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Mathematical modelling of drug transport in tumour multicell spheroids and monolayer cultures. (English) Zbl 1014.92021
Summary: We adapt an avascular tumour growth model to compare the effects of drug application on multicell spheroids and on monolayer cultures. The model for the tumour is based on nutrient driven growth of a continuum of live cells, whose birth and death generates volume changes described by a velocity field. The drug is modelled as an externally applied, diffusible material capable of killing cells, both linear and Michaelis–Menten kinetics for drug action on cells being studied. Numerical solutions of the resulting system of partial differential equations for the multicell spheroid case are compared with closed form solutions of the monolayer case, particularly with respect to the effects on the cell kill of the drug dosage and of the duration of its application.
The results show an enhanced survival rate in multicell spheroids compared to monolayer cultures, consistent with experimental observations, and indicate that the key factor determining this is drug penetration. An analysis of the large time tumour spheroid response to a continuously applied drug at fixed concentration reveals up to three stable large time solutions, namely the trivial solution (i.e., a dead tumour), a travelling wave (continuously growing tumour) and a sublinear growth case in which cells reach a pseudo-steady-state in the core. Each of these possibilities is formulated and studied, with the bifurcations between them being discussed. Numerical solutions reveal that the pseudo-steady-state solutions persist to a significantly higher drug dose than travelling wave solutions.

92C50 Medical applications (general)
35Q92 PDEs in connection with biology, chemistry and other natural sciences
92C45 Kinetics in biochemical problems (pharmacokinetics, enzyme kinetics, etc.)
35B32 Bifurcations in context of PDEs
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[1] Folkman, J., Tumour angiogenesis, Adv. canc. res., 19, 331, (1974)
[2] Jain, R.K., Physiological barrier to delivery of monoclonalantibodies and other macromolecules in tumours, Canc. res., 50, 814, (1990)
[3] Jones, A.C.; Stratford, I.J.; Wilson, P.A.; Peckham, N.J., In vitro cytotoxic drug sensitivity testing of human tumour xenografts grown as multicellular tumour speroids, Br. J. canc., 46, 870, (1982)
[4] Mueller-Klieser, W., Multicell spheroids–a review on cellular aggregate in cancer research, J. canc. res. clin. oncol., 113, 101, (1987)
[5] Folkman, J.; Hochberg, M., Self-regulation of growth in three dimensions, J. exp. med., 138, 745, (1973)
[6] Kerr, D.J.; Wheldon, T.E.; Hydns, S.; Kaye, S.B., Cytotoxic drug penetration studies in multicellular tumour spheroids, Xenobiotica, 18, 641, (1988)
[7] Knuchel, R.; Hofstadter, F.; Jenkins, W.E.; Masters, J.R.W., Sensitivities of monolayers and spheroids of the human bladder cancer cell line MGH-U1 to the drugs used for intravesical chemotherapy, Canc. res., 49, 1397, (1989)
[8] Kwok, T.T.; Twentyman, P.R., The response to cytotoxic drugs of EMT6 cells treated either as intact or disaggregated spheroids, Br. J. canc., 51, 211, (1985)
[9] Wibe, E., Resistance to vincristine of human cells grown as multicellular spheroids, Br. J. canc., 42, 937, (1980)
[10] Oloumi, A.; MacPhail, S.H.; Johnston, P.J.; Banáth, J.P.; Olive, P.L., Changes in subcellular distribution of topoisomerase IIα correlate with etoposide resistance in multicell spheroids and xenograft tumors, Canc. res., 60, 5747, (2000)
[11] Kwok, T.T.; Twentyman, P.R., The relationship between tumour geometry and the response of tumour cells to cytotoxic drugs–an in vitro study using EMT6 multicellular spheroids, Br. J. canc., 35, 675, (1985)
[12] Durand, R.E., Flow cytometry studies of intracellular adriamycin in multicell spheroids in vitro, Canc. res., 41, 3495, (1981)
[13] Erlichman, C.; Vidgen, D., Cytotoxicity of adriamycin in mgh-ul cells grown as monolayer cultures, spheroids and xenografts in immune-deprived mice, Canc. res., 44, 5369, (1984)
[14] Nederman, T., Effects of vinblastine and 5-fluorouracil on human glioma and thyroid cancer cell monolayers and spheroids, Canc. res., 44, 254, (1984)
[15] Rauth, A.M.; Mohindra, J.K.; Tannock, I.F., Activity of mitomycin C for aerobic and hypoxic cells in vitro and in vivo, Canc. res., 43, 4154, (1983)
[16] Cojocaru, L.; Agur, Z., A theoretical analysis of interval drug dosing for cell-cycle-phase specific drugs, Math. biosci., 109, 85, (1992) · Zbl 0746.92012
[17] Panetta, J.C.; Adam, J.A., A mathematical model of cycle-specific chemotherapy, Math. comput. mod., 22, 67, (1995) · Zbl 0829.92011
[18] Usher, J.R.; Henderson, D., Some drug resistant models for cancer chemotherapy part 1: cycle-nonspecific drug, IMA J. math. appl. med. biol., 13, 99, (1996) · Zbl 0856.92011
[19] Costa, M.I.S.; Boldrini, J.L., Hemotherapeutic treatments: a study of the interplay among drug resistance, toxicity and recuperation from side effects, Bull. math. biol., 59, 205, (1997) · Zbl 0899.92024
[20] Levin, V.A.; Patlak, C.S.; Landahl, H.D., Heuristic modelling of drug delivery to malignant brain tumours, J. pharm. biopharm., 8, 257, (1980)
[21] Tracqui, P.; Cruywagen, G.C.; Woodward, D.E.; Bartoo, G.T.; Murray, J.D.; Alvord, E.G., A mathematical model of glioma growth–the effect of chemotherapy on spatiotemporal growth, Cell prolif., 28, 17, (1995)
[22] Wenning, L.A.; Murphy, R.M., Coupled cellular trafficking and diffusional limitations in delivery of immunotoxins to multicell tumor spheroids, Biotech. bioeng., 62, 562, (1999)
[23] Jackson, T.L.; Byrne, H.M., A mathematical model to study the effects of drug resistance and vasculature on the response of solid tumours to chemotherapy, Math. biosci., 164, 17, (2000) · Zbl 0947.92013
[24] J.P. Ward, Mathematical modelling of avascular tumour growth, PhD thesis, University of Nottingham, 1997 · Zbl 0866.92011
[25] Hicks, K.O.; Ohms, S.J.; van Zijl, P.L.; Hunter, P.J.; Wilson, W.R., An experimental and mathematical model for the extravascular transport of a DNA intercalator in tumours, Br. J. canc., 76, 894, (1997)
[26] Düchtung, W.; Vogelsaenger, T., Modelling and simulation of growing spheroids, Rec. res. canc. res., 95, 168, (1984)
[27] Ward, J.P.; King, J.R., Mathematical modelling of avascular tumour growth, IMA J. math. appl. med. biol., 14, 39, (1997) · Zbl 0866.92011
[28] Ward, J.P.; King, J.R., Mathematical modelling of avascular tumour growth II: modelling growth saturation, IMA J. math. appl. med. biol., 16, 171, (1999) · Zbl 0943.92019
[29] Kwok, C.S.; Cole, S.E.; Liao, S.K., Uptake kinetics of monoclonal-antibodies by human-malignant melanoma multicell spheroids, Canc. res., 48, 1856, (1988)
[30] Luk, C.K.; Veinot-Drebot, L.; Tannock, I.F., Effect of transient hypoxia on sensitivity to doxorubicin in human and murine cell lines, J. nat. canc. inst., 82, 684, (1990)
[31] Olive, P.L.; Banáth, J.P.; Evans, H.H., Cell Killing and DNA damage by etoposide in Chinese hamster v79 monolayers and spheroids: influence of growth kinetics, growth environment and DNA packaging, Br. J. canc., 67, 522, (1993)
[32] Sano, Y.; Hoshino, T.; Barker, M.; Deen, D.F., Response of 91 rat brain tumour multicell spheroids to single and fractionated doses of 1,3-bis(2-chloroethyl)-1-nitrosourea, Canc. res., 44, 571, (1984)
[33] E.S. Norris, Modelling the growth of avascular tumours and their response to chemotherapy, PhD thesis, University of Nottingham, 2002
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