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A Hybrid Discrete–Continuum Modelling Approach to Explore the Impact of T-Cell Infiltration on Anti-tumour Immune Response


We present a spatial hybrid discrete–continuum modelling framework for the interaction dynamics between tumour cells and cytotoxic T cells, which play a pivotal role in the immune response against tumours. In this framework, tumour cells and T cells are modelled as individual agents while chemokines that drive the chemotactic movement of T cells towards the tumour are modelled as a continuum. We formally derive the continuum counterpart of this model, which is given by a coupled system that comprises an integro-differential equation for the density of tumour cells, a partial differential equation for the density of T cells and a partial differential equation for the concentration of chemokines. We report on computational results of the hybrid model and show that there is an excellent quantitative agreement between them and numerical solutions of the corresponding continuum model. These results shed light on the mechanisms that underlie the emergence of different levels of infiltration of T cells into the tumour and elucidate how T-cell infiltration shapes anti-tumour immune response. Moreover, to present a proof of concept for the idea that, exploiting the computational efficiency of the continuum model, extensive numerical simulations could be carried out, we investigate the impact of T-cell infiltration on the response of tumour cells to different types of anti-cancer immunotherapy.

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Data Availability

The datasets generated and analysed during the current study are available from the corresponding author on reasonable request.


  • Aguadé-Gorgorió G, Solé R (2020) Tumour neoantigen heterogeneity thresholds provide a time window for combination immunotherapy. J R Soc Interface 17(171):20200736

    Google Scholar 

  • Al-Tameemi M, Chaplain M, d’Onofrio A (2012) Evasion of tumours from the control of the immune system: consequences of brief encounters. Biol Direct 7(1):31

    Google Scholar 

  • Almeida L, Audebert C, Leschiera E, et al (2021) Discrete and continuum models for the coevolutionary dynamics between CD8+ cytotoxic T lymphocytes and tumour cells. arXiv:2109.09568

  • Almuallem N, Trucu D, Eftimie R (2021) Oncolytic viral therapies and the delicate balance between virus-macrophage-tumour interactions: A mathematical approach. Math Biosci Eng 18(1):764–799

    MathSciNet  MATH  Google Scholar 

  • Angell H, Galon J (2013) From the immune contexture to the immunoscore: the role of prognostic and predictive immune markers in cancer. Curr Opin Immunol 25(2):261–267

    Google Scholar 

  • Atsou K, Anjuère F, Braud VM et al (2020) A size and space structured model describing interactions of tumor cells with immune cells reveals cancer persistent equilibrium states in tumorigenesis. J Theor Biol 490(110):163

    MathSciNet  MATH  Google Scholar 

  • Basu R, Whitlock BM, Husson J et al (2016) Cytotoxic T cells use mechanical force to potentiate target cell killing. Cell 165(1):100–110

    Google Scholar 

  • Boissonnas A, Fetler L, Zeelenberg IS et al (2007) In vivo imaging of cytotoxic T cell infiltration and elimination of a solid tumor. J Exp Med 204(2):345–356

    Google Scholar 

  • Bubba F, Lorenzi T, Macfarlane FR (2020) From a discrete model of chemotaxis with volume-filling to a generalized Patlak-Keller-Segel model. Proc R Soc Lond A 476(2237):20190871

    MathSciNet  MATH  Google Scholar 

  • Byrne H, Drasdo D (2009) Individual-based and continuum models of growing cell populations: a comparison. J Math Biol 58(4):657–687

    MathSciNet  MATH  Google Scholar 

  • Cattani C, Ciancio A, d’Onofrio A (2010) Metamodeling the learning-hiding competition between tumours and the immune system: a kinematic approach. Math Comput Model Dyn Syst 52(1):62–69

    MathSciNet  MATH  Google Scholar 

  • Champagnat N, Ferrière R, Méléard S (2008) From individual stochastic processes to macroscopic models in adaptive evolution. Stoch Models 24(sup1):2–44

    MathSciNet  MATH  Google Scholar 

  • Chisholm RH, Lorenzi T, Desvillettes L et al (2016) Evolutionary dynamics of phenotype-structured populations: from individual-level mechanisms to population-level consequences. Z Angew Math Phys 67(4):100

    MathSciNet  MATH  Google Scholar 

  • Christophe C, Müller S, Rodrigues M, et al (2015) A biased competition theory of cytotoxic T lymphocyte interaction with tumor nodules. PloS ONE 10(3)

  • Cooper AK, Kim PS (2014) A cellular automata and a partial differential equation model of tumor-immune dynamics and chemotaxis. In: Eladdadi A, Kim P, Mallet D (eds) Mathematical models of tumor-immune system dynamics. Springer, New York, New York, NY, pp 21–46

    Google Scholar 

  • Coulie PG, Van den Eynde BJ, Van Der Bruggen P et al (2014) Tumour antigens recognized by T lymphocytes: at the core of cancer immunotherapy. Nat Rev Cancer 14(2):135–146

    Google Scholar 

  • Delitala M, Lorenzi T (2013) Recognition and learning in a mathematical model for immune response against cancer. Discrete Contin Dyn Syst B 18(4)

  • Eftimie R, Bramson JL, Earn DJ (2011) Interactions between the immune system and cancer: a brief review of non-spatial mathematical models. Bull Math Biol 73(1):2–32

    MathSciNet  MATH  Google Scholar 

  • Galon J, Bruni D (2019) Approaches to treat immune hot, altered and cold tumours with combination immunotherapies. Nat Rev Drug Discov 18(3):197–218

    Google Scholar 

  • Galon J, Costes A, Sanchez-Cabo F et al (2006) Type, density, and location of immune cells within human colorectal tumors predict clinical outcome. Science 313(5795):1960–1964

    Google Scholar 

  • Galon J, Fox B, Bifulco C, et al (2016) Immunoscore and Immunoprofiling in cancer: an update from the melanoma and immunotherapy bridge 2015

  • Gandhi L, Rodríguez-Abreu D, Gadgeel S et al (2018) Pembrolizumab plus chemotherapy in metastatic non-small-cell lung cancer. N Engl J Med 378(22):2078–2092

    Google Scholar 

  • Gong C, Milberg O, Wang B et al (2017) A computational multiscale agent-based model for simulating spatio-temporal tumour immune response to PD1 and PDL1 inhibition. J R Soc Interface 14(134):20170320

    Google Scholar 

  • Gorbachev AV, Kobayashi H, Kudo D et al (2007) Cxc chemokine ligand 9/monokine induced by ifn-\(\gamma \) production by tumor cells is critical for t cell-mediated suppression of cutaneous tumors. J Immunol 178(4):2278–2286

    Google Scholar 

  • Griffiths JI, Wallet P, Pflieger LT et al (2020) Circulating immune cell phenotype dynamics reflect the strength of tumor-immune cell interactions in patients during immunotherapy. Proc Natl Acad Sci USA 117(27):16072–16082

    Google Scholar 

  • Halle S, Keyser KA, Stahl FR et al (2016) In vivo killing capacity of cytotoxic T cells is limited and involves dynamic interactions and cooperativity. Immunity 44(2):233–245

    Google Scholar 

  • Handel A, La Gruta NL, Thomas PG (2020) Simulation modelling for immunologists. Nat Rev Immunol 20(3):186–195

    Google Scholar 

  • Hegde PS, Karanikas V, Evers S (2016) The where, the when, and the how of immune monitoring for cancer immunotherapies in the era of checkpoint inhibition. Clin Cancer Res 22(8):1865–1874

    Google Scholar 

  • Hellmann MD, Ciuleanu TE, Pluzanski A et al (2018) Nivolumab plus ipilimumab in lung cancer with a high tumor mutational burden. N Engl J Med 378(22):2093–2104

    Google Scholar 

  • Huang AC, Postow MA, Orlowski RJ et al (2017) T-cell invigoration to tumour burden ratio associated with anti-PD-1 response. Nature 545(7652):60–65

    Google Scholar 

  • Hughes BD et al (1995) Random walks and random environments: random walks, vol 1. Oxford University Press

    MATH  Google Scholar 

  • Iwai Y, Ishida M, Tanaka Y et al (2002) Involvement of PD-L1 on tumor cells in the escape from host immune system and tumor immunotherapy by PD-L1 blockade. Proc Natl Acad Sci USA 99(19):12293–12297

    Google Scholar 

  • Jarrett AM, Faghihi D, Hormuth DA et al (2020) Optimal control theory for personalized therapeutic regimens in oncology: background, history, challenges, and opportunities. J Clin Med 9(5):1314

    Google Scholar 

  • Johnston ST, Simpson MJ, Baker RE (2015) Modelling the movement of interacting cell populations: a moment dynamics approach. J Theor Biol 370:81–92

    MathSciNet  MATH  Google Scholar 

  • Kather JN, Poleszczuk J, Suarez-Carmona M et al (2017) In silico modeling of immunotherapy and stroma-targeting therapies in human colorectal cancer. Cancer Res 77(22):6442–6452

    Google Scholar 

  • Kato D, Yaguchi T, Iwata T et al (2017) Prospects for personalized combination immunotherapy for solid tumors based on adoptive cell therapies and immune checkpoint blockade therapies. Nihon Rinsho Meneki Gakkai Kaishi 40(1):68–77

    Google Scholar 

  • Kim PS, Lee PP (2012) Modeling protective anti-tumor immunity via preventative cancer vaccines using a hybrid agent-based and delay differential equation approach. PLoS Comput Biol 8(10):e1002742

    MathSciNet  Google Scholar 

  • Kolev M (2003) Mathematical modeling of the competition between acquired immunity and cancer. Int J Appl Math Comput Sci 13:289–296

    MathSciNet  MATH  Google Scholar 

  • Konstorum A, Vella AT, Adler AJ et al (2017) Addressing current challenges in cancer immunotherapy with mathematical and computational modelling. J R Soc Interface 14(131):20170150

    Google Scholar 

  • Kuznetsov VA, Knott GD (2001) Modeling tumor regrowth and immunotherapy. Math Comput Model 33(12):1275–1287

    MathSciNet  MATH  Google Scholar 

  • Kuznetsov VA, Makalkin IA, Taylor MA et al (1994) Nonlinear dynamics of immunogenic tumors: parameter estimation and global bifurcation analysis. Bull Math Biol 56(2):295–321

    MATH  Google Scholar 

  • Leschiera E, Lorenzi T, Shen S, et al (2022) A mathematical model to study the impact of intra-tumour heterogeneity on anti-tumour CD8+ T cell immune response. J Theor Biol, p 111028

  • Lin Erickson AH, Wise A, Fleming S et al (2009) A preliminary mathematical model of skin dendritic cell trafficking and induction of T cell immunity. Discret Contin Dyn Syst - B 12:323–336

    MathSciNet  MATH  Google Scholar 

  • Lorenzi T (2022) Cancer modelling as fertile ground for new mathematical challenges. comment on" improving cancer treatments via dynamical biophysical models" by m. kuznetsov, j. clairambault & v. volpert. Phys Life Rev 40:3–5

    Google Scholar 

  • Lorenzi T, Chisholm RH, Melensi M et al (2015) Mathematical model reveals how regulating the three phases of T-cell response could counteract immune evasion. Immunology 146(2):271–280

    Google Scholar 

  • Łuksza M, Riaz N, Makarov V et al (2017) A neoantigen fitness model predicts tumour response to checkpoint blockade immunotherapy. Nature 551(7681):517–520

    Google Scholar 

  • Macfarlane FR, Lorenzi T, Chaplain MA (2018) Modelling the immune response to cancer: an individual-based approach accounting for the difference in movement between inactive and activated T cells. Bull Math Biol 80(6):1539–1562

    MathSciNet  MATH  Google Scholar 

  • Macfarlane FR, Chaplain MA, Lorenzi T (2019) A stochastic individual-based model to explore the role of spatial interactions and antigen recognition in the immune response against solid tumours. J Theor Biol 480:43–55

    MathSciNet  MATH  Google Scholar 

  • Macfarlane FR, Chaplain MA, Lorenzi T (2020) A hybrid discrete-continuum approach to model turing pattern formation. Math Biosci Eng 17(6):7442–7479

    MathSciNet  MATH  Google Scholar 

  • Maini P, Painter K, Chau HP (1997) Spatial pattern formation in chemical and biological systems. J Chem Soc Faraday Trans 93(20):3601–3610

    Google Scholar 

  • Makaryan SZ, Cess CG, Finley SD (2020) Modeling immune cell behavior across scales in cancer. Wiley Interdiscip Rev Syst Biol Med 12(4):e1484

    Google Scholar 

  • Mallet DG, De Pillis LG (2006) A cellular automata model of tumor-immune system interactions. J Theor Biol 239(3):334–350

    MathSciNet  MATH  Google Scholar 

  • MATLAB (2020) (R2020b). The MathWorks Inc., Natick, Massachusetts

  • Matzavinos A, Chaplain MA, Kuznetsov VA (2004) Mathematical modelling of the spatio-temporal response of cytotoxic T-lymphocytes to a solid tumour. Math Med Biol 21(1):1–34

    MATH  Google Scholar 

  • McGranahan N, Furness AJ, Rosenthal R et al (2016) Clonal neoantigens elicit T cell immunoreactivity and sensitivity to immune checkpoint blockade. Science 351(6280):1463–1469

    Google Scholar 

  • Miller MJ, Wei SH, Cahalan MD et al (2003) Autonomous T cell trafficking examined in vivo with intravital two-photon microscopy. Proc Natl Acad Sci USA 100(5):2604–2609

    Google Scholar 

  • Motzer RJ, Tannir NM, McDermott DF, et al (2018) Nivolumab plus ipilimumab versus sunitinib in advanced renal-cell carcinoma. N Engl J Med

  • Painter KJ (2019) Mathematical models for chemotaxis and their applications in self-organisation phenomena. J Theor Biol 481:162–182

    MathSciNet  MATH  Google Scholar 

  • Painter KJ, Hillen T (2002) Volume-filling and quorum-sensing in models for chemosensitive movement. Can Appl Math Quart 10(4):501–543

    MathSciNet  MATH  Google Scholar 

  • Pitt J, Marabelle A, Eggermont A et al (2016) Targeting the tumor microenvironment: removing obstruction to anticancer immune responses and immunotherapy. Ann Oncol 27(8):1482–1492

    Google Scholar 

  • Rabinovich GA, Gabrilovich D, Sotomayor EM (2007) Immunosuppressive strategies that are mediated by tumor cells. Annu Rev Immunol 25:267–296

    Google Scholar 

  • Ribas A, Wolchok JD (2018) Cancer immunotherapy using checkpoint blockade. Science 359(6382):1350–1355

    Google Scholar 

  • Slaney CY, Kershaw MH, Darcy PK (2014) Trafficking of t cells into tumors. Cancer Res 74(24):7168–7174

    Google Scholar 

  • Spranger S, Bao R, Gajewski TF (2015) Melanoma-intrinsic \(\beta \)-catenin signalling prevents anti-tumour immunity. Nature 523(7559):231–235

    Google Scholar 

  • Takayanagi T, Ohuchi A (2001) A mathematical analysis of the interactions between immunogenic tumor cells and cytotoxic T lymphocytes. Microbiol Immunol 45(10):709–715

    Google Scholar 

  • Tian L, Goldstein A, Wang H et al (2017) Mutual regulation of tumour vessel normalization and immunostimulatory reprogramming. Nature 544(7649):250–254

    Google Scholar 

  • Topalian SL, Hodi FS, Brahmer JR et al (2012) Safety, activity, and immune correlates of anti-PD-1 antibody in cancer. N Engl J Med 366(26):2443–2454

    Google Scholar 

  • Tumeh PC, Harview CL, Yearley JH et al (2014) PD-1 blockade induces responses by inhibiting adaptive immune resistance. Nature 515(7528):568–571

    Google Scholar 

  • Van Allen EM, Miao D, Schilling B et al (2015) Genomic correlates of response to CTLA-4 blockade in metastatic melanoma. Science 350(6257):207–211

    Google Scholar 

  • Wang Z, Hillen T (2007) Classical solutions and pattern formation for a volume filling chemotaxis model. Chaos 17(3):037108

    MathSciNet  MATH  Google Scholar 

  • Wieland A, Kamphorst AO, Adsay NV et al (2018) T cell receptor sequencing of activated CD8 T cells in the blood identifies tumor-infiltrating clones that expand after PD-1 therapy and radiation in a melanoma patient. Cancer Immunol Immunother 67(11):1767–1776

    Google Scholar 

  • Wilkie KP (2013) A review of mathematical models of cancer-immune interactions in the context of tumor dormancy. In: Enderling H, Almog N, Hlatky L (eds) Systems biology of tumor dormancy. Springer, New York, New York, NY, pp 201–234.

    Chapter  Google Scholar 

  • Wolchok JD, Chiarion-Sileni V, Gonzalez R et al (2017) Overall survival with combined nivolumab and ipilimumab in advanced melanoma. N Engl J Med 377(14):1345–1356

    Google Scholar 

  • van der Woude LL, Gorris MA, Halilovic A et al (2017) Migrating into the tumor: a roadmap for t cells. Trends Cancer 3(11):797–808

    Google Scholar 

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E.L. has received funding from the European Research Council (ERC) under the European Union’s Horizon2020 research and innovation programme (grant agreement No 740623). T.L. gratefully acknowledges support from the Italian Ministry of University and Research (MUR) through the grant ‘Dipartimenti di Eccellenza 2018-2022’ (Project no. E11G18000350001) and the PRIN 2020 project (No. 2020JLWP23) ‘Integrated Mathematical Approaches to Socio–Epidemiological Dynamics’ (CUP: E15F21005420006). L.A., E.L. and T.L. gratefully acknowledge support from the CNRS International Research Project ‘Modélisation de la biomécanique cellulaire et tissulaire’ (MOCETIBI).

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Appendix A Formal Derivation of the Continuum Model

Building on the methods employed in Bubba et al. (2020), we carry out a formal derivation of the deterministic continuum model given by the IDE-PDE-PDE system (3.2) for \(d = 1\). Similar methods can be used in the case where \(d = 2\).

1.1 A.1 Formal Derivation of the IDE for the Density of Tumour Cells n(xt)

When tumour cell dynamics are governed by the rules described in Sects. 2.1.1 and 2.1.2, considering \(i \in [0,\mathcal {N}]\), between time-steps k and \(k+1\) the principle of mass balance gives the following difference equation for the tumour cell density \(n_{i}^{k}\):

$$\begin{aligned} n_{i}^{k+1}=\left[ 2 \, \tau \alpha _n + 1-\tau (\alpha _n+\zeta _n K_{i}^{k}+\mu _n\rho _n^k \right] n_{i}^{k}. \end{aligned}$$

Using the fact that the following relations hold for \(\tau \) and \(\chi \) sufficiently small

$$\begin{aligned}{} & {} t_k\approx t, \quad t_{k+1}\approx t+\tau , \quad x_i\approx x, \quad x_{i\pm 1}\approx x\pm \chi , \end{aligned}$$
$$\begin{aligned}{} & {} n_{i}^{k}\approx n(x,t), \quad n_{i}^{k+1}\approx n(x,t+\tau ), \quad c_{i}^{k}\approx c(x,t), \end{aligned}$$
$$\begin{aligned}{} & {} \rho _n^k\approx \rho _n(t):=\int _\Omega n(x,t)\,\textrm{d}x, \quad K_{i}^{k}\approx K(x,t):=\int _\Omega \eta (x,x^\prime ;\theta )c(x^\prime ,t)\,\textrm{d}x^\prime , \end{aligned}$$

where the function \(\eta \) is defined via (2.12), Eq. (A1) can be formally rewritten in the approximate form

$$\begin{aligned} n(x,t+\tau )-n(x,t)=\tau \left( \alpha _n-\zeta _n K(x,t)-\mu _n\rho _n(t) \right) n(x,t). \end{aligned}$$

If, in addiction, the function n(xt) is continuously differentiable with respect to the variable t, starting from Eq. (A5), and letting the time-step \(\tau \rightarrow 0\), one formally obtains the following IDE for the tumour cell density n(xt):

$$\begin{aligned} \partial _t n(x,t) =\alpha _n n(x,t)-\mu _n\rho _n(t)\, n(x,t)-\zeta _nK(x,t) n(x,t)\quad (x,t)\in \Omega \times (0,t_f].\end{aligned}$$

1.2 A.2 Formal Derivation of the PDE for the Density of T Cells c(xt)

When T cell dynamics are governed by the rules described in Sect. 2.3, considering \(i \in [1,\mathcal {N}-1]\), between time-steps k and \(k+1\) the principle of mass balance gives the following difference equation for the T cell density \(c_{i}^{k}\):

$$\begin{aligned} \begin{aligned} c_i^{k+1}&=c_i^k(1-\tau \,\mu _c\rho _c^k)+\tau \,\alpha _cr^k_i \\&\quad + \frac{\lambda }{2}\psi (w_i^k)\Big (c_{i+1}^k+c_{i-1}^k\Big ) \\&\quad -\frac{\lambda }{2}\Big (\psi (w_{i-1}^k)+\psi (w_{i+1}^k)\Big )c_i^k\\&\quad +\frac{\nu }{2\phi _{\max }}\psi (w_i^k)\Big [\Big (\phi _i^k-\phi _{i-1}^k\Big )_{+}c_{i-1}^k\Big ]\\&\quad +\frac{\nu }{2\phi _{\max }}\psi (w_i^k)\Big [\Big (\phi _i^k-\phi _{i+1}^k)\Big )_{+}c_{i+1}^k\Big ] \\&\quad -\frac{\nu }{2\phi _{\max }}\psi (w_{i+1}^k)\Big [\Big (\phi _{i+1}^k-\phi _i^k\Big )_{+}c_i^k\Big ]\\&\quad -\frac{\nu }{2\phi _{\max }}\psi (w_{i-1}^k)\Big [\Big (\phi _{i-1}^k-\phi _i^k\Big )_{+}c_i^k\Big ]. \end{aligned} \end{aligned}$$

Using the fact that relations (A2A3)–(A4) and the following relations

$$\begin{aligned}{} & {} c_i^k\approx c(x,t), \quad c_{i\pm 1}^k\approx c(x\pm \chi ), \quad \rho _c^k\approx \rho _c(t):=\int _\Omega c(x,t)\,\textrm{d}x, \\{} & {} \phi _i^k\approx \phi (x,t), \quad \phi _i^{k+1}\approx \phi (x,t+\tau ), \quad \phi _{i\pm 1}^k\approx \phi (x\pm \chi ), \\{} & {} w_i^k\approx w(x,t), \text { with } w(x,t)=n(x,t)+c(x,t), \quad w_{i\pm 1}^k\approx w(x\pm \chi ), \\{} & {} r_i^k\approx r(x,t):=\phi _{tot}(t)\mathbb {1}_\omega (x), \text { with } \phi _{tot}(t):=\displaystyle \int _\Omega \phi (x,t)\,\textrm{d}x \end{aligned}$$

hold for \(\tau \) and \(\chi \) sufficiently small, Eq. (A6) can be formally rewritten in the approximate form

$$\begin{aligned} c(x,t+\tau )&=c(x,t)(1-\tau \,\mu _c\rho _c(t))+\tau \,\alpha _cr(x,t) \\&\quad + \frac{\lambda }{2}\psi (w(x,t))\Big (c(x+\chi ,t)+c(x-\chi ,t)\Big )\\&\quad -\frac{\lambda }{2}\Big (\psi (w(x-\chi ,t))+\psi (w(x+\chi ,t))\Big )c(x,t) \\&\quad +\frac{\nu }{2\phi _{\max }}\psi (w(x,t))\Big [\Big (\phi (x,t)-\phi (x-\chi ,t)\Big )_{+}c(x-\chi ,t)\Big ]\\&\quad +\frac{\nu }{2\phi _{\max }}\psi (w(x,t))\Big [\Big (\phi (x,t)-\phi (x+\chi ,t)\Big )_{+}c(x+\chi ,t)\Big ] \\&\quad -\frac{\nu }{2\phi _{\max }}\psi (w(x+\chi ,t))\Big [\Big (\phi (x+\chi ,t)-\phi (x,t)\Big )_{+}c(x,t)\Big ]\\&\quad -\frac{\nu }{2\phi _{\max }}\psi (w(x-\chi ,t))\Big [\Big (\phi (x-\chi ,t)-\phi (x,t)\Big )_{+}c(x,t)\Big ]. \end{aligned}$$

Building on the methods employed in Bubba et al. (2020), letting \(\tau \rightarrow 0\) and \(\chi \rightarrow 0\) in such a way that

$$\begin{aligned} \frac{\lambda }{2}\frac{\chi ^2}{\tau }\rightarrow \beta _c \in \mathbb {R}_*^+ \quad \text {and} \quad \frac{\nu }{2\phi _{\max }}\frac{\chi ^2}{\tau }\rightarrow \gamma _c \in \mathbb {R}_*^+ \quad \text {as } \tau \rightarrow 0, \, \chi \rightarrow 0, \end{aligned}$$

after a little algebra, considering \((x,t)\in \Omega \setminus \partial \Omega \times (0,t_f]\), we find

$$\begin{aligned} \partial _t c- \partial _x\Big [\beta _c \psi (w) \partial _x c-\gamma _c \psi (w)c\partial _x \phi -\beta _c c\psi ^\prime (w)\partial _x w\Big ]= -\mu _c\rho _c(t)c+\alpha _cr \end{aligned}$$

where \(\psi \) is given by (2.22) and \(w:=n+c\). Moreover, zero-flux boundary conditions easily follow from the fact that T-cell moves that require moving out of the spatial domain are not allowed.

1.3 A.3 Formal Derivation of the Balance Equation for the Chemoattractant Concentration \(\phi (x,t)\)

The formal derivation of the balance equation for the chemoattractant concentration \(\phi (x,t)\) is obtained using the methods employed in Bubba et al. (2020).

Appendix B Details of Numerical Simulations

The numerical simulations of our hybrid and continuum models are carried out on a two-dimensional domain and are performed in Matlab.

1.1 B.1 Details of Numerical Simulations of the Hybrid Model

The flowchart in Fig. 13 illustrates the general computational procedure to carry out simulations of the hybrid model in one-dimensional settings, while the flowchart in Fig. 14 provides further details of the computational procedure to simulate cell dynamics in one-dimensional settings. Analogous strategies are used in two-dimensional settings. All random numbers mentioned in Fig. 14 are real numbers drawn from the standard uniform distribution on the interval (0, 1), which in our case are obtained using the built-in Matlab function rand.

As summarised by Fig. 14, at any time-step, each T cell undergoes a three-phase process: Phase A) undirected, random movement according to the probabilities defined via (2.26) and (2.27); Phase B) chemotaxis according to the probabilities defined via (2.23) and (2.24); Phase C) death according to the probabilities defined via (2.17) and (2.21). We let then each tumour cell proliferate with the probability defined via (2.8), die due to intra-tumour competition with the probability defined via (2.10) or die due to immune action with the probability defined via (2.13). Finally, the tumour cell density at every lattice site is computed via (2.1) and inserted into (2.14) in order to update the concentration of the chemoattractant.

In a two-dimensional setting, the positions of the single T cells are updated following a procedure analogous to that illustrated in Figs. 13 and 14, with the only differences being that: T cells are allowed to move up and down as well; the concentration of the chemoattractant is updated through the two-dimensional analogue of (2.14), where the operator \(\mathcal {L}\) is defined as the finite-difference Laplacian on a two-dimensional regular lattice of step \(\chi \); the tumour and T cell densities are, respectively, computed via (2.1) and (2.2).

Fig. 13
figure 13

Flowchart illustrating the computational procedure to simulate the hybrid model in one-dimensional settings. A detailed summary of steps 2) and 3) is provided by the flowchart in Fig. 14. A similar procedure is used in two-dimensional settings (Color figure online)

Fig. 14
figure 14

Flowchart illustrating the detailed computational procedure followed to update the positions of every T cell, as well as the fate of each tumour cell and T cell in one-dimensional settings. Analogous strategies are used in two-dimensional domains (Color figure online)

1.2 B.2 Details of Numerical Simulations of the Continuum Model

To construct numerical solutions of the IDE-PDE-PDE system (3.2), we use a uniform discretisation consisting of \(N^2 = 3721\) points of the square \(\Omega := [0, 1]^2\) as the computational domain of the independent variable \({\textbf {x}}\equiv (x, y)\) (i.e. \((x_{i}, y_{j}) = (i\Delta x, j\Delta x)\) with \(\Delta x=0.016\) and \(i, j = 0,\dots ,N)\) Moreover, we choose the time step \(\Delta t= 10^{-4}\) and, unless stated otherwise, we perform numerical simulations for \(15\times 10^4\) time-steps (i.e. the final time of simulations is \(t_f=15\)).

The method for constructing numerical solutions of the IDE-PDE-PDE system (3.2) is based on a finite difference scheme whereby the discretised dependent variables are

$$\begin{aligned}n_{i,j}^{k}:=n(x_{i},y_{j},t_k), \quad c_{i,j}^{k}:=c(x_{i},y_{j},t_k) \quad \text {and} \quad \phi _{i,j}^{k}:=\phi (x_{i},y_{j},t_k) . \end{aligned}$$

We solve numerically the IDE (3.2)\(_1\) for n and the PDE (3.2)\(_3\) for \(\phi \) using the following schemes

$$\begin{aligned} \frac{n_{i,j}^{k+1}-n_{i,j}^{k}}{\Delta t}=\left( \alpha _n-\mu _n\rho _n^k-\zeta _nK_{i,j}^{k}\right) n_{i,j}^{k} \quad i,j=0,\dots ,N, \end{aligned}$$


$$\begin{aligned} \begin{aligned} \frac{\phi _{i,j}^{k+1}-\phi _{i,j}^{k}}{\Delta t}=&\beta _{\phi }\frac{\phi _{i-1,j}^k+\phi _{i+1,j}^k-2\phi _{i,j}^{k}}{(\Delta x)^2}+\beta _{\phi }\frac{\phi _{i,j-1}^k+\phi _{i,j+1}^k-2\phi _{i,j}^{k}}{(\Delta x)^2}\\&+\alpha _{\phi }n^k_{i,j}-\kappa _{\phi }\phi _{i,j}^{k}, \quad i,j=1,\dots ,N-1, \end{aligned} \end{aligned}$$

and impose zero-flux boundary conditions for \(\phi \) by letting

$$\begin{aligned}{} & {} \phi _{0,j}^{k+1}=\phi _{1,j}^{k+1} \quad \text {and} \quad \phi _{N,j}^{k+1}=\phi _{N-1,j}^{k+1}, \quad j=0,\dots , N \\{} & {} \phi _{i,0}^{k+1}=\phi _{i,1}^{k+1} \quad \text {and} \quad \phi _{i,N}^{k+1}=\phi _{i,N-1}^{k+1}, \quad i=0,\dots , N \end{aligned}$$

Moreover, we solve numerically the PDE (3.2)\(_2\) for c using the following explicit scheme, which is the same as the one used in Bubba et al. (2020),

$$\begin{aligned} \frac{c_{i,j}^{k+1}-c_{i,j}^{k}}{\Delta t}-\frac{F^{k}_{i+\frac{1}{2},j}+F^{k}_{i-\frac{1}{2},j}}{\Delta x}-\frac{F^{k}_{i,j+\frac{1}{2}}+F^{k}_{i,j-\frac{1}{2}}}{\Delta x}=r(\phi ^k_{i,j})-\mu _c\rho _c^kc_{i,j}^{k} \end{aligned}$$

for \(i,j=0,\dots , N\), where

$$\begin{aligned} \begin{aligned}&F^{k}_{i+\frac{1}{2},j}:=\beta _c \psi \left( w^k_{i+\frac{1}{2},j}\right) \frac{c^k_{i+1,j}-c_{i,j}^{k}}{\Delta x}-\beta _c c^k_{i+\frac{1}{2},j}\psi ^\prime \left( w^k_{i+\frac{1}{2},j}\right) \frac{w^k_{i+1,j}-w_{i,j}^{k}}{\Delta x}\\&\quad -b^{k,+}_{i+\frac{1}{2},j}c_{i,j}^k\psi \left( w_{i+1,j}^k\right) +b^{k,-}_{i+\frac{1}{2},j}c_{i+1,j}^{k}\psi \left( w_{i,j}^k\right) , \quad i=0, \dots N-1, \; j=0, \dots N, \\&F^{k}_{i,j+\frac{1}{2}}:=\beta _c \psi \left( w^k_{i,j+\frac{1}{2}}\right) \frac{c^k_{i,j+1}-c_{i,j}^{k}}{\Delta x}-\beta _c c^k_{i,j+\frac{1}{2}}\psi ^\prime \left( w^k_{i,j+\frac{1}{2}}\right) \frac{w^k_{i,j+1}-w_{i,j}^{k}}{\Delta x}\\&\quad -b^{k,+}_{i,j+\frac{1}{2}}c_{i,j}^k\psi \left( w_{i,j+1}^k\right) +b^{k,-}_{i,j+\frac{1}{2}}c_{i,j+1}^{k+1}\psi \left( w_{i,j}^k\right) , \qquad i=0, \dots N, \; j=0, \dots N-1, \end{aligned} \end{aligned}$$


$$\begin{aligned}{} & {} w^k_{i+\frac{1}{2},j}:=\frac{w^k_{i+1,j}+w^k_{i,j}}{2}, \qquad w^k_{i,j+\frac{1}{2}}:=\frac{w^k_{i,j+1}+w^k_{i,j}}{2},\\{} & {} c^k_{i+\frac{1}{2},j}:=\frac{c^k_{i+1,j}+c^k_{i,j}}{2}, \qquad c^k_{i,j+\frac{1}{2}}:=\frac{c^k_{i,j+1}+c^k_{i,j}}{2},\\{} & {} b^{k}_{i+\frac{1}{2},j}:=\gamma _c\frac{\phi ^k_{i+1,j}-\phi ^k_{i,j}}{\Delta x}, \; b^{k,+}_{i+\frac{1}{2},j}=\max \left( 0,b^{k}_{i+\frac{1}{2},j}\right) , \; b^{k,-}_{i+\frac{1}{2},j}=\max \left( 0,-b^{k}_{i+\frac{1}{2},j}\right) \end{aligned}$$


$$\begin{aligned}b^{k}_{i,j+\frac{1}{2}}:=\gamma _c\frac{\phi ^k_{i,j+1}-\phi ^k_{i,j}}{\Delta x}, \; b^{k,+}_{i,j+\frac{1}{2}}=\max \left( 0,b^{k}_{i,j+\frac{1}{2}}\right) , \; b^{k,-}_{i,j+\frac{1}{2}}=\max \left( 0,-b^{k}_{i,j+\frac{1}{2}}\right) \end{aligned}$$

The discrete fluxes \(F^k_{i-\frac{1}{2},j}\) for \(i=1,\dots , N,\, j=0,\dots , N\) and \(F^k_{i,j-\frac{1}{2}}\) for \(i=0,\dots , N, \,j=1,\dots , N\) are defined in an analogous way, and we impose zero-flux boundary conditions by using the definitions

$$\begin{aligned}{} & {} F^k_{0-\frac{1}{2},j}:=0 \quad \text {and} \quad F^k_{N+\frac{1}{2},j}:=0, \quad \text {for} \quad j=0,\dots ,N,\\{} & {} F^k_{i,0-\frac{1}{2}}:=0 \quad \text {and} \quad F^k_{i,N+\frac{1}{2}}:=0, \quad \text {for} \quad i=0,\dots ,N. \end{aligned}$$

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Almeida, L., Audebert, C., Leschiera, E. et al. A Hybrid Discrete–Continuum Modelling Approach to Explore the Impact of T-Cell Infiltration on Anti-tumour Immune Response. Bull Math Biol 84, 141 (2022).

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  • Hybrid models
  • Continuum models
  • Numerical simulations
  • Tumour–immune cell interactions
  • T-cell infiltration
  • Immunotherapy