L. G. de Pillis and A. Radunskaya, A mathematical model of immune response to tumor invasion, in Computational Fluid and Solid Mechanics 2003. Proceedings of the second M. I. T. conference on computational fluid dynamics and solid mechanics (ed. K. J. Bathe), Elsevier Science, (2003), 1661-1668.

[13]

L. G. de PillisA. E. Radunskaya and C. I. Wiseman, A validated mathematical model of cell-mediated immune response to tumor growth, Cancer Res., 65 (2005), 7950-7958.

[14]

A. d'Onofrio, Metamodeling tumor-immune system interaction, tumor evasion and immunotherapy, Math. Comput. Model., 47 (2008), 614-637. doi: 10.1016/j.mcm.2007.02.032.

[15]

J. Duderstadt and W. Martin, Transport Theory, Wiley, New York, 1979.

[16]

R. EftimieJ. L. Bramson and D. J. D. Earn, Interactions between the immune system and cancer: A brief review of non-spatial mathematical models, Bull. Math. Biol., 73 (2011), 2-32. doi: 10.1007/s11538-010-9526-3.

[17]

R. EftimieJ. J. Gillard and D. A. Cantrell, Mathematical models for immunology: Current state of the art and future research directions, Bull. Math. Biol., 78 (2016), 2091-2134. doi: 10.1007/s11538-016-0214-9.

[18]

J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer, Berlin, 1983.

[19]

K. P. Hadeler and C. Castillo-Chavez, A core group model for disease transmission, Math. Biosci., 128 (1995), 41-55. doi: 10.1016/0025-5564(94)00066-9.

[20]

M. IoriG. Nespi and G. Spiga, Analysis of a kinetic cellular model for tumor-immune system interaction, Math. Comput. Model., 29 (1999), 117-129. doi: 10.1016/S0895-7177(99)00075-8.

[21]

E. Jäger and L. A. Segel, On the distribution of dominance in population of social organisms, SIAM J. Appl. Math., 52 (1992), 1442-1468. doi: 10.1137/0152083.

[22]

D. Kirschner and J. C. Panetta, Modeling immunotherapy of the tumor-immune interaction, J. Math. Biol., 37 (1998), 235-252. doi: 10.1007/s002850050127.

[23]

M. Kolev, Mathematical modelling of the competition between tumors and immune system considering the role of the antibodies, Math. Comput. Model., 37 (2003), 1143-1152. doi: 10.1016/S0895-7177(03)80018-3.

[24]

M. Kolev, A mathematical model of cellular immune response to leukemia, Math. Comput. Model., 41 (2005), 1071-1081. doi: 10.1016/j.mcm.2005.05.003.

[25]

Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, Applied Mathematical Sciences, Vol. 112, Springer Science + Business Media, New York, 1998.

[26]

K. L. LiaoX. F. Bai and A. Friedman, Mathematical modeling of Interleukin-35 promoting tumor growth and angiogenesis, PLoS One, 9 (2014), e110126. doi: 10.1371/journal.pone.0110126.

[27]

R. H. Martin Jr, Nonlinear Operators and Differential Equations in Banach Space, Wiley, New York, 1976.

[28]

A. MerolaC. Cosentino and F. Amato, An insight into tumor dormancy equilibrium via the analysis of its domain of attraction, Biomed. Signal Process. Control, 3 (2008), 212-219. doi: 10.1016/j.bspc.2008.02.001.

[29]

L. Preziosi, From population dynamics to modelling the competition between tumors and immune system, Math. Comput. Model., 23 (1996), 135-152. doi: 10.1016/0895-7177(96)00023-4.

[30]

H. R. Thieme, Asymptotically autonomous differential equations in the plane, Rocky Mt. J. Math., 24 (1993), 351-380.

Figure 1.  Phase portrait of system (15) for $A>1/X$
Figure 2.  Comparison of the time evolution of solutions to system (14) and (15) (thickest curves) for $G = 5$ (left) and $G = 50$ (right)
Figure 3.  Nullcline surfaces of system (15)
Figure 4.  Comparison between the trajectories of the nonautonomous system (18) (solid curves) and of the limit system (19) (dashed curves) respectively; the dotted line represents the intersection of the tangent plane to the stable manifold in $E_2$ with the plane $Y_3 = 0$, that can be considered an approximation of the right boundary of $R$; the curve $\gamma$ is dash-dotted
Figure 5.  Solutions for increasing values of D
Figure 6.  Qualitative bifurcation diagram versus $A$ for $C^*<BG/F+G/(FX)$ : forward bifurcation of equilibria (parameter values used: $B = 1, C^* = 4.5, F = 1, G = 1, X = 1/5$)
Figure 7.  Qualitative bifurcation diagram versus $A$ for $C^*>BG/F+G/(FX)$ : backward bifurcation of equilibria (parameter values used: $B = 1, C^* = 9, F = 1, G = 1, X = 1/5$)
Figure 8.  Phase portrait, representative of the case $C^*>BG/F+G/(FX)$ and $A^*<A<1/X$
Figure 9.  Comparison of the time evolution of solutions to system (14) and (21) (thickest curves) for $D = 1$ (left) and $D = 10$ (right), with $D/E = 1.5$
Table 1.  Threshold values $D^*$ versus initial data $Y_{10}$
$Y_{10}$ $0.2$ $0.3$ $0.4$ $0.5$ $0.6$ $0.7$ $0.8$ $1.0$ $2.0$
$D^*$ $1.43$ $2.7$ $4.08$ $5.52$ $7.02$ $8.54$ $10.1$ $13.26$ $29.67$
$Y_{10}$ $0.2$ $0.3$ $0.4$ $0.5$ $0.6$ $0.7$ $0.8$ $1.0$ $2.0$
$D^*$ $1.43$ $2.7$ $4.08$ $5.52$ $7.02$ $8.54$ $10.1$ $13.26$ $29.67$
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