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Mathematical model of tumour spheroid experiments with real-time cell cycle imaging. (English) Zbl 1458.92043

Summary: Three-dimensional (3D) in vitro tumour spheroid experiments are an important tool for studying cancer progression and potential cancer drug therapies. Standard experiments involve growing and imaging spheroids to explore how different conditions lead to different rates of spheroid growth. These kinds of experiments, however, do not reveal any information about the spatial distribution of the cell cycle within the expanding spheroid. Since 2008, a new experimental technology called fluorescent ubiquitination-based cell cycle indicator (FUCCI) has enabled real-time in situ visualisation of the cell cycle progression. Observations of 3D tumour spheroids with FUCCI labelling reveal significant intratumoural structure, as the cell cycle status can vary with location. Although many mathematical models of tumour spheroid growth have been developed, none of the existing mathematical models are designed to interpret experimental observations with FUCCI labelling. In this work, we adapt the mathematical framework originally proposed by J. P. Ward and J. R. King [IMA J. Math. Appl. Med. Biol. 14, No. 1, 39–69 (1997; Zbl 0866.92011)] to produce a new mathematical model of FUCCI-labelled tumour spheroid growth. The mathematical model treats the spheroid as being composed of three subpopulations: (i) living cells in G1 phase that fluoresce red; (ii) living cells in S/G2/M phase that fluoresce green; and (iii) dead cells that are not fluorescent. We assume that the rates at which cells pass through different phases of the cell cycle, and the rate of cell death, depend upon the local oxygen concentration. Parameterising the new mathematical model using experimental measurements of cell cycle transition times, we show that the model can qualitatively capture important experimental observations that cannot be addressed using previous mathematical models. Further, we show that the mathematical model can be used to qualitatively mimic the action of anti-mitotic drugs applied to the spheroid. All software programs required to solve the nonlinear moving boundary problem associated with the new mathematical model are available on GitHub (https://github.com/wang-jin-mathbio/Jin2021).

MSC:

92C50 Medical applications (general)

Citations:

Zbl 0866.92011

Software:

GitHub
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[1] Bajar, BT; Lam, AJ; Badiee, RK; Oh, Y-H; Chu, J.; Zhou, XX; Kim, N.; Kim, BB; Chung, M.; Yablonovitch, AL; Cruz, BF; Kulalert, K.; Tao, JJ; Meyer, T.; Su, X-D; Lin, M-Z, Fluorescent indicators for simultaneous reporting of all four cell cycle phases, Nat Methods, 13, 993-996 (2016) · doi:10.1038/nmeth.4045
[2] Beaumont, KA; Mohana-Kumaran, N.; Haass, NK, Modeling melanoma in vitro and in vivo, Healthcare, 2, 27-46 (2014) · doi:10.3390/healthcare2010027
[3] Beaumont, KA; Hill, DS; Daignault, SM; Lui, GY; Sharp, DM; Gabrielli, B.; Weninger, W.; Haass, NK, Cell cycle phase-specific drug resistance as an escape mechanism of melanoma cells, J Investig Dermatol, 136, 1479-1489 (2016) · doi:10.1016/j.jid.2016.02.805
[4] Breward, CJ; Byrne, HM; Lewis, CE, The role of cell-cell interactions in a two-phase model for avascular tumour growth, J Math Biol, 45, 125-152 (2002) · Zbl 1012.92017 · doi:10.1007/s002850200149
[5] Byrne, HM; King, JR; McElwain, DLS; Preziosi, L., A two-phase model of solid tumour growth, Appl Math Lett, 16, 567-573 (2003) · Zbl 1040.92015 · doi:10.1016/S0893-9659(03)00038-7
[6] Chan, FK; Shisler, J.; Bixby, JG; Felices, M.; Zheng, L.; Appel, M.; Orenstein, J.; Moss, B.; Lenardo, MJ, A role for tumor necrosis factor receptor-2 and receptor-interacting protein in programmed necrosis and antiviral responses, J Biol Chem, 278, 51613-51621 (2003) · doi:10.1074/jbc.M305633200
[7] Chaplain, MA; Graziano, L.; Preziosi, L., Mathematical modelling of the loss of tissue compression responsiveness and its role in solid tumour development, Math Med Biol, 23, 197-229 (2006) · Zbl 1098.92037 · doi:10.1093/imammb/dql009
[8] Collis, J.; Hubbard, ME; O’Dea, RD, Computational modelling of multiscale, multiphase fluid mixtures with application to tumour growth, Comput Methods Appl Mech Eng, 309, 554-578 (2016) · Zbl 1439.76003 · doi:10.1016/j.cma.2016.06.015
[9] Collis, J.; Hubbard, ME; O’Dea, RD, A multi-scale analysis of drug transport and response for a multi-phase tumour model, Eur J Appl Math, 28, 499-534 (2017) · Zbl 1375.92030 · doi:10.1017/S0956792516000413
[10] Crivelli, JJ; Földes, J.; Kim, PS; Wares, JR, A mathematical model for cell cycle-specific cancer virotherapy, J Biol Dyn, 6, 104-120 (2012) · Zbl 1447.92193 · doi:10.1080/17513758.2011.613486
[11] Deakin, AS, Model for the growth of a solid in vitro tumor, Growth, 39, 159-165 (1975)
[12] de Pillis, LG; Gu, W.; Radunskaya, AE, Mixed immunotherapy and chemotherapy of tumors: modeling, applications and biological interpretations, J Theor Biol, 238, 841-862 (2006) · Zbl 1445.92135 · doi:10.1016/j.jtbi.2005.06.037
[13] Enderling, H.; Chaplain, MAJ, Mathematical modeling of tumor growth and treatment, Curr Pharm Des, 20, 4934-4940 (2014) · doi:10.2174/1381612819666131125150434
[14] Flegg, JA; Nataraj, N., Mathematical modelling and avascular tumour growth, Resonance, 24, 313-325 (2019) · doi:10.1007/s12045-019-0782-8
[15] Friedrich, J.; Seidel, C.; Ebner, R.; Kunz-Schughart, LA, Spheroid-based drug screen: considerations and practical approach, Nat Protoc, 4, 309-324 (2009) · doi:10.1038/nprot.2008.226
[16] Greenspan, HP, Models for the growth of a solid tumor by diffusion, Stud Appl Math, 51, 317-340 (1972) · Zbl 0257.92001 · doi:10.1002/sapm1972514317
[17] Haass, NK; Sproesser, K.; Nguyen, TK; Contractor, R.; Medina, CA; Nathanson, KL; Herlyn, M.; Smalley, KSM, The mitogen-activated protein/extracellular signal-regulated kinase kinase inhibitor AZD6244 (ARRY-142886) induces growth arrest in melanoma cells and tumor regression when combined with docetaxel, Clin Cancer Res, 14, 230-239 (2008) · doi:10.1158/1078-0432.CCR-07-1440
[18] Haass, NK; Beaumont, KA; Hill, DS; Anfosso, A.; Mrass, P.; Munoz, MA; Kinjyo, I.; Weninger, W., Real-time cell cycle imaging during melanoma growth, invasion, and drug response, Pigment Cell Melanoma Res, 27, 764-776 (2014) · doi:10.1111/pcmr.12274
[19] Haass, NK; Gabrielli, B., Cell cycle-tailored targeting of metastatic melanoma: challenges and opportunities, Exp Dermatol, 26, 649-655 (2017) · doi:10.1111/exd.13303
[20] Jin, W.; Shah, ET; Penington, CJ; McCue, SW; Chopin, LK; Simpson, MJ, Reproducibility of scratch assays is affected by the initial degree of confluence: experiments, modelling and model selection, J Theor Biol, 390, 136-145 (2016) · Zbl 1343.92074 · doi:10.1016/j.jtbi.2015.10.040
[21] Jin, W.; Shah, ET; Penington, CJ; McCue, SW; Maini, PK; Simpson, MJ, Logistic proliferation of cells in scratch assays is delayed, Bull Math Biol, 79, 1028-1050 (2017) · Zbl 1368.92112 · doi:10.1007/s11538-017-0267-4
[22] Jin, W.; McCue, SW; Simpson, MJ, Extended logistic growth model for heterogeneous populations, J Theor Biol, 445, 51-61 (2019) · Zbl 1397.92164 · doi:10.1016/j.jtbi.2018.02.027
[23] Kienzle, A.; Kurch, S.; Schlöder, J.; Berges, C.; Ose, R.; Schupp, J.; Tuettenberg, A.; Weiss, H.; Schultze, J.; Winzen, S.; Schinnerer, M.; Koynov, K.; Mezger, M.; Haass, NK; Tremel, W.; Jonuleit, H., Dendritic mesoporous silica nanoparticles for pH-stimuli-responsive drug delivery of TNF-alpha, Adv Healthc Mater, 6, 1700012 (2017) · doi:10.1002/adhm.201700012
[24] Kunz-Schughart, LA; Kreutz, M.; Knuechel, R., Multicellular spheroids: a three-dimensional in vitro culture system to study tumour biology, Int J Exp Pathol, 79, 1-23 (1998) · doi:10.1046/j.1365-2613.1998.00051.x
[25] Landman, KA; Please, CP, Tumour dynamics and necrosis: surface tension and stability, Math Med Biol, 18, 131-158 (2001) · Zbl 0973.92017 · doi:10.1093/imammb/18.2.131
[26] Lewin, TD; Maini, PK; Moros, EG; Enderling, H.; Byrne, HM, The evolution of tumour composition during fractionated radiotherapy: implications for outcome, Bull Math Biol, 80, 1207-1235 (2018) · Zbl 1394.92061 · doi:10.1007/s11538-018-0391-9
[27] Lewin, TD; Maini, PK; Moros, EG; Enderling, H.; Byrne, HM, A three phase model to investigate the effects of dead material on the growth of avascular tumours, Math Model Nat Phenom, 15, 22 (2020) · Zbl 1467.92061 · doi:10.1051/mmnp/2019039
[28] Loessner, D.; Flegg, JA; Byrne, HM; Clements, JA; Hutmacher, DW, Growth of confined cancer spheroids: a combined experimental and mathematical modelling approach, Integr Biol, 5, 597-605 (2013) · doi:10.1039/c3ib20252f
[29] Maeda, S.; Wada, H.; Naito, Y.; Nagano, H.; Simmons, S.; Kagawa, Y.; Naito, A.; Kikuta, J.; Ishii, T.; Tomimaru, Y.; Hama, N.; Kawamoto, K.; Kobayashi, S.; Eguchi, H.; Umeshita, K.; Ishii, H.; Doki, Y.; Mori, M.; Ishii, M., Interferon-\( \alpha\) acts on the S/G2/M phases to induce apoptosis in the G1 phase of an IFNAR2-expressing hepatocellular carcinoma cell line, J Biol Chem, 289, 23786-23795 (2014) · doi:10.1074/jbc.M114.551879
[30] Maini, PK; McElwain, DLS; Leavesley, DI, Traveling wave model to interpret a wound-healing cell migration assay for human peritoneal mesothelial cells, Tissue Eng, 10, 475-482 (2004) · doi:10.1089/107632704323061834
[31] Maini, PK; McElwain, DLS; Leavesley, D., Travelling waves in a wound healing assay, Appl Math Lett, 17, 575-580 (2004) · Zbl 1055.92025 · doi:10.1016/S0893-9659(04)90128-0
[32] McElwain, DLS; Morris, LE, Apoptosis as a volume loss mechanism in mathematical models of solid tumor growth, Math Biosci, 39, 147-157 (1978) · doi:10.1016/0025-5564(78)90033-0
[33] McElwain, DLS; Callcott, R.; Morris, LE, A model of vascular compression in solid tumours, J Theor Biol, 78, 405-415 (1979) · doi:10.1016/0022-5193(79)90339-4
[34] Nath, S.; Devi, GR, Three-dimensional culture systems in cancer research: focus on tumor spheroid model, Pharmacol Ther, 163, 94-108 (2016) · doi:10.1016/j.pharmthera.2016.03.013
[35] Norton, L.; Simon, R.; Brereton, HD; Bogden, AE, Predicting the course of Gompertzian growth, Nature, 264, 542-545 (1976) · doi:10.1038/264542a0
[36] Pettet, GJ; Please, CP; Tindall, MJ; McElwain, DLS, The migration of cells in multicell tumor spheroids, Bull Math Biol, 63, 231-257 (2001) · Zbl 1323.92037 · doi:10.1006/bulm.2000.0217
[37] Sakaue-Sawano, A.; Kurokawa, H.; Morimura, T.; Hanyu, A.; Hama, H.; Osawa, H.; Kashiwagi, S.; Fukami, K.; Miyata, T.; Miyoshi, H.; Imamura, T.; Ogawa, M.; Masai, H.; Miyawaki, A., Visualizing spatiotemporal dynamics of multicellular cell-cycle progression, Cell, 132, 487-498 (2008) · doi:10.1016/j.cell.2007.12.033
[38] Santini, MT; Rainaldi, G., Three-dimensional spheroid model in tumor biology, Pathobiology, 67, 148-157 (1999) · doi:10.1159/000028065
[39] Sarapata, EA; de Pillis, LG, A comparison and catalog of intrinsic tumor growth models, Bull Math Biol, 76, 2010-2024 (2014) · Zbl 1300.92042 · doi:10.1007/s11538-014-9986-y
[40] Simpson, MJ; Landman, KA; Clement, TP, Assessment of a non-traditional operator split algorithm for simulation of reactive transport, Math Comput Simul, 70, 44-60 (2005) · Zbl 1087.65599 · doi:10.1016/j.matcom.2005.03.019
[41] Simpson, MJ; Jin, W.; Vittadello, ST; Tambyah, TA; Ryan, JM; Gunasingh, G.; Haass, NK; McCue, SW, Stochastic models of cell invasion with fluorescent cell cycle indicators, Phys A Stat Mech Its Appl, 510, 375-386 (2018) · doi:10.1016/j.physa.2018.06.128
[42] Simpson, MJ; Baker, RE; Vittadello, ST; Maclaren, OJ, Parameter identifiability analysis for spatiotemporal models of cell invasion, J R Soc Interface, 17, 20200055 (2020) · doi:10.1098/rsif.2020.0055
[43] Smalley, KS; Lioni, M.; Noma, K.; Haass, NK; Herlyn, M., In vitro three-dimensional tumor microenvironment models for anticancer drug discovery, Expert Opin Drug Discov, 3, 1-10 (2008) · doi:10.1517/17460441.3.1.1
[44] Spill, F.; Andasari, V.; Mak, M.; Kamm, RD; Zaman, MH, Effects of 3D geometries on cellular gradient sensing and polarization, Phys Biol, 13, 036008 (2016) · doi:10.1088/1478-3975/13/3/036008
[45] Spoerri L, Beaumont KA, Anfosso A, Haass NK (2017) Real-time cell cycle imaging in a 3D cell culture model of melanoma. In: 3D cell culture. Humana Press, New York, NY, pp 401-416. doi:10.1007/978-1-4939-7021-6_29
[46] Spoerri L, Tonnessen-Murray CA, Gunasingh G, Hill DS, Beaumont KA, Jurek RJ, Vanwalleghem GC, Fane ME, Daignault SM, Matigian N, Scott EK, Smith AG, Stehbens SJ, Schaider H, Weninger W, Gabrielli B, Haass NK (2020) Functional melanoma cell heterogeneity is regulated by MITF-dependent cell-matrix interactions. doi:10.1101/2020.06.09.141747
[47] Stehn, JR; Haass, NK; Bonello, T.; Desouza, M.; Kottyan, G.; Treutlein, H.; Zeng, J.; Nascimento, PRBB; Sequeira, VB; Butler, TL; Allanson, M.; Fath, T.; Hill, TA; McCluskey, A.; Schevzov, G.; Palmer, SJ; Hardeman, EC; Winlaw, D.; Reeve, VE; Dixon, I.; Weninger, W.; Cripe, TP; Gunning, PW, A novel class of anticancer compounds targets the actin cytoskeleton in tumor cells, Clin Cancer Res, 73, 5169-5182 (2013) · doi:10.1158/0008-5472.CAN-12-4501
[48] Sutherland, RM; McCredie, JA; Inch, WR, Growth of multicell spheroids in tissue culture as a model of nodular carcinomas, J Natl Cancer Inst, 46, 113-120 (1971) · doi:10.1093/jnci/46.1.113
[49] Vittadello, ST; McCue, SW; Gunasingh, G.; Haass, NK; Simpson, MJ, Mathematical models for cell migration with real-time cell cycle dynamics, Biophys J, 114, 1241-1253 (2018) · doi:10.1016/j.bpj.2017.12.041
[50] Vittadello, ST; McCue, SW; Gunasingh, G.; Haass, NK; Simpson, MJ, Mathematical models incorporating a multi-stage cell cycle replicate normally-hidden inherent synchronization in cell proliferation, J R Soc Interface, 16, 20190382 (2019) · doi:10.1098/rsif.2019.0382
[51] Vittadello, ST; McCue, SW; Gunasingh, G.; Haass, NK; Simpson, MJ, Examining go-or-grow using fluorescent cell-cycle indicators and cell-cycle-inhibiting drugs, Biophys J, 118, 1243-1247 (2020) · doi:10.1016/j.bpj.2020.01.036
[52] Ward, JP; King, JR, Mathematical modelling of avascular-tumour growth, Math Med Biol, 14, 39-69 (1997) · Zbl 0866.92011 · doi:10.1093/imammb/14.1.39
[53] Ward, JP; King, JR, Mathematical modelling of avascular-tumour growth II: modelling growth saturation, Math Med Biol, 16, 171-211 (1999) · Zbl 0943.92019 · doi:10.1093/imammb/14.1.39
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