A mathematical model of doxorubicin penetration through multicellular layers

C. J. Evans, R. M. Phillips, P. F. Jones, P. M. Loadman, B. D. Sleeman, C. J. Twelves, S. W. Smye

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Abstract

Inadequate drug delivery to tumours is now recognised as a key factor that limits the efficacy of anticancer drugs. Extravasation and penetration of therapeutic agents through avascular tissue are critically important processes if sufficient drug is to be delivered to be therapeutic. The purpose of this study is to develop an in silico model that will simulate the transport of the clinically used cytotoxic drug doxorubicin across multicell layers (MCLs) in vitro. Three cell lines were employed: DLD1 (human colon carcinoma), MCF7 (human breast carcinoma) and NCI/ADR-Res (doxorubicin resistant and P-glycoprotein [Pgp] overexpressing ovarian cell line). Cells were cultured on transwell culture inserts to various thicknesses and doxorubicin at various concentrations (100 or 50 μM) was added to the top chamber. The concentration of drug appearing in the bottom chamber was determined as a function of time by HPLC-MS/MS. The rate of drug penetration was inversely proportional to the thickness of the MCL. The rate and extent of doxorubicin penetration was no different in the presence of NCI/ADR-Res cells expressing Pgp compared to MCF7 cells. A mathematical model based upon the premise that the transport of doxorubicin across cell membrane bilayers occurs by a passive "flip-flop" mechanism of the drug between two membrane leaflets was constructed. The mathematical model treats the transwell apparatus as a series of compartments and the MCL is treated as a series of cell layers, separated by small intercellular spaces. This model demonstrates good agreement between predicted and actual drug penetration in vitro and may be applied to the prediction of drug transport in vivo, potentially becoming a useful tool in the study of optimal chemotherapy regimes.

Original languageEnglish
Pages (from-to)598-608
Number of pages11
JournalJournal of Theoretical Biology
Volume257
Issue number4
Early online date4 Jan 2009
DOIs
Publication statusPublished - 21 Apr 2009
Externally publishedYes

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doxorubicin
Penetration
Doxorubicin
Drugs
Theoretical Models
mathematical models
Cells
Mathematical Model
Mathematical models
drugs
Glycoproteins
Cell
Pharmaceutical Preparations
Glycoprotein
Chemotherapy
Flip flop circuits
P-Glycoprotein
Cell membranes
Drug delivery
Cell culture

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Evans, C. J., Phillips, R. M., Jones, P. F., Loadman, P. M., Sleeman, B. D., Twelves, C. J., & Smye, S. W. (2009). A mathematical model of doxorubicin penetration through multicellular layers. Journal of Theoretical Biology, 257(4), 598-608. https://doi.org/10.1016/j.jtbi.2008.11.031
Evans, C. J. ; Phillips, R. M. ; Jones, P. F. ; Loadman, P. M. ; Sleeman, B. D. ; Twelves, C. J. ; Smye, S. W. / A mathematical model of doxorubicin penetration through multicellular layers. In: Journal of Theoretical Biology. 2009 ; Vol. 257, No. 4. pp. 598-608.
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abstract = "Inadequate drug delivery to tumours is now recognised as a key factor that limits the efficacy of anticancer drugs. Extravasation and penetration of therapeutic agents through avascular tissue are critically important processes if sufficient drug is to be delivered to be therapeutic. The purpose of this study is to develop an in silico model that will simulate the transport of the clinically used cytotoxic drug doxorubicin across multicell layers (MCLs) in vitro. Three cell lines were employed: DLD1 (human colon carcinoma), MCF7 (human breast carcinoma) and NCI/ADR-Res (doxorubicin resistant and P-glycoprotein [Pgp] overexpressing ovarian cell line). Cells were cultured on transwell culture inserts to various thicknesses and doxorubicin at various concentrations (100 or 50 μM) was added to the top chamber. The concentration of drug appearing in the bottom chamber was determined as a function of time by HPLC-MS/MS. The rate of drug penetration was inversely proportional to the thickness of the MCL. The rate and extent of doxorubicin penetration was no different in the presence of NCI/ADR-Res cells expressing Pgp compared to MCF7 cells. A mathematical model based upon the premise that the transport of doxorubicin across cell membrane bilayers occurs by a passive {"}flip-flop{"} mechanism of the drug between two membrane leaflets was constructed. The mathematical model treats the transwell apparatus as a series of compartments and the MCL is treated as a series of cell layers, separated by small intercellular spaces. This model demonstrates good agreement between predicted and actual drug penetration in vitro and may be applied to the prediction of drug transport in vivo, potentially becoming a useful tool in the study of optimal chemotherapy regimes.",
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author = "Evans, {C. J.} and Phillips, {R. M.} and Jones, {P. F.} and Loadman, {P. M.} and Sleeman, {B. D.} and Twelves, {C. J.} and Smye, {S. W.}",
note = "Also see corrigendum at dx.doi.org/10.1016/j.jtbi.2010.12.030. ''Three misprints were identified in Evans et al. (2009): First, the maximal transport rate due to Pgp pumping was stated as rp=0.0035 μmol s−1 in Table 1 (p. 601). The value found in Eytan (2005) is rp=0.0035 nmol s−1. The latter of these was used for the computations. Second, the nondimensional constants αp and αm associated with the Michaelis–Menten Pgp-pump term were stated on page 602 as αp≈10−4 and αm≈50. These values should be αp≈10−1 and αm≈1/50. Third, the concentrations were nondimensionalised with respect to the initial concentration in the top compartment, i.e. T0, with the exception of the concentrations of drug bound within each cell layer; these were nondimensionalised with respect to the initial concentration of free binding sites in the cell layer, i.e. C0. For consistency, these concentrations should also be nondimensionalised with respect to T0. This change trivially alters the kinetic equations only in the terms associated with binding, so that details are not discussed any further. The above stated misprints have no effect on the presented results in Evans et al. (2009), because the computations were based on the ODE system in its dimensional form, in which the nondimensional constants do not appear. Hence, all results and conclusions drawn from the model remain valid. Although the numerical results are unaffected by the misprints, the approximate solution derived in Section 2.5 of Evans et al. (2009) is based on the pumping rate αp being small, and is therefore only valid for αp⪡1. The effect of a large pumping rate αp≥1 is the subject of ongoing research. Finally, it should be noted that the model equations derived in Evans et al. (2009) are bespoke to the geometry of the transwell system used in the experiments.''",
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Evans, CJ, Phillips, RM, Jones, PF, Loadman, PM, Sleeman, BD, Twelves, CJ & Smye, SW 2009, 'A mathematical model of doxorubicin penetration through multicellular layers', Journal of Theoretical Biology, vol. 257, no. 4, pp. 598-608. https://doi.org/10.1016/j.jtbi.2008.11.031

A mathematical model of doxorubicin penetration through multicellular layers. / Evans, C. J.; Phillips, R. M.; Jones, P. F. ; Loadman, P. M.; Sleeman, B. D.; Twelves, C. J.; Smye, S. W.

In: Journal of Theoretical Biology, Vol. 257, No. 4, 21.04.2009, p. 598-608.

Research output: Contribution to journalArticle

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AU - Evans, C. J.

AU - Phillips, R. M.

AU - Jones, P. F.

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N1 - Also see corrigendum at dx.doi.org/10.1016/j.jtbi.2010.12.030. ''Three misprints were identified in Evans et al. (2009): First, the maximal transport rate due to Pgp pumping was stated as rp=0.0035 μmol s−1 in Table 1 (p. 601). The value found in Eytan (2005) is rp=0.0035 nmol s−1. The latter of these was used for the computations. Second, the nondimensional constants αp and αm associated with the Michaelis–Menten Pgp-pump term were stated on page 602 as αp≈10−4 and αm≈50. These values should be αp≈10−1 and αm≈1/50. Third, the concentrations were nondimensionalised with respect to the initial concentration in the top compartment, i.e. T0, with the exception of the concentrations of drug bound within each cell layer; these were nondimensionalised with respect to the initial concentration of free binding sites in the cell layer, i.e. C0. For consistency, these concentrations should also be nondimensionalised with respect to T0. This change trivially alters the kinetic equations only in the terms associated with binding, so that details are not discussed any further. The above stated misprints have no effect on the presented results in Evans et al. (2009), because the computations were based on the ODE system in its dimensional form, in which the nondimensional constants do not appear. Hence, all results and conclusions drawn from the model remain valid. Although the numerical results are unaffected by the misprints, the approximate solution derived in Section 2.5 of Evans et al. (2009) is based on the pumping rate αp being small, and is therefore only valid for αp⪡1. The effect of a large pumping rate αp≥1 is the subject of ongoing research. Finally, it should be noted that the model equations derived in Evans et al. (2009) are bespoke to the geometry of the transwell system used in the experiments.''

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N2 - Inadequate drug delivery to tumours is now recognised as a key factor that limits the efficacy of anticancer drugs. Extravasation and penetration of therapeutic agents through avascular tissue are critically important processes if sufficient drug is to be delivered to be therapeutic. The purpose of this study is to develop an in silico model that will simulate the transport of the clinically used cytotoxic drug doxorubicin across multicell layers (MCLs) in vitro. Three cell lines were employed: DLD1 (human colon carcinoma), MCF7 (human breast carcinoma) and NCI/ADR-Res (doxorubicin resistant and P-glycoprotein [Pgp] overexpressing ovarian cell line). Cells were cultured on transwell culture inserts to various thicknesses and doxorubicin at various concentrations (100 or 50 μM) was added to the top chamber. The concentration of drug appearing in the bottom chamber was determined as a function of time by HPLC-MS/MS. The rate of drug penetration was inversely proportional to the thickness of the MCL. The rate and extent of doxorubicin penetration was no different in the presence of NCI/ADR-Res cells expressing Pgp compared to MCF7 cells. A mathematical model based upon the premise that the transport of doxorubicin across cell membrane bilayers occurs by a passive "flip-flop" mechanism of the drug between two membrane leaflets was constructed. The mathematical model treats the transwell apparatus as a series of compartments and the MCL is treated as a series of cell layers, separated by small intercellular spaces. This model demonstrates good agreement between predicted and actual drug penetration in vitro and may be applied to the prediction of drug transport in vivo, potentially becoming a useful tool in the study of optimal chemotherapy regimes.

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