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April  2022, 21(4): 1447-1479. doi: 10.3934/cpaa.2022025

## On a macrophage and tumor cell chemotaxis system with both paracrine and autocrine loops

 1 School of Mathematical Sciences, Chongqing Normal University, Chongqing, 401131, China 2 Department of Mathematics, University of Miami, Coral Gables, FL33146, USA 3 Sylvester Comprehensive Cancer Center, University of Miami Miller School of Medicine, Miami, FL 33136, USA

*Corresponding author

Received  July 2021 Revised  November 2021 Published  April 2022 Early access  February 2022

Fund Project: Research was partially supported by Chongqing Science and Technology Commission Project (No. sctc2020jcyj-msxmX0560), Research Project of Chongqing Education Commission (No. CXQT21014), China Postdoctoral Science Foundation (No. 2017M622990, No. 2018T110956), and NSFC (No.11701461, No. 11771168)

In this paper, we consider a homogeneous Neumann initial-boundary value problem (IBVP) for the following two-species and two-stimuli chemotaxis model with both paracrine and autocrine loops:
 \begin{equation*} \label{IBVP} \left\{ \begin{aligned} &u_t = \nabla\cdot(D_1(u)\nabla u-S_1(u)\nabla v), &\qquad x\in\Omega, \, t>0, \\ & \tau_1 v_t = \Delta v- v+w, &\qquad x\in\Omega, \, t>0, \\ &w_t = \nabla\cdot(D_2(w)\nabla w-S_2(w)\nabla z-S_3(w)\nabla v), &\qquad x\in\Omega, \, t>0, \\ & \tau_2 z_t = \Delta z- z+ u, &\qquad x\in\Omega, \, t>0, \end{aligned} \right. \end{equation*}
where
 $u(t, x)$
and
 $w(t, x)$
denote the density of macrophages and tumor cells at time
 $t$
and location
 $x\in \Omega,$
respectively,
 $v(t, x)$
and
 $z(t, x)$
represent the concentration of colony stimulating factor 1 (CSF-1) secreted by the tumor cells and epidermal growth factor (EGF) secreted by macrophages at time
 $t$
and location
 $x\in \Omega,$
respectively.
 $\Omega\subset \mathbb{R}^n$
is a bounded region with smooth boundary,
 $\tau_i\ge 0 \; (i = 1, 2)$
,
 $D_i(s)\ge d_i(s+1)^{m_i-1}$
with parameters
 $m_i\ge 1 \; (i = 1, 2)$
and
 $S_j(s)\lesssim (s+1)^{q_j}$
with parameters
 $q_j>0 \;(j = 1, 2, 3)$
. For the case without autocrine loop (i.e.,
 $S_3(w) = 0$
), it is shown that when
 $q_j\le 1 \; (j = 1, 2)$
, if one of
 $q_j$
is smaller than one or one of
 $m_i$
is larger than one, then the IBVP has a global classical solution which is uniformly bounded. Moreover, when
 $m_1 = m_2 = q_1 = q_2 = 1$
, an inequality involving the product
 $d_1d_2$
and the product of the two species' initial mass is obtained which guarantees the existence of global bounded classical solutions. More specifically, it allows one of
 $d_i$
to be small or one of the species initial mass to be large. For the case with autocrine loop (i.e
 $S_3(w)\ne 0$
), similar results hold only if
 $q_3<1$
. If
 $q_3 = 1$
, solutions to the IBVP exist globally only when
 $d_2$
is suitably large or the mass of species
 $w$
is suitably small.
Citation: Li Xie, Shigui Ruan. On a macrophage and tumor cell chemotaxis system with both paracrine and autocrine loops. Communications on Pure and Applied Analysis, 2022, 21 (4) : 1447-1479. doi: 10.3934/cpaa.2022025
##### References:
 [1] N. D. Alikakos, $L^p$ bounds of solutions of reaction-diffusion equations, Commun. Partial Differ. Equ., 4 (1979), 827-868.  doi: 10.1080/03605307908820113. [2] A. H. Beck, I. Espinosa, B. Edris, R. Li, K. Montgomery, S. Zhu, S. Varma, R.J. Marinelli, M. van de Rijn and R. B. West, The macrophage colony-stimulating factor response signature in breast carcinoma, Clin. Cancer Res., 15 (2009), 778-787. [3] N. Bellomo, A. Bellouquid, Y. Tao and M. Winkler, Toward a mathematical theory of Keller-Segel models of pattern formation in biological tissues, Math. Models Methods Appl. Sci., 25 (2015), 1663-1763.  doi: 10.1142/S021820251550044X. [4] P. Biler, Global solutions to some parabolic-elliptic systems of chemotaxis, Adv. Math. Sci.Appl., 9 (1999), 347-359. [5] P. Biler, W. Hebisch and T. Nadzieja, The Debye system: Existence and large time behavior of solutions, Nonlinear Anal., 23 (1994), 1189-1209.  doi: 10.1016/0362-546X(94)90101-5. [6] X. Cao, Global bounded solutions of the higher-dimensional Keller-Segel system under smallness conditions in optimal spaces, Discrete Contin. Dyn. Syst., 35 (2015), 1891-1904.  doi: 10.3934/dcds.2015.35.1891. [7] T. Cieślak and M. Winkler, Finite-time blow-up in a quasilinear system of chemotaxis, Nonlinearity, 21 (2008), 1057-1076.  doi: 10.1088/0951-7715/21/5/009. [8] T. Cieślak and C. Stinner, Finite-time blowup and global-in-time unbounded solutions to a parabolic-parabolic quasilinear Keller-Segel system in higher dimensions, J. Differ. Equ., 252 (2012), 5832-5851.  doi: 10.1016/j.jde.2012.01.045. [9] T. Cieślak and C. Stinner, New critical exponents in a fully parabolic quasinear Keller-Segel system and applications to volume filling models, J. Differ. Equ., 258 (2015), 2080-2113.  doi: 10.1016/j.jde.2014.12.004. [10] C. Conca, E. Espejo and K. Vilches, Remarks on the blowup and global existence for a two species chemotactic Keller-Segel system in $\mathbb{R}^2$, Eur. J. Appl. Math., 22 (2011), 553-580.  doi: 10.1017/S0956792511000258. [11] E. Espejo Arenas, A. Stevens and J. Velázquez, Simultaneous finite time blow-up in a two-species model for chemotaxis, Analysis (Munich), 29 (2009), 317-338.  doi: 10.1524/anly.2009.1029. [12] A. Friedman, Partial Differential Equations, Holt, Rinehart and Winston, New York, 1969. [13] A. S. Harney, Real-time imaging reveals local, transient vascular permeability, and tumor cell intravasation stimulated by TIE2hi macrophage-derived VEGFA, Cancer Discovery, 5 (2015), 932-943. [14] M. Herrero and J. Velázquez, A blow-up mechanism for a chemotaxis model, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 24 (1997), 633-683. [15] T. Hillen and K. J. Painter, Global existence for a parabolic chemotaxis model with prevention of overcrowding, Adv. Appl. Math., 26 (2001), 280-301.  doi: 10.1006/aama.2001.0721. [16] D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system, J. Differ. Equ., 215 (2005), 52-107.  doi: 10.1016/j.jde.2004.10.022. [17] D. Horstmann and G. Wang, Blow-up in a chemotaxis model without symmetry assumptions, Eur. J. Appl. Math., 12 (2001), 159-177.  doi: 10.1017/S0956792501004363. [18] R. Kowalczyk and Z. Szymańska, On the global existence of solutions to an aggregation model, J. Math. Anal. Appl., 343 (2008), 379-398.  doi: 10.1016/j.jmaa.2008.01.005. [19] H. Knútsdóttir, E. Pálsson and L. Edelstein-Keshet, Mathematical model of macrophage-facilitated breast cancer cells invasion, J. Theor. Biol., 357 (2014), 184-199.  doi: 10.1016/j.jtbi.2014.04.031. [20] E.Y. Lin, V. Gouon-Evans, A. V. Nguyen and J. W. Pollard, The macrophage growth factor CSF-1 in mammary gland development and tumor progression, J. Mammary Gland Biol. Neoplasia, 7 (2002), 147-162. [21] K. Lin and T. Xiang, On boundedness, blow-up and convergence in a two-species and two-stimuli chemotaxis system with/without loop, Calc. Var., 59 (2020), 35 pp. doi: 10.1007/s00526-020-01777-7. [22] C. Mu, L. Wang, P. Zheng and Q. Zhang, Global existence and boundedness of classical solutions to a parabolic-parabolic chemotaxis system, Nonlinear Anal. Real World Appl., 14 (2013), 1634-1642.  doi: 10.1016/j.nonrwa.2012.10.022. [23] T. Nagai, Blow-up of radially symmetric solutions to a chemotaxis system, Adv. Math. Sci. Appl., 5 (1995), 581-601. [24] T. Nagai, Blow-up of nonradial solutions to parabolic-elliptic systems modeling chemotaxis in two-dimensional domains, J. Inequal. Appl., 6 (2001), 37-55.  doi: 10.1155/S1025583401000042. [25] T. Nagai, T. Senba and K. Yoshida, Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkcial. Ekvac., 40 (1997), 411-433. [26] A. Patsialou, J. Wyckoff, Y. Wang, S. Goswami, E. R. Stanley and J. S. Condeelis, Invasion of human breast cancer cells in vivo requires both paracrine and autocrine loops involving the colony-stimulating factor-1 receptor, Cancer Res., 69 (2009), 9498-9506. [27] K. J. Painter and T. Hillen, Volume-filling and quorum sensing in models for chemosensitive movement, Canad. Appl. Math. Quart., 10 (2002), 501-543. [28] Y. Tao and Z. Wang, Competing effects of attraction vs. repulsion in chemotaxis, Math. Models Methods Appl. Sci., 23 (2013), 1-36.  doi: 10.1142/S0218202512500443. [29] Y. Tao and M. Winkler, Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity, J. Differ. Equ., 252 (2012), 692-715.  doi: 10.1016/j.jde.2011.08.019. [30] Y. Tao and M. Winkler, Boundedness vs. blow-up in a two-species chemotaxis system with two chemicals, Discrete Contin. Dyn. Syst. B., 20 (2015), 3165-3183.  doi: 10.3934/dcdsb.2015.20.3165. [31] Y. Tao and M. Winkler, Energy-type estimates and global solvability in a two-dimensional chemotaxis-haptotaxis model with remodeling of non-diffusible attractant, J. Differ. Equ., 257 (2014), 784-815.  doi: 10.1016/j.jde.2014.04.014. [32] L. Wang, Y. Li and C. Mu, Boundedness in a parabolic-parabolic quasilinear chemotaxis system with logistic source, Discrete Contin. Dyn. Syst., 34 (2014), 789-802.  doi: 10.3934/dcds.2014.34.789. [33] L. Wang, C. Mu and P. Zheng, On a quasilinear parabolic-elliptic chemotaxis system with logistic source, J. Differ. Equ., 256 (2014), 1847-1872.  doi: 10.1016/j.jde.2013.12.007. [34] M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, J. Differ. Equ., 248 (2010), 2889-2905.  doi: 10.1016/j.jde.2010.02.008. [35] M. Winkler, Does a 'volume-filling effect' always prevent chemotactic collapse?, Math. Methods Appl. Sci., 33 (2010), 12-24.  doi: 10.1002/mma.1146. [36] M. Winkler and K. C. Djie, Boundedness and finite-time collapse in a chemotaxis system with volume-filling effect, Nonlinear Anal. TMA, 72 (2010), 1044-1064.  doi: 10.1016/j.na.2009.07.045. [37] M. Winkler, Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system, J. Math. Pures Appl., 100 (2013), 748-767.  doi: 10.1016/j.matpur.2013.01.020. [38] M. Winkler, Global classical solvability and generic infinite-time blow-up in quasilinear Keller-Segel systems with bounded sensitivities, J. Differ. Equ., 266 (2019), 8034-8066.  doi: 10.1016/j.jde.2018.12.019. [39] J. B. Wyckoff, Direct visualization of macrophage-assisted tumor cell intravasation in mammary tumors, Cancer Res., 67 (2007), 2649-2656. [40] L. Xie, On a fully parabolic chemotaxis system with nonlinear signal secretion, Nonlinear Anal. Real World Appl., 49 (2019), 24-44.  doi: 10.1016/j.nonrwa.2019.02.005. [41] L. Xie and Y. Wang, Boundedness in a two-species chemotaxis parabolic system with two chemicals, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 2717-2729.  doi: 10.3934/dcdsb.2017132. [42] L. Xie and Z. Xiang, Boundedness in quasilinear Keller-Segel equations with nonlinear sensitivity and logistic source, Discrete Contin. Dyn. Syst., 35 (2015), 3503-3531.  doi: 10.3934/dcds.2015.35.3503. [43] T. Xiang, Boundedness and global existence in the higher-dimensional parabolic-parabolic chemotaxis system with/without growth source, J. Differ. Equ., 258 (2015), 4275-4323.  doi: 10.1016/j.jde.2015.01.032. [44] C. Yang, X. Cao, Z. Jiang and S. Zheng, Boundedness in a quasilinear fully parabolic Keller-Segel system of higher dimension with logistic source, J. Math. Anal. Appl., 430 (2015), 585-591.  doi: 10.1016/j.jmaa.2015.04.093. [45] H. Yu, W. Wang and S. Zheng, Criteria on global boundedness versus finite time blow-up to a two-species chemotaxis system with two chemicals, Nonlinearity, 31 (2018), 502-514.  doi: 10.1088/1361-6544/aa96c9. [46] Q. Zhang and Y. Li, Boundedness in a quasilinear fully parabolic Keller-Segel system with logistic source, Z. Angew. Math. Phys., 66 (2015), 2473-2484.  doi: 10.1007/s00033-015-0532-z. [47] J. Zheng, Boundedness in a two-species quasi-linear chemotaxis system with two chemicals, Topol. Methods Nonlinear Anal., 49 (2017), 463-480.  doi: 10.12775/tmna.2016.082.

show all references

##### References:
 [1] N. D. Alikakos, $L^p$ bounds of solutions of reaction-diffusion equations, Commun. Partial Differ. Equ., 4 (1979), 827-868.  doi: 10.1080/03605307908820113. [2] A. H. Beck, I. Espinosa, B. Edris, R. Li, K. Montgomery, S. Zhu, S. Varma, R.J. Marinelli, M. van de Rijn and R. B. West, The macrophage colony-stimulating factor response signature in breast carcinoma, Clin. Cancer Res., 15 (2009), 778-787. [3] N. Bellomo, A. Bellouquid, Y. Tao and M. Winkler, Toward a mathematical theory of Keller-Segel models of pattern formation in biological tissues, Math. Models Methods Appl. Sci., 25 (2015), 1663-1763.  doi: 10.1142/S021820251550044X. [4] P. Biler, Global solutions to some parabolic-elliptic systems of chemotaxis, Adv. Math. Sci.Appl., 9 (1999), 347-359. [5] P. Biler, W. Hebisch and T. Nadzieja, The Debye system: Existence and large time behavior of solutions, Nonlinear Anal., 23 (1994), 1189-1209.  doi: 10.1016/0362-546X(94)90101-5. [6] X. Cao, Global bounded solutions of the higher-dimensional Keller-Segel system under smallness conditions in optimal spaces, Discrete Contin. Dyn. Syst., 35 (2015), 1891-1904.  doi: 10.3934/dcds.2015.35.1891. [7] T. Cieślak and M. Winkler, Finite-time blow-up in a quasilinear system of chemotaxis, Nonlinearity, 21 (2008), 1057-1076.  doi: 10.1088/0951-7715/21/5/009. [8] T. Cieślak and C. Stinner, Finite-time blowup and global-in-time unbounded solutions to a parabolic-parabolic quasilinear Keller-Segel system in higher dimensions, J. Differ. Equ., 252 (2012), 5832-5851.  doi: 10.1016/j.jde.2012.01.045. [9] T. Cieślak and C. Stinner, New critical exponents in a fully parabolic quasinear Keller-Segel system and applications to volume filling models, J. Differ. Equ., 258 (2015), 2080-2113.  doi: 10.1016/j.jde.2014.12.004. [10] C. Conca, E. Espejo and K. Vilches, Remarks on the blowup and global existence for a two species chemotactic Keller-Segel system in $\mathbb{R}^2$, Eur. J. Appl. Math., 22 (2011), 553-580.  doi: 10.1017/S0956792511000258. [11] E. Espejo Arenas, A. Stevens and J. Velázquez, Simultaneous finite time blow-up in a two-species model for chemotaxis, Analysis (Munich), 29 (2009), 317-338.  doi: 10.1524/anly.2009.1029. [12] A. Friedman, Partial Differential Equations, Holt, Rinehart and Winston, New York, 1969. [13] A. S. Harney, Real-time imaging reveals local, transient vascular permeability, and tumor cell intravasation stimulated by TIE2hi macrophage-derived VEGFA, Cancer Discovery, 5 (2015), 932-943. [14] M. Herrero and J. Velázquez, A blow-up mechanism for a chemotaxis model, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 24 (1997), 633-683. [15] T. Hillen and K. J. Painter, Global existence for a parabolic chemotaxis model with prevention of overcrowding, Adv. Appl. Math., 26 (2001), 280-301.  doi: 10.1006/aama.2001.0721. [16] D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system, J. Differ. Equ., 215 (2005), 52-107.  doi: 10.1016/j.jde.2004.10.022. [17] D. Horstmann and G. Wang, Blow-up in a chemotaxis model without symmetry assumptions, Eur. J. Appl. Math., 12 (2001), 159-177.  doi: 10.1017/S0956792501004363. [18] R. Kowalczyk and Z. Szymańska, On the global existence of solutions to an aggregation model, J. Math. Anal. Appl., 343 (2008), 379-398.  doi: 10.1016/j.jmaa.2008.01.005. [19] H. Knútsdóttir, E. Pálsson and L. Edelstein-Keshet, Mathematical model of macrophage-facilitated breast cancer cells invasion, J. Theor. Biol., 357 (2014), 184-199.  doi: 10.1016/j.jtbi.2014.04.031. [20] E.Y. Lin, V. Gouon-Evans, A. V. Nguyen and J. W. Pollard, The macrophage growth factor CSF-1 in mammary gland development and tumor progression, J. Mammary Gland Biol. Neoplasia, 7 (2002), 147-162. [21] K. Lin and T. Xiang, On boundedness, blow-up and convergence in a two-species and two-stimuli chemotaxis system with/without loop, Calc. Var., 59 (2020), 35 pp. doi: 10.1007/s00526-020-01777-7. [22] C. Mu, L. Wang, P. Zheng and Q. Zhang, Global existence and boundedness of classical solutions to a parabolic-parabolic chemotaxis system, Nonlinear Anal. Real World Appl., 14 (2013), 1634-1642.  doi: 10.1016/j.nonrwa.2012.10.022. [23] T. Nagai, Blow-up of radially symmetric solutions to a chemotaxis system, Adv. Math. Sci. Appl., 5 (1995), 581-601. [24] T. Nagai, Blow-up of nonradial solutions to parabolic-elliptic systems modeling chemotaxis in two-dimensional domains, J. Inequal. Appl., 6 (2001), 37-55.  doi: 10.1155/S1025583401000042. [25] T. Nagai, T. Senba and K. Yoshida, Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkcial. Ekvac., 40 (1997), 411-433. [26] A. Patsialou, J. Wyckoff, Y. Wang, S. Goswami, E. R. Stanley and J. S. Condeelis, Invasion of human breast cancer cells in vivo requires both paracrine and autocrine loops involving the colony-stimulating factor-1 receptor, Cancer Res., 69 (2009), 9498-9506. [27] K. J. Painter and T. Hillen, Volume-filling and quorum sensing in models for chemosensitive movement, Canad. Appl. Math. Quart., 10 (2002), 501-543. [28] Y. Tao and Z. Wang, Competing effects of attraction vs. repulsion in chemotaxis, Math. Models Methods Appl. Sci., 23 (2013), 1-36.  doi: 10.1142/S0218202512500443. [29] Y. Tao and M. Winkler, Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity, J. Differ. Equ., 252 (2012), 692-715.  doi: 10.1016/j.jde.2011.08.019. [30] Y. Tao and M. Winkler, Boundedness vs. blow-up in a two-species chemotaxis system with two chemicals, Discrete Contin. Dyn. Syst. B., 20 (2015), 3165-3183.  doi: 10.3934/dcdsb.2015.20.3165. [31] Y. Tao and M. Winkler, Energy-type estimates and global solvability in a two-dimensional chemotaxis-haptotaxis model with remodeling of non-diffusible attractant, J. Differ. Equ., 257 (2014), 784-815.  doi: 10.1016/j.jde.2014.04.014. [32] L. Wang, Y. Li and C. Mu, Boundedness in a parabolic-parabolic quasilinear chemotaxis system with logistic source, Discrete Contin. Dyn. Syst., 34 (2014), 789-802.  doi: 10.3934/dcds.2014.34.789. [33] L. Wang, C. Mu and P. Zheng, On a quasilinear parabolic-elliptic chemotaxis system with logistic source, J. Differ. Equ., 256 (2014), 1847-1872.  doi: 10.1016/j.jde.2013.12.007. [34] M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, J. Differ. Equ., 248 (2010), 2889-2905.  doi: 10.1016/j.jde.2010.02.008. [35] M. Winkler, Does a 'volume-filling effect' always prevent chemotactic collapse?, Math. Methods Appl. Sci., 33 (2010), 12-24.  doi: 10.1002/mma.1146. [36] M. Winkler and K. C. Djie, Boundedness and finite-time collapse in a chemotaxis system with volume-filling effect, Nonlinear Anal. TMA, 72 (2010), 1044-1064.  doi: 10.1016/j.na.2009.07.045. [37] M. Winkler, Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system, J. Math. Pures Appl., 100 (2013), 748-767.  doi: 10.1016/j.matpur.2013.01.020. [38] M. Winkler, Global classical solvability and generic infinite-time blow-up in quasilinear Keller-Segel systems with bounded sensitivities, J. Differ. Equ., 266 (2019), 8034-8066.  doi: 10.1016/j.jde.2018.12.019. [39] J. B. Wyckoff, Direct visualization of macrophage-assisted tumor cell intravasation in mammary tumors, Cancer Res., 67 (2007), 2649-2656. [40] L. Xie, On a fully parabolic chemotaxis system with nonlinear signal secretion, Nonlinear Anal. Real World Appl., 49 (2019), 24-44.  doi: 10.1016/j.nonrwa.2019.02.005. [41] L. Xie and Y. Wang, Boundedness in a two-species chemotaxis parabolic system with two chemicals, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 2717-2729.  doi: 10.3934/dcdsb.2017132. [42] L. Xie and Z. Xiang, Boundedness in quasilinear Keller-Segel equations with nonlinear sensitivity and logistic source, Discrete Contin. Dyn. Syst., 35 (2015), 3503-3531.  doi: 10.3934/dcds.2015.35.3503. [43] T. Xiang, Boundedness and global existence in the higher-dimensional parabolic-parabolic chemotaxis system with/without growth source, J. Differ. Equ., 258 (2015), 4275-4323.  doi: 10.1016/j.jde.2015.01.032. [44] C. Yang, X. Cao, Z. Jiang and S. Zheng, Boundedness in a quasilinear fully parabolic Keller-Segel system of higher dimension with logistic source, J. Math. Anal. Appl., 430 (2015), 585-591.  doi: 10.1016/j.jmaa.2015.04.093. [45] H. Yu, W. Wang and S. Zheng, Criteria on global boundedness versus finite time blow-up to a two-species chemotaxis system with two chemicals, Nonlinearity, 31 (2018), 502-514.  doi: 10.1088/1361-6544/aa96c9. [46] Q. Zhang and Y. Li, Boundedness in a quasilinear fully parabolic Keller-Segel system with logistic source, Z. Angew. Math. Phys., 66 (2015), 2473-2484.  doi: 10.1007/s00033-015-0532-z. [47] J. Zheng, Boundedness in a two-species quasi-linear chemotaxis system with two chemicals, Topol. Methods Nonlinear Anal., 49 (2017), 463-480.  doi: 10.12775/tmna.2016.082.
Schematic of EGF/CSF-1 paracrine and autocrine interactions between tumor cells and macrophages. Macrophages secrete EGF and have CSF-1 receptors. Tumor cells secrete CSF-1 and have both EGF/CSF-1 receptors. When CSF-1R on macrophages are activated, the macrophages respond by secreting EGF and chemotact in the direction of the CSF-1 gradient. When EGFR on tumor cells are activated, the tumor cells respond by secreting CSF-1 and chemotact up the EGF gradient. This paracrine signaling loop enables tumor cells to migrate alongside macrophages away from the primary tumor. Moreover, the tumor cells not only secrete CSF-1 but also respond to the concentration gradient of CSF-1 and this is the CSF-1/CSF-1R autocrine signalling loop (Patsialou et al. [26]). EGFR$=$EGF receptors, CSF-1R$=$CSF-1 receptors
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