Research article

A phase-field model for non-small cell lung cancer under the effects of immunotherapy


  • Received: 17 June 2023 Revised: 22 August 2023 Accepted: 23 August 2023 Published: 07 October 2023
  • Formulating mathematical models that estimate tumor growth under therapy is vital for improving patient-specific treatment plans. In this context, we present our recent work on simulating non-small-scale cell lung cancer (NSCLC) in a simple, deterministic setting for two different patients receiving an immunotherapeutic treatment. At its core, our model consists of a Cahn-Hilliard-based phase-field model describing the evolution of proliferative and necrotic tumor cells. These are coupled to a simplified nutrient model that drives the growth of the proliferative cells and their decay into necrotic cells. The applied immunotherapy decreases the proliferative cell concentration. Here, we model the immunotherapeutic agent concentration in the entire lung over time by an ordinary differential equation (ODE). Finally, reaction terms provide a coupling between all these equations. By assuming spherical, symmetric tumor growth and constant nutrient inflow, we simplify this full 3D cancer simulation model to a reduced 1D model. We can then resort to patient data gathered from computed tomography (CT) scans over several years to calibrate our model. Our model covers the case in which the immunotherapy is successful and limits the tumor size, as well as the case predicting a sudden relapse, leading to exponential tumor growth. Finally, we move from the reduced model back to the full 3D cancer simulation in the lung tissue. Thereby, we demonstrate the predictive benefits that a more detailed patient-specific simulation including spatial information as a possible generalization within our framework could yield in the future.

    Citation: Andreas Wagner, Pirmin Schlicke, Marvin Fritz, Christina Kuttler, J. Tinsley Oden, Christian Schumann, Barbara Wohlmuth. A phase-field model for non-small cell lung cancer under the effects of immunotherapy[J]. Mathematical Biosciences and Engineering, 2023, 20(10): 18670-18694. doi: 10.3934/mbe.2023828

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    [1] R. C. Rockne, J. G. Scott, Introduction to mathematical oncology, JCO Clin. Cancer Inform., 3 (2019), 1–4. https://doi.org/10.1200/CCI.19.00010 doi: 10.1200/CCI.19.00010
    [2] R. A. Weinberg, The Biology of Cancer, W.W. Norton & Company, (2006). https://doi.org/10.1201/9780203852569
    [3] T. A. Graham, A. Sottoriva, Measuring cancer evolution from the genome, J. Pathol., 241 (2017), 183–191. https://doi.org/10.1002/path.4821 doi: 10.1002/path.4821
    [4] D. Hanahan, R. A. Weinberg, Hallmarks of cancer: The next generation, Cell, 144 (2011), 646–674. https://doi.org/10.1016/j.cell.2011.02.013 doi: 10.1016/j.cell.2011.02.013
    [5] R. D. Schreiber, L. J. Old, M. J. Smyth, Cancer immunoediting: Integrating immunity's roles in cancer suppression and promotion, Science, 331 (2011), 1565–1570. https://doi.org/10.1126/science.1203486 doi: 10.1126/science.1203486
    [6] Y. Zhang, Z. Zhang, The history and advances in cancer immunotherapy: Understanding the characteristics of tumor-infiltrating immune cells and their therapeutic implications, Cell. Mol. Immunol., 17 (2020), 807–821. https://doi.org/10.1038/s41423-020-0488-6 doi: 10.1038/s41423-020-0488-6
    [7] A. Rounds, J. Kolesar, Nivolumab for second-line treatment of metastatic squamous non-small-cell lung cancer, Am. J. Health-Syst. Pharm., 72 (2015), 1851–1855. https://doi.org/10.2146/ajhp150235 doi: 10.2146/ajhp150235
    [8] G. M. Keating, Nivolumab: A review in advanced nonsquamous non-small cell lung cancer, Drugs, 76 (2016), 969–978. https://doi.org/10.1007/s40265-016-0589-9 doi: 10.1007/s40265-016-0589-9
    [9] Y. Iwai, J. Hamanishi, K. Chamoto, T. Honjo, Cancer immunotherapies targeting the PD-1 signaling pathway, J. Biomed. Sci., 24 (2017), 26. https://doi.org/10.1186/s12929-017-0329-9 doi: 10.1186/s12929-017-0329-9
    [10] N. Ghaffari Laleh, C. M. L. Loeffler, J. Grajek, K. Staňková, A. T. Pearson, H. S. Muti, et al., Classical mathematical models for prediction of response to chemotherapy and immunotherapy, PLOS Comput. Biol., 18 (2022), 1–18. https://doi.org/10.1371/journal.pcbi.1009822 doi: 10.1371/journal.pcbi.1009822
    [11] I. Ezhov, K. Scibilia, K. Franitza, F. Steinbauer, S. Shit, L. Zimmer, et al., Learn-Morph-Infer: A new way of solving the inverse problem for brain tumor modeling, Med. Image Anal., 83 (2023), 102672. https://doi.org/10.1016/j.media.2022.102672 doi: 10.1016/j.media.2022.102672
    [12] A. K. Laird, Dynamics of tumour growth: Comparison of growth rates and extrapolation of growth curve to one cell, Br. J. Cancer, 19 (1965), 278–291. https://doi.org/10.1038/bjc.1965.32 doi: 10.1038/bjc.1965.32
    [13] L. Norton, A Gompertzian model of human breast cancer growth, Cancer Res., 48 (1988), 7067–7071.
    [14] S. Benzekry, C. Lamont, A. Beheshti, A. Tracz, J. M. L. Ebos, L. Hlatky, et al., Classical mathematical models for description and prediction of experimental tumor growth, PLoS Comput. Biol., 10 (2014), e1003800. https://doi.org/10.1371/journal.pcbi.1003800 doi: 10.1371/journal.pcbi.1003800
    [15] M. Bilous, C. Serdjebi, A. Boyer, P. Tomasini, C. Pouypoudat, D. Barbolosi, et al., Quantitative mathematical modeling of clinical brain metastasis dynamics in non-small cell lung cancer, Sci. Rep., 9 (2019), 13018. https://doi.org/10.1038/s41598-019-49407-3 doi: 10.1038/s41598-019-49407-3
    [16] P. Schlicke, C. Kuttler, C. Schumann, How mathematical modeling could contribute to the quantification of metastatic tumor burden under therapy: Insights in immunotherapeutic treatment of non-small cell lung cancer, Theor. Biol. Med. Model., 18 (2021), 1–15. https://doi.org/10.1186/s12976-021-00142-1 doi: 10.1186/s12976-021-00142-1
    [17] S. Benzekry, C. Sentis, C. Coze, L. Tessonnier, N. André, Development and validation of a prediction model of overall survival in high-risk neuroblastoma using mechanistic modeling of metastasis, JCO Clin. Cancer Inf., 5 (2021), 81–90. https://doi.org/10.1200/CCI.20.00092 doi: 10.1200/CCI.20.00092
    [18] S. Benzekry, P. Schlicke, P. Tomasini, E. Simon, Mechanistic modeling of brain metastases in NSCLC provides computational markers for personalized prediction of outcome, medRxiv preprint, 2023. https://doi.org/10.1101/2023.01.10.23284189
    [19] F. Bray, J. Ferlay, I. Soerjomataram, R. L. Siegel, L. A. Torre, A. Jemal, Global cancer statistics 2018: GLOBOCAN estimates of incidence and mortality worldwide for 36 cancers in 185 countries, CA. Cancer J. Clin., 68 (2018), 394–424. https://doi.org/10.3322/caac.21492 doi: 10.3322/caac.21492
    [20] C. Zappa, S. A. Mousa, Non-small cell lung cancer: Current treatment and future advances, Transl. Lung Cancer Res., 5 (2016). https://doi.org/10.21037/tlcr.2016.06.07
    [21] W. D. Travis, E. Brambilla, A. G. Nicholson, Y. Yatabe, J. H. Austin, M. B. Beasley, et al., The 2015 world health organization classification of lung tumors: Impact of genetic, clinical and radiologic advances since the 2004 classification, J. Thorac. Oncol., 10 (2015), 1243–1260. https://doi.org/10.1097/JTO.0000000000000630 doi: 10.1097/JTO.0000000000000630
    [22] J. S. Lowengrub, H. B. Frieboes, F. Jin, Y. L. Chuang, X. Li, P. Macklin, et al., Nonlinear modelling of cancer: Bridging the gap between cells and tumours, Nonlinearity, 23 (2009). https://doi.org/10.1088/0951-7715/23/1/r01
    [23] V. Cristini, J. Lowengrub, Multiscale Modeling of Cancer: An Integrated Experimental and Mathematical Modeling Approach, Cambridge University Press, (2010). https://doi.org/10.1017/cbo9780511781452
    [24] O. Clatz, M. Sermesant, P. Y. Bondiau, H. Delingette, S. K. Warfield, G. Malandain, et al., Realistic simulation of the 3-D growth of brain tumors in MR images coupling diffusion with biomechanical deformation, IEEE Trans. Med. Imaging, 24 (2005), 1334–1346. https://doi.org/10.1109/tmi.2005.857217 doi: 10.1109/tmi.2005.857217
    [25] S. Subramanian, A. Gholami, G. Biros, Simulation of glioblastoma growth using a 3D multispecies tumor model with mass effect, J. Math. Biol., 79 (2019), 941–967. https://doi.org/10.1007/s00285-019-01383-y doi: 10.1007/s00285-019-01383-y
    [26] H. J. Bowers, E. E. Fannin, A. Thomas, J. A. Weis, Characterization of multicellular breast tumor spheroids using image data-driven biophysical mathematical modeling, Sci. Rep., 10 (2020), 1–12. https://doi.org/10.1038/s41598-020-68324-4 doi: 10.1038/s41598-020-68324-4
    [27] P. Friedl, D. Gilmour, Collective cell migration in morphogenesis, regeneration and cancer, Nat. Rev. Mol. Cell Biol., 10 (2009), 445–457. https://doi.org/10.1038/nrm2720 doi: 10.1038/nrm2720
    [28] H. Garcke, K. F. Lam, E. Sitka, V. Styles, A Cahn–Hilliard–Darcy model for tumour growth with chemotaxis and active transport, Math. Models Method Appl. Sci., 26 (2016), 1095–1148. https://doi.org/10.1142/s0218202516500263 doi: 10.1142/s0218202516500263
    [29] S. Frigeri, M. Grasselli, E. Rocca, On a diffuse interface model of tumour growth, Eur. J. Appl. Math., 26 (2015), 215–243. https://doi.org/10.1017/s0956792514000436 doi: 10.1017/s0956792514000436
    [30] H. G. Lee, Y. Kim, J. Kim, Mathematical model and its fast numerical method for the tumor growth, Math. Biosci. Eng., 12 (2015), 1173–1187. https://doi.org/10.3934/mbe.2015.12.1173 doi: 10.3934/mbe.2015.12.1173
    [31] M. Ebenbeck, H. Garcke, Analysis of a Cahn–Hilliard–Brinkman model for tumour growth with chemotaxis, J. Differ. Equations, 266 (2019), 5998–6036. https://doi.org/10.1016/j.jde.2018.10.045 doi: 10.1016/j.jde.2018.10.045
    [32] M. Ebenbeck, H. Garcke, On a Cahn–Hilliard–Brinkman model for tumor growth and its singular limits, SIAM J. Math. Anal., 51 (2019), 1868–1912. https://doi.org/10.1137/18m1228104 doi: 10.1137/18m1228104
    [33] M. Fritz, E. Lima, J. T. Oden, B. Wohlmuth, On the unsteady Darcy–Forchheimer–Brinkman equation in local and nonlocal tumor growth models, Math. Models Method Appl. Sci., 29 (2019), 1691–1731. https://doi.org/10.1142/S0218202519500325 doi: 10.1142/S0218202519500325
    [34] K. F. Lam, H. Wu, Thermodynamically consistent Navier–Stokes–Cahn–Hilliard models with mass transfer and chemotaxis, Eur. J. Appl. Math., 29 (2018), 595–644. https://doi.org/10.1017/s0956792517000298 doi: 10.1017/s0956792517000298
    [35] G. Lorenzo, A. M. Jarrett, C. T. Meyer, V. Quaranta, D. R. Tyson, T. E. Yankeelov, Identifying mechanisms driving the early response of triple negative breast cancer patients to neoadjuvant chemotherapy using a mechanistic model integrating in vitro and in vivo imaging data, arXiv preprint, (2022), arXiv: 2212.04270. https://doi.org/10.48550/arXiv.2212.04270
    [36] H. Garcke, K. F. Lam, R. Nürnberg, E. Sitka, A multiphase Cahn–Hilliard–Darcy model for tumour growth with necrosis, Math. Models Method Appl. Sci., 28 (2018), 525–577. https://doi.org/10.1142/s0218202518500148 doi: 10.1142/s0218202518500148
    [37] J. T. Oden, A. Hawkins, S. Prudhomme, General diffuse-interface theories and an approach to predictive tumor growth modeling, Math. Models Method Appl. Sci., 20 (2010), 477–517. https://doi.org/10.1142/s0218202510004313 doi: 10.1142/s0218202510004313
    [38] S. M. Wise, J. S. Lowengrub, H. B. Frieboes, V. Cristini, Three-dimensional multispecies nonlinear tumor growth – I: Model and numerical method, J. Theor. Biol., 253 (2008), 524–543. https://doi.org/10.1016/j.jtbi.2008.03.027 doi: 10.1016/j.jtbi.2008.03.027
    [39] V. Cristini, X. Li, J. S. Lowengrub, S. M. Wise, Nonlinear simulations of solid tumor growth using a mixture model: Invasion and branching, J. Math. Biol., 58 (2009), 723–763. https://doi.org/10.1007/s00285-008-0215-x doi: 10.1007/s00285-008-0215-x
    [40] H. B. Frieboes, F. Jin, Y. L. Chuang, S. M. Wise, J. S. Lowengrub, V. Cristini, Three-dimensional multispecies nonlinear tumor growth – II: Tumor invasion and angiogenesis, J. Theor. Biol., 264 (2010), 1254–1278. https://doi.org/10.1016/j.jtbi.2010.02.036 doi: 10.1016/j.jtbi.2010.02.036
    [41] E. Lima, J. T. Oden, R. Almeida, A hybrid ten-species phase-field model of tumor growth, Math. Models Method Appl. Sci., 24 (2014), 2569–2599. https://doi.org/10.1142/s0218202514500304 doi: 10.1142/s0218202514500304
    [42] A. Hawkins-Daarud, K. G. van der Zee, J. T. Oden, Numerical simulation of a thermodynamically consistent four-species tumor growth model, Int. J. Numer. Meth. Biol., 28 (2012), 3–24. https://doi.org/10.1002/cnm.1467 doi: 10.1002/cnm.1467
    [43] M. Fritz, P. K. Jha, T. Köppl, J. T. Oden, A. Wagner, B. Wohlmuth, Modeling and simulation of vascular tumors embedded in evolving capillary networks, Comput. Methods Appl. Mech. Eng., 384 (2021), 113975. https://doi.org/10.1016/j.cma.2021.113975 doi: 10.1016/j.cma.2021.113975
    [44] M. Fritz, P. K. Jha, T. Köppl, J. T. Oden, B. Wohlmuth, Analysis of a new multispecies tumor growth model coupling 3D phase-fields with a 1D vascular network, Nonlinear Anal. Real World Appl., 61 (2021), 103331. https://doi.org/10.1016/j.nonrwa.2021.103331 doi: 10.1016/j.nonrwa.2021.103331
    [45] G. Lorenzo, M. A. Scott, K. Tew, T. J. Hughes, Y. J. Zhang, L. Liu, et al., Tissue-scale, personalized modeling and simulation of prostate cancer growth, Proc. Natl. Acad. Sci., 113 (2016), E7663–E7671. https://doi.org/10.1073/pnas.1615791113 doi: 10.1073/pnas.1615791113
    [46] G. Song, T. Tian, X. Zhang, A mathematical model of cell-mediated immune response to tumor, Math. Biosci. Eng., 18 (2021), 373–385. https://doi.org/10.3934/mbe.2021020 doi: 10.3934/mbe.2021020
    [47] D. Kirschner, A. Tsygvintsev, On the global dynamics of a model for tumor immunotherapy, Math. Biosci. Eng., 6 (2009), 573–583. https://doi.org/10.3934/mbe.2009.6.573 doi: 10.3934/mbe.2009.6.573
    [48] K. R. Fister, J. H. Donnelly, Immunotherapy: An optimal control theory approach, Math. Biosci. Eng., 2 (2005), 499–510. https://doi.org/10.3934/mbe.2005.2.499 doi: 10.3934/mbe.2005.2.499
    [49] A. Soboleva, A. Kaznatcheev, R. Cavill, K. Schneider, K. Stankova, Polymorphic gompertzian model of cancer validated with in vitro and in vivo data, bioRxiv preprint, 2023. https://doi.org/10.1101/2023.04.19.537467
    [50] G. G. Powathil, D. J. Adamson, M. A. Chaplain, Towards predicting the response of a solid tumour to chemotherapy and radiotherapy treatments: Clinical insights from a computational model, PLoS Comput. Biol., 9 (2013), 1–14. https://doi.org/10.1371/journal.pcbi.1003120 doi: 10.1371/journal.pcbi.1003120
    [51] C. Wu, D. A. Hormuth, G. Lorenzo, A. M. Jarrett, F. Pineda, F. M. Howard, et al., Towards patient-specific optimization of neoadjuvant treatment protocols for breast cancer based on image-guided fluid dynamics, IEEE Trans. Biomed. Eng., 69 (2022), 3334–3344. https://doi.org/10.1109/tbme.2022.3168402 doi: 10.1109/tbme.2022.3168402
    [52] A. M. Jarrett, D. A. Hormuth, S. L. Barnes, X. Feng, W. Huang, T. E. Yankeelov, Incorporating drug delivery into an imaging-driven, mechanics-coupled reaction diffusion model for predicting the response of breast cancer to neoadjuvant chemotherapy: theory and preliminary clinical results, Phys. Med. Biol., 63 (2018), 105015. https://doi.org/10.1088/1361-6560/aac040 doi: 10.1088/1361-6560/aac040
    [53] R. C. Rockne, A. D. Trister, J. Jacobs, A. J. Hawkins-Daarud, M. L. Neal, K. Hendrickson, et al., A patient-specific computational model of hypoxia-modulated radiation resistance in glioblastoma using 18F-FMISO-PET, J. R. Soc. Interface, 12 (2015). https://doi.org/10.1098/rsif.2014.1174
    [54] P. Colli, H. Gomez, G. Lorenzo, G. Marinoschi, A. Reali, E. Rocca, Optimal control of cytotoxic and antiangiogenic therapies on prostate cancer growth, Math. Models Method Appl. Sci., 31 (2021), 1419–1468. https://doi.org/10.1142/s0218202521500299 doi: 10.1142/s0218202521500299
    [55] M. Fritz, C. Kuttler, M. L. Rajendran, L. Scarabosio, B. Wohlmuth, On a subdiffusive tumour growth model with fractional time derivative, IMA J. Appl. Math., 86 (2021), 688–729. https://doi.org/10.1093/imamat/hxab009 doi: 10.1093/imamat/hxab009
    [56] S. A. Quezada, K. S. Peggs, Exploiting CTLA-4, PD-1 and PD-L1 to reactivate the host immune response against cancer, Br. J. Cancer, 108 (2013), 1560–1565. https://doi.org/10.1038/bjc.2013.117 doi: 10.1038/bjc.2013.117
    [57] A. Ribas, Tumor immunotherapy directed at PD-1, N. Engl. J. Med., 366 (2012), 2517–2519. https://doi.org/10.1056/NEJMe1205943 doi: 10.1056/NEJMe1205943
    [58] D. M. Pardoll, The blockade of immune checkpoints in cancer immunotherapy, Nat. Rev. Cancer, 12 (2012), 252–264. https://doi.org/10.1038/nrc3239 doi: 10.1038/nrc3239
    [59] E. N. Rozali, S. V. Hato, B. W. Robinson, R. A. Lake, W. J. Lesterhuis, Programmed death ligand 2 in cancer-induced immune suppression, Clin. Dev. Immunol., 2012 (2012), 1–8. https://doi.org/10.1155/2012/656340 doi: 10.1155/2012/656340
    [60] S. P. Patel, R. Kurzrock, PD-L1 expression as a predictive biomarker in cancer immunotherapy, Mol. Cancer Ther., 14 (2015), 847–856. https://doi.org/10.1158/1535-7163.MCT-14-0983 doi: 10.1158/1535-7163.MCT-14-0983
    [61] Y. Viossat, R. Noble, A theoretical analysis of tumour containment, Nat. Ecol. Evol., 5 (2021), 826–835. https://doi.org/10.1038/s41559-021-01428-w doi: 10.1038/s41559-021-01428-w
    [62] T. Hillen, K. J. Painter, M. Winkler, Convergence of a cancer invasion model to a logistic chemotaxis model, Math. Models Method Appl. Sci., 23 (2013), 165–198.
    [63] B. Gompertz, On the nature of the function expressive of the law of human mortality, and on the new mode of determining the value of life contingencies, Philos. Trans. R. Soc., 115 (1825), 513–585. https://doi.org/10.1098/rstl.1825.0026 doi: 10.1098/rstl.1825.0026
    [64] K. Erbertseder, J. Reichold, B. Flemisch, P. Jenny, R. Helmig, A coupled discrete/continuum model for describing cancer-therapeutic transport in the lung, PLOS ONE, 7 (2012), 1–17. https://doi.org/10.1371/journal.pone.0031966 doi: 10.1371/journal.pone.0031966
    [65] H. Garcke, K. F. Lam, Well-posedness of a Cahn–Hilliard system modelling tumour growth with chemotaxis and active transport, Eur. J. Appl. Math., 28 (2017), 284–316. https://doi.org/10.1017/S0956792516000292 doi: 10.1017/S0956792516000292
    [66] H. Garcke, K. F. Lam, Analysis of a Cahn–Hilliard system with non-zero Dirichlet conditions modeling tumor growth with chemotaxis, Discrete Contin. Dyn. Syst. Ser. A, 37 (2017), 4277–4308. https://doi.org/10.3934/dcds.2017183 doi: 10.3934/dcds.2017183
    [67] P. Colli, H. Gomez, G. Lorenzo, G. Marinoschi, A. Reali, E. Rocca, Optimal control of cytotoxic and antiangiogenic therapies on prostate cancer growth, Math. Models Method Appl. Sci., 31 (2021), 1419–1468. https://doi.org/10.1142/s0218202521500299 doi: 10.1142/s0218202521500299
    [68] D. J. Eyre, Unconditionally gradient stable time marching the Cahn–Hilliard equation, MRS Online Proc. Lib., 529 (1998), 39–46. https://doi.org/10.1557/proc-529-39 doi: 10.1557/proc-529-39
    [69] S. C. Brenner, A. E. Diegel, L. Y. Sung, A robust solver for a mixed finite element method for the Cahn–Hilliard equation, J. Sci. Comput., 77 (2018), 1234–1249. https://doi.org/10.1007/s10915-018-0753-3 doi: 10.1007/s10915-018-0753-3
    [70] S. Balay, S. Abhyankar, M. F. Adams, S. Benson, J. Brown, P. Brune, et al., PETSc/TAO} Users Manual, Technical Report ANL-21/39 - Revision 3.18, Argonne National Laboratory, 2022.
    [71] A. Logg, K. A. Mardal, G. Wells, Automated Solution of Differential Equations by the Finite Element Method: The FEniCS Book, Springer Science & Business Media, (2012). https://doi.org/10.1007/978-3-642-23099-8
    [72] R. Kikinis, S. D. Pieper, K. G. Vosburgh, 3D Slicer: A platform for subject-specific image analysis, visualization, and clinical support, in Intraoperative Imaging and Image-guided Therapy, Springer, (2013), 277–289. https://doi.org/10.1007/978-1-4614-7657-3_19
    [73] Blender, Accessed: 2022-12-02. Available from: https://www.blender.org/.
    [74] C. Geuzaine, J. F. Remacle, Gmsh: A 3-D finite element mesh generator with built-in pre-and post-processing facilities, Int. J. Numer. Meth. Eng., 79 (2009), 1309–1331. https://doi.org/10.1002/nme.2579 doi: 10.1002/nme.2579
    [75] The Vascular Modeling Toolkit website, Accessed: 2022-12-02. www.vmtk.org.
    [76] L. Antiga, M. Piccinelli, L. Botti, B. Ene-Iordache, A. Remuzzi, D. A. Steinman, An image-based modeling framework for patient-specific computational hemodynamics, Med. Biol. Eng. Comput., 46 (2008), 1097–1112. https://doi.org/10.1007/s11517-008-0420-1 doi: 10.1007/s11517-008-0420-1
    [77] E. Eisenhauer, P. Therasse, J. Bogaerts, L. Schwartz, D. Sargent, R. Ford, et al., New response evaluation criteria in solid tumours: Revised RECIST guideline (version 1.1), Eur. J. Cancer, 45 (2009), 228–247. https://doi.org/10.1016/j.ejca.2008.10.026 doi: 10.1016/j.ejca.2008.10.026
    [78] L. Hanin, J. Rose, Suppression of metastasis by primary tumor and acceleration of metastasis following primary tumor resection: A natural law, Bull. Math. Biol., 80 (2018), 519–539. https://doi.org/10.1007/s11538-017-0388-9 doi: 10.1007/s11538-017-0388-9
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