Investigating the steady state of multicellular spheroids by revisiting the two-fluid model
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1.
Dipartimento di Matematica "U. Dini", Università di Firenze, Viale Morgagni 67/A, 50134 Firenze
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2.
Dipartimento di Matematica "U. Dini", Universita' di Firenze, Viale Morgagni 67/A, 50134 Firenze
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3.
Istituto di Analisi dei Sistemi ed Informatica "A. Ruberti" - CNR, Viale Manzoni 30, 00185 Roma
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Received:
01 March 2010
Accepted:
29 June 2018
Published:
01 April 2011
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MSC :
Primary: 92C50; Secondary: 92C17, 76D27, 35R35.
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In this paper we examine the steady state of tumour spheroids considering
a structure in which the central necrotic region contains an inner liquid
core surrounded by dead cells that keep some mechanical integrity.
This partition is a consequence of assuming that a finite delay
is required for the degradation of dead cells into liquid.
The phenomenological assumption of constant local volume fraction of
cells is also made. The above structure is
coupled with a simple mechanical model that views the cell component as a
viscous fluid and the extracellular liquid as an inviscid fluid. By imposing
the continuity of the normal stress throughout the whole spheroid, we show
that a steady state can exist only if the forces on cells at the outer
boundary (provided e.g. by a surface tension) are intense enough, and in such
a case we can compute the stationary radius. By giving reasonable values to
the parameters, the model predicts that the stationary radius decreases with
the external oxygen concentration, as expected from experimental observations.
Citation: Antonio Fasano, Marco Gabrielli, Alberto Gandolfi. Investigating the steady state of multicellular spheroids by revisiting the two-fluid model[J]. Mathematical Biosciences and Engineering, 2011, 8(2): 239-252. doi: 10.3934/mbe.2011.8.239
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Abstract
In this paper we examine the steady state of tumour spheroids considering
a structure in which the central necrotic region contains an inner liquid
core surrounded by dead cells that keep some mechanical integrity.
This partition is a consequence of assuming that a finite delay
is required for the degradation of dead cells into liquid.
The phenomenological assumption of constant local volume fraction of
cells is also made. The above structure is
coupled with a simple mechanical model that views the cell component as a
viscous fluid and the extracellular liquid as an inviscid fluid. By imposing
the continuity of the normal stress throughout the whole spheroid, we show
that a steady state can exist only if the forces on cells at the outer
boundary (provided e.g. by a surface tension) are intense enough, and in such
a case we can compute the stationary radius. By giving reasonable values to
the parameters, the model predicts that the stationary radius decreases with
the external oxygen concentration, as expected from experimental observations.
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